FUZZY TRANSPORTATION PROBLEM WITH ADDITIONAL RESTRICTIONS

Transcription

1 VOL. 5, NO. 2, FEBRUARY 200 ISSN ARPN Joural of Egieerig ad Applied Scieces Asia Research Publishig Network (ARPN). All rights reserved. FUZZY TRANSPORTATION PROBLEM WITH ADDITIONAL RESTRICTIONS Debashis Dutta ad A. Satyaarayaa Murthy Departet of Matheatics, Natioal Istitute of Techology Waragal, A.P, Idia ABSTRACT This paper deals with the trasportatio proble with additioal ipurity restrictios where costs are ot deteriistic ubers but iprecise oes. Here, the eleets of the cost atrix are suboral fuzzy itervals with strictly icreasig liear ebership fuctios. By the Max-Mi criterio suggested by Bella ad Zadeh [7], the fuzzy trasportatio proble ca be treated as a ixed iteger oliear prograig proble. We show that this proble ca be siplified ito a liear fractioal prograig proble. This fractioal prograig proble is solved by the ethod give by Kati Swarup [2]. Keywords: fuzzy ubers, oliear prograig, fractioal prograig, ipurity costraits.. INTRODUCTION The trasportatio proble is oe of the earliest applicatios of liear prograig probles. The basic trasportatio proble was origially developed by Hitchcock [5]. The objective of the trasportatio proble is to deterie the optial aouts of a coodity to be trasported fro various supply poits to various dead poits so that the total trasportatio cost is a iiu. The uit costs i.e. the cost of trasportig oe uit fro a particular supply poit to a particular dead poit, the aouts available at the supply poits ad the aouts required at the dead poits are the paraeters of the trasportatio proble. I practice, the paraeters of the trasportatio proble are ot always exactly kow ad stable. This iprecisio ay follow fro the lack of exact iforatio or ay be a cosequece of a certai flexibility the give eterprise has i plaig its capacities. A frequetly used eas to express the iprecisio are the fuzzy ubers. A algorith to obtai a iteger optial solutio has bee preseted i [8]. This algorith requires solvig a paraetric trasportatio proble with a paraeter i the dead ad supply values. A procedure for solvig a fuzzy solid trasportatio proble was preseted by Jieez ad Verdegay [4]. Fuzzy prograig ad additive fuzzy prograig techiques for ulti-objective trasportatio probles were discussed i [, 2]. A geoetric prograig approach for a ultiobjective trasportatio proble was cosidered by Isla ad Roy [0]. A ethod for solvig the fuzzy assiget proble was give by Li ad We [3]. The ulti-objective tie trasportatio proble with additioal ipurity restrictio was studied by Sigh ad Saxea [6]. I the preset paper, the trasportatio proble with fuzzy costs with strictly icreasig liear ebership fuctios ad with additioal ipurity costraits is trasfored ito a liear fractioal prograig proble. This proble is solved by the Kati Swarup s ethod ad coputatioal results show that the proposed ethod gives a optial solutio to the proble. The optial shipets fro the origis to various destiatios are itegers, provided the supply ad dead values are itegers. 2. MATHEMATICAL FORMULATION OF THE PROBLEM The atheatical forulatio of the fuzzy trasportatio proble with additioal restrictios is Mi c ij x ij x ij = a i, (,2,,) x ij = b j, (,2,,) (2.) f i x ij p j, (,2,,) Here a i is the aout of coodity available at the i th supply poit ad b j is the requireet of the coodity at the j th dead poit. Oe uit of the coodity at the i th supply poit cotais f i uits of ipurity. Dead poit j caot receive ore tha p j uits of ipurities ad x ij is the aout of coodity trasported fro the i th supply poit to the j th dead poit. The fuzzy costs c ij = (α ij,β ij ) (i =,2,,, j =,2,,) are suboral fuzzy ubers havig strictly icreasig liear ebership fuctios. The ebership fuctio of c ij is q ij if c ij = β ij, x ij > 0 36

2 VOL. 5, NO. 2, FEBRUARY 200 ISSN ARPN Joural of Egieerig ad Applied Scieces Asia Research Publishig Network (ARPN). All rights reserved. µ ij (c ij ) = q ij (c ij -α ij ) / (β ij -α ij ) if α ij c ij β ij, x ij > 0(2.2) 0 otherwise The coditio x ij > 0 is added to (2.2) because there is o real expese if x ij = 0 i ay feasible solutio X of (2.). We use the otatio < α ij, β ij > to deote c ij. Matrix c ij is show as follows [c ij] = [< α ij, β ij >] Matrix [q ij ] is defied by [q ij ] = [q ij ] Let c T deote the total cost ad the ubers a ad b are defied as the lower ad upper bouds of the total cost, respectively. We defie the ebership fuctio of c T as the liear ootoically decreasig fuctio i (2.3) ad use the otatio < a, b > to deote fuzzy iterval c T. Nubers a ad b are costats ad subjectively chose by the aager. We ay take a as the iiu cost of the trasportatio proble with α ij s as costs ad b as the axiu cost of the trasportatio proble with β ij s as costs, the dead ad supply values i both cases beig sae as those of proble (2.). The ebership fuctio of the total cost is, if c T a µ T (c T ) = µ T ( cijx ij ) = (b- cijx ij )/ (b-a), if a c T b (2.3) 0, c T b 3. SOLUTION OF THE PROBLEM Followig the Bella-Zadeh s criterio [7], we axiize the iiu of the ebership fuctios correspodig to that solutio i.e. Max-Mi(µ ij (,2,,,,2,,),µ T (c T )) (3.) Where x ij is a eleet of a feasible solutio X of (2.). The we ca represet the proble as follows Max-Mi (µ ij,µ T (c T )) x ij > 0 x ij = a i, (,2,,) x ij = b j, (,2,,) (3.2) f i x ij p j, (,2,,) x ij 0 for,2,,,,2,, By ebership fuctios of (2.2) ad (2.3) we ca further represet (3.2) as the followig equivalet odel. Max λ λx ij q ij (c ij λ - α ij )x ij / (β ij - α ij ) for,2,,,,2,, λ (b- cijx ij )/(b-a), x ij = a i, (,2,,) x ij = b j, (,2,,) f i x ij p j, (,2,,) c ij λ x ij β ij x ij for,2,,,,2,, x ij 0 for,2,,,,2,, (3.3) Where c ij λ deotes the λ-cut of c ij. I (3.3), sice x ij, c ij λ, ad λ are all decisio variables, it ca be treated as a ixed iteger oliear prograig odel. We defie the set E as the set of all pairs (i,j) where x ij is a eleet of the feasible solutio X of (2.) ad cofie our discussio based o E. The, we ca siplify (3.3) as follows Max λ λ λ q ij (c ij - α ij ) / (β ij - α ij ) for (i,j) E λ (b- c λ ij x ij )/ (b-a), (3.4) c ij λ β ij for (i,j) E We let d ij = β ij -c ij λ 0. The (3.4) ca be expressed as follows Max λ (3.5.0) 37

3 VOL. 5, NO. 2, FEBRUARY 200 ISSN ARPN Joural of Egieerig ad Applied Scieces Asia Research Publishig Network (ARPN). All rights reserved. λ q ij (β ij - α ij -d ij ) / (β ij - α ij ) for (i,j) E (3.5.) λ (b- (βij d ij ) x ij )/ (b-a), (3.5.2) d ij, λ 0 for (i,j) E (3.5.3) Theore Let λ x be the optial value of (3.5.0)-(3.5.3). Suppose b < ( (βij - d ij )x ij - a i{q ij /(i,j) E})/( i{q ij /(i,j) E}) The λ x = q ij (β ij - α ij -d ij ) / (β ij - α ij ) for (i,j) E = (b- (βij d ij ) x ij )/ (b-a) Proof The proble (3.5.0) - (3.5.3) ca be writte ito a liear prograig odel as Max λ (3.6.0) d ij + λ(β ij - α ij )/ q ij (β ij - α ij ) for (i,j) E (3.6.) - dijx ij + (b-a)λ b- λ, d ij 0 for ( i,j) E βijx ij (3.6.2) We obtai the dual proble of the above proble as Mi (βij - α ij ) w i + (b- β ij x ij ) w + (3.7.0) w i - w + x ij 0 for( i,j) E (3.7.) ((β ij - α ij ) /q ij )w i + (b-a)w + (3.7.2) w i 0 for,2,,+ Let s, s 2, s + be the slack variables of (3.6.) ad (3.6.2) respectively. Siilarly, let u, u 2,,u + be the surplus variables of (3.7.) ad (3.7.2) respectively. sice b < ( (βij - d ij )x ij - a i{q ij / (i,j) E})/(- i{q ij /(i,j) E}) We have i {q ij / (i,j) E} > (b- (βij - d ij )x ij )/(b-a) By (3.5.2) we have λ < i {q ij / (i,j) E} ad d ij > 0 for (i,j) E. Based o the copleetary slackess theore, we obtai u = u 2 =. =u + = 0 Hece w i w + x ij = 0 for, 2 + Now, if w + = 0 the w =w 2 = =w +- = 0. This is a cotradictio to (3.7.2) Hece w + > 0 Assuig that the solutio is o-degeerate w i = w + x ij > 0 i.e w >0, w 2 >0,,w + >0 Agai by copleetary slackess theore s =s 2 =. = s + =0 Therefore λ x = q ij (β ij - α ij -d ij ) / (β ij - α ij ) for (i,j) E = (b- (β ij d ij ) x ij )/ (b-a) I ost of the real world probles, the upper boud coditio of the total cost c T i.e b < ( (β ij - d ij )x ij - a i{q ij / (i,j) E})/(-i{q ij /(i,j) E}) ca be just satisfied. Therefore, we cocetrate our discussio i this situatio. Theore 2 Let λ x be the optial value of (3.5.0) - (3.5.3) ad b < ( (β ij - d ij )x ij - a i{q ij /(i,j) E})/(- i{q ij / (i,j) E}). Also let γ ij = (β ij - α ij )/q ij for,2,,,,2,,. The λ x = (b- α ij x ij ) / (b-a + γ ijx ij ) Proof: By theore, assuig the solutio to be Nodegeerate, we have λ x = ((β ij - α ij -d ij )x ij ) / (γ ij x ij ) for (i,j) E = (b- (β ij d ij ) x ij )/ (b-a) Hece, by copoedo ad dividedo, we get λ x = (b- (β ij - d ij )x ij + (βij - α ij -d ij )x ij ) / (b-a + = (b- γ ij x ij ) α ij x ij ) / (b-a + γijx ij ) (3.8) 4. THE FRACTIONAL PROGRAMMING MODEL By Theore 2 ad (3.8), (3.3) ca be restated as 38

4 VOL. 5, NO. 2, FEBRUARY 200 ISSN ARPN Joural of Egieerig ad Applied Scieces Asia Research Publishig Network (ARPN). All rights reserved. Max (b- α x ij = a i, (,2,,) ijx ij ) / (b-a + γ ijx ij ) x ij = b j, (,2,,) (4.) f i x ij p j, (,2,,) x ij 0 for,2,,,,2,, This is a liear fractioal prograig proble ad its optial solutio ay be obtaied by Kati Swarup s algotith [2]. Now d ij for (i,j) E ca be obtaied fro λ x = (β ij - α ij -d ij )/γ ij for (i,j) E The the fuzzy costs correspodig to the axial value of λ are give by c ij λ = β ij - d ij 5. NUMERICAL EXAMPLE Cosider the proble Miiize 3 3 c ijx ij x + x 2 + x 3 = 4 x 2 + x 22 + x 23 = 5 x 3 + x 32 + x 33 = 6 x + x 2 + x 3 = 5 x 2 + x 22 + x 32 = 5 (5.) x 3 + x 23 + x 33 = 5 2x + x x 3 4 2x 2 + x x 32 2x 3 + x x 33 9 x ij 0 for i,j =,2,3 < 4, 3 > <3, 2 > < 2, 6 > Where c ij = < 4, 3 > < 6, 4 > < 7, 5 > < 7, 0> < 4, 8 > < 6, 2 > [q ij ] = The we have [α ij ] = [β ij ] = [γ ij ] = a is take as the iiu cost of the trasportatio proble with costs as α ij s (a = 54) b is take as the axiu cost of the trasportatio proble with costs as β ij s (b = 92) Hece, by (4.), proble (5.) ca be forulated as Max (92-4x - 3x 2-2x 3-4x 2-6x 22-7x 23-7x 3-4x 32-6x 33) / (38 + 0x + 5x 2 + 5x 3 + 0x 2 + 0x 22 0x x 3 + 5x x 33) x + x 2 + x 3 = 4 x 2 + x 22 + x 23 = 5 x 3 + x 32 + x 33 = 6 x + x 2 + x 3 = 5 x 2 + x 22 + x 32 =5 (5.2) x 3 + x 23 + x 33 =5 2x + x x 3 4 2x 2 + x x 32 2x 3 + x x 33 9 x ij 0 for i,j =,2,3 The optial solutio of proble (5.2) is obtaied by usig Kati Swarup s ethod as x 3 = 4, x 2 = 4, x 23 =, x 3 =, x 32 = 5 with ax λ x = For (i,j) E, we have λ x = (β ij - α ij - d ij )/γ ij so that d ij = β ij - α ij - λ x γ ij We have d 3 = 4 - (0.563) (5) =.85 d 2 = 9 - (0.563) (0) = 3.37 d 23 = 8 - (0.563) (0) = 2.37 d 3 = 3 - (0.563) (5) = 0.85 d 32 = 4 - (0.563) (5) =.85 The fuzzy costs correspodig to λ = are c ij λ = β ij - d ij for (i,j) E We have c = = 4.85 c = = 9.63 c = = 2.63 c = = 9.85 c = = 6.85 Total trasportatio cost = cij x ij = CONCLUSIONS I this paper, a fuzzy trasportatio proble with additioal ipurity restrictios is forulated ito a ixed Noliear prograig proble by Bella Zadeh approach. This proble is the fraed ito a liear 39

5 VOL. 5, NO. 2, FEBRUARY 200 ISSN ARPN Joural of Egieerig ad Applied Scieces Asia Research Publishig Network (ARPN). All rights reserved. fractioal prograig proble (LFPP). This LFPP is solved usig the Kati Swarup s Method. The optial degree of satisfactio was obtaied as The costs at the optial degree of satisfactio were obtaied usig kow expressios. I cotrast to the classical trasportatio proble, the existece of a feasible solutio of the fuzzy trasportatio proble cosidered i this paper is ot guarateed. The o existece of a feasible solutio is due to the ipurity restrictios. I such cases, feasibility ca be attaied by icreasig the ipurity liits of the dead poits. ACKNOWLEDGEMENTS The secod author expresses his sicere thaks to the Coucil of Scietific ad Idustrial Research (CSIR), Idia for providig fiacial support for this research i the for of a seior research fellowship (Grat No.09/922(000)/2006-EMR- [9] S. Chaas, W. Kolosziejczyj A. Machaj, A fuzzy approach to the trasportatio proble. Fuzzy Sets ad Systes. 3: [0] Shahidul Isla, Tapa Kuar Roy A ew fuzzy ulti-objective prograig: Etropy based geoetric prograig ad its applicatio of trasportatio probles. Europea Joural of Operatioal Research. 73: [] Waeil F. Abd El-Wahed A Multi-objective trasportatio proble uder fuzziess. Fuzzy Sets ad Systes. 7: [2] Kati Swarup Liear fractioal fuctioal prograig. Operatios Research.2: REFERENCES [] A.K. Bit, M.P. Biswal, S.S. Ala Fuzzy prograig approach to ulti criteria decisio akig trasportatio proble. Fuzzy Sets ad Systes. 50: [2] A.K. Bit, M.P. Biswal, S.S. Ala A additive fuzzy prograig odel for ulti objective trasportatio proble. Fuzzy Sets ad Systes. 57: [3] Chi-Je Li, Ue-Pyg We A labelig algorith for the fuzzy assiget proble. Fuzzy Sets ad Systes. 42: [4] F.Jieez, J.L. Verdegay Ucertai solid trasportatio probles. Fuzzy Sets ad Systes. 00: [5] F.L. Hitchcock. 94. The distributio of a product fro several sources to uerous desiatios. J. Math. Phys. 20: [6] Preetvati Sigh, P.K. Saxea The ultiobjective tie trasportatio proble with additioal restrictios. Europea Joural of Operatioal Research. 46: [7] R.R Bella ad L.A. Zadeh Decisio akig i a fuzzy eviroet. Maageet Sci. B [8] S. Chaas, D.Kuchta Fuzzy iteger trasportatio proble. Fuzzy Sets ad Systes. 98: