A New Approach for Solving Balanced and/or Unbalanced Intuitionistic Fuzzy Assignment Problems

 

M. Jayalakshmi

Assistant  Professor, Department of Mathematics, School of Advanced Sciences, VIT University, Vellore-14, India.

 

ABSTRACT:

In this paper a new algorithm is proposed to find the optimal solution and total optimal intuitionistic fuzzy cost for the given intuitionistic fuzzy assignment (IFA) problems using crisp linear programming (LP) problem. In this proposed method, all the parameters are represented by triangular fuzzy number. The fuzzy optimal solution of the IFA problems obtained by the proposed method, do not require Hungarian algorithm and ranking method. The proposed algorithm is very easy to understand and apply to find intuitionistic fuzzy optimal cost taking place in the real life situations.

 

 

 

1 INTRODUCTION:

The assignment problem (AP) is a special type of LP problem in which our objective is to assign n number of jobston number of machines (persons) at a minimum cost. To find solution to assignment problems, several algorithm such as Hungari analgorithm⁽⁷⁾ and ranking method^{(1,2,6,8,9,11,12)} have been developed.

 

However, in real life situations, the parameters of AP are inexact numbers instead of fixed real numbers because time/cost for doing a job by a facility (machine/person) might differ due to different reasons. The theory of fuzzy set introduced by Zadeh⁽¹³⁾ in 1965 has achieved successful applications in various fields. More recently, Kalaiarasi et al.⁽⁵⁾ solved FA model with triangular  fuzzy numbers. Sriniwas and Ganesan⁽¹⁰⁾ introduced a new branch and bound technique to solve FA problems. Anchal Choudhary et.al⁽²⁾

 

Solved FA problem by ranking function and by branch and bound method. Thorani and Ravi Sankar⁽¹²⁾ proposed a new algorithm in classical LP for FA problem based on the ranking method. Amit Kumar et.al⁽¹⁾  sovled FA using ranking method. Kirubha⁽⁶⁾ used ranking method to defuzzy FA problem. 

 

The concept of  Intuitionistic  Fuzzy (IF) setsproposed by Atanassov⁽³⁾ in 1986 is found to be highly useful to deal with vagueness. Here in this paper, we investigate a more realistic problem, namely an intuitionistic fuzzy assignment (IFA) problem. Let be the intuitionistic fuzzy cost of assigning the jth job to the ithmachine. We assume that one machine can be assigned exactly one job; also each machine can do at most one job. The problem is to find an optimal assignment so that the total intuitionistic fuzzy cost of performing all jobs is minimum or the total intuitionistic fuzzy profit is maximum.

 

More recently, Senthil Kumar and Jahir Hussain⁽⁹⁾ solved using Hungarian method by transforming IFA problem into crisp AP. Sagaya Roseline et al.⁽⁸⁾ considered Fuzzy Approximation method to find the optimal assignment of IFA, using ranking method. Srinivas and Ganesan⁽¹¹⁾ introduced a new ranking method and then by branch and bound method to solve IFA problem.

 

2. Preliminaries:

We need the following mathematical orientated definitions of IF set, triangular IF number and membership function and non-membership function of an IF set/number which can be found^(3,8,9,11).

 

3.2 Algorithm to solve intuitionistic fuzzy assignment using linear programming:

In this section, a new algorithm is proposed to find minimum total intuitionisticfuzzy cost,maximum totalintuitionisticfuzzy cost and to find the minimum total intuitionisticfuzzy cost whenrestrictions are made in the IFA problems using linear programming.

 

The steps of the proposedmethod are as follows:

 

Step 1: First test whether the given intuitionisticfuzzy cost matrix of a IFA problem is a balanced one or not. 

 

If it is balanced one (i.e, number of persons equal to the number of works) then go to step 3.  If it is an unbalanced one (i.e, number of persons are not equal to the number of works) then go to step 2.

 

Step 2: Introduce dummy rows and/or column with zero intuitionisticfuzzy costs so as to form a balanced one.

 

Step 3: Formulate the fuzzy intuitionisticassignment problem based on the chosen condition into the following fuzzy linear programming problems.

 

 

4. CONCLUSION:

 In this paper, a new algorithm has been proposed to find the optimal assignment and total optimal intuitionistic fuzzy cost for aintuitionistic fuzzy assignment problem using crisp linear programming problem. Hungarian algorithm⁽⁷⁾ and ranking method^{(1,2,6,8,9,11,12)}were not used.Since, the proposed method is based only on crisp LP problem it is very easy to solve IFAproblems having more number of fuzzy variables with help of existing computer software.

 

5. REFERENCES:

1.       Amit Kumar, Anila Gupta and Amarpreet Kumar. Method for Solving Fully Fuzzy Assignment Problems Using Triangular Fuzzy Numbers. International Journal of Computer and Information Engineering. 3(4); 2009.

2.       Anchal Choudhary, Jat R.N, Sharma S.C and Sanjay Jain. A new algorithm for solving fuzzy assignment problem using branch and bound method. International Journal of Mathematical. 7(3); 2016: 5-11.

3.       Atanassov K.T. Intuitionistic fuzzy sets.Fuzzy Sets and Systems. 20(1); 1986: 87- 96.

4.       Chen M.S. On a fuzzy assignment problem. Tamkang J. 22; 1985: 407-411.

5.       Kalaiarasi K., Sindhu S and Arunadevi M. Optimization of fuzzy assignment model with triangular fuzzy numbers. International Journal of Innovative Science, Engineering & Technology. 1(3); 2014: 10-15.

6.       Kirubha G. Prompting Method for Solving Fuzzy Assignment Problem under Fuzzy Environment. Imperial Journal of Interdisciplinary Research. 2(4); 2016: 682-688.

7.       Kuhn H.W. The Hungarian method for the assignment problem. Naval Research Logistics Quarterly. 2; 1955: 83-97.

8.       SagayaRoseline and Henry Amirtharaj. Methods to Find the Solution for the Intuitionistic Fuzzy Assignment Problem with Ranking of Intuitionistic Fuzzy Numbers. International Journal of Innovative Research in Science, Engineering and Technology. 4(7); 2015: 10008-10014.

9.       Senthil Kumar P and Jahir Hussain R. A Method for Finding an Optimal Solution of an Assignment Problem under Mixed Intuitionistic Fuzzy Environment. International Conference on Mathematical Sciences. 2016: 417-421.

10.     Sriniwas B and Ganesan G. Method for solving branch and bound technique for assignment problem using triangular and trapezoidal fuzzy numbers. International Journal of Management and Social Science. 3(3); 2015: 167-176.

11.     Srinivas B and Ganesan S. A Method for Solving Intuitionistic Fuzzy Assignment Problem using Branch and Bound Method. International Journal of Engineering Technology Management and Applied Sciences. 3(2); 2015: 227-237.

12.     Thorani Y.L.P and Ravi Sankar N. Fuzzy Assignment Problem with Generalized Fuzzy Numbers. Applied Mathematical Sciences. 7(71); 2013: 3511-3537. 

13.     Zadeh L.A. Fuzzy sets. Information and Computation. 8; 1965:338-353.

 

 

Accepted on 07.11.2016        © RJPT All right reserved

Research J. Pharm. and Tech 2016; 9(12):2382-2388.