Session 1 运筹学Operations Research1-1运筹学运筹学1 IntroductionOperations ResearchOperations Origins of Operations Research1 Introduction¾One problems is a tendency for the many components of an organization to grow into relatively autonomous The Origins of Operations Researchempires with their own goals and value systems, thereby losing sight of how their activities and objectives mesh with those of the overall organization. The Nature of Operations Research¾A related problem is that as the complexity and specialization increase, it becomes more and more difficult The impact of Operations Researchto allocate the available resources to the various activities in a way that is most effective for the organization as a whole. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog1y, JiangXiUniversity of Finance & Economics©2006School of Informat2ion Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学Operations Origins of Operations ResearchOperations Origins of Operations ResearchTwo factors that played a key role in the rapid ¾The root of OR can be traced back many decades, when growth of ORearly attempts were made to use a scientific approach in the management of organizations. ¾One was the substantial progress that was made early in improving the techniques to OR. ¾The beginning of the activity called operations researchhas, however, generally been attributed to the military servicesA prime example is the simplex methodfor solving linear early in World War Ⅱ. programming problems, developed by George Dantzigin 1947. Many of the standard tools of OR, such as linear programming, ¾By the early 1950s, a growing number of people had dynamic programming, queuing theory, and inventory theory, introduced the use of OR to a variety of organizations in were relatively well developed before the end of the , industry, and government. The rapid spread of OR soon followed.¾A second factor that gave great impetus to the growth of the field was the onslaught of the computer revolution.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog3y, JiangXiUniversity of Finance & Economics©2006School of Informat4ion Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学1 IntroductionOperations ResearchOperations The Nature of Operations The Nature of Operations ResearchThe characteristics of OR are:Operations research means “research on operations”. Thus ¾OR has its broad research is applied to problems that concern how to conduct and coordinate the operations within an organization.¾OR frequently attempts to find a best solution for the Therefore, the nature of the Operations Researchis:problem under consideration. We say a best instead of the best solution because there ¾OR is essentially immaterial and the breadth of application may be multiple solutions tied as unusually wide.“Search for optimality”is an important theme in OR. ¾OR uses an approach that resembles the way research is conducted in established scientific fields. ¾OR experts are a team to do all the things.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo5l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Informat6ion Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 1 运筹学Operations Research1-2运筹学运筹学1 IntroductionOperations ResearchOperations The impact of Operations The impact of Operations Research¾It appears that the impact of OR will continue to example, upon entering the 1990s, the . Bureau of ¾OR has had an impressive impact on improving the Labor Statistics predicted that OR will be the third-fastest-efficiency of numerous organizations around the world. It has growing career area for . college graduates from 1990 to made a significant contribution to increasing the productivity 2005. of the economies of various also predicted that 100,000 people will be employed as ¾There now are more than 30 member countries in the operations research analysts in the United States by the year international federation of operational research societies 2005(IFORS), with each country having a national operations reseaYour can see some applications of operations research rch society. from the table in P5.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog7y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog8y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 2 运筹学Operations Research2-1运筹学运筹学 of the operationsOperations ResearchOperations Researchresearch modeling approach2 Overview of the operations Defining the problem research modeling approachand gathering data¾Define the Problem and Gathering DataThe first order OR team is to study the relevant system and develop a well-defined statement of the problem to be ¾Formulating a Mathematical Modelconsidered.¾Deriving Solutions from the ModelThis includes determining such things as the appropriate ¾Testing the Model objectives, constraints on what can be done, interrelationships between the area to be studied and other areas of the ¾Prepare for Apply the Modelorganization, possible alternative courses of action, time limits ¾Implementationfor making a decision, and so on. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangX1iUniversity of Finance & Economics©2006Schoo2l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Defining the problem and gathering Defining the problem and gathering dataaOperations ResearchOperations ResearchThe five parties generally affected by a business firm : ¾A number of studies of . corporations have found that ¾the owners (stockholders, etc.), who desire profits management tends to adopt the goal of satisfactory profits, (dividends, stock appreciation, and so on);combined with other obejctives, instead of focusing on long-run ¾the employees, who desire steady employment at reasonable profit maximization. wages ;¾Typically, some of these other objectives might be to ¾the customers, who desire a reliable product at a reasonable maintain stable prices, improve worker morale, maintain price ;family control of the business, and increase company prestige. ¾the suppliers, who desire integrity and a reasonable selling ¾Furthermore, there are additional considerations price for their goods ;involving social responsibilities that are distinct from the profit ¾the government and hence the nation, which desire motive. payment of fair taxes and consideration of the national interest. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo3l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of In4formation Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 of the operationsOperations Defining the problem and gathering dataOperations Researchresearch modeling approachGathering relevant data about the Formulating a Mathematical ModelExamples The mathematical model of a business problem is the An OR study done for the San Francisco Police system of equations and related mathematical in the development of a computerized system for optimally scheduling and deploying police patrol ¾nrelated quantifiable decisions to be made are called officers. The new system provided annual savings of $11 decision variables( say,x,x,x,…,x).123nmillion, an annual $3 million increase in traffic citation ¾The appropriate measure of performance is then revenues, and a 20 percent improvement in response times. In expressed as a mathematical function of these decision assessing the appropriate obejctivesfor this study, three fundamental objective were identified:variables. The function is called the obejctive function.¾Maintain a high level of citizen safety.¾Any restrictions on the values that can be assigned to ¾Maintain a high level of officer morale. these decision variables are also expressed, which called¾Minimizconstraints( for example, x3+xx2+x1<0).e the cost of operations. 1122江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog5y, JiangXiUniversity of Finance & Economics©2006School of6 Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 2 运筹学Operations Research2-2运筹学运筹学 Formulating a Mathematical Formulating a Mathematical ModelOperations ResearchOperations ResearchThe constants (namely, the coefficients and right-hand sides) in the constraints and the objective function are called¾Determining the appropriate values to assign to the parametersof the model. parameter of the model is both a critical and a challenging An OR model is often expressed aspart of the model-building process. follow:The mathematical ¾The value assigned to a parameter often is, of necessity, max(ormin)Z=cx+cx+...+cx1122nnmodel might then say only a rough estimate. Because of the uncertainty about the true value of the parameter. ax+ax+...+ax≤(=,≥that the problem is to )b⎧1111221nn1⎪choose the values of ax+ax+...+ax≤(=,≥)b¾To analyze how the solution would change if the value 2112222nn2⎪the decision variables ⎪assigned to the parameter were change to other plausible .......⎨so as to maximize the values (sensitivit yanalsyis.)⎪ax+ax+...+ax≤(=,≥)bobjective function, m11m22mnnm⎪subject to the ¾It is even possible that two or more completely different x≥0(j=1,2,...,n)⎪j⎩specified of models may be developed to help analyze the same problem.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo7l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo8l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Formulating a Mathematical Formulating a Mathematical ModelOperations ResearchOperations ResearchThis model has over 1,000 Examples: decision variables R, and its ijAn OR study done for Monsanto Corpwas concerned with use let to annual savings of rsoptimizing production operations in Monsantos’ chemical approximately $2 =cR∑∑ijijplants to minimize the cost of meeting the target for the i=1j=1amount of a certain chemical product (maleicanhydride) to rs⎧cc=ost for reactor i at settingj,ijpR≥T∑∑ijijbe produced in a given month. The decisions to be made⎪ are i=1j=1⎪pp=roduction of reactor I at setting j ;ijthe dial setting for each of the catalytic reactors used to s⎪⎪=1,fori=1,2,L,rTp=roduction target ;produce this product, where the setting determines both the ⎨∑ijj=1⎪amount produced and the cost of operating the reactor. The rn=umber of reactors ;⎪R=0,or,1ijform of the resulting mathematical model is as follows: ⎪⎪sn=umber of settings.⎩⎧1,ifreactoriisoperatedatsettingjwhereR=⎨ij0,otherwise⎩江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog9y, JiangXiUniversity of Finance & Economics©2006Schoo10l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学 of the operations运筹学 Deriving solutions from the modelOperations ResearchOperations Researchresearch modeling Deriving solutions from the modelGoals may be set to establish minimum satisfactory levels of performance in various areas, based perhaps on ¾Sometimes it is a relatively simple step, which can use past levels of performance or on what the competition is the standard algorithms, sometimes it is difficult. A common achieving. If a solution is found that enables all these theme in OR is the search for an optimal, or the best goals to be met, it is likely to be adopted without further . Such is the mature of satisfying.¾EEminent management scientist and Nobel Laureate in economicsH eHrbert Simon points out thatsatisfiyng is much more prevalent than optimizing in actual practice. In coining the term satisfying as a combination of the wordssatisfactoryzgyand optimiizng, 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog11y, JiangXiUniversity of Finance & Economics©2006Schoo12l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 2 运筹学Operations Research2-3运筹学 of the operations运筹学 of the operationsOperations ResearchOperations Researchresearch modeling approachresearch modeling Testing the Preparing to apply the model¾The process of testing and improving a model to After the testing phase, the next step is to install a well-increase its validity is commonly referred to as model documented system for applying the model as prescribed by . This system will include the model, solution ¾A systematic approach to testing the model is to procedure and operating procedures for implementation. use a retrospective test. When it is applicable, this test And this system usually is using historical data to reconstruct the past and then determining how well the model and the resulting solution would have performed if they had been used. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo13l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog14y, JiangXiUniversity of Finance & Economics©2006运筹学 of the operations运筹学 of the operationsOperations ResearchOperations Researchresearch modeling approachresearch modeling ¾It is important for the OR team to participate in launching In this chapter, we only focus on the overall process of this OR. We have tried to emphasize that this constitutes only ¾The success of the implementation phase depends a great a portion of the overall process involved in conducting a deal upon the support of both top management and operating OR study. In the rest of the book, we will focuses primarily on constructing and solving mathematical models.¾The implementation phase involves several steps. Firstthe OR team gives operating management a careful explanation of the new system to be adopted and how it relates to operating realities. Next, these two parties share the responsibility for developing the procedures required to put this system into operation. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Informatn Technology, JiangXiUniversity of Finance & Econom15ioics©2006School of Information Technology, JiangXiUniversity of Finance & Econom16ics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 3 运筹学Operations Research3-1运筹学运筹学Operat3 Introduction to Linear Programmingions ResearchOperations ResearchIntroduction 3 Introduction to Linear ProgrammingThe development of linear programming has been ranked thamong the most important scientific advances of the mid-20¡ Prototype xEamplecentury, and we must agree with this assessment. Its impact since just 1950 has been extraordinary.¡ The Linear Programming Model Today it is a standard tool that has saved many thousands or ¡ Solving LP by xEcel Solvermillions of dollars for most companies or businesses of even ¡ Solving LP by Graphic Methodmoderate size in the various industrialized countries of the world ;and its use in other sectors of society has been spreading ¡ Assumptions f Linear Programmingrapidly.¡ Additional xEamples江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo1l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo2l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学Operat3 Introduction to Linear Programmingions ResearchOperat3 Introduction to Linear Programmingions Prototype example¾Linear programming involves the general problem of The Background :allocating limited resourcesamong competing activities in a ¾The WKYNDOR GLASS CO. produces high-quality glass best possible way. products, including windows and glass doors. It has three plants. Aluminum frames and hardware are made in Plant1, ¾It use s a mathematical model to describe the problem of wood frames are made in Plant 2, and Plant 3 produces the glass and assembles the products. concern. And the remarkably efficient solution procedure to ¾Unprofitable products are being discontinued, releasing the linear programming is called the simplex capacity to launch two new products having large sales potential:Product1: an 8-foot glass door with aluminum framing Product2: A 4*6 foot double-hung wood-famed window江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo3l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog4y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学Operations Prototype Prototype exampleions ResearchProduction time per batch, hoursProduction time Production time per batch,plantavailable per week, Production product 1product 2hourshourstime available plant1104productper week, hours2021212332181104Profit per batch $3,000$5,0002021233218The decision variable is : x=? x?=12Profit per batch $3,000$5,000maxz=3x+5xThe objective is maximize the profit: 12The constraints are :x+0x≤4Plant 1Question:Determine what the production rates should be for 12the two products in order to maximiez their total profit ?Plant 20x+2x≤12 33x+2x≤18Plant1212江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo Technology, JiangXiUniver5l of Informationsity of Finance & Economics©2006School of Information Technolog6y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 3 运筹学Operations Research3-2运筹学运筹学Operations Prototype exampleOperations Prototype exampleProduction time per batch, hoursProduction time plantavailable per week, Regional Planningproduct 1product 2hours1104THE SOUTHER CONFEDERATION KIBBUTIZM is a 20212group of three kibbutizm(communal farming communities) in 33218Israel. This office is planning agricultural production for the Profit per batch $3,000$5,000coming year. maxz=3x+5x12KibbutzUsable Land (Acres)Water Allocation (Acre Feet)This is a linear 1x+0x≤4⎧121400600programming⎪0x+2x≤12⎪.⎨3x+2x≤18330037512⎪⎪x,x≥0⎩12江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo7l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo8l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学Operations Prototype Prototype exampleions ResearchBecause of the limited water available for irrigation, the The crops suited for this region include sugar beets, cotton, Southern Confederation of Kibbutzim will not be able to use and sorghum, and these are the three being considered for the all its irrigable land for planning crops in the upcoming upcoming season. These crops differ primarily in their season. To ensure equity between the three kibbutzim, it has expected net return per acre and their consumption of water. In addition, the Ministry of Agriculture has set a maximum been agreed that every kibbutz will plant the same proportion quota for the total acreage that can be devoted to each of these of its available irrigable by the Southern Confederation of Kibbutzim, as shown in Table question is:Maximum Water consumption Net return How man yacres to deovte to each crop at the respectiev cropquota (acres)(acre feet/acre)($/acre)ikbbutizm whiel satisfying the gievn restrictions. The objectievis Sugar beets60031,000to maximiez the tota lnet return to the southern oCnfederation of Cotton5002750Kibbutizm as a 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog9y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog10y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学Operations Prototype Prototype exampleions ResearchTheobjectiveis to maximize the Net ReturnSo, the decision variablesare MaxZ=1000(x+x+x)+750(x+x+x)+250(x+x+x)CropAllocation usable land to Kibbutz 111213212223313233123The constraints are more complicated, Sugarxxxx+x+x≤400⎧we divided them into a few kinds111213112131⎪oCttonxxxx+x+x≤600⎨ land for each kibbutz:⎪Sorghumx+x+x≤300xxx⎩1332333132333x+2x+x≤600⎧ allocation for each kibbutz:112131Maximum Water consumption Net return ⎪crop3x+2x+x≤800quota (acres)(acre feeta/cre)($a/cre)⎨122232⎪Sugar beets60031,0003x+2x+x≤375⎩132333oCtton5002750江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Sorghum3251250School of Information Technolog11y, JiangXiUniversity of Finance & Economics©2006School of Information TechnologUn12y, JiangXiiversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 3 运筹学Operations Research3-3运筹学运筹学Operations Prototype exampleOperations Prototype examplex+x+x≤600⎧ acreage for each ⎪So, we can see, any linear programming model institutes of x+x+x≤500crop⎨:212223three parts:⎪x+x+x≤325⎩313233Decision variables, Objective function, Constraints x+x+xx+x+x⎧ proportion of land =And to formulate a model of linear programming ⎪400600planted:⎪according to three stepsx+x+xx+x+x⎪122232132333=⎨600300•Determine decision variables⎪⎪x+x+xx+x+x132333112131=⎪•Determine the objective function300400⎩：•Determine the constraintsx≥0,forj=1,2,L,9j江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of13 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog14y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学Operat3 Introduction to Linear The Linear Programming Modelions ResearchOperations The Linear Programming ModelResource usage per unit of activityobjective functionhTe standard formAmount of resource ActivityResource availableMaximizeZ=cx+cx+L+cx1122nn1 2 …. nfunctional ⎧ax+ax+L+ax≤b1111221nn1constraints1aaLab11121n1⎪ax+ax+L+ax≤b2112222nn2⎪2⎪baaLa221222nsubjecttoM⎨⎪MMnonnegative Max+ax+L+ax≤bm11m22mnnm⎪constraintsm⎪x≥0b,x≥0,L,x≥0,aaLa⎩12nmm1m2mnoCntribution to ZAny situation whose mathematical formulation ficcLcts this per unit activity12nmodel is a linear programming problem (LP).江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo15l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Economics©200616运筹学运筹学 The Linear Proions Resgramming The Linear Programming Modelions ResearchCyybCertaint ysmybols are commonyl yused to denote the various Other Formscomponents of a programming smybols are We now hasten to add that the preceding does not actually listedb: below:wfit the natural form of some linear programming problems. The model poses the The other legitimate formsare the following:Zv=alue of overall measure of problem in terms of rather than maximizing the objective function: decisions about Minimize =Zcx+cx+…+cx1122nnxthe levels of the activities, l=evel of activityj(for j1=,2, …,n) function constraints with a greater-than-or-equal-to so x,x, ,… xare called 12nci=ncrease in Zthat would result from inequality:jthe decision variables. each unit increase in level of activity +ax+…+axbfor some values of ≥iThe values of c, b, and function constraints in equation form:ba=mount of resource i that is iaare also referred to as ijax+ax+…+ax=bfor some values of for allocation to activities (for i11i22innithe parametersof the the nonnegative constraints foe some decision i1=,2, …,m).:aa=mount of resource i consumed by ijxunrestedin sign for some values of unit of activity j.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information TechnologangXiUniversity of Finance & Economics©200617y, JiSchoo18l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 3 运筹学Operations Research3-4运筹学运筹学 Solving Linear Programming by using LECEXOperat3 Introduction to Linear Programmingions ResearchOperations Research(2) Input the data to data Solving Linear Programming by using LCEXEmaxz=3x+5x12(3) Determine the location of decision variables (1) Add-in Excel Solver1x+0x≤4⎧12⎪0x+2x≤12Wyndor Glass Co. Product-Mix Prob⎪.⎨3x+2x≤1812⎪oDorsWinodsw⎪x,x≥0⎩12Unit Profit3$005$00oHrusoHrus Use dPer Unit ProcudedAvailalbePlant 1104Plant 20212Plant 33218oDorsWinodswUnits Procuded00江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo19l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Informaton Technology, JiangXiUniversity of Finance & Econom20iics©2006运筹学运筹学 Solving Linear Programming by using Solving Linear Programming by using LECEXOperations ResearchOperations Research(5) Input the left function of each constraint (4) In the objective cell, input the objective functionDoorsWindowsDoorsWindowsUnit Profit$300$500Unit Profit$300$500HoursHoursHoursHours Used Per Unit ProducedAvailableHours Used Per Unit ProducedUsedAvailablePlant 1101<=4Plant 1104Plant 2022<=12Plant 20212Plant 3325<=18Plant 33218DoorsWindowsTotal ProfitDoorsWindowsTotal ProfitUnits Produced11$800Units Produced11$800E5HoursG6Used11Total Profit12=SUMPRODUCT(UnitProfit,UnitsProduced)7=SUMPRODUCT(C7:D7,UnitsProduced)8=SUMPRODUCT(C8:D8,UnitsProduced)9=SUMPRODUCT(C9:D9,UnitsProduced)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©200621School of Information Technolog22y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Solving Linear Programming by using Solving Linear Programming by using LECEXOperations ResearchOperations Research(6) Launch Solver (7) Answer the parameters of SolverBCDEFG3DoorsWindows4Unit Profit$300$5005HoursHours6Hours Used Per Unit ProducedUsedAvailable7Plant 1101<=18Plant 2022<=129Plant 3325<=181011DoorsWindowsTotal Profit江西财经大学信息管理学院©2006江西财经大学信息管理学院©200612Units Produced11$800School of Informaton Technology, JiangXiUniversity of Finance & Econom23iics©2006School of Information Technolog24y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 3 运筹学Operations Research3-5运筹学运筹学 Solving Linear Programming by using Solving Linear Programming by using LECEXOperations ResearchOperations Research(8) Add constrains(9) Finish the Solver dialog boxBCDEFG3DoorsWindows4Unit Profit$300$5005HoursHours6Hours Used Per Unit ProducedUsedAvailable7Plant 1101<=18Plant 2022<=129Plant 3325<=181011DoorsWindowsTotal Profit江西财经大学信息管理学院©2006江西财经大学信息管理学院©200612Units Produced11$800Schoo25l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog26y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Solving Linear Programming by using Solving Linear Programming by using LECEXECEXOperations ResearchOperations ResearchExercise: To solve the Regional Planning (10) Solve the problem and get the optimal solutionRegional PlanningDoorsWindowswater consumptionlAlocation land to KibbtzUnit Profit$300$500Crop(cAre FeetcA/re)123HoursHoursSugar =<060Hours Used Per Unit =<050Plant 1102<=1Sorghum10=<235Plant 20212<= 33218<=18<=<=<=sUable aLnd (cAres) <=<=<=Units Produced26$3,600aWter allocation(cAre Feet) optimal solution is:produced doors2=, windows6= uqEal Proportion of land PlantedlPant of aRte of Kibbutz 1lPant of aRte of Kibbutz 2Plant of aRte of Kibbutz 3maximum profit3===江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog27y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog28y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学Operat3 Introduction to Linear Solving LP by Graphic Methodions Researchions Solving LP by Graphic MethodThe steps of Graphic Method: maxz=3x+5x12The optimal solution •Draw each constraints linex=2, x=61x+0x≤412⎧x=6122•Determine the feasible regionZ=3600⎪0x+2x≤12⎪.⎨•Draw objective functional line3x+2x≤1812⎪•Solving equations to gain the optimal solutioneFasible ⎪x,x≥0⎩12eRgion•Determine the optimal objective valuesZ=3x5+x12x=413x+2x=1812江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of29 Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo30l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 3 运筹学Operations Research3-6运筹学运筹学 Solving LP by Graphic MethodOperations Solving LP by Graphic Methodions ResearchearchSome other possible situations : Exercise: To solving the following LP (1)Multiple Optimal SolutionsB(0,13)maxz=4+3xx123x+2x=2612maxZ=3x+2x12x⎧2+3≤24xx212Q(0,8)⎪vEery point on this 3Q(,64)x≤42⎧+2≤26⎨xx126red line segment is ⎪2x+3x=24⎪122x≤12,≥0⎪2xx⎩12optimal, each with ⎨1=Z83x+2x≤1812⎪Q(2,3/60)A(12,0)1⎪x,x≥02⎩12The optimal solution xx06,=x4=1122463=Z江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo31l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog32y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Solving LP by Graphic Methodions ResearchOperations Solving LP by Graphic Methodearch(1) No Optimal Solutions. Terminology for Solution of the ModelmaxZ=3x+5x12¾A feasible solutionis a x2⎧x≤41solution for which all the ⎪2x≤12maxZ=3x+5x2constraints are ⎪⎪+2x≤18⎨¾An infeasible solutionis a ⎧x≤.⎨⎪solution for which at least one 63x+5x≥50x,x≥012⎩12⎪constraints is violated.⎪x≥0,x≥0⎩124¾The feasible regionis the eFasible collection of all feasible solutions. And it is possible for eRgion2a problem to have no feasible (2) the constraints do not improving the ¾An optimal solutionis a 02value of the objective 4xfeasible solution that has the 1function (Z) indefinitely in most favorable valueof the (1) it has no feasible solutionsthe favorable function.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo33l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Informaton Technology, JiangXiUniversity of Finance & Econom34iics©2006运筹学运筹学OperatP by Graphic MethodOperat3 Introduction to Linear Programmingions Solving Learchions Assumptions of Linear ProgrammingRelationship between optimal A corner-point feasible solutions and CPF solutions:(PCF)solutionis that lies (1) Proportionalityat a corner of the feasible PrPoportionalityyis an assumption about both (1) The problem must posses PCF objective function and the functional solutions and at least one optimal maxZ=3x+2xconstraints, as summarized . 2Proportionality assumption:x≤4⎧(2) The best PCF solution must be 16The contribution of each activity to the value ⎪an optimal solution. 2x≤12of the ojebctive f⎪jbunctionZ Zis proportional to the (3) If a problem has exactly one ⎨level of the activityyx,as represented by the cxjjjjjj3x+2x≤18optimal solution, it must be a CPF 12⎪term in the objective function. solution. ⎪2x,x≥0Similarly, the contribution of each activity to ⎩12(4) If the problem has multiple the left--hand side of each functional constraint is optimal solutions, at least two must 0proportional to the level of the activityy x,as jj24x1be CPF by the axterm in the 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog35y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog36y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 3 运筹学Operations Research3-7运筹学运筹学 Assumptions of Linear Assumptions of Linear ProgrammingOperations ResearchOperations ResearchTo illustrate this assumption, consider the WyndorGlass Case 1: Would arise if there were start-up costs associated Co. problem. with initiating the production of product 1. Profit from product 1 ($000 per week)For example, there might be costs involved with setting up Proportthe production facilities. There might alionalso be costs associated ity xProportionality violated 1satisfied with arranging the distribution of the new product. Because Case 1Case 2Case 3these are one-time costs, they would need to be amortized on a per-week basis to be commensurable with .Z Suppose that this 00000amortization were done and that the total start-up cost 13233amounted to reducing Zby 1, but that the profit without 26575398126considering the start-up cost would be 3x. This would mean 141211186that the contribution from product 1 to Zshould be 3x-1 for 1x0>, whereas the contribution would be 3x0= when x0=.111江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo37l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog38y, JiangXiUniversity of Finance & Economics©2006Setu-p cost运筹学运筹学 Assumptions of Linear Assumptions of Linear ProgrammingOperations ResearchOperations ResearchCase 2:There now is an increasing marginal return ;., the slope of the profit functionfor product 1 keeps increasing as iVolates xis increased. Satisfies1This violation of proportionality might occur because of economics of scale that can sometimes be achieved at higher levels of production, ., through the use of more efficient high-volume machinery, longer production runs, quantity discounts for large purchases of raw materials, and the iVolatesCase 1learning-curve effect whereby workers become more efficient as they gain experience with a particular mode of production. As the incremental cost goes down, the incremental profit will go up (assume constant marginal revenue).Case 2江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog39y, JiangXiUniversity of Finance & Economics©2006Schoo40l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Assumptions of Linear Assumptions of Linear ProgrammingOperations ResearchOperations ResearchAdditiveE Assumption: Every function in a linear programming Case 3:There is a decreasing marginal return. In this case, the model (whether the objective function or the function on the slope of the profit functionfor product 1 keeps decreasing as x1left-hand side of a function constraint) isthe sum of the individual contributionsof the respective For Example:This violation of aVlue of Zproportionality might occur (x,x)Additivity violated b12ecause the marketing costs need Additivitysatisfied Case 1Case 2to go up more than proportionally to attain increases in the level of iVolates (1,0)333sales.(0,1)555(1,1)897江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo41l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog42y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 3 运筹学Operations Research3-8运筹学运筹学 Assumptions of Linear Assumptions of Linear ProgrammingOperations ResearchOperations ResearchCase 2:Violates the additivityassumption because of the extra Case 1:Corresponds to an objective function of term in its objective function, 3=Zx5+x-xx. As the reverse of 12123=Zx5+x+xx, thereby violating the additive assumption. 1212the first case, case 2 would arise if the two products were This case would arise if the two products were competitivein some waythat decreased their joint in some way that increases profit. For For example, suppose that both products need to use the example, suppose that a major advertising campaign would be same machinery and equipment. Producing both products required to market either new product produced by itself, but would require switching the production processes back and forth, with substantial time and cost involved in temporarily that the same single campaign can effectively promote both shutting down the production of one product and setting up products if the decision is made to produce both. for the other.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog43y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog44y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Assumptions of Linear Assumptions of Linear ProgrammingOperations ResearchOperations ResearchCase 3:the production time used by the two products is given Case 4:the function for production time used is 3x2+x-12by the function 3x2+x0+.5xx,which violates the additive , so violates the additive of resource used Amount of resource used (x,x)Additivity violated (x,x)Additivity violated 1212Additivitysatisfied Additivitysatisfied Case 3Case 4Case 3Case 4(2,0)666(2,0)666(0,3)666(0,3)666(2,3)(2,3)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of45 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog46y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Assumptions of Linear Assumptions of Linear ProgrammingOperations ResearchOperations ResearchDivisibility assumption: Decision variables in a linear CCertainty assumptions:The value assigned to each programming model are allowed to have any values, parameter of a linear programming model is assumed to be a including non-integer values, that satisfy the known constant..functional and non-negativity constraints. Thus, these variables are not restricted to just inter values. Since In real applications, the certainty assumption is seldom each decision variable represents the level of some satisfied precisely. Linear programming models usually are activity, it is being assumed that the activities can be formulated to select some future course of action. Therefore, run at fractional parameter values used would be based on a prediction of In certain situations, the divisibility assumption future conditions, which inevitably introduces some degree of does not hold because some of or all the decision variables must be to integer valueuncertainty..江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technologr47y, JiangXiUnivesity of Finance & Economics©2006Schoo48l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 3 运筹学Operations Research3-9运筹学运筹学 Additional xEamplesOperat3 Introduction to Linear Programmingions ResearchOperations Research3. 6 Additional xEamplesDesign of Radiation TherapyMary has just been diagnosed as having a cancer at a The applicability of linear programming is very widely fairly advanced stage. Specifically, she has a large malignant used. In this section, we begin broadening our horizons. As tumor in the bladder area (a “whole bladder lesion”).you study the following examples, note that it is their She will receive the most advanced medical care available underlying mathematical model rather than their context to give her every possible chance for survival. This care will including extensive radiation therapyy. that characterizes them as linear programming problems. Radiation therapyinvolves using an external beam These examples are scaled-down versions of actual treatment machine to pass ionizing radiation through the ’’s body, damaging both cancerous and healthy tissues. Normally, several beams are precisely administered from different angles in a two-dimensional plane.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo49l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo50l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Additional Additional xEamplesOperations ResearchOperations ResearchTable .3 7 aDta for the eDsign of aMrys’ aRdiation Therapy Due to attenuation, each beam delivers more radiation to the tissue near the entry point than to the tissue near to the exit rFaction of nEtry oDse bAsorbed point. Scatter also causes some delivery of radiation to tissue by rAea (vAerage) eRstriction on Total outside the direct path of the beam. Because tumor cells are rAea eBam 1 eBam 2 vAerage oDsage,iKlorads typically microscopically interspersed among healthy cells, the eHalthy anatomy 0. 4 0. 5 iMnimize radiation dosage throughout the tumor region must be large rCitical tissues 0. 3 2=<. 7enough to kill the malignant cells. Which are slightly more Tumor region 0. 5 0. 5 6=radiosensitive, yet small enough to spare the healthy cell. eCnter of tumor 0. 6 0. 4 6=>eBam 2133The goal of the design is to select the combination 2of beams to be used, andthe intensity of each one, to generate the best possible dose 1江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog51y, JiangXiUniversity of Finance & Economics©2006Schoo52l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Additional Additional xEamplesOperations ResearchOperations ResearchFORMULATION :oCntrollin giAr oPllutionThe two decision variables xand xrepresent the dose (in 12kilorads) at the entry point for beam 1 and bean 2, The&EEC NORI &LEES CO.,one of the major producers of steel respectively. in its part of the world,is located in the city of Steeltownand is the only large employer there. Steeltown has grown and MinimizeZ=+ along with the company, which now employs nearly 1250,000 residents. But uncontrolled air pollution from the +≤⎧12⎪company’’s furnaces is running the appearance of the city and +=6⎪ the health of its residents.⎨+≥612⎪The directors are discussing what to do about the air ⎪x,x≥6⎩12pollution.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©200653School of Information Technolog54y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 3 运筹学Operations Research3-10运筹学运筹学 Additional Additional xEamplesOperations ResearchOperations ResearchIn both cases the engineers have decided that the most The three main types of pollutants in this airside effective types of abatement methods are:are particulate matter, sulfur oxides, and hydrocarbons. The new standards require that the (1)Increasing the height of the smokestacks;company reduce its annual emission of these (2)Using filter devices (including gas traps) in the smokestacks;pollutants by the amounts shown in Table . (3)Including cleaner, high-grade materials among the fuels for Pollution Required reduction in annual emission rate the furnaces.(million pounds)Particulates60Each of these methods has a technological limit on how Sulfur oxides150heavily it can be used(.,a maximum feasible increase in the yHdrocarbons125height of the smokestacks), but there also is considerable flexibility for using the method at a fraction of its technological limit. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo55l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog56y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Additional Additional xEamplesOperations ResearchOperations ResearchA method’s annual cost includes increased operating and maintenance expense as well as reduced revenue due to any Table shows how much emission (in millions of pounds loss in the efficiency of the production process caused by usingper year) can be eliminated from each type of furnace by fully the any abatement method to its technological limit. The other major cost is the start-up cost required to install Taller Smokestacks iFlters Better fuelsthe method. The total annual cost estimates (in million of dollars) are given in Table for using the methods at their full oPllutant Blast Open-Blast Open-Blast Open-abatement capacities. furnaceshearth furnaceshearth furnaceshearth furnacesfurnacesfurnacesAbatement method Blast furnacespOenh-earth furnacesaPrticulates12925201713Sulfur oxides354218315649Taller smokestacks810hydrocarbons375328242920Filters76eBtter fules119江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom57ics©2006School of Information Technolog58y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Additional Additional xEamplesOperations ResearchOperations ResearchFORMULATION:Minimize 8=Zx1+0x7+x6+x1+1x9+x,123456Subject to the following constraints:Question:This plan specifies which types of reduction:abatement methods will be used and at what 12x9+x2+5x2+0x1+7x1+3x≥06123456fractions of their abatement capacities for (1) the 35x4+2x1+8x3+1x5+6x4+9x≥15012343637x+53x+28x+24x+29x+20x≥125123456 blast furnaces and (2) the open-hearth furnaces the limit: x≤1, for j1=,2,…, with the smallest possible : x≥0, for j1=,2,…,江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog59y, JiangXiUniversity of Finance & Economics©2006Schoo60l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 3 运筹学Operations Research3-11运筹学运筹学 Additional Additional xEamplesOperations ResearchOperations ResearchReclaimin gSolid aWstes Product Data for Save-it SAE-VIT COMPANY operates a reclamation center Grade SpecificationAmalgamationSelling price per that collects four types of soCost per pound ()$lid waste materials and treats them pound()$so that they can be amalgamated into a salable product. Three Material 1:Not more than 30% of totalAdifferent grads of this product can be made depending upon the Material 2:Not less than 40% of 3:Not more than 50% of totalmix of the materials used. Although there is some flexibility inMaterial 4:xEactly 20% of totalthe mix for each the proportion of a material in the product Material 1:Not more than 50% of totalBMaterial 2:Not less than 10% of totalgrade. For each of the two higher grades, a fixed percentage is 4:xEactly 10% of totalspecified for one of the 1:not more than 70% of 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of61 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog62y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Additional Additional xEamplesOperations ResearchOperations ResearchThe reclamation center collects its solid waste materials from The Save-It Co. is solely owned by Green Earth, Green regular sources and so is normally able to maintain a steady rate Earth has raised contributions and grants, amounting to for treating them. $30,000 per week, to be used exclusively to cover the entire Solid waste material data for the save-it cost for the solid waste materials. The board of aMterial Pounds per Treatment cost Additional restrictions directors of Green Earth has instructed the management of week availableper pound($)Save-It to divide this money among the materials in such a way each material, at least that at least half of the amountavailable of each material is 22000600half of the pounds per week actually collected and should be collected QQuestion:Management wants to determine the amount of 41000500and treated .each product grade to produce and the exact mix of materials to 2.$30,000 per week should be used for each grade so as to maximize the total weekly be used to treat these materials. profit( total sales income minus total amalgamation and treatment cost).江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom63ics©2006School of Information Technolog64y, JiangXiUniversity of Finance & Economics©2006Let x= Pounds of material jallocated to product iper week (i =A, B, C; j =1, 2, 3, 运筹i学j运筹学 Additional xEamples4O).perations ResearchOperations ResearchMaximize Profit =(x +x +x +x) +(x +x +x +x) +(x +xA1A2A3A4B1B2B3B41C2C +x +x)3C4Csubject toMixture Specifications:x≤ (x +x +x +x)A1A1A2A3A4Personnel Schedulingx≥ (x +x +x +x) A2A1A2A3A4x≤ (x +x +x +x) A3A1A2A3A4x = (x +x +x +x) A4A1A2A3A4x≤ (x +x +x +x)B1B1B2B3B4UNION AIRWAYSis adding more flights to and from its x≥ (x +x +x +x)B2B1B2B3B4hub airport, and so it needs to hire additional customer x = (x +x +x +x)B4B1B2B3B4x≤ (x +x +x +x) 1C1C2C3C4Cservice agents. HoHwever, it is not clear just how many more Availability of Materials:x +x +x≤3,000A1B11Cx +x +x≤2,000should be hired. Management recognize the need for cost A2B22Cx +x +x≤4,000A3B33Ccontrol while also consistently providing a satisfactory level x +x +x≤1,000A4B44CRestrictions on amount treated:x +x +x≥1,500A1B11Cof service to customers. Therefore, an OR team is studying x +x +x≥1,000A2B22Chow to scheduling the agents to provide satisfactory service x +x +x≥2,000A3B33Cx +x +x≥500A4B44Cwith the smallest personnel on treatment cost:3(x +x +x) +6(x +x +x)A1B11CA2B22C +4(x +x +x) +5(x +x +x) =A3B33CA4B44C30,000江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006and x≥0 (i =A, B, C; j =1, 2, 3, 4).Schooijl of Information Technolog65y, JiangXiUniversity of Finance & Economics©2006Schoo66l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 3 运筹学Operations Research3-12运筹学运筹学 Additional Additional xEamplesOperations ResearchOperations ResearchTime periods coveredMinimum number of Based on the new schedule of flights, an analysis has been Time periodShiftagents neededmade of the minimumnumber of customer service agents that 1 2 3 4 56:00 . to 8:00 .√48need to be on duty at different times of the day to provide a 8:00 . to 10:00 .√√79satisfactory level of service. 10:00 . to 12:00 .√√6512:00 . to 2:00 .√√√87The other entries in this reflect one of the provisions in the 2:00 . to 4:00 .√√644:00 . to 6:00 .√√73company’s current contract with the union that the 6:00 . to 8:00 .√√82represents the customer service agents. The provision is that 8:00 . to 10:00 .√4310:00 . to 12:00 .√√52each agent work an 8-hours shift 5 days per week, and the 12:00 . to 6:00 .√15authorized shifts are in the following cost per agent $170 $160 $175 $180 1$95江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog67y, JiangXiUniversity of Finance & Economics©2006Schoo68l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Additional Additional xEamplesOperations ResearchOperations ResearchMinimize=Z+ 1=Z70x16+60x+1+75x+18+08x+1+95x,12345FORMULATION:Subject to Linear programming problems always involve finding the x>=86>=4 8 (6-8)81best mix of activit ylevels. The key to formulating this particular x++x>=>=79 8 (8-10)12x++x>=6p>=65 (10-12)roblem is to recognize the nature of the activities. This 12x++x+>=8+x>=87 (12-14)problem involves finding the best mix of shifts siezs, the five 123x+>=6+x>=64 (14-166)23decision variables here arex++x>=>=73 (166-188)34 xn=umber of agents assigned to shift j, for j1=,2,3,4,+>=8+x>=82 (188-20)34The main restrictions on the values of these decision x>=>=43 (20-22)4variables are that the number of agents working during each x++x>=>=52 (22-24)45time period must satisfy the >=>=15 (24-6)6 5x=>0==>, for j1=,2,3,4,江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog69y, JiangXiUniversity of Finance & Economics©2006Schoo70l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 4 运筹学Operations Research4-1运筹学运筹学4 Solving LP Problems: Simplex MethodOperations ResearchOperations The sEsence of the Simplex Method4 Solving Linear Programming Problems: The Simplex MTEOHDThe simplex method is an algebraic procedure. However, its underlying concepts are these geometric ¡ The Essence Of the Simplex Methodconcepts provides a strong intuitive feeling for how the simplex¡ Setting pU the Simplex Methodmethod operates and what makes it so efficient. ¡ The Algebra of the Simplex MethodFor a simplex method, we first find its cornerp-oint solutions (CPF SOLUTIONS), then we will provide a very useful way of ¡ The Simplex Method in Tabular Formchecking whether a PCF solution is an optimal solution.¡ Tie Breaking in the Simplex Method Now we are ready to apply the simplex method to the ¡ Adapting to Other Model Formexample.¡ oPstOp-timality Analysis江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of1 Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo2l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Essence of the Simplex The Essence of the Simplex MethodOperations ResearchOperations Research¾Initialization:hCoose (0,0) as the initial PCF solution to Solving the xEampleexamine. (This is a convenient choice because no calculation eHre is an outline of what the simplex xare required to identify this PCF solution.)2method does(from a geometric viewpoint) (0,9)to solve theyWndorGlass oC. problem. tA ¾Optimal Test:oCnclude that (0,0) is not an optimal solution. each step, first the conclusion is stated and (Adjacent CPF solutions are better.)then the reason is given in parentheses.(4,6)(2,6)¾Iteration 1:Move to a better adjacent PCF solution, (0,6), CCPF Its adjacent CCPF (0,6)6by performing the following three •Between the two edges of the feasible region that emanate from 4(0,0)(0,66) and (4,0)(4,3)(0,0), choose to move along the edge that leads up the xaxis. 2(0,66)(2,66) and (0,0)(2,•Stop at the first new constraint boundary: 2x=12. 66)(4,3) and (0,66)22(4,3)(4,0) and (2,66)(,60)(0,0•Solve for the intersection of the new set of constraint )(4,0)(4,0)(0,0) and (4,3)boundaries:(0,)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Informat3ion Technology,JiangXiUniversity of Finance & Economics©2006Schoo4l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Essence of the Simplex The Essence of the Simplex MethodOperations ResearchOperations ResearchOptimal Test:Co6Cnclude that (2,6) is an optimal solution,so ¾Op:C6Otimal Test:Conclude that (0,6) is not an optimal solution. stop.(None of the adjacent CCPF solutions are better.)(AAnCPF adjacent CPF solution is better.)¾Iteration 2::MCPFu6Move to a better adjacent CPF soltion,(2,6), by The KeKy SoClution Conceptsperforming the following three steps. Solution concept 1:the simplex method focuses solely on CCPF solutions. For any problem with at least one optimal •B•Between the two edges of the feasible region that emanate solution,finding one required only finding a bestC CPF from6 (0,6),choose to move along the edge that leads to the . Solution concept 2:the simplex method is an iterative ••Stop at the first new constraint boundary encountered when alggorithm(a systematic solution procedure that keeps moving in that direction3:x+3:+2x==12. 12repeating a fixed series of steps, called an iteration, until a •desired result has been obtained) with the following •Solve for the intersection of the new set of constraint :ies :(26,6). 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo5l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technolog6y,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 4 运筹学Operations Research4-2运筹学运筹学 The Essence of the Simplex The Essence of the Simplex MethodOperations ResearchOperations ResearchSolution concept 5:After the current CPF solution is Solution concept 3:Whenever possible, the initialization of identified, the simplex method examines each of the edges of the simplex method chooses the origin(all decision variables the feasible region that emanate from this CPF solution. equal to zero)to beC the initial PCF solution. Each of these edges leads to an adjacent CPF solution at the other end, but the simplex method does not even take the time to solve for the adjacent PCF solution. Solution concept 4:GivenC a CPF solution, it is much quicker computationally to gather information about its adjacent Instead, it simply identifies the rate of the improvement in Zthat would be obtained by moving along the CCPF solutions than aboutC other CPF the edges with a positive rate of improvement ,each time the simplex method performs in ,Zit then chooses to move along the one with the largest an iteration to move from the current CCPF solution to a rate of improvement in .Zbetter one, it always chooses a CCPF solution that is adjacent to the current one.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog7y,JiangXiUniversity of Finance & Economics©2006School of8 Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Essence of the Simplex Method4 Solving LP Problems: Simplex MethodOperations ResearchOperations ResearchSolution concept 6:A positive rate of improvement in Setting Up the Simplex Methodimplies that the adjacent PCF solution is better than the current CPF solution(since we are assuming The preceding section stressed the geometric concepts maximization), whereas a negative rate of improvement in that underlie the simplex method. In this section,we introduce the algebraic language of the simplex method and Zimplies that the adjacent CPF solution is worse. relate it to the concepts of the preceding , the optimality test consists simply of checking whether any of the edges give a positive rate of improvement in none do, the current CPF solution is The algebraic procedure is based on solving systems of equations. Therefore,the first step in setting up the simplex is to convert the functional inequality constraints to equivalent equality constraints. This conversion is accomplished by introducing slack variables.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo9l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technology,JiangXiUniversity of Finance & Econom10ics©2006运筹学运筹学 Setting Up the Simplex Setting Up the Simplex MethodOperations ResearchOperations ResearchFor example,EExamplemaxz=3x+5x :12x4=<11x+0x≤4⎧12Original Form of the ModelThe slack variable for this constraint is defined to be ⎪0x+2x≤12⎪12x4=⎨3x+2x≤1812Whi⎪ch is the amount of slack in the left-hand side of the ⎪inequality. Thusx,x≥0⎩12x+x4=13Given this equation, x4=< if and only if 4-x=x0=>. maxz=3x+5x+0x+0x+0x113Augmented Form of the Model12345Therefore, the original constraint x4=< is entirely equivalent 11x+0x+x=4⎧123to the pair of constraints⎪0x+2x+x=12x⎪+x4= and x0=>.⎨3x+2x+x=18125⎪⎪x,x,x,x,x≥0⎩12345江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo11l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technolog12y,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 4 运筹学Operations Research4-3运筹学运筹学 Setting Up the Simplex Setting Up the Simplex MethodOperations ResearchOperations ResearchAn augmented solution：is a solution for the original A basic solution has the following properties:variables(the decision variables)that has been augmented by ¾EEach variables is designated as either anonbasicvariable the corresponding values of the slack a basic example, augmenting the solution (3,2) in the ¾The number of basic variables equals the number of example yields the augmented solution (3,2,1,8,5) because the functional constraints (now equations).Therefore,the corresponding values of the slack variables are x1=,x8=,and 34number of non-basic variables equals the total number of x5=.5variables minus the number of functional constraints.¾The non-basic variablesare set equal to basic solutionis an augmented corner-point solution.¾The values of the basic variablesare obtained as the simultaneous solution of the system of equationsA basic feasible (BF)solutionis an augmented CPF solution.¾If the basic variables satisfy the non-negativity constraints, the basic solution is a BF solution.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technologn13y,JiangXiUiversity of Finance & Economics©2006School of Informatn Technology,JiangXiUniver14iosity of Finance & Economics©2006运筹学运筹学 The Algebra of the Simplex Method4 Solving LP Problems: Simplex MethodOperations ResearchOperations The Algebra of the Simplex MethodThe choice of xand xto be thenonbasicvariables for the initial 12FB solution. eW set x=0, x=0 12We illustrate the procedure with following example:maxz=3x+5x12x=4⎧3⎧1x+0x+x=4123⎪⎪maxmaxz=3x+5x+0x+0x+0xz=3x+5x12123450x+2x+x=12x=12⎪12⎨.⎨3x+2x+x=18⎪1x+0x≤41x+0x+x=4⎧⎧125⎪12123x=18⎩5⎪⎪⎪x,x,x,x,x≥0⎩123450x+2x≤120x+2x+x=12⎪⎪ will soon see that when the set of basic variables changes, ⎨⎨3x+2x≤183x+2x+x=1812125⎪⎪the simplex method uses an algebraic procedure to convert the ⎪⎪x,x≥0x,x,x,x,x≥0equations to this same convenient form for reading every ⎩12⎩12345subsequent BF solution as well. This form is called proper form from Gaussian elimination.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology,JiangXiUniversity of Finance & Econom15ics©2006School of Information Technology,JiangXiUn16iversity of Finance & Economics©2006运筹学运筹学 The Algebra of the Simplex The Algebra of the Simplex MethodOperations ResearchOperations ResearchZ=3xx5+Optimality Test12Increase x? Rate of improvement in Z=3The objective function is 3Z=x+5x112Increase x? Rate of improvement in Z=5So 0Z= for the initial FB solution. 235,> so choose xto , we conclude that (0,0,4,12,18) is not indicated next, we call xthe entering basic variable for 2iteration any iteration of the simplex method, the purpose of step 1 is to choose onenonbasicvariable to increase from zero. Increasing Determining the direction of movement (step 1 of an iteration)thisnonbasicvariable from zero will convert it to a basic variable step 1:Increasing onenonbasicvariable from zero corresponds to for the next FB solution. Therefore, this variable is called themoving along one edge emanating from the CPF solution. entering basic variablefor the current iteration.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology,JiangXiUniversity of Finance & Econom17ics©2006Schoo18l of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 4 运筹学Operations Research4-4运筹学运筹学 The Algebra of the Simplex The Algebra of the Simplex MethodOperations ResearchOperations ResearcheW need to check how far xcan be increased without violating Determining where to stop (step 2 of an iteration)2thenonnegativityconstraints for the basic 2:addresses the question of how far to increase the entering basic variable xbefore stopping. Increasing xincreases Z, so we 22want to go as far as possible without leaving the feasible region. no upper bound on x2x4=≥03x≤122/ =6 (minimum)x1=22-x≥0242Increase x2Keep x0=1x1=82-x≥052x≤182/=92(1) x+x=4 x=4133•2x+x1=2 Thus, xcan be increased just to 6, at which point xhas x=122-x224442dropped to 0. (3) 3x+2x+x=18 x=182-x12552江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information TechnologngX19y,JiaiUniversity of Finance & Economics©2006School of Information Technology,JiangX20iUniversity of Finance & Economics©2006运筹学运筹学 The Algebra of the Simplex The Algebra of the Simplex MethodOperations ResearchOperations ResearchDetermine the leaving basic variable(0) Z3-x5-x=012(1) x+x4 = 13At any iteration of the simplex method, step 2 uses the (2) 2x+x=12 24minimum ratio test to determine which basic variable drops to zero (3) 3x+2x+x1=8 first as the entering basic variable is increased. Decreasing this 125basic variable to zero will convert it to anonbasicvariable for the On the original system of equations, we next FB solution. Therefore, this variable is called the leaving shall elementary algebraic operations to basic variablefor the current the current pattern of coefficients of x(0,0,1,0) as the new coefficients of x. 42Solving for the new FB solution (step 3 for an iteration)(0) Z3-x+52/x=301 4Since x=x=0, the new BF 14(1) x+x4 = step 3:Initial FB solutionNew FB solution13solution is (2) x+12/x6 = (x,x,x,x,x)=(0,,64,0,6), 2412345Nonbasicvariables :x=0,x=0 x=0,x=01214which yields Z=30.(3) 3x-x+x=6 1 45Basic variables : x=4,x=12,x1=8 x=?,x=6,x?=345325江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology,JiangXiUniversity of Finance & Econom21ics©2006Schooof22l Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Algebra of the Simplex The Algebra of the Simplex MethodOperations ResearchOperations Research3Optimally test for the new BF solution(0)Z+x+x=3645x==2, x=6=6,x==221233=3Z0+3x52-/xxincrease, ZZ will be 11141(1)x+x−x=2x===0,x=03454545increased33So, the new FB solution 1Z=36Z=36(2)x+x=624(x,x,x,x,x)=(0,,64,0,6) is 212345not optimal 11(3)x−x+x=214533Iteration 2: Increase x, xwill drop to 015Optimality test:whenever xand xincrease, the obZjective Z will xen45tering basic 451(0) Z3-x+52/x=30 1 not increased , so iteration is stopped4var5iable, x5 leaving basic (1) x+x =4 x≤4 131So, the new FB solution (x,x,x,x,x)=(2,6,2,0,0) is optimal. The variable12345(2) x+12/x6 = x6 = 242iteration proceeding finished. (3) 3x-x+x=6 x=63-x≥0,x≥21 45511江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo23l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technolog24y,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 4 运筹学Operations Research4-5运筹学运筹学 The Simplex Method in Tabular Form4 Solving LP Problems: Simplex MethodOperations ResearchOperations The Simplex Method in Tabular FormSummary of the Simplex MethodWe recommend the tabular formdescribed in this section Initialization:to solve slack variables. Select the decision variablesto be The tabular form of the simplex method records only the the initialnonbasicvariablesand the slack variablesto be the essential information ,namely,initial basic variables.(1) the coefficients of the variablesFor the example: We also take theWyndorGlass (2) the constants on the right-hand side of the for =3x+5x+0x+0x+0x12345(3) the basic variable appearing in each equation.⎧1x+0x+x=4123⎪0x+2x+x=12⎪.⎨First,we summarize the tabular form of the simplex 3x+2x+x=18125⎪method.⎪x,x,x,x,x≥0⎩12345江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of25 Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technolog26y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Simplex Method in Tabular The Simplex Method in Tabular FormOperations ResearchOperations ResearchBecause the coefficient of x, xin row 0 is negative, the of :12Basic Right sidecurrent BF solution is not (0)1-3-50000Step 1:Determine the entering basic variable by selecting x(1)01010043the variable with the negative coefficient having the largest x(2)002010124absolute value inEq(0). Put a box around the column below x(3)0320011885this coefficient, and call this the pivot the initial BF solution is (0,0,4,12,18).CoCefficient of :Basic sideOptimality Test: ABF solution is optimal if and only if variableZxxxxxZ12345every coefficient in row 0 is nonnegative(>0=).ZZ(0)1-3-50000If it is,stop;x(1)01010043Otherwise, go to an iteration to obtain the next BF (2)002010124x5(3)032001188江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo27l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technolog28y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Simplex Method in Tabular The Simplex Method in Tabular FormOperations ResearchOperations ResearchStep 2:Determine the leaving basic variables by applying the minimum Step 3: Solve for the new BF solution by using elementary ratio operations to construct a new simplex Ratio Test(1) Pick out each coefficient in the pivot column that is strictly positive(0>).variableZxxxxx12345(2) Divide each of these coefficients into the rihgt sideentry for the same Z(0)(1)01010043(3) Identify the row that has the smallest of these (2)002010124(4) The basic variable for that row is the leaving basic (3) of :Basic Right sidevariableZZxxxxx12345variableZxxxxx12345ZZ(0)1-3-50000Z(0)1-3005/2030x(1)01010043x(1)01010043x(2)002010124x(2)00101/2062x(3)0320011885x(3)0300-1165江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology,JiangXiUniversity of Finance & Econom29ics©2006School of Information Technolog30y,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 4 运筹学Operations Research4-6运筹学运筹学 The Simplex Method in Tabular Form4 Solving LP Problems: Simplex MethodOperations ResearchOperations ResearchIteration 2 for theE Example and the Resulting Optimal The Breaking in the Simplex MethodvariableZZx1x2x3x4x5In this section, we will discuss what to do if the various ZZ(0)1-3005//2030choice rules of the simplex method do not lead to a clear––cut x3(1)0101004x2(2)00101//(3)03001166Tie for the EEntering Basic VaVriablevariableZZx1x2x3x4x5Step 1 of each iteration chooses thenonbasicvariable ZZ(0)10003//21366having the negative coefficient with the largest absolute value x3(1)00011//3-1//32in the currentEEq.(0) as the entering basic variable. Now x2(2)00101//2066x1(3)0100-1//31//32suppose that two or morenonbasicvariable are tied for having the largest negative , the new BF solution is (2,6,2,0,0,),with 3= to the optimality test, we find that this solution is optimal The answer is that the selection between these contenders because none of the coefficients in row 0 is negative,so the may be is finished.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog31y,JiangXiUniversity of Finance & Economics©2006School of32 Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学Operations The Breaking in the Simplex MethodOperations The Breaking in the Simplex MethodTie for the Leaving Basic VaVriable--DegeneracyThird, if Zmay remain the same rather than increase at Now suppose that two or more basic variables tie for being each iteration, the simplex method may then go around in a the leaving basic variable in step 2 of an iteration. Does it loop, repeating the same sequence of solutions periodically matter which one is chose?? rather than eventually increasing Ztoward an optimal solution. First, all the tied basic variables reach zero simultaneously In fact, examples have been artificially constructed so that they as the entering basic variable is increased. Therefore, the one do become entrapped in just such a perpetual loop. or ones not chosen to be the leaving basic variable also will have a value of zero in the new BF solution. Second, if one of Fortunately, although a perpetual loop is theoretically these degenerate basic variables retains its value of zero untilpossible, it has rarely been known to occur in practical it is chosen at a subsequent iteration to be leaving basic , the corresponding entering basic variable also must remain zero, so the value ofZ Zmust remain unchanged. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of33 Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technolog34y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学Operations The Breaking in the Simplex MethodOperations The Breaking in the Simplex MethodmaxZ=x+xNo Leaving Basic VariableU—nbounded ZFor example12−2x+2x+3x≤6⎧123In step 2 of an iteration,if there is no variable qualified ⎪−3x+x−x≤5⎨to be the leaving basic outcome would occur if 123⎪the entering basic variable could be increased indefinitely x,x,x≥0⎩123without giving negative values to any of the current basic Right variableZZxxxxxvariables. In tabular form, this means that every coefficient 12345sidein the pivot column (excluding row 0) is either negative or ZZ0(0)1-1-1000x66zero. 4(1)0-22310x55(2)0-31-101The objective function Zwould increase indefinitely, so the simplex method would stop with the message that Zis In this problem, Zis unbounded, so it has no optimal .江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoof35l o Information Technology,JiangXiUniversity of Finance & Economics©2006Schoonom36l of Information Technology,JiangXiUniversity of Finance & Ecoics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 4 运筹学Operations Research4-7运筹学运筹学Operations The Breaking in the Simplex MethodOperations The Breaking in the Simplex MethodMultiple Optimal Solutionsmaxz=3x+2x12For example1x+0x+x=4⎧123A problem can have more than one optimal solution. The ⎪0x+2x+x=12⎪124simplex method automatically stops after one optimal BF .⎨3x+2x+x=18solution is found. But you can find them as follows:125⎪⎪x,x,x,x,x≥0⎩12345Whenever a problem has more than one optimal BF BasicZZxxxxxRight side12345solution,at least one of thenonbasicvariables has a coefficient ZZ(0)1-3-20000of zero in the final row 0,so increasing any such variable will x(1)01010043not change the value of .ZTherefore,these other optimal BF x(2)002010124solutions can be identified(if desired) by performing x(3)0320011885additional iterations of the simplex method,each time choosing anonbasicvariable with a zero coefficient as the entering basic variable.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog37y,JiangXiUniversity of Finance & Economics©2006Schoo38l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学Operations The Breaking in the Simplex The Breaking in the Simplex Method ResearchZZ(0)10-230012ZZ(0)100001188x(1)01010041x(1)01010041x(2)002010124x(2)00031-1664x(3)002-301665x(3)001-3/2/01/2/32ZZ(0)100001188Choose xas entering basic variable, and xleaving basic 34x(1)01010041variable, we get another optimal (2)00031-1664x(3)001-3/2/01/2/32ZZ(0)100001188x(1)0100-1/3/1/3/21In this table, the BF solution is an (2)00011/3/-1/3/23And wx(3)00101/2/066e notice, non-basic variable x3, its objective 2coefficient is zero, it means there is multiple solution for this problem. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo3940l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Adapting to other Model Forms4 Solving LP Problems: Simplex MethodOperations ResearchOperations Adapting to other Model FormsThis technique constructs a more convenient artificial Thus far we have presented the details of the simplex problem by introducing a dummy variable into each meconstraint that needs under the assumptions that the problem is in our standard form. In this section we point out how to make the adjustments required for other legitimate forms of the linear •Equality Constraintsprogramming model. Consider an equality constraints The only serious problem introduced by the other forms for ax+ax…++axb=functional constraints lies in identifying an initial BF solution. i11i22inniBefore, this initial solution was found very conveniently by Actually it is equivalent to a pair of inequality constraints:letting the slack variables be the initial basic variables, so that ax+ax +…+axb=<each one just equals thenonegativeright-hand side of its i11i22inniax+ax +…+axb=>equation. Now, something else must be done. The standard i11i22inniapproach that is used for all these cases is the artificial-variable technique. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog41y,JiangXiUniversity of Finance & Economics©2006Schoo42l of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 4 运筹学Operations Research4-8运筹学运筹学 Adapting to other Model Adapting to other Model FormsOperations ResearchOperations ResearchZ−3x−5x= the artificial-variable techniqueby introducing aFor example:theWyndor12nonegativeartificial variableinto it, just as if it were a slack Glass oC. problem is modified x+x=413variable. to2x+x= an overwhelming penalty to having x0> by i3x+2x=18changing the objective function12Z=ax+xbto Z=ax+bx-Mx1212iWhere Msymbolically represents a huge positive number.(it Unfortunately, these equations do not have an obvious is called the big Mmethod.)initial BF solution because there is no longer a slack variable to Z=3x+5x−Mxuse as the initial basic variable forqE.(3). It is necessary to find 125This system is not yet in an initial BF solution to start the simplex method. x+x=413proper form from Gaussian 2x+x=12elimination because xhas a We construct an artificial problemthat has the same optimal 245nonzero coefficient inEq(0).solution as the real problem by making two modifications of 3x+2x+x=18125the real problem.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of43 Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technolog44y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Adapting to other Model Adapting to other Model FormsOperations ResearchOperations ResearchNegative Right-Hand SidesTherefore, before the simplex method can apply the optimality test and find the entering basic variable, we must For dealing with an equality constraint with a negative obtain this form by subtracting fromqE(3).right-hand side,we multiply through both sides of an inequality by –1 also reverses the direction of the example:Z=3x+5x−Mx125Z=−18M+(3M+3)x+(2M+5)x12x+x=41x+x=4313Z=3x+5xZ=3x+5x12122x+x=122x+x=122424x≥4−x≤−411x+x+x=3x+2x+x=1832181212552x≤122x≤12223x+2x≤183x+2x≤181212江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo45l of Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo46l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Adapting to other Model Adapting to other Model FormsOperations ResearchOperations ResearchFunctional Constraints in >= FormminZ=+=++Mx++≤⎧++x=⎧123If there is a constraint⎪⎪+=6⎪++x=6⎪124ax++ax+…+ +…+ax=>b=>⎨.⎨+≥+−x+x≥6Then how we solve it?⎪?1256⎪⎪x≥0,x≥0⎪For example, ⎩12x≥0,x≥0⎩+0+.4x6=>6=>12Note that the coefficients of the artificial variables in the F+irst,we let = x06=.6x0=>+.4x-6, (x0=>, xis called a 31233surplus variable)objective function are M+ instead of –M, because we now are minimizing .Z thus, even though x0> and o/r x0> is possible Second, we introduce an artificial variable x,Then the 464constraint gives to for a feasible solution for the artificial problem, the huge unit penalty of M+ prevents this from occurring in an optimal +-x+x=6(x>=0, x>=0)123434solution.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog47y,JiangXiUniversity of Finance & Economics©2006School of Information Technolog48y,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 4 运筹学Operations Research4-9运筹学运筹学 Adapting to other Model Adapting to other Model FormsOperations ResearchOperations ResearchSUMMARY OF T EHTWO-PAHS EMTEOHDThe Two-Phase MethodInitialization:For example :RRevise the constraints of the original problem by introducing BMM:eZ=4+M+MBig M methodM:inimiz Z=+++Mx+Mx124646artificial variables as needed to obtain an obvious initial BFBF Since the first two coefficients are negMligible to ,M the two--phase solution for the artificial is able to drop M by using the following two objective functions with completely different definitions ofZ Z in 1:UUse the simplex method to solve the linear programming problem::MMinimize Z=Z=∑artificial variables,Two-P-PhaseM Method::subject to revised :Phase 1=iM:nimizeZ= Z=x++x(un=til x=0,x=0)46464646P:MZ=4Phase 2 :Minimize Z=+05+.5x(with x===0,x=0)124646The optimal solution obtained for this probZ=lem(with Z=0) will bea BFBF solution for the real problem.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo49l of Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo50l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Adapting to other Model Adapting to other Model FormsOperations ResearchOperations ResearchminZ=x+x46minZ=+++x=⎧++x= example⎧123⎪⎪++x=6⎨+=6⎨12⎪⎪+−x+x=6⎩+−x=656⎩125The only differences between these two problems are in the Phase 2:DDrop the artificial variables (they are all zero objective function and in the inclusion (phase 1) or exclusion (phase 2) of the artificial variables xandx. iWthout the artificial now anyway).StartFBing from the F Bsolution obtained at 46variables, the phase 2 problem does not have an obvious initial BF the end of phase 1, use the simplex method to solve the solution. The sole purpose of solving the phase 1 problem is to real a FBsolution with x=0 and x=0, so that thissolution can 46be used as the initial FB solution for phase 2.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog51y,JiangXiUniversity of Finance & Economics©2006Schoo52l of Information Technology,JiangXiUniversity of Finance & Economics©2006minZ=x+x运筹学运筹学 Adapting to other Model FormsOperations ResearchOperations ++x=⎧123⎪++x=6For example⎨ of :124Basic Right side⎪+−x+x=6Zx1x2x3x4x5x6⎩1256Z(0)-100-5/30-5/38/ of :Basic x1(1)010203/05/3-5/38Right sidevariablex4(2)0005/315/3-5/(3)001-100-553Z(0)-12x3(1)(2)(3)-116Z(0)-10001010x1(1)0100-4-556x3(2)00013/(0)-10-16/30113/(3)001065-56x1(1)011/3103/0009x4(2)001/3-5/(3) xhave been leaved basic variable, phase 1 6江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog5354y,JiangXiUniversity of Finance & Economics©2006School of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 4 运筹学Operations Research4-10运筹学运筹学 Adapting to other Model Adapting to other Model FormsOperations ResearchOperations ResearchPhase 2:delete column xx, and change objective to original 46Z(0) functionx1(1)0100-56x3(2) of :x2(3)001056Basic Right sidevariableZZx1x2x3x4x5x66ZZ(0)-10001010Z(0)(1)0100-4-5566x1(1)0100-56x4(2)00013//(2)(3)0010665-566x2(3)001056Z(0)(0)-100000x1(1)(1)0100-56x5(2)(2)(3)(3)001056江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog55y,JiangXiUniversity of Finance & Economics©2006School of Information Technolog56y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Adapting to other Model Adapting to other Model FormsOperations ResearchOperations ResearchIt is interesting to compare the big M and two-phase methods. Begin with their objective functions. Z(0)(1) M method:Minimize =+.5x+Mx+Mx1246x5(2)(3)-Phase Method:Phase 1:Minimize =Zx+x46Phase 2: Minimize =+ the Mxand Mxterms dominate the optimal test, this table is optimal, the optimal terms in the objective function for the big M method, this objective function is essentially equivalent to the phase 1 objective function as long as xand /or xis greater than zero. Then, when both x0= and 464solution is (x,x,x,x)=(, , 0, ) 1235x=0, the objective function for the big M method becomes 6completely equivalent to the phase 2 objective function.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo57l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technolog58y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Adapting to other Model Adapting to other Model FormsOperations ResearchOperations ResearchNo feasible SolutionsThe twop-hase method streamlines the Big M method by You have seen how the artificialv-ariable technique can be used to construct an artificial problem and obtain an initial Bfusing only the multiplicative factors in phase 1 and by droppingsolution for this artificial problem instead. sUe of either the iBg Mmethod or the two-phase method then enables the simplex method the artificial variables in phase 2. For these reasons, the two-to begin its pilgrimage toward the FB solutions, and ultimately phase method is commonly used in computer codes. toward the optimal solution, for the real problem. If the original problem has no feasible solutions, then either the big M method or phase 1 of the two-phase method yields a final solution that has at least one artificial variables greater ,they all equal zero.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo59l of Information TechnologSchoo60y,JiangXiUniversity of Finance & Economics©2006l of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 4 运筹学Operations Research4-11运筹学运筹学 Adapting to other Model Adapting to other Model FormsOperations ResearchOperations ResearchVariables with a bound on the negative values allowed:Variables Allowed to Be NegativeConsider any decision variablesxthat is allowed to have jnegative values which satisfy a constraint of the formIn most practical problems, negative values for the decision x≥L,jjvariables would have no physical meaning, so it is necessary to WhereLis some negative constraint can be jincludenonnegativityconstraints in the formulations of their converted to anonegativityconstraint by making the change of variableslinear programming models. ′′x=x−L,sox≥0jjjjSo any problem containing variables allowed to be negative musqt be converted to an equivalent problem involving only′ x+LThus would be substituted forxthroughout the jjjmodel,so that the redefined decision variablesx’cannot be jnonnegative variables before the simplex method is applied. negativeFFortunately, this conversion can be done.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo61l of Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo62l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Adapting to other Model Adapting to other Model FormsOperations ResearchOperations Research+−Set x=x−x111Variables with no bound on the negative values allowed:In the case wherexdoes not- have a lower-bound constraint in jthe model formulated,anotherq approach is required::xis replaced j+_throughout the model by the difference of the two nonnegative maxZ=3x+5xmaxZ=3x−3x+5x12112variables+−x≤4⎧⎧x−x≤4+-+-+-, +-111x==x->=-xwherex>=0,x>=>=⎪⎪2x≤122x≤12⎪2⎪.⎨⎨+_+-+-3x+2x≤18Sincexandxcan have any nonnegative va3x−3x+2x≤18lues, this 12⎪112jj⎪+-+-differencex--xcan have any value (positive or negative), so it is ⎪+−⎪jjxunsigned,x≥0x≥0,x≥0,,x≥0⎩12⎩112a legitimate substitute forxin the modeBl. uBt after such jsubstitutions, the simplex method can proceed with just nonnegative variables.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog63y,JiangXiUniversity of Finance & Economics©2006School of Information Technolog64y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Adapting to other Model Forms4 Solving LP Problems: Simplex MethodOperations ResearchOperations ResearchFrom a computational viewpoint, this approach has the Post-Optimality Analysisdisadvantage that the new equivalent model to be used has more variables than the original model. In fact, if all the original variables The posto-ptimality analysist-he analysis done after an optimal lack lower-bound constraints, the new model will have twice as many variables. Fortunately, the approach can be modified slightly solution is obtained for the initial version of the model—so that the number of variables is increased by only one, regardless constitutes a very major and very important part of most of how many original variables need to be replaced. This modification is done by replacing each such variablexby joperations research fact that posto-ptimality analysis ′′′′′′is very important is particularly true for typical linear x=x−x,wherex≥0,x≥0jjjprogramming this section,we focus on the simplex method in performing this , where ′ ′ is the same variable for all relevant j. x′′x>0江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo65l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technolog66y,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 4 运筹学Operations Research4-12运筹学运筹学4 Solving LP Problems: Simplex Method4 Solving LP Problems: Simplex MethodOperations ResearchOperations ResearchRe-optimizationShadow pricesAfter having found an optimal solution for one version of a Information on the economic contribution of the resources to linear programming model, we frequently must solve again for a the measure of performance would be useful. The simplex method slight difference version of the this information in the form of shadow pricesfor the nOe efficient approach is to reo-ptimization. eRo-ptimization respective deducing how changes in the model get carried along to *the final simplex tableau. This revised tableau and the optimal The shadow pricefor resource i(denoted byy) measures the isolution for the prior model are then used as the initial tableau margina lavuleof this resource, ., the rate at which Z could be and the initial basic solution for the new model. If this solution is increased by(slightly) increasing the amount of this resource(b) ifeasible for the new model, then the simplex method is applied in being made available. The simplex method identifies this shadow the usual way, starting from this initial FB solution. If the *price byy=coefficient of theithof the slack variable in row 0 of isolution is not feasible, a related algorithm called the dua lthe final simplex methodprobably can be applied to find the new optimal solution, starting from this initial basic solution.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Informat67ion Technology,JiangXiUn68iversity of Finance & Economics©2006School of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学4 Solving LP Problems: Simplex Method4 Solving LP Problems: Simplex MethodOperations ResearchOperations ResearchmaxZ=3x+5x12To illustrate, for theyWndorGlass Co. problem, b=hour of TheyWndorGlass Co. problemix+x=413production time per week being made available in plantifor 2x+x=1224these new optimal simplex tableau3x+2x=1812Providing a substantial amount of production time for the new variableZx1x2x3x4x5products would reuqire adjusting production times for the current products, so choosing the bvalue is a difficult managerial iZ(0)10003/2136decision. The tentative initial decision has beenx3(1)00011/3-1/32x2(2)00101/206b,=4 b1=2, b1=8x1(3)0100-1/31/32123However, management now wishes to evaluate the effect of y*=0 ,shadow price for resource 1 ;1changing any of the bvalues. The shadow prices for these three i y*=0 ,shadow price for resource 2 ;resources provide just the information that management needs. 2y*=0 ,shadow price for resource 3 3江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog69y,JiangXiUniversity of Finance & Economics©2006School of Information Technolog70y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学4 Solving LP Problems: Simplex Method4 Solving LP Problems: Simplex MethodOperations ResearchOperations Research¾The kind of information provided by shadow prices clearly ¾Now note why y0*=. Because the constraint on resource 1, 1is valuable to management when it considers reallocations of x≤4, is not binding on the optimal solution (2,6), there is a resources within the of this resource. Therefore, increasing bbeyond 4cannot 1¾It also is very helpful when an increase in bcan be iyield a new optimal solution with a larger value of Z. achieved only by going outside the organization to purchase moreof the resource in the marketplace. ¾By contrast, the constraints on resources 2 and 3, 2x≤12 2¾For example, suppose that Z represents profit and that the and 3x+2x≤18, are binding the limited 12unit profits of the activities include the costs of all the resources supply of these resources (b=12,b=18) binds Z from being 23consumed. Then a positive shadow price ofy* for resource 1 iincreased further, they have positive shadow prices. Economists means that the total profit Z can be increased byy* by irefer to such resources as scarce goods, whereas resources purchasing 1 more unit of this resource at its regular price. available in surplus are free goods. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology,JiangXiUniversity of Finance & Economics©200671Schoo72l of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 4 运筹学Operations Research4-13运筹学运筹学4 Solving LP Problems: Simplex Method4 Solving LP Problems: Simplex MethodOperations ResearchOperations ResearchWhWat is sensitive parameter??Sensitivity AnalysisA Asensitive parameter is the parameter that will need to be monitored particularly closely as the study is discussing the certainty assumption for linear programming, we point out that the values used for the model oHw are the sensitive parameters identified? In the case of the parameters are just estimates of quantities whose values will bi, you have just seen that this information is given by the shadow not become known until the linear programming study is prices provided by the simplex method. In particular, ify*>0, then implemented at some time in the optimal solution changes if bi is changed, so bi is a sensitive parameter. oHwever,y*=0 implies that the optimal solution is not iSo, the main purpose of sensitivity analysis is to identify sensitive to at least small changes in bi. oCnsequently, if the value the sensitive for bi is an estimate of the amount of the resource that will be available, then the bi values that need to be monitored more closely are those with positive shadow pricese—specially those with large shadow prices. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Informat73ion Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technolog74y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学4 Solving LP Problems: Simplex Method4 Solving LP Problems: Simplex MethodOperations ResearchOperations ResearchParametric linear programmingThe easiest way to analyze the sensitivity of each of theaijSensitivity analysis involves changing one parameter at a time parameters graphically is to check whether the corresponding in the original model to check its effect on the optimal solution. constraint is binding at the optimal solution. Because x≤4 is not 1a binding constraint, any sufficiently small change in its However, a more important application is the investigation of coefficients (a=0,a=0) is not going to change the optimal 1112tradeo-ffs in parameter values. For example, if thecvalues jrepresent the unit profits of the respective activities, it may be solution, so these are not sensitive parameters. nO the other hand, possible, if thecvalues represent the unit profits of the respective jboth 2x≤12 and 3x+2x≤18 are binding constraints, so 212activities, it may be possible to increase some of thecvalues at jchanging any one of their coefficients is going to change the the expense of decreasing others by an appropriate shifting of personnel and equipment among solution, and therefore these are sensitive parameters. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information TechnologangXiUniversity of Finance & Economics©200675y,JiSchoo76l of Information Technology,JiangXiUniversity of Finance & Economics© The Interior-Point Approach to Solving Linear 运筹学运筹学4 Solving LP Problems: Simplex MethodOperations ResearchOperations ResearchProgramming The Interior-Point Approach to Solving Linear Comparison with the Simplex MethodProgramming Problems1. Its algorithm is a polynomial time algorithm,but the simplex ¾During 1890sKarmarkardiscovered the interp-oint approach method’s algorithm is an exponential time solve linear programming approach appears to have great potential for solving huge linear programming 2. A key advantage of interiorp-oint algorithms is that large problems,but It still has not been fully do not require many more iterations than small ¾It is an iterative gets started by identifying a trail each iteration,it moves from the current trial solution to a better trial solution in the feasible region. It 3. The interior-point approach currently has very limited then continues this process until it reaches a trial solution that is capability in the posto-ptimality analysis .¾FForKKarmakar’’s algorithm,the trial solutions are interior points,.,points inside the boundary of the feasible region.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology,JiangXiUniversity of Finance & Econom77ics©2006School of Information Technolog78y,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 4 运筹学Operations Research4-14运筹学运筹学 Conclusions4 Solving LP Problems: Simplex MethodOperations ResearchOperations oCnclusions¾eW presented the full algebraic form of the simplex method to ¾The simplex method is an efficient and reliable algorithm for convey its logic,and then we streamlined the method to a more solving linear programming also provides the basis convenient tabular form. To set up for starting the simplex for performing the various par-ts of post-optimality analysis method,it is sometimes necessary to use artificial variables to very an initial FB solution for an artificial ¾so,either the big M method or the twop-hrase method is used AAlthough it has a useful geometric interpretation,the simplex method is an algebraic each iteration,it moves to ensure that the simplex method obtains an optimal solution from the currenBFBFt BF solution to a better,adjacent BF solution byfor the real both an entering basic variable and a leaving basic variable and then using Gaussian elimination to solve a system of linear the current solution has no adjacent ¾Interiorp-oint algorithms provide a powerful new tool for FBFB solution that is better,the current solution is optimal and the algorithm very large problems.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo79l of Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo80l of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 5 运筹学Operations Research5-1运筹学运筹学55 The Theory of the Simplex MMethodOperations ResearchOperations Research55 The Theory of the SimpMlex Method¾ oFundations of the Simplex MeMthodhCapter 4 introduced the basic mechanics of the ¾ The ReRvised SimpMlex eMthodsimplex method. Now we shall delve a little more ¾ A Fundamental Insightdeeply into this algorithm by examining some of its ¾ oCnclusionsunderlying theory.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo1l of Information Technology,JiangXiUniversity of Finance & Economics©2006School ofat2 Informion Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学55 TheM Theory of the Simplex Foundations of the Simplex MMethodOperations ResearchOperations Foundations of the Simplex eMthod¾The constraint boundary equation,which is obtained by QuQestions we must study further into:replacing= its≤,.=or≥s=ign by an =sign. CConsequently, the form of a constraint boundary ¾CCPF was done by only two decision varHiables ,How do equation is ax+a+x+…++…+ax=b=for functional constraints i11i22……innithese concepts generalize to higher dimensions when we deal andx==0 for nonnegative larger problems??¾The boundary of the feasible region, which contains just ¾Does it has a straightforward geometric interpretation those feasible solutions that satisfy one or more of the when we have more than 2 variables??constraint boundary equations.¾A corner-point feasCible(PCF) solution,which is a feasible To introduce some basic terminology for any linear solution that does not lie on any line segment connecting two programming problem with ndecision feasible solutions.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schooce & Economics©20063l of Information Technology,JiangXiUniversity of FinanSchool of Information Technolog4y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Founda5Ftions of the Simplex Foundations of the SimpMlex MethodOperations ResearchOperations ResearchajdAcent FPC SoltuionsoCnclusions we can draw:What meaning for Adjacent CPF solution?¾ n=2Each iteration of the simplex method moves from the current The feasible region being a polygon, the constraint CPF solution to an adjacent one. hWat is the path followed in boundaries being lines ,the path followed in an iteration from this process? hWat really is meant by adjacent CPF solution? one end of to other ,ie. from one CPF to an adjacent CPF as ¾illustrating in Fig . to comprehend adjacent CPF solution with n decision variables¾ n=3?The feasible region being a polyhedron, the constraint For any linear programming problem with n decision variables, boundaries being planes rather lines, any CPF lies in the each CPF solution lies at the intersection of n constraint intersection of three constraint boundaries, so in the next boundaries ;. it is the simultaneous solution of a system of n iteration it chooses one of the three edges until it reach the first constraint boundary constraint Boundaries .江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo5l of Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo6l of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 5 运筹学Operations Research5-2运筹学运筹学 Founda5Ftions of the SimpMlex Foundations of the Simplex MMethodOperations ResearchOperations Research¾ n3>,xThe constraint boundaries now are hyper planes. Instead 3of planes. nA edge of the feasible region is a feasible line segment that lies at the intersection of n1- constraint boundaries,where each endpoint lies on one additional constraint boundary(so xthat these endpoints are CPF solutions).1Two PCF solutions are adjacent if the line segment x2connecting them is an edge one leading to one of the n adjacent CPF iteration of the simplex method moves from the current CPF solution to an adjacent one by Feasible region and CPF solutions for a three –variable moving along one of these n programming problem江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo7l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technolog8y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Foundations of the Simplex Foundations of the Simplex MethodOperations ResearchOperations ResearchEExplanation about property 1:PrPoperties oPCf FPC SolutionsProperty 1:¾PProperty 1 is a rather intuitive one from a geometric (a) If there is exactly one optimal solution, then it must viewpoint. beCPF a CPF solution.¾FCFirst consider Case (a),for any problem having just one (b) If there are multiple optimal solutions(and a bounded optimal solution,it always is possible to keep raising the feasible region),then at least two must beCPF adjacent CPF objective function line until it just touches one point(the optimal ) at a corner of the feasible region.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology,JiangXiUniversity of Finance & Econom910ics©2006School of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 FFoundations of the SimpMlex Foundations of the SimpMlex MethodOperations ResearchOperations ResearchPC()Proof of aCse (a) of PProperty:1 y:1AA proof by contradiction::¾L''Let the vectors x' and x' denote these two other feasible ¾AAssuming that there is exactly one optimal solution also not soZlutions, and let Z1 and ZZ2 denote their respective beCPFWing a CPF then try to infer a contradiction and sowe objective function it .¾LLike each other point on the line segment connecting x'''' X*-X*-the solution assumed to be optimal;; and x'',Z*-Z*-its objective function value.¾Since we have assumed that the optimal solution x** is not a x*=*''+-'=x ''(+1-) x'PCFPCF solution,this implies that there must be two other feasible solutions such that the line segment connecting them contains the optimal solution.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo11l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technolog12y,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 5 运筹学Operations Research5-3运筹学运筹学 Foundations of the Simplex Foundations of the SimpMlex MethodOperations ResearchOperations ResearchCContradictions we have::¾FFor some value of such< that 0<< <. The first possibility implies that x''and x''''also are optimal ,which contradicts the assumption that there is exactly Thus*=ZZ+- *=Z Z2+(1-) optimal solution.¾Since the weights and 1--add to 1, the only possibilities for 2. BoBth the latter possibilities contradict the assumption that howZ*ZZ ,Z*Z1,and 2Z compare arex*CPF*(not a CPF solution) is optimal. The resulting conclusion is that *Z=Z=Z2Z<Z<*Z*1Z=Z=Z, and (2) Z13Z>*ZZ><Z<2*Z , and (3)Z1>*ZZ> is impossible to have a single optimal soCPFlution that is not aCPF property 1(a) holds!!江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo13l of Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo14l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Founda5FMtions of the Simplex Foundations of the Simplex MethodOperations ResearchOperations ResearchP:CPFProperty 2:There are only a finite number of CPF of Case (b)Py1: (b)of Property 1:¾ExEplana:CPFtion :each CPF solution is the simultaneous solution ¾Interpretation::of a system of n out of the m+q+n constraint boundary equations. NCNow consider Case (b),an objective functionhyperplaneThe number of different combinations+q of m+n equations taken n would keep getting raised until it contained the line at a time issegment(s) connecting two (or more) adjacentCPF CPF m+n⎛⎞(m+n)! a consequence,all optimal solutions can be ⎜⎟=⎜⎟nm!n!obtained as weighted averages of optCPFimal CPF solutions.⎝⎠¾Significance ::m: :the number of the constraint :T :the number of thenonnegativityfunctions,he realP significance of Property 1 is that it greatly simplifies the search for a optimal solution because now onlyCPF CPF solutions need to be considered.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology,JiangXiUnr15ivesity of Finance & Economics©2006School of Information Technolog16y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Founda5Ftions of the Simplex Foundations of the SimpMlex MethodOperations ResearchOperations Research¾P3:Property 3:QQuestion and Significance: :If a CPFCCPFPF solution has no adjacent CPF solutions that are ¾FFor a rather small linear programming problem with==5 m=n=50 299 better, then there are no better PCFPCF solutions anywhere. would have 100/!(5/!50!)!≈10systems of eqquations to be solved! !ThereforeCPF, such a CPF solution is guaranteed to be an optimal BBy contrast the simplex method would need to examine only solution, assuming only that the problem possesses at least one about 100 CPFCPF solutions for a problem of this size .optimal solution.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo17l of Information TechnologySchoo18,JiangXiUniversity of Finance & Economics©2006l of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 5 运筹学Operations Research5-4运筹学运筹学 Foundations of the Simplex FFoundations of the SimpMlex MethodOperations ResearchOperations ResearchEAExtension to the AugmenFPted Form of the Problemafter EExplanationslack variables are as follows::The basic reasonP that Proper3ty 3 holds for any linear ax++ax+…++=+…+ax+x=b1111221nnn++11ax+…+=+ax++…+ax++x=b2112222nnn++212programming problem is that the feasible region always has the ………………property of being a convex set, which means that its boundary ax++ax+…+=+…+ax++x=bm11m22mnnn++mmcannot“ “bend outward””beyond an adjacent CPFCPF solution to give WhWerex…,x, …,xare slack variablesn++1n++2n++m .Thus the original solutions (x1,x2,…,xn) turns to be (x1, x2,…,xn,anotherCPF CPF solution that lies on the favorable side of thexn+1,xn2+,…,xnm+).hyperplane.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo19l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technolog20y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 FMFoundations of the Simplex Foundations of the Simplex MMethodOperations ResearchOperations Research¾TheE: answer is E:ach constraint has an indicating variables that ¾RRecall that each corner--point solution is the simultaneous completely indicates (by whether its value is zero) whether solution of a system of n constraint boundaryq equations, called that constrain’t’s boundaryq equation is satisfied by the current its defining .¾The keyq:H question is :How to tell whether a particular ¾Indicating variable==0 ,then constraint boundaryq equation consqqtraint boundary equation is one of the defining equations satisfied; ;when the problem is in augmented form??Indicating variable not==0 ,then constraintq boundary equation viola;ted ;江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo21l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technolog22y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Founda5Ftions of the Simplex Foundations of the SimpMlex MethodOperations ResearchOperations ResearchThe conclusions are::AbAout the basic feasible solutions¾EEach basic solution has mbasic variables, and the rest of ¾D:Definition :Itis a basic solution where all mbasic variables are non--negative(>=>=0).the variables are non--basic variables seqt equal to zero.¾D:Degeneration soluBFtion:a BF solution is said to be degenerate ¾The values of the basic variables are given by the if any of these m variables eqquals solution of the system ofq mequations for the ¾A:tAtention:it is possible for a variable to be zero and still not problem in augmented a non--basic variable for the currenBFt BF solution. Therefore, ¾This basic solution is the augmented corner--point solution it is necessary to keep track of which is the current set of non--whose ndefining eqquations are those indicated by the non--basic basic variables rather than to reply upon their zero .江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo23l of Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo24l of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 5 运筹学Operations Research5-5运筹学运筹学5M5 The Theory of the Simplex The Revised Simplex MMethodOperations ResearchOperations The eRvised Simplex MeMthod¾N:MNecessity :Many resources are wasted!!The steps we: take:The simplex method is not the most efficient computational procedure for computers because it computes and stores many ¾DeDscribe the problem in that are not needed at the current iteration and that may be irrelevant for decision making at subseqquent iterations. ¾Study the problem in the light of matrix.¾M:HMethod :How to improve? it?It would be very useful to have a procedure that could obtain the necessary information efficiently without computing and storing all the other--- coefficients-----using matrix method.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo25l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of26 Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 TheR Revised Simplex The Revised Simplex MMethodOperations ResearchOperations Describe the problem in matrix.(2) To obtain the augmented form of the problem, introduce (1) rWite the matrix form for the problemthe column vector of slack ⎡⎤n+1maxZ=cx⎢⎥wherexC=[c,c,L,c]n+212nX⎡⎤⎢⎥X=Ax≤b⎧s≥0⎢⎥M⎢⎥⎨X⎣s⎦⎢⎥x≥o⎩x⎣n+m⎦0⎡⎤aaKaxb⎡⎤⎡⎤⎡⎤n1111121So that the constraints become⎢⎥⎢⎥⎢⎥⎢⎥0baaKax21222n22⎢⎥⎢⎥⎢⎥⎢⎥0=Αb==X=X⎡⎤⎢⎥M⎢⎥⎢KKKK⎥M⎢⎥M[Α,Ι]=b⎢⎥⎢⎥X⎢⎥⎢⎥⎢⎥⎣s⎦0xaaKabm⎣⎦⎣n⎦⎣⎦⎣m1m2mn⎦江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of27 Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo28l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The5R Revised Simplex The Revised Simplex MMethodOperations ResearchOperations Research(2) Find the matrix form for X2. SolvBing for a Basic Feasible Solution(1) eDscribe the problem in matrix basis matrixBBKB⎡⎤11121m⎢⎥BBKBX⎡⎤21222m⎢⎥B=[Α,Ι]=b⎢⎥⎢KKKK⎥X⎣s⎦⎢⎥BBKB⎣m1m2mm⎦DDenotedBXB= byBXB =b1-x⎡⎤To solve BXb=, both sides arepremultipliedby B:B1B⎢⎥xB2⎢⎥1-1-X=hWere the vector of basic variablesΒBXB= Bb,B⎢⎥M⎢⎥1-xSo X= Bb.⎣Bm⎦B江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006SchooSchoo30l of Information Technology,JiangXiUniversity of Finance & Econom29ics©2006l of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 5 运筹学Operations Research5-6运筹学运筹学 The Revised Simplex The Revised Simplex MethodOperations ResearchOperations Researchor examplemaxZ=3x+5xF123F(3) iFnd the matrix form for objective ≤4⎧1⎪WCiWth Cbe the coefficienXts of ≤12BB⎨ 2⎪3x+2x≤18⎩12--1So C=ZXC=B C=ZXC= BvariableZZxxxxx12345ZZ(0)10003/2/1366x(1)00011/3/-1/3/23x(2)00101/2/0662x(3)0100-1/3/1/3/21江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schooom31l of Information Technology,JiangXiUniversity of Finance & Econics©2006Schoo32l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Revised Simplex The Revised Simplex MethodOperations ResearchOperations ResearchMaFMtrix oFrm of the CuCrrent Set of qEuqEations¾Step2::premultiplyboth sides of the original seqt of equations --1byB B.¾Step 1: :FoFr the original set of equqations, the --1--1¾S3:tep3a:fter any iterationX=B, X=Bb andZ=CB- Z=CBb,so the right-BBBBmatrix form isand sides ofq the new set of equations have becomeΖ⎡⎤−1−1Ζ⎡0⎡⎤1cΒ⎤⎡cb⎤⎡⎤ΒBB1−c00⎡⎤⎡⎤==⎢⎥⎢⎥⎢−1⎥⎢⎥⎢⎥−1x=xΒb0Βb⎢⎥⎢⎥⎣B⎦⎣⎦⎣⎦⎣⎦⎢⎥0ΑΙb⎣⎦⎣⎦⎢⎥x⎣s⎦江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog33Schoo34y,JiangXiUniversity of Finance & Economics©2006l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Revised SimpMlex The Revised Simplex MMethodOperations ResearchOperations Research¾WiWth the same way------matrixpremultiplyInitial and later simplex tableaux in matrix of :−1−1−1⎡⎤1−c0⎡⎤1cB⎡⎤1cBΑ−ccBasic BSlack BBBiterationRight sideOriginal =variable⎢⎥⎢⎥⎢⎥ZZvariable−1−1−1variables0ΑΙ0B0BΑB⎣⎦⎣⎦⎣⎦sZZ(0)1-c000XX(1,2,……,m)0AIbBΖ⎡⎤−1−1−1⎡1cΒΑ−ccΒ⎤⎡cΒb⎤⎢⎥BBB………………………………x=⎢⎥⎢⎥−1−1−1⎢⎥0ΒΑΒΒb⎣⎦⎣⎦-1-1-1⎢x⎥AnyZZ1CCCCBA-CCCBBCBBb(0) ⎣s⎦B-1-1-1(1,2,……,m)XX0BABBbB江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo35l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of In36formation Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 5 运筹学Operations Research5-7运筹学运筹学 The Revised SimpMlex The Revised Simplex MMethodOperations ResearchOperations ResearchSummary of the ReRvised SimplexM eMthodThe Overall Procedure1. Initialization: :Same as for the original simplex -1-¾1. Only Bis needed:only Bneeds to be derived to be able 2. Iteration: : to calculate all the numbers in the simplex tableau from the SStep 1D: 1eDtermine the entering basic variablesS:ame as for the original parameters (A,b,C) of the simplex 2StepD: 2Determine the leaving basic variablesS:ame as for the ¾2. The second is :any one of these numbers can be obtained original simplex method,expect calculate only the numbers by performing only a vector multiplication instead of a reqquired to do matrix multiplication. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of37 Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technologr38y,JiangXiUnivesity of Finance & Economics©2006运筹学运筹学 The Revised Simplex The Revised Simplex MMethodOperations ResearchOperations ResearchGeneral ObOservations--1S:DF:DBStep 3 3:eDtermine the mew BBF solution :Derive Band setThe advantages of the revised simplex method::--1x=B=Bb. (CCalculatingxis optimal unless the optimality test finds BBBB¾The number of arithmetic computations may be reduced.¾The amount of information that must be stored at each it to be optimal.)iteration is :3. Optimality test:Same as for the original simplex method, ¾It permits the control of the rounding errors inevitably generated by computers. expecqt calculate only the numbers required to do this test, ., the ¾Some of the pos-t-optimality analysis problems can be coefficients of the non--basic variablesEq inEq.(0).handled more conveniently with the revised simplex method.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog39y,JiangXiUniversity of Finance & Economics©2006Schoo40l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 A uFndamental Insight55 The Theory of the SimpMlex MethodOperations ResearchOperations Research53AFuQestions we may ask: A Fundamental Insight¾First, Suppose you are given the initial tableau, t and T, and ¾OOur new focus:W :eW shall now focus on a property of the simplex method that has been revealed by the revised just y* and S* from the final tableau. oHw can this information simplex method in the preceding section. This fundamental alone be used to calculate the rest of the final tableau? insight provides the key to both duality theory and sensitivity ¾Second, how y* and S* themselves can be calculated analysis, two very important parts of linear -1-(y*=CBand S=B*) by knowing the current set of basic ¾VeBVrbal description of fundamental:A insight :fAter any iteration, the coefficients of the slack variables in each eqquation variables and so the current basis matrix B? immediaqtely reveal how that equation has been obtained from the initialqlequations.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of41 Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo42l of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 5 运筹学Operations Research5-8运筹学运筹学 A uFndamental Insight55 The Theory of the Simplex MMethodOperations ResearchOperations ResearchAbout these ,there are lots of interesting work to do ,which oCnclusionsis good for our students to study it intensively. we summarized ¾AAlthough the simplex method is an algebraic procedure, it is based on some fairly simple geometric concepts as following: enable one to use the algorithm to examine only a relatively small Fundamental Insight:number ofBF BF solutions before reaching and identifying an optimal solution.(1) t*=t +y*T= [y*A-c y* y*B.](2) T*= S*T =[S*AS * Sb*.]江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo43l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technolog44y,JiangXiUniversity of Finance & Economics©2006运筹学 ConclusionsOperations Research¾The revised simplex method provides an effective way of adapting the simplex method for computer implementation.¾The final simplex tableau includes complete information on how it can be algebraically reconstructed directly from the initial simplex tableau. This fundamental insight has some very important applications, especially for post-optimality analysis.江西财经大学信息管理学院©2006Schoo45l of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 6 运筹学Operations Research6-1运筹学运筹学66 Dual theory and sensitivity analysisOperations ResearchOperations Research¾Important discoveries66 Dual theory and sensitivity analysisOne of the most important discoveries in the early development of linear programming was the concept of duality ¾Dual theory revealed that every linear and its many important ramifications. This discovery revealed that every linear programming problem has associated with it programming problem has associated with it another linear programming problem called the dual. The another linear programming called the dual . so relationships between the dual problem and the original problem prove to be extremely useful in a variety of duality theory and Sensitivity analysis are just starting from the relationship between the dual and¾ SignificanceOne of the key uses of duality theory lies in the prime and implementation of sensitivity analysis.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology,JiangXiUniversity of Finance & Econom1ics©2006School of Information Technology,JiangXiUniversity of Finance & Econom2ics©2006运筹学运筹学66 Dual theory and sensitivity analysis66 Dual theory and sensitivity analysisOperations ResearchOperations The sEsence of Duality TheoryCoCntents¾ The essence of duality theoryPrimal problemDual problem¾ Economic interpretation of dualitynmmaxZ=cx∑¾jj6miny= Primal-dual relationships0∑iij=1i=1n¾ Adapting to other primal forms⎧max≤b⎧∑⎪ijjiay≥cj=.⎪∑ijij⎨¾ The role of duality theory in sensitivity .⎨i=1⎪x≥0j⎩⎪y≥0¾66⎩ The essence of sensitivity analysismaxZ=CX¾ Applying sensitivity analysisminy=yb0AX≤b⎧yA≥c¾⎧ .⎨.⎨X≥0y≥0⎩⎩江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog3y,JiangXiUniversity of Finance & Economics©2006School of Informat4ion Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Essence of Duality The Essence of Duality TheoryOperations ResearchOperations ResearchxEamples in algebraic formPrimal Problem in Dual Problem in rPimal rPoblemmatrix formmatrix formDual Problem4⎡⎤x⎢⎥⎡⎤MaxZ=3x+5x1MinY=[yyy]1212MaxZ=[12335]⎢⎥⎢⎥MinY=x4y+12y+18y⎣2⎦013⎢18⎥⎣⎦x≤4⎧1⎧104⎡⎤⎡⎤⎪y+3y≥3⎧10⎧⎡⎤1x3⎡⎤⎪2⎢⎥1⎢⎥x≤12⎪2⎪≤12⎢⎥⎪⎨⎢⎥⎢⎥⎢⎥.[yyy]02≥[35].⎨x123⎨⎢⎥⎣2⎦y+3y≥5⎪⎨23⎢⎥⎢3218⎥⎪3x+2x≤18⎣⎦⎣⎦⎩12⎢⎥32⎪⎣⎦⎩⎪y,y,y≥0⎪⎩123x,x≥0⎩12江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo5l of Information Technolog6y,JiangXiUniversity of Finance & Economics©2006School of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 6 运筹学Operations Research6-2运筹学运筹学 The Essence of Dua6lity The EEssence of Duality TheoryOperations ResearchOperations ResearchThe primal-dual tablefor linear programmingA direct correspondence between these entities in the two problemsprimal problem ¾The parameters for a constraint in either problem are the xxLx12ncoefficients of a variable in the other problem. yaaLa1≤b11121n1¾The coefficients for the objective function of either Dual aaLay21222n≤b22problem are the right sides for the other problem. problemMMMMM¾EEntities in Primal and Dual ProblemsaaLam1m2mn≤bymnCConstraint i variable iccLcObjective function Right sides12n江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo067l of Information Technology,JiangXiUniversity of Finance & Economics©20Schoo8l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Essence of Duality The Essence of Duality TheoryOperations ResearchOperations Research¾The original Model¾optimality test:mmy=by∑0iii=1y=yb=by∑Conditionfor optimality0iimi=1⎧ay≥c⎪∑=1⎨⎪y≥0z=yAsoz=ay∑⎩ijijii=1z−c≥0This goal is expressed jjsymbolically as follows:y≥0i江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog9y,JiangXiUniversity of Finance & Economics©2006Schoo10l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Essence of Duality The Essence of Duality TheoryOperations ResearchOperations ResearchQQuestionsThe optimality testSo we can find y from¾why we study the dual problem of the prime for the prime problemThe optimality is the relationship between the pair of the i=1problems?? my≥0⎧iay≥c∑⎪ijij¾Only apply the simplex method to the prime =1⎨problem, then get the key for the dual ,can we??⎪y≥0⎩i江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo00611l of Information Technology,JiangXiUniversity of Finance & Economics©2Schoo12l of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 6 运筹学Operations Research6-3运筹学运筹学 The Essence of Dua6Elity The Essence of Duality TheoryOperations ResearchOperations ResearchrPoveuDality TheorySupposeB is aif prime problem: max z feasible base,C= ;X A+XXb= ;,XX>0=SS≥then AB(,=N);dual problem: min wY=b ; YA-Y C;=Y,Y0=>SSThenthe prime problem iwll be :then the optimality test of the prime problem consist with thema xz =CBXB+CNXNbasBXB+ NXN+ Xs=b ic solution of dual problem .XB,X, Xs≥0Its correspondence can be seen in the below dual problem iwll be:XxXSNBminωb=Y-1B-YSY1C=B1--CB0BC-CBNNBN-YSY2C=N,YSY1,SY2 ≥0Y-YS1S2-YHereiYS(=Y1, SY2)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog13y,JiangXiUniversity of Finance & Economics©2006Schoo14l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Essence of Duality The Essence of Duality TheoryOperations ResearchOperations of :Basic Right Whenfind one solve of the prime X:=B-1b Bvariableside-1-1Zxx……xxx……x12nn1+n2+nm+ThenC-CBN and -CBbe the NBBoptimality (0)1z-cz-c……z-cyy……yy1122nn12m0-1DenoteC=YB, connect it iwth BB-YY=Cand N-YYC=Primal problemDual problemS1BS2NIterationy0Row 0yyyz-cz-c1231122SoY0=S1-1-YC=-CBN0000-3-50[S2 [-3, -5 0 , 0 0 0 ] ] NB105/2/0-3030[[-3 , 0 0, 5//2, 0 30]] [0/[ , 0 0, 32/, 1 36]6]203/2/100366江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schooof15l Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo6l of Information Technolog1y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Essence of Duality The Essence of Duality TheoryOperations ResearchOperations ResearchComplementary-solutions propertyCoCmplementary-solutions property¾The complementary-solutions property also holds at the ¾At the final iteration, the simplex method final iteration of the simplex method, where an optimal simultaneously identifies an optimal solution x* for the solution is found for the primal problem. primal problem and a complementary optimal solution cx=by=byy* for the dual problem (found in row 0,the ¾If x is not optimal for the primal problem, then y is not coefficients of the slack variables), wherefeasible for the dual problem. cx*y=*b¾Symmetry propertyThe y* are the shadow prices for the primal any primal problem and its dual problem, all ¾If x is not optimal for the primal problem,then y is not relation ships between them must be symmetric because the feasible for the dual problem is this primal problem.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo17l of Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo18l of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 6 运筹学Operations Research6-4运筹学运筹学 The Essence of Dua6lity The EEssence of Duality TheoryOperations ResearchOperations ResearchDuality theoremApplications¾If one problem has feasible solutions and a bounded objective function (and so has an optimal solution),then so does¾Another important application is its use in the economic the other problem,so both the weak and strong duality interpretation of the dual problem and the resulting insights properties are analyzing the primal problem.¾If one problem has feasible solutions and an unbounded ¾One important application of duality theory is that the objective function(and so no optimal solution),then the other problem has no feasible problem can be solved directly by the simplex method ¾If one problem has no feasible solutions,then the other in order to identify an optimal solution for the primal problem has either no feasible solution or an unbounded function.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog19y,JiangXiUniversity of Finance & Economics©2006School of Information Technolog20y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学66 Dual theory and sensitivity The Essence of Duality TheoryOperations ResearchOperations Economics Interpretation of Duality¾The weak and strong duality properties describe key ¾The interpretation of the primal problem leads to an relationships between the primal and dual interpretation for the dual problem, note in table ¾One useful application is for evaluating a proposed that y0 is the value of Zat the current iteration. Becausesolution for the primal problems. y=by+by+L+by01122mm¾Each biyican thereby be interpreted as the current ¾For example, suppose that x is a feasible solution that has contribution to profit having bi units of resource i available been proposed for implementation and that a feasible for the primal y has been found by inspection for the dual ¾Thus aVriableyiis interpreted as the contribution to profit per unit of resource i (i1=,2,,…m), when the current set problem such thatcx==yb. In this case, x must be optimal of basic variables is used to obtain the primal solution. In without the simplex method even being applied. other words, theyivalues are just the show prices. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo21l of Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo22l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Economics Interpretation of Economics Interpretation of DualityOperations ResearchOperations ResearchThe interpretation of marginal value of resource¾This interpretation of the dual variables leads to our interpretation of the overall dual problem. Specifically, ¾The objective can be viewed as mmsince each unit of activity j in the primal problem consumes inimizing the total implicit value of miny=by0∑iiunits of resource i,the resources consumed by the i=∑¾So,ijii=1¾This interpretation can be sharpened mis interpreted as the current contribution to profit of somewhat by differentiating between ay=c,ifx>0∑ijijji=1the mix of resources that would be consumed if 1 unit of basic andnonbasicvariables in the y=0,ifx>0in+iactivity j were used .(j1=,2,,…n).primal problem for any given BF solution.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schooe & Economics©200623l of Information Technology,JiangXiUniversity of FinancSchool ofn24 Iformation Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 6 运筹学Operations Research6-5运筹学运筹学 Economics Interpretation of Economics Interpretation of DualityOperations ResearchOperations Research¾If an original variablexjisnonbasicso that activity j Interpretation of the simplex methodis not used, then the current contribution to profits of ¾The interpretation of the dual problem also provides an the resources that would be required to undertake each economic interpretation of what the simplex method does in unit of activity jmthe primal problem. The goal of the simplex method is to find ay∑ijii=1how to use the available resources in the most profitable feasible way. ¾It may be either smaller()< or large(≥) than the unit ¾To attain this goal, we must reach a BF solution that profitcjobtainable from the activity. If it is smaller, so thatsatisfies all the requirements on profitable use of the zj-cj0< in row 0 of the simplex tableau, then these resources resources. can be used more profitably initiating this activity.¾These requirements comprise the condition for optimality for the algorithm. For any given BF solution,江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of2526 Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学66 Dual theory and sensitivity Economics Interpretation of DualityOperations ResearchOperations Research¾If it is larger, then these resources already Primal-Dual Relationshipsare being assigned elsewhere in a more ¾Because the dual problem is a linear programming profitable way, so they should not be diverted to problem it also has corner-point solutions. activity j¾By using the augmented form of the problem, we can ¾Therefore, what the simplex method does is express these corner-point solutions as basic solutions. to examine all thenonbasicvariables in the ¾This surplus iscurrent BF solution to see which one can provide ma more profitable use of the resources by being z−c=ay−c∑jjijiji=1increased. ¾Thuszj-cjplays the role of the surplus variable for constraint j.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo27l of Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo28l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Primal-Dual Primal-Dual RelationshipsOperations ResearchOperations Research¾Because of symmetry property , the correspondence Complementary basic solutionsbetween basic solutions in the prime and dual is a symmetric one , a pair of Complementary basic solutions has the same ¾One of the important relationships between the primal objective function dual problems is a direct correspondence between their basic solutions. So a complete solution for the dual problem Primal variableAssociated dual variablecan be read directly from row 0. (original variable)xz-z-c(surplus variable)j=…, j1=,2,…,njjjjjj(sylack variable)x=y(original variable) i1…=,2…,mn+i+i¾A key insight here is that the dual solution read from row 0 must also be a basic solution.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog29y,JiangXiUniversity of Finance & Economics©2006School of Information Technolog30y,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 6 运筹学Operations Research6-6运筹学运筹学 Primal-Dual Primal-Dual RelationshipsOperations ResearchOperations ResearchComplementary basic solutions property¾Each basic solution in the primal problem has a oCmplementary slackness relationship for Complementary basic solutions complementary basic solution in the dual problem, where their respective objective function values ( Zand y0) are slackness property¾Given the association between variables in table , the variables in the primal basic solution and the complementary dual basic solution satisfy the complementary slackness relationship shown as below.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo31l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of32 Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Primal-Dual Primal-Dual RelationshipsOperations ResearchOperations ResearchComplementary optimal basic solutions propertyRelationships between complementary ¾Each optimal basic solution in the primal problem has a basic solutionscomplementary optimal basic solution in the dual problem, ¾Only when the pair of the Complementary basic solutions where their respective objective function values (z and y0) are feasible in the mean time ,the solutions are equal. ¾Explanation: it is supported the strong duality property ¾To review the reasoning behind this property, note that that optimal and dual solutions have Y=Z0 the dual solution (y*,z*-c) must be feasible for the dual ¾Feasible feasible problem because the condition for optimality for the primal problem requires that all these dual variables be nonnegative. optimalSince this solution is feasible, it must be optimal for the dualproblem by the weak duality property.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo33l of Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo34l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学66 Dual theory and sensitivity Primal-Dual RelationshipsOperations ResearchOperations Research¾One is the condition for feasibility, namely, whether all the Adapting to Other Primal Formsvariables in the augmented solution are nonnegative. ¾The other is the condition for optimality, namely, whether ¾Any linear programming problem, whether in standard all the coefficients in row 0 are nonnegative. form or not, possesses a dual problem. This section focuses on how the dual problem changes for other primal condition for optimality?¾We can convert each nonstandard form to an equivalent Feasible?yesnoform. Hence you always have the option of converting any yesOptimal Suboptimal model to our standard form and then constructing its dual noproblem in the usual way. Superoptimal Neither feasible norsuperoptimal江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog35y,JiangXiUniversity of Finance & Economics©2006School of36 Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 6 运筹学Operations Research6-7运筹学运筹学 Adapting to Other Primal Adapting to Other Primal FormsOperations ResearchOperations ResearchCConversions to Standard FormThe symmetry property of the Primal and dualNonstandard FormEqEuivalent Standard FormmnMinimize ZZMaximize (-Z)Zcx=Zy=by∑∑jj0iij=1i=1Suboptimal nSuperoptimal n−ax≤−b∑ijjiax≥b∑j=1ijjij=1nnnpOtimal Z*−ax≤−bax≤bandy∑ijji*0 (optimal)∑ijjiax=bj=1∑ijjij=1j=1+−+−x−x,x≥0,x≥0jjjjSuperoptimal Suboptimal 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo37l of Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo38l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Adapting to Other Prima6l Adapting to Other Primal FormsOperations ResearchOperations Research¾Since any pair of primal and dual problems can be ¾One shortcut is that an equality constraint in the prime converted to these forms, this fact implies that the dual of theproblem should be treated just like a =<constraint in the dual problem is the prime problem, all relationships constructing the dual problem except that the correspondingbetween them must be unconstrained in sign . ¾One consequence is that the symmetry property is that all the statements made earlier in the chapter about the ¾Another shortcut involves functional constraints in =>relationships of the dual problem to the primal problem also form for a maximum straightforward approach hold in begin by converting each such constraint to =<form¾Another consequence is that it is immaterial which nnproblem is called the primal and which is called the dual. ax≥b→−ax≤−b∑ijji∑=1j=1江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog39y,JiangXiUniversity of Finance & Economics©2006School of40 Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Adapting to Other Primal Adapting to Other Primal FormsOperations ResearchOperations ResearchCCorresponding Primal-Dual Forms---symmetry property¾How to find the corresponding prime –dual formsMaximize Z Z(or y) Minimize y(or Z)Z00¾Choose one column for the primal problem according to CConstraint i: VaVriabley(or x)ii=<=<formy=>0=>iwhether this problem is in maximization form(left column) or ==form Unconstrained=>f=>ormy’=<’0=<iminimization form (right column).VaVriablex(ory): CConstraint j:jjx=>0=> >>=f=ormjUnconstrained =f=orm¾Then read off the form of the dual problem from the x’=<’0=< =<f=<ormjother column.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo41l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technology,JiangXiUniversity of Finance & Econom42ics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 6 运筹学Operations Research6-8运筹学运筹学 Adapting to Other Pr6imal Adapting to Other Primal FormsOperations ResearchOperations ResearchEExample¾When artificial variables are used to help the simplex Primal ProblemDual Problemmethod solve a primal problem, the duality interpretation of Maximize -=Z=, Minimize y2=.7y6+y6+y'120123row 0 of the simplex tableau is the tosubject to ¾After all the artificial variables becomenonbasic, we are +0=<+.1x2=<.7 y0=>121back to the real primal and dual problems. +0+.5x6=6=yunconstrained in sign122 ¾With the two-phase method, the artificial variables would +06=>+.4x6= >y0=<’123to be retained in phase 2 in order to read off the complete dualandand x=>0=>+.5y0solution from row 0. With the Big M method, since M has been +.6y '-=> x=>0=>+.5y0+.4y '-=> to the coefficient of each artificial variable in row 0, the 2123 current value of each corresponding dual variable is the current coefficient of this artificial variable minus M.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information TechnologUn43y,JiangXiiversity of Finance & Economics©2006School of Informatn Technolog4y,JiangXiUniver4iosity of Finance & Economics©2006运筹学运筹学 The role of duality theory in sensitivity analysis6 Dual theory and sensitivity analysisOperations ResearchOperations ResearchCCChanges in the Coefficients of The role of duality theory in sensitivity ¾Suppose that the changes made in the original model occur analysisin the coefficients of a variable that wasnonbasicin the original ¾Changing parameter values in the primal problem also optimal solution. changes the corresponding values in the dual problem. ¾Because the variable involved isnonbasic, changing its coefficients cannot affect the feasibility of the solution. Therefore, you have your choice of which problem to use to Therefore, the open question in this case is whether it is stillinvestigate each change. Because of the primal-dual optimal. relationships ,it is easy to move back and forth between the ¾This question can be answered simply by checking whether two problems as desired. this complementary basic solution still satisfies this revised ¾It is more convenient to analyze the dual problem constraint. directly in order to determine the complementary effect on the primal problem. We begin by considering two such cases.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo45l of Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo46l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The role of duality theory in sensitivity The role of duality theory in sensitivity analysisOperations ResearchOperations ResearchmaxZ=3x+5x+4xIntroducVFor exampletion of a New aVriable12new¾The decision variables in the model typically represent x+2x≤4⎧1newthe levels of the various activities, then would adding any of ⎪2x+3x≤12⎪2newthese activities to the model change the original optimal .⎨3x+2x+x≤18solution??12new⎪⎪¾Adding anothex≥0r activity amounts to introducing a new j⎩variable into the model. The only resulting change in the dual problem is to add a new constraint. variableZZxxxxx1234512345¾This question can be answered simply by checking whether this complemenZZ(0)10003/2/1366tary basic solution satisfies one consx(1)00011/3/-1/3/2traint, which in this case is the new constraint for the 33dual problemx(2)00101/2/(3)0100-1/3/1/3/211江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo647l of Information Technology,JiangXiUniversity of Finance & Economics©200Schoo48l of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 6 运筹学Operations Research6-9运筹学运筹学 The role of duality theory in sensitivity The role of duality theory in sensitivity analysisOperations ResearchOperations Research¾The original optimal solution beOther Applications¾(x1,x2,x3,x4,x5)=(6=2,6,2,0,0)When we investigate the effect of changing the bi or the, avalues, the original optimal solution may become ij¾Is it along withXnXew=0=, still optima?l? asuperoptimalbasic solution instead. 2y1+3+y2+y+3≥4¾If we want toreoptimize. to identify the new optimal solution, the dual simplex method should be applied. ¾Plugging in this solution:¾It is more efficient to solve the dual problem by the dual 2(0)+3/++(32/)(+1) ≥4simplex method to identify an optimal solution ,in this The solution along wX=ithnXew0= is still sensitivity analysis is conducted and then inferring the complementary effects on the prime problem.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo49l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technology,JiangX50iUniversity of Finance & Economics©2006运筹学运筹学66 Dual theory and sensitivity The Essence of Sensitivity AnalysisOperations ResearchOperations ResearchThe basic idea of sensitivity analysis: The Essence of Sensitivity Analysis¾To reveal how any changes in the original model would why we need sensitivity analysis? ?change the numbers in the final simplex tableau. ¾Actually, the parameter values used in the model normally are ju¾stAfter revising this tableau, check whether the original estimates based on a prediction of future conditions. So it is important to perform sensitivity optimal BF solution is nownonoptimal(or infeasible). If so , analysis to investigate the effect on the optimal solution if just from it to restart the simplex method (or dual simplex the parameters ) to find the new optimal solution. purpose of sensitivity analysis¾The main purpose of sensitivity analysis is to determine the ¾If the changes in the model are not major, only a very few range of values over which the optimal BF solution will iterations should be required to reach the new optimal solution remain feasible . from this “advanced”initial basic solution.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of51 Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo52l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Essence of Sensitivity The Essence of Sensitivity AnalysisOperations ResearchOperations Researchxx⎡⎤⎡⎤11Converting the revised final simplex tableau to proper formFor examplemaxZ=[35]maxZ=[45]⎢⎥⎢⎥xx⎣2⎦⎣2⎦variableZZx1x2x3x4x5⎧104⎧104⎡⎤⎡⎤⎡⎤⎡⎤xx⎡⎤⎡⎤⎪⎪⎢⎥1⎢⎥⎢⎥1⎢⎥ZZ(0)1-2003/2/≤≤24⎨⎨⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥xx⎣2⎦⎣2⎦⎪⎪x3(1)01/3/011/3/-1/3/66⎢⎥⎢⎥⎢22⎥⎢18⎥3218⎣⎦⎣⎦⎣⎦⎣⎦⎩⎩x2(2)00101/2/012Final tableaux1(3)02/3/00-1/3/1/3/-2Row (0)t*=[0003/2136]=[z*−cy*Z*]⎡0011/3−1/32⎤variableZZx1x2x3x4x5Other ⎢⎥T*=0101/206=[A*S*b*]⎢⎥ZZ(0)10001/2/2488rows ⎢100−1⎥/31/32⎣⎦x3(1)00011/2/-1/2/7t*z*−cy*Z*⎡⎤⎡⎤=x2(2)00101/2/012CCombined⎢⎥⎢⎥T*A*S*b*⎣⎦⎣⎦x1(3)0100-1/2/1/2/-3江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology,JiangXiUniversity of Finance & Economics©200653School of Information Technologr54y,JiangXiUnivesity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 6 运筹学Operations Research6-10运筹学运筹学 The Essence of Sensitivity The Essence of Sensitivity AnalysisOperations ResearchOperations Research¾EExtrapolation to determine the range of values for a Summary of Procedure for Sensitivity Analysisgiven parameter over which the final basic solution remains ¾1. Revision of the feasible and optimal.¾2. Revision of final tableau.¾CConvert the tableau to proper form. In particular, the basic variable for row1 must have a coefficient of 1 in that ¾3C. Conversion to proper form from Gaussian and a coefficient of 0 in every other row for the tableau. ¾4. Feasibility test.¾If the changes have violated this requirement, further changes must be made to restore this form . ¾5. Optimality test.¾Algebraic operations may also cause further changes in ¾ right side column, so the current basic solution can be read from column only when the proper form has been fully restored.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology,JiangXiUniversity of Finance & Econom55ics©2006Schoo56l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学66 Dual theory and sensitivity Applying Sensitivity AnalysisOperations ResearchOperations ResearchRight side of final row 0: y*=Z*, Applying Sensitivity AnalysisRight side of final rows 1,2,…,m: b*S=*CCase 1–C–Changes in biCCoefficient of:Right side¾Suppose one or more of the bparameters(i=1=,2…,…,m) iZZOriginal variablesSlack variablesNew initial change, the only resulting changes in the final simplex 1−C00tableautableau are in the rigght side column. 0IbA¾Therefore, both the conversion to proper form from Revised 1zcyAcZ*=y*b*−=*−y*Gaussian elimination and the optimalyl ytest steps of the final 0S*A*=S*Ab*=S*btableaugeneral procedure can be skipped.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog57y,JiangXiUniversity of Finance & Economics©2006School of Information Technolog58y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Applying Sensitivity Applying Sensitivity AnalysisOperations ResearchOperations Research44⎡⎤⎡⎤Example:¾Therefore, the current basic solution has become⎢⎥⎢⎥b=12→b=24⎢⎥⎢⎥¾(x1, x2, x3,x4,x5)=(=-2,12,66,0,0), ⎢18⎥⎢18⎥⎣⎦⎣⎦¾which fails the feasibility test because of the negative valueso4⎡⎤¾The dual simple method now can be applied, starting with 3⎡⎤⎢⎥**Z=yb=0124=54,⎢⎥⎢⎥2⎣⎦⎢18⎥this revised simplex tableau, to find the new optimal ⎣⎦11⎡⎤1−solution. ⎢⎥3346x6⎡⎤⎡⎤⎡⎤⎡⎤3⎢⎥1⎢⎥⎢⎥⎢⎥⎢⎥**b=Sb=⎢00⎥24=12,sox=12.¾After doing that, the new optimal solution solution is2⎢⎥⎢⎥⎢⎥⎢⎥2⎢⎥⎢18⎥⎢−2⎥⎢x⎥⎢−2⎥11⎣⎦⎣⎦⎣1⎦⎣⎦⎢⎥0−¾(x1, x2, x3,x4,x5)=(6=Z=0,9,4,,60), with 4=Z5.⎢⎥33⎣⎦江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog59y,JiangXiUniversity of Finance & Economics©2006Schoo60l of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 6 运筹学Operations Research6-11运筹学运筹学 Applying Sensitivity Applying Sensitivity AnalysisOperations ResearchOperations ResearchThe solution remains feasible, as long as all three quantities AlthoughΔb=122remain to be too large an increase in b2 to retain feasibility 11with the basic solution where x1, x2 and x3 are the basic 2+Δb≥0⇒Δb≥−2⇒Δb≥−6,222variables, the above incremental analysis shows how large an 33increase is feasible. In particular, note that116+Δb≥0⇒Δb≥−6⇒Δb≥−12,222221*b=2+Δb,121132−Δb≥0⇒2≥Δb⇒Δb≤*b=6+Δb,2221*b=2−Δb,323江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology,JiangXiUniversity of Finance & Economics©200661Schoo62l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Applying Sensitivity Applying Sensitivity AnalysisOperations ResearchOperations ResearchCase 2a --Change in the Coefficients of aNonbasicaVriable¾Thb=12+Δberefore, since ,the solution remains 22c→candA→Asupposejjjjfeasible only if−6≤Δb≤6⇒6≤b≤¾If the complementary basic solut*ion y* in the dual problem ¾This range ofwb values for bbis referred to as its allowable 2still satisfies the signal dual constraint then the original rangybge to sta solution in the primal problem remains optimal.¾For any bb, its allowable range to stay feasibleis the range i¾If* y* violates this dual constraint, then the primal solution isof values which the optimal BF solution remains longer optimal. ¾Simply apply the fundamental insight to revise thexjcolumn (the only one hat has changed) in the final simplex tableau.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology,JiangXiUniversity of Finance & Economics©200663School of Information Technolog64y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Applying Sensitivity Applying Sensitivity AnalysisOperations ResearchOperations Research¾EXAEEXMPLE:¾The formulas reduce to the following:¾The set of changes wou=ld be to reset c14= and a31=2=. *CCoefficient ofxjin final row 0: z−c=y*A−cjjJjWith x1nonbasicin the current optimal solution ,*A=S*AjjCCoefficient ofxjin final rows 1 to m: 11⎡⎤⎡⎤⎢⎥⎢⎥¾With the current basic solution no longer optimal, the new c=3→c=4,A=0→A=⎢⎥⎢⎥*value of now will be the one negative coefficient z−cin jj⎢⎥⎢⎥32⎣⎦⎣⎦row 0, so the simplex method withxjas the initial entering basic variable.¾Both this revised constraint and the current y** are shown 5below.***y=0,y=0,y=,12325y+3y≥3→y+2y≥4,即0+2()≥413132江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology,JiangX65iUniversity of Finance & Economics©2006Schoo66l of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 6 运筹学Operations Research6-12运筹学运筹学 Applying Sensitivity Applying Sensitivity AnalysisOperations ResearchOperations Research¾.Since y** still satisfies the revised constraint, the current1⎡⎤5⎢⎥**z−c=yA−c=[0,0,]0−4= solution is still optimal. Nevertheless, we do so jjjj⎢⎥2⎢2⎥⎣⎦below for illustrative purposes.⎡100⎤11⎡⎤⎡⎤⎢1⎥⎢⎥⎢⎥**A=SA=000=⎢⎥⎢⎥⎢⎥2⎢⎥⎢2⎥⎢−2⎥01−1⎣⎦⎣⎦⎣⎦ZZ(0)19/2/0005/2/45x3(1)0101004¾The fact that confirms the optimality of the current x2(2)03/2/1001/2/9x4(3)0-3001-166solution.*z−c≥0jj江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo67l of Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo68l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Applying Sensitivity Applying Sensitivity AnalysisOperations ResearchOperations Research¾CCase— 2b—Introduction of a New VaVriable¾The allowable range to stay optimal:¾This is done by pretending that the new variablexjactually ¾For anycj, its allowable range to stay optimal is the values was in the original model with all its coefficients equal to where the current optimal solution remains optimal. Whenzero and thatxjis anonbasicvariable in the current BF xjis anonbasicvariable for this solution, the solution solution. ¾remains optimaIf we change these zero coefficients to their actual values for l as long as. *the new variable, the procedure does indeed become z=y*Ajjidentical to that for CaCse 2a. ¾ChCeck whether the current solution still is optimal is to *hwerez−c≥0unaffectedjjcheck whether the complementary basic solution y** satisfies the one new dual constraint.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo69l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technologr70y,JiangXiUnivesity of Finance & Economics©2006运筹学运筹学 Applying Sensitivity Applying Sensitivity AnalysisOperations ResearchOperations Researchc¾CCC6Case3——Change in the Coefficients of aV Basic aVriable⎡⎤⎡⎤6⎢⎥⎢⎥a2Introduction a new variable x16¾Now suppose that the variablexjis a basic variable in the 6⎢⎥⎢⎥=⎢a⎥⎢1⎥26optimal solution shown by the final simplex tableau.⎢⎥⎢⎥a2⎣36⎦⎣⎦¾CCase 3 assumes that the only changes in the current model variableZZx1x2x3x4x5x66are made to the coefficients of this (0)110005/2/-145¾CCCase 3 differs from Case 2a in that a simplex tableau x3(1)01010024x2(2)011001/2/½½9be in proper form from Gaussian elimination. So the basic x4(3)0-2001-1-166variablexjmust have a coefficient of 1 in its row of the simplex tableau and a coefficient of 0 in every other row. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo71l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technolog72y,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 6 运筹学Operations Research6-13运筹学运筹学 Applying Sensitivity Applying Sensitivity AnalysisOperations ResearchOperations Research¾A00fter the changes in thexjcolumn of the final simplex ⎡⎤⎡⎤⎢⎥⎢⎥tableau have been calculated, it is necessary to restore this EXAMPLEc=5→c=3,A=2→A=32222⎢⎥⎢⎥form. ⎢2⎥⎢⎥4⎣⎦⎣⎦¾In turn, this step may change the value of the currentbasic solution and may make it either infeasible ornonoptimal. ¾The old solution is no longer optimal, because the revised ¾Before Gaussian elimination is applied, the formulas for objective function of=Z+ 3=Zx13+x2 now yields a new optimal revising thexjcolumn are theC Case 2b, as summarized solution of (x1, x2)=(=4/,32/). below.¾Because the only changes in the model are in the *¾z−c=y*A−cCCoefficient ofxjin final row 0:coefficients of x2, the only resulting changes in the final jjjjsimplex tableau are in the x2 column.*¾CCoefficient ofxjin final rows 1 to m:A=S*Ajj江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology,JiangXiUniversity of Finance & Economics©200673School of Information Technolog74y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Applying Sensitivity Applying Sensitivity AnalysisOperations ResearchOperations Research0⎡⎤variableZZx1x2x3x4x55⎢⎥Soz−c=y*A−c=[0,0,]3−3=7ZZ(0)19/2/7005/2/452222⎢⎥2x3(1)0101004⎢⎥4⎣⎦x2(2)03/2/2001/2/9⎡100⎤00x4(3)0-3-101-166⎡⎤⎡⎤⎢1⎥⎢⎥⎢⎥**A=SA=003=222⎢⎥⎢⎥⎢⎥2⎢⎥⎢⎥⎢−⎥4101−1⎣⎦⎣⎦variableZZx1x2x3x4x5⎣⎦ZZ(0)1003/4/03/4/33/2/x1(1)0101004¾Now after calculating,the column of x2 does not fit the x2(2)001-3/4/01/4/3/2/optimal form, so the conversion is (3)0009/4/1-3/4/39/2/江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology,JiangXiUniversity of Finance & Economics©200675School of Information Technolog76y,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Applying Sens6itivity Applying Sensitivity AnalysisOperations ResearchOperations ResearchAnother examplemaxZ=3x+5xvariableZZx1x2x3x4x512ZZ(0)19/2/-5/2/-CC005/2/452x≤4⎧1x3(1)0101004⎪≤24⎨2x2(2)03/2/2001/2/9⎪x4(3)0-3-101-1663x+2x≤18⎩12variableZZx1x2x3x4x5variableZZx1x2x3x4x5ZZ(0)13-3/4/CC0001/4/CC4522ZZ(0)19/2/0005/2/45x3(1)0101004x3(1)0101004x2(2)03/2/1001/2/9x2(2)03/2/1001/2/9x4(3)0-3001-166x4(3)0-3001-166江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo7778l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 6 运筹学Operations Research6-14运筹学运筹学 Applying6 Sensitivity Applying Sensitivity AnalysisOperations ResearchOperations ResearchCC¾Case 4––Introduce of a New ConstraintThe conversion to proper form from Gaussian elimination is applied next, and then thereoptimizationstep is applied ¾In the last case, a new constraint muse be introduced to the in the usual after it has already been solved.¾The procedure for CCase 4 is a streamlined version of the ¾To see if the current optimal solution would be affected by a general procedure .The only question to be addressed for new constraint, all you have to do is to check directly this case is whether the previously optimal solution still is whether the optimal solution satisfies the , so step 5 and 4 has been deleted. ¾If the new constraint does eliminate the current optimal ¾Step 4 has been replaced buy a much quicker test of solution, then introduce this constraint into the final feasibility to be performed right after step 1. It is only if simplex tableau just as if this were the initial tableau, where this test provides a negative answer, and you wish tothe usual additional variable is designated to be the basic reoptimize, that step 2, 3, and 66 are for this new row. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo79l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technology,JiangXn80iUiversity of Finance & Economics©2006运筹学运筹学 Applying Sensitivity Applying Sensitivity AnalysisOperations ResearchOperations ResearchEXAEEXMPLE:¾Systematic Sensitivity Analysis——Parametric ¾Suppose that the introduced new constraint beProgramming2x1+3+x2=<2=<4¾Another common approach to sensitivity analysis is to vary one or more parameters continuously over some ¾To find a new solution, add the new constraint to the intervals to see when the optimal solution final simplex tableau, with the slack variable x66 as its initial basic variable.¾For example, with theWyndorGlassC , we set¾The new basic so=lution x34=,=6=6= x29=, x46=, x6=-3 (x1=0=, b=12+θ2x5=0=).It is infeasible, so the dual simplex method is ¾Then vary θcontinuously from 0 to to get the proper form. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo81l of Information Technology,JiangXiUniversity of Finance & Economics©2006Schoo82l of Information Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Applying Sensitivity Applying Sensitivity AnalysisOperations ResearchOperations ResearchFor example¾we can investigate the effect on the optimal solution of supposevarying several parameters .When we vary just the bi 4⎡⎤parameters, we express the new value bi in terms of the b=12+θ31⎢⎥2Z*=y*b=[0,,1]12+θ=36−θ⎢⎥22original value bi as follows:b=18−2θ⎢18−2θ⎥3⎣⎦11⎡⎤b=b+αθ,fori=1,2,K,miii1−⎢⎥⎡332+θ⎤4⎡⎤⎢⎥1⎢1⎥⎢⎥b*=S*b=⎢00⎥12+θ=6+θ⎢⎥¾Wh⎢⎥ere theαivalues are input constants specifying the 22⎢⎥⎢⎥⎢18−2θ⎥112−θ⎣⎦⎢⎥⎣⎦0−desired rate of increase of the corresponding right-hand ⎢⎥33⎣⎦side as θis θ≤2, the optimal solution is still optimal江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo83l of Information Technology,JiangXiUniversity of Finance & Economics©2006School of Information Technology,JiangXiUniversity of Finance & Econom84ics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 6 运筹学Operations Research6-15运筹学运筹学 Applying Sensitivity Applying Sensitivity AnalysisOperations ResearchOperations Research¾Now we consider both c1 and c2 changes, by setting¾The approach to varying severalcjparameters simultaneously is similac=3+θandc=5−2θr .In this case, we express the new 12value in terms of the original value ofcjas¾Where the value of θmeasures the fraction of the c=c+αθ,forj=1,2,K,njjjmaximum possible change that is made. So,¾Where theαjare input constants specifying the desired rate Z(θ)=(3+θ)x+(5−2θ)x12of increase ofcjas θis increased.¾So the optimization now can be performed for any desired ¾To illustrate, reconsider the sensitivity analysis of c1 and c2 (fixed) value of θbetween 0 and theWyndorGlass CCo. problem.¾HHow to solve: P237——江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Informaton Technology,JiangXiUniversity of Finance & Econom85iics©2006School of Infomat86rion Technology,JiangXiUniversity of Finance & Economics©2006运筹学运筹学66 Dual theory and sensitivity ConclusionsOperations ResearchOperations Conclusions¾Sensitivity analysis needs to be performed to investigates ¾There are a number of very useful relationships between what happens if these estimates are wrong. The fundamental the original problems and its dual problem that enhance our insight provides the key to performing this investigation ability to analyze the primal problem. efficiently.¾For example, the economic interpretation of the dual ¾The general objectives are to identify the sensitive problem gives shadow prices and provides an parameters that affects the optimal solution, to try to Interpretation of the simplex these sensitive parameters more closely, and then to ¾The simplex method can be applied to save considerable select a solution that remains good over the range of likely computational effort. Also it plays a major role in sensitivity values of the sensitive parameters. analysis.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo87Schoo88l of Information Technology,JiangXiUniversity of Finance & Economics©2006l of Information Technology,JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 7 运筹学Operations Research7-1运筹学运筹学Operations ResearchOperations Prefailnearg oap lrogramming7 Other Algorithms for Linear hWere, rather than having a singel objectievProgramming(maximize or minimize Z) as for linear programming, Prefathe problem has several goasltoward which we must ¾variants of the simplex method. strive simultaneously. eCrtain formulation techniuqes ¾The dual simplex method (a modification particularly useful for sensitivity analysis), parametric linear enable us to convert a linear goal programming problem programming (an extension for systematic sensitivity back to a linear programming problem so that solution analysis)¾The upper bound technique (a streamlined version of the procedures based on the simplex method can still be used. simplex method for dealing with variables having upper Section describes these techniques and ).江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo1l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo2l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学7 Other Algorithms for Linear The Dual Simplex MethodOperations ResearchOperations The Dual Simplex MethodThe dual simplex method is very useful in certain special types of situations. Ordinarily it is easier to find an initial dual simplexWhich can be thought of as the mirror imaggeof the simplex method. The simplex method deals basic feasible (BF) solution than an initial superoptimalbasic directlyb with suboptimalbasic solutions and moves toward an solution. oHwever, it is occasionally necessary to introduce optimal solution by striving to satisfy the opytimalit ytest. many artificial variables to construct an initial BF solution By contrast, the dual simplex method deals directly with artificially. In such cases it may be easier to begin with a super super optimalbasic solutions and moves toward an optimal solution by striving to achieve feasbyibility. optimal basic solution and use the dual simplex method. Furthermore, the dual simplex method deals with a Furthermore, fewer iterations may be required when it is not problem as if the simplex method were being applied necessary to drive many artificial variables to to its dual problem. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom3ics©2006School of Information Technolog4y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Dual Simplex The Dual Simplex MethodOperations ResearchOperations ResearchThe basic solutions will be infeasible (except for the last AAnother important primary application of the dual simplex one) only because some of the variables are negative. The method is its use in conjunction with sensitivity analysis. method continues to decrease the value of the objective To start the dual simplex method (for a maximization function, always retaining nonneggative coefficientsinEEq.(0), problem), we must have all the coefficients inEqEq.(0) nonnegatvievuntil all the variabblesare nonnegative. Such a basic solution is feasible (it satisfies all the equations) and is ,therefore, (so that the basic solution is super optimal).optimal by the simplex method criterion of nonnegative coefficients inEEq.(0).江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School ofnom5 Information Technology, JiangXiUniversity of Finance & Ecoics©2006School of Information Technolog6y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 7 运筹学Operations Research7-2运筹学运筹学 The Dual Simplex The Dual Simplex MethodOperations ResearchOperations ResearchSummary of the Dual Simplex Method¾Iteration:¾Initiazliaztion::After converting any functional constraints Step 1Determine the leaving basic variable: Select the in≥form to ≤form (by multiplying through both sides by -1), negativebasic variable that has the largest absolute slack variables as needed to construct a set of Step 2Determine the entering basic variableequations describing the problem. Find a basic solution such Step 3Determine the ne wbasic solution: Starting fro the that the coefficients inEEq.(0) are zero for basic variables and current set of equations , solve for the basic variables in terms nonnegative for nonbasicvariables (so the solution is optimal of the nonbasicvariables by Gaussian elimination. When we if it is feasible). Go to the feasibility the nonbasicvariables equal to zero, each basic variable ¾Feasby:Cibilit ytest:Check to see whether all the basic variables (and )Z equals the new right-hand side of the one equation in are nonnegative .if they are ,then this solution is feasible, and which it appears (with a coefficient of 1+). Return to the therefore optimal, so stop. Otherwise ,go to an test.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo7l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Informat8ion Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Dual Simplex The Dual Simplex MethodOperations ResearchOperations ResearchIn maximCoCefficient of::ization form, the problem to be solved is Iteration sideZy345Zy1 y2 3 y4 y5 maxZ=−4y−12y−18y1230 ZZ(0)144 12 188 0 00y+3y≥3⎧y44(1)0--3-1 0 -3- 3 1 .⎨y55(2)00 0 --2 0 1-5-5y+3y≥5⎩231 ZZ(0)14 6 6 0 66-340 -30y44 (1)0--3-1 0 -3-3 1 0-3Table Dual Simplex Method Applied to the WyndorGlass y2 (2)00 1 1 0 --1/2//52/5CCo. Dual Problem2 ZZ(0)120 0 2 66-36-36y33(1)013/3 / 0 1 -/3-1/3 01y2(2)0-3/3/-/-13 / 1 0 13 / -12//32/3江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©20069Schoo10l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Dual Simplex The Dual Simplex MethodOperations ResearchOperations ResearchWe simply convert the functional constraints to ≤form The next leaving basic variable is y, and the entering 4and introduce slack variables to play this role . The resulting basic variable6/</ is y(63/4<1/), which leads to the final set of 3initial set of equations is that shown for iteration 0 in Table equations in Table . Notice that all the coefficients inEEq.(0) are nonnegative,so The corresponding basic so==/lution= is y0=, y3=2/, y1=, 123the solution is optimal if it is =0=, y=0=Z=, with =Z-366, which is feasible and therefore 45The initial basic solution=== is y0=,y0=,y=0==,y=-3,y=-5, =Zith 0=Z, which is not feasible because of the negative values. Notice that the optimal solution for the dual of this The leaving basic variable is y(5>3>), and the entering basic 5**********problem is x=26====,x,6=x2=,x0=,x=0=, as was obtained in 12345variable is y(12/2/>1>8/82/). Which leads to the second set of 2Table8 4. 8by the simplex method . We suggest that you now equations, labeled as iteration 1 in Table . The trace through Tables8 and 4. 8simultaneously and compare corresponding basic= solution is y0=/==,y5=2/,y0=,y==-1234the complementary steps for the two mirror-image ,y=0=,with=Z =Z-30,which is not 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo11l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo12l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 7 运筹学Operations Research7-3运筹学运筹学7 Other Algorithms for Linear Parametric Linear ProgrammingOperations ResearchOperations Parametric Linear ProgrammingFor any given value of θ, the optimal solution of the corresponding linear programming problem can be obtained by the SystematicC Changes in the cParametersjFFor the case where the cparameters are being changed, the simplex method. This solution may have been obtained already forjobjective function of the ordinary linear programming model the original problem where θ=0. oHwever, the objective is to find nZ=cx∑jjthe optimal solution of the modified linear programming problem j=1n=+is replaced byZ(θ)(cαθ)x ∑ m[aximize z(θ) subject to the original constraints ]as a function of jjjj=1Whθ. Therefore, in the solution procedure you need to be able to ere the αj are given input constants representing the relativerates at which the coefficients are to be changed. determine when and how the optimal solution changes (if it does)Therefore, gradually increasing θfrom zero changes the as θincreases form zero to any specified positive at these relative rates.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo13l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo14l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Parametric Linear Parametric Linear ProgrammingOperations ResearchOperations Research**FZigure illustrates how Z(θ), the objective function ThusF7 depicts a problem where three different value for the optimal solution (given θ), changes as θ**increases . In fact Z, Z(θ) always has this piecewise linear and solutions are optimal for different values of θ,the first for 0≤convex form (see -10) , the corresponding optimal θ≤θ,the second for θ≤θ≤θ,and the third for θ≥θ. 1122solution changes (as θincreases) just at the values of θwhere **the slope of the ZZ(θ) function changes. BBecause the value of each xjremains the same within each of these intervals for θ*Z, the value of *Z(θ) varies with θonly Z*(θ)because the coefficients of the xjare changing as a linear function of θ. The solution procedure is based directly upon the sensitivity analysis procedure for investigating changes in the cjθθθ12parameters (case 2a and 33,).江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog15y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog16y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Parametric Linear Parametric Linear ProgrammingOperations ResearchOperations ResearchThe optimality test says that the current FB solution will remain To illustrate, suppose that α=2 and α=1- for the original 12optimal as long as these coefficients of the nonbasicvariables WyndorGlass presented in , so that remain nonnegative:3/2-76/3/-76/ θ≥0, for 0≤θ≤97/97/,Z(θ)=(3+2θ)x+(5-θ)x,121+2+3/ 3/θ≥0, for all θ≥ with the final simplex tableau for θ=0(Table ), Therefore, after θis increased past θ=97/,x 4would need to be we see that itsEq.(0)(0) 2/Z+3xx+=.3645the entering basic variable for another iteration of the simplexoWuld first have these changes from the original (θ=0) method to find the new optimal solution. Then θwould be coefficients added into it on the lefth-and side:increased further until another coefficient went negative ,and so on until θhad been increased as far as -θx+θx2+3/xx+=361245This entire procedure is now summarized and the example is Because both xand xare basic variables a[ppearing inqsE.(3) 12completed in Table (2), respectively,] they both need to be eliminated algebraically Table the cjParametric Programming Procedure Applied to fromEq.(0).the yWndorglass Co. Example江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Informaton Technology, JiangXiUniversity of Finance & Econom17iics©2006Schoo18l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 7 运筹学Operations Research7-4Range 运筹学运筹学CCoefficient of : Parametric Linear ProgrammingOBpaesraictions ResearchRight OptimalOperations θSide solution x1 x2 x3 x4 x5Systematic Changes in the bParametersVaVriablei(0)0 0 0 (9-7θ)6/6 / (9-7θ)6/6/366-2θx4=0=, Z(Zθ) For the case where the bparameters change ix5=0= 0≤θ≤9/7/systematically , the one modification made in the original linear 2 x3==2(1)0 0 1 1/3/ -1/3/x3 6=6programming model is that bi is replaced by b+αθ, for 6 x2=6x2 ( 2 ) 0 1 0 1/2/ 0iix1 2 x1==2(3)1 0 0 -1/3/ 1/3/i1=,2,,…m, where the αare given input constants. Thus the i(0)0 0 (-9+7+θ)/2/ 0 (5-θ)/2/27+5+θx3=0=Z(Zθ) problem becomes9/7/≤θ≤5 nx5=0= Z(θ)=cxMaxim∑ize jjx4 (1)0 0 3 1 -16=66 x4=6j=1x2(2)0 1 -2/3/ 0 1/2/3 x2==3x1n(3)1 0 1 0 04 x1==4a≤b+αθSubject to ∑ i j i i for i1=,2,…,m(0)0 -5+ +θ3+2+ θ0 0 128+8+θx2=0=j=1Z(Zθ)x3=0= andx≥0 for i=1,2,…,n.θ≥5 j(1)0 2 0 1 012 x4=1=2x4 The goal is to identify the optimal solution as a function of θ.x5 (2)0 2 -3 0 166 x5=6=6x1 (3)1 0 1 0 04 x1==4江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo19l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo20l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Parametric Linear Parametric Linear ProgrammingOperations ResearchOperations ResearchWith this formulation , the corresponding objective **Summary of the Parametric Programming Procedure functZion value Z(θ) always hasw the piecewise linear and concaveform shown in (see -11.) The set of for Systematic hCanges in the cParametersjbasic variables in the optimal solution still changes (as θ**increases) only where the slope ofZ Z(θ) changesH. However, 1=. Solve the problem with θ0= by the simplex contrast to the preceding case, the values of these variables2. Use the sensitivity analysis procedure (CCase 2a and now change as a (linear) function of θbetween the slope Δc=αθ3,) to introduce the change intoEEq.(0). jjchanges. The reason is that increasing θthe right-hand sides in the initial set of equations, which then causes changes in 3. Increase θuntil one of the nonbasicvariables has its the right-hand sides in the final set of equations, ., in the coeffEicient inEq.(0) go negative (or until θhas been increased values of the final set of basic variables . as far as desired ).4. Use this variable as the entering basic variable for iteration of the simplex method to find the new optimal solution . Return to step 3.θθθ12江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo21l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of22 Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Parametric Linear Parametric Linear ProgrammingOperations ResearchOperations ResearchThe following solution procedure summary is very similar Fito that just presented for systematic cgure depicts a problhanges in the cjem with three sets of basic parameters. The reason is that changing the bi values is variables that are optimal for different values of θ, the first equivalent to changing the coefficients in the objective function of the dual model. Therefore , the procedure for the for 0≤θ≤θ1, the second for θ1≤θ≤θ2, and the third primal problem is exactly complementary to applying simultaneously the procedure for systematic changes in the cjof θ≥θ2. Within each of these intervals of θ, the value of parameters to the dual problem . Consequently, the dual *Z*(Zθ) varies with θdespite the fixed coefficients cjbecause simplex method (see ) now would be used to obtain each new optimal solution, and the applicable sensitivity analysis the xjvalues are (see ) now is Case 1,but these differences are the only major differences.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog23y, JiangXiUniversity of Finance & Economics©2006Schoo24l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 7 运筹学Operations Research7-5运筹学运筹学 Parametric Linear Parametric Linear ProgrammingOperations ResearchOperations ResearchSummary of the parametric programming To illustrate this procedure in a way that demonstrates its procedure for systematic changes in the bparametersi duality relationship with the procedure for systematic changes the cjparameters, we now apply it to the dual ¾Solve the problem with= θ0= by the simplex method.¾Usproblem for the WyndorGlass Co. (see Table ). In e the sensitivity analysis procedureC (Case 1, ) to introduce the changes to the right side column.Δb=αθparticular, suppose that α2= and α-=1 so that the functional ii12constraints becomeIncrease θuntil one of the basic variables has its value in y3+y≥32+θor -y-3y≤-3-2θ1313the right side column go negative (or until θhas been increased as far as desired).2y2+y≥5-θor -2y-2y≤-5+θ.2323Use this variable as the leaving basic variable for an Thus the dual of this problem is just the example iteration of the dual simplex method to find the new optimal solution . Return to step in Table .江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo25l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of26 Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Parametric Linear Parametric Linear ProgrammingOperations ResearchOperations ResearchRange CoCefficient of:This problem with θ0= has already been solved in . BasicRight Optimalso we begin with the final simplex tableau given there. Using Of θ Solution yyyyy12345variable the sensitivity analysis procedure for Case 1,, we findZZthat the entries in the right side column of the tableau change to 0≤θ≤9//7 Z(Zθ)(0)12 0 0 2 66-3+62+6θy=y===y=0= 145the values given (1)01/3/ 0 1 -1//3 0(3+2+θ)/3/ y=(+/=32+θ)3/33−3−2θy(2)0-1//3 1 0 1//3 -1/2/(9-7θ)6/ 6/ y=(=9-7θ)6/6/**22y=yb=[2,6][]=−36+2θ09/7/≤θ≤5 Z(Zθ)(0)10 66 0 4 3-27-5θy=y==4y=5=0−5+θ=2y(1)00 1 1 0 -1/2/(5-θ)/2/ y=(=5-θ)/2/ 3312θy(2)01 -3 0 -1 3/2/(-9+7+θ)/2/ y=(=-9+7+θ)/2/ −01 2⎡⎤−3−2θ⎡1+⎤⎡⎤33**b=sb==θ≥5 Z(Zθ)(0)10 12 66 4 0-12-88θy=y===3y=40=⎢⎥⎢⎥⎢⎥21137θ−−5+θ−⎣⎦⎣32⎦⎣26⎦y(1)00 -2 -2 0 1-5++θy=+=-5+θ55y(2)01 0 3 -1 03+2+θy=3+=2+θ11江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog27y, JiangXiUniversity of Finance & Economics©2006School of Informaton Technology, JiangXiUniversity of Finance & Econom28iics©2006运筹学运筹学7 Other Algorithms for Linear Parametric Linear ProgrammingOperations ResearchOperations The Upper Bound TechniqueTherefore, the two basic variables in this tableauIt is fairly common in linear programming problems for some of or all the individual xjvariables to have upper bound +θ9−7θ32andy=y=3263constraints xj≤uj,Where uis a positive constant representing the maximum jfeasible value of x. We pointed out in that the most jRemain nonnegative for 0≤θ≤97/,increasing θ9=7/ important determinant of computation time for the simplex requires making ya leaving basic variable for another 2iteration of the dual simplex method, and so on ,as method is the number of functional constraints, whereas the summarized in Table of nonnegativityconstraints is relatively unimportant. Therefore, having a large number of upper bound constraints among the functional constraints greatly increases the computational effort required.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo29l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog30y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 7 运筹学Operations Research7-6运筹学运筹学 The Upper Bound The Upper Bound TechniqueOperations ResearchOperations ResearchThe upper bound technique avoids this increased effort by To implement this idea , note that a decision variable xjremoving the upper bound constraints from the functional with an upper bound constraint xj≤ujcan always be replaced constraints and treating them separately, essentially like by nonegativityconstraints . Removing the upper bound xju=j-yj,constraints in this way causes no problems as long as none of the Where yjwould then be the decision variable . In other variables gets increased over its upper bound. The only time thewords, you have a choice between letting the decision variable simplex method increases some of the variables is when the be the amount above zero (xj) or the amount below uj(yju=j-entering basic variable is increased to obtain a new BF solution. xj). (We shall refer to xjand yjas complementary decision Therefore , the upper bound technique simply applies the variables .) Because simplex method in the usual way to the remainder of the 0≤xj≤ujproblem (., without the upper bound constraints ) but with the It also follows that one additional restriction that each new BF solution must satisfy the upper bound constraints in addition to the usual lower 0≤yj≤ (nonnegativity) constraints.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of31 Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo32l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Upper Bound The Upper Bound TechniqueOperations ResearchOperations Research¾Use x, where 0≤x≤ that the simplex method selects as the leaving basic ¾Replace xby u-y, where 0≤y≤ the one that would be the first to become infeasible by¾The upper bound technique use the following rule to make going negative as the entering basic variable is increased. The this choice:modification now made is to select instead the variable that Rule:Begin with choice be the first to become infeasible in any way, either by Whenever x=0=, use choice 1, so xbis nonbasic..jjgoing negative or by going over the upper bound, as the Whenever x==u, use choice 2, so y=0b= is basic variable is increased. (Notice that one possibility Switch choices only when the other extreme value of xis jis that the entering basic variable may become infeasible first reachedby going over its upper bound, so that its complementary Therefore, whenever a basic variable reaches its upper decision variable becomes the leaving basic variable.) If the bound, you should switch choices and use its complementary leaving basic variable reaches zero ,then proceed as usual with decision variable as the new nonbasicvariable (the leaving the simplex method. oHwever, if it reaches its upper bound basic variable) for identifying the new BF solution. Thus the instead, then switch choices and make its complementary one substantive modification being made in the simplex method decision variable the leaving basic in the rule for selecting the leaving basic variable.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of33 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUner34ivsity of Finance & Economics©2006运筹学运筹学 The Upper Bound The Upper Bound TechniqueOperations ResearchOperations ResearchTo illustrate, consider this problem:Initial Set of MFMaximum Feasible value of x1EqEquationsmaxZ=2x+x+2x123x≤44 (since u=4=4)11(0) ZZ--2x=2=014x+x=12⎧12x≤124=3/4=3/1⎪(1) 44x+x=+1=.−2x+x=4⎨13x≤(66-4/=-4)2/=1 minimum (because 1(2)--2x++x=4=4 ⎪133u6=0≤x≤6)=4,0≤x≤15,0≤x≤633⎩123Z 2-xx-2-x= 0123 (+x4+x1=2) 12 +2(2-x+ x =4)13Z 2-x=江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©200635School of Information Technolog36y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 7 运筹学Operations Research7-7运筹学运筹学7 Other Algorithms for Linear The Upper Bound TechniqueOperations ResearchOperations Research-+=4-2x+ x= An Interior-Point Algorithm133--2x+6-+6-y=4= 4133WeW now introduce the nature ofK Karmarkar’’s approach by --2x--y=-= -2describing a relatively elementary variant (the affine or 133 affinescalingvariant) of his algorYOithmRC. (Your OR Courseware also x+/+(12/)y== 1133Pinclude )this variant in the rPocedure mmeneun uu nudnedre rt hteh et itiltel eS Solovlve Therefore ,after we eliminate x1 algebraically from the other AuAtomatically by the InPAterior--Point lAgorithm.)eqqquations , the second complete set of equations becomesThroughout this section we shalKl focus on Karmarkar’’s main (0) Z+= Z + y2=2ideas on an intuitive level while avoiding mathematical 33particular , we shall bypass certain details that are needed forthe (1) =8 x--2y=8233full implementation of the algorithm (., how to find an initial (2) += x+(1/2/)y= trial solution) but are not central to a basic conceptual The resultingFB= FB solution is xunderstanding. The ideas to be described can be summarized as =1,x=8==8,y=0B. yB the optimality 1233follows::test, it also is an optimal solu=tion , so x=1, x=8-=6=8,x6=6=-y=6 is the 123333desired solution for the original problem.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo37l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity ofinance & Econom38 Fics©2006运筹学运筹学 An Interior-Point An Interior-Point AlgorithmOperations ResearchOperations ResearchConcept 1:Shoot through the interior of the feasible region toward an optimal :Eample:Maximize=Z+ x=Z1+ 2:Move in a direction that improves the objective Subject to function value at the fastest possible +x+2≤88AndConcept 3:Transform the feasible region to place the current x1≥0, x2≥ solution near its center , thereby enabling a large improvement when concept 2 is problem is depicted graphically in . where the optimal solution is seen to be (x1,x2)=(8=Z=0,8) with 16=Z6.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog39y, JiangXiUniversity of Finance & Economics©2006Schoo40l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 An Interior-Point An Interior-Point AlgorithmOperations ResearchOperations ResearchThTo begin implementing concepts 1 and 2, note in that e Relevance of the GradientC for Concepts 1 and 2the direction of movement from (2,2) that increasesZ Zat the The algorithm begins with an initial trial solution that (like all fastest possible rate is perpendicular to (and toward ) the subseqquent trial solutions) lies in the interior of the feasibleregion, objective function line =Z16==Z6x+=2+x. We have shown this 12direction by the arrow from (2,2) to (3,4). Using vector addition, ., inside the boundary of the feasible region. Thus, for thewe haveexample, the solution must not lie on any of the three lines (3=+,4)(=2,2)(+1,2),(x===8=0,x=0,x+x+=)8 that form the boundary of this region in 1212Where the vector (1,2) is the gradient of the objective . (A trial solution that lies on the boundary cannot be used function. (We will discuss gradients further in in the broader context of nonlinear programming. Where algorithms because this would lead to the undefined mathematical operation similar toK Karmarkar’’s have long been used.) The component of division by zero at one point in the algorithm.) we have of (1,2) are just the coefficients in the objective function . Thus, arbitrarily chosen (x,x)==(2,2) to be the initial trial one subsequent modification, the gradient (1,2) defines theideal direction to which to move, where the question of the distance to move will be considered later.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo41l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog42y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 7 运筹学Operations Research7-8运筹学运筹学 An Interior-Point An Interior-Point AlgorithmOperations ResearchOperations ResearchIn matrix notation (slightly different fromC The algorithm actually operates on linear programming because the slack variable now is incorporated into the problems after they have been rewritten in augmented form. notation), the augmented form can be written in general asLetting x3 be the slack variable for the functional constraint Maximize =Z=ZcTx,Subject to Ax=b=of the example, we see that this form is And x≥0,Maximize=Z+ x=Z12+x2,Where Subject to 1x0⎡⎤⎡⎤⎡⎤1x1+x+8=+2x+38=⎢⎥⎢⎥⎢⎥c=2, x=x, A=[111],b=[8], 0=02⎢⎥⎢⎥⎢⎥And⎢0⎥⎢x⎥⎢0⎥⎣⎦⎣3⎦⎣⎦x1≥0, x2≥0, x3≥0.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo43l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog44y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 An Interior-Point An Interior-Point AlgorithmOperations ResearchOperations ResearchUsinThe perpendicular line through (3,4,4) is given by the g the Projected Gradient to Implement CoCncepts 1 equationand 2(x,x,x)=(=3,4,4)-θ(1,1,1),123In augmented form, the initial trial solution for the Where θis a scalar. Since the triangle satisfies the equation example is (x1,x2,x3)=(=2,2,4). Adding the gradient (1,2,0) leadsx+x+8=+x+8=, this perpendicular line intersects the triangle at 123to (2,3,3). Because (3=+,4,4)(=2,2,4)+(1,2,0).(2=+,3,3)(=2,2,4)(+0,1,-1),HoHwever, now there is a complication. The algorithm The projected gradient of the objective function (the cannot move from (2,2,4) toward (3,4,4), because (3,4,4) is gradient projected onto the feasible region) is (0,1,-1). infeasible!== !When x13= and x24=,8= then x38=-x1-x2=1= instead of A formula is available for computing the projected gradient 4. The point (3,4,4) lies on the near side as you look down on directly . By defining the projection matrix P as the feasible triangle in . therefore, to remain feasible,TT-1P=I=-A(AA) A,the algorithm (indirectly) projects the point (3,4,4) down onto The projected gradient (in column form) isthe feasible triangle by dropping a line that is perpendicular to C=CP= triangle. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo45l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo46l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 An Interior-Point An Interior-Point AlgorithmOperations ResearchOperations Research211so⎡⎤For example−−⎢⎥33310⎡⎤⎡⎤−1⎢⎥1211001⎛1⎞⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥c=⎢−−⎥2=1.⎜⎟p⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥333p=010−1[111]1[111]⎢⎥⎜⎟⎢⎥⎢⎥⎢⎥⎢0⎥⎢−1⎥112⎣⎦⎣⎦⎢⎥⎜⎟⎢−−001⎥⎢1⎥⎢1⎥⎣⎦⎣⎦⎣⎦⎝⎠⎢⎥333⎣⎦1001⎡⎤⎡⎤Moving from (2,2,4) in the direction of the projected ⎢⎥⎢⎥1=010−1[111]gradient (0,1,-1) in volvesincreasing αfrom zero in the 3⎢⎥⎢⎥formula⎢001⎥⎢1⎥⎣⎦⎣⎦211220⎡⎤⎡⎤⎡⎤100111⎡⎤⎡⎡−−⎤⎤333⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥1121=010−111=−−x=2+4αc=2+4α1,3333p⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥112⎢001⎥⎢111⎥⎢⎥−−⎣⎦⎣⎦⎣333⎦⎢⎥⎢⎥⎢−⎥441⎣⎦⎣⎦⎣⎦江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog47y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog48y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 7 运筹学Operations Research7-9运筹学运筹学 An Interior-Point An Interior-Point AlgorithmOperations ResearchOperations ResearchHHowever, the problem with a value too close to 1 is that the Where the coefficient 4 is used simply to give an upper bound next trial solution then is jammed against a constraint boundary, of 1 for αto maintain feasibility (allxj≥0). Note that increasing thereby making it difficult to take large improving steps duringαto α=1= would cause xto decrease to=+ x4=4+(1)(-1)=0=, where 33subsequent iterations. α>1<> yields x0<. Thus αmeasures the fraction used of the 3Therefore , it is very helpful for trial solutions to be near the distance that could be moved before the feasible region is of the feasible region and not too close to any constraint HoHw large should αbe made for moving to the next trial boundary. With thisK in mind, Karmarkarhas stated for his solution??algorithm that a value as large= as α0=.25 should be “safe.”In Because the increase inZ Zis proportional to α, a value close practice, much larger values (for= example, α0=.9)sometimes are to the upper bound 1 is good for giving a relatively large step used. For the purposes of this example, we have chosen toward optimality on the current iteration. α=0C=.5.(Your OR Courseware uses= α0=.5 as the default value, but also has α=0=.9 available under the option menu.)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo49l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog50y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 An Interior-Point An Interior-Point AlgorithmOperations ResearchOperations ResearchFor the example, there are three constraint boundaries in , A CCentering Scheme for Implementing CConcept 3each one corresponding to a zero value for one of the three variables We now have just one more step to complete the description of the problem in augmented form,namely,x1=0=,x2=0=,and x3==0. of the algorithm, namely, a special scheme for transforming the In ,see how these three constraint boundaries intersect the Ax==b(x1+x+2+x+3=8=8) plane to form the boundary of the feasible region. feasible region to place the current trial solution near its center. The initial trial solution is (x1,x2,x3)==(2,2,4),so this solution is 2 units We have just described the benefit of having the trial solution away from the x1==0 and x2=0= constraint boundaries and 4 units away near the center, but another important benefit of this centeringfrom the x3=0= constraint boundary, when the units of the respective scheme is that it keeps turning the direction of the projected variables are used .HHowever ,whatever these units are in each case, gradient to point more nearly toward an optimal solution as the they are quite arbitrary and can be changed as desired without changing the problem, Therefore ,let us rescale the variables asalgorithm converges toward this solution. The basic idea of the follows:centering scheme is straightforward——simply change the scale xxx123~~~x=, x =, x=123(units) for each of the variables so that the trial solution 224becomes equidistant from the constraint boundaries in the new coordinate system. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©200651Schoo52l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 An Interior-Point An Interior-Point AlgorithmOperations ResearchOperations ResearchSummary and illustration of the algorithmIn these new coordinates, A and c have becomeIteration1: 1:Given the initial trial solution= (x,x,x) (=2,2,4), 123let D be the corresponding diagonal matrix200⎡⎤~⎢⎥200⎡⎤A=AD=[111]020=[224],⎢⎥⎢⎥D=020⎢00⎥4⎣⎦⎢⎥⎢00⎥4⎣⎦20012⎡⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥~c=Dc=0202=4The resca⎢⎥⎢⎥⎢⎥led variables then are the components of ⎢1x00⎥⎢40⎥⎢0⎥⎡⎤⎡⎤1⎣⎦⎣⎦⎣⎦00⎢⎥⎢⎥22x⎡⎤1⎢⎥⎢⎥1x−1⎢⎥2~x=Dx=⎢00⎥x=⎢⎥2⎢⎥22⎢⎥⎢⎥⎢x⎥1x⎣3⎦⎢⎥⎢⎥300⎢⎥⎢⎥44⎣⎦⎣⎦江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog53y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog54y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 7 运筹学Operations Research7-10运筹学运筹学 An Interior-Point An Interior-Point AlgorithmOperations ResearchOperations ResearchSo that the projected gradient is Therefore, the projection matrix is511−−21⎡⎤⎡⎤⎡⎤~~~~TT−1663P=I−A(AA)A⎢⎥⎢⎥⎢⎥~151−1c=Pc=−−4=⎛2⎞p⎡⎤⎡⎤⎡⎤663⎢⎥⎢⎥⎢⎥⎜⎟⎢⎥⎢⎥⎢⎥111=010−2[224]2[224]⎜⎟⎢⎥⎢⎥⎢⎥⎢⎥−−⎢⎥⎢0−⎥2⎣333⎦⎣⎦⎣⎦⎜⎟⎢001⎥⎢⎥⎢⎥44⎣⎦⎣⎦⎣⎦⎝⎠511100448⎡⎤⎡⎡−−⎤⎤6631⎢⎥⎢⎥⎢⎥151=010−448=−−,Define v as the absolute value of the negative component of cp 663⎢⎥⎢⎥⎢⎥24111⎢having the largest absolute value , so that v = − 2 = 2 inthis 001⎥⎢⎥⎢8816−−⎥⎣⎦⎣⎦⎣333⎦case.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo55l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog56y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 An Interior-Point An Interior-Point AlgorithmOperations ResearchOperations ResearchAs shown in (The definition of v has been chosen to ~make the smallest component of equal to zero when α=1= in xCConsequently, in the current coordinates, the algorithm now this equation for the next trial solution.) In the original ~~~moves from the current trial solution (x,x,x)=(1,1,1)123coordinates, this solution is to the next trial solution55x200⎡⎤⎡⎤⎡⎤⎡⎤5142111⎡⎤⎡⎤⎡⎤⎡⎤4⎢⎥⎢⎥⎢⎥⎢⎥~77α=Dx=020=.⎢⎥⎢⎥⎢⎥⎢⎥~7242⎢⎥⎢⎥⎢⎥⎢⎥x=1+c=1+3=,p4⎢⎥⎢⎥⎢⎥⎢⎥1v2⎢⎥⎢⎥⎢⎥⎢⎥x00421⎣3⎦⎣⎦⎣2⎦⎣⎦⎢⎥⎢1⎥⎢−2⎥⎢⎥1⎣⎦⎣⎦⎣⎦⎣2⎦These steps can be summarized as follows for any iteration.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo57l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog58y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 An Interior-Point An Interior-Point AlgorithmOperations ResearchOperations Researchsummary of the interior-point algorithm¾Identify the negative component of cp having the largest absolute value, and set v equal to this absolute value. Then ¾Given the current trial solution… (x,x,…,x), set 12ncalculatex00L0⎡⎤11⎡⎤⎢⎥0⎢⎥x0L02⎢⎥1α~⎢⎥x=+c,D=⎢00xL0⎥p3⎢⎥M⎢⎥vLLLLL⎢⎥⎢⎥1⎢⎥000Lx⎣⎦⎣n⎦Where αis a selected constant between 0 and 1 (for ~¾C~Calculate example= ,α0=.5).A=AD and c=Dc.~~~~~TT−1~P=I−A(AA)A a nd c=Pc¾x=DxCCalculate as the trial solution for the next iteration ¾CCalculate p(step 1) . (If this trial solution is virtually unchanged from the preceding one, then the algorithm has virtually converged to an optimal solution, so stop.)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog59y, JiangXiUniversity of Finance & Economics©2006Schoo60l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 7 运筹学Operations Research7-11运筹学运筹学 An Interior-Point An Interior-Point AlgorithmOperations ResearchOperations ResearchNow let us apply this summary to iteration 2 for the example:So that the BF solutions in these new coordinates areStep 1:16800⎡⎤⎡⎤⎡⎤⎡⎤5Given the current tr=//ial solution (x1,x2,x3)(=52/,72/,2),set−1⎢⎥⎢⎥−1⎢⎥⎢⎥~~16x=D0=0, x=D8=, 75⎢⎥⎢⎥⎢⎥⎢⎥⎡00⎤2⎢⎥⎢⎥00⎢0⎥⎢0⎥⎢⎥⎣⎦⎣⎦⎣⎦⎣⎦7D=⎢⎥and⎢⎥002⎣⎦00⎡⎤⎡⎤Note that the rescaled variables are−1⎢⎥⎢⎥~x=D0=0,⎢⎥⎢⎥22x00xx⎡⎤⎡⎤⎡⎤⎡⎤11155⎢⎥⎢⎥84⎣⎦⎣⎦⎢⎥−1⎢⎥⎢⎥⎢⎥22x=Dx=00x=x,22277⎢⎥⎢⎥⎢⎥⎢⎥11⎢x⎥⎢⎥⎢x⎥⎢⎥00xAs depicted in fig ⎣3⎦⎣2⎦⎣3⎦⎣23⎦江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo61l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo62l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 An Interior-Point An Interior-Point AlgorithmOperations ResearchOperations ResearchStep 4:Step 2:4111415−>−, so v= and ⎡⎤2151215~⎢⎥~57A=AD=[2] and c=Dc=⎡−⎢⎥⎡⎤⎤⎡⎤⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥0~133461⎣⎦x=1+=≈⎢⎥41⎢⎥⎢⎥⎢⎥15411⎢1⎥⎢⎥⎢⎥−⎢0⎥.50St⎣⎦ep⎣⎦⎣⎦⎣⎦ 3:152Step 5:137211⎡−−⎤⎡-⎤13651818912⎡⎤⎡⎤656⎢⎥⎢⎥74114133P=−− and cp=.18904560⎢⎥⎢⎥~⎢⎥⎢⎥3227x=Dx=≈⎢⎥⎢⎥⎢⎥⎢⎥−−-⎣94545⎦⎣15⎦⎢⎥1⎢1⎥.00⎣⎦⎣⎦is the trial solution for iteration 3.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom63ics©2006Schoo64l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 An Interior-Point An Interior-Point AlgorithmOperations ResearchOperations ResearchHoHwever, do not let this contrast fool you into downgrading To conclude, we need add a comment to place the algorithm the efficiency of the interior-point algorithm. This algorithm is into better perspective. For our extremely small example, the designed for dealing with big problems having many hundreds algoror thousands of functional constraints. The simplex method ithm requires relatively extensive calculations and typically requires thousands of iterations on such problems . Bythen ,after many iterations, obtains only an approximation of “shooting ”through the interior of the feasible region, the the optimal solution. By contrast, the graphical procedure of interior-point algorithm tends to require a substantially smaller finds the optimal solution in immediately, and the number of iterations (although with considerably more work simplex method requires only one quick iteration.) Therefore, as discussed in , iteration-point algorithms similar to the one presented here should play an important role in the future of linear programming.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo65l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog66y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 7 运筹学Operations Research7-12运筹学运筹学7 Other linear goal programminggorithms for Linear ProgrammingOperations ResearchOperations linear goal programming and its solution proceduresGoal programming problems can be categorized according to the type of mathematical programming model (linear programming, Goal programminginteger programming, nonlinear programming, etc.) that it fits lower, one-sided goal sets a lower limit that we do not except for having multiple goals instead of a single objective. In want to fall under (but exceeding the limit is fine).this book, we only consider linear goal programming–those goal upper, one-sided goal sets an upper limit that we do not programming problems that fit linear programming otherwise want to exceed (but falling under the limit is fine).(each objective function is linear, etc.) and so we will drop the two-sided goal sets a specific target that we do not want to adjective linear from now on .miss on either side.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo67l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo68l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 linear goal linear goal programmingOperations ResearchOperations Researchnonpreemptivegoal programmingPrototype example fornonpreemptivegoal AnAother categorization is according to how the goals programmingcompare in importance . The DEEWRIGHHT CCOMPANY is considering three new preemptive goal programmingproducts to replace current models that are being discontinued ,so their OR department has been assigned the task of AAll the goals are of roughly comparable importance. determining which mix of these products should be produced .Management wants primary consideration given to there is a hierarchy of priority levels for the goals, so that three factors: long-run profit , stability in the workforce, and the goals of- primary importance receive first-priority attention, the level of capital investment that would be required now for those of secondary importance receive second--priority attention, new equipment. In particular , management has established the and so forth (if there are more than two priority levels). goals of (1) achieving a long-run profit (net present value) of at ¾we begin with an example that illustrates the basic features least $$125 million from these products.(2) maintaining the ofnonpreemptivegoal programming and then discuss the current employment level of 4000 employees, and (3) holding preemptive capital investment to less than$ $55 million. are shown in Table , along with the goals and penalty weights.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo69l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog70y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 linear goal linear goal programmingOperations ResearchOperations ResearchFormulation:the Dewrightcompany problem includes all oHwever, management realizes that it probably will not be three possible types of goals:a lower, one-sided goal (long-run possible to attain all these goals simultaneously, so it has profit); ;a two-sided goal (employment; level) ;and an upper, one-sided goal (capital investment). Letting the decision discussed priorities with the OR department. This discussion variables x1,x2,x3 be the production rates of products 1,2 and has led to setting penalty weights of 5 for missing the profit 3, respectively, we see that these goals can be stated as goal (per $1 million under), 2 for going over the employment profit goalgoal (per 100 employees), 4 for going under this same goal , employment goaland 3 for exceeding the capital investment goal (per $1 million investment ). Each new product’s contribution to profit, employment level, and capital investment level is proportional to the rate of 12x+9x+15x≥125123production. These contributions per unit rate of production5x+7x+8x≤551235x+3x+4x=40123江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo71l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo72l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 7 运筹学Operations Research7-13运筹学运筹学 linear goal linear goal programmingOperations ResearchOperations ResearchZlet Z be the numbber of penalyt ypointsincurred by missing Table75D for the DewrightCCo. nonpreemptivegoal these goals. The overall objective then is to choose the values of programming problemx1,x2,and x33 so as to M5Z=Minimize 5Z(=amount under the long--run profit goal)Unit ++2(amount over the employment level goal)contribution+4+4(amount under the employmentlevel goal)Penalty+3FactGoal (units)+3(amount over the capital investment goal) ,orProduct:weightWhWere no penalty points are incurred for being over the long ––1 2 3run profit goal or for being under the capital investment goal. To express this overall objective mathematically, we introduce someLong-run profit≥125(millions of dollars)12 9 155auxiliary variables (extra variables that are helpful for formulating Em=Eployment level4=0(hundreds of the model) y,y, and ydefined as follows::5 3 42(+)+,4(-)1233employees)CCapital 5 7 8 8 3y=9+=12x9x+5-++15x-125-5 (long-run profit minus the target).11233investment≤55(millions of dollars)y=53+= 5x3x+4-4++ 4x-40 (employment level minus the target).21233y=57++855= 5x7x++ 8x-- 55 (capital investment minus the target).333123江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo73l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Econom74ics©2006运筹学运筹学 linear goal linear goal programmingOperations ResearchOperations ResearchSince each ycan be either positive or negative, we next use the Given these new auxiliary variables, the overall objective can itechnique described at the end of for dealing with such be expressed mathematically as−+−+variables ;namely ,we replace each one by the difference of two Minimize Z=5y+2y+4y+3y,1223nonnegative variables:Which now is a legitimate objective function for a linear +−+−y=y−y, where y≥0,y≥0,programming model. (Because there is no penalty for exceeding 11111the profit goal of 125 or being under the investment goal of +−+−y=y−y, where y≥0+−,y≥0,22222yy55,neither nor should appear in this objective function31+−+−representing the total penalty for deviations from the goals.)y=y−y, where y≥0,y≥0,33333To complete the conversion of this goal programming As discussed in , for any BF solution, these new problem to a linear programming model, we must incorporate auxiliary variables have the interpretation+−yythe above definitions of the j and j directly into the model. (It ⎧y if y≥0, +j jy= is not enough to simply record the definitions, as we just did, ⎨j0 otherwise; ⎩because the simplex method considers only the objective ⎧function and constraints that constitute the model. )y if y≤0, −jjy=⎨j0 otherwise; ⎩江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of75 Information Technology, JiangXiUniversity of Finance & Economics©2006Schoof76l o Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 linear goal linear goal programmingOperations ResearchOperations Research−+−+Minimize Z=5y+2y+4y+3y,12x3Applying the simplex method to this formulation yields an optima=/=/subject tol solution x2=53/, x=0=, x=5/3, with 123+−+−+−25+−y=0,y=0,y=,y=0, y =0, and y=0,12x+9x+15x-(y-y)=125112233312311+− 5x+3x+4x-(y-y)=4012322Therefore ,y=0=/=,y2=53/.and y=0=, so the first and third +−123 5x+7x+8x-(y-y)=5512333goals are fully satisfied , but the employment level goal of 40 is 1andexceeded8 by 8 (833 employees). The resulting penalty for 32+−deviating from the goals is Z=16x≥03, y≥0, y≥0 (j=1,2,3;k=1,2,3).jkk (If the original problem had any actual linear programming constraints, such as constraints on fixed amounts of certain resources being available, these would be included in the model.)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of7 Information Technolog7y, JiangXiUniversity of Finance & Economics©2006Schoo78l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 7 运筹学Operations Research7-14运筹学运筹学 linear goal linear goal programmingOperations ResearchOperations ResearchPreemptive goal programmingWhen we deal with goals on the samepriority level, our In the preceding example we assume that all the goals are of approach is just like the one described fornonpreemptivegoal roughly comparable importance. Now consider the case of programming. Any of the same three types of goals (lower one-preemptivegoal programming , where there is a hierarchy of sided, two-sided, upper one-sided) can arise. Different penalty priority levels for the goals. Such a case arises when one or weights for deviations from different goals still can be included, more of the goals clearly are far more important than the if desired. The same formulation technique of introducing auxiliary variables again is used to reformulate this portion ofothers. Thus the initial focus should be on achieving as closelythe problem to fit the linear programming possible these firs-t-prioryitygoals. The other goals also might There are two basic methods based on linear programming naturally divide further into second-priority goals, third-for solving preemptive goal programming problems . One is priority goals, and so on. After we find an optimal solution with called the sequential procedure , and the other is streamlined respect to the first– –priority goals, we can break any ties for the procedure. We shall illustrate these procedures in turn by optimal solution by considering the second-priority goals. Any solving the following that remain after this reoptimizationcan be considering the third-priority goals, and so on . 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog79y, JiangXiUniversity of Finance & Economics©2006School of Information TechnologangXiUniversity of Finance & Economics©200680y, Ji运筹学运筹学 linear goal linear goal programmingOperations ResearchOperations ResearchEExample::Faced with the unpleasant recommendation to Based on these considerations, management has concluded increase the company’’s workforce by more than 20 percent, the that a preemptive goal programming approach now should be management of the DewrightCCompany has reconsidered the used, where the two goals just discussed should be the first-original formulation of the problem that was summarized in priority goals, and the other two original goals (exceeding 1$25Table increase in workforce probably would be a million in long-run profit and avoiding a decrease in the rather temporary one , so the very high cost of training 8833 newemployment level) should be the second-priority goals. Within employees would be largely wasted, and the large (undoubtedly the two priority levels, management feels that the relative well-publicized) layoffs would make it more difficult for the penalty weights still should be the same as those given in the company to attract high-quality employees in the future. CConsequently, management has concluded that a very high rightmost column of Table . This reformulation is priority should be placed on avoiding an increase in the summarized in , where a factor of M (representing a workforce. Furthermore , management has learned that raising huge positive number) has been included in the penalty weights more than $$55 million for capital investment for the new for the first-priority goals to emphasize that these goals products would be extremely difficult, so a very high priority preempt the second-priority goals. (the portions of Table alsoshould be placed on avoiding capital investment above this that are not included in Table are unchanged.)level.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangX81iUniversity of Finance & Economics©2006School of Information Technolog82y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 linear goal linear goal programmingOperations ResearchOperations ResearchThe sequential procedure for preemptive goal programmingThus ,in this case, all these auxiliary variables now can be The sequential procedure solves a preemptive goal programming completely deleted from the model, where the equality problem by solving a sequence of linear programming the first stage of the sequential procedure, the only goalsconstraints that contain these variables are replaced by the included in the linear programming model are the first-priority mathematical expressions (inequalities or equations) for these goals , and the simplex method is applied in the usual way. If the first-priority goals , to ensure that they continue to be fully resulting optimal solution is unique, we adopt it immediately without considering any additional goals.*achieved. On the other hand , if Z0>, the second-stage model HHowever, if there are multiple optimal solutions with the same simply adds the second-priority goals to the first-stage model **optimal value of Z Z(call it ZZ),we prepare to break the tie among these solutions by moving to the second stage and adding the second-(as if these additional goals actually were first-priority goals **priority goals to the model. If Z=Z=0, all the auxiliary variables from the second-stage objective function). After we apply the representing the deviations from first-priority goals must equal zero simplex method again, if there still are multiple optimal (full achievement of these goals) for the solutions remaining under consideration. solutions, we repeat the same process for any lower-priority goals.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUn83iversity of Finance & Economics©2006Schoo84l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 7 运筹学Operations Research7-15运筹学运筹学 linear goal linear goal programmingOperations ResearchOperations ResearchEExample::we now illustrate this procedure by applying it to By using the simplex method (or inspection), an optimal ++the exampy=yle summarized in Tab6le 7..6solution for this linear programming= model has 0=, 23**with=ZZ= 0=Z(so Z0=), because there are innumerable solutions for Table 6Revised Formulation for the Dewright(x,x,x) that satisfy the Goal Programming Problem1235x+3x+4x≤40123Priority Level Factor GoalPenalty Weight5x+7x+8x≤55123EmEployment level≤402MFirst priorityAs well as the nonnegativityconstraints. Therefore, these CCapital investment≤553Mtwo first-priority goals should be used as constraints hereafter. Long-run profit≥1255++Using them as constraints will force y and to remain zero Second priorityy23EmEployment level≥404and thereby disappear from the model automatically.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUner85ivsity of Finance & Economics©2006School of Information Technolog86y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 linear goal linear goal programmingOperations ResearchOperations Research++If we drop and yy but add the second priority goals ,23Because this solution is unique (or because there are no the second-stage linear programming model becomes more priority levels), the procedure can now stop, with −−Minimize Z=5y+4y,12(x=,xx)(=5,0,) as the optimal solution for the overall Subject to12,3+−12+9x+15x−(y−y) =1252311problem. This solution fully achieves both first-priority goals −5x+3x+4x +y = 401232as well as one of the second-priority goals (no decrease in −5x+7x+8x +y=551233employment level), and it falls short of the other second-+−x≥0, y≥0, y≥0 (j=1,2,3;k=1,2,3)jkkpriority goal (long-run profit≥125)8 by just the simplex method to this model yields the uniqueoptimal solution x1=5=,x2=0=,+−−−333x=3,y=0,y=8,y=0,and y=0,with Z=江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo87l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo88l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 linear goal linear goal programmingOperations ResearchOperations ResearchThe streamlined procedure for preemptive goal The linear programming formulation for the streamlined programmingprocedure with two priority levels would include all the goals in Instead of solving a sequence of linear programming models, the model in the usual manner, but with basic penalty weights like the sequential procedure, the streamlined procedure finds of M and 1 assigned to deviations from first-priority and an optimal solution for a preemptive goal programming problem by solving just one linear programming model. Thus, second-priority goals ,respectively. If different penalty weights the streamlined procedure is able to duplicate the work of the are desired within the same priority level , these basic penaltysequential procedure with just one run of the simplex method. weights then are multiplied by the individual penalty weights This one run simultaneously finds optimal solutions based just assigned within the level. This approach is illustrated by theon first-priority goals and breaks ties among these solutions by considering lower priority goals. However, this does require a following modification of the simplex method.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo89l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo09l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 7 运筹学Operations Research7-16运筹学运筹学 linear goal linear goal programmingOperations ResearchOperations ResearchEFDExample:For the goal programming More than two priority levels:problem summarized in Table76 , note that (1) different penaltyWhen there are more than two priority levels (say, p of weights are assigned within each of the two priority levels and (2) the individual penalty weights (2 and 33) for the firs-t-priority goals them ), the streamlined procedure generalizes in a have been muMltiplied by .M These penalty weights yield the straightforward way. The basic penalty weights for the following single linear programming model that incorporates all respective levels now are M,M…,…,M,1,where Mis the -11−+−+Z=5y+2My+4y+3Myrepresents a number that is vastly larger than M,Mis vastly MMinimize 122223Subject to larger than M…,…,and Mis vastly larger than1E. Each 3p-1+−12x1+9x2+15x3−(y−y)=12511coefficient in row 0 of each simplex tableau is now a linear +−function of all these quantities, where the multiplicative factor 5x1+3x2+ 4x3−(y−y)=4022+−of Mus used o make the necessary decisions, with tie 1 5x1+7x2+ 8x3−(y−y)=5533breakers beginning with the multiplicative factor of Mand +−2x≥0,y≥0,y≥0(j=1,2,3;k=1,2,3).AAndjkkending with the additive term.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of19 Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo29l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学7 Other Algorithms for Linear ConclusionsOperations ResearchOperations ResearchVaVrious other special-purpose algorithms also have been oCnclusionsdeveloped to exploit the special structure of particular types of linear programming problems (such as those to be The dual simplex method and parametric linear programming discussedC8 in and 9). Much research is currently are especially valuable for posto-ptimality analysis, although they being done in this can be very useful in other ’’s interior-point algorithm has been an exciting The upper bound technique provides a way of streamlining the recent development in linear programming . This algorithm simplex method for the common situation in which many of or all and its variants hold much promise as a powerful new the variables have explicit upper bounds. It can greatly reduce the approach for efficiently solving some very large effort for large goal programming and its solution procedures aMthematicalp-rogramming computer packages usually include provide an effective way of dealing with problems where all three of these procedures, and they are widely used. Becausemanagement wishes to strive toward several goals their basic structure is based largely upon the simplex method as simultaneously. The key is a formulation technique of presented in , they retain the exceptional computational introducing auxiliary variables that enable one to convert the efficiency to handle very large problems of the sizes described in problem to a linear programming .江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog39y, JiangXiUniversity of Finance & Economics©2006School of49 Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 8 运筹学Operations Research8-1运筹学运筹学Operations ResearchOperations ResearchTransportationT3A2B1118 Transportation and Assignment 74T1164145Problems5T24AB7B14268343167722T2B386T5A3江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo1l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo2l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Transportation The Transportation ProblemOperations ResearchOperations ResearchAssignment The transportation problemEaEch task required time (hours).82 A streamlined simplex method for the workersTask 1Task 2Task 3Task 4Wage per hourtransportation problemAn354An354127740$44$14 The assignment problemIan4745354745325112Jn9Joa356364339563631343 oCnclusionsSean3532125546554615江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog3y, School of Information Technolog4JiangXiUniversity of Finance & Economics©2006y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Transportation The Transportation ProblemOperations ResearchOperations Research1. Prototype xEample .81 The Transportation ProblemP&T COMPANY is canned peas. The peas are prepared at hTe transportation problem: is defined as from three canneries (near Bellingham ;Eugene ;and Albert Lea;) and some sources to shipment goods to destinations, then shipped by truck to four distributing warehouses in the and determining how to optimally transport goods western United States (Sacramento, California ;Salt Lake City, to minimize the total shipping .)Because the shipping costs are a major expense, management The algorithms to solved transportation problemis initiating a study to reduce them as much as possible. The is :transportation simplex method or the network problem now is to determine which plan for assigning these simplex to the various cannery-warehouse combinations would minimiez the total shipping cost.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©20065Schoo6l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 8 运筹学Operations Research8-2运筹学运筹学 The Transportation The Transportation ProblemOperations ResearchOperations ResearchPT& oCmpany Distribution ProblemCANNERY 1 BellinghamWAREHOUSE 3 CANNERY 2 Rapid CityEugeneCANNERY 3 Shipping DataAlbert LeaWAREHOUSE 2 Salt Lake CityWAREHOUSE 1 SacramentoCaCnneryOutputWarehouseAllocationWAREHOUSE 4 AlbuquerqueBBellingham7575 truckloadsSacramento880 truckloadsEEugene125adsL5 truckloSalt aLke CCity6565 truckloadsAlLAbert eLa100 truckloadsRRapid CiCty770 truckloadsTotal3 truckloadsAlqAbuquerq300que8585 truckloadsTotal3300 truckloadsTotal supplyT=otal demand 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom7ics©2006Schoo8l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Transportation The Transportation ProblemOperations ResearchOperations ResearchPT& oCmpany Distribution ProblemPT& oCmpany Distribution ProblemNetwork RepresentationDemandsSuppliesShipping oCst per TruckloadDestinationsSources(Sacramento)D081CaCnneryWarehouse464513(eBllingham)57S1To\\FromSacramentoSalt Lake CiCtyRapid CiCtyAlbuquerque654867D56(Salt Lake iCty2352416SBm64Bellingha$$53$564$687$64$153$564$687(Eugene)1252690791EEugene35466979325416609719D07(aRpid iCty)3682995388AlL5Abert eLa9968388658959628388658(Albert Lea)100S3685D58(Albuuqerque)4江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo9l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo10l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Transportation The Transportation ProblemOperations ResearchOperations ResearchPT& oCmpany Distribution ProblemPT& oCmpany Distribution ProblemThe Transportation Problem is an LPCurrent Shipping PlanLet x =the number of truckloads to ship from cannery ito warehouse jij(i= 1, 2, 3; j= 1, 2, 3, )4CaCnneryWarehouseiMnimize oCst =464$x+ $513x+ $654x+ $867x11121314+ $352x+ $416x +$690x+ $791xToSacramentoSalt Lake CiCtyRapid CiCtyAlbuquerque21222324\\From+ $995x+ $682x+ $388x+ $685x31233343BBellingham7575000subject toCannery 1:x+ x+ x +x =75Supply constraintsE11121314Eugene56555565550Cannery 2:x+ x+ x +x =12521222324AlAbert LeLa001558585Cannery 3:x+ x+ x +x =10031323343aWrehouse 1:x+ x+ x =80112131Total shipping cost =(57)464$ +(5253$) + 65($416) + 55($690) +15($388) + 85($685)aWrehouse 2:x+ x+ x =65122232= $165,595aWrehouse :3x+ x+ x =70132333Demand constraintsaWrehouse :4x+ x+ x =85142434and x≥0 (i =1, 2, ;3 j =1, 2, ,3 4)ij江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog11y, JiangX12iUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 8 运筹学Operations Research8-3运筹学运筹学 The Transportation The Transportation ProblemOperations ResearchOperations ResearchPT& oCmpany Distribution ProblemPT& oCmpany Distribution ProblemBCDEFGHIJ3Unit CostDestination (Warehouse)4SacramentoSalt Lake CityRapid CityAlbuquerqueCaCnnery5SourceBellingham$464$513$654$867Warehouse6(Cannery)Eugene$352$416$690$7917Albert Lea$995$682$388$685To8SacramentoSalt Lake CiCtyRapid CiCtyAlbuquerque\\From910Shipment QuantityDestination (Warehouse)11(Truckloads)SacramentoSalt Lake CityRapid CityAlbuquerqueTotal ShippedSupplyBBellingham7575(0)0(20)00(5555)12SourceBellingham02005575=7513(Cannery)Eugene804500125=125EEugene5865545(80))5565(5455(0)014Albert Lea007030100=10015Total Received80657085AlAbert LeLa00155(70)853785(03)16====Total Cost17Demand80657085$152,535Revised H12=sum(D12G:12)shipmentD15=sum(D12D:14) ……Save H13=sum(D13G:13)$15,0601J7=sumproduct(D5G:7,D12G:14)H14=sum(D14G:14)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of13 Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo14l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Transportation The Transportation ProblemOperations ResearchOperations Researchc11oCst per Unit Distributed-[d]sS1D1[]c1112cDestination1nSupplyc1 2 …n21S2D2-[d]s2[]2c122cccs1112 1n1Source2cccs2122 2n2c2n．．．．．．...………………………...mcc…csm1m2 mnmcm2cm1demanddd…d12ncSmmns[]-[d]mDn3江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo15l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo16l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Transportation The Transportation ProblemOperations ResearchOperations ResearchThe general model of transportation problemInteger solutions property:For transportation problems where mmeverysanddhave an integer value, all the basic variables ijMinimizeZ=cx∑∑ijij(allocations) in every basic feasible (FB) solution (including an i=1j=1optimal one ) also have integer ⎧x=sfori=1,2,....m∑iji⎪j=1⎪mFeasible solutions property:A necessary and sufficient condition for ⎪=dforj=1,2,...,na transportation problem to have any feasible solution is that⎨∑ijji=1⎪⎪x≥0,foralliandjmnij⎪s=d∑i∑j⎩i=1j=1江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of17 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Econom18ics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 8 运筹学Operations Research8-4运筹学运筹学 The Transportation The Transportation ProblemOperations ResearchOperations ResearchExample—Production Scheduling: The NORTHERN AIRPLANE COMPANY builds commercial airplanes for various airline Cost Unit Distributed1=.03+8× around the world. The last stage in the production process 1=.00+ to produce the jet engines and then to install them in the completed DestinationSupply airplane frame. The company has been working under some contracts Source1 2 34to deliver a considerable number of airplanes in the near future, and production of the jet engines for these planes must now be 2? for the next 4 ? ? cost*Unit cost* of 4? ?? production1 10 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog19y, JiangXiUniversity of Finance & Economics©2006Schoo20l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Transportation The Transportation ProblemOperations ResearchOperations ResearchNorthern Airplane Co. Production-Scheduling ProblemProduction CostprocudtionStorage CostCost Unit Distributed($millions)cost($millions per month)Month Source1 2 3 4 5(D)Month CostMonth 025($millions) M M M 010Month InstalledMaximmuUnits Produced1234ProcudedProcudtionDemand 10 1525203010100525<=25Month2503Procuded30025530<=3040001010<=10Instaled115252====Total oCstSchelude d Installations10152520m$(illions)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of In21formation Technology, JiangXiUniversity of Finance & Economics©2006Schoo22l of Information Technology, JiangXiUniversity of Finance & Economics©2006Exam运p筹le学D—istribution of Water Resources :METRO ATWER 运筹学 The Transportation ProblemOperations ResearchOperations ResearchDEISTRIC is an agency that administers water distribution in a large geographic region. The region is fairly arid, so the district Must purchase oCst (Tens of Dollars) per Acre Footand bring in water from outside the region. The sources of this imported BerdooLos DevilsSan GooHllyglassSupply water are the Colombo,Sacron, and aClorie river. The district then resells the water to users in its region. Its main customers arethe water oClombo River1613221750departments of the cities ofBerdoo, Los Devils, San Go, River1413191560aClorie River192023__50oCst (Tens of Dollars) per Acre FootMinimum needed3070010(in units of 1 million BerdooLos DevilsSan GooHllyglassSupply acre feet)Requested507030∞oClombo River1613221750Sacron River1413191560Question:aMnagement wishes to allocated all the available water from aClorie River192023__50the three rivers to the four cities in such a way as to at leastmeet the essential needs of each city while minimizing the total cost to the needed3070010(in units of 1 million acre feet)Requested507030∞江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo23l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniverf24sity o Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 8 运筹学Operations Research8-5运筹学运筹学 The Transportation The Transportation ProblemOperations ResearchOperations ResearchMetro Water District Distribution ProblemCoCst (Tens of Millions of Dollars) per Unit DistrictedUnit Cost ($millionBerodooLs eDvilsSan oGoHllyglasslmb Rver1613217DestinationSacron i4195Clie Rver192023-Supply Berdoo Berdoo Los Devils San Go HoHllyglass(min) (extra)Water DistributionTotalCoClombo River1 1 66 166 13221750(million acre-feet)BerodooLs eDvilsSan oGoHllyglassFrom RiverAvailalbeSacron River 2 14 14 13 1915660oClomob River0500050<=50CaClorie River 3Sacron River302001060<=6019 19 2023 M50Clie 5Dummy 4(D)M 0 M 0 0503070010Demand 30<=<=<=<= 20 7030 660Total To iCty3070010<=<=<=<=Total oCsteNeedd50703060m$(illion)1,480江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog25y, JiangXiUniversity of Finance & Economics©2006School of Informat26ion Technology, JiangXiUniversity of Finance & Economics© A Streamlined Simplex Method for 运筹学运筹学8 the transportation and assignment problemsOperations ResearchOperations Researchthe Transportation A Streamlined Simplex Method for the Setting Up the Transportation Simplex MethodTransportation ProblemTo highlight the streamlining achieved by the transportation WeW shall refer to this streamlined procedure as the simplex method, let us first review how in the general transportation simplex method.(unstreamlined) simplex method the transportation problem AAs you read on, note particularly how the special structure is would be set up in tabular formA. After constructing the table ofexploited to achieve great computational savings. Then bear in C86Constraint coefficients (see Table ), converting the objective mind that comparable savings sometimes can be achieved by function to maximization formexploiting other types of special structures as well, including those described later in the chapter.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo27l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo28l of Information Technology, JiangXiUniversity of Finance & Economics© A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation Problemand using theBM Big M method to introduce artificial variables u=g= multiple of originalrow i that has been subtracted iz, z…, …zinto the m ++q n respective equality constraints (see 12m+n+(directly or indirectly) from orgiginal row 0 by the simplex method ), we see that typical columns of the simplex tableau during all iterations leading to the current simplex have the form shown in Table83 , where all entries not v== mugltiple oforiginalrow+ m+ j that has been subtracted jshown in these columns are zeros.(direcOtly Or indirectly) from original row 0 by the simplex method during all iterations leading to the current simplex tableau.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo29l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog30y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 8 运筹学Operations A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemYYou might recognize uand vfrom as being the dual InFB the initialization, an initial FB solution must be obtained, ijvariabbles. If xis a nonbasicvariable, c–––u–vis interpreted as which is done artificially by introducing artificial variables as the ijijijthe rate at whZich Z will change as xis basic variables and setting thenq equal to siand dj. The ijoptimality test and step 1 of an iteration (selecting an entering To lay the groundwork for simplifying this setup, recall what basqic variable) require knowing the current row 0, which is information is needed by the simplex by subtracting a certain multiple of another row from the preceding row 0.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog31y, JiangXiUniversity of Finance & Economics©2006Schoo32l of Information Technology, JiangXiUniversity of Finance & Economics© A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemTable Original Simplex Tableau before Simplex Method Is Step 2 (determining the leaving basic variable) must Applied to Transportation Problemidentify the basic variable that reaches zero first as the entering CoCefficient of :Basic VbaaVriablesic variable is increased , which is done by comparing the Z……Z…. x……. z…. zij im+j+sidecurrent coefficients of the entering basic variable and the ZZ(0)corresponding right side. cM 0ij (1)-1( i )St1 1 sep3 3 must determineBF the new BF solution , which is found iZZ(m+j+)0 iby subtracting certain multiples of one row from the other rows 1 1djZ(m+Zn+)0m+j+in the current simplex tableau江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog33y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog34y, JiangXiUniversity of Finance & Economics© A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemSecond, the current row 0wgy can e obtained iwthout using an yNNow, how does the transportation simplex methodobtain other rowwsimply by calculating the current values of uand vijdirectly. Since each basic variable must have a coefficient of zero the same ?information in much simpler ways? This story will in row 0, the current uand vare obtained by solving the set of ijunfold fully in the coming pages , but here are some preliminaryeqquationsc––u––v0=0=for each i and j such that xis a basic ,FFirst, no artificial variabblesare neededwhich can be done in a straightforward way., because a simplex Third, the leaving basic variable can be identified in a simple and convenient procedure (with several variations) is available wayW iWthout (explicitly) using the coefficients of the entering basic constructing an initial FBFB solution.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniver635sity of Finance & Economics©200Schoo36l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 8 运筹学Operations A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemTable Original Simplex Tableau before Simplex Method Is InzitialiaztionApplied to Transportation ProblemReRcall that the objective of the initialization is to obtain an initial FBFBB solution. eBcause all the functional constraints in the CoCefficient of:Basic transportation problem areq equality constraints, the simplex SideVaVriableZ……xij ……zi ……m+j+ …Z…MMethod would obtain this solution by introducing artificial mnvariables and using them as the initial basic variables, as −su−dv∑ii∑jjZZ(0)-1cij-ui––vj M-ui M-vji=1j=1described in .江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo3l of7 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Informat38ion Technology, JiangXiUniversity of Finance & Economics© A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemTable Format of a Transportation Simplex TableauGePCgGneral Procedure for oCnstructing an InBitial FB SolutionDestination 1. From the rows and columns still under consideration, select 1 ply 2 ．．．nsupUicccs111121nthe next basic variable (allocation) according to some ．．．2. Make that allocation large enough to exactly use up the ccc21222n．．．remaining supply in its row or the remaining demand in its 2s2．．．．．．．．．．．．source column (whichever is smaller).ccCCm1m2mnm．．．smDemandd1 d2 ．．．dn=Z=Zvj江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of39 Information Technology, JiangXiUniversity of Finance & Economics©2006Schoon40l of Iformation Technology, JiangXiUniversity of Finance & Economics© A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemDeDstination3E3. Eliminate that row or column (whichever had the smaller supply ui1 2 33 44 55remaining supply or demand) from further consideration. (If the 116616613322177550row and column have the same remaining supply and demand, then arbitrarily select the row as the one to e eliminated. The column will be used later to provide a degenerate basic variable, ., a circled allocation of zero.)660Source 44. If only one row or only one column remains under 19919920233MM33060CConsideration, then the procedure is completed by selecting every 550remaining variable (., those variables that were neither M0MMM00p4D5reviously selected to be basic nor eliminated from consideration 4()D50103010by eliminating their row or column) associated with that row or column to e basic with the only feasible allocationO. tOherwise, demand36=Z30 207370247+3060=Z0471+0return to step 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Informat41ion Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUn42iversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 8 运筹学Operations A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemAAlternative CCriteria for Step corner rule::BBegin by selecting x( that is, start 11EExample::To make this description more concrete, we now in the northwest corner of the transportation simplex illustrate the general procedure on the MeMtroWD aWter iDstrict tableau). Thereafter , if xwas the last basic variable ijselected, then next select x(that is , move one column to problem (see8 Table ) with the northwest corner rule being i,j++1the rOight) if source i has any supply remaining. Otherwise, usedB=4 in step 1. Because m=4 and n=5=5 in this case, the procedure next select x(that is , move one row down).i++1,jwould findFB+- an in-1=8itial FB solution having m+n=8 basic variables.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog43y, JiangXiUniversity of Finance & Economics©2006Schoon44l of Iformation Technology, JiangXiUniversity of Finance & Economics© A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemA86=3As shown in Table , the first allocation is x11 0=3, CContinuing in this manner , we eventually obtain the entire which exactly uses up the demand in column 1 ( and eliminates FB86initial FB solution shown in Table , where the circled this column from further consideration). This first iteration leaves numbers are the values of the basic3… variables (for x11 == 30, …a supply of 20, and this row is eliminated from further consideration. (RRow 1 is chosen fro elimination rather than x4545 5=50=) and all the other3 variables (x13, etc.) are nonbasiccolumn 2 because of the parenthetical instruction in step3 3.)variables eqqual to , select x1++1,2 == x22 next. BBecause the remaining demand of6 0 in column 2 is less than the supply of 60 in row 2, alloca=te x22 =0 and eliminate column 2.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo45l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo46l of Information Technology, JiangXiUniversity of Finance & Economics© A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemADArrows have been added to show the order in which the basic Defined as the arithmetic differencebw betewen the smallest variables (allocations) wereZ selected. The value of Z for this and next---to-the--smallest unit cost cstill remagining in thawt ro wijj solution is at column..(If two unit costs tie for begin the smallest remaining in a row or column, then the difference is 0.) In that row or Z=6(3+6+…30) + 16(20)+ + …+0 (50=74+MZ= 165) = 2,704 + 10Mcolumn having theg largest difference, select the variable having 2. VVoge’l’s approximation method:: FFor each row and column the smallest unit cost.(Ties for the largest difference, remaining under consideration, calculate its difference, which is or for the smallest remaining unit cost, may be broken arbitrarily.)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo47l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo48l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 8 运筹学Operations A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemTable Initial BF Solution from VoVgel’’s Approximation MethodENExample:Now let us apply the general procedure to the DRDestinationoRwMWDMetro aWter District problem by using the criterV’ion fro Vogel’s Supply DDifference1 2 33 44 551166166 133 22 177 5350 3approximation method to select the next basic variable in step 1. Source 2 144 144 133 199 155660 1 353199 199 20 233 M M50 0WiWth this criterion , it is more convenient to work with cost and 4M M4(DD) M 0 M 0 0550 0reqquirements tables (rather than with complete transportation D37Demand30 20 70 330 660Select x=3=30444419Co difference2 145EClumn4 0 15 lEiminate column 44simplex tableaux), beginning with the one shown in Table.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo49l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog50y, JiangXiUniversity of Finance & Economics© A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemTable8 (continued)Table (continued)DRDestinationoRwDRDestinationoRwSupply DDifferenceSupply DDifference1 2 33 551 2 33 551166166 133 177 550 331166166 177 550 133Source 2 144 14 133 1564 560 1 Source 2 144 144 133 156560 1 33199 199 20 M 5M50 03199 1939 20 M 5M50 04DM4(D) M 0 MM550 00D37Demand30 20 70 404Select x=5=50133D3Demand30 20 770 606Select x==204545CoClumn difference2 2 0 2 EEliminate row 1Coumn difference2 14ECl4 0 lEiminate column 4D4(D)15江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo51l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo52l of Information Technology, JiangXiUniversity of Finance & Economics© A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemTable8 (continued)Table8 (continued)DnRDestinatiooRwSupply DDifferenceDestinationRow1 2 33 55Supply DifferenceSource 2 144 1464 133 60 1 353199 199 20 M M50 01 2 3 15DDemand330 20 20 404Select x=4=40255Source 2 14 14 20 1 Co557EClumn difference5 5 7 lEiminate column 55319 19 20 50 0131-M5Demand30 20 20 Select x=2=023CoClumn difference5 5 EEl i m i nate row 27江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo0653l of Information Technology, JiangXiUniversity of Finance & Economics©20Schoo54l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 8 运筹学Operations A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemTable 8initial BF solution from Russell’’s approximation methodTable8 (continued) LargestIterationAAllocationu1 u2 u3 u4v v v v v12345Negative ΔijDDestination122 199 M MMM M M 199 M M 233 MMX=5X=50Δ=−2M454545Supply 222 199 M M 199 199 20 23 3MMX=X=101551 2 33 Δ=−5−M15322 1939 233199 199 20 233X=4X=40133Δ=−29Source 33199 1199 199 20 23993 1 91 9 959 20 50 134X=34199 233X=30233233Δ=−265X=35199 233X=302321D3Demand30 20 0 Select x=3=303316X=6X=0Δ=−24*33121x=2=Z46=0=2Z460332X=X=20332x=0Irrelevant=3333X=3X=303434=Z57=2Z570江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo55l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo56l of Information Technology, JiangXiUniversity of Finance & Economics© A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemTable8 initial transportation simplex tableau (before weobtainc-u-v) from Russe’ll’s approximation methodOptimality TestijijIterationDDestinationUUsing the notation of Table84 , we can reduce the standard u0supplyi1 2 33 44 55optimality test for the simplex method to the following for the116616613322177550transportation problem:4010:6602144144133199155OptimaAl test:BF-A BF solution is optimal if and only if c-u--Source 3030ijj550319919920233MM3v≥0 for every (i,j) such that xis (DD)M0MMM0050Dnd36Dema302073703060v=Z57=2Z570j江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of57 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog58y, JiangXiUniversity of Finance & Economics© A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemX:=3+3K=38=-5X21 :1=u31+v3. nKow v=13,8 so u1=-5,To demonstrate, we give each equqation that corresponds to a X5:=7X15 :1=u71+5K1=-+=5 nKow u=-5, so v2=52,basic variable in our inFBitial FB :=4+5K5=4=-X45 :0u=v4+5. nKow v25=2, so u4=-22,X:9=+==9X :19=uv+. Set u0=, so v=19,333131331Setting u3=3=0 (since row 33 of89 Table has the largest X:9=+X :19=uv+. v=19=9,3332322number of a3llocation——3) and moving down theq equations one at X:3=+X :23=uv=3+. v2=3,3434434344a time immediately give the derivation of values for the X:4=+X :14=uvK=9=-5+. Know v=19, so u=-5,212112unknowns shown toq the right of the :X :13=3+=uvK5=+. Know u=-=-5, so v=江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo59l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo60l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 8 运筹学Operations A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemStep- 1:Since c-u--vrepresents the rate at which the objective ijijAn iterationfunction will change as the nonbasicvariable xis increased, the ijAsA with the fu–ll –fledged simplex method , an iteration for entering basic variable must have a negative c--u--vvalue to ijijthis streamlined version must determine an entering basic decrease the total cost ZZ. Thus the candidates in are x255variable (step 1), a leaving basic variable (step 2), and the and x. To choose between the candidates, select the one having 4444idenFBtify the resulting new FB solution (step3 3).the larger (in absolute terms) negative va-lue of c-u--vto be the ijijentering basic variable, which is xin this 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo61l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo62l of Information Technology, JiangXiUniversity of Finance & Economics© A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemTab8le completed initial transportation simplex tableauStep 2:Increasing the entering basic variable from zero sets IterationDeDstination of0Supplyuf a chain reaction of compensating changes in other basic i1 2 33 44 55variables (allocations) , in order to continue satisfying the supply 116616613322177+2+4+5+2-5+4+50-54010and demand constraints. The first basic variable to be decreased2144144133199155source6-560-5to zero then becomes the leaving basic +1-+-2 3030335500Step 3:The new FBF Bsolution is identified simply by adding 19919920233MM44(D)D+2M-020+ 30M-225-50-22the value of the leaving basic variable (before any change) to the MM0MM00allocation for each recipient cell and subtracting this same M+33+M+4-M+3 3+M+4-150amount from the allocation for each donor =Z257=Z057j江西财经大学信息管理学院©2006199199江西财经大18学8信息管理2学33院©200622School of Information Technolog63y, JiangXiUniversity of Finance & Economics©2006Schoo64l of Information Technology, JiangXiUniversity of Finance & Economics© A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemTable part of initial transportation simplex tableau showing the Table part of second transportation simplex tableau showingthe chain reaction caused by increasing the entering basic variable x25changes in the BF solutionDeDstinationDeDstinationSupply343 4 55343 4 55Supply13322177133221771…5…501…5…50+-504010SourceSource4+4+1331991552 6601331991552 660…………-2010+1-+-230+…………………………………………De73Dmand6703060江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Deand736Dm703060School of Information Technology, JiangXiUniversity of Finance & Economics©200665Schoo66l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 8 运筹学Operations A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemTable complete set of transportation simplex tableaux for the metro Table (continued)water district problemIterationDeDstination SupplyuiIterationDeDstination 11 2 33 44 550Supplyui1 2 33 44 55116616613322177166166133 ++22177 --+2+4++5+2+504+2+50-5-51+2+40105-5+24++4+50-52144 --14413319915+5 +source6-560-514414413-3 19-915520Source 2106-560-50 +1+3030330 +1-+-2 +305500199 ++19920233 -M-M35350019919920233MM4D4()D020+2M-+ M-200203030+2 M-+M-225-50-204D4()D5-50-22MMM0M00 MM0MM00M++M+-3-M+1 1+M+2-3-50M+33+M+4-M+3 +3+M+4-150Deman3Dd3020703673060De3Dmand2073630703060=Z255=Z055=Z257=Z057vj19919918823320199199江西财经大188学信息管理2学33院22v©2006江西财经大学信息管理学院©2006jSchool of Information TechnologUn67y, JiangXiiversity of Finance & Economics©2006School of Information Technolog68y, JiangXiUniversity of Finance & Economics© A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemTable (continued)Table (continued) IterationDeDstination IterationDeDstination 2SupplyuiSupplyui1 2 33 44 551 2 33 44 55331166166133221771166166133221775+5+7++-85+5+507+2 5+50-8504+4+7++54+4+7+2+50-7-72144144133 --19915+5 +2source6-860-81441441331991553+3+3+ 204+403+ 4+source6-760-7332040+2+4++2+ 4+3355001991992023 -M3-M550019919920233MM4D4()D3020-M-3-1 M-23+0302004D4()D+1M-+ M-225-350-235-50-22MMM00 +M+0 --MMM0M00M+4 4+M+M+4 4+M+2M+33+M+3020M+3 3+M+23020Deand3Dm2073630703060Deand3Dm3020736703060=Z246=Z046=Z246=Z046v19919921233233vj江西财经大学信息管理学院©2006j江西财经大学信息管理学院©2006199199202222SchooSchoo70l of Information Technology, JiangX69iUniversity of Finance & Economics©2006l of Information Technology, JiangXiUniversity of Finance & Economics© A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemItera:tion:Summary of the transportation simplex method1D:. eDtermine the entering basic variable :Select the nonbasicInCFBitiazliaztion::oCnstruct an initial FB solution by the variable xhaving the largest (in absolute terms) negative value of ijprocedure outlined earlier in this section. Go to the ---u-v. ijij2D:. eDtermine the leaving basic variable :Identify the chain Opytimalit ytest:D:Derive uand vby selecting the row having ijreacqtion required to retain feasibility when the entering basic the largest number of allocations, setting= its u=0, and then solving ivariable is increasedF . From the donor cells, select the basic variable theq set of equations c==u++vfor each (i,j) such that xis basic. If ijijijhaving the smallest valuec--u--v≥0 for every (i,j) such that xis nonbasic, then the current ijijij3D3. eDtermineFB: the new FB solution :add the value of the leaving solution is optimal, so stopO. tOherwise, go to an variable to the allocation for each recipient cell. Subtract this value from the allocation fro each donor cell.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUn71iversity of Finance & Economics©2006Schoo72l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 8 运筹学Operations A Streamlined Simplex Method for A Streamlined Simplex Method for 运筹学运筹学Operations Researchthe Transportation ProblemOperations Researchthe Transportation ProblemSecond, another degenerate basic variable (x) arises in the 3434NoNte three special points that are illustrated by this tableau because the basic variables for two donor cells inthe second tableau, cells (2,1) and (343,4), tie for having the smallest FFirsFBt,the initial FB solution is degenerate because the basic va3lue (30). (This tie is broken arbitrarily by selecting xas the 21variab=Hle x=0. oHwever, this degenerate basic variable causes no 331leaving basic variab;le ;if xhad been selected instead, then x434321would have become the degenerate basic variable.)comp3lication, because cell (3,1) becomes a recipient cell in theThird , because none of the c---u-vturned out be negative in ijijsecond tableau, which increases xto a value greater than fourth tableau, the equivalent set of allocations in the third tableau is optimal also. Thus the algorithm executed one more iteration than was necessary. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of73 Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo74l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学8 the transportation and assignment The Assignment ProblemOperations ResearchOperations The Assignment Problem55. The objective is to determine how all n assignments To fit the definition of an assignment problem, these kinds of applications need to be formulated in a way that satisfy the should be made in order to minimize the total . The number of assignees and the number of tasks are the AAny problem satisfying all these assumptions can be same. (This number is denoted by n.)2E. Each assignee is to be assigned to exactly one extremely efficiently by algorithms designed 3E3. Each task is to be performed by exactly one . There is a cost cassociated with assignee i (i==1…,2,…,n) ijspecifically for assignment task j (j==1,2,……,n).江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom75ics©2006School of Information Technolog76y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Assignment The Assignment ProblemOperations ResearchOperations ResearchTable8 materials-handing cost data for job shop ExEampleTheOBJHOPCOMPA OBJ SHOP NYCOMPANY has purchased three new LLocationmachines of different types. There are four available locations in 1 2 33 44the shop where a machine could be installed. Some of these 1133 16 6 12 11locations are more desirable than others for particular machinesMaMchine 2 155 ----13 3 20because of their proximity to work centers that will have a heavy 3573 65 7 10 6work flow to and from these machines. (There will be no work flow between the new machines.)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo77l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo78l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 8 运筹学Operations Research8-14运筹学运筹学 The Assignment The Assignment ProblemOperations ResearchOperations ResearchTable cost table for the job shop co. assignment problemThe assignment problem model and solution proceduresThe mathematical model for the assignment problem uses the Task (location)following decision variables::1 2 3 41 i f assigne ei perform stask j,⎧x=⎨ij1166 12 110 if not.⎩Assignee 2 15 M 13 20(Machine) 35 7 10 66for i==1=…,2…,…,n and j=1,2, …,n. Thus each xijis a binary variable 4(D)0 0 0 0(it has value 0 or 1).江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©200679School of Information TechnologangXiUniversity of Finance & Economics©200680y, Ji运筹学运筹学 The Assignment The Assignment ProblemOperations ResearchOperations Researchc11A1[]1T11-[]cA12As discussed at length in the chapter on integer programming , binary variables are importantOR in OR for/ representing yesn/o c1ndecisions. In this case/, the yesn/o decision is: :should assigneei perform? task j?c21c22BZBy letting Z denote the total cost, the assignment problem A2T1-[]1[]2model isc2nnnMinimize Z =cx,∑∑ijiji=1j==1 for i=1,2,K,n .∑ijj=1cn21[]nAnT1-[]ncx=1 for j=1,2,K,n.∑nniji=1iFgure .8 4eNtwork representation of the assignment problem江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom81ics©2006Schoon82l of Iformation Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Assignment The Assignment ProblemOperations ResearchOperations ResearchTable .82 6cost and requirements table for the assignment problem formulated as a transportation problem, illustrated by the job shop co. xEampleE—Example—assigning products to plants(a) General CaCse(b) JJob Shop CoC. EExampleTheBE TBETERRODPUCCOMPANYER RODPUCTS COMPANY has decided to CoCCst per Unit oCst per Unit DistributedDistributedinitiate the production of four new products, using three plantsthat currently have excess production capacity. The producqts require a DestinationDestination (Location)supplySupply1 2 ……n1 2 3 4 comparable production effort per unit, so the available production 1cc……c1113166 12 11111121nSource 2cc……c1Source 2 14M capacity of the plants is measured by the number of units of any 13 20121222n………………………………(Machine) 35 7 10 661m=n=cc……c14(D) 0 0 0 0 1n1n2nnproduct that can be produced per day, as given in the rightmost column87 of Table . Demand1 1 ……1Demand1 1 1 1江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo83l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog84y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 8 运筹学Operations Research8-15运筹学运筹学 The Assignment The Assignment ProblemOperations ResearchOperations ResearchTable CCost and Requirements Table for the Transportation Problem Formulation of Option 1 for the Better Products CoC. ProblemTable Data for the Better ProductsC oC. ProblemCCost per Unit DistributedUnit CCost for ProductCCapacityDestination (Product)SupplyAvailable1 2 3 41 2 3 4 5(D) 14127 288 2475141 27 288 24 075Plant 240 29 --2375Source 240 29 M 23 0753 37 30 27 21 45337 30 27 21 0 45Production rate20 30 30 40 Demand20 30 30 40 75 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom85ics©2006School of86 Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Assignment The Assignment ProblemOperations ResearchOperations ResearchTable CoCst Table for the Assignment Problem Formulation of Option 2 The optimal solution for this transportation problem has for the Better Products CoC. Problembasic variab=3les (allocations) x0=3, x=3=30, x=5=15, x=15=6=5, x0=6, 12133155244255Task (PrPoduct)x==20, and x=5=25, So 3433143PPlant 1produces all of products 2 and 2 3 3 4 4 55(DD)P375Plant 2produces percent of produc4t 3produces percent of product 4and all of 1a8884820 810 840 96960 0product 810 840 960 0A8Assignee 2a800 87870 M M 9290 0(PPlant) 2b887M9800 870 M 290 0The total cost is 3=Z263= 900 810 840 MM江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo87l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog88y, JiangXiUniversity of Finance & Economics©2006运筹学8 the transportation and assignment problemsOperations ConclusionsThe linear programming model encompasses a wide variety of specific types of problems. The general simplex methodis a powerful algorithm that can solve surprisingly large versions of any of these problems. HHowever, some of these problem types have such simple formulations that they can be solved much more efficiently by streamlined algorithms that exploit their special structure. These streamlined algorithms canq cut down tremendously on the computer time required for large problems, and they sometimes make it computationally feasible tosolve huge problems. 江西财经大学信息管理学院©2006Schoo89l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 9 运筹学Operations Research9-1运筹学运筹学9 Network analysis, including PERT-CPMOperations ResearchOperations ResearchCoCre problem:¾The shortest--path problem9 Network analysis, ¾The minimum spanning tree problemincluding PERT-CPM¾The maximum flow problem¾The minimum cost flow problem¾The project planning and controREPl with TREP and 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog1y, JiangXiUniversity of Finance & Economics©2006Schoo2l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学9 Network analysis, including PrototypeE ExampleOperations ResearchOperations ProEtotype xEampleSEEVKEERAVDA PARK has recently been set aside for a Alimited amount of sCightseeing and backpack hiking. aCrs are 75not allowed into the park, but there is a narrow, winding road 2D2Tsystem for trams and for jeeps driven by the park rangers. 41OBThi5s road system is shown (without the curves) in , 37where location O is the entrance into the park; ;other letters 14Edesignate the locations of ranger stations (and other limited 4Cfacilities). The numbers give the distances of these winding roads in miles.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo3l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog4y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学9 Network analysis, including The Terminology of NetworksOperations ResearchOperations The Terminology of Networks¾Directed network:a network that has only directed ¾Network:a network consists of a set of points and a arcs is called a directed network ;(undirected network) set of lines connecting certain pairs of the points;¾Path:a path between two nodes is a sequence of ¾Node:the points of a network are called nodes;distinct arcs connecting these nodes;¾Arc:the lines of a network are called arcs;¾Directed path:a directed pathefrom node ito node jis ¾Directed arc:if flow through an arc is allowed in only a seuqence of connecting arcs whose direction is toward one direction, the arc is said to be a directed arc ;node j, so that flow from node ito node jalong this path (undirected arc)is feasible ;(undirected path)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo5l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of6 Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 9 运筹学Operations Research9-2运筹学运筹学 The Terminology of The Terminology of NetworksOperations ResearchOperations Research¾oCnnected:two nodes are said to be connected if the The nodes network contains at least one undirected path between without arcsthem;AD¾oCnnected network:a connected network is a network Cwhere every pair of nodes is connected;AD¾Tree:a tree is a connected network (for some subset of BEthe n nodes) that contains no undirected cycles;A tree with ¾Spanning tree:a spanning tree is a connected network one arcfor all n nodes that contains no undirected cycles;江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog7y, JiangXiUniversity of Finance & Economics©2006Schoo8l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Terminology of The Terminology of NetworksOperations ResearchOperations Research¾AArc capacity::the maximum amount of flow (possible infinity ) A tree with that can be carried on a directed arc is referred to as the arc three arcsADcapacity;;AD¾Supply: node :a supply node has the property that the flow out of the node exceeds the flow into the node;;C¾DDemand node: :a demand node has the property that the flow CEinto the node exceeds the flow out of the node;;EB¾Transshipmen:t node :a transshipment node satisfies conservaqtion of flow, so flow in equals flow spanning tree江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of9 Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo10l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学9 Network analysis, including The Shortest-Path ProblemOperations ResearchOperations ResearchAgPblAgorithm for the shortest--Path problem The Shortest-Path Problem1. Objective of nth iteration: Find the nth nearest node to the CoCnsider an undirected and connected networkwith origin (to be repeated for n=1=,2,……until the nth nearest node is the destination. two special nodes called the originand the . Input for nth iteration: n-1 nearest nodes to the origin AsAsociated with each of the links (undirected arcs) is a (solved for at the previous iterations), including their shortest path and distance from the origin. (These nodes, nonnegative distance. The objective is to find the plus the origin, will be called solved nodes; ;the others are shortest path (the path with the minimum total distance) unsolved nodes.) from the origin to the destination.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo11l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo12l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 9 运筹学Operations Research9-3运筹学运筹学 The Shortest-Path The Shortest-Path ProblemOperations ResearchOperations ResearchErvadaPPark management needs to find the Example::The See3. C Candidates for nth nearest node:E Each solved node that is shortest path from the park entrance (nodeO O) to the scenic directly connected by a link to one or more unsolved wonder (note T) through the road system shown in following nodes provides one candidate——the unsolved node with network. the shortest connecting link. A4. CaClculation of nth nearest node: for each such solved 75node and its candidate, add the distance between them 2D2Tand the distance of the shortest path from the origin to 41this solved node. The candidate with the smallest such OB537total distance is the nth nearest node .14E4C江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo13l of Information TechnologSchoo14y, JiangXiUniversity of Finance & Economics©2006l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Shortest-Path The Shortest-Path ProblemOperations ResearchOperations ResearchSolution:AApplying the algorithm to this problem yields the results shown in following table. The shortest path from the destination to the origin Solved nodes CClosest Total nth Last Minimum ndirectly connected connected distance nearest connecdistanceto unsolved nodesunsolved nodeinvolvednodetioncan now be traced back through the last column of Table as either T→D→EE→B→A→O or T→D→B→A→O. OCC4CC4OCC2,3Therefore, the two alternates for the shortest path form AB2+2+=4=B4ABAD2+7+=9=the origin to the destination have been identified as 4BE4+3+=E7=E7BEEECECE4+4+8=8=O→A→B→EE→D→T and O→A→B→D→T, with a total AD2+7+=9=5BD4+4+8=8=D88BDEED7+1+8=8=D8E8DEdistance of 13 miles on either +=85+1=3T13DT66EET7+7+=1=4江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of15 Information Technology, JiangXiUniversity of Finance & Economics©2006School of16 Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Shortest-Path Problem9 Network analysis, including PERT-CPMOperations ResearchOperations ResearchOAther The Minimum Spanning Tree ProblemMany of these applications require finding the shortest The minimum spanning tree problem bears some dsimilarities to the main version of the shortest-path problem irected path from the origin to the destination throughadirected network. The algorpresented in the preceding section. In both cases, an ithm already presented can be easily modified to deal just with directed paths at each undirected and connected networkis being considered, where the given information includes some measure of the positive iteration. In particular, when candidates for the nth nearest node are identified, only directed arcs from a solved nodelength (distance, cost, time, etc.) associated with each link. to an unsolved node are considered.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog17y, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Econom18ics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 9 运筹学Operations Research9-4运筹学运筹学 The Minimum Spanning Tree The Minimum Spanning Tree ProblemOperations ResearchOperations ResearchAgMgTPbAlgorithm for the Minimum Spanning Tree Problem3:3. Tie breaking :Ties for the nearest distinct node (step 1) or the closest unconnected node (step 2) 1. Select any node arbitrarily, and then connect it (., may be broken arbitrarily, and the algorithm add a link) to the nearest distinct still yHield an optimal solution . oHwever, such ties are a signal that there may be (but need 2. Identify the unconnected node that is closest to a not be ) multiple optimal solutions. AAll such connected node, and then connect these two nodes optimal solutions can be identified by pursuing (., add a link between themR). eRpeat this step all ways of breaking ties to their all nodes have been connected.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo19l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo20l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Minimum Spanning Tree The Minimum Spanning Tree ProblemOperations ResearchOperations ResearchSo:lution :E:PExample: The SeervadaPark management (see Sec. ) needs to determine under which roads telephone 1AO. Arbitrarily select node Oto start. The unconnected lines should be installed to connect all stations with a node closest toOAC node O is node A. oCnnect nodeA Ato mUinimum total length of line. Using the data given in node 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo21l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Econom22ics©2006运筹学运筹学 The Minimum Spanning Tree The Minimum Spanning Tree ProblemOperations ResearchOperations The unconnected node closest to either node Oor 33. The unconnected node closesOABt to node O,A, or Bis node AABAC is node B(closest to A). oCnnect node B Bto node C C(closest toBCB )B. oCnnect node C Cto node .Bnode 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo23l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoon24l of Iformation Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 9 运筹学Operations Research9-5运筹学运筹学 The Minimum Spanning Tree The Minimum Spanning Tree ProblemOperations ResearchOperations Research4OABC4. The unconnected node closest to node O,A, Bor Cis 55. The unconnected node closestOABCE to node O,A,,B Cor Enode E E(cBCElosest to )B. oCnnect node Eto node node DDECDE (closest to )E. oCnnect node D to node .EAA77552D2D2T2T441OB15OB537371414EE44CC江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog25y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog26y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Minimum Spanning Tree Problem9 Network analysis, including PERT-CPMOperations ResearchOperations Research66. The only remaining unconnected node is node T. It The Maximum Flow Problemis closest to node DDC. oCnnect node T to node . Residual networkA:After some flows have been 752assigned to the arcs of the original network, the D2T4residual networkshows the remaining arc capacities 1OB5(called residual capacities) for assigning additional nodes are now connected, so this solution to the problem OBis the desired (optimal) one. The total length of the links4 1 4miles.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo27l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog28y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Maximum Flow The Maximum Flow ProblemOperations ResearchOperations ResearchTAggPAgMwThe Augmenting Path Algorithm for the Maximum Flow PbProblem2. Augmenting PaAth: nA augmenting pathis a directed path from the supply node to the demand 1. Identify an augmenting path by finding some directed node in the residual network such that every arc on this path has strictly positive residual path from the supply node to the demand node in the minimum of these residual capacities is called the residual network such that every arc on this path has residual capacity of the augmenting path because it strictly positive residual capacity..represents the amount of flow that can feasibly be added to the entire path . Therefore, each augmenting 2. Identify the residual capaci*ty c* of this augmenting path provides an opportunity to further augment the path by finding the minimum of the residual capacities flow through the original the arcs on this path. Increase the flow in this path by* c*.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog29y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog30y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 9 运筹学Operations Research9-6运筹学运筹学 The Maximum Flow The Maximum Flow ProblemOperations ResearchOperations ResearchExEample:ApPAplying this algorithm to the SeervadaPark 3D*3. Decreaseby c* the residual capacity of each arc on this problem yields the results summarized next. Starting with the initial residual netwoF96rk given in , we give the augmenting path. Increase by c* *the residual capacity new residual network after each one or two iterations, of each arc in the opposite direction on this augmenting where the total amount of flow fromO O to T achieved thus paRth. eRturn to step inO boldface (where to nodes O and T).3ARemark0:WhWen step 1 is carried out, there often will be a number 1090Dof alternative augmenting paths from which toA choose. Although 0T050the algorithmic strategy for making this selection is important for 4007OBthe efficiency of large--scale implementations, we shall not delve 2514into this relatively specialized topic. 006E00C4江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom31ics©2006Schoo32l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Maximum Flow95 The Maximum Flow ProblemOperations ResearchOperations ResearchSolution:Itera:A3tion 2 :Assign a flow of 3 to the augmenting path Iteration 1:F96 :In , one of several augmenting paths is OO→AA→DD→T. The resulting residual network isOO→BEB→E→T, which has a residual capacity of{756}B min{7,5,6}. By assigning a flow of5 5 to this path, the resulting residual network is 0A313633DA08T0012009045D5205OBT805020144552OB5501E201400C5401E00C4江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog33y, JiangXiUniversity of Finance & Economics©2006Schoo34l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Maximum Flow The Maximum Flow ProblemOperations ResearchOperations ResearchI3:Ateration 3 :Assign a flow of 1 to the augmenting path I5:Ateration 5 :Assign a flow of 1 to the augmenting OO→AA→BB→DD→→CC→EE→DD→ 4:4A :Assign a flow of 2 to the augmenting path I6:Ateration 6 :Assign a flow of 1 to the augmenting path OBOO→B→DD→T. The resulting residual network is O→CC→EE→T. The resulting residual network is 00AA4400333267DD311313TT1110111156770011OB13OB22001042550100EE0202CC42江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo35Schoo36l of Information Technology, JiangXiUniversity of Finance & Economics©2006l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 9 运筹学Operations Research9-7运筹学运筹学 The Maximum Flow The Maximum Flow ProblemOperations ResearchOperations ResearchI7:Ateration 7 :Assign a flow of 1 to the augmenting path There are no more augmenting paths, so the current flow pattern OO→CC→EBE→B→DD→T. The resulting residual network isis optimal. A03A841D034T1418D4414T1111OB147067014OB4621013E4003EC33C1江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of37 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog38y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Maximum Flow The Maximum Flow ProblemOperations ResearchOperations : A cut may be defined as any set of directed Theorem(The max--flow min--cut theorem)arcs containing at least one arc from every directed path from the supply node to the demand any network with a single supply node and demand 4C:4. uCt value :The cut value is the sum of the arc node , the maximum feasible flow from the supply node capacities of the arcs (in the specified direction ) of the the demand node equqals the minimum cut value for all cuts of the network. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog39y, JiangXiUniversity of Finance & Economics©2006Schoof40l o Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学9 Network analysis, including The Minimum CCost Flow ProblemOperations ResearchOperations ResearchFormulation: The Minimum CoCst Flow ProblemCConsider a directed and connected network, where the n nodes include at least one supply node and at The minimum cost flow problem holds a central least one demand node. The decision variables areposition among network optimization models, both X=Xf=low through are i it encompasses such a broad class of applications and the given information includesand because it can be solved extremely efficiently. Like C=C=cost per unit flow through arc i maximum flow problem, it considers flow through a u=a=rc capacity for arc i jijb==net flow generated at node with limited arc 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog41y, JiangXiUniversity of Finance & Economics©2006School of42 Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 9 运筹学Operations Research9-8运筹学运筹学 The MCinimum Cost Flow The Minimum CCost Flow ProblemOperations ResearchOperations ResearchThe value of bi depends on the nature of node i, where nnMinimize bi>0> if node i is a supply nodeZ=cx∑∑ijiji−1j=1bi<0< if node i is a demand nodebi=0= if node i is a transshipment node. objective is to minimize the total cost of sending the nnavailable supply through the network to satisfy the given x−x=b∑ij∑jiifor each node =1j= using the convention that summations are taken only 0≤xij≤uij, for each arc i→jover existing arcs, the linear programming formulation of this problem is 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo43l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo44l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Minimum CCost Flow The Minimum CCost Flow ProblemOperations ResearchOperations ResearchFeasible solutions property:A necessary condition for a Integer solution property:For minimum cost flow problems minimum cost flow problem to have any feasible solutions where every bi and uijhave integer values , all the basic is thatnvariables in every basic feasible (BF) solution (including an b=0∑ii=1optimal one ) also have integer is, the total flow being generated at the supply nodes equals the total flow being absorbed at the demand nodes.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo45l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Informat Technology, JiangXiUniver46ionsity of Finance & Economics©2006运筹学运筹学 The Minimum CCost Flow The Minimum CCost Flow ProblemOperations ResearchOperations ResearchAn example of a minimum cost flow problem is shown in The linear programming model for this example is following figure. b=5[0]Minimize=Z+ 2=ZxAB4+xAC+9C+xAD+3+xBC+xC+CE+C3E+xDE+2E+xEEDA3-0[]C=9DASubject to ADx++=+x+x5=0ABACCAD40[]-x+=+x4=0ABBCC223C-x-x++x=0=ACCBCCECEC(u=10)BA-x++x-x==-30ADDEEECEC31B-x-x++x==-660ECECECDEEEC4[0](u0=8)CEEAnd 8 x≤10, x≤08, all xij≤0ABECEC6-0[]江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog47y, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Econom48ics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 9 运筹学Operations Research9-9运筹学运筹学 The Minimum CCost Flow Problem9 Network analysis, including PERT-CPMOperations ResearchOperations The Network Simplex MethodSpeciaCl Cases¾The network simplex method is a highly streamlined The Transportation Problemversion of the simplex method for solving minimum cost flow ¾The Assignment ProblemproblemsA. As such, it goes through the same basic steps at each iteration——finding the entering basic variable, ¾The Transshipment Problemdetermining the leaving basic variable, and solving for the new FBFB solution——in order to move fFrom the currentB FB¾The Shortest-Path Problemsolution to a better adjacent one. ¾The Maximum Flow Problem江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of49 Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo50l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Network Simplex The Network Simplex MethodOperations ResearchOperations ResearchTo illustrate this process, consider the minimum cost flow IncorporatgUBTing the Upper Bound Techniqqueproblem shown in following [0]The first concept is to incorporate the upper bound techniqque ADdescribed in to deal efficiently with the arc capacity 4constraints xij≤uij. Thus, rather than these constraints being treated 3au1=0s functional constraints, they are handled just as nonnegativity3-0[]BA9Cconstraints are. Therefore, they are considered only when the 2-leaving basic variable is determined. 0[]32BEu=80CE6-0[]5[0]1江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo51l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo52l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Network Simplex The Network Simplex MethodOperations ResearchOperations ResearchCbwBbCorrespondence between BF solutions and Feasible O96On the left is the set of node constraints given in . 6Spanningg TTreesafter 10--yis substituted for x, where the basic ABABABABvariables are shown in nO the right, starting at AA spanning treesolution is obtained as follows::the top and moving down,q is the sequence of steps for 1F. For the arcs not in the spanning tree (the nonbasicarcs), setting or calculating the values of the the corresponding variables (xijo ryqij) equal to zero.--y++x++x=4=4=40 x=40BACADABACA DADADA2F. For the arcs that are in the spanning tree (the basic arcs ),y++x =50=5 =5 x=50BABCBCBABCBCsolve for the corresponding variables (xijor yij) in the system of --+-x-x x=+ =0 so x =5=50CACBECCECACBECCElinearq equations provided by the node constraints. -+-x+x-=-3 -x =-30 so x =1=0DA EDEDEDDAEDEDED--x --x +x=-6R+ =-60 RedundantECEDEDECED ED江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog53y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog54y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 9 运筹学Operations Research9-10运筹学运筹学 The Network Simplex The Network Simplex MethodOperations ResearchOperations Research4[0]3-0[]SelectinggE the nEteringgBVb aBsic aVriable9(40)ADTo begin an iteration of the network simplex method, recall 0[]that the standard simplex method criterion for selecting the 3C(50)entering basic variable is to choose the nonbasicvariable which, (10)(50)3when increasedv from zero, will improev Z at the fastest rate. 1BNNow let us see how this is done without having a simplex 5[0]u=-0[]江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo5l of Information Technolog5y, JiangXiUniversity of Finance & Economics©2006Schoo56l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Network Simplex The Network Simplex MethodOperations ResearchOperations Research4[0]3-0[]−7,if.Δx=1⎧9(40)ACAD⎪ΔZ=6,if.Δy=1⎨AB0[]⎪5,if.Δx=1⎩ED3C(50)(50)NZNow what is the incremental effect on Z (total flow cost )from 31adding the flow өto arc AA→C?F96C ? Figure shows most of the Banswer by giving the unit cost times the change in the flow fro 5[0]u=80CEEeach arc ofF95 Therefore, the overall incremenZt in Z is ΔZ=+6-0[]Z=cθ+cθ++c(--θ)++c(--θ)CAECEDDACAECEDDAhWen x= θ, ΔZ=cθ+cθ+c(-θ)c+(-θ)4=θ+θ3-θ9-θ7-=θ=4ACCAECDEAD=4θ++θ-3-3θ-9-9θhWen x= θ, ΔZ=2-θ+9θ+3θ+1(-θ)+3(-θ)= 6θAB=-7=-7θhWen x= θ, ΔZ=2θ+3θ=5θAB江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangX57iUniversity of Finance & Economics©2006School of Information Technolog58y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Network Simplex The Network Simplex MethodOperations ResearchOperations ResearchSetting ө=1= then gives the rate of change of ZZ as x ACACWeW now need to perform the same analysis for the is increased, namelyother nonbasicvariables before we make the final Δ=Z-7=Z-7, when θ=1=.selection o f the entering basic variable. The only BeBcause the objective is to minimize Z,Z this large other nonbasicvariables are y and x , ACEDACEDrate of decrease Z in Zby increasing x is very ACACcorresponding to the two other nonbasicarcsBA B→ Adesirable, so x becomes a prime candidate to be CACAand EE→DD inF93 . the entering basic variable.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo59l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo60l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 9 运筹学Operations Research9-11运筹学运筹学 The Network Simplex The Network Simplex MethodOperations ResearchOperations ResearchTherefore, Therefore.Δ=Z-9+=Z-2θ9+θ3+Δ=Z2=Zθ3+3+θ5=5=θ3+θ+1+(-3+--θ6=)(3+-θ)6=5=5= whenθ=1=6=6= when θ=1=So xis ruled out as a candidate to be the entering basic The fact thatZED Zincreases rather than decreases ED y (flow through the reverse arc BB→AA ) is ABABTo summarize,increased from zero rules out this variable as a candidate to be the entering basic variable. −7,if.Δx=1⎧ACR(Remember that increasing y from zero really ⎪ABABΔZ=6,if.Δy=1⎨ABmeans decreasing x , flow through the real arc AB⎪AB5,if.Δx=1⎩EDABA→,B from its upper bound of 10.) 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog61y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog62y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Network Simplex The Network Simplex MethodOperations ResearchOperations ResearchFFor those arcs whose flow decreases with θ(arcs DD→E EFgindinBVbNB gthe LeLavingg aBsic aVriable and the Next FB Solutionand AA→DD), only the lower bound of 0 needs to be considered::FFor those arcs whose flow increase with θinF95 A(arcs A→C CandC C→E)E, only the upper bounds =(u =∞ACACx=- =10-θ≤0, so θ≤10DEDEand u8=80=) need to be considered::ECECx=4- =04-θ≤04, so θ≤40ADADx==θ≤∞ACACx=5+8 =05+θ≤08, so θ≤303ACAC江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo63l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo64l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Network Simplex Method9 Network analysis, including PERT-CPMOperations ResearchOperations Project Planning and CoCntrol wEith PRETC-MCPSetting θ=1=0 in this figure thereby yields the flows The successful management of large--scale projects through the basic arcs in he nextFB: F Bsolution:requqires calreful palnning, scheduluilng, and coordinatingof numerous interrelated activities. To aid in these x==θ=1=0,ACACtasks, fokrmal procedures based on the use of networs k and networkk uqtechniuqeswere developed beginning in x5=0+6=5=+θ06=,ECECthe late 159590s. The most prominent of these procedures x4=0-4=-θ3=30=ADare REPTREP (program evaluation and review qtechniuqe) ADand MPC MPC(critical--path method), although there have x5=05=CBCBbeen many variants under different names. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo65l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of66 Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 9 运筹学Operations Research9-12运筹学运筹学 Project Planning and CoCntrol with Project Planning and CoCntrol with PERET-CCMPOperations ResearchOperations Research100最早开工序200¾A PEERT-type system is designed to aidin planning 2工时间总时差20and control, so it may not involve much direct optimizzation. 204工序最迟开360¾All PEERT-type systems use a projejcwkt networkto 610单时差工时间160portray graphically the interrelationships among the 46162244elements of a ¾In the terminology of PEERT, eacharcof the project 52942508733network represents an activitythat is one of the tasks 2598330384required by the 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©200667Schooof68l Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Project Planning and CoCntrol with Project Planning and CoCntrol with PERET-CCMPOperations ResearchOperations Research100200220The latest timefor an v ( )l eevnt is the e(stimated )alst time 204360610at160 which the eveuvnt can occur wulyithout dealying the 4616224420062650207comple 5ltiono ft he p ro yjectb eoynd i ts e arlilest t ime..29425087332598330384109335423804371123838211064404413江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©200669Schoo70l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Project Planning and CoCntrol with Project Planning and CoCntrol with PERET-CCMPOperations ResearchOperations Research3A3. lAl events having zero slack must lie on a critical AA critical path for a project is a path through the network path, whereas no events having slack greater than such that all the activities on this path have z lkzeros can lie on a critical of rCiPtical aPths4A4. A path through eh network such that the events on this path have zero slack need not be a critical path Aproject network always has a critical path , and sometimes there is more than one or more activities on the path can have slack greater than activities having zeroslack must lie on a critical path , whereas no activities having slack greater than zerocan lie on a critical path.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoon71l of Iformation Technology, JiangXiUniversity of Finance & Economics©2006School of72 Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 9 运筹学Operations Research9-13运筹学运筹学 Project Planning and CoCntrol with Project Planning and CoCntrol with PERET-CCMPOperations ResearchOperations ResearchTEAThePETT PERT Three--Estimate ApproacheBta distribution¾The most likely estimate, denoted bym, is intended to be the most realistic estimate of the mode(the highest point) of the probability distribution for the activity ¾The optimistic estimate, denoted by a, is intended to be the unilek bylut possible time if eevrtyhing goes assumptions are made to convert m, a, and b to estimates of ¾2The pessimistic estimate, denoted byb , is intended to the expected value tand variance σof the elapsed time required e be t heu nu lilkek ylb ylutu p ossiblel t ime i fevy yl evrtyhingg oesb the activity.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo73l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog74y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Project Planning and CoCntrol with Project Planning and CoCntrol with PERET-CCMPOperations ResearchOperations ResearchAu2:Assmuption 2:The probability distribution of each Assumption 1:The spread between a(the optimistic activity time is (at least approximately ) a beta estimate) and b(the pessimistic estimate) is 6 standard , that is 6σ=b-a. conseuqently, the variance of an activity time is11t=[2m+(a+b)]e32122σ=[(b−a)]6江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo5l of Information Technology, JiangXiUn7iversity of Finance & Economics©2006School of Information Technolog76y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Project Planning and CoCntrol with PERET-CCMPOperations ResOeaprtcihmistic Most likely Pessimistic Expected Operations ResearchaVrianceActivity Estimate aEstimate mEstimate baVlue t2eσ23(1,2)1219/ AvyAssump3:tion 3:The actiivt ytimes are statisticayll yll3 ½8(2,3)241independen tr andomv av (3,4)61044 ½5(4,5)1449/5 ½10A4:Assumption 4:AAs an approximation,u ,assmue that the (4,6)4617 ½9critica l ( )ly qu lpath b(ased on expected times )awlasy requires (4,7)371410al ol nger t ota lel lalpsedt ime t han a ny .yo therp ath.(5,7)4516 ½11(,68)57199(7,9)381AAssumption 5:5:The probablyuiilt ydistribution of project 817(8,10)59444time is ( a( l yl ) t elast approximate yl )a norma lu ldistribution.(9,11)4405 ½7(9,12)15123(10,13)1219/5 ½9(12,13)5619/江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo77l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Informaton Technology, JiangXiUniversity of Finance & Econom78iics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 9 运筹学Operations Research9-14运筹学运筹学 Project Planning and CoCntrol with Project Planning and CoCntrol with PERET-CCMPOperations ResearchOperations ResearchActivities on rCiticalxEpected aVlueaVrianceTCPMMT-CTThe CPM Method of Time-Cost Trade--offs(1,2)219/(2,3) MPCassumes that activity times are deterministic, (3,4)104(4,5)449/so that the th-ree-estimate approach just described is (5,7)51not needed. (7,9) than primarily emphasizing time (explicitly), (9,12)51(12,13)649/MPCq MPCplaces euqal emphasis on time and cost. Project time449江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©200679Schoo80l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Project Planning and CoCntrol with Project Planning and CoCntrol with PERET-CCMPOperations ResearchOperations ResearchKCrashijCrash costCd¾Thus there is one decision variable xfor each activity, ijijbut there is none for those values of i and j that do not NormalNormal costCDijhave a corresponding activity.¾To express the direct cost for activity (i ,j) as a (linear) Dxdijijijfunction of xij, denote the slope of the line through Crash Normal the normal and crash points for activity (i, j) bytimetimeD= lDij=norma ltime for activivt y( iy( , ,j.).)C−CC= l( ) v y( ,.)C=norma ld(irect )cost fro actiivt yi( ,j.)DdDijijDijs=dij= v=crash time for actiivt y( iy(, ,.) j.)ijD−dijijC= (C=crash d(irect ) )cost v y(i( , for actiivt y ,j))dijThe decision v avriablels for the problelm are the xij, , where xij= u = druation time for a ctivivt y( yi( , ,j.).)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog81y, JiangXiUniversity of Finance & Economics©2006Schoo2l of Information Technolog8y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Project Planning and CoCntrol with Project Planning and CoCntrol with PERET-CCMPOperations ResearchOperations ResearchLinear programming formulation:Maximize....Z=(−rCash all activities S)x∑ijijTotal cost(i,j)on critical pathIndirect x≥d⎫ijijcostlAl activities ⎪normal x≤(i,j)⎬ijijiDrect Optimum⎪costuDration of project y+x−y≤0iijj⎭uDration of project y≤Tn江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006SchooSchoo84l of Information TechnologangXiUniversity of Finance & Economics©200683y, Jil of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006Minimum total direct cost projectMinimum cost for project
Session 9 运筹学Operations Research9-15运筹学运筹学 Project Planning and CoCntrol with PERET-C9 Network analysis, including PERT-CPMCMPOperations ResearchOperations CConclusionsCgbwPETChoosing between PERT and PMCPMC¾The minimum cost flow problem plays a central 1. PERPETR is particularly appropriate when there is considerable role among these network optimization models, both uncertainty in predicting activity times and when it is important to because it is so broadly applicable and because it can effectively control the project schedule. be solved extremely efficiently by the network simplex 2. CPMCPM is particularly appropriate when activity times can be method. predicted well but these times can be adjusted readily, and whenit ¾The minimum spanning treeP Problem is a is important to plan an appropriate trade--off between project time prominent example of a model for optimizing the and of a new network.¾The most widely usedq network technique has been the REPTREP--type system for project planning and control.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom85ics©2006School of Information Technolog86y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 10 运筹学Operations Research10-1运筹学运筹学Operations ResearchOperations Res10 Dynamic A Prototype example for dynamic CCharacteristics of dynamic programming10 Dynamic Deterministic Dynamic Probabilistic Dynamic CoCnclusions江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog1y, JiangXiUniversity of Finance & Econom2ics©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 A Prototype exampleOperat10 Dynamic Programmingions ResearchOperations A Prototype example for dynamic 3EB14programming46Example 1T----he Stagecoach ProblemH23There was a mythical fortune seeker in iMssouiwho decided 36thto go west to join the gold rush in California during the mid1-92FC4Acentury. The journey would require traveling by marauders. 34JAlthough his starting point and destination were serious danger of attack by marauders. Although his starting point and destination34were fixed, he had considerable choice as to which states to travel I413through en route. The possible routes are shown , where eachstate is represented by a circled letter and the direction of travel is 3DGalways from left to right in the diagram. Thus four stages were 5required to travel from his point of embarkation in sate A to his destination in state .J江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo3l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog4y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 A Prototype A Prototype exampleOperations ResearchOperations Research¾The cost for the standard policy on the stagecoach run from state I to state j, which will be denoted by fortune seeker was a prudent man who was quite ¾eW shall now focus on the uqestion of which route minimizes concerned about his safety. After some thought, he came up with a the total cost of the clever way of determining the safest route. Life insurance Solving the rPoblempolices were offered to stagecoach passengers. Because the cost of the policy for taking any given stagecoach run was based on a f*(s)=f(s,x)=C+f(s,x)nminnnminsxnnnxxnncareful evaluation of the safety of that run, the safest route should ={immediatecost(stagen)+minbe the one with the cheapest total life insurance +1onward)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog5y, JiangXiUniversity of Finance & Economics©2006Schoo6l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 10 运筹学Operations Research10-2运筹学运筹学 A Prototype A Prototype exampleOperations ResearchOperations Researchf(s, x)=x =CCsx++f**(x)222232sf*(s)x***44f*(*s)x**22s4N=EEFGHJH3J2N=B11111211EE or FI**4JJCC79107EED88118E888E or Ff(s, x)= =CCsx++f*x*(x)333333f*()x**s*33sHHI3N=EE4884HHf(s, x)= =CCsx++f*x*(x)111121f*(*s)x**F977I11sBCCDG66H676H1N=A13111111CC or D江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog7y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog8y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 A Prototype exampleOperat10 Dynamic Programmingions ResearchOperations ResearchThen the last result is followed: Characteristics of dynamic programming problemsEB1These basic features that characterize dynamic programming problems is:¾The problem can be divided into stages, with a policy H33decisionrequired at each ¾Each stage has a number of states associated with the beginning 4A3Jof that stage. ¾The effect of the policy decision at each stage is to transform the 34Icurrent state to a state associated with the beginning of the next 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of9 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog10y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Characteristics of DP Characteristics of DP problemsOperations ResearchOperations Research¾The solution procedure is designed to find an optimal ¾A recursive relationship that identifies the optimal policyfor the overall problem, , a prescription of the policy for stage n, given the optimal policy for stage optimal policy decision at each stage for each of the n1+, is states.¾¾hWen we use this recursive relationship, the solution Given the current state, an optimal policy for the remaining stages is independentof the policy decisions procedure starts at the end and moves backward stage by on only the current state and not on how you got there. stage e---ach time finding the optimal policy for that This is the principle of optimalityfor dynamic stage-u--ntil if finds the optimal policy starting at the programming. initial stage. ¾The solution procedure begins by finding the optimal policy for the last stage.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog11y, JiangXiUniversity of Finance & Economics©2006Schoo12l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 10 运筹学Operations Research10-3运筹学运筹学10 Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations Deterministic Dynamic ProgrammingOne way of categorizingdeterministic dynamic The section further elaborates upon the dynamic programming programming problems is by the form of the objective function. approach to deterministic problems, where the state at the next stage For example, the objective might be to minimize the sum of the is completely determined by the state and policy decision at thecontributions from the individual stages, or to maximize such a current stage. sum, or to minimize a product of such terms, and so dynamic programming can be described diagrammatically as shown in the next figure:nAother categorizationis in terms of the nature of the set of Stage nStage n+1states for the respective stages. In particular, states smight be noCntribution representableby a discrete state variable or by a continuous state SnSn1+of xvariable, or perhaps a state vector is (s,x)f*(s)nnnn+1n+1江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of13 Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo14l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations ResearchxEample 2:Distributing eMdical Teams to oCuntries¾The measure of performance being used is additional Theperson--years ofF life. (oFr a particular country, this oWrld Health Council is devoted to improving health care in the underdeveloped countries of the world. It nowmeasure equqals the increased life expectancy in years has five medical teams available to allocate among three such times the country’’s population.)Table the countries to improve their medical care, health education, and estimated additional PePrson--years of life (in multiples of training programs. Therefore, the council needs to determine 1000) for each country for each possible allocation of how many teams (if any) to allocate to each of these countries to medical teams. WhWich allocation maximizes the measure maximize the total effectiveness of the five teams. The teams musof performance??t be kept intact, so the number allocated to each country must be an integer.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog15y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog16y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations ResearchFormulation:Thousands of Additional Person-Years of LifeMedical CCountry(1) Stage :N=1,2,3Teams123(2) Decision variables : x---a-re the number of teams to allocate n0000to stage (3) State :nA appropriate choice for the s“tate of the system”is 39075880S--n---umber of medical teams still available for allocation to n4105110100remaining countries5120150130S5=, S5-=xS=-x, S3S=-x1211122江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006SchooSchoo18l of Information Technology, JiangXiUn17iversity of Finance & Economics©2006l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 10 运筹学Operations Research10-4运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations ResearchSolution procedure:(4) Decision effect :p(x)----the measure of performance from iiallocating xmedical teams to country i, its value is in .¾MeMthod 1::solve with graphic method. The detail i(5) Recursiverelationship:process shows in the following graphic.**⎧f(s)={p(x)+f(s)}maxnnnnn+1n+1⎪xn⎨*⎪f(s)=0⎩44江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUn19iversity of Finance & Economics©2006School of Information Technolog20y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations Research000eMthod 2:solve in algebraic with the last stage10**01f(s)={p(x)+f(s)}maxhWen n=333334445x3200502={p(x)}max331202105x75345709020sf*(s)x***333370030800005753110454515012010027020047538803411013015041004452051305055江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoof21l o Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo22l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations Research**x=0,s=2⎧23f(s)={p(x)+f(s)}hWen n2=max222233⎪xS2=22x=1,s=1⎨23S0⎪=,x0=, s0=**223x=2,s=0f(s)={p(x)+f(s)}=0max⎩23222233x20+70⎧⎫⎪⎪**⎧x=0,s=1f(s)={p(x)+f(s)}=20+50=7023⎨⎬22max2233⎨S21=,x2⎪⎪x=1,s=0⎩245+03⎩⎭⎧0+50⎫**f(s)={p(x)+f(s)}==50max⎨⎬222233x20+02⎩⎭江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangX23iUniversitySchool of Information Technolog24 of Finance & Economics©2006y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 10 运筹学Operations Research10-5运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations Researchx=0,s=4⎧23x=0,s=3⎧⎪23S4=x=1,s=3223⎪⎪⎪x=1,s=2⎪2x=2,s=23⎨S23=23⎨⎪x=3,s=1x=2,s=12323⎪⎪⎪x=4,s=0⎪⎩23x=3,s=0⎩230+100⎧⎫⎪⎪⎧0+80⎫20+80⎪⎪⎪⎪20+⎪⎪70**⎪⎪**f(s)={p(x)+f(s)}=45+70=125f(s)=max{p(x)+f(s)}==95max⎨⎬⎨⎬222222223333xx45+5022⎪⎪⎪⎪75+50⎪⎪75+0⎪⎪⎩⎭⎪110+0⎪⎩⎭江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog25y, JiangXiUniversity of Finance & Economics©2006Schoo26l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations ResearchhWen n1=x=0,s=5⎧23⎪0+160⎧⎫x=1,s=423⎪⎧0+130⎫⎪⎪⎧x=0,s=51245+125⎪⎪⎪20+100⎪⎪x=2,s=3⎪⎪23⎪⎪x=1,s=412⎪⎪⎨⎪⎪70+9545+⎪80S25=⎪⎪⎪⎪****S5=f(s)={p(x)+f(s)}==1601x=3,s=22222⎨⎬maxf(s)={p(x)+f(s)}==17033max2⎪⎨⎬3x=2,s=3111122⎪75+70⎪x122⎪⎪x90+701⎨⎪⎪⎪⎪⎪110+50x=4,s=1x=3,s=22312⎪⎪⎪⎪⎪105+50⎪⎪150+0⎪⎩⎭⎪x=4,s=1⎪⎪⎪x=5,s=012⎩23⎪⎪120+0⎪⎩⎭⎪x=5,s=0⎩12江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog27y, JiangXiUniversity of Finance & Economics©2006School of28 Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations ResearchA Prevalent Problem Type—The distribution of So, the optimal solution is x11=*,x23,=*x1=*3, the fEfort Problemoptimal objective function value is 170. That is (1,3,1) The preceding example illustrates a particularly common type of dynamic programming problem called the distribution of effort allocation of medical teams to the three countries will this type of problem, there is just one kind of resource that is to be allocated to a number of activities. The yield an estimated total of 107,000 additional person-objective is to determine how to distribute the effort (the resource) among the activities most effectively. For the oWrld Health years of life, which is at least 5,000 more than for any Council example, the resource involved is the medical terms, andother allocation. the three activities are the health care work in the three countries.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo29l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog30y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 10 运筹学Operations Research10-6运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations On the other hand, the distribution of effort Assumptions:The allocating resources to activities is problem is far more general than linear programming in similar to linear programming, but there are some key other waysC. oCnsider the four assumptions of linear difference between them. programming presented in :3. 3:proportionality, 1. nOe key difference is that the distribution of effort additivity, divisibility, and cePrtainty. rPoportionality is problem involves only one resource, whereas linear routinely violated by nearly all dynamic programming programming can deal with hundreds or even thousands of problems, including distribution of effort problems. resources. (In principle, dynamic programming can handle DDivisibility also is often violated, as in example 2, slightly more than one resource, as we shall illustrate in where the decision variables must be integers. In fact, example 5by solving the threer-esource yWndorGlass oC. dynamic programming calculations become more problem, but it quickly becomes very inefficient when the complex when divisibility does hold . number of resources is increased.)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniverf31sity o Finance & Economics©2006Schoo32l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations ResearchxEample D—i3stributing Scientists to eRsearch Of the four assumptions of linear programming, the Teamsonly one needed by the distribution of effort problem (or other dynamic programming problems) is additivity(or A government space project is conducting research its analog for functions involving a product of terms). on a certain engineering problem that must be solved This assumption is needed to satisfy the principle of before people can fly safely to aMrs. Three research optimality for dynamic are currently trying three different circumstances, the probability that the respective teamsc—all them 1,2, The detailed oFrmulationis left to you as an w—3ill not succeed is ,,,respectively. Thus the current probability that all three teams will fail is ()()()0=.129. eBcause the objective is to minimize the probability of failure, two more top scientists have been assigned to the project.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog33y, JiangXiUniversity of Finance & Economics©2006Schoo34l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations ResearchThe following table gives the estimated probability that the Formulation:respective teams will fail when 0,1,or 2 additional scientists are ¾Stage N=1,2,3, corresponds to research team nadded to that team. The problem is to determine how to allocate the two additional scientists to minimize the probability that all three ¾State Si—s the number of new scientists still available for nteams will to the remaining teams.¾Decision variables x—are the number of additional scientists nProbability of failureallocated to team scientistsTeam¾State transfer equation—S=S-xn1+¾bOjective function—p(x) denote the probability of failure for I if it is assigned xadditional scientists, as given by table i .江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schooce & Economics©200635l of Information Technology, JiangXiUniversity of FinanSchool of Information Technolog36y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 10 运筹学Operations Research10-7运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations ResearchSolvexEample 4S—cheduling mEployment Levelsby graphic method: workload for the LoLCAOBJ SHOP is subject to seasonal fluctuation. However, machine operators are to hire and costly to train, so the manager is reluctant to off workers during the slack seasons. He is likewise reluctant maintain his peak season payroll when it is not required. , he is definitely opposed to overtime work on a 2regular basis. Since all work is done to custom orders, it is not to build up inventories during slack seasons. Therefore, manger is in a dilemma as to what his policy should be employment levels.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of37 Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo38l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations ResearchThe following estimates are given for the minimum EEmployment will not be permitted to fall below these levels. employment reuqirements during the four seasons of the AAny employment above these levels is wasted at an approximate year for the foreseeable future :cos$t of $2,000 per person per season. It is estimated that the hiring and firing costs are such that the total cost of changingthe Season Spring Summer Autumn Winter Spring level of employment from one season to the next$ is $200 times Requirements 255220240200255(rtheqF square of the difference in employment levels. Fractional )nlevels of employment are possible because of a few- part-time employees, and the cost data also apply on a fractional basis.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom39ics©2006School of Information Technolog40y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations Research(3) State variable S—the preceding employment level, Formulation:On the basis of the data available, it is not nworthwhile to have the employment level go above the peak seasonState S=xnn1-requirements of 255. Therefore, spring employment should be at hWen n=1, S=x=x=, and the problem is reduced to finding the employment level 104for the other three seasons. (4) bOjective function p(x)—the cost of employment levelii(1)Stage N=1,2,3, 4;stage 1=summer, stage 2=autumn, stage 2p(x)=200(x−x)+2000(x−r)iiii−1ii3=winter, stage 4=spring .(2)Decision variables xe—mployment level for stage n;n(5) The recursive relationship :**let r=minimum employment requirement for stage n ⎧nf(s)={p(x)+f(s)}minnnnnn+1n+1⎪xnthen ⎨r≤x≤255nn*⎪f(s)=0⎩55江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo41l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo42l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 10 运筹学Operations Research10-8运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations ResearchSolution procedure :Stage3 (n3)=:Stage 4:**f(s)=beginning at the last stage (n=4), we already know thatmax{p(x)+f(s)}333344x3x*=255, so the necessary results are 42*=max{200(x−s)+2000(x−200)+f(s)}33344200≤x≤2553**f(s)={p(x)+f(s)}2244min4455=max{200(x−s)+2000(x−200)+200(255−x)}x33334200≤x≤2553={p(x)}min44x42In order to determining the minimize value, we usecacluuls method. ={200(x−s)+2000(x−255)}min444x42=200(255−s)where200≤s≤25544江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006SchooSchoo44l of Informat43ion Technology, JiangXiUniversity of Finance & Economics©2006l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations Research∂Stage2 (n2=) :f(s,x)=400(x−s)+2000−400(255−x)333333∂x3**f(s)={p(x)+f(s)}max222233=x400(2x−s−250)2332*=0={200(x−s)+2000(x−240)+f(s)}max22233240≤x≤2552s+2503so,x*=3222f*(s)=50(250−x)+50(260−x)+1000(x−150)33222江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technologr45y, JiangXiUnivesity of Finance & Economics©2006School of Information Technolog46y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations Research¾This value of x2 is the desired minimizing value if it ∂f(s,x)=400(x−s)+2000−100(250−x)222222is feasibel(204≤x2 ≤25)5. Over the possible ∂x2s2values (220≤s2 ≤255) , this solution actually is −100(260−x)+10002feasible only if 240≤s2 ≤255.=200(3x−2s−240)22=0¾Therefore, we still nee to solve for the feasible value 2s+240of x2 that minimizes f2(s2,x2) when 220≤s2 2<04. 2so,x*=23The key to analyzing the behavior of f2(s2,x2) over the feasible region for x2 again is the partial derivative of f2(s2,x2) . hWen s22<40江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Info47rmation Technology, JiangXiUniversity of Finance & Economics©2006Schoo48l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 10 运筹学Operations Research10-9运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations ResearchStage 1(n1=):∂f(s,x)>0,for240≤x≤25522222*∂xf(s,x)=200(x−s)+2000(x−220)+f(x)211111221⎧So thatx=240 is the desired minimizing ⎪200(x−s)+2000(x−220)+200(240−x)+1150001111⎪if220≤x≤2401⎪⎪2f(s,x)=200(x−s)+2000(x−220)+⎨111111sf*(s)x***2222⎪2002222⎪[(240−x)+(255−x)+(270−x)]+2000(x−195)220≤s≤2440200(2404--s)+5+11500024401111229⎪⎪if240≤x≤255⎩12222440≤s≤25555200[9/[9/(2404--s)+(+25555-+]-s)+(+2707--s)2-+]000(s-19595)(2s+24+04)3/3/222222江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of49 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog50y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations ResearchhWen 240 ≤x≤255Considering first the case where 220≤x1≤240, we have12∂f(s,x)=200(x−s)+2000(x−220)+111111f(s,x)=400(x−s)+2000−400(240−x)111111∂x1200222[(240−x)+(255−x)+(270−x)]+2000(x−195)1111=400(2x−s−235)911400=(4x−3s−225)11It is known that s1=255, so that3=0∂f(s,x)=800(x−245)<01111∂x1So, 3s+225for all x1≤240. Therefore, x1=240 is the minimizing value 1x=1of f1(s1,x1) over the region 220 ≤x1≤江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog51y, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Econom52ics©2006运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations ResearchBecause s=255, it follows that x=247. 5minimizes f(s,x) 11111*over the region 240 ≤x≤(255)12=200(−255)+2000(−220)Note that this region (240 ≤x≤255) includes x=240, so that 11200222f(s,240)>f(s,), and we found that x=240 minimizes +[(240−)+(255−)+(270−)]111119f(s,x) over the region 204 ≤x≤240. Consequently, we now 1111+2000(−195)can conclude that x= also minimizes f(s,x) over the entire =185,0001111feasible region 240 ≤x≤255. Hence 1江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog53y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog54y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 10 运筹学Operations Research10-10运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations ResearchSolution:xEample 5—WyndorGlass oCmpany ProblemState S=amount of respective resources still available for 2Consider the following linear programming problem:allocation to remaining =(R,R,R)2123hWere Riis the amount of resourcei remaining to be allocatedmaxZ=3x+5x12x≤4⎧s=(4,12,18)11⎪x≤12⎪s=(4−x,12,18−3x)⎨3x+2x≤1812⎪⎪x,x≥0However, when we begin by solving for stage 2, we do not yet ⎩12know the value the value of x, and so we use S=(R,R,R)at that 12123point. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo55l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo56l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Dynamic Deterministic Dynamic ProgrammingOperations ResearchOperations ResearchTo solve it, we need to introduce the usual dynamic f*(4,12,18)={3x+f*(4−x,12,18−3x)}1max1211x≤41programming notation. Thus 3x≤181x≥01f*(R,R,R)={5x}2123max21218−3x⎧⎫12x≤R22=3x+5min(,)⎨⎬max12x≤R22220≤x≤4⎩⎭1x≥026,if0≤x≤2f*(4,12,18)={3x+f*(4−x,12,18−3x)}⎧11max12111218−3x⎪1x≤41min(,)=⎨33x≤181229−xif2≤x≤4x≥011⎪1⎩2x*f*(RR,R)(R,R,R)221,233x+30,if0≤x≤2⎧123112N=⎧RR⎫1218−3x⎪231⎧RR⎫2min,3⎨⎬R≥0,R≥05min3x+5min(,)=,⎨⎬2⎨93221⎩⎭22⎩⎭2245−xif2≤x≤411⎪⎩2江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUnr57ivesity of Finance & Economics©2006School of58 Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Dynamic ProgrammingOperations ResearchOperat10 Dynamic Programmingions ResearchBecause both Probabilistic Dynamic Programming9⎧⎫{3x+30}max145−xmax⎨⎬And10≤x≤212≤≤2x4⎩⎭1Achieve their maximum at x=2, at follows that x*1=2 and 1Stage n+1that this maximum is 36, as given in the following tableContribution 1probabilityfrom stagenCfState:*(RR,R)(R,R,R)111,23x*1231p12decisionpCsx36(4,12,1228)2nnp3C3x*=2,R=4−2=2,R=12,R=18−3(2)=121123s1N=x*=62江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog59y, JiangXiUniversity of Finance & Economics©2006Schoo60l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 10 运筹学Operations Research10-11运筹学运筹学 Probabilistic Dynamic Probabilistic Dynamic ProgrammingOperations ResearchOperations ResearchxEample D6—etermining Reject AllowanceseBcause of the probabilistic structure, the relationship betweenf(s,x) and the f(s) necessarily is somewhat more nnnn1+n1+The HITA-NDM-ISS manufacturing company has received an complicated than that for deterministic dynamic programming. Theprecise form of this relationship will depend upon the form of the order to supply one item of a particular type. However, the overall objective has specified such stringent quality requirements that the To illustrate, suppose that the objective is to minimize the manufacturer may have to produce more than one item to obtain anexpected sum of the contributions from the individual stages. Thenitem that is acceptable. The number of extra items produced in as*production run is called the reject allowance. Including a reject f(s,x)=∑p[C+f(s)]nnniin+1n+1i=1allowance is common practice when one is producing for a custom order, and it seemsadvisable in this case.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technologer61y, JiangXiUnivsity of Finance & Economics©2006Schoo62l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Probabilistic Dynamic Probabilistic Dynamic ProgrammingOperations ResearchOperations ResearchThe manufacturer estimates that each item of this In addition, a setup cost of $030 must be incurred whenever the production process is set up for this product, and a completely new type that is produced will be acceptable with setup at this same cost is required for each subsequent production probability½ ½and defective (without possibility for run if a lengthy inspection procedure reveals that a completed lot has rework) with probab½ility ½. Thus the number of not yielded an acceptable item. The manufacturer has time to make acceptable items produced in a lot of size L Lwill have a no more than three production runs. If an acceptable item has not b;inomial distribution ;., the probability of producing been obtained by the end of the third production run, the cost to the no acceptable items in such a lot is (1/2/L) in lost sales income and penalty costs will be $1600. ¾MaMrginal production costs for this product are The objective is to determine the policy regarding the lot size (1+reject allowance) for the required production run that minimizes estimated to be $$100 per item (even if defective), and total expected cost for the items are worthless.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUn63iversity of Finance & Economics©2006School of Information Technolog64y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Probabilistic Dynamic Probabilistic Dynamic ProgrammingOperations ResearchOperations ResearchFormulation:A dynamic programming formulation for this ¾With 1$00 as the unit of money, the contribution to problem is as follows:cost from stage n is [K(x)+x ]regardless of the next nnstage n=production run nstate, where (Kx) is a function of xsuch thatnndecision variable x=lot size for stage n nstate s=number of acceptable items still needed (1 or 0) at 0,ifx=0⎧nnK(x)=⎨nbeginning of stage ,ifx>0⎩nstated objective f(s,x)t=otal expected cost for stages n,.. 3if nnnsystem starts in state sat stage n, immediate decision is x, and nnoptimal decisions are made thereafter. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of65 Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo66l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 10 运筹学Operations Research10-12运筹学运筹学 Probabilistic Dynamic Probabilistic Dynamic ProgrammingOperations ResearchOperations ResearchSolution Procedure:The recursive relationshipStagen 3(=3:)x3f(1,x)K=(K=x)++x+16x+(61/2/)for s1=33333nf(s***)x*333s0123453xxnn⎧⎡⎤11⎛⎞⎛⎞**0000⎪f(1,x)=K(x)+x+f(1)+1−f(0)⎜⎟⎢⎜⎟⎥nnnnn+1n+12211661298817/8882/83 or 4⎝⎠⎝⎠⎪⎢⎥⎣⎦⎪xnStagen2( =:2)⎪1⎛⎞*=K(x)+x+f(1)⎜⎟⎨nnn+1x22⎝⎠f(1,x)K=(K=x)++x+f*+(*1)(1/x2/)⎪222223f(s**)x**222*⎪s012342f(1)=164⎪0000⎪188887715/2/72 or 3⎩江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Informat67ion Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog68y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Probabilistic Dynamic Probabilistic Dynamic ProgrammingOperations ResearchOperations ResearchxEample 7W—inning in Las eVgasStagen1( =:1)nA enterprising young statistician believes that she has x1f(1,x)K=(K=x)++x+f*/x+(*1)(12/)111112developed a system for winning a popular Las Vegas game. Her f(s**)x**111s012341colleagues do not believe that her system works, so they have 1715/2/27/4/558/8/119/1/6627/4/2made a large bet with her that if she starts with three chips, she Thus the optimal policy is to produce two items on the first will not have at least five chips after three plays of the run; if none is acceptable, then produce either two or Each play of the game involves betting any desired number of three items on the second production run; if none is acceptable,then available chips and then either winning or losing this number ofproduce either three or four items on the third production run. The chips. The statistician believes that her system will give her a total excepted cost for this policy is $ of 23/ of winning a given play of the game. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog69y, JiangXiUniversity of Finance & Economics©2006Schoo70l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Probabilistic Dynamic Probabilistic Dynamic ProgrammingOperations ResearchOperations ResearchThe recursive relationshipFormulation:the dynamic programming formulation for The expression for f(s,x) must reflect the fact that it may still nnnthis problem is be possible to accumulate five chips eventually even if the statistician should lose the next play. If she loses, the state at the Stagen= nth play of game (n=1,2,3)next stage will be s-x, and the probability of finishing with at least nnDecision variable x=number of chips to bet at stage nnfive chips will then be f*(s-x). If she wins the next play instead, n1+nnthe state will becomes+x, and the corresponding probability will nnState n= number of chips in hand to begin stage f*(s+x) . eBcause the assumed probability of winning a given n1+nnbOjective function:play is 23/, it now follows that f(s,x)=probability if finishing three plays with at least five nnn12⎧**chips, given that statistician starts stage nin state s, make nf(s,x)=f(s−x)+f(s+x)⎪nnnn+1nnn+1nnimmediate decision ⎨*⎪f(s)=0⎩44江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo71l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of72 Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 10 运筹学Operations Research10-13运筹学运筹学 Probabilistic Dynamic Probabilistic Dynamic ProgrammingOperations ResearchOperations Research**xf(s,x)=1=/3-+/3f*(*s-x) +1/3/3f*(*s++x)f(s)x**22223333332332222sSolution Procedure:20123434000----Stagen2( :2)=1000----sf*(*s)x**3333204/94/94/94/94/94/91 or 200--332/3/34/94/92/3/32/3/32/3/30,2,or 33Stagen3( :3)=10--4/3/32/3428/98/92/3/3/32/3/38/98/91≥55110 (or ≤s-5-5)202--32/3/2 (or more)42/3/1 (or more)≥510 (or ≤s-5)xf(s,x)=1=/3-+/3f*(*s-x) +1/3/3f*(+*s+x)31111211211**Stagen1( :1)=f(s)x**111s10123332/3/320/2/772/3/32/33/320/2/771江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of73 Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo74l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学Operations Res10 Dynamic oCnclusionsDynamic programming is a very useful technique for making a sequence of interrelated decisions. It requires formulating an appropriate recursive relationship for each individual problem. However, it provides great computational saving over using exhaustive enumeration to find the best combination of decisions, especially for large problems. For example, if a problem has 10 stages with 10 states and 10 possible decisions at each stage, then exhaustive enumeration must consider up to 10 billion combinations, whereas dynamic programming need make no more than a thousand calculations. 江西财经大学信息管理学院©2006School of Information Technolog75y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 11 运筹学Operations Research11-1运筹学运筹学Operations ResearchOperations Res11 Game Theoryearch11 Game TheoryContents1、Ze-Zro-Sum Games Game theory is a mathematical theory that deals 2、APEA Prototype Example with the general features of competitive situations like these in a formal, abstract way. It places particular 33、Games wiMth iMxed Strategiesemphasis on the decision--making processes of the 44、Graphical Solution PrPocedure 、Solving byLP Linear Programming江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo1l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo2l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Formulation of Two-Person, eZro-Sum GamesOperations Res11 Game TheoryearchOperations The Formulation of Two-Person, Player 2¾The payoff tableshows Strategy eZro-Sum Gamesthe gain (positive or 1 2negative) for player 1 that 11 -1¾To illustrate the basic characteristics of two--person, zero--Player 1would result from each sum games, consider the game called odds and evens. This 2-1 1game consists simply of each player simultaneously combination of strategies showing either one finger or two the two players. It is In general, a two--person game ¾If the number of fingers matches, so that the total number given only for player 1 is characterized by for both players is even, then the player taking evens (say, because the table for player pl¾ayerThe strategies of player 1 1) wins the bet (say$, $1) from the player 2. 2 is just the negative of this ¾Thus each player has two: strategies: to show either one ¾The strategies of player 2one, due to the zero-sum finger or two fingers. The resulting payoff to player 1 in ¾The payoff table nature of the is shown in the following payoff table.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom3ics©2006Schoo4l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The Formulation of Two-Person, eZro-Sum GamesOperations ResearchOperat11 Game Theoryions Solving Simple Games-A Prototype AA primary objective of game theory is the development xEampleof rational criteria for selecting a strategy. Two key ¾Two politicians are running against each other for the . are made::¾CCampaign plans must now be made for the final 2 days, which are expected to be crucial because of the closeness of ¾BBoth players are race. Therefore, both politicians want to spend these days campaigningB in two key cities, Bigtownand ¾BBoth players choose their strategies solely to promote MMegalopolis. ¾To avoid wasting campaign time, they plan to travel at night their own welfare (no compassion for the opponent). and spend either 1 full day in each city or 2 full days in just one of the cities. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo5l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog6y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 11 运筹学Operations Research11-2运筹学运筹学 Solving Simple Games-A Prototype Solving Simple Games-A Prototype xEampleOperations ResearchOperations ResearchFFormulation::¾HHowever, since the necessary arrangements must be made ¾AAs the problem has been stated, each player has following in advance, neither politician will learn his (or her) three strategies::opponent’’s campaign schedule until after he has finalized ¾Strategy= 1=spend 1 day in each cityhis own.¾Strategy 2==spend both daysB in Bigtown ¾Therefore, each politician has asked his campaign ¾Strategy3=M 3=spend both days in in each of these cities to assess what the impact Player 2would be (in terms of votes won or lost) from the various Strategy 1 2 3possible combinations of days spent there by himself and 1 2 4by his opponentH. He then wishes to use this information to 11 0 5Player 12choose his best strategy on how to use these 2 1 -1江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog7y, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangX8iUniversity of Finance & Economics©2006运筹学运筹学 Solving Simple Games-A Prototype Solving Simple Games-A Prototype xEampleOperations ResearchOperations Research¾Specifically, a strategy can be eliminated from further ¾UUsing the form given in table , we give three alternative consideration if it is dominated by another strategy, ., if sets of data for the payoff table to illustrate how to solve three there is another strategy that is always at least as good different kinds of of what the opponent does.¾VaVriation13 1:Given that table is the payoff table for the two politicians (players), which strategy should each select??Player 2Strategy ¾This situation is a rather special one, where the answer can be 1 2 3obtained just by applying the concept of dominated strategies 1-3 -2 1to rule out a succession of inferior strategies until only one Player 123 0 2choice remains. 35 -2 -4江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School ofn9 Iformation Technology, JiangXiUniversity of Finance & Economics©2006Schoo10l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Solving Simple Games-A Prototype Solving Simple Games-A Prototype xEampleOperations ResearchOperations ResearchVaNVriation2:2:Now suppose that the current data give table ¾The end product of this line of reasoning is that each player 114. 4as the payoff table for the politicians (players). This game does not have dominated strategies, so it is not obvious what should play in such a way as to minimize his maximum losses the players should do. WhWat line of reasoning does game theory whenever the resulting choice of strategy cannot be exploited say they should use??by the opponent to then improve his 2Strategy ¾NoNtice the interesting fact that the same entry in this payoff 1 2 3table yields both the maxminand minimaxvalues. The reason 1-3 -2 66is that this entry is both the minimum in its row and the Player 122 0 2maximum of its column. The position of any such entry is 35 -2 -4called a saddle point.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo11l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog12y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 11 运筹学Operations Research11-3运筹学运筹学 Solving Simple Games-A Prototype Solving Simple Games-A Prototype xEampleOperations ResearchOperations Research¾Va3:VriationL 3:Late developments in the campaign result in the ¾In this case, rather than applying some known criterion for final payoff table for the two politicians given by table . determining a single strategy that will definitely be used, it is HHow should this game be played??necessary to choose among alternative acceptable strategies Player 2Strategy on some kind of randomB basis. By doing this, neither player 1 2 3knows in advance which of his own strategies will be used, 0 -2 21let alone what his opponent will 125 4 -332 3 -4¾¾This suggests, in very general terms, the kind of approach In short, the originally suggested solution (play 1 to play thaqt is required for games lacking a saddle point. In the next strategy 1 and player 2 to play strategy3 3) is an unstable section we discuss the approach more fully. solution, so it is necessary to develop a more satisfactory solutionB. uBt what kind of solution should it be??江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technologr13y, JiangXiUnivesity of Finance & Economics©2006School of Information Technolog14y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Games with Mixed StrategiesOperat11 Game Theoryions ResearchOperations Games with Mixed Strategies¾WhWere m and n are the respective numbers of available strategies. Thus player 1 would specify her plan for ¾WhWenever a game does not possess a saddle point, game playing the game by assigning values to x…,x,…,x. 12mtheory advises each player to assign a probability distribution ¾BeBcause these values are probabilities, they would need to over her set of strategies. To express this mathematically, let be nonnegative and add to 1. Similarly, the plan for player ¾x=p=robability that player 1 will use strategy i i2 would be described by the values she assigns to her (i==1,2…,…,n)decision variaby…yles yy,y,…,y. These plans (x,x…,…,x) 12n12m¾y=yp=robability that player 2 will use strategy j jand (yy…yy,y,…,y) are usually referred to as mixed strategies, (j==1,2…,…,n),12nand the original strategies are then called pure strategies. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog15y, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangX16iUniversity of Finance & Economics©2006运筹学运筹学 Games with Mixed Games with Mixed StrategiesOperations ResearchOperations Research¾To illustrate, mnsuppose that players 1 and 2 in variation 33 of ∑∑pxy¾ijijEExpected payoff for player= 1=the political campaign problem select the mixed strategies i=1j=1(xyy=//,x=//(y,x)=(12/,12/,0) and y,y,y)=(0,12/,12/), respectively. 12331233¾This selection would say that player 1 is giving anq equal ¾In this context, the minimaxcriterion says that a given player chance (probab½ility of ½) to choosing either (pure) strategy 1 should select the mixed strategy that minimizes the maximum or 2, but he is discarding strategy3 3 entirely. expected loss to hEqimself. Equivalently, when we focus on ¾Similarly, player 2 is randomly choosing between his last two payoffs (player 1) rather than losses (player 2), this criterionpure strategies. To play the game, each player could then flip a coin to determine which of his two acceptable pure says to maximininstead, ., maximize the minimum strategies he will actually payoff to the player.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom17ics©2006School of18 Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 11 运筹学Operations Research11-4运筹学运筹学 Games with Mixed StrategiesOperations ResearchOperations Res11 Game Graphical Solution Procedure¾M:iMnimaxtheorem :If mixed strategies are allowed, the pair of mixed strategies that is optimal according to the minimax¾CConsider any game with mixed strategies such that, after criterion provides a stable solution with (the value of the dominated strategies are eliminated, one of the players has only two pure strategies. To be specific, let this player be game), so that neither player can do better by nuilaterallyplayerB 1. Because her mixed strategies are (x=,x)and x=1--122changing her or his , it is necessary for her to solve only for the optimal 1value of 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo19l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog20y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Graphical Solution Graphical Solution ProcedureOperations ResearchOperations Research¾To illustrate this procedure, cons3ider variation 3 of the (yyyEPy,y,y)Expected Payoff 1233political campaign problem. NoNtice that the third pure (1,0,0)0x+5-5+5(1--x)=5=5-5x111strategy for player 1 is dominated by her second, so the (0,1,0)--6-2x+4+4(1--x)=4=4-6x111payoff table can be reduced to the form given in table (0,0,1)2x-33+5-3(1--x)==--3+5x111Therefore, for each of the pure strategies available to player 2, ¾E=yExpected payoff for player 1 = y(55-5-5x)++y6+yy(44--6x)+y(-3+5-3+5x)the expected payoff for player 1 will be1121331¾AcAcording to the minimax(or maxmin) criterion, player 1 wants Player 2Probability yyyyyyto maximize this minimum expected payoffCq. Consequently, player 1231 should select the value of x1 where the bottom line peaks, ., ProbabilityPure Strategy1 2 3Player1where the (-3+56-3+5x) and (44--6x) lines intersect, which yields an 11x10 -2 21expected payoff of 1-x25 4 -31江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo21l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog22y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Graphical Solution Graphical Solution ProcedureOperations ResearchOperations ResearchAccording to the dev=v=max{min{finition of the minimaxvalue vand the −3+5x,4−6x}}110≤x≤11minimaxtheorem, the expected payoff resulting from the optimal strategy (y,y,y)=(y*,y*,y*) will satisfy the −3+5x=4−6x12312311condition7so that x = 1211y*(5−5x)+y*(4−6x)+y*(−3+5x)≤v=v=11213111thus74(x,x)=(,)12is the optimal mixed strategy for player 1, For all values of x1 (0≤x1≤1). Furthermore, when player 1 111172is playing optimally (that is x1=71/1), this inequality will be an =v=−3+5()=is the value of the −1111equality (by the minimaxtheorem), so that 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School ofg23 Information Technoloy, JiangXiUniversity of Finance & Economics©2006Schoo24l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 11 运筹学Operations Research11-5运筹学运筹学 Graphical Solution Graphical Solution ProcedureOperations ResearchOperations Research2020202y*+y*+y*=v=12311111111Therefore, y*=0. eBcause y*>0 would vilatethe next-11Because (y,y,y) is a probability distribution ,tol-ast it is also known that y*+y*+y*=1123Hence2⎧≤for0≤x≤151-⎪5x⎪11aMximinpoint14y*(4−6x)+y*(−3+5x)-6⎨2131x127⎪=forx=1⎪⎩1111x15+3-江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo25l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog26y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Graphical Solution ProcedureOperat11 Game Theoryions ResearchOperations ResearchHence, to solve for y* and y*, select two values of x(say, Solving by Linear Programming0 and 1), and solve the resulting two simultaneous equations. Thus¾First, consider how to find the optimal mixed strategy for 25⎧4y*−3y*=y*=player 1. As indicated in Sec. ⎪11⎪11⇒⎨26⎪mn−2y*+2y*=y*=233⎪Expected payoff for player 1=∑∑pxy11⎩11ijiji=1j=156mnTherefore (y,y,y)=(0,,)123pxy≥v=v1111∑∑ijiji=1j=1江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006SchooSchoo28l of Information Technolog27y, JiangXiUniversity of Finance & Economics©2006l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Solving by Linear Solving by Linear ProgrammingOperations ResearchOperations Researchmaxxn+1and the strategy (x,x…,…,x) is optimal if px+px+L+px−x≥0⎧12m111122m1mm+1⎪px+px+L+px−x≥0121222m2mm+1⎪¾⎪for all opposing strategies (yy…yy,y,…,y).⎨⎪px+px+L+px−x≥01n12n2mnmm+1¾To summarize, player 1 would find his optimal mixed ⎪⎪x+x+L+x=1⎩12mstrategy by using the simplex method to solve the linear By proceeding in a way that is completely analogous to programming problem::that just described, player 2 would conclude that his optimal mixed strategy is given by an optimal solution to the linear programming problem:江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Info29rmation Technology, JiangXiUniversity of Finance & Economics©2006School of30 Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 11 运筹学Operations Research11-6运筹学运筹学 Solving by Linear Solving by Linear ProgrammingOperations ResearchOperations ResearchminyTo illustrate this linear programming approach, consider n+1again variation 3 of the political campaign problem after py+py+L+py−y≥0⎧1111221nnn+1dominated strategy 3 for player 1 is eliminated. Because there ⎪py+py+L+py−y≥01212222nnn+1⎪are some negative entiresin the reduced payoff table, it is ⎪⎨unclear at the outset whether the value of the game v is ⎪nonnegative. For the moment, let us assume that v≥0 and py+py+L+py−y≥0m11m22mnnn+1⎪proceed without making any of the adjustments discussed in the ⎪y+y+L+y=1⎩12npreceding is easy to show that this linear programming 2y2yy05x−x≥0⎧⎧−+−≤22343⎪⎪−2x+4x−x≥05y+4y−3y−y≤0⎪⎪1212343problem and the one given for player 1 are dual to .⎨⎨2x−yyy3x−x≥0++=1121233⎪⎪each other.⎪⎪x+x=1y,y,y≥0⎩12⎩123江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog31y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog32y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 12 运筹学Operations Research12-1运筹学运筹学Operat12 Integer Programmingions ResearchOperations Research¾ PPrototype example¾ Some other formulation possibilities with 12 Integer Programmingbinary variables¾ Some formulation examples¾ Some perspectives onP solving IP problems¾-Bq5 The rBanch-and ––oBund technique and its applications to binaryP IP¾-6 A rBanch-and-B-oBund algorithm for mixed IPP江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog1y, JiangXiUniversity of Finance & Economics©2006Schoo2l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学Operat12 Integer Programming12 Integer Programmingions ResearchOperations ResearchiWth just two choices, we can represent such rPefacedecisions by decision variables that are restricted to just If reuqiring integer values is the only way in two values, say 0 and 1. Thus the jthyeso-r-no decision which a problem deviates from a linear programming would be represented by, say, xsuch thatjformulation, then it is an integer programming 1,ifdecisionjisyes⎧probelm. x=⎨j0,ifdecisionjisno⎩If all variables are reuqired to have integer values, this model is referred to as prue integer Such variables are called binary avriabels(or 01- ). oCnseuqently, IP problems that contain only If only some of the variables are required to binary variables sometimes are called binar yinteger have integer values, this model is referred to as mixed programming(IBP)problems (or 01- integer integerp rogramming (IM)P. programming problems).江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo3l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo4l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Prototype ExampleOperat12 Integer Programmingions ResearchOperations Prototype ExampleNet Decision Decision CCapital Yes-or-no questionpresent The California Manufacturing Company is considering numbervariablerequiredvalueexpansion by building a new factory in either Los Angeles or SanFrancisco, or perhaps even in both cities. It also is considering Build factory in Los 1Angeles??building at most one new warehouse, but the choice of location is X1$X$9 million$6$6 millionBuild factory in San restricted to a city where a new factory is being built. The net2Francisco??X2$X5$ million$3$ millionpresent value of each of these alternatives is shown in the fourth BuXild warehouse in Los 36$X 6$million$5$ millioncolumn of table . The rightmost column gives the capital 3Angeles??X4$X4$ million$2$ millionrequired for the respective investments, where the total capitalBuild warehouse in San available is $10 million. The objective is to find the feasible 4Francisco??combination of alternatives that maximizes the total net presentC$Capital available: 1$0 millionvalue.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo5l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog6y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 12 运筹学Operations Research12-2运筹学运筹学 Prototype Prototype ExampleOperations ResearchOperations ResearchGroups of yeso-r-no decisions often constitute Solution:groups of mtuau yllexcsuliev atlernatievssuch that only one decision in the group can be yes. aEch group maxZ=9x+5x+6x+4x1234reuqires a constraint that the sum of the corresponding binary variables must be euqal to 1 or less than or euqal ⎧6x+3x+5x+2x≤101234⎪to +x≤134⎪⎪cOcasionally, decisions of the yeso-rn-o type are .−x+x≤0⎨13contingent decision, ., decisions that depend upon ⎪−x+x≤024⎪previous decisions. For example, one decision is said to ⎪xisbinary,forj=1,2,3,4j⎩be contingent on another decision if it is allowed to be yes only if the other is yes. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog7y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog8y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Formulation Possibilities with B..VOperat12 Integer Programmingions ResearchOperations ResearchTo illustrate the approach to such situations, suppose that Some Other Formulation Possibilities one of the requirements in the overall problem is thatwith Binary VariablesiEtheriEther-or-Constraints3x+2x≤1812orConsider the important case where a choice can be made between two constraints, so that only one (either one) must holdx+4x≤1612(whereas the other one can hold but is not required to do so). For Let Mbe a very large positive number. Then this requirement example, there may be a choice as to which of two resources to use can be rewritten asfor a certain purpose, so that it is necessary for only one of the two resource availability constraints to hold mathematically. 3x+2x≤18+My12x+4x≤16+M(1−y)12江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog9y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog10y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Formulation Possibilities with B.. Formulation Possibilities with B..VOperations ResearchOperations ResearchThis case is a direct generalization of the preceding case, K out of N constraints must holdwhich had K=1 and N=2. Denote the Npossible constraints by oCnsider the case where the overall model includes a set of f(x,x,...,x)≤d+My112n11N possible constraints such that only some K of these constraints f(x,x,...,x)≤d+Mymust hold. (Assume that K<N). Part of the optimization process 212n22is to choose the combination of K constraints that permits the Mobjective function to reach its best possible value. The NK- f(x,x,...,x)≤d+MyN12nNNconstraints not chosen are, in effect, eliminated from the Nproblem, although feasible solutions might coincidentally still y=N−K∑ii=1satisfy some of them. andyisbinary,fori=1,2,Knjwhere M is an extremely large positive number.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo11l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog12y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 12 运筹学Operations Research12-3运筹学运筹学 Formulation Possibilities with B.. Formulation Possibilities with B..VOperations ResearchOperations ResearchFunctions with N Possible aVluesThe Fixed-Charge ProblemConsider the situation where a given function is required to take on any one of N given values. Denote this requirement byIt is uqite common to incur a fixed charge or setup cost when one is undertaking an activity. For example, f(x,x,...,x)=dord,L,ord12n12Nsuch a charge occurs when a production run to produce a The equivalent IP formulation of this requirement is the following :Nbatch of a particular product is undertaken and the f(x,x,...,x)=∑dy12niii=1reuqired production facilities must be set up to initiate the N∑y=1run. In such cases, the total cost of the activity is the sum ii=1of a variable cost related to the level of the activity and the So this new set of constraints would replace this requirement inthe setup cost reuqired to initiate the activity. statement of the overall problem.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo13l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo14l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Formulation Possibilities with B.. Formulation Possibilities with B..VOperations ResearchOperations ResearchFrequently the variable cost will be at least roughly To formulate the overall model, suppose that there proportional to the level of the activity. If this is the case, the total costof the activity can be represented by a function of the formare nactivities, each with the preceding cost structure and that the problem is to k+cx,ifx>0⎧⎪jjjjnf(x)=⎨jjminZ=(cx+ky)0,ifx=0⎪∑jjjjj⎩j=1theoriginalconstraints⎧where xdenotes the level of activity j, kdenotes .⎨the setup cost, and cdenotes the cost for each incremental jx−My≤0jj⎩unit. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo15l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo16l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Formulation Possibilities with B.. Formulation Possibilities with B..VOperations ResearchOperations Research¾Binary Representation of general Integer Specifically, if the bounds on an integer variable xareVariables0≤x≤uSuppose that you have a pure IP problem where most of the And if Nis defined as the integer such that variables are binar yvariables,but the presence of a few general NN+1integer variables prevents you from solving the problem by one 2≤u≤2of the very efficient BIP algorithms now available. A nice way to Then the binarr yepresentationof xis circumvent this different is to use the binar yrepresentationfor Nieach of these general integer =∑2yii=0江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo617l of Information Technology, JiangXiUniversity of Finance & Economics©200School of Information Technology, JiangXiUniversity of Finance & Econom18ics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 12 运筹学Operations Research12-4运筹学运筹学Operat12 Integer Programmingions ResearchOperations Some formulation examplesearchExample 1 :aMking Choices hWen the Decision Variables are Some formulation examples¾The research and development division of a manufacturing We now present a series of examples that illustrate a variety of formulation techniuqes with binary company has developed three possible new products. However, variables, including those discussed in the preceding two sections. oFr the sake of clarity, these examples to avoid undue diversification of the companys’ product line, have been kept very small. In actual applications, these management has imposed the following restriction:formulations typically would be just a small part of a vastly larger model.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog19y, JiangXiUniversity of Finance & Economics©2006Schoo20l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Some formulation ResearchOperations Some formulation examplesearchRestriction :1rFom the three possible new products, at Productmost two should be chosen to be hours per 1 2 3weekRestriction :2Just one of the two plants should be chosen to produce the new 13 4 2 3024 66 240And the data are given in the next table. Unit profit5 7 3(thousands of dollars)Sales potential7 5 9(units per week)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo21l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo22l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Some formulation examplesions ResearchOperations Some formulation examplesearchFormulation with auxiliary binary variables:xEample 2: Violating ProportionalityMaxZ=5x+7x+3x123⎧x≤71A corporation is developing its marketing plans for ⎪x≤52next years’ new products. oFr three of these products, it ⎪⎪x≤93is considering purchasing a total of five TV spots for ⎪x≤My⎪11commercials on national television networks. The ⎪x≤My⎪22sproblem we will focus on is how to allocate the five .t.⎨x≤My33⎪spots to these three products, with a maximum of three ⎪y+y+y≤2123⎪spots (and a minimum of zero) for each product. 3x+4x+2x≤30+yM⎪1234⎪4x+6x+2x≤30+(1−y)M1234⎪⎪x≥0,x≥0,x≥0,y=0or1⎩123i江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo23l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog24y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 12 运筹学Operations Research12-5运筹学运筹学 Some formulation Some formulation examplesions ResearchOperations Research¾eNxt table shows the estimated impact of allocating Profitzero, one, two, or three spots to each product. This Number of TV VProduct spots impact is measured in terms of the profit from the 1 2 3additional sales that would result from the spots, 00 0 011 0 -1considering also the cost of producing the commercial 23 2 2and purchasing the spots. The objective is to allocate 33 3 4five spots to the products so as to maximize the total profit.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog25y, JiangXiUniversity of Finance & Economics©2006Schoo26l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Some formulation examplesions ResearchOperations Some formulation examplesearchThenOne formulation with auxiliary binary variables:let x,x,xbe the number of TV spots allocated to 123MaxZ=y+2y+2y+y−y+3y+2y11122223313233the respective products. y−y≤0⎧1211we introduce an auxiliary binary variable yfor ⎪ijy−y≤01312⎪each positive integer value of xj= (j1=,2,)3, where yiij⎪y−y≤02221has the interpretation ⎪−y≤0⎨2322⎪y−y≤013231,ifx=j⎧i⎪y=⎨ij⎪y−y≤033320,otherwise⎩⎪y+y+y+y+y+y+y+y+y=5⎩111213212223313233江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog27y, JiangXiUniversity of Finance & Economics©2006School of28 Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Some formulation examplesions ResearchOperations Some formulation examplesearchFlight Feasible sequence of flights xEample 3: oCvering All Characteristics1 2 3 4 5 6 6 7 8 8 9 10 11 12San Francisco to Los 1 1 1 1nA airline needs to assign its crews to cover all its upcoming Angeles1 1 1 1flights. eW will focus on the problem of assigning three crews San Francisco to Denver1 1 1 1based in San Francisco (SF) to the flights listed in the first column San Francisco to Seattle2 2 3 2 3 of the next table. The other 12 column shows the 12 feasible Los Angeles to ChCicago2 3 5 5sequences of flights for a crew. Exactly three of the sequences Los Angeles to San 3 3 4Francisconeed to be chosen on such a away that every flight is covered. 3 3 3 3 4CChicago to Denver2 4 4 5 The objective is to minimize the total cost of the three crew ChCicago to Seattle2 2 2assignments that cover all the flights. Denver to San Francisco2 4 4 5Denver to CChicago2 2 4 4 2Seattle to San FranciscoSeattle to Los Angeles江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Ct, $$1,0002 3 4 66 7 5 7 8 8 9 9 8Cos 8 9School of Information Technolog29y, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUn30iversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 12 运筹学Operations Research12-6运筹学运筹学 Some formulation Some formulation examplesions ResearchOperations ResearchminZ=2x+3x+4x+6x+7x+5x+7x+8x+9x123456789Formulation with binary variables: ⎧⎪x+x+x+x≥114710⎪x+x+x+x≥1with 12 feasible seu⎪qences of flights, we have 12 25811⎪x+x+x+x≥136912yes-orn-o decisions:⎪⎪x+x+x+x+x≥14791012⎪x+x+x≥1459⎪⎪+x+x+x+x≥1⎨781011121,ifsequencejisassignedtoacrew,⎧⎪x+x+x+x≥12459⎪x=⎨j⎪x+x+x≥158110,otheriwse⎩⎪x+x+x+x≥137812⎪⎪x+x+x+x+x≥169101112⎪12⎪x=3∑j⎪j=1⎩江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©200631School of Information Technolog32y, JiangXiUniversity of Finance & Economics© Some Perspectives on Solving运筹学运筹学Operat12 Integer Programmingions ResearchOperations ResearchInteger Programming Some Perspectives on Solving Integer nOe pitfall is that an optimal linear Programming Problemsprogramming solution is not necessari ylfeasibelafter it is round. oFr example:Linear programming problems generally are much easier to solve than IP problems, so it is tempting to use the approximatemaxz=x2procedure of simply applying the simplex method to the LP 1⎧relaxation and thenroundingthe noni-nteger values to integers in −x+x≤12⎪the resulting solution. 2⎪⎪1This approach may be adequate for some applications, +x≤3⎨122especially if the values of the variables are quite large so that ⎪rounding creates relatively little error. However, you should ⎪x,x≥0,andareintegers12⎪beware of two pitfalls involved in this approach.⎩江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo33l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog34y, JiangXiUniversity of Finance & Economics© Some Perspectives on Some Perspectives on Solving运筹学运筹学Operations ResearchOperations ResearchInteger Programming ProblemsInteger Programming ProblemsnAother pitfall is that even an optimal solution for the LP For examplerelaxation is rounded successfully, there is no guarantee that this The rounded solutions are not feasiblerounded solution will be the optimal integer solution . For example:(3/2,2)maxz=x+5x12pOtimal solution for the PLrelaxationx+10x≤20⎧12⎪≤2⎨1⎪x,x≥0,andareintegers⎩12江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo35l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Econom36ics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 12 运筹学Operations Some Perspectives on Some Perspectives on Solving运筹学运筹学Operations ResearchOperations ResearchInteger Programming ProblemsInteger Programming ProblemsThe most popular mode for IP algorithms is to use pOtimal IP solutionpOtimal solution for the LP the brancha-ndb-onudtechnique and related ideas to relaxationimpilcit ylenmueratethe feasible integer solutions, and we shall focus on this approach. The next section presents the branch-andb-ound technique in a general context and illustrates it with a basiRounded solutionc brancha-ndb-ound algorithm for IBP problems.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo37l of Information Technology, JiangXiUniversity of Finance & Econom38ics©2006School of Information Technology, JiangXiUniversity of Finance & Economics© The Branch-and-Bound Technique and运筹学运筹学Operat12 Integer Programmingions ResearchOperations Researchits Application to Binary Integer The Branch-and-Bound Technique and its eW shall now describe in turn these three basic steps—Application to Binary Integer Programmingbranching, bounding, and fathominga—nd illustrate them by applying a branch—and –bound algorithm to the next example:The basic concept underlying the brancha-ndb-ound technique is to divideand conquer. Since the original l“arge”problem is too difficult to be solved directly, it is divided into maxZ=9x+5x+6x+4x1234smaller and smaller subproblemsuntil these subproblemscan be 6x+3x+5x+2x≤10⎧conquered. The dividing (branching) is done by partitioning the 1234⎪entire set of feasible solutions into smaller and smaller +x≤1⎪.⎨The conquering f(athoming)is done partially by bounding −x+x≤013⎪how good the best solution in the subset can be and then ⎪−x+x≤0⎩24discarding the subset it its bound indicates that it cannot possibly contain an optimal solution for the original problem.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo39l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo40l of Information Technology, JiangXiUniversity of Finance & Economics© The Branch-and-Bound Technique The Branch-and-Bound Technique and运筹学运筹学Operations ResearchOperations Researchits Application to Binary Integer Programmingits Application to Binary Integer Programming¾BranchingFathoming¾hWen you are dealing with binary variables, the most straightforward way to partition the set of feasible solutions into we conduct our search for an optimal solution by retaining subsets is to fix the value of one of the variables (say, x1) atx1=0 for further investigation only those subp-roblems that could for one subset and at x1=1 for the other subset. ¾Boundingpossibly have a feasible solution better than the current ¾The standard way of doing this is to quickly solve a simpler of the subp-roblem. In most cases, a relaxationof a problem is obtained simply by deleting(r“elaxing)” one set of constraints that had made the problem difficult to solve.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo41l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo42l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 12 运筹学Operations The Branch-and-Bound Technique The Branch-and-Bound Technique and运筹学运筹学Operations ResearchOperations Researchits Application to Binary Integer Programmingits Application to Binary Integer ProgrammingZ=9X=(0,1,0,1)4Z=16Z=13Summary of Fathoming Tests55X=(,1,0,1)4X=(1,0,,0)A subp-roblem is fathomed ( dismissed from further 6x0=51Z=14consideration ) if Allx0=X=(1,1,0,0)2Z=16Test 1 :its bound ≤*Z1X=(1,1,0,)x0=0 ≤x≤1Or24ix1=1¾Test 2 :its PLrelaxation has no feasible solutions, x0=Z=163Or 44X=(1,,0,)x1=x0=2554¾Test 3 :The optimal solution for its pLrelaxation is Z=16No feasible integer. ( if this solution is better than the incumbent, it 1x1=solution3X=(1,1,0,)becomes the new incumbent, and test 1 is reapplied to all 2unfathomed subproblemswith the new large *)Z No feasible solution江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of In43formation Technology, JiangXiUniversity of Finance & Economics©2006Schoo44l of Information Technology, JiangXiUniversity of Finance & Economics© The Branch-and-Bound Technique The Branch-and-Bound Technique and运筹学运筹学Operations ResearchOperations Researchits Application to Binary Integer Programmingits Application to Binary Integer ProgrammingSummary of the BIP Branch-and Bound Algorithm¾aFthoming:oFr each new subp-roblem, apply the three -Initialization :set =*Z∞. fathoming tests summarized above, and discard those Steps for each iterationsubp-roblems that are fathomed any of the tests.:¾Branching:Among the remaining (unfathomed) subp-roblems, select the one that was created most recently. Branch from the Optimality test:Stop when there are no remaining sub-node for this subp-roblem to create two new subp-roblems by problems ;the current incumbent is optimal. Otherwise, fixing the next variable (the branching variable) at either 0 to perform another iteration.¾Bounding:for each new subp-roblem, obtain its bound by applying the simplex method to its LP relaxation and rounding down the value of Z for the resulting optimal solution.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Informaton Technology, JiangXiUniversity of Finance & Econom45iics©2006School of Information Technolog46y, JiangXiUniversity of Finance & Economics© The Branch-and-Bound Technique The Branch-and-Bound Technique and运筹学运筹学Operations ResearchOperations Researchits Application to Binary Integer Programmingits Application to Binary Integer ProgrammingBranchingalways involves selecting one remaining sub-Other Options with the Branch-and-Bound problem and dividing it into smaller BIP algorithm selected the most recently created sub-problem, because this is very efficient for reo-ptimizing each LP relaxation from the preceding branch-andb-ound algorithm has the same three basic ¾Selecting the subp-roblem with the best bound is the other most steps of branching, bounding , and flexibility popular rule, because it tends to lead more quickly to better lies in how these steps are and so more fathoming. ¾The dividing typically is done by choosing a branching variable and assigning it either individual values or ranges of values.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of In47formation Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog48y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 12 运筹学Operations The Branch-and-Bound Technique The Branch-and-Bound Technique and运筹学运筹学Operations ResearchOperations Researchits Application to Binary Integer Programmingits Application to Binary Integer ProgrammingFathominggenerally is done pretty much as described for Boundingusually is done by solving a relaxation. oHwever, the BIP algorithm. The three fathoming criteria can be stated inthere are a variety of ways to form relaxations. For example, more general terms as the Lagrangianrelaxation,where the entire set of functional constraints AX≤b is deleted and then the objective ¾Summary:A sub-problem is fathomed if an analysis of its function relaxation reveals that ¾Criterion 1:Feasible solutions of the subp-roblem must have Maximize Z=CXZ≤Z*, or is replaced by ¾Criterion 2:The subp-roblem has no feasible solutions, or Maximize Z=CX-λ(Ab-X)R¾Criterion 3:An optimal solution of the subp-roblem has been the fixed vector λ≥0.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo49l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog50y, JiangXiUniversity of Finance & Economics© The Branch-and-Bound Technique The Branch-and-Bound Technique and运筹学运筹学Operations ResearchOperations Researchits Application to Binary Integer Programmingits Application to Binary Integer ProgrammingDefinition:A solution is g“ood enough”if its Z is c“lose To find a solution that is close enough to being optimal, enough”to the value of Zfor an optimal solution (call it Z**). only one change is needed in the usual brancha-ndb-ound Close enough can be defined in either of two ways as eitherprocedure. This change is to replace the usual fathoming test 1 for a **−K≤Z*or(1−α)Z**≤ZBound ≤Z* ? by either oBundK-≤Z* ?for a specified (positive) constant K or α. or For example, if the second definition is chosen and (1-α)bound ≤ Z* ?α=, then the solution is required to be within 5percent nAd then perform this test after test optimal.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo51l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Economics© A Branch-and-Bound Algorithm运筹学运筹学Operat12 Integer Programmingions ResearchOperations Researchfor Mixed Integer A Branch-and-Bound Algorithm for Mixed nmaxZ=∑cxInteger Programmingjjj=1n⎧eW shall now consider the general MIP problem, where ∑ax≤bijjij=1⎪some of the variables (say, I of them) are restricted to integer⎪≥0(j=1,2,L,n)⎨values (but not necessarily just 0 and 1). But the rest are j⎪ordinary continuous variables. For notational convenience, we xisintegerforj=1,2,...,I;I≤nj⎪shall order the variables so that the first I variables are the ⎩integer-restricted variables. Therefore, the general form of the problem being considered is 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo53l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Informaton Technology, JiangXiUniversity of Finance & Econom54iics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 12 运筹学Operations A Branch-and-Bound A Branch-and-Bound Algorithm运筹学运筹学Operations ResearchOperations Researchfor Mixed Integer Programmingfor Mixed Integer ProgrammingSteps for each iteration:: oFr each new subproblem, obtain its :among the remaining (unfathomed) bound by applying the simplex method to its pLsubproblems, select the one that was created most recently. Among the integerr-estricted variables that have a nonintegervalue in the relaxation and using the value of Zfor the resulting optimal solution for the LP relaxation of the subproblem, choose optimal solution. the first one in the natural ordering of the variables to be thebranching variable. 3. Fathoming: For each new subproblem, apply the Let xjbe this variable and xj* its value in this solution. rBanch three fathoming tests given below, and discard those from the node for the subproblemto create two new subproblemssubproblemsthat are fathomed by any of the adding the respective constraints xj≤x[j* ]and xj≥x[j*+]1.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom55ics©2006Schoo56l of Information Technology, JiangXiUniversity of Finance & Economics© A Branch-and-Bound A Branch-and-Bound Algorithm运筹学运筹学Operations ResearchOperations Researchfor Mixed Integer Programmingfor Mixed Integer ProgrammingTest 1:Its bound ≤Z*,where Z* is the value of Z for the current IM PxEample:Test2:Its LP relaxation has no feasible =4x−2x+7x−x1234x+5x≤10⎧13Test 3:The optimal solution for its LP relaxation has integer ⎪x+x−x≤1123⎪values for the integer-restricted variables.⎪6x−5x≤0⎪.⎨−x+2x−2x≤3124⎪Optimality test:Stop when there are no remaining ⎪x≥0,forj=1,2,3jsubproblems ;the current incumbent is optimal. Otherwise, perform ⎪xisaninteger,forj=1,2,3⎪another ⎩江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog57y, JiangXiUniversity of Finance & Economics©2006Schoo58l of Information Technology, JiangXiUniversity of Finance & Economics© A Branch-and-Bound A Branch-and-Bound Algorithm运筹学运筹学Operations ResearchOperations Researchfor Mixed Integer Programmingfor Mixed Integer Programming11Z=14Z=131126Z=14Z=141561511x0=Z=1411X=(0,0,2,)X=(,1,,0)6951141Z=14266X=(1,,,0)X=(,1,,0)Z=14555x≤1665374269X=(,,,0)X=(1,,,0)Optimal solution42453755X=(,,,0)x≤12424x≤11x≤x1=1111No feasible solutionsAllZ=12x≥262All511x≥2X=(,2,,0)2661x≥11Z=12uCtting branch6x≥1511No feasible solutions1oN feasible X=(,2,,0)66solutions江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo5960l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 13 运筹学Operations Research13-1运筹学运筹学13 Nonlinear ProgrammingOperations ResearchOperations ResearchIn this chapter, we turn our attention to this important area, Nonlinear ProgrammingmaxZ=f(X)In one general form, the 13 Nonlinear Programmingnonlinear programming problem is g(X)≤b⎧.⎨to find x=(x1,x2,,…xn) so as to X≥0⎩Where f(X) and the g(X) are given functions of the n idecision do present a few sample applications and then introduce some of the basic ideas for solving certain important types of nonlinear programming problems.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School ofo1 Infrmation Technology, JiangXiUniversity of Finance & Econom2ics©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学13 Nonlinear Sample ApplicationsOperations ResearchOperations ResearchIn , where p(x) is the price required in order to be Sample Applicationsable to sell x units. The firms’ profit from producing and selling xunits of the product then is the sales revenue xp(x) The Product-Mix Problem with Price lEasticityminus the production and distribution costs. Therefore, if the unit cost for producing and distributing the product is fixed atc , the firm’s profit from producing and selling x units is given In produce-mix problems, such as the WyndorGlass Co. by the nonlinear functionproblem of , the resulting objective function is linear. p(x)P(x)=xp(x)−cxHowever, in many productmixproblems, certain factors priceintroduce nonlinearitiesinto the objective function. For cUnit costxDemandexample, a large manufacturer江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technologer3y, JiangXiUnivsity of Finance & Economics©2006Schoo4l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Sample Sample ApplicationsOperations ResearchOperations ResearchIf each of the firm’s n products has a similar profit Nonlinearities also may arise in the gj(x) constraint function, say, p(x) for producing and selling xunits of functions in a similar fashion. For example, if there is a budget jjjproduct j(j=1,2,..,n), then the overallobjective nonlinear constraint on total production cost, the cost function will be function is nonlinear if the marginal cost of production varies as just nf(X)=∑P(x)jjdescribed. For constraints on the other kinds of resources, j=1gi(x) will be nonlinear whenever the use of the corresponding Another reason that nonlinearities can arise in the objective resource is not strictly proportional to the production levels of function is due to the fact that the marginal cost of producing the respective unit of a given product varies with the production level. For example, the marginal cost may decease when the The Transportation Problem with oVlume Discounts on production level is increased because of a learning-curve CostOn the other hand, it may increase instead, because special measures such as overtime or more expensive production facilities may be needed to increase production further.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo5l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog6y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 13 运筹学Operations Research13-2运筹学运筹学 Sample Sample ApplicationsOperations ResearchOperations ResearchAs illustrated by the P&T oCmpany example in , a typical application of the transportation problem is to determine an optimal plan for shipping goods from various aMrginal costsources to various destinations, given supply and demand constraints, in order to minimize total shipping cost. It was Amount shippedassumed in Chap. 8 that the cost per unit shipped from a given source to a given destination is fixed, regardless of the . In actuality, this cost may not be fixed. (x)(Px)=x[p(x)c-]江西财经大学信息管理学院©2006Amou江n西t s财h经ip大p学ed信息管理学院©2006Schoo7l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUn8iversity of Finance & Economics©2006运筹学运筹学 Sample Sample ApplicationsOperations ResearchOperations ResearchSuppose that n stocks are being considered for inclusion in Portfolio Selection with Risky Securitiesthe portfolio, and let the decision variables xj(j1=,2,…n) be the number of shares of stock j to be included. Let μj and σjjInvestors are concerned about both the expected return be the mean and variance, respectively, of the return on each (gain) and the risk associated with their investments, share of stock j. For I1=,2,…,n (i≠j), let be the covariance of nonlinear programming is used to determine a portfolio that, the return on one share each of stock I and stock j. then the under certain assumptions, provides an optimal trade-off expected value R(x) and the variance (Vx) of the total return between these two the entire portfolio are This approach is based largely on path-breaking research nnndone by aHrry Markowitzand williamsharp that helped them R∑and (X)=μxV(X)=∑∑σxxjjjjijj=1i=1j=1win the 1991 Nobel Prize in V()X measures the risk associated with the A nonlinear programming model can be formulated for portfolio. this problem as follows. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of9 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog10y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Sample Sample ApplicationsOperations ResearchOperations ResearchWhere the parameter βis Under certain assumptions about the investors’ utility The device used to nonegativeconstant that reflects function ,it can be shown that an optimal solution for this consider the trade-off between the investors’ desired trade-off nonlinear programming problem maximizes the investor’s these two factors is to combine between expected return and expected in the objective function risk. β=0 implies that risk to be maximizedshould be ignored completely, one drawback of the preceding formulation is that, whereas a large value for because R()X and V()X are somewhat incommensurable, it is f(X)=R(X)−βV(X)βplaces a heavy weight on relatively difficult to choose an appropriate value for β. minimizing risk .Therefore, rather than stopping with one choice of β, it is common to use a parametric programming approach to The complete nonlinear programming model might be nnngenerate the optimal solution as a function of βover a wide maxf(X)=∑μx−β∑σxxjjjjijj1i=1jrange of values β. The next step is to examine the values of nWhere Pjis the price for each ⎧R()X and (VX) for these solutions that are optimal for some ∑Px≤β⎪jjj=1share of stock j and B is the .⎨value of βand then to choose the solution that seems to given ⎪x≥0amount of money budgeted for j⎩the best trade-off between these two quantities. the portfolio.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo1l of Information Technolog1y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog12y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 13 运筹学Operations Research13-3运筹学运筹学 Sample Applications13 Nonlinear ProgrammingOperations ResearchOperations Graphical Illustration of Nonlinear This procedure often is referred to as generating the Programming Problemssolutions on the efficient frontier of the two-dimensional graph of (R()X,(V)X) points for feasible .X the reason is that the (R(X),(V)X) point for an optimal Xlies on the frontier of MaxZ=3x+5x12the feasible points. Furthermore, each optimal Xis efficient inx≤4⎧1the⎪ sense that no other feasible solution is at least equally +5x≤216⎨12good with one measure (R or V) and strictly better with the ⎪x≥0,x≥012⎩other measure (smaller Vor large R).江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo13l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo14l of Information Technology, JiangXiUniversity of Finance & Economics© Graphical Illustration of Nonlinear Graphical Illustration of Nonlinear 运筹学运筹学Operations ResearchOperations ResearchProgramming ProblemsProgramming Problems22Maxz=54x−9x+78x−13x1122⎧x≤41⎪1=Z172x≤12⎪.⎨3x+2x≤1812⎪⎪x,x≥0126⎩12=Z1=Z981=Z89 Afunction with several local maxima江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoor15l of Information Technology, JiangXiUnivesity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of16 Finance & Economics© Graphical Illustration of Nonlinear Graphical Illustration of Nonlinear 运筹学运筹学Operations ResearchOperations ResearchProgramming ProblemsProgramming ProblemsmaxZ=3x+5xoCncave function(0,7) =optimal solution12x≤4⎧1⎪−x+14x−x≤49⎨1122⎪x,x≥012⎩oCnvex function(,43) =local maximumTherefore, to guarantee that a local maximum is a global maximum for a nonlinear programming problem with constraints gi(x)≤bi(i-1=,2,…,m) and x≥0,the objective function f(x) must be concave and each gi(x) must be convex. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo17l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog18y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 13 运筹学Operations Research13-4运筹学运筹学13 Types of Nonlinear Programming Problear ProgramminemsgOperations ResearchOperations ResearchUnconstrained OptimizationUnconstrained optimization problems have no Types of Nonlinear Programming constraints, so the objective is simply to Maximize f(x).ProblemsOver all values X(=x1,x2,…,xn). As reviewed in Appendix 3, the necessary condition that a particular solution =*XX be Optimal when f(x) is a differentiable function is ∂f=0atX=X*,forj=1,2,...,n∂xiWhen a variable xjdoes have a nonnegativityconstraintxj≥0, the preceding necessary and sufficient condition changes slightly to ≤0atX=X*,ifx*=0⎧∂f⎪j⎨∂x=0atX=X*,ifx*>0⎪jj⎩江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo19l of Information Technology, JiangXiUniversitySchoo20 of Finance & Economics©2006l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Types of Nonlinear Programming Types of Nonlinear Programming ProblemsOperations ResearchOperations ResearchWhen f(x) is concave, then solving for *X reduces to for each such j. this condition is illustrated in ,11, solving the system of n equations obtained by setting the n where the optimal solution for a problem with a single partial derivatives equal to zero. For nonlinear functions f()X,variable is at 0=X even though the derivative there is negative these equations often are going to be nonlinear as well, in rather than zero. Because this example has a concave function which case you are unlikely to be able to solve analytically forto be maximized subject to a nonnegativityconstraint, having their simultaneous solution. What then? Sections and the derivatives less than or equal to 0 at 0=X is both a describe algorithmic search procedures for finding *,X necessary and sufficient condition for X0= to be for n1= and then for n1>. These procedures also play an A problem that has some nonnegativityconstraints but no important role in solving many of the problem types functional constraints is one special case (m0=) of the next described next, where there are constraints. The reason is that class of problems. many algorithms for constrained problems are designed so that they can focus on an unconstrained version of the problem during a portion of each iteration.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Informaton Technology, JiangXiUniversity of Finance & Econom21iics©2006School of Information Technolog22y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Types of Nonlinear Programming Types of Nonlinear Programming ProblemsOperations ResearchOperations ResearchQuadratic ProgrammingLinearly Constrained OptimizationQuadratic programming problems again have linear Linearly constrained optimization problems are constraints, but now the objective function f()X must be characterized by constraints that completely fit linear quadratic. Thus, the only difference between them and a programming, so that all the gi(x) constraint functions are linear programming problem is that some of the terms in the linear, but the objective function f()X is nonlinear. The objective function involve the square of a variable or the problem is considerably simplified by having just one product of two function to take into account, along with a linear Many algorithms have been developed for this case under programming feasible region. A number of special algorithms the additional assumption that f(X) is concave. Section based upon extending the simplex method to consider the presents an algorithm that involves a direct extension of the nonlinear objective function have been method.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo23l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Infomat24rion Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 13 运筹学Operations Research13-5运筹学运筹学 Types of Nonlinear Programming Types of Nonlinear Programming ProblemsOperations ResearchOperations ResearchConvex ProgrammingAll the f()X and g(X) functions are separable programming covers a broad class of problems nthat actually encompasses as special cases all the preceding f(X)=∑f(x)jjj=1types when f(X) is concave. The assumptions are that 1. f()X is each fj(xj) function includes only the terms giinvolving just xj. In the terminology of linear ()X is (see Sec. ), separable programming problems satisfy the assumption of additives but not the Separable Programmingassumption of proportionality (for nonlinear functions). Separable programming is a special case of convex programming, where the one additional assumption is that江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo25l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Econom26ics©2006运筹学运筹学 Types of Nonlinear Programming Types of Nonlinear Programming ProblemsOperations ResearchOperations ResearchNonconvexProgrammingNonconvexprogramming encompasses all nonlinear It is important to distinguish these problems from other programming problems that do not satisfy the assunptionsof convex programming. Now, even if you are successful in convex programming problems, because any separable finding a local maximum, there is no assurance that it also willbe a global maximum. Therefore, there is no algorithm that programming problem can be closely approximated by a will guarantee finding an optimal solution for all such linear programming problem so that the extremely efficient problems. However, there do exist some algorithms that are relatively well suited for finding local maxima, especially whensimplex method can be used. the forms of the nonlinear functions do not deviate too strongly from those assumed for convex programming. One such algorithm is presented in Sec. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom27ics©2006School of Information Technolog28y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Types of Nonlinear Programming Types of Nonlinear Programming ProblemsOperations ResearchOperations ResearchThis case is where all the cicoefficients in each function Geometric programmingare strictly positive, so that the functions are generalized When we apply nonlinear programming to engineering positive polynomial—(now call polynomial)—and the objective design problems, the objective function and the constraint function is to be minimized. The equivalent convex functions frequently take the formprogramming problem with decision variables y1,y2,…,ynis Naaai1i2inthen obtained by setting g(X)=∑cP(X)whereP(X)=xxLxiin12ii=1yjx=e,forj=1,2...,njIn such case, the ciand aijtypically represent physical constants, and the xjare design variables. These functions Throughout the original model, so now a convex generally are neither convex nor concave, so the techniques of programming algorithm can be applied. Alternative solution convex programming cannot be applied directly to these procedures also have been developed for solving these geometric programming problems. polynomial programming problems, as well as for geometric programming problems of other types. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo29l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo30l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 13 运筹学Operations Research13-6运筹学运筹学 Types of Nonlinear Programming Types of Nonlinear Programming ProblemsOperations ResearchOperations ResearchFractional ProgrammingWhen it can be done, the most straightforward approach to solving a Suppose that the objective function is in the form of a fractional programming problem is to fraction, ., the ratio of two functions, CX+C0transform it to an equivalent problem of a f(X)=dX+df(X)0standard type for which effective solution 1maxf(X)=procedures already are available. To f(X)2illustrate, suppose that f(X) is of the linear Such fractional programming problems arise, ., when fractional programming formone is maximizing the ratio of output to person-hours expended (productivity), or profit to capital expended (rate of return), or expected value to standard deviation of some Where c and d are row vectors, Xis a column vector, and measure of performance for an investment portfolio c0 and d0 are scalars. Also assume that the constraint (returnr/isk). Some special solution procedure have been functions gi()X are linear, so that the constraints in matrix developed for certain forms of f1()X and f2() are AX≤b and X≥0.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo31l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog32y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Types of Nonlinear Programming Types of Nonlinear Programming ProblemsOperations ResearchOperations ResearchUnder mild additional assumptions, we can transform the The oCmplimentarily problemproblem to an equivalent linear programming problem by hWen we deal with quadratic programming in , you will letting1Xsee one example of how solving certain nonlinear programming andt=y=dX+ddX+dproblems can be reduced to solving the complimentarily problem. 00Given variables w,w,…,wand z,z,…,z, the complementarity12p12pSoproblem is to find a feasible solution for the set of constraints that y=X/t. this result yields: w=F(z),w≥0,z≥0WhiHere, w and z are column vectors, F ch can be solved by the simplex MaxZ=cy+ct0is a given vector-valued function, method. More generally, the same kind of Ay−bt≤0⎧That also satisfies the and the superscript T denotes the transformation can be used to convert a ⎪complementarities constrainttranspose. The problem has no +dt=1⎨fract0ional programming problem with objective function, so technically it ⎪concave f1()X, convex f2()X, and convex Ty≥0,t≥0⎩wz=0is not a full-fledged nonlinear gi(X) to an equivalent convex programming programming problem. It is called complimentarily problem 江西财经大学信息管理学院©2006江西财经大学信b息e管ca理u学se院 o©f2 0th06e complimentary Schoo33l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog34y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Types of Nonlinear Programming Problems13 Nonlinear ProgrammingOperations ResearchOperations ResearchRelationships that eitherAn important special case is the One-aVriable Unconstrained Optimizationlinear complimentarily problem, eW now begin discussing how to solve some of the types of w=0orz=0(orboth)iiwhereproblems just described by considering the simplest case—foreachi=1,2,...,pF(z)=q+Mzunconstrained optimization with just a single variable x(n=1), where the differentiable function f(x) to be maximized is concave. Where q is a given column vector and M is a given p×p Thus the necessary and sufficient condition for particular solution x=x *to be optimal (a global maximum) is matrix. Efficient algorithms have been developed for solving dfthis problem under suitable assumptions about the properties =0,atx=x*dxof the matrix M. one type involves pivoting from one basic sA depicted in Fig. . If this euqation can solved directly for feasible (BF) solution to the next, much like the simplex x*, you are done. oHwever, if f(x) is not a particularly simple function, so the derivative is not just a linear or quadratic function, you may not method for linear able to solve the euqation analytically. If not, the oned-imensional search procedure provides a straightforward way of solving the problem numerically.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo35l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo36l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 13 运筹学Operations Research13-7运筹学运筹学 One-aVriable Unconstrained One-aVriable Unconstrained OptimizationOperations ResearchOperations ResearchThe One-Dimensional Search ProcedureAs long as a reasonable rule is used to select each trial Like other search procedures in nonlinear programming, solution in this way, the resulting sequence of trial solutions the one-dimensional search procedure finds a sequence of trial must converge to x*. In practice, this means continuing the solutionsthat leads toward an optimal solution. At each sequence until the distance between the bounds is sufficiently iteration, you begin at the current trial solution to conduct a small that the next trial solution must be within a prespecifiedsystematic search that culminates by identifying a new error tolerance of x*.improved trial :If the derivative evaluated at a particular value of x is x'c=urrent trial solution,positive, then x* must be larger than this x, so this x becomes a xc=urrent lower bound on x*lower bound on the trial solutions that need to be considered xc=urrent upper bound on x*thereafter. Conversely, if the derivative is negative, then x* must be smaller than this x, so xwould become an uppeεe=rror tolerance for x*.r 江西财经大学信息管理学院©2006bound江西财经大学信息管理学院© of Information TechnologgX37y, JianiUniversity of Finance & Economics©2006School of Information Technology, JiangX38iUniversity of Finance & Economics©2006运筹学运筹学 One-aVriable Unconstrained One-aVriable Unconstrained OptimizationOperations ResearchOperations Researchdf(x)Summary of the One-Dimensional Search procedureIteration:=x'dxInitialization:Select ε. Find an initial x and x by df(x)≥0,resetx=x'inspection (or by respectively finding any value of x at which dxthe derivative is positive and then negative). df(x)≤0,resetx=x'dxSelect an initial trial solutionx+=2x+xx=2Stopping rule :If , so that the new ′must be x−x≤2εxwithin εof x*, stop. Otherwise , perform another iteration.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog39y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog40y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 One-aVriable Unconstrained One-aVriable Unconstrained OptimizationOperations ResearchOperations ResearchExample:Suppose that the function to be maximized isvalues of x, but it is negative for x0< or x2>. Therefore, 46f(x)=12x−3x−2xx0= and x2= can be used as the initial bounds, with their midpoint x1= as the initial trial solution. (The existence of Its first two derivatives are these bounds ensures that a global maximum exists, since this 2df(x)df(x)2435=−12(3x+5x)=12(1−x−x)rules out the only contrary possibilities that the first 2dxdxderivative is either positive everywhere or negative Because the second derivative is nonpositiveeverywhere, everywhere.) Let be the error tolerance for x* in the f(x) is a concave function, so the one-dimensional search stopping rule, so the final procedure can be applied safety to find its global maximum x-x≤ with the final x at the midpoint. Applying the (assuming a global maximum exists). A quick inspection of this one-dimensional search procedure then yields the sequence of function (without even constructing its graph as shown in Fig. results shown in table ) indicates that f(x) is positive for small positive江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo41l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Econom42ics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 13 运筹学Operations Research13-8运筹学运筹学 One-aV13 Nonlinear Programmingriable Unconstrained OptimizationOperations ResearchOperations Researchdf(x)xIteration xeNw′x′f(x) Multivariable Unconstrained Sec. , the value of the ordinary derivative was used ++. select one of just two possible directions (increase x or +. x) in which to move from the current trial solution to +. next one. The goal was to reach a point eventually where stopthis derivative is 0. Now, there are innumerable possibleThe conclusion is that x*≈ in which to move;<x*<江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo43l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo44l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Multivariable Unconstrained Multivariable Unconstrained OptimizationOperations ResearchOperations ResearchThe objective function f(x) is assumed to be differentiable, it possesses a gradient, denoted by∇f(x) , at each point x. In The gradient Search Procedureparticular, the gradient ata specific point is the vector whose elements are the respective partial derivatives ′′evax=xx=xluated aaEch iteration involves changing the current trial solution t , so thatas follows:⎛∂f∂f∂⎞f∇⎜⎟()=,,...,at′fxx=x⎜⎟∂x∂x∂x⎝12n⎠∇f(x)∇f(x*)=0∂f∂x′xj江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo45l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo46l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Multivariable Unconstrained Multivariable Unconstrained OptimizationOperations ResearchOperations ResearchUnder the assumption that the hill f[(x1,x2) ]is concave, ′eRsetx=x+t∗∇f(x)you must then be essentially at the top of the hill.′hfx+t∇fx′Were t*is the positive value of t valuemaximiezs (())The most difficult part of the gradient search procedure usually is to find t*, the value of t that maximizes f in the ′Tha(+∗∇′=′+∇′fxtf(x))fxtfxtmax(()) ist≥0direction of the gradient, at each iter∇ation. Because x and f(x)have fixed values for the maximization, and because f(x) is ′+′f(xt∇f(x))Note that is simply f(x) whereconcave, this problem should be viewed as maximizing a concave functionof a single variable t. ⎛⎞∂f⎜⎟for j1=,2,,…=x+tjjTherefore, it can be solved by the one-dimensional search ⎜⎟∂xj⎝⎠x=x′procedure of Sec. (where the initial lower bound on t must be nonnegative because of the t≥0 constraint). Alternatively, This problem is a twov-ariable problem, where (x,x) 12if f is a simple function, it may be possible to obtain an represents the coordinates (ignoring height) of your current analytical solution by setting the derivative with respect to t location. The function f(x,x) gives the height of the hill at (x,x). equal to zero and 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo47l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo48l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 13 运筹学Operations Research13-9运筹学运筹学 Multivariable Unconstrained Multivariable Unconstrained OptimizationOperations ResearchOperations ResearchStopping∇f(x′)′ rule:Evaluate at x = x . hCeck if Summary of the Gradient Search ProcedurenIitiailaztion:Select εand any initial trial solution ′ . go first to x∂fthe stopping rule.≤εoFr all j,=n1,2,…∂xjtIeration:1. xEpress ′ ′ as a function of t by setting f(x+t∇f(x))If so, stop with the current ′x as the desired approximation of ⎛⎞∂fan optimal solution x.* tOherwise, perform another iteration⎜⎟x=′x+tfor jn,…,2,1=.jj⎜⎟∂xj⎝⎠x=x′Example:Consider the following twov-ariable problemand then substituting these expressions into f(x).22maximizef(x)=2xx+2x−x−2x122122. Use the oned-imensional search procedure (or calculus) to find ∂ft=t* that maximizes ′ + ∇ ′ over t≥0. f(xtf(x))=2x−2x21∂xThus1 ′f(x+t*∇f(x′))3. Reset . Then go to the stopping rule.∂f=2x+2−4x12∂x2江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006SchooSchoo50l of Information Technolog49y, JiangXiUniversity of Finance & Economics©2006l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Multivariable Unconstrained Multivariable Unconstrained OptimizationOperations ResearchOperations ResearchAs Fig. suggests, the gradient search procedure zigzags to the optimal solution rather than moving in a straight line. Some ionx′+∇′t*f(x)Iteratx′x′+′t∇f(x)∇f(x′)f(x′+t∇f(x′))t*modifications of the procedure have been developed that accelerate movement toward the optimum by taking this zigzag 21(0,0)(0,2)(0,2t)4t8-t¼(0,1/2)behavior into (0,1/2)(1,0)(t,1/2)tt-+1/21/2(1/2,12/)If f(x) were not a concave function, the gradient search procedure still would converge to a local maximum. The only x*=(1,1)change in the description of the procedure for this case is thatt* now would correspond to the first local maximum of (,4/3)8/7(,8/7)8/7′+∇′f(xtf(x))at t is increased from 0.(12/,)4/3(,4/3)4/3If the objective were to minimize f(x) instead, one change in the procedure would be to move in the opposite direction of the gradient at each iteration. In other words, the rule for obtaining (0,12/)(12/,12/)the next point would be′=′′xx−t*∇f(x)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo51l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo52l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Multivariable Unconstrained Optimization13 Nonlinear ProgrammingOperations ResearchOperations The Tarush-Kuhn-Tucker (KTK) Conditions only other change is that t* now would be the nonegativevalue of t that minimizesfor Constrained OptimizationWe now focus on the question of how to recognize an ′′f(x−t∇f(x))optimal solution for a nonlinear programming problem (with differentiable functions). What are the necessary and that is(perhaps) sufficient conditions that such a solution must satisfy? ′(t*∇′′′fx−f(x))=minf(x−t∇f(x))In the preceding sections we already noted these conditions t≥0for unconstrained optimiaztion, as summarized in the first two rows of table (X)g(X)≤b⎧.⎨X≥0⎩江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©200653School of Informat54ion Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 13 运筹学Operations Research13-1013. 6 The Tarush-uKhn-Tucker (TKK) oCnditions 13. 6 The Tarush-uKhn-Tucker (TKK) oCnditions 运筹学运筹学Operations ResearchOperations Researchfor Constrained Optimizationfor Constrained OptimizationEarly in we also gave these conditions for the slight Table necessary and sufficient conditions for optimalityextension of unconstrained optimization where the only constraints are nonnegativityconstraints. These conditions are shown in the third row of table sA indicated in the last row Necessary conditions of the table, the conditions for the general case are called theProblemAlso sufficient if:for optimalityKarush-kuhn_Tucker conditions (or KKT conditions), dfbecause they were derived independently by aKrushand by uKhn One-variable =0f(x) concaveand Tucker. Their basic result is embodied in the following ∂ff(x) concave=0Theorem:Assume that f(x),g(x),.…,g(x) are differentiabel1nunconstrained∂xfunctions satisfying certain regularity conditions. Then Constrained, ∂f=0nonegativityconstraints f(x) concaveX*=(x*,x*,....,x*)12n∂xonlycan be an optimal solution for the nonlinear programming problemGeneral constrained aKrush-uKhn-Tucker f(x) concave and g(x) iproblemconditionsconvexonly if there exist mnumbers u, u,,…usuch that all the 12mfollowing TKK conditions are satisfied:江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog55y, JiangXiUniversity of Finance & Economics©2006Schoo56l of Information Technology, JiangXiUniversity of Finance & Economics©200613. 6 The Tarush-uKhn-Tucker (TKK) oCnditions 13. 6 The Tarush-uKhn-Tucker (TKK) oCnditions 运筹学运筹学Operations ResearchOperations Researchfor Constrained Optimizationfor Constrained Optimization∂mf∂gi1.−∑u≤0Consequently, condition 4 can be combined with condition 3 ii=1∂x∂xjjto express them in another euqivalent form asat X=X,*for j1=,2,,…n⎛m⎞∂f∂gi⎜⎟*−∑u=0(3,4)g(X*)−b=0or(≤0ifu=0),fori=1,2,...,mjiiii⎜⎟i=1∂x∂xjj⎝⎠(X*)−b≤0for i1=,2,,…mSimilarly, condition 2 can be combined with condition 1 [g(X*)−b]=0miii∂f∂gfor j1=,2,,…ni(1,2)−∑u=0or(≤0ifx*=0),forj=1,2,...,*≥0i=1j∂x∂xjjfor i1=,2,m…≥0ihWen m0= (no functional constraints), this summation drops out and the combined condition (1,2) reduces to the condition oNte that both conditions 2 and 4reuqire that the product of given in the third row of table . Thus, for m0>, each term in two uqantities be zero. Therefore, each of these conditions really is the summation modifies the m0= condition to incorporate the saying that at least one of the two uqantities must be zero. effect of the corresponding functional constraint. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog57y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog58y, JiangXiUniversity of Finance & Economics©200613. 6 The Tarush-uKhn-Tucker (TKK) oCnditions 13. 6 The Tarush-uKhn-Tucker (TKK) oCnditions 运筹学运筹学Operations ResearchOperations Researchfor Constrained Optimizationfor Constrained OptimizationIn conditions 1,2,4, and 6, the ucorrespond to the dual Corollary:sAsume that f(x) is a concave function and that ivariables of linear programming (we expand on this g(x),g(x),,…g(x) are convex functions (., this problem is a 12mcorrespondence at the end of the section), and they have a convex programming problem), where all these functions satisfy comparable economic interpretation. oHwever, the uactually the regularity conditions. Then (=*Xx*,x*,.…,x)* is an optimal i12narose in the mathematical derivation as aLgrange mtluipilers. solution if and only ifall the conditions of the theorem are oCndition 3and 5do nothing more than ensure the feasibility solution. The other conditions eliminate most of the feasible Example:To illustrate the formulation and application of the solutions as possible candidates for an optimal conditions, we consider the following twov-ariable nonlinear However, note that satisfying these conditions does not programming problem:guarantee that the solution is optimal. As summarized in the maxf(x)=ln(x+1)+xrightmost column of table ,3 certain additional convexity 12Where lndenotes the natural logarithm. Thus m1= (one assumptions are needed to obtain this guarantee. These 2x+x≤3⎧12functional constraint) and .⎨assumptions are spelled out in the following extension of the x≥0,x≥0g(x)2=xx+, so g1(x) is convex. ⎩12112theorem.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of59 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog60y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 13 运筹学Operations Research13-1113. 6 The Tarush-uKhn-Tucker (TKK) oCnditions 13. 6 The Tarush-uKhn-Tucker (TKK) oCnditions 运筹学运筹学Operations ResearchOperations Researchfor Constrained Optimizationfor Constrained OptimizationThe steps in solving the TKK conditions for this particular Furthermore, it can be easily verified that f(x) is concave. example are outlined the corollary applies, so any solution that satisfies the TKK conditions will definitely be an optimal solution. pAplying the ≥1, from condition 1(j=1) x1≥0, from condition 5formulas given in the theorem yields the following KKT ,conditions for this example:−2u<0111(j=1).−2u≤0x+11∂mf∂g1ix+11.−∑u≤01ii=1∂x∂, x0=, from condition 2(j=1)jj112(j=1).x(−2u)=011x+1⎛m⎞1∂f∂≠0 implies that 2x+x=-03, from condition ⎜⎟*−∑u=0ji1(j=2).1−u≤0⎜⎟i=11∂x∂xjj⎝⎠5. Steps 3 and 4 imply that x3=22(j=2).x(1−u)=(X*)−b≤+x−3)≤06. x≠0 implies that u1=, from condition 2(j=2)[g(X*)−b]=(2x+x−3)=01127. oN conditions are violated by x0=, x3,= and u=1 satisfy all *≥≥0,x≥012the conditions. oCnsequently, x*=(0,3) is an optimal solution for this ≥≥0i1problem.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog61y, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity ofFinance & Econom62 ics©200613. 6 The Tarush-uKhn-Tucker (TKK) oCnditions 13. 6 The Tarush-uKhn-Tucker (TKK) oCnditions 运筹学运筹学Operations ResearchOperations Researchfor Constrained Optimizationfor Constrained OptimizationThe particular progression of steps needed to solve the KKT +0≤3(All the other conditions are redundant.)conditions will differ from one problem to the next. hWen the sA listed below, the other three cases where u=0 also give 1logic is not apparent, it is sometimes helpful to consider separately immediate contradictions in a similar way, so no solution is the different cases where each xand uare specified to be either to or greater than 0 and then trying each case until one leads to a solution. In the example, there are eight such cases aCse x=0, x>0, u=0 contradicts conditions 1(j=1), 1(j=2), 121corresponding to the eight combinations of x=0 versus x>0, x0= and 2(j=2).112versus x0> and u=0 versus u>0 Each case leads to a simpler 211aCse x>0, x=0, u=0 contradicts conditions 1(j=1), 2(j=1), 121statement and analysis of the conditions. To illustrate, consider and 1(j=2).first the case shown next, where x=0, x0=, and u= x>0, x>0, u=0 contradicts conditions 1(j=1), 1(j=1), 121TKK conditions for the case x=0,x0=,u0=1211(j=2)and 2(j=2).11(j=1).≤0contradictionThe case x>0, x0>, u0> enables one to delete these 0+1121nonzero multipliers from conditions 2(j=1),2(j=2), and (j=2).1−0≤0contradiction江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©200663School of64 Information Technology, JiangXiUniversity of Finance & Economics©200613. 6 The Tarush-uKhn-Tucker (TKK) oCnditions 13. 6 The Tarush-uKhn-Tucker (TKK) oCnditions 运筹学运筹学Operations ResearchOperations Researchfor Constrained Optimizationfor Constrained Optimization11(j=1).−2u=0TKK conditions for the case 1x+11Therefore, x=0,x3,=u=1. Having found a solution, we know 121x0>,x>0,u>01212(j=2).1−u=01that no additional cases need be +x3-0=2There also many valuable indirect appilcationsof the TKK (All the other conditions are redundant.)1conditions. nOe of these applications arises in the duailt ytheory1(j=1).−2u=010+1Therefore, u=1, so x1=-2/, which 11that has been developed for nonlinear programming to parallel the 2(j=2).1−u=01contradicts x>+x−3=0duality theory for linear programming presented in chapter 6. In2Now suppose that the case x=0, 1particular, for any given constrained maximization problem, the x0>,u0> is tired conditions for the case x=0, KKT conditions can be used to define a closely1x0>,u>021江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog65y, JiangXiUniversity of Finance & Economics©2006Schoo66l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 13 运筹学Operations Research13-1213. 6 The Tarush-uKhn-Tucker (TKK) oCnditions 运筹学运筹学13 Nonlinear ProgrammingOperations ResearchOperations Researchfor Constrained OptimizationAssociated dual problem that is a constrained minimization problem. The variables in the dual problem consist of both the Quadratic Programminglagrangemultipliers ui(I1=,2,,…m) and the primal variables xj(j=1,2,,…n). In the special case where the primal problem is a linear As indicated in , the quadratic programming problem programming problem of linear programming. hWen the primal differs from the linear programming problem only in that the problem is a convex programming problem. It is possible to establish 2objective function also includes xand xxterms. Thus, if we use jijrelationships between the primal problem and the dual problem that matrix notation like that introduced at the beginning of , are similar to those for linear programming. the problem is to find x so as toFor example, the strong dauilt ypropertyof sec. , which 1Tstates that the optimal objective function values of the two problems maxf(x)=CX−XQX2are equal, also holds here. Furthermore, the values of the uivariables AX≤b⎧in an optimal solution for the dual problem can again be interpreted .⎨X≥0as shadow price ;., they give the rate at which the optimal ⎩objective function value for the primal problem could be increased by increasing the righth-and side of the corresponding constraint.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of67 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog68y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Quadratic Quadratic ProgrammingOperations ResearchOperations ResearchTo illustrate this notation, 22minf(x,x)=15x+30x+4xx−2x−4x12121212Before describing the algorithm, we shall develop this consider the following ⎧x+2x≤3012convenient of a quadratic .⎨x≥0,x≥0⎩12programming problemThe TKK oCnditions for uQadratic 1(j=1).15+4x−4x−u≤0211Programming2(j=1).x(15+4x−4x−u)=01211Several algorithms have been developed for the special case 1(j=2).30+4x−8x−2u≤0121Consider the above example whose of the quadratic programming problem where the objective 2(j=2).x(30+4x−8x−2u)=02121KKT conditions are the is a concave function. eW shall describe one of these +2x−30≤012algorithms, the modified simplex method, that has been quite (x+2x−30)=0112popular because it reuqires using only the simplex method with ≥0,x≥012slight modification. ≥01江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog69y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog70y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Quadratic Quadratic ProgrammingOperations ResearchOperations ResearchFor each of these pairs(—x1,y1),(x2,y2),(u1,v1)t—he two eW move the constants in variables are called complementary variables, because only one of conditions 1(j=1), 1(j=2), and 3 to 1(j=1).4x−4x−u+y=−15the two variables can be nonzero. Since all six variables are 2111the righth-and side and then 1required to be nonnegative, these new forms of conditions 2(j=1), (j=2).4x−8x−2u+y=−301212introduce nonnegative slack 2(j=2), and 4 can be combined into one +2x+v=30variables to conve121rt these inequalities to +xy+uv=0112211aClled the complimentary that condition 2(j1=) can now be expressed as simply fAter multiplying through the euqations for conditions 1(j1=) requiring that either x10= or y1=0 ;that is, 2(j=1). x1y1=0and (j=2) by –1 to obtain nonnegative righth-and sides, we now In just the same way, conditions 2(j=2) and 4 can be replaced have the desired convenient form for the entire set of conditions by 2(j=2). x2y2=0 4. u1v10=shown here :江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo71l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo72l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 13 运筹学Operations Research13-13运筹学运筹学 Quadratic Quadratic ProgrammingOperations ResearchOperations Research4x−4x−u+y=−152111This form is particularly convenient because, except 4x−8x−2u+y=−301212hWere the elements of the column vector u are the uiof the for the complementarityx+2x+v=30121constraint, these conditions preceding section and the elements of the column vectors Yand are linear prorgammin gx≥0,x≥0,u≥0,y≥0,y≥0,v≥0121121constraints. Vare slack +xy+uv=0112211 Xis optimal if and only if there exist values of ,Y,U and V TTQX+AU−Y=Csuch that all four vectors together satisfy all these conditions. The In matrix notation, this Ax+v=boriginal problem is thereby reduced to the equivalent problem ofgeneral form isTTXY+UV=0finding a feasible solution to these constraints. X≥0,U≥0,Y≥0,V≥0江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo73l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo74l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Quadratic Quadratic ProgrammingOperations ResearchOperations ResearchThe Modified Simplex MethodsA we discussed in , finding such an initial FBsolution is relatively straightforward. In the simple case where TC≤0 and The modified simplex methodexploits the key fact that, with b≥0, the initial basic variables are the elements of Yand Vso that the exception of the complementarityconstraint, the TKK the desired solution is X=0,U=0,TCY=-,V=b. Otherwise, you need to revise the problem by introducing an artificial variable intoeach conditions in the convenient form obtained above are nothing of the equations where cj>0 or bj<0, in order to use these artificial more than linear programming constraints. Furthermore, the variables as initial basic variables for the revised problem. complementarityconstraint simply implies that it is not Next, use phase 1 of the twop-hase method to find a BF permissible forboth complementary variables of any pair to be solution for the real problem ;.,apply the simplex method to the basic variables when FBsolutionsare considered. Therefore, the following linear programming problemproblem reduces to finding an initial FBsolution to any linear Subject to the linear programming programming problem that has these constraints, subject to this minZ=∑zconstraints obtained from the TKK conditions, but jjadditional restriction on the identity of the basic variables. with these artificial variables included.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo75l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog76y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Quadratic Quadratic ProgrammingOperations ResearchOperations ResearchExample:eW shall now illustrate this approach on the Restrictede-ntry Rule :when you are choosing an entering example given at the beginning of the variable exclude from consideration any nonbasicvariable whose complementary variable already is a basic variable ;the The starting point for choice should be made from the other nonbasicvariables solving this example is its according to the usual criterion for the simplex =z+z12KKT conditions in the This rule keeps the complementarityconstraint satisfied convenient form obtained ⎧4x−4x+u−y+z=1512111⎪throughout the course of the algorithm. hWen an optimal solutionearlier in the section. −4x+8x+2u−y+z=3012122⎪⎪fAter the needed artificial x+2x+v=30⎨121X*,U*,Y*,V*,Z10=,…,Zn0=variables are introduced, ⎪x≥0,x≥0,u≥0,y≥0,y≥0,v≥0121121⎪is obtained for the phase 1 problem, X* is the desired optimal the linear programming ⎪xy+xy+uv=0⎩112211solution for the original quadratic programming problem. Phase 2problem to be addressed of the twop-hase method is not by the modified simplex method then is江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of77 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog78y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 13 运筹学Operations Research13-14运筹学运筹学13 Nonlinear Separable ProgrammingOperations ResearchOperations ResearchThus only the objective function can be nf(X)=∑f(x)expressed as a sum of concanefunctions of 13. 8 Separable Programmingjjj=1individual variablesSo that each fj(xj) has shape such as the one shown in fig. As indicated in section in separable programming it is 5over the feasible range of values of xj. Because f(X) assumed that the objective function f(x) is concave, that each of represents the measure of performance for all the activities the constraint functions gi(x) is convex, and that all these together, fj(xj) represents the contribution to profit from activity j when it is conducted at level xj. The condition of f()X being functions are separable functions. However, to simplify the separable simply implies additivity ;, there are no interactions discussion, we focus here on the special case where the convex between the activities that affect total profit beyond their independent contributions. The assumption that each fj(xj) is and separable gi(x) are, in fact, linear functions, just as for linear concave says that the marginal profitability either stays the same programming. or decrease as xjis increased.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo79l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Economics©200680运筹学运筹学 Separable Separable ProgrammingOperations ResearchOperations ResearchThis approximation is very convenient because a piecewise This kinds of situations can lead either type of profit curve linear function of a single variable can be rewritten as a linear function of several variables, with one special restriction on the shown in . In case 1, the slope decrease only at certain value of these variables, as described , so that fj(xj) is a pecewiselinear function. oFr case 2, Reformulation as a Linear Programming Problemthe slope may decrease continuously as xjincreases, so that fj(xj) The key to rewriting a piecewise linear function as a linear function is to use separate variable for each line segment. To is a general concave function. Any such function can be illustrate, consider the piecewise linear function fj(xj) shown in ,5 case 1, which has three line segments over the feasible approximated as closely as desired by a piecewise linear functions, range of values of xj. Introduce the three new variables x, x, and j1j2xand setand this kind of approximation isj3 used as needed for separable x=x+x+xjj1j2j3progrwhere0≤x≤u,0≤x≤u,0≤x≤uamming 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog81y, JiangXiUniversity of Finance & Economics©2006Schoo82l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Separable Separable ProgrammingOperations ResearchOperations ResearchThen use the slopes sj1, sj2, and sj3 to rewrite fj(xj) asand so on, whes>s>srej1j2j3With the special restriction oHwever, the special restriction permits only the first f(x)=sx+sx+sxthatjjj1j1j2j2j3j3possibility, which is the only one giving the correct value forxj20= whenever xj1u<j1xj30= whenever xj2u<j2fj(1).nUfortunately, the special restriction dies not fit into the To see why this special restriction is required, suppose that required format for linear programming constraints, so some xj=1, where ujk1> (k=1,2,3), so thatfi(1)=sj1. Note that piecewise linear functions cannot be rewritten in a linear x+x+x=1j1j2j3programming format. However, our fj(xj) are assumed to be concave, so sj1>sj2,>… so that an algorithm for maximizing f()X x=1,x=0,x=0⇒f(1)=sautomatically gives the highest priority to using xj1 when pejjjjjrm1231itsincreasing xjfrom zero, the next highest priority to using xj2, and x=0,x=1,x=0⇒f(1)=sj1j2j3jj2so on, without even including the special restriction explicitlyin x=0,x=0,x=1⇒f(1)=sj1j2j3jj3the model. This observation leads to the following key property.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom83ics©2006School of Information Technolog84y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 13 运筹学Operations Research13-15运筹学运筹学 Separable Separable ProgrammingOperations ResearchOperations ResearchnKey Property of Separable ProgrammingjoWuld be substituted throughout the f(x)=∑sxjjjkjkoriginal model andhWen f(X) and the gi()X satisfy the assumptions of separable k=1programming, and when the resulting piecewise linear functions oWuld be substituted into the objective function for are rewritten as linear functions, deleting the special restriction j=1,2,,…n. the resulting model isgives a linear programming model whose optimal solution nautomatically satisfies the special =∑∑sxjkjkj=1k=1eW shall elaborate further on the logic behind this key nproperty later in this section in the context of a specific ⎧⎛⎞∑a⎜∑x⎟≤b,fori=1,2,...,m⎪ijjkij=1k=1To write down the complete linear programming model in the ⎝⎠⎪⎪above notation, let nbe the number of line segments in fj(xj), so ≤u,fork=1,2,...,n,j=1,2,...,nj⎨jkjkjthat n⎪jx≥0,fork=1,2,...,n,j=1,2,...,njkjx=∑x⎪jjkk=1⎪⎩江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog85y, JiangXiUniversity of Finance & Economics©2006Schoo86l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Separable Programming13 Nonlinear ProgrammingOperations ResearchOperations ResearchIf some original variable xjhas no upper bound, then u=∞, Convex programmingso the constraint involving this quantity will be efficient way of solving this model is to use the eW already have discussed some special cases of convex streamlined version of the simplex method for dealing with upperprogramming in , and (unconstrained problems), bound constraints. After obtaining an optimal solution for this (quadratic objective function with linear constraints), andmodel, you then would (separable functions). You also have seen some theory for the njgeneral case (necessary and sufficient conditions for optimality) in x=∑xsec . In this section, we briefly discuss some types of jjkk=1approaches used to solve the general convex programming problem w[here the objective function f(x) to be maximized is oFr j1=,2,,…n in order to identify an optimal solution for the concave and the g(x) constraint functions are convex,] and then ioriginal separable programming present one example of an algorithm for convex programming.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo87l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo88l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 oCnvex oCnvex programmingOperations ResearchOperations ResearchAs one example of a sequential-approximation algorithm, we present there the Franko-Wlfe algorithmfor the case of linearly The first category is gradient algorithms, where the gradient constrained convex programming. This procedure is particularly search procedure of sec. is modified in some way to keep the straightforward ;it combines linear approximations of the objective function with the oned-imensional search procedure of sec path from penetrating any constraint boundary. For A Sequential Linear Approximation Algorithm( Franko-Wlfe)example, one popular gradient method is the generalized reduced Given a feasible trial solution , the linear approximationused ′xgradient (GRG) the objective function f(x) is the firsto-rder Taylor series expansion of f(x) around x= ,′x namely,The second category---sequential unconstrained algorithms--nincludes penalty function and barrier function methods. ∂′f(X)f(X)≈′f(X)+(x−x′)=′f(X)+∇′f(X)(X−′X)∑jj∂xj=1j江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog89y, JiangXiUniversity of Finance & Economics©2006Schoo09l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 13 运筹学Operations Research13-16运筹学运筹学 oCnvex oCnvex programmingOperations ResearchOperations ResearchSummary of the rFankoW-lfe lAgorithmwe can dropped to given an equivalent linear objective function(0)nIitiailaztion:find a feasible initial trial solution X, for nexample, by applying linear programming procedures to find an g(X)=∇′f(X)X=cx∑jjinitial FBsolution. Set k=1,j=1tIeration∂f(X)∂f(X)(k−1)wherec=atX=′Xc=atX=Xj1. For jj=1,2,,…n, evaluate∂x∂xjnjmaxg(X)=cx∑2. Find an optimal solution for the following jjj=1linear programming ≤b⎧.⎨3. oFr the variable t (0≤t≤1) setX≥0⎩(k)(k−1)(k−1)h(t)=f(X)forX=X+t(X−X)Lp江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo19l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo29l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 oCnvex oCnvex programmingOperations ResearchOperations ResearchExample:we can dropped to given an equivalent linear objective function22maxf(X)=5x−x+8x−2xn∂f∂f1122∂f(X)=5−2x,=8−4xwherec=atX=′Xg(X)=∇′f(X)X=cx12∑jjj3x+2x≤6∂x⎧∂x∂x12jj=.⎨x≥0,x≥0⎩12sUe some procedure such as the oned-imensional search (k)So that the unconstrained maximum X=(25/,2) violates the procedure to maximize h(t) over 0≤t≤1 , and set Xequal to the functional constraint. Thus more work is needed to find the corresponding X. go to the stopping maximum.(k1-)(k)Stopping reul:if Xand Xare sufficiently close, stop and (o)(1)(k1-)(k)eBcause (=X0,0) is clearly feasible, let us choose it as the use X(k) (or some extrapolation of X,X,…,X,X) as your initial trial solution (X0) for the rFanko-Wlfe algorithm. estimate of an optimal solution. Otherwise, reset k=k+1 and Plugging x1=0 and x2=0 into the expressions for the partial perform another gives c1 =5and c2=8, so that g(x)=5x1+8x2 is the oNw let us illustrate this linear approximation of the objective function. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of39 Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo49l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 oCnvex oCnvex programmingOperations ResearchOperations ResearchGraphically, solving this linear programming problem yieldsNote that X(1)=(0,2) gives maxg(X)=5x1XLp(1)=(0,3). For step 3 of the first iteration, the points on the (2)3x+2x≤6c125-=(0)5=⎧X=(2,0)Lp12line segment between (0,0) and (0,3) are expressed by .⎨x≥0,x≥0⎩12c248(-=2)0=(x,x)=(0,0)+t[(0,3)−(0,0)]=(0,3t)for0≤t≤112X=(0,2)+t[(2,0)−(0,2)]=(2t,2−2t)22h(t)=f(0,3t)=8(3t)−2(3t)=24t−18tdh(t)So thath(t)=f(2t,2−2t)=24−dh(t)36t=0=10−24tdt22=5(2t)−(2t)+8(2−2t)−2(2−2t)dtSo that t*2=.3/ This result yields the next trial solution2=8+10t−12t2(1)X=(0,0)+[(0,3)−(0,0)]=(0,2)3557(2)Yields t*=15/2. Hence X=(0,2)+[(2,0)−(0,2)]=(,)h1266Wich completes the first iteration.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo59l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo69l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 13 运筹学Operations Research13-17运筹学 oCnvex programmingOperations ResearchoYu can see in fig how the trial solutions keep alternating between two trajectories that appear to intersect atapproximately the point X=(1,23)/. This point is, in fact, the optimal solution, as can be verified by applying the KKT conditions from sec example illustrates a common feature of the rFank-oWlfe algorithm, namely, that the trial solutions alternate between two trajectories. hWen they alternate in this way, we can extrapolate the trajectories to their approximate point of intersection to estimate an optimal solution. This estimate tends to be better than using the last trial solution generated. The reason is that the trial solutions tend to converge rather slowly toward an optimal solution, so the last trial solution may still be quite far from optimal.江西财经大学信息管理学院©2006Schoo79l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 15 运筹学Operations Research15-1运筹学运筹学15 Queuing TheoryOperations ResearchOperations Prototype Example15 Queuing TheoryThe emergency room of county hospital provides quick medical care for emergency cases brought to the hospital by ¾Prototype Exampleambulance or private automobile. At any hour there is always one doctor on duty in the emergency room. However, because ¾Basic Structure of Queuing Modelsof a growing tendency for emergency cases to use these ¾The role of the Exponential Distributionfacilities rather than go to a private physician, the hospital has been experiencing a continuing increase in the number if ¾The birth-and-death Processemergency room visits each year. As a result, it has become ¾QueueingModel Based on the Birth-and-quite common for patients arriving during peak usage hours Death Process(the early evening) to have to wait until it is their turn to betreated by the doctor. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog1y, JiangXiUniversity of Finance & Economics©2006Schoo& Econom2l of Information Technology, JiangXiUniversity of Finance ics©2006运筹学运筹学 Prototype Example15 Queuing TheoryOperations ResearchOperations Basic Structure of Queuing ModelsTherefore, a proposal has been made that a second doctor should be assigned to the emergency room during these hours, The basic Queuing Processso that two emergency cases can be treated simultaneously. The basic process assumed by most queuing models is the The hospital’s management engineer has been assigned to following. Customersrequiring service are generated over study this by an input source. These customers enter the queuing The management engineer began by gathering the systemrelevant historical data and then projecting these data into And join a queue. At certain times, member of the queue is the next year. Recognizing that the emergency room is a selected for service by some rule known as the queue discipline. queuing system, she applied several alternative queuing The required service is then performed for the customer by theory models to predict the waiting characteristics of the the service mechanism, after which the customer leaves the system with one doctor and with two doctors, as you will see queuing system. This process is depicted in the latter sections of this chapter.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUn3iversity of Finance & Economics©2006Schoo4l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Basic Structure of Queuing Basic Structure of Queuing ModelsOperations ResearchOperations ResearchThe basic queuing processInput Source (Calling Population)uQeueingsystemOne characteristic of the input source is its size. Thesizeis the total number of customer that might require service from Served Service customersInput Queutime to time, ., the total number of distinct potential customersmechanissourcecustomers. This population from which arrivals come is m ereferred to as the calling population. The size may be assumed to be either infinite or finite(so that the input source also is said to be either unlimited or limited). Many alternative assumptions can be made about the various elements of the queuing process; they are discussed next.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School ofo5 Infrmation Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog6y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 15 运筹学Operations Research15-2运筹学运筹学 Basic Structure of Queuing Basic Structure of Queuing ModelsOperations ResearchOperations ResearchThe statistical pattern by which customers are generated Queueover time must also be specified. The common assumption is that they are generated according to a Poisson process; ., A queue is characterized by the maximum permissible the number of customers generated until any specific time has a Poisson distribution. As we discuss in , this case is number of customers that it can contain. Queues are called the one where arrivals to the queuing system occur randomly infinite or finite，according to whether this number is infinite but at a certain fixed mean rate, regardless of how many of finite. The assumption of an infinite queueis the standard customers already are there (so thesizeof the input source is one for most queuing models, even for situations where there infinite).actually is a (relatively large) finite upper bound on the An equivalent assumption is that the probability permissible number of customers, because dealing with such distribution of the time between consecutive arrivals is an an upper bound would be a complicating factor in the analysis. exponential distribution. The time between consecutive However, for queueingsystems where this upper bound is arrivals is referred to as the interarrivaltime. Any unusual assumptions about the behavior of the customers must also be small enough that it actually would be reached with some specified. One example is balking，where the customer frequency, it becomes necessary to assume a finite to enter the system and is lost if the queue is too long. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo7l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo8l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Basic Structure of Queuing Basic Structure of Queuing ModelsOperations ResearchOperations ResearchQueue DService MechanismisciplineThe service mechanism consists of one or more service The queue discipline refers to the order in which facilities, each of which contains one or more parallel members of the queue are selected for service. For service Channels, called servers. If there is more than one example, it may be first-come-first-served, random, service facility, the customer may receive service from a according to some priority procedure, or some other. sequence of these (service channels in series). At a given facility , the customer enters one of the parallel service First-come-first-served usually is assumed by queuing channels and is completely serviced by that server. models, unless it is stated otherwise.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of9 Information TechnologSchoo10y, JiangXiUniversity of Finance & Economics©2006l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Basic Structure of Queuing Basic Structure of Queuing ModelsOperations ResearchOperations ResearchServed An Elementary Queuing ProcesscustomersAsQueuing system we have already suggested, queuing theory has been applied to many different types of waiting-line situations. SHowever, the most prevalent type of situation is the following: uQeue CuCstomersA single waiting line (which may be empty at times) forms in ServicCS C C C C CCe the front of a single service facility, within which are stationed CSfacilitone or more severs. CySEach customer generated by an input source is serviced by one of the serves, perhaps after some waiting in the queue Served (waiting line). The queuing system involved is depicted in .江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog11y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog12y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 15 运筹学Operations Research15-3运筹学运筹学 Basic Structure of Queuing Basic Structure of Queuing ModelsOperations ResearchOperations ResearchIt is not necessary for three actually to be a physical waiting Notice that the queuing process in the illustrative line forming in front of a physical structure that constitutes the example of sec. is of this type. The input source service facility; ., the members of the queue may be scattered generates customers in the form of emergency cases requiring medical care. The emergency roomthroughout an area, waiting for a server to come to them, ., is the service facility, and the doctors are the waiting to be repaired. The server or group of servers assigned to a given area constitutes the service facility for that For example, they may be items waiting for a area. Queuing theory still gives the average number waiting, thecertain operation by a given type of machine, or they average waiting time, and so on, because it is irrelevant whether may be cars waiting in front of a customers wait together in a group. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo13l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog14y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Basic Structure of Queuing Basic Structure of Queuing ModelsOperations ResearchOperations ResearchExcept for , all the queuingmodels discussed in this Terminology and Notationchapter are of the elementary type depicted in Fig. . Many Unless otherwise noted, the following standard terminology of these models further assume that all interarrivaltimesare and notation will be and identically distributed and that all service State of system=number of customers waiting for servicetimesare independent and identically distributed. Such models =state of system minus number of customers being served conventionally are labeled as follows: iDstribution of service N(t)=number of customers in queuing system at times t (t≥0)timesPn(t)=probability of exactly n customers in queuing system at Number of serverstime t, given number at time e=Mxponential iDstribution of interarrivals=number of servers (parallel service channels) in queuing =Degenerate distribution λn=mean arrival rate (expected number of arrivals per unit E=Erlangdistributionktime) of new customers when n customers are in =eneral distribution江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog15y, JiangXiUniversity of Finance & Economics©2006Schoo16l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Basic Structure of Queuing Basic Structure of Queuing ModelsOperations ResearchOperations Researchμ=mean service rate (expected number of customers nCertain notation also is required to describe steady-state completing service per unit time) when n customers are in results. When a queuing system has recently begun operation, system. Note: represents combined rate at which all busy the state of the system will be greatly affected by the initial state servers (those serving customers) achieve service by the time that has since elapsed. The system is said to bein a transient condition. However, after sufficient time has When λn is a constant for all n, this constant is denoted by elapsed, the state of the system becomes essentially independentλ. When the mean service rate per busy server is a constant of the initial state and the elapsed time. The system has now for all n≥1, this constant is denoted by μ.(in this case, μ=sμnessentially reached a steady-state condition, where the when n≥s, that is, when all s servers are busy.) Under these probabiltydistribution of the state of the system remains the circumstances, 1/λand 1/ μare the expected interarrivaltime same (the steady-stateor stationary distribution) over time. and the expected service time, respectively. Also, ρ= λ/(sμ) is Queueingtheory has tended to focus largely on the steady-state the utilization factor for the service facility, .,the expected condition, partially because the transient case is more difficult fraction of time the indiviualservers are busy, because λ/(sμ) analytically. The following notation assumes that the system is represents the fraction of the system’s service capacity (sμ) in steady-state condition:that is being utilized on the average by arriving customers(λ). 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo17l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog18y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 15 运筹学Operations Research15-4运筹学运筹学 Basic Structure of Queuing Basic Structure of Queuing ModelsOperations ResearchOperations ResearchP=probability of exactly n customers in queuing (Because provided the first rigorous proof, this L=expected number of customers in queuing sometimes is referred to as Little ’s formula.) L=expected queue length (excludes customers being served).pFurthermore , the same proof also shows that W=’waiting time in system (includes service time) for each individual =λWqq=wWaiting time in queue (excludes service time) for each individual customer.•If the λ are not equal ,then λ can be replaced in these n= WE( W’)equations by ,the averagearrival rate over the long run . λqW=waiting time in uqeue (excludes service time) for each (We shall show later how can be determined for some basicλindividual .)q=WE(q)W•Now assume th1μat the mean service time is a constant , Relationship between L W Lpand Wqfor all n>=1. It then follows thatAssumthat is a constant for all n. It has been proved that in λλna steadys-tate queuing process,1W=Wq+μL=λW江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog19y, JiangXiUniversity of Finance & Economics©2006Schoo20l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学15 Queuing Examples of Real Queuing SystemsOperations ResearchOperations ResearchAnother important class is transportation service Examples of Real Queuing some of these systems the vehicle are the customers ,such as cars waiting at a tollbooth or traffic light (the Our description of queuing systems in the preceding section may server) ,a truck or a ship waiting to be loaded or unloaded by aappear relatively abstract and applicable to only rather specialcrew(the server),and airplanes waiting to land or take of from apractical situations .On the contrary , queuing systems are surprising prevalent in a wide variety of contexts. To broaden your horizons on runway (the server ),(An unusual example of this kind is a the applicability of queuing theory ,we shall briefly mention various parking lot ,where the cars are the customers and the parking examples of real queuing are the servers ,but there is no queue because arriving One important class of queuing systems that we shall encounter customers go trucks ,and elevators ,are the servers. In recent in our daily lives is commercial service systems,where outside years, queuing theory probably has been applied most to business customers receive service from commercial organizations .Many ofthese involve person-to-person service at a fixed location ,such as a industrial internal service systems,where the customers receiving barber shop (the barbers are the servers) ,bank teller service service are internal to the organization .,checkout stands at a grocery store ,and a cafeteria in (servicechannels in series) .江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo21l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo2l of Information Technolog2y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Examples of Real Queuing Systems15 Queuing TheoryOperations ResearchOperations The role of the Exponential DistributionThere is now growing recognition that queuing theory The operating characteristics of queuingsystems are largely by two statistical properties, namely, the probability also is applicable to social service example ,a distribution of interarrivaltimes(see “Input Source”in judges (or panels of judges )are servers ,and the network )and the probability distribution of service times(see “Service Mechanism ”in ).For real queuing systems, ,where the customer are the bills waiting to be processed these distributions can take on almost any form .(The only .Various health-care hospital emergency room) ,but you restriction is that negative values cannot occur.) However ,to formulate a queuingtheory modelas a representation of the can also view ambulances ,x-ray machines ,and hospital real system. It is necessary to specify the assumed form of eachbeds as servers in their own queuing of these distributions. To be useful ,the assumed form should be,families waiting for low-and moderate-income housing sufficiently realisticthat the model provides reasonable predictionswhile, at the same time, being sufficiently simple,or other social services,can be viewed as customers in a that the model is mathematically tractable. Based on the theory queuing system .these considerations, the most important probability distribution in queuingtheory is the exponential distribution.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog23y, JiangXiUniversity of Finance & Economics©2006Schoo24l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 15 运筹学Operations Research15-5运筹学运筹学 The role of the Exponential The role of the Exponential DistributionOperations ResearchOperations ResearchSuppose that a random variable Trepresents either ft()Property 1: T is a strictly decreasingfunction of t(t≥0)。interarrivalor service times .(We shall refer to the occurrences One consequence of Property 1 is thatmarking the end of these times-arrivals or service completion-as events. )This random variable is said to have an exponential P{0≤T≤Δt}>P{t≤T≤t+Δt}distribution with parametera if its probability density function isfor any strictly positive values of △tand t .[This consequence −at⎧aefort≥0,follows from the fact that these probabilities are the area under f(t)=⎨T0fort<0,⎩the curve over the indicated interval of length △t,and the average height of the curve is less for the second probability and the expected value and as shown in this case , than for the first.] Therefore ,it is not only possible but alsovariance of Tare , respectively,the cumulative probabilities arerelatively likely that T will take on a small value near zero .In 1ET=()fact,−ataP{T≤t}=1−e111P{0≤T≤}=−atvarT=()2aP{T>t}=e(t≥0)2a江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog25y, JiangXiUniversity of Finance & Economics©2006Schoo26l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The role of the Exponential The role of the Exponential DistributionOperations ResearchOperations ResearchIf represents interarrivaltimes ,Property 1 rules out Whereassituations where potential customers approaching the queuing 1131P{≤T≤}= tend to postpone their entry if they see another customer2a2aentering ahead of them .so that the value T takes on is more likely to be”On the other hand ,it is entirely consistent with the common small”[ . ,less than half of E(T)] than “near ”its expected phenomenon of arrivals occurring “randomly ,”described by value [ . ,no further away than half of E(T)] ,even though subsequent properties .Thus ,when arrival times are plotted on the second interval is twice as wide as he time line ,they sometimes have the appearance of being Is this really a reasonable property for T in a queuing clustered with occasional large gaps separating clusters ,because model ?If T represents service times ,the answer depends of the substantial probability of small interarrivalstimes and upon the general nature of the service involved ,as discussed small probability of large interarrivaltimes ,but an irregular is part of true randomness.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog27Schoo28y, JiangXiUniversity of Finance & Economics©2006l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The role of the Exponential The role of the Exponential DistributionOperations ResearchOperations ResearchP{T>Δt,T>t+Δt}Property 2:lack of moneyP{T>t+Δt|T>Δt}=.P{T>Δt}This property can be stated mathematically asP{T>t+Δt}=P{T>Δt}−a(t+Δt)P{T>t+Δt|T>Δt}=P{T>t}e=−aΔte−atfor any positive quantities tand △tIn other words ,the =eprobability distribution of the remaining time until the event For interarrivaltimes,this property describes the common (arrival or service completion )occurs always is the situation where the time until the next arrival is completely same ,regardless of how much time (△t) already has uninfluenced by when the last arrival occurred .For service passed .In effect ,the process ”forgets ”its history .This times ,the property is more difficult to interpret .We should not surprising phenomenon occurs with the exponential expect it to hold in a situation share the server must perform the distribution becausesame fixed sequence of the operations for each customer ,because then a long elapsed service should imply that probably little remains to be done .江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo29l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo30l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 15 运筹学Operations Research15-6运筹学运筹学 The role of the Exponential The role of the Exponential DistributionOperations ResearchOperations ResearchHowThus ,if Ti represents the time until a particular kind of ever ,in these case ,if considerable service has already event occurs ,then U represents the time until the first of the n elapsed for a customer ,the only implication may be that this different events occurs . Now note that for any t>=0,particular customer requires more extensive service than 3:The minimumof the several independent P{U>t}=P{T>t,T>t,L,T>t}12nexponential random variables has an exponential distribution.=P{T>t}P{T>t}LP{T>t}12n−at−atat12nTo state this property mathematically ,let T1 T2 ....Tnbe =eeLenindependentexponential random variable with parameters =exp(−at)∑irespectively. Also let U be the random variable that takes on i=1the value equal to the minimumof the values actually taken on so that U indeed has an exponential distribution with parameterby T1 T2 ....Tn;that is,na=a∑iU=min{TTL,T}1,2,ni=1江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo31l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog32y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The role of the Exponential The role of the Exponential DistributionOperations ResearchOperations ResearchProperty 4: Relationship to the Poisson property has some implications for interarrivaltimes in queuing models .In particular ,suppose that there are severalSuppose that the timebetween consecutive occurrences of some particular kind of event (. ,arrivals or service (n) differenttypes of the customers ,but the interarrivaltimes completions by a continuously busy server) has an exponential for each type (type i) have an exponential distribution with distribution with parameter α. parameter α(i=1,2,…,n).i Property 4 then has to do with the resulting implication By property 2,the remaining time from any specified about the probability distribution of the numberof times this instant until the next arrival of a customer of type i has this kind of event occurs over a specified time .In particular ,let X(t) same distribution .Therefore ,let Tbe this remaining time , ibe the number of occurring by time t(t≥0),where time 0 measured from the instant a customer of any type arrives .designates the instant at which the count begins .The Property 3 then tells us that U ,the interarrivaltimes for implication is thatqueuing system as a whole ,has an exponential distribution with parameter αdefined by the last equation .As a result ,you n−at(at)ecan choose to ignore the distinction between customers and sP{X(t)=n}=forn=0,1,2,K;tilln!have the exponential interarrivaltimes for the queuing model.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo33l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Ino34frmation Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The role of the Exponential The role of the Exponential DistributionOperations ResearchOperations Researchthat is,X(t) has a Poisson distribution with parameter αt. For example ,with n=0,This property provides useful information about service −atcompletionswhen service-time have an exponential distribution P{X(t)=0}=e,with parameter µ. We obtain this information by in defining which is just the probability from the exponential distribution thaX(t) as the number of the service completions achieved by a t the first event occurs after t ,The mean of the Poisson distribution is continuously busy server in elapsed time t ,where α=µ. For E{X(t)}=at,multiple-serverqueuing models,X(t) can also be defined as the so that the expected number of events per unit time is α. number of the service completions achieved by n continuously Thus αis said to be the mean rate at which the events occur. When the events are continuing basis ,the counting process busy servers in elapsed time t ,where α=nµ.{X(t) t≥0} is said to be a Poisson processwith parameter α(the mean rate).江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo35l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog36y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 15 运筹学Operations Research15-7运筹学运筹学 The role of the Exponential The role of the Exponential DistributionOperations ResearchOperations ResearchProperty 5: For all positive values of t,Such queuing models also as assuming a Poisson input. Arrivals sometimes are said to occur randomly, meaning PT{≤t+Δt|T>t}=P{T≤}tΔthat they occur in accordance with a Poisson input process. for small △tContinuing to interpret T as the time from the One intuitive interpretation of this phenomenon is than every last event of a certain type (arrival or service completion) until the next such event, we suppose that a time t already has time period of fixed length has the same chance of having an elapsed without the event’s occurring. We know from Property 2 that probability that the event will occur within the next time arrival regardless of when the preceding arrival occurred, as interval of fixed length △tis a constant(identified in the next suggested by the following property. paragraph), regardless of how large or small t is. Property 5 goes further to say that when the value of △tis small, this constant probability can be approximated very closely by . α t江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniver37sity of Finance & Economics©2006Schoo38l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The role of the Exponential The role of the Exponential DistributionOperations ResearchOperations ResearchFurthermore, when considering different small values of △t, xFor any t >= 0, Therefore, because the series expansion of e this probability is essentially proportional to △t, with for any exponent xisααproportionality factor . In fact, is the mean rateat which the events occur(see Property 4), so that the expected number of∞nxevents in the interval of length △tis exacα ttly . xe=1+x+∑The only reason that the probability of an events’ occurring n!n=2differs slightly from this value is the possibility that more than oneevent will occur, which has negligible probability when △tis It follows see why Property 5 holds mathematically, note that the ∞n(−aΔt)constant value of our probability (oFr a fixed value of △t) is just PT{≤t+Δt|T>t}1=−+aΔt−∑n!n=2P{T≤t+Δt|T>t}=P{T≤Δt}≈at,forsmalt−aΔt=1−e江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog39y, JiangXiUniversity of Finance & Economics©2006School of40 Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The role of the Exponential The role of the Exponential DistributionOperations ResearchOperations ResearchProperty 6:Unaffected by aggregation or disaggregating .Because the summation terms become relatively negligible for sufficiently small values of △tThis property is relevant primarily for verifying that the input processis Poisson .Therefore ,we shall describe it in these terms ,although it also applies directly to the exponential p{T≤t+Δt|T>t}lim=adistribution (exponential interarrivaltimes) because of Δt→0Δtproperty that there are several (n) different typesof Because T can be represent either interarrivalor service times customers ,where the customers of each type (type i) arrive in queuing models ,this property provides a convenient approximation of the probability that the event of interest according to a Poisson input process with parameter occurs in the next small intervals △tof time . An analysis λ(i=1,2,…,n). Assuming that these are independent Poisson ibased on this approximation also can be made exact by taking process ,the property also must be Poisson ,with parameter appropriate limits as △t →0.(arrival rate).λ=λ+λ...+λ2江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo41l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog42y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 15 运筹学Operations Research15-8运筹学运筹学 The role of the Exponential Distribution15 Queuing TheoryOperations ResearchOperations ResearchAssuming that each arriving customer has a fixed The birth-and-death Processprobability λof being type I ( i=1,2,…,n ) ,withASSUMPTION 1: Give N (t) =n, the current probability ndistribution of theremainingtime until the next birth λ=pλandp=1(arrival) is exponential with parameter λ(n=0,1,2…)ii∑ini=1ASSUMPTION 2: GiveN (t)=n, the current probability the property says that the input process for customers of distribution of the remaining time until the next death type I also must be Poison with parameter other i(service completion) is exponential with parameter μnwords ,having a Poisson process is unaffected by desegregations.(n=1,2,…)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of In43formation Technology, JiangXiUniversity of Finance & Economics©2006School of Informat44ion Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The birth-and-death The birth-and-death ProcessOperations ResearchOperations ResearchBecause of these assumptions, the birth-and-death process ASSUMPTION 3：The random variable of assumption is a special type of continuous time Markov chain. (See 1(the remaining time until the next birth) and the random for a description of continuous time Markov chains and their varproperties,including an introduction to the general procedure iable of assumption 2 (the remaining time until the next for finding steady-state probabilities that will be applied in the death) are mutually independent. The next transition in the remainder of this section.) Queuing models that can be state of the process is either represented by a continuous time Markov chain are far more n →n+1 (a single birth)tractable analytically than any other .or Because Property 4 for the exponential distribution (see n →n+1 (a single death)) implies that the λand μare mean rates, we can nndepending on whether the former or latter random summarize these assumptions by the rate diagram shown in .iable is small .江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUn45iversity of Finance & Economics©2006Schoon46l of Iformation Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The birth-and-death The birth-and-death ProcessOperations ResearchOperations ResearchExcept for a few special cases ,analysis of the birth-and-Consider any particular state of the system n(n=1,2,..). death process is very difficult when the system is in a transientStarting at time 0,suppose that a count is made of the number condition ,Some results about the probability Distribution of of time that the process enters this state and the number of N(t) have been obtained , but they are too complicated to be of times it leaves this state ,as denoted below much practical use. On the other hand, it is relatively En (t)=number of time that process enters state n by time to derive this distribution after the system hasLn(t)= number of time that process leaves state n by time a steady-state condition (assuming that this condition can be reached ). This derivation can be done directly from the rate diagram, as outlined next.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo47l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo48l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 15 运筹学Operations Research15-9运筹学运筹学 The birth-and-death The birth-and-death ProcessOperations ResearchOperations ResearchDividing En (t) and Ln(t) by t gives the actual rate (number Because the two types of events (entering and leaving) of events per unit time) at which these two kinds of events must alternate, these two numbers must always either be equal have occurred, and letting t→∞then gives the mean rate (expected number of events per unit time):or differ by just 1; that is, Dividing through both side sides by t and then Et()nlim=mean rate at which process enters state t→∞givet→∞tE(t)−L(t)≤1nnL(t)nlim=mean rate at which process leaves state →∞tE(t)L(t)1E(t)L(t)nnnn−≤solim−=0t→∞tttttThese results yield the following key principle:江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of49 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog50y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The birth-and-death The birth-and-death ProcessOperations ResearchOperations Researchthe expected number of times that it would leave state 1 RATE IN =RATE OUT PRINCIPLE: For any state of the to enter 1 to enter state 0 is μ) From any other state ,this system n (n=0,1,2,…)，mean entering rate =mean leaving rate is , the overall mean rate at which the The equation expressing this principle is called the balances process leaves its current state to enter state 0(the mean equationfor state n, After constructing the balance equations entering rate ) isfor all the states in terms of the unknown Pnprobabilities must sum to 1) to find these probabilities.μp+0(1−p)=μp11111To illustrate a balance equation, consider state 0, The process enter this state only from state 1. Thus the steady-state By the same reasoning ,the mean leaving rate must be probability of being in state 1(P1) represents the proportion of****, so the balance equation for state 0 istime that it would be possible for the process to enter state that the process is in state 1, the mean rate of enteringμp=λp1100state 0 isμ ( In other words for each cumulative unit time that nthe process spends in state 1江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Informat51ion Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog52y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 The birth-and-death The birth-and-death ProcessOperations ResearchOperations ResearchApplying this procedure yields the following yields the λλLλTo simplifyn−n− notation, let120C=forn=1,2,L,nfollowing results; μμLμnn−11stateλ0And then define Cn=1 for n=0. Thus the steady-state 0:P=P10μ1probabilities areP=CPforn=0,1,2,Lnn0λ1λλλ11101:p=P+(μp−λp)=p=p21110010μμμμμ22222∞∞λ1λλλλThe requirement that22210P=1(C)P=12:p=P+(μp−λp)=p=p∑n32221120∑n0μμμμμμ333322n=0n=0MM∞λ1λλλLλn−1n−1n−1n−20So that−1 n−1:p=P+(μp−λp)=p=pnn−1n−1n−1n−2n−P=(C)2n−100∑nμμμμμLμnnnnn−11n=0λ1λλλLλnnnn−10n:p=P+(μp−λp)=p=pn+1nnnn−1n−1n0∞μμμμμLμn+1n+1n+1n+1n1Given this information L=nP∑nMMn=0江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog53y, JiangXiUniversity of Finance & Economics©2006School of Information TechnologangXiUniversity of Finance & Economics©200654y, Ji江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 15 运筹学Operations Research15-10运筹学运筹学 The birth-and-death The birth-and-death ProcessOperations ResearchOperations ResearchAWhere is the average arrival rate over the long run . lso, because the number of servers srepresents the λnumber of customers that can be served (and thus are not in the Because λis the mean arrival rate while the system is in nqueue) simultaneously,state n (n=0,1,2,…) and Pnis proportion of time that the system is in this state ,∞∞L=(n−s)P∑qnλ=λPn=s∑nnn=0Furthermore, the relationships given in Sec. yieldSeveral of the expressions just given involve summations with an infinite number of terms. Fortunately, these LLqsummations have analytic solutions for a number of W=W=qλλinteresting interesting special cases.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog55y, JiangXiUniversity of Finance & Economics©2006School of56 Information Technology, JiangXiUniversity of Finance & Economics© Queuing Model Based on 运筹学运筹学15 Queuing TheoryOperations ResearchOperations Researchthe Birth-and-Death ProcessAs described in ,the M/M/s model assumes that all interarrivaltimes are independently and identically distributed Queuing Model Based on the Birth-according to an exponential distribution (. ,the input process is and-Death ProcessPoisson ),that all service times are independent and identicallydistributed according to another exponential distribution ,and that the number of servers is s (and positive integer ) .Consequently ,this model is just the special case of the birth-and-The s/M/M oMdeldeathprocess where the queuing system’s mean arrival rate and mean service rate per busy severer are constant regardless of the state of the system .When the system has just a single server (s=1),the implication is that the parameters for the birth-and-death process areThe resulting rate diagram is shown in .λ=λn=0,1,2...andμ=μn=1,2...()()江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog57y, JiangXiUniversity of Finance & Economics©2006Schoo58l of Information Technology, JiangXiUniversity of Finance & Economics© Queuing Model Based on Queuing Model Based on 运筹学运筹学Operations Researchthe Birth-and-Death ProcessOperations Researchthe Birth-and-Death ProcessHowever ,when the system has multiple servers (s>1),the When the maximum mean service rate sμexceeds the mean can not be expressed this simply .arrival rate λ,that is whenKeep in mind that μrepresents the mean service rate for nthe overall queuing system (. ,the mean rate at which serviceλρ=<1completions occur ,so that customers leave the system) when sμthere are n customers currently in the system .As Mentioned for Property 4 of the exponential distribution (see A queuing system fitting this model will eventually reach a ),when the mean service rate per busy server is μthe steady-state results derived in for the general birth-overall rate for n busy servers must be nμ. and-death process are directly applicable .However ,these results simplify considerably for this model and yield Therefore ,when n<=s,whereas μ=sμwhen n>=sso that nclosed=form expression for Pn,L , Lq,and so forth ,as shown all sservers are busy .The rate diagram for this case is shown in .江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo59l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of In60formation Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 15 运筹学Operations Queuing Model Based on Queuing Model Based on 运筹学运筹学Operations ResearchOperations Researchthe Birth-and-Death Processthe Birth-and-Death ProcessResults For The Single-server Case(M/M/1):Consequently,For s=1,the Cnfactors for the birth-and-death process ∞nL=n(1−ρ)ρreduce ton∑⎛λ⎞n=0n∞C=⎜⎟=ρ,forn=0,1,2,Ln⎜⎟∞∞μ⎝⎠L=(n−1)Pd⎛⎞n∑qn=(1−ρ)ρ⎜ρ⎟∑∑n=1Thereforendρn=0⎝n=0⎠P=ρP,forn=0,1,2,Ln0=L−1(1−P)0−1∞∞d⎛1⎞⎛⎞n2P=⎜ρ⎟n∑=(1−ρ)ρ⎜⎟∑⎜⎟λwhere⎝n=0⎠dρ1−ρn=0⎝⎠=−1⎛1⎞μ(μ−λ)=⎜⎟ρλ⎜⎟1−ρ⎝⎠===1−ρ1−ρ1−λnThusP=(1−ρ)ρ,forn=0,1,2,Ln江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo61l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog62y, JiangXiUniversity of Finance & Economics© Queuing Model Based on Queuing Model Based on 运筹学运筹学Operations Researchns Researchthe Birth-and-Death ProcessOperatiothe Birth-and-Death ProcessAssuming again that λ>μ, we now can derive the When λ>μ,so that the mean arrival rate exceeds the probability distribution of the waiting time in the system mean service rate, the preceding solution “blows up ”(because (including service) W for a random arrival when the queue the summation for computing P0 diverges). For this case, the discipline is first_come_first_served. If this arrival finds n queue would “explode ”and grow without bound. If the customers already in the system, then the arrival will have to queueingsystem begins operation with no customers present, wait through n+1 exponential service times, including his or the server might succeed in keeping up with arriving her own .(For the customer currently being served, recall the customers over a short period of time, but this is impossible inlack_of_memory property for the exponential distribution the long run .(Even when λ=μ,the expected number of discussed in Sec. .)Therefore, let T1,T2…be independent customers in the queueingsystem slowly grows without bound service_time random variables having an exponential over time because, even though a temporary return to no distribution with parameter μ, and letcustomers present always is possible, the probabilities of huge numbers of customers present become increasingly significant S=T+T+L+T,forn=0,1,2,L,over time.)n+112n+1江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog63y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog64y, JiangXiUniversity of Finance & Economics© Queuing Model Based on Queuing Model Based on 运筹学运筹学Operations Researchthe Birth-and-Death ProcessOperations Researchthe Birth-and-Death ProcessThe surprising conclusion is that W has an exponential So that Sn+1 represents the conditional waiting time given distribution with parameter μ（1-ρ）. Therefore,n customers already in the system. As discussed in 11ο,Sn+1 is known to have an Erlangdistribution. Because W=E(w)==μ(1−ρ)μ−λthe probability that the random arrival will find n customers already in the system is Pn, it follows thatThese results include service time in the waiting time. In some contexts (., the County Hospital emergency room ∞problem), the more relevant waiting time is just until service οP{w>t}=PP{S>t}∑nn+1begins. Thus consider the waiting time in the queue (so n=0including service time)** for a random arrival when the queue Whidiscipline is first_come_first_served. If this arrival finds no ch reduces after considerable manipulation (see already in the system, then the arrival is served 6_15) toimmediately, so thatο−μ(1−ρ)tοP{w>t}=e,fort≥0P{w=0}=P=1−ρq0江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog65y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog66y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 15 运筹学Operations Queuing Model Based on Queuing Model Based on 运筹学运筹学Operations Researchthe Birth-and-Death ProcessOperations Researchthe Birth-and-Death ProcessIf this arrival finds n>0 customers already there instead, Note that Wpdose not quite have an exponential then the arrival has to wait through n exponential service timesdistribution, because P {Wq=0} However, the conditional until his or her own service begins, so that distribution of Wq, given that Wq>0, dose have an exponential distribution with parameter μ（1-ρ）, just as W dose, ∞οP={w>t}=PP{S>t}because∑qnnn=1ο∞P{w>t}qοο−μ(1−ρ)tnP{w>t|w>0}==e,fort≥0=(1−ρ)ρP{S>t}qq∑nοP{w>0}n=1q∞=ρPP{S>t}By deriving the mean of the (unconditional) distribution of ∑nn+1n=0Wq(or applying either Lp=λWqor Wq=W-1/ μ),ο=ρP{w>t}λο−μ(1−ρ)tW=E(w)==ρeqq,fort≥0μ(μ−λ)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of67 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Informaton Technology, JiangXiUniversity of Finance & Econom68iics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 16 运筹学Operations Research16-1运筹学运筹学166 The AppQlication of uQeuing TheoryOperations ResearchOperations Research16 6 The Application of QuQeuing TheoryThis chapter discusses the application of queuing theory QuQeuing theory has enjoyed a prominent place among the in the broader context of an overall OR study. It begins by modern analytical techniques of ORH. oHwever, the emphasis introducing three examples that will be used for illustration thus far has been on developing a descriptive mathematical throughout the chapter. Section discusses the basic theory. Thus queuing theory is not directly concerned with achieving the goal of OR: optimal decision making. Rather, it considerations for decision making in this context. The develops information on the behavior of queuing systems. This following two sections then develop decision models for the theory provides part of the information needed to conduct an OR optimal design of queuing attempting to find the best design for a queuing system.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo1l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog2y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学166 The ApplicationQ of uQeuing E ExamplesOperations ResearchOperations ExEamplesThe time until any given operating machine breaks down E—Example 1H?—oHw Many Repairers?has an exponential distribution, with a mean of 2 days. Until Simulation, Inc., a small company that makes widgets for analog computers, has 10 widget-making machines. now the company has had just one repairer to fix these HoHwever, because these machines break down and require machines, which has frequently resulted in reduced repair frequently, the company has only enough operators to operate eight machines at a time, so two machines are productivity because fewer than eight machines are operating. available on a standby basis for use while other machines Therefore, the company is considering hiring a second are down. Thus eight machines are always operating machines is reduced by 1 for each additional machine repairer, so that two machines can be repaired waiting to be .江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo3l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo4l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 E E ExamplesOperations ResearchOperations ResearchThus the queuing system to be studied has the repairer as its servers and the machines requiring repair as its customers,E aEch repairer costs the company approximately $$2880 per where the problem is to choose between having one or two servers. (Notice the analogy between this problem and the day. HHowever, the estimated lost profit from having fewer CHCounty oHspital emergency room problem described in ) with one slight exception, this system fits the finitethan eight machines operating to produce w$idgets is $400 per calling population// variation of the MM/s/ model presented in =.6, where N=10=/ machines, λ1=2/0 customer per day (for day for each machine down. (The company can sell the full each operating machine), and=/ μ1=2/ customer per day. The exception is that the λ0 and λ1 parameters of the birth-and-output from eight operating machines, but not much more.)death process are changed form λ=1=0λand λ=9=λto λ8=8=λ010and λ8=8=λ. (All the other parameters are the same as those 1The analysis of this prob6lem will be pursued in given6C in ) Therefore, the Cfactors for calculating the nPprobabilities change accordingly (see ).nand .江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog5y, JiangXiUniversity of Finance & Economics©2006Schoo6l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 16 运筹学Operations Research16-2运筹学运筹学 E E ExamplesOperations ResearchOperations ResearchE—Example 2—Which computer??Thus the queuing system of concern has the computer as its EEEMERALD UNIEVERVSITYis making plans to lease a supercomputer to be used for scientific research by the faculty (single) server and the jobs to be run as its customers. Furthermore, and students. Two models are being considered: one from the MBICthis system fits the M//M//1 model presented at the beginning of Corporation and the otherC from the CRAB CCompany. The MBI computer costs more but is somewhat faster than CCRAB , with 1 day as the unit of time, λ=2=0 customers per day, computer. In particular, if a sequence of typical jobs were run continuously for one 24-hour day, the number completed would and μ=3=0 and 25 customers per day with the MBI and the CRCAB have a Poisson distribution with a mean of 30 and 25 for the MBI and the CCRAB computers , respectively. It is estimated computers, respectively. You will see in and how the that an average of 20 jobs will be submitted per day and that the time from one submission to the next will have an exponential choice was made between the two with a mean of day. The leasing cost per day would be$$ 5$000 for the MBI computer and $3750 for theC CRAB computer.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo7l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo8l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 E E ExamplesOperations ResearchOperations ResearchE—HCExample 3—How Many Tool Cribs??EEach tool crib constitutes a queuing system, with the clerks TheMECHANCAL MECHANICAL COMPANYCOMPANY is designing a new plant. This as its servers and the mechanics as its customers. Based on plant will need to include one or more tool cribs in the factory area to store tools requqired by the shop mechanics. The tools will beprevious experience, it is estimated that the time required by handed out by clerks as the mechanics arrive and reqquest them and a tool crib clerk to service a mechanic has an exponential will be returned to the clerks when they are no longer needed. In distribution, with a mean of ½J½minute. Judging from the existing plants, there have beenq frequent complaints from anticipated number of mechanics in the entire factory area, it supervisors that their mechanics have had to waste too much timeis also predicted that they would require this service traveling to tool cribs and waiting to be served, so it appears that randomly but at a mean rate of 2 mechanics per minute. there should be more tool crOibs and more clerks in the new plant. On Therefore, it was decided to// use the MM/S/ model of the other hand, management is exerting pressure to reduce overhead to represent each queuing system . With 1 hour as the unit of in the new plant, and this reduction would lead to fewer tool cribs time , μ=1=20. If only one tool crib were to be provided, λalso and fewer clerks. To resolve these conflicting pressures, anOR OR study is to be conducted to determine just how many tool cribs and clerks would be 120. with more than one tool crib , this mean arrival the new plant should would be divided among the different queuing systems.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom9ics©2006School of Information Technolog10y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 E Examples166 The ApplicationQ of uQeuing TheoryOperations ResearchOperations ResearchThe total cost to the company of each tool crib clerk is Decision Makingabout $$20 per hour. The capital recovery costs, upkeep costs , QQueuing-type situations that require decision making and so forth associated with each tool crib provided are arise in a wide variety of contexts. For this reason, it is not possible to present a meaningful decision-making procedure estimated to be$6 $16 per working hour. While a mechanic is that is applicable to all these situations. Instead, this section busy , the value to the company of his or her output averages attempts to give a broad conceptual picture of a typical $8 $4 8per a queuing system typically involves making Sect66ions and include discussions of how this one or a combination of the following decisions:(and additional) information was used to make the required of servers at a service of the of service facilities.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of11 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog12y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 16 运筹学Operations Research16-3运筹学运筹学 Decision Decision MakingOperations ResearchOperations ResearchThe number of service facilities is directly related to λThe first kind of decision is particularly common in because , assuming a uniform workload among the facilities , practiceH. oHwever, the other two also arise frequently, λequals the total mean arrival rate to all facilities divided byparticularly for the business-industrial internal service the number of described in . One example illustrating a Refer to and note how the three examples there decision on the efficiency of the servers is the selection of the respectively illustrate situations involving these three type of materials-handling equipment (the servers) to decisions. In particular, the decision facing simulation, Inc., is purchase to transport certain kinds of loads (the customers). how many repairers (server) is needed. The problem facing Another such example is the determination of the size of a MechanicalC Company is how many tool cribs (service maintenance crew (where the entire crew is one server). facilities) to install as well as how many clerks (servers) to Other decisions concern the number of service facilities, such provide at each restrooms, first-aid centers, drinking fountains, storage areas ,and so on , to distribute throughout an area.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of13 Information Technology, JiangXiUniversity of Finance & Economics©2006School of14 Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Decision Decision MakingOperations ResearchOperations ResearchThese two considerations create conflicting pressures on All the specific decisions discussed here involve the the decision maker. the objective of reducing service costs general question of the appropriate level of service to provide recommends a minimal level of service. On the other hand , in a queuing system. As mentioned at the beginning of long waiting times are undesirable, which recommends a high ,decisions regarding the amount of service capacity level of service. Therefore, it is necessary to strive for some to provide usually are based primarily on two considerations: type f compromise. To assist in finding this compromise, (1) the cost incurred by providing the service as shown in and may be combined, as shown in . , and (2) the amount of waiting for that service , as The problem is thereby reduced to selecting the point on the suggested66 in . can be obtained by using the curve of6 that gives the best balance between the appropriate waiting time equation from queuing delay in being serviced and the cost of providing that service. Reference to and indicates the corresponding level of service.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of15 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Infomat16rion Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Decision Decision MakingOperations ResearchOperations ResearchObtaining the proper balance between delays and service One broad category is where the customers are external costs requires answers to such questions asH , oHw much expenditure on service is equivalent (in its detrimental impact)to the organization provid;ing the service ;., they are to a customer’’s? being delayed 1 unit of time? Thus , to outsiders bringing their business to the organization. compare service costs and waiting times, it is necessary to CConsider first the case of profit-making organizations adopt (explicitly or implicitly) a common measure of their (typified by the commercial service systems described in impact. The natural choice for this common measure is cost, which then requires estimation of the cost of ). From the viewpoint of the decision make, the cost Because of the diversity of waiting-line situations, no single of waiting probably consists primarily of the lost profit from process for estimating the cost of waiting is generally lost . oHwever, we shall discuss the basic considerations involved for several type of situations.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of17 Information Technology, JiangXiUniversity of Finance & Economics©2006School of18 Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 16 运筹学Operations Research16-4运筹学运筹学 Decision Decision MakingOperations ResearchOperations ResearchNoNw consider the type of situation where service is provided This loss of business may occur immediately (because the on a nonprofit basis to customers external to the organization customer grows impatient and leaves) or in the future (typical of social service systems and some transportation service (because the customer is sufficiently irritated that he or she does not come again). This kind of cost is quite difficult to systems described in ). In this case, the cost of waiting estimate, and it may be necessary to revert to other criteria, usually is a social cost of some kind. Thus it is necessary to such as a tolerable probability distribution of waiting times. evaluaqte the consequences of the waiting for the individuals When the customer is not a human being, but a job being /involved ando/r for society as a whole and to try to impute a performed on order, there may be more readily identifiable costs incurred, such as those caused by idle in-process monetary value to avoiding theseqO consequences. Once again, this inventories or increased expediting and administrative of cosqt is quite difficult to estimate, and it may be necessary to revert to other criteria.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog19y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog20y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Decision Decision MakingOperations ResearchOperations ResearchA situation may be more amenable to estimating waiting Given that the cost of waiting has been evaluated costs if the customers are internal to the organization explicitly, the remainder of the analysis is conceptually providing the service (as for business-industrial internal straightforward. The objective is to determine the level of service systems). For example, the customers may be service that minimizes the total of the expected cost of servicemachines (as Ein ExampleE 1) or employees (as in Example 3) of and the expected cost of waiting for that service. This concept a firm. Therefore, it may be possible to identify directly some is depicted6C in , where WC denotes waCiting cost, SC of or all the costs associated with the idleness of these denotes servCice cost, and T Cdenotes total cost. Thus the customers. Typically, what is being wasted by this idleness is mathematical statement of the objective is to productive output, in which case the waiting cost becomes the MinimECE=ize E(TC)E(=SCC)E+CE+(WC).lost profit from all lost productivity.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©200621School of Information Technolog22y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Decision Making166 The Application ofQ uQeuing TheoryOperations ResearchOperations Formulation of WaCiting-Cost Functionsthe next two sections are concerned with the application of th6is concept to various types of problems. Thus ToE express E(WCC) mathematically, we must first formulate a waiting-cost function that describes how the describes howEC E(WC) can be expressed mathematically. Section actual waiting cost being incurred varies with the current behavior of the queuing system. The form of this function then focus on E(SC) to formulate the overall objective depends on the context of the individual problemH. However, function EE(TCC) for several basic design problems (including most situations can be represented by one of the two basic forms described with multiple decision variables, so that the level-of-service axis6 in then requires more than one dimension).江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog23y, JiangXiUniversity of Finance & Economics©2006Schoof24l o Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 16 运筹学Operations Research16-5运筹学运筹学 FormulationC of Waiting-Cost Formulation of WaCiting-Cost FunctionsOperations ResearchOperations ResearchThe g(N) FormThe g(NN) function is constructed for a particular situation by CConsider first the situation discussed in the preceding section estimating g(n), the waiting-N=-cost rate incurred when N=n, for where the qqueuing system customers are internal to the n==1,2,……, where g(0)=A=0. After computing theP Pprobabilities for norganization providing the service, and so the primary cost of a given desqign of the queuing system, we can calculate waiting may be the lost profit from lost productivity. The rate at ECW=EE(C)W=E(g(NN)).which productive output is lost sometimes is essentially BNBecause N is a random variable, this calculation is made by proportional to the number ofq customers in the queuing system. using the expression for the expected value of a function of a HHowever, in many cases there is not enough productive work discrete random variableavailable to keep all the members of the calling population ∞continuously busy. Therefore, little productive output may be lost E(WC)=g(n)P.∑nby having just a few members idle, waiting for service in the n=0qqueuing system, whereas the loss may increase greatly if a few more members are made idle because theyq require service. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog25y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog26y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Formulation of WaCiting-Cost Formulation of Waiting-CCost FunctionsOperations ResearchOperations ResearchEHExample— 1—oHw Many Repa?irers?When g(N) is a linear function (, when the waiting-ForE Example 1 of6 , Simulation, Inc., has two cost rate is proportional to N), thenstandby widget-making machines, so there is no lost productivity as long as the number of customers (machines g(N)=C=CN,wrequiring repair) in the system does not exceedH 2. However, Where CCis the cost of waiting per unit time for each wfor each additional customer (up to the maximum of 10 total), customer. In thisEC case, E(WC) reduces tothe estimated lost profit is $$400 per day. Therefore,∞0 f or n=0,1,2,⎧E(WC)=CnP=∑nwg(n)=⎨n=0400(n-2) for n=3,4,L,10, ⎩江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©200627Schoo28l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Formulation of WaCiting-Cost Formulation of Waiting-CCost FunctionsOperations ResearchOperations ResearchThe b(ω) FormasNow consider the cases disc shown in6CE Table . Consequently, E(WCC) isussed6 in where the queuing system customers are external to the organization calculated by summing the rightmost column of Table providing the service. Three major types of queuing systems for each of the two cases of interest, namely, having one described in Sec .——commercial service systems, transportation service systems,— and social service systems—repairer (s=1=) or two repairers (s=2=).typically fall into this category. In the case of commercial service systems, the primary cost of waiting may be the lost profit from lost future business. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog29y, JiangXiUniversity of Finance & Economics©2006Schoo30l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 16 运筹学Operations Research16-6运筹学运筹学 Formulation of Waiting-CCost Formulation of Waiting-CCost FunctionsOperations ResearchOperations ResearchTable CaClculation of E(EWC)C for ExEample 1For transportation service systems and social systems, the s=1=s=2=primary cost of waiting may be in the form of a social cost. N=n=g(n)Pg(n)PPg(n)PnnnnHoHwever, for either type of cost, its magnitude tends to be greatly by the size of the waiting times experienced customers. Thus the primary property of the queuing that determines the waiting cost currently being is ω, the waiting time in the system for the individual -4customers. CConsequently, the form of the waiting-cost function ****100for this kind of situation66 is that illustrated in , namely, -4-669288007**1004**100a function of ω. We shall denote this form by h(ω).-5-71032007**1002**100EC$8E(WC) per day$8$281$48 per day江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo31l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog32y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Formulation ofC Waiting-Cost FormulatCion of Waiting-Cost FunctionsOperations ResearchOperations Researchwhere fω(w) is the probability density function of ω. One way of constructing the h(ω) function is to estimate HoHwever, because EE(h(ω)) is the expected waiting cost per h(w) (the waiting cost incurred when a customer’’s waiting customer andEC E(WC) is the expected waiting cost per unit time ω=w=) for several values of w and then to fit a polynomial time, these two quantities are not equal in this case. To relateto these points. The expectation of this function of a them, it is necessary to muEltiply (Eh(ω)) by the expected continuous random variable is then defined asnumber of customers per unit time entering the queuing ∞system. In particular, if the mean arrival rate is a constant λ, E(h(ω))=h(w)f(w)dw,ω∫0then∞E(WC)=λE(h(ω))=λh(w)f(w)dw.ω∫0江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog33y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog34y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Formulation ofC Waiting-Cost Formulation of Waiting-CCost FunctionsOperations ResearchOperations ResearchThe scientists also pointed out that that a second major ECExample 2——Which oCmputer??Because the faculty and consequence of waiting is a break in the continuity of the research. students of EEmerald University would experience different Although a short delay (a fraction of a day) causes little problem in turnaround times with the two computers under consideration (seethis regard, a longer delay causes significant wasted time in having to ), the choice between the computers required an evaluation gear up to resume the research. The scientists estimated that this of the consequences of making them wait for their jobs to be run. wasted time would be roughly proportional to the square of the delay Therefore, several leading scientists on the faculty were asked to time. Dollar figures of $$100 and $$400 were then imputed to the value evaluate these being able to avoid this consequence entirely rather than having a The scientists agreed that one major consequence is a delay in wait of ½½day and 1 day, respectively. Therefore, this component of getting research done. Little effective progress can be made while the waiting cost was estimated to be 400 ω is awaiting the results from the run. The scientists estimated This analysis yieldsthat it would be worth $$500 to reduce this delay by 1 day. h(ω)=5=00 ω+4+00 ω2Therefore, this component of waiting cost was estimated to be $$500 Becauseper day, that is, 500ω, where ωis expressed in ω(w) ==μ(1-ρ)e-μ(1-ρ)w江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog35y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog36y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 16 运筹学Operations Research16-7运筹学运筹学 FormulatCion of Waiting-Cost Formulation of Waiting-CCost FunctionsOperations ResearchOperations Researchfor the M/M//1/6 model (see ) fitting this single-EEvaluating the integral for these two cases yieldsserver queuing system,58 for MBI computer,⎧∞E(h(ω))=2−μ(1−ρ)w⎨E(h(ω))=(500w+400w)μ(1−ρ) for CRAB computer.⎩∫0The result represents the expected waiting cost (in dollars) By using the fact that μ(1-ρ)==μ-λfor a single-server for each scientist arriving with a job to be run. Because λ=2=0, system, the values of μand λpresented in givethe total expected waiting cost per day becomes10 for MBI computer,⎧$1160 per day for MBI computer,⎧μ(1−ρ)=E(WC)=⎨⎨5 for CRAB computer.$2640 per day for CRAB computer .⎩⎩江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of37 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog38y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Formulation of WaCiting-Cost FormulatCion of Waiting-Cost FunctionsOperations ResearchOperations ResearchThe LinearC Case: When h(ω) is a linear function EExampleHC? 3——How Many Tool Cribs? As indicated in h(ω)==CCwω,, the value to the Mechanical Company of a busy thenEC E(WC) reduces to mechanic’’s output averages$8 about $48 per hourC=. Thus Cw48=8. CEConsequently, for each tool crib the expected waiting cost per (C=EW)C=λEE(cwω)==CCw(λW)== isNote that this result is identical to the result when g(N) is EaE(WCC)=48=8L, linear functionC. Consequently, when the total waiting cost incurred by the queuing system is simply proportional to the where L represents the expected number of mechanics total waiting time, it does not matter whether the g(N) or the waiting (or being served) at the tool (ω) form is used for the waiting-cost function.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo39l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Informaton Technology, JiangXiUniversity of Finance & Econom40iics©2006运筹学运筹学166 The Application ofQ uQeuing Decision ModelsOperations ResearchOperations Decision ModelsFORMULATION OF MODEEL 1We mentioned in that three common decision Definition: CCs==marginal cost of a server per unit in designing queuing system are s (number of Given: μ,λC,), μ(mean service rate for each server), and λ(mean To find: sarrival rate at each service facility). We shall now formulate Objective: MinimizeEC E(TC)==sCCs+EC+E(WC).models for making some of these only a few alternative values of s normally need Model 1——Unknown sto be considered, the usual way of solving this model is to Model 1 is designed for the case where both μand λare caEClculate E(TC) for these values of s and select the fixed at a particular service facility, but where a decision minimizing be made on the number of serves to have on duty at the facility.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo41l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog42y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 16 运筹学Operations Research16-8运筹学运筹学 Decision Decision ModelsOperations ResearchOperations ResearchModel 2——Unknown μand sEHEE?Example 1——HOW MANY REPAIRERS?Model 2 is designed for the case where both the For example 1 of , each repairer (server) costs efficiency of service, measured by μ, and the number of Simulation, Inc., approximately$8 $208 per day. Thus , with 1 servers s at a service facility need to be values of μmay be available because there is a day as the unit of timeC=8, Cs2=80. usECing the values of E(WC) choice on the quality of the servers. For example, when the calculated in Table then yields the results shown in Table servers will be materials-handing units, the quality of the , which indicate that the company should continue having units to be purchased affects their service rate for moving one possibility is that the speed of the servers can be adjusted mechanically. For example, the speed of machines frequently can be adjusted by changing the amount of power consumed, which also changes the cost of operation.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo43l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Informaton Technology, JiangXiUniversity of Finance & Econom44iics©2006运筹学运筹学 Decision Decision ModelsOperations ResearchOperations ResearchStill other type of example is the selection of the number Table CaCEClculation of E(TC) forE Example 1of crews (the serves) and the size of each crew (which determines μ) for jointly performing a certain task. The task ssCCEWCC)EE(TCE(C)smight be maintenance work, or loading and unloading operations, or inspection work, or setup of machines, and so $6$561 per day minimumIn many case, only a few alternative values of μare 25660488$68$608 per dayavailable, ., the efficiency of the alternative types of ≥3≥8840≥0≥$84$80 per daymaterials-handing equipment or the efficiency of alternative crew sizes.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo45l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo46l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Decision Decision ModelsOperations ResearchOperations ResearchEXAEEXMPLE 2——WHCHCEHICH COMPUTER??FORMULATION OF MODEEL 2As indicated in= , μ3=0, for the MBI computer and Definitions:= f(μ)m=arginal cost of server per unit time μ=C=25 for the CRAB computer, where 1 day is the unit of time. when mean service rate is μ.These computers are the only two being considered by A=s=et of feasible values of μ.EEmerald University, soGiven : λ,f(μ),{={2=5,30}}.To find: μ, the leasing cost per$C day is $3750 for the CRAB Objective: MinECE+imize E(TC)==sf(μ)(E+WCC), subject to computer (μ=2$=5) and $5000 for the MBI computer (μ=3=0),μ∈ for μ=25,⎧f(μ)=⎨5000 for μ=30.⎩江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog47y, JiangXiUniversity of Finance & EconomSchoo48ics©2006l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 16 运筹学Operations Research16-9运筹学运筹学 Decision Decision ModelsOperations ResearchOperations ResearchThe supercomputer chosen will be the only one available CConsequently, the decision was made to lease the MBI to the faculty and students, so the number of servers supercomputer.(supercomputers) for this queuing system is restricted= to s1=. HeHnce This example illustrates a case where the number of EC=E(TC)f=(μ)E+(E+WCC),feasible values of μis finite but the value of s is fixed. If s were Where EE(WCC) is given in for the two not fixed, a two-stage approach could be used to solve such a alternatives . Thusproblem. First, for each individual value of μ, set CCs=f=(μ), and solve for the value of s that minimizesEC E(TC) for model 1. 3750+2640=$6390 per day f or CRAB computer⎧E(TC)=Second , compare these minECimum E(TC) for the alternative ⎨5000+1160=$6160 per day f or MBI computer.⎩values of μ, and select the one giving the overall minimum.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo49l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog50y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Decision Decision ModelsOperations ResearchOperations ResearchWhen the number of feasible values of μis infinite (such This analytical approach frequently is relatively as when the speed of a machine or piece of equipment is set straightforward for the case of= s1= (see -15). mechanically within some feasible interval), another two-HoHwever, because far fewer and less convenient analytical stage approach sometimes can be used to solve the problem. results are available for multiple-server versions of queuing First, for each individual value of s , analytically solve for the models, this approach is either difficult (requiring computer value of μthat minimizesEC[ E(TC). T[his approach requires calculations with numerical methods to solve the equation for setting to zero the derivative ofEC (ET)C with respect to μand μ) or comp>letely impossible when s1>. Therefore, a more then solving this equation for μ, which can be done only when practical approach is to consider only a relatively small analytical expressions are available for both f(μ) andEC (EW)C.]] number of representative values of μand to use available Second, compare these minECimum E(TC) for the alternative tabulated results for the appropriate queuing model to obtain values of s, and select the one giving the overall minimum.(or approximate) E(CET)C for these μvalues.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of51 Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo52l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Decision Decision ModelsOperations ResearchOperations ResearchFortunately, under certain fairly common circumstances In effect, this optimality result indicates that it is described next=, s=1 (and its minimizing value of μ) must yield better to concentrate service capacity into one fast the overaECll minimum E(TC) for model 2, so s>1> cases need server rather than dispersing it among several slow not be considered at oCndition 2 says that this concentrating of a Optimality of a single server: under certain conditions, given amount of service capacity can be done s=1= necessarily is optimal for model increasing the cost of service . CCondition 1 The primary conditions are thatsays that it must be possible to make μsufficiently 1. The value of μminEC=imizing (ET)C for s1= is that a single server can be used to full 2. Function f(μ) is either linear or concave (as defined in Appendix 2)advantage..江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo53l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo54l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 16 运筹学Operations Research16-10运筹学运筹学 Decision Decision ModelsOperations ResearchOperations ResearchThis table shows that the service efficiency* of the (s**,μ*) To understand why this result holds, consider any other solution sometimes is worse but never is better then for the solution to model 2, (s,μ)=(=s**,μ**>*), where s*1>. The service (1,s**μ**) solution because it can use the full service only when capacity of this system (as measured by the mean rate of there are at* least s* customers in the system, whereas the service completions when all servers are working) is s**μ**. We single-server solution uses the full capacity whenever there shall now compare this solution with the corresponding are any customers in the system. Because this lower service single-server solution (s,μ)=(=1*,s*μ**) having the same service efficiency can only increase waiting in the systemE, E(WCC) must be larger for (s***,μ*) than for (1,s**μ**). Furthermore, the capacity. In particular, Table compares the mean rate at expected service cost must be at least as large because which service completions occur for each given number of condition[=] 2 a[nd f(0)0= ]implies that** s*f(μ*)≥f (s**μ**).customers in the system= Nn=. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo55l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniver56sity of Finance & Economics©2006运筹学运筹学 Decision Decision ModelsOperations ResearchOperations ResearchTherefore, ECE(TC) is large for (s**,μ**) than (1*,s*μ**). This result is still of some use even when one or both Finally, note that condition 1 implies that there is a feasible solution w=**ith s1= that is at least as good as (1,s*μ*). The conditions fail to hold. If μcannot be made sufficiently conclusion is> that any s1> solution cannot be optimal for large to permit a single server, it still suggests that a few model 2, so s=1= must be servers should be preferred to many slow ones. If Table Comparison of Service Efficiency for Model 2 Solutionscondition 2 does not hold, we still know thatEC E(WC) is Mean Rate of Service CoCmpletionsminimized by concentrating any given amount of service N=n=********(s,μ)=(=s,μ) versus (s,μ)=(=1,sμ)capacity into a single server, so the best= s1= solution must n=0=0=0=be at least nearly optimal unless it causes a substantial ********n==1,2,……,s-1nμ<s<μ**********n ≥ssμ=s=μincrease in service cost.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schooof57l Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Econom58ics©2006运筹学运筹学 Decision Decision ModelsOperations ResearchOperations ResearchIt may sometimes be clear that only a single server should Model 3——Unknown λand sbe provided at each facility (., one drinking fountain or oneModel 3 is design especially for the case where it is copy machine), but s often is also a decision to select both the number of service facilities and To simplify our presentation, we shall require in model 3 the number of servers at each facility. In the typical situation, that λand s be the same for all service facilities. HoHwever, it a population (such as the employees in an industrial building) should be recognized that a slight improvement in the must be provided with a certain service, so a decision must be indicated solution might be achieved by permitting minor made as to what proportion of the population (and therefore deviations in these parameters at individual facilities. This what value of λ) should be assigned to each service facility. should be investigated as part of the detailed analysis that ExEamples of such facilities include employee facilities generally follows the application of the mathematical model.(drinking fountains, vending machines, and restrooms), storage facilities, and reproduction equipment facilities.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog59y, JiangXiUniversity of Finance & Economics©2006Schoo60l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 16 运筹学Operations Research16-11运筹学运筹学 Decision Decision ModelsOperations ResearchOperations ResearchTo find: λ, OF MODEEL 3Objective: MinimizeEC E(TC), subject to λ=/ =λpn/, where Definitions:C= Cs=marginal cost of server per unit =1=,2,…….C=Cff=ixed cost of service per service facility per unit might appear at first glance that the appropriate expression for the expected total cost per unit time of all the λp=m=ean arrival rate for entire calling should be n=n=umber of service facilities= =λp/ /λ.?E(TC)=n[(C+sC)+E(WC)],Given: μCC,Cs, Cf, λ江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of61 Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo62l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Decision Decision ModelsOperations ResearchOperations Researchwhere EE(WCC) here represents the expected waiting cost per Because there are many situations where it obviously unit time for each facilityH. oHwever, if his expression actuallywould not be optimal to have just one service facility (., the were va=lid, it would imply that n1= necessarily is optimal for number of restrooms in a 50-story building), something must model 3. The reasoning is completely analogous to that for the be wrong with this expression. Its deficiency is that it optimality of a single-server result for mode;l 2 ;namely, any considers only the cost of service and the cost of waiting at the solution (n,s)==(n***>*,s*) with n*1> has higher service costs than service facilities while totally ignoring the cost of the time the (n=**,s)(=1,n*s*) solution , and it also has a higher expected wasted in traveling to and from the facilities. Because travel waiting cost because it sometimes makes less effective use of the time would be prohibitive with only one service facility for a available service capacity . In particular, it sometimes has idle servers at one facility while customers are waiting at another large population, enough separate facilities must be facility, so the mean rate of service completions would be less distributed throughout the calling population to hold travel than if the customers had access to all the servers at one time down to a reasonable facility.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School ofe & Economics©200663 Information Technology, JiangXiUniversity of FinancSchool of Information Technology, JiangXiUniversity of Finance & Economics©200664运筹学运筹学 Decision Decision ModelsOperations ResearchOperations ResearchWe shall also assume that the travel-time cost is Thus, letting the random variable T be the round-trip proportional to T, where CCt is the cost of each unit of travel time travel time for a customer coming to and going back from for each customer. For ease of presentation, suppose that the one of the service facilities, we see that the total time lost by probability distribution of T is the same for each service facility. the customer actua+Clly is ωT+. (Recall from that ωis So thatCE tCE(T) is the expected travel cost for each arrival at any the waiting time in the queuing system after the customer of the service facilities. The resultingC expression for E(ET)C isarrives.) Therefore, a customer’’s total cost for time lost EC=[E(TC)n=[(C+Cf+sCCs)+E+E(WCC)++λCtECE(T)]]should be+ based on ωT+ rather than just ω. To simplify the because λis the expected number of arrivals per unit time analysis, let us separate this total cost into the sum of the at each facilityC. oCnsequently, by evaluating (or estimating ) waiting-time cost based on ω(or N) and the travel-time cost EE(T) for each ease of interest, model 3 can be solved by based on E(TC) for various values of s for each n and then selecting the solution giving the overall minimum.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo65l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog66y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 16 运筹学Operations Research16-12运筹学运筹学 Decision Decision ModelsOperations ResearchOperations ResearchEHExample 3——How Many Tool CCribs??The three basic alternatives being considered are these:For the new plant being designed for the Mechanical Alternative 1:H Have one tool crib——use location (see ), the layout of the portion of the Alternative 2:H Have two tool cribs——use location 1 and area where the mechanics will work is shown . Alternative 3:H aHve three tool cribs——use location 1,2,and The three possible locations for tool cribs are identified as 1,2,and 3, where access to these locations will be The mechanics will be distributed quite uniformly provided by a system of orthogonal aisles parallel to the sides throughout the area shown, and each mechanic will be of the indicated area. The coordinates are given in units of assigned to the nearest tool crib. feet.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog68y, JiangXiUniversity of Finance & Economics©200667School of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Decision Decision ModelsOperations ResearchOperations ResearchIt is estimated that the mechanics will walk to and from a Table CCalculatEHion of E(T), Dollars per Hour, for tool crib at an average speed of slightly less than 3 miles per EExample3hourE. Based on this information, E(T) is estimated to be , nλsLE)CE(TC+CEC+s(WCC)λCEECECE(T)E(TC), and hour for alternatives 1,2 and 3, ∞∞∞The stage now is set for using model 3 to choose from alternatives. Most of the data required for this model given in , namely,μ==120 per hour, CCf$=16$=6 per hour,$=Cs$=20 per hour,λp=1=20 per hourC$=8, Ct$=48 per hour,江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUnivernce & Economics©200669sity of FinaSchoo70l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Decision Decision ModelsOperations ResearchOperations Researchwhere the M/M///s model given in is used to The resulting caEClculation of E(TC) for various s values calculate L and so on . In addition, the end of gives for each n is given in Tab6le , which indicates that the EC=8E(WC)4=8L in dollars per hour. Therefore,overall minimumEC$ E(TC) of $ per hour is obtained by hav=ing three tool cribs (so λ4=0 for each), with one clerk at 120each tool (TC)=n[(16+20s)+48L+48E(T)].n江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo7172l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 16 运筹学Operations Research16-13运筹学运筹学166 The AppQlication of uQeuing Theory6Operations ResearchOperations C ConclusionsearchHoHwever, there are many other possible decision variables CoCnclusions(., the size of a waiting room for a queuing system) and many more complicated situations (., designing a priority This chapter has discussed the application of queuing queuing system) that can also be analyzed in a similar for designing queuing systemsE. Every individual Another useful area for the application of queuing theory problem has its own special characteristics, so no standard is the development of policies for controlling queuing systems, procedure can be prescribed to fit every situation. Therefore, ., for dynamically adjusting the number of servers or the service rate to compensate for changes in the number of the emphasis has been on introducing fundamental customers inC the system. Considerable research is being considerations and approaches that can be adapted to most conducted in this . We have focused on three particularly common QQueuing theory has proved to be a very useful tool, and we decision variables (s,μ,and λ) as a vehicle for introducing and anticipate that its use will continue to grow as recognition of illustrating these concepts. the many guises of queuing systems grows.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo7374l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 17 运筹学Operations Research17-1运筹学运筹学Operat17 Inventory Theoryions ResearchOperations ResearchoHw does such a facility decide upon its “inventory 17 Inventory Theorypolicy,”., its policy for whento replenish inventory and by how much?Inevntory necessityProcedures of inevntory decision-makingKeeping an inventory (stock of goods) for ¾Formulate a mathematical model describing the behavior of the inventory sale or use is very common in business. Retail ¾Derive an optimal inventory policy with respect to this firms, wholesalers, manufacturing companies—and blood banksg—enerally have a stock of goods ¾Frequently use a computer to maintain a record of the Inventory levels and to signal when and how much to on .江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom1ics©2006Schoo2l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学Operat17 Inventory TheoryOperat17 Inventory Theoryions Researchions xEampleMain contents in chapter17We present one example in context (a manufacturer) where an inventory policy needs to be developed.¾ the example is presentedExample :Manufacturing Speakers for T VSets¾ development and analysis of deterministic A television manufacturing company produces its inventory models, ., models where the future demand own speakers, which are used in the production of its is assumed to be sets. The television sets are assembled on a ¾ development and analysis of stochastic continuous production line at a rate of ,8000 per month, models where this demand is a random one speaker needed per set.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom3ics©2006Schoo4l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Exampleions Exampleions ResearchThe company is interested in determining when to produce a batch of speakers and how many speakers to produce in each batch. we will develop the inventory policy for Several costs must be considered:this example with the help of the first 1. Each time a 2. The unit production cost of a batch is produced, a single speaker (excluding the setup cost) setup cost of $12,000 is $10, independent o f the batch size inventory model presented in incurred. . The estimated 4. Shortage cost of each speaker is holding cost of $ per month. keeping a speaker in stock is $ per month.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo5l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog6y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 17 运筹学Operations Research17-2运筹学运筹学Operat17 Inventory Theoryions Deterministic Modelsions ResearchA simple model representing this situation is the Deterministic Modelsfollowing economic order quantitymodel or , for This section is concerned with inventory problems where short, the actual demand in the future is assumed to be known. The costs to be considered are:Several models are considered, including the well-known economic order quantity formulation.¾Ks=etup cost for producing or ordering one Continuous eRivewU—niform Demandbatch,The most common inventory problem faced by ¾c=unit cost for producing or purchasing each manufactures, retailers, and wholesalers is that stock levels are over time and then are replenished by the arrival of ¾hh=olding cost per unit of time held in units.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo7l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog8y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Deterministic Modelsions Researchions ResearchThe following assumptions are madeThe objectiev of inevntory policy¾Units of the product are drawn from inventory The objective is to determine when and by how much continuously at a known constant rate, denoted by a ;that to replenish inventory so as to minimize the sum of these is , the demand is a units per unit per unit time.¾Inventory is replenished when needed by producing or Shortages Not Permitted:ordering a batch of fixed size (Qunits ), where all Qunits arrive simultaneously at the desired depicts the resulting pattern of inventory levels over time when we start at time 0 by producing or ¾We assume continuous review, so that inventory can be ordering a batch of Q units in order to increase the initial replenished whenever the inventory level drops sufficiently low. inventory level from 0 to Q.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo9l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Econom10ics©2006运筹学运筹学 Deterministic Deterministic Modelsions Researchions ResearchInventory level¾The total cost per cycle Tis obtained from the Qfollowing -at¾Production or ordering cost per cycle=cK+Q¾The average inventory level during a cycle is (Q0+)/2 Q/aunits per unit time, and the corresponding cost is hQ2/ per 2Q/aTime time. Because the cycle length is Q/a, The cycle length is Q/a¾oHlding cost per cycle =hQ2(/2a)QFor the speaker example, a cycle can be viewed as the Q-attime between production runs. If 24,000 speakers are produced in each production run and are used at the rate of 8,000 per month, then the cycle Q/2Q/alength is 24,0008/,0003= month..江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schooe & Economics©200611l of Information Technology, JiangXiUniversity of FinancSchool of Information Technolog12y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 17 运筹学Operations Research17-3运筹学运筹学 Deterministic Modelsions ResearchOperations Deterministic ModelsearchTherefore, total cost per cycleSimilarly, the time it takes to withdraw this optimal value K= +c Q+hQ2(/2a)of *,Q say t*, is given bySo the total cost per unit time is 2K+cQ+hQ/2aaKhQQ*2K2aKT==+ac+t*==Q*=Q/aQ2aahhdTaKh=−+=0As setup cost Kincreases, both *Qand t* increase (fewer 2dQ2Qsetups). When the unit holding cost h increases, both *Q and 2aKt* decrease (smaller inventory levels). As the demand rate a Q*=hincreases, *Q increases (larger batches) but t* decreases (more frequent setups).Above equation is the well-known economic lot-size formula.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog13y, JiangXiUniversity of Finance & Economics©2006Schoo14l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Deterministic Modelsions Researchions ResearchThese formulas for Q* and t* will now be applied to the speaker , the optimal solution is to set up the The appropriate parameter values from arek1=2,000, h=, a8=,000production facilities to produce speakers once 2ak(2)(,8000)(12,000)Q*==every months and to produce 25,298 = each * =25,9288/,000 = months江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog15y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog16y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Deterministic Modelsions Researchions ResearchThe resulting pattern of inventory levels over time is shown in when one starts at time 0 with an Shortages Permitted:inventory level of may be worthwhile to permit small shortages to Inventory levelSoccur because the cycle length can then be increased with a S-atresulting saving in setup tps=hortage cost per unit short per unit time =nventory level just after a batch of Qunits is added Q-S =shortage in inventory just before a batch of Qinventory,units is added返回24页江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom17ics©2006School of Information Technolog18y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 17 运筹学Operations Research17-4运筹学运筹学 Deterministic Modelsions Deterministic Modelsions ResearchSimilarly, shortages occur for a time (Q-S)/a. The average The total cost per unit time now is obtained from amount of shortages during this time is (0+Q-S)2/= (Q-S)2/ the following , and the corresponding cost is p(Q-S)2/ per unit time. Production or ordering cost per cycle = K +cQInventory levelThe average inventory level during this time is SS-at(S0+)2/S=2/ units, and the corresponding cost is hS2/ per unit t2S-athSShSholding cost per cycle ==Time cost per 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo1920l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Determinions Resistic Modelsions ResearchearchIn this model , there are two decision variables (Sand Q)Total cost per cycle 22 =K +cQ+hS(/2a)+p(Q-S)(/2a)∂ThSpQ(S−)=−=0∂SQQThe total cost per unit time is 22TaKhSpQS(−pQ)S(−)22=−−+−=0K+cQ+hSa/(2)p+Q(S−a)(/2)222∂QQQQQT=Qa/22Solving these equations simultaneously leads toaKhSpQ(S−)=+ac++QQ2Q22aKp2aKp+h2S*=,Q*=aKhSpQ(S−)hp+hhp=+ac++QQ2Q2江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo21l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo22l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Deterministic Modelsions Researchions ResearchFurther, from , the fraction of time that no 2aKp2aKp+hS*=,Q*=shortage exists is given by hp+hhpS*/ap=The optimal cycle length t* is given byQ*/ap+hQ*2Kp+ht*==In particular, when p→∞with h constant (so shortage aahpcosts dominate holding costs), Q*-S*→0 whereas both Q* and t* converge to their values for the preceding model The maximum shortage is (shortages not permitted). Even though the current model permits shortages, p→∞implies that having them is not *−S*=pp+h江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog23y, JiangXiUniversity of Finance & Economics©2006Schoo24l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 17 运筹学Operations Research17-5运筹学运筹学 Deterministic Modelsions Deterministic Modelsions ResearchIf shortages are permitted in the speaker example, the On the other hand, when h→∞with pconstant (so shortage cost is estimated in as p1=. costs dominate shortage costs), S*→0. thus, having As before,h→∞makes it uneconomical to have positive inventory levels, K1=2,000, h0=.30 a8=.000so each new batch of Q* units goes no further than removing So nowthe current shortage in */ap2aKp2aKp+h=S*=,Q*=Q*/ap+hhp+hhpInventory level(2)(8,000)(12,000)*==22,+ tS-at(2)(8,000)(12,+*==28,江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School ofSchoo26 Information Technolog25y, JiangXiUniversity of Finance & Economics©2006l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Deterministic Modelsions Researchions Research*Q2,8045uQantity Discounts, Shortages Not Permitted:===.36mnha,8000Suppose the unit cost varies with the batch size. For eHnce , the production facilities are to be set up every months to produce 28,540 speakers. example, suppose the unit cost for every speaker isc$=11 if less than 10,000 speakers are produced, Note that Q* and t* are not very different from the 1no-shortage $=10 if production falls between 10,000 and 08,000 2speakers, andQ*=$= if production exceeds 80,000 speakers..3t* = months江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog27y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog28y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学Operations Deterministic Deterministic Modelsions ResearchT1From the results for the first economic lot-size model 105000(shortages not permitted), the total cost per unit Tif the junit cost is cis given byjT1000002aKhQ95000T=+ac+,for j=1,2,.3jjQ2T390000For K1=2,000, h0=.30, and a8=,000, 85000The value of Q that minimizes Tjis82500Q2ak(2)(,8000)(12,000)Q*== =江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom29ics©2006School of In30formation Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 17 运筹学Operations Research17-6运筹学运筹学 Deterministic Modelsions ResearchOperations Deterministic ModelsearchThiPeriodic eRivew—A eGneral Model for s number 25298 is a feasible value for the cost function T. Production Planning2For any fixed Q, T <T,for all j, so T1 can be jj1-The preceding analysis explored the economic lot-size eliminated from further consideration. oHwever, Tcannot 3model. The results were dependent upon the assumption of be immediately discarded. a constant demand rate. Its minimum feasible value (which is occurs at Q8=0,000)must be compared to Tevaluated at 25298(which When this assumption is relaxed, ., when the 2amounts required from period to period are allowed to is $87,589). vary, the economic lot-size formula no longer ensures a Because Tevaluated at 80,000 equals $89,200, it is 3minimum cost to produce in quantities of 25,298, so this quantity is the optimal value for this set of quantity discounts.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technologr31y, JiangXiUnivesity of Finance & Economics©2006School of Information Technolog32y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Modelsions Deterministic Modelsions ResearchConsider the following model.¾The costs included in this model are similar to those for the first economic lot-size model:Model assumptions¾Planning is to be done for the next n periods regarding Ks=etup cost for producing or ordering any how much (if any) to produce or order to replenish units to replenish inventory at beginning of in each of the =nit cost for producing or ordering each unit,hh=olding cost for each unit left in inventory at ¾The demands for the respective periods are the end of (but not the same in every period ) and are denoted byrd=emand in period i , for i=1,2,… objective of inventory policyThe objective is to minimize the total cost over n periods.¾These demands must be met on time. ¾There is no stock on hand initially, but there is still We will illustrate this model by the following variation of time for a delivery at the beginning of period speaker example introduced in .江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo33l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Infomat34rion Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Deterministic Modelsions Researchions ResearchExample :A market survey conducted by the television In these units, the demands for speakers at the manufacturer has indicated that the demand for television beginning of the four upcoming periods (seasons) are sets is seasonal rather than =, r2=, r3=, r2=In particular, sales of 30,000 sets is forecast for the 1234Christmas season (October to December), 20,.000 for the winter slack season (The revised cost estimates for the speakers now are: January to March), 30,000 for the “new model “season (April to June), and 20,000 for the summer K=20,000, c=1, h0=.20season (July to September).The objective of inventory policyTo meet these sales in the respective seasons, the speakers must be available for assembly into the television set at the The objective is to determine how many speakers to beginning of the (if any) for the beginning of each of the four Since all the sales forecasts are integer multiples of periods in order to minimize the total ,000, it has been decided to produce the speakers in integer multiples of 10,000 as well.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom35ics©2006School ofn36 Iformation Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 17 运筹学Operations Research17-7运筹学运筹学Operat Deterministic Modelsions earchOperations Deterministic ModelsearchOne feasible inventory policy (but not an optimal one) is The inventory policy in is producing depicted in ,000 speakers at each beginning of the first period (Christmas season ), Inventory 6level60,000 speakers at the beginning of the second period, 510,000 speakers at the beginning of the fourth period. 43How can the optimal production schedule be found ? For 2this model in general, production (or ordering) is automatic 1in period 1,but a decision on whether on produce must be 1made for each Of the other n-1 periods. method to solving this model is described next in general terms.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo37l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo38l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Deterministic Modelsions Researchions ResearchPeriodic Review—Production Planning: An algorithmThis policy can easily be adjusted to satisfy the above characterization by simply producing one less unit in period The key to developing an efficient algorithm for finding 2 and one more unit in period optimal inventory policy (or equivalently, an optimal production schedule) for the above model is the following This adjusted policy (called it B) is shown by the dashed insight into the nature of an optimal in wherever B differs from A (the solid line). Now note that policy B must have less total cost than policy A. An optimal policy (production schedule) produces only The setup and the production costs for both policies are the when the inventory level is zerosame . To illustrate why this is true, consider the policy shown oHwever, The holding cost is smaller for B than for A in ,for the example. (Call it policy A) Policy A because B has less inventory than A in periods 2 and 3 (and violates the above characterization of an optimal policy the same inventory in the other periods). Therefore, B is because production occurs at the beginning of period 4 better than A, so A cannot be he inventory level is greater than zero.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo39l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog40y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Modelsions ResearchOperations Deterministic ModelsearchInevntor6ylevelA5Because it implies that the only choices for the 4amount produced at the beginning of the ithperiod are B3A0, r,r+r,……or r+r+….+r, it can be exploited to iii1+ii1+n2 Aand B Aand BBobtain an efficient algorithm that is related to the 11deterministic dynamic programming approach 1234perioddescribed in 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo41l of Information TechnologSchoo42y, JiangXiUniversity of Finance & Economics©2006l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 17 运筹学Operations Research17-8运筹学运筹学 Deterministic Modelsions ResearchOperations Deterministic ModelsearchC=C{K+cr+(++.r..+)+mimimmj+1i+1juIn particular, definej=i,i+1,....nh[r2r+3r.(+j−)i}]ri+1i+2i+3jCt=otal cost of an optimal policy for periods i, ii1+,.….n, when period i starts with zero inventory (before Notice:When j=n, the term C0=.n1+producing), for i1=,2,…. algorithm for solving the modelBy using the dynamic programming approach of The algorithm for solving the model consists basically of solving backward period by period by period,solving for C, C, ….Cin turn. For i1=, the minimizing nn-11value of jthen indicates that the production in period 1 should cover the demand through period j, so the second production These Cvalues can be found by first finding C, then inwill be in period j1+. For i+j=1, the new minimizing value of jfinding C, and so on. Thus, after C, C, …., Care n-1nn-1i1+identifies the time interval covered by the second production , found, then Ccan be found from the recursive iand so forth to the will illustrate this approach with the example.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo43l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog44y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic Deterministic Modelsions Researchions Research(3)The policy associated with Ccalls for producing only for Example:Returning to the speaker example, first we 3period 3and then following the optimal policy for period 4, consider the case of finding C, the cost of the optimal policy 4(4)whereas the policy associated with Ccalls for producing for from the beginning of period 4 to the end of the planning 3(3)periods 3 and 4. the cost Cis then the minimum of Cand horizon:33=(4)C. These cases are reflected by the policies given in =C2+1+(2)0=2+2+4=.0 345CTo find C, we must consider two cases, namely, the first 3Inevntory Inevntory 455leevlleevltime after period 3 when the inventory reaches a zero level, +with occurs at (1) the end of the third period or (2) the end of424the fourth period. +33In the recursive relationship for C, these two cases 31correspond to (1) j3= and (2) 22j4=.(Denote the corresponding costs (the right-hand side of the 311(3)(4)recursive relationship with this j) by Cand C, )=4江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo45l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog46y, JiangXiUniversity of Finance & Economics©2006+运筹学运筹学 Deterministic Deterministic Modelsions Researchions ResearchIn the recursive relationship for C, these cases correspond 2Therefore, if the inventory level drops to zero upon to (1) j2=, (2) j3=, and (3) j4=, where the corresponding costs (2)(3) (4)are C, Cand C, respectively. The cost Cis then the 2222entering period 3 (so production should occur then), the (2)(3)(4)minimum of C, C, and C,222production in period 3 should cover the demand for both (2)C=C2+1+(2)7=.42+2+1= 3periods 3 and 4.(3)C=C2+1+(23+)(+3)5/4=2+5+0+.61=(4)C=C2+1+(23+2+)3[+2+(2)]5/25To find C, we must consider three cases, namely, the first 0=2+7+1+.41==in{,11.,6 }1= period 2 when inventory reaches a zero level, which Consequently,if production occurs in period 2 (because occurs at (1) the end of the second period, (2) the end of the the inventory level drops to zero), this production should third period. Or (3) the end of the fourth period. cover the demand for all the remaining periods.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo47l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo48l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 17 运筹学Operations Research17-9运筹学运筹学 Deterministic Deterministic Modelsions Researchions Research(2)(4)Finally, to find C, we must consider four cases, namely, 1Note that C, and C, as the minimum, giving C. 111the first time after period 1 when the inventory reaches zero, (2)This means that the policies corresponding to C, and 1which occurs at the end of (1) the first period. (2) the second (4)C, as being the optimal , (3) the third period, or (4) the fourth period. These (4)(1)(2)The C, policy says to produce enough in period 1 to cases correspond to j1=,2,3,4 and to the costs C, C, 111(2)(3)(4)cover the demand for all four periods. The C, policy C,C, respectively. The cost Cis then the minimum of 1111(1)(2)(3)(4)C, C, C,and only the demand through period 2. Since the latter policy has the inventory level drop to (1)C=C2+1+(3)1=+3+1= at the end of period 2, the Cresult is used next, 3(2)C=C2+1+(32+)(=2)5/7=.42+5+0+.41=, produce enough in period 3 to cover the demand (3)C=C2+1+(32+3+)2[+2+(3)5/]4=+281++.61= periods 3 and 4. (4)C=C2+1+(32+3+2+)2[+2+(3)3+(2)5/]0=2+1++02+.81= resulting production schedules are summarized Cm=in {, 14.,8 15.,6 }=.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo49l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog50y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Deterministic ModelsOperat17 Inventory Theoryions Researchions Stochastic ModelsOptimal Production SchedulesThis section is concerned with inevntory 100,000 speakers in period 1problems where the demand for a period is a random avriable haivng a known probability Total cost $=148,. Both single-period and multi-period models are . Produce 50,000 speakers in period 1 and 50,000 speakers in period Single-Period Model with No Setup CostTotal cost$=148,:江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo51l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog52y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Stochastic Modelsions ResearchOperations Stochastic Modelsearch¾Planning is being done for just a single period.¾The objective is to minimize the expected total cost, where ¾The demand D in this period is a random variable with the cost componentsare a known probability = =unit cost for purchasing or producing each unit,¾There is no initial inventory on = =holding cost per unit remaining at the end of period ¾The decision to be made is the value of y, the number (includes storage cost minus salvage value),of units to purchase or produce for inventory at the p= =shortage cost per unit of unsatisfied demand beginning of the period.(includes lost revenue and cost of loss of customer goodwill).江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo53l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog54y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 17 运筹学Operations Research17-10运筹学运筹学 Stochastic Stochastic Modelsions ResearchOperations ResearchAnalysis of The Model:This single-period model may represent the inventory of ¾The decision on the value of y, the amount of inventory to an item that acquire, depends heavily on the probability distribution of ¾becomes obsolete quickly, such as the bicycle or a daily demand D. More than the expected demand may be desirable, but probably less than the maximum possible newspaper; ;demand. ¾spoi¾ls quickly, such as vegetables; A trade-off is needed between (1) the risk of being short ;and thereby incurring shortage costs and (2) the risk of ¾is stocked only once, such as spare parts for a single having an excess and thereby incurring wasted costs of ordering and holding excess units. production run of a new model airplane; ;¾This is accomplished by minimizing he expected value (in ¾or has a future that is uncertain beyond a single statistical sense) of the sum of these costs.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog55y, JiangXiUniversity of Finance & Economics©2006School of Informat56ion Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Stochastic Modelsions ResearchOperations Stochastic ModelsearchìDifD y<ïThis cost is also a random variable. The expected cost is ïThe amount sold is given byMin{D,y}=íïyifD y³then given by C(y),whereïî∞C(y)=E[C(D,y)]=(cy+pmax{0,d−y}+hmax{0,y−d})P(d)eHnce the cost incurred if the demand is D and y is ∑Dd=0stocked is given by ∞y−1C(D,y) =cy +p max {0, D-y} +h max {0,y-D.}=cy+p(d−y)P(d)+h(y−d)P(d)∑∑DDd=yd=0Because the demand is a random variable w[ith probability distribution P(d).]DUnless otherwise stated ,continuous demand is assumed throughout the remainder of this chapter.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School ofo57 InfSchool of Information Technolog58rmation Technology, JiangXiUniversity of Finance & Economics©2006y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Stochastic Modelsions Stochastic Modelsions ResearchSo, the expected cost CC(yy) is then expressed as oFr this continuous random variable D,∞ let j(x) =probability density function of DDC(y)=E[C(D,y)]=C(ξ,y)ϕ(ξ)dξD∫0 and∞=(cy+pmax{0,ξ−y}+hmax{0,y−ξ}ϕ(ξ)dξD∫0 F()= cumulative distribution function of ,∞=cy+p(ξ−y)ϕ(ξ)dξD∫ya=cy+L(y)Φ(a)=ϕ(ξ)dξD∫0Where L(y) is often called the expected shortage plus holding cost.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog60y, JiangXiUniversity of Finance & Economics©200659School of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 17 运筹学Operations Research17-11运筹学运筹学 Stochastic Stochastic Modelsions ResearchOperations Research0A similar result for the optimal order quantity is It then becomes necessary to find the value of y, say y, 0obtained. In particular, the optimal quantity to order y, is which minimizesC(y). First we give the answer, and then the smallest integer such thatwe will show the derivation a little −c0F(y)≥p−cD0The optimal quantity to order p+hΦ(y)=0p+hy, is that value which satisfiesModel with Initial stock Level:In the above model we assume that there is no initial If Dis assumed to be a binventory. As a slight variation, suppose that the initial stockF(b)=P(d)discrete random variable having ∑DDlevel is given by x, and the decision to be made is the value of d=0the cumulative distribution y, the inventory level after replenishment by ordering (or functionproducing) additional units. 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom61ics©2006Schoo62l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Stochastic Modelsions ResearchOperations Stochastic ModelsearchThe optimal inventory policy is the following:Thus, x-yis to be ordered, so that000ìï<y ordery x to yAmount available (y) =initial stock (x)ï +amount ordered If x í0ï³y do not order (x-y).ïîp−c00Φ(y)=Th hWere ysatisfiese cost equation presented earlier remains identical p+hexcept for the term that was previously cy. This term now Derivation of The Optimal Policy:becomes c(x-y), so that minimizing the expected cost is given We start by assuming that the initial stock level is ∞For any positive constants cand c, define g(ξ,y) as 1 2[c(y−x)+p(ξ−y)ϕ(ξ)dξminD∫yc(y–ξ) if y>ξy≥x1yg(ξ,y)= +h(y−ξ)ϕ(ξ)dξ]D∫0c(ξ-y) if y≤ξ2江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniver & Econom63sity of Financeics©2006School of Information Technolog64y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Stochastic Modelsions ResearchOperations Stochastic Modelsearch∞c−c0let 2Gy()g(yξ,)ϕd(ξ)cξy+Solving this expression results in DΦ(y)=∫0c+c2100Then G(y) is minimized at y=y, where yis the solution to0The solution of this equation minimizes G(y) becausec−c022Φ(y)=dG(y)c+c=(c+c)ϕ(y)≥02112D2dy0To see why this value of yminimizes G(y),note that ,by To derive the results for the case where the initial stock definition,level is x0>, recall that it is necessary to solve the y∞relationshipG(y)=c(y−ξ)ϕ(ξ)dξ+c(ξ−y)ϕ(ξ)dξ+cy1D2D∫∫0yy∞dG(y)∞y=cϕ(ξ)dξ−cϕ(ξ)dξ+c=01D2D0y{−cx+[p(ξ−y)ϕ(ξ)dξ+h(y−ξ)ϕ(ξ)dξ+cy]}dyminDD∫∫y0y≥x00cΦ(y)−c[1−Φ(y)]+c=012江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoor65l of Information Technology, JiangXiUnivesity of Finance & Economics©2006School of Informatn Technology, JiangXiUniversity of Finance & Econom66ioics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 17 运筹学Operations Research17-12运筹学运筹学 Stochastic Stochastic Modelsions ResearchOperations ResearchModel with Nonlinear oCsts:Note that the expression in brackets has the form of G(y) , with c=h, c=p, and c=c. hence the cost function to be Similar results for these models can be obtained for 12other than linear holding and shortage costs. Denote the minimized can be written asholding cost by{−cx+G(y)}miny≥xh [D-y] if y≥DHence, the valued of y0 that minimizes G(y) satisfies0 if y<−c0Φ(y)=p+hWhere h.[ ]is amathematical function, 2Furthermore, G(y) must be a dG(y)not necessarily linear.≥02convex function, becausedy江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom67ics©2006School of Information Technolog68y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Stochastic Modelsions ResearchOperations Stochastic ModelsearchSimilarly, the shortage cost can be added by then the total expected cost can be written as c( y–x) +L (y).p[D–y] if D≥y0 if D <yThe optimal policy is obtained by minimizing this expression, subject to the constraint that y≥x, that ishWere p[. ]is also a function , not necessarily {c(y−x)+L(y)} the total expected cost is given byminy≥xG(y)∞(y−x)+p(ξ−y)ϕ(ξd)ξ+hy(−ξ)ϕd(ξ)ξDD∫∫y0where x is the amount on ()If L(y) is defined as the expected shortage plus holding cost, .,0y∞yyL(y)=cy+p(ξ−y)ϕ(ξ)dξ+h(y−ξ)ϕ(ξ)dξDD∫∫y0江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog69y, JiangXiUniversity of Finance & Economics©2006School of Information Technolog70y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Stochastic Modelsions ResearchOperations Stochastic ModelsearchA Single-Period Model with a Setup CostIfLy[ L(y) is strictly convex [a sufficient condition being that the ϕshortage and holding costs each are convex and (ξ)>0>]], Dthen the optimal policy is the following In general, the setup cost will be denoted by K. To begin, 000<y <yorder y–y–x to bring inventory level up toy ythe shortage and holding costs will each be assumed to be If xlinear. Their resulting effect is then given by L (y), where0 ≥yydo not order∞y0dL(y)where yis the value of y L(y)=cy+p(ξ−y)ϕ(ξ)dξ+h(y−ξ)ϕ(ξ)dξDD+c=0∫∫y0that satisfies the expressiondy江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom71ics©2006School of72 Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 17 运筹学Operations Research17-13运筹学运筹学Operations Stochastic ModelsearchOperations Stochastic ModelsearchFrom ,it can be seen thatThus the total expected cost incurred by bringing the If x >S, then K +c y +L(y) >cx +L(x),for all x>yso thatinventory level up to yis given byK +c ( y–x ) +L (y) >L (x).K +c(y–x) +L(y) if y>xcy+Ly()L(x) if y=xDefine Sas the valued of ythat minimizes cy +L(y), and Kydefine s as the smallest value of yfor which cs +L(s) =K+cSsS +L(S). .江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog73y, JiangXiUniversity of Finance & Economics©2006Schoo74l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学Operations Stochastic ModelsearchThe optimal inventory policy is the following:<s order S-xto bring inventory level up If xtoS≥s do not orderp−cThe value of S is obtained fromΦ(S)=p+h江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom75ics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations Research18-118 Forecasting管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearch18 ForecastingInventoryTheory, was concerned with finding optimal inventory policies are , in part ,dependent upon some forecast of sales or use of the items of 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©200612interest. Forecasting is anessential 18 Forecasting18 Forecasting管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchTwo distinct quantitative techniques JgTJudmgental eTchnqiuqesbased on conventional statistical methods are uJdgmental techniques are,b ytheir ver yused in forecasting, namel,y time series nature, subjective, and the ymay involve such analsyis and regression analsyis .A statistical qualities as intuition, expert opinion, and time series is a series of numerical values that experience. hTe ygenerally lead to forecasts 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©200634a random variable takes on over a period of . Judgmental . Judgmental TTechniques管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchOne commonly used technique is the expert group technique. Perhaps the most important judgmental The Delphi method begins with the technique is called the Delphi method. Like the expert group panel of experts filling out a technique, the Delphi method utilizes a group of experts, but upon the results,a not in a meeting setting. second questionnaire is developed and sent 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©200656to the same panel of experts,together with 江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations . Judgmental TTechniques18 Forecasting管理运筹学Management pOerations eRsearch管理运筹学Management pOerations TSiTme eSriesOf course,the success of the Delphi method A time series can be viewed as representation of the outcomes of a random hinges on the quality of the design of the variable of concern,usuall ytaken at equall yspaced intervals, over a fixed period .For time questionnaires. Occasionally, more than two series,as illustrated in quarterl yrounds of questionnaires may be deemed sales of a particular product over the last three quarters also comprise a time situation occurs when there is behavior of a time series can be displaeyd in 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©200678graphical form,bar graphical form,or tabular considerabledisparitybetweentheresultsof18..2 TiTme Series18..2 TiTme Series管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchBecause a time series is a description of the past, a In most realistic situations,we do not have logical procedure for forecasting the future is to make use of these historical data. If the past data are indicative of what complete knowledge of the exact form of the we can expect in the future, we can postulate an underlying model that generates the time series,so an mathematical model that is representative of the process. approximate model must be chosen .Frequentl,y The model can then be used to generate choice is made by observing the outcomes 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©2006910fthtiiti18..2 TiTme Series18..2 TiTme Series管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchSeveral typical time series patterns are shown in Finall, shows a time series that shows a time series that might be might be observed if the generating process observed if the generating process were represented by a were represented b ya constant level constant level superimposed with random fluctuations. Figure shows a time series that might be observed if superimposed with a seasonal effect together the generating process were represented by a linear trend with random are man ysuper imposed with random fluctuations. 江西财经大学信息学院©2006江西财经大学信息学院©2006other plausible representations,but these three JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20061112江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations Research18-318..2 TiTme Series18..2 TiTme Series管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchSuch a representation can be given by x=A+ettOnce the form of the model is chosen,a where xis the random variable observed at time t, Ais t the constant level of the model, and eis the random error tmathematical representation of the generating occurring at time t (assumed to have expected value equal to zero and constant variance). process of the time series can be given .For Let Fdenote the forecast of the values of the time series tat time t+1,It is reasonable to expect that Fwill be a t1+function of some of, or all, the observed values of the time example, suppose that the generating process is series prior to time t+ as a constant-level model 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©2006superimposed with rand fluctuations, as 131418..3 Forecasting Procedures 18 Forecasting管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchfor a Constant--Level ModelXtLast-Value Forecasting ProcedureDenote by xthe value that the random variable takes on. FogFrecastin gProcedures for a tBy interpreting t as the current time, the last-value ConstantL-eLvel Modelforecasting procedure uses the value of the time series observed at time t(x) as the forecast at time 1t+1(x) F1t1+We now present four alternative Therefore, F=xt1+tFor example, if xrepresents the sales of a particular tproduct in the quarter just ended, this procedure uses forecasting procedures for the constant-level these sales as the forecast of the sales for the next quarter. model introduced in the preceding paragraph.江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©2006151618..3 Forecasting Procedures 18..3 Forecasting Procedures 管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchfor a Constant--Level Modelfor a Constant--Level ModelThis simple technique is frequently compared with the This forecasting procedure has the results of using a more sophisticated. but more umberome, disadvantage of being imprecise ;ie.,.its technique (such as the three described below) to assess variance is large because it is based upon a whether the sophisticated technique is of size is worth considering only if (1) the underlying assumption about the constant-level model is “shaky”and the 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20061718process is changing so rapidly that anything 江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations Research18-418..3 Forecasting Procedures 18..3 Forecasting Procedures 管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchfor a Constant--Level Modelfor a Constant--Level ModelThis estimate is an excellent one if the process is entirely AverageFgg oFrecasting Procedurestable , ., if the assumptions about the underlying model aRther than discarding all but the most are correct. However, frequently there exists skepticism about the persistence of the underlying model over an recent observation. this procedure averages all extended time. Conditions inevitably change eventually. the observations as the forecast for the next txiBecause of a natural reluctance to use very old data, this F=t+1∑ti=1period .ie.,.procedure generally is limited to young processes.江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©2006192018..3 Forecasting Procedures 18..3 Forecasting Procedures 管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchfor a Constant--Level Modelfor a Constant--Level ModelNote that this forecast is easily updated from period to Movingg-AverageFg oFrecasting gProcedureperiod. All that is needed each time is to lop off the first observation and add the last than using very old data that ma yno The moving-average estimator combines the advantages longer be relevant,this procedure averages the of the last value and average estimators in that it uses only recent history and it uses Multiple observations. A tdata for onl ythe last n periods as the forecast for xiF=t+1∑disadvantage of this procedure is that it places as much ni=t−n+1the next period,ie.,.weight onxas on -1+t江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©2006212218..3 Forecasting Procedures 18..3 Forecasting Procedures 管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchfor a Constant--Level Modelfor a Constant--Level ModelExSEponential mSoothing gFoFrecasting gProcedurehTis procedure uses the formulaIntuitively, one would expect a good F=αx+(1−α)Ft+1ttprocedure to place more weight on the most 2F=αx+α(1−α)x+α(1−α)x+Lt+1tt−1t−2recent observation than on older observations that ma ybe less representative of current 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©200624conditions .uOr next procedure does ujst 江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations Research18-518..3 Forecasting Procedures 18..3 Forecasting Procedures 管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchfor a Constant--Level Modelfor a Constant--Level ModelAnother alternative form for the exponential smoothing A measure of effectiveness of technique is given byF=F+α(x−F)t+1tttexponential smoothing can be obtained under which gives a heuristic ujstification for the assumption that the process is completely this procedurenI. particular, the forecast of stable, so that x1,x2,… independent, 22the time series at time t+1 is just the ασσvar[F]≈=t+12−α(2−α)αidentically distributed random variables with preceding forecast at time t plus the product 江西财经大学信息学院©2006江西财经大学信息学院©20062JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©200625varice σ .tI then follows tat (for large t )2618..3 Forecasting Procedures 18..3 Forecasting Procedures 管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchfor a Constant--Level Modelfor a Constant--Level Modelso that the variance is statistically equivalent to a moving average with （ 2 − α ） observations. If /ααis chosen equal to ,then .An important drawback of exponential （2−α）/α=19Thus the exponential smoothing technique with this value of is equivalent to a moving-average smoothing is that it lags behind a α trendi;e.,.if the constantl-evel model is incorrect and the mean is 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©200627increasing steadily, then the forecast will be 2818..3 Forecasting Procedures 18..3 Forecasting Procedures 管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchfor a Constant--Level Modelfor a Constant--Level ModelAnother disadvantage of exponential smoothing is that fI αis chosen to be small,response to it is difficult to choose an appropriate smoothing constant change is slow,with resultant smooth αExponential smoothing can be viewed as a statistical filler that inputs raw data from a stochastic process and estimatorsnO. the other hand, if is chosen to outputs smoothed estimates of a mean that varies with large, response to change is fast, with resultant large variabilit yin the , 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20062930there is a need to compromise, depending 江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations Research18-618..3 Forecasting Procedures 18..3 Forecasting Procedures 管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchfor a Constant--Level Modelfor a Constant--Level ModelIt has been suggested that αshould not exceed and that a reasonable choice for αis approximately 01. This where some initial estimate of the value can be increased temporarily if a change in the process is expected or when one is just starting the constant level A must be obtainedfI. past forecasting. At the start, a reasonable approach is to choose data are available,such an estimate ma ybe the forecast for period 2 according tothe average of these =αx+(1−α)(initialestimate)21江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information© A Forecasting Procedure 18 Forecasting管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchfor a Linear Trend A oFrecastin gProcedure for a iLnear where Xis the random variable that is tTTrend Modelobserved at time t,A is a constant,B is the Suppose that the generating process of the trend factor,and is the random error occurring at time t (assumed to have expected observed time series can be represented by a value equal to zero and constant variance)nI. the constant-level model, the forecast for linear trend superimposed with random period t1+ (based upon data through period t) fluctuations,as illustrated in .eDnote also provides the best current forecast for x=A+Bt+ettperiods t1+m+, for m1=,2,……… linear 江西财经大学信息学院©2006江西财经大学信息学院©2006the slope of the linear trend by B,where the JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20063334trend models,such a statement no longer holds . A Forecasting Procedure A Forecasting Procedure 管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchfor a Linear Trend Modelfor a Linear Trend ModelA smoothed lUnfortunately, the trend (slope) B is unknown so it evel at time t will be a linear must be estimated, and exponential smoothing can again be combination of and the smoothed value at used for this purpose , .,period t-1 corrected by adding the trend(slope) B=β(S+S)+(1−β)BS=αx+(1−α)(S+B)110t−111t−10S=αx+(1−α)(S+B）t1t−1t−1to indicate theF pas=saSge+ oBf a unit of time, +111江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20063536江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations A Forecasting Procedure A Forecasting Procedure 管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchfor a Linear Trend Modelfor a Linear Trend Modeland the forecast for m periods ahead (m=1,2………) is given smoothed values of the trend at the end of period t is given byF=S+mBt+mttB=β(S+S)+(1−β)B1tt−1t−1Summary of the Forecasting Procedure with a Linear Trend1. the smoothed level of the time series at time t is given forecast of the time series for m periods ahead(m1=,2………) is given byS=αx+(1−α)(S+B)1tt−1t−1F=S+mBt+mtt江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information© A Forecasting Procedure for a Constant Level18 Forecasting管理运筹学Management pOerations eRsearch管理运筹学Management pOerations . A Forecasting Procedure for a Constant Level with Seasonal-E-Effects ModelSuppose that the generating process of the observed time series can be represented by a nI man yforecasting problems,there exists seasonal effects that must be accounted for in constant level superimposed with seasonal the model .For example,in the above case of a ∗effects and randomX f=luAIctu+aetions,as illustrated in tttwholesale distributor of bicycles, it would be Fig . .Such a model can be represented byreasonable to have lower sales during the bad 江西财经大学信息学院©2006江西财经大学信息学院©2006weather of winter than later in the eyar, and JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information© A Forecasting Procedure for a Constant A Forecasting Procedure for a Constant Level管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchsmoothed levels at time of t for both of these The forecast for the next period then is given byfactors are useful prior to making a forecast . Exponential smoothing can again be used for this F=SIt+1tt−p+1purpose ;.,xtS=α+(1−α)Stt−1Where p is the number of periods in the seasonal cycle. It−pxtI=γ+(1−γ)Itt−pSt江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20064142江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations A Forecasting Procedure for a Constant A Forecasting Procedure for a Constant Level管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchThe forecast for m periods ahead (m=1,2,…….)is given byFor example ,if the seasonal periods are F=SIt+mtt−p+mwinter ,spring, summer, and autumn, then p=4. Note that Irepresents the smoothed value of the seasonal t-p For example, if a forecast for autumn 1998 is desired index for period t computed for the same season p and no data later than the spring 1995 are available, the periods ago; ., the seasonal index for autumn 1995 is seasonal index for autumn 1994 is used in place of that for based upon autumn 1994 data. autumn 1997.江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information© A Forecasting Procedure for a Constant A Forecasting Procedure for a Constant Level管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchThe forecast of the time series for m periods ahead the smoothed level of the time series at time t is given by (m=1,2,……)is given by xtS=α+(1−α)Stt−1F=SIt+mtt−p+mIt−pthe updated smoothed level of the seasonal index for period t is given byA routine for this forecasting procedure is included in xyour OR Course –ware. tI=γ+(1−γ)Itt−pSt江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information© A Forecasting Procedure for a Constant A Forecasting Procedure for a Constant Level管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchExample: Consider the bicycle example of Sec. . Upon examination of past data, the linear trend model has been found to be inappropriate and has been replaced by A description of the initialization procedure the seasonal model described above. The seasonal sales figures for both last year and this year (to date) are shown is best given in the context of an example,as in Table . The time index t to be used for each season also is given in the table. (The fact that t=1 for winter of this done indicates that this winter’s sales provide the first data point江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20064748江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations A Forecasting Procedure for a Constant A Forecasting Procedure for a Constant Level管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchA reasonable way to initially estimate the seasonal x12=,800 for the exponential indices is to divide last year’s seasonal average sales over smoothing,although last eyar’s sales figures the last year , that is (2,786+2,928+3,025+3,061)/4 =2,950. These values are shown in the rightmost column of will be used for initialization and for Table . The initial estimate of the constant level A is chosen to be the average of the four seasons over the past determining the initial seasonal indices.) The 295γ=.S=0α=, that is . We will use and .0goal now is to use the above procedure to 江西财经大学信息学院©2006江西财经大学信息学院©2006develop a forecast F4 for autumn of this eyar .JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information© A Forecasting Procedure for a Constant A Forecasting Procedure for a Constant Level管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchWith these initial estimates, the forecast for this InIitial seasonal SeasonLasYearThYt YTis eYarInIdexWinterwinter would have been=-t=-3 2786==-t1= 2800t=-3 =SI=2950()=2785=-t=-2 29298=95=-t=2 229t=-2 −3Summe=-=t=-1 3025t=3 3040=-t=-1 obtain the forecast for this spring quarter, the r=t0= 3061=t0= level of the time series S1 must be obtained. 11800The smoothed level of the seasonal index for the next year’s winter quarter forecast is also obtained:江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information© A Forecasting Procedure for a Constant A Forecasting Procedure for a Constant Level管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchThe forecast for this summer quarter F2 ,we need =α+(1−α)S=+()= calculate S2 and I2−3x2925x280021S=α+(1−α)S=+(2952)=295121I=γ+(1−γ)I=+()=−−2S29521x29252I=γ+(1−γ)I=+()=−2hTe forecast for this spring then would S29512have beenF=SI=2951()=302532−1F=SI=2952()=293121−2江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20065354江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations A Forecasting Procedure for a Constant A Forecasting Procedure for a Constant Level管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchTherefore, the desired forecast isSo finall yto obtain the forecast for this autumn quarter F4, we calculate S3 F=SI=2952(1038)=3064 and 3I 430x30403S=α+(1−α)S=+(2951)=295232L1025−1Similarl,y the current forecast for next x30403winter isI=γ+(1−γ)I=+(1025)=10263−1S2952F=SI=2952()=27903531江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information© A Forecasting Procedure for a Constant Level18 Forecasting管理运筹学Management pOerations eRsearch管理运筹学Management pOerations FoEFrecast rErorSeveral forecasting techniques have been This example now has been considered from the presented together with different underlying viewpoint of having a linear trend( the preceding section) or models of the time series .oHw does one having seasonal effects(here). It also is possible to construct a compare these techniques , especially if the model that has both a linear trend and seasonal effects,but we generating process is unknown ,as is frequentl ywill not delve further into this case in practice? Some measure of E=x−Fttt江西财经大学信息学院©2006江西财经大学信息学院©2006performance is of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©. Forecast . Forecast EError管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchFor the bicycle example, using this year’s spring and The forecast error is also referred to as the residual . A summer sales as the observations together with the forecasting procedure that produces “small”absolute values of appropriate forecasts for these seasons, the following are mean Et is desirable. A measure of small is the mean square square errors for three forecasting procedures discussed error(MSE) associated with the forecasting technique. If there earlier:are n time periods, then 22(2925−2800)+(3040+2925)last−valueforecastingMSE==14424222(2925−2945)+(3040−3042)LineartrendforecastingMSE==2022222E+E+L+E12n22MSE=(2925−2931)+(3040−3025)SeasonalforecastingMSE==131n2江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20065960江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations . Forecast Erro18 Forecastingr管理运筹学Management pOerations eRsearch管理运筹学Management pOerations Box-eJnkins MethodTherefore based upon the MSE measure of nI practice, a forecasting procedure often performance,the seasonal forecasting procedure is chosen without adequately checking is the best of the three. Last –value forecasting whether the underlying model is an performs much worse than the other two .appropriate one for the application .hTe oHwever ,these results are based upon onl y two beaut yof the Boxe-Jnkins method is that it pieces of past data (sales figures for two past 江西财经大学信息学院©2006江西财经大学信息学院©2006carefull ycoordinates the model and the JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20066162quarters ), so there is no assurance that similar . oBx--Jenkins . oBx-J-Jenkins Method管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchThe historical data are used to test the validity of the hTe oBxe-Jnkins method is iterative in model. The model also generates an appropriate forecasting nature .First ,a model is chosen .oT choose accomplish all this the Box –Jenkins method requires this model , we must compute autocorrelations a great amount of past data (a minimum of 50 time period), and partial autocorrelations and examine their so it is used only a conceptual overview of the method.( See patterns .An autocorrelations measures the Select References 2 and 5 for further details.)江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006corJrieanlgaXitiUoninve rbsiteyt owf eFienanc et i& mEcoeno msiecsr, iSechso ovl aofl uInefosrm ation©. oBx-J-Jenkins -. oBxJ-Jenkins Method管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchTherefore, the autocorrelation for a lag of two periods forward a fixed number of periods , holding measures the correlation between every other observation; the effect of the other lagged times fixed. Good ., it is the correlation between the original time series andestimates of both the autocorrelations and the the same series moved forward two periods. The partial autocorrelation is a conditional autocorrelation between the partial autocorrelations for all lags can be original time series and the same series movedobtained by using a computer to calculate the 江西财经大学信息学院©2006江西财经大学信息学院©2006sample autocorrelations and the sample partial JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20066566江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations . oBx--Jenkins . oBx--Jenkins Method管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchFrom the autocorrelations and the partial auto Similarl,y we can examine he behavior of correlations, we can identify the functional form of one or the estimated parameters .fI both the residuals more possible models because a rich class of model is characterized by these quantities. Next we must estimate and the estimated parameters behave as the parameters associated with the model by using the expected under the presumed model, the historical data. Then we can compute the residuals and model appears to be validated .fI the ydo examine their behavior. 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©200667not,then the mdel should be modified ad the68 . oBx--Jenkins . oBx--Jenkins Method管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchFor example , suppose that the sample autocorrelations with the time series data,we then obtain the and the sample partial autocorrelations have the patterns residualsx−(b+bx+bx)shown in Fig . The sample autocorrelations appear to t01t−12t−2decrease exponentially as a function of the time lags followed by values that seem to be of negligible magnitude. This behavior is characteristic of the functional formX=B+BX+BX+et01t−12t−2t江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©-J. oBx-Jenkins -. oBxJ-Jenkins Method管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchIf the assumed functional form is adequate,the residuals hTe oBxeJ-nkins procedure appears to be and the estimated parameters should behave in a predictable particular,the sample residuals should behave a complex one,and it , computer approximately as independent, normally distributed random variables,each having mean 0 and variance (assuming that software is a variables . hTe programs the random error at time period has mean 0 and variance ).The estimated parameters should be uncorrelated calculate the sample auto correlations and the and significantly different from tests are sample partial autocorrelations necessar yfor available for this diagnostic checking.江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006ideJniatngiXfiiynUngiv etrshitey offo Frimnan coe &f Etchoneom imcs,o SdchoeolhT. ofe yIanflosrmoat ion©20067172江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations . oBx-J-Jenkins Method18 Forecasting管理运筹学Management pOerations eRsearch管理运筹学Management pOerations LRgiLnear eRrgessionThese programs, however, cannot accurately identify one or more models that are compatible with the Statistical problems often are concerned autocorrelations and the partial autocorrelations. Expert human judgment is required. This expertise can be acquired, with data where exists a relationship between but its beyond the scope of this text. Although the Box-Jenkins method is complicated, the resultant forecasts are two variables. This section highlights the extremely accurate and, when the time horizon is short, better than most other forecasting techniques. Furthermore, results when the relationship is linear .For the procedure produces a measure of the forecast error. example, suppose that a publisher of textbooks 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©2006737418..8 Linear RRegression18..8 Linear RRegression管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchThis latter method uses an extensive advertising hTus,if the number of mail order sales campaign through publishing media and direct mail. The for a book is denoted b yX and the number of advertising campaign is conducted prior to the publication of the book. The sales manager has noted that bookstore sales by Y ,then the random there is a rather interesting linear relationship between variables Xand Yexhibit a degree of the number of mail orders and the number sold through there is no functional bookstores during the first year. He suggests that this relationship be exploited to determine the initial press run relationship between these two random for subsequent i;e.,. given the number of mail order 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20067576sales,one does not expect to determine exactly 18..8 Linear RRegression18..8 Linear RRegression管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchWhat, then, is meant by statement , “The sales manager has noted that there is a rather interesting linear relationship between the number of mail orders and the hTus,if the number of mail order sales number sold through bookstores during the first year ”? Such a statement implies that the expected values of the is x for man ydifferent books, the average number of bookstore sales is linear with respect to the number of mail order sales, .,number of corresponding bookstore sales E[Y|X=]=A+BE[Y|X=x]=A+Bxwould tend to be approximately A+Bx .This 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006relJaiatnigoXinUsnihveirpsi tby eoft Fwineanecen & XEaconomicds, Y Sicsho orle offe Irnrfoermdat iton© 2a0s06 7778江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations Research18-1418..8 Linear RRegression18..8 Linear RRegression管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchOther examples of this degree of The degree of association model is not the only model of interest. In some cases ,there exists a functional relationship association model can easily be between two variables that may be linked linearly. In a educator may be interested in the relationship forecasting context ,one of the two variables is time, while between a student’s performance on the the other is the variable of interest. In , college entrance examination and subsequent performance in engineer ma ybe 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20067980interested in the relationship between tensile 18..8 Linear RRegression18..8 Linear RRegression管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchwhere A is a constant,B is the slope ,and is the random error,assumed to have expected value equal to zero and constant variance.(The symbol can also be read as X given t or as Xt|.)It follows thatsuch an example was mentioned in the E(x)=A+Bttcontext of the generating process of the time Note that both the degree of association model and the x=A+Bt+ettseries being represented b ya linear trend exact functional relationship model lead to the same linear relationship,and their subsequent treatment superimposed with random fluctuations,.,江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©2006818218..8 Linear RRegression18..8 Linear RRegression管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchwith t taking on integer values starting with 1,leads to certain simplified expressions. In the standard notation of is almost the publishing regression analysis,X represents the independent variable and Y represents the dependent variable of interest. example will be explored further to illustrate Consequently ,the notational expressions for this special time series model now becomes E(x)=A+Btthow to treat both kinds of models,although Y=A+Bt+ettthe special structure of the model江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20068384江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations Research18-1518..8 Linear RRegression18..8 Linear RRegression管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchMeLqSthod of eLast uqSaresis drawn through the data. A measure of how well this Suppose that bookstore sales and mail line fits the data can be obtained by computing the sum of order sales are given for 15 books .hTese data squares of the vertical deviations of the actual points from appear in Table ,and resulting plot is given the fitted line. The proposed measure of fit is then given byin Fig. on is evident that the points in Fig. 152222~~~~Q=(y−y)+(y−y)+L+(y−y)=(y−y)11221515∑iido not lie on a straight line .Hence it is not i=1~clear where the line should be drawn to show Y=a＋bxthe linear relationship .Suppose that an 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20068586arbitraryline,givenbytheexpression18..8 Linear RRegression18..8 Linear RRegression管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchnnnn(x−x)(y−y)xy−(xy)n∑ii∑ii∑i∑iThe usual method for identifying the “best”fitted line is i=1i=1i=1i−1b==n2nn2⎛⎞2(x−x)∑ix−⎜x⎟the method of least squares. This method chooses that line ∑i∑ini=1i=1⎝i=1⎠a+bXthat makes Q a minimum. Thus a and b are obtained −−Anda=y−bxsimply by setting the partial derivatives of Q with respect to an−xix=∑and b equal to zero and solving the resultant equations. This whereni=1n−ymeithod yields the solutiony=∑andni=1江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©2006878818..8 Linear RRegression18..8 Linear RRegression管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchFor the publishing example,the date in Table and Fig yieldeHnce ,the leasts-quares estimate of −−y==~bookstore saYle=s −w1i9t0h4 ＋il1 sales x is 15−−⎛⎞⎛⎞x−xy−y=⎜⎟⎜⎟∑ii⎝⎠⎝⎠i=1given byand this is the line drawn in Fig. . Such a line is 215−a=−⎛⎞referred to as a regression line. x−x=11966⎜⎟∑i⎝⎠i=1b=江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20068909江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations Research18-1618..8 Linear RRegression18..8 Linear RRegression管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchThe decision maker may be interested in some measure of A routine called Method of Least Squares is available in uncertainty that is associated with this forecast. This measure your OR Courseware for calculating a regression line in this is easily obtained provided that certain assumptions can be made. Therefore ,for the remainder of this section ,it is thatThis regression line is useful for forecasting purposes. A random sample of n pairs (1X,1Y), For a given value of x, the corresponding value of y (X2,Y2) ．．．．．．（nX,Yn）is to be represent the forecast .taken .江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20061929hTe are normall ydistributed with mean 18..8 Linear RRegression18..8 Linear RRegression管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchThe assumption that is normally distributed is not a ConIEEY|fidence nIterval sEtimate of E(xY|==x**)critical assumption in determining the uncertainty in the forecast, but the assumption of constant variance is crucial. A ver yimportant reason for obtaining Furthermore ,an estimate of this variance is required .the relationship between two variable is to 22An unbiased estimate of σis given by S, wherexy|n2~use the line to for future decision making .2(y−y)ii=∑Sy|xn−2i=1From the regression line, it is possible to 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20063949tit(E|Y)bittit(th18..8 Linear RRegression18..8 Linear RRegression管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchThe endpoints of a(100)(1-α) percent confidence For example ,the publisher might want to use this interval for E(YX|=X*) are given byapproach to estimate the expected number of bookstore 2−⎛⎞x−x⎜⎟+1sales corresponding to mail order sales of ,say, 1,400,by ⎝⎠a＋bx−ts1+++α2;n−2yx2n−n⎛⎞x−x⎜⎟∑iboth a point estimate and a confidence interval estimate ⎝⎠i=12for forecasting purpose.−⎛⎞x−x⎜⎟+1⎝⎠a＋bx+ts1+++α2;n−2yx2n−n⎛⎞x−x⎜⎟∑i⎝⎠i=1江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©20065969江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations Research18-1718..8 Linear RRegression18..8 Linear RRegression管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearch21()Lower=6060−()+hTe fact that the confidence interval was 1511966obtained at a data point (x=1,400)is purely 21()upper=6060+()+ Method of Least Squares routine in 2n−2⎛⎞x−x⎜⎟∑iSy|x⎝⎠i=1江西财经大学信息学院©2006江西财经大学信息学院©2006oyur ROCourseware dose most of the JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©2006798918..8 Linear RRegression18..8 Linear RRegression管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchPredictionsThe two endpoints of a prediction interval are given by the expressionshTe confidence interval statement for 2−⎛⎞x−x⎜⎟+the expected number of bookstore sales 1⎝⎠a＋bx−ts1+++α2;n−2yx2n−n⎛⎞x−x⎜⎟∑corresponding to mail order sales of 1,400 may i⎝⎠i=1And be useful for budgeting purposes,but it is not 2−⎛⎞x−x⎜⎟+1too useful for making decisions about the ⎝⎠a＋bx+ts1+++α2;n−2yx2n−n⎛⎞x−x⎜⎟∑iactual press run. nIstead of obtaining bounds ⎝⎠i=1on the expected number of bookstore 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©200699100salesthiskindofdecisionrequiresboundson18..8 Linear RRegression18..8 Linear RRegression管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchinterval for the number of bookstore For an ygiven value of x(denoted here sales is given by 6060±315 , which is b y +X),the probabilit yis 1-αthat the naturall ywider than the confidence interval value of the future Y+ associated with +Xfor the expected number of bookstore in this the publisher can find an Thus ,in the publishing example,if +Xinterval that will contain bookstore sales 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©2006is 1,400 ,then the corresponding95pecent 101102corresponding to particular mail order sales 江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations Research18-1818..8 Linear RRegression18..8 Linear RRegression管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchTpossibly unknown ,even this technique he reason is that these statements would all be based upon the same based upon the (incorrect ) assumption of statistical data, so that the statements would not beindependence would be useless. statistically independent. If the statements were independent and if k future bookstore sales are predicted, This difficulty can be overcome by using with each statement being made with probability 1-α,then simultaneous tolerance intervals. Using this the probability is that all k predictions of future bookstoretechnique ,the publisher can take the mail order sales are correct. However ,if k is large orsales of an ybook,find an interval(based on the 江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©2006103previou104sl ydetermined linear regression 18..8 Linear RRegression18..8 Linear RRegression管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchFurthermore ,the probability is p that all these 2−⎛⎞predictions are correct. An alternative interpretation is as x−x⎜⎟+1**⎝⎠a＋bx−cs1+++yx2follows . If every publisher followed this procedure ,each n−n⎛⎞x−x⎜⎟∑iusing his or her own linear regression line, then 100p percent ⎝⎠i=12−of the publishers (on average) would find that at least 100⎛⎞x−x⎜⎟+1**⎝⎠（1-α）percent of their bookstore sales fell into the a＋bx+cs1+++yx2n−n⎛⎞x−x⎜⎟predicted intervals. The expression for the endpoints of each ∑i⎝⎠i=1such tolerance interval is given by江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©200610510618..8 Linear RRegression18..8 Linear RRegression管理运筹学Management pOerations eRsearch管理运筹学Management pOerations eRsearchHence ,the number of bookstore sales corresponding to hTus the publisher can predict that the mail order sales of 1,400 books is predicted to fall in the bookstore sales corresponding to known mail interval 6060±3759 . If another book had mail order sales order sales will fall in these to learn it intervals .of 1,353,the bookstore sales are predicted to fall in the Such statements can be made for as many books interval 5258±390 ,and so on. At least 95percent of the bookstore sales will fall into their predicted intervals, and as the publisher desires .Furthermore ,the these statements are made with confidence y is p that at least 100（1-α）江西财经大学信息学院©2006江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006JiangXiUniversity of Finance & Economics, School of Information©2006107108percent of bookstore sales corresponding to mail 江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 18 运筹学Operations Research18-1918..8 Linear RRegression管理运筹学Management pOerations eRsearchTo summarize, we now have described three measures of forecast uncertainty . The first (in the preceding subsection) is a confidence interval on the expected value of the random variable Y (for example, bookstore sales) given the observed value x of the independent variable X (for example, mail order sales) . The second is a prediction interval on the actualvalue that Ywill take on, ,given x. The third is simultaneous tolerance intervals on a succession of actual values that Y willtake on given a succession of observed values of X.江西财经大学信息学院©2006JiangXiUniversity of Finance & Economics, School of Information©2006109江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 20 运筹学Operations Research20-1运筹学运筹学Operations ResearchOperations Research20 Decision A Prototype ExEampleD20 Decision analysis¾TheEKEC GOFERBROKE COMPANY owns a tract of land that ¾DDecision analysyis provides a framework and may contain oil. A consulting geologist has reported to management that she believes there is 1 chance in 4 of for rational decision making in this context.¾Because of this prospect,another oil company has offered to purchase the land for 90$00,0H90$00,0. oHwever, Goferbrokeis ¾Decision analysis divides decision making between the considering holding the land in order to drill for oil itself.¾If oil is found, the company’’s expected profit will be cases ofw iwthout experimentationandwwith experimentation..approximate$700,$10000,0ly $700, loss of $10000,0will be incurred if the land is dry (no oil). Table summarizes these data. Thiscompany is operating without much capital, so a loss of $10000,0$10000,0would be quences of the various possible outcomes.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Informaton Technology, JiangXiUniversity of Finance & Econom1iics©2006Schoo2l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学Operations ResearchOperations A PrototypeE Example20 Decision Decision Making withoutStatsu ofLandExEperimentationDryAlternativeOil¾The decision makekrneeds to choose one of the possibleDrill for oil 7$00,000 1$00,000actions..Sell the land09$,000 9$0,000¾NNaturethen would choose one of the possiblestates of hCance of statsu1 in 4 3 in 4nature.¾EEach combination of an action a and state of nature θTable prospective profits forwould result in a payoff, p(a ,θ),which is given as one of the the GoferbrokeCompanyentries in a payoyffb table..¾This payoff table should be used to find an optimal actionfor the decision maker according to an appropriate criterion.江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006SchooSchoo4l of Informat3ion Technology, JiangXiUniversity of Finance & Economics©2006l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Decision Making without Decision Making without ExEperimentationions Researchions ResearchFPEFormulation of the Prototype Example in This State of θθ12NatureDryFFrameworkcAtionOil¾As indicated in Table ,the GoferbrokeCCo. has two aDrill for oil 700 -1001possible action under consideration:drill for oil or sell the land. aSell the land90 902¾The possible states of nature are that the land contains oil andrPior it does not,as designated in the column headings of Table oil and the consulting geologist has estimated that there is 1 chance in 4 of oil (and so 3 chances in 4 of no oil), Table prior probabilities of the two states of nature are ,repectively. ¾Therefore, with the payoff in units of thousands of dollars of profit, the payoff table can be obtained directly from Table , as shown in Table 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of5 Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Econom6ics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 20 运筹学Operations Research20-2运筹学运筹学 Decision Making without ExEperimentationions Decision Making without ExEperimentationions ResearchState of θθ12The MPMaximCinPayoff rCiterionNatureDrycAtionOilMinimuma-100Drill for oil 700 -100¾Maximinpayoff criterion: For each possible action, find 1S90Maximinvalueaell the land90 902the minimum payoff over all possible states of nature. Next, Table the maximum of these minCimum payoffs. Choose the State of θθ12Natureaction whose minimum payoff hives this ¾Table shows the application of this criterion to the aDrill for oil 700 -1001aMximum in this aSell the land90 902columnprototype example. Thus, since( the minimum payoff for a(90) 22rPior larger that fora(-100,)100,)action a(sell the land) will be 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo7l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog8y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Decision Making without Decision Making without ExEperimentationions Researchions ResearchBaBye’Ds’eDcision RuRleThe MaMximum LiLkelihood CrCiterion¾Bayes’’decision rule: Using the best available estimates ¾MaMximum likelihood criterion::Identify the most likely state of of the probabilities of the respective states of nature nature (the one with the largest prior probabFility). For this state (currently the prior probabilities ), calculate the expected of nature, find the action with the maximum payoffC. Choose this value of the payoff for each of the possible actions. action.¾For the prototype example, these expected payoffs are calculated directly from Table as follow:Applying this criterion to the example, table indicates that θ has the largest prior probability. In the θ column, 22E[P(a,θ)]=(700)+(−100)=1001 a has the maximum payoff, so the choice is to sell the [P(a,θ)]=(90)+(90)=902Since 100 is larger than 90, the action selected is a1(drill for oil).江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo9l of Information Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technolog10y, JiangXiUniversity of Finance & Economics©2006运筹学运筹学Operations ResearchOperations Decision Making with ExEperimentationearch20 Decision analysisCContinuing the Prototype Decision Making with ¾A seismic survey obtains seismic soundings that indicate ExEperimentationwhether the geological structure is favorable to the presence of oil. To quantify this process, we let the random ¾We first updateC the GoferbrokeCo. example to variable Sandincorporate experimentation, then describe how to derive the posterior probabilities, and finally discuss how to ¾Its possible values be defined as follows:decide whether it is worthwhile to conduct S = =statisbtic obtained from seismic =0:Ub =0 : Unfavorableg;yk seismic soundings ;oil fairl =1:bg;yk =1 : Favorable seismic soundings ;oil is fairl 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog11y, JiangXiUniversity of Finance & Economics©2006School of In12formation Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 20 运筹学Operations Research20-3运筹学运筹学 Decision Making with Decision Making with ExEperimentationions ResearchOperations ResearchPosterior ProbabilitiesBased upon past experience, if there is oil (.,the true state of nature is θ),then the probability of S = =00is 1¾Proceeding now in general terms, P(=0|)=,PS( = 0|θ==θ ) =,1we let soP(=1|PS( = 1|θ==θ)== ) = == =number of possible states of nature;;Similarly, if there is no oil,( .,the true state of natureis P(P(θ==θ) = =prior probability that true state of nature iθ), then=0 the probability ofS =0is estimated to be 2isθ,iP(=0|PS( =)=, 0|θ= =θ ) =,2for=i =1,2,……,n;;S=zg( =statistic summariizng results of experimentation (a soP(=1|PS( = 1|θ=)=1- =θ= ) = = variabble );); S= =one possibble value of S;;江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of13 Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo14l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Decision Making with Decision Making with ExEperimentationions Researchions Research¾Therefore, for each1=2,,…,, i 1=2,,…,n,the desired formula for the P(|P(θ= =θS=)=| =s) =posterior probability that true state icorresponding posterior probability is of nature is θi, given that P(S=s|θ=θ)P(θ=θ)iiS =,=1,2,…, =s ,for i =1,2,…,nP(θ=θ|S=s)=inPS=s|θ=θ(Pθ=θ)¾The question currently being addressed is the following:∑jjj=1Given P(=P(θ =θ)P(=| )and PS( =s |θ=,) =θ ,)for =i = ii¾then the posterior probabilities are1,2,…,(P=|=)?1,2,…,n,, what is(Pθ= S| = s) ? (=)Pθθ|S=()1iP(θ=θ|S=0)==1P(θ=θ|S=s)=()+()7()P|S=s16nP(θ=θ|S=0)=1−=277P(|S=s)=P(θ=θ|S=s)∑ij=1)Pθ=θ|S=s=Pθ=θ|S=sP(θ=θ)iii江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo15l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoo16l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Decision Making with Decision Making with ExEperimentationions Researchions Research¾Similarly, if the seismic survey gives S1=1=, then ¾Since the objective is to maximize the expected payoff, ()1P(θ=θ|S=0)==these yield the optimal policy shown in Table ()+()2¾EXPECEVEECEEXECTED VALUE OF PERFCTEINFORMATION: 11P(θ=θ|S=1)=1−=222Suppose now that the experiment could definitely identify ¾When you are dealing with larger problems ,the tabular what the true state of nature is, thereby providing “perfect”algorithm provides a convenient way of organizing these information. whichever state .computations. ¾This calculation is shown in 6for the prototype example, where the expected payoff with perfect information is .江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©200617School of Information Technolog18y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 20 运筹学Operations Research20-4运筹学运筹学 Decision Making with Decision Making with ExEperimentationions ResearchOperations ResearchEpxected aPyoffEpxected aPyoffState of θResulθt12Optimal ActionEcxludingIncluding NatureSeismicDrycAtionOilCost of SurveyCost of SurveySurveya1Drill for oil 700 -100S0=Sell the land90 60a2Sell the land90 90S1=Drill for oil300270Maximum payoff700 90rPior payoff iwth perfect information0=(700)0+(902)= 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo19l of Information Technology, JiangXiUniversity of Finance & Economics©2006Schoolog20l of Information Technoy, JiangXiUniversity of Finance & Economics©2006运筹学运筹学 Decision Making with xEperimentationions ResearchOperations Research20 Decision analysis¾The expected value of perfect information, abbreviated Decision TreesEVPEVI, is calculated as ¾The corresponding decision tree are referred to as forks, and the arcs are called =ywPIEVe=xpected paoyff with perfect informationAk,byq,A decision fork, represented b ya square, indicates that –yw–expected paoyff without experimentation..a decision needs tob be made at that point in the . A ¾For the prototype example, we found in that the chance fork,k, represented byb, ya circle, indicates that a random event occurs. at that payoff without experimentation (under Bayes’’decision00 rule) is ,PEV2=-01=.0452PIEV2=-01=.0452江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of In21formation Technology, JiangXiUniversity of Finance & Economics©2006Schoo22l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学Operations ResearchOperations Decision Decision Trees670)f7¾/Following the open paths from left to right in 1l( lliirOD60now yields the following policy.-130DFigure ()6ac/70=r)(SoOptimal policy:vasfelln60UDo the seismic survey.)2/1( liO123670270If the result is unfavorable(0,=) s(0,=)sell the ( the result, is favorable(1=) sa)(1,=)drill for -130a(1/b2()SaPyoffled=The expected payoff (including the cost of 1)123experimentat123ion) is )¾For any decision tree, this backward induction procedure 4/1( liO700100always will lead to the optimal policy (or policies) after hllirDthe probabilities are computed for the branches D100ry-100 (3/4)emanating from a chance 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog23y, JiangXiUniversity of Finance & Economics©2006Schoo24l of Information Technology, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006Do seismic survey()yevrus cimsies oN
Session 20 运筹学Operations Research20-5运筹学运筹学Operations ResearchOperations Research20 Utility Theorycision Utility TheoryUtility Functions for Money¾For example, suppose that an individual is offered the choice of (1) accepting a 50:50 chance of winning001,$00 001,$00¾Figure shows a typical utility function u(M) for money or nothing or (2) receiving $400,0$400,0with certainty. Many M. it indicates that an individual having this utility function people would prefer the $40,000$40,000even though the expected payoff on the 50:50 chance of winning1$0000,0 1$0000,0is $0,50$0,50. a would value obtaining $01,00$01,00twice as much as$300,0. $300, may be unwilling to invest a large¾When a utility function for money is incorporated into a ¾sum of money in a new product even when the expected decision analysis approach to a problem, this utility function profit is substantial, if there is a risk of losing its investment must be constructed to fit the preferences and values of the and thereby becoming bankrupt. People buy insurance even decision maker involved .(the decision maker can be either a though it is a poor investment from the viewpoint of the expected individual or a group of people .)江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog25y, JiangXiUniversity of Finance & Economics©2006Schoo26l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学Operations ResearchOperations Utility Utility Theoryu (M)4¾The key to constructing the utility function for money to fit the decision maker is the following fundamental 3property of utility ¾To illustrate, suppose that the decision maker has the utility function shown in . Further suppose that 1the decision maker is offered an opportunity0$100,000M$10,000$30,000$60,000fiugre 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technolog27y, JiangXiUniversity of Finance & Economics©2006Schoo28l of Information Technology, JiangXiUniversity of Finance & Economics©2006运筹学运筹学Operations ResearchOperations Utility Utility TheoryApplying Utility Theory to the ProEtotype xEampleTo obtain either 00$0100,(y=4)00$01 00,(utility =4)with probability p ¾To apply the word’’s (decision maker’’s) utility function for or noth(y0=)ing u(tilit y0=)with probability1- 1-p. Thus,money to the problem as described in , it is (Eu)y=4,(Etilit)y=4p, for to identify the utilities for all the possible monetary ¾Therefore, the decision maker is indifferent between payoffs. Again, in units of thousands of dollars, these possibleeach of the following three pairs of alternatives:payoffs and the corresponding utilities are given in Table offer with0=.25(E[1=])y p0=.25 u(E[tilit1=])yor definitely ¾Suppose you have only the following two alternatives. obtaining$01,00(u1=)y $01,00(tilit1=)yAlternatives 1 is to do(y0.=)y nothing p(aoyff and utilit0.=)ythe offer with p0=.5E[(2=])y0=.5 E[u(tilit2=])yor definitely Alternative 22 is to have a probability p of a700 payoff of 700and obta3$00,00(2=)yining 3$0u0,00(tilit2=)ythe offer70(5=E.[u3=)y] with p70(5=E.[tilit3=)y]a probab1-310(130.)ility 1-p of a payoff of- -3 10l(oss of 130.)What value or def$00,60(0u3=)yinitely obtaining $00,60(0tilit3=)yof p makes you indifferent between two alternatives?? 江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Econom29ics©2006School of Information Technolog30y, JiangXiUniversity of Finance & Economics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
Session 20 运筹学Operations Research20-6运筹学运筹学Operations ResearchOperations Utility Utility TheoryApplyingE Utility Theory to the Prototype xEampleTable Utilities for the dec’ision makker’s:15=/ choice :p15=/Monetary PayoffUtility¾These expected utilities lead to the same decisions at forks -130-150as in , but the decision at fork e now switches to sell -100-105instead of drill. HoHwever, the backward induction procedure 6060still leaves fork on a closed path. 9090670580700600江西财经大学信息管理学院©2006江西财经大学信息管理学院©2006Schoo1l of In3formation Technology, JiangXiUniversity of Finance & Economics©2006School of Information Technology, JiangXiUniversity of Finance & Econom32ics©2006江西财经大学信息管理学院©2006School of Information Technology, JiangXiUniversity of Finance & Economics©2006
