Reliable hub-and-spoke systems with multiple capacity levels and flow dependent discount factor

In this paper we investigate the reliable single allocation p-hub location problem with multiple capacity levels and flow dependent discount factor. We first present new and novel MIP formulations that are built upon the well-known uncapacitated FLOWLOC model proposed by O'Kelly and Bryan (1998). The proposed reliable models aim at simultaneously determining (a) the optimal location of the hubs, (b) the allocation of demand to these hubs,(c) the backup facilities for each demand point, (d) the hub capacity level to handle the normal flow, (e) the additional capacity to handle excessive rerouted flows due to possible hub disruption, (f) the values of discount factor for inter-hub links at normal and (g) the discount factor to be applied on inter-hub links should volume of flow increases because of hub disruption. The proposed mathematical models could solve small instances to optimality using a commercial optimiser such as CPLEX. To solve large instances we propose a variant of the VNS algorithm, namely, the reduced VNS. We present computational results including lower and upper bounds of the optimal solutions to problems with 15, 20 and 25 nodes and the upper bounds of the solutions to larger problems up to 170 nodes. Managerial insights for the reliable hub location problem with and without the use of flow dependent discount factors are presented and recommendations on the use of trade-off curves between the two objectives of minimizing the network cost in normal and disrupted conditions are also provided.