Exact procedures for solving the discrete ordered median problem

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Abstract

The discrete ordered median problem (DOMP) integrates classical discrete location problems, such as the N-median, N-center and Uncapacitated Facility Location problems. It was introduced by Nickel (In: Fleischmann B, Lasch R, Derigs U, Domschke W, Rieder U, editors. Operations Research Proceedings 2000, Berlin: Springer, 2001. p. 71–76), who formulated it as both a nonlinear and a linear integer program. We propose an alternative integer linear programming formulation for the DOMP, discuss relationships between both integer linear programming formulations, and show how properties of optimal solutions can be used to strengthen these formulations. Moreover, we present a specific branch and bound procedure to solve the DOMP more efficiently. We test the integer linear programming formulations and this branch and bound method computationally on randomly generated test problems.

Introduction

Discrete location problems have been widely studied, last but not least due to their importance in practical applications. A number of survey articles and textbooks have been written on these problems, see, for example, [1], [2], [3], and references therein. Discrete location problems typically involve a finite set of sites at which facilities can be located, and a finite set of clients, which demand requests to be supplied from facilities. Even for the simplest of such problems, such as the Capacitated or Uncapacitated Facility Location Problem, which are relatively tractable, new methods and results continue to emerge (see for example [4], [5]) in an effort to solve larger problems, faster. Variations, such as hub location, are much more challenging, and it is still difficult to solve even modest sized problems to optimality (see for example [6], [7], [8]). Whilst many problem variations have been considered in the literature, we will focus on problems in which a fixed number of facilities must be located at sites chosen from among the given set of candidate sites, and in which a given client can only be supplied from a single facility. For each client-site pair, there is a given cost for meeting the demand of the client from a facility located at the site.

An interesting feature of discrete location problems is the variety of objective functions that have been considered. The median objective is to minimize the sum of the costs of supplying all clients from the facilities at the selected sites. The center objective is to minimize the maximum cost of supplying a client, from amongst the sites chosen, over all clients. The centdian objective is a convex combination of the median and center objectives; it aims to keep both the total cost and largest cost low. These are the three objectives most frequently encountered in the literature. Despite their common economic model very little is known on the common mathematical structure of the aforementioned location problems. Firstly, it is clear that the objective functions must depend on distances (or allocation costs) but the way these local information affect the location of the new facilities has been hardly studied. Secondly, very little is known about the connection between the difficulty of solving discrete location problems and the type of objective which was chosen.

The increasing need for discrete location models in strategic supply chain planning, see for example [9], has made it necessary to develop new and flexible location models. To that end, [10] introduced a new type of objective function which generalized the most popular objective functions mentioned above. This objective function applies a penalty to the cost of supplying a client which is dependent on the position of that cost relative to the costs of supplying other customers. For example, a different penalty might be applied if the cost of supplying the client was the fifth-most expensive such cost rather than the second-most expensive. This adds a “sorting” problem to the underlying facility location problem, making formulation and solution much more challenging.

For planar and network location problems the generalized model was studied in [11], [12], [13], [14], [15]. In [10], a formulation of the discrete case, called the discrete ordered median problem (DOMP), is discussed. A nonlinear integer programming formulation is developed, and a linearization with number of variables and constraints proportional to the number of sites, cubed, is proposed. However, no computation is attempted in [10], and there is no attempt to determine how effective integer programming approaches can be in solving the DOMP.

In this paper, we look at two different integer linear programming (ILP) formulations for the DOMP. Both have O(M2) constraints, but one uses O(M3) variables, while the other uses only O(M2) variables, where M is the number of clients (and sites). For hub location problems, it was found in [6] that using properties of optimal solutions to eliminate variables and add constraints to the formulation could strengthen it significantly. Here, we are able to strengthen both formulations using properties of optimal solutions. In order to compare the two formulations we generate test problems from eight representative classes. These eight classes attempt to represent the variety of objective functions possible for DOMP, for example, one of the classes uses a median objective while another class uses a trimmed mean objective. By testing the performance of the two ILP formulations on problems from these eight classes, we can get some measure of the degree of difficulty of the different classes and can compare the ILP formulations’ performance on each class. Finally, we develop a branch and bound (B&B) algorithm designed using the special structure of our model, using relatively simple lower bounds rather than lower bounds determined by solving an LP. We test the B&B algorithm on a large battery of problems, comparing its performance with that of the ILP approach using a standard solver.

Of the three approaches investigated, we would expect the O(M3) ILP formulation would yield the strongest lower bounds, followed by the O(M2) ILP formulation, with the easily calculated lower bounds used in the branch and bound procedure weakest. (In fact this was not uniformly the case, as we report in detail in Section 5.) However, the order is reversed if we base it on least computational effort required to calculate the bound: the branch and bound method requires only a very easy calculation, followed by the O(M2) ILP formulation, with the O(M3) formulation requiring most effort. The three methods thus explore the trade-off between the quality of bound and computational effort required at each node of the branch and bound tree. In the context of hub location, it was found in [6] that an intermediate position gave the best trade-off, but here that is not the case, as we shall discuss in detail in Sections 5 and 7.

The remainder of the paper is organized as follows. In the next section, we formally describe the DOMP and review the nonlinear formulation of [10]. In Section 3, we present the integer linear formulation already introduced in [10]. Moreover, we introduce an alternative ILP formulation for the DOMP. We go on in Section 4 to discuss properties of optimal solutions, which allow us to strengthen the formulations. In Section 5, we describe our random problem generator, and test problem sets, and present the results of computational experiments with all ILP formulations. A branch and bound method is introduced in Section 6 and numerical results comparing two branching schemes are given. Further computational results for this method and comparison of its performance with that of the integer programming linearizations are given in Section 7. Finally, in Section 8, we summarize our conclusions and give suggestions for further research.

Section snippets

Problem definition

Let A denote the given set of M sites, and identify these with the integers 1,,M, so A={1,,M}. Let C=(cij)i,j=1,,M be the given non-negative M×M cost matrix, where cij denotes the cost of satisfying the demand of client i from a facility located at site j. As is customary for location problems, we assume without loss of generality that the set of candidate sites is identical to the set of clients. If this is not the case we can simply introduce dummy clients. Let NM be the number of

Linearizations

First, we recall the linearization given in [10] which is analogous to that of Lawler [17] for the quadratic assignment problem. We use binary variableszikj=sikykj,i,j,k=1,,M,so zikj=1 if client k is supplied by a facility located at site j and is the ith cheapest client supplied; zikj=0 otherwise. The linearization of the DOMP is thus as follows:minsP,(x,y)Ni=1Mk=1Mj=1Mλickjzikjs.t.k=1Mj=1Mckjzikjk=1Mj=1Mckjzi+1,kji=1,,M-1,i=1Mzikj=ykjk,j=1,,M,j=1Mzikj=siki,k=1,,M,k=1Mi=1M

Strengthening the formulations

In this section, we will present some results to improve the linearizations. These improvements consist of additional constraints, strengthened forms of original constraints, or preprocessing steps, such as fixing some variables to zero, or relaxing integrality requirements on some variables. These help to reduce the computing time required to solve the DOMP, either by reducing the gap between the optimal objective function value and the relaxed LP solution, or by reducing the number of

Comparing the linearizations

In order to get a good comparison between the linearizations (LDOMP1) and (LDOMP2) we have developed an experimental design with the following factors and levels:

  • Size of the problem: As we have seen the number of sites, M, determines the dimensions of the cost matrix (C) and the λ-vector. Moreover, it is an upper bound of the number of facilities (N) to be located. For these reasons we consider M as a factor in our design and we propose four levels: M=8,10,12,15.

  • New facilities: N is the second

A branch and bound method

The driving variables for the DOMP are the binary xj variables, indicating which sites have been selected for facility location. Once these are known, the objective value is easy to calculate. All the other variables are in the ILP formulations to enable the costs to be calculated. It thus makes sense to build a B&B method based entirely on the xj variables, i.e. on decisions of whether or not a site is selected for facility location.

We develop a B&B in which each node represents a disjoint

Computational results

In this section, we compare the computational performance of the B&B method described in Section 6, with the max-regret branching rule, with that of the best linearization (for type T2 problems this is (LDOMP2) and for all other problem types it is (LDOMP1)). Again all codes were run on a Pentium III 800 Mhz with 1 GB RAM, and tested on problem instances with the structure described in Section 5. Upper bounds for the B&B method were initialized by a heuristic method based on variable neighborhood

Conclusions

We have discussed two integer linear programming (ILP) formulations for the DOMP, one having O(M3) variables (from [10]) and a new one having O(M2) variables (both have O(M2) constraints). We go on to prove a variety of properties of optimal solutions, which allow us to strengthen the formulations, via either additional constraints or preprocessing (in particular, fixing the values of some variables). After defining eight problem classes, according to objective function, we compare the

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Supported by Spanish grants numbers: BFM01-2378, HA2003:0121 and MTM04-0909.

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