The complete vertex p-center problem

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Abstract

The vertex p-center problem consists of locating p facilities among a set of M potential sites such that the maximum distance from any demand to its closest located facility is minimized. The complete vertex p-center problem solves the p-center problem for all p from 1 to the total number of sites, resulting in a multi-objective trade-off curve between the number of facilities and the service distance required to achieve full coverage. This trade-off provides a reference to planners and decision makers, enabling them to easily visualize the consequences of choosing different coverage design criteria for the given spatial configuration of the problem. We present two fast algorithms for solving the complete p-center problem: one using the classical formulation but trimming variables while still maintaining optimality and the other converting the problem to a location set covering problem and solving for all distances in the distance matrix. We also discuss scenarios where it makes sense to solve the problem via brute-force enumeration. All methods result in significant speedups, with the set covering method reducing computation times by many orders of magnitude.

Introduction

Let N be the set of discrete demands to be served, represented by the index i=1, 2, …, n. Let M be the set of discrete potential facility sites that may serve those demands, represented by the index j = 1, 2, …, m. Let dij be the distance from demand i to site j. The vertex p-center problem consists of locating p facilities among a set of m potential sites such that the maximum distance from any demand to its closest located facility is minimized. This article considers the problem in the Euclidean space, where the “service distance” is a real number radius of coverage around the located facilities.

The p-center (PC) problem was originally formulated by Hakimi (1964), and it is often used for equitable location of public services such as fire stations, police stations, ambulance depots (Daskin 2013; Marianov and ReVelle 1995; Shier 1977) since it minimizes the furthest distance of the service to any constituent. Variations include locating facilities anywhere in the Euclidean space (Drezner 1984), also known as the continuous or non-vertex problem; locating facilities on a network (Kariv and Hakimi 1979); covering continuous area demands rather than point demands (Suzuki and Drezner 1996); and partial coverage variants (Daskin and Owen 1999). Many other variations exist, as thoroughly cataloged in Calik et al. (2015) and Contardo et al. (2019). The formulations in this article expand upon the vertex p-center problem with discrete point facility site locations and demands, and it can be assumed henceforth that we always refer to this vertex p-center problem as opposed to any other variant.

The p-center problem is strongly NP-hard, since its formulation is of the form of a location-allocation mixed-integer programming problem. Location-allocation problems are typically not integer friendly (ReVelle 1993) and often require relaxations or heuristics to solve for large problem sizes (Dzator and Dzator 2013; Gadegaard et al. 2018; Mladenović et al. 2003; Rosing et al. 1979; Teitz and Bart 1968). Minieka (1970) was the first to realize the close relationship between the p-center problem and the location set covering problem (LSCP), where the LSCP consists of minimizing the number of facilities to completely cover a set of demands given each facility has a fixed covering radius. He used this relationship to heuristically solve the p-center problem by solving numerous LSCP problems, adjusting the coverage distance until a suitable bound was found. More recently, Elloumi et al. (2004) used the fact that optimal solutions to the p-center problem will always be distances that exist in the distance matrix for the problem (Daskin 2013). They developed a clever two-stage formulation that first converts the problem from a MIP to a binary radius formulation problem by replacing the continuous covering radius variable with a set of binary variables that represent “coverage disks” with radii of unique distances Dk that exist in the distance matrix. They then used an LP relaxation of the formulation to determine distance lower and upper bounds for a given p, and iterated an LSCP in-between the bounds to optimality. This formulation is very complex, but harnesses numerous deep insights into the p-center problem to create a faster solution algorithm.

Further refinements to finding tighter bounds were proposed by Calik and Tansel (2013), who presented a method similar to that of Elloumi et al. (2004) but at each iteration could update both the upper and lower bounds, thus converging on the solution more quickly. The most recent work on solving the PC problem comes from Contardo et al. (2019), who developed an improvement on the relaxation-based iterative algorithm by Chen and Chen (2009). The relaxation approach considers only local subsets of the demands within the coverage matrix. It assigns demands to facilities in the local subsets, and if all demands are allocated, then the solution is globally optimal. If some demands are not allocated to any facility, then the local subsets are expanded with a fixed number of demands, and the process repeats until all demands are assigned and the optimal solution is found. The method by Contardo et al. (2019), which they call CIK2018, is memory efficient by avoiding the computation and storage of the complete covering matrix, allowing it to solve for single p-center problems of up to one million points, and more quickly than previous methods. Their algorithm was developed for integer distance matrices and for small values of p, from 2 to 30 in their paper.

Some studies have modeled bi-objective center problems where the number of facilities p and the covering distance are varied. One such model, by Niblett and Church (2015), is intended for retail site location. It trades off the number of facilities with the separation distance between facilities in order to maximize retail profits, while adding some amount of over-coverage in order to prevent competition from other franchises to profitably enter a market. Other bi-objective franchise location models that use p-center for one objective include a greedy metaheuristic by Salazar-Aguilar et al. (2013), and the p-center p-dispersion bi-criteria model by Tutunchi and Fathi (2019).

Daskin (2013) outlines the close relationship that exists between the three primary types of covering location problems, namely the PC problem, the LSCP, and the maximal covering location problem (MCLP) (Church and ReVelle 1974). All three of these contain elements of the amount of coverage, the number of facilities, and the facility covering distance, in which one sets two and optimizes for the third. The relationship is as follows.

  • 1.

    The p-Center Problem.

  • Fixed number of facilities and achieve complete coverage.

  • Minimize the coverage distance.

  • 2.

    The Location Set Covering Problem.

  • Fixed coverage distance and achieve complete coverage.

  • Minimize the number of facilities.

  • 3.

    The Maximal Covering Location Problem.

  • Fixed number of facilities and coverage distance.

  • Maximize the amount of coverage.

The MCLP typically is solved in instances when the number of facilities and the coverage distance cannot achieve complete coverage. As such, it maximizes the weighted demand that can be covered. The close relationship between these three covering problems is clear when plotting the results of the complete p-center problem.

Section snippets

Problem formulation

The complete p-center problem solves the p-center problem for all p from 1 to m. Mathematically, this can be formulated with the classical mixed-integer programming formulation for the p-center problem to create a model we call CPC. Let yj = 1 if a facility is located at site j, 0 if not. Let xij = 1 if demand i is allocated to facility j, 0 if not. Let zp be the minimum coverage radius require to cover all demands with p facilities. We already defined dij to be the distance from demand i to

Trivial solutions

The CPC problem has a few trivial solutions, namely when p=1, when p=m−1, and when p=m. Let Dk be a sorted list of the k=0, …, K unique distances that exist in the symmetric distance matrix, where D0=0 and DK is the maximum distance between any two points in the data set. When p=m, the solution is zm=0, since every facility can cover itself. When p=m−1, the solution is zm−1=D1, where the two closest demands are covered by one facility with a covering radius that is the smallest distance in the

Computational experiments

We tested the efficiency of the various CPC algorithms on a variety of spatial data sets. For simplicity, the experiments here solve the symmetric variant where every demand location is also a potential facility site, i.e., N = M. All derived distances are Euclidean real number distances unless noted otherwise.

Conclusions

We introduced the complete p-center problem, and how it could be useful to planners and decision makers in determining the optimal trade-off between the number of facilities and the maximum distance from any demand to such a facility. We developed a basic formulation for the problem iterating the classical p-center MIP formulation over all m potential p values, called CPC-MIP. We then showed a modification called CPC-MIPUB that trims many variables and constraints of the CPC-MIP formulation

Publisher's Note

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Acknowledgements

We thank the Editor-in-Chief, guest editors, and two anonymous referees whose extensive comments and suggestions greatly helped to improve this manuscript. We also thank Greg Glockner and Gurobi for helping tooptimize our code.

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