Incomplete hub covering network design problem
Bilkent University Industrial Engineering Department
Hubs are special facilities that serve as switching, transshipment and sorting points in many-to-
many distribution systems. Instead of serving each origin-destination pair directly, hub facilities
consolidate flow in order to take advantage from the economies of scale. The hub location problem
is concerned with locating hub facilities and allocating demand nodes to these located hub facilities
in order to route the traffic (flow) between origin-destination pairs. The hub location problem arises
especially in transportation (air passenger, cargo) and telecommunication network design. In this
study we focus on cargo applications.
In the existing hub location literature, it is commonly that the hub network is complete. In a
complete hub network there is a direct hub link present between every pair of hub nodes, i.e. every
hub sends its flow to another hub directly. There are only a few studies in the literature relaxing this
complete hub network assumption. However, we observed that almost all of the cargo firms
operating in Turkey do not employ a complete hub network structure. Figure 1 schematically
presents a comparison between a complete network, a complete hub network and an incomplete hub
network of 9 demand nodes.
a. b. c.
Figure 1: a. A complete network with 9 demand nodes, b. a complete hub network with 4 hubs and 9
demand nodes c. an incomplete hub network with 4 hubs and 9 demand nodes
Most of the literature on hub location focuses on sending flow with minimum cost and overlook the
service time. In cargo applications, time is of major concern. The hub covering problem is one of
the hub location problems that considers travel time between origin-destination pairs. The hub
covering problem deals with finding the necessary number and the location of hubs in order to
provide service within a given time limit between every origin-destination pairs.
In this study we focused on designing a hub network, which is not necessarily complete, for the
single allocation (where each demand center is allocated to a single hub) hub covering problem. We
do not impose any structure on the hub network other than connectivity. We present an efficient
mathematical formulation for this problem with O(n3) binary variables. The model minimizes the
cost of establishing hubs and hub links while guaranteeing service between every origin-destination
pair within a given time bound.
We have also explored the performance of some valid inequalities for our model. The model is then
tested on the various instances of the 25 node CAB data set (Figure 2) – a benchmark data set
introduced in the hub location literature – and on the 81 node Turkish network (Figure 3). We have
solved our mathematical model with the optimization solver CPLEX 10.1 to optimality. With our
model, we were able to solve all instances within 1 hour of CPU time to optimality. Figures 2 and 3
depict solutions of some instances in the two networks respectively.
Figure 2: A CAB data set solution with 4 hubs
Figure 3: A Turkish network solution with 5 hubs
In this study we were able to show that, in some instances, the service that is provided with a
complete hub network can also be provided with an incomplete hub network. Thus, the fundamental
assumption in the hub location literature of building complete hub networks is not essential.