Uncapacitated Location Problem Research Articles

Uncapacitated (Facility) Location Problem: A Hybrid Genetic-Tabu Search Approach

In this paper the uncapacitated location problem (ULP) with restriction on number of facilities is considered. The ULP generalizes the uncapacitated facility location problem and the p-Median problems. A hybrid genetic algorithm and tabu search (HGA-TS) is presented for solution of the problem. The genetic algorithm is based on random-key (RK) applied to sequencing problems. The tabu search (TS) is based on adaptive critical event TS (CETS). The RK component serves as a diversification generation within the CETS. Effectiveness of the heuristic is tested on benchmark problems available on the Internet and compared with a leading algorithm based on GRASP.

A kernel search heuristic for a fair facility location problem

We consider an uncapacitated location problem where p facilities have to be located in order to serve a given set of customers, and we assume that a customer requesting for a service has to reach a facility at his/her own cost. In this setting, a central issue is that of fairness among customers for the accessibility to the services provided. Every choice regarding the location of facilities corresponds to a distance distribution of customers to reach an open facility. Minimizing the average of this distribution would lead to a p-median problem, where system efficiency is optimized but the fair treatment of users is neglected. Minimizing the maximum (worst-case) of the distribution would lead to a p-center problem, where the unfair treatment of users is mitigated but system efficiency is neglected. To compromise between these two extremes, we minimize the conditional β-mean, i.e., the average distance traveled by the 100×β% of customers farther from a facility. We call Fair Facility Location Problem (FFLP(β)) the resulting optimization problem, which is formulated as a Mixed-Integer linear Program (MIP) with a proven integer-friendly property. We propose a heuristic framework to produce a set of representative solutions to the FFLP(β). The framework is based on Kernel Search, a heuristic scheme that has been shown to obtain high-quality solutions for a number of MIPs. Computational experiments are reported to validate the quality of the solutions found by the proposed solution algorithm, and to provide some general guidelines regarding the trade-off between average and worst-case optimization. Finally, we report on a case study stemming from the screening activities related to the pandemic triggered by the SARS-CoV-2 virus. The case study regards the optimal location of a number of drive-thru temporary testing sites for collecting swab specimens.

The Methods for Approximation of Principal Points for Binary Distributions on the Basis of Submodularity

Principal points for binary distributions are able to be defined based on Flury’s principal points (1990). However, finding principal points for binary distributions is hard in a straightforward manner. In this article, a method for approximating principal points for binary distributions is proposed by formulating it as an uncapacitated location problem. Moreover, it is shown that the problem of finding principal points can be solved with the aid of submodular functions. It leads to a solution whose value is at least (1 − 1/e) times the optimal value.

EWGLA XVII: the XVII Meeting of the Euro Working Group on Locational Analysis

The XVII Meeting of the Euro Working Group on Locational Analysis (EWGLA XVI) was held in Elche, Spain, from September 17 to 19, 2008. The Euro Working Group on Locational analysis (EWGLA) is a Working Group devoted to Locational Analysis within EURO (webpage: http://ewgla.fc.ul.pt/). It was created during the first EURO Summer Institute, held at Brussels, July/August 1984, devoted to the subject of Location Theory. The main aim of starting up the Working Group was to provide a network of people working together in the field of Locational Analysis. Since its creation, the group holds yearly meetings organized by one of its members. Various countries have hosted the meeting: France, Yugoslavia, Greece, Belgium, Germany, United Kingdom, Portugal and Spain. Each time, the meeting is designed to provide a dynamic forum for critical interchange of ideas and experiences about locational analysis and related fields. EWGLA XVII was a fruitful workshop: more than 80 participants coming from 12 different countries took part in the meeting and a total of 56 scientific papers were presented and discussed there. The fact that fifteen percent of the assistants were students evidence that locational analysis is an interesting field for people starting research and also that the EWGLA group is concerned about promoting interest in the location analysis field. The topics presented at the meeting covered many different areas in locational analysis. Papers in this special issue represent the work discussed in EWGLA XVII. Robustness is the topic of the paper “Conditional Median as a Robust Solution Concept for Uncapacitated Location Problems” by W. Ogryczak. The papers “The Ordered Capacitated Facility Location Problem” by A.M. Rodriguez-Chia, J. Kalcsics,

Conditional median as a robust solution concept for uncapacitated location problems

While making location decisions, one intends to increase effects (reduce distances) for the service recipients (clients). A conventional optimization approach to location problems considers only the optimality of locational decisions for specific clients data. Real-world applications inevitably involve errors and uncertainties in the operating conditions, and thereby the resulting performance may be lower than expected. In particular, a distribution system design is very sensitive to the varying demands for goods, and the demand changes may deteriorate drastically the system efficiency when optimized for different demand structure. Several approaches have been developed to deal with uncertain or imprecise data. The approaches focused on the quality or on the variation (stability) of the solution for some data domains are considered robust. Frequently, uncertainty is represented by limits (intervals) on possible values of demand weights varying independently rather than by scenarios for all the weights simultaneously. In this paper we show that a solution concept of the conditional median can be used to optimize effectively such robust location problems. The conditional median is a generalization of the minimax solution concept extended to take into account the number of services (the portion of demand) related to the worst performances. Namely, for a specified portion of demand, we take into account the corresponding portion of the maximum results, and we consider their average as the worst conditional mean to be minimized. Similar to the standard minimax approach, the minimization of the worst conditional mean can be defined by a linear objective and a number of auxiliary linear inequalities.

Dynamic multi-level capacitated and uncapacitated location problems: an approach using primal-dual heuristics

In this paper several dynamic multi-level location problems are formulated as mixed-integer linear programs. Both uncapacitated and capacitated versions of the problem are studied. The models presented are more complete than the ones known from the literature: they are dynamic and consider the possibility of a facility being opened, closed and reopened more than once during the planning horizon. They may include both upper and lower limits on the used capacity of each facility and may also consider the situation where there is no flow conservation in the intermediate facilities. Primal-dual heuristics were developed to solve the proposed models, having as main objective the capability of finding good primal solutions in reasonable computational times. Computational results are presented and discussed.

Lagrangean Relaxation Based Approximate Approach to the Capacitated Location Problem

When a distribution system is to be designed, limits on terminal capability often must be taken into account. These capacity constraints in this and other facility location problems constitute severe obstacles in exact solving processes. Within this paper, we focused on study of approximate methods based on Lagrangean relaxation of the capacity constraints, which has several advantageous properties. The first of them is that the relaxed problem, known as the uncapacitated location problem, can be solved exactly even for real sized instances [6], [4]. The second useful property of the Lagrangean relaxation is that the objective function value of the optimal solution of the relaxed problem provides lower bound of the optimal solution of the original problem. We present two methods for obtaining suitable values of Lagrangean multipliers. The classical one is based on a sub-gradient method applied on capacity constraints after their special adjustment. The second method is designed as an adaptive method with random experiments for determination of candidates for move from the current solution to the next one. These two methods were tested, compared and the associated results are reported in the concluding part of this paper.

Solution methods for the bi-objective (cost-coverage) unconstrained facility location problem with an illustrative example

The Colombian coffee supply network, managed by the Federacion Nacional de Cafeteros de Colombia (Colombian National Coffee-Growers Federation), requires slimming down operational costs while continuing to provide a high level of service in terms of coverage to its affiliated coffee growers. We model this problem as a biobjective (cost-coverage) uncapacitated facility location problem (BOUFLP). We designed and implemented three different algorithms for the BOUFLP that are able to obtain a good approximation of the Pareto frontier. We designed an algorithm based on the Nondominated Sorting Genetic Algorithm; an algorithm based on the Pareto Archive Evolution Strategy; and an algorithm based on mathematical programming. We developed a random problem generator for testing and comparison using as reference the Colombian coffee supply network with 29 depots and 47 purchasing centers. We compared the algorithms based on the quality of the approximation to the Pareto frontier using a nondominated space metric inspired on Zitzler and Thiele's. We used the mathematical programming-based algorithm to identify unique tradeoff opportunities for the reconfiguration of the Colombian coffee supply network. Finally, we illustrate an extension of the mathematical programming-based algorithm to perform scenario analysis for a set of uncapacitated location problems found in the literature.

Limits of the Multiplier Adjustment Approach to Capacitated Location Problem

This contribution deals with a distribution system design problem, in which the located facilities satisfy all customer demands under limited abilities. The objective is to minimize the total costs, including both fixed charges and service costs. A special approach based on Lagrangean relaxation will be discussed here for its ability to cope with limited capacities of located facilities. Using the Lagrangean relaxation, the model of the original problem is rearranged to a model, which can be solved by exact algorithms for considerably large size. The capacities of the located facilities bring serious difficulties concerning solving technique, in the cases, where real-sized facility location problems are solved. In contrast to an uncapacitated facility location problem, which can be solved exactly in reasonable time for real-sized case containing hundreds of possible locations and thousands of customers.In this paper, we shall discuss a transformation of the capacitated location problem into the uncapacitated location problem by means of Lagrangean relaxation of capacity constraints. To demonstrate the efficiency of the studied approach, numerical experiments were performed and their results are reported in the concluding part of this paper.

The Exact and Near Optimal Solution of the Competitive Uncapacitated Location Problem

This paper is dedicated to the recent unprecedented boom of new supermarkets and hypermarkets in middle Europe. The motivation is to provide for the newcomers the tool for decision support, and to help answer the questions: "Is still economically advantageous to build new shopping malls and where to locate them?" We introduce the four versions of data correcting algorithm and three heuristics: simple exchange heuristic, exchange heuristics enhanced by simulated annealing metaheuristic, and genetic algorithm. All these methods were examined at the benchmarks of practical nature. This research showed the data correcting method as useable for practical instances of this problem. There were also identified interesting dependencies between the computational time and the number of competitors.

Uncapacitated facility location problems: contributions

The objective of the present paper is to review my personal contributions in the field of uncapacitated facility location problems. These contributions took place throughout my academic career, from the time I was a Ph.D. student at Imperial College to the present day. They cover approximately 30 years, from 1973 to 2003; they address: algorithms developed for the p-median problem and for a general formulation of uncapacitated location problems; the study of dynamic location models; covering and hierarchical location problems; queuing-based probabilistic location models. The contributions encompass theoretical developments, computational algorithms and practical applications. All work took place in an academic environment, with the invaluable collaboration of colleagues (both in Brazil and abroad) and research students at COPPE. Each section in the paper is dedicated to a topic that involves a personal contribution. Every one of them is placed within the context of the existing literature.

Comparing dual access approaches for exact solution of location problems

The paper deals with exact solution of two uncapacitated location problems. The location problem consists in placing some facilities of unrestricted capacity in the formed network. The facility can have various natures. It could be aservice centre, whose location can be done at anode of the network, or it can be adirect train, which can be placed into a transport plan. In comparison with the former case, this facility has the nature of a graph edge.The key-stone of the computational success of the often used branch and bound method is the lower bound tightness. This paper, reports numerical experiments with the dual access approach used for the solution of the node location problem in a large network and compares several possible approaches to lower bound enumeration for the edge location problem.

Exact solution methods for uncapacitated location problems with convex transportation costs

In this paper we study exact solution methods for uncapacitated facility location problems where the transportation costs are nonlinear and convex. An exact linearization of the costs is made, enabling the formulation of the problem as an extended, linear pure zero–one location model. A branch-and-bound method based on a dual ascent and adjustment procedure is developed, and compared to application of a modified Benders decomposition method. The specific application studied is the simple plant location problem (SPLP) with spatial interaction, which is a model suitable for location of public facilities. Previously approximate solution methods have been used for this problem, while we in this paper investigate exact solution methods. Computational results are presented.

A perturbation heuristic for a class of location problems

A constructive heuristic which has the flexibility of perturbing the solution while guiding the search toward a better solution is developed. Interesting findings are obtained when this heuristic is tested against reputable methods on a class of uncapacitated location problems.

A method for solving to optimality uncapacitated location problems

In this paper we develop a method for solving to optimality a general 0–1 formulation for uncapacitated location problems. This is a 3-stage method that solves large problems in reasonable computing times.

Maximising Real-Valued Submodular Functions: Primal and Dual Heuristics for Location Problems

Recently it has been shown how various results on the behaviour of exact and approximate primal algorithms for certain discrete location problems can be obtained by studying the more general problem of maximising a submodular set function. Unfortunately, this more general model tells us little about an important aspect of the discrete location problems, their linear programming relaxations and the possible use of dual solutions in obtaining bounds. Here we examine a slightly more specialised model that again generalises the location problems of interest, but now also includes the continuous aspects of the problems missing from the earlier model, namely the problem: max{w(y): ∑j=1najyj ≤ b, 0 ≤ yj < 1} of maximising a real-valued nondecreasing piecewise linear, concave submodular function subject to a knapsack constraint. We show that a continuous greedy heuristic always attains at least (1 − e−1) × 100% of the optimal value, and that for the discrete problem an adapted greedy heuristic always attains 35% of the optimal value. Specialised to the capacitated and uncapacitated location problems these results permit us to strengthen earlier results, and to obtain new results on dual greedy heuristics.

Note—On “Location of Bank Accounts to Optimize Float: An Analytic Study of Exact and Approximate Algorithms”

In the course of the deliberations of the 1977 Lanchester Prize Committee, Alan J. Goldman brought to our attention an error in the proof of Lemma 1 of our paper (Cornuejols, G., M. L. Fisher, G. L. Nemhauser. 1977. Location of bank accounts to optimize float: an analytic study of exact and approximate algorithms. Management Sci. 23 789–810.). The lemma, however, is true and the original correct, but long and intricate, proof was provided to the Committee, see (Cornuejols, G., M. L. Fisher, G. L. Nemhauser. 1977. On the uncapacitated location problem. Ann. Discrete Math. 1 163–178.) for details.

An analysis of approximations for maximizing submodular set functions—I

LetN be a finite set andz be a real-valued function defined on the set of subsets ofN that satisfies z(S)+z(T)źz(SźT)+z(SźT) for allS, T inN. Such a function is called submodular. We consider the problem maxSźN{a(S):|S|≤K,z(S) submodular}. Several hard combinatorial optimization problems can be posed in this framework. For example, the problem of finding a maximum weight independent set in a matroid, when the elements of the matroid are colored and the elements of the independent set can have no more thanK colors, is in this class. The uncapacitated location problem is a special case of this matroid optimization problem. We analyze greedy and local improvement heuristics and a linear programming relaxation for this problem. Our results are worst case bounds on the quality of the approximations. For example, whenz(S) is nondecreasing andz(0) = 0, we show that a "greedy" heuristic always produces a solution whose value is at least 1 ź[(K ź 1)/K]K times the optimal value. This bound can be achieved for eachK and has a limiting value of (e ź 1)/e, where e is the base of the natural logarithm.

A Parametric Decomposition Approach for the Solution of Uncapacitated Location Problems

This paper presents a new approach to determining the optimal facility locations for uncapacitated location problems in two stages. First, it is shown that a subset of all solutions to the uncapacitated public facility location problem can be obtained by considering a closely related private location problem. The exact nature of the problem is established using the Generalized Lagrange Multiplier results given by Everett. In the second section of the paper a particularly efficient parametric decomposition algorithm, which is based on the results of the first section, is presented. Computational results are given which summarize the current level of experience with the algorithm.