Classical and Inverse Median Location Problems under Uncertain Environment

A hybridisation of a clustering-based technique and of a variable neighbourhood search (VNS) is designed to solve large-scale $$p$$p-median problems. The approach is based on a multi-stage methodology where learning from previous stages is taken into account when tackling the next stage. Each stage is made up of several subproblems that are solved by a fast procedure to produce good feasible solutions. Within each stage, the solutions returned are put together to make up a new promising subset of potential facilities. This augmented $$p$$p-median problem is then solved by VNS. As these problems used aggregation, a cost evaluation based on the original demand points instead of aggregation is computed for each of the `aggregation'-based solution. The one yielding the least cost is then selected and its chosen facilities included into the next stages. This multi-stage process is repeated several times until a certain criterion is met. This approach is enhanced by an efficient way to aggregate the data and a neighbourhood reduction scheme when allocating demand points to their nearest facilities. The proposed approach is tested, using various values of $$p$$p, on the largest data sets from the literature with up to 89,600 demand points with encouraging results.

Now a day there is much more amplified curiosity in combinatorial optimization. The p-median problem which is a facility location problem, deals with discrete data and hence it is characterized as a combinatorial optimization problem. It is NP-Hard in nature that ascertains the specified number of locations as facilitators which serves the maximum locations. The p-median problem will be productive in several applications areas such as mounting marketing strategies in Management Sciences and recognition of server positions in computer networks. A new Metaheuristic approach with Neighbourhood Search (NS) technique has been proposed in the present paper which unveil all possible combinations with the elements in the neighbourhood of individual elements in the solution and recognizes the optimal solution i.e., which serves the maximum locations so that the sum of the total distance from the each element to the facilities is minimized. The proposed metaheuristic approach is an iterative one which contains two phases. Construction phase is the first phase that structure the initial solution and based on the initial solution the second phase explore for the optimal solution based on NS approach and then the probable solution space is computed to obtain the optimal solution.

Mixed planar/network facility location problems

There is a number of models that take into account the market competition in the location theory. Aboulian R. et al. formulated the location and design problem (CFLDP), in which the share of the market being served elastically varies depending on the location. The models with elastic demand are considered to be rather complicated and advantageous. We developed this problem and proposed the new one which combines CFLDP and classical p-median problem. In the given formulation a number of new facilities is fixed and is equal to p, so competitive p-median facility location and design problem (CPFLDP) has been derived. Therefore, the task can be considered as a clustering problem. In this paper we develop local search algorithms for CPFLDP. A computational experiment is carried out, the results are discussed.

p-Median based formulations with backbone facility locations

The Discrete Ordered Median Problem (DOMP) unifies discrete facility location problems in the sense that classical location problems, such as the N-median, N-centre and Uncapacitated Facility Location problems, are particular cases of the DOMP. In addition, new location problems, which are interesting from a practical point of view (e.g. the k 1 + k 2 trimmed mean problem, see Section 2.1.1), can be formulated as special cases of the DOMP. Therefore, the study of the DOMP is interesting not only from a theoretical point of view (unification of discrete facility location problems) but also from a practical one (possibility to formulate new interesting problems).KeywordsLocation ProblemVariable Neighbourhood SearchFacility Location ProblemSupply Chain PlanningUncapacitated Facility LocationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Applied p-median and p-center algorithms for facility location problems

The classical p-median problem is discussed, together with methods for its solution. The multi-median problem, a generalization of the p-median problem in which more than one type of facility is allowed, is introduced and methods of solution developed. Numerical results are presented.

This paper considers two planar facility location problems while employing the Manhattan travel metric. We first consider the p-median problem in the presence of arbitrarily shaped barriers and convex forbidden regions. For this problem we establish that the search for an optimal solution can be restricted to a finite set of easily identifiable points. Next, we consider the stochastic queue median problem in the presence of arbitrarily shaped barriers. A procedure to obtain a global optimum solution for this problem is established. The results of the paper are illustrated via numerical examples. Finally, we comment on a connection between network location problems and planar location problems which use the Manhattan travel metric.

This paper considers the p-median problem on a general network with link demands. We construct the network in such a way that only transportation intersections are taken to be nodes and, therefore, both continuous and discrete link demands are allowed in our model. For such a model, we show that a nodal solution can be used to approximate the true optimal solution and an error bound which involves only the demands on a single link is given for the error caused by such an approximation. Based on nodal solutions, we demonstrate that a model with continuous link demands can be transformed into an equivalent discrete link demand model. Further, we propose a method to aggregate demands on each link in solving the p-median problem on a general network without introducing any aggregation errors to the problem solution. The implementation of the proposed approach with some heuristics is discussed. © 2000 John Wiley & Sons, Inc.

When a location problem is too large to be computed in a reasonable time or if it is impossible to store it in the computer memory, an aggregation is commonly used tool that allows for transforming it to smaller size. Typically, an aggregation method is used only once, in the initial phase, before the solving process. An unavoidable consequence of the aggregation is the loss of optimality due to aggregation errors. The effects of aggregation errors might be significant, especially, when solving spatially large problems with huge number of potential customers. Here, we propose new simple re-aggregation approach, which minimizes sources of the aggregation errors. Our method aggregates the original problem to the size that can be solved by the used optimization algorithm, and in each iteration, the aggregated problem is adapted to achieve more precise location of facilities. The proposed approach is versatile and thus it can be easily extended to other location problems. To investigate the optimality error, we use benchmarks that can be computed exactly and to test the applicability of our approach, we study large benchmarks reaching 670 000 customers. Numerical experiments reveal that the re-aggregation heuristic works well for commonly used, as well as for extremely large sizes of the p-median problem. The approach allows for finding solutions of higher quality than the exact method that is applied to the aggregated problem.

Background: Possible solutions to overcome the many challenges of home delivery are collection and delivery points (CDPs). In addition to commercial facilities, the role of CDPs can also be played by users’ households, providing a crowd storage service. Key decisions regarding CDPs relate to their location, as well as the allocation of users to selected locations, so that the distance of users from CDPs is minimal. Methods: In this paper, the described problem is defined as a p-median problem and solved for the area of the city of Belgrade, using the heuristic “greedy” and the simulated annealing algorithm. Results: Fifty locations of CDPs were selected and the users allocated to them were distributed in over 950 zones. The individual distances between users and the nearest CDPs and the sum of these distances, multiplied by the number of requests, were obtained. An example of modification of the number of CDPs is presented as a way of obtaining solutions that correspond to different preferences of operators and/or users in terms of their distances from the CDPs. Conclusions: User households can be used as CDPs to achieve various benefits. Locating CDPs, i.e., selecting households, can be solved as a p-median problem, using a combination of heuristic and metaheuristic algorithms. In addition, by modifying the number of medians, the total and average distances between users and CDPs can be better managed. The main contributions of the paper are the establishment of users’ households as potential locations of CDPs, the establishment of a framework for analysis of impact of the number of CDPs on the sum and average distances from the customers, as well as the creation of a basis for upgrading and modifying the model for implementation in the business practice.

Many distributed services in computer networks rely on a set of active facilities that are selected among a potentially large number of candidates. The active facilities then contribute and cooperate to deliver a specific service to the users of the distributed system. In this scenario graph partitioning or clustering is often adopted to determine the most efficient locations of the facilities. The identification of the optimal set of facility locations is known as the p-median problem in networks, is NP-hard and is typically solved by using heuristic methods. The goal is to select p locations among all candidate network nodes such that some cost function is minimised. A typical example of such a function is the overall communication cost to deliver the service to the users of the distributed system. Locating facilities in near-optimal locations has been extensively studied for different application domains. Most of these studies have investigated sequential algorithms and centralised approaches. However, centralised approaches are practically infeasible in large-scale and dynamic networks, where the problem is inherently distributed or because of the large communication overhead and memory requirements for gathering complete information about the network topology and the users. In this work distributed approaches to the p-median problem are investigated. Two solutions are proposed for addressing the facility locations problem in a fully distributed environment. Two different iterative heuristic approaches are applied to gradually improve a random initial solution and to converge to a final solution with a local minimum of the overall cost. While the first approach adopts a fine granularity by identifying a single change to improve the solution at each iteration, the second approach applies changes to every component of the solution at each iteration. An experimental comparative analysis based on simulations has shown that the approach with a finer granularity is able to deliver a better optimisation of the overall cost with longer convergence time. Both approaches have excellent scalability and provide an effective tool to optimise the facility locations from within the network. No prior knowledge of the system is required, no data needs to be gathered in a centralised server and the same process is used to identify and to deploy the facility locations solution in the network since the process is fully decentralised.

The p-median problem is to select p facility sites from among n locations to minimize the average distance from the populations at the n locations to their nearest facility. A set of linear constraints and a linear objective function describe the problem. By varying the way that the objective function coefficients are derived, many other location problems can be defined as special cases of the same general mathematical form of the p-median model. These models include maximum distance-covering problems, problems with facility costs, and problems having multiple objectives. The diversity of these special cases suggests the use of the model as the core of a computer software system for location—allocation and spatial analyses.

Solutions to combinatorial optimization problems frequently rely on heuristics to minimize an intractable objective function. The optimum is sought iteratively and pre-setting the number of iterations dominates in operations research applications, which implies that the quality of the solution cannot be ascertained. Deterministic bounds offer a mean of ascertaining the quality, but such bounds are available for only a limited number of heuristics and the length of the corresponding interval may be difficult to control in an application. A small, almost dormant, branch of the literature suggests using statistical principles to derive statistical bounds for the optimum. We discuss alternative approaches to derive statistical bounds. We also assess their performance by testing them on 40 test $$p$$p-median problems on facility location, taken from Beasley's OR-library, for which the optimum is known. We consider three popular heuristics for solving such location problems; simulated annealing, vertex substitution, and Lagrangian relaxation where only the last offers deterministic bounds. Moreover, we illustrate statistical bounds in the location of 71 regional delivery points of the Swedish Post. We find statistical bounds reliable and much more efficient than deterministic bounds provided that the heuristic solutions are sampled close to the optimum. Statistical bounds are also found computationally affordable.