General network design: A unified view of combined location and network design problems

This paper presents a unified framework for the general network design problem which encompasses several classical problems involving combined location and network design decisions. In some of these problems the service demand relates users and facilities, whereas in other cases the service demand relates pairs of users between them, and facilities are used to consolidate and re-route flows between users. Problems of this type arise in the design of transportation and telecommunication systems and include well-known problems such as location-network design problems, hub location problems, extensive facility location problems, tree-star location problems and cycle-star location problems, among others. Relevant modeling aspects, alternative formulations and possible algorithmic strategies are presented and analyzed.     

Simulated N-Body: New Particle Physics-Based Heuristics for a Euclidean Location-Allocation Problem

The general facility location problem and its variants, including most location-allocation and P-median problems, are known to be NP-hard combinatorial optimization problems. Consequently, there is now a substantial body of literature on heuristic algorithms for a variety of location problems, among which can be found several versions of the well-known simulated annealing algorithm. This paper presents an optimization paradigm that, like simulated annealing, is based on a particle physics analogy but is markedly different from simulated annealing. Two heuristics based on this paradigm are presented and compared to simulated annealing for a capacitated facility location problem on Euclidean graphs. Experimental results based on randomly generated graphs suggest that one of the heuristics outperforms simulated annealing both in cost minimization as well as execution time. The particular version of location problem considered here, a location-allocation problem, involves determining locations and associated regions for a fixed number of facilities when the region sizes are given. Intended applications of this work include location problems with congestion costs as well as graph and network partitioning problems.     

Multiple voting location problems

The facility voting location problems arise from the application of criteria derived from the voting processes concerning the location of facilities. The multiple location problems are those location problems in which the alternative solutions are sets of points. This paper extends previous results and notions on single voting location problems to the location of a set of facility points. The application of linear programming techniques to solve multiple facility voting location problems is analyzed. We propose an algorithm to solve Simpson multiple location problems from which the solution procedures for other problems are derived.     

选址问题研究的若干进展

中值问题、覆盖问题、中心问题是选址研究中的三个经典问题，它们的应用非常广泛，也是迄今为止大多数选址理论研究的坚实基础。本文综述了近年来它们的研究进展，包括模型、求解方法以及相关问题，最后，指出这一领域未来研究的一些问题与方向。     

A 1-center problem on the plane with uniformly distributed demand points

Center problems or minimax facility location problems are among the most active research areas in location theory. In this paper, we find the best unique location for a facility in the plane such that the maximum expected weighted distance to all random demand points is minimized.     

On a Constrained Optimal Location Algorithm

In problems of optimal location, one seeks a position or location that optimizes a particular objective function; this objective function typically relates location and distances to a fixed point set. When one's search is restricted to a given set, we refer to this as a constrained optimal location problem. For a finite point set A, there exist numerous finite algorithms to solve optimal location problems. In this paper we demonstrate how an algorithm, solving optimal location problems in inner-product spaces, can be modified to solve certain constrained optimal location problems. We then apply this modification to a particularly simple (and easily implemented) algorithm and investigate the complexity of the result. In particular we improve a known estimate from exponential to polynomial.     

Fixed Point Optimality Criteria for the Location Problem with Arbitrary Norms

The single-facility location problem in continuous space is considered, with distances given by arbitrary norms. When distances are Euclidean, for many practical problems the optimal location of the new facility coincides with one of the existing facilities. This property carries over to problems with generalized distances. In this paper a necessary and sufficient condition for the location of an existing facility to be the optimal location of the new facility is developed. Some computational examples using the condition are given.     

A computational evaluation of a general branch-and-price framework for capacitated network location problems

The purpose of this paper is to illustrate a general framework for network location problems, based on column generation and branch-and-price. In particular we consider capacitated network location problems with single-source constraints. We consider several different network location models, by combining cardinality constraints, fixed costs, concentrator restrictions and regional constraints. Our general branch-and-price-based approach can be seen as a natural counterpart of the branch-and-cut-based commercial ILP solvers, with the advantage of exploiting the tightness of the lower bound provided by the set partitioning reformulation of network location problems. Branch-and-price and branch-and-cut are compared through an extensive set of experimental tests.     

On the convergence of the generalized Weiszfeld algorithm

In this paper we consider Weber-like location problems. The objective function is a sum of terms, each a function of the Euclidean distance from a demand point. We prove that a Weiszfeld-like iterative procedure for the solution of such problems converges to a local minimum (or a saddle point) when three conditions are met. Many location problems can be solved by the generalized Weiszfeld algorithm. There are many problem instances for which convergence is observed empirically. The proof in this paper shows that many of these algorithms indeed converge.     

Integer linear programming models for grid-based light post location problem

Selecting optimal location is a key decision problem in business and engineering. This research focuses to develop mathematical models for a special type of location problems called grid-based location problems. It uses a real-world problem of placing lights in a park to minimize the amount of darkness and excess supply. The non-linear nature of the supply function (arising from the light physics) and heterogeneous demand distribution make this decision problem truly intractable to solve. We develop ILP models that are designed to provide the optimal solution for the light post problem: the total number of light posts, the location of each light post, and their capacities (i.e., brightness). Finally, the ILP models are implemented within a standard modeling language and solved with the CPLEX solver. Results show that the ILP models are quite efficient in solving moderately sized problems with a very small optimality gap.     

GASUB: finding global optima to discrete location problems by a genetic-like algorithm

In many discrete location problems, a given number s of facility locations must be selected from a set of m potential locations, so as to optimize a predetermined fitness function. Most of such problems can be formulated as integer linear optimization problems, but the standard optimizers only are able to find one global optimum. We propose a new genetic-like algorithm, GASUB, which is able to find a predetermined number of global optima, if they exist, for a variety of discrete location problems. In this paper, a performance evaluation of GASUB in terms of its effectiveness (for finding optimal solutions) and efficiency (computational cost) is carried out. GASUB is also compared to MSH, a multi-start substitution method widely used for location problems. Computational experiments with three types of discrete location problems show that GASUB obtains better solutions than MSH. Furthermore, the proposed algorithm finds global optima in all tested problems, which is shown by solving those problems by Xpress-MP, an integer linear programing optimizer (21). Results from testing GASUB with a set of known test problems are also provided.     

An extended covering model for flexible discrete and equity location problems

To model flexible objectives for discrete location problems, ordered median functions can be applied. These functions multiply a weight to the cost of fulfilling the demand of a customer which depends on the position of that cost relative to the costs of fulfilling the demand of the other customers. In this paper a reformulated and more compact version of a covering model for the discrete ordered median problem (DOMP) is considered. It is shown that by using this reformulation better solution times can be obtained. This is especially true for some objectives that are often employed in location theory. In addition, the covering model is extended so that ordered median functions with negative weights are feasible as well. This type of modeling weights has not been treated in the literature on the DOMP before. We show that several discrete location problems with equity objectives are particular cases of this model. As a result, a mixed-integer linear model for this type of problems is obtained for the first time.     

Group decision-making based on concepts of ideal and anti-ideal points in a fuzzy environment

Decision-making problems (location selection) often involve a complex decision-making process in which multiple requirements and uncertain conditions have to be taken into consideration simultaneously. In evaluating the suitability of alternatives, quantitative/qualitative assessments are often required to deal with uncertainty, subjectiveness and imprecise data, which are best represented with fuzzy data. This paper presents a new method of analysis of multicriteria based on the incorporated efficient fuzzy model and concepts of positive ideal and negative ideal points to solve decision-making problems with multi-judges and multicriteria in real-life situations. As a result, effective decisions can be made on the basis of consistent evaluation results. Finally, this paper uses a numerical example of location selection to demonstrate the applicability of this method, with its simplicity in both concept and computation. The results show that this method can be implemented as an effective decision aid in selecting location or decision-making problems.     

Improving solution of discrete competitive facility location problems

We consider discrete competitive facility location problems in this paper. Such problems could be viewed as a search of nodes in a network, composed of candidate and customer demand nodes, which connections correspond to attractiveness between customers and facilities located at the candidate nodes. The number of customers is usually very large. For some models of customer behavior exact solution approaches could be used. However, for other models and/or when the size of problem is too high to solve exactly, heuristic algorithms may be used. The solution of discrete competitive facility location problems using genetic algorithms is considered in this paper. The new strategies for dynamic adjustment of some parameters of genetic algorithm, such as probabilities for the crossover and mutation operations are proposed and applied to improve the canonical genetic algorithm. The algorithm is also specially adopted to solve discrete competitive facility location problems by proposing a strategy for selection of the most promising values of the variables in the mutation procedure. The developed genetic algorithm is demonstrated by solving instances of competitive facility location problems for an entering firm.     

Error bounds for the approximative solution of restricted planar location problems

Facility location problems in the plane play an important role in mathematical programming. When looking for new locations in modeling real-world problems, we are often confronted with forbidden regions, that are not feasible for the placement of new locations. Furthermore these forbidden regions may have complicated shapes, even if we require them to be convex. It may be more useful or even necessary to use approximations of such forbidden regions when trying to solve location problems. In this paper, we develop error bounds for the approximative solution of restricted planar location problems using the so called sandwich algorithm. The number of approximation steps required to achieve a specified error bound is analyzed. As examples of these approximation schemes, we discuss round norms and polyhedral norms. Computational tests are also included.     

Using tropical optimization to solve minimax location problems with a rectilinear metric on the line

Methods of tropical (idempotent) mathematics are applied to the solution of minimax location problems under constraints on the feasible location region. A tropical optimization problem is first considered, formulated in terms of a general semifield with idempotent addition. To solve the optimization problem, a parameter is introduced to represent the minimum value of the objective function, and then the problem is reduced to a parametrized system of inequalities. The parameter is evaluated using existence conditions for solutions of the system, whereas the solutions of the system for the obtained value of the parameter are taken as the solutions of the initial optimization problem. Then, a minimax location problem is formulated to locate a single facility on a line segment in the plane with a rectilinear metric. When no constraints are imposed, this problem, which is also known as the Rawls problem or the messenger boy problem, has known geometric and algebraic solutions. For the location problems, where the location region is restricted to a line segment, a new solution is obtained, based on the representation of the problems in the form of the tropical optimization problem considered above. Explicit solutions of the problems for various positions of the line are given both in terms of tropical mathematics and in the standard form.     

Multiple criteria facility location problems: A survey

This paper provides a review on recent efforts and development in multi-criteria location problems in three categories including bi-objective, multi-objective and multi-attribute problems and their solution methods. Also, it provides an overview on various criteria used. While there are a few chapters or sections in different location books related to this topic, we have not seen any comprehensive review papers or book chapter that can cover it. We believe this paper can be used as a complementary and updated version.     

Relaxed voting and competitive location under monotonous gain functions on trees

We examine competitive location problems where two competitors serve a good to users located in a network. Users decide for one of the competitors based on the distance induced by an underlying tree graph. The competitors place their server sequentially into the network. The goal of each competitor is to maximize his benefit which depends on the total user demand served. Typical competitive location problems include the (1,X₁)-medianoid, the (1,1)-centroid, and the Stackelberg location problem.An additional relaxation parameter introduces a robustness of the model against small changes in distance. We introduce monotonous gain functions as a general framework to describe the above competitive location problems as well as several problems from the area of voting location such as Simpson, Condorcet, security, and plurality.In this paper we provide a linear running time algorithm for determining an absolute solution in a tree where competitors are allowed to place on nodes or on inner points. Furthermore we discuss the application of our approach to the discrete case.     

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.The 3-stage method is composed of a primal-dual algorithm, a subgradient optimization to solve a Lagrangean dual and a branch-and-bound algorithm. It has a hierarchical structure, with a given stage being activated only if the optimal solution could not be identified in the preceding stage.The proposed method was used in the solution of three well-known uncapacitated location problems: the simple plant location problem, thep-median problem and the fixed-chargep-median problem. Computational results are given for problems of up to the size 200 customers ×200 potential facility sites.