Hub and spoke network design with single-assignment, capacity decisions and balancing requirements

In this paper, an extension of the capacitated single-allocation hub location problem is considered in which the capacity of the hubs is part of the decision making process and balancing requirements are imposed on the network. The decisions to be made comprise (i) the selection of the hubs, (ii) the allocation of the spoke nodes to the hubs, (iii) the flow distribution through the sub network defined by the hubs and (iv) the capacity level at which each hub should operate. In the latter case, for each potential hub, a set of available capacities is considered among which one can be chosen. The objective is to minimize the total cost, which includes the setup cost for the hubs as well as the flow routing cost. Economies of scale are assumed for the costs. Balancing requirements are imposed to the network. In particular, a value is considered for the maximum difference between the maximum and minimum number of spoke nodes that are allocated to the hubs. Two mixed-integer linear programming formulations are proposed and analyzed for this problem. The results of a set of computational experiments using an off-the-shelf commercial solver are presented. These tests aim at evaluate the possibility of solving the problem to optimality using such a solver with a particular emphasis to the impact of the balancing requirements. The tests also allow an analysis of the gap of the bounds provided by linear relaxation.