Single Facility Location Problem Research Articles

A unified approach to network location problems

In this paper, we introduce a new type of single-facility location problem on networks which includes as special cases most of the classical criteria in the literature. Structural results as well as a finite dominating set for the optimal locations are developed. Also, the extension to the multifacility case is discussed. The frontiers for finding easy finite dominating sets are shown by a counterexample. © 1999 John Wiley & Sons, Inc. Networks 34: 283–290, 1999

A unified approach to network location problems

In this paper, we introduce a new type of single-facility location problem on networks which includes as special cases most of the classical criteria in the literature. Structural results as well as a finite dominating set for the optimal locations are developed. Also, the extension to the multifacility case is discussed. The frontiers for finding easy finite dominating sets are shown by a counterexample. © 1999 John Wiley & Sons, Inc. Networks 34: 283–290, 1999

MINIMAX LOCATION PROBLEM WITH A-DISTANCE

In this paper a single facility location problem for an ambulance service station in a polygonal area X is considered. Our objective is to locate an ambulance service station so as to minimize the maximum distance of the route which passes from the facility to the hospital by way of the scene of accident. In this paper, we consider A-distance which is a generalization of rectilinear distance and was introduced by Widmayer et al. Assuming m hospitals at the points Hh H-^, - -, Hm and denoting the nearest hospital to a point Q of X with S(Q), the following problem PM is considered. PM : Minimize max R(P*, Q) = {^(P*, Q) + s(Q))}, P* QeX where P* = (X*, y*) is the location of an ambulance service station to be determined. Then we show PM can be reduced to the messenger boy problem with A-distance. Utilizing this result, we propose an efficient solution procedure by extending Elzinga & Hearn Algorithm to A-distance case.

An Efficient Algorithm for Minimizing a Sum of Euclidean Norms with Applications

In recent years rich theories on polynomial-time interior-point algorithms have been developed. These theories and algorithms can be applied to many nonlinear optimization problems to yield better complexity results for various applications. In this paper, the problem of minimizing a sum of Euclidean norms is studied. This problem is convex but not everywhere differentiable. By transforming the problem into a standard convex programming problem in conic form, we show that an $\epsilon$-optimal solution can be computed efficiently using interior-point algorithms. As applications to this problem, polynomial-time algorithms are derived for the Euclidean single facility location problem, the Euclidean multifacility location problem, and the shortest network under a given tree topology. In particular, by solving the Newton equation in linear time using {\em Gaussian elimination on leaves of a tree}, we present an algorithm which computes an $\epsilon$-optimal solution to the shortest network under a given full Steiner topology interconnecting N regular points, in $O(N \sqrt{N}(\log(\bar c/\epsilon)+\log N))$ arithmetic operations where $\bar c$ is the largest pairwise distance among the given points. The previous best-known result on this problem is a graphical algorithm which requires O(N2) arithmetic operations under certain conditions.

The single facility location problem with minimum distance constraints

We consider the problem of locating a single facility (server) in the plane, where the location of the facility is restricted to be outside a specified forbidden region (neighborhood) around each demand point. Two models are discussed. In the restricted 1-median model, the objective is to minimize the sum of the weighted rectilinear distances from the n customers to the facility. We present an O( n log n) algorithm for this model, improving upon the O( n 3) complexity bound of the algorithm by Brimberg and Wesolowsky (1995). In the restricted 1-center model the objective is to minimize the maximum of the weighted rectilinear distances between the customers and the serving facility. We present an O( n log n) algorithm for finding an optimal 1-center. We also discuss some related models, involving the Euclidean norm.

A single facility location problem on a tree with unreliable edges

A single facility location problem on a tree with unreliable edges

A single facility location problem on a tree with unreliable edges

This paper addresses the problem of determining a single facility location on an undirected tree with n nodes and edges that fail independently with probability (1 − pe) so that service provided to demand points can be achieved with reliability. Termed the relisum problem, the objective is to find a network location that maximizes the expected number of nodes reachable by operational paths. A decomposition formula is developed for which several properties are analytically derived and realistic vertex optimality conditions are identified. Two polynomial algorithms are presented: an O(n3) algorithm which is an adaptation of the Floyd-Warshall algorithm for finding all pairwise shortest paths in a graph and an O(n2) algorithm based on a depth-first node traversal and the decomposition nature of an operational path. Computational results are provided, and sensitivity analysis ranges and marginal values for pe are analytically derived. Properties and algorithms for the weighted version of the problem are discussed, as well as a polynomial heuristic for finding a relisum node on a cyclic network. © 1996 John Wiley & Sons, Inc.

A D.C. optimization method for single facility location problems

The single facility location problem with general attraction and repulsion functions is considered. An algorithm based on a representation of the objective function as the difference of two convex (d.c.) functions is proposed. Convergence to a global solution of the problem is proven and extensive computational experience with an implementation of the procedure is reported for up to 100,000 points. The procedure is also extended to solve conditional and limited distance location problems. We report on limited computational experiments on these extensions.

A discretizing algorithm for location problems

A new and simple methodology is proposed to solve both constrained and unconstrained planar continuous single-facility location problems. As particular instances, the classical location problems with mixed gauges can be solved. Theoretical convergence is proved, and numerical examples are given, showing a fast convergence in a small number of steps.

Single facility location problem with region-dependent distance metrics

A single facility location problem where the distance is measured differently in different regions on the plane is considered. For example, if some demand points are in a city with streets located as horizontal or vertical lines on the map and if other demand points are outside the city where travel in a straight line is possible using, e.g. helicopters, we obtain a mixed distance problem and the current model becomes applicable. We first formulate the problem as a mixed integer non-linear programming problem. Next, we prove the non-convexity of the cost function by showing that it is discontinuous along the line that divides the two regions. Bounds on the value of the cost function are provided. We propose a heuristic, as a modified version of the Weiszfeld algorithm, to solve the problem and compare its performance with a global optimization method. A numerical example and sensitivity analyses are discussed comparing the efficiency of the modified algorithm with the results of the global optimization method

Technical Note—Algorithms for Weber Facility Location in the Presence of Forbidden Regions and/or Barriers to Travel

We describe algorithms for optimal single facility location problems with forbidden regions and barriers to travel. The former are those where location is not permitted, but one can travel through them, e.g., a lake. The latter are the regions where neither location nor travel is permitted, e.g., large parks in a city. Using the convexity properties of the objective function, in the first case, we develop an algorithm for finding the optimal solution. The objective function in the barrier case is shown to be non-convex. We use the concept of visibility to create a network with the location point as the source and use Dijkstra's algorithm to compute the shortest distance to all the other demand points. Using simulated annealing we find an approximate optimal solution. Numerical examples illustrate the implementation of the algorithms.

A polygonal upper bound for the efficient set for single-facility location problems with mixed norms

In this paper it is shown that, in a multiobjective single-facility location problem in which distances are measured by arbitrary mixed norms, the set of efficient points is a subset of apolygonal hull of the demand points.

On the Convergence of a Hyperboloid Approximation Procedure for the Perturbed Euclidean Multifacility Location Problem

For the Euclidean single facility location problem, E. Weiszfeld proposed a simple closed-form iterative algorithm in 1937. Later, numerous authors proved that it is a convergent descent algorithm. In 1973, J. Eyster, J. White and W. Wierwille extended Weiszfeld's idea and proposed a Hyperboloid Approximation Procedure (HAP) for solving the Euclidean multifacility location problem. They believed, based on considerable computational experience, that the HAP always converges. In 1977, Ostresh proved that the HAP is a descent algorithm under certain conditions. In 1981, Morris proved that a variant of the HAP always converges. However, no convergence proof for the original HAP has ever been given. In this paper, we prove that the HAP is a descent algorithm and that it always converges to the minimizer of the objective function from any initial point.

Deriving an optimal dwell point of the storage/retrieval machine in an automated storage/retrieval system

Abstract The selection of the dwell point of the storage/retrieval (S/R) machine when it becomes idle in an automated storage/retrieval system (AS/RS) is discussed. Two existing models for dynamic control of the S/R machine are adopted for the analysis. They are transformed to the single-facility location problem with Tchebyshev distance, and the Tchebyshev minimax facility location problem, respectively. Algorithms are presented utilizing well known research results in location theory. Through computational tests, the algorithms are shown to be quite suitable for real-world applications.

A globally convergent algorithm for the Euclidean multiplicity location problem

The Euclidean single facility location problem (ESFL) and the Euclidean multiplicity location problem (EMFL) are two special nonsmooth convex programming problems which have attracted a large literature. For the ESFL problem, there are algorithms which converge both globally and quadratically. For the EMFL problem, there are some quadratically convergent algorithms, but for global convergence, they all need nontrivial assumptions on the problem.

GBSSS: The generalized big square small square method for planar single-facility location

Big Square Small Square is a geometrical branch-and-bound algorithm, devised by P. Hansen et al. (OR 33, 1251–1265), for the solution of constrained planar minisum single-facility location problems with L p norms and continuous non-decreasing costfunctions. The method basically works by splitting the studied planar region into squares, and either rejecting or further processing these squares by the evaluation of a lower bound. We present a modified version of this algorithm aimed at correcting a small failure to converge, accelerating the calculations, minimising the information to be stored, and, most importantly, determining a region of near-optimality. Furthermore the method is applicable to any planar single-facility problem with distances measured by mixed norms and as an objective any continuous function of the distances. This includes nearly all the models which have been proposed in the literature.

Locating an obnoxious facility

We demonstrate that a formulation of the obnoxious single facility location problem is equivalent to the complex moment problem. The equivalence is understood in the sense that the critical points of the potential function defining the location problem are equal to the zeros of the function defining the complex moment problem.

On the Convergence of Miehle's Algorithm for the Euclidean Multifacility Location Problem

For the Euclidean single facility location problem, E. Weiszfeld proposed a simple iterative algorithm in 1937. Later, it was proved by numerous authors that it is a convergent descent algorithm. W. Miehle extended Weiszfeld's algorithm to solve the Euclidean multifacility location problem. Then, L. M. Ostresh proved that Miehle's algorithm is a descent algorithm. Recently, F. Rado modified Miehle's algorithm and provided several sets of sufficient conditions for the modified algorithm to converge. He also indicated that the convergence of Miehle's algorithm was an open problem. In this paper, the relationship between Miehle's multifacility location algorithm and Weiszfeld's single facility location algorithm is analyzed. Counterexamples show that Miehle's algorithm may converge to nonoptimal points for well structured problems.

Global optimization of a nonconvex single facility location problem by sequential unconstrained convex minimization

The problem of maximizing the sum of certain composite functions, where each term is the composition of a convex decreasing function, bounded from below, with a convex function having compact level sets arises in certain single facility location problems with gauge distance functions. We show that this problem is equivalent to a convex maximization problem over a compact convex set and develop a specialized polyhedral annexation procedure to find a global solution for the case when the inside function is a polyhedral norm. As the problem was solved recently only for local solutions, this paper offers an algorithm for finding a global solution. Implementation and testing are not treated in this short communication.

Computational Comparison of Two Algorithms for the Euclidean Single Facility Location Problem

The best known first order and second order methods for the Euclidean single facility location problem are Weiszfeld's algorithm and Newton's algorithm, respectively. With only a linear rate of convergence, Weiszfeld's algorithm is a simple closed form iteration formula. With a quadratic rate of convergence, Newton's algorithm is more complicated and requires a line search at each iteration. In this paper, computational comparisons were made between these two algorithms. Both sequential and parallel versions of each algorithm were compared. These results show that the second order method is more efficient for low dimensional problems while the first order method is more efficient for high dimensional problems. A parallel version of each algorithm was implemented on a NCUBE hypercube with 64 processors and tested on randomly generated problems with space dimension from 2 to 9, and the number of existing facilities from 1000 to 9000. A speedup of as high as 58 was achieved with 64 processors. INFORMS Journal on Computing, ISSN 1091-9856, was published as ORSA Journal on Computing from 1989 to 1995 under ISSN 0899-1499.