International Journal of Computer Applications (0975 – 8887) Volume 63– No.5, February 2013 38 Fuzzy Multi Objective Assignment Linear Programming Problem based on L-R fuzzy Numbers Y.L.P.Thorani Dept.of Applied Mathematics GIS,GITAM University Visakhapatnam N. Ravi Shankar Dept.of Applied Mathematics GIS,GITAM University Visakhapatnam ABSTRACT Transportation and assignment models play significant role in logistics and supply chain management for reducing cost and time, for better service. In this paper, a fuzzy multi objective assignment problem using linear programming model is developed. The reference functions of L-R fuzzy numbers of fuzzy multi objective assignment problem are considered being linear and non-linear functions. This paper develops a procedure to derive the fuzzy objective value of the fuzzy multi objective assignment problem, in that the fuzzy cost coefficients, the fuzzy time and fuzzy quality are L-R fuzzy numbers. The method is illustrated with an example by various cases. Keywords Multi objective assignment; Yager’s ranking index; L-R fuzzy numbers; linear programming. 1. INTRODUCTION Assignment problem is used worldwide in solving real world problems. It plays an important role in industry and is used very often in solving problems of engineering, management science and it has many other applications. Project management is designed to control organization resources on a given set of activities, within time, cost and quality. Therefore, the limited resources must be utilized efficiently such that the optimal available resources can be assigned to the most needed tasks so as to maximize and minimize the profit and cost respectively. The assignment problem is one of the most important problem in mathematical programming in which a number of jobs (tasks or works) assigned to an equal number of machines (persons), so as to perform the jobs depend on their efficiency. It can be viewed as a balanced transportation problem in which all supplies and demands equal to 1, and the number of rows and columns in the transportation matrix are identical. Hiller and Libermann[1], Taha[2], Murthy[3] and Swarup et al.[4] discussed a single objective function in crisp environment for different type of assignment problems. Ravindran et al. [5] solved the assignment problem by using the transportation simplex method, but due to the high degree of degeneracy in the problem, it is often inefficient and not recommended to attempt to solve it by simplex method. Ravindran et al. [6] utilized another technique called Hungarian method to solve the minimizing assignment problem. The multi-objective assignment problem in crisp environment studied by Bao et al. [7]. Labeling algorithm to solve assignment problem with fuzzy interval cost proposed by Lin and Wen [8]. Chen [9] proposed fuzzy assignment model by considering that all the individuals involved have the same skills and Wang[10] solved a similar model by graphical approach. Mukherjee and Basu [11] resolved an assignment problem with fuzzy cost by Yager’s ranking method [12] which transforms the fuzzy assignment problem into a crisp assignment problem. Geetha et al. [13] expressed the cost-time minimizing assignment as the multicriteria problem. The fuzzy programming technique with linear membership function applied to solve the multi-objective transportation problem by Bit et al. [14]. Tsai et al [15] solved a balanced multi-objective decision making problem which is related with cost, time and quality in fuzzy environment. The multi-objective assignment problem as a vector minimum problem was resolved by Kagade and Bajaj [16] using linear and non-linear membership functions. In the paper [17] the solution procedure to the multi-objective assignment problem where the cost coefficients of the objective functions are interval values and the equivalent transformed problem explained using fuzzy programming techniques. Verma et al. [18] worked out a multi-objective transportation problem by some non-linear (hyperbolic and exponential) membership functions using fuzzy programming method. Dhingra et al. [19] defined other types of the non-linear membership functions and relate them to an optimal design problem. In this paper, we consider multi-objective assignment problem with fuzzy parameters for the case of construction process. Here, we let ij c ~ to be fuzzy payment to i th person for doing j th work, ij t ~ to be fuzzy time for i th person for doing j th work and ij q ~ to be fuzzy quality of i th person for doing j th work. Here, the fuzzy cost coefficients, the fuzzy time and fuzzy quality are L-R fuzzy numbers. Due to different unit of fuzzy cost, fuzzy time, and fuzzy quality, it is not possible to merge each other until we normalize them. Therefore, for normalization purpose divide fuzzy cost, fuzzy time and fuzzy quality by their corresponding maximum ranking index. By assigning a weight to the objectives according to their priorities the single objective function is obtained. Then, by ranking method, transform a newly formed single objective fuzzy assignment problem to a crisp assignment problem in linear programming problem form and it can be solved by any conventional method. The rest of the paper is organized as follows : In section 2, preliminaries of L-R fuzzy numbers,  -cut of L-R fuzzy number, reference functions and Yager’s ranking approach for various linear and non-linear functions using L-R fuzzy numbers are presented. In section 3, the proposed linear programming model for fuzzy multi objective assignment problem for various linear and non-linear functions, where the