APPLICATION OF MULTI OBJECTIVE FUZZY LINEAR PROGRAMMING IN SUPPLY PRODUCTION PLANNING PROBLEM

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1 APPLICATION OF MULTI OBJECTIVE FUZZY LINEAR PROGRAMMING 37 Jurnl Teknolog, 40(D) Jun. 2004: Unverst Teknolog Mlys APPLICATION OF MULTI OBJECTIVE FUZZY LINEAR PROGRAMMING IN SUPPLY PRODUCTION PLANNING PROBLEM PANDIAN M VASANT* Astrct. The ojectve of ths pper s to estlsh the usefulness of modfed s-curve memershp functon n lmted supply producton plnnng prolem wth contnuous vrles. In ths respect, fuzzy prmeters of lner progrmmng re modeled y non-lner memershp functons such s s- curve functon. Ths pper egns wth n ntroducton nd constructon of the modfed s-curve memershp functon. A numercl rel lfe exmple of supply producton plnnng prolem s then presented. The computtonl results show the usefulness of the modfed s-curve memershp functon wth fuzzy lner progrmmng technque n optmsng ndvdul ojectve functons, compred to non-fuzzy lner progrmmng pproch. Furthermore, the optml soluton helps to conclude tht y ncorportng fuzzness n lner progrmmng model through the ojectve functon nd constrnts, etter level of stsfctory soluton wll e provded n respect to vgueness, compred to non-fuzzy lner progrmmng. Keywords: Vgueness, s-curve memershp functon, degree of stsfcton, fuzzy lner progrmmng, mult-ojectves 1.0 INTRODUCTION In the rel-world of decson-mkng processes n engneerng nd usness, decson mkng theory hs ecome one of the most mportnt felds. It uses the optmzton methodology connected to sngle crter, ut lso stsfyng concepts of multple crter. Decson processes wth multple crter del wth humn judgment. Ths s relly hrd to e modeled. The humn judgment element s n the re of preferences defned y the decson mker [1-3]. Frst, ttempts to model decson processes wth multple crter n usness nd engneerng led to concepts of mult ojectve fuzzy lner progrmmng [4-6]. In ths pproch, the decson mker underpns ech ojectve wth numer of gols tht should e stsfed [7-8]. The term stsfyng requres fndng soluton to mult crteron prolem, whch s preferred, understood, nd mplemented wth confdence. The confdence tht the est soluton hs een found s estmted through the del soluton. Ths s the soluton whch optmzes ll crter smultneously. Snce ths s prctclly unttnle, decson mker consders fesle solutons closest to the del soluton [9-10]. * Senor Mthemtcs Lecturer, Nl Interntonl College, Nl, Mlys. E-ml: pndn_vsnt@yhoo.com JTJUN40D[05].pmd 37

2 38 PANDIAN M VASANT In mthemtcl progrmmng, prolems re expressed s optmzng some ojectve functon gven certn constrnts. So, the methods of soluton were drected towrds sngle ojectve mthemtcl progrms such s the smplex method for lner progrmmng. In pplyng mthemtcl progrmmng, the decson-mkers relzed tht there re rel-lfe prolems tht consder multple ojectves. To e le to come up wth model tht cn e solved y the developed soluton methods for sngleojectve mthemtcl progrms, these multple ojectves must e comned n some wy to ecome one sngle ojectve. Mny prolems n opertons reserch, decson scence, engneerng nd mngement hve mnly een studed from optmzng ponts of vew. As decson mkng s much nfluenced y the dsturnces of socl nd economcl crcumstnces, optmzton pproch s not lwys the est. It s ecuse under such nfluences, mny prolems re ll-structured. Therefore, stsfcton pproch my e much etter thn n optmzton one. In ths regrds, t s cceptle tht the sprton level on the treted prolem s resolved on the se of pst experences nd current knowledge possessed y decson mker, n the cse where the sprton level of decson mker should e consdered to solve prolem from the perspectve of stsfcton strtegy. Therefore, t s more nturl tht the vgueness n the fuzzy system s denoted y fuzzy numers y the decson mker. Vrous types of memershp functons were used n fuzzy lner progrmmng prolems nd some of them re lner memershp functon [11-12], tngent type of memershp functon [13], n ntervl lner memershp functon [14], n exponentl memershp functon [15], nverse tngent memershp functon [16], logstc type of memershp functon [17], concve pecewse lner memershp functon [18] nd pecewse lner memershp functon [19]. The tngent type, memershp functon, exponentl memershp functon, nd hyperolc memershp functon re non-lner functons. A fuzzy mthemtcl progrmmng defned wth non-lner memershp functon results n non-lner progrmmng. Usully lner memershp functon s employed n order to vod non-lnerty. Nevertheless, there re some dffcultes n selectng the soluton of prolem wrtten n lner memershp functon. Therefore, n ths pper modfed s-curve memershp functon s employed to overcome such defcences whch lner memershp functon hs. Furthermore, s-curve memershp functon s more flexle enough to descre the vgueness n the fuzzy prmeters for the supply producton plnnng prolems. In ths pper, the new methodology of modfed s-curve memershp functon [20-22] usng fuzzy lner progrmmng n supply producton plnnng nd ther pplctons to decson mkng s proposed. Especlly, the fuzzy lner progrmmng sed on vgueness n the fuzzy prmeters such s resource vrles gven y decson mker s nlyzed. JTJUN40D[05].pmd 38

3 APPLICATION OF MULTI OBJECTIVE FUZZY LINEAR PROGRAMMING MODIFIED S-CURVE MEMBERSHIP FUNCTION The modfed s-curve memershp functon s prtculr cse of the logstc functon wth specfed prmeters. These prmeter vlues re to e found out. Ths logstc functon s gven y equton (1) nd depcted n Fgure 1 s ndcted s s-shped memershp functon y Gonguen [23] nd Zdeh [24]. We defne here, modfed s-curve memershp functon s follows: 1 x < x x = x µ = B ( x) x < x < x + α x 1 Ce x = x 0 x > x (1) where µ s the degree of memershp functon. The term α determnes the shpe of memershp functon µ(x), where α > 0. The lrger the vlue of prmeter α, the lesser s the vgueness. It s necessry tht prmeter α, whch determnes the shpe of memershp functons, should e heurstclly nd experentlly decded y experts. In equton (1), B nd C re constnts, ther vlues re gven n equton (10). Fgure 1 shows the modfed S-curve. The memershp functon s redefned s µ(x) Ths rnge s selected ecuse n supply producton, the revenue nd hrmful polluton need not e lwys 100% of the requrement. At the sme tme the totl revenue nd totl hrmful polluton wll never e 0%. Therefore, there s suggested rnge etween x nd x s µ(x) Ths concept of rnge of µ(x) s used n ths reserch pper. µ(x) x 0 x x x Fgure 1 Modfed s-curve memershp functon JTJUN40D[05].pmd 39

4 40 PANDIAN M VASANT Accordng to Wtd [17], trngulr or trpezodl memershp functon shows lower level nd upper level for µ t ther grdes 0 nd 1, respectvely. On the other hnd, concernng non-lner memershp functon such s logstc functon, lower level nd upper level my e pproxmted t the ponts wth grdes nd 0.999, respectvely. We rescle the x xs s x = 0 nd x = 1 n order to fnd the vlues of B, C, nd α. Novkowsk [25] hs performed such resclng n hs work of socl scences. The vlues of B, C, nd α re otned from equton (1) s: B = (1 + C) (2) B = α 1 + Ce By susttutng equton (2) nto equton (3): (1 + C) = α (4) 1 + Ce Rerrngng equton (4): α = ln (5) C Snce, B nd α depend on C, we requre one more condton to get the vlues for B, C, nd α. x + x Let, when x 0 =, µ(x 0 ) = 0.5. Therefore, 2 B = α 05. (6) 1+ Ce 2 nd hence, 2B 1 α = 2ln (7) C Susttutng equton (5) nd equton (6) nto equton (7), we otn, 2(0. 999)(1 + C ) ln = ln (8) C C Rerrngement of equton (8) yelds, ( C) 2 = C( C) (9) Solvng equton (9): (3) JTJUN40D[05].pmd 40

5 APPLICATION OF MULTI OBJECTIVE FUZZY LINEAR PROGRAMMING ± C = (10) Snce C hs to e postve, equton (10) gves C = nd from equton (2) nd (5), B = 1 nd α = The modfed s-curve memershp functon hs smlr shpe s the logstc functon [17] nd tht of the tngent hyperolc functon employed y Leerlng [13]; ut t s more esly hndled thn the tngent hyperol. In ddton, trpezodl nd trngulr memershp functons re n pproxmton from logstc functon. Therefore, the s- functon s consdered much more pproprte to denote vgue gol level whch decson mker consders for the soluton mplementton. Furthermore, t s possle tht the modfed s-curve memershp functon chnges ts shpe ccordng to the prmeter vlues [17]. Then, decson mker s le to pply hs strtegy to fuzzy supply producton plnnng usng these prmeters. Hence, the modfed s-curve memershp functon s much more convenent thn the lner ones. It should e noted tht trngulr or trpezodl memershp functon shows lower level nd upper level t ther memershp vlues of 0 nd 1 respectvely; on the other hnd, concernng non-lner memershp functon such s modfed s-curve functon, lower level nd upper level my e pproxmted wth memershp vlues of nd 0.999, respectvely. The de of ths pproch s dopted from Wtd [17]. 3.0 FUZZY RESOURCE PARAMETER Frst, we derve the equtons for the fuzzy resource prmeters. These equtons wll e used to generte fuzzy vlues for the respected prmeters. Fuzzy Resource Prmeter. From equton (1), for n ntervl < <, Rerrngng exponentl term, µ = 1 + Ce B α Tkng log e oth sdes, α e 1 B = 1 C µ 1 B α = ln 1 µ C JTJUN40D[05].pmd 41

6 42 Hence, PANDIAN M VASANT 1 B = + ln 1 α C µ (11) Snce s the fuzzy resource vrle n equton (11), t s denoted y. Therefore, µ = µ = + 1 B ln 1 α C µ (12) The memershp functon for µ nd the fuzzy ntervl, Fgure 1. 0 to 1 for re gven n 4.0 NUMERICAL EXAMPLE OF FUZZY MULTI-OBJECTIVE LINEAR PROGRAMMING The lmted Supply Producton Plnnng (SPP) prolem wth contnuous vrles re stted s: A certn compny hs fctory, whch produces 3 products. To produce 1 ton of product A, t needs 2 tons of mterl X, 3 tons of mterl Y, nd 4 tons of mterl Z. To produce 1 ton of product B, t needs 8 tons of mterl X, nd 1 ton of mterl Y. To produce 1 ton of product C, t needs 4 tons of mterl X, 4 tons of mterl Y, nd 2 tons of mterl Z. At current prces, the compny expects to sell product A t rte of 5 mllon/ton, product B t rte of 10 mllon/ton, nd product C t rte 12 mllon/ton. However, durng producton process, producng 1 ton of product A genertes 1 ton of hrmful polluton, producng 1 ton of product B genertes 2 tons of hrmful polluton, nd producng 1 ton of product C produces 1 ton of hrmful polluton. The compny s ojectve s to mxmze totl revenue nd mnmze totl hrmful polluton produced. Ths prolem cn e expressed s the followng mult-ojectve lner progrm: Mx z 0 = cx nd Mn z 1 = dx s.t. Ax, x 0 where: x T = (x 1, x 2, x 3 ), c = (5, 10, 12), d = (1, 2, 1) A = [(2,8,4), (3,1,4), (4,0,2)], T = (100, 50, 50) Solvng ech ojectve seprtely we get: () for Mx cx s.t. Ax, x 0. z 0 = 200, z 1 = JTJUN40D[05].pmd 42

7 APPLICATION OF MULTI OBJECTIVE FUZZY LINEAR PROGRAMMING 43 () for Mn cx s.t. Ax, x 0. z 0 = 0, z 1 = 0 In () we get mxmum revenue of 200 mllon ut ths produces ton of hrmful polluton. In () we get the mnmum 0 ton of hrmful polluton ut ths would men 0 mllon of revenue! As we cn see, these two ojectves re n conflct wth ech other. When we mxmze revenue, we ncrese polluton. When we mnmze polluton, we decrese revenue. To come up wth compromse soluton n respect to vgueness nd degree of stsfcton, the compny defnes the followng rules: Gol 1: Must retn t lest 75% of mxmum revenue (150 mllon), ut would prefer 100% of mxmum revenue (200 mllon). Gol 2: Must not exceed 30 ton, of totl polluton, ut prefer tht no polluton s produced. Gol 3: The rnge for totl revenue nd totl hrmful polluton re to e mnmum. Ths rnge s clled s fuzzy nd. These frst two gols cn e modeled nto fuzzy lner progrmmng wth modfed s-curve memershp functon. The fuzzy lner progrmmng model for the ove SPP prolem s gven s: Suject to: Mx 3 j= 1 c x j j 4 = 1 x j j 1 B + α ln 1 C µ where x 0, j = 1,2,3., 0 < µ < 1, 0 < α <. j (13) C = , B = 1, nd α = In equton (13), fter trde-off etween fuzzy resource prmeter ~ nd non-fuzzy prmeters j, nd c j, the est vlue for the ojectve functon t the fxed level of µ s reched when [15]: µ = µ = µ for = 1,2,3,4.; j = 1,2,3. (14) j Solvng equton (13) usng lner progrmmng technque, we otn the followng result. Seres of tertons re crred out n order to otn the mnmum rnge for the totl revenue nd totl hrmful polluton. The fuzzy nd for totl revenue defned s R R R P P P R z = zmx z mn nd for the totl hrmful polluton z = z z. z mx s mx mn JTJUN40D[05].pmd 43

8 44 PANDIAN M VASANT R mxmum totl revenue, z mn s mnmum totl revenue nd z R s rnge for totl P P revenue. z mx s mxmum hrmful polluton, z mn s mnmum hrmful polluton nd z P s rnge for hrmful polluton. The nput dt for revenue nd hrmful polluton re gven n Tle 1 nd Tle 2 respectvely. Tle 1 Input dt for revenue c j j [0,30] Tle 2 Input dt for polluton c j j [33.33,173.33] We hve to stop t terton 3. Ths s ecuse, the mnmum revenue hs lredy exceeded 30 tons even though the fuzzy nd z P = s smllest, menng gol 2 s volted. Therefore, terton 3 gves the good enough outcome for the mxmum totl revenue nd mnmum totl hrmful polluton. The result n Tle 3 shows tht the totl mxmum revenue s nd the totl mnmum hrmful polluton s t 99.9% degree of stsfcton wth the vgueness α = Tle 3 Fuzzy nd for totl revenue Iterton Mnmum revenue Mxmum revenue Fuzzy nd z R Accordng to Zmmermnn [26] nd Crlsson [15], the relstc possle soluton exst t 50% degree of stsfcton n fuzzy envronment. The followng Tles 5 nd 6 provde the fuzzy solutons for optml revenue nd optml hrmful polluton n respect to vgueness α nd the the degree of stsfcton. The followng result s otned for terton 3. JTJUN40D[05].pmd 44

9 APPLICATION OF MULTI OBJECTIVE FUZZY LINEAR PROGRAMMING 45 Tle 4 Fuzzy nd for totl polluton Iterton Mnmum polluton Mxmum polluton Fuzzy nd z P Tle 5 Optml polluton nd degree of stsfcton (α = 13.81) D.S O.P D.S: Degree of stsfcton (%) O.P: Optml polluton (ton) Tle 6 Optml revenue nd degree of stsfcton (α = 13.81) D.S O.R D.S: Degree of stsfcton (%) O.R: Optml revenue (ton) From Tle 5, we cn see tht the totl hrmful polluton t 50% degree of stsfcton s ton, whch stsfes gol two. The totl revenue t 50% degree of stsfcton s mllon, whch stsfes gol one. Ths result explns tht the optmum nd stsfed soluton hs een otned. It s possle to otn the totl hrmful polluton less thn tons f the vgueness s less thn Ths cn hppen most lkely n less fuzzy stuton. The ove results re comprle to Krsch [27]. 5.0 CONCLUSION In generl, ths reserch work hs cheved ts ojectves n formultng new form of memershp functon nd nvestgtng ts pplctons n lmted supply producton plnnng prolem. Ths memershp functon s modfed form of the generl set of s-curve memershp functons. The flexlty of ths memershp functon n pplyng to rel world prolem hs een proved through n nlyss. Bsed on the nlyss of the results of rel world supply producton plnnng prolem of mxmzng revenue nd mnmzng hrmful polluton, the followng conclusons re drwn: JTJUN40D[05].pmd 45

10 46 PANDIAN M VASANT 1. Fuzzy Lner Progrmmng (FLP) s smple nd sutle tool for mult-ojectve optmzton prolems, compred to other methods. 2. The model cn e extended to ny numer of ojectves y ncorportng only one ddtonl constrnt n the constrnt set for ech ddtonl ojectve functon. 3. The model cn e extended to ny stuton not only supply producton plnnng ut ny feld of engneerng wth lttle modfctons. 4. Anlyss of results ndcted tht totl revenue nd totl hrmful polluton n FLP re ncresed y 9.164% nd decresed y % respectvely s compred to mxmum memershp vlues n the fuzzy decson Lner Progrmmng (LP) model [27]. ACKNOWLEDGMENT The uthor would lke to thnk the referees for ther vlule comments nd suggestons n revewng ths pper. REFERENCES [1] Chnkong, V., nd Y. Y. Hmes Mult Ojectve Decson Mkng: Theory And Methodology. New York: North-Hollnd. [2] Skw, M., K. Kto, nd I. Nshzk An Interctve Fuzzy Stsfyng Method For Mult Ojectve Stochstc Lner Progrmmng Prolems Through An Expectton Model. Europen Journl of Opertonl Reserch. 145(3): [3] Klszewsk, I Dynmc Prmetrc Bounds On Effcent Outcomes In Interctve Multple Crter Decson Mkng Prolems. Europen Journl of Opertonl Reserch. 147(1): [4] Ignzo, J. P Gol Progrmmng And Extenson. Lexngton: Lexngton Books. [5] Lreuche, C., nd M. Grsch The Choquet Integrl For The Aggregton Of Intervl Scles In Mult Crter Decson Mkng. Fuzzy Sets nd Systems. 137(1): [6] Merno, G. G., D. D. Jones, D. L. Clements., nd D. Mller Fuzzy Compromse Progrmmng Wth Precedence Order In The Crter. Appled Mthemtcs nd Computton. 134: [7] Lootsm, F. A Optmzton Wth Multple Ojectves. In Ir M nd Tne K (eds.). Mthemtcl Progrmmng, Recent Developments And Applctons. Tokyo: KTK Scentfc Pulshers. [8] Kuotn, H., nd K. Yoshmur Performnce Evluton Of Acceptnce Prolty Functons For Mult-Ojectve AS. Computers nd Opertons Reserch. 30(3): [9] Zeleny, M Multple Crter Decson Mkng. New York: McGrw-Hll Book Compny. [10] Borges, A. R., nd C. H. Antunes A Fuzzy Multple Ojectve Decson Support Model For Energy- Economy Plnnng. Europen Journl of Opertonl Reserch. 145(2): [11] Zmmermnn, H. J Descrpton And Optmzton Of Fuzzy Systems. Interntonl Journl Generl Systems. 2: [12] Zmmermnn, H. J Fuzzy Progrmmng nd Lner Progrmmng Wth Severl Ojectve Functons. Fuzzy Sets And Systems. 1: [13] Leerlng, H On Fndng Compromse Solutons In Multcrter Prolems Usng The Fuzzy Mn- Opertor. Fuzzy Sets And Systems. 6: [14] Hnnn, E. L Lner Progrmmng Wth Multple Fuzzy Gols. Fuzzy Sets nd Systems. 6: [15] Crlsson, C., nd P. Korhonen A Prmetrc Approch To Fuzzy Lner Progrmmng. Fuzzy Sets nd Systems. 20: [16] Skw, M Interctve Computer Progrms For Fuzzy Lner Progrmmng Wth Multple Ojectves. Interntonl Journl of Mn-Mchne Studes 18(5): JTJUN40D[05].pmd 46

11 APPLICATION OF MULTI OBJECTIVE FUZZY LINEAR PROGRAMMING 47 [17] Wtd, J Fuzzy Portfolo Selecton And Its Applctons To Decson Mkng. Ttr Mountns Mthemtcs Pulcton. 13: [18] Inuguch, M., Ichhsh., nd Y. Kume A Soluton Algorthm For Fuzzy Lner Progrmmng Wth Pecewse Lner Memershp Functons. Fuzzy Sets And Systems. 34: [19] Hu, C. F., nd S. C. Fng Solvng Fuzzy Inequltes Wth Pecewse Lner Memershp Functons. IEEE Trnsctons On Fuzzy Systems. 7: [20] Pndn, V., R. Ngrjn., nd Szl Yco Fuzzy Lner Progrmmng: A Modern Tool for Decson Mkng. Jurnl Teknolog. 37: [21] Pndn, V., R. Ngrjn., nd Szl Yco Decson Mkng Usng Modfed s-curve Memershp Functon n Fuzzy Lner Progrmmng Prolem. Journl of Informton nd Communcton Technology. 1(2): [22] Pndn, V., R. Ngrjn., nd Szl Yco Decson Mkng n n Industrl Producton Plnnng Usng Fuzzy Lner Progrmmng. IMA Journl of Mngement Mthemtcs. Forthcomng [23] Goguen, J. A The Logc Of Inexct Concepts. Syntheses. 19: [24] Zdeh, L. A. Smlrty Reltons nd Fuzzy Orderngs. Informton Scence : [25] Nowkowsk, N Methodologcl Prolems Of Mesurement Of Fuzzy Concepts In The Socl Scences. Behvorl Scence. 2: [26] Zmmermmn, H. J Applcton Of Fuzzy Set Theory To Mthemtcl Progrmmng. Informton Scences. 36: [27 ] Krsch, D. A Fuzzy Sets n Mult Ojectve Optmzton. Fuzzy Mult Ojectve Lner Progrmmng. Htm. JTJUN40D[05].pmd 47

International Journal of Pure and Applied Sciences and Technology

International Journal of Pure and Applied Sciences and Technology Int. J. Pure Appl. Sc. Technol., () (), pp. 44-49 Interntonl Journl of Pure nd Appled Scences nd Technolog ISSN 9-67 Avlle onlne t www.jopst.n Reserch Pper Numercl Soluton for Non-Lner Fredholm Integrl