International Journal of Computer Applications (0975 – 8887) Volume 115 – No. 3, April 2015 10 Fuzzy Goal Programming Approach to Solve Linear Multilevel Programming Problems using Genetic Algorithm Papun Biswas Department of Electrical Engineering JIS College of Engineering Kalyani-741235 West Bengal, India Bijay Baran Pal Department of Mathematics University of Kalyani Kalyani-741235 West Bengal, India ABSTRACT This paper introduces a priority based fuzzy goal programming (FGP) method for modelling and solving multilevel programming problem (MLPP) through genetic algorithm (GA). In model formulation, the individual best solution of objectives of each of the decision makers (DMs) is determined by using the GA method for fuzzy description of the objectives. Then, tolerance membership functions of the defined fuzzy goals are constructed for measuring the degree of satisfaction of goal achievement and there by degree of optimality of the decision vectors controlled by the higher level DMs. In the executable FGP model, minimization of the under-deviational variables of the defined membership goals with highest membership value (unity) as the aspiration levels of them on the basis of pre- emptive priority is taken into consideration in the decision making context. In the solution process, sensitivity analysis with variations of priority structure of model goals is performed and then Euclidean distance function is used to identify the appropriate priority structure under which the most satisfactory decision can be reached in the decision making horizon. In the proposed GA scheme, roulette-wheel selection scheme, single point crossover and uniform mutation are adopted in the decision search process with regard to reach a satisfactory solution in the proposed hierarchical decision system. The effective use of the proposed approach is illustrated through a numerical example. Performance comparisons are also made to highlight the superiority of the proposed approach over the approaches studied previously. Keywords Euclidean Distance, Fuzzy Programming, Fuzzy Goal Programming, Genetic Algorithm, Goal Programming, Multilevel Programming. 1. INTRODUCTION In the field of mathematical programming (MP), multilevel programming (MLP) [1] was developed to solve decentralized planning problems with multiple DMs in a hierarchical decision making organization. In MLPP, the execution of the decision power is sequential from a higher level to lower level, and each DM tries to optimize his own benefit under a conflicting environment in the hierarchical levels. The concept of hierarchical decision problem as a special field of study in the area of MP was first suggested by Burton and Obel [2] in 1977. During 1980s, a considerable number of solution approaches for MLPPs as well as bilevel programming problems (BLPPs) [3] as a special case of MLPP have been deeply studied in [4,5,6] by pioneer researchers in this field. But, in the real-life decision situations, it may be mentioned that the previous approaches are computationally not very efficient, especially for large and complex hierarchical decision problems. In most of the classical approaches for solving hierarchical decision problems developed so far, it was found that the decision power of a higher level DM is often dominated by a lower level DM. However, in hierarchical decision structure of a decentralized decision system, it is generally assumed that the DMs cooperative each other to reach a minimum level of satisfaction for smooth running the activities of the organization. In such a situation, the fuzzy programming (FP) [7] approach based on the concept of fuzzy set theory (FST) [8] has been introduced to solve hierarchical decision problems [9]. But, due to conflicting in nature of objectives, there is a possibility of rejecting the solution again and again by followers and re-evaluation of the problem with elicited membership values of the membership functions is repeatedly involved in the solution search process. As a result, decision deadlock often arises in a decision making situation. Thereafter, the supervised search procedure with use of max-min operator introduced by Bellman and Zadeh [10] was studied in [11] to solve such decision problems. The conventional FP approach have been further extended by Shih and Lee [12] in 2000 to solve hierarchical decision problems from the view point of making a balance of decision powers of DMs in the decision making context. In using such an approaches, the elicited membership functions for the fuzzy goals are also need be redefined again and again to reach a satisfactory decision in the solution search process. To avoid the computational difficulty with a FP approach, FGP approach [13] as an extension of conventional goal programming (GP) [14] that is based on the ‘goal satisficing philosophy’ [15], has been studied in [16] for making decision with regard to achievement of multiple fuzzy goals in uncertain environment. The FGP based solution approach to BLPPs has been studied in [17] and further extended to solve MLPPs in