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Network design problems are at the heart of several applications in domains such as transportation, telecommunication, energy and natural resources. This paper proposes a new multi-period network design problem variant, in which modular capacities can be added or reduced along the planning horizon in order to adapt to demand changes. The problem further allows to represent economies of scales in function of the total arc capacity, a detail that has typically been overlooked in the literature. This paper particularly emphasizes the different alternatives to formulate the problem. We propose two different mixed-integer programming formulations and analyze further modeling alternatives. We theoretically compare the strength of all formulations and evaluate their computational performance in extensive experiments. The results suggest that a recent modeling technique using more precise decision variables yields the strongest formulation, which also results in significantly faster solution times. The use of this formulation may therefore be beneficial when considering similar problem variants. Finally, we also evaluate the economical benefits of the features introduced in this new problem variant, indicating that both the selection of the capacity levels and the capacity adjustment along time are likely to result in significant cost savings.

Mathematical formulations for multi-period network design with modular capacity adjustments