Branch-and-bound for biobjective mixed-integer linear programming

TL;DR

This paper introduces a generic branch-and-bound algorithm for biobjective mixed-integer linear programming, focusing on finding all Pareto solutions with new dual bounds, fathoming checks, and gap measurement techniques.

Contribution

It presents novel algorithms for dual bounds, fathoming, and gap measurement, and implements a decision space search approach with comparisons to existing objective space methods.

Findings

Effective in solving literature and generated instances

Outperforms triangle splitting method in certain scenarios

Provides comprehensive computational experiments

Abstract

We present a generic branch-and-bound algorithm for finding all the Pareto solutions of a biobjective mixed-integer linear program. The main contributions are new algorithms for obtaining dual bounds at a node, for checking node fathoming, presolve and duality gap measurement. Our branch-and-bound is predominantly a decision space search method since the branching is performed on the decision variables, akin to single objective problems, although we also sometimes split gaps and branch in the objective space. The various algorithms are implemented using a data structure for storing Pareto sets. Computational experiments are carried out on literature instances and also on a new set of instances that we generate using the MIPLIB benchmark library for single objective problems. We also perform comparisons against the triangle splitting method from literature, which is an objective space search algorithm.