Fuzzy Assignment Method for Finding an Optimal Solution for Fuzzy Assignment Problem

INTRODUCTION

Fuzzy sets were introduced by L.A.Zadeth [7] as an enter of classical notion of the set. Later many researchers [2, 3, 5] also use fuzzy set theory and fuzzy numbers in different field. After this J.J. Buckly [1] used triangular fuzzy numbers in linear programming. P.Pandian and G.Natarajan[4] use fuzzy zero point method in trapezoidal fuzzy numbers for finding a fuzzy optimal solution for a fuzzy transportaion problem. In this paper, we propose a new method called fuzzy assignment method for finding a fuzzy optimal solution for a fuzzy assignment problem where all parameters are triangular fuzzy numbers. The optimal solution of fuzzy assignment problem by the fuzzy aasignment method is a triangular fuzzy number. The solution procedure is illustrated with the numerical examples. When we use the assignment method for finding an optimal solution for a fuzzy assignment problem, we have the following advantages. • We don‘t use linear programming techniques. • We don‘t use goal and parametric programming techniques. • The optimal solution is a fuzzy number. • The proposed method is very easy to understand and to apply.

We use the following definitions of triangular fuzzy number and membership function and also, definitions of basic arithmetic operation on fuzzy triangular numbers. Definition 6.2.1. A fuzzy number x~is a triangular fuzzy number denoted by (x1, x2, x3) where x1, x2 and x3 are real numbers and its membership function )(~ax is given below. Definition 6.2. 2. Let a~= (a1, a2, a3) and b~= (b1, b2, b3) be two triangular fuzzy numbers. Then (i) (iii) (iv) (v) where t1= min. {a1b1, a1b3, a3b1, a3b3}; t2 = a2b2 and t3 = max. { a1b1, a1b3, a3b1, a3b3}; (vi) where b1, b2 and b3 are all non-zero positive real numbers. (vii) where b1, b2 and b3 are all non-zero positive real numbers. Definition 6.2.3 The magnitude of the triangular fuzzy numberu~= (a, b, c) is given by Definition 6.2.4. Let u~ and v~ be two triangular fuzzy numbers. The ranking of u~ and v~by the Mag (.) on E is defined as follows:

Consider the following fuzzy assignment problem, Subject to the constraints: only one person.

such that

The fuzzy assignment method proceeds as follows:- Step-1 if the number of rows are not equal to the number of columns, the as required a dummy row or dummy column must be added. The cost elements in dummy cells are always zero. Step-2 Row reduction: Subtract each row entries of the fuzzy assignment table from the corresponding row minimum. Step-3 Column reduction: After completion of step-2, subtract each column entries of the fuzzy assignment table from the corresponding column minimum. Step-4 Remember that each row and each column now have at least one zero. Step-5 Zero assignment: In the modified matrix obtained in the step-3, search for the optimal assignment as follows: (a) Examine the rows successively until a row with a single zero is found. Rectangle this row (□) and cross off (×) all other zeros in its column. Continue in this manner until all rows have been taken care of. (b) Repeat the procedure for each column of the reduced matrix. (c) If a row and/or column have two or more zeros and one cannot be chosen by inspection then assign arbitrary any one of

(d) Repeat (a) through (c) above successively until the assigning (□) or cross (×) ends. Step-6 If the number of assignment (□) is equal to n (the order of the cost matrix), an optimal solution is reached. If the number of assignment is less than n (the order of the matrix), go to the next step. Step-7 Draw the minimum number of lines to cover zero‟s In order to cover all the zero‘s at least once you may use the following procedure. (i) Marks( ) to all rows in which the assignment has not be done. (ii) See the position of zero in marked () row and then mark () to the corresponding column. (iii) See the marked () column and find the position of assigned zero‘s and then mark () to the corresponding rows which are not marked till now. (iv) Repeat the procedure (ii) and (iii) till the completion of marking. (v) Draw the lines through unmarked rows and marked columns. Step-8 Select the smallest element from the uncovered elements. (a) Subtract this element from all uncovered elements and add the same to all the elements laying at the intersection of any two lines. Step-9 Go to step-5 and repeat the procedure until an optimum solution is attained.

The proposed method is illustrated by the following examples. Example 1. Let three persons A, B and C are to be assigned three jobs I, II and III. The cost matrix is given as under, find the proper assignment.

In order to find the proper assignment, we apply the fuzzy assignment method as follows: (I) Row reduction

(II) Column reduction

(III) Zero assignment

In this way all the zero‘s are either crossed out or assigned. Also total assigned zero‘s = 3 (i.e., number of rows or columns). Thus, the assignment is optimal. So the minimum cost = (4, 12, 20).

1. Buckly, J.J. (1998). Possibilistic linear programming with triangular fuzzy numbers, fuzzy sets and system, Vol.26, pp. 135-138. 2. Chen S.H. (2013). Operation on fuzzy numbers with function principle, Tamkang Journal of Management Sciences, Vol.6, pp.-13-25. 3. Kaufmann A. (1996). Introduction to the theory of fuzzy sets, Vol. 1, Academic press, New York, (1996). 4. Pandian P. and Natarajan G. (2013). Anew algorithm for finding a fuzzy optimal solution for fuzzy transportation problems, Applied mathematical sciences, Vol.4, pp. 79-90.

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