A Transportation Problem Analysed by a New Ranking Method

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1 (IJIRSE) Interntionl Journl of Innovtive Reserch in Science & Engineering ISSN (Online) 7-07 A Trnsporttion Problem Anlysed by New Rnking Method Dr. A. Shy Sudh P. Chinthiy Associte Professor PG Scholr Deprtment of mthemtics Nirml College Women Coimbtore Indi Abstrct.This pper describes rnking method of hegonl fuzzy numbers using the re of trpezium. This rnking is used in trnsporttion problem in which the vlues of trnsporttion costs re represented s hegonl fuzzy numbers nd then the fuzzy trnsporttion problem(ftp) is converted into crisp vlued Trnsporttion Problem using new rnking. A numericl emple is pplied to obtin n initil Bsic Fesible solution of trnsporttions in minimizing trnsporttion time by North-West Corner method. Keywords- Rnking; Hegonl fuzzy numbers; Fuzzy Trnsporttion Problem. I. INTRODUCTION In fuzzy environment rnking fuzzy numbers is n importnt decision mking procedure. Most of the rel world problems tht eist re fuzzy in nture thn probbilistic or deterministic. Fuzzy sets ws first given by Zdeh [] in 9. Rnking fuzzy numbers ws first proposed by Jin in the yer 97 decision mking in fuzzy situtions by representing in fuzzy set. Jin [] proposed method using the concept of mimizing set to order the fuzzy numbers nd the decision mker considers only the right side membership function. Yger [] proposed four indices to order fuzzy quntities in [0 ]. Liou nd Wng [9] presented rnking fuzzy numbers with integrl vlue. Centroid bsed distnce method to rnk fuzzy numbers ws given by Cheng [] in 998. Wng nd Kerre [] clssified the eisting rnking procedures into three clsses. The first clss consists of rnking procedures bsed on fuzzy men nd spred nd second clss consists of rnking procedures bsed on fuzzy scoring wheres the third clss consists of methods bsed on preference reltions nd concluded tht the ordering procedures ssocited with first clss re reltively resonble the ordering of fuzzy numbers specilly. The rnking procedure presented by Admo [] stisfies ll the resonble properties the ordering of fuzzy quntities. The Rnking of fuzzy numbers by re between the centroid point nd the originl point ws given by Chu nd Tso [] in 00.The fuzzy risk nlysis bsed on rnking of generlized trpezoidl fuzzy numbers ws given by Chen nd Chen []. Abbsbndy nd Hjjri [] proposed new pproch rnking trpezoidl fuzzy numbers. Stephen Dingr nd Kmlnthn [] defined rnking bsed on re method. Since then severl methods hve been proposed by vrious reserchers which includes rnk mode divergence nd spred [8] Rjrjeshwri nd Shy Sudh [0] defined Rnking of Hegonl fuzzy numbers using centroid. II. PRILIMINARIES. Definition [ 7] Let X be nonempty set. A fuzzy set A in X is chrcterized by its membership function A: X [0 ] where A() is interpreted s the degree of membership of element in fuzzy A ech X.. Definition: [7] The chrcteristic function of crisp set A X ssigns vlue of either or 0 to ech individul in the universl set X.This function cn be generlized to function such tht the vlue ssigned to the element of the universl set X fll within specified rge (i.e.) : X [0 ]. The ssigned vlue indictes the IJIRSE/07/Vol. Iss. / Pge 9

2 (IJIRSE) Interntionl Journl of Innovtive Reserch in Science & Engineering ISSN (Online) 7-07 membership grde of the element in the set A. The function is clled the membership function nd the set à = {( ))/ X} defind by ech X is clled fuzzy set. Definition: [7] A fuzzy set defined on the universl set of rel numbers R is sid to be fuzzy number of its membership function hs the following chrcteristics (i) is continuous mpping from R to the closed intervl [0] (ii) = 0 ll [- ] [ ] (iii) is strictly incresing on [ ] nd strictly decresing on [ ] (iv) = ll [ ] where. Definition: [7] A fuzzy set defined on the universl set of rel numbers R is sid to be generlized fuzzy number of its membership function hs the following chrcteristics (i) : R [0] is continuous (ii) = 0 ll [- ] [ ] (iii) is strictly incresing on [ ] nd strictly decresing on [ ] (iv) = w ll [ ] where. Definition: [7] A fuzzy number A is trpezoidl fuzzy number denoted by ( given below where ) nd its membership function is ()=. Definition: [7] A generlized fuzzy number = ( ) is sid to be generlized trpezoidl fuzzy number if its membership function is given by ()=.7 Definition: [] Hegonl Fuzzy Numbers A fuzzy number is hegonl fuzzy number denoted by ( ) where re rel numbers nd its membership function () is given below. IJIRSE/07/Vol. Iss. / Pge 0

3 (IJIRSE) Interntionl Journl of Innovtive Reserch in Science & Engineering ISSN (Online) 7-07 IJIRSE/07/Vol. Iss. / Pge otherwise H A 0 ) ( ~.8 Definition: [] A generlized fuzzy number is hegonl fuzzy number denoted by ( ) where ( ) re rel numbers nd its membership function () is given below. otherwise w w w H A 0 0 ) ( ~ III. PROPOSED APPROACH In this pper we hve used the re of trpezium s new rnking procedure generlized trpezoidl fuzzy numbers. Divide the hegon into two trpezium nd nd ech trpezium hs two bses prllel to ech other nd two legs joining these bses t the points of re ( ) ( ) ( )( ) nd re( 0) ( 0) ) ( ).

4 (IJIRSE) Interntionl Journl of Innovtive Reserch in Science & Engineering ISSN (Online) 7-07 Are of trpezium = ( ) = distnce between B ( ) nd E ( ) = ) = distnce between C ( ) nd D ( ) = ) = [ )+ )] Are of trpezium = ( ) = distnce between B ( ) nd E ( ) = = distnce between A ( ) nd F ( ) = = [ )+ )] Hence the rnking function is given by R ( ) = + R ( ) = [ ) + )] + [ ) + )] R ( ) = [ ( ) + ) + )] (.) IV. MATHEMATICAL FORMULATION OF FUZZY TRANSPORTATION PROBLEM The mthemticl models of fuzzy trnsporttion problem is to minimize the totl trnsporttion cost from m Mchine to n Foundries is s follows IJIRSE/07/Vol. Iss. / Pge

5 Minimize Z Subject m nd n i j to m j n i m ~ c~ ~ ~ ~ i i j 0 (IJIRSE) Interntionl Journl of Innovtive Reserch in Science & Engineering ISSN (Online) 7-07 b n i j b j ll i nd i.... m j.... n Where c ~ the fuzzy unit trnsporttion cost from i th Mchines to the j th Foundries. The fuzzy trnsporttion tble is: j. M A C H I N E S Tble.: Fuzzy Trnsporttion Tble FOUNDRIES N Supply ~c ~c ~c ~c c~ n c~ n..... M ~ c ~ Demnd c m m b b c ~ mn m b n V. THE ALGORITHM FOR THE PROPOSED WORK IS AS FOLLOWS Step : Construct the fuzzy trnsporttion tble the given problem nd then convert into blnced if it is not. Step : Using the bove rnking method (.) the fuzzy trnsporttion problem is converted into crisp problem. Step : Apply North West Corner (NWC) method to obtin n initil bsic fesible solution. Step : ) Select the north-west (upper left hnd) corner cell of the trnsporttion tble nd llocte s much s possible so tht either the cpcity of the first row or first column is stisfied i.e. = min ( ). ) If > we move down verticlly to the second row nd mke the second lloction of mgnitude = min ( ) in the cell ( ). If < we move right horizontlly to the second column nd mke the second lloction of mgnitude = min ( ) in the cell ( ). If = there is tie the second lloction. We cn mke the second lloction of mgnitudes = min ( ) = 0 in the cell ( ). Or = min ( ) = 0 in the cell ( ). IJIRSE/07/Vol. Iss. / Pge

6 (IJIRSE) Interntionl Journl of Innovtive Reserch in Science & Engineering ISSN (Online) 7-07 Step : Repet the bove steps moving down towrds the lower/ right corner of the trnsporttion tble until the optiml solution is obtined. VI. NUMERICAL EXAMPLE A compny mnufctures certin type of products in three different Mchines. The compny hs to eecute the trnsporttion of the three Mchines to four different Foundries. The inmtion bout the cost of trnsporttion is imprecise nd here hegonl fuzzy numbers re used to represent the cost. The fuzzy trnsporttion problem is given in tble. Step : Construct the fuzzy trnsporttion tble the given fuzzy trnsporttion problem nd then convert it into blnced one if it is not. Foundries TABLE. FUZZY TRANSPORTATION TABLE Supply Mchines (7 (88 (08 (0 9;0.) 0;0.7) ;0.) 8;0.8) (99 (79 (70 (00 9;0.) 0;0.) 7;0.) 0;0.) (7 (79 ( (78 0;0.7) 8;0.) 79;0.) 7;0.9) Demnd Step : Using rnking method (.) in the fuzzy trnsporttion problem is converted into trnsporttion problem. R(7 9 : 0) R(999;0.) R(70;0.7) R(000000) R(880;0.7) R(790;0.) R(798;0.) R(000000) R(08;0.) R(707;0.) R(79;0.) R(000000) R(08;0.8) R(000;0.) R(787;0.9) R(000000) R ( ) = [ ( ) + ) + )] R(79;0.) Similrly R(880;0.7).7 R(08;0.) 8. R(08;0.8) 8. R(999;0.).8 R(790;0.). R(707;0.).7 R(000;0.) 9. R(70;0.7).7 R(798;0.) 0.8 R(79;0.). R(787;0.9).0 R(000000;0) 0 R(000000;0) 0 R(000000;0) 0 R(000000;0) 0 IJIRSE/07/Vol. Iss. / Pge

7 (IJIRSE) Interntionl Journl of Innovtive Reserch in Science & Engineering ISSN (Online) 7-07 Foundries Mchines TABLE. TRANSPORTATION TABLE Supply Demnd Step : Applying NWC method the initil bsic fesible solution is obtined TABLE. INITIAL BASIC FEASIBLE SOLUTION Foundries Mchines Demnd Supply 0 Step : The trnsporttion cost ccording to the bove route is given by Z = Z =.9 RESULT AND CONCLUSION: The overll cost of the process is.9 which is the minimum cost. This cost is rrived when we supply mchines of nd 8 mchines of to foundries demnd of units. Similrly demnd of 8 units foundry is supplied with 8 mchines of. Then demnd of 0 mchines in foundry is supplied with 0 units of mchine nd with 0 units of mchine. Then demnd of 0 units foundry is supplied with 0 units of mchine.this is the optiml cost on considertion of vrious prmeters. In this pper we hve proposed method where hegonl fuzzy rnking is obtined using n re of trpezium. This method cn be used in Trnsporttion problem where number of prmeters re involved nd the fuzzy hegonl vlues re converted into crisp vlue nd the solution is obtined s eplined in the emple. Future Directions: This method cn lso be used esily different industril s well s socil pplictions where the prmeters to be nlysed re more in number. REFERENCES [] Abbsbndy S. nd Hjjri T. A new pproch rnking of trpezoidl fuzzy numbers Computer nd Mthemtics with Appliction 7(009)-9. [] Admo J.M. Fuzzy Decision trees Fuzzy sets nd systems (980)07-9. [] Chen S.M. nd Chen J.H. Fuzzy risk nlysis bsed on the rnking of generlised trpezoidl fuzzy numbers Applied Intelligence (007)-. IJIRSE/07/Vol. Iss. / Pge

8 (IJIRSE) Interntionl Journl of Innovtive Reserch in Science & Engineering ISSN (Online) 7-07 [] Cheng C.H. A new pproch rnking fuzzy numbers by Distnce method Epert system ppliction 9 (998) [] Chu T.C. nd Tso C.T. Rnking fuzzy numbers with n re between the centroid point nd originl point Computers nd Mthemtics with Applictions (00) -7. [] Jin R. Decision mking in the presence of fuzzy vribles IEEE Trnsctions on systems Mn nd Cybernetics (97) [7] Kufmnn A. nd Gupt M.M. Fuzzy mthemticl models in engineering nd mngement science Elsvier Science Publishes (988). [8] Kumr A. Singh p. Kur A. Kur P. Rnking of generlised trpezoidl fuzzy numbers bsed on rnk mode spred nd divergence Turkish Journl of Fuzzy Systems (00)-. [9] Liou T.S nd Wng M.J. Rnking Fuzzy numbers with integrl vlue Fuzzy Sets nd Systems 0(99) 7-. [0] Rjrjeswri P. nd Shy Sudh A. Rnking of Hegonl Fuzzy numbers using centroid AARJMD (0) -77. [] Rjrjeshwri P.Shy Sudh A. nd Krthik R. A New Opertion on Hegonl Fuzzy Number IJFLS (0)-. [] Stephen Dingr D. nd Kmlnthn S. A method rnking of fuzzy numbers using new re method IJFMA (0) -7. [] Wng X nd Kerre E.E. Resonble properties the ordering of fuzzy quntities (I) Fuzzy sets nd systems 8(00) 7-8. [] Yger R.R. A procedure ordering fuzzy subsets of the unit intervl Inmtion Sciences (98)-. [] Zdeh L.A. Fuzzy sets Inmtion nd Control 89 (9) 8-. IJIRSE/07/Vol. Iss. / Pge