FUZZY APPROACHES TO THE PRODUCTION PROBLEMS: THE CASE OF REFINERY INDUSTRY

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1 FUZZY APPROACHES TO THE PRODUCTION PROBLEMS: THE CASE OF REFINERY INDUSTRY Mustafa GÜNES Unv. Of Dokuz Eylül Fac. Of Econ. & Adm. Scences Department of Econometrcs Buca Izmr TURKEY Abstract: The Fuzzy prncple states that everythng s a matter of degree. So far many busness producton problems are solved by Operatonal Research Optmzaton Technques, under the consderatons of some assumptons. In the current lterature, stll we have several applcatons of fuzzy lnear, nteger, goal and other programmng applcatons. The man am of ths study s to add new applcaton to the lterature and to solve the refnery producton problem by usng the fuzzy prncples. In applcaton, the real refnery model has been developed and an alternatve fuzzy model solutons crtczed to determne whch one s better than the others. Fnally, comparng the classcal soluton by the one of the obtaned best soluton of the fuzzy models, one can obtan more sutable output of the models than tradtonals. Keywords: Fuzzy consderaton, Multvalued Logc, Fuzzy Lnear Programmng, Fuzzy Optmzaton. 1-INTRODUCTION Snce Zadeh [1] ntroduced the concept of possblty, t s realzed that a type of mpresson can be expressed by a possblty dstrbuton. The possblty space whch s called pattern space, has been dscussed n detal by Nahmas [2] and Sugeno [3]. On the other hand, fuzzy arthmetc wth fuzzy numbers or operatons of fuzzy numbers by the extensons prncple [4,5,6] have been studed n many studes. Snce the poneer work on fuzzy lnear programmng (FLP) by Tanaka and Zmmerman, n the last past years several knds of FLP applcatons have appeared n the lterature and obvously wth them dfferent approaches of resoluton have been proposed too [7]. All mathematcal programmng technques, such as LP, IP, DP and the others have the same objectve: maxmzaton or mnmzaton, generally optmzaton. But n a realty, snce everythng s matter of degree, the coeffcent of objectve functons or the values of other components such as rght hand sde values of model may be concdered as a multvalence. It means three or more optons, perhaps an nfnte spectrum of optons, nstead of just two extremes. The structure of the conventonal lnear programmng (LP) problem Subject to Ax b,...(1) X 0, where A s an (m,n) matrx, m n, and c,x R m, b R m columns vectors. As we do know from the lterature that the frst approach to the Fuzzy Lnear Programmng (FLP) made by Bellman and Zadeh[8]. After ths study, many dfferent types of FLP applcatons appared on ths area, lke crtcal dscussons on membershp functons for fuzzy lnear programmng problems.[9]. The fuzzness of the Classcal Lnear Programmng problems can appear ether n the constrant set or n the coeffcents takng part n the defnton of ths set. Namely, we may consder the Fuzzy Lnear

2 Programmng Problems, manly, n two alternatves: FLP problems wth fuzzy constrants and FLP problems wth fuzzy coeffcents. 1.1 Fuzzy LP problems wth fuzzy constrants The base of the FLP problems wth fuzzy constrants depends on the tolerance of volaton n the accomplsment of the constrants. In other words the decson makers permts the constrants to be satsfed as well as possble for each constrant n the constrant set. Ths assumpton can be represented by a χ < b,... = 1,2,3,...m and modeled by means of a membershp functon µ (x) ={ 1 f a x b, f (a x) f b a x b + b...(2) 0 f a x b + b Those knd of membershp functons expresses that the decson maker tolarates volatons n each constrants up the value b + b, =1,2,...,m. At the same tme f functons are assumng as a nondecreasng and contnuous. The graphcal representaton of the stuaton as shown n the followng Fgure 1. LP problems [10,11,12]. Fuzzy soluton to (3) can be obtaned nvolvng as partcular values and s found from the soluton of the parametrc LP problem.... (4) s.t Ax g (α ), x ε 0, α (0,1] where g(α ) s a column vector defned by the nverse functons of the f, = 1,2,,m. Here,the lnearty characterstc of (3) s preserved and, n partcular, f the f are lnear, the new form of the above system(4) becomes such as below:... (5) s.t Ax b + b ( 1 α ) x 0, α [0,1] where b = ( b 1, b 2,..., b m ) 1.2 FLP Problems wth fuzzy coeffcents In many applcatons, the decson makers do not know exactly the values of the coeffcents takng part n a certan problem and, moreover, that vaguness s not of a probablstc knd, one can model those nexact values by means of fuzzy numbers. In ths stuaton, the FLP system may be represented as followng µ St aj x b, = 1,2,3...m, x j 0, j = 1,2,3,...n, b b + b Fgure 1: Graphcal Representaton of µ (x) After ths defnton, the assocated problem s usually represented by the followng form of the LP structure. St Ax b,...(3) X 0 So far, many dfferent approaches has been developed to solve the fuzzfed model on dfferent f a where a j, b N(R), = 1,2,...,m, j= 1,2,...,n, are defned by respectve membershp functons µ j and µ. 2. Defnton and Soluton of The Producton Problem The applcaton area of ths research belongs to a certan producton problem on petro-chemstry ndustry. The producton problem whch s modeled n a LP structure has 44 varables. Refnery model has 4 dfferent crude ol nputs: Kerkük, Lght Iran, Lght Arab and Domestc producton. The developed Refnary LP model has also 36 constrants whch denotes capabltes of market and current system. The full formulated case of the

3 producton problem and ts restrctons such as below: The objectve functon dependng on the varables whch are descrbed above s as below: Max: X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X 44 = K Here are 36 consrants of the model descrbed unt by unt such as follows: (1) X Ton/Month, Avalable Kerkuk (2) X Ton/Month, Avalable L.Iran (3) X Ton/Month, Avalable L.Arab (4) X Ton/Month, Avalable Domestc Unt 100 (5) X 1 + X 2 + X 3 + X Ton/Month, Capacty of Unt 100 (6) 0.012X X X X 4 = ton, From Crude O.to L.P.G. (7) 0.196X X X X 4 = ton, From Crude O.to Naphta (8) 0.115X X X X 4 = ton, From Crude O.to Kerosne (9) 0.241X X X X 4 = ton,from Crude O.to Motorne (10) 0.436X X X X 4 = , From Crude O.to Atm.Rezdu Unt 200 (11) X 10 + X 11 + X 12 + X 13 + X 14 = ton naphta, as a nput to unt 200 (12) 0.41X 6 = ton-naphta- Fnal Product Unt 300 (13) X 15 + X 16 = ton nput (H2 and Naphta) to unt 300 Unt 400 (14) X 17 + X 18 = ton nput (H2 and Kerosne); to unt 400 Unt 600 (15) 0.49X 11 + X 34 = ton LPG nput to unt 600 (16) X ton Prophane fnal product (17) X X ton LPG fnal product Unt 700 (18) X 13 +X 15 +X 17 +X ton Sulphur Unt 1200 (19) X25 +X ton Vacum Rezdu,nput to unt 1200 Unt 1300 (20) X 27 + X ton Subproduct of Dst. And Brt.Dst Unt 1400 (21) X29 + X ton Rafnates nput Unt 1500 (22) 0.10X 10 + X ton Fnal products of M.Ols Unt 1100 (23) X 22 + X 23 +X ton Atm.Rez. nput (24) 0.035X X9+0.20X X X28+X36+X ton Fuel ol Unt 1900 (25) 0.697X X X 24 +X X ton Fnal products of Asphalt. Unt F.F.C. (26) X X X X 28 + X ton Subproducts, H 2 S,L.P.G., C.Gasolne, L.C.O., C.L.O.. Unt 600 (27) 0.098X X 34 - X 19 = 0 Fnal product Prophane (28) 0.51 X 11 +X 20 X 38 = 0 Fnal product LPG (29) 0.41X 6 - X 39 = 0 Fnal Product Naphta (30) X 16 + X 18 X 42 = 0 Fnal Product of Kerosne (31) X 12 + X 14 + X 35 - X 43 - X 44 = 0 Fnal Product of Normal and Super Gasolne. (32) 0.65X X 8 X 41 = 0 Fnal Product Motorn (33) 0.05X X X X X 28 +X 3 6+X 3 - X 40 = 0 Fnal product F.Ol (34) X 13 +X 15 +X 17 +X 33 - X 21 = 0 Fnal Product Sülphür (35) 0.10X10 + X29 - X31 = 0 Fnal Product M.Ols (36) X 24 +X 26 - X 32 = 0 Fnal Product Asphalt The method whch s usng for formulaton of ths problem s that the total amounts of nput and output at each unt are equal. Some of the coeffcents of objectve functon are postve and negatve. The coeffcents of frst four varables whch represent the crude ols and the subproducts, are negatve because of cost values. Postve coeffcents belongng to the fnal products are sale prces for per ton. It has been also used the productvty nformaton on formulaton of Unt 100. As a decson varable X 1 (Kerkük Crude Ol) and X 3 (Lght Arab) have been chosen by normal soluton. On the other hand, optmum value of the

4 objectve fucton, determned by selected varables, s as below; K = $ /Month The senstvty analyss results of the objectve functon to coeffcents for Kerkük Crude Ol and Lght Arab are between ( ) and ( Infnte) respectvely. Accordng to the normal soluton, maxmum value of prces of two knd of crude ols $ / Ton and $ / Ton respectvely. The other mean of the senstvty analyss ndcates that the crude ols must be bought as cheap as possble. 3- Fuzzy Solutons Instead of conventonal solutons of LP problems, snce the real lfe applcatons has dynamc structure, the decson makers have to consder altenatve models wth fuzzy constrants or fuzzy coeffcents. To transform the orgnal model to the alternatve fuzzy models, the frst fve constrants, dsplayed above, has been nvolved n fuzzness operaton. Snce the frst fve constrants have been consdered t values as a fuzzy numbers fxed by decson maker, determnes maxmum volaton n the accomplshment of the th constrant. Thus t makes sense to change that th constrant by the followng one: a x < b + t ( 1- α ), = 1,...5, α [0,1] whch expresses that for α = 1 the consrant s completely verfed wth respect to the wshes of the decson maker. Moreover, the smaller α s, the smaller the accomplsment degree for the decson maker shall be [7]. By followng ths logc, the trangular fuzzy values of coeffcents of the model, ntroduced above, for frst fve coeffcents, decded such as follows: 1 = ( 1,0.5,1.5) = ( , , ) 1 = ( 1,0.5,1.5) = ( , , ) 1 = ( 1,0.75,1.25) = ( , , ) 1 = ( 1,0.25,1.75) = ( ,75.000, ) 1 = (1,0.5,1.5), 1 =(1,0.75,1.25), 1 =(1,0.30,1.70), 1 =(1,0.40,1.60) = ( , , ) t 1 =(40.000,35.000,50.000), t 2 =(20.000,10.000,30.000), t 3 =(30.000,20.000,40.000), t 4 =(50.000,40.000,60.000), t 5 =(30.000,10.000,50.000) b 1 + t 1 (1- α) = [ (1- α), (1- α), (1- α)] b 2 + t 2 (1- α) =[ (1- α), (1- α), (1- α)] b 3 + t 3 (1- α)=[ (1- α), (1- α), (1- α)] b 4 + t 4 (1- α) = [ ( 1- α), (1- α), (1-α)] b 5 +t 5 (1- α)=[ (1- α), (1- α), (1- α)] After these arrangements, we may choose two or three dfferent auxlary models to solve such as below: Model-1 X (1- α) X (1- α) X (1- α) X (1- α) X 1 +X 2 +X 3 +X 4 +X (1- α) Model -2 X (1- α) X (1- α) X (1- α) X (1- α) X 1 +X 2 +X 3 +X 4 +X (1- α) Model -3 X (1- α) X (1- α) X (1- α) X (1- α) X 1 +X 2 +X 3 +X 4 +X (1- α)

5 Solutons of these models wth α = 0.80 and ther selected decson varables by algorthm lsted below; Models Solutons Selected Var. Orgnal $ ,89 X 1 and X 3 Fuzzy Mod.1 $ ,24 X 1 and X 3 Fuzzy Mod2 $ ,12 X 1 and X 3 Fuzzy Mod3 $ ,64 X 1 and X 3 ** 4- Concluson [10] H.Tanaka, T.Okudaand K.Asa,n Fuzzy Mathematcal Programmng, J. Cybernet.3 (1974) [11] J.L. Verdegay, Fuzzy Mathematcal Programmng, n: M.M. Gupta and E.Sanches, Eds., Fuzzy Informaton and Decson Processes (North Holland, Amsterdam, 1982) [12] H.J. Zmmerman,Optmzaton n Fuzzy Envronment, Presented at XXI Intern. TIMS and 46 th ORSA Conference, San Juan, Puerto Rco (1974). As a result, we have seen from the above alternatve soluton lst that maxmum value of the objectve functon belongs to the 3 rd Fuzzfed Model. The approach has been gven s based on the concept of a comparason relaton between fuzzy numbers. In applcaton, top manegers always have the ablty to mprove ther nput capactes and system restrctons. In ths applcaton, only three alternatve fuzzy model compared by the orgnal one. It ýs very obvous that there are unlmted verson of the fuzzfed alternatves. So that ýt s necessary powerfull software support to fnd out and crtces best soluton between many logcal models, representng orgnal systems. References [1] L.A. Zadeh, Fuzzy Sets as a bass for a theory of possblty, Fuzzy Sets and Systems 1(1978), [2] S.Nahmas, Fuzzy Varables, Fuzzy Sets and Systems 1(2) (1978) [3] M. Sugeno, Fuzzy Theory IV (Lecture Note), J. Of SICE 22(1983) [4] C.V. Negota and D.A. Ralescu, Applcaton of Fuzzy Sets to Systems Analyss (Brkhaüser, Basel- Boston,1975) [5] H.T. Nguyen, A note on the extenton prncple for fuzzy sets, Jourrnal of Math. Anal. Appl.64(1978) [6] L.A. Zadeh, The Concept of a Lngustc Varable and It s Applcaton to Approxmate Reasonng-I, Informaton Sc.8(1975) [7] M.Delgado, J.L. Verdegay, M.A. Vla, A General Model For Fuzzy Lnear Programmng, Fuzzy Sets and Systems 29(1989) [8] R. Bellman and L.A.Zadeh,Decson Makng n Fuzzy Envronment, Management Sc. B.17(1970) [9] C.Garca-Aguado, J.L.Verdegay, On the senstvty of membershp functons for fuzzy lnear programmng problems, Fuzzy Sets and Systems 56(1993)