Hub Problems with Bounded Path Lengths
Résumé
In this paper, we consider the problem of designing a two level telecommunications network with service quality considerations. We are given a set of users or demand nodes and each of these nodes wants to communicate with all others. A fixed central hub is given and $p$ additional hubs should be chosen among the user nodes. Then each hub is connected by direct links to the central hub and each of the remaining nodes is connected directly to exactly one hub. The resulting network is a two level star/star network, where the network connecting the hub nodes to the central hub, called the backbone network, is a star, and each of the networks connecting user nodes to a hub node, called an access network, is a star.
We define two separate but related design problems for star/star networks. These problems are different from those existing in the literature as they incorporate a measure of service quality. First observe that in a star/star network, there exists a single simple path between any pair of demand nodes. If two demand nodes are connected to the same hub, then this path starts at the origin, goes directly to the hub, and then ends at the destination. If these nodes are connected to two different hubs, then the path starts at the origin, goes to the hub of the origin, then to the central hub, then to the hub of the destination, and ends at the destination. The length of the path connecting these two nodes is taken as a measure of the quality of service for this pair of nodes.
In our first problem, we are interested in optimizing the poorest service quality in the network. Hence, our aim is to select the location of hubs and assign the remaining nodes to hubs in such a way that the longest path between two distinct nodes in the resulting network has the smallest possible value. In other words, we would like to minimize the maximum length of the path connecting any pair of distinct demand nodes. We call this problem {\it Star $p$-hub Center Problem} and abbreviate with {\it SpHCP}.
In our second problem, we also incorporate the cost of routing into the design process. Our aim is to find a network such that the total cost of routing in the network is minimum and the length of the path connecting any pair of distinct nodes does not exceed a predetermined value. This yields a solution with a given level of service quality and minimum cost. We call this problem {\it Star $p$-hub Median Problem with Bounded Path Lengths} and abbreviate with {\it SpHMP-BP}. If there is no limit on the path lengths and the cost of routing the traffic between the hubs and the central hub is null, then our problem is equivalent to the problem of locating $p$ hubs and allocating the remaining nodes to these hubs to minimize the total cost of allocation. Hence the $p$-median problem is a special case of our problem. As the $p$-median problem is NP-hard, {\it SpHMP-BP} is also NP-hard.
The closest problem to our first problem {\it SpHCP} is the so-called $p$-hub center problem, which aims at minimizing the maximum transportation cost. The $p$-hub center problem is introduced by O'Kelly and Miller \cite{om91} and Campbell \cite{c94}.
Our second problem {\it SpHMP-BP} has features of both the $p$-hub median problem and the hub covering problem. The path length constraints are covering type constraints and are used in the hub covering problem. But in the hub covering problem, the objective is to minimize the number of hubs, whereas the objective of {\it SpHMP-BP} is to minimize the total routing cost in the network, and this is the same as the objective of the $p$-hub median problem.
In this paper, we propose formulations for {\it SpHCP} and {\it SpHMP-BP} and enhancements of these formulations by valid inequalities. Then, we discuss the outcomes of a computational study where we compare the performances of these formulations.