A New Efficient Approach to Solve Multi-Objective Transportation Problem in the Fuzzy Environment (Product approach)

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1 Itertol Jourl of Appled Egeerg Reserch IN Volue, Nuer 8 (08) pp Reserch Id Pulctos A New Effcet Approch to olve Mult-Oectve Trsportto Prole the Fuzzy Evroet (Product pproch) M Afwt AE, AAM l, N Frou,* Professor of Mth, ept of Physcs d Mths, Fculty of egeerg, gzg Uversty, Egypt octor of Mth, ept of Physcs d Mths, Fculty of egeerg, gzg Uversty, Egypt Assstt Lecturer of Egeerg Mth, Fculty of egeerg, gzg Uversty, Egypt (Correspodg Author*) Astrct I ths pper, we preset ew ethod to solve ultoectve trsportto prole clled (product pproch) We use fuzzy progrg to covert the oectves whch hve dfferet uts to eershp vlue the ggregte the y product Our pproch s esy d fst ethod to fd solutos close to the optu soluto Flly, uercl eple s troduced to llustrte the ethod AM suect clssfctos: 90B06, 90C0, 90C70 Keywords Mult-oectve trsportto prole, fuzzy eershp fucto, Vogel s pproto ethod INTROUCTION Trsportto prole (TP) s specl ctegory of ler progrg, whch s led up wth dy-to-dy ctvtes our ctul lfe d ly dels wth logstcs It ds resolvg proles of dstruto d trsportto of resources fro oe posto to soe other The coodtes re trsported fro lot of sources (eg, fctory) to set of ddresses (eg, wrehouse) to cofor cert ded I our recet stutos (TP) ot del wth sgle oectve ut owdys cot uer of coflctg d coesurle oectve fuctos whch c e clled ult-oectve trsportto prole (MOTP) The of solvg the (MOTP) s to ze severl peltes such s (trsportto cost, delvery te etc), etwee sources d desttos to fd sc fesle or optu soluto to the prole My clsscl ethods re used to solve the prole such s orth west corer ethod, u cost ethod d Vogel s pproto ethod The trsportto prole ws frstly studed y Htchcoc 97 [] Effcet solutos usg pplcto of fuzzy ler progrg to (MOTP) were troduced y Bt et l [] olvg (MOTP) usg trust-rego glolzto strtegy ws preseted y Yousr et l [7] Kudu et l [9] odeled ult-oectve, ult-te sold trsportto prole wth fuzzy coeffcets for the oectves d costrts, the wored y two dfferet ethods Yeol et l [] troduced ethod to solve (MOTP) usg fuzzy progrg techque I ths pper, we preset ew proposed pproch to solve (MOTP) fuzzy evroet to get sc fesle soluto Our ew pproch s esy d ucoplcted ethod We llustrte the ew pproch wth uercl eple d copre our results wth the other ethods MATHEMATICAL MOEL () Nottos Before troducg (MOTP) thetcl odel We wll clrfy soe ottos Let the followg:,,, dcte () sources,, dcte () desttos (=,, ) deote the vlle uts t ech sources ( =,, ) deote the ded uts for ech desttos s the uer of uts tht wll e trsported fro to 6 C std for y oectve(pelty) such s [trsportto cost, trsportto te, etc] per ut fro to 660

2 Itertol Jourl of Appled Egeerg Reserch IN Volue, Nuer 8 (08) pp Reserch Id Pulctos () Model The thetcl odel of (MOTP) c e epressed s follows: M ( ) C =,,,p uect to for,,, for,,, ( r codto) 0 Where superscrpt () represets the uer of oectve fuctos FUY PRELIMINARIE I ths pper, we wll fuzzfy our peltes (oectves) such s (trsportto cost, delvery te etc), to covert the fro crsp rego to fuzzy rego (soluto spce) to ze set of (p) oectves, the eershp fucto used for tht s defed s follows: Note tht: 0 U U ( ) L U () U L L L s the sllest crsp vlue of lrgest crsp vlue of If d U s the U L the ( ) for ll vlues of METHOOLOGY tep : Clculte the eershp vlues usg equto () for ech pelty tle tep : Costruct ew tle whch ech cell s the product of ll eershp vlues of the correspodg cells the three peltes tles tep : Clculte the dfferece ( ) etwee the hghest eershp vlue d the et hghest eershp vlue for ech row d colu tep : elect the u ( ) d serch for the hghest eershp vlue tht row or colu Whe fdg tht cell we llocte t wth the u vlue of or tep : Whe g llocto cell d ether row or colu or oth re stsfed we ccel tht row or Colu or oth d ecluded fro the et clcultos tep 6: After elto of the stsfed row or colu or oth, we clculte g ( ) for the reg rows d colus d g llocto for the u eershp vlue cell tep 7: Repet the prevous steps utl or colus d rows re stsfed tep 8: fro llocto we get the vlues of Notes: susttute the oectve fuctos the If there s te the u ( ) we serch for the u eershp vlue the correspodg row or colu If there s further te the u( ) d u eershp vlue We select the cell wth the hghest NUMERICAL EXAMPLE Let us cosder the followg uercl ult-oectve Trsportto eple preseted y Ae d Nr []; Rguest d Rs [] to llustrte our pproch The prole hs the followg chrcterstcs ed Tle : Pelty ( P ) estto

3 Itertol Jourl of Appled Egeerg Reserch IN Volue, Nuer 8 (08) pp Reserch Id Pulctos ed ed Tle : Pelty ( P ) estto Tle : Pelty ( P ) estto Accordg to the frst step we evlute the eershp vlues of the frst pelty ( the usg the equto () we get the followg tle ed p ) tle, where U K d L K Tle : Meershp vlues for Pelty ( P ) estto By slrty we do tht for (P ) d (P ) we get: ed ed Tle : Meershp vlues for Pelty ( P ) estto Tle 6: Meershp vlues for Pelty ( P ) estto Now fro the secod step we clculte the product of ll eershp vlues for the peltes we get the followg tle ed Tle 7: The product of eershp vlues estto

4 Itertol Jourl of Appled Egeerg Reserch IN Volue, Nuer 8 (08) pp Reserch Id Pulctos Now pplyg the reg steps our proposed pproch we get: Tle 8: Proposed product pproch estto // /0 / /8// ed //0 //0 6/0 /0 / The set of soluto s {,,,,,, 6} Now clculte the correspodg oectve fuctos we get ( ) 7, ( ) 7 d ( ) 86 Tle 9: Coprso etwee dfferet pproches The Ne of pproch ( ) ( ) ( ) The fuzzy pproch [] Iterctve pproch [6] Trust Rego Approch [7] 0 7 Proposed prllel ethod [] Our proposed (product) pproch Idel oluto

5 Itertol Jourl of Appled Egeerg Reserch IN Volue, Nuer 8 (08) pp Reserch Id Pulctos CONCLUION I ths pper, we propose ew pproch e t wth (product pproch) to solve (MOTP) We use fuzzy progrg to covert dfferet pelty uts (cost, te, etc) to eershp vlue, the ggregte the y product The fetures of our pproch c e surzed s follows: It c solve (MOTP) wth hgher desos It s esy d ucoplcted ethod As copred to other pproches, we see ts results rther close to the optu soluto [] MC, Y d VA, J,06, olvg Mult-Oectve Trsportto Prole Usg Fuzzy Progrg Techque-Prllel Method, Itertol Jourl of Recet cetfc Reserch, 7 (), pp8-87 REFERENCE [] Htchcoc, FL, 9, The struto of Product fro everl s to Nuerous Locltes, Jourl of Mthetcs d Physcs, 0 (-), pp -0 [] Ae, YP d Nr, KPK, 979, Bcrter Trsportto prole, Mgeet cece, (), pp7-78 [] Rguest, JL d Rs, B, 987, Iterctve solutos for the ler ult-oectve trsportto prole, Europe Jourl of Opertol Reserch, pp [] Bt, AK, Bswl, MP d Al,, 99, Fuzzy progrg pproch to ultcrter decso g trsportto prole, Fuzzy ets d ystes, 0 (), pp- [] El-Whed, WFA, 00, A ult-oectve trsportto prole uder fuzzess, Fuzzy ets d ystes, 7 (), pp7- [6] Ad El-Whed, WF d Lee, M, 006, Iterctve fuzzy gol progrg for ultoectve trsportto proles, Oeg, (), pp8-66 [7] Ao-elg, Y, El-soy, B d hed, H, 0, Trust Rego Algorth for Mult-oectve Trsportto, Assget, d Trsshpet Proles, Lfe cece Jourl, 9(), pp [8], A, Mous, AAA, Geeed, HM d Elewy, AY, 0 Effcet Multoectve Geetc Algorth for olvg Trsportto, Assget, d Trsshpet Proles, Appled Mthetcs, 0 (0), pp 9-99 [9] Kudu, P, Kr, d Mt, M, 0, Multoectve ult-te sold trsportto prole fuzzy evroet, Appled Mthetcl Modellg, 7 (), pp08-08 [0] Jh, PJ 0 Opertos Reserch Mcgrw Hll Pulshg Co Ltd 66