TurkJMath26(2002),375396.cTUBITAKSolvingFuzzyLinearProgrammingProblemswithLinearMembershipFunctionsRafail N. Gasimov, K ursat YenilmezAbstractIn this paper, we concentrate on two kinds of fuzzy linear programming problems:linearprogrammingproblemswithonlyfuzzytechnological coecientsandlinearprogramming problems inwhichboththe right-handside andthe technologicalcoecients arefuzzynumbers. Weconsider hereonlythecaseof fuzzynumberswithlinearmembershipfunctions. Thesymmetricmethodof BellmanandZadeh[2] is usedfor a defuzzicationof theseproblems. The crisp problemsobtainedafterthedefuzzicationarenon-linearandevennon-convexingeneral. Weproposeherethemodiedsubgradientmethodanduseitforsolvingtheseproblems. Wealsocomparethenewproposedmethodwithwell knownfuzzydecisivesetmethod.Finally,wegiveillustrativeexamplesandtheirnumericalsolutions.Key Words: Fuzzy linear programming; fuzzy number; modied subgradientmethod; fuzzydecisivesetmethod.1. IntroductionIn fuzzy decision making problems, the concept of maximizing decision was proposedbyBellmanandZadeh[2]. This concept wasadoptedtoproblems of mathematicalprogrammingbyTanakaetal. [13]. Zimmermann[14]presentedafuzzyapproachtomultiobjectivelinearprogrammingproblems. Healsostudiedthedualityrelationsinfuzzylinearprogramming. FuzzylinearprogrammingproblemwithfuzzycoecientswasformulatedbyNegoita[8]andcalledrobustprogramming. DuboisandPrade[3]investigatedlinear fuzzyconstraints. Tanaka and Asai [12] also proposeda formulationoffuzzylinear programming withfuzzyconstraintsandgave amethodfor itssolutionwhichbasesoninequality relationsbetweenfuzzynumbers. Shaocheng[11] considered2000 Mathematical Subject Classication: 90C70,90C26.375GASIMOV,YENILMEZthe fuzzy linear programming problem with fuzzy constraints and defuzzicated it by rstdetermining an upper bound for the objective function.Further he solved the so-obtainedcrisp problem by the fuzzy decisive set method introduced by Sakawa and Yana [10].In this paper, we rst consider linear programming problems in which only technolog-ical coecients are fuzzy numbers and then linear programming problems in which bothtechnological coecients and right-hand-side numbers are fuzzy numbers. Each problemis rstconvertedinto an equivalent crispproblem. This is a problem of nding a pointwhich satises the constraints and the goal with the maximum degree. The idea of thisapproach is due to Bellman and Zadeh [2]. The crisp problems, obtained by such a man-ner,can be non-linear (even non-convex), where the non-linearity arises in constraints.Forsolving theseproblems weuseandcomparetwomethods. Oneofthemcalledthefuzzydecisivesetmethod, asintroducedbySakawaandYana[10]. Inthismethodacombination withthebisectionmethodandphaseoneofthesimplexmethodoflinearprogrammingisusedtoobtainafeasiblesolution. Thesecondmethodweuse, isthemodiedsubgradientmethodsuggestedbyGasimov [4]. For bothkinds ofproblemsweconsider, thesemethodsare appliedtosolve concreteexamples. Theseapplicationsshowthattheuseof modiedsubgradientmethodis more eectivefrompoint ofviewthe number of iterations required for obtaining the desired optimal solution.The paper is outlined as follows.Linear programming problem with fuzzy technolog-ical coecients is considered in Section 2. In section 3, we study the linear programmingproblem in which both technological coecients and right-hand-side are fuzzy numbers.The general principles of the modied subgradient method are presented in Section 4. InSection 5, we examine the application of modied subgradient method and fuzzy decisiveset method to concrete examples.2. Linearprogramming problems withfuzzytechnological coecientsWe consider a linear programming problem with fuzzy technological coecientsmaxnj=1cjxjsubject tonj=1 aijxj bi, 1 i mxj 0, 1 j n(2.1)where at least onexj> 0.We will accept some assumptions.376GASIMOV,YENILMEZAssumption 1.aij is a fuzzy number with the following linear membership function:aij(x) =

1 if x < aij,(aij +dij x)/dijif aij x < aij + dij,0 if x aij +dij,where x Randdij>0forall i =1, ..., m, j =1, ..., n. Fordefuzzicationof thisproblem, we rst fuzzify the objective function. This is done by calculating the lower andupperbounds oftheoptimal values rst. The boundsof theoptimal values, zlandzuare obtained by solving the standard linear programming problemsz1 = maxnj=1cjxjsubject tonj=1aijxj bi, i = 1, ..., m,xj 0,j = 1, ..., n,(2.2)andz2 = maxnj=1cjxjnj=1(aij + dij)xj bixj 0.(2.3)The objective function takes values betweenz1 and z2 while technological coecientsvary betweenaijandaij + dij. Letzl = min(z1, z2) andzu = max(z1, z2). Then, zlandzuare called the lower and upper bounds of the optimal values, respectively.Assumption 2. The linear crisp problems (2.2) and (2.3) have nite optimal values.In this case the fuzzy set of optimal values,G, which is a subset of Rn, is dened as(see Klir and Yuan [6]);G(x) =

0 ifnj=1cjxj< zl,(nj=1cjxj zl)/(zu zl) if zl nj=1cjxj< zu,1 ifnj=1cjxj zu.(2.4)377GASIMOV,YENILMEZThe fuzzy set of thei th constraint,Ci, which is a subset of Rm, is dened byCi(x) =

0 , bi