M U L T I-L A Y E R S I N G L E A L L O C A T I O N H U B S E T C O V E R I N G P R O B L E M

Finding the location of hub facilities and the allocation of non-hub nodes to these located hub facilities are the aim of hub location problems. Commodities ow in the hub and spoke network in three phases; 1) Collecting: they move along their origin nodes to the assigned hub nodes. 2) Transferring: commodities ow through the hub arcs if necessary. 3) Distributing: commodities depart the hub network and arrive at destination nodes. Typical applications of hub locations include: airline passenger travel, telecommunication systems and postal networks. The hub location problem was originally introduced by OKelly (1986). Campbell (1994) provided the hub set and hub maximal covering problem with single and multiple allocations. In this work, we propose a multi-layer single allocation hub set covering problem over fully interconnected hub networks, and provide a formulation to this end. The postal service can be a multi-layer hub covering application. Postal companies oer dierent delivery time pledges, such as next day delivery, to their customers. However, due to geographical distribution of cities and the structure of highways, delivery within 24 hours between all city pairs is impossible if only ground transportation is employed. Chie y, due to competitiveness, it is better for postal companies to check the feasibility of including airlines in their distribution networks. This issue motivates us to introduce a multi layer model for hub covering problems, which can determine whether a ground or air route for each link is better in the hub network, in which the delivery time bound is guaranteed, as the covering radius. Trade hubs are another real application of the proposed approach. The trade growth of each country can occur if trade hubs are designed and developed properly. On the other hand, trade hubs connect most trade routes with some facilities to decrease total transportation costs with lowest delivery times, so, according to their geographic position, they should employ dierent modes of transportation system. We provide a clear example to introduce the model. For better illustration of the proposed model, a numerical example with four nodes is provided and solved by the CPLEX solver. Moreover, we test the performance of the model on the AP data set. Results of the AP data set for problems of size n = 10, 20, 25, 40 and 50, are given. Since the AP data set does not consider multi-layer data, we consider two layers for these benchmarks as assumptions. The computed gap from the lower bound, using the CPLEX solver, shows the eciency of the proposed approach. The results show that the problem lower bounds increase in a tighter covering radius, and the number of hub locations decreases in a looser covering radius.