An efficient branch-and-bound algorithm to optimize a function over a nondominated set

This study introduces an algorithm based on the branch-and-bound approach for optimizing a main function Psi over the nondominated set of a multiobjective integer programming (MOIP) problem. Initially, Psi is optimized within the feasible solution set of the MOIP. A new efficiency test combining Benson's test with Psi is then developed using an auxiliary optimization program. This program provides both an efficient solution and a lower bound for Psi. Moreover, this solution is the best one for Psi when compared to its alternative solutions for MOIP. Subsequently, efficient cuts are incorporated into the criteria space to eliminate dominated points. Furthermore, the algorithm is tailored to handle scenarios where the objective involves optimizing a linear combination of multiobjective programming criteria over the nondominated set. The study concludes by showcasing the superior performance of the proposed two algorithms through comparison with existing approaches on well-known problem instances from the literature.