A note on the solution of fuzzy transportation problem using fuzzy linear system

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1 2013 ( Available online at wwwispacscom/fsva Volume 2013, Year 2013 Article ID fsva-00138, 9 Pages doi:105899/2013/fsva Research Article A note on the solution of fuzzy transportation problem using fuzzy linear system P Senthilkumar 1, S Vengataasalam 1 (1 Department of Mathematics, Kongu Engineering College, Erode, Tamilnadu, India Copyright 2013 c P Senthilkumar S Vengataasalam This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, reproduction in any medium, provided the original work is properly cited Abstract In this paper, we discuss the solution of a fuzzy transportation problem, with fuzzy quantities The problem is solved in two stages In the first stage, the fuzzy transportation problem is reduced to crisp system by using the lower upper bounds of fuzzy quantities In the second stage, the crisp transportation problems are solved by usual simplex method The procedure is illustrated with numerical examples Keywords: Fuzzy number; Fuzzy linear system; Fuzzy linear programming; Fuzzy Transportation model 1 Introduction Transportation model deals with transportation of a product available at several origin (sources to a number of different destinations (obs This model can be used for a wide variety of situations such as scheduling, production, investment, plant location, inventory control, employment scheduling many others The total movement from each origin the total movement to each destination are given it is desired to find how the associations can be made the limitations on total In such problems, sources can be divided among the obs or obs may be done with a combination of sources The obective is to minimize the cost of transportation while meeting the requirement at the destination The parameters unit cost of transportation, supply dem is not always exactly known stable This paper deals with the case when the unit costs are known exactly, but the supply dem values are only imprecise We adopt to express the supply dem is fuzzy number in parametric form The first formulation of fuzzy linear programming is proposed by Zimmermann [13] Thereafter, many authors considered various types of the fuzzy linear programming problems proposed several approaches for solving these problems [4, 5, 9, 11, 12] Usually, most of the methods are based on the concept of comparison of fuzzy numbers by use of ranking function [3, 8] In this paper, we propose a new method algorithm to solve fuzzy transportation problem Initially, the fuzzy transportation problem is divided into four crisp transportation problems The crisp transportation problems are solved by usual simplex method The optimal solution of fuzzy transportation problem is obtained from the solutions of crisp transportation problem 2 Preliminaries We first review some known definitions which are relevant to this work Corresponding author address: raendranpsk@gmailcom

2 Page 2 of 9 Definition 21 [7] A fuzzy number is a map u : R I = [0,1] which satisfies: (i u is upper semi-continuous (ii u(x = 0 outside some interval [c, d] R (iii There exists real numbers a, b such that c a b d where 1 u(x is monotonic increasing on [c,a] 2 u(x is monotonic decresing on [b,d] 3 u(x = 1, a x b Also u is called symmetric fuzzy number if u(u c + x = u(u c x, x R, where u c = a + b 2 Definition 22 [2] A fuzzy number à is called non-negative (positive, if its membership function µã(x satisfies µã(x = 0, x 0 (x < 0 Definition 23 [1] An arbitrary fuzzy number in parametric form is represented by an ordered pair of functions (u(r,u(r, 0 r 1, which satisfy the following requirements: 1 u(r is a bounded left-continuous non-decreasing function over [0,1] 2 u(r is a bounded left-continuous non-increasing function over [0,1] 3 u(r u(r, 0 r 1 Also u = (u,u is called a symmetric fuzzy number in parametric form if u c (r = u + u is a real constant for all 2 0 r 1 Definition 24 [1] A crisp number α is simply represented by u(r = u(r = α, 0 r 1 Definition 25 [6] The n n linear system a 11 x 1 + a 12 x a 1n x n = y 1 a 21 x 1 + a 22 x a 2n x n = y 2 a n1 x 1 + a n2 x a nn x n = y n (21 where the coefficients matrix A = (a i, 1 i, n is a crisp n n matrix each y i, 1 i n, is fuzzy number in parametric form is called a fuzzy linear systems in parametric form Remark 21 [1] For any arbitrary fuzzy numbers x = (x(r,x(r,y = (y(r,y(r in parametric form scalar k, 1 x = y iff x(r = y(r x(r = y(r 2 x + y = (x(r + y(r,x(r + y(r 3 kx = (kx(r,kx(r if k is non-negative kx = (kx(r,kx(r if k is negative Definition 26 [10] Consider the following linear programming problem Max / Min z = c x A x (= or b (22 x 0 where the coefficient matrix A = [a i ] m n the vector c = (c 1,c 2,,c n are a nonnegative crisp matrix vector respectively x = ( x, b = b i are non-negative fuzzy vectors such that x, b i, 1 n, 1 i m are fuzzy number in parametric form is called a fuzzy linear programming problem in parametric form

3 Page 3 of 9 Definition 27 [10] A fuzzy vector x is a fuzzy feasible solution of A x = b, x 0 where A b are defined in (22, if x satisfies in system Definition 28 [10] A feasible solution fuzzy vector x is optimizes the obective function z = c x then it s said to be optimal solution to fuzzy linear programming problem Definition 29 Transportation Problem in Fuzzy Environment: In many practical situations, the constraints like supply dem in transportation models are uncertain; they cannot be treated as crisp terms In such situation, it is desirable to use some type of fuzzy transportation model One of the mathematical models of transportation problem to minimize the total transportation cost from m sources of supply to n destinations is stated as follows: Destination W 1 W 2 W n Supply X 11 X 12 X 1n F 1 C 11 C 12 C 1n a 1 = (a 1,a 1 Source X 21 X 22 X 2n F 2 C 21 C 22 C 2n a 2 = (a 2,a 2 X m1 X m2 X mn F m C m1 C m2 C mn a m = (a m,a m Dem b 1 = (b 1,b 1 b 2 = (b 2,b 2 b n = (b n,b n Min Z = m n i=1 =1 C i X i m i=1 n =1 X i = (b,b, = 1,2,,n X i = (a i,a i, i = 1,2,,m X i 0, i = 1,2,,m, = 1,2,,n 3 Solution of Fuzzy Transportation Model The supply dem are triangular fuzzy number in parametric form Fuzzy transportation model is transformed into equivalent two different transportation model by replacing r as 0 1 since 0 r 1 Proposition 31 When r = 0, the fuzzy transportation model is transformed into transportation model with crisp values in supply dem The crisp transportation model yields the solution as Xi 0 X i 0 for all i = 1,2,,m = 1,2,,n

4 Page 4 of 9 Proposition 32 When r = 1, the fuzzy transportation model is transformed into transportation model with crisp values in supply dem The crisp transportation model yields the solution as Xi 1 X i 1 for all i = 1,2,,m = 1,2,,n From the above results, we obtain the optimal solution for fuzzy transportation model is ( X i = X i = Xi 1 Xi 0 ( Xi 1 X i 0 r + X 0 i r + X 0 i The method to solve fuzzy transportation problem consists of the following steps: Algorithm 31 Step 1 In the parametric form of fuzzy transportation model, replace r as zero in each supply (a i (r,a i (r dem (b (r,b (r, for i = 1,2,,m = 1,2,,n Step 2 Divide the fuzzy transportation model into two different problems such that the supply dem for the first problem are a i (0 b (0 For the second problem, the supply dem are a i (0 b (0, for i = 1,2,,m = 1,2,,n Step 3 Solving the above two different fuzzy transportation problems, we obtain Xi 0 Xi 0, i = 1,2,,m = 1,2,,n Step 4 Replace r by 1 in fuzzy transportation model repeating the steps 2 3, we get X 1 i X 1 i, i = 1,2,,m = 1,2,,n Step 5 From 4 th 5 th step, we have ( X i = X i = Xi 1 Xi 0 ( Xi 1 X i 0 r + X 0 i r + X 0 i for i = 1,2,,m = 1,2,,n Step 6 The optimal solution of the fuzzy transportation model is X i = (X i,x i for i = 1,2,,m = 1,2,,n 4 Examples Example 41 Consider the following fuzzy transportation problem in parametric form W 1 W 2 Supply Factory F (3 + r,5 r F (8 + r,10 r Dem (4 + r,6 r (7 + r,9 r

5 Page 5 of 9 The equivalent Linear programming problem for the above fuzzy transportation model is Min Z = 5X X X X 22 X 11 + X 12 = (3 + r,5 r X 21 + X 22 = (8 + r,10 r X 11 + X 21 = (4 + r,6 r X 12 + X 22 = (7 + r,9 r X 11,X 12,X 21,X 22 0 The above fuzzy transportation model is first divided into two models as, Model 1 (r=0 Model 2 (r=1 W 1 W 2 Supply Factory F (3,5 F (8,10 Dem (4, 6 (7, 9 W 1 W 2 Supply Factory F (4,4 F (9,9 Dem (5, 5 (8, 8 Model 1 further classified into two linear programming problems such as, Problem 1 Problem 2 Min Z = 5X X X X 0 22 X X 0 12 = 3 X X 0 22 = 8 X X 0 21 = 4 X X 0 22 = 7 X 0 11,X 0 12,X 0 21,X Min Z = 5X X X X 0 22

6 Page 6 of 9 X X 12 0 = 5 X X 22 0 = 10 X X 21 0 = 6 X X 22 0 = 9 X11 0,X 12 0,X 21 0,X Solving the above two problems, we have X11 0 = 0, X 12 0 = 3, X 21 0 = 4, X 22 0 = 4 X11 0 = 0, X 12 0 = 5, X 21 0 = 6, X 22 0 = 4 Similarly the Model 2 also further divided into two models then solving that model, we have X11 1 = 0, X12 1 = 4, X21 1 = 5, X22 1 = 4 X11 1 = 0, X 12 1 = 4, X 21 1 = 5, X 22 1 = 4 From the above solutions, we have the following optimal solution X 11 = 0, X 12 = 3 + r, X 21 = 4 + r, X 22 = 4 X 11 = 0, X 12 = 5 r, X 21 = 6 r, X 22 = 4 That is X 11 = (0,0,X 12 = (3 + r,5 r,x 21 = (4 + r,6 r X 22 = (4,4 The above solution can be represented graphically as follows: Figure 1: The minimum transportation cost for the given problem is Min Z = 5(0,0 + 4(3 + r,5 r + 2(4 + r,6 r + 6(4,4 = (44 + 6r,56 6r

7 Page 7 of 9 Example 42 Consider the following fuzzy transportation problem in parametric form W 1 W 2 W 3 Supply Factory F (13 + 2r,17 2r F (8 + 3r,11 Dem (3 + 2r,7 2r (7 + 2r,9 (11 + r,12 The equivalent Linear programming problem for the above fuzzy transportation model is Min Z = 9X X X X X X 23 X 11 + X 12 + X 13 = (13 + 2r,17 2r X 21 + X 22 + X 23 = (8 + 3r,11 X 11 + X 21 = (3 + 2r,7 2r X 12 + X 22 = (7 + 2r,9 X 13 + X 23 = (11 + r,12 X 11,X 12,X 13,X 21,X 22,X 23 0 The above fuzzy transportation model is first divided into two models as, Model 1 (r=0 Model 2 (r=1 W 1 W 2 W 3 Supply Factory F (13,17 F (8,11 Dem (3, 7 (7, 9 (11, 12 W 1 W 2 W 3 Supply Factory F (15,15 F (11,11 Dem (5, 5 (9, 9 (12, 12 Model 1 further classified into two Linear programming problems such as, Problem 1 Min Z = 9X X X X X X 23 0 X X X 13 0 = 13 X X X 23 0 = 8 X X 21 0 = 3 X X 22 0 = 7 X X 23 0 = 11

8 Page 8 of 9 Problem 2 X 0 11,X 0 12,X 0 13,X 0 21,X 0 22,X Min Z = 9X X X X X X 23 0 X X X 13 0 = 17 X X X 23 0 = 11 X X 21 0 = 7 X X 22 0 = 9 X X 23 0 = 12 X11 0,X 12 0,X 13 0,X 21 0,X 22 0,X Solving the above two problems, we have X11 0 = 0,X 12 0 = 7,X 13 0 = 6,X 21 0 = 3,X 22 0 = 0,X 23 0 = 5 X11 0 = 0,X 12 0 = 9,X 13 0 = 8,X 21 0 = 7,X 22 0 = 0,X 23 0 = 4 Similarly the Model 2 also further divided into two models then solving that model, we have X11 1 = 0,X12 1 = 9,X13 1 = 6,X21 1 = 5,X22 1 = 0,X23 1 = 6 X11 1 = 0,X 12 1 = 9,X 13 1 = 6,X 21 1 = 5,X 22 1 = 0,X 23 1 = 6 From the above solutions, we have the following optimal solution X 11 = 0,X 12 = 7 + 2r,X 13 = 6,X 21 = 3 + 2r,X 22 = 0,X 23 = 5 + r X 11 = 0,X 12 = 9,X 13 = 8 2r,X 21 = 7 2r,X 22 = 0,X 23 = 4 + 2r That is X 11 = (0,0, X 12 = (7 + 2r,9, X 13 = (6,8 2r X 21 = (3 + 2r,7 2r, X 22 = (0,0, X 23 = (5 + r,4 + 2r The above solution can be represented graphically as follows: Figure 2: The minimum transportation cost for the given problem is Min Z = ( r,225 14r

9 Page 9 of 9 5 Conclusions We proposed a new method algorithm to solve fuzzy transportation problem Initially, the fuzzy transportation problem was divided into four crisp transportation problems These crisp transportation problems are solved by existing simplex methods The optimal solution of the fuzzy transportation problem was obtained from the solutions of crisp transportation problems References [1] S Abbasby, M Alavi, A method for solving fuzzy linear systems, Iranian ournal of fuzzy systems, 2 ( [2] A Kumar, N Bhtia, Strict sensitivity analysis for fuzzy linear programming problem, Journal of fuzzy set valued analysis, 2012 ( [3] L Campos, J L Verdegay, Linear programming problems ranking of fuzzy numbers, Fuzzy Sets Systems, 32 ( [4] M Delgado, J L Verdegay, MA Vila, A general model for fuzzy linear programming, Fuzzy Sets Systems, 29 ( [5] S C Fang, C F Hu, Linear programming with fuzzy coefficients in constraint, Computers Mathematics with Applications, 37 ( [6] M Friedman, M Ming, A Kel, Fuzzy linear systems, Fuzzy sets systems, 96 ( [7] G J Klir, V S Clair, B Yuan, Fuzzy set theory: Foundations Applications, Prentice-Hall Inc (1997 [8] N Mahdavi-Amiri, S H Nasseri, Duality results a dual simplex method for linear programming problems with trapezoidal fuzzy variables, Fuzzy Sets Systems, 158 ( [9] H R Maleki, M Tata, M Mashinchi, Linear programming with fuzzy variables, Fuzzy Sets Systems, 109 ( [10] S H Nasseri, E Ardil, Simplex method for fuzzy variable linear programming problems, World academy of science, engineering technology, 8 ( [11] H Rommelfanger, R Hanuscheck, J Wolf, Linear programming with fuzzy obective, Fuzzy Sets Systems, 29 ( [12] P Senthilkumar, G Raendran, On the solution of fuzzy linear programming problem, International Journal of Computational Cognition, 8 ( [13] H J Zimmermann, Fuzzy programming linear programming with several obective functions, Fuzzy Sets Systems, 1 (