FLEXIBILITY IN ASSIGNMENT PROBLEM USING FUZZY NUMBERS WITH NONLINEAR MEMBERSHIP FUNCTIONS ATUL KUMAR TIWARI 1 , ANUNAY TIWARI 2 , CHERIAN SAMUEL 3 & SATISH KUMAR PANDEY 4 1,3 Department of Mechanical Engineering, Indian Institute of Technology, Banaras Hindu University, Varanasi, India 2 School of Management Sciences, Indira Gandhi National Open University, New Delhi, India 4 Computational Laboratory, Tapasthali Vidyashram Society, Varanasi, India ABSTRACT Assignment Problem (AP), as extensively discussed in operations research, industrial engineering and computational mathematics has been addressed under different headings and using different approaches. One of the approaches to solve an AP under uncertainty and in real world scenarios is using fuzzy theory. Fuzzy multi-objective linear programming usually deals with flexible aspiration levels that are indicative of optimality when considering all objectives or goals simultaneously with possible deviation in objectives or constraints in vague, imprecise and uncertain environment. Therefore in this study we propose an algorithm for solving multi-objective assignment problem (MOAP) with linear and nonlinear membership functions. The proposed model will give a compromised solution for best optimality and higher satisfaction level for parameters being considered in uncertain environment. The result obtained by using a linear membership function has been compared with the solution obtained by using some non-linear membership functions. A Numerical example is taken to illustrate the solution procedure. KEYWORDS: Multi Objective Assignment Problem (MOAP), Vague Parameters, Fuzzy Multi Objective Linear Programming, Non Linear Fuzzy Numbers INTRODUCTION The Assignment Problem (AP) is among the most-studied, well-known problem in operations research, industrial engineering and mathematical programming in which objective is to assign a number of jobs (tasks) to an equal number of machines (workers) so as to minimize the total cost incurred or to minimize the total time consumed for execution of all the jobs (tasks).It was first introduced by Votaw (1952) in military operation contexts. Assignment Problem can be categorized into three groups, Assignment Problem model with at most one task per agent, Assignment Problem model with multiple tasks per agent and Assignment Problem model for multidimensional assignment problem. The general solution techniques can be found in the literature survey of Pentico (2007) on assignment problems. Gilbert and Hofstra (1988),Kennington and Wang Z. (1992), Geetha and Nair (1993), Volgenant (1996), Amico and Martello (1997), Aora and Puri(1998) and Caron et al. (1999) give good of examples of these techniques applied to many situation. Hansen (2000) proposed some heuristics to solve very large and complex assignment problems. Tiwari et al. (2012) used fuzzy logic with non- linear membership functions to solve a multi-objective problem with three different ctiteria in a realistic situation. Tsai et al (1999) provided a solution for balanced multi-objective decision making problem associated with cost,time and quality by fuzzy concept. Zadeh(1965) first introduced the concept of fuzzy set theory. Then, Zimmermann (1978) first applied a suitable linear membership functions to solve linear programming problem with several objective functions. He showed that solutions obtained by fuzzy linear programming are always efficient. Tsai et.al. (1999) used fuzzy multi objective program for deployment of manpower. International Journal of Industrial Engineering & Technology (IJIET) ISSN 2277-4769 Vol. 3, Issue 2, Jun 2013, 1-10 © TJPRC Pvt. Ltd.