Network expansion problems are a special class of multi-period network design problems in which arcs can be opened gradually in different time periods but can never be closed. Motivated by practical applications, we focus on cases where demand between origin-destination pairs expands over a discrete time horizon. Arc opening decisions are taken in every period, and once an arc is opened it can be used throughout the remaining horizon to route several commodities. Our model captures a key timing trade-off: the earlier an arc is opened, the more periods it can be used for, but its fixed cost is higher, since it accounts not only for construction but also for maintenance over the remaining horizon. An overview of practical applications indicates that this trade-off is relevant in various settings. For the capacitated variant, we develop an arc-based Lagrange relaxation, combined with local improvement heuristics. For uncapacitated problems, we develop four Benders decomposition formulations and show how taking advantage of the problem structure leads to enhanced algorithmic performance. We then utilize real-world and artificial networks to generate 1080 instances, with which we conduct a computational study. Our results demonstrate the efficiency of our algorithms. Notably, for uncapacitated problems we are able to solve instances with 2.5 million variables to optimality in less than two hours of computing time. Finally, we provide insights into how instance characteristics influence the multi-period structure of solutions.

Benders decomposition, Heuristics, Lagrange relaxation, Multi-period, Network design,
Transportation Research. Part B: Methodological

Fragkos, I, Cordeau, J.-F. (Jean-François), & Jans, R.F. (2021). Decomposition methods for large-scale network expansion problems. Transportation Research. Part B: Methodological, 144, 60–80. doi:10.1016/j.trb.2020.12.002