Elsevier

Computers & Operations Research

Volume 75, November 2016, Pages 1-11
Computers & Operations Research

An adaptive perturbation-based heuristic: An application to the continuous p-centre problem

https://doi.org/10.1016/j.cor.2016.04.018Get rights and content

Highlights

  • Two new perturbation heuristics are proposed for the continuous p-centre problem.

  • A novel local search is designed based on covering circles.

  • A learning process is introduced making the heuristics self-adaptive and effective.

  • New best results including new optimal solutions are found.

Abstract

A self-adaptive heuristic that incorporates a variable level of perturbation, a novel local search and a learning mechanism is proposed to solve the p-centre problem in the continuous space. Empirical results, using several large TSP-Lib data sets, some with over 1300 customers with various values of p, show that our proposed heuristic is both effective and efficient. This perturbation metaheuristic compares favourably against the optimal method on small size instances. For larger instances the algorithm outperforms both a multi-start heuristic and a discrete-based optimal approach while performing well against a recent powerful VNS approach. This is a self-adaptive method that can easily be adopted to tackle other combinatorial/global optimisation problems. For benchmarking purposes, the medium size instances with575nodes are solved optimally for the first time, though requiring a large amount of computational time. As a by-product of this research, we also report for the first time the optimal solution of the vertex p-centre problem for these TSP-Lib data sets.

Introduction

Continuous location problems are concerned with the location of one or more facilities in the plane. These are characterised by the number of possible sites being infinite and hence the unconstrained location of new facilities can be anywhere. In other words, any point is considered as a potential location for a new facility. The objective of the p-centre problem is to minimise the maximum distance between all customers (demand points or fixed points) and their nearest facilities. This problem is particularly useful in locating emergency facilities, such as fire stations, police stations and hospitals, where it is aimed to minimise the longest response time.

For completeness, we cite a few pcentre related real life applications spanning over the last 25 years. One of the earliest applications considers the location of fifteen fire stations in the Belgian rural province of Luxembourg. This problem was investigated by Richard et al. [23] who used villages, sparsely populated hamlets and some roads in the country side as demand points, some of which also served as potential sites. The location of a number of health resources such as geriatric and diabetic health care clinics in the rural area of Burgos in Spain was examined by Pacheco and Casado [21] using scatter search. A study to locate a number of bicycle stations in the city of Isfahan, Iran, was conducted by Kavesh and Nasr [17] using harmony search. A real life application that aims to minimise the number of emergency warning sirens in Dublin (Ohio) was explored by Wei et al. [29] who adopted an enhanced Voronoi-based approach to cover the entire area with the minimum number of facilities. A humanitarian aid problem to locate a number of urgent relief distribution centres to help with the casualties due to an earthquake in Taiwan that measured at 7.3 on the Richter scale, and caused over 2500 deaths and 8000 injuries, was recently investigated by Lu [18] using simulated annealing.

The continuous (or planar) p-centre problem has a succinct geometrical interpretation. For example, the single unweighted facility location problem (i.e., p=1) corresponds to finding the smallest circle that encloses all n points (customers), with the centre being the location of the new facility. Equivalently, the continuous p-centre problem (p>1) aims to cover a set of customers in the plane with p circles where the radius of the largest circle is minimised.

The (weighted) p-centre problem can be formulated as follows (Drezner [6]).MinimizeX1,...,Xp{Max1in[Min1jpDi(Xj)]}where

Xj=(xj,yj)2: the coordinates of facility j(j=1,...,p)

(ai,bi): the coordinates of demand point i(i=1,...,n)

wi: the weight associated with demand point i(i=1,...,n)

Di(Xj)=wi[(xjai)2+(yjbi)2]1/2: the weighted Euclidean distance between the jth facility and the ith demand point (i=1,...,n;j=1,...,p).

For variable p, the continuous p-centre problem is known to be NP-hard (see Megiddo and Supowit [19]), whereas for fixed p, Drezner [6] shows that the problem can be solved in O(n2p+4) though it is computationally unattractive for large p.

The single facility minimax location problem (1-centre) in the continuous space has a long history, having been posed originally in 1857 by the English mathematician James Joseph Sylvester (1814–1897) who also proposed in 1860 an algorithm to solve it. Elzinga and Hearn [11] proposed an efficient geometrical-based algorithm for solving optimally the problem. Other authors attempted some enhancements to speed up the search, such as Xu et al. [30] and Elshaikh et al. [10] and references therein. For more details on the continuous 1-centre problem including a fascinating history on this topic, the reader will find the chapter by Drezner [9] to be informative.

Drezner [7] proposed two algorithms for the solution of the two-centre and two-median location problems with Euclidean distances on the plane. The idea is that the two customer sets in any solution can be separated by a straight line (i.e., n(n1)2possibilities). Since the optimal facility location in each of the two sets (p=1) can be easily found due to the convexity of the objective function, the problem reduces to finding an efficient way of defining all these straight lines and hence these corresponding subset pairs.

There is, however, a relatively small number of authors who have studied the p-centre problem; see Plastria [22] and the references therein. One of the commonly used approaches is based on Cooper's [5] locate–allocate procedure. In brief, the idea is to choose initially p facility points randomly or using a heuristic and assign each demand point to its nearest facility making p subsets. In each cluster the optimal single facility location is found using Elzinga–Hearn or an equivalent method. The allocation is then performed again followed by the optimal solution of p 1-centre problems. This is repeated until there is no improvement in the allocation. Drezner [6] presents two methods, namely, a multi-start similar to Cooper's locate allocate adapted to the p-centre problem (referred to as (H1)) followed by a composite heuristic made up of H1 and a post optimiser that allocates the critical points between the clusters (called (H2)). Eiselt and Charlesworth [12] propose three constructive and improvement-based heuristics. Their first one resembles the locate–allocate procedure of Cooper, the second uses the vertex substitution of Teitz and Bart [28] with the critical points used for reallocation, and their third one is based on the drop method. As the latter will be used in our computational results section, we briefly describe it here. The idea is to start with all n demand sites as potential sites and then combine the two nearest points to make up a new centre leading to n1 clusters. This process of exploring the two nearest centres to make up a combined centre continues until p clusters with their corresponding centres are found. The ‘locate–allocate’ process is then activated as an optional improvement step. A more flexible version is to allow a certain number of pairs with their corresponding customers to be explored and the pair corresponding to the combined cluster with the lowest radius is chosen instead of selecting the pair with the closest distance. A control parameter β(0<β1) is introduced to select these pairs with a value of 0.5 empirically shown to produce the best results. This flexible variant, known as STEPDOWN, outperforms their other two methods. Very recently, Elshaikh et al. [10] devise an enhanced version of the Elzinga and Hearn algorithm for the 1-centre problem which is then embedded within a powerful VNS-based heuristic to solve the p-centre problem. The results from H1, H2 and STEPDOWN heuristics will be used alongside those given in [10] for comparison purposes in Section 5.2.

For the case of area coverage, which can be of interest, for example, to agriculture, environment and mobile phone coverage technology, a Voronoi diagram-based heuristic, using an iterative procedure based on the locate–allocate principle, was proposed by Suzuki and Okabe [27]. This was then applied by Drezner and Suzuki [8] who added a post-optimiser using nonlinear programming to cover a square with p circles. Wei et al. [29] extended the above Voronoi-based approach to account for irregular and non-convex shapes, including the possibility of forbidden regions where the new facilities cannot be sited. Though the area and the point coverage problems are related, these preceding approaches should not be used directly for point coverage given that the results can be misleading as demonstrated by Murray and Wei [20].

Few papers deal with exact methods for the planar p-centre problem. Drezner [6] put forward an interesting idea of enumerating all the maximum sets given a threshold (the radius of the largest circle at a given iteration) to be used within a covering-based model. If the problem is feasible, the obtained feasible solution is then used to get a new threshold. The process is repeated until the covering problem has no feasible solution leading to the current threshold being the optimal solution. Results for small instances up to n=40 and p=5 were tested starting with the initial solution (threshold) found by the Drezner's heuristic H2 [6]. This optimal method will be revisited in the computational results section as it is found to be not as slow as originally mentioned in the literature (see Section 5.2). Excellent results for both the discrete and the continuous cases are found by Chen and Chen [3] who extended the work of Chen and Handler [4] in several interesting ways. The authors used three types of relaxation methods. One is to solve optimally for a small subset of the original problem, while gradually adding additional demand points (usually the farthest from the service points of the current feasible solution) until the solution becomes feasible for the original problem, and hence, may be considered as the optimal solution of the original problem. Two further relaxations were developed. These include a reverse relaxation where a lower bound is first found which is then gradually increased until the optimal solution is reached, and a binary relaxation where both upper and lower bounds are updated accordingly. The only optimal solutions for the planar p-centre problem reported by the authors are for the TSP data set with n=439.For comparison purposes, these optimal results will also be used in our computational results section (see Section 5.2).

It is worth noting that the proposed perturbation heuristic is, to our knowledge, the second only metaheuristic that is developed to investigate this class of location problem. This is an adaptive method that can easily be modified to tackle a variety of combinatorial and/or global optimisation problems. In addition, this approach solves large data sets with more than 1300 demand points with encouraging results. For benchmarking purposes, we have also implemented Drezner's optimal method and report, for the first time, the optimal solutions for medium size instances (i.e., n=575) though the computational time required was excessively large especially for small values of p (mostly exceeding 10 h of CPU time with a few that required nearly 24 h.

Though the pcentre problem can be seen as an old and well-established combinatorial problem, in our view it serves as an interesting and useful base to test innovative ideas which can then be extended and adapted for other related and more complex continuous location problems such as those with restricted non convex regions with and without capacity restriction, presence of fixed cost, just to cite a few.

The contributions of the study include

  • (1)

    The design of a powerful perturbation-based metaheuristic that uses an adaptive degree of perturbation and can be adapted to a variety of other combinatorial and global optimisation problems.

  • (2)

    A novel local search that is based on the concept of a ‘covering circle’ whose neighbourhood is dynamically adjusted.

  • (3)

    The incorporation of learning within the search, which we consider to be an invaluable ingredient in heuristic search design in general and in this new perturbation metaheuristic in particular.

  • (4)

    The generation of high quality results for large planar p-centre problems (some instances with more than 1300 customers) including the optimal solutions for the first time for n=575, as well as all the optimal solutions for their discrete counterpart problems.

The paper is organised as follows: the next section discusses the basic perturbation heuristic. In Section 3, the two local searches including a novel swap-based scheme using the concept of covering circles are first described, followed by the two new perturbation-based heuristics that use a dynamic level of perturbation. In Section 4, learning is introduced within the search. Computational experiments are given in Section 5 and our conclusions and suggestions for future research are summarised in the last section.

Section snippets

A brief on the basic perturbation-based heuristic

This approach guides the search by introducing some perturbations into the problem. For the p-centre problem these can be achieved by allowing the number of facilities of a solution to go over and under the required number of facilities (p) by a certain value (q). In other words, the solution is allowed to be infeasible in terms of the number of open facilities. In brief, the method works as follows: an initial solution of the p-centre problem is first found, and then the number of open

The new perturbation-based heuristic

In this study we extend the perturbation metaheuristic given in [24] by

  • (1)

    introducing flexibility in the level of perturbation using a variable value of q that is adaptively determined instead of being fixed throughout the search as initially used in the literature [24], [31].

  • (2)

    Tailoring the swap, add and drop moves to the p-centre problem.

  • (3)

    Examining two new local searches. One relates to the case when the solution is infeasible (i.e., the number of facilities in the solution is p±s,s=1,...,q) where

The integration of learning into the search

In this section we incorporate learning into our perturbation-based heuristics. The aim is to identify the most promising values of q,qMax and the depth of the covered area (i.e., the destination region that we insert the added facilities in).

The learning process consists of two phases. In the first phase, the information that is mentioned above is recorded during a certain time period (say for instance 25% of the total CPU time) which we call the learning phase. In the second phase, we use the

Computational results

The perturbation-based heuristics are coded in C++ and executed on a laptop computer with an Intel Core 2 Duo processor, 2.0 GHz CPU and 4 G memory. For the vertex p-centre problem, the IBM ILOG CPLEX12.5 Concert library is used. The proposed heuristics are tested on TSP-Lib data sets (n=439, 575, 783, 1002 and 1323) using values of p ranging from p=10 to 100 with an increment of 10.

To be consistent with previous results given in [10], we also used the CPU times corresponding to 10,000 iterations

Conclusion and suggestions

A new perturbation-based heuristic is designed to solve the continuous p-centre problem. The idea is to allow the number of facilities to be higher and lower than p in order to act as a filtering process where the promising facility locations tend to stay in the chosen set. We also guide the search by allowing the amount of perturbation to vary adaptively instead of being a constant throughout the search as originally proposed in the literature. A novel local search that uses the concept of

Acknowledgements

We would like to thank the referees and the area editor for their constructive comments that improved both the content as well as the presentation of the paper. The first author is also grateful to the University of Misurata for his PhD studentship.

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    This research has been supported in part by the UK Research Council EPSRC (EP/I009299/1), the Natural Sciences & Engineering Research Council of Canada Discovery Grant (NSERC #20541 – 2008), the Russian Federation Grant RFS 14-41-00039, the National Council for Scientific and Technological Development – CNPq/Brazil grant number 400350/2014-9, and the Spanish Ministry of Economy and Competitiveness, research project MTM2015-70260-P.

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