Abstract
We consider a cost allocation problem arising from a hub network problem design. Finding an optimal hub network is NP-hard, so we start with a hub network that could be optimal or not. Our main objective is to divide the maintenance and/or building cost of such network among the nodes. We consider two cases. In the one-way flow case, we assume that the cost paid by a set of nodes depends only on the flow they send to other nodes (including nodes outside the set), but not on the flow they receive from nodes outside. In the two-way flow case, we assume that the cost paid by a set of nodes depends on the flow they send to other nodes (including nodes outside the set) and also on the flow they receive from nodes outside. In both cases, we study the core and the Shapley value of the corresponding cost game.
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Notes
Assume that you are from Spain but you are in Argentina. If you receive a phone call from Spain, some phone companies consider it as an international call. The cost of this international call is divided by your company into two parts. The people who phone you pay the cost of a local call (inside Spain) and you pay the difference between the cost of an international call and the cost of a local call.
A generalization would be to assume that these costs are given by another cost matrix \(C^{h}=\left( c_{ij}^{h}\right) _{i,j\in N}\) with \(c_{ij}^{h}\le c_{ij}\) for all \(i,j\in N\). In our case, \(C^{h}=\alpha C\).
These tables correct some irrelevant typos presented in Matsubayashi et al. (2005).
Due to the symmetry of the matrices, the fact that all flows are positive, and the low congestion rate, both the one-way case and two-way case give very similar values for the Shapley rule.
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This work is partially supported by research grants ECO2014-52616-R from the Spanish Ministerio de Economía y Competitividad, ECO2017-82241-R from the Spanish Ministerio de Economía, Industria y Competitividad, and ED431B 2019/34 from Xunta de Galicia.
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Bergantiños, G., Vidal-Puga, J. One-way and two-way cost allocation in hub network problems. OR Spectrum 42, 199–234 (2020). https://doi.org/10.1007/s00291-020-00573-1
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DOI: https://doi.org/10.1007/s00291-020-00573-1