Vertex P-center Problem Research Articles

Robust MILP formulations for the two-stage weighted vertex [formula omitted]-center problem

The weighted vertex p-center problem (PCP) consists of locating p facilities among a set of available sites such that the maximum weighted distance (or travel time) from any demand node to its closest located facility is minimized. This paper studies the exact solution of the two-stage robust weighted vertex p-center problem (RPCP2). In this problem, the location of the facilities is fixed in the first stage while the demand node allocations are recourse decisions fixed once the uncertainty is revealed. The problem is modeled by box uncertainty sets on both the demands and the distances. We introduce five different robust reformulations based on MILP formulations of (PCP) from the literature. We prove that considering a finite subset of scenarios is sufficient to obtain an optimal solution of (RPCP2). We leverage this result to introduce a column-and-constraint generation algorithm and a branch-and-cut algorithm to efficiently solve this problem optimally. We highlight how these algorithms can also be adapted to solve the single-stage problem (RPCP1) which is obtained when no recourse is considered. We present a numerical study to compare the performances of these formulations on randomly generated instances and on a case study from the literature.

Estimations of Covering Functionals of Convex Bodies Based on Relaxation Algorithm

Estimating covering functionals of convex bodies is an important part of Chuanming Zong’s program to attack Hadwiger’s covering conjecture, which is a long-standing open problem from convex and discrete geometry. In this paper, we transform this problem into a vertex p-center problem (VPCP). An exact iterative algorithm is introduced to solve the VPCP by making adjustments to the relaxation-based algorithm mentioned by Chen and Chen in 2009. The accuracy of this algorithm is tested by comparing numerical and exact values of covering functionals of convex bodies including the Euclidean disc, simplices, and the regular octahedron. A better lower bound of the covering functional with respect to 7 of 3-simplices is presented.

Robust Milp Formulations for the Two-Stage Weighted Vertex P-Center Problem

Robust Milp Formulations for the Two-Stage Weighted Vertex P-Center Problem

Finding the absolute and vertex center of a fuzzy tree

ABSTRACT The p-center problem is the problem of finding p best places on a network to locate facilities for providing customers' demands such that the maximum distance between the demand points and facilities is minimized. If these facilities are located on the edges and vertices of the network, the problem is called the absolute p-center problem. Whereas, if the facilities can just be located on the vertices, the problem is called the vertex p-center problem. In this paper, both absolute and vertex 1-center problems are studied on tree graphs, where the lengths of edges are fuzzy numbers. Based on some presented properties, fuzzy algorithms are proposed for finding absolute and vertex center of a fuzzy tree. The performance of presented algorithms are shown by solving some numerical examples. is considered and the absolute center problem and the vertex center problem are studied, respectively. Also, the weights of all vertices are equal to one and the lengths of edges are fuzzy numbers. Based on some presented properties, fuzzy algorithms are proposed for finding the absolute center and vertex center of a fuzzy tree. The performance of our presented algorithm is shown by solving two numerical examples. Also, to illustrate the applicability and efficiency of the presented algorithm, a real life example is solved.

Tractable approximations for the distributionally robust conditional vertex p-center problem: Application to the location of high-speed railway emergency rescue stations

This article introduces a variation of the p-center problem (PCP), called distributionally robust conditional vertex p-center problem. This problem differs from the conventional PCP in the sense that (i) some key centers in a given set of candidates are designated, and (ii) a distributionally robust optimization (DRO) method is developed. We present a distributionally robust chance-constrained model to formulate this problem. In terms of tractability, we propose a safe tractable approximation method to reformulate the original DRO model as mixed-integer second-order cone programs under the bounded and Gaussian perturbation ambiguous sets. We further use the branch-and-cut algorithm to solve the tractable counterpart models. The application of the DRO model is illustrated for locating emergency rescue stations in the high-speed railway network by two different sized case studies. Finally, we demonstrate the advantages of the DRO model in comparison with the traditional robust optimisation model and the nominal stochastic programming model.

The complete vertex p-center problem

The vertex p-center problem consists of locating p facilities among a set of M potential sites such that the maximum distance from any demand to its closest located facility is minimized. The complete vertex p-center problem solves the p-center problem for all p from 1 to the total number of sites, resulting in a multi-objective trade-off curve between the number of facilities and the service distance required to achieve full coverage. This trade-off provides a reference to planners and decision-makers, enabling them to easily visualize the consequences of choosing different coverage design criteria for the given spatial configuration of the problem. We present two fast algorithms for solving the complete p-center problem, one using the classical formulation but trimming variables while still maintaining optimality, the other converting the problem to a location set covering problem and solving for all distances in the distance matrix. We also discuss scenarios where it makes sense to solve the problem via brute-force enumeration. All methods result in significant speed-ups, with the set covering method reducing computation times by many orders of magnitude.

A scalable exact algorithm for the vertex p-center problem

The vertex p-center problem consists in selecting p centers among a finite set of candidates and assigning a set of clients to them, with the aim of minimizing the maximum dissimilarity between a client and its associated center. State-of-the-art algorithms for the problem are based on the solution of a series of covering subproblems, and are particularly efficient when additional reduction rules are used to limit the size of the subproblems. In fact, these algorithms do not scale well when the cardinality of the dissimilarity matrix grows above a few million entries, as the time and space required to compute and store the matrix start dominating those required for solving the subproblems. We introduce a scalable relaxation-based iterative algorithm that does not rely on the computation of the entire matrix, but rather relies on the computation of a typically much smaller sub-matrix that is only enlarged if deemed necessary. This method can solve to proven optimality p-center problems derived from the TSP library and containing up to one million clients for small but realistic values of p.

Greedy Randomized Adaptive Search Procedure with Path-Relinking for the Vertex p-Center Problem

The p-center problem consists of choosing a subset of vertices in an undirected graph as facilities in order to minimize the maximum distance between a client and its closest facility. This paper presents a greedy randomized adaptive search procedure with path-relinking (GRASP/PR) algorithm for the p-center problem, which combines both GRASP and path-relinking. Each iteration of GRASP/PR consists of the construction of a randomized greedy solution, followed by a tabu search procedure. The resulting solution is combined with one of the elite solutions by path-relinking, which consists in exploring trajectories that connect high-quality solutions. Experiments show that GRASP/PR is competitive with the state-of-the-art algorithms in the literature in terms of both solution quality and computational efficiency. Specifically, it virtually improves the previous best known results for 10 out of 40 large instances while matching the best known results for others.

Ubicación óptima de unidades de servicio bajo condiciones de capacidad limitada mediante un método metaheurístico

Introducción. El problema de localización de p-centro capacitado consiste en ubicar p instalaciones y asignar usuarios a cada una de ellas, de tal manera que se minimice la distancia máxima entre cualquier usuario y su instalación asignada, sujeto a la capacidad en la demanda restringida por cada instalación. Este trabajo propone una metodología heurística para la solución del problema; los resultados de la experimentación demuestran la calidad de la heurística propuesta en relación con los métodos existentes en la literatura. Método. Se propone una metodología heurística para la solución de este problema, la cual integra varios componentes, tales como un método voraz-adaptativo con una selección probabilística, búsqueda local voraz iterada y una búsqueda descendente por entornos variables. Resultados. La evidencia empírica sobre un conjunto de instancias de localización usualmente utilizadas en la literatura, revela el impacto positivo de cada uno de los componentes desarrollados y de la calidad de la heurística propuesta en relación con los métodos existentes. Por ejemplo, la heurística propuesta pudo encontrar soluciones factibles a todas las instancias probadas, excepto a dos; mientras que el mejor de los otros tres métodos probados falló en 18 de las instancias. Conclusión. Se encontró empíricamente que la heurística propuesta supera a la mejor heurística existente para este problema en términos de calidad de la solución, tiempo de ejecución y confiabilidad en la búsqueda de soluciones factibles en instancias difíciles.

Improving the quality of heuristic solutions for the capacitated vertex p-center problem through iterated greedy local search with variable neighborhood descent

The capacitated vertex p-center problem is a location problem that consists of placing p facilities and assigning customers to each of these facilities so as to minimize the largest distance between any customer and its assigned facility, subject to demand capacity constraints for each facility. In this work, a metaheuristic for this location problem that integrates several components such as greedy randomized construction with adaptive probabilistic sampling and iterated greedy local search with variable neighborhood descent is presented. Empirical evidence over a widely used set of benchmark data sets on location literature reveals the positive impact of each of the developed components. Furthermore, it is found empirically that the proposed heuristic outperforms the best existing heuristic for this problem in terms of solution quality, running time, and reliability on finding feasible solutions for hard instances.

New methods for solving a vertex p-center problem with uncertain demand-weighted distance: A real case study

Article history: Received July 11 2014 Received in Revised Format October 23 2014 Accepted October 26 2014 Available online October 26 2014 Vertex and p-center problems are two well-known types of the center problem. In this paper, a pcenter problem with uncertain demand-weighted distance will be introduced in which the demands are considered as fuzzy random variables (FRVs) and the objective of the problem is to minimize the maximum distance between a node and its nearest facility. Then, by introducing new methods, the proposed problem is converted to deterministic integer programming (IP) problems where these methods will be obtained through the implementation of the possibility theory and fuzzy random chance-constrained programming (FRCCP). Finally, the proposed methods are applied for locating bicycle stations in the city of Tabriz in Iran as a real case study. The computational results of our study show that these methods can be implemented for the center problem with uncertain frameworks. © 2015 Growing Science Ltd. All rights reserved

Enhancements to Two Exact Algorithms for Solving the Vertex P-Center Problem

Enhancements to two exact algorithms from the literature to solve the vertex P-center problem are proposed. In the first approach modifications of some steps are introduced to reduce the number of ILP iterations needed to find the optimal solution. In the second approach a simple enhancement which uses tighter initial lower and upper bounds, and a more appropriate binary search method are proposed to reduce the number of subproblems to be solved. These ideas are tested on two well known sets of problems from the literature (i.e., OR-Lib and TSP-Lib problems) with encouraging results.

On alternativep-center problems

LetG=(V, E) be an undirected connected graph with positive edge lengths. The vertexp-center problem is to find the optimal location ofp centers so that the maximum distance to a vertex from its nearest center is minimized, where the centers may be placed at the vertices. Kariv and Hakimi have shown that this problem is NP-hard. We will consider two modifications of this problem in which each center may be located in one of two predetermined vertices. We will show the NP-completeness of their recognition versions.

Algorithms for finding P‐centers on a weighted tree (for relatively small P)

AbstractAlgorithms of O(p. n. log n) time to find an (absolute or vertex) p‐center on a weighted tree of n vertices are presented. Those algorithms represent improvement over previous known results in the range p < log n (for the vertex p‐center problem) and in the range p < log n. log log n (for the absolute p‐center problem).