interactive multiobjective fuzzy random linear programming: maximization of possibility and...

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European Journal of Operational Research 188 (2008) 530–539

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Decision Support

Interactive multiobjective fuzzy random linearprogramming: Maximization of possibility and probability

Hideki Katagiri *, Masatoshi Sakawa, Kosuke Kato, Ichiro Nishizaki

Graduate School of Engineering, Hiroshima University, 1-4-1, Kagami-Yama, Higashi-Hiroshima, Hiroshima 739-8527, Japan

Received 18 August 2004; accepted 9 February 2007Available online 19 April 2007

Abstract

This paper considers multiobjective linear programming problems with fuzzy random variables coefficients. A new deci-sion making model is proposed to maximize both possibility and probability, which is based on possibilistic programmingand stochastic programming. An interactive algorithm is constructed to obtain a satisficing solution satisfying at leastweak Pareto optimality.� 2007 Elsevier B.V. All rights reserved.

Keywords: Fuzzy random variable; Multiobjective linear programming; Probability maximization model; Stochastic programming; Po-ssibilistic programming; Interactive algorithm

1. Introduction

In the real-world decision making problems, one often needs to make a decision under uncertainty. Sto-chastic programming [1,2,4] and fuzzy programming [6,10] were developed as useful tools for decision makersto determine a solution. However, decision makers are faced with environments in which both fuzziness andrandomness are included. In order to construct a framework of decision making models under fuzzy and sto-chastic environments, fuzzy random variables [25,27,34] are brought to the attention of researchers.

Let us consider the situation that a profit per unit crop acreage is dependent on weathers, i.e., fine, cloudy,rain, etc., which are considered as scenarios that occur randomly. When a realized profit under each scenario isestimated as a fuzzy set or fuzzy number, it turns out that the profit is expressed by a fuzzy random variable.In such a case, a linear programming problem is formulated to maximize total profit, in which the coefficientsof objective function are fuzzy random variable.

Fuzzy random linear programming problems [11,33,38] were investigated to provide decision making mod-els and methodologies under fuzzy stochastic environments.

0377-2217/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.ejor.2007.02.050

* Corresponding author. Tel.: +81 824 24 7709; fax: +81 824 22 7195.E-mail address: [email protected] (H. Katagiri).

mailto:[email protected]

H. Katagiri et al. / European Journal of Operational Research 188 (2008) 530–539 531

Fuzzy random variables have large potential to be applied to various decision making problems. Someapplications of fuzzy random variables were discussed in previous studies; bottleneck spanning tree problems[13,19,20], inventory problems [5], pricing European options [40], stopping games [41], American put options[42] and value at risk methodology for index portfolio [45], linear regression [24] and 0-1 programming [14].

When considering real-world decision making problems, it is natural to deal with multiobjective cases. Forinstance, in crop planning problems, working time is often dependent on weathers, and a decision maker pre-fers not only to maximize the total profit but also to minimize total working time. Since the realized workingtime under each weather is estimated as a fuzzy number, then the working time is also represented by a fuzzyrandom variable. In this way, a two-objective fuzzy random linear programming problem is formulated tooptimize both total profit and total working time.

In this paper, we shall consider a multiobjective fuzzy random programming (FRP) problem and propose amodel based both on a Stochastic Programming (SP) model and on a Possibilistic Programming (PP) model.

This paper is organized as follows. The next section devotes to introducing the definition of fuzzy randomvariables. In Section 3, we introduce previous studies on SP, PP and FRP. Section 4 formulates multiobjectiveFRP problem and shows that a proposed model reduces the original problem to a deterministic multiobjectiveproblem. In Section 5, we take a goal programming approach to the reduced multiobjective problem and con-struct an interactive algorithm to derive a satisficing solution for a decision maker among a Pareto optimalsolution set through interaction. Section 6 provides a numerical example to demonstrate the interactive pro-cess of deriving a satisficing solution. Finally, in Section 7, we conclude this paper.

2. Fuzzy random variable

A fuzzy random variable was first defined by Kwakernaak [27]. The mathematical basis was established byPuri and Ralescu [34]. Kruse and Meyer [25] provides a slightly different definition. Since this article utilizes asimple one, we define fuzzy random variables as follows:

Definition 1 (Fuzzy random variable). Let (X,B,P) be a probability space, F ðRÞ the set of fuzzy numbers withcompact supports and X a measurable mapping X! F ðRÞ. Then X is a fuzzy random variable if and only ifgiven x 2 X, Xa(x) is a random interval for any a 2 (0,1], where Xa(x) is a a-level set of the fuzzy set X(x).

The above definition of fuzzy random variables corresponds to a special case of those given by Kwakernaakand Puri-Ralesu. The definitions of Kwakernaak and Puri-Ralescu are equivalent for the above case because afuzzy number is a convex fuzzy set [8]. Though it is a simple definition, it would be useful for various appli-cations. The hybrid number introduced by Kaufman and Gupta [22], is a special case of the definition above.

Recently, some authors discuss vigorously definitions of fuzzy random variables [28,31,32]. The readersinterested in more details of the definitions, refer to these literatures.

3. Fuzzy random programming

Wang and Qiao [35,38] firstly considered fuzzy random programming and its distribution problem. SinceFRP problems are generally ill-defined problems, various decision making models can be considered to satisfyvariety of preferences of decision makers.

In order to construct a decision making model which takes account of both fuzzy and random information,we shall incorporate a SP approach with the possibilistic programming approach.

3.1. Stochastic programming and possibilistic programming

In SP, Beale [1] and Dantzig [4] considered two-stage problems. Charnes and Cooper [2] proposed thechance constrained programming models such as the E-model (expectation optimization model), the V-model(variance minimization model) and the P-model (probability maximization model). Kataoka [21] and Geoff-rion [7] individually proposed the fractile criterion model, which we call F-model.

On the other hand, in possibilistic programming, the concepts of possibility and necessity measures [44]were introduced to deal with the ambiguity included in the objective function and/or constraints. Dubois

532 H. Katagiri et al. / European Journal of Operational Research 188 (2008) 530–539

and Prade [6] proposed possibilistic programming and considered a degree of possibility or necessity that fuzzygoals for the objective function and/or constraints are satisfied. Inuiguchi and Ramik [10] viewed various typesof PP models and developed modality constrained programming models.

Here, we especially pay attention to the difference of vagueness and ambiguity examined by Klir and Folger[23]. Vagueness is a notion representing the fuzziness of the degree to which the element of a set belongs to theset, and ambiguity is related to fuzziness of the value itself. From this point of view, fuzzy random variablesare considered as the concepts dealing with ambiguity in respect to the realization of a random variablebecause the realization of a random variable is fuzzy. On the other hand, fuzzy event, which was introducedby Zadeh [43], is the concept related to vagueness with respect to the realization of a random variable becausethe realization is not fuzzy but crisp, and the degree to which an element belongs to a fuzzy set is imprecise.

Since fuzzy random variables are related to the ambiguity of random variables, a possibilistic programmingapproach is appropriate for constructing a FRP model.

3.2. FRP model based on the fusion of PP and SP models

Katagiri et al. [11] firstly introduced a FRP model by incorporating PP models and SP models. They con-sidered a simple linear programming problem in which only the right-hand side of a single constraint is a fuzzyrandom variable. Their approach consists of two steps: (1) firstly they focused on a degree of possibility thatthe constraint is satisfied, which is based on a possibilistic programming approach, (2) secondly, they applied aSP approach to the problem by considering the fact that the degree of possibility varies randomly.

Their FRP model was extended to deal with a linear programming problem where the coefficients of anobjective function are fuzzy random variables, which are based on the fusion of PP model and F-model[12]. A fuzzy random chance-constrained programming model proposed by Liu [29,30] also focused on pos-sibility and probability simultaneously.

Katagiri and Sakawa [18] constructed other new models based on the fusion of PP model and E-model/V-model.

4. Fuzzy random multiobjective linear programming problem

4.1. Problem formulation

Consider the following multiobjective linear programming problem:

minimize ~�C ix; i ¼ 1; . . . ; k

subject to x 2 X , fx 2 RnjAx 6 b; x P 0g

): ð1Þ

In this paper, we denote randomness and fuzziness of the coefficients by the ‘‘dash above’’ and ‘‘wave above’’i.e., ‘‘–’’ and ‘‘�’’, respectively. In (1), x is an n-dimensional decision variable column vector, A is an m · n

coefficient matrix and b is an m-dimensional column vector. The coefficients ~�Cij; j ¼ 1; . . . ; n of the vector

Fig. 1. Membership function of a fuzzy random variable coefficient.


~�C i ¼ ð~�Ci1; . . . ; ~�CinÞ are fuzzy random variables that take fuzzy numbers under the occurrence of each elemen-tary event x, which are characterized by the following membership functions (see Fig. 1):

l~�CijðxÞðsÞ ¼L

�dijðxÞ�s�aijðxÞ

� �ðs 6 �dijðxÞ 8xÞ;

R s��dijðxÞ�bijðxÞ

� �ðs > �dijðxÞ 8xÞ; i ¼ 1; . . . ; k; j ¼ 1; . . . ; n;

8><>: ð2Þ

where ~�CijðxÞ is a realization of the fuzzy random variable ~�Cij under the occurrence of each elementary event x.The function LðtÞ¼M maxf0; lðtÞg is a real-valued continuous function from [0,1) to [0, 1], and l(t) is a strictlydecreasing continuous function satisfying l(0) = 1. Also, RðtÞ¼M maxf0; rðtÞg satisfies the same conditions. Letus assume that the parameters �dij, �aij and �bij are random variables expressed as �dij ¼ d1

ij þ�tid2ij, �aij ¼ aij þ�tia2

ij

and �bij ¼ b1ij þ�tib

2ij, respectively, where �ti is a random variable with mean mi and variance r2

i , and dlij, aij and

bij, l = 1, 2 are constants. It should be noted here that �aijðxÞ and �bijðxÞ are to be positive for any x becausethey are spread parameters of L–R fuzzy numbers. Therefore, let us give the assumption that aij þ�tiðxÞa2

ij > 0and bij þ�tiðxÞb2

ij > 0 for any x.Since all the coefficients of each objective function are L–R type fuzzy random variables, we can apply the

operation on the fuzzy numbers induced by Zadeh’s extension principle to the realizations ~�C iðxÞx; i ¼ 1; . . . ; kof the objective functions. Namely, the realizations become fuzzy random variables characterized by the fol-lowing membership functions:

l ~�C iðxÞxðyÞ ¼L

�d iðxÞx�y�aiðxÞx

� �ðy 6 �d iðxÞx 8xÞ;

R y��d iðxÞx�biðxÞx

� �ðy > �d iðxÞx 8xÞ; i ¼ 1; . . . ; k:

8><>: ð3Þ

4.2. FRP model: Possibility and probability maximization criterion

Considering the imprecise nature of the decision maker’s judgment, it is natural to assume that the decisionmaker may have imprecise or fuzzy goals for each of the objective functions in problem (1). In a minimizationproblem, a goal stated by the decision maker may be to achieve ‘‘substantially less than or equal to somevalue.’’ This type of statement can be quantified by eliciting a corresponding membership function. For eachobjective function, we introduce a fuzzy goal characterized by the following membership function to considerthe imprecise of human’s judgment (see Fig. 2).

l~GiðyÞ ¼

1; y 6 g1i ;

giðyÞ; g1i 6 y 6 g0

i ;

0; g0i 6 y;

8><>: ð4Þ

where gi is a strictly decreasing continuous function. By using a concept of possibility measure, the degrees ofpossibility that the objective function values satisfy the fuzzy goal are represented as

P ~�C ixð~GiÞ ¼ sup

ymin l ~�C ix

ðyÞ;l~GiðyÞ

n o; i ¼ 1; . . . ; k: ð5Þ

Fig. 2. Membership function of a fuzzy goal.


Accordingly, when a decision maker prefers to maximize a degree of possibility with respect to the fuzzy goal,problem (1) is reformulated as

maximize P ~�C ixð~GiÞ; i ¼ 1; . . . ; k

subject to x 2 X

): ð6Þ

It should be noted that the objective function value P ~�C ixð ~GiÞ varies randomly due to the randomness of l ~�C ix

even if the decision vector x is determined. In other words, problem (6) is a stochastic programming problem,and it is even an ill-defined problem. Therefore, we take the fuzzy random programming approach based onthe model by Katagiri and Ishii [12].

In particular, we apply the probability maximization model to problem (6) and obtain the followingproblem:

maximize Pr x P ~�C iðxÞxð~GiÞP hi

��h i; i ¼ 1; . . . ; k

subject to x 2 X

); ð7Þ

where Pr[ Æ ] denotes a probability measure, and hi is a positive constant given by a decision maker. It should benoted here that problem (7) is based on the fusion of PP model and P-model. Katagiri et al. [15–17] consideredseveral FRP models, which are the fusion of PP model and other optimization criteria such as E-model [15], V-model [16] and F-model [17].

In the above problem, P ~�C iðxÞxð~GiÞP hi i ¼ 1; . . . ; k are transformed into the following:

supy

minimize l ~�C iðxÞxðyÞ; l~GiðyÞ

n oP hi;

() 9y : l ~�C iðxÞxðyÞP hi; l~GiðyÞP hi;

() 9y : L�d iðxÞx� y

�aiðxÞx

� �P hi; R

y � �d iðxÞx�biðxÞx

� �6 hi; l~Gi

ðyÞP hi;

() 9y : f�d iðxÞ � L�ðhiÞ�aiðxÞgx 6 y 6 f�d iðxÞ þ R�ðhiÞ�biðxÞgx; y 6 l�~GiðhiÞ;

() f�d iðxÞ � L�ðhiÞ�aiðxÞgx 6 l�~GiðhiÞ;

where L*(hi) and l�~GiðhiÞ are pseudo inverse functions defined by

L�ðhiÞ ¼ sup rjLðrÞP hif g; l�~GiðhiÞ ¼ sup rjl~Gi

ðrÞP hi

� �:

In addition, provided that fd2i � L�ð0Þa2

i gx > 0; i ¼ 1; . . . ; k hold for any x 2 X, we have

Pr x P ~�C iðxÞxð~GiÞP hi

��h i¼ Pr f�d iðxÞ � L�ðhiÞ�aiðxÞgx 6 l�~Gi

ðhiÞh i

¼ Pr x ðd1i þ�tiðxÞd2

i Þx� L�ðhiÞða1i þ�tiðxÞa2

i Þx 6 l�~GiðhiÞ

��h i

¼ Pr x �tiðxÞ 6fL�ðhiÞa1

i � d1i gxþ l�~Gi

ðhiÞfd2

i � L�ðhiÞa2i gx

��" #

¼ T i

fL�ðhiÞa1i � d1

i gxþ l�~GiðhiÞ

fd2i � L�ðhiÞa2

i gx

!, piðxÞ:

Consequently, problem (7) is rewritten by

maximize piðxÞ; i ¼ 1; . . . ; k

subject to x 2 X

: ð8Þ


According to Dubois and Prade [6], possibility theory has four indices about possibility and necessity mea-sures. Although we have only focused on a possibility measure as described in (6), our approach can be easilyapplied to the models using other three induces.

5. Interactive algorithm to obtain a Pareto optimal solution

Since problem (8) is a multiobjective programming problem, a complete optimal solution that simulta-neously optimizes all of the multiple objective functions does not always exist. Thus, instead of a completeoptimal solution, a Pareto optimal solution is reasonable for a multiobjective case. In the proposed model,a Pareto optimal solution is defined as follows:

Definition 1. x* is said to be a Pareto optimal solution if and only if there does not exist another x such thatpi(x) P pi(x*) for all i and pj(x) 5 pj(x*) for at least one j.

As can be seen from the definition, a Pareto optimal solution consists of an infinite number of points. Inaddition to Pareto optimality, the following weak Pareto optimality is defined as a slightly weaker solutionconcept than Pareto optimality.

Definition 2. x* is said to be a weak Pareto optimal solution if and only if there does not exist another x 2 X

such that piðxÞ > piðx�Þ; i ¼ 1; . . . ; k.

Several computational methods such as scalarization methods [26] and goal programming approaches [3,9]have been proposed for obtaining a Pareto optimal solution.

For each of the multiple conflicting objective function piðxÞ; i ¼ 1; . . . ; k, assume that the decision makercan specify the so-called reference point p̂ ¼ ðp̂1; . . . ; p̂kÞ which reflects in some sense the desired values ofthe objective functions of the decision maker. This type of approach was originally proposed by Wierzbicki[39]. In this paper, we call p̂i reference probability level. Also assume that the decision maker can change thereference probability levels interactively due to his/her learning or improved understanding during the solutionprocess. When the decision maker specifies the reference probability levels p̂i; i ¼ 1; . . . ; k, the correspondingPareto optimal solution, which is, in the minimax sense, nearest to the reference probability levels or better thanthose if the reference probability levels are attainable, is obtained by solving the following minimax problem:

minimize maxi¼1;...;k

fp̂i � piðxÞg

subject to x 2 X

): ð9Þ

The above problem is equivalent to

minimize v

subject to p̂i � piðxÞ 6 v; i ¼ 1; . . . ; k

x 2 X

9>=>;: ð10Þ

Using the definition of pi(x), problem (10) is rewritten by

minimize v

subject tofL�ðhiÞa1

i �d1i gxþl�

~GiðhiÞ

fd2i �L�ðhiÞa2

i gxP T �i ðp̂i � vÞ; i ¼ 1; . . . ; k

x 2 X

9>>=>>;; ð11Þ

where T*(s) is the pseudo inverse function defined by

T �ðsÞ ¼ inffrjT iðrÞP sg; i ¼ 1; . . . ; k:

Since problem (11) is a nonconvex programming problem, an optimal solution of the problem is not necessar-ily obtained by usual nonlinear programming techniques. Instead, we take another approach to solve theproblem strictly. It should be noted that the constraints of the problem are reduced to a set of linear inequal-ities if the value of v is fixed. This means that an optimal solution of problem (11) is obtained by combined useof the bisection method and the first-phase of the two-phase simplex method of linear programming.

Fig. 3. Flowchart of the proposed algorithm.


Other nonlinear programming techniques [36,37] are also applied to solve problem (9) or (11). However, itshould be emphasized that usual nonlinear programming techniques do not necessarily reach an optimalsolution.

Now we are ready to construct an interactive algorithm for fuzzy random multiobjective linear program-ming in order to derive a satisficing solution. In order to make readers intuitively understand the outline of ouralgorithm, the flowchart of our algorithm is illustrated in Fig. 3. More details of our algorithm are describedas follows:

An interactive satisficing method for fuzzy random multiobjective linear programming problems

Step 1: Calculate the individual minimum minx2X E½�di�x and the individual maximum maxx2X E½�di�x under thegiven constraints.

Step 2: Elicit membership functions l~Gi; i ¼ 1; . . . ; k from the decision maker for the objective functions.

Step 3: Ask the decision maker to set the aspiration levels hi; i ¼ 1; . . . ; k.Step 4: Set the initial reference probability levels p̂i; i ¼ 1; . . . ; k to 1.Step 5: For the reference probability levels, solve problem (11) to obtain the minimum value of v.Step 6: If the decision maker is satisfied with the current levels of piðxÞ; i ¼ 1; . . . ; k, then stop. The current

optimal solution is a satisficing solution for the decision maker. Otherwise, ask the decision makerto update the current reference probability levels and return to Step 5.

It should be stressed to the decision maker that any improvement of one objective function value can beachieved only at the expense of at least one of the other objective function values. Since problem is solvedoptimally by conventional computational methods, the obtained satisficing solution absolutely satisfies at leastthe weak Pareto optimality defined in this section.

6. Numerical example

In this section, we provide a numerical example and specifically demonstrate the decision making process ofthe proposed interactive algorithm. Consider the following problem:

Table 1Interactive process

1 2 3

p̂1 1.00 1.00 0.90p̂2 1.00 0.80 0.80p1(x) 0.56 0.67 0.61p2(x) 0.56 0.47 0.51x1 6.49 7.79 7.14x2 13.19 13.68 13.43x3 19.30 17.45 18.37


minimize ~�C11x1 þ ~�C12x2 þ ~�C13x3

minimize ~�C21x1 þ ~�C22x2 þ ~�C23x3

subject to 2x1 þ 6x2 þ 3x3 6 150; 6x1 þ 3x2 þ 5x3 6 175

5x1 þ 4x2 þ 2x3 6 160; 2x1 þ 2x2 þ 3x3 P 90

x1 P 0; x2 P 0

9>>>>>>=>>>>>>;: ð12Þ

Assume that the parameters included in (12) are estimated by some expert as

d11 ¼ ð2; 1; 3Þ; d2

1 ¼ ð1:3; 1:1; 1:2Þ; ð13Þd1

2 ¼ ð�7;�7;�9Þ; d22 ¼ ð1:1; 1:2; 1:1Þ; ð14Þ

a11 ¼ ð0:5; 0:4; 0:5Þ; a2

1 ¼ ð0:05; 0:04; 0:05Þ; ð15Þa1

2 ¼ ð0:3; 0:5; 0:4Þ; a22 ¼ ð0:05; 0:04; 0:05Þ; ð16Þ

b11 ¼ ð0:6; 0:5; 0:6Þ; b2

1 ¼ ð0:06; 0:05; 0:06Þ; ð17Þb1

2 ¼ ð0:4; 0:5; 0:5Þ; b22 ¼ ð0:06; 0:06; 0:05Þ ð18Þ

and that �ti; i ¼ 1; 2 are the normal random variable N(0,1).Table 1 shows the interactive process of deriving a satisficing solution. According to Steps 1 and 2 in the

proposed algorithm, minx2X E½�di�x and maxx2X E½�di�x; i ¼ 1; 2 are calculated to elicit the membership functionsof fuzzy goals. Here, assume that the membership functions are elicited by Zimmermann’s method [46]. Next,according to Steps 3 and 4, a decision maker sets h1 = h2 = 0.80, and the minimax problem is solved for theinitial reference probability levels p̂1 ¼ p̂2 ¼ 1:0. The result is shown in the second column of Table 1. Then thedecision maker prefers to enlarge p1(x) and p3(x) at the expense of p2(x) and update the reference probabilitylevels. The result of solving the minimax problem for the updated reference probability levels is shown in thethird column of Table 1. Then, since the decision maker feels that p2(x) becomes a little too small, he/sheupdates the reference probability levels so as to enlarge p2(x) at the expense of p1(x). The result of solvingthe minimax problem is shown in the forth column of Table 1. Since the decision maker is satisfied withthe obtained probability levels, the interactive process is terminated.

7. Conclusion

In this paper, we have considered a multiobjective linear programming problem where the coefficients of theobjective function are fuzzy random variables. We have proposed a fuzzy random multiobjective program-ming model based both on a stochastic programming model and on a PP model. To solve the formulated mul-tiobjective problem, we have constructed an interactive algorithm to derive a satisficing solution for a decisionmaker.

It should be noted here that in general, the problems involving fuzzy random variables are a little complexcompared with those involving only fuzzy sets or random variables. This means that when the original prob-lems are reduced to deterministic equivalent ones through some optimization criterion, the resulting determin-istic problems to be solved are usually difficult to solve even if the optimization criteria are reasonable. On the


other hand, in the proposed interactive algorithm, the minimax problem can be solved not only easily but alsooptimally by combined use of conventional solution methods, i.e., the first phase of the two-phase simplexmethod and the bisection method. It should be stressed here that the obtained solution necessarily satisfiesat least weak Pareto optimality. As far as this point of view is concerned, the proposed model and algorithmhave an advantage over other FRP models.

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