An L1-norm method for generating all of efficient solutions of multi-objective integer linear programming problem

Subject Areas : Mathematical Optimization

Ghasem Tohidi ¹ , Shabnam Razavyan ²

1 - 1Assistant professor Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran
2 - Assistant professor Department of Mathematics, Islamic Azad University, South Tehran Branch, Tehran, Iran

Received: 2020-10-01 Accepted : 2020-10-01 Published : 2012-01-01

Keywords: Multi-objective integer linear programming, Recession direction, L1-Norm, Single objective integer linear programming, Efficient solution,

Abstract :

This paper extends the proposed method by Jahanshahloo et al. (2004) (a method for generating all the efficient solutions of a 0–1 multi-objective linear programming problem, Asia-Pacific Journal of Operational Research). This paper considers the recession direction for a multi-objective integer linear programming (MOILP) problem and presents necessary and sufficient conditions to have unbounded feasible region and infinite optimal values for objective functions of MOILP problems. If the number of efficient solution is finite, the proposed method finds all of them without generating all feasible solutions of MOILP or concluding that there is no efficient solution. In any iteration of the proposed algorithm, a single objective integer linear programming problem, constrained problem, is solved. We will show that the optimal solutions of these single objective integer linear programming problems are efficient solutions of an MOILP problem. The algorithm can also give subsets of efficient solutions that can be useful for designing interactive procedures for large, real-life problems. The applicability of the proposed method is illustrated by using some numerical examples.