The stratified p-center problem

This work presents an extension of the p-center problem. In this new model, called Stratified p-Center Problem (SpCP), the demand is concentrated in a set of sites and the population of these sites is divided into different strata depending on the kind of service that they require. The aim is to locate p centers to cover the different types of services demanded minimizing the weighted average of the largest distances associated with each of the different strata. In addition, it is considered that more than one stratum can be present at each site. Different formulations, valid inequalities and preprocessings are developed and compared for this problem. An application of this model is presented in order to implement a heuristic approach based on the Sample Average Approximation method (SAA) for solving the probabilistic p-center problem in an efficient way.Comment: 32 pages, 1 pictur

Hybrid Meta-heuristics with VNS and Exact Methods: Application to Large Unconditional and Conditional Vertex p-Centre Problems

Large-scale unconditional and conditional vertex p-centre problems are solved using two meta-heuristics. One is based on a three-stage approach whereas the other relies on a guided multi-start principle. Both methods incorporate Variable Neighbourhood Search, exact method, and aggregation techniques. The methods are assessed on the TSP dataset which consist of up to 71,009 demand points with p varying from 5 to 100. To the best of our knowledge, these are the largest instances solved for unconditional and conditional vertex p-centre problems. The two proposed meta-heuristics yield competitive results for both classes of problems

Neighbourhood Reduction in Global and Combinatorial Optimization: The Case of the p-Centre Problem

Neighbourhood reductions for a class of location problems known as the vertex (or discrete) and planar (or continuous) p-centre problems are presented. A brief review of these two forms of the p-centre problem is first provided followed by those respective reduction schemes that have shown to be promising. These reduction schemes have the power of transforming optimal or near optimal methods such as metaheuristics or relaxation-based procedures, which were considered relatively slow, into efficient and exciting ones that are now able to find optimal solutions or tight lower/upper bounds for larger instances. Research highlights of neighbourhood reduction for global and combinatorial optimisation problems in general and for related location problems in particular are also given

Optimal solutions for the continuous p-centre problem and related α-neighbour and conditional problems: A relaxation-based algorithm

This paper aims to solve large continuous p-centre problems optimally by re-examining a recent relaxation-based algorithm. The algorithm is strengthened by adding four mathematically supported enhancements to improve its efficiency. This revised relaxation algorithm yields a massive reduction in computational time enabling for the first time larger data-sets to be solved optimally (e.g., up to 1323 nodes). The enhanced algorithm is also shown to be flexible as it can be easily adapted to optimally solve related practical location problems that are frequently faced by senior management when making strategic decisions. These include the α-neighbour p-centre problem and the conditional p-centre problem. A scenario analysis using variable α is also performed to provide further managerial insights

A review of network location theory and models

Cataloged from PDF version of article.In this study, we review the existing literature on network location problems. The study has a broad scope that includes problems featuring desirable and undesirable facilities, point facilities and extensive facilities, monopolistic and competitive markets, and single or multiple objectives. Deterministic and stochastic models as well as robust models are covered. Demand data aggregation is also discussed. More than 500 papers in this area are reviewed and critical issues, research directions, and problem extensions are emphasized.Erdoğan, Damla SelinM.S

An Investigation into Exact Methods for the Continuous p?Centre Problem and its Related Problems

This thesis will analyse, investigate and develop new and interesting ideas to optimally solve a location problem called the continuous p?centre problem. This problem wishes to locate p facilities in a plane or network of n demand points such that the maximum distance or travel time between each demand point and its closest facility is minimised. Several difficulties are identified which are to be overcome to solve the continuous p?centre problem optimally. These relate to producing a finite set of potential facility locations or decreasing the problem size so that less computational time and effort is required. This thesis will examine several schemes that can be applied to this location problem and its related version with the aim to optimally solve large problems that were previously unsolved. This thesis contains eight chapters. The first three chapters provide an introduction into location problems, with a focus on the p?centre problem. Chapter 1 begins with a brief history of location problems, followed by the various classifications and methodologies used to solve them. Chapter 2 provides a review of the methods that have been used to solve the p?centre problem, with a focus on the continuous p?centre problem. An overview of the location models used in this research is given in Chapter 3, alongside an initial investigative work. The next two chapters enhance two well-known optimal algorithms for the continuous p?centre problem. Chapter 4 develops an interesting exact algorithm that was first proposed over 30 years ago. The original algorithm is reexamined and efficient neighbourhood reductions which are mathematically supported are proposed to improve its overall computational performance. The enhanced algorithm shows a substantial reduction of up to 96% of required computational time compared to the original algorithm, and optimal solutions are found for large data sets that were previously unsolved. Chapter 5 develops a relatively new relaxation-based optimal method. Four mathematically supported enhancements are added to the algorithm to improve its efficiency and its overall computational time. The revised reverse relaxation algorithm yields a vast reduction of up to 87% of computational time required, which is then used to solve larger data sets where n ? 1323 optimally. Chapter 6 creates a new relaxation-based matheuristic, called the relaxed p' matheuristic, by combining a well-known heuristic and the optimal method developed in Chapter 5. The unique property of the matheuristic is that it deals with the relaxation of facilities rather than demand points to establish a sub-problem. The matheuristic yields a good, but not necessarily optimal, solution in a reasonable time. To guarantee optimality, the results found from the matheuristic are embedded into the optimal algorithms developed in Chapters 4 and 5. Chapter 7 adapts the optimal algorithm developed in Chapter 5 to solve two related location problems, namely the ??neighbour p?centre problem and the conditional p?centre problem. The ??neighbour p?centre problem is investigated and solved where ? = 2 & 3. A scenario analysis is also conducted for managerial insights by exploring changes in the number of facilities required to cover each demand point. Furthermore, an existing algorithm for the conditional p?centre problem is enhanced by incorporating the optimal algorithm proposed in Chapter 5, and it is used to solve large data sets where the number of preexisting facilities is 20. This chapter therefore demonstrates that an algorithm developed in this research can be adapted or used to enhance existing algorithms to optimally solve more practical and challenging related location problems. Finally, Chapter 8 summarises the findings and main achievements of this research, and outlines any further work that could be worthwhile exploring in the future