FUZZY GOAL PROGRAMING APPROACH FOR SOLVING MULTI- LEVEL MULTI OBJECTIVE QUADRATIC PROGRAMMING PROBLEMS WHEN ROUGHNESS EXISTS IN CONSTRAINT

Abstract

In this thesis, a fuzzy goal programing (FGP) approach has been used for solving concave multi-level multi-objective quadratic programming problem (ML-MOQPP) when a constraint is a rough set. To avoid the complexity of the problem two crisp concave multi-level multi objective quadratic programing problems ML-MOQPP1 and ML-MOQPP2 were formulated; the first is a ML-MOQPP1 where the set of constraints is the upper approximation set, and the second is a ML-MOQPP2 where the set of constraints is the lower approximation set. An algorithm has been developed based on FGP approach to solve concave ML-MOQPP when a constraint is a rough set, by constructing the membership functions and fuzzy goals for both crisp problems. In doing so the individual best and worst solution of each objective functions subject to the system constraints were determined. The membership functions were defined as flexible membership goals by introducing under- and over-deviational variables and assigning highest membership value (unity) as aspiration level to each of them. Then the FGP models were formulated to obtain the solutions for both ML-MOQPP1 and ML-MOQPP2 by minimizing the sum of the negative deviational variables. Based on these crisp problems solution, the original problem solution(s) has been classified as surely or possibly solution. Illustrative examples given to demonstrate the convergence of algorithm near to ideal (desired) values of each objective functions. All solutions of the problem were obtained by using LINGO (17.0) mathematical software. The result showed that FGP approach was suitable to solve ML-MOQPP1 and ML-MOQPP2. The main advantage of the proposed FGP approach presented here is that the computational load with re-evaluation of the problem again and again by re-defining the membership values of the decision makers (DMs) for searching higher degree of satisfaction does not arise in the solution search process.