Classical P-median Problem Research Articles

The Classical p-median Problem as a Surrogate Model in Hub Location

The Classical p-median Problem as a Surrogate Model in Hub Location

Classical and Inverse Median Location Problems under Uncertain Environment

In this paper, we first consider the classical p-median location problem on a network in which the vertex weights and the distances between vertices are uncertain variables. The uncertainty distribution of the optimal objective value of the p-median problem is given and the concepts of the α-p-median, the most p-median and the expected p-median are introduced. Then, it is shown that the uncertain p-median problem is NP-hard on general networks. However, if the underlying network is a tree, an efficient algorithm for the uncertain 1-median problem with linear time complexity is proposed. Finally, we investigate the inverse 1-median problem on a tree with uncertain vertex weights and present a programming model for the problem. Then, it is shown that the proposed model can be reformulated into a deterministic programming model.

When closest is not always the best: The distributed p-median problem

The classical p-median problem assumes that service to customers is always provided by the closest facility, while in practice, customers often interact for a variety of reasons with several of the facilities (not just the closest). In this article, we examine the concept of a distribution rule for modelling a more general case where the demand of a customer is not entirely satisfied by its closest facility, but rather is split into different flows to different facilities according to the given rule. We use this concept to formulate a new class of median problems, which we call the “distributed” p-median problem. Different types of distribution rules are investigated leading to some interesting properties. For example, if the weights are increasing (ie, assigned flows are greater to facilities that are further away), the problem can be solved in polynomial time as a 1-median problem. For decreasing weights, we obtain new and efficient generalizations of the standard continuous and discrete p-median models, which in turn lead to a broader interpretation of median points and a generalization of Cooper’s well-known locate–allocate heuristic. Some small numerical examples and computational results are given to illustrate the concepts.

Simulated Annealing for Competitive p-Median Facility Location Problem

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.

An analysis of p-median location problem: Effects of backup service level and demand assignment policy

Any solution to facility location problems will consider determining the best suitable locations with respect to certain criteria. Among different types of location problems, involving emergency service system (ESSs) are one of the most widely studied in the literature, and solutions to these problems will mostly aim to minimize the mean response time to demands. In practice, however, a demand may not be served from its nearest facility if that facility is engaged in serving other demands. This makes it a requirement to assign backup services so as to improve response time and service quality. The level of backup service is a key, strategic-level planning factor, and must be taken into consideration carefully. Moreover, in emergency service operations conducted in congested demand regions, demand assignment policy is another important factor that affects the system performance. Models failing to adopt sufficient levels of backup service and realistic demand assignment policies may significantly deteriorate the system performance.Considering the classic p-median problem (pMP) location model, this paper investigates the effects of backup service level, demand assignment policy, demand density, and number of facilities and their locations on the solution performance in terms of multiple metrics. For this purpose, we adopt a combined optimization and simulation approach. We will first modify the classic pMP to account for distances to backup services. Next, we employ a discrete event simulation to evaluate the performance of location schemes obtained from the deterministic mathematical model. Our results provide insights for decision-makers while planning ESS operations.

Fuzzy Majority Algorithms for the 1-Median and 2-Median Problems on a Fuzzy Tree

ABSTRACTIn the classical p-median problem, we want to find a set Y containing p points in a given graph G such that the sum of weighted distances from Y to all vertices in V is minimised. In this paper, we consider the 1-median and 2-median problems on a tree with fuzzy weights. We show that the majority property holds for fuzzy 1-median problem on a tree. Then based on a proposed ranking function and the majority property, a fuzzy algorithm is presented to find the median of a fuzzy tree. Finally, the algorithm is extended to solve 2-median problem on fuzzy trees.

P-Median based formulations with backbone facility locations

In this paper, we propose new mixed integer linear programming (MILP) models for the p-Median problem subject to ring, tree and star backbone topology constraints on the facility locations. More precisely, we minimize simultaneously the total connection cost distances between customers and facilities, and cost distances among facilities when connected under a ring, tree or a star network topology. In principle, all our proposed models can be used in any application related with the classical p-Median problem. Application examples include wired and wireless network design, computer networks, transportation, water supply and electrical networks, just to name a few. The proposed models arise as a combination of the classical p-Median problem with the traveling salesman, spanning tree, and star network problems, respectively. We prove the correctness of each proposed model. Then, we further propose variable neighborhood search (VNS) meta-heuristic algorithms, one for each topology. Our numerical results indicate that the ring models are harder to solve with CPLEX than the tree and star ones. Whilst VNS algorithms proved to be highly efficient when compared to the optimal solutions of the problem for small, medium and large size instances. Moreover, we obtain better feasible solutions than CPLEX for the large instances and in significantly less computational cost.

Incorporating activity space and trip chaining into facility siting for accessibility maximization

Location models have been widely used to support locational decisions for various service provision. One common objective of location models has been to ensure maximal accessibility of sited facilities to demand populations. Accessibility evaluation in location models often assumes that trips originate from fixed locations (usually home) and are single purpose. These assumptions contradict the empirical evidence that suggests trips also commonly originate from non-home locations and may involve multiple stops. In this study, a new multi-objective location model is developed that extents the classic p-median problem (PMP) to account for a more realistic assessment of accessibility. Based on the individual accessibility assessment, notions of trip chaining and activity space are incorporated into the model development. In addition to fixed home locations, stops along chained trips are allowed for potential service site visits, and activity space is introduced as an additional dimension to evaluate accessibility of alternative opportunities. The effectiveness of the new model is demonstrated using an application in Tucson, AZ.

A Lagrangian search method for the P-median problem

In this paper, we propose a novel algorithm for solving the classical P-median problem. The essential aim is to identify the optimal extended Lagrangian multipliers corresponding to the optimal solution of the underlying problem. For this, we first explore the structure of the data matrix in P-median problem to recast it as another equivalent global optimization problem over the space of the extended Lagrangian multipliers. Then we present a stochastic search algorithm to find the extended Lagrangian multipliers corresponding to the optimal solution of the original P-median problem. Numerical experiments illustrate that the proposed algorithm can effectively find a global optimal or very good suboptimal solution to the underlying P-median problem, especially for the computationally challenging subclass of P-median problems with a large gap between the optimal solution of the original problem and that of its Lagrangian relaxation.

A hybrid artificial bee colony algorithm for the p-median problem with positive/negative weights

In the classical p-median problem, the objective is to find a set Y of p vertices on an undirected graph \(G=(V,E)\) in such a way that \( Y \subseteq V\) and the sum of distances from all the vertices to their respective closest vertices in Y is minimized. In this paper, we have considered the weighted case where every vertex in G has either a positive or a negative weight under two different objective functions, viz. the sum of the minimum weighted distances and the sum of the weighted minimum distances. In this paper, we have proposed a hybrid artificial bee colony (ABC) algorithm for the positive/negative weighted p-median problem where each solution generated by ABC algorithm is improved by an interchange based randomized local search. In addition, an interchange based exhaustive local search is applied on some of the best solutions obtained after the execution of ABC algorithm in a bid to further improve their quality. We have compared our approach with the state-of-the-art approaches available in the literature on the standard benchmark instances. Computational results demonstrate the effectiveness of our approach.

A new Semi-Lagrangean Relaxation for the k-cardinality assignment problem

Recently Beltran-Royo, Vial & Alonso-Ayuso (2012) presented a semi-Lagrangean relaxation for the classical p-median location problem and for the incapacitated facility location problem. The results, obtained using the semi-Lagrangean relaxation approach, were quite impressive. In this paper we use a semi-Lagrangean relaxation to obtain an efficient solution method for the k-cardinality assignment problem. The method has only one semi-Lagrangean multiplier that can only take on a limited number of values, making the search for the optimal multiplier easy. Since the semi-Lagrangean relaxation closes the duality gap, this leads to an extremely reliable and easily implementable method for finding k-cardinality assignments in large-scale cases. The method is computationally tested on test examples commonly used in the literature.

Locating bioenergy facilities using a modified GIS-based location–allocation-algorithm: Considering the spatial distribution of resource supply

This paper proposes a modification to the classic p-median problem that considers the spatial distribution of supply resources and competition for them by potential facility locations. It is illustrated with a simplified case study to optimally locate community scale anaerobic digesters (ADs) in an area in the East Midlands in the UK. The modification evaluates the spatial distribution of the feedstocks needed by each potential AD unit and only includes new locations if their supply catchments do not overlap with the catchments of the current set of locations being considered. The modified algorithm is implemented using the Teitz and Bart search heuristic. This starts with an initial set of locations that it seeks to improve by swapping poor performing locations with better ones. In this case the modification takes account of the spatial distribution of the feedstocks for a typical AD recipe whilst seeking minimise demand weighted distance. The results demonstrate large improvements over the classic p-median model in location selection that eliminate the overlap of facility catchments. Some wider points relating to the robustness of many analyses reported in the bioenergy literature are discussed, with some observations about commonly found methodological deficiencies, before some areas for further research are suggested.

A unified approach for location-allocation analysis: integrating GIS, distributed computing and spatial optimization

Location-allocation modeling is an important area of research in spatial optimization and GIScience. A large number of analytical models for location-allocation analysis have been developed in the past 50 years to meet the requirements of different planning and spatial-analytic applications, ranging from the location of emergency response units (EMS) to warehouses and transportation hubs. Despite their great number, many location-allocation models are intrinsically linked to one another. A well-known example is the theoretical link between the classic p-median problem and coverage location problems. Recently, Lei and Church showed that a large number of classic and new location models can be posed as special case problems of a new modeling construct called the vector assignment ordered median problem (VAOMP). Lei and Church also reported extremely high computational complexity in optimally solving the best integer linear programming (ILP) formulation developed for the VAOMP even for medium-sized problems in certain cases.In this article, we develop an efficient unified solver for location-allocation analysis based on the VAOMP model without using ILP solvers. Our aim is to develop a fast heuristic algorithm based on the Tabu Search (TS) meta-heuristic, and message passing interface (MPI) suitable for obtaining optimal or near-optimal solutions for the VAOMP in a real-time environment. The unified approach is particularly interesting from the perspective of GIScience and spatial decision support systems (DSS) as it makes it possible to solve a wide variety of location models in a unified manner in a GIS environment. Computational results show that the TS method can often obtain in seconds, solutions that are better than those obtained using the ILP-based approach in hours or a day.

Reliable Facility Systems Based on the p-Median Problem Subject to Edge Failures

The reliability of a facility system is to a large degree adversely affected by the edge facility failures. In this paper, we consider the design of reliable facility systems based on the classical p-median problem subject to edge failures. A reliable model is formulated and a scenario based algorithm is applied for computing an instance. The results show that the reliability of a facility system will increase when considering the edge facility failures in locating decision.

Vector Assignment Ordered Median Problem

The vector assignment p-median problem (VAPMP) and the ordered p-median problem (OMP) are important extensions of the classic p-median problem. The VAPMP extends the p-median problem by allowing assignment of a demand to multiple facilities, and a wide variety of multi-assignment and backup location problems are special cases of this problem. The OMP optimizes a weighted sum of service distances according to their relative ranks among all demands. The OMP is well known as it represents a generalization of both the p-median and the p-center problems. In this article, a new model is developed which extends both the VAPMP and OMP problems. In addition, beyond median, center, and vector assignment, this new model can resolve problems where the system objective involves maximizing distance. The new model also gives rise to meaningful special-case problems, such as a “reliable p-center” problem. Different integer linear programming (ILP) formulations of the new problem are presented and tested. It is demonstrated that an efficient formulation for a special case of the VAOMP problem can solve medium sized problems optimally in a reasonable amount of time.

Analysis of Facility Systems’ Reliability Subject to Edge Failures: Based on the p-Median Problem

We view a facility system as a kind of supply chain and model it as a connected graph in which the nodes represent suppliers, distribution centers or customers and the edges represent the paths of goods or information. The efficiency, and hence the reliability, of a facility system is to a large degree adversely affected by the edge failures in the network. In this paper, we consider facility systems' reliability analysis based on the classical p-median problem when subject to edge failures. We formulate two models based on deterministic case and stochastic case to measure the loss in efficiency due to edge failures and give computational results and reliability envelopes for a specific example.

Constructs for Multilevel Closest Assignment in Location Modeling

In the classic p-median problem, it is assumed that each point of demand will be served by his or her closest located facility. The p-median problem can be thought of as a ‘‘single-level’’ allocation and location problem, as all demand at a specific location is assigned as a whole unit to the closest facility. In some service protocols, demand assignment has been defined as ‘‘multilevel’’ where each point of demand may be served a certain percentage of the time by the closest facility, a certain percentage of the time by the second closest facility, and so on. This article deals with the case in which there is a need for ‘‘explicit’’ closest assignment (ECA) constraints. The authors review past location modeling work that involves single-level ECA constraints as well as specific constraint constructs that have been proposed to ensure single-level closest assignment. They then show how each of the earlier proposed ECA constructs can be generalized for the ‘‘multilevel’’ case. Finally, the authors provide computational experience using these generalized ECA constructs for a novel multilevel facility interdiction problem introduced in this article. Altogether, this article proposes both a new set of constraint structures that can be used in location models involving multilevel assignment as well as a new facility interdiction model that can be used to optimize worst case levels of facility disruption.

Facility Reliability Issues in Network p-Median Problems: Strategic Centralization and Co-Location Effects

In this paper, we analyze a facility location model where facilities may be subject to disruptions, causing customers to seek service from the operating facilities. We generalize the classical p-median problem on a network to explicitly include the failure probabilities, and analyze structural and algorithmic aspects of the resulting model. The optimal location patterns are seen to be strongly dependent on the probability of facility failure, with facilities becoming more centralized, or even co-located, as the failure probability grows. Several exact and heuristic solution approaches are developed. Results of numerical experiments are reported.

A polynomial method for the pos/neg weighted 3-median problem on a tree

Let a connected undirected graph G = (V, E) be given. In the classical p-median problem we want to find a set X containing p points in G such that the sum of weighted distances from X to all vertices in V is minimized. We consider the semi-obnoxious case where every vertex has either a positive or negative weight. In this case we have two different objective functions: the sum of the minimum weighted distances from X to all vertices and the sum of the weighted minimum distances. In this paper we show that for the case p = 3 an optimal solution for the second model in a tree can be found in O(n5) time. If the 3-median is restricted to vertices or if the tree is a path then the complexity can be reduced to O(n3).

A genetic algorithm for the p-median problem with pos/neg weights

Let a connected undirected graph G = ( V, E) be given. In the classical p-median problem we want to find a set X containing p points in G such that the sum of weighted distances from X to all vertices in V is minimized. We consider the semi-obnoxious case where every vertex has either a positive or negative weight. In this case we have two different objective functions: the sum of the minimum weighted distances from X to all vertices and the sum of the weighted minimum distances. In this paper we propose a genetic algorithm for both problems. Computational results are compared with a previously investigated variable neighborhood search algorithm.