Using Ones Assignment Method and. Robust s Ranking Technique

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1 Applied Mathematical Sciences, Vol. 7, 2013, no. 113, HIKARI Ltd, Method for Solving Fuzzy Assignment Problem Using Ones Assignment Method and Robust s Ranking Technique A. Srinivasan Department of Mathematics P.R Engineering College Thanjavur , Tamilnadu, India yeacheenu4phd@gmail.com G. Geetharamani Department of Mathematics Anna University Chennai, BIT Campus, Tiruchirappalli , Tamilnadu, India geeramdgl@rediffmail.com Copyright 2013 A. Srinivasan and G. Geetharamani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract In this paper Ones Assignment Method is adopted to solve Fuzzy Assignment Problem (FAP). In this problem C i, j denotes the cost for assigning the n jobs to the n workers and C i, j has been considered to be triangular and trapezoidal

2 5608 A. Srinivasan and G. Geetharamani number denoted by C % i, j which are more realistic and general in nature. For finding the optimal assignment, we must optimize total cost this problem assignment. In this paper first the proposed fuzzy assignment problem is formulated to the crisp assignment problem in the linear programming problem (LPP) form and solved by using Ones Assignment Method [1] and using Robust s ranking method [4] for the fuzzy numbers. Numerical examples show that the fuzzy ranking method offers and effective tool for handling the fuzzy assignment problem (FAP) with imprecise render and requirement condition. The algorithm of this approach is presented, and explained briefly with numerical instance to show its efficiency. Keywords: Assignment Problem, Ones Assignment Algorithm, Fuzzy Number, Robust s ranking Method. 1. Introduction Fuzzy sets, introduced by Zadeh in 1965[6].Provide us a new mathematical tool to deal with uncertainty of information. Since then, fuzzy set theory has been rapidly developed. In this paper, we will review basic concepts of fuzzy sets, fuzzy number and fuzzy linear programming (FLP) which will be used in the rest section of the paper. The Assignment Problem (AP) is a special type of linear programming problem (LPP)[2] in which our objective is to assign a number of origins to the equal number of destinations at a minimum cost (or maximum profit). The mathematical formulation of the problem suggests that this is a 0-1 programming problem and highly degenerate all the algorithms developed to find optimal solutions of Assignment Problem. However due to its highly degeneracy nature a specially designed algorithm, known as Ones Assignment Method. In this paper, we provided a method to solve Fuzzy Assignment Problem (FAP), with fuzzy cost C % i, j.since the objectives are to minimize the total cost or to maximize the total profit, Subject to some crisp constraints, the objective function is considered also as a fuzzy number. First, to rank the objective values of the objective function by Robust s ranking method [4] for transform the Fuzzy Assignment Problem to a crisp one so that the conventional solution methods may be applied to solve Assignment problem. This idea is to transform a problem with fuzzy parameters to a crisp version and solve it by the Ones Assignment method.

3 Method for solving fuzzy assignment problem 5609 Different kinds of Assignment Problem are solved. Dominance of Fuzzy numbers can be explained by many ranking methods of these, Robust s ranking method [4] which satisfies the properties of compensation, linearity and additively. In this paper we have applied Robust s ranking technique. 2. Preliminaries 2.1. Fuzzy Set: A fuzzy set is characterized by a membership function mapping element of a domain, space, or the universe of discourse X to the unit interval [0, 1] i.e. {(, µ A ( )); } A = x x x X. Here µ A : X [0, 1] is a mapping called the degree of membership function of the fuzzy set A and ( x ) µ is called the membership value of x X in the fuzzy set A. These membership grades are often represented by real numbers ranging from [0, 1]. A 2.2. Normal fuzzy set: A fuzzy set A of the universe of discourse X is called a normal fuzzy set µ =1. implying that there exist at least one x X such that ( x ) 2.3. Convex: A fuzzy set A is convex if and only if, for any x 1, x 2 X, the membership function of A satisfies the inequality. ( x1 ( x2 )) ( ( x1 ) ( x2 )) µ λ + 1 λ min µ, µ,0 λ 1. A A A 2.4. Triangular Fuzzy Number: For a triangular fuzzy number A(x), it can be represented by A (a,b,c;1) with membership function µ(x) given by A µ ( x) x a, a x b b a 1, x = b = c x, b x c c b 0, otherwise

4 5610 A. Srinivasan and G. Geetharamani 2.5. Trapezoidal fuzzy number: For a trapezoidal fuzzy number A(x), it can be represented by A (a,b,c,d;1) with membership function µ(x) given by µ ( x) x a, a x b b a 1, b x c = d x, c x d d c 0, otherwise 2.6. α-cut: The α-cut of a fuzzy number A( x ) is defined as A( α ) = { x / µ ( x) α, α [ 0,1] }. 2.7 Arithmetic operations between two triangular and trapezoidal fuzzy numbers fuzzy numbers: Addition and Subtraction of two triangular fuzzy numbers can be performed as A% + B% = a + b, a + b, a + b ( ) (,, ) A% B% = a b a b a b Addition and Subtraction of two trapezoidal fuzzy numbers can be performed A% + B% = ( a1 + b1, a2 + b2, a3 + b3, a4 + b4,) as A% B% = a b, a b, a b, a b ( ) Robust s Ranking Technique [4] Robust ranking technique which satisfy compensation, linearity, and additively properties and provides results which are consist human intuition. If ã is a fuzzy number then the Robust Ranking is defined by 1 R ( a% ) 0.5 ( al, au ) dα = Where (, ) l u 0 a a is the α-level cut of the fuzzy number a % In this paper we use this method for ranking the objective values. The Robust ranking index R(ã) gives the representative value of fuzzy number ã.

5 Method for solving fuzzy assignment problem Ones Assignment Algorithm [1] Step1. In a minimization (maximization) case, find the minimum (maximum) element of each row in the assignment matrix( say ) and write it on the right hand side of the n a1,1 a1,2 a1,3... a1, n a1 a2,1 a2,2 a2,3... a 2, n a an,1 an,2 an,3... a n, n an Then divide each element of row of the matrix by.these operations create at least one ones in each rows. In term of ones for each row and column do assignment, otherwise go to step 2. Step2. Find the minimum (maximum) element of each column in assignment matrix ( ), and write it below column. Then divide each element of column of the matrix by.these operations create at least one ones in each columns. Make assignment in terms of ones. If no feasible assignment can be achieved from step (1) and (2) then go to step n 1 a1,1 / a1 a1,2 / a1 a1,3 / a1... a1, n / a1 2 a2,1 / a2 a2,2 / a2 a2,3 / a2... a2, n / a n an,1 / a2 an,2 / a2 an,3 / a2... an, n / a 2 b b b... b Note: In a maximization case, the end of step 2 we have a matrix, which all elements are along to [0,1], and the greatest element is one. Step 3. Draw the minimum number of lines to cover all the ones of the matrix. If the number of drawn lines less than n, then the complete assignment is not possible, while if the number of lines is exactly equal to n, then the complete n

6 5612 A. Srinivasan and G. Geetharamani assignment is obtained. Step 4. If a complete assignment program is not possible in step 3, then select the smallest (largest) element (say ) out of those which do not lie on any of the lines in the above matrix. Then divide by each element of the uncovered rows or columns, which d lies on it. This operation creates some new ones to this row or column. If still a complete optimal assignment is not achieved in this new matrix, then use step 4 and 3 iteratively. By repeating the same procedure the optimal assignment will be obtained. Priority plays an important role in this method, when we want to assign the ones. Priority rule, for minimization (maximization) assignment problem, assign the ones on the rows which have smallest (greatest) element on the right hand side, respectively. 5. Numerical example To illustrate the proposed method a fuzzy assignment problem is solved by using the proposed method. Example 5.1 Three persons are available to do three different jobs. From past records, the cost (in dollars) that each person takes to do each job is known and is represented by triangular fuzzy numbers and is shown in following. Fuzzy costs (in dollars) Person A B C job 1 (1,5,9) (8,9,10) (2,3,4) 2 (7,8,9) (6,7,8) (6,8,10) 3 (5,6,7) (6,10,14) (10,12,14) Find the assignment of persons to jobs that will minimize the total fuzzy cost.

7 Method for solving fuzzy assignment problem 5613 Solution: In Conformation to model the fuzzy assignment problem can be formulated in the following Min{R(1,5,9) +R(8,9,10) +R(2,3,4) +R(7,8,9) +R(6,7,8) +R(6,8, 10) +R(5,6,7) +R(6,10,14) +R(10,12,14) } Subject to + + =1 + + =1 + + =1 + + =1 + + =1 + + =1 ϵ[0,1] Now we calculate R(1,5,9) by applying Robst s ranking method. The membership function of the triangular fuzzy number (1,5,9) is µ ( x) x 1, 1 x 5 4 1, x = 5 = 9 x, 5 x 9 4 0, otherwise l u The α-cut of the fuzzy number (1,5,9) is ( a, a ) ( 4 1,9 4 ) 1 1 (% l u R a 1,1 ) = R ( 1,5, 9) = 0.5 ( aα, aα ) dα = 0.5(10) dα = α α = α + α for which Proceeding similarly, the Robust s ranking indices for the fuzzy costs aij are calculated as: R a% =9, R a =3, R a =8, R a =7, R a =8, R a =6, R a =16, R a =12 ( ) (% ) (% ) (% ) (% ) (% ) (% ) (% 1,2 1,3 2,1 2,2 2,3 3,1 3,2 3,3 ) We replace these values for their corresponding assignment problem in the linear programming problem. a% ij in which result in a convenient We solve it by ones assignment method to get the following optimal solution. Step 1 In a minimization case, find the minimum element of each row in the assignment matrix (say ) and write it on the right hand side of the matrix. Then divide each

8 5614 A. Srinivasan and G. Geetharamani element of row of the matrix by. These operations create at least one ones in each rows. In term of ones for each row and column do assignment, otherwise go to step 2. min / / / / 3 2 Step 2: Find the minimum element of each column in assignment matrix ( ), and write it below column. Then divide each element of column of the matrix by.these operations create at least one ones in each columns. 5 / / / / 3 2 min / / / / 3 2 Step 3: Make assignment in terms of ones ( ) 5 / / 7 ( 1) 8 / 7 ( 1) 5 / 3 2 We can assign the ones and the solution is (1, 2), (2, 2) and (3, 1) The fuzzy optimal total cost a % % % 13 + a22 + a31 = R(2,3,4) + R(6,7,8)+ R(5,6,7) = R (13, 16, 19). Example 5.2 Let us consider a Fuzzy Assignment Problem with rows representing 4 persons A,B,C,D and columns representing 4 jobs, job1, job2, job3, job4. The cost matrix a ij is given whose elements are trapezoidal fuzzy

9 Method for solving fuzzy assignment problem 5615 numbers. The problem is to find the optimal assignment so that the total cost of job assignment becomes minimum ( ) ( ) 3,5, 6, 7 5,8,11,12 (9,10,11,15) (5,8,10,11) % ( 7,8,10,11) ( 3,5, 6, 7) ( 6,8,10,12 ) ( 5,8,9,10 ) ai, j = ( 2, 4,5, 6) ( 5, 7,10,11) ( 8,11,13,15 ) ( 4, 6, 7,10) ( 6,8,10,12) ( 2,5, 6, 7) ( 5, 7,10,11) ( 2, 4,5, 7) Solution: In Conformation to model the fuzzy assignment problem can be formulated in the following ( ) ( ) ( ) ( ) ( ) ( ) ( 2, 4,5, 6) ( 5, 7,10,11) ( 8,11,13,15 ) ( 4,6, 7,10) ( 6,8,1 ) x + R( ) x + R( ) x + R( ) x Min{ R 3,5,6, 7 x + R 5,8,11,12 x + R(9,10,11,15) x + R(5,8,10,11) x R 7,8,10,11 x + R 3,5, 6,7 x + R 6,8,10,12 x + R 5,8,9,10 x R x + R x + R x + R x + R ,12 2,5, 6, 7 5, 7,10,11 2, 4,5, 7 } Subject to x + x + x + x = 1 x + x + x + x = x + x + x + x = 1 x + x + x + x = x + x + x + x = 1 x + x + x + x = x + x + x + x = 1 x + x + x + x = Now we calculate R( 3,5,6,7) by applying the Robst s ranking method. The membership function of the trapezoidal fuzzy number (3, 5, 6, 7) is µ ( x) The x 3, 3 x 5 2 1, 5 x 6 = 7 x, 6 x 7 1 0, otherwise α cut of the fuzzy number (3,5.6.7) is l u ( aα, aα ) = ( 2α + 3,7 α ) (% l u l u R a11 ) = 0.5 ( aα, aα ) dα = 0.5 ( aα + aα ) dα = 0.5 ( 10 + α ) dα = Proceeding similarly, the Robust s Ranking indices for fuzzy cost aij are

10 5616 A. Srinivasan and G. Geetharamani calculated as: R( a% ) =9, R( a% ) =11.25, R( a% ) =8, R( a% ) =9, 1,2 1,3 1,4 2,1 R( a% ) =5.25, R( a% ) =9, R( a% ) =8, R( a% 2,2 2,3 2,4 3,1 ) =4.25, R( a% ) =8.25, R( a% ) =11.75, R( a% ) =6.75, R( a% 3,2 3,3 3,4 4,1 ) =9, R( a% ) =5, R % 4,2 ( a 4, ) =8.25, R( a% 3 4,4 ) =4.5 We replace these values for their corresponding a% ij in (3) which result in a convenient assignment problem in the linear programming problem We solve it by ones assignment method to get the following optimal solution. Step 1 In a minimization case, find the minimum element of each row in the assignment matrix( say ) and write it on the right hand side of the matrix. Then divide each element of row of the matrix by.these operations create at least one ones in each rows. In term of ones for each row and column do assignment, otherwise go to step 2. min

11 Method for solving fuzzy assignment problem 5617 Step 2: Find the minimum element of each column in assignment matrix ( ), and write it below column. Then divide each element of column of the matrix by.these operations create at least one ones in each columns min Step 3: Draw the minimum number of lines to cover all the ones of the matrix. If the number of drawn lines less than n, then the complete assignment is not possible, while if the number of lines is exactly equal to n, then the complete assignment is obtained ( ) ( 1) ( 1) Step 4. If a complete assignment program is not possible in step 3, then select the smallest (largest) element (say ) out of those which do not lie on any of the lines in the above matrix. Then divide by each element of the uncovered rows or columns, which lies on it. This operation creates some new ones to this row or column. Make assignment in terms of ones

12 5618 A. Srinivasan and G. Geetharamani ( ) ( 1) ( 1) ( 1) We can assign the ones and the solution is (1,3),(2,2),(3,1) and (4,4). ie, A 3,B 2,C 1,D 4. The fuzzy optimal total cost a % + a % + a % + a % ( ) ( ) ( ) = R(9,10,11,15) x + R 3, 5, 6, 7 x + R 2, 4, 5, 6 x + R 2, 4, 5, 7 x = R(16,23,27,35) Conclusion In this paper, a simple yet effective method was introduced to solve fuzzy assignment problem by using ranking of fuzzy numbers. This method can be used for all kinds of fuzzy assignment problem, whether triangular and trapezoidal fuzzy numbers.the new method is a systematic procedure, easy to apply and can be utilized for all type of assignment problem whether maximize or minimize objective function. References [1] Hadi Basirzadeh Ones Assignment Method for Solving Assignment Problems (2012). [2] Hamdy A. Taha, Operations Research, An introduction 8 th Ed.(2007). [3] H.J. Zimmermann, Fuzzy set theory and its Applications, third Ed., Kluwer Academic, Boston [4] A. Solairaju and R. Nagarajan Computing Improved Fuzzy Optimal Hungarian Assignment Problems with Fuzzy Costs under Robust Ranking Techniques (2010).

13 Method for solving fuzzy assignment problem 5619 [5] Kadhirvel. K, Balamurugan. K, Method For Solving Hungarian Assignment Problems Using Triangular And Trapezoidal Fuzzy Number (2012). [6] A. Zadeh, Fuzzy Set (1965). [7] M. Hellmann Fuzzy logic Introduction, preprint. Received: July 10, 2013