Large region location problems

The paper describes two algorithms for solving single facility location problems in which the planar assumption is not appropriate. Transformations on the non-Euclidean spherical space are combined with efficient solution techniques in Eⁿ. An example problem illustrates the possible magnitude of error due to a planar assumption for a non-Euclidean space. Due to the nature of the problem (non convexity) a local optimum is obtained. Some computational experience is reported.     

Planar geometric location problems

We present anO(n ² log³ n) algorithm for the two-center problem, in which we are given a setS ofn points in the plane and wish to find two closed disks whose union containsS so that the larger of the two radii is as small as possible. We also give anO(n ² log⁵ n) algorithm for solving the two-line-center problem, where we want to find two strips that coverS whose maximum width is as small as possible. The best previous solutions of both problems requireO(n ³) time.Pankaj Agarwal has been supported by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), an NSF Science and Technology Center, under Grant STC-88-09648. Micha Sharir has been supported by the Office of Naval Research under Grants N00014-89-J-3042 and N00014-90-J-1284, by the National Science Foundation under Grant CCR-89-01484, by DIMACS, and by grants from the US-Israeli Binational Science Foundation, the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development. A preliminary version of this paper has appeared inProceedings of the Second Annual ACM-SIAM Symposium on Discrete Algorithms, 1991, pp. 449–458.     

Aggregation in hub location problems

Because of their widespread use in real-world transportation situations, hub location models have been extensively studied in the last two decades. Many types of hub location problems are NP-hard and remain unmanageable when the number of nodes exceeds 200. We present a way to tackle large-sized problems using aggregation, explore the resulting error, and show how to reduce it. Furthermore, we develop a heuristic based on aggregation for k-hub center problems and present computational results.     

Geometric complexity of some location problems

Given a set ofn demand points with weightW _i ,i = 1,2,...,n, in the plane, we consider several geometric facility location problems. Specifically we study the complexity of the Euclidean 1-line center problem, discrete 1-point center problem and a competitive location problem. The Euclidean 1-line center problem is to locate a line which minimizes the maximum weighted distance from the line (or the center) to the demand points. The discrete 1-point center problem is to locate one of the demand points so as to minimize the maximum unweighted distance from the point to other demand points. The competitive location problem studied is to locate a new facility point to compete against an existing facility so that a certain objective function is optimized. An Ω(n logn) lower bound is proved for these problems under appropriate models of computation. Efficient algorithms for these problems that achieve the lower bound and other related problems are also given.     

Colony location algorithm for assignment problems

A novel algorithm called Colony Location Algoritban (CLA) is proposed. It mimics the phenomena in biotic community that colonies of species could be located in the places most suitable to their growth. The factors working on the species location such as the nutrient of soil, resource competition between species, growth and decline process, and effect on environment were considered in CLA via the nutrient function, growth and decline rates, environment evaluation and fertilization strategy. CLA was applied to solve the classical assignment problems. The computation results show that CLA can achieve the optimal solution with higher possibility and shorter running time.     

On solving unreliable planar location problems

In this paper, we generalize conventional P-median location problems by considering the unreliability of facilities. The unreliable location problem is defined by introducing the probability that a facility may become inactive. We proposed efficient solution methods to determine locations of these facilities in the unreliable location model. Space-filling curve-based algorithms are developed to determine initial locations of these facilities. The unreliable P-median location problem is then decomposed to P 1-median location problems; each problem is solved to the optimum. A bounding procedure is used to monitor the iterative search, and to provide a consistent basis for termination. Extensive computational tests have indicated that the heuristics are efficient and effective for solving unreliable location problems.Scope and purposeThis paper addresses an important class of location problems, where p unreliable facilities are to be located on the plane, so as to minimize the expected travel distance or related transportation cost between the customers and their nearest available facilities. The unreliable location problem is defined by introducing the probability that a facility may become inactive. Potential application of the unreliable location problem is found in numerous areas. The facilities to be located can be fire station or emergency shelter, where it fails to provide service during some time window, due to the capacity or resource constraints. Alternatively, the facilities can be telecommunication posts or logistic/distribution centers, where the service is unavailable due to breakdown, repair, shutdown of unknown causes. In this paper, we prescribed heuristic procedures to determine the location of new facilities in the unreliable location problems. The numerical study of 2800 randomly generated instances has shown that these solution procedures are both efficient and effective, in terms of computational time and solution quality.     

Approximation algorithms for hierarchical location problems

We formulate and (approximately) solve hierarchical versions of two prototypical problems in discrete location theory, namely, the metric uncapacitated k-median and facility location problems. Our work yields new insights into hierarchical clustering, a widely used technique in data analysis. For example, we show that every metric space admits a hierarchical clustering that is within a constant factor of optimal at every level of granularity with respect to the average (squared) distance objective. A key building block of our hierarchical facility location algorithm is a constant-factor approximation algorithm for an “incremental” variant of the facility location problem; the latter algorithm may be of independent interest.     

Geometric complexity of some location problems

Given a set ofn demand points with weightW _i ,i = 1,2,...,n, in the plane, we consider several geometric facility location problems. Specifically we study the complexity of the Euclidean 1-line center problem, discrete 1-point center problem and a competitive location problem. The Euclidean 1-line center problem is to locate a line which minimizes the maximum weighted distance from the line (or the center) to the demand points. The discrete 1-point center problem is to locate one of the demand points so as to minimize the maximum unweighted distance from the point to other demand points. The competitive location problem studied is to locate a new facility point to compete against an existing facility so that a certain objective function is optimized. An (n logn) lower bound is proved for these problems under appropriate models of computation. Efficient algorithms for these problems that achieve the lower bound and other related problems are also given.Supported in part by the National Science Foundation under Grants ECS 83-40031 and DCR 84-20814.     

Single-allocation ordered median hub location problems

The discrete ordered median location model is a powerful tool in modeling classic and alternative location problems that have been applied with success to a large variety of discrete location problems. Nevertheless, although hub location models have been analyzed from the sum, maximum and coverage point of views, as far as we know, they have never been considered under an alternative unifying point of view. In this paper we consider new formulations, based on the ordered median objective function, for hub location problems with new distribution patterns induced by the different users’ roles within the supply chain network. This approach introduces some penalty factors associated with the position of an allocation cost with respect to the sorted sequence of these costs. First we present basic formulations for this problem, and then develop stronger formulations by exploiting properties of the model. The performance of all these formulations is compared by means of a computational analysis.     

A branch-and-price approach to p-median location problems

This paper describes a branch-and-price algorithm for the p-median location problem. The objective is to locate p facilities (medians) such as the sum of the distances from each demand point to its nearest facility is minimized. The traditional column generation process is compared with a stabilized approach that combines the column generation and Lagrangean/surrogate relaxation. The Lagrangean/surrogate multiplier modifies the reduced cost criterion, providing the selection of new productive columns at the search tree. Computational experiments are conducted considering especially difficult instances to the traditional column generation and also with some large-scale instances.     

On the distribution approach to location problems

While making location decisions, the distribution of travel distances among the service recipients (clients) is an important issue. It is usually tackled with the minimax (center) or the minisum (median) solution concepts. Both concepts minimize only simple scalar characteristics of the distribution: the maximal distance and the average distance, respectively. In this paper, all the distances for the individual clients are considered as a set of multiple uniform criteria to be minimized. This results in a multiple criteria model taking into account the entire distribution of distances. Our analysis of the multiple criteria problem focuses on the symmetrically efficient solutions which comply with minimization of distances as well as with impartial consideration of the clients. Various solution concepts generating symmetrically efficient location patterns are analyzed. Finally, the reference distribution approach is developed as an interactive technique which enables us to identify a satisfactory symmetrically efficient location pattern by evolving a reference (target) distribution of distances.     

Airline network design and hub location problems

Due to the popularity of hub-and-spoke networks in the airline and telecommunication industries, there has been a growing interest in hub location problems and related routing policies. In this paper, we introduce flow-based models for designing capacitated networks and routing policies. No a priori hub-and-spoke structure is assumed. The resulting networks may suggest the presence of “hubs”, if cost efficient. The network design problem is concerned with the operation of a single airline with a fixed share of the market. We present three basic integer linear programming models, each corresponding to a different service policy. Due to the difficulty of solving (even small) instances of these problems to optimality, we propose heuristic schemes based on mathematical programming. The procedure is applied and analyzed on several test problems consisting of up to 39 U.S. cities. We provide comments and partial recommendations on the use of hubs in the resulting network structures.     

Bilevel competitive facility location and pricing problems

We propose new models for competitive facility location and pricing as bilevel Boolean linear programming problems. We obtain results that characterize the complexity of the problem where a monopolist’s profit on each of the markets is defined with a monotone nonincreasing function of the servicing cost. For this problem, we also propose two approximate algorithms based on the ideas of alternating heuristics and local search. We give results of a computational experiment that show a possibility for fast computation of approximate solutions.     

An integrated approach to facility location problems

An integrated analysis approach to facility location problems is described. The approach is based on integrating analytical location models and a multicriteria decision model.     

A heuristic method for large-scale multi-facility location problems

A heuristic method for solving large-scale multi-facility location problems is presented. The method is analogous to Cooper's method (SIAM Rev. 6 (1964) 37), using the authors’ single facility location method (Comput. Optim. Appl. 21 (2002) 213) as a parallel subroutine, and reassigning customers to facilities using the heuristic of nearest center reclassification. Numerical results are reported. Scope and purpose We study the multiple facility location problem (MFLP). The objective in MFLP is to locate facilities to serve optimally a given set of customers. MFLPs have many applications in Operations Research, and a rich literature, see Drezner (Location Sci. 3(4) (1995) 275) for a recent survey.MFLPs involve, in addition to the location decision, also the assignment of customers to facilities. The MFLP is therefore a special clustering problem, the clusters here are the sets of customers assigned to the same facility.We propose a parallel heuristic method for solving MFLPs, using ideas from cluster analysis (nearest mean reclassification (Cluster Analysis, 3rd Edition, Edward Arnold, London, 1993)), and the authors’ Newton bracketing method for convex minimization (Comput. Optim. Appl. 21 (2002) 213) as a subroutine. The method is suitable for large-scale problems, as illustrated by numerical examples.     

Covering problems in facility location: A review

In this study, we review the covering problems in facility location. Here, besides a number of reviews on covering problems, a comprehensive review of models, solutions and applications related to the covering problem is presented after Schilling, Jayaraman, and Barkhi (1993). This survey tries to review all aspects of the covering problems by stressing the works after Schilling, Jayaraman, and Barkhi (1993). We first present the covering problems and then investigate solutions and applications. A summary and future works conclude the paper.     

Aggregation method experimentation for large-scale network location problems

We present demand point aggregation procedures for the p-median and p-center network location models. A coarse aggregation structure is initially obtained by partitioning the demand points according to a grid imposed over the demand region. A “row-column’’ aggregation algorithm is used to determine the spacing of rows and columns of the grid to exploit the problem structure. A second step involves locating aggregate demand points on the subnetworks induced by the cells of the grid partitioning. The aggregate demand point set so obtained then defines an approximating location model; alternatively, it may initialize an iterative network location–allocation procedure to find the aggregate demand points. We have tested our procedures on data sets based on maps from the TIGER/Line database of the United States Census Bureau, and report on our computational experience.     

Solving bus terminal location problems using evolutionary algorithms

Bus terminal assignment with the objective of maximizing public transportation service is known as bus terminal location problem (BTLP). We formulate the BTLP, a problem of concern in transportation industry, as a p-uncapacitated facility location problem (p-UFLP) with distance constraint. The p-UFLP being NP-hard (Krarup and Pruzan, 1990), we propose evolutionary algorithms for its solution. According to the No Free Lunch theorem and the good efficiency of the distinctive preserve recombination (DPX) operator, we design a new recombination operator for solving a BTLP by new evolutionary and memetic algorithms namely, genetic local search algorithms (GLS). We also define the potential objective function (POF) for the nodes and design a new mutation operator based on POF. To make the memetic algorithm faster, we estimate the variation of the objective function based on POF in the local search as part of an operator in memetic algorithms. Finally, we explore numerically the performance of nine proposed algorithms on over a thousand randomly generated problems and select the best two algorithms for further testing. The comparative studies show that our new hybrid algorithm composing the evolutionary algorithm with the GLS outperforms the multistart simulated annealing algorithm.