Dynamic facility location when the total number of facilities is uncertain: A decision analysis approach

Models developed to analyze facility location decisions have typically optimized one or more objectives, subject to physical, structural, and policy constraints, in a static or deterministic setting. Because of the large capital outlays that are involved, however, facility location decisions are frequently long-term in nature. Consequently, there may be considerable uncertainty regarding the way in which relevant parameters in the location decision will change over time. In this paper, we propose two approaches for analyzing these types of dynamic location problems, focussing on situations where the total number of facilities to be located in uncertain. We term this type of location problem NOFUN (Number Of Facilities Uncertain). We analyze the NOFUN problem using two well-established decision criteria: the minimization of expected opportunity loss (EOL), and the minimization of maximum regret. In general, these criteria assume that there are a finite number of decision options and a finite number of possible states of nature. The minisum EOL criterion assumes that one can assign probabilities for the occurrence of the various states of nature and, therefore, find the initial set of facility locations that minimize the sum of expected losses across all future states. The minimax regret criteria finds the pattern of initial facility locations whose maximum loss is minimized over all possible future states.     

Condorcet winner configurations of linear networks

We consider the problem of locating a fixed number of facilities along a line to serve n players. We model this problem as a cooperative game and assume that any locational configuration can be eventually disrupted through a strict majority of players voting for an alternative configuration. A solution of such a voting location problem is called a Condorcet winner configuration. In this article, we state three necessary and one sufficient condition for a configuration to be a Condorcet winner. Consequently, we propose a fast algorithm which enables us to verify whether a given configuration is a Condorcet winner, and can be efficiently used also for computing the (potentially empty) set of all Condorcet winner configurations.     

Location of an undesirable facility on a network: A bargaining approach

We examine a model where, on a line network, individuals collectively choose the location of an undesirable public facility through bargaining with the unanimity rule. We show the existence of a stationary subgame perfect equilibrium and the characterization of stationary subgame perfect equilibria when the discount factor is sufficiently large. Furthermore, we show that as the discount factor tends to 1, the equilibrium location can converge to a location that is least desirable according to both the Benthamite and Rawlsian criteria.     

On an open problem in spherical facility location

In this paper we partially resolve an open problem in spherical facility location. The spherical facility location problem is a generalization of the planar Euclidean facility location problem. This problem was first studied by Katz and Cooper and by Drezner and Wesolowsky where a Weszfeld-like algorithm was proposed. This algorithm is very simple and does not require a line search. However, its convergence has been an open problem for more than ten years. In this paper, we prove that the sequence generated by the algorithm converges to the unique optimal solution under the condition that the oscillation of the sequence converges to zero. We conjecture that the algorithm is a descent algorithm and prove that the sequence generated by the algorithm converges to the optimal solution under this conjecture.     

Online facility location with facility movements

In the online facility location problem demand points arrive one at a time and the goal is to decide where and when to open a facility. In this paper we consider a new version of the online facility location problem, where the algorithm is allowed to move the opened facilities in the metric space. We consider the uniform case where each facility has the same constant cost. We present an algorithm which is 2-competitive for the general case and we prove that it is 3/2-competitive if the metric space is the line. We also prove that no algorithm with smaller competitive ratio than \({(\sqrt{13}+1)/4\approx 1.1514}\) exists. We also present an empirical analysis which shows that the algorithm gives very good results in the average case.     

大型群体决策问题中的简单多数规则（3个评选对象场合）

一个群体决策问题取决于两个因素,一个是群体决策的规则,另一个是投票。当选定群体决策规则时,一个群体决策问题由投票完全决定,此时,群体决策问题与投票之间一一对应。简单多数规则是个简单且被广泛采用的群体决策规则,但它有缺陷,我们可举出些群体决策问题使用简单多数规则没法从投票得到最后群体决策的结果。这里我们将给出一个简单多数规则的有趣性质,即在3个评选对象场合,使用简单多数规则没法从投票得到最后群体决策结果的n个评选人的群体决策问题的个数与所有n个评选人的群体决策问题的个数之比当评选人个数n趋向无穷时趋于零,这说明3个评选对象的大型群体决策场合,简单多数规则的缺陷不严重。     

Medi-centre Location Problems

In this paper we consider two medi-centre location problems. One is the m-medi-centre problem in which we add to the m-median problem uniform distance constraints. The other problem is the uncapacitated medi-centre facility location problem where we include the fixed costs of establishing the facilities and thus the number of facilities is also a decision variable. For the two problems we present algorithms and discuss computational experience.     

Robust strategies for facility location under uncertainty

This paper considers a stochastic facility location problem in which multiple capacitated facilities serve customers with a single product, and a stockout probabilistic requirement is stated as a chance constraint. Customer demand is assumed to be uncertain and to follow either a normal or an ambiguous distribution. We study robust approximations to the problem in order to incorporate information about the random demand distribution in the best possible, computationally tractable way. We also discuss how a decision maker’s risk preferences can be incorporated in the problem through robust optimization. Finally, we present numerical experiments that illustrate the performance of the different robust formulations. Robust optimization strategies for facility location appear to have better worst-case performance than nonrobust strategies. They also outperform nonrobust strategies in terms of realized average total cost when the actual demand distributions have higher expected values than the expected values used as input to the optimization models.     

Location of retail facilities under conditions of uncertainty

Models for the optimal location of retail facilities are typically premised on current market conditions. In this paper we incorporate future market conditions into the model for the location of a retail facility. Future market conditions are analyzed as a set of possible scenarios. We analyze the problem of finding the best location for a new retail facility such that the market share captured at that location is as close to the maximum as possible regardless of the future scenario. The objective is the minimax regret which is widely used in decision analysis. To illustrate the models an example problem is analyzed and solved in detail.     

Location of Preventive Health Care Facilities

This paper is focused on the problem of locating preventive health care facilities. The aim is to maximize participation to prevention programs. We assume that distance is a major determinant of participation and people would go to the closest facility for preventive health care. Each facility is required to have more than a predetermined number of clients because of the direct relationship between volume and quality of preventive services. We provide a mathematical formulation and present alternative solution approaches for this new location problem. We report on computational performance of the proposed methods in locating public health centers in Fulton County, Georgia and mammography screening centers in Montreal, Quebec.     

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.     

有容量限制的可靠性固定费用选址问题

设施网络可能面临各种失灵风险,而设施选址属于战略决策问题,短期内难以改变,因而在选址设计时需要充分考虑设施的非完全可靠性。本文针对无容量限制的可靠性固定费用选址问题进行扩展,进一步考虑设施的容量约束,基于非线性混合整数规划方法建立了一个有容量限制的可靠性固定费用选址问题优化模型。针对该模型的特点,应用线性化技术进行模型转化,并设计了一种拉格朗日松弛算法予以求解。通过多组算例分析,验证了算法的性能。算例分析结果表明设施失灵风险和设施容量对于选址决策有显著影响,因而在实际的选址决策过程中有必要充分考虑设施的失灵风险及容量约束。     

Location Among Regions with Varying Norms

This paper investigates a new variation in the continuous single facility location problem. Specifically, we address the problem of locating a new facility on a plane with different distance norms on different sides of a boundary line. Special cases and extensions of the problem, where there are more than two regions are also discussed. Finally, by investigating the properties of the models, efficient solution procedures are proposed.     

Local search algorithm for universal facility location problem with linear penalties

The universal facility location problem generalizes several classical facility location problems, such as the uncapacitated facility location problem and the capacitated location problem (both hard and soft capacities). In the universal facility location problem, we are given a set of demand points and a set of facilities. We wish to assign the demands to facilities such that the total service as well as facility cost is minimized. The service cost is proportional to the distance that each unit of the demand has to travel to its assigned facility. The open cost of facility i depends on the amount z of demand assigned to i and is given by a cost function \(f_i(z)\). In this work, we extend the universal facility location problem to include linear penalties, where we pay certain penalty cost whenever we refuse serving some demand points. As our main contribution, we present a (\(7.88+\epsilon \))-approximation local search algorithm for this problem.     

Single-peaked orders on a tree

Inada (1969) and Sen and Pattanaik (1969) have characterized the sets of preference orders which ensure the transitivity of the strict majority rule, no matter how each voter selects his own order in the set. But a problem remains untouched: which domains of orders guarantee the existence of a majority winner without necessarily ensuring the transitivity of the strict majority rule. We provide in this paper domains, called sets of single-peaked linear orders on a tree, which enjoy such a property. They appear as a generalization of the well-known sets of single-peaked linear orders.     

Exact solution methods for uncapacitated location problems with convex transportation costs

In this paper we study exact solution methods for uncapacitated facility location problems where the transportation costs are nonlinear and convex. An exact linearization of the costs is made, enabling the formulation of the problem as an extended, linear pure zero–one location model. A branch-and-bound method based on a dual ascent and adjustment procedure is developed, and compared to application of a modified Benders decomposition method. The specific application studied is the simple plant location problem (SPLP) with spatial interaction, which is a model suitable for location of public facilities. Previously approximate solution methods have been used for this problem, while we in this paper investigate exact solution methods. Computational results are presented.     

An interior-point Benders based branch-and-cut algorithm for mixed integer programs

We present an interior-point branch-and-cut algorithm for structured integer programs based on Benders decomposition and the analytic center cutting plane method (ACCPM). We show that the ACCPM based Benders cuts are both pareto-optimal and valid for any node of the branch-and-bound tree. The valid cuts are added to a pool of cuts that is used to warm-start the solution of the nodes after branching. The algorithm is tested on two classes of problems: the capacitated facility location problem and the multicommodity capacitated fixed charge network design problem. For the capacitated facility location problem, the proposed approach was on average 2.5 times faster than Benders-branch-and-cut and 11 times faster than classical Benders decomposition. For the multicommodity capacitated fixed charge network design problem, the proposed approach was 4 times faster than Benders-branch-and-cut while classical Benders decomposition failed to solve the majority of the tested instances.     

基于可信度模糊费用多设施选址模型和算法

设施选址在整个物流网络中是一个十分重要的决策问题，它决定了整个物流系统的模式，结构和形状。设施选址方法尤其是多设施选址方法的研究已经成为一个备受人们关注的研究领域。本文首先介绍了设施选址的重要性，然后在模糊环境中根据不同的决策标准，建立了三种不同类型的模型，并设计了一个遗传算法来解决其中一个模型。最后给出了一个数值例子。     

A discrete competitive facility location model with variable attractiveness

We consider the discrete version of the competitive facility location problem in which new facilities have to be located by a new market entrant firm to compete against already existing facilities that may belong to one or more competitors. The demand is assumed to be aggregated at certain points in the plane and the new facilities can be located at predetermined candidate sites. We employ Huff's gravity-based rule in modelling the behaviour of the customers where the probability that customers at a demand point patronize a certain facility is proportional to the facility attractiveness and inversely proportional to the distance between the facility site and demand point. The objective of the firm is to determine the locations of the new facilities and their attractiveness levels so as to maximize the profit, which is calculated as the revenue from the customers less the fixed cost of opening the facilities and variable cost of setting their attractiveness levels. We formulate a mixed-integer nonlinear programming model for this problem and propose three methods for its solution: a Lagrangean heuristic, a branch-and-bound method with Lagrangean relaxation, and another branch-and-bound method with nonlinear programming relaxation. Computational results obtained on a set of randomly generated instances show that the last method outperforms the others in terms of accuracy and efficiency and can provide an optimal solution in a reasonable amount of time.     

Optimality of the Expert Rule Under Partial Information

We study the uncertain dichotomous choice model. In this model a group of decision makers is required to select one of two alternatives. The applications of this model are relevant to a wide variety of areas, such as medicine, management and banking. The decision rule may be the simple majority rule; however, it is also possible to assign more weight to the opinion of members known to be more qualified. The extreme example of such a rule is the expert decision rule. We are concerned with the probability of the expert rule to be optimal. Our purpose is to investigate the behaviour of this probability as a function of the group size for several rather general types of distributions. One such family of distributions is that where the density function of the correctness probability is a polynomial (on the interval [1/2,1]). Our main result is an explicit formula for the probability in question. This contains formerly known results as very special cases.