Efficient cuts for generating the non-dominated vectors for Multiple Objective Integer Linear Programming

While most of the published exact methods for solving multiobjective integer linear programming (MOILP) problems are based on solving many integer linear programs (ILP), we propose a branch and bound multi-objective method to find the whole non-dominated set. Two types of nodes are created in the tree. The first type corresponds to not integer optimal solution of solved (LP) problems . In this case, a branching procedure is activated to generate integer solutions. Each node of type two corresponds to a found integer solution. For this type of nodes, efficient cuts are established with the aim to remove dominated integer vectors whenever it is possible to improve at least one criterion. Otherwise, the current node is fathomed. The algorithm stops when all nodes are fathomed. In order to evaluate the performance of our method we have implemented two exact methods found in the literatue dedicated to general MOILP problems with two, three and four criteria. A comparative study is done on randomly generated problems. Moreover, tests are performed on several well-known problem instances for set packing problem on bi-objective case.