A Solution to Three Objective Transportation Problems Using Fuzzy Compromise Programming Approach

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1 A Solution to Three Objective Transportation Problems Using Fuzzy Compromise Programming Approach *Doke D.M. 1 Ph.D. Student Science college Nanded Nanded, Maharashtra India d.doke@yahoo.co.in Dr.V.A.Jadhav 2 Research Guide Science College Nanded, Maharashtra,India vinayakjadhav2261@gmail.com Abstract To solve multiobjective linear transportation problem here we have used arithmetic mean of global evaluation and obtained solution of three objective linear transportation problem. Using different weights for aggregation operator we obtained 4 different solution to the given 3 objective linear transportation problem. Keywords: Modern Sciences, Engineering and Technology. 1.INTRODUCTION: Commonly occurring problem in industry is that to distribute goods produced at several locations to different destinations. This problem involves availability of good at each source; we will call it origin of goods and requirement of goods at each destination. Also there is penalty associated with each rout for transferring a unit quantity. The problem is to minimize total penalty such that demands of all destinations are fulfilled and goods available at each of the source is completely shipped. This problem is known as Transportation Problem (TP). TP is originally developed by Hitchcock [1]. Standard TP has one linear objective function which is to be optimized i.e. minimized in case of cost of transportation and maximized in case of profit of transportation. Also standard TP has a set of linear constraints arising out of demand of destination and supply of origin. This type of problem can be solved by Simplex Method [2]. Simplex method is used when linear constraints are either equation, less than or greater than inequality. But in transportation problem all constraints are equations hence some special techniques are used to solve TP. Modified distribution method (MODI) is one such method to optimize single objective TP. Another method is stepping stone method. Further while transporting goods from one place to another there are several objectives such as minimize transportation cost, minimize damages to the product, minimize total time of shipping etc. In such type of situation we have to optimize several objectives hence the problem becomes multi objective transportation problem. If the multi objective transportation problem has linear objectives and linear constraints then the problem is called Multi Objective Linear Transportation Problem (MOLTP). Large numbers of algorithms have been developed by several scholars to solve MOLTP. Aneja and Nair [3] have given method to solve Bicriteria Transportation Problem. Alexandra I Tkacenko [ 4] [5] have developed algorithm to solve multi objective fractional transportation problem. Diaz JA [6], [7],Ringuest J-L, Rinks DB [8] etc have suggested algorithms to solve MOLTP. For a MOLTP with K objective functions the algorithm developed by Climate et al. [9] and Ringuestet al. [ 10 ] gave more than K non dominated solutions to MOLTP. Often it happens that the solution at which one objective is optimal other objective may not be optimal. For that matter it may be worst solution. To optimize several objectives simultaneously is not possible practically. Thus we have to find compromise solution for MOLTP. 9

2 Fuzzy set theory proposed by Zadeh[11] is used in several fields. Using fuzzy cost coefficients a concept of optimal solution of TP is given by Chanas and Kuchta [12]. V.J.Sudhakar and V.Navaneetha Kumar [13], W. Ritha and J. Merline Vinotha [14] gave approach to solve MOLTP in two stages. Lushu Li and K.K. Lai [15] used aggregation approach using Fuzzy method and solved MOLTP. In this paper we will provide solution to three objective TP using weighted arithmetic mean as aggregation operator. 2. MULTI OBJECTIVE LINEAR TRANSPORTATION PROBLEM (MOLTP): Standard MOLTP is to transfer goods from several origins to different destination subject to linear constraints such that all objectives are optimal. Suppose there are m origins of goods denoted by O 1, O 2, O 3,...,O m having supplies a 1, a 2,, a m and n destinations denoted by D 1, D 2,, D n having demands b 1, b 2,, b n. We assume that total demand is equal to total supply. Mathematically ai=bj For each of the objective, c (k) ij be the cost or penalty of transferring one unit from i th origin to j th destination for all i and j. The MOLTP is that to find x ij a quantity to be transferred from i th origin to j th destination such that Z k is minimum for k= 1,2,,K. Thus MOLTP is as under. Minimize Z k = cij(k) xij k= 1,2, K ----(1) Subject to = b j j= 1,2,,n ----(2-a) = i= 1,2,..,m - ---(2-b) ai= bj ----(2-c) x ij 0 i and j ----(2 -d) Definition 1: A feasible solution x* = { x ij } X is said to be non dominated solution of (1) and (2-a to 2-d)if there exists no other solution x = { x ij } X such that (k). x ij c ij. x (*) (k) ij c ij and (k). x ij c ij. x (*) (k) ij c ij for at least one k. The set of all non dominated solution is called complete solution. Definition 2: Fuzzy Number: A real fuzzy number is a fuzzy subset of the real number R with membership function () satisfying the following conditions. 1. is continuous from R to the closed interval [ 0,1] 2. is strictly increasing and continuous on [ a 1, a 2 ] 3. is strictly decreasing and continuous on [ a 1,a 2 ] Where a 1,a 2,a 3,a 4 and real numbers, and the fuzzy number denoted by. 3.FUZZY COMPROMISE APPROACH FOR MOLTP: Consider MOLTP Minimize Z (x) =[ Z 1 (x), Z 2 (),, Z k (x) ] (3) Subject to x where X set of feasible solutions. As stated earlier solution to (3) are often conflicting as several objectives cannot be optimized simultaneously. To find compromise solutions first solve each of the objective function as marginal or single objective function. 10

3 Doke D.M.et.al / International Journal of Modern Sciences and Engineering Technology (IJMSET) Suppose x k * is optimal solution of k th objective function. Find values of each objective at optimal solution of k th objective for all k= 1,2,,K. Thus we have matrix of evaluation of objectives. 4.MARGINAL EVALUATION FOR SINGLE OBJECTIVE: For each particular objective we define marginal evaluation function k : X [0,1] as given below 1 k (x) = < () < 0 () Where U k = Max Z k (x) L k = Min Z k (x) k=1,2,,k k=1,2,,k According to fuzzy sets, k is fuzzy subset describing fuzzy concept of optimum for objective Z k on feasible solution space X. Then to find compromise solution maximize aggregation operator (x) = w [1(), 2(),, ()] L Li K.K. Lai [15] used weighted arithmetic mean and weighted quadratic mean to find global evaluation of multiple objectives. Here we will solve 3 objective linear transportation problems using weighted A.M. as aggregation operator. Once we choose aggregating operator w and obtain the global evaluation: X [0,1] for all the objectives then we can convert this problem into single objective transportation problem. This single objective transportation can be solved using standard software like TORA. In this method lot of matrix multiplication and addition is involved which is done using SCILAB a software. 5.ILLUSTRATION : Consider 3 objective linear transportation problem as under, Minimize Z 1 = 3 x x x x x x x x x 31 + x x x 34, Minimize Z 2 = 2 x x x x x x x x x x x x 34, Minimize Z 3 = 8 x x x x x x x x x x x x 34. Subject to x 11 + x 12 + x 13 + x 14 10, x 21 + x 22 + x 23 + x

4 x 31 + x 32 + x 33 + x 34 40, x 11 + x 21 + x x 12 + x 22 + x 32 15, x 13 + x 23 + x x 14 + x 24 + x 34 20, x ij 0 = 1,2,3 = 1,2,3,4 Using TORA optimal solution of each of the marginal objectives is as shown below. X1* =( x 11 = 10, x 23 = 20, x 31 = 5, x 32 = 15,) X2* =( x 11 =10, x 24 = 20, x 31 = 5, x 32 = 15) X3* = ( x 13 = 10,x 24 = 20,x 31 = 15) Evaluating values of Z k for k= 1,2,3 objective we have following table. Objective Function solution of Z 1 i.e. at x 1 * solution of Z 2 i.e. at x 2 * solution of Z 3 i.e. at x 3 * Upper bound for Z k Lower bound for Z k i.e. U k i.e. L k Z Z Z Thus define k (x ij ) = () for k = 1,2,3 To find compromise solution we define (x ij ) = w [ 1 ( x ij ), 2 (x ij ), 3 (x ij ) ] Using weighted AM with weights w 1,w 2,w 3, as aggregating operator we have (x ij ) = w 1 1 ( x ij ) + w 2 2 (x ij ) + w 3 3 (x ij ) (x ij ) = w 1 (Z 1-175)/(-60) + w 2 (Z 2 305)/(-20) + w 3 (Z 3 265)/(-120) We have to maximize above function subject to linear constraints stated above. When w 1,w 2,w 3 are fixed then maximizing (x ij ) is essentially same as minimizing Z 1 w 1 /60 + Z 2 w 2 / 20 + Z 3 w 3 /120 subject to given set of conditions. Selecting different values of weights and Using SCILAB to find constraint matrix and Using TORA to solve single objective transportation problem we have following solutions. Solution No. Weights for aggregation Values of Objectives W 1 W 2 W 3 Z 1 Z 2 Z

2 0.1.01 0.8 235 325 265 3 0.8 0.1 0.1 175 325 285 4.45.1.45 205 335 305 6.

5 CONCLUSIONS: For multi objective transportation problem it is not easy to find optimal solution which can optimize all objectives simultaneously. A Fuzzy compromise approach used in this paper helps in getting compromise solution. Using different weighting system we can get many more solutions. We have shown some of them in this paper. 7. REFERENCES: [1]. Hitchcock FL. The distribution of a product from several sources to numerous localities. Journal of mathematical Physics 1941 ; 20: [2]. Dantzig GB. Linear programming and Extensions.Princeton,NJ: Princeton University Press,1963. [3]. Aneja YP, Nair KPK. Bicriteria Transportation Problems. Management Science, 1979;25:73-8. [4]. Alexandra I. Tkacenko The generalized algorithm for solving the fractional multi objective transportation problem.romai J.,1(2006), [5].Alexandra I. Tkacenko The Multiple criteria transportation model. (Special case ) Recent advances in applied mathematical and computational and information Science- Volume I [6].Diaz JA. Solving multi objective transportation problem..economicko Mathematicky Obzor. 1978;14: [7].Diaz JA Finding a complete description of all efficient solutions to a multi objective transdportation. Economicko Mathematicky Obzor1979;15; [8].Ringuest J-L,Rinks DB. Interactive solution for multi objective transportation problem. European Journal of Operations Reaserch. 1987; 32: [9].Climaco JN,Antunes CH, Alves MJ. Interactive Decision support for multi objective transportation problem. European Journal of Operations Reaserch.1993;65:4-19. [10].Ringuest J-L,Rinks DB. Interactive solution for multi objective transportation problem. European Journal of Operations Reaserch. 1987; 32: [11].Zadeh LA.Fuzzy Sets.Information control. 1965;8:338:53. [12].Stefan Chanas,Dorota Kuchta. A concept of the optimal solution of the transportation problem with fuzzy cost coefficients. [13].V.J.Sudhakar and V.Navaneetha Kumar. A different approach for solving two stage fuzzy transportation problem. Int.J. Contemp. Math.Sciences, Vol. 6,2011,no. 11, [14].W.Ritha and J.Merline Vinotha.Multi objective two stage fuzzy transportation problem. Journal of Physical Sciences, Vol.13,2009, [15].Lushu Li, K. K. Lai. A Fuzzy approach to the multi objective transportation problem. Computers and operations research. 27 (2000) AUTHOR S BRIEF BIOGRAPHY Doke D.M. : He is Ph.D. student at Science College Nanded and Associate Professor of Statistics at M.L.Dahanukar College of Commerce Mumbai. Have published papers in statistics in different journals and also authored text books in the subject of Statistics, Operations Research and Quantitative Techniques. Dr. V.A Jadhav: He is research a guide at Science College Nanded and Head of Statistics Department at Science College Nanded, Swami Ramand Tirth University,Nanded. 13

Solving Multi-objective Generalized Solid Transportation Problem by IFGP approach

778 Solving Multi-objective Generalized Solid Transportation Problem by IFGP approach Debi Prasad Acharya a,1, S.M.Kamaruzzaman b,2, Atanu Das c,3 a Department of Mathematics, Nabadwip Vidyasagar College,