Fuzzy Transportation Problem Using Triangular Membership Function-A New approach

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1 Vol3 No.. PP 8- March 03 ISSN: 3 006X Trasportatio Proble Usig Triagular Mebership Fuctio-A New approach a S. Solaiappaa* K. Jeyaraab Departet of Matheatics Aa UiversityUiversity College of Egieerig Raaathapura Tail Nadu Idia solaipudur@yahoo.co.i b DeaSciece ad Huaities PSNA college of Egieerig ad Techology Didigul 646 Tail Nadu Idiajaya_jaaki07@yahoo.co.i Abstract I this Paper a Trasportatio Proble is ivestigated usig Triagular ebership fuctio. vogel s approxiatio ethod is used to obtai iitial basic feasible solutio ad optiu solutio is obtaied usig fuzzy odified distributio ethod. The ethod is illustrated by a exaple. Key Words: triagular fuzzy ubers fuzzy triagular ebership fuctio fuzzy Vogel s approxiatio ethod fuzzy odified distributio ethod. I. INTRODUCTION The trasportatio proble is a special class of liear prograig probles. I a typical proble a product is to be trasported fro several sources to uerous locatios at iiu cost. Suppose that there are warehouses where a coodity is stocked ad trasported to arkets where it is eeded. Let the availability i the warehouses be a a...a ad the s at the arkets be b b...b respectively. I additio there is a pealty c associated with trasportig uit of product fro the source i to the destiatio j. This pealty ay be the cost or delivery tie or safety of delivery etc. A variable x represets the ukow quatity to be shipped fro the source i to the destiatio j. The basic trasportatio proble was origially stated i [4 ad later discussed i [7. A liear prograig proble usig L-R fuzzy uber was give i [9 a operator theory of paraetric prograig for Geeralized Trasportatio Proble (GTP) was preseted by [. I [ the cocept of decisio akig i fuzzy eviroet is proposed ad i [3 the situatio where all the paraeters are fuzzy is cosidered. I [5 H.Isera itroduced a algorith for solvig this proble which provides a effective solutio based o iterval ad fuzzy coefficiets. Further developet o Triagular ebership fuctios i solvig Trasportatio proble uder fuzzy eviroet had bee elaborated by [4 0. This paper is orgaized as follows; i sectio () soe basic defiitios o fuzzy set theory are listed. I sectio (3) the ethod of fuzzy odified distributio to fid out the optial solutio for the total fuzzy trasportatio of iiu cost is discussed. I Sectio (4) a uerical exaple is worked out. II. FUZZY BASCIS.. Defiitio [ If X is a collectio of objects ad a fuzzy subset A of X is a set of ordered pairs. i.e. A (x A(x) / x X where A ( x ) is called the ebership fuctio for the fuzzy set A.The ebership fuctio aps each eleet of X to a ebership grade (or) ebership value betwee 0 ad.. Defiitio [8 The cut (or) - level set of fuzzy subset A is a set cosistig of those eleets of the uiverse X whose ebership values exceed the threshold level. i.e. A x / A ( x).3. Defiitio [8 It A triagular fuzzy uber is represeted with the three poits as A ( a a.a 3 ). This represetatio is iterpreted as ebership fuctio ad holds the followig coditios (i) a to a is icreasig fuctio (ii) a to a3 (iii) a a a3 0 x a a a A ( x) a3 x a3 a 0 is decreasig fuctio for x a for a x a for a x a3 for x a3 8

2 Vol3 No.. PP 8- March 03 ISSN: 3 006X.3 [8 =[ = [ be two triagular fuzzy ubers the the arithetic operatio is } [ = {i( : } + = {[ ) + [ + [ + = {[ [ ax( ).4Defiitio (i) feasible solutio: If x 0 which satisfies the row ad colu su is a fuzzy feasible solutio. (ii) basic feasible solutio: A feasible solutio is a fuzzy basic feasible solutio if the uber of o-egative allocatio is atost (+-) where is the uber of rows ad is the uber of colus i the trasportatio table. (iii) o-degeerate basic feasible solutio: A feasible solutio to a trasportatio proble cotaiig origis ad destiatios is said to be fuzzy o-degeerate if it cotais exactly (+-) occupied cells with o zero values. (iv) degeerate basic feasible solutio: If a basic feasible solutio cotais less tha (+-) o-egative allocatio it is said to be degeerate. III. FUZZY TRANSPORTATION PROBLEM: Cosider a Trasportatio proble with fuzzy origis (row) ad fuzzy destiatios (colus).let c [ c c c is the cost of trasportatio oe uit of the product fro ith fuzzy origi to jth destiatios = [ be the quatity of coodity at fuzzy origi i = [ be the quatity of coodity required at fuzzy destiatios j X is quatity trasported fro ith fuzzy origi to jth fuzzy destiatios. The liear prograig odel represetig the Trasportatio proble is give by i z [ c c c [ X X X Subject to the costraits [ =[ [ =[ For i= 3... rows For j = 3... colus for all X 0 the give fuzzy trasportatio proble is said to be balaced if ai b j. 3.Solutio of Trasportatio Proble: The Solutio of FTP ca be solved i two stages. (i) Iitial Solutio (ii) Optial Solutio. For fidig the iitial solutio of FTP fuzzy Vogel s approxiatio ethod is preferred over the other ethods. Sice the iitial fuzzy basic feasible solutio obtaied by this ethod is either optial (or) very closed to the optial solutio. We are goig to discuss fuzzy Vogel s approxiatio ethod ad verify its solutio i the ature of fuzzy triagular ebership fuctio. 3. Vogel s Approxiatio Method Algorith: Step (i). Fid the fuzzy pealty cost aely the fuzzy differece betwee the sallest ad ext sallest fuzzy costs i each variable i each row ad each colu. Step (ii). Aog the fuzzy pealties foud i step (i) choose the fuzzy axiu pealty by rakig ethod. If this axiu pealty is attaied i ore tha oe cell choose ay oe arbitrary. Step (iii). If the selected row (or) colu by step (ii) fid out the cell havig the least fuzzy cost by usig the fuzzy rakig ethod. Allocate to this cell as uch as possible depedig o the fuzzy capacity ad fuzzy s. Step (iv). Delete the row (or) colu which is truly exhausted. Agai copute colu ad row fuzzy pealties for the reduced fuzzy trasportatio table ad the go to step (i) Repeat the procedure util all the s are satisfied. Oce the iitial fuzzy feasible solutio is coputed the ext step i the proble is to deterie whether the solutio obtaied is optial or ot. optiality test ca be coducted to ay fuzzy iitial basic feasible solutio of a fuzzy trasportatio proble provided such allocatio has exactly ( + -) o-egative allocatio where is the uber of fuzzy origis ad is the uber of fuzzy destiatios. These allocatio cells are idepedet. The optiality procedure is give below 3.3 Modified Distributio Method: This proposed ethod is used for fidig the optial basic feasible solutio i the fuzzy eviroet ad the followig step by step procedure is used to fid out the sae. 9

3 Vol3 No.. PP 8- March 03 ISSN: 3 006X Step. Fid out the set of fuzzy triagular Table 4. The basic fuzzy trasportatio proble [ui ui ui ad [vi vi vi for each ubers D D capacity [ui ui ui + O [-0 [0 [-0 [0 [vi vi vi = [ c c c for each occupied cell. [0 O [48 [470 [46 [46 [46 [468 [468 [470 [468 [04 row ad colu satisfyig To start with we assig a fuzzy zero to ay row (or) colu havig axiu uber of allocatios. If this axiu uber of allocatios is ore tha oe choose ay oe arbitrary. Step. For each epty (uoccupied) cell fid [ui ui ui ad [vi vi vi i = [ c c c - [ui ui ui + [vi vi vi. uique solutio exists. (b) if Z 0 the solutio is optial but a alterate solutio exists. (C) if Z 0 the solutio is ot fuzzy optial. Step 5. The above step yields a better solutio by akig oe (or) ore occupied cell as epty cells. For the ew set of fuzzy basic feasible allocatio repeat fro step till a fuzzy optial basic feasible solutio is obtaied 4.Nuerical Exaple To solve the followig fuzzy trasportatio proble of iial cost we start with the iitial fuzzy basic feasible solutio obtaied by Vogel s Approxiatio ethod for which fuzzy cost ad fuzzy requireet table is give below the give proble is balaced fuzzy trasportatio proble ad the supply ad costs are syetric fuzzy triagular ubers (FTN). j j = [6 6 = [3 9 =33 capacity [-0 [0 [46 [468 [-4 [46 [-35 [468 [-406 [04 D D O [-0 [0 [0 [-0 O [48 [470 [46 I this case we go to ext step to iprove the solutio ad iiize the cost. path drawig horizotal ad vertical lies with corers beig occupied cells. Assig sigs positive ad egative alteratively ad fid out the fuzzy iiu allocatio cell havig egative sig. This allocatio should be added to the allocatio of the other cells havig egative sig. i the proble is balaced fuzzy trasportatio proble. There exists a fuzzy iitial basic feasible solutio. Table 4. It represets the iitial basic fuzzy feasible solutio by VAM ethod This step gives the optiality: (a) if Z 0 the solutio is optial ad a Step 4. Select the epty cell havig the ost o egative value Z fro this cell we draw a closed a b Sice Step 3. Fid out for each epty cell the et evaluatio; [ z z z [470 Sice the uber of occupied cell is +- = 6 ad the occupied cells are idepedet. The iitial solutio is a o-degeerate fuzzy basic feasible solutio. Therefore the iitial fuzzy trasportatio iiu cost is [ z z z = [ (I) 4.3Movig to the optial solutio: Here C are fuzzy cost coefficiet ad X (i = 3 j = 34) are fuzzy allocatios. We have to fid fuzzy ebership fuctios of C ad X for each cell (i j).the ebership trasportatio cost C ( x ) fuctio of fuzzy is foud as follows x 0 0 x 0 C ( x) x x To copute the iterval of cofidece for each level the triagular shapes will be described by fuctios of i the followig aer 0 [468

4 Vol3 No.. PP 8- March 03 ISSN: 3 006X ( x 0) x ( x ) x C [ x x [ (i) Here The ebership fuctios of fuzzy allocatio to the cell () X ( x ) as follows x 0 X ( x ) 0 x 0 x x ( x 0) x Here (x ) x X [ x x [ (ii) Fro (i) ad (ii) we get C X [ ( ) (A) Exactly i the siilar way C 3 X 3 [( )(4 3) (6 )(5 4 ) (B) C4 X 4 [( ) (5 ) (C) i. cos t. Z x ). 8 {( 8 ) 4 ( 7 X )} / 56 / 66 {(66 ) 40 (66 X )} 70 7 x 3 3 x 69 This is the required fuzzy ebership fuctio of fuzzy trasportatio iiu cost Z (usig the equatio uber ). 4.4 Deteriatio of fuzzy optial solutio: Applyig the fuzzy odified distributio ethod we deterie a set of triagular fuzzy ubers [ui ui ui ad [vi vi vi each row ad colu such that [ c c c = [ui ui ui + [vi vi vi for each occupied cell. Sice the 3rd row has axiu uber of allocatio we give [u3 u3 u3 = [- 0. The fuzzy uber reaiig ubers ca be obtaied as give by: [ v v v = [3 4 5 [ v v v = [3 6 7 [ v 3 v 3 v 3 = [3 6 7 [u u u = [-5-3. [v4 v4 v4 = [- 0 [ u u u = [-7-5- We fid for each epty cell the su of [ui ui ui C 3 X 3 [( )( ) (5 )(6 ) ad [vi vi vi (D) Next the et evaluatio of [ z z z are foud C 3 X 3 [( 4)(3 ) (8 )( 4 3 ).Usig the table below: (E) C33 X 33 [( 4)(4 4) (8 )(6 6 ) Table 4.5 It represets the uoccupied cells D D O [-0 *[-6-8 [0 [ X 0 O [48 *[-466 [470 *[-63 [-0 *[-8-6 [46 [ X 0 [46 [468 [-4 [468 [-406 [ (F) A +B+C+D+E+F gives the iiu cost: Z [ (II) Solvig the equatios: (G) We get (H) [ 8 {(3) 4( 7 X )} / 56 [66 {( 66 ) 40 (66 X )} / 70 capacit y [-0 [0 *[040 [4 6 [470 *[363 Sice [ z z z >0 the solutio is fuzzy optial ad uique. Miiu cost of FTP is [ z z z [ [46 8

5 Vol3 No.. PP 8- March 03 ISSN: 3 006X Applied Matheatical Sciece Vol.6 () (0) pp55. Usig exactly i the sae procedure as i the fuzzy iitial basic feasible solutio. We could fid fuzzy ebership fuctio of optial solutio. V. CONCLUSION I this paper we obtaied a optial solutio for Trasportatio Proble of iial cost usig Triagular Mebership Fuctio. The ew arithetic operatios of Triagular fuzzy ubers are eployed to get the fuzzy optial solutios. The optial solutio obtaied by usig the fuzzy odified distributio ethod ca also be verified i the Triagular Mebership Fuctio. This would be a ew attept i solvig the Trasportatio Proble i fuzzy eviroet. I our future extesio we would like to utilize the ew optiizatio techiques i the literature. Aticipatig the valuable coets ad suggestios [9 A. Nagoor Gai C. Duraisay ad C. Veeraai A Note o liear Prograig proble usig L-R fuzzy uber Iteratioal joural of Algoriths Coputig ad Matheatics vol. (3)(009). [0 P. Stepa Dlaagar K. Palaivel O Trapezoidal Mebership Fuctio i solvig trasportatio proble uder fuzzy eviroet Iteratioal Joural of Coputatioal Physical Scieces vol (3) (009) pp93. REFERENCES [ V. Balachadra G.L.Thopso A operator theory of paraetric prograig for the geeralized trasportatio proble: I Basic Theory 975 Nav. Res. Log. Quart [ R.E. Bella L.A. Zadeh Decisio akig i a fuzzy eviroet Maageet sci.7 (970) pp 4-64 [3 S. Chaas D. Kuchta iteger trasportig proble sets ad systes 98(998) pp 9-98 [4 F.L.Hitchock Distributio of product fro several sources to uerous localities Joural of Math Physics.vol. No.3 (978). [5 H. Isera The eueratio of all efficiet solutio for a liear ultiple- objective trasportatio proble Naval Research Logistics Quarterly 6 (979) pp.3-73 [6 L.V. Katrovich O the Traslocatio of asses Doklerdly Akad Nauk SSR vol.37 (94) pp7. [7 C.Koopas Optiu utilizatio of the trasportatio syste Ecooetrica vol.7 suppleet 949 [8 A.Nagoor Gai S.N.Mohaed Assarudee A New Operatio o Triagular Nuber for Solvig Liear Prograig Proble