Uncapacitated single and multiple allocation p-hub center problems

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Abstract

The hub median problem is to locate hub facilities in a network and to allocate non-hub nodes to hub nodes such that the total transportation cost is minimized. In the hub center problem, the main objective is one of minimizing the maximum distance/cost between origin destination pairs. In this paper, we study uncapacitated hub center problems with either single or multiple allocation. Both problems are proved to be NP-hard. We even show that the problem of finding an optimal single allocation with respect to a given set of hubs is already NP-hard. We present integer programming formulations for both problems and propose a branch-and-bound approach for solving the multiple allocation case. Numerical results are reported which show that the new formulations are superior to previous ones.

Introduction

Hubs form critical elements in many airline, transportation, postal and telecommunications networks. They are centralized facilities in these networks whose functions are to consolidate, switch and sort flows. Flow concentration and consolidation on the arcs that connect hub nodes (“hub arcs”) allow us to exploit transportation flow economies. It is also possible to eliminate many expensive direct connection arcs between origin destination pairs.

Typical applications of hub location include airline passenger travel [1], telecommunication systems [2] and postal networks [3]. Reviews of the hub location literature including theory and applications can be found in [4], [5]. Two major classes of objective functions are considered, median and center objectives.

The hub median problem is to locate hub facilities in a network and to allocate non-hub nodes to hub nodes such that the total transportation cost is minimized. It is applicable, for instance, in airline and telecommunication systems. This model of hub network design can sometimes lead to unsatisfactory results when worst-case origin–destination distances are excessively large. In order to avoid this drawback, hub center problems provide one option. Here the main objective is to minimize the maximum distance or cost between origin–destination pairs. This objective is particularly important for the delivery of perishable or time sensitive items.

While the hub median problem has been well studied in the literature—including several problem variants like the latest arrival hub location problem by Kara and Tansel [6]—the hub center problem has attained much less attention. It was introduced in [7], [8]. Campbell [8] formulates it as a quadratic program and reformulates it as a linear program. Several linearizations of the quadratic program are proposed by Kara and Tansel [9], who also provide an NP-completeness proof for the single allocation case and numerical comparisons for the linearizations. A single-relocation algorithm with tabu search is developed for a hub center problem considering flow volumes in Pamuk and Sepil [10] where extensive numerical experiments are also carried out. Based on a previous version of our paper from 2002, Hamacher and Meyer [11] proposed a solution approach for the hub center problem which consists of an iterative solution of hub covering problems. A polyhedral analysis of the hub center polytope can be found in [12].

In this paper we study the uncapacitated single allocation p-hub center problem (USApHCP) and uncapacitated multiple allocation p-hub center problem (UMApHCP) defined on a complete, symmetric network G=(N,E) with node set N={1,2,,N} and arc set E. Each arc [i,j] has infinite capacity and cost cij=cji satisfying the triangle inequality. We also assume that cij0 and often cii=0 though this is not required. Both USApHCP and UMApHCP require the selection of a complete subnetwork of p hubs where each hub has infinite node capacity for flow collection, transfer and distribution. The cost of flow between hub nodes is discounted by a factor α[0,1], such that the cost on arcs [m,k] connecting two hub nodes is reduced to αcmk. Depending on the application the costs could measure travel time or monetary costs. We use cost or path length interchangeably in this paper.

Note that in general there will be multiple equivalent optimal solutions since only the length of the longest path matters for the objective and hence there are often multiple choices for paths between other i and j pairs. However, we will generally assume that in this case the optimal solution is chosen to keep path lengths as short, e.g. by not including more than two hubs even if a third or fourth hub could be included between some pairs of nodes in the optimal solution.

In USApHCP, each non-hub node is allocated to a unique hub, whereas in UMApHCP, non-hub nodes can be allocated to more than one hub. Once the hub nodes have been selected, transportation between origin–destination pairs [i,j] can only take place via allocated hub nodes. Hence any transportation path has the form (i,k,l,j), where i is allocated to hub k and j is allocated to hub l.

The goal in the hub center problem is to minimize the discounted cost of the largest origin–destination path, i.e. minmaxi,jN(cik+αckm+cmj)where k and m are the hub nodes allocated to i and j, respectively. This represents the situation where the performance of the network is to be optimized (e.g. the amount of time to move goods between any pair of points in a mail network or supply chain). The discount on hub arcs in this case represents the fact that faster links, for example faster planes or higher throughput telecommunication links, can be employed between major hubs than would be economical to use over the other edges.

Some authors have commented in the context of p-hub median problems that the use of a completely interconnected hub network, where discounts apply only on inter-hub arcs, is somewhat unrealistic. Alternative models are possible. See for example [13], [14]. The virtue of the p-hub center model presented here is that it provides a simple model that allows alternative approaches and solution methods to be investigated.

In this paper, we present integer programming formulations for USApHCP and UMApHCP in Sections 2 and 3, respectively. The former is based on the innovative concept of hub radius and yields a two-index formulation with a linear objective function. For UMApHCP, which had not been studied before in the literature, we give two alternative three-index formulations. In Section 4, we prove that both problems are NP-hard and that the single allocation problem with respect to a given set of hubs is already NP-hard. We describe a shortest path based branch-and-bound method for solving UMApHCP in Section 5. Extensive numerical tests with regard to all formulations and the new branch-and-bound algorithm are presented in Section 6. They show that the proposed solution approaches are very efficient and outperform existing approaches by an order of magnitude. The results of our paper are summarized in the concluding Section 7.

Section snippets

The USApHCP

Define a binary variable Xik such that Xik=1 if and only if node i is allocated to hub k, and Xkk=1 if and only if k is a hub node. Let z be the maximum transportation cost between all origin–destination pairs. USApHCP is defined as a quadratic integer program in Campbell [8]:minmaxi,j,k,mN(cik+αckm+cmj)XikXjms.t.k=1NXik=1,i=1,,NXikXkk,i,k=1,,Nk=1NXkk=pXik{0,1},i,k=1,,NHere, the objective is to minimize the maximum transportation cost between all origin–destination pairs, the first

The UMApHCP

Since in UMApHCP a node can be allocated to several different hubs, the binary variable Xik needed in USApHCP is no longer required. Subsequently, we give a four-index formulation for UMApHCP. Let yijkm be a binary variable such that yijkm=1 if and only if for the flow between i and j, i is allocated to k and j to m. Let Zk be a binary variable defined by Zk=1 if and only if node k is selected to be a hub. Then UMApHCP is to find an optimal solution of the following optimization problem:minzs.t.

Complexity of p-hub center problems

In the last two sections, we have formulated USApHCP and UMApHCP as (mixed) integer programs. In this section we show that both USApHCP and UMApHCP are NP-hard. The first result is known from Kara and Tansel [9] who use a transformation from the dominating set problem. We present in the following a very simple proof which also shows the relation between hub center and general (non-hub) other problems and which can be generalized to a second version of hub problems.

Proposition 4.1

USApHCP and UMApHCP are

A branch-and-bound approach for UMApHCP

There is some similarity between the uncapacitated multiple allocation p-median problem (UMApHMP) and UMApHCP. Both are NP-hard problems, and their allocation problems can be solved in polynomial time by solving a series of the shortest path problems (see the last paragraph of Section 3). Ernst and Krishnamoorthy [22] propose an efficient branch-and-bound algorithm for solving UMApHMP by implicitly exploring all possible hub combinations in the node set N. When the number of hubs is small, the

Numerical experiments

We tested our algorithms for both USApHCP and UMApHCP with the CAB data set [24], [25] and with the AP data set [23]. The CAB data set is generated from the Civil Aeronautics Board Survey of 1970 passenger data in the United States. The AP data set is derived from the real-world application of a postal delivery network.

In test problem a.b.c of CAB, there are a nodes and b hubs, and c represents the economic discount factor for the cost of transfer of flow between hub nodes ranging from 0.2 to

Conclusions

In this paper we have studied USApHCP and UMApHCP. We have developed a new mixed integer programming formulation for USApHCP and two integer programming formulations for UMApHCP. Both problems are proved to be NP-hard even when the economic discount factor is zero. We also showed that the allocation sub-problem of USApHCP is NP-hard. A shortest path based branch-and-bound method is proposed similar to that developed in [22] for UMApHMP.

We have carried out numerical experiments using well-known

Acknowledgements

The authors are grateful to Mark Horn for carefully proof reading an earlier draft of this paper. We are thankful to two referees for their constructive comments which have helped to improve the presentation of the paper significantly.

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