Fuzzy Multi Objective Transportation Model. Based on New Ranking Index on. Generalized LR Fuzzy Numbers

Transcription

1 Aled Mathematcal Scences, Vol 8, 4, no 8, HIKARI Ltd, wwwm-hkarcom htt://dxdoorg/988/ams4486 Fuzzy Mult Obectve Transortaton Model Based on New Rankng Index on Generalzed LR Fuzzy Numbers Y L P Thoran and N Rav Shankar Det of Aled Mathematcs, GIS GITAM Unversty, Vsakhaatnam, Inda Coyrght 4 Y L P Thoran and N Rav Shankar Ths s an oen access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch ermts unrestrcted use, dstrbuton, and reroducton n any medum, rovded the orgnal work s roerly cted Abstract In ths aer, we resent a new method for analyzng a fuzzy mult obectve transortaton roblem usng a lnear rogrammng model based on a new method for rankng generalzed LR fuzzy numbers Frst, we resent a new method for rankng generalzed LR fuzzy numbers from ts λ-cutto the best of our knowledge tll now there s no method n the lterature to fnd the rankng order of the generalzed LR fuzzy numbers to varous lnear and non-lnear functons of LR fuzzy numbers from ts -cut based on area, mode and sread Our method can effcently rank varous LR fuzzy numbers, ther mages and crs numbers whch are consder to be a secal case of fuzzy numbers (normal/nonnormal, trangular/traezodal, and general), whch could not be ranked by the exstng rankng methods Our rankng method also satsfes the lnearty roerty condton For the corroboraton, we used some comaratve examles of dfferent exstng methods to llustrate the advantages of the roosed methodthe reference functons of LR fuzzy numbers of fuzzy mult obectve transortaton roblem are consdered to be lnear and non-lnear functons Ths aer develos a rocedure to derve the fuzzy obectve value of the fuzzy mult obectve transortaton roblem, n that the fuzzy cost coeffcents and the fuzzy tme are LR fuzzy numbers The method s llustrated wth an examle by varous cases

2 685 Y L P Thoran and N Rav Shankar Keywords: Mult obectve transortaton; rankng ndex; LR fuzzy numbers; lnear rogrammng; -cut Introducton Transortaton models lay a sgnfcant role n logstcs and suly chan management for reducng cost and tme, for better servce In today s dynamc market the demands on organzatons to fnd mroved ways to create and delver value to customers becomes stronger How and when to send the roducts to the customers n the quanttes whch they want n a cost-effectve manner, become more challengng To meet ths challenge transortaton models endow wth a owerful framework They ensure the cometency movement and tmely accessblty of raw materals and fnshed goods Effcent algorthms had been develoed for solvng the transortaton roblem when the cost coeffcents and the suly and demand quanttes are known exactly In real world alcatons, these arameters may not be resented n a recse manner due to uncontrollable factors Bellman and Zadeh [] and Zadeh [] ntroduced the noton of fuzzness to deal wth mrecse nformaton n makng decsons The solutons obtaned by fuzzy lnear rogrammng were always effcent s roved by Zmmermann [] and ts fuzzy lnear rogrammng had develoed nto several fuzzy otmzaton methods for solvng transortaton roblems Several researchers had been makng rgorous nvestgatons on multobectve transortaton roblem Otmzaton of multobectves transortaton roblems was resented by Lee and Moore [4] A bcrtera transortaton roblem was resented by Anea and Nar [5] Multobectve lnear transortaton roblem rocedure whch generates all nondomnated solutons were resented by Daz [6,7] and Isermann [8] Bhata et al[9,], Guta[], Prasad et al[], L et al [], Lu and Zhang [4] studed the tme-cost mnmzng transortaton roblem The fractonal functon s one of the obectves n the multobectve transortaton roblem Swaru [5], Sharama and Swaru [6], Chandra and Saxena [7] studed that the multobectve transortaton roblem whch also relates to the fractonal transortaton roblem Bt et al [8] consder a k-obectve transortaton roblem fuzzfed by fuzzy numbers and used -cut to obtan a transortaton roblem n the fuzzy sense exressed n lnear rogrammng form Chanas and Kuchta[9] used fuzzy numbers of the tye L-L to fuzzfy cost coeffcents n the obectve functon and -cut to exress the obectve functon n the form of an nterval Hussen [] studed the comlete set of - ossblty effcent solutons of mult obectve transortaton roblem wth ossblstc co-effcents of the obectve functons L and La [] roosed a fuzzy comromse rogrammng aroach to a mult obectve lnear transortaton roblem For ratng and rankng multle asect alternatves usng fuzzy sets was

3 Fuzzy mult obectve transortaton model 685 resented by Bass and Kwakeernaak [7] For the rankng of fuzzy numbers a number of rankng aroaches had been roosed A rankng rocedure for orderng fuzzy sets n whch a rankng ndex was calculated for the LR fuzzy number from ts λ- cut was roosed by Yager [8], t cannot rank generalzed LR fuzzy numbers, but by the roosed method we can overcome the lmtatons and shortcomngs of the exstng method and the roosed method s relatvely smle n comutaton and s easly logcal Lou and Wang [6] resented a method for rankng fuzzy numbers wth ntegral values Abbasbandy and Asady [], Asady and Zendehnam [8] roosed that a fuzzy number s maed to a real number based on the area measurement A new aroach for rankng fuzzy numbers by the dstance method and the coeffcent varance (CV) ndex method roosed by Cheng [7] had some drawbacks(for account see Examles and as mentoned n secton 4 of ths aer) For rankng fuzzy numbers wth an area between the centrod onts of fuzzy numbers and the orgnal ont was roosed by Chu and Tsao [] to over come the drawbacks that are exstng n Cheng [7],but ther aroach stll had some drawbacks (for account see Examle as mentoned n secton 4 of ths aer) Wang and Lee [9] made a revson on rankng fuzzy numbers wth an area between the centrod and orgnal onts to mrove Chu and Tsao s aroach [] Though some mrovements are made, Wang and Lee s aroach cannot stll dfferentate two fuzzy numbers wth the same centrod ont (see Examle as mentoned n secton 4) Wang et al [] roosed an aroach to rankng fuzzy numbers based on lexcograhc screenng rocedure and summarzed some lmtatons of the exstng methods (see Table n Ref []) However, Wang et al s aroach can not dfferentate these knds of fuzzy numbers as shown n Examle n Secton 4 of ths aer, as most of the exstng aroaches do To overcome the lmtatons of the exstng studes and smlfy the comutatonal rocedures, we defne a new rankng aroach to varous lnear and non-lnear functons of LR fuzzy numbers from ts - cut and also uses mode and sreads n those cases where the dscrmnaton s not ossble s roosed Based on the rooed fuzzy rankng method, we resent a method for dealng wth fuzzy mult obectve transortaton model uses lnear rogrammng In ths aer, we consder fuzzy mult-obectve transortaton roblem wth fuzzy arameters n the case of the transortaton rocess Here, we let to be a fuzzy cost for shng one unt roduct from th source to th destnaton, c to be fuzzy tme from th source to th destnaton Here, the fuzzy cost t coeffcents, the fuzzy tmes are LR fuzzy numbers By assgnng a weght to the obectves accordng to ther rortes the sngle obectve functon s obtaned Then, by usng the roosed rankng method, transform a newly formed sngle obectve fuzzy transortaton roblem to a crs transortaton roblem n the lnear rogrammng roblem form and t can be solved by any conventonal method

4 685 Y L P Thoran and N Rav Shankar The rest of the aer s organzed as follows : In secton relmnares of LR fuzzy numbers, -cut of LR fuzzy number, reference functons and ther nverses are resented In secton new rankng aroach to varous lnear and non-lnear functons of LR fuzzy numbers from ts -cut and also uses mode and sreads n those cases where the dscrmnaton s not ossble s gven and an mortant result lke lnearty of rankng functon whch s the bass for defnng the rankng rocedure n secton s dscussed and roved In secton 4, we comare the rankng results of the roosed method for varous cases wth the exstng methodsin secton 5, lnear rogrammng model for fuzzy mult obectve transortaton model to varous lnear and non-lnear functons wth generalzed LR fuzzy numbers s dscussed and, the roosed method to solve the total otmal fuzzy cost, and fuzzy tme for fuzzy mult-obectve transortaton roblem usng lnear rogrammng s dscussed wth a numercal examle and total otmal fuzzy cost, and fuzzy tme for varous cases are resented Fnally the concluson s gven n secton 6 Prelmnares In ths secton, LR fuzzy numbers, -cut of LR fuzzy number, reference functons and ther nverses are resented LR fuzzy numbers and reference functons In ths secton, LR fuzzy number, -cut of LR fuzzy number, reference functons and ther nverses are revewed [8] Defnton A fuzzy number LR A m,n,, s sad to be an LR fuzzy number f m x L, x m,, x n (x) R, x n,, A, otherwse where L and R are contnuous, non-ncreasng functons that defne the left and rght shaes of (x) resectvely and L() = R() = A Lnear reference functons and nonlnear reference functons wth ther nverses are resented n Table I

5 Fuzzy mult obectve transortaton model 685 Table I: Reference functons and ther nverses Functon Name Reference Functon (RF) Inverse of Reference Lnear x max,- x functon, RF RF x x Exonental RF x e, RF x In/ Power RF x max, x, Exonental ower RF RF x x e x, RF x In Ratonal RF x x, RF x Defnton Let A m,n,, LR be an LR fuzzy number and be a real number n the nterval [,] Then the crs set A x X: x [m L,n R ] s sad to be -cut of A A New rankng aroach to varous lnear and non-lnear functons of LR fuzzy Numbers from ts -cut based on area, mode and sread In ths secton, new rankng aroach s resented for the rankng of LR fuzzy numbers Ths method nvolves a rocedure for orderng fuzzy sets n whch a rankng aroach A m,n,, from ts -cut A Proosed rankng Method R s calculated for the fuzzy number LR The Centrod of a traezod s consdered as the balancng ont of the traezod (Fg) Dvde the traezod nto three trangles as APC, QCD and PQC resectvely Let G be the Centrod of the trangle APC, G be the Centrod of the trangle QCD and G be the Centrod of the trangle PQC The Centrod of the Centrods of these three trangles s taken as the ont of reference to defne the rankng of generalzed traezodal fuzzy numbers The reason for selectng ths ont as a ont of reference s that each Centrod ont s a balancng ont of each ndvdual trangle, and the Centrod of these Centrod onts s a much more balancng ont of a generalzed LR fuzzy number Therefore, ths ont would be a better reference ont than the Centrod ont of the traezod

6 6854 Y L P Thoran and N Rav Shankar Consder a generalzed LR fuzzy number A m,n,, ; w LR (Fg) The Centrods of the three trangles are m n w n w m n w G,, G, andg, resectvely w Equaton of the lne G G s y and G does not le on the lne G G Therefore G, G and G are non-collnear and they form a trangle G x, y of the trangle wth vertces G, G and G of the We defne the Centrod generalzed A A m,n,, ; w as LR fuzzy number LR m 6n 4w x,y, G A 9 9 () As a secal case, for trangular L-R fuzzy number A m,n,, ; w LR e, m = n the Centrod of Centrods s gven by 9m 4w G x, y, A 9 9 () The rankng functon of the generalzed LR fuzzy number A m,n,, ; w LR whch mas the set of all fuzzy numbers to a set of real numbers s defned as: m 6n 4w RA x y () 9 9 G x, y as defned n Eq() Ths the Area between the Centrod of the Centrods and the orgnal ont A

7 Fuzzy mult obectve transortaton model 6855 A m,n,, ;w of The rankng functon of the generalzed LR fuzzy number LR Eq() from ts cut s defned as m L d 6n R d 4w R A (4) 8 Snce R A s calculated from the extreme values of -cut of A, rather than ts membersh functon, t s not requred knowng the exlct form of the membersh functons of the fuzzy numbers to be ranked That s unlke most of the rankng methods that requre the knowledge the membersh functons of all fuzzy numbers to be ranked Ths centrod of centrod ndex s stll alcable even f the exlct form the membersh functon of the fuzzy numbers s unknown Case () R A 8w m 6n 8 The rankng ndexes for Eq(4) wth varous lnear and non-lnear functons are: L x R x max,- x L x R x e 8w m 6n 8 L x max,- x x Case () R A and R x e x Case () R A 8w m 6n 8 x L x e and R x Case (v) max, x R A 8w m 6n 8 L x e and R x Case (v) x max, x R A 8w m 6n 8 A m,n,, ; w s The Mode (M) of the generalzed LR fuzzy number LR defned as: w w m ndx m n M A (5)

8 6856 Y L P Thoran and N Rav Shankar A m,n,, ; w s defned The Sread (S) of the generalzed LR fuzzy number LR as: w n mdx w n m S A (6) A m,n,, ; w s The Left sread (LS) of the generalzed LR fuzzy number LR defned as: A LS w dx w (7) A m,n,, ; w s The Rght sread (RS) of the generalzed LR fuzzy number LR defned as: A RS w dx w (8) An algorthm for fndng the rankng of two generalzed LR fuzzy numbers A m,n,, ; w LR and A m,n,, ; w LR, by usng the rankng ndexes for Eq(4) wth varous lnear and non-lnear functons, and usng the equatons from (5-8), s defned as follows Ste : Fnd R A and RA If RA RA then A A If RA RA then A A If RA RA Ste : Fnd M A and MA If MA MA thena A If MA MA thena A If MA MA Ste : Fnd S A and SA If SA SA thena A If SA SA thena A If SA The comarson s not ossble, then go to ste The comarson s not ossble, then go to ste A The comarson s not ossble, then go to ste 4

9 Fuzzy mult obectve transortaton model 6857 Ste 4: Fnd LS A and LSA If LSA LSA thena A If LSA LSA thena A If LSA A The comarson s not ossble, then go to ste 5 Ste 5: Examne w and w If w w then A A If w w then A A If w w thena A In ths secton an mortant result whch s the bass for defnng the rankng rocedure n secton s dscussed and roved Prooston The rankng functon of centrod of centrods (Eq4) s a lnear functon of the normal LR fuzzy number A m,n,, ; w LR If A m,n,, ; w LR and A m,n,, ; w LR are two normalzed LR fuzzy numbers, then Rk A ka kra kra R A RA R A A Proof () Case () Let k, k > k A ka = (km + km, kn + kn, kα + kα, kβ + kβ) Usng Eq (4), we get 8 R k A ka km km k k L 6 kn 8 k n k k R 8 8 k m 6n L k m 6n R 8 8 kra kra Rk A ka kra kra

10 6858 Y L P Thoran and N Rav Shankar Smlarly, the result can be roved for the case () k >, k < and case () k <, k > Proof () A m,n,, ; w and Let LR A,, m, n; 8 A R x L R R A R A = m 6n L x R x 6n md Proof () R A A RA R A RA RA by() by() 4 A comarson of the rankng results of the roosed method for varous cases wth the exstng methods In ths secton, some numercal examles from the exstng lterature and two roosed examles and the comaratve study wth fgures are used to llustrate the roosed method for rankng generalzed LR fuzzy numbers usng λ-cut, for varous cases s gven Examle,,,; Consder set wth two fuzzy numbers [], LR B,, 9,;, as shown n Fg LR A and

11 Fuzzy mult obectve transortaton model 6859 By usng varous rankng ndexes wth varous lnear and nonlnear functons as defned n secton the rankng order of A and B n all the cases s A B, the rankng results by the roosed method s gven n Table II From Fg of set, t can be seen that the result obtaned by our aroach s relable wth human nstnct However, by the CV ndex roosed by Cheng [7], the rankng order s A B, whch s unreasonable Examle Consder set wth three fuzzy numbers [], 6,6,, ; LR, B 6,6,,; LR, and C 6,6,,; as shown n Fg LR By usng varous rankng ndexes wth varous lnear and nonlnear functons as defned n secton the rankng order of A, B and C n case(,, and v) s C B A and n case(v) the rankng order s B C A The rankng results by the roosed method s gven n Table II However, by Chu and Tsao s aroach [], the rankng order s B C A Cheng [7], roosed CV ndex, through s aroach the rankng order s A B C From Fg, of set, t s easy to see that the rankng results obtaned by the exstng aroaches [, 7] are unreasonable and are not consstent wth human ntuton On the other hand, by Abbasbandy and Asady s aroach [], the rankng results s C B A, whch s the same as the one obtaned by our aroach However, our aroach s smler n the comutaton rocedure Examle Consder set wth two fuzzy numbers [4], 6,6,, ; LR, B 6, 6,,; LR as shown n Fg By usng varous rankng ndexes wth varous lnear and nonlnear functons as R A R B defned n secton the rankng order of A and B n case(-) s therefore by usng Eq 6 and Eq 7 we get the result as B A and n case (-v) the rankng order s A B, the rankng results by the roosed method s gven n Table II Because fuzzy numbers A and B have the same mode and symmetrc sread, most of exstng aroaches fal Asady and Zendehnam [8], Chu and Tsao [], Wang and Lee [9] and Yao and Wu [5], get the same rankng order as A B Abbasbandy and Asady s [] aroach the rankng order of fuzzy numbers A A

12 686 Y L P Thoran and N Rav Shankar s A B and A B for dfferent ndex values e, when = and = resectvely By Wang et al [], aroach the rankng order s A B Tran-Ducken [] get the varance results when Dmax and Dmn are used resectvely Lu [], Matarazzo and Munda [], Yao and Ln [4], aroaches gave dfferent rankng order when dfferent ndces of otmsm are taken But by our rankng aroach the rankng results n all the cases s same e, A B Examle 4 Consder two sets set 4 and set 5 wth three fuzzy numbers from Yao and Wu [5], as shown n Fg Set 4:A 5, 5,, 5;, B 7, 7,, ; and C 9, 9, 5, ; ; LR By usng varous rankng ndexes wth varous lnear and nonlnear functons as defned n secton the rankng order of A, B and C n all the cases s C B A The rankng results by the roosed method s gven n Table II The rankng result by the roosed method and the rankng results of the exstng methods [,5,7,,5] the rankng order s C B A whch s the same as the one obtaned by our aroach However, our aroach s smler n the comutaton rocedure The rankng results of other aroaches are gven n Table III By the CV ndex aroach [7], the rankng order s A C B, whch s counter-ntuton (see set 4 of Fg ) Set 5:A 4,7,,;, B 7,7,4,; and C 7,7,,; LR By usng varous rankng ndexes wth varous lnear and nonlnear functons as defned n secton the rankng order of A, B and C n case(,, and v) s C B A and n case(v) the rankng order s B C A The rankng results by the roosed method s gven n Table II The rankng results of other aroaches are gven n Table III By Bortolan-Degan [5] the rankng order s C B A and by the CV ndex [7] the rankng order s A B C, both the rankng orders s counter-ntuton (see set 5 of Fg ) Examle 5 Consder set 6 wth two fuzzy numbers [6],,,, ; LR, B,,,; as shown n Fg LR LR LR A LR LR

13 Fuzzy mult obectve transortaton model 686 By usng varous rankng ndexes wth varous lnear and nonlnear functons as defned n secton the rankng order of A and B n all the cases s A B, the rankng results by the roosed method s gven n Table II Lou and Wang [6] aroach obtaned the dfferent rankng order when dfferent otmstc ndces are adoted Chu and Tsao [] aroach obtan the rankng order as B A whch s same as the roosed aroach But our aroach s smler n the comutaton rocedure By Deng et al [6] aroach the rankng order s B A set 6, we can conclude that B A s more consstent wth human ntuton From Fg of Table III: Comaratve results of Examle 4 Methods Set 4 Set 5 A B C A B C Bortolan-Degan [5] Results C B A C B A Chu-Tsao [] Results C B A C B A Yao-Wu [5] Results C B A C B A Sgn dstance ( = ) [] Results C B A C B A Sgn dstance ( = ) [] Results C B A C B A Cheng dstance [7] Results C B A C B A Cheng CV unform dstrbuton [7] Results Cheng CV roortonal dstrbuton [7] Results A C B A B C A C B A B C Asady-Zendehnam dstance [8] Results C B A C B A Note: ndcates ncorrect rankng results

14 686 Y L P Thoran and N Rav Shankar Examle 6 Consder set 7, the mages of set 6,,,, ; LR, B,,, ; LR as shown n Fg By usng varous rankng ndexes wth varous lnear and nonlnear functons as defned n secton the rankng order of A and B n all the cases s A B, the rankng results by the roosed method s gven n Table II From set 6 and set 7, A B A B Therefore the roosed rankng method can rank mages also A Examle 7 Consder set 8 wth two fuzzy numbers,,, ; 8 LR,,,,; LR A B as shown n Fg By usng varous rankng ndexes wth varous lnear and nonlnear functons as defned n secton the rankng order of A and B n all the cases s A B, the rankng results by the roosed method s gven n Table II Therefore the roosed rankng method can rank fuzzy numbers wth dfferent heghts and t can rank crs numbers also Yager [8], cannot rank generalzed LR fuzzy numbers, but by the roosed method we can overcome the lmtatons and shortcomngs of the exstng method and the roosed method s relatvely smle n comutaton and s easly logcal Tll now n the exstng lterature there are no rankng methods to rank L-R fuzzy numbers wth dfferent heghts

15 Fuzzy mult obectve transortaton model A B Set : A,,, B,,9, LR LR A B C Set : A 6,6,,; LR B 6,6,,; LR C 6,6,,; LR A B C A B Set : A 6,6,, ; LR, B 6, 6,,; LR Set 4:A 5,5,, 5; LR, B 7,7,, LR and C 9,9,5, LR A B C 6 4 A B Set 5:A 4,7,,; LR, 7,7,4, LR and 7,7,, LR B C Set 6: A,,, ; LR, B,,,; LR B A Set : 7 A A B,,, ; LR,,,, ; LR B Set : 8 A,,, ; 8 LR, B,,,; LR Fg: Eght sets of fuzzy numbers

16 6864 Y L P Thoran and N Rav Shankar Table II: Rankng values of the roosed rankng method Set Set Fuzzy number Case() Case() Case() Case(v) Case(v) A,,,; B A B C,,9,; Result 6,6,, ; LR 6,6,,; 6,6,,; Result LR LR LR LR A B A B A B A B A B C B A C B A C B A B C A C B A Set Set 4 Set 5 Set 6 Set 7 A A B 6,6, 5,5,, ; LR 6,6,,; Result LR, 5; LR B 7,7,,; C 9,9,5,; A Result 4,7,,; B 7,7,4,; C A B A B 7,7,,; Result,,, ; LR LR,,,; Result,,, LR LR LR LR LR,,, ; Result ; LR LR A B A B A B A B A B C B A C B A C B A C B A C B A C B A C B A C B A B C A C B A A B A B A B A B A B A B A B A B A B A B Set 8 A,,, ; 8 LR B,,,; Result LR A B A B A B A B A B

17 Fuzzy mult obectve transortaton model Lnear Programmng Model for Fuzzy Mult Obectve Transortaton Model In ths secton, Fuzzy multobectve transortaton model wth fuzzy cost and fuzzy tme, mathematcal formulaton of fuzzy multobectve transortaton model s resented and we aly the roosed rankng method to solve the total otmal fuzzy cost, and fuzzy tme for fuzzy multobectve transortaton roblem usng lnear rogrammng to varous lnear and non-lnear functons wth LR fuzzy numbers, so as to mnmze the fuzzy cost, and fuzzy tme Generally the fuzzy transortaton roblem s to transort varous amounts of a sngle homogeneous commodty that are ntally stored at varous sources, to dfferent destnatons n such a way that the total fuzzy transortaton cost s a mnmum Let there be m sources, th sources ossessng a fuzzy suly unts of a certan roduct, n destnatons (n may or may not be equal to m) wth destnaton requrng fuzzy demand unts Cost of shng of an tem from each of m b sources to each of the n destnatons are known ether drectly or ndrectly n terms of mleage, shng hours, etc If the obectve of a transortaton roblem s to mnmze fuzzy cost, and fuzzy tme, then ths tye of fuzzy roblem s treated as a fuzzy multobectve transortaton roblem Here, we consder fuzzy transortaton roblem wth two obectves n the followng form of m n fuzzy matrx (Table IV) where each cell havng a fuzzy cost, and fuzzy tme c Table IV: Fuzzy mult obectve transortaton model wth fuzzy cost and fuzzy tme t Source/ Destnaton A B I M Demand n Suly c ; t c ; c ; t c ; t t c ; t c ; t n n c ; t c ; t n n a a c ; c ; c ; c ; a n t t c ; c ; m m t n t t t c ; t c ; t m m m m mn mn b b b b n a m

18 6866 Y L P Thoran and N Rav Shankar 5 Mathematcal formulaton of fuzzy multobectve transortaton model Mathematcally, the fuzzy multobectve transortaton roblem n Table IV can be stated as: m n Mnmze k z x, subect to k n x m x a b where z, z,, k z k,,, m,,,n z s a vector of k-obectve functons If the obectve functon z denotes the fuzzy cost functon, m n Mnmze z cx, If the obectve functon z denotes the fuzzy tme functon, m n Mnmze z tx, Then t s a two obectve fuzzy transortaton roblem Use weghts to consder the rortes of the obectve m n m n z w c x w t Mnmze subect to n x x m x a b,,, m,,,n and w + w = x =,, m ; =,, n c m,n,, : Fuzzy cost from th source to th destnaton t m,n,, LR : Fuzzy tme form th source to th destnaton a m,n,, : Fuzzy suly from th source to th destnaton LR b m,n, : Fuzzy demand from th source to th destnaton where LR, LR

19 Fuzzy mult obectve transortaton model 6867 ( ) L(x) Rx max,- x x Lx max,- xand Rx e, x v Lx e and Rx max, x x,, x v Lxe and Rx max, x L x R x e, L(x) = left shae functons; R(x) = rght shae functons All c, t, a, b denotes a non-negatve LR fuzzy numbers m n c x : Total fuzzy cost for shng from th source to th destnaton m n t x : Total fuzzy tme for shng from th source to th destnaton 5 The total otmal fuzzy soluton for fuzzy multobectve transortaton roblem In ths secton, the roosed method to solve the total otmal fuzzy cost, and fuzzy tme for fuzzy multobectve transortaton roblem usng lnear rogrammng as follows: Ste : Frst test whether the gven fuzzy multobectve transortaton roblem s a balanced one or not If t s a balanced one (e, sum of suly unts equal to the sum of demand unts) then go to ste If t s an unbalanced one (e sum of suly unts s not equal to the sum of demand unts) then go to ste Ste : Introduce dummy rows and/or columns wth zero fuzzy costs, and tme so as to form a balanced one Ste : Consder the fuzzy lnear rogrammng model as roosed n secton 5 Ste 4: Convert the fuzzy multobectve transortaton roblem nto the followng crs lnear rogrammng roblem m n m n z w R c x w R t Mnmze n x subect to x Ra,,, m m x and w + w = R b,,, n

20 6868 Y L P Thoran and N Rav Shankar x =,, m; =,, n Ste 5: Based on the case chosen n secton, calculate the values of R c, R t,r a and R b, by usng the rankng rocedure as mentoned n secton for the chosen fuzzy multobectve transortaton roblem Ste 6: For the values obtaned n ste 5, and usng the assgned values of w and w, the mult obectve transortaton roblem s converted nto a sngle obectve crs transortaton roblem as follows: Mnmze m n z Q x n subect to x Ra,,, m m x R b,,, n x =,,,m; =,,,n where Q s a constant Ste 7: To fnd the otmal soluton {x }, solve the crs lnear rogrammng roblem obtaned n ste 6 by usng software lke TORA Ste 8: Fnd the otmal total fuzzy transortaton cost, and total fuzzy transortaton tme by substtutng the otmal soluton obtaned n ste 7 n the obectve functon of ste In the followng, we use the roosed method to deal wth the fuzzy multobectve transortaton roblem To llustrate the roosed model, consder a case of transortaton rocess wth three sources and three destnatons, as a fuzzy mult obectve transortaton roblem so as to mnmze the fuzzy cost, and fuzzy tme The cost coeffcents, the tme, the suly and demand n fuzzy mult obectve transortaton roblem are consdered as LR fuzzy numbers The fuzzy multobectve transortaton roblem wth fuzzy cost, fuzzy tme, and fuzzy suly and fuzzy demand s shown n Table V and t s solved by usng varous cases TABLE V: Fuzzy Mult Obectve Transortaton Problem wth Fuzzy Cost and Fuzzy Tme Note: Source/ Destnaton suly A (6,7,,) (9,,,) (5,7,,) (9,,,) (8,,,) (4,5,,) B (,5,,) (7,8,,) (8,7,,4) (7,,,) (,4,,) (5,8,,) C (8,,,) (7,,,) (8,9,,) (4,5,,) (5,7,,) (7,9,,) Demand b (5,8,,4) (8,9,4,) (4,6,,) ndcates fuzzy tme a (5,7,4,) (7,8,,) (5,8,,)

21 Fuzzy mult obectve transortaton model 6869 The total otmal fuzzy cost and fuzzy tme for fuzzy multobectve transortaton roblem usng fuzzy lnear rogrammng for varous cases as follows: Case () Lx Rx max,- x Ste : The gven fuzzy mult-obectve transortaton roblem s a balanced one Ste : Usng ste of the roosed model, the gven fuzzy mult-obectve transortaton roblem s converted nto a sngle fuzzy obectve roblem as follows z 5 c x 5 t x Mnmze subect to x a,, x b,, Here, w = 5, w = 5 Ste : The fuzzy mult-obectve transortaton roblem s converted nto the followng crs lnear rogrammng roblem z 5 R c x 5 R t x Mnmze subect to x R a,, x R b,, are Ste 4: Usng secton, the values of Rc, Rt,Ra and Rb, calculated and gven n Table VI TABLE VI: Ranks of Fuzzy Cost and Fuzzy Tme of case () Source/ Destnaton suly A B C Demand Note: ndcates rank of fuzzy tme

22 687 Y L P Thoran and N Rav Shankar Ste 5: Usng ste 6 of the roosed method convert the chosen fuzzy multobectve transortaton roblem nto the followng crs lnear rogrammng Mnmze: (755)x + (77)x + (69)x + (597)x + (95)x + (69)x + (64)x + (74)x + (76)x subect to: x + x + x = 55 ; x + x + x = 676 ; x + x + x = 6 ; x + x + x = 6 ; x + x + x = 755 ; x + x + x = 474 x, for all =,, and =,, Ste 6: Solve the crs lnear rogrammng roblem, obtaned n ste 5, by usng TORA software the otmal soluton obtaned s: x =, x = 4, x = 55, x =, x = 474 Ste 7: Usng ste 8 of the roosed model, the mnmum fuzzy transortaton cost and tme resectvely are (4, 558, 599, 48) and (789, 464, 897, 5) x Case () Lx Rx e Ste : The gven fuzzy multobectve transortaton roblem s a balanced one Ste : Usng ste of the roosed model, the gven fuzzy multobectve transortaton roblem s converted nto a sngle fuzzy obectve roblem as follows z 5 c x 5 t x Mnmze subect to x a,, x b,, Here, w = 5, w = 5 Ste : The fuzzy multobectve transortaton roblem s converted nto the followng crs lnear rogrammng roblem z 5 R c x 5 R t x Mnmze subect to x R a,, x R b,,

23 Fuzzy mult obectve transortaton model 687 Ste 4: Usng secton, the values of R c, R t,r a and Rb, are calculated and gven n Table VII TABLE VII: Ranks of Fuzzy Cost and Fuzzy Tme of case () Source/ Destnaton suly A B C Demand Note: ndcates rank of fuzzy tme Ste 5: Usng ste 6 of the roosed method convert the chosen fuzzy multobectve transortaton roblem nto the followng crs lnear rogrammng Mnmze: (755)x + (7)x + (66)x + (6)x + (98)x + (64)x + (66)x + (745)x + (765)x subect to: x + x + x = 54 ; x + x + x = 67 ; x + x + x = 64 ; x + x + x = 64 ; x + x + x = 74 ; x + x + x = 474 x, for all =,, and =,, Ste 6: Solve the crs lnear rogrammng roblem, obtaned n ste 5, by usng TORA software the otmal soluton obtaned s: x = 97, x = 444, x = 54, x = 97, x = 474 Ste 7: Usng ste 8 of the roosed model, the mnmum fuzzy transortaton cost and tme resectvely are (6, 56, 6, 4) and (75, 454, 89, 5) L x max,- x and R x e x Case () Ste : The gven fuzzy multobectve transortaton roblem s a balanced one Ste : Usng ste of the roosed model, the gven fuzzy multobectve transortaton roblem s converted nto a sngle fuzzy obectve roblem as follows

24 687 Y L P Thoran and N Rav Shankar z 5 Mnmze c x 5 subect to x a,, x b,, t Here, w = 5, w = 5 Ste : The fuzzy multobectve transortaton roblem s converted nto the followng crs lnear rogrammng roblem z 5 R c x 5 R t x Mnmze x subect to x R a,, x R b,, are Ste 4: Usng secton, the values of Rc, Rt,Ra and Rb, calculated and gven n Table VIII TABLE VIII: Ranks of Fuzzy Cost and Fuzzy Tme of case () Source/ Destnaton suly A B C Demand Note: ndcates rank of fuzzy tme Ste 5: Usng ste 6 of the roosed method convert the chosen fuzzy multobectve transortaton roblem nto the followng crs lnear rogrammng Mnmze: (765)x + (7)x + (69)x + (67)x + (95)x + (65)x +

25 Fuzzy mult obectve transortaton model 687 (64)x + (75)x + (77)x subect to: x + x + x = 56 ; x + x + x = 686 ; x + x + x = 646 ; x + x + x = 65 ; x + x + x = 76 ; x + x + x = 48 x, for all =,, and =,, Ste 6: Solve the crs lnear rogrammng roblem, obtaned n ste 5, by usng TORA software the otmal soluton obtaned s: x =, x = 448, x = 56, x = 98, x = 48 Ste 7: Usng ste 8 of the roosed model, the mnmum fuzzy transortaton cost and tme resectvely are (5, 589, 674, 487) and (958, 4855, 946, 59) x Case (v) Lx e and Rx max,- x Ste : The gven fuzzy multobectve transortaton roblem s a balanced one Ste : Usng ste of the roosed model, the gven fuzzy multobectve transortaton roblem s converted nto a sngle fuzzy obectve roblem as follows z 5 c x 5 t x Mnmze subect to x a,, x b,, Here, w = 5, w = 5 Ste : The fuzzy multobectve transortaton roblem s converted nto the followng crs lnear rogrammng roblem z 5 R c x 5 R t x Mnmze subect to x R a,, x R b,, are Ste 4: Usng secton, the values of Rc, Rt,Ra and Rb, calculated and gven n Table IX

26 6874 Y L P Thoran and N Rav Shankar TABLE IX: Ranks of Fuzzy Cost and Fuzzy Tme of case (v) Source/ Destnaton suly A B C Demand Note: ndcates rank of fuzzy tme Ste 5: Usng ste 6 of the roosed method convert the chosen fuzzy multobectve transortaton roblem nto the followng crs lnear rogrammng Mnmze: (785)x + (747)x + (656)x + (6)x + (97)x + (668)x + (646)x + (765)x + (775)x subect to: x + x + x = 6 ; x + x + x = 7 ; x + x + x = 646 ; x + x + x = 66 ; x + x + x = 84 ; x + x + x = 5 x, for all =,, and =,, Ste 6: Solve the crs lnear rogrammng roblem, obtaned n ste 5, by usng TORA software the otmal soluton obtaned s: x = 7, x = 444, x = 6, x =, x = 5 Ste 7: Usng ste 8 of the roosed model, the mnmum fuzzy transortaton cost and tme resectvely are (6, 6469, 788, 4577) and (58, 5647, 58, 754) x Case (v) Lx e and Rx max,- x Ste : The gven fuzzy multobectve transortaton roblem s a balanced one Ste : Usng ste of the roosed model, the gven fuzzy multobectve transortaton roblem s converted nto a sngle fuzzy obectve roblem as follows z 5 c x 5 t x Mnmze

27 Fuzzy mult obectve transortaton model 6875 subect to x a,, x b,, Here, w = 5, w = 5 Ste : The fuzzy multobectve transortaton roblem s converted nto the followng crs lnear rogrammng roblem z 5 R c x 5 R t x Mnmze subect to x R a,, x R b,, are Ste 4: Usng secton, the values of Rc, Rt,Ra and Rb, calculated and gven n Table X TABLE X: Ranks of Fuzzy Cost and Fuzzy Tme of case (v) Source/ Destnaton suly A B C Demand Note: ndcates rank of fuzzy tme Ste 5: Usng ste 6 of the roosed method convert the chosen fuzzy multobectve transortaton roblem nto the followng crs lnear rogrammng Mnmze: (76)x + (77)x + (64)x + (6)x + (94)x + (645)x + (69)x + (747)x + (765)x subect to: x + x + x = 558 ; x + x + x = 68 ; x + x + x = 69 ; x + x + x = 64 ; x + x + x = 758 ; x + x + x = 479

28 6876 Y L P Thoran and N Rav Shankar x, for all =,, and =,, Ste 6: Solve the crs lnear rogrammng roblem, obtaned n ste 5, by usng TORA software the otmal soluton obtaned s: x =, x = 49, x = 558, x =, x = 479 Ste 7: Usng ste 8 of the roosed model, the mnmum fuzzy transortaton cost and tme resectvely are (49, 574, 67, 458) and (875, 4748, 9, 556) TABLE XI: Total Otmal Fuzzy Transortaton Cost and Tme Lnear and non-lnear functons Otmal soluton Total Otmal Fuzzy Transortaton Cost and Tme L x Rx,- x x Rx e L x max,- x L x R x e x =, x = 4, x = 55, x =, x = 474 x = 97, x = 444, x = 54, x = 97, x = 474 x =, x = 448, x = 56, x = 98, x x = 48 x L x e x = 7, x = 444, x Rx max, x = 6, x =, x = 5 L x x e x =, x = 49, max, x R x x = 558, x =, x = 479 (4, 558, 599, 48) and (789, 464, 897, 5) (6, 56, 6, 4) and (75, 454, 89, 5) (5, 589, 674, 487) and (958, 4855, 946, 59) (6, 6469, 788, 4577) and (58, 5647, 58, 754) (49, 574, 67, 458) and (875, 4748, 9, 556) Note: denotes total otmal fuzzy transortaton tme 6 Concluson In ths aer, we have resent a new method for rankng generalzed LR fuzzy numbers to varous lnear and non-lnear functons of generalzed LR fuzzy numbers from ts -cut based on area, mode and sread The roosed rankng method can effcently rank varous LR fuzzy numbers, ther mages and crs numbers whch are consder to be a secal case of fuzzy numbers and can overcome the drawbacks of the exstng rankng methods We also have resented a fuzzy mult obectve

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