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Uncertain random goal programming#
QIN Zhongfeng
*
(School of Economics and Management, Beihang University) 5
Foundations: Specialized Research Fund for the Doctoral Program of Higher Education of China Grant
().
Brief author introduction:QIN Zhongfeng, Associate Professor, His main research interests include uncertain
portfolio optimization, uncertainty modelling and decisions. E-mail: qin@
Abstract: Goal programming provides an efficient technique to deal with decision making problems
with multiple conflicting objectives. This paper aims to provide a new goal programming called
uncertain random goal programming to model the multiobjective programming involving uncertain
random variables. Some equivalent deterministic forms are derived on the condition that the set of
parameters consist of uncertain variables and random variables. Finally, an example is given to 10
illustrate the applications of these approaches.
Key words: Goal programming; Uncertainty theory; Uncertain random variable; Uncertain random
programming.
0 Introduction 15
Goal programming was introduced by Charnes and Cooper [1], and subsequently developed by
Ijiri [2] to efficiently solve multiobjective programming. It is essentially a compromise method by
ordering or weighing the unwanted deviations with a number of priority levels. Since then, goal
programming has been widely employed in many and diverse fields since its simplicity and ease
of use. 20
Considering that the parameters in the practical problem are always nondeterministic, goal
programming with incomplete information is introduced and widely investigated. Contini [3] first
studied the goal programming with random parameters. Retzlaff-Roberts [4] applied the goal
programming method to solve a stochastic allocative data envelopment analysis. Ballestero [5]
proposed a stochastic goal programming model leading to a structure of mean-variance 25
minimization. Liu [6] provided a theoretical framework of the third type of stochastic
programming named dependent-chance goal programming.
For the case lack of historical data, model parameters are often estimated by experienced
experts. These parameters are always regarded as fuzzy variables. Tiwari et al. [7] first introduced
fuzzy goal programming. Liu [8] provided a spectrum of dependent-chance programming such as 30
dependent-chance goal programming models with fuzzy instead of crisp decisions. Kumar et al. [9]
applied a fuzzy goal programming approach to a vendor selection problem in the supply chain
management.
Different from randomness or fuzziness, Liu [10] proposed uncertain variable and found
uncertainty theory to describe the indeterminacy phenomena which behaves neither randomness 35
nor fuzziness. At present, uncertainty theory has been applied to many fields such as uncertain
calculus [11], uncertain control [12], uncertain risk analysis [13], and uncertain logic [14]. In
particular, Liu [10] and Liu and Chen [15] introduced an uncertain goal programming and its
applications to capital budget problem.
More generally, randomness and uncertainty simultaneously appear in a complex system. In 40
order to deal with this problem, Liu [16] introduced the concept of uncertain random variable and
employed it to characterize some complex parameters. Surrounding the subject, some researches
have been done such as Ke [17], Zhou et al. [18]. In order to handle the multiobjective
programming involving uncertain random variables, this paper aims to provide a new goal
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programming called uncertain random goal programming. After each model, we discuss the 45
equivalent deterministic forms on the condition that the set of parameters consist of uncertain
variables and random variables. Finally, we calculate an example to illustrate the applications of
these models.
The rest of the paper is organized as follows. Section 2 recalls some basic concepts and
fundamental conclusions. In Section 3, three kinds of uncertain random goal programming are 50
presented and their deterministic equivalent forms are given. Section 4 illustrates the applications
of the proposed goal programming models. Finally, some conclusions are given in Section 5.
1 Preliminaries
In 2007, Liu [19] proposed the concept of uncertain measure and founded uncertainty theory. Let
Γ be a nonempty set, and let L be a σ-algebra over Γ. Each element Λ∈L is called an event. It is 55
necessary to assign to each event Λ a number M{Λ} which indicates the chance that Λ will occur.
Liu [19] proposed the following four axioms to ensure that the number M{Λ} satisfying certain
mathematical properties,
Axiom 1. (Normality) M{Γ} = 1.
Axiom 2. (Monotonicity) M{Λ1} ≤ M{Λ2} whenever Λ1⊂ Λ2. 60
Axiom 3. (Self-Duality) M{Λ} + M{Λ
c
} = 1 for any event Λ.
Axiom 4. (Countable Subadditivity) For every countable sequence of events {Λi}, we have
.}{}{
1
1
i
ii i
MM (1)
The triplet (Γ,L,M) is called an uncertainty space. An uncertain variable ξ is defined by Liu
[19] as a measurable function from an uncertainty space (Γ,L,M) to the set of real numbers, ., 65
for any Borel set B of real numbers, the set {ξ∈B} = {γ∈Γ |ξ(γ)∈B} is an event. An uncertain
variable ξ can be characterized by an uncertainty distribution which is a function Φ : R → [0,1]
defined by Liu [19] as Φ(t) = M{ξ ≤ t}.
Definition 1. (Liu [16]) An uncertain random variable is a function ξ from a probability space
(Ω,A,Pr) to a collection of uncertain variables such that M{ξ(ω)∈B} is a measurable function of 70
ω for any Borel set B of real numbers.
Example 1. Let n ,,, 21 be uncertain variables and (Ω, A, Pr) a probability space with Ω =
{ω1,ω2,··· ,ωn}. Then the function
nn if
if
if
,
,
,
)(
22
11
is just an uncertain random variable. 75
Definition 2. (Liu [16]) Let ξ be an uncertain random variable, and let B be a Borel set of real
numbers. Then the chance measure of uncertain random event ξ∈B is defined by
.}})({|Pr{}{
1
0 drrBMBCh (2)
Theorem 1. (Liu [20]) Let n ,,, 21 be independent random variables with probability
distributions n ,,, 21 , and q ,,, 21 be uncertain variables with uncertainty 80
distributions q ,,, 21 , respectively. Then the uncertain random variable
),,,,,,,( 2121 qnh has a chance distribution
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中国科技论文在线
nR nnn ydydydyyyxFx )()()(),,,;()( 221121 (3)
where F(x;y1,y2,··· ,yn) is the uncertainty distribution of the uncertain variable
),,,,,,,( 2121 qnh and is determined by its inverse function 85
))(,),(),(,,,,(),,,;( 112
1
12121
1 pnn yyyhyyyF
provided that ),,,,,,,( 2121 qnh is a strictly increasing function with respect to
q ,,, 21 .
Definition 3. (Liu [16]) Let ξ be an uncertain random variable. Then its expected value is defined
by 90
0
0
}{}{][ drrChdrrChE (4)
provided that at least one of the two integrals is finite.
Theorem 2. (Liu [20]) Let n ,,, 21 be independent random variables with probability
distributions n ,,, 21 , respectively. Then the uncertain random variable
),,,,,,,( 2121 qnh has an expected value 95
nR nnqn ydydydyyyhEE )()()()],,,,,,,([][ 22112121 (5)
where )],,,,,,,([ 2121 qnhE is the expected value of the uncertain variable
),,,,,,,( 2121 qnh for any given real numbers y1,y2,··· ,yn.
Theorem 3. (Liu [20]) Let n ,,, 21 be independent random variables with probability
distributions n ,,, 21 , and q ,,, 21 be uncertain variables with uncertainty 100
distributions q ,,, 21 , respectively. Then the uncertain random variable
),,,,,,,( 2121 qnh has an expected value
)()()(
))](,),(),(,,,,([][
2211
1
0
11
2
1
121
nn
R
qn
ydydydd
yyyhEE
n
(6)
where provided that ),,,,,,,( 2121 qnh is a strictly increasing function or a strictly
decreasing function with respect to q ,,, 21 . 105
2 Goal Programming - Basic Form
Assume that x is a decision vector, and ξ is a vector representing the model parameters in
decision making problem. The return function ),( ξxif is the ith objective faced by a
decision-maker for i = 1, 2, ···, m. In general, these m objectives are always conflicting with
different importance. Therefore, the decision-maker may establish a hierarchy of importance 110
among these incompatible goals so as to be satisfied as many as possible in the order specified.
Suppose that ),( ξxjg are constraint functions for j = 1, 2, · · · , p.
For simplicity, we use the following notations in the rest of the paper.
m: the number of goal constraints;
p: the number of system constraints; 115
l: the number of priorities;
bi : the target value according to goal i;
Pj : the preemptive priority factor which shows relative importance of various goals with
1 jj PP for all j;
iju : the weighting factor corresponding to positive deviation for goal i with priority j 120
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assigned;
ijv : the weighting factor corresponding to negative deviation for goal i with priority j
assigned.
Then a general form of goal programming is written as follows,
midd
pjg
mibddfts
dvduP
ii
j
iiii
l
j
m
i
iijiijj
,,2,1,0,
,,2,1,0)(
,,2,1,)(..
)(min
1 1
ξx,
ξx, (7) 125
where
id is the positive deviation from the target of goal i, defined as 0])([
iii bfd ξx, ,
and
id is the negative deviation from the target of goal i, defined as 0)]([
ξx,iii fbd .
In general, as the model parameters, the vector ξ as is always difficult to be accurately
estimated, such as the demand quantity in the supply chain management, the security return in
portfolio selection. If ξ is considered as random vector, fuzzy vector, uncertain variable, then 130
Model (7) will be formulated as random goal programming [1], fuzzy goal programming,
uncertain goal programming [15], respectively.
3 Expected Value Goal Programming
In this section, ξ is considered as an uncertain random vector. According to the operational
law, the return function ),( ξxif and the constraint function ),( ξxjg are also uncertain random 135
variables, respectively, for i = 1, 2, ···, m, j = 1, 2, ···, p. Different from the situation of real
numbers, there does not exist a natural order between uncertain random variables. However, we
may employ their expected values to rank uncertain random variable. Thu, we propose the
following uncertain random expected value goal programming (UREVGP),
midd
pjgE
mibddfEts
dvduP
ii
j
iiii
l
j
m
i
iijiijj
,,2,1,0,
,,2,1,0)]([
,,2,1,)]([..
)(min
1 1
ξx,
ξx, (8) 140
where
id is the positive deviation from the target of goal i, defined as
0])]([[ iii bfEd ξx, , and
id is the negative deviation from the target of goal i, defined as
0)]]([[ ξx,iii fEbd .
Sometimes, the objective function in Model (8) is written as follows,
m
i
iiliil
m
i
iiii
m
i
iiii dvdudvdudvdulex
11
22
1
11 )(,,)(,)(min 145
where lexmin represents lexicographically minimizing the objective vector, or is set as lexmin{S}
where S is a subset of midd ii ,,2,1,, 1 with a given order.
In order to simplify the proposed model, we consider a special case in which uncertain
random vector ξ is represented by ),,,,,,,( 2121 qn ξ . Here n ,,, 21 are
independent random variables with probability distributions n ,,, 21 , and q ,,, 21 150
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are uncertain variables with uncertainty distributions q ,,, 21 , respectively. Further,
suppose that ),,,,,,,,( 2121 qnif x and ),,,,,,,,( 2121 qnjg x are
strictly increasing or strictly decreasing function with respect to uncertain variables q ,,, 21 .
In this situation, Model (8) can be converted into the following crisp goal programming,
.,,2,1,0,
,,2,1,0)()()(
))](,),(),(,,,,,([
,,2,1,)()()(
))](,),(),(,,,,,([..
)(min
2211
1
0
11
2
1
121
2211
1
0
11
2
1
121
1 1
midd
pjydydydd
yyygE
mibddydydydd
yyyfEts
dvduP
ii
nn
R
qnj
iiinn
R
qni
l
j
m
i
iijiijj
n
n
x
x
(9) 155
4 Chance-Constrained Goal Programming
Minimin Chance-Constrained Goal Programming
Expected value is the average value of uncertain random variable in the sense of chance
measure. Thus, it is weak that a constraint bfE )],([ ξx holds only in the sense of expected
value. A natural alternative is to require that a uncertain constraint holds with given confidence 160
level. For example, the constraints 0)],([ ξxjgE in Model (8) are replaced with
jjgCh }0),({ ξx where j are confidence levels for j = 1, 2, ···, p. If the decision-maker
wishes the chance of the event that the expected value of the ith goal ),( ξxjf is as close as the
target value ib is at least some given confidence level, then we can formulate the uncertain
random decision system as a minimin chance-constrained goal programming according to the 165
priority structure and target levels set by the decision-maker,
midd
pjgCh
midfbCh
midbfChts
dvduP
ii
jj
iiii
iiii
l
j
m
i
iijiijj
,,2,1,0,
,,2,1,}0)({
,,2,1,})({
,,2,1,})({..
)(min
1 1
ξx,
ξx,
ξx,
(10)
where
id is the
i -optimistic positive deviation from the target of goal i, defined as
iii dbfChd })({|0min{ ξx, , and
id is the
i -optimistic negative deviation from
the target of goal i, defined as
iii dfbChd })({|0min{ ξx, . 170
Similarly, we consider the special case assumed by Section 3. In this situation, Model (10)
can be converted into the following crisp goal programming,
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midd
pjydydydyyyG
miydydydyyyH
miydydydyyyHts
dvduP
ii
j
R
nnnj
i
R
nnni
i
R
nnni
l
j
m
i
iijiijj
n
n
n
,,2,1,0,
,,2,1,)()()(),,,;(
,,2,1,)()()(),,,;(
,,2,1,)()()(),,,;(..
)(min
221121
221121
221121
1 1
x
x
x
(11)
where ),,,;( 21 nj yyyG x is the root v of
,0))(,),(),(,,,,,( 112
1
121
vvvyyyg pnj x 175
),,,;( 21 ni yyyH x
and ),,,;( 21 ni yyyH x
are the roots and of
,))(,),(),(,,,,,( 112
1
121
iipni dbyyyf x
and
,))(,),(),(,,,,,( 112
1
121
iipni dbyyyf x
respectively. 180
In a deterministic goal programming, we have 0
ii dd which implies at most one of
positive deviation and negative deviation takes a positive value. However, it is possible that both
id and
id are positive in Model (10).
Minimax Chance-Constrained Goal Programming
Different from minimizing the optimistic return, to maximize the pessimistic return is another 185
commonly used decision criteria. If the priority structure and target levels are set by the
decision-maker, then we can also propose a minimax chance-constrained goal programming as
follows,
pjgCh
midfbCh
midbfChts
dvduP
jj
iiii
iiii
l
j
m
i
i
d
iji
d
ijj
ii
,,2,1,}0)({
,,2,1,})({
,,2,1,})({..
0max0maxmin
1 1
ξx,
ξx,
ξx, (12)
where
id is the
i -pessimistic positive deviation from the target of goal i, defined as 190
iii dbfChd })({|0min{ ξx, , and
id is the
i -pessimistic negative deviation from
the target of goal i, defined as
iii dfbChd })({|0min{ ξx, .
5 A Numerical Example
In order to illustrate the modelling idea, a simpler numerical example is presented in this
section. Assume that the uncertain random vector ),,,( 2121 ξ where 1 and 2 are 195
independent random variables N(2, 1), N(4, 1), 1 and 2 are independent uncertain variables
L(0, 4), L(4, 8). The return functions 22111 )( xxf ξx, and 12212 )( xxf ξx, . The
constraint functions 10)( 22111 xxg ξx, and 20)( 2
2
22
2
12 xxg ξx, . In addition,
the number l of priorities is set as 1, and 15,10 21 bb . For this situation, Model (8) can be
converted into the following uncertain random UREVGP, 200
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.2,1,0,
2022
1022
1524
1062..
)()(min
2
2
2
1
21
2221
1121
22221111
idd
xx
xx
ddxx
ddxxts
dvdudvdu
ii
(13)
Model (13) is a crisp goal programming without uncertain parameters. If we set
,,0, 2211 vuvu , then Model (13) becomes a single-objective programming
problem and can be solved by Lingo. The optimal solution is ,
*
2
*
1 xx and
0,,0, 2211
dddd . The value of the first objective is and the one of the 205
second objective is .
If the decision maker donot know how to determine the values of the weighting factors
211 ,, uvu and 2v , then we can lexicographically minimize the subset of }2,1,0,{
idd ii as
the objective function. For example, we set the objective function as },,,min{ 2211
ddddlex .
Then the optimal solution is ,
*
2
*
1 xx and 0,,0,0 2211
dddd . The 210
value of the first objective is 10 and the one of the second objective is .
6 Conclusions
In this paper, we proposed two kinds of uncertain random goal programming for the problem
with randomness and fuzziness. The equivalent forms are obtained for a special case in which the
uncertain random vector is composed of random variables and uncertain variables. For the purpose 215
of an illustration, a numerical example is introduced to show the modeling idea.
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[18] ZHOU J, YANG F, WANG K. Multi-objective optimization in uncertain random environments, 250
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DOI:
255
不确定随机目标规划
秦中峰
（北京航空航天大学经济管理学院）
摘要：实际的决策问题往往含有多个互相冲突的目标函数，目标规划为这类问题的建模提供260
了有效技巧。本文提出了所谓不确定随机目标规划这样一类新的目标规划方法，目的在于为
涉及不确定随机参数的多目标规划建立模型。在模型参数集仅仅由随机变量和模糊变量组成
的时候，一些等价形式被推导。最后，一个数值例子被给出以距离说明方法的应用。
关键词：目标规划；不确定理论；不确定随机变量；不确定随机理论
中图分类号： 265
