Speeding up the optimal method of Drezner for the p-centre problem in the plane

This paper revisits an early but interesting optimal algorithm first proposed by Drezner to solve the continuous p-centre problem. The original algorithm is reexamined and efficient neighbourhood reductions which are mathematically supported are proposed to improve its overall computational performance. The revised algorithm yields a considerably high reduction in computational time reaching, in some cases, a decrease of 96%. This new algorithm is now able to find proven optimal solutions for large data sets with over 1300 demand points and various values of p for the first time

An elliptical cover problem in drone delivery network design and its solution algorithms

Given n demand points in a geographic area, the elliptical cover problem is to determine the location of p depots (anywhere in the area) so as to minimize the maximum distance of an economical delivery trip in which a delivery vehicle starts from the nearest depot to a demand point, visits the demand point and then returns to the second nearest depot to that demand point. We show that this problem is NP-hard, and adapt Cooper’s alternating locate-allocate heuristic to find locally optimal solutions for both the point-coverage and area-coverage scenarios. Experiments show that most locally optimal solutions perform similarly well, suggesting their sufficiency for practical use. The one-dimensional variant of the problem, in which the service area is reduced to a line segment, permits recursive algorithms that are more efficient than mathematical optimization approaches in practical cases. The solution also provides the best-known lower bound for the original problem at a negligible computational cost

The multi-period pp-center problem with time-dependent travel times

This paper deals with an extension of the $p$-center problem, in which arc traversal times vary over time, and facilities are mobile units that can be relocated multiple times during the planning horizon. We investigate the relationship between this problem and its single-period counterpart. We also derive some properties and a special case. The insight gained with this analysis is then used to devise a two-stage heuristic. Computational results on instances based on the Paris (France) road graph indicate that the algorithm is capable of determining good-quality solutions in a reasonable execution time

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

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

The continuous single source location problem with capacity and zone-dependent fixed cost: Models and solution approaches

The continuous capacitated single-source multi-facility Weber problem with the presence of facility fixed cost is investigated. A new mathematical model which incorporates multi-level type capacity (or design) and facility fixed cost that is capacity-based and zone-dependent is introduced. As no data set exists for this new location problem, a new data set based on convex polygons using triangular shape is constructed. A generalised two stage heuristic scheme that combines the concept of aggregation, an exact method, and an enhanced Cooper’s alternate location-allocation method is put forward. A framework that embeds Variable Neighbourhood Search is also proposed. Computational experiments show that these matheuristics produce encouraging results for this class of location problems. The proposed approaches are also easily adapted to cater for a recently studied variant namely the single-source capacitated multi-facility Weber problem where they outperform those recently published solution method

Adaptive Heuristic Methods for the Continuous p-Centre Location Problems

This research studies the p-centre problem in the continuous space. This problem is particularly useful in locating emergency facilities, such as fire-fighting stations, police stations and hospitals where it is aimed to minimise the worst-case response time. This problem can be divided into a single facility minmax location problem (1-centre) and multi-facility minmax location problem (p-centre). The solution of the 1-centre location problem can be found optimally in polynomial time by using the well known Elzinga-Hearn algorithm for both the weighted and the unweighted case. The objective of the p-centre problem is to locate p facilities (p>1) so as to minimise the radius of the largest circle. However, in this case, we cannot always guarantee optimality as the problem is known to be NP hard. The aim of the research is to develop and analyse powerful meta-heuristics including the hybridisation of exact methods and heuristics to solve this global optimisation problem. To our knowledge this is the first study that meta-heuristics are developed for this problem. In addition larger instances previously used in the literature are tested .This is achieved by designing an efficient variable neighbourhood search, adapting a powerful perturbation method and extending a newly developed reformulation local search. Large instances are used to evaluate our approaches with promising results

Addressing the Location Problem of a Perishables Redistribution Center in the Middle of Europe

This work aims to contribute to the debate on practical utilization of different location models for consolidation, redistribution, and repackaging centers in a supply chain to optimize shipments, thereby reducing food loss and waste, within the framework of quality of customer service improvement. The scenario in question is the creation of a redistribution center for highly perishable products (fruits and vegetables) from southeast Spain—the leading European supplier—for customers throughout Europe. It is estimated that 10% of exports (more than 530,000 metric tons) from this area are returned by customers due to minor defects. These products cannot be reused and are therefore wasted. Regarding the methodology, comparisons were made between the p-median, gravity p-median, and p-center models. Scenarios of change in demand and randomness in distances were also tested. In addition, the modelling used included the cost and time within a multicriteria optimization framework to assess the possibility of a transport mode change. It was observed, for example, that the gravity p-median model proved useful for perishable products and the logistics strategy chosen. Furthermore, the p-median model displayed strong robustness against long-term changes in demand and random distances. In general, it was demonstrated that this strategy would successfully reduce the response time and distance of shipment from the distribution center to the customers and thereby improve sustainability of the service, reducing the waste related to direct shipments. Furthermore, this research also demonstrated the difficulty of using intermodality in this context, mainly due to transit time, which would undoubtedly increase the waste generate