Uncertain models for single facility location problems on networks

In practical location problems on networks, the vertex demand is usually non-deterministic. This paper employs uncertainty theory to deal with this non-deterministic factor in single facility location problems. We first propose the concepts of satisfaction degree for both vertices and the whole network, which are used to evaluate products assignment. Based on different network satisfaction degree, two models are constructed. The solution to these models is based on Hakimi’s results, and some examples are given to illustrate these models.     

A D.C. optimization method for single facility location problems

The single facility location problem with general attraction and repulsion functions is considered. An algorithm based on a representation of the objective function as the difference of two convex (d.c.) functions is proposed. Convergence to a global solution of the problem is proven and extensive computational experience with an implementation of the procedure is reported for up to 100,000 points. The procedure is also extended to solve conditional and limited distance location problems. We report on limited computational experiments on these extensions.This research was supported in part by the National Science Foundation Grant DDM-91-14489.     

Competitive facility location models

Two classes of competitive facility location models are considered, in which several persons (players) sequentially or simultaneously open facilities for serving clients. The first class consists of discrete two-level programming models. The second class consists of game models with several independent players pursuing selfish goals. For the first class, its relationship with pseudo-Boolean functions is established and a novel method for constructing a family of upper and lower bounds on the optimum is proposed. For the second class, the tight PLS-completeness of the problem of finding Nash equilibriums is proved.     

A single facility stochastic location problem undera-distance

In this paper we consider a stochastic facility location model in which the weights of demand points are not deterministic but independent random variables, and the distance between the facility and each demand point isA-distance. Our objective is to find a solution which minimizes the total cost criterion subject to a chance constraint on cost restriction. We show a solution method which solves the problem in polynomial order computational time. Finally the case of rectilinear distance, which is used in many facility location models, is discussed.     

求解无容量设施选址问题的半拉格朗日松弛新方法

无容量设施选址问题Un-capacitated Facility Location, UFL是应用于诸多领域的经典组合优化难题, 半拉格朗日松弛方法是求解UFL问题的一种精确方法. 分析了半拉格朗日松弛方法在求解UFL问题时所具有的性质, 在此基础上, 对求解UFL问题的半拉格朗日松弛方法进行了一定的理论完善, 并探讨了提高半拉格朗日松弛方法求解性能的有效途径.数值计算结果表明：改进方法具有明显的可行性和有效性.     

Some proposals for stochastic facility location models

The aim of this paper is to introduce stochastic features into a facility location model to describe both the total demand for facilities and the trip pattern of the customers. The usefulness of stochastic programming tools in formulating and solving problems of this type is explored. Numerical stochastic nondifferentiable optimization techniques are outlined, and optimality conditions and practical computations are discussed.     

A general model for the undesirable single facility location problem

In this paper, a finite set in which an optimal solution for a general Euclidean problem of locating an undesirable facility in a polygonal region, is determined and can be found in polynomial time. The general problem we propose leads us, among others, to several well-known problems such as the maxisum, maximin, anticentdian or r-anticentrum problem.     

A projection method for the uncapacitated facility location problem

Several algorithms already exist for solving the uncapacitated facility location problem. The most efficient are based upon the solution of the strong linear programming relaxation. The dual of this relaxation has a condensed form which consists of minimizing a certain piecewise linear convex function. This paper presents a new method for solving the uncapacitated facility location problem based upon the exact solution of the condensed dual via orthogonal projections. The amount of work per iteration is of the same order as that of a simplex iteration for a linear program inm variables and constraints, wherem is the number of clients. For comparison, the underlying linear programming dual hasmn + m + n variables andmn +n constraints, wheren is the number of potential locations for the facilities. The method is flexible as it can handle side constraints. In particular, when there is a duality gap, the linear programming formulation can be strengthened by adding cuts. Numerical results for some classical test problems are included.     

A new approximation algorithm for the multilevel facility location problem

In this paper we propose a new integer programming formulation for the multilevel facility location problem and a novel 3-approximation algorithm based on LP-rounding. The linear program that we use has a polynomial number of variables and constraints, thus being more efficient than the one commonly used in the approximation algorithms for these types of problems.     

On models for continuous facility location with partial coverage

This paper presents a new concept of partial coverage distance, where demand points within a given threshold distance of a new facility are covered in the traditional sense, while non-covered demand points are penalized an amount proportional to their distance to the covered region. Two single facility location models, based on the minisum and minimax criteria, are formulated with the new distance function, and the structure of the models is analysed.     

Existence theory for spatially competitive network facility location models

Models for locating a firm's production facilities while simultaneously determining production levels at these facilities and shipping patterns so as to maximize the firm's profits are presented. In these models, existing firms, are assumed to act in accordance with an appropriate model of spatial equilibrium. A proof of existence of a solution to the combined location-equilibrium problem is provided.     

Scatter search for the single source capacitated facility location problem

This paper considers the Single Source Capacitated Facility Location Problem (SSCFLP). We propose a Scatter Search approach to provide upper bounds for the optimal solution of the problem. The proposed approach uses GRASP to initialize the Reference Set. Solutions of the Reference Set are combined using a procedure that consists of two phases: (1) the initialization phase and (2) the improvement phase. During the initialization phase each client is assigned to an open facility to obtain a solution that is then improved with the improvement phase. Also, a tabu search algorithm is applied. In order to evaluate the proposed approach we use different sets of test problems. According to the results obtained we observe that the method provides good quality solutions with reasonable computational effort.     

A capacitated facility location and inventory management problem with single sourcing

The current paper addresses the integrated location and inventory problem with capacity constraints. Adopting more realistic assumptions comes at the cost of increased complexity and inability to solve the model with existing methods, mainly due to the non-linear terms that arise. We attempt to render the extended formulation solvable by linearizing its non-linear terms. Certain terms are replaced by exact reformulations while for the rest a piecewise linearization is implemented. The contribution of this work is not only the development of a formulation that is more practical, but also the reformulation that enables its solving with commercial software. We test our proposed approach on a benchmark dataset from the literature, including both small and large instances of the problem. Results clearly demonstrate the superiority of this approach in terms of both solution quality and computational time.     

The single facility location problem with average-distances

This paper considers a location problem in ℝ ⁿ , where the demand is not necessarily concentrated at points but it is distributed in hypercubes following a Uniform probability distribution. The goal is to locate a service facility minimizing the weighted sum of average distances (measured with ℓ _p norms) to these demand hypercubes. In order to do that, we present an iterative scheme that provides a sequence converging to an optimal solution of the problem for p∈[1,2]. For the planar case, analytical expressions of this iterative procedure are obtained for p=2 and p=1, where two different approaches are proposed. The paper ends with a computational analysis of the proposed methodology, comparing its efficiency with a standard minimizer.      

A comparison of linear reformulations for multinomial logit choice probabilities in facility location models

In the last decade several papers appeared on facility location problems that incorporate customer demand by the multinomial logit model. Three linear reformulations of the original non-linear model have been proposed so far. In this paper, we discuss these models in terms of solvability. We present empirical findings based on synthetic data.     

A BSSS algorithm for the single facility location problem in two regions with different norms

Suppose the plane is divided by a straight line into two regions with different norms. We want to find the location of a single new facility such that the sum of the distances from the existing facilities to this point is minimized. This is in fact a non-convex optimization problem. The main difficulty is caused by finding the distances between points on different sides of the boundary line. In this paper we present a closed form solution for finding these distances. We also show that the optimal solution lies in the rectangular hull of the existing points. Based on these findings then, an efficient big square small square (BSSS) procedure is proposed.     

An exact cooperative method for the uncapacitated facility location problem

In this paper, we present a cooperative primal-dual method to solve the uncapacitated facility location problem exactly. It consists of a primal process, which performs a variation of a known and effective tabu search procedure, and a dual process, which performs a lagrangian branch-and-bound search. Both processes cooperate by exchanging information which helps them find the optimal solution. Further contributions include new techniques for improving the evaluation of the branch-and-bound nodes: decision-variable bound tightening rules applied at each node, and a subgradient caching strategy to improve the bundle method applied at each node.     

An exact algorithm for the capacitated facility location problems with single sourcing

Facility location problems are often encountered in many areas such as distribution, transportation and telecommunication. We describe a new solution approach for the capacitated facility location problem in which each customer is served by a single facility. An important class of heuristic solution methods for these problems are Lagrangian heuristics which have been shown to produce high quality solutions and at the same time be quite robust. A primal heuristic, based on a repeated matching algorithm which essentially solves a series of matching problems until certain convergence criteria are satisfied, is incorporated into the Lagrangian heuristic. Finally, a branch-and-bound method, based on the Lagrangian heuristic is developed, and compared computationally to the commercial code CPLEX. The computational results indicate that the proposed method is very efficient.     

A matheuristic for the stochastic facility location problem

Journal of Heuristics - In this paper, we describe a matheuristic to solve the stochastic facility location problem which determines the location and size of storage facilities, the quantities of...