Optimizing a linear function over an efficient set

In this work, we deal with a global optimization problem (P) for which we look for the most preferred extreme point (vertex) of the convex polyhedron according to a new linear criterion, among all efficient vertices of a multi-objective linear programming problem. This problem has been studied for decades and a lot has been done since the 70’s. Our purpose is to propose a new and effective methodology for solving (P) using a branch and bound based technique, in which, at each node of the search tree, new customized bounds are established to delete uninteresting areas from the decision space. In addition, an efficiency test is performed considering the last simplex tableau corresponding to the current visited vertex. A comparative study shows that the proposed method outperforms the most recent and performing method dedicated to solve (P).