Optimization of a quadratic programming problem over an integer efficient set

Multi-objective programming problem often contains numerous efficient solutions, which con-fuses the decision-maker. To assist in selecting the most desirable solution, optimizing a function over the efficient set becomes crucial. In this paper, we present a novel method for optimizing a general quadratic function over the efficient set of a multi-objective integer linear programming problem. To solve this problem, a ranking approach and efficiency test is utilized. The proposed methodology obtains a globally optimal solution by systematically scanning ranked solutions of an integer quadratic programming problem until the efficiency condition is satisfied. For generating ranked solutions, we construct a related integer linear programming problem. Then, ranked solutions of the integer linear programming problem are used for enumerating ranked solutions of the integer quadratic programming problem. The convergence of our algorithm is established theoretically, and its steps are illustrated using a numerical example. Aparticular case of the proposed method for optimizing a linear function over the efficient set of a multi-objective integer linear programming problem is also discussed. Further, extensive computational results demonstrate the effectiveness of our method for solving problems with large number of constraints, variables, and objective functions. Moreover, comparative analysis shows that the developed algorithm came out to be computationally more efficient as compared to the existing state-of-the-art algorithms.