A NEW APPROACH FOR SOLVING LINEAR FUZZY FRACTIONAL TRANSPORTATION PROBLEM

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1 Internatonal Journal of Cvl Engneerng and Technology (IJCIET) Volue 8, Issue 8, August 217, pp , Artcle ID: IJCIET_8_8_12 Avalable onlne at ISSN Prnt: and ISSN Onlne: IAEME Publcaton Scopus Indeed A NEW APPROACH FOR SOLVING LINEAR FUZZY FRACTIONAL TRANSPORTATION PROBLEM S. Mohanaselv, K. Ganesan Departent of Matheatcs, SRM Unversty, Kattankulathur, Tal Nadu, Inda ABSTRACT Ths paper deals wth a soluton procedure for solvng lnear fuzzy fractonal transportaton proble (LFFTP) whch s a specal type of lnear fuzzy fractonal prograng proble (LFFPP). Fuzzy verson of proved Vogel s approaton ethod and Fuzzy verson of Mod ethod are used to obtan the fuzzy optal soluton to the gven LFFTP by wthout reforulatng the orgnal proble nto an equvalent crsp proble. Also a nuercal eaple s dscussed for supportng the soluton theory developed n ths paper. Key words: Trapezodal fuzzy nuber; Fuzzy rankng; Fuzzy fractonal prograng proble; Fuzzy fractonal transportaton proble. Cte ths Artcle: S. Mohanaselv, K. Ganesan, A New Approach for Solvng Lnear Fuzzy Fractonal Transportaton Proble. Internatonal Journal of Cvl Engneerng and Technology, 8(8), 217, pp INTRODUCTION In varous applcatons such as producton plannng, fnancal plannng etc., the decson aker ay be nterested n optzng an objectve functon havng rato of lnear functon. These types of probles can be handled by usng lnear fractonal prograng proble (LFPP) technques. In lterature LFPP was frst developed and studed etensvely by Matros.B et al (196) [8]. Later on several authors such as Bajalnov, E. B.(23), Charnes and Cooper (1962), Odor, A. O.(212), Pandey P and Punnen A P.(27)[ 1, 3,14,15 ] proposed dfferent approaches for solvng LFPP. Transportaton technque n LFPP was frst ntroduced by Swarup.K (1966) [17]. Also Gupta V et al (1993) [5] studed about parado n lnear fractonal transportaton probles (LFTP) wth ed constrants. Josh D et al (211) [6] nvestgated the transportaton proble wth fractonal objectve functon when the deand and supply quanttes are varyng. Moanta, D.(27) [1] proposed a sple ethod technque for solvng three densonal transportaton proble whose objectve functon s the rato of two postve lnear functons. Nuran Guzel (212)[13] presented an Taylor seres approaton and nterval arthetc based procedure for the soluton of nterval fractonal transportaton proble. Bheean Radhakrshnan (214) [2] proposed a copensatory edtor@aee.co

2 S. Mohanaselv, K. Ganesan approach to LFFTP by usng Werner s fuzzy or operator. Narayanaoorthy.S et al (215) [12] presented a soluton procedure to solve LFFTP by usng dual sple ethod. Kalyan. S et al (216) [7] converted the gven LFFTP nto two fuzzy transportaton proble and presented the soluton. By solvng par of lnear progras at a specfc α-cut Shang-Ta Lu (216) [16] dscussed a soluton procedure for LFFTP. V.A.Jadhav et al (216) [5] presented a copensatory approach for LFFTP by solvng each of the fractonal objectve functon ndependently. In ths paper a soluton procedure for solvng fuzzy fractonal transportaton proble s dscussed wthout reforulatng the orgnal proble nto classcal type. In secton 2 prelnares necessary for supportng our work s presented. In secton 3 atheatcal forulaton of LFFTP s defned and an algorth for solvng t s presented. In secton 4 a nuercal eaple s dscussed for supportng the above sad algorth. In secton 5 concluson of ths paper s presented. 2. PRELIMANARIES In ths secton, we ehbt soe eleental defntons, whch are used all through ths paper. Defnton 1 A fuzzy set A defned on the set of real nubers R s sad to be a fuzzy nuber f ts ebershp functon A: R [,1] has the followng characterstcs: A s conve A s noral A s pecewse contnuous. Defnton 2 A fuzzy nuber A on R s sad to be a trapezodal fuzzy nuber (TrFN) f ts ebershp functon A: R [,1] has the followng characterstcs: a1, for a1 a2 a2 a 1 1, for a2 a3 a a3 a4 a4 a3, elsewhere 4 A =, for We denote the trapezodal fuzzy nuber by A = ( a1, a2, a3, a 4). Defnton 3 A fuzzy nuber can also be epressed as a par A A ( r ), A ( r ) A ( r ) for r 1 whch satsfes the followng requreents: A ( r ) s a bounded onotonc ncreasng left contnuous functon. of functons A ( r ) and edtor@aee.co

3 A New Approach for Solvng Lnear Fuzzy Fractonal Transportaton Proble A ( r ) s a bounded onotonc decreasng left contnuous functon. A ( r) A ( r), r 1. Defnton 4 The trapezodal fuzzy nuber A = ( a1, a2, a3, a4) can be hence represented by A A ( r ), A ( r ) a a r a, a a a r. Defnton The dpont of a fuzzy nuber A A ( r ), A ( r ), where r 1 s defned by A A (1)+A (1) Arthetc operaton on Fuzzy Nubers: Mng Ma et al [16] has proposed a new fuzzy arthetc operator based on paraetrc for of fuzzy nubers. The followng arthetc operaton on fuzzy nubers based on the paraetrc A, A ( r ), A ( r ) s used n ths paper. trplet A A A For any two fuzzy nubers A A ( r ), A ( r ) and B B ( r ), B ( r ),,,, are defned as: operatons A* B, a A ( r ), B ( r ), a A ( r ), B ( r ) A B A B A B the arthetc 2.2. Rankng of Fuzzy Nubers In decson akng probles rankng of fuzzy nubers s an essental part to ake a best decson. In ths paper the agntude of a fuzzy nuber s calculated as follows to rank the fuzzy nubers. Defnton 1 Mag(A) 1 A r r ( )+ A ( ) 2 A r dr Two fuzzy nubers A A ( r ), A ( r ) and B B ( r ), B ( r ) are sad to be equvalent f and only f Mag(A) Mag(B). That s A B f and only f Mag(A) Mag(B). And they are sad to be equal that s A B f and only f, A B A ( r ) = B ( r) and A ( r) B ( r). A B A B 3. MATHEMATICAL FORMULATION OF LFFTP The lnear fuzzy fractonal transportaton proble (LFFTP) s a part of logstcs and supply chan anageent probles for provng the proft of the organzaton whle consderng the other factors whch affects the proft. Let there be sources fro whch goods have to be C c be the fuzzy cost atr where c s the cost spend suppled to n destnatons. Let n n transportng the goods fro a source to destnaton j. Let P p n be the fuzzy proft atr where p s the fuzzy proft ganed f a unt of good s transported fro a source to edtor@aee.co

4 S. Mohanaselv, K. Ganesan destnaton j. Let be the nuber of unknown quanttes of goods to be transported fro the source to the destnaton j. Let c and p be the gven fed fuzzy cost and fuzzy proft. Then the atheatcal forulaton of LFFTP s gven by: Mn Q( ) n c c =1 j=1 n p P( ) p p =1 j=1 n j =1 =1 a j C( ) c,, j b a b j Snce we are consderng the fracton of lnear fuzzy functon t ay be possble that for soe the denonator ay be equal to zero. To avod that stuaton we assue that always n p p =1 j=1 Defnton 1 A set of non-negatve allocatons whch satsfes the row and the colun restrctons (n the sense equvalent) s known as fuzzy feasble soluton to (1). Defnton 2 A fuzzy feasble soluton to (1) s sad to be a fuzzy basc feasble soluton f the nuber of postve allocatons ade are n 1. If the nuber of allocatons n a fuzzy basc feasble soluton are less than n 1, then t s called as a fuzzy degenerate basc feasble soluton. Defnton 3 A fuzzy basc feasble soluton to (1) s sad to be fuzzy optal soluton f t nzes the objectve functon. 4. PROPOSED METHOD TO FIND THE FUZZY OPTIMAL SOLUTION TO THE GIVEN LFFTP The soluton procedure to obtan the fuzzy optal soluton to the gven LFFTP nvolves two steps. In the frst step the ntal fuzzy basc feasble soluton s obtaned by usng fuzzy verson of proved Vogel s approaton ethod and n the second step the ntal basc feasble soluton s proved by usng fuzzy verson of Mod ethod to obtan the fuzzy optal soluton. Mustafa Svr et al (211)[11] has proposed a proved transportaton algorth for solvng LFTP. We have etended ths algorth for LFTP wth fuzzy nubers Fuzzy verson of proved vogel s approaton ethod () Calculate the dfference between two lowest fuzzy costs c n all rows and coluns. Slarly calculate the dfference between two lowest fuzzy profts p n all rows and coluns edtor@aee.co

5 A New Approach for Solvng Lnear Fuzzy Fractonal Transportaton Proble () Calculate the su of the dfferences of c and p for each row and colun. () Identfy the row or colun that has the greatest su of the dfferences copared wth all other su of the dfferences. (v) Suppose that th row has the greatest su of dfference. Deterne the row for whch s nu and ake au allotent n t. (v) Repeat the process untl all goods n the sources are transported Optalty condton to a fuzzy Transportaton Proble After deternng the ntal fuzzy basc feasble soluton by the proposed algorth, we have to test the current ntal fuzzy basc feasble soluton for optalty by usng fuzzy verson of odfed dstrbuton ethod. Let u 1,u 2,...,u and u 1,u 2,...,u be the ultplers for fuzzy cost and fuzzy proft to the constrants and let v 1,v 2,...,v n and v 1,v 2,...,v n be the ultplers for fuzzy cost and fuzzy proft to the n constrants. We can calculate u,u and v,v for the allocated cells usng the relaton c u v and p u v by settng a ultpler to zero whch s assocated wth the row or colun of the transportaton table that contans the au nuber of allocated cells. If c u v and p u v, then crteron for optalty s gven by where Q( ), j for the unallocated cells of fuzzy fractonal transportaton table. 5. NUMERICAL EXAMPLE Consder a nuercal eaple dscussed by [12] (, 2, 4, 6) (1, 2, 6, 7) (1, 4, 5, 6) (3, 4, 5, 8) Mn (, 1, 3, 4) (2, 3, 5, 6) (1, 3, 5, 7) (2, 6, 7, 9) Representng the gven proble n paraetrc trplet and applyng the algorth eplaned n the prevous secton we have Table 1 Fuzzy fractonal transportaton proble (3,3-2 r,3-2 r) (2,2- r,2- r) (4.5,3.5-3 r,1.5- r) (4,3-2 r,3-2 r) (4,3- r,3- r) (4,2- r,2- r) 6 (4.5,1.5- r,3.5-3 r) (6.5,4.5-4 r,2.5-2 r) The dfference between the lowest fuzzy cost cells for both c and p n 1 st row s gven by (1,3- r,3- r )and(2,2- r,2- r ). Then ther su of dfferences s gven by (3,3- r,3- r ). Slarly c p edtor@aee.co

6 S. Mohanaselv, K. Ganesan for the 2 nd row the su of the dfferences s gven by (2.5,4.5-4 r,3.5-3 r ). For 1 st colun the su of the dfferences s gven by (3.5,3.5-3 r,3-2 r) and 2 nd colun su of the dfferences s gven by (3,4.5-4 r,3.5-3 r ). The greatest su of the dfferences copared wth all other su of the dfferences s avalable n the 1 st colun. Hence fndng n (1.5,3-2 r,3-2 r ),(1.125,3.5-3 r,3-2 r ) we have the frst allocaton 45 n 2 nd row 1 st poston. Slarly all other allocatons are ade. Table 2 Fuzzy optal soluton to the gven LFFTP (3,3-2 r,3-2 r) 5 (2,2- r,2- r) (4.5,3.5-3 r,1.5- r) 45 (4,3-2 r,3-2 r) (4,3- r,3- r) 55 (4,2- r,2- r) (4.5,1.5- r,3.5-3 r) (6.5,4.5-4 r,2.5-2 r) Hence the ntal fuzzy basc feasble soluton (IBBFS) to the gven LFFTP s obtaned as, 55and (437. 5, r, 3 2r) Q( ) (1. 67, r, 3 2r). (41, 3 2r, 3 2r) Now we wll check whether the IFBFS obtaned s optal or not by usng the proved fuzzy Mod ethod. By assung u 1 we calculate u 2,v 1,v 2 for the allocated cells for the fuzzy cost atr. Slarly by assung u 1 we calculate u 2,v 1,v2for the allocated cells for the fuzzy proft atr. Now we calculate c u v ( 1. 5, r, r)and p u v (. 5, r, r) for the unallocated cell. The crtera for optalty Q( ) are satsfed. Hence the obtaned fuzzy ntal fuzzy basc feasble soluton s optal. Mn Q( ) = (1. 67, r, 3 2r) 6. CONCLUSIONS In ths paper a drect ethod for solvng a lnear fractonal transportaton proble wth fuzzy coeffcent s consdered. Wthout reforulatng the orgnal proble nto a classcal proble and by usng the entoned fuzzy arthetc and fuzzy rankng technque we have obtaned the fuzzy optal soluton to the gven LFFTP. A nuercal eaple s llustrated for descrbng the soluton procedure eplaned n ths paper. REFERENCES [1] Bajalnov, E. B. Lnear-fractonal-Prograng Theory, Methods, Applcatons and Software, Boston: Kluwer Acadec publshers,23. [2] Bheean Radhakrshnan and Paraan Anukokla, A copensatory approach to fuzzy fractonal transportaton proble, Internatonal Journal of Matheatcs n Operatonal Research,6(2), 214, pp edtor@aee.co

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