FUZZY GOAL PROGRAMMING VS ORDINARY FUZZY PROGRAMMING APPROACH FOR MULTI OBJECTIVE PROGRAMMING PROBLEM

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1 Internatonal Conference on Ceramcs, Bkaner, Inda Internatonal Journal of Modern Physcs: Conference Seres Vol. 22 (2013) World Scentfc Publshng Company DOI: /S FUZZY GOAL PROGRAMMING VS ORDINARY FUZZY PROGRAMMING APPROACH FOR MULTI OBJECTIVE PROGRAMMING PROBLEM Int. J. Mod. Phys. Conf. Ser : Downloaded from KAILASH LACHHWANI Department of Mathematcs, Government Engneerng College, Bkaner , Rajasthan, Inda kalashclachhwan@yahoo.com Ths paper presents the comparson between two soluton methodologes Fuzzy Goal Programmng (FGP) and ordnary Fuzzy Programmng (FP) for multobjectve programmng problem. Ordnary fuzzy programmng approach s used to develop the soluton algorthm for multobjectve functons whch works for the mnmzaton of the perpendcular dstances between the parallel hyper planes at the optmum ponts of the objectve functons. Sutable membershp functon s defned as the supremum perpendcular dstance and a compromse optmum soluton s obtaned as a result of mnmzaton of supremum perpendcular dstance. Whereas, In the FGP model formulaton, frstly the objectves are transformed nto fuzzy goals (membershp functons) by means of assgnng an aspraton level to each of them and sutable membershp functon s defned for each objectves. Then achevement of the hghest membershp value of each of fuzzy goals s formulated by mnmzng the negatve devatonal varables. Keywords: Multobjectve programmng problem; Dstance functon; Membershp functon; Fuzzy goal programmng. A fuzzy approach for solvng mult-objectve lnear fractonal programmng problems (MOLFPP) was presented by by Sakawa and Kato 1, Dutta et al. 2,3 and Htosh and Takahash 4. Gauss and Roy 5 dscussed the compromse hyper sphere for mult objectve lnear programmng problem. Gupta and Chakraborty 6 suggested fuzzy programmng algorthm for the soluton of multobjectve programmng problem. Then agan Gupta and Chakraborty 7 modfed the concept of fuzzy approach and gave fuzzy mathematcal programmng for mult objectve lnear fractonal programmng problem. Jan and Lachhwan 8 proposed a soluton methodology for multobjectve lnear fractonal programmng problem.e. f ( X ) form n MOLFP based on fuzzy programmng g ( X ) approach. Afterwards Jan and Lachhwan 9 suggested a fuzzy programmng approach for soluton of mult objectve quadratc program. Here we propose a useful method for solvng a set of quadratc mult objectve functons. Numerous methods for mult objectve optmzaton problems have been suggested n the lterature. Each method appears to have some advantages as well as dsadvantages. In the context of each applcaton, some of the methods seem more approprate than others. However, the ssue of choosng a proper method n a gven context s stll a subject of actve research. A 757

2 758 K. Lachhwan Int. J. Mod. Phys. Conf. Ser : Downloaded from number of researchers have worked for fuzzy mathematcal programmng problem usng fuzzy goal programmng approach lke Pal and Motra et al. 10 suggested a goal programmng procedure for fuzzy multobjectve lnear fractonal programmng problem. Chao-Fang et al. 11 proposed a generalzed varyng doman optmzaton method usng fuzzy goal programmng for mult objectve optmzaton problem wth prortes. Pramank and Roy 12 gave a procedure for solvng mult level programmng problems n a large herarchcal decentralzed organzaton through lnear fuzzy goal programmng approach. Ibrahm 13 presented fuzzy goal programmng (FGP) algorthm for solvng decentralsed b-level mult objectve (DBL-MOP) problems wth a sngle decson maker at the upper level and multple decson makers at the lower level. L and Hu 14 proposed a satsfyng optmzaton method based on goal programmng for fuzzy mult objectve optmzaton problem wth the am of achevng the hgher desrable satsfyng degree. The am of ths paper s to present comparson between FGP approach ntroduced by Mohamed 15 and ordnary fuzzy programmng approach n context of multobjectve programmng (MOP) problem. The paper s organzed as follows: In secton 2, we dscuss formulaton of MOPP, membershp functon and related defntons. In next secton, we dscuss comparson between these two methodologes and formulate correspondng mathematcal models. Concludng remarks are gven n the last secton. 1. Problem Formulaton The general format of classcal mult objectve programmng problem can be stated as: { } Max. Z ( X ), Z ( X ),..., Z ( X ) (1) 1 2 n Subject to, X m S = X R AX = b, X 0, b R (2) Here C are row vectors wth n-components. X and b are column vectors wth n and m components respectvely. Let us consder some related defntons as follows: Defnton 1: X 0 S s an effcent soluton to problem (1) (2) f and only f there exsts 0 0 no other X S such that Z Z for all =1,2,,k and Z > Z for at least one. For our purpose, we defne deal soluton (deal pont) of sngle objectve and compromse effcent soluton for mult objectve programmng problems. Defnton 2: For problem (1) - (2), a compromse optmal soluton s an effcent soluton selected by the decson maker (DM) as beng the best soluton where the selecton s based on the DM s explct or mplct crtera. k

3 Fuzzy Goal Programmng vs Ordnary Fuzzy Programmng Approach Comparson between FGP and Ordnary FP 2.1. Ordnary Fuzzy Programmng (FP) Int. J. Mod. Phys. Conf. Ser : Downloaded from Now, n the feld of fuzzy programmng, the fuzzy goals are characterzed by ther assocated membershp functons. The membershp functon for the th fuzzy goal can be defned accordng to Gupta and Chakraborty 6 as: 0 f d X p p d( X) µ ( d( X) ) = f 0< d( X) < p (3) p 1 f d ( X ) 0 Where the dstance functon d wth unt weght as: d ( X ) = Z Z ( X ), = 1,2,..., k Ths dstance depends upon X. At X = X (deal pont n X-space), d = 0 and at (nadr pont n X-space), Z ( X )= Z, we get the maxmum value of d ( X ) as: and = sup { } So, If θ.e. be the mnmum of all ( d ( X ) ) Now the problem reduces to X = X d = Z Z (4) p d (5) µ, then d X pθ + p Z Z (X) pθ + p Maxθ subject to, Z (X) + p. θ p Z = 1,2,..., k and X, θ 0 = 1,2,..., k (6) whch s a non-lnear programmng problem and can be solved usng non-lnear technques or software package lke LINDO Goal Programmng Formulaton Regardng the presently avalable procedures, a FGP approach seems to be most approprate for the problem and consdered n ths paper. In the fuzzy goal programmng approach, the hghest degree of membershp s 1. So, as n Mohamed 15 for the defned

4 760 K. Lachhwan membershp functon n (3), the flexble membershp goals wth aspraton level 1 can be expressed as: + Z + Z X + pd pd = 0, = 1,2,..., k (7).e. Int. J. Mod. Phys. Conf. Ser : Downloaded from where D ( 0) and + D ( 0) wth D + D = 0 represent the under and over devatonal varables respectvely from the aspred levels. In conventonal GP, the under and/or over devatonal varables are ncluded n the achevement functon for mnmzng them and that depends upon the type of the objectve functons to be optmzed. In ths approach, only the under devatonal varables D s requred to be mnmzed to acheve the aspred levels of the fuzzy goals. The th membershp goal wth aspred level 1 can be presented as: + Z + Z X + pd pd = 0, = 1,2,..., k (8) However, for model smplfcaton the expresson (8) can be consdered as a general form of goal expresson of the above stated membershp goals. Now, f the most wdely used and smplest verson of GP (.e. mnsum GP) s ntroduced to formulate the model of the problem under consderaton, then GP model formulaton becomes: Model I Fnd X so as to Mnmze λ = k D = 1 + Subject to, Z ( X ) - Z + pd pd = 0 n X m S = X R AX = b, X 0, b R and D, D + 0, = 1,2,..., k (9) where λ represents the fuzzy achevement functon consstng of the sum of under devatonal varables. 3. Concludng Remarks An effort has been made to compare fuzzy goal programmng approach and ordnary fuzzy programmng n context of multobjectve programmng problem. The FGP technque s effcent and requres less computatonal work than ordnary fuzzy programmng approach. References 1. M. Sakawa and K. Kato, Fuzzy sets and systems 97, (1988). 2. D. Dutta, R.N. Twar and J.R. Rao, Fuzzy sets and systems 122, (2001). 3. D. Dutta, R.N. Twar and J.R. Rao, Fuzzy sets and systems 54, (1993). 4. M.S. Htosh and Y.J. Takahash, Informaton Scences 63, (1992). 5. S.I. Gass and P.G. Roay, European Journal of Operatonal Research 144, (2003).

5 Fuzzy Goal Programmng vs Ordnary Fuzzy Programmng Approach 761 Int. J. Mod. Phys. Conf. Ser : Downloaded from 6. S. Gupta and M. Chakraborty, Internatonal journal of management and system 13(2), (1997). 7. M. Chakraborty and S. Gupta, Fuzzy sets and systems 125, (2005). 8. S. Jan and K. Lachhwan, Proceedngs of Natonal Academy of Scences, Inda-Physcal scences (Secton- A) 79 (III), (2009). 9. S. Jan and K. Lachhwan, AMSE Perodc: Advances n modelng & analyss seres A 46(2), 1-10 (2009). 10. B.B. Pal, B.N. Motra and U. Maulk, Fuzzy Sets and Systems 139, (2003). 11. H. Chao-Fang, T. Chang-Jun and L. Shao-Yuan, European Journal of Operatonal Research. 176, (2007). 12. S. Pramank and T.K. Roy, European Journal of Operatonal Research 176, (2007). 13. A.B. Ibrahm, Fuzzy Sets and Systems 160, (2009). 14. S. L and C. Hu, European Journal of Operatonal Research. 193, (2009). 15. R.H. Mohamed, Fuzzy Sets and System 89, (1997).