On solving unreliable planar location problems

In this paper, we generalize conventional P-median location problems by considering the unreliability of facilities. The unreliable location problem is defined by introducing the probability that a facility may become inactive. We proposed efficient solution methods to determine locations of these facilities in the unreliable location model. Space-filling curve-based algorithms are developed to determine initial locations of these facilities. The unreliable P-median location problem is then decomposed to P 1-median location problems; each problem is solved to the optimum. A bounding procedure is used to monitor the iterative search, and to provide a consistent basis for termination. Extensive computational tests have indicated that the heuristics are efficient and effective for solving unreliable location problems.Scope and purposeThis paper addresses an important class of location problems, where p unreliable facilities are to be located on the plane, so as to minimize the expected travel distance or related transportation cost between the customers and their nearest available facilities. The unreliable location problem is defined by introducing the probability that a facility may become inactive. Potential application of the unreliable location problem is found in numerous areas. The facilities to be located can be fire station or emergency shelter, where it fails to provide service during some time window, due to the capacity or resource constraints. Alternatively, the facilities can be telecommunication posts or logistic/distribution centers, where the service is unavailable due to breakdown, repair, shutdown of unknown causes. In this paper, we prescribed heuristic procedures to determine the location of new facilities in the unreliable location problems. The numerical study of 2800 randomly generated instances has shown that these solution procedures are both efficient and effective, in terms of computational time and solution quality.     

Dynamic facility location: The progressive p-median problem

A dynamic p-median problem is considered. Demand is changing over a given time horizon and the facilities are built one at a time at given times. Once a new facility is built, some of the customers will use its services and some of the customers will patronize an existing facility. At any given time, customers patronize the closest facility. The problem is to find the best locations for the new facilities. The problem is formulated and the two facilities case is solved by a special algorithm. The general problem is solved using the standard mathematical programming code AMPL.     

BEAMR: An exact and approximate model for the p-median problem

The p-median problem is perhaps one of the most well-known location–allocation models in the location science literature. It was originally defined by Hakimi in 1964 and 1965 and involves the location of p facilities on a network in such a manner that the total weighted distance of serving all demand is minimized. This problem has since been the subject of considerable research involving the development of specialized solution approaches as well as the development of many different types of extended model formats. One element of past research that has remained almost constant is the original ReVelle–Swain formulation [ReVelle CS, Swain R. Central facilities location. Geographical Analysis 1970;2:30–42]. With few exceptions as detailed in the paper, virtually no new formulations have been proposed for general use in solving the classic p-median problem. This paper proposes a new model formulation for the p-median problem that contains both exact and approximate features. This new p-median formulation is called Both Exact and Approximate Model Representation (BEAMR). We show that BEAMR can result in a substantially smaller integer-linear formulation for a given application of the p-median problem and can be used to solve for either an exact optimum or a bounded, close to optimal solution. We also present a methodological framework in which the BEAMR model can be used. Computational results for problems found in the OR_library of Beasley [A note on solving large p-median problems. European Journal of Operational Research 1985;21:270–3] indicate that BEAMR not only extends the application frontier for the p-median problem using general-purpose software, but for many problems represents an efficient, competitive solution approach.     

Variable neighborhood search for the pharmacy duty scheduling problem

In this paper, we study on the Pharmacy Duty Scheduling (PDS) problem, where a subset of pharmacies should be on duty on national holidays, at weekends and at nights in order to be able to satisfy the emergency drug needs of the society. PDS problem is a multi-period p-median problem with special side constraints and it is an NP-Hard problem. We propose four Variable Neighborhood Search (VNS) heuristics. The first one is the basic version, BVNS. The latter two, Variable Neighborhood Decomposition Search (VNDS) and Variable Neighborhood Restricted Search (VNRS), aim to obtain better results in less computing time by decomposing or restricting the search space. The last one, Reduced VNS (RVNS), is for obtaining good initial solutions rapidly for BVNS, VNDS and VNRS. We test BVNS, VNRS and VNDS heuristics on randomly generated instances and report the computational test results. We also use VNS heuristics on real data for the pharmacies in central İzmir and obtain significant improvements.     

The p-median problem in a changing network: the case of Barcelona

In this paper a p-median-like model is formulated to address the issue of locating new facilities when there is uncertainty in demand, travel times or distance. Given several possible scenarios, the planner would like to choose a set of locations that will perform as well as possible over all future scenarios. This paper presents a discrete location model formulation to address this p-median problem under uncertainty. The model is applied to the location of fire stations in Barcelona.     

A decomposition approach for the p-median problem on disconnected graphs

The p-median problem seeks for the location of p facilities on the vertices (customers) of a graph to minimize the sum of transportation costs for satisfying the demands of the customers from the facilities. In many real applications of the p-median problem the underlying graph is disconnected. That is the case of p-median problem defined over split administrative regions or regions geographically apart (e.g. archipelagos), and the case of problems coming from industry such as the optimal diversity management problem. In such cases the problem can be decomposed into smaller p-median problems which are solved in each component k for different feasible values of p_k, and the global solution is obtained by finding the best combination of p_k medians. This approach has the advantage that it permits to solve larger instances since only the sizes of the connected components are important and not the size of the whole graph. However, since the optimal number of facilities to select from each component is not known, it is necessary to solve p-median problems for every feasible number of facilities on each component. In this paper we give a decomposition algorithm that uses a procedure to reduce the number of subproblems to solve. Computational tests on real instances of the optimal diversity management problem and on simulated instances are reported showing that the reduction of subproblems is significant, and that optimal solutions were found within reasonable time.     

New heuristic algorithms for solving the planar p-median problem

In this paper we propose effective heuristics for the solution of the planar p-median problem. We develop a new distribution based variable neighborhood search and a new genetic algorithm, and also test a hybrid algorithm that combines these two approaches. The best results were obtained by the hybrid approach. The best known solution was found in 466 out of 470 runs, and the average solution was only 0.000016% above the best known solution on 47 well explored test instances of 654 and 1060 demand points and up to 150 facilities.     

Defining tabu list size and aspiration criterion within tabu search methods

An investigation to explicitly define two key elements in tabu search methods is attempted. In this study a functional representation of the tabu list size is presented and a softer aspiration criterion is put forward. Experiments are conducted on a set of p-median problems.Scope and purposeTabu search is a metaheuristic that proved successful in finding good solutions to difficult combinatorial problems that were hard to find otherwise. In this study, we attempt to help the user in the choice of some of the parameters used in this type of heuristics. We based our analysis on the tabu list size and on an implementation on how to define the aspiration criterion. This added information can be valuable to those users who apply these methods in a near systematic manner without relying heavily on experimentations. As an example we used a simple location problem to test the usefulness of these ideas.     

Optimal switch location in mobile communication networks using hybrid genetic algorithms

The optimal positioning of switches in a mobile communication network is an important task, which can save costs and improve the performance of the network. In this paper we propose a model for establishing which are the best nodes of the network for allocating the available switches, and several hybrid genetic algorithms to solve the problem. The proposed model is based on the so-called capacitated p-median problem, which have been previously tackled in the literature. This problem can be split in two subproblems: the selection of the best set of switches, and a terminal assignment problem to evaluate each selection of switches. The hybrid genetic algorithms for solving the problem are formed by a conventional genetic algorithm, with a restricted search, and several local search heuristics. In this work we also develop novel heuristics for solving the terminal assignment problem in a fast and accurate way. Finally, we show that our novel approaches, hybridized with the genetic algorithm, outperform existing algorithms in the literature for the p-median problem.     

Local Search Heuristics for Capacitated p-Median Problems

The search for p-median vertices on a network (graph) is a classical location problem. The p facilities (medians) must be located so as to minimize the sum of the distances from each demand vertex to its nearest facility. The Capacitated p-Median Problem (CPMP) considers capacities for the service to be given by each median. The total service demanded by vertices identified by p-median clusters cannot exceed their service capacity. Primal-dual based heuristics are very competitive and provide simultaneously upper and lower bounds to optimal solutions. The Lagrangean/surrogate relaxation has been used recently to accelerate subgradient like methods. The dual lower bound have the same quality of the usual Lagrangean relaxation dual but is obtained using modest computational times. This paper explores improvements on upper bounds applying local search heuristics to solutions made feasible by the Lagrangean/surrogate optimization process. These heuristics are based on location-allocation procedures that swap medians and vertices inside the clusters, reallocate vertices, and iterate until no improvements occur. Computational results consider instances from the literature and real data obtained using a geographical information system.     

Discrete approximation heuristics for the capacitated continuous location–allocation problem with probabilistic customer locations

The capacitated continuous location–allocation problem, also called capacitated multisource Weber problem (CMWP), is concerned with locating m facilities in the Euclidean plane, and allocating their capacity to n customers at minimum total cost. The deterministic version of the problem, which assumes that customer locations and demands are known with certainty, is a nonconvex optimization problem. In this work, we focus on a probabilistic extension referred to as the probabilistic CMWP (PCMWP), and consider the situation in which customer locations are randomly distributed according to a bivariate probability distribution. We first formulate the discrete approximation of the problem as a mixed-integer linear programming model in which facilities can be located on a set of candidate points. Then we present three heuristics to solve the problem. Since optimal solutions cannot be found, we assess the performance of the heuristics using the results obtained by an alternate location–allocation heuristic that is originally developed for the deterministic version of the problem and tailored by us for the PCMWP. The new heuristics depend on the evaluation of the expected distances between facilities and customers, which is possible only for a few number of distance function and probability distribution combinations. We therefore propose approximation methods which make the heuristics applicable for any distance function and probability distribution of customer coordinates.     

A computational study of a nonlinear minsum facility location problem

A discrete location problem with nonlinear objective is addressed. A set of p plants is to be open to serve a given set of clients. Together with the locations, the number p of facilities is also a decision variable. The objective is to minimize the total cost, represented as the transportation cost between clients and plants, plus an increasing nonlinear function of p.     

New Genetic Algorithms Based Approaches to Continuous <Emphasis Type="Italic">p</Emphasis>-Median Problem

We proposed new genetic algorithms (GAs) to address well-known p-median problem in continuous space. Two GA approaches with different replacement procedures are developed to solve this problem. To make the approaches more efficient in finding near-optimal solution two hybrid algorithms are developed combining the new GAs and a traditional local search heuristic. The performance of the newly developed models is compared to that of the traditional alternating location-allocation heuristics by numerical simulation and it is found that the models are effective in finding optimum facility locations.     

An aggregation heuristic for large scale p-median problem

The p-median problem (PMP) consists of locating p facilities (medians) in order to minimize the sum of distances from each client to the nearest facility. The interest in the large-scale PMP arises from applications in cluster analysis, where a set of patterns has to be partitioned into subsets (clusters) on the base of similarity.In this paper we introduce a new heuristic for large-scale PMP instances, based on Lagrangean relaxation. It consists of three main components: subgradient column generation, combining subgradient optimization with column generation; a “core” heuristic, which computes an upper bound by solving a reduced problem defined by a subset of the original variables chosen on a base of Lagrangean reduced costs; and an aggregation procedure that defines reduced size instances by aggregating together clients with the facilities. Computational results show that the proposed heuristic is able to compute good quality lower and upper bounds for instances up to 90,000 clients and potential facilities.     

Manufacturing cell formation using an improved P-median model

The p-median model objective function is modified for the cell formation problem to minimize the variability between parts in a group by considering part similarity to all other parts in the group instead of similarity to an arbitrary median. The heuristic vertex substitution method for solution of the part grouping problem is adapted for this objective function and then modified to provide improved starting points. The theoretical lower bound for the heuristic is developed and shown to be valid for all solutions. Worst case run time is shown to be O(n²) or O(n³) for distance matrix or network inputs respectively. Tests on published problems show that the proposed p-median model method provides as good or better objective function value (OFV) than the OFV of a p-median model in which parts are grouped to an arbitrary median. Likewise the new p-median model is shown, for these published problems, to give as good or better OFV than the algorithms reported by the original authors of the problem. The test problems suggest that other measures of solution quality such as bottlenecks and duplicate machines in addition to OFV become important measures of solution quality for dense problems.     

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

Facility location problems with the objective to minimize the sum of the setup cost and transportation cost are studied in this paper. The setup and transportation costs are considered as a function of the number of opened facilities. Three methods are introduced to solve the problem. The facility location model with bounds for the number of opened facility is constructed in this work. The relationship between setup cost and transportation cost is studied and used to build these methods based on greedy algorithm, p-median algorithm and p-center algorithm. The performance of the constructed methods is tested using 100 random data sets. In addition, the networks representing the road transportation system of Chiang Mai city and 5 provinces in Northern Thailand are illustrated and tested using all presented methods. Simulation results show that the method developed from greedy algorithm is suitable for solving problems when the setup cost is higher than transportation cost while the opposite cases are more efficiently solved with the method developed by the p-median problem.     

Improved Algorithms for Some Competitive Location Centroid Problems on Paths, Trees and Graphs

We consider a common scenario in competitive location, where two competitors (providers) place their facilities (servers) on a network, and the users, which are modeled by the nodes of the network, can choose between the providers. We assume that each user has an inelastic demand, specified by a positive real weight. A user is fully served by a closest facility. The benefit (gain) of a competitor is his market share, i.e., the total weight (demand) of the users served at his facilities. In our scenario the two providers, called the leader and the follower, sequentially place p and r servers, respectively. After the leader selects the locations for his p servers, the follower will determine the optimal locations for his r servers, that maximize his benefit. An (r,p)-centroid is a set of locations for the p servers of the leader, that will minimize the maximum gain of the follower who can establish r servers. In this paper we focus mainly on the cases where either the leader or the follower can establish only one facility, i.e., either p=1, or r=1. We consider two versions of the model. In the discrete case the facilities can be established only at the nodes, while in the absolute case they can be established anywhere on the network. For the (r,1)-centroid problem, we show that it is strongly NP-hard for a general graph, but can be approximated within a factor e/(e?1). On the other hand, when the graph is a tree, we provide strongly polynomial algorithms for the (r,p)-centroid model, whenever p is fixed. For the (1,1)-centroid problem on a general graph, we improve upon known results, and give the first strongly polynomial algorithm. The discrete (1,p)-centroid problem has been known to be NP-hard even for a subclass of series-parallel graphs with pathwidth bounded by 6. In view of this result, we consider the discrete and absolute (1,p) centroid models on a tree, and present the first strongly polynomial algorithms. Further improvements are shown when the tree is a path.     

The 2-facility centdian network problem

The p-facility centdian network problem consists of finding the p points that minimize a convex combination of the p-center and p-median objective functions. The vertices and local centers constitute a dominating set for the 1-facility centdian; i.e., it contains an optimal solution for all instances of the problem. Hooker et al. (1991) give a theoretical result to extend the dominating sets for the 1-facility problems to the corresponding p-facility problems. They claim that the set of vertices and local centers is also a dominating set for the p-facility centdian problem. We give a counterexample and an alternative finite dominating set for p=2. We propose a solution procedure for a network that improves the complexity of the exhaustive search in the dominating set. We also provide a very efficient algorithm that solves the 2-centdian on a tree network with complexity O(n²).     

A new directed branching heuristic for the pq-median problem

The pq-median problem, of Serra and ReVelle seeks to locate hierarchical facilities at two levels so as to obtain a coherent structure. Coherence requires that the entire area assigned to a facility at the inferior level be assigned to one and the same facility at the next higher level of the hierarchy. Although optimal solutions can be obtained by means of solving biobjective integer linear programs, large problems are likely to require heuristics. Here we present a new heuristic that combines the generation of points, by means of a “directed” branching procedure with the final selection of points, using the FDH-technique. We further compare our new heuristic with the two most relevant heuristics proposed by Serra and ReVelle.     

Heuristics for a continuous multi-facility location problem with demand regions

We consider a continuous multi-facility location allocation problem where the demanding entities are regions in the plane instead of points. The problem can be stated as follows: given m (closed, convex) polygonal demand regions in the plane, find the locations of q facilities and allocate each region to exactly one facility so as to minimize a weighted sum of squares of the maximum Euclidean distances between the demand regions and the facilities they are assigned to.We propose mathematical programming formulations of the single and multiple facility versions of the problem considered. The single facility location problem is formulated as a second order cone programming (SOCP) problem, and hence is solvable in polynomial time. The multiple facility location problem is NP-hard in general and can be formulated as a mixed integer SOCP problem. This formulation is weak and does not even solve medium-size instances. To solve larger instances of the problem we propose three heuristics. When all the demand regions are rectangular regions with their sides parallel to the standard coordinate axes, a faster special heuristic is developed. We compare our heuristics in terms of both solution quality and computational time.