FUZZY DIAGONAL OPTIMAL ALGORITHM TO SOLVE INTUITIONISTIC FUZZY ASSIGNMENT PROBLEMS

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1 International Journal of Civil Engineering and Technology IJCIET Volume 9, Issue 11, November 2018, pp , Article ID: IJCIET_09_11_037 Available online at ISSN Print: and ISSN Online: IAEME Publication Scopus Indexed FUZZY DIAGONAL OPTIMAL ALGORITHM TO SOLVE INTUITIONISTIC FUZZY ASSIGNMENT PROBLEMS S. Dhanasekar, A. Manivannan, V. Parthiban VIT, Tamilnadu, Chennai, India ABSTRACT In this article, we propose a new approach to solve intuitionistic fuzzy assignment problem. Classical assignment problem deals with deterministic cost. In practical situations it is not easy to determine the parameters. The parameters can be modeled to fuzzy or intuitionistic fuzzy parameters. This paper develops an approach based on diagonal optimal algorithm to solve an intuitionistic fuzzy assignment problem. A new ranking procedure based on combined arithmetic mean is used to order the intuitionistic fuzzy numbers so that Diagonal optimal algorithm [22] can be applied to solve the intuitionistic fuzzy assignment problem. To illustrate the effectiveness of the algorithm numerical examples were given.. Mathematics Subject Classification: 90C08, 90C70, 90B06, 90C29, 90C90 Key words: Intuitionistic Fuzzy number, Intuitionistic Triangular fuzzy number, Intuitionistic Trapezoidal fuzzy number, Intuitionistic Fuzzy arithmetic operations, Intuitionistic Fuzzy assignment problems, Intuitionistic Fuzzy optimal solution. Cite this Article: S. Dhanasekar, A. Manivannan, V. Parthiban, Fuzzy Diagonal Optimal Algorithm to Solve Intuitionistic Fuzzy Assignment Problems, International Journal of Civil Engineering and Technology IJCIET 911, 2018, pp INTRODUCTION Assignment Problem AP arises in the situations of assigning the things to other things for example assigning of persons to jobs, or classes to rooms, operators to machines, drivers to trucks, trucks to routes, or problems to research teams, etc. The assignment problem is a kind of linear programming problem LPP where the objective is to minimize the assignment cost. Various algorithms have been proposed to find solution to assignment problems, such as linear programming [9,10,14, 18], Hungarian algorithm [16], neural network [13], genetic algorithm [8]. But, in real life, the parameters of assignment problem are not crisp numbers. The fuzzy set theory introduced by Zadeh [1] in 1965 has been applied successfully in various fields. In 1970, Belmann and Zadeh [2] applied concepts of fuzzy set theory into the decisionmaking problems with uncertainty and imprecision. Many authors [ 3,4,19,21,23] developed many algorithms to solve fuzzy assignment problems. Different kinds of fuzzy assignment problems are solved in the papers [5, 12, 13, 14, 21]. Atanassov [7] proposed the Intuitionistic editor@iaeme.com

2 S. Dhanasekar, A. Manivannan, V. Parthiban Fuzzy Sets IFSs in 1986 is found to more useful to overcome the vagueness than fuzzy set theory. Here we investigate a intuitionistic fuzzy assignment problem. Senthil Kumar et al..[24] applied the Hungarian method to solve Intuitionistic fuzzy assignment problem. Nagoor Gani et al..[25] solved Intuitionistic fuzzy linear bottle neck assignment problem by using the perfect matching algorithm. In this paper, ranking based on combined arithemetic mean is applied for the comparison of the intuitionistic fuzzy numbers. Finally an Intuitionistic Fuzzy diagonal optimal method may be applied to solve an IFAP. In Section 2 gives some basic terminology and ranking of triangular and trapezoidal intuitionistic fuzzy numbers, Section 3, provides the proposed algorithm. Numerical examples are solved in Section 4. The concluding remarks are given in Section SECTION-1 The fuzzy set can be obtained by assigning to each element in the universe of discourse a value representing its grade of membership in the fuzzy set. The fuzzy number is a fuzzy set whose membership function continuous, convex and normal. is a piecewise The intuitionistic fuzzy set, in the universe of discourse X is the set of ordered triples, * + Where is a function from X to [0,1] such that,. Where membership represent the degree of membership and the degree of non membership of the element to respectively. A intuitionistic fuzzy number is called to be a intuitionistic trapezoidal fuzzy number if its membership function is of the form = and = { { Where and If then is called Intuitionistic triangular fuzzy number. Operations on Intuitionistic trapezoidal number and Intuitionistic triangular number: Addition: editor@iaeme.com

3 Fuzzy Diagonal Optimal Algorithm to Solve Intuitionistic Fuzzy Assignment Problems Subtraction: If is a intuitionistic triangular fuzzy number then the ranking of is given by,. If, then. This ranking technique satisfies Compensation, Linearity and additive properties. Intuitionistic Fuzzy assignment problem is defined by subject to where {. The above problem can also be depicted as follows Where the cost per unit in transporting from i th place to j th place. 2. PROPOSED ALGORITHM 1 Find the intuitionistic fuzzy penalty by subtracting the minimum intuitionistic fuzzy cost and the next minimum intuitionistic fuzzy cost and put it against the corresponding row. Do the same procedure for the column too and put it against the corresponding column. 2 Choose the maximum intuitionistic fuzzy penalty among all. If it is along the row locate the minimum intuitionistic fuzzy cost in that corresponding row and remove the corresponding row and the corresponding column of the intuitionistic fuzzy element. If it is along the column locate the minimum intuitionistic fuzzy cost in that corresponding column and remove the corresponding row and the corresponding column of the intuitionistic fuzzy element. 3 Let be the assigned cost for all the columns. Subtracting from each entry of the corresponding column of assignment matrix. 4 For each unassigned cell, form a loop such that one corner contains negative intutionistic Fuzzy penalty and remaining two corners are assigned intuitionistic cost values in corresponding row and column. Find the sum of diagonal cells of unassigned element, say editor@iaeme.com

4 S. Dhanasekar, A. Manivannan, V. Parthiban Choose the most negative process until all I d ~ ij. If any and exchange the assigned cell of diagonals. Repeat this I I d ~ ~ ij 0, then exchange the cells of diagonals at the end. SECTION-3 Example 1 Consider the following interval integer Assignment problem. Applying the proposed algorithm Choose the maximum intuitionistic Fuzzy penalty which is in the first row. Choose the minimum intuitionistic fuzzy value in that row. Strikeout the corresponding row and column. Repeating the same procedure for the remaining matrix Choose the maximum intuitionistic Fuzzy penalty which is in the second row. Choose the minimum intuitionistic fuzzy value in that row. Strike out the corresponding row and column Repeating the same procedure for the remaining matrix Choose the maximum intuitionistic Fuzzy penalty which is in the second row. Choose the minimum intuitionistic fuzzy value in that row. Strike out the corresponding row and column The assignments are * Checking the optimum of these assignments editor@iaeme.com

5 Fuzzy Diagonal Optimal Algorithm to Solve Intuitionistic Fuzzy Assignment Problems Subtracting the each element of the column from the corresponding assignment For the non-assigned cells *. = = = = =, = For all the non assigned cells The optimum solution is. So the assignments are optimum. =27,34,7315,44,67. SECTION-4 3. CONCLUSIONS In this paper, we applied diagonal optimal method to find a solution of an assignment problem in which parameters are triangular and trapezoidal intuitionistic fuzzy numbers. The total intuitionistic optimal cost obtained by this method remains same as that obtained by converting the total intuitionistic fuzzy cost by applying the ranking method. Also the membership and non-membership values of the intuitionistic fuzzy costs are derived. This algorithm can also be used in solving other types of assignment problems like, unbalanced, prohibited assignment problems. REFERENCES [1] Lotfi A.Zadeh, Fuzzy sets, Information Control [2] Richard E.Bellman and Lotfi A.Zadeh, Decision making in a fuzzy environment, Management Sciences B141-B164. [3] Amit Kumar and Anila Gupta, Assignment and Travelling Salesman Problems with Coefficients as LR Fuzzy Parameters, International Journal of Applied Science and Engineering [4] Amit Kumar, Amarpreet Kaur, Anila Gupta, Fuzzy Linear Programming Approach for Solving Fuzzy Transportation probles with Transshipment, J Math Model Algor., editor@iaeme.com

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