Fuzzy approach to solve multi-objective capacitated transportation problem

Transcription

1 Internatonal Journal of Bonformatcs Research, ISSN: , Volume, Issue, 00, -0-4 Fuzzy aroach to solve mult-objectve caactated transortaton roblem Lohgaonkar M. H. and Bajaj V. H.* * Deartment of Statstcs, Dr. B. A. M. nversty, Aurangabad, MS, vhbajaj@gmal.com, mhlohgaonkar@gmal.com Abstract: The lnear mult-objectve caactated transortaton roblem n whch the suly and demand constrants are equalty tye, caacty restrcton on each route are secfed and the objectves are non commensurable and conflct n nature. The fuzzy rogramng technque (Lnear, Hyerbolc and Exonental s used to fnd otmal comromse soluton of a mult-objectve caactated transortaton roblem has been resented n ths aer. An examle s llustrate the methodology. Also comarson s taken out, usng same examle. Keyword: Mult-crtera Decson Makng, Caactated Transortaton Problem, Lnear Membersh Functon, Non-lnear Membersh Functon.. Introducton A transortaton roblem wth caacty restrcton s a lnear rogrammng roblem. A basc soluton to a caactated transortaton roblem may contan more than m+n- ostve values due to the caacty constrants whch are addtonal to the m+n- ndeendent equatons. Fuzzy lnear rogrammng occurs when fuzzy set theory s aled to lnear multcrtera decson makng roblem. In fuzzy set theory, an element x has a degree of membersh n a set A, denoted by a membersh functon (. The range of the membersh functon s [0, ]. Degree of the membersh functon for each objectve reresents ts satsfacton level. If the membersh functon of an objectve s one or zero then objectve s fully acheved or not at all acheved, resectvely. If the membersh functon of the objectve les n (0, then the objectve s artally acheved. adeh [] ntroduced the concet of fuzzy set theory. mmermann [4] frst aled the fuzzy set theory concet wth some sutable membersh functon to solve Mult-objectve lnear rogrammng roblems. He showed that solutons obtaned by fuzzy lnear rogrammng effcent. Rnguest and Rnks [] have mentoned the exstng soluton rocedures for Mult-objectve transortaton roblem. Bt [,] have shown the alcaton of fuzzy rogrammng wth lnear membersh functon to the multcrtera decson makng sold transortaton roblem and classcal transortaton roblem. Leberlng [0] has develoed algorthms for obtanng comromse soluton n multcrtera roblems usng the mnoerator. In ths aer, we resent fuzzy rogrammng wth lnear, hyerbolc and exonental membersh functon for solvng mult-objectve caactated transortaton roblem.. Mult-objectve caactated transortaton roblem Consder m orgns ( =,,,m and n destnatons (j =,,,n at each orgn O, let a be the amount of a homogeneous roduct whch we want to transort to n destnatons D j to satsfy the demand for b j unts of the roduct there. A enalty c j s assocated wth transortaton of a unt of the roduct from source to destnaton j for the - th crteron. The enalty could reresent transortaton cost, delvery tme, quantty of goods delvered, under used caacty. A varable j reresents the unknown quantty to be transorted from orgn O to destnaton D j. Let r j be the caacty restrctons on route, j for caactated transortaton roblem. A mult-objectve caactated transortaton roblem can be reresented as: m n Mnmze = c j x j =,,...,P j= Subjectto n x =a, j= j =,,...,m ( m x j =b j j=,,...,n 0 x j r j forall, j (4 Where the subscrt on and suerscrt on c j denote -th enalty crteron; a > 0 for all b j > 0 for all j, r j 0 for all, j m n And a = b as balanced condton. j= j Ths balanced condton s necessary condton for the roblem to have a feasble soluton, however, ths s not suffcent because of the condton (4. For =, roblem become to a sngle objectve caactated transortaton roblem. It may be consdered as a secal case of lnear rogrammng roblem. Coyrght 00, Bonfo Publcatons, Internatonal Journal of Bonformatcs Research, ISSN: , Volume, Issue, 00

2 Fuzzy aroach to solve mult-objectve caactated transortaton roblem. Fuzzy rogrammng technque for the mult-objectve caactated transortaton roblem Ste : Solve the mult-objectve caactated transortaton roblem as a sngle objectve caactated transortaton roblem P tmes, by takng one of the objectves at a tme. Ste : From the results of ste, calculate the values of all the P objectve functons. Then a ay off matrx s formed. The dagonal of the matrx consttutes ndvdual otmum mnmum values for the objectves. ( (... ( (.. (P... ( Ste : From ste, we fnd for each objectve, the lower bound (L and uer bound ( corresondng to the sets of solutons, where, =max(,,..., and L = =,,...,P An ntal fuzzy model of the roblem -(4 can be stated as: - canbestatedas Fnd j =,,..,m j =,,.,n, (6 ~ soastosatsfy <L =,,.,P (-(4 Ste 4: Case ( Defne Hyerbolc membersh functon f L ( ( { - (x}α -{ - (x}α H e -e (x= + f L < < ( ( { - (x}α -{ - (x}α e +e 0 f (7 Case ( Defne Lnear membersh functon for the th objectve functon as follows: f ( L - ( (= f L < < (8 0 f Ste 5: Fnd an equvalent crs model by usng a lnear membersh functon for the ntal fuzzy model Maxmze λ λ - ( subjectto (-(4 Ste 6: Solve the crs model by an arorate mathematcal rogrammng algorthm. Maxmze λ Subjectto m n C j j +λ( j= =,,...,P (0 Subjectto (-(4 n j = j = a =,,..., m m j = b j j =,,... n = j r j f o r a l l, j The foregong lnear rogrammng roblem that can be solved by lnear rogrammng algorthm to fnd an otmal comromse soluton. Case Now, by usng exonental membersh functon for the th objectve functon and s defned as, f L ( E e -e (x=, fl < < -e 0, f P Where, (= =,,...,P S s a non zero arameter, rescrbed by the decson maker Then an equvalent crs model for fuzzy model can be formulated as Maxmze λ -sψ ( x -s e -e λ -s -e =,,---,P subject to (-(4 6. Numercal Examle: Mnmze = Mnmze = ( Mnmze = ( (9 Internatonal Journal of Bonformatcs Research, ISSN: , Volume, Issue, 00

3 Lohgaonkar MH and Bajaj VH =0 j= j ; =45 j= j ; =95 j= j =80 ; =00 ; =80 0,,., j=,,. j ( Caacty restrctons of the routes are gven as: 45, 60, 00 90, 00, 80S 5, 85, 0 (4 Ste and ste. Otmal solutons for mnmzng the frst objectve constrants ( and (4 are as follows x = 0, x = 60, x = 40, x = 5, x = 40, x = 80, x = 5, x = 60 and other decson varable are zero and = 660 Otmal solutons for mnmzng the second objectve constrants ( and (4 are as follows x = 45, x = 5, x = 40, x = 5, x = 0, x = 80, x = 5, x = 60 and other decson varable are zero and = 805 Otmal solutons for mnmzng the thrd objectve constrants ( and (4 are as follows x = 0, x = 60, x = 40, x = 60, x = 5, x = 80, x = 5, x = 60 and other decson varable are zero and = 80 ( Now for we can fnd out (, ( =95 Now for we can fnd out, ( =940 Now for we can fnd out, ( =570 Now for we can fnd out, ( =90 Now for we can fnd out, ( =670 ( Now for we can fnd out (, ( =50 The ay off matrx s ( = 940, = 90, = 50 L = 660, L = 805, L = 80 Fnd { j } x, =,, ; j =,, so as satsfy 660, 805, 80 and constrants, ( % % % Ste4. Wth 6 α =,α = =,α = = α = =, = 800, 50 = , = 455 We get the membersh functons H H H (, (, ( for the objectves, and resectvely, are as follows: Case (: Hyerbolc membersh functon, f (x 660 H 6 ( = tanh[(800- (x ]+, f660 (x , f (x 940, f (x 805 H 6 ( = tanh[( (x ]+, f 805 (x , f (x 90, f (x 80 H 6 ( = tanh[(455 - (x ]+, f 80 (x , f (x 50 Coyrght 00, Bonfo Publcatons, Internatonal Journal of Bonformatcs Research, ISSN: , Volume, Issue, 00

4 Fuzzy aroach to solve mult-objectve caactated transortaton roblem Maxmze x+ α (+ α ( mn+ ( mn+ ( , f ( (, f 80 < ( < 50 (= , f ( mn Now, α (+ α ( mn+ ( mn+ ( mn+ 985 And α (+ α ( mn+ ( mn+ ( mn+ 470 The roblem was solved by usng the lnear nteractve and dscrete otmzaton (LINDO software, the otmal comromse soluton s mn+ = 0.04 x =0,x = 60, x =40, x = , x =.0449, * = x =8.0449,x = x =60 = ; =75.0 and = λ = 0.55 Lnear Membersh Functon, f ( (, f 660 < ( < 940 (= , f ( 940, f ( (, f 805 < ( < 90 (= , f ( 90 Fnd an equvalent crs model Maxmze λ, (+80λ λ 940 and Maxmze λ, (+85λ λ 90 Maxmze λ, λ 50 (+50λ 50 x =0, x = 60, x =40, x = , x =.0449, * = x =8.0449,x = x =60 = ; =75.0 and = λ = 0.57 Exonental Membersh Functon, f ( - E e -e < < -e 0, f 940 (x=, f , f ( - E e -e (x=, f 805 < < 90 -e 0, f 90, f 80 - ( - E e -e (x=, f 80 < < 50 -e 0, f 50 Then an equvalent crs model for fuzzy model can be formulated as Maxmze λ Internatonal Journal of Bonformatcs Research, ISSN: , Volume, Issue, 00

5 Lohgaonkar MH and Bajaj VH -ψ ( x - e -e λ, - =,, -----, P and -e subject to (7-( (= = = (= = = (= = = ( = ( / 80 ( = ( / 85 ( = ( / 50 Then the roblem can be smlfed as Maxmze λ -( - - -( -( e -(-e λ e e -(-0.68λ 0.68 e -(0.6λ ( - - -( -( e -(-e λ e e -(-0.68λ 0.68 e -(0.6λ ( - - -( -( e -(-e λ e e -(-0.68λ 0.68 e -(0.6λ 0.68 The roblem s solved by the (LINGO software * x =0, x =00, x =65, x =5, x =80. = rest all x j are zero's =880 ; =790 and =40 λ= And Ideal soluton s {660, 805, 80} Also set of non-domnated solutons {660, 570, 50}; {95, 805, 50}; {940, 90, 80}. mnmum roblems. Ths algorthm can be aled to the varants of mult-objectve transortaton roblems smlar lnear multobjectve rogrammng roblems. Ths aer s to be seen as a frst ste to ntroduce nonlnear membersh functons to a multobjectve caactated transortaton roblem. The value of membersh functon of an objectve reresents the satsfacton level of the objectve. 8. References [] Bt A. K. (004 OPSEARCH 4, [] Bt A.K., Bswal M.P. and Alam S. S. (99 Fuzzy sets and systems 50, 5-4. [] Charnes A. and Cooer W. W. (954 Management scence, [4] Dantzg G. B. (95 Alcaton of the smlex method to a transortaton roblems, Chater II n Actvty Analyss of Producton and allocaton (T. C. Koomans, Ed., Wley, New York. [5] Daz J. A. (978 Ekonomckomatematcky Obzor 4, [6] Daz J. A. (979 Ekonomckomatematcky Obzor 5, 6-7. [7] Dhngra A.K. and Moskowtz H. (99 Euroean journal of Oeratonal Research 55, [8] Htchcock F. L. (94 Journal Of Mathematcs and Physcs 0, 4-0. [9] Isermann H. (979 Naval Research Logstcs Quarterly 6, -9. [0] Leberlng H. (98 Fuzzy sets and systems 6, [] Rnguest J. L. and Rnks D. B. (987 Euroean Journal Of oeratonal Research, [] Verma Rakesh, Bswal M.P. and Bswas A. (997 Fuzzy sets and systems 9, 7-4. [] adeh, L. A. (965 Informaton and Control 8, 8-5. [4] mmermann H. J. (978 Fuzzy sets and system, Concluson We have obtaned same otmal comromse soluton by our roosed algorthm and fuzzy algorthm wth membersh functons (Bt et al. [] for the mult-objectve caactated transortaton roblem. For a mult-objectve caactated transortaton roblem wth objectve functons, the fuzzy rogrammng wth hyerbolc, lnear and exonental membersh functon gves non-domnated (effcent solutons and an otmal comromse soluton. The fuzzy rogrammng algorthm wth hyerbolc membersh functons s alcable to mult-objectve caactated sold transortaton roblems and the vector 4 Coyrght 00, Bonfo Publcatons, Internatonal Journal of Bonformatcs Research, ISSN: , Volume, Issue, 00