The transit time constrained fixed charge multi-commodity network design problem

Abstract

This paper introduces the transit time constrained fixed charge multi-commodity network design problem. Transit times are origin-to-destination time limits for the commodities, which appear for example in transport systems with perishable goods. We discuss how to model the problem and present three different formulations of it. The first formulation is an exponential size path formulation, which we solve with a branch-and-price algorithm. Several speed up techniques from the literature on fixed charge multi-commodity network design problems are implemented, such as lifted cover inequalities and the recently proposed deep dual-optimal inequalities. In an extensive set of computational experiments, we show that these inequalities significantly improve the performance of the algorithm. The other two formulations are of polynomial size: one uses path indices and the other uses time indices. While the branch-and-price algorithm outperforms solving the compact formulations with a general-purpose mixed-integer programming solver, the study of compact models helps better understand the problem, and we can use them as benchmarks. A detailed sensitivity analysis of the branch-and-price algorithm shows that longer transit times and an increased ratio of fixed charge to flow cost increase the difficulty of solving the problem whereas the arc capacity has less impact. We further discuss in-depth implementational details.