Enhancing Branch-and-Bound for Multi-Objective 0-1 Programming

TL;DR

This paper improves multi-objective branch-and-bound algorithms for 0-1 programming by introducing enhanced objective branching, probing, cut generation, and node selection techniques, leading to faster solutions and better performance on benchmark problems.

Contribution

The paper introduces novel enhancements to objective branching, probing, cut generation, and node selection in multi-objective branch-and-bound, significantly improving efficiency and solution quality.

Findings

Significant CPU time speed-ups with enhanced objective branching.

Outperforms existing branch-and-bound methods on benchmark problems.

Achieves results comparable or better than state-of-the-art objective space search methods.

Abstract

In the bi-objective branch-and-bound literature, a key ingredient is objective branching, i.e. to create smaller and disjoint sub-problems in the objective space, obtained from the partial dominance of the lower bound set by the upper bound set. When considering three or more objective functions, however, applying objective branching becomes more complex, and its benefit has so far been unclear. In this paper, we investigate several ingredients which allow to better exploit objective branching in a multi-objective setting. We extend the idea of probing to multiple objectives, enhance it in several ways, and show that when coupled with objective branching, it results in significant speed-ups in terms of CPU times. We also investigate cut generation based on the objective branching constraints. Besides, we generalize the best-bound idea for node selection to multiple objectives and we show that the proposed rules outperform the, in the multi-objective literature, commonly employed depth-first and breadth-first strategies. We also analyze problem specific branching rules. We test the proposed ideas on available benchmark instances for three problem classes with three and four objectives, namely the capacitated facility location problem, the uncapacitated facility location problem, and the knapsack problem. Our enhanced multi-objective branch-and-bound algorithm outperforms the best existing branch-and-bound based approach and is the first to obtain competitive and even slightly better results than a state-of-the-art objective space search method on a subset of the problem classes.