SOLVING FUZZY SOLID TRANSPORTATION PROBLEM BASED ON EXTENSION PRINCIPLE WITH INTERVAL BUDGET CONSTRAINT

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1 Iteratoa Joura of Avace Research I Scece A Egeerg IJARSE Vo. No.4 Issue 04 Apr 05 ISSN-9-854(E) SOVING FY SOID TRANSPORTATION PROBEM BASED ON EXTENSION PRINCIPE WITH INTERVA BDGET CONSTRAINT Debashs Dutta G. Nsh uar Departet of Maeatcs Natoa Isttute of Techoogy Waraga T.S (Ia) ABSTRACT The so trasportato probe cosers e suppy e ea a e coveyace satsfyg e trasportato requreet a cost-effectve aer. Ths paper eveops a eo at s abe to erve e fuzzy obectve vaue of e fuzzy so trasportato probe whe e cost coeffcets e suppy a ea quattes coveyace capactes are fuzzy ubers a atoa costrats o e tota buget at each estato whch s terva type. We ae use of Hu a Wag s Approach base o terva rag. Base o e eteso prcpe e fuzzy so trasportato probe s trasfore to a par of aeatca progras at s epoye to cacuate e ower a upper bous of e fuzzy tota trasportato cost at possbty eve. Fro fferet vaues of e ebershp fucto of e obectve vaue s costructe. Sce e obectve vaue s fuzzy e vaues of e ecso varabes erve s paper are fuzzy as we. A eape s ustrate for s oe. Keywors: Eteso Prcpes Fuzzy Nubers So Trasportato Probe. I INTRODCTION The tratoa trasportato probe (TP) s a we-ow optzato probe operatoa research whch two s of costrats are tae to coserato.e. source costrat a estato costrat. But e rea syste we aways ea w oer costrats beses of source costrat a estato costrat such as prouct type costrat or trasportato oe costrat. I such case e tratoa TP turs to e so trasportato probe (STP). As a geerazato of tratoa TP e STP was trouce by Haey []. I s paper we vestgate a souto of e fuzzy so trasportato probe w terva vaue buget at each estato. A assesset of fferet resuts of e oe s aso presete. Bea a aeh [] trouce e oto of fuzzess. Sce e trasportato probe s essetay a ear progra oe ufory appy e estg fuzzy ear prograg techques (Bucy [] Chaas et a. [] A Ha Basrzaeh [4]) to e fuzzy trasportato probe. fortuatey ost of e estg techques [ a 4] oy prove crsp soutos. The eo of Jue [5] a Parra et a. [6] s abe to f e possbty strbuto of e obectve vaue prove a e equaty costrats are of type or type. However ue to e structure of e trasportato probe soe cases er eo requres e refeet 6 P a g e

2 Iteratoa Joura of Avace Research I Scece A Egeerg IJARSE Vo. No.4 Issue 04 Apr 05 ISSN-9-854(E) of e probe paraeters to be abe to erve e bous of e obectve vaue. There are aso stues scussg e fuzzy trasportato probe. Obvousy whe e cost coeffcets suppy a ea quattes are fuzzy ubers e tota trasportato cost w be fuzzy as we. I s paper we eveop a souto proceure at s abe to cacuate e fuzzy obectve vaue of e fuzzy so trasportato probe where a e paraeters are fuzzy ubers. The ea s to appy aeh s eteso prcpe [7]. A par of twoeve aeatca progras s foruate [8 9] to cacuate e ower a upper bous of e α-eve cut of e obectve vaue. I secto we trouce e crsp coverso of e costrats of e respectve oe usg a fferet orer reato of e tervas such as Hu a Wag s Approach [0]. The ebershp fucto of e fuzzy obectve vaue s erve uercay by eueratg fferet vaues of α. It has bee observe at a very ess research wor s oe o e fuzzy trasportato probe to ze e trasportato cost usg pubcy avaabe ata whch shou be ore auetc a reabe as copare to crsp ata. I e foowg sectos we frst cocsey escrbe e fuzzy so trasportato probe. The a par of aeatca progras s foruate to cacuate e fuzzy tota trasportato cost bous at a specfc α eve. A eape s ustrate to epa e propose eo. Fay soe cocusos are raw. II FY SOID TRANSPORTATION PROBEM WITH INTERVA BDGET CONSTRAINT Coser sources a estatos a so trasportato probe. At each source et be e aout of a hoogeeous prouct we wat to trasport to estatos to satsfy e ea for uts of e prouct. Here cae coveyace eotes e uts of s prouct at ca be carres by fferet oes of trasportato terva buget at e estato such as e a trasportato by car or tra a ocea shppg. A peaty vaue of e ut shppg cost represets by of a prouct fro org to estato by eas of e coveyace. We ee to etere a feasbe way of shppg e avaabe aouts to satsfy e ea such at e tota trasportato cost s ze. et eote e uber of uts to be trasporte fro Source to Destato rough Coveyace capactes. The aeatca for of e so trasportato probe w terva vaue buget costrat trasportato costs avaabtes a coveyace capactes s gve beow: c s. t. s... 6 P a g e

3 Iteratoa Joura of Avace Research I Scece A Egeerg IJARSE Vo. No.4 Issue 04 Apr 05 ISSN-9-854(E)... e... () c B Itutvey f ay of e paraeters or s fuzzy e tota trasportato cost becoes fuzzy as we a buget costrat s tae w terva vaue B [ b b ]. The e Moe () turs to e fuzzy so R trasportato probe w terva vaue buget costrat. Suppose e ut shppg cost suppy ea coveyace capacty a buget tervas are approatey ow. They t ca be represete by e cove fuzzy ubers a respectvey w ebershp fuctos a : C { ( c ( c )) c S )} C S { s ( s )) s S )} S D { ( ( )) S )} D () where ( ) E { ( e ( e )) e S )} E S C S ) ( ) S D a S ) are e supports of C S D a E whch eote e uverse sets of e ut shppg cost e quatty suppe by e org e quatty requre by e estato a e capacty carre by e coveyace respectvey. The fuzzy obectve fucto C whch s to be ze togeer w e foowg costrats costtutes e fuzzy so trasportato probe: sg Hu a Wag s approach [7] o buget costrat we have e foowg crsp coverso. ( b b R ) C... () Wout oss of geeraty a e suppy a ea quattes a coveyace capactes are assue to be cove fuzzy ubers as e crsp vaues ca be represete by egeerate ebershp fuctos whch have 64 P a g e

4 Iteratoa Joura of Avace Research I Scece A Egeerg IJARSE Vo. No.4 Issue 04 Apr 05 ISSN-9-854(E) oy oe vaue er oas. I e et secto we sha eveop e souto proceure for fuzzy so trasportato probe w fuzzy suppy requreet a coveyace capacty. III THE SOTION PROCEDRE We are tereste ervg e ebershp fucto of e tota trasportato cost. Sce s a fuzzy uber stea of a crsp uber t caot be ze recty. To tace s probe oe ca trasfor e fuzzy so trasportato probe whch s base o aeh s eteso prcpe to a fay of aeatca progras to be sove. Base o e eteso prcpe e ebershp fucto ca be efe as: ( z ) s u p { ( c ) ( s ) ( ) ( e ) z ( c s e )} C D S E (4) Where ( c s e ) s efe Moe (). The appcato of e eteso prcpe to ay be vewe as e appcato of s eteso prcpe to e -cuts of. et us eote e -cuts of E as C S D a ) { c S ) ( c ) } [ ) ) ] (5.) C ) { s S ) ( s )) } [ ) ) ] (5. ) S ) { S ) ( ) } [ ) ) ] (5.) D ) { ( e S ) ( e ) } [ ) ) ] (5.4 ) E These tervas cate where e ut shppg cost suppy ea a coveyace e at possbty eve. I Eq. (4) severa ebershp fuctos are vove. To erve cose for s hary possbe. Accorg to (4) s e u of a. C S ee ( c ) ( s )) ( ) C S or ( e ) oe ( c ) ( s ) ( ) ( e ) C D S D E E D E We a at east equa to such at z ( c s e ) to satsfy ( z ). To f e ebershp fucto whch s equvaet to fg e ower bou u of ( c s e ) a t suffces to f e eft shape fucto a rght shape fucto of a upper bou of e -cuts of. Sce s e au of ( c s e ) ey ca be epresse as: { ( c s e ) ) c ) ) s ) ) ) ) e ) } a { ( c s e ) ) c ) ) s ) ) ) ) e ) } Ths ca be reforuate as e foowg par of two-eve aeatca progras: s e 65 P a g e

5 Iteratoa Joura of Avace Research I Scece A Egeerg IJARSE Vo. No.4 Issue 04 Apr 05 ISSN-9-854(E) ) c ) ) s ) ) ) ) e ) a ) c ) ) s ) ) ) ) e ) c s. t. s e... ( b b R ) C c s. t. s e... ( b b R ) C I Moe (6a) e er progra cacuates e obectve vaue for each c s (6b) (6a) a e specfe by e outer progra whe e outer progra eteres e vaues of c s obectve vaue a e at geerate e saest. The obectve vaue s e ower bou of e obectve vaue for Moe (). By e sae toe e er progra of Moe (6b) cacuates e obectve vaue for each gve vaue of c s a e whe e outer progra eteres e vaues of c s a e at prouce e argest obectve vaue. The obectve vaue s e upper bou of e obectve vaue for Moe (). Sce e vaue of vares Moe (6a a 6b) t ca aso be regare as a par of paraetrc prograg oe. A ecessary a suffcet coto for Moe (6a a 6b) to have feasbe soutos s s a e. I e frst eve of Moe (6a a 6b) s [) ) ] [ ) ) ] a e are aowe to vary e rage of a [ ) ) ] respectvey. However to esure e trasportato 66 P a g e

6 Iteratoa Joura of Avace Research I Scece A Egeerg IJARSE Vo. No.4 Issue 04 Apr 05 ISSN-9-854(E) probe of e seco eve to be feasbe t s ecessary at e costrat s a e be pose e outer progra. here B b b. Hece Moe (6a a 6b) becoes: ) c ) ) s ) ) ) ) e ) s e c s. t. s e... c B (7a) a ) c ) ) s ) ) ) ) e ) s e c s. t. s e... c B (7b) I above Moe (7a a 7b) w be feasbe whe S 0 D for ay α eve. I oer wors a 0 fuzzy trasportato probe s feasbe f e upper bou of e tota fuzzy suppy s greater a or equa to e ower bou of e tota fuzzy ea. To erve e ower bou of e obectve vaue Moe (7a) we ca recty set c to ts ower bou ( ) Hece Moe (7a) ca be reforuate as: C to f e u obectve vaue. 67 P a g e

7 Iteratoa Joura of Avace Research I Scece A Egeerg IJARSE Vo. No.4 Issue 04 Apr 05 ISSN-9-854(E) ) c ) ) s ) ) ) ) e ) s e c s. t. s e... c B [ b b ] (8) Sce Moe (8) s to f e u of a e u obectve vaues oe ca cobe e costrats of er progra a outer progra togeer a spfy e two-eve aeatca progra to e covetoa oe-eve progra as foows: ) s. t. s e... (9) s ) B... e ) s ) ) ) ) e ) 0. Ths oe s a ear progra whch ca be sove easy. I s oe sce a have bee set to e ower bous of er -cuts at s ( c ) s assures ( z ) as requre by (4). C To sove Moe (7b) s ot so straghtforwar as Moe (7a). The outer progra a er progra of Moe (7b) have fferet rectos for optzato oe for azato a aoer for zato. A 68 P a g e

8 Iteratoa Joura of Avace Research I Scece A Egeerg IJARSE Vo. No.4 Issue 04 Apr 05 ISSN-9-854(E) trasforato s requre to ae a souto obtaabe. The ua of er progra s foruate to becoe a azato probe to be cosstet w e azato operato of outer progra. It s we ow fro e uaty eore of ear prograg at e pra oe a e ua oe have e sae obectve vaue. Thus Moe (7b) becoes: a ) c ) ) s ) ) ) ) e ) s e a s u v e w B y s.. t u v w y c u v w 0 (0) Sce ) c ) Moe (0) oe ca erve e upper bou of e obectve vaue by settg c to ts upper bou because s gves e argest feasbe rego. Thus we ca reforuate Moe (0) as: a ) c ) ) s ) ) ) ) e ) s e a s u v e w s. t. u v w ( c ) u v w 0 () Now sce bo outer progra a er progra perfor e sae azato operato er costrats ca be cobe to for e foowg oe-eve aeatca progra: (.0) a s u v e w s. t. u v w ( c ) (.) s (.) e (.) 69 P a g e

9 Iteratoa Joura of Avace Research I Scece A Egeerg IJARSE Vo. No.4 Issue 04 Apr 05 ISSN-9-854(E) (.4) ) s )... (.5) ) )... (.6) ) e )... 0 (.7)Ths oe s a eary costrae oear progra. There are severa effectve a effcet eos for sovg s Moe (.0.7). Sar to Moe (9) sce a c have bee set to e upper bous of er -cuts at s ( c ) s assures ( z ) as requre by (4). C If e tota suppy a e tota coveyace capacty are greater a e tota ea at a vaues respectvey.e. ) 0 ) a 0 ) 0 ) e e costrats 0 s ca be eete fro Moe (.0.7). Mutpyg costrats (.4) (.6) by u v a w respectvey a substtutg su by p v by q a e w trasfore to e foowg ear progra: by r Moe (.0.7) s a p q r s. t. u v w ( c ) ) u p ) u... ) v q ) v... () ) w r ) w... p q r 0 I s case e upper bou of e tota trasportato cost at eve ca be fou ore easy. Probes (7a) a (7b) are assure to be feasbe f e ower bou of e tota fuzzy ea s saer a bo of e upper bou of e tota fuzzy suppy a e upper bou of e tota coveyace capacty.e. a ) 0 ). 0 ) 0 ( ) S 0 EXAMPE As a ustrato of e propose approach coser a fuzzy so trasportato probe w two fuzzy suppes ree fuzzy eas two coveyace capactes a ree buget tervas ature. The otatos use s eape s (a b c ) for a trapezoa fuzzy uber w a b c a as e coorates of e four vertces of e trapezo a ( y z) for e traguar fuzzy uber w y z as e coorates of e ree vertces of e trage. The probe has e foowg aeatca for: M P a g e

10 Iteratoa Joura of Avace Research I Scece A Egeerg IJARSE Vo. No.4 Issue 04 Apr 05 ISSN-9-854(E) s.t ( ) ( ) ( ) ( ) ( ) ( ) [ ] [ ] [ ] 0. The tota Suppy S S S ( ) e tota ea D D D D ( ) a e tota coveyace capacty E E E ( ) a e tervas of bugets are [ ] [585 65] a [ ]. Sce S D E Probe has feasbe soutos. Accorg to oes (9) a () e ower a upper bous of at possbty eve ca be foruate as: s.t s s e e s s e e s s e e a s u s u v v v e w e w y y y s.t. u v w 0 y P a g e

11 Iteratoa Joura of Avace Research I Scece A Egeerg IJARSE Vo. No.4 Issue 04 Apr 05 ISSN-9-854(E) u v w 7 0 y 7 0 u v w 6 0 y 6 0 u v w 0 y 0 u v w 5 0 y 5 0 u v w 0 y 0 u v w (0 0 ) y 0 0 u v w 4 0 y 4 0 u v w 0 y 0 u v w 5 0 y 5 0 u v w 4 0 y 4 0 u v w 5 0 y 5 0 s s e e s s e e u u v v v w w 0. We sove e above two probes by usg go []. Tabe. sts e -cuts of e tota trasportato cost at stct vaues: a Fg. epct e ebershp fucto of e tota trasportato cost of s eape. The vaue cates e eve of possbty a egree of ucertaty of e obtae forato. The greater e vaue e greater e eve of possbty a e ower e egree of ucertaty s. Tabe : The -cuts of e tota trasportato cost P a g e

12 Iteratoa Joura of Avace Research I Scece A Egeerg IJARSE Vo. No.4 Issue 04 Apr 05 ISSN-9-854(E) Fg.. The ebershp fucto of e tota trasportato cost Sce e fuzzy tota trasportato cost es a rage ts ost ey vaue fas betwee 600 a 400 a ts vaue possbe to fas outse e rage of 800 a For =0 e ower bou of =800 occurs at =40 =0 =0 w s = 0 s =90 =0 =40 =0 e =00 e =90 a e oer ecso varabes are 0. The upper bou of =5700 occurs at =40 =0 =70 =0 =60 =70 e =00 e =80 a e oer ecso varabes are 0. At oer etree e of = e ower bou of =600 occurs at =50 w s = 0 s =60 =50 =0 =40 =0 =0 w s = 80 s =70 =0 =50 =40 e =80 e =70 a e oer ecso varabes are 0. The upper bou of =400 occurs at =0 =0 =60 =0 =40 w s = 80 s =70 =40 =50 =60 e =80 e =70 a e oer ecso varabes are 0. Notaby e vaues of e ecso varabes erve s eape are aso fuzzy. IV CONCSION Trasportato oes have we appcatos ogstcs a suppy cha aageet for provg servce a reuce e cost. We have eveope e souto proceure for a fuzzy so trasportato probe w fuzzy suppy requreet coveyace capacty a buget terva a we put souto usg Hu a Wag s approach a fuzzy prograg approach. I e preset stuy we sove aeatca probes usg go software. I frae wor w geue fe probe e techque cou be use as very effectua a prosg a vew of a practca sgfcace. The probe ca be etee or appe to oer sar ucerta oes oer areas such as vetory cotro ecoogy sustaabe for aageet etc. REFERENCES [] R.E. Bea.A. aeh Decso-ag a fuzzy evroet Maageet Scece [] K.B. Haey The so trasportato probe Operatos Research P a g e

13 Iteratoa Joura of Avace Research I Scece A Egeerg IJARSE Vo. No.4 Issue 04 Apr 05 ISSN-9-854(E) [] J.J. Bucy A Possbstc ear prograg w traguar fuzzy ubers Fuzzy Sets a Systes [4] S. Chaas D. Kuchta A cocept of e opta souto of e trasportato probe w fuzzy cost coeffcets Fuzzy Sets a Systes [5] Ha Basrzaeh A Approach for Sovg Fuzzy Trasportato Probe Appe Maeatca Sceces 5() [6] A. Baya.K. Bera M. Mat Mut-te terva vaue so trasportato probe w safety easure uer fuzzy-stochastc evroet Joura of Trasportato Securty [7] M.A. Parra A.B. Tero a M.V.R. ra Sovg e utobectve possbstc ear prograg probe Europea Joura of Operatoa Research [8].A. aeh Fuzzy sets as a bass for a eory of possbty Fuzzy Sets a Systes [9] R.R. Yager A characterzato of e eteso prcpe Fuzzy Sets a Systes [0] H.J. era Fuzzy Set Theory a Its Appcatos (r e. Kuwer-Nhoff Bosto 996). [] B.Q. Hu S. Wag A ove approach ucerta prograg part : New aretc a orer reato for terva ubers. Joura of Iustra a Maageet Optzato (4) [] go ser_s Gue (INDO Systes Ic. Chcago 999). 74 P a g e