An efficient algorithm for facility location in the presence of forbidden regions

This paper investigates a constrained form of the classical Weber problem. Specifically, we consider the problem of locating a new facility in the presence of convex polygonal forbidden regions such that the sum of the weighted distances from the new facility to n existing facilities is minimized. It is assumed that a forbidden region is an area in the plane where travel and facility location are not permitted and that distance is measured using the Euclidean-distance metric. A solution procedure for this nonconvex programming problem is presented. It is shown that by iteratively solving a series of unconstrained problems, this procedure terminates at a local optimum to the original constrained problem. Numerical examples are presented.     

Error bounds for the approximative solution of restricted planar location problems

Facility location problems in the plane play an important role in mathematical programming. When looking for new locations in modeling real-world problems, we are often confronted with forbidden regions, that are not feasible for the placement of new locations. Furthermore these forbidden regions may have complicated shapes, even if we require them to be convex. It may be more useful or even necessary to use approximations of such forbidden regions when trying to solve location problems. In this paper, we develop error bounds for the approximative solution of restricted planar location problems using the so called sandwich algorithm. The number of approximation steps required to achieve a specified error bound is analyzed. As examples of these approximation schemes, we discuss round norms and polyhedral norms. Computational tests are also included.     

Multicriteria planar location problems

Given Q different objective functions, three types of single-facility problems are considered: lexicographic, Pareto and max ordering problems. After discussing the interrelation between the problem types, a complete characterization of lexicographic locations and some instances of Pareto and max ordering locations is given. The characterizations result in efficient solution algorithms for finding these locations. The paper relies heavily on the theory of restricted locations developed by the same authors, and can be further extended, for instance, to multi-facility problems with several objectives. The proposed approach is more general than previously published results on multicriteria planar location problems and is particularly suited for modelling real-world problems.     

The location of median paths on grid graphs

In this paper we consider the location of a path shaped facility on a grid graph. In the literature this problem was extensively studied on particular classes of graphs as trees or series-parallel graphs. We consider here the problem of finding a path which minimizes the sum of the (shortest) distances from it to the other vertices of the grid, where the path is also subject to an additional constraint that takes the form either of the length of the path or of the cardinality. We study the complexity of these problems and we find two polynomial time algorithms for two special cases, with time complexity of O(n) and O(nℓ) respectively, where n is the number of vertices of the grid and ℓ is the cardinality of the path to be located. The literature about locating dimensional facilities distinguishes between the location of extensive facilities in continuous spaces and network facility location. We will show that the problems presented here have a close connection with continuous dimensional facility problems, so that the procedures provided can also be useful for solving some open problems of dimensional facilities location in the continuous case.     

Location on a One-Way Rectilinear Grid

This paper considers two basic location problems — the Weber problem and the minimax problem-on a regular grid of alternating one-way routes or streets. Both the facility to be located and the demand points are restricted to any point on the network. The one-way restriction is often used for efficiency in traffic flow, but complicates the distances in the system.     

The multi-source Weber problem with constant opening cost

A constant fixed cost of establishing a facility is introduced within the framework of minisum facility location in the continuous space. The solution method developed uses a multi-phase heuristic that first solves a discrete version of the problem by existing methods to obtain an estimate of the optimal number of facilities. Some results are presented for test problems taken from the literature and compared with best-known solutions of the multi-source Weber problem with the addition of the appropriate fixed costs.     

Minisum location problem with farthest Euclidean distances

The paper formulates an extended model of Weber problem in which the customers are represented by convex demand regions. The objective is to generate a site in R ² that minimizes the sum of weighted Euclidean distances between the new facility and the farthest points of demand regions. This location problem is decomposed into a polynomial number of subproblems: constrained Weber problems (CWPs). A projection contraction method is suggested to solve these CWPs. An algorithm and the complexity analysis are presented. Three techniques of bound test, greedy choice and choice of starting point are adopted to reduce the computational time. The restricted case of the facility is also considered. Preliminary computational results are reported, which shows that with the above three techniques adopted the algorithm is efficient.The authors were supported by the dissertation fund of Nari-Relays Corporation.     

Simulated N-Body: New Particle Physics-Based Heuristics for a Euclidean Location-Allocation Problem

The general facility location problem and its variants, including most location-allocation and P-median problems, are known to be NP-hard combinatorial optimization problems. Consequently, there is now a substantial body of literature on heuristic algorithms for a variety of location problems, among which can be found several versions of the well-known simulated annealing algorithm. This paper presents an optimization paradigm that, like simulated annealing, is based on a particle physics analogy but is markedly different from simulated annealing. Two heuristics based on this paradigm are presented and compared to simulated annealing for a capacitated facility location problem on Euclidean graphs. Experimental results based on randomly generated graphs suggest that one of the heuristics outperforms simulated annealing both in cost minimization as well as execution time. The particular version of location problem considered here, a location-allocation problem, involves determining locations and associated regions for a fixed number of facilities when the region sizes are given. Intended applications of this work include location problems with congestion costs as well as graph and network partitioning problems.     

Strategic facility location: A review

Facility location decisions are a critical element in strategic planning for a wide range of private and public firms. The ramifications of siting facilities are broadly based and long-lasting, impacting numerous operational and logistical decisions. High costs associated with property acquisition and facility construction make facility location or relocation projects long-term investments. To make such undertakings profitable, firms plan for new facilities to remain in place and in operation for an extended time period. Thus, decision makers must select sites that will not simply perform well according to the current system state, but that will continue to be profitable for the facility's lifetime, even as environmental factors change, populations shift, and market trends evolve. Finding robust facility locations is thus a difficult task, demanding that decision makers account for uncertain future events. The complexity of this problem has limited much of the facility location literature to simplified static and deterministic models. Although a few researchers initiated the study of stochastic and dynamic aspects of facility location many years ago, most of the research dedicated to these issues has been published in recent years. In this review, we report on literature which explicitly addresses the strategic nature of facility location problems by considering either stochastic or dynamic problem characteristics. Dynamic formulations focus on the difficult timing issues involved in locating a facility (or facilities) over an extended horizon. Stochastic formulations attempt to capture the uncertainty in problem input parameters such as forecast demand or distance values. The stochastic literature is divided into two classes: that which explicitly considers the probability distribution of uncertain parameters, and that which captures uncertainty through scenario planning. A wide range of model formulations and solution approaches are discussed, with applications ranging across numerous industries.     

Decision/forecast horizon for a single facility dynamic location/relocation problem

Procedures to solve finite horizon dynamic location/relocation problems have been reported in the literature by many authors. This paper provides several decision/forecast horizon results for a single facility dynamic location/relocation problem; these results are helpful in finding optimal initial decisions for the infinite horizon problem by using information only for a finite horizon.     

A mixed integer programming formulation for the 1-maximin problem

In this paper, I present a mixed integer programming (MIP) formulation for the 1-maximin problem with rectilinear distance. The problem mainly appears in facility location while trying to locate an undesirable facility. The rectilinear distance is quite commonly used in the location literature. Our numerical experiments show that one can solve reasonably large location problems using a standard MIP solver. We also provide a linear programming formulation that helps find an upper bound on the objective function value of the 1-maximin problem with any norm when extreme points of the feasible region are known. We discuss various extension alternatives for the MIP formulation.     

Fixed Point Optimality Criteria for the Location Problem with Arbitrary Norms

The single-facility location problem in continuous space is considered, with distances given by arbitrary norms. When distances are Euclidean, for many practical problems the optimal location of the new facility coincides with one of the existing facilities. This property carries over to problems with generalized distances. In this paper a necessary and sufficient condition for the location of an existing facility to be the optimal location of the new facility is developed. Some computational examples using the condition are given.     

2-Facility manifold location routing problem

The location routing problem (LRP), known to be the combination of the facility location and vehicle routing problems, is solved in the literature by either assuming planar or spherical surfaces. In this work, the manifold location routing problem (MLRP), that is an LRP on Riemannian manifold surfaces, is explained for the 2-facility (2-MLRP) case with the corresponding heuristic algorithm solution. The 2-MLRP problem is a mixed integer non-linear programming problem that is determined to be NP-hard. Special cases of MLRP include LRP on planar surfaces, when the manifold’s curvature is 0, and LRP on spherical surfaces when the curvature of the manifold is 1.     

General network design: A unified view of combined location and network design problems

This paper presents a unified framework for the general network design problem which encompasses several classical problems involving combined location and network design decisions. In some of these problems the service demand relates users and facilities, whereas in other cases the service demand relates pairs of users between them, and facilities are used to consolidate and re-route flows between users. Problems of this type arise in the design of transportation and telecommunication systems and include well-known problems such as location-network design problems, hub location problems, extensive facility location problems, tree-star location problems and cycle-star location problems, among others. Relevant modeling aspects, alternative formulations and possible algorithmic strategies are presented and analyzed.     

A survey of recent research on location-routing problems

The design of distribution systems raises hard combinatorial optimization problems. For instance, facility location problems must be solved at the strategic decision level to place factories and warehouses, while vehicle routes must be built at the tactical or operational levels to supply customers. In fact, location and routing decisions are interdependent and studies have shown that the overall system cost may be excessive if they are tackled separately. The location-routing problem (LRP) integrates the two kinds of decisions. Given a set of potential depots with opening costs, a fleet of identical vehicles and a set of customers with known demands, the classical LRP consists in opening a subset of depots, assigning customers to them and determining vehicle routes, to minimize a total cost including the cost of open depots, the fixed costs of vehicles used, and the total cost of the routes. Since the last comprehensive survey on the LRP, published by Nagy and Salhi (2007), the number of articles devoted to this problem has grown quickly, calling a review of new research works. This paper analyzes the recent literature (72 articles) on the standard LRP and new extensions such as several distribution echelons, multiple objectives or uncertain data. Results of state-of-the-art metaheuristics are also compared on standard sets of instances for the classical LRP, the two-echelon LRP and the truck and trailer problem.     

Lagrangian-relaxation-based solution procedures for a multiproduct capacitated facility location problem with choice of facility type

This paper presents exact and heuristic solution procedures for a multiproduct capacitated facility location (MPCFL) problem in which the demand for a number of different product families must be supplied from a set of facility sites, and each site offers a choice of facility types exhibiting different capacities. MPCFL generalizes both the uncapacitated (or simple) facility location (UFL) problem and the pure-integer capacitated facility location problem. We define a branch-and-bound algorithm for MPCFL that utilizes bounds formed by a Lagrangian relaxation of MPCFL which decomposes the problem into UFL subproblems and easily solvable 0-1 knapsack subproblems. The UFL subproblems are solved by the dual-based procedure of Erlenkotter. We also present a subgradient optimization-Lagrangian relaxation-based heuristic for MPCFL. Computational experience with the algorithm and heuristic are reported. The MPCFL heuristic is seen to be extremely effective, generating solutions to the test problems that are on average within 2% of optimality, and the branch-and-bound algorithm is successful in solving all of the test problems to optimality.     

Algorithms for central-median paths with bounded length on trees

The location of path-shaped facilities on trees has been receiving a growing attention in the specialized literature in the recent years. Examples of such facilities include railroad lines, highways and public transit lines. Most of the papers deal with the problem of locating a path on a tree by minimizing either the maximum distance from the vertices of the tree to the facility or of minimizing the sum of the distances from all the vertices of the tree to the path. However, neither of the two above criteria alone capture all essential elements of a location problem. The sum of the distances criterion alone may result in solutions which are unacceptable from the point of view of the service level for the clients who are located far away from the facilities. On the other hand, the criterion of the minimization of the maximum distance, if used alone, may lead to very costly service systems. In the literature, there is just one paper that considers the problem of finding an optimal location of a path on a tree using combinations of the two above criteria, and efficient algorithms are provided. In particular, the cases where one criterion is optimized subject to a restriction on the value of the other are considered and linear time algorithms are presented. However, these problems do not consider any bound on the length or cost of the facility. In this paper we consider the two following problems: find a path which minimizes the sum of the distances such that the maximum distance from the vertices of the tree to the path is bounded by a fixed constant and such that the length of the path is not greater than a fixed value; find a path which minimizes the maximum distance with the sum of the distances being not greater than a fixed value and with bounded length. From an application point of view the constraint on the length of the path may refer to a budget constraint for establishing the facility. The restriction on the length of the path complicates the two problems but for both of them we give O(n log² n) divide-and-conquer algorithms.     

Adaptation of the plant location model for regional environmental facilities and cost allocation strategy

This paper is divided into two parts. In the first part of the paper, the plant location model is adapted for the special case of siting wastewater treatment facilities when the wastewater sources and treatment facilities are arranged in a chain or linear configuration. In this problem, flows or shipments may be merged in common pipes that provide economies of scale in transport. In order to apply the plant location model, an appropriate definition of the additional cost incurred when a waste source joins a regional facility is required. In addition, sequential priority constraints are developed in the siting model in order to make possible proper accounting of transport costs. The new siting model can be conveniently solved by linear programming.In the second part of the paper, a dual of the plant location model is explored as a cost allocation method for the fixed charge facility siting problem. The constraint sets of the dual model can be shown to imply the core conditions of the related cost game; hence, a set of the dual variables from the dual problem can be regarded as rational cost allocations. The analysis places both facility siting and cost allocation in a common framework.