Sum of Linear and Fractional Multiobjective Programming Problem under Fuzzy Rules Constraints

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1 Australan Journal of Basc and Appled Scences, 2(4): , 2008 ISSN Sum of Lnear and Fractonal Multobjectve Programmng Problem under Fuzzy Rules Constrants 1 2 Sanjay Jan and Kalash Lachhwan 1 Department of Mathematcs, Government College, Ajmer, Inda. 2 Department of Mathematcs, Engneerng College, Bkaner, Inda. Abstract: Ths paper deals wth the soluton procedure of the sum of lnear and fractonal multobjectve programmng problem n whch the fractonal relatonshp between the decson varables and the objectve functons s not completely known. Our knowledge base conssts of a block of fuzzy If-then rules, where antecedent part of the rules contans some lngustc value of the decson varables and the consequence part s lnear combnaton of the crsp value of the decson varable. We suggest the use of Takag and Sugeno fuzzy reasonng method to determne the crsp functonal relatonshp between the objectve functon and the decson varables under the assumpton that the denomnator of the fractonal part of the objectve functons s non-zero on the constrants set and fnally solvng the resultng programmng problem to fnd a far optmal soluton of the orgnal problem. Key words: Multobjectve programmng, Lnear plus fractonal programmng, If-then rules, Fuzzy reasonng method. INTRODUCTION Fuzzy programmng problem can be stated and solved n many dfferent ways as gven by Zmmermann (1975) and (1992). Earler researchers, consders programmng problem of the form * where f or / and X are defned by fuzzy terms. After that searchng for a crsp x whch maxmzes/ mnmzes f on X. Smlarly fuzzy lnear programmng problems (FLP) can be stated as gven by Herrera (1992), Kovacs (1991) and Rommelfanger (1996) as (1) where the fuzzy terms are denoted by tlde. Carlsson and Fuller (2001) have consdered constraned fuzzy optmzaton problem wth sngle objectve of the form (2) wth R (x): f x 1 A 1 and,...,and x n s A n then f(x) s c where A j an c are fuzzy numbers and they have suggested the use of Tsukemoto s fuzzy reasonng method gven by Tuskamoto (1979) to determne the crsp value of f. Correspondng Author:Dr. Sanjay Jan, Department of Mathematcs, Government College, Ajmer E-mal: drjansanjay@gmal.com 1204

2 Later DadashZadeh and Nmse (2005) extended the prncple for multple objectve optmzaton problem under fuzzy If-then rules (1998). and consdered the multple objectve optmzaton problem of the form (3) Such that about the value of determne the crsp value of the k-th objectve functon by the fuzzy reasonng method gven by Takag and Sugeno (1985) and obtan an optmal soluton of (3) by solvng the resultng multobjectve nonlnear mathematcal programmng problem wthout constrants and c The present paper s organzed as follows: In secton 2, we descrbe the problem and notatons. In secton 3, we propose the soluton procedure for multobjectve lnear plus fractonal programmng problem under fuzzy If-then rules and related defntons and propertes. An example s gven n last to support our proposed model. The Problem: We consder the multobjectve lnear plus fractonal programmng problem of the form where are lngustc varables and Such that about the value of and Here we restrct our self for 1205

3 only the values for whch denomnator of the fractonal part of the objectve functons s non-zero. Fnally we determne the crsp value of the k-th objectve functon by the fuzzy reasonng method gven by Takag and Sugeno (1985) and ths reduces the problem nto multobjectve fractonal programmng problem wthout constrants n the form Ths problem can be solved usng fuzzy programmng method gven by Gupta and Chakraborty (1997) and a compromse optmal soluton of problem Mark eq. no. can be obtaned. Multobjectve Lnear plus Fractonal Programmng Problem under Fuzzy If-then Rules: As per Zadeh (1975), lngustc varable can be regarded ether as a varable whose value s a fuzzy number or as a varable whose value s defned n lngustc terms. Defnton: A t-norm T s a functon T:[0,1] [0,1] [0,1] havng the followng four propertes. For any t-norm T wth use of property (), we have For obtanng a far optmal soluton to the fuzzy optmzaton problem Subject to, (5) wth fuzzy If-then rules of form (4), we determne the crsp value of the k-th objectve functon f k at from the fuzzy rule base R usng the fuzzy reasonng method gven by Takag and Sugeno (1985) as Where the frng levels of the rules are computed by And the ndvdual rule outputs denoted by are dvded from the relatonshp 1206

4 Here s an nput vector and to determne the frng level of the rules, we suggest the use of the product t-norm. In ths way our constraned multobjectve optmzaton problem (5) turns nto the followng crsp unconstraned multobjectve fractonal programmng problem. (6) Example: Consder the sum of lnear and fractonal multobjectve optmzaton problem Suppose that s an nput vector. where T s a product t-norm. 1207

5 Usng non-lnear fuzzy programmng technques as gven by Gupta and Chakraborty (1997) the compromse optmal soluton of ths problem be. REFERENCES Carlsson, C. and R. Fuller, Optmzaton under fuzzy f then rules, Fuzzy sets and systems, 119: Carlsson, C. and R. Fuller, Multobjectve Optmzaton wth Lngustc Varables (n Proceedng of the Sxth European Congress on Intellgent Technques and Soft Computng (EUFIT, 98), Aachen, September 7-10, 1998), Verlag Manz, Aachen, II: Dadashzadeh, R. and S.B. Nmse, Multobjectve Optmzaton under Fuzzy rule constrants, Rajasthan Academy of Physcal Scences, 4(3): Fuller, R. and H.J. Zmmerman, Fuzzy reasonng for solvng fuzzy mathematcal programmng problems, Fuzzy sets and systems, 60: Herrera, F., M. Kovacs and J.L Verdegay, Fuzzy Lnear Programmng Problems wth Homogeneous Lnear Fuzzy Functons (n Proc. Of IPMU 1992), Unverstat de les lles Balears, Kovacs, M. 1991, Lnear Programmng wth cetered fuzzy numbers, Annales, Unv. Sc. Budapest (Secto Computatorca), 12: Gupta, S. and M. Chakraborty, Mult objectve lnear programmng: A fuzzy programmng approach, Internatonal journal of management and system, 13(2): Rommelfanger, H., Fuzzy Lnear Programmng and Applcatons, European Journal of Operatonal Research, 92: Takag, T. and M. Sujgeno, Fuzzy Identfcaton of Systems and ts applcatons to modelng nd control, IEEE Trans. Systems, MAN Cybernet, Tuskamoto, Y., An approach to Fuzzy reasonng method (n M.M. Gupta, R.K. Ragode and R.R. Yager (eds.)- Advances n Fuzzy Set Theory and Applcatons, North-Holland, New York). Zadeh, L.A., The concept of Lngustc Varable and ts applcatons to approxmate reasonng, Pts. I, II, III, Informaton Scences, 8: ; 8: ; 9: Zmmerman, H.J., Descrpton and optmzaton of fuzzy systems nternet, J.General Systems 2: Zmmerman, H.J., Fuzzy mathematcal programmng (n Struct C.Shapr (ed.) - Encyclopeda of Artfcal Intellgence, John Wlley and Sons, New York),

GSLM Operations Research II Fall 13/14

GSLM Operations Research II Fall 13/14 GSLM 58 Operatons Research II Fall /4 6. Separable Programmng Consder a general NLP mn f(x) s.t. g j (x) b j j =. m. Defnton 6.. The NLP s a separable program f ts objectve functon and all constrants are