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.     

Covering problems in facility location: A review

In this study, we review the covering problems in facility location. Here, besides a number of reviews on covering problems, a comprehensive review of models, solutions and applications related to the covering problem is presented after Schilling, Jayaraman, and Barkhi (1993). This survey tries to review all aspects of the covering problems by stressing the works after Schilling, Jayaraman, and Barkhi (1993). We first present the covering problems and then investigate solutions and applications. A summary and future works conclude the paper.     

The heuristic concentration-integer and its application to a class of location problems

We propose a new metaheuristic called heuristic concentration-integer (HCI). This metaheuristic is a modified version of the heuristic concentration (HC), oriented to find good solutions for a class of integer programming problems, composed by problems in which p   elements must be selected from a larger set, and each element can be selected more than once. These problems are common in location analysis. The heuristic is explained and general instructions for rewriting integer programming formulations are provided, that make the application of HCI to these problems easier. As an example, the heuristic is applied to the maximal availability location problem (MALP), and the solutions are compared to those obtained using linear programming with branch and bound (LP+B&B)$(\mathrm{LP}+\mathrm{B}\&\mathrm{B})$. For one-third of the instances of MALP, LP+B&B$\mathrm{LP}+\mathrm{B}\&\mathrm{B}$ can be allowed to run until the computer is out of memory without termination, while HCI can find good solutions to the same instances in a reasonable time. In one such case, LP-IP was allowed to run for nearly 100 times longer than HCI and HCI still found a better solution. Furthermore, HCI found the optimal solution in 33.3% of cases and had an objective value gap of less than 1% in 76% of cases. In 18% of the cases, HCI found a solution that is better than LP+B&B. Therefore, in cases where LP+B&B$\mathrm{LP}+\mathrm{B}\&\mathrm{B}$ is unreasonable due to time or memory constraints, HCI is a valuable tool.     

Solving continuous location–districting problems with Voronoi diagrams

Facility location problems are frequent in OR literature. In districting problems, on the other hand, the aim is to partition a territory into smaller units, called districts or zones, while an objective function is optimized and some constraints are satisfied, such as balance, contiguity, and compactness. Although many location and districting problems have been treated by assuming the region previously partitioned into a large number of elemental areas and further aggregating these units into districts with the aid of a mathematical programming model, continuous approximation, on the other hand, is based on the spatial density of demand, rather than on precise information on every elementary unit. Voronoi diagrams can be successfully used in association with continuous approximation models to solve location–districting problems, specially transportation and logistics applications. We discuss in the paper the context in which approximation algorithms can be used to solve this kind of problem.     

Competitive facility location: the Voronoi game

We consider a competitive facility location problem with two players. Players alternate placing points, one at a time, into the playing arena, until each of them has placed n points. The arena is then subdivided according to the nearest-neighbor rule, and the player whose points control the larger area wins. We present a winning strategy for the second player, where the arena is a circle or a line segment. We permit variations where players can play more than one point at a time, and show that the first player can ensure that the second player wins by an arbitrarily small margin.     

The gravity multiple server location problem

Models for locating facilities and service providers to serve a set of demand points are proposed. The number of facilities is unknown, however, there is a given number of servers to be distributed among the facilities. Each facility acts as an M/M/k queuing system. The objective function is the minimization of the combined travel time and the waiting time at the facility for all customers. The distribution of demand among the facilities is governed by the gravity rule. Two models are proposed: a stationary one and an interactive one. In the stationary model it is assumed that customers do not consider the waiting time at the facility in their facility selection decision. In the interactive model we assume that customers know the expected waiting time at the facility and consider it in their facility selection decision. The interactive model is more complicated because the allocation of the demand among the facilities depends on the demand itself. The models are analyzed and three heuristic solution algorithms are proposed. The algorithms were tested on a set of problems with up to 1000 demand points and 20 servers.     

Optimal location policy for three competitive facilities

Competitive facility location problems have been investigated in many papers. In most, authors have applied location models with two competitors. In this paper three companies, which are mutually competitive, intend to locate their facilities in a linear market. It is well-known that Nash equilibrium solution for location problem does not include three competitive facilities. In this paper we present the optimal location strategies for three facilities. In our model we assume that the demands are continuously distributed in a linear market and the facilities are locating according to a specific order of sequence, A, B and C. We apply the Stackelberg equilibrium solutions for competitive location problems with three facilities. In our model, we consider the decision problems in three stages. In the first stage, we decide the optimal location of facility A, which is located optimally in respect to the remaining two facilities B and C. In the second stage, we determine the optimal location of facility B which is optimally located in respect to facility C, by utilizing the information on the location of facility A. Finally in the third stage problem we decide the location of facility C, optimally located by utilizing the information on the location of A and B. In the first stage, we need the optimal solutions of the second and third stages. In the second stage we need the optimal solution of the third stage problem. Therefore, first we solve the third stage problem which is the simplest. After that, we solve the second stage problem utilizing the optimal solution strategy of the third stage problem. In this paper we present the optimal location strategies for three facilities.     

Use of distributed data sources in facility location

Facility location decisions are usually determined by cost and coverage related factors although empirical studies show that such factors as infrastructure, labor conditions and competition also play an important role in practice. The objective of this paper is to develop a multi-objective facility location model accounting for a wide range of factors affecting decision-making. The proposed model selects potential facilities from a set of pre-defined alternative locations according to the number of customers, the number of competitors and real-estate cost criteria. However, that requires large amount of both spatial and non-spatial input data, which could be acquired from distributed data sources over the Internet. Therefore, a computational approach for processing input data and representation of modeling results is elaborated. It is capable of accessing and processing data from heterogeneous spatial and non-spatial data sources. Application of the elaborated data gathering approach and facility location model is demonstrated using an example of fast food restaurants location problem.     

A note on point location in arrangements of hyperplanes

We give an algorithm for point location in an arrangement of n hyperplanes in E^d with running time poly(d,logn) and space O(n^d). The space improves on the O(n^d+ε) bound of Meiser's algorithm [Inform. and Control 106 (1993) 286] that has a similar running time.     

Local search heuristics for the mobile facility location problem

In the mobile facility location problem (MFLP), one seeks to relocate (or move) a set of existing facilities and assign clients to these facilities so that the sum of facility movement costs and the client travel costs (each to its assigned facility) is minimized. This paper studies formulations and develops local search heuristics for the MFLP. First, we develop an integer programming (IP) formulation for the MFLP by observing that for a given set of facility destinations the problem may be decomposed into two polynomially solvable subproblems. This IP formulation is quite compact in terms of the number of nonzero coefficients in the constraint matrix and the number of integer variables; and allows for the solution of large-scale MFLP instances. Using the decomposition observation, we propose two local search neighborhoods for the MFLP. We report on extensive computational tests of the new IP formulation and local search heuristics on a large range of instances. These tests demonstrate that the proposed formulation and local search heuristics significantly outperform the existing formulation and a previously developed local search heuristic for the problem.     

Multi-dimensional dynamic facility location and fast computation at query points

We present O(nlogn) time algorithms for the minimax rectilinear facility location problem in R¹ and R². The algorithms enable, once they terminate, computing the cost of any given query point in O(logn) time. Based on these algorithms, we develop a preprocessing procedure which enables solving the following two problems: Fast computation of the cost of any query point in Rd, and fast solution for the dynamic location problem in R² (namely, in the presence of an additional facility). Finally, we show that the preprocessing always gives a bound on the optimal value, which allows us in many cases to find the optimum fast (for both the traditional and the dynamic location problems in Rd for any d).     

An integrated approach to facility location problems

An integrated analysis approach to facility location problems is described. The approach is based on integrating analytical location models and a multicriteria decision model.     

Solving the uncapacitated facility location problem using tabu search

A tabu search heuristic procedure is developed to solve the uncapacitated facility location problem. Tabu search is used to guide the solution process when evolving from one solution to another. A move is defined to be the opening or closing of a facility. The net cost change resulting from a candidate move is used to measure the attractiveness of the move. After a move is made, the net cost change of a candidate move is updated from its old value. Updating, rather than re-computing, the net cost changes substantially reduces computation time needed to solve a problem when the problem is not excessively large. Searching only a small subset of the feasible solutions that contains the optimal solution, the procedure is computationally very efficient. A computational experiment is conducted to test the performance of the procedure and computational results are reported. The procedure can easily find optimal or near optimal solutions for benchmark test problems from the literature. For randomly generated test problems, this tabu search procedure not only obtained solutions completely dominating those obtained with other heuristic methods recently published in the literature but also used substantially less computation time. Therefore, this tabu search procedure has advantage over other heuristic methods in both solution quality and computation speed.     

A genetic algorithm for the uncapacitated single allocation planar hub location problem

Given a set of n interacting points in a network, the hub location problem determines location of the hubs (transfer points) and assigns spokes (origin and destination points) to hubs so as to minimize the total transportation cost. In this study, we deal with the uncapacitated single allocation planar hub location problem (PHLP). In this problem, all flow between pairs of spokes goes through hubs, capacities of hubs are infinite, they can be located anywhere on the plane and are fully connected, and each spoke must be assigned to only one hub. We propose a mathematical formulation and a genetic algorithm (PHLGA) to solve PHLP in reasonable time. We test PHLGA on simulated and real life data sets. We compare our results with optimal solution and analyze results for special cases of PHLP for which the solution behavior can be predicted. Moreover, PHLGA results for the AP and CAB data set are compared with other heuristics.     

Capacitated facility location problem with general setup cost

This paper presents an extension of the capacitated facility location problem (CFLP), in which the general setup cost functions and multiple facilities in one site are considered. The setup costs consist of a fixed term (site setup cost) plus a second term (facility setup costs). The facility setup cost functions are generally non-linear functions of the size of the facility in the same site. Two equivalent mixed integer linear programming (MIP) models are formulated for the problem and solved by general MIP solver. A Lagrangian heuristic algorithm (LHA) is also developed to find approximate solutions for this NP-hard problem. Extensive computational experiments are taken on randomly generated data and also well-known existing data (with some necessary modifications). The detailed results are provided and the heuristic algorithm is shown to be efficient.     

The multi-facility median problem with Pos/Neg weights on general graphs

In this paper we discuss the multi-facility location problem on networks with positive and negative weights. As the finite dominating set for the single facility problem does not carry over to the multi-facility problem, we derive a new finite dominating set. To solve the problem, we present a straight-forward algorithm. Moreover, for the problem with just two new facilities, we show how to obtain a more efficient solution procedure by using planar arrangements. We present computational results to underline the efficiency of the improved algorithm and to test some approximations which are based on a reduced candidate set.     

Modeling health care facility location for moving population groups

Locating public services for nomadic population groups is a difficult challenge as the locations of the targeted populations seasonally change. In this paper, the population groups are assumed to occupy different locations according to the time of the year, i.e., winter and summer. A binary integer programming model is formulated to determine the optimal number and locations of primary health units for satisfying a seasonally varying demand. This model is successfully applied to the actual locations of 17 seasonally varying nomadic groups in the Middle East. Computational tests are performed on different versions of the model in order analyze the tradeoffs among different performance measures.     

Determining the timing of project control points using a facility location model and simulation

Projects are usually performed in relatively unstable environments. As such, changes to the baseline schedules of projects are inevitable. Therefore, project progress needs to be monitored and controlled. The control process can be assumed as a continuum in which one side is continuous control and the other side is no-control. Continuous control and no-control strategies are cost-wise prohibited. Hence, project progress should be controlled at some discrete points in time during the project׳s duration. The optimal number and timing of control points are the main issues that should be addressed. In this paper, taking a dynamic view to the project control, for the first time we use an adapted version of the facility location model (FLM) to find the optimal timing of project control points. Initially, the adapted FLM determines the optimum timing of the control points in the project׳s duration. A simulation model is then used to predict the possible disruptions in the time period between the beginning of the project and the first control point (monitoring phase). If no disruptions are observed, the project׳s progress is monitored in the second control point, otherwise possible corrective actions are taken using an activity compression model. Whenever due to disruptions, the baseline schedule is to be updated, the FLM is run again to determine the new timing of the control points for the rest of the project׳s duration. In an iterative manner, this process continues until the timing of the last control point is determined.     

Aggregation in hub location problems

Because of their widespread use in real-world transportation situations, hub location models have been extensively studied in the last two decades. Many types of hub location problems are NP-hard and remain unmanageable when the number of nodes exceeds 200. We present a way to tackle large-sized problems using aggregation, explore the resulting error, and show how to reduce it. Furthermore, we develop a heuristic based on aggregation for k-hub center problems and present computational results.     

A facility location allocation model for reusing carpet materials

Re-using the huge quantities of carpet waste that are yearly generated has become a must. A facility location–allocation model for the collection, preprocessing and redistribution of carpet waste is presented. This model differs from other mathematical models for supporting the design of the logistic structure of reuse networks among others in a completely free choice of the locations for preprocessing and in explicitly taking into account depreciation costs. Two applications of the model, one in Europe and one in the United States of America, are described.