Keywords - Fuzzy sets, Fuzzy numbers, Fuzzy transportation problem, Fuzzy ranking, Fuzzy arithmetic

Transcription

1 M. Shaugasudar, K. Gaesa / Iteratoal Joural of Egeerg Research ad Applcatos (IJERA) ISSN: Vol. 3, Issue 1, Jauary -February 2013, pp A Novel Approach for the fuzzy optal soluto of Fuzzy Trasportato Proble M. Shaugasudar *, K. Gaesa ** *Departet of Matheatcs, Faculty of Scece Huates, SRM Uversty, Kattakulathur, Chea , Ida. ** Departet of Matheatcs, Faculty of Egeerg ad Techology, SRM Uversty, Kattakulathur, Chea , Ida ABSTRACT I ths paper we propose a ew ethod for the Fuzzy optal soluto to the trasportato proble wth Fuzzy paraeters. We develop Fuzzy verso of Vogels ad MODI algorths for fdg Fuzzy basc feasble ad fuzzy optal soluto of fuzzy trasportato probles wthout covertg the to classcal trasportato probles. The proposed ethod s easy to uderstad ad to apply for fdg Fuzzy optal soluto of Fuzzy trasportato proble occurrg real world stuato. To llustrate the proposed ethod, uercal exaples are provded ad the results obtaed are dscussed. Keywords - Fuzzy sets, Fuzzy ubers, Fuzzy trasportato proble, Fuzzy rakg, Fuzzy arthetc I. INTRODUCTION The trasportato proble s a specal type of lear prograg proble whch deals wth the dstrbuto of sgle product (raw or fshed) fro varous sources of supply to varous destato of dead such a way that the total trasportato cost s zed. There are effectve algorths for solvg the trasportato probles whe all the decso paraeters,. e the supply avalable at each source, the dead requred at each destato as well as the ut trasportato costs are gve a precse way. But real lfe, there are ay dverse stuatos due to ucertaty oe or ore decso paraeters ad hece they ay ot be expressed a precse way. Ths s due to easureet accuracy, lack of evdece, coputatoal errors, hgh forato cost, whether codtos etc. Hece we caot apply the tradtoal classcal ethods to solve the trasportato probles successfully. Therefore the use of Fuzzy trasportato probles s ore approprate to odel ad solve the real world probles. A fuzzy trasportato proble s a trasportato proble whch the trasportato costs, supply ad dead are fuzzy quattes. Bella ad Zadeh [3] proposed the cocept of decso akg Fuzzy evroet. After ths poeerg work, several authors such as Shag-Ta Lu ad Chag Kao[23], Chaas et al [4], Pada et.al [19], Lu ad Kao [17] etc proposed dfferet ethods for the soluto of Fuzzy trasportato probles. Chaas ad Kuchta [4] proposed the cocept of the optal soluto for the Trasportato wth Fuzzy coeffcet expressed as Fuzzy ubers. Chaas, Kolodzeckzy, Macha[5] preseted a Fuzzy lear prograg odel for solvg Trasportato proble. Lu ad Kao [17] descrbed a ethod to solve a Fuzzy Trasportato proble based o exteso prcple. L troduced a geetc algorth to solve Trasportato wth Fuzzy obectve fuctos. Nagoor Ga ad Abdul Razak [12] obtaed a fuzzy soluto for a two stage cost zg fuzzy trasportato proble whch supples ad deads are trapezodal fuzzy ubers. A. Nagoor Ga, Edward Sauel ad Auradha [9] used Arshakha s Algorth to solve a Fuzzy Trasportato proble. Pada ad Nataraa [19] proposed a Fuzzy zero pot ethod for fdg a Fuzzy optal soluto for Fuzzy trasportato proble where all paraeters are trapezodal fuzzy ubers. I geeral, ost of the exstg techques provde oly crsp solutos for the fuzzy trasportato proble. I ths paper we propose a sple ethod, for the soluto of fuzzy trasportato probles wthout covertg the to classcal trasportato probles. The rest of ths paper s orgazed as: I secto II, we recall the basc cocepts of Fuzzy ubers ad related results. I secto III, we defe Fuzzy trasportato proble ad prove the related theores. I secto IV, we propose Fuzzy Verso of Vogels Approxato Algorth (FVAM) ad fuzzy verso of MODI ethod (FMODI). I secto V, uercal exaples are provded to llustrate the ethods proposed ths paper for the fuzzy optal soluto of fuzzy trasportato probles wthout covertg the to classcal trasportato probles ad the results obtaed are dscussed P a g e

2 M. Shaugasudar, K. Gaesa / Iteratoal Joural of Egeerg Research ad Applcatos (IJERA) ISSN: Vol. 3, Issue 1, Jauary -February 2013, pp II. PRELIMINARIES The a of ths secto s to preset soe otatos, otos ad results whch are of useful our further cosderato. 2.1 Fuzzy ubers A fuzzy set à defed o the set of real ubers R s sad to be a fuzzy uber f ts ebershp fucto : R 0,1 A has the followg characterstcs () às oral. It eas that there exsts a xr such that A x 1 () A s covex. It eas that for every x 1, x2 R, A A A x 1 x x, x, 0,1 () A s upper se-cotuous. (v) supp ( A ) s bouded R Tragular fuzzy ubers A fuzzy uber A R s sad to be a tragular fuzzy uber f ts ebershp fucto A : R 0,1 has the followg characterstcs. x a1 a x a 1 2 a a x a2 x A a x 3 a x a 2 3 a a otherwse We deote ths tragular fuzzy uber by A a,a,a. We use F(R) to deote the set of all tragular fuzzy ubers. Also f a2 represets the odal value or dpot, a2 a1 represets the left spread ad a3 a2 represets the rght spread of the tragular fuzzy uber A a,a,a, the the tragular fuzzy uber A ca be represeted by the trplet That s A a,a,a,,. A,, Rakg of tragular fuzzy ubers Several approaches for the rakg of fuzzy ubers have bee proposed the lterature. A effcet approach for coparg the fuzzy ubers s by the use of a rakg fucto based o ther graded eas. That s, for every A a,a,a F R, the rakg fucto : F(R) R by graded ea s defed as a1 4a2 a3 (A) 6 For ay two fuzzy tragular ubers A a,a,a B b,b,b F(R), we ad have the followg coparso (). A B f ad oly f (A) (B). (). A B f ad oly f (A) (B). (). A B f ad oly f (A) (B). (v). A B 0 f ad oly f (A) (B) 0. A tragular fuzzy uber A a,a,a FR s sad to be postve f (A) 0 ad deoted by A 0. Also f (A) 0, the A 0 ad f (A) 0, the A 0. I f (A ) ( B ), the the tragular ubers A ad B are sad to be equvalet ad s deoted by A B Arthetc operatos o tragular fuzzy ubers Mg Ma et al. [9] have proposed a ew fuzzy arthetc based upo both locato dex ad fuzzess dex fuctos. The locato dex uber s take the ordary arthetc, whereas the fuzzess dex fuctos are cosdered to follow the lattce rule whch s least upper boud the lattce L. That s for a, bl we defe a b ax a, b ad a b a,b. For arbtrary tragular fuzzy ubers A (a,a,a ),,, B (b,b,b ) 1 1 1,, ad 2 2 2,,,, the arthetc operatos o the tragular fuzzy ubers are defed by A B,,,, (, ax{, }, ax{, } (,, ) P a g e

3 Sources M. Shaugasudar, K. Gaesa / Iteratoal Joural of Egeerg Research ad Applcatos (IJERA) ISSN: Vol. 3, Issue 1, Jauary -February 2013, pp I partcular for ay two tragular fuzzy ubers where c s the fuzzy ut trasportato cost fro th A (a,a,a ),, 1 1 1, B (b,b,b ) source to the th destato. The obectve s to,, 2 2 2, we defe: ze the total fuzzy trasportato cost, ths paper the fuzzy trasportato proble s solved by (). Addto fuzzy verso of Vogel s ad MODI ethod. Ths A B a,a,a b,b,b fuzzy trasportato proble s explctly represeted by the followg fuzzy trasportato,,,, table.,ax,,ax, Table 3.1 Fuzzy trasportato table (). Subtracto A B a,a,a b,b,b,,,, ,ax 1 2, 1 2,ax, 1 2 ().Multplcato AB a,a,a b,b,b,,,, ,ax 1 2, 1 2,ax, 1 2 (v). Dvso A a,a,a 1,ax(, ),ax(, ), B b,b,b 2 f, B b, b, b s o zero fuzzy uber. III. MAIN RESULTS Cosder a fuzzy trasportato wth sources ad destatos wth tragular fuzzy ubers. Let a 0 be the fuzzy avalablty at source ad b, (b 0) be the fuzzy requreet at destatos. Let c (c 0) be the ut fuzzy trasportato cost fro source to destato. Let x deote the uber of fuzzy uts to be trasfored fro source to destato. Now the proble s to fd a feasble way of trasportg the avalable aout at each source to satsfy the dead at each destato so that the total trasportato cost s zed Matheatcal forulato of fuzzy trasportato proble The atheatcal odel of fuzzy trasportato proble s as follows Mze Z c x subect to x a, 1,2,3,..., x b, 1,2,3,..., (3.1) a b, 1,2,3,...,; 1,2,3,..., ad x 0 for all ad Basc Theores Theore 3.1. (Exstece of a fuzzy feasble soluto) The ecessary ad suffcet codto for the exstece of a fuzzy feasble soluto to the fuzzy trasportato proble (3.1) s, a b (3.2) 1 1 (Total supply Ttotal dead).. Proof: (Necessary codto) Let there exst a fuzzy feasble soluto to the fuzzy trasportato proble Mze Z c x 1 1 subect to x a, 1,2,3,...,. 1 1 x b, 1,2,3,...,. ad x 0 for all ad. 1 (3.3) Fro x a,( 1,2,3,...,), we have x a Also fro x b, ( 1,2,3,...,), we have Destato 1 2 Supply 1 c 11 c 12 c 1 a 1 2 c 21 c 22. c 2 a 2 Dead b 1 b 2 c 1 c 2. c a x b (3.5) Fro equatos (3.4) ad (3.5), we have a b 1 1 b 1418 P a g e

4 M. Shaugasudar, K. Gaesa / Iteratoal Joural of Egeerg Research ad Applcatos (IJERA) ISSN: Vol. 3, Issue 1, Jauary -February 2013, pp (Suffcet codto) Sce all a ad b postve. Therefore equato (3.2) yelds a feasble soluto. are postve, x ust be all Theore 3.2. The deso of the bass of a fuzzy trasportato are (+-1)(+-1).That a fuzzy trasportato proble has oly (+-1) depedet structural costrats ad ts basc feasble soluto has oly (+-1) postve copoets. wll be atleast 2 ad 2 respectvely. Suppose f the total uber of basc varables s D, the obvously D 2, D 2. Case (). If <, the + < + + < 2 + < 2 < D + < D. Case(). If, the D D. Proof: Cosder a fuzzy trasportato proble wth sources ad destatos, Mze Z c x subect to x a, 1, 2,3,...,. x b, 1, 2,3,...,. ad x 0 for all ad. Let us assue that a fuzzy trasportato has rows (supply costrat equatos) ad colus (dead costrat equatos). Therefore there are totally (+) costrat equatos. Ths s due to the codto that a b 1 1 whch s the last requreet costrat. Therefore oe of (+) costrats ca always be derved fro the reag (+-1). Thus there exsts oly (+-1) depedet costrats ad ts basc feasble soluto has oly (+-1) postve copoets. Theore 3.3 The values of the fuzzy basc feasble soluto are all dffereces betwee the partal su of a ad the partal su of b. That s 1 1 x r a s b, where r,s are ether 1 1,1,1 or 0,0 0 = 0,. Theore 3.4 The fuzzy trasportato proble has a tragular bass. Proof: We ote that every equato has a basc varable, otherwse, the equato caot be satsfed for a 0, b 0. Suppose every row ad colu equatos has atleast two basc varables, sce there are rows ad colus, the total uber of basc varables row equatos ad colu equatos Case(). If =, the + = + + = + 2 = + + = 2 + = 2 < D + < D. Thus all cases D +. But the uber of basc varables fuzzy trasportato proble s (+-1) whch s a cotradcto. Therefore there s atleast oe equato, ether row or colu havg oly oe basc varable. Let the r th equato have oly oe basc varable. Let x rs be the oly basc varable the r th row ad s th colu,the x rs a r. Elate r th row fro the syste of equato ad substtute x rs a r s th colu equato ad replace b s by b s b s a r. After elatg the r th row, the syste has (+- 2) learly depedet costrats. Hece the uber of basc varables s (+-2). Repeat the process ad coclude that the reduced syste of equatos there s a equato whch has oly oe basc varable. But f ths s s th colu equato, the t wll have two basc varables. Ths cocludes that our orgal syste of equatos, there s a equato whch has atleast two basc varables. Cotue the process to get the theore. IV. PROPOSED ALGORITHMS 4.1 Fuzzy Verso of Vogels Approxato Algorth (FVAM) Step 1: Fro the Fuzzy Trasportato table, detere the pealty for each row or colu. The pealtes are calculated for each row colu by substtutg the lowest cost eleet that row or colu fro the ext cost eleet the sae row or colu. Wrte dow the pealtes below ad asde of the rows ad colus respectvely of the table P a g e

5 Dead (100,150,200) (100,150,200) (200,300,400) M. Shaugasudar, K. Gaesa / Iteratoal Joural of Egeerg Research ad Applcatos (IJERA) ISSN: Vol. 3, Issue 1, Jauary -February 2013, pp Step 2: Idetfy the colu or row wth largest fuzzy pealty. I case of te, break the te arbtrarly. Select a cell wth u fuzzy cost the selected colu (or row, as the case ay be) ad assg the axu uts possble by cosderg the dead ad supply posto correspodg to the selected cell. Step 3: Delete the colu/row for whch the supply ad dead requreets are et. Step 4: Cotue steps 1 to 3 for the resultg fuzzy trasportato table utl the supply ad dead of all sources ad destatos have bee et. 4.2 Fuzzy Verso of MODI Algorth (FMODI) Step 1: Gve a tal fuzzy basc feasble soluto of a fuzzy trasportato proble the for of allocated ad uallocated cells of fuzzy trasportato table. Assg the auxlary varables u, =1,2,3,, ad v, =1,2,3,, for rows ad colus respectvely. Copute the values of u ad v usg the relatoshp c u v for all, for all occuped cells. Assue ether u or v as zero arbtrarly for the allocatos row/colu. Step 2: For each uoccuped cell (, ), copute the fuzzy opportuty cost usg c ( u v ). Step 3: () If all 0, the the curret soluto uder the test s optal () If at least oe 0, the the curret soluto uder the test s ot optal ad proceeds to the ext step. Step 4: Select a uoccuped cell (, ) wth ost egatve opportuty cost aog all uoccuped cells. Step 6: O the closed path, detfy the corers wth θ. Select the sallest allocato aog the corers wth θ whch dcate the uber of uts that ca be shfted to soe other uoccuped cells. Add ths quatty to those corers arked wth +θ ad subtract ths quatty to those corers arked wth θ o the closed path ad cheek whether the uber of oegatve allocatos s (+-1) ad repeat step 1 to step 7, tll we reach 0 for all ad. 4.3 Ubalaced Fuzzy Trasportato Proble. Suppose the fuzzy trasportato proble s ubalaced oe,. e, f a b, the 1 1 covert ths to a balaced oe by troducg a duy source or duy destato wth zero fuzzy ut trasportato costs. Solve the resultg balaced fuzzy trasportato proble usg above sad algorths. V. NUMERICAL EXAMPLES The followg two uercal exaples are take fro the paper Splex type algorth for solvg fuzzy trasportato proble by Edward Sauel et.al [9]. Exaple 5.1 A copay has two factores O 1, O 2 ad two retal stores D 1, D 2. The producto quattes per oth at O 1, O 2 are (150, 201, 246) ad (50, 99, 154) tos respectvely. The deads per oth for D 1 ad D 2 are (100,150,200) ad (100,150,200) tos respectvely. The trasportato cost per to c, =1, 2; =1, 2 are c 11 =(15,19,29), c 12 =(22,31,34), c 21 =(8,10,12) ad c 22 =(30,39,54). Soluto: Trasportato table of the gve fuzzy trasportato proble s Table 5.1 Fuzzy Trasportato Proble D 1 D 2 Supply O 1 (15,19,29) (22,31,34) (150,201,246) O 2 (8,10,12) (30,39,54) (50,99,154) Step 5: Draw a closed path volvg horzotal ad vertcal les for the uoccuped cells startg ad edg at the cell obtaed step 4 ad havg ts other corers at soe allocated cells. Assg +θ ad θ alterately startg wth +θ for the selected uoccuped cells. To apply the proposed algorths ad the fuzzy arthetc, let us express all the tragular fuzzy ubers based upo both locato dex ad fuzzess dex fuctos. That s the for of,, gve table P a g e

6 Dead (400,448,508) (300,351,396) Dead (150, 50-50r, 50-50r) (150, 50-50r, 50-50r) (10, 2-2r, 2-2r) (39, 9-9r, 15-15r) (19, 4-4r, 10-10r) (31, 9-9r, 3-3r) M. Shaugasudar, K. Gaesa / Iteratoal Joural of Egeerg Research ad Applcatos (IJERA) ISSN: Vol. 3, Issue 1, Jauary -February 2013, pp Table 5.2 Balaced fuzzy trasportato proble whch all the tragular ubers are of the for,, D 1 D 2 Supply The correspodg fuzzy optal trasportato cost s gve by Mze Z c x 1 1 where 0 r 1 ca be sutably chose by the decso aker. 6609, 50 50r, 55 55r, O 1 O 2 By applyg fuzzy verso of Vogel s Approxato ethod (FVAM)), the tal fuzzy basc feasble soluto s gve table 5.3. Table 5.3 Ital fuzzy basc feasble soluto O 1 O 2 The correspodg tal fuzzy trasportato cost s gve by IFTC 19, 4 4r, 10 10r51, 50 50r, 55 55r 31, 9 9r, 33r150, 50 50r, 50 50r 10, 2 2r, 2 2r99, 49 49r, 55 55r 6609, 50 50r, 55 55r. By applyg fuzzy verso of MODI ethod (FMODI), t ca be see that the curret tal fuzzy basc feasble soluto s optal. Hece the fuzzy optal soluto ters of locato dex ad fuzzess dex s gve by x x , 50 50r, 55 55r ; 150, 50 50r, 50 50r ; (201, 51-51r, 45-45r) (99, 49-49r, 55-55r) D 1 D 2 (19, 4-4r, 10-10r) (31, 9-9r, 3-3r) (51, 50-50r, 55-55r) (150, 50-50r, 50-50r) (10, 2-2r, 2-2r) (99, 49-49r, 55-55r) (39, 9-9r, 15-15r) x 99, 49 49r, 55 55r, w here 0 r 1. Case(). Whe r=0, the fuzzy optal trasportato cost ters of the for (, α, β) s (6609, 50, 55). The correspodg fuzzy optal trasportato of the for a 1,a 2,a 3 s (6559, 6609, 6664) ad ts defuzzyfed trasportato cost s Case(). Whe r = 0.5, the fuzzy optal trasportato cost ters of the for (, α, β) s (6609, 25, 27.5). The correspodg fuzzy optal trasportato cost of the for a 1,a 2,a 3 s (6584, 6609, ) ad ts defuzzyfed trasportato cost s Case(). Whe r=1, the fuzzy optal trasportato cost ters of the for (, α, β) s (6609,0,0). The correspodg fuzzy optal trasportato cost of the for a 1,a 2,a 3 s (0, 6609, 0) ad ts defuzzyfed trasportato cost s Optu fuzzy trasportato cost s (1150, 6609, 12882). Defuzzfed fuzzy trasportato cost s Exaple 5.2 Fd a fuzzy optu soluto for the ubalaced fuzzy trasportato proble gve below. Table 5.4. Ubalaced Fuzzy Trasportato Proble B1 B2 Supply A1 (1,3,5) (3,5,13) (300,399,504) A2 (2,3,10) (3,4,11) (250,301,346) A3 (5,6,13) (1,3,5) (300,399,504) Soluto: Itroduce a duy destato B3 wth (150,300,450) as ts fuzzy dead ad as ts fuzzy trasportato costs per ut. Hece the balaced fuzzy trasportato wth a duy destato becoes 1421 P a g e

7 (448,48-48r,60-60r) (351,51-51r,45-45r) (300, r, r) (6,1-r,7-7r) (3,2-2r,2-2r) (399,99-99r, r) (3,1-r,7-7r) (4,1-r,7-7r) (301,51-51r,45-45r) (3,2-2r,2-2r) (5,2-2r,8-8r) (399,99-99r, r) (6,1-r,7-7r) (48,99-99r, r) (3,2-2r,2-2r) (351,51-51r,45-45r) (3,1-r,7-7r) (1, r,150-50r) (4,1-r,7-7r) Dead (400,448,508) (300,351,396) (150,300,450) (3,2-2r,2-2r) (399,99-99r, r) (5,2-2r,8-8r) M. Shaugasudar, K. Gaesa / Iteratoal Joural of Egeerg Research ad Applcatos (IJERA) ISSN: Vol. 3, Issue 1, Jauary -February 2013, pp Table 5.5 Balaced Fuzzy Trasportato Proble B1 B2 B3 Supply A1 (1,3,5) (3,5,13) (300,399,504) A2 (2,3,10) (3,4,11) (250,301,346) A3 (5,6,13) (1,3,5) (300,399,504) By applyg fuzzy verso of Vogels Approxato ethod, the tal fuzzy basc feasble soluto s gve by Table 5.7 Ital fuzzy basc feasble soluto B1 B2 B3 A1 Now the balaced fuzzy trasportato proble whch all the tragular ubers are of the for,, s gve table 5.6. Table 5.6 Balaced Fuzzy Trasportato Proble whch all the tragular ubers are of the for,, A2 B 1 B 2 B 3 Supply A 1 A3 A 2 A 3 Dead By applyg fuzzy verso of MODI ethod, t ca be see that the curret tal fuzzy basc feasble soluto s optal. The fuzzy optal soluto ters of locato dex ad fuzzess dex s gve by x 399, 99 99r, r ; x x x , r, r ; 252, r, r ; 351, 51 51r, 45 45r ad x 48,99 99r, r, where 0 r 1. The correspodg fuzzy optal trasportato cost s gve by Mze Z c x , r, r where 0 r 1 ca be sutably chose by the decso aker. Case(). Whe r = 0, the fuzzy optal trasportato cost ters of the for (, α, β) s (2397, 150,150). The correspodg fuzzy optal 1422 P a g e

8 M. Shaugasudar, K. Gaesa / Iteratoal Joural of Egeerg Research ad Applcatos (IJERA) ISSN: Vol. 3, Issue 1, Jauary -February 2013, pp trasportato cost of the for A (a,a,a ) s (2217, 2397, 2547). Its defuzzyfed trasportato cost s Case(). Whe r = 0.5, the fuzzy optal trasportato cost ters of the for (, α, β) s (2397, 75, 75). The correspodg fuzzy optal trasportato cost of the for A (a,a,a ) s (2322, 2397, 2472). Its defuzzyfed trasportato cost s Case(). Whe r =1, the fuzzy optal trasportato cost ters of the for (, α, β) s (2397, 0, 0). The correspodg fuzzy optal trasportato cost of the for A (a,a,a ) s (0, 2397, 0). The optu fuzzy trasportato cost = (-548, 2397, 7120). Defuzzed fuzzy trasportato cost s VI. CONCLUSION I ths paper, the trasportato costs are cosdered as precse ubers descrbed by tragular fuzzy ubers whch are ore realstc ad geeral ature. We proposed a fuzzy verso of VAM ad MODI algorths to solve fuzzy trasportato proble wthout covertg the to classcal trasportato probles. Two uercal exaples are solved usg the proposed algorths ad obtaed results are better tha the exstg results. AKNOWLEDGEMENTS The authors are grateful to the aoyous referees ad the edtors for ther costructve coets ad suggestos. REFERENCES [1]. Aarpreet Kau, At Kuar, A ew approach for solvg fuzzy trasportato probles usg geeralzed trapezodal fuzzy ubers, Appled Soft Coputg, 12 (2012) [2]. H. Arsha ad A. B. Kah, A splex type algorth for geeral trasportato probles: A alteratve to steppgstoe, Joural of Operatoal Research Socety, 40 (1989), [3]. R. E. Bella ad L. A. Zadeh, Decso akg a fuzzy evroet, Maageet scece, 17(1970), [4]. S. Chaas, D. Kuchta, A cocept of optal soluto of the trasportato wth Fuzzy cost co effcet, Fuzzy sets ad systes, 82(9) (1996), [5]. S. Chaas, W. Kolodzeczyk ad A. Macha, A fuzzy approach to the trasportato proble, Fuzzy Sets ad Systes, 13(1984), [6]. A. Chares, W. W. Cooper, The steppg stoe ethod for explag lear prograg calculato trasportato, Maageet scece, 1 (1954), [7]. G. B. Datzg, M. N. Thapa, Sprger: L.P:2: Theory ad Extesos, Prceto uversty Press New Jersey, [8]. D. Dubos ad H. Prade, Fuzzy Sets ad Systes, Theory ad applcatos, Acadec Press, New York,1980. [9]. A. Edward Sauel ad A. Nagoor Ga, Splex type algorth for solvg fuzzy trasportato proble, Tasu oxford oural of forato ad atheatcal sceces, 27(1) (2011), [10]. Fag. S. C, Hu.C. F, Wag. H. F ad Wu.S.Y, Lear Prograg wth fuzzy coeffcets costrats, Coputers ad Matheatcs wth applcatos, 37 (1999), [11]. Fegad. M. R, Jadhav. V. A ad Muley. A. A, Fdg a optal soluto of trasportato proble usg terval ad tragular ebershp fuctos, Europea Joural of Scetfc Research, 60 (3) (2011), [12]. A. Nagoor Ga, K. A. Razak, Two stage fuzzy trasportato proble, Joural of Physcal Sceces, 10 (2006), [13]. L. S. Gass, O solvg the trasportato proble, Joural of operatoal research Socety, 41 (1990), [14]. E. Kaucher, Iterval aalyss exteded terval space IR, Coput. Suppl, 2(1980), [15]. L. J. Kraewsk, L. P. Rtza ad M. K. Malhotra, Operatos aageet process ad value chas, Upper Saddle Rver, NJ: Pearso / Pretce Hall, [16]. Lous.T.S. ad Wag.M.J, Rakg fuzzy ubers wth tegral value, Fuzzy sets ad systes, 50 (3) (1992), [17]. S. T. Lu, C. Kao, Solvg Fuzzy trasportato proble based o exteso prcple, Europea Joural of Operatos Research, 153 (2004), [18]. T.Nrala, D.Datta, H. S. Kushwaha ad K. Gaesa, Iverse Iterval Matrx: A New Approach, Appled Matheatcal Sceces, 5(13) (2011) [19]. Pada. P ad Nagaraa. G, A ew algorth for fdg a fuzzy optal soluto for fuzzy trasportato proble, Appled Matheatcs Sceces, 4 (2) (2010),79-90 [20]. G. Raesh ad K. Gaesa, Iterval Lear prograg wth geeralzed terval arthetc, Iteratoal Joural of Scetfc & Egeerg Research, 2(11) (2011), P a g e

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