Information Sciences 239 (2013) 191–200

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Information Sciences

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Interactive random fuzzy two-level programming throughpossibility-based probability model

0020-0255/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.ins.2013.03.024

⇑ Corresponding author. Tel.: +81 82 424 7694; fax: +81 82 422 7195.E-mail address: sakawa@hiroshima-u.ac.jp (M. Sakawa).

Masatoshi Sakawa ⇑, Takeshi MatsuiFaculty of Engineering, Hiroshima University, Higashi-Hiroshima 739-8527, Japan

a r t i c l e i n f o

Article history:Received 22 September 2011Received in revised form 7 March 2013Accepted 14 March 2013Available online 21 March 2013

Keywords:Two-level programmingRandom fuzzy programmingPossibilityProbability maximizationInteractive programming

a b s t r a c t

This paper considers interactive decision making methods for random fuzzy two-level lin-ear programming problems. Assuming that the decision makers concern about the proba-bilities that their own objective function values are smaller than or equal to certain targetvalues, fuzzy goals of the decision makers for the probabilities are introduced. Then, thepossibility-based probability model to maximize the degrees of possibility with respectto the attained probability is considered. Interactive fuzzy nonlinear programming toobtain a satisfactory solution for the decision maker at the upper level in considerationof the cooperative relation between decision makers is presented. An illustrative numericalexample demonstrates the feasibility and efficiency of the proposed method.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

Decision making problems in hierarchical managerial or public organizations are often formulated as two-level mathe-matical programming problems [21,33]. In the context of two-level programming, the decision maker at the upper level firstspecifies a strategy, and then the decision maker at the lower level specifies a strategy so as to optimize the objective withfull knowledge of the action of the decision maker at the upper level. In conventional multi-level mathematical program-ming models employing the solution concept of Stackelberg equilibrium, it is assumed that there is no communicationamong decision makers, or they do not make any binding agreement even if there exists such communication[2,14,33,34]. Compared with this, for decision making problems in such as decentralized large firms with divisional indepen-dence, it is quite natural to suppose that there exists communication and some cooperative relationship among the decisionmakers [21].

For two-level linear programming problems or multi-level ones such that decisions of decision makers in all levels aresequential and all of the decision makers essentially cooperate with each other, Lai [7] and Shih et al. [32] proposed fuzzyinteractive approaches. In their methods, the decision makers identify membership functions of the fuzzy goals for theirobjective functions, and in particular, the decision maker at the upper level also specifies those of the fuzzy goals for thedecision variables. The decision maker at the lower level solves a fuzzy programming problem with a constraint with respectto a satisfactory degree of the decision maker at the upper level. Unfortunately, there is a possibility that their method leadsa final solution to an undesirable one because of inconsistency between the fuzzy goals of the objective function and those ofthe decision variables. In order to overcome the problem in their methods, by eliminating the fuzzy goals for the decisionvariables, Sakawa et al. have proposed interactive fuzzy programming for two-level or multi-level linear programming prob-lems to obtain a satisfactory solution for decision makers [24,25]. Extensions to two-level linear fractional programming

192 M. Sakawa, T. Matsui / Information Sciences 239 (2013) 191–200

problems [27], decentralized two-level linear programming problems [19,28], two-level linear fractional programmingproblems with fuzzy parameters [26], and two-level nonconvex programming problems with fuzzy parameters [20] wereprovided. The subsequent works on two-level or multi-level programming have been appearing [1,8,15,17,35] includingthe recent extension to fuzzy random two-level programming problems [18,23,29–31]. A recent survey paper of Sakawaand Nishizaki [22] is devoted to reviewing and classifying the numerous major papers in the area of so-called cooperativemulti-level programming.

It should be noted here that fuzzy random variables [6,16,37] are considered to be random variables whose realized val-ues are not real values but fuzzy numbers or fuzzy sets. Studies on linear programming problems with fuzzy random variablecoefficients, called fuzzy random linear programming problems, were initiated by Wang and Qiao [37] and mathematicalprogramming problems with fuzzy random variables have been developed [5,9,23,38].

On the other hand, from a viewpoint of ambiguity and randomness different from fuzzy random variables [6,16,37], byconsidering the experts’ ambiguous understanding of means and variances of random variables, a concept of random fuzzyvariables was proposed [10]. Mathematical programming problems with random fuzzy variables were formulated and havealso been developed [9,11,23,38–40].

Under these circumstances, in this paper, as a first attempt to tackle decision making problems in hierarchical organiza-tions under random fuzzy environments, assuming cooperative behavior of the decision makers, we first consider two-levellinear programming problems involving random fuzzy variables. To deal with the formulated random fuzzy two-level linearprogramming problems, we assume that the decision makers concern about the probabilities that their own objective func-tion values are smaller than or equal to certain target values. By considering the imprecise nature of the human judgments,we introduce the fuzzy goals of the decision makers for the probabilities. Then, assuming that the decision makers are willingto maximize the degrees of possibility with respect to the attained probability, we consider the possibility-based probabilitymodel for random fuzzy two-level programming problems. Interactive fuzzy nonlinear programming to obtain a satisfactorysolution for the decision maker at the upper level in consideration of the cooperative relation between decision makers ispresented. It is shown that all of the problems to be solved in the proposed interactive fuzzy nonlinear programming becomenonlinear programming problems and approximate optimal solutions can be obtained through the use of particle swarmoptimization for nonlinear programming (PSONLP) [12]. An illustrative numerical example is provided to demonstrate thefeasibility and efficiency of the proposed method.

2. Random fuzzy variables

In the framework of stochastic programming [4,36], it is implicitly assumed that the uncertain parameter which well rep-resents the stochastic factor of real systems can be definitely expressed as a single random variable. This means that the real-ized values of random parameters under the occurrence of some event are assumed to be definitely represented with realvalues. Depending on the situations, however, it is natural to consider that the possible realized values of these randomparameters are often only ambiguously known to the experts. In this case, it may be more appropriate to interpret the ex-perts’ ambiguous understanding of the realized values of random parameters as fuzzy numbers. From such a point of view, afuzzy random variable was first introduced by [6], and its mathematical basis was constructed by [16]. An overview of thedevelopments of fuzzy random variables was found in the recent article of [3].

From the expert’s experimental point of view, however, the experts may think of a collection of random variables to beappropriate to express stochastic factors rather than only a single random variables. In this case, reflecting the expert’s con-viction degree that each of random variables properly represents the stochastic factor, it would be quite reasonable to assignthe different degrees of possibility to each of random variables. For handling such an uncertain parameter, a random fuzzyvariable was defined by Liu [10] as a function from a possibility space to a collection of random variables, which is consideredto be an extended concept of fuzzy variable [13]. It should be noted here that the fuzzy variables can be viewed as anotherway of dealing with the imprecision which was originally represented by fuzzy sets. Although we can employ Liu’s defini-tion, for consistently discussing various concepts in relation to the fuzzy sets, we define the random fuzzy variables byextending not the fuzzy variables but the fuzzy sets.

Definition 1 (Random fuzzy variable). Let C be a collection of random variables. Then, a random fuzzy variable eC is definedby its membership function

leC : C! ½0;1�: ð1Þ

In Definition 1, the membership function leC assigns each random variable �c 2 C to a real number leC ð�cÞ. It should be noted

here that if C is defined as R, then (1) becomes equivalent to the membership function of an ordinary fuzzy set. In this sense,a random fuzzy variable can be regarded as an extended concept of fuzzy sets. On the other hand, if C is defined as a sin-

gleton C ¼ f�cg and leC ð�cÞ ¼ 1, then the corresponding random fuzzy variable eC can be viewed as an ordinary random

variable.

When taking account of the imprecise nature of the realized values of random variables, it would be appropriate to em-ploy the concept of fuzzy random variables. However, it should be emphasized here that if mean and/or variance of random

M. Sakawa, T. Matsui / Information Sciences 239 (2013) 191–200 193

variables are specified by the expert as a set of real values or fuzzy sets, such uncertain parameters can be represented by notfuzzy random variables but random fuzzy variables.

As a simple example of random fuzzy variables, we consider a Gaussian random variable whose mean value is not def-initely specified as a constant. For example, when some random parameter �c is represented by the Gaussian random variableN(si, 102), where the expert identifies a set {s1, s2, s3} of possible mean values as (s1, s2, s3) = (90,100,110), if the membershipfunction leC is defined by8

leC ð�cÞ ¼0:5 if �c � Nð90;102Þ0:7 if �c � Nð100;102Þ0:3 if �c � Nð110;102Þ0 otherwise;

>>>><>>>>:

then eC is a random fuzzy variable. More generally, when the mean values are expressed as fuzzy sets or fuzzy numbers, thecorresponding random variable with the fuzzy mean is represented by a random fuzzy variable.

3. Random fuzzy two-level programming

Consider the random fuzzy two-level linear programming problems formulated as

minimizefor DM1

z1ðx1; x2Þ ¼ eC 11x1 þ eC 12x2

minimizefor DM2

z2ðx1; x2Þ ¼ eC 21x1 þ eC 22x2

subject to A1x1 þ A2x2 6 bx1 P 0; x2 P 0;

9>>>>>>=>>>>>>;ð2Þ

where the two objective functions z1(x1, x2) and z2(x1, x2) are those of DM1 and DM2, respectively, and ‘‘minimizefor DM1

’’ and

‘‘minimizefor DM2

’’ mean that DM1 and DM2 are minimizers for their objective functions. Moreover, x1 is an n1 dimensional decision

variable column vector for the decision maker at the upper level (DM1), x2 is an n2 dimensional decision variable columnvector for the decision maker at the lower level (DM2), Aj, j = 1, 2 are m � nj coefficient matrices, and b is an m dimensionalcolumn vector.

Observing that the real data with uncertainty are often distributed normally, from the practical point of view, we assumethat each of eCljk; k ¼ 1;2; . . . ;nj of eC lj, l = 1, 2, j = 1, 2 is the Gaussian random variable with fuzzy mean value eMljk which isrepresented by an L–R fuzzy number characterized by the membership function

leMljkðsÞ ¼

L mljk�saljk

� �if mljk P s;

R s�mljk

bljk

� �if mljk < s;

8><>: ð3Þ

where the shape functions L and R are nonincreasing continuous functions from [0,1) to [0,1], mljk is the mean value, and aljk

and bljk are positive numbers which represent left and right spreads. Fig. 1 illustrates an example of the membership functionleMljk

ðsÞ.

Let C be a collection of all possible Gaussian random variables N(s,r2), where s 2 (�1,1) and r2 2 (0,1). Then, eCljk isexpressed as a random fuzzy variable with the membership function

Fig. 1. An example of the membership function leMljkðsÞ.

194 M. Sakawa, T. Matsui / Information Sciences 239 (2013) 191–200

leC ljk

ð�cljkÞ ¼ leMljkðsljkÞ j �cljk � N sljk;r2

ljk

� �� �; 8�cljk 2 C: ð4Þ

Through the Zadeh’s extension principle, in view of (4), the membership function of a random fuzzy variable correspondingto each of objective functions zl(x1,x2), l = 1, 2 is given as

leC lxð�ulÞ ¼ sup

�cl

min16k6nj ; j¼1;2

leC ljk

ð�cljkÞ �ul ¼X2

j¼1

Xnj

k¼1

�cljkxjk

�����( )

¼ supsl

min16k6nj ; j¼1;2

leMljkðsljkÞ �ul � N

X2

j¼1

Xnj

k¼1

sljkxjk;VlðxÞ �����

!( ); ð5Þ

where �cl ¼ ð�cl11; . . . ; �cl1n1 ; �cl21; . . . ; �cl2n2 Þ, sl ¼ ðsl11; . . . ; sl1n1 ; s121; . . . ; sl2n2 Þ, and

VlðxÞ ¼X2

j¼1

Xnj

k¼1

r2ljkx2

jk:

Assuming that the decision makers (DMs) concern about the probabilities that their own objective function values eC lx aresmaller than or equal to certain target values fl, l = 1, 2, we introduce the probabilities Pðx j eC lðxÞx 6 flÞwhich are expressedas fuzzy sets ePl with the membership functions

lePlðplÞ ¼ sup

�ul

leCeC lxð�ulÞ j pl ¼ Pðx j ulðxÞ 6 flÞ

� �; ð6Þ

where fl, l = 1, 2 are target values specified by the DMs as constants.Considering the imprecise nature of the DMs’ judgments for the probabilities ePl with respect to the random fuzzy objec-

tive function values eC lx, l = 1, 2, we introduce the fuzzy goals eGl; l ¼ 1;2 such as ‘‘ePl should be greater than or equal to acertain value’’. Such fuzzy goals eGl; l ¼ 1;2 can be quantified by eliciting corresponding membership functions

leGlðpÞ ¼

0 if p 6 p0l

glðpÞ if p0l 6 p 6 p1

l ; l ¼ 1;21 if p1

l 6 p;

8><>: ð7Þ

where gl(p), l = 1, 2 are nondecreasing functions. Fig. 2 illustrates a possible shape of the membership function for the fuzzygoal eGl.

Recalling that the membership function is regarded as a possibility distribution, the degree of possibility that the prob-ability ePl attains the fuzzy goal eGl is expressed as

PePlðeGlÞ ¼ sup

pl

minflePlðplÞ; leGl

ðplÞg; l ¼ 1;2: ð8Þ

Fig. 3 illustrates the degree of possibility PePlðeGlÞ.

Now, assuming that the DMs are willing to maximize the degrees of possibility with respect to the attained probability,we consider the possibility-based probability model for random fuzzy two-level programming problems formulated as

maximizefor DM1

ZP;P1 ðx1; x2Þ ¼ PeP1

ðG1Þ

maximizefor DM2

ZP;P2 ðx1; x2Þ ¼ PeP2

ðG2Þ

subject to A1x1 þ A2x2 6 bx1 P 0; x2 P 0

9>>>>>=>>>>>;ð9Þ

Fig. 2. An example of a membership function leGlðyÞ of a fuzzy goal eGl .

Fig. 3. The degree of possibility PeP lðeGlÞ.

M. Sakawa, T. Matsui / Information Sciences 239 (2013) 191–200 195

or equivalently

maximizefor DM1

h1

maximizefor DM2

h2

subject to PeP1ðfG1ÞP h1

PeP2ðfG2ÞP h2

A1x1 þ A2x2 6 bx1 P 0; x2 P 0:

9>>>>>>>>>>>>=>>>>>>>>>>>>;ð10Þ

From (8), the constraints PePlð eGlÞP hl; l ¼ 1;2 in (10) is equivalently replaced by the condition that there exists a p such that

lePlðplÞP hl and leGl

ðplÞP hl, namely

sup�sl

min16k6nj ;j¼1;2

leMljkðsljkÞ pl ¼ Pðxj�ulðxÞ 6 flÞ; �ul � N

X2

j¼1

Xnj

k¼1

sljkxjk;VlðxÞ !�����

( )P hl; ð11Þ

and pl P lHeGl

ðhlÞ; l ¼ 1;2, where lHeGl

ðhlÞ are pseudo inverse functions defined as lHeGl

ðhlÞ ¼ inffpljleGlðplÞP hlg; l ¼ 1;2. This

implies that there exists a vector ðpl; sl; �ulÞ; l ¼ 1;2 such that

min16k6nj ;j¼1;2

leMljkðsljkÞP hl; �ul � N

X2

j¼1

Xnj

k¼1

sljkxjk;VlðxÞ !

;

pl ¼ Pðxj�ulðxÞ 6 flÞ;pl P lHeGl

;

which can be equivalently transformed into the condition that there exists a vector ðsl; �ulÞ such that

leMljkðsljkÞP hl; �ul � N

X2

j¼1

Xnj

k¼1

sljkxjk;VlðxÞ !

; Pðxj�ulðxÞ 6 flÞP lHeGl

; l ¼ 1;2; j ¼ 1;2; k ¼ 1; . . . ;nj: ð12Þ

In view of (3), it follows that

leMljkðsljkÞP hl () sljk 2 ½mljk � LHðhlÞaljk;mljk þ RHðhlÞbljk�;

where Lw(hl) and Rw(hl) are pseudo inverse functions defined as Lw(hl) = sup{tjL(t) P hl} and Rw(hl) = sup{tjL(t) P hl}. Hence,(12) is rewritten as the equivalent condition that there exists a �ul such that

Pðxj�ulðxÞ 6 flÞP lHeGl

; �ul � NX2

j¼1

Xnj

k¼1

fmljk � LHðhlÞaljkgxjk;VlðxÞ !

: ð13Þ

Since Prðxj�ulðxÞ 6 flÞ is transformed into

P x�ul �

P2j¼1

Pnj

k¼1fmljk � LHðhlÞaljkgxjkffiffiffiffiffiffiffiffiffiffiffiVlðxÞ

p 6

fl �P2

j¼1

Pnj

k¼1fmljk � LHðhlÞaljkgxjkffiffiffiffiffiffiffiffiffiffiffiVlðxÞ

p����� !

;

in consideration of

�ul �P2

j¼1

Pnj

k¼1fmljk � LHðhlÞaljkgxjkffiffiffiffiffiffiffiffiffiffiffiVlðxÞ

p � Nð0;1Þ:

196 M. Sakawa, T. Matsui / Information Sciences 239 (2013) 191–200

(13) is equivalently transformed as

Ufl �

P2j¼1

Pnj

k¼1fmljk � LHðhlÞaljkgxjkffiffiffiffiffiffiffiffiffiffiffiVlðxÞ

p !P lHeGl

ðhlÞ; ð14Þ

where U is a probability distribution function of the standard Gaussian random variable N (0,1).From the monotone increasingness of U, (14) is rewritten as

X2

j¼1

Xnj

k¼1

mljk � LHðhlÞaljk�

xjk þU�1 lHeGl

ðhlÞ � ffiffiffiffiffiffiffiffiffiffiffi

VlðxÞp

6 fl; ð15Þ

where U�1 is the inverse function of U.From (11)–(15), it holds that

PePlðeGlÞP hl ()

X2

j¼1

Xnj

k¼1

fmljk � LHðhlÞaljkgxjk þU�1 lHeGl

ðhlÞ � ffiffiffiffiffiffiffiffiffiffiffi

VlðxÞp

6 fl: ð16Þ

Consequently, (9) is equivalently transformed into

maximizefor DM1

h1;

maximizefor DM2

h2;

subject toX2

j¼1

Xnj

k¼1

fm1jk � LHðh1Þa1jkgx1jk þU�1 lHeG1

ðh1Þ � ffiffiffiffiffiffiffiffiffiffiffiffi

V1ðxÞp

6 f1;

X2

j¼1

Xnj

k¼1

fm2jk � LHðh2Þa2jkgx2jk þU�1 lHeG2

ðh2Þ � ffiffiffiffiffiffiffiffiffiffiffiffi

V2ðxÞp

6 f2;

A1x1 þ A2x2 6 b;

x1 P 0; x2 P 0:

9>>>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>>>;

ð17Þ

In order to obtain an initial candidate for an overall satisfactory solution to (9) or (17), it would be useful for DM1 to find asolution which maximize the smaller degree of satisfaction between the two DMs by solving the maximin problem

maximize min ZP;P1 ðx1; x2Þ; ZP;P

2 ðx1; x2Þn o

;

subject to A1x1 þ A2x2 6 b;x1 P 0; x2 P 0;

9>>=>>; ð18Þ

or equivalently

maximize vsubject to ZP;P

1 ðx1; x2ÞP v ;ZP;P

2 ðx2; x2ÞP v ;A1x1 þ A2x2 6 b;x1 P 0; x2 P 0:

9>>>>>>=>>>>>>;ð19Þ

Similar to the equivalent transformation (16), in view of ZP;Pl ðx1; x2Þ ¼ PePl

ðGlÞ, it holds that

ZP;Pl ðx1; x2ÞP v ()

X2

j¼1

Xnj

k¼1

fmljk � LHðvÞaljkgxjk þU�1 lHeGl

ðvÞ � ffiffiffiffiffiffiffiffiffiffiffi

VlðxÞp

6 fl; ð20Þ

where U�1 is the inverse function of the probability distribution function U of the standard Gaussian random variable N(0,1),and lHeGl

ðvÞ ¼ supfpl j leGlðplÞP vg and Lw(v) = sup{tjL(t) P v} are pseudo inverse functions.

Consequently, (19) is equivalently transformed as

M. Sakawa, T. Matsui / Information Sciences 239 (2013) 191–200 197

maximize v ;

subject toX2

j¼1

Xnj

k¼1

fm1jk � LHðvÞa1jkgxjk þU�1 lHeG1

ðvÞ � ffiffiffiffiffiffiffiffiffiffiffiffi

V1ðxÞp

6 f1;

X2

j¼1

Xnj

k¼1

fm2jk � LHðvÞa2jkgxjk þU�1 lHeG2

ðvÞ � ffiffiffiffiffiffiffiffiffiffiffiffi

V2ðxÞp

6 f2;

A1x1 þ A2x2 6 b;x1 P 0; x2 P 0:

9>>>>>>>>>>>>=>>>>>>>>>>>>;ð21Þ

Although (21) is nonconvex, an approximate solution can be obtained through particle swarm optimization for nonlinearprogramming (PSONLP) [12].

If DM1 is satisfied with the degrees ZP;Pl x�1; x

�2

� ; l ¼ 1;2, the corresponding optimal solution x�1; x

�2

� to (21) is regarded as

the satisfactory solution. However, if DM1 is not satisfied, by introducing the constraint that ZP;P1 ðx1; x2Þ is larger than or

equal to the minimal satisfactory level d 2 (0,1) specified by DM1, we consider the maximization problem formulated as

maximize ZP;P2 ðx1; x2Þ;

subject to ZP;P1 ðx1; x2ÞP d;

A1x1 þ A2x2 6 b;x1 P 0; x2 P 0;

9>>>>=>>>>; ð22Þ

or equivalently

maximize h;

subject to ZP;P2 ðx1; x2ÞP h;

ZP;P1 ðx1; x2ÞP d;

A1x1 þ A2x2 6 b;x1 P 0; x2 P 0:

9>>>>>>=>>>>>>;ð23Þ

Similar to the equivalent transformation (16), it follows that

ZP;P2 ðx1; x2ÞP h ()

X2

j¼1

Xnj

k¼1

fm2jk � LHðhÞa2jkgxjk þU�1 lHeG2

ðhÞ � ffiffiffiffiffiffiffiffiffiffiffiffi

V2ðxÞp

6 f2;

ZP;P1 ðx1; x2ÞP d ()

X2

j¼1

Xnj

k¼1

fm1jk � LHðdÞa1jkgxjk þU�1 lHeG1

ðdÞ � ffiffiffiffiffiffiffiffiffiffiffiffi

V1ðxÞp

6 f1:

9>>>>>=>>>>>;ð24Þ

Consequently, (23) is rewritten as

maximize h;

subject toX2

j¼1

Xnj

k¼1

fm2jk � LHðhÞa2jkgxjk þU�1 lHeG2

ðhÞ � ffiffiffiffiffiffiffiffiffiffiffiffi

V2ðxÞp

6 f2;

X2

j¼1

Xnj

k¼1

fm1jk � LHðdÞa1jkgxjk þU�1 lHeG1

ðdÞ � ffiffiffiffiffiffiffiffiffiffiffiffi

V1ðxÞp

6 f1;

A1x1 þ A2x2 6 b;x1 P 0; x2 P 0:

9>>>>>>>>>>>>=>>>>>>>>>>>>;ð25Þ

In view of the nonconvexity of (25), quite similar to (21), particle swarm optimization for nonlinear programming (PSONLP)[12] can be applied for solving this problem.

Realizing that the objective functions of DM1 and DM2 often conflict with each other, it should be noted here that thelarger the minimal satisfactory level d for the DM1’s objective function value is specified, the smaller the DM2’s objectivefunction value becomes, which may lead to the unbalanced satisfactory degrees of DM1 and DM2 due to the large differencebetween the objective function values of both DMs. In order to derive the satisfactory solution which has well-balancedobjective function values of DM1 and DM2, by introducing the ratio D between the satisfactory degrees of both DMs ex-pressed as

D ¼ ZP;P2 ðx1; x2Þ

ZP;P1 ðx1; x2Þ

; ð26Þ

we assume that DM1 specifies the lower bound Dmin and the upper bound Dmax of D, which are used for evaluating theappropriateness of the ratio D. To be more specific, if it holds that

Table 1Values

a1

a2

a3

a4

198 M. Sakawa, T. Matsui / Information Sciences 239 (2013) 191–200

D 2 ½Dmin;Dmax�;

then DM1 regards the corresponding solution as a promising candidate for the satisfactory solution with well-balancedmembership function values.

Now can we construct a procedure of interactive fuzzy programming for the possibility-based probability model in orderto derive an overall satisfactory solution.

3.1. Interactive fuzzy programming in the possibility-based probability model

Step 1: Ask each DM to specify the membership function leGland the target values fl, l = 1, 2.

Step 2: For the current target values fl, l = 1, 2, solve the maximin problem (18).Step 3: DM1 is supplied with the objective function values ZP;P

1 x�1; x�2

� and ZP;P

2 x�1; x�2

� for the optimal solution x�1; x

�2

� obtained in step 2. If DM1 is satisfied with the current membership function values, then stop. If DM1 is not satisfiedand prefers to update fl, l = 1, 2, ask DM1 to update fl, and return to step 2. Otherwise, ask DM1 to specify the minimalsatisfactory level d and the permissible range [Dmin,Dmax] of D.

Step 4: For the current minimal satisfactory d, solve (22).Step 5: DM1 is supplied with the current values of ZP;P

1 x�1; x�2

� , ZP;P

2 x�1; x�2

� and D. If D 2 [Dmin, Dmax] and DM1 is satisfied

with the current objective function values, then stop. Otherwise, ask DM1 to update the minimal satisfactory leveld, and return to step 4.

4. Numerical example

To demonstrate the feasibility and efficiency of the proposed method, consider the following two-level linear program-ming problem involving random fuzzy variable coefficients:

minimizefor DM1

z1ðx1; x2Þ ¼ eC 11x1 þ eC 12x2;

minimizefor DM2

z2ðx1; x2Þ ¼ eC 21x1 þ eC 22x2;

subject to a11x1 þ a12x2 6 b1;

a21x1 þ a22x2 6 b2;

a31x1 þ a32x2 6 b3;

a41x1 þ a42x2 6 b4;

x1 ¼ ðx11; x12; x13ÞT P 0;

x2 ¼ ðx21; x22; x23ÞT P 0:

9>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>;

ð27Þ

Table 1 shows values of coefficients of constraints ai, i = 1, 2, 3, 4 and bi, i = 1, 2, 3, 4 and Table 2 shows values of parametersof random fuzzy variables mljk, aljk and r2

ljk; l ¼ 1;2; j ¼ 1;2; k ¼ 1; . . . ;6.Through the use of this numerical example, it is now appropriate to illustrate the proposed interactive fuzzy nonlinear

programming.The parameter values of particle swarm optimization for nonlinear programming (PSONLP) are set as swarm size N = 50,

maximal search generation number Tmax = 3000, c1 = 2.0, c2 = 2.0, w0 = 1.2 and wTmax ¼ 0:1.Although the membership function does not always need to be linear, for the sake of simplicity, we adopt the linear mem-

bership function defined as

llðzlðx1; x2ÞÞ ¼

1; if zlðx1; x2Þ < z1l ;

zlðx1 ;x2Þ�z0l

z1l�z0

l; if z1

l 6 zlðx1; x2Þ 6 z0l ;

0; if zlðx1; x2Þ > z0l :

8>><>>:

Here, the parameter z1

l is determined as z1l ¼ zmin

l ¼ zl xlo1 ; x

lo2

� , and the parameter z0

l is specified as z0l ¼ zm

l ¼ zl xjo1 ; x

jo2

� �; l – j,

where xlo1 ; x

lo2

� is a feasible solution minimizing zl(x1,x2). Through PSONLP, the parameter values characterizing the linear

membership functions are determined as z11 ¼ �169:23; z0

1 ¼ �168:40; z12 ¼ �117:20, and z0

2 ¼ �112:08.

of coefficients in constraints.

al11 al12 al13 al21 al22 al23 b

25 15 17 0 0 0 12000 0 0 7 9 10 4004 3 5 3 2 2 1505 5 4 3 4 3 220

Table 2Values of mljk, aljk and r2

ljk .

�~cl11�~cl12

�~cl13�~cl21

�~cl22�~cl23

m1jk �4.30 �3.50 �5.00 �1.30 �1.50 �1.80m2jk �2.90 �2.50 �2.30 �1.50 �1.20 �1.00a1jk 1.20 0.80 0.90 0.40 0.60 0.30a2jk 0.70 0.50 1.10 0.40 0.70 0.60r2

1jk0.30 0.90 1.00 0.50 0.70 1.10

r21jk

0.70 0.60 1.20 0.70 1.00 0.90

Table 3Interaction process.

Interaction 1st 2nd 3rd

d̂ – 0.80 0.90

x11 0.40 15.61 17.99x12 17.20 22.38 20.46x13 8.91 0.30 0.78x21 0.68 0.88 0.18x22 9.13 3.65 0.76x23 2.78 2.59 2.72

ZP;P1 ðxÞ 0.74 0.80 0.90

ZP;P2 ðxÞ 0.74 0.77 0.68

D 1.0 0.96 0.76Iime (sec) 12.14 11.83 9.44

M. Sakawa, T. Matsui / Information Sciences 239 (2013) 191–200 199

The target values fl; p0l ; p1

l ; l ¼ 1;2 are set as f1 ¼ �160:00; f 2 ¼ �115:00; p01 ¼ 0:20; p0

2 ¼ 0:10; p11 ¼ 0:65; p1

2 ¼ 0:60.The obtained result is shown at the column labeled ‘‘1st’’ in Table 3. DM1 is not satisfied with this solution, but he doesnot desire to update fl, l = 1, 2. Thus, DM1 determines the minimal satisfactory level d̂ ¼ 0:80 to improve ZP;P

1 ðx1; x2Þ at theexpense of ZP;P

2 ðx1; x2Þ. Furthermore, DM1 specifies the upper bound Dmax = 0.90 and the lower bound Dmin = 0.70 for the ra-tio of objective functions D ¼ ZP;P

2 ðx1; x2Þ=ZP;P1 ðx1; x2Þ.

For the updated value of d̂, (27) is solved by PSONLP. The obtained result is shown at the column labeled ‘‘2nd’’ in Table 3.DM1 considers that ZP;P

1 ðx1; x2Þ is improved but D is greater than Dmax. Hence, DM1 is not satisfied with this solution andupdates the minimal satisfactory level d̂ from 0.80 to 0.90. (27) is solved for the updated value of d̂, and the obtained resultis shown at the column labeled ‘‘3rd’’ in Table 3. Since D exists in the interval [Dmin, Dmax] and DM1 is satisfied with thebalance between ZP;P

1 ðx1; x2Þ and ZP;P2 ðx1; x2Þ, the interactive algorithm is terminated.

In the proposed interactive fuzzy nonlinear programming, through a series of update procedures of the minimal satisfac-tory level d̂ and the target values fl, l = 1, 2, it can be possible to obtain a satisfactory solution, where the satisfactory degree ofDM1 is guaranteed to be greater than or equal to the minimal satisfactory level d̂ and is well balanced with that of DM2.

Finally, it is appropriate to point out here that our numerical experiments were performed on a personal computer (PC)with Intel Core 2 Duo 6300 (1.86 GHz) and, as shown in Table 3, the computational times of PSONLP were 12.14, 11.83 and9.44 s, respectively.

5. Conclusion

In this paper, interactive decision making methods for random fuzzy two-level linear programming problems have beenconsidered. Through the introduction of the possibility-based probability model, the original random fuzzy two-level pro-gramming problem was reduced to a deterministic one. In order to obtain a satisfactory solution for the decision makerat the upper level in consideration of the cooperative relation between decision makers, interactive fuzzy nonlinear pro-gramming for random fuzzy two-level linear programming problems was proposed. It was shown that all of the problemsto be solved in the proposed interactive fuzzy nonlinear programming can be solved through particle swarm optimization fornonlinear programming (PSONLP). An illustrative numerical example demonstrated the feasibility and efficiency of the pro-posed method. Extensions to other stochastic programming models will be considered elsewhere. Also extensions to randomfuzzy two-level linear programming problems with two decision makers under noncooperative environments will be re-ported in the near future.

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