Fuzzy Approaches for Multiobjective Fuzzy Random Linear Programming Problems Through a Probability Maximization Model

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1 Fuzzy Approaches for Multobjectve Fuzzy Random Lnear Programmng Problems Through a Probablty Maxmzaton Model Htosh Yano and Kota Matsu Abstract In ths paper, two knds of fuzzy approaches are proposed for not only multobjectve stochastc lnear programmng problems, but also multobjectve fuzzy random lnear programmng problems through a probablty mzaton model. In a probablty mzaton model, t s necessary for the decson maker to specfy permssble values of objectve functons n advance, whch have a great nfluence on the correspondng dstrbuton functon values. In our proposed methods, the decson maker does not specfy permssble values of objectve functons, but sets hs/her membershp functons for permssble values. By assumng that the decson maker adopts the fuzzy decson as an aggregaton operator of fuzzy goals for the orgnal objectve functons and dstrbuton functons, a satsfactory soluton of the decson maker s easly obtaned based on lnear programmng technque. Index Terms multobjectve stochastc lnear programmng, multobjectve fuzzy random lnear programmng, fuzzy decson, a probablty mzaton model. I. INTRODUCTION In the real world decson makng stuatons, we often have to make a decson under uncertanty. In order to deal wth decson problems nvolvng uncertanty, stochastc programmng approaches and fuzzy programmng approaches have been developed. In stochastc programmng approaches, two stage problems [3] and chance constraned programmng models [1] have been nvestgated n varous ways, and they were extended to multobjectve stochastc programmng problems [11],[14]. In fuzzy programmng approaches, varous types of fuzzy programmng problems have been formulated and nvestgated [8],[10],[17]. As a natural extenson, multobjectve fuzzy programmng technque frst proposed bu Zmmermann [16], and many methods have been proposed [12],[17]. From a dfferent pont of vew, mathematcal programmng problems wth fuzzy random varables have been proposed [6],[9],[15], whose concept ncludes both probablstc uncertanty and fuzzy one smultaneously. Snce such fuzzy random programmng problems are usually ll-defned, t s necessary to utlze not only stochastc programmng technque but also fuzzy programmng technque to construct a decson makng model. Recently, n order to deal wth probablstc uncertanty and fuzzy one smultaneously, the hybrd approaches of stochastc programmng and fuzzy programmng have been H. Yano s wth Graduate School of Humantes and Socal Scences, Nagoya Cty Unversty, Nagoya, , Japan, e-mal: yano@hum.nagoya-cu.ac.jp K. Matsu s wth Graduate School of Humantes and Socal Scences, Nagoya Cty Unversty, Nagoya, , Japan, e-mal: c094716@ed.nagoya-cu.ac.jp proposed [5]. Especally, Sakawa et al. [13] proposed an nteractve method for multobjectve fuzzy lnear programmng problem wth random varable coeffcents. Katagr et al. [7] proposed an nteractve method for multobjectve lnear programmng problem wth fuzzy random varable coeffcents. They also adopted a probablty mzaton model to transform stochastc programmng problems nto well-defned mathematcal programmng ones. However, usng a probablty mzaton model, t s necessary that, n advance, the decson maker specfes permssble levels for objectve functons n hs/her subjectve manner. It seems to be very dffcult to specfy such values n advance, because there exst conflcts among permssble levels and the correspondng dstrbuton functon values. From such a pont of vew, n ths paper, assumng that the decson maker adopts the fuzzy decson to ntegrate membershp functons, two types of fuzzy approaches are proposed for both multobjectve fuzzy lnear programmng problem wth random varable coeffcents and fuzzy random varable coeffcents. In secton 2, a fuzzy approach s proposed for multobjectve fuzzy lnear programmng problem wth random varable coeffcents. In secton 3, a fuzzy approach s proposed for multobjectve fuzzy lnear programmng problem wth fuzzy random varable coeffcents. Secton 4 provdes a numercal example to demonstrate the proposed fuzzy approach for multobjectve fuzzy lnear programmng problem wth fuzzy random varable coeffcents. Fnally, n secton 5, we conclude ths paper. II. A FUZZY APPROACH FOR MULTIOBJECTIVE STOCHASTIC LINEAR PROGRAMMING PROBLEMS In ths secton, we focus on multobjectve programmng problem nvolvng random varable coeffcents n objectve functons. Such a problem can be formally formulated as follows. [MOSP1] mn zx = z 1x,, z k x 1 where x = x 1, x 2,, x n T s an n dmensonal decson varable column vector, z x = c x + α, = 1,, k, are objectve functons nvolvng random varable coeffcents, c s a n dmensonal random varable row vector expressed by c = c 1 + t c 2, where t s a random varable, α s a random varable row vector expressed by α = α 1 +t α 2. In the followng, we assume that T s a dstrbuton functon of a random varable t, whch s strctly monotone ncreasng and contnuous. X s a lnear constrant set wth respect to x.

2 Snce MOSP1 contans random varable coeffcents n objectve functons, mathematcal programmng technques can not be drectly appled. In order to deal wth such multobjectve stochastc programmng problems, we make use of a probablty mzaton model, whch ams to mze the probablty that each objectve functon z x s less than or equal to a certan permssble objectve level f. Such a probablty x, f can be defned as follows. x, f def = Prω z x, ω f, = 1,, k 2 where Pr denotes a probablty measure, ω s an event, and z x, ω s a realzaton of the random objectve functon z x under the occurrence of each elementary event ω. The decson maker subjectvely specfes a certan permssble objectve level f for each objectve functon z x, ω. Let us denote a k dmensonal vector of certan permssble objectve levels as f = f 1,, f k. Then, MOSP1 can be transformed nto the tradtonal multobjectve programmng problem MOP1f, where probablty functons x, f, = 1,, k are adopted as objectve functons nstead of z x, ω, and each of them s mzed. [MOP1f] p 1x, f 1,, p k x, f k 3 Under the assumpton that c 2 x + α2 0, = 1,, k for any x X, usng dstrbuton functons T, = 1,, k we can rewrte the objectve functon x, f as the followng form. x, f = Prω z x, ω f = Prω c ωx + α ω f = Pr ω t ω f c 1 x + α1 c 2 x + α2 f c 1 = T x + α1 c 2 x + α2 In order to deal wth MOP1f, we consder the feasble regon P f = {p 1 x, f 1,, p k x, f k R k x X}. In the feasble regon P f, we can defne Pareto optmal soluton to MOP1f. Defnton 1. x X s sad to be a Pareto optmal soluton to MOP1f, f and only f there does not exst another x X such that x, f x, f, = 1,, k, wth strct nequalty holdng for at least one. Sakawa et al. [13] formulated a probablty mzaton model for MOSP1, and proposed an nteractve method to obtan the satsfactory soluton of the decson maker. In ther nteractve method, after the decson maker specfes permssble objectve levels f, = 1,, k for each objectve functon z x, ω, the canddate of the satsfactory soluton s obtaned from among M-Pareto optmal soluton set whch s Pareto optmal solutons n membershp space. However, n general, the decson maker seems to prefer not only the less value of permssble objectve level f, but also the larger value of probablty functon x, f. Snce these values conflct wth each other, the less values of permssble objectve level f results n the less value of probablty functon x, f. From such a pont of vew, we consder the followng multobjectve programmng problem whch can be regarded as a natural extenson of MOP1f. [MOP2] p 1x, f 1,, p k x, f k, f 1,, f k,f,=1,,k 4 Consderng the mprecse nature of the decson maker s judgment, t s natural to assume that the decson maker have fuzzy goals for each objectve functon n MOP2. In ths secton, t s assumed that such fuzzy goals can be quantfed by elctng the correspondng membershp functons. Let us denote a membershp functon of probablty functon x, f as µ p x, f, and a membershp functon of permssble objectve level f as µ f f respectvely. Then, MOP2 can be transformed as the followng multobjectve programmng problem. [MOP3],f,=1,,k µ p1 p 1 x, f 1,, µ pk p k x, f k, µ f1 f 1,, µ fk f k 5 Throughout ths secton, we make the assumptons that µ f f, = 1,, k are strctly monotone decreasng and contnuous wth respect to f, and µ p x, f, = 1,, k are strctly monotone ncreasng and contnuous wth respect to x, f. For example, we can defne the doman of µ p x, f as follows. Consderng the ndvdual mnmum and mum of Ez x, the decson maker subjectvely specfes the suffcently satsfactory mum value f mn and the acceptable mnmum value f. Then, the doman of µ f f s defned as: F = [f mn, f ]. 6 Correspondng to the doman F, denote the doman of µ p x, f as: P F = [mn, ]. 7 can be obtaned by solvng the followng problem. = x, f 8 It should be noted here that the above problem s equvalent to the followng lnear fractonal programmng problem [2] because dstrbuton functon T s strctly monotone ncreasng and contnuous. f c 1 x + α1 c 2 x + α2 On the other hand, n order to obtan mn, we frst solve x, f mn, and denote the correspondng optmal soluton as x. Usng the optmal solutons x, = 1,, k, mn can be obtaned as follows. mn = 9 mn x l, f mn 10 l=1,,k,l If the decson maker adopts the fuzzy decson as aggregaton operator for MOP3, the satsfactory soluton s obtaned by solvng the followng mn problem. [MAXMIN1] λ 11,f F,=1,,k,λ [0,1] subject to µ p x, f λ, = 1,, k 12 µ f f λ, = 1,, k 13

3 α 2 Accordng to the assumpton for µ p x, f and c 2 x+ > 0, the constrants 12 can be transformed as: µ p x, f λ, x, f λ, f c 1 T x + α1 c 2 x + α2 λ, f c 1 x + α 1 T 1 µ p 1 λ c 2 x + α 2, 14 where and T 1 are pseudo-nverse functons wth respect to µ p and T respectvely. Moreover, from the constrants 13 and the assumpton for µ f f, t holds that f f λ. Therefore, the constrant 14 can be reduced to the followng nequalty where a permssble objectve level f s removed. f λ c 1 x + α 1 T 1 λ c 2 x + α 2 15 Then, MAXMIN1 s equvalently transformed nto the followng problem. [MAXMIN2] λ 16,λ [0,1] subject to f λ c 1 x + α 1 T 1 λ c 2 x + α 2, = 1,, k 17 It should be noted here that the constrants 17 can be reduced to a set of lnear nequaltes for some fxed value λ. Ths means that an optmal soluton x, λ of MAXMIN2 s obtaned by combned use of the bsecton method wth respect to 0 λ 1 and the frst-phase of the two-phase smplex method of lnear programmng. The relatonshp between the optmal soluton x, λ of MAXMIN2 and Pareto optmal solutons to MOP1f can be characterzed by the followng theorem. Theorem 1. If x, λ s a unque optmal soluton of MAXMIN2, then x X s a Pareto optmal soluton to MOP1f, where f = f 1 λ,, f k λ. Proof Snce an optmal soluton x, λ satsfes the constrants 17, t holds that f λ c 1 x + α 1 T 1 λ c 2 x + α 2, µ 1 f T λ c 1 x + α 1 c 2 x + α 2 = x, f λ λ, = 1,, k. Assume that x X s not a Pareto optmal soluton to MOP1f, where f = f 1 λ,, f k λ, then there exsts x X such that x, f λ x, f λ λ, = 1,, k. Then there exsts x X such that f λ c 1 x + α 1 T 1 λ c 2 x + α 2, = 1,, k, whch contradcts the fact that x, λ s a unque optmal soluton of MAXMIN2. III. A FUZZY APPROACH FOR MULTIOBJECTIVE FUZZY RANDOM LINEAR PROGRAMMING PROBLEMS In ths secton, we focus on multobjectve programmng problems nvolvng fuzzy random varable coeffcents n objectve functons called multobjectve fuzzy random lnear programmng problem MOFRLP. [MOFRLP] mn Cx = c 1 x,, c k 18 where x = x 1, x 2,, x n T s an n dmensonal decson varable column vector, X s a lnear constrant set wth respect to x. c = c 1,, c n, = 1,, k, are coeffcent vector of objectve functon c x, whose elements are fuzzy random varables The symbols "-" and " " mean randomness and fuzzness respectvely, and the concept of fuzzy random varable n ths secton s defned precsely n [7]. Under the occurrence of each elementary event ω, c j ω s a realzaton of the fuzzy random varable c j, whch s a fuzzy number whose membershp functon s defned as follows. dj ω s L s = α j ω s d j ω ω, µ cj ω R s > d j ω ω, s d j ω β j ω where the functon Lt def = {0, lt} s a real-valued contnuous functon from [0, to [0, 1], and lts a strctly decreasng contnuous functon satsfyng l0 = 1. Also, Rt def = {0, rt} satsfes the same condtons. Let us assume that the parameters d j, α j, β j are random varables expressed as d j = d 1 j + t d 2 j, α j = αj 1 + t αj 2, β j = βj 1 +t βj 2 respectvely, where t s a random varable whose dstrbuton functon T s contnuous and strctly monotone ncreasng, and d l j, αl j, βl j, l = 1, 2 are constants. It should be noted that α j ω, β j ω are postve for any ω because of a property of spread parameters of LR-type fuzzy numbers. Therefore, let us gve the assumptons that αj 1 + t ωαj 2 > 0, βj 1 + t ωβj 2 > 0, for any ω. As shown n [7], the realzatons c ωx becomes a fuzzy number characterzed by the followng membershp functons. L µ c ωx y = R dωx y α ωx y d ωx β ωx y d ωx ω, y > d ωx ω, Smlar to the prevous secton, t s assumed that the decson maker has fuzzy goals for the objectve functons n MOFRLP, whose membershp functons µ G y, = 1,, k are contnuous and strctly monotone decreasng. By usng a concept of possblty measure [4], the degree of possblty that the objectve functon value c x satsfes the fuzzy goal G s expressed as follows. Π c x G def = sup mn{µ c x y, µ G y} 19 y Usng the above possblty measure, MOFRLP can be transformed to the followng multobjectve stochastc programmng problem MOSP2.

4 [MOSP2] Π c 1 x G 1,, Π c k x G k 20 Katagr et al. [7] frst formulated MOFRLP as the followng multobjectve programmng problem through a probablty mzaton model. [MOP4h] Prω Π c 1 ωx G 1 h 1,, Prω Π c k ωx G k h k 21 where h = h 1,, h k are permssble degrees of possblty measure specfed by the decson maker. In MOP4h, the constrant Π c ωx G h can be transformed as follows. sup y mn{µ c y, ωx µ G y} h, y : µ c y h, y h ωx µ G, y : L d ωx y α ωx h, R y d ωx β ωx h, y h µ G, y : d ω L 1 h α x y d ω + R 1 h β x, y h, G d ω L 1 h α ωx h G where L 1 and R 1 are pseudo-nverse functon correspondng to L and R. Usng a dstrbuton functon T of t, each objectve functon of MOP4h s transformed as below. Prω Π c G h ωx = Pr ω d ω L 1 h α ωx h G = Pr ω d 1 + t ωd 2 x L 1 h α 1 + t ωα 2 x h G = Pr ω d 1 x L 1 h α 1 x +t ωd 2 x L 1 h α 2 x h G h d 1 G x L 1 h α 1 = Pr ω x t ω d 2 x L 1 h α 2 x = T µ 1 h d 1 G x L 1 h α 1 x d 2 x L 1 h α 2 x def = x, h 22 where d 2 L 1 0α 2 x > 0, = 1,, k for any x X. Usng x, h, = 1,, k, MOP4h can be expressed as the followng smple form. [MOP5h] p 1x, h 1,, p k x, h k 23 In order to deal wth MOP5h, we defne Pareto optmal solutons n the feasble set P h = {p 1 x, h 1,, p k x, h k [0, 1] k x X}. Defnton 2. x X s sad to be a Pareto optmal soluton to MOP5h, f and only f there does not exst another x X such that x, h x, h, = 1,, k, wth strct nequalty holdng for at least one. Katagr et al. [7] proposed an nteractve method to obtan a satsfactory soluton from among Pareto optmal soluton set to MOP5h, where permssble values of possblty measure h = h 1,, h k must be set n advance by the decson maker n hs/her subjectve manner. However, n general, the decson maker seems to prefer not only the larger value of permssble value of possblty measure h but also the larger value of probablty functon x, h. From such a pont of vew, we consder the followng multobjectve programmng problem whch can be regarded as a natural extenson of MOP5h. [MOP6] p 1x, h 1,, p k x, h k, h 1,, h k,h [0,1],=1,,k 24 Smlar to the prevous secton, we assume that the decson maker has fuzzy goals for x, h, = 1., k, and such fuzzy goals can be quantfed by elctng the correspondng membershp functons µ p x, h. Then MOP6 can be replaced by the followng form. [MOP7],h [0,1],=1,,k µ p1 p 1 x, h 1,, µ pk p k x, h k, h 1,, h k 25 Throughout ths secton, we assume that µ p x, h, = 1,, k are strctly monotone ncreasng and contnuous wth respect to x, h. For example, we can defne the doman of µ p x, h as follows. Consderng the ndvdual mnmum and mum of Ed x, the decson maker subjectvely specfes the suffcently satsfactory mum value and the acceptable mnmum value for the orgnal objectve functons n MOFRLP, and defnes membershp functon µ G y. For the possblty measure 19 based on µ G y, the decson maker subjectvely specfes the suffcently satsfactory mum value h and the acceptable mnmum value h mn. Then, the nterval for permssble value h s defned as: H = [h mn, h ]. 26 Correspondng to the nterval H, let us denote the doman of µ p x, h as: P H = [mn, ]. 27 can be obtaned by solvng the followng problem. = x, h mn 28 In order to obtan mn, we frst solve x, h, = 1,, k, and denote the optmal soluton as x. Usng the optmal solutons x, = 1,, k, we can obtan mn as follows. mn = mn x l, h 29 l=1,,k,l If the decson maker adopts the fuzzy decson as an aggregaton operator for MOP7, a satsfactory soluton s obtaned by solvng the followng mn problem. [MAXMIN3] λ 30,h H,=1,,k,λ [0,1]

5 subject to µ p x, h λ, = 1,, k 31 h λ, = 1,, k 32 Snce there exst pseudo-nverse functons and T 1 wth respect to µ p and T, the constrants 31 can be transformed as: µ p x, h λ, x, h λ, T µ 1 h d 1 G x L 1 h α 1 x d 2 x L 1 h α 2 x λ, h d 1 G x L 1 h α 1 x λ d 2 x L 1 h α 2 x h d 1 G x + T 1 λd 2 x L 1 h α 1 x + T 1 λα 2 x 33 On the other hand, because of the constrant 32, t holds that h λ, L 1 h L 1 λ. Snce t s G G guaranteed that α 1 x + T 1 λ α 2 x > 0, the rght hand sde of the constrant 33 can be transformed as the followng form. d 1 x + T 1 λd 2 x L 1 h α 1 x + T 1 λα 2 x d 1 x + T 1 λd 2 x L 1 λα 1 x + T 1 λα 2 x = d 1 x L 1 λα 1 x λ d 2 x L 1 λα 2 x 34 From the nequaltes 33 and 34, the followng nequalty can be easly obtaned. λ h d 1 G G x L 1 λα 1 x µ p 1 λ d 2 x L 1 λα 2 x As a result, MAXMIN3 can be transformed nto MAXMIN4 where permssble degrees of possblty measure h, = 1,, k have dsappeared. [MAXMIN4] λ 35,0 λ 1 subject to G λ d 1 x L 1 λα 1 x λ d 2 x L 1 λα 2 x, = 1,, k 36 It should be noted here that the constrants 36 can be reduced to a set of lnear nequaltes for some fxed value λ. Ths means that an optmal soluton x, λ of MAXMIN4 s obtaned by combned use of the bsecton method wth respect to 0 λ 1 and the frst-phase of the two-phase smplex method of lnear programmng. The relatonshp between the optmal soluton x, λ of MAXMIN4 and Pareto optmal solutons to MOP5h can be characterzed by the followng theorem. Theorem 2. TABLE I PARAMETERS OF FUZZY RANDOM NUMBERS cj j j d 1 1j d 2 1j d 1 2j d 2 2j α 1 1j α 2 1j α 1 2j α 2 2j β1j β1j β2j β2j If x, λ s a unque optmal soluton of MAXMIN4, then x Xs a Pareto optmal soluton of MOP5λ, where λ = λ,, λ. Proof Snce an optmal soluton x, λ satsfes the constrants 36, t holds that G λ d 1 x L 1 λ α 1 x T µ 1 λ d 2 x L 1 λ α 2 x λ d 1 G x L 1 λ α 1 x d 2 x L 1 λ α 2 x = x, λ λ, = 1,, k Assume that x X s not a Pareto optmal soluton of MOP5λ, where λ = λ,, λ, then there exsts x X such that x, λ x, λ λ, = 1,, k. Then there exsts x X such that G λ d 1 x L 1 λ α 1 x = 1,, k λ d 2 x L 1 λ α 2 x, whch contradcts the fact that x, λ s a unque optmal soluton of MAXMIN4. IV. NUMERICAL EXAMPLE In ths secton, n order to demonstrate the feasblty of our proposed method, we consder the followng multobjectve fuzzy random lnear programmng problem MOFRLP whch was formulated by Katagr et al. [7]. [MOFRLP] mn mn c 1 x = c 11 x 1 + c 12 x 2 + c 13 x 3 c 2 x = c 21 x 1 + c 22 x 2 + c 23 x 3 where X = {x = x 1, x 2, x 3 2x 1 + 6x 2 + 3x 3 150, 6x 1 + 3x 2 + 5x 3 175, 5x 1 + 4x 2 + 2x 3 160, 2x 1 + 2x 2 + 3x 3 90}, the parameters of fuzzy random number c j, = 1, 2, j = 1, 2, 3 are gven n Table 1, and t, = 1, 2 are Gaussan random varables N0, 1. By calculatng mn Ed x, = 1, 2 and Ed x, = 1, 2,

6 the hypothetcal decson maker sets hs/her lnear membershp functons of fuzzy goals G, = 1, 2 for the orgnal objectve functons n MOFRLP as follows. y µ G1 1 = y y 2 y µ G2 2 = For the elcted membershp functons µ G y, = 1, 2, the hypothetcal decson maker sets the ntervals H = [h mn, h ], = 1, 2 as [0.4, 0.75] and [0.4, 0.75] respectvely n hs/her subjectve manner. Then, usng 28 and 29, the correspondng domans P H = [mn, ], = 1, 2 are calculated. The correspondng lnear membershp functons of fuzzy goals for x, h, = 1, 2 n MOP5h are obtaned as follows. µ p1 p 1 x, h 1 = p 1x, h µ p2 p 2 x, h 2 = p 2x, h For these membershp functons µ G y, µ p x, h, = 1, 2, MAXMIN4 s formulated and solved by combned use of the bsecton method wth respect to 0 λ 1 and the frst-phase of the two-phase smplex method of lnear programmng. The optmal soluton s obtaned as x 1, x 2, x 3, λ = , , , Snce two constrants for 36 are actve at the optmal soluton, we can get the followng optmal values of the correspondng membershp functon. def µ p1 p 1 x, λ = µ p2 p 2 x, λ = µ G1 y 1 = µ G2 y 2 = where y = d 1 x L 1 λ α 1 x + T 1 λ d 2 x L 1 λ α 2 x, = 1, 2. At the optmal soluton, the proper balance between the membershp functons µ p x, λ, = 1, 2 and y µ G, = 1, 2 n a probablty mzaton model s attaned through the fuzzy decson. V. CONCLUSION In ths paper, two knds of fuzzy approaches are proposed to obtan a satsfactory soluton of the decson maker, where the frst one s for multobjectve stochastc lnear programmng problems, and the second one s for multobjectve fuzzy random lnear programmng problems. Both of them are formulated on the bass of a probablty mzaton model. In our proposed methods for such two knds of multobjectve programmng problems, t s not necessary that the decson maker specfes permssble levels n a probablty mzaton model. Instead of that, by adoptng the fuzzy decson as an aggregaton operator of fuzzy goals for both permssble levels and dstrbuton functons, a satsfactory soluton of the decson maker s easly obtaned based on lnear programmng technque. Although a probablty mzaton model s one of the most effcent tool to transform stochastc programmng problems nto well-defned mathematcal programmng ones, approprate permssble levels are not known for the decson maker n advance. In order to resolve such a problem, we have proposed fuzzy approaches for both multobjectve stochastc lnear programmng problems and multobjectve fuzzy random lnear programmng problems under the assumpton that the decson maker adopts the fuzzy decson as an aggregaton operator of fuzzy goals. REFERENCES [1] Charnes,A. and Cooper,W.W.: Chance Constraned Programmng, Management Scence, 6, pp.73-79, [2] Charnes,A. and Cooper,W.W.: Programmng wth Lnear Fractonal Functons, Naval Research Logstc Quarterly, pp , Vol.9, [3] Danzg, G.B.: Lnear Programmng under Uncertanty, Management Scence, 1, pp , [4] Dubos, D. and Prade, H.: Fuzzy Sets and Systems, Academc Press, [5] Hulsurkar, S. Bswal, M.P. and Shnha, S.B.: Fuzzy Programmng Approach to Mult-Objectve Stochastc Lnear Programmng Problems, Fuzzy Sets and Systems, 88, pp , [6] Katagr, H., Ish, H. and Itoh, T.: Fuzzy Random Lnear Programmng Problem, n: Proceedngs of Second European Workshop on Fuzzy Decson Analyss and Neural Networks for Management, Plannng and Optmzaton, pp , [7] Katagr, H., Sakawa, M., Kato, K. and Nshzak, I.: Interactve Multobjectve Fuzzy Random Lnear Programmng: Maxmzaton of Possblty and Probablty, European Journal of Operatonal Research, 188, pp , [8] La, Y.J. and Hwang, C.L.: Fuzzy Mathematcal Programmng, Sprnger, Berln [9] Luhandjula, M.K. and Gupta, M.M.: On Fuzzy Stochastc Optmzaton, Fuzzy Sets and Systems, 81, pp.47-55, [10] Rommelfanger, H.: Fuzzy Lnear Programmng and Applcatons, European Journal of Operatonal Research, 92, pp , [11] Stancu-Mnasan,I.M.: Overvew of Dfferent Approaches for Solvng Stochastc Programmng Problems wth Multple Objectve Functons, Stochastc Versus Fuzzy Approaches to Multobjectve Mathematcal Programmng under Uncertanty, Slownsk R. and Teghem J.Eds., Kluwer Academc Publshers, pp , [12] Sakawa, M.: Fuzzy Sets and Interactve Multobjectve Optmzaton, Plenum Press, [13] Sakawa,M., Kato,K. and Katagr,H.: An Interactve Fuzzy Satsfcng Method for Multobjectve Lnear Programmng Problems wth Random Varable Coeffcents Through a Probablty Maxmzaton Model, Fuzzy Sets and Systems, 146, pp , [14] Teghem, J. Dufrane, D. and Thauvoye, M.: STRANGE: An Interactve Method for Mult-Objectve Lnear Programmng Under Uncertanty, European Journal of Operatonal Research, 26, pp.65-82, [15] Wang, G.-Y. and Qao, Z.: Lnear programmng wth Fuzzy Random Varable Coeffcents, Fuzzy Sets and Systems, 57, pp , [16] Zmmermann, H.-J.: Fuzzy Programmng and lnear programmng wth several objectve functons, Fuzzy Sets and Systems, 1, pp.45-55, [17] Zmmermann, H.-J.: Fuzzy Sets, Decson-Makng and Expert Systems, Kluwer Academc Publshers, Boston 1987.