An approximation algorithm for k-center problem on a convex polygon

This paper studies the constrained version of the k-center location problem. Given a convex polygonal region, every point in the region originates a service demand. Our objective is to place k facilities lying on the region’s boundary, such that every point in that region receives service from its closest facility and the maximum service distance is minimized. This problem is equivalent to covering the polygon by k circles with centers on its boundary which have the smallest possible radius. We present an 1.8841-approximation polynomial time algorithm for this problem.     

Social Optimal Location of Facilities with Fixed Servers,Stochastic Demand,and Congestion

We consider two capacity choice scenarios for the optimal location of facilities with fixed servers, stochastic demand, and congestion. Motivating applications include virtual call centers, consisting of geographically dispersed centers, walk‐in health clinics, motor vehicle inspection stations, automobile emissions testing stations, and internal service systems. The choice of locations for such facilities influences both the travel cost and waiting times of users. In contrast to most previous research, we explicitly embed both customer travel/connection and delay costs in the objective function and solve the location–allocation problem and choose facility capacities simultaneously. The choice of capacity for a facility that is viewed as a queueing system with Poisson arrivals and exponential service times could mean choosing a service rate for the servers (Scenario 1) or choosing the number of servers (Scenario 2). We express the optimal service rate in closed form in Scenario 1 and the (asymptotically) optimal number of servers in closed form in Scenario 2. This allows us to eliminate both the number of servers and the service rates from the optimization problems, leading to tractable mixed‐integer nonlinear programs. Our computational results show that both problems can be solved efficiently using a Lagrangian relaxation optimization procedure.     

A location analysis approach for military maintenance scheduling with geographically dispersed service areas

This research uses a two-stage maximal covering location problem (MCLP) model to develop Inter continental ballistic missile (ICBM) maintenance schedules for the US Air Force. Solutions are compared to actual missile maintenance activities accomplished at F. E. Warren Air Force Base (AFB), Wyoming in May 2005. Sensitivity analysis is performed to determine the impact of altering security response times and the number of security patrol areas on the quality of daily maintenance schedules and personnel usage. Results indicate marked improvement over traditional Air Force scheduling methods. In addition sensitivity analysis identifies the levels at which the quality and quantity of maintenance performance is impacted.     

Location and reliability problems on a line: Impact of objectives and correlated failures on optimal location patterns

In this paper we study a class of locations models where facilities are not perfectly reliable and failures may be correlated. We analyze problems with Median and Center objectives under complete and incomplete customer information regarding the state of facilities. The goal is to understand how failure probabilities, correlations, availability of information, and problem objective affect the optimal location patterns. In particular, we want to find analytical confirmations for location patterns observed in numerical experiments with network location models. To derive closed-form analytical results the analysis is restricted to a simple (yet classic) setting: a 2-facility problem on a unit segment, with customer demand distributed uniformly over the segment (results can be extended to other demand distributions as well). We derive explicit expressions for facility trajectories as functions of model parameters, obtaining a number of managerial insights. In addition we provide the decomposition of the optimal cost into the closed form components corresponding to the cost of travel, the cost of facility unreliability and the cost of incomplete information. Most of the theoretical insights are confirmed via numerical experiments for models with larger (3–5) number of facilities.     

A FACILITY LOCATION PROBLEM WITH DISTANCE-DEPENDENT DEMAND

This paper examines network systems where demand for the services of a facility originates at the nodes of the network and its magnitude depends on the shortest distance to a service-providing facility. Service systems with such features include bank branches, fastfood outlets, and grocery stores. With the assumption that demand is a Poisson-distributed random variable whose mean is an exponentially decreasing function of distance, possible locations based on two important performance measures are characterized: the expected value and the variance of the demand. Two procedures are proposed: one to find the locations with the minimum and maximum expected demand and the other to find the location(s) that provide a given level of expected demand. The procedures are illustrated by two examples.     

具有遗憾值约束的鲁棒供应链网络设计模型研究

考虑不确定性环境,研究战略层次的供应链网络鲁棒设计问题,目标是设计参数发生摄动时,供应链性能能够保持稳健性。基于鲁棒解的定义,建立从上游供应商选择到下游设施选址-需求分配的供应链网络设计鲁棒优化模型;提出确定遗憾值限定系数上限和下限的方法,允许决策者调节鲁棒水平,选择多种供应链网络结构;通过模型分解与协调,设计了供应链节点配置的禁忌搜索算法。算例的计算结果表明了禁忌搜索算法具有良好的收敛特性,以及在处理大规模问题上的优越性;同时也反映了利用鲁棒优化模型进行供应链网络设计,可以有效规避投资风险。     

Developing and Testing a Facility Location Model for Canadian Nuclear Fuel Waste

The purpose of this paper is to develop and test a facility location model for the siting of a nuclear fuel waste disposal facility in Canada. The model is based on successful Canadian siting processes related to hazardous waste and low level radioactive waste facilities, as well as attributes of facility siting found in the literature. The proposed model was presented to a sample of participants in the federal environmental assessment review of the technical feasibility of the Canadian Nuclear Fuel Waste Disposal Concept (CNFWDC) held throughout Canada in 1990. Results demonstrate that despite the fact that over half of the survey respondents did not support the CNFWDC during the public hearings, the majority favorably rated the proposed facility location model. Components of the model that were tested included siting criteria and goals, decision-making groups, and siting safeguards. On the basis of these results, it is concluded that the siting of a nuclear fuel waste disposal facility must make the decentralization of decision-making authority to local communities and governments a priority.     

The P-Hub maximal covering problem and extensions for gradual decay functions

The p-hub maximal covering problem aims to find the best locations for hubs so as to maximize demands within a coverage distance with a predetermined number of hubs. Classically, the problem is defined in the framework of binary coverage only; an origin–destination pair is covered if the cost (time, etc.) is lower than the critical value, and not covered at all if the cost is greater than the critical value. In this paper, we extend the definition of coverage, introducing “partial coverage”, which changes with distance. We present new and efficient mixed-integer programming models that are also valid for partial coverage for single and multiple allocations. We present and discuss the computational results with different data sets.     

Siting Noxious Facilities: A Test of the Facility Siting Credo

Over the past decade it has become increasingly difficult to site noxious facilities, despite the fact that there is a growing need to do so. To address this problem, a set of guidelines for a fairer, wiser, and more workable siting process—the Facility Siting Credo—was developed during a National Facility Siting Workshop in 1990. This paper presents an empirical test of these guidelines. A questionnaire based on the Credo was completed by stakeholders in 29 waste facility siting cases, both successful and unsuccessful, across the United States and Canada. Using an independent determination of outcome (success), a preliminary ranking of the importance of various Credo principles was obtained. The data reveal that establishing trust between the developer and host community is an important factor in facilitating the siting process. The siting process is most likely to be successful when the community perceives the facility design to be appropriate and to satisfy its needs. Public participation also is seen to be an important process variable, particularly if it encourages a view that the facility best meets community needs. Moreover, a siting process where communities volunteer to host facilities is an approach that holds promise for meeting many of these key success criteria.     

Multiple facility location on a network with linear reliability order of edges

We study the problem of locating facilities on the nodes of a network to maximize the expected demand serviced. The edges of the input graph are subject to random failure due to a disruptive event. We consider a special type of failure correlation. The edge dependency model assumes that the failure of a more reliable edge implies the failure of all less reliable ones. Under this dependency model called Linear Reliability Order (LRO) we give two polynomial time exact algorithms. When two distinct LRO’s exist, we prove the total unimodularity of a linear programming formulation. In addition, we show that minimizing the sum of facility opening costs and expected cost of unserviced demand under two orderings reduces to a matching problem. We prove NP-hardness of the three orderings case and show that the problem with an arbitrary number of orderings generalizes the deterministic maximum coverage problem. When a demand point can be covered only if a facility exists within a distance limit, we show that the problem is NP-hard even for a single ordering.     

基于二阶段随机规划的城市医疗废弃物回收网络设计

城市医疗废弃物日益增加,且回收需求量受诸多因素的影响,难以准确预测,假定回收需求为确定值的医疗废弃物网络优化设计不能与实际需求相匹配。本文考虑了离散随机参数环境下,医疗回收网络设计中选址规划、分配计划及运输规划的协同优化问题,建立了以选址成本、运输成本最小为目标,设施与车辆能力限制为约束的二阶段随机规划模型。根据模型特点,设计了基于Benders decomposition的求解算法,同时,设计了一系列加速技术用于提高算法的求解效率。最后,以国内某城市医疗回收网络为背景设计算例,检验本文模型和求解策略的可行性和有效性。结果表明：相比确定性规划,随机规划的解能够节约总成本,结合一系列加速技术的Benders decomposition方法比CPLEX与纯的Benders decomposition更有优势。     

服务水平保证下应急抢修点选址模型及求解算法研究

本文研究了一类故障率低但重要性较高设备的应急抢修点选址问题。设备的故障发生过程和从应急抢修点到故障设备的通行时间是随机的,每个设备被分配给一个应急抢修点进行抢修,并且整个应急抢修系统的服务水平要大于给定标准。本文以应急抢修点总开设成本最小作为目标,同时考虑了设备覆盖约束、抢修分配关系约束和抢修系统服务水平约束,在合理的假设下证明设备发生故障且应急抢修小组迟到的总次数服从泊松分布,最终将应急抢修点选址问题描述为一个0-1整数规划模型。通过对模型中的覆盖约束和抢修系统服务水平约束进行松弛,设计了相应的拉格朗日启发式算法。最后通过对大量随机算例进行计算,证明了该模型和算法的有效性。     

Risk based facility location by using fault tree analysis in disaster management

Determining the locations of facilities for prepositioning supplies to be used during a disaster is a strategic decision that directly affects the success of disaster response operations. Locating such facilities close to the disaster-prone areas is of utmost importance to minimize response time. However, this is also risky because the facility may be disrupted and hence may not support the demand point(s). In this study, we develop an optimization model that minimizes the risk that a demand point may be exposed to because it is not supported by the located facilities. The purpose is to choose the locations such that a reliable facility network to support the demand points is constructed. The risk for a demand point is calculated as the multiplication of the (probability of the) threat (e.g., earthquake), the vulnerability of the demand point (the probability that it is not supported by the facilities), and consequence (value or possible loss at the demand point due to threat). The vulnerability of a demand point is computed by using fault tree analysis and incorporated into the optimization model innovatively. To our knowledge, this paper is the first to use such an approach. The resulting non-linear integer program is linearized and solved as a linear integer program. The locations produced by the proposed model are compared to those produced by the p-center model with respect to risk value, coverage distance, and covered population by using several test problems. The model is also applied in a real problem. The results indicate that taking the risk into account explicitly may create significant differences in the risk levels.     

Fixed or variable demand? Does it matter when locating a facility?

In most competitive location models available in the literature, it is assumed that the demand is fixed independently of market conditions. However, demand may vary depending on prices, distances to the facilities, etc., especially when the goods are not essential. Taking variable demand into consideration increases the complexity of the problem and, therefore, the computational effort needed to solve it, but it may make the model more realistic. In this paper, a new planar competitive location and design problem with variable demand is presented. By using it, it is shown numerically for the first time in the literature that the assumption of fixed demand influences the location decision very much, and therefore the selection of the type of demand (fixed or variable) must be made with care when modeling location problems. Finally, two methods are presented to cope with the new model, an exact interval branch-and-bound method and an evolutionary algorithm called UEGO (Universal Evolutionary Global Optimizer).     

Locating Facilities in the Presence of Disruptions and Incomplete Information*

In this article, we analyze a location model where facilities may be subject to disruptions. Customers do not have advance information about whether a given facility is operational or not, and thus may have to visit several facilities before finding an operational one. The objective is to locate a set of facilities to minimize the total expected cost of customer travel. We decompose the total cost into travel, reliability, and information components. This decomposition allows us to put a value on the advance information about the states of facilities and compare it to the reliability and travel cost components, which allows a decision maker to evaluate which part of the system would benefit the most from improvements. The structure of optimal solutions is analyzed, with two interesting effects identified: facility centralization and co‐location; both effects appear to be stronger than in the complete information case, where the status of each facility is known in advance.     

A local search approximation algorithm for the uniform capacitated <Emphasis Type="Italic">k</Emphasis>-facility location problem

In the uniform capacitated k-facility location problem (UC-k-FLP), we are given a set of facilities and a set of clients. Every client has a demand. Every facility have an opening cost and an uniform capacity. For each client–facility pair, there is an unit service cost to serve the client with unit demand by the facility. The total demands served by a facility cannot exceed the uniform capacity. We want to open at most k facilities to serve all the demands of the clients without violating the capacity constraint such that the total opening and serving cost is minimized. The main contribution of this work is to present the first combinatorial bi-criteria approximation algorithm for the UC-k-FLP by violating the cardinality constraint.     

On multi-criteria chance-constrained capacitated single-source discrete facility location problems

This work aims at investigating multi-criteria modeling frameworks for discrete stochastic facility location problems with single sourcing. We assume that demand is stochastic and also that a service level is imposed. This situation is modeled using a set of probabilistic constraints. We also consider a minimum throughput at the facilities to justify opening them. We investigate two paradigms in terms of multi-criteria optimization: vectorial optimization and goal programming. Additionally, we discuss the joint use of objective functions that are relevant in the context of some humanitarian logistics problems. We apply the general modeling frameworks proposed to the so-called stochastic shelter site location problem. This is a problem emerging in the context of preventive disaster management. We test the models proposed using two real benchmark data sets. The results show that considering uncertainty and multiple objectives in the type of facility location problems investigated leads to solutions that may better support decision making.     

Locating Emergency Warning Sirens

In this paper, we report on the application of set covering and maximal covering location models to the problem of locating emergency warning sirens in a midwestern city. Two siren types are available, each having different costs and covering radii. Using a modified version of the set covering location model, we analyze the cost implications of several policy options being considered by the city's planners. Results of the study indicate that location covering models can be powerful and efficient tools in the design of such systems, and their use can lead to significant cost savings. In addition, such models provide decision makers the flexibility to examine the inherent costs associated with various policy options.     

Applications and Implementation A SEPARABLE PROGRAMMING APPROACH TO EXPECTED COVERAGE: AN APPLICATION TO AMBULANCE LOCATION

The maximum expected covering location problem (MEXCLP) is reformulated using a separable programming approach. The resulting formulation—nonlinear maximum expected covering location problem (NMEXCLP)—guarantees optimality and also solves more quickly than previous heuristic approaches. NMEXCLP allows two important extensions. First, minor formulation changes allow the specification of the minimum number of times each node is to be covered in order to satisfy expected coverage criteria. Second, coverage matrices can be constructed that consider two different types of coverage simultaneously. Both extensions are useful for ambulance location problems and are demonstrated in that setting.     

基于失效情景的应急设施选址评估指标体系与模型

突发事件具有巨大的破坏性、不确定性,应急设施有可能失效,本文研究基于失效情景的应急设施选址评估指标体系与评估模型。首先基于定义的失效情景研究应急设施选址评估指标。构建全局性、可靠性、时效性、均衡性、经济性评估指标。可靠性作为重要评估指标有助于提高应急设施对需求区域的物资保障程度,保证系统的稳定性,采用多重覆盖率刻画。然后,设计具有不同侧重评估目标的一般情景评估指标体系、设施失效情景评估指标体系以及多区域情景评估指标体系。最后,由于应急设施选址评估具有多影响因素特征,涉及输入和输出多个指标的测度,选取处理多输入\多输出问题具有优势的评估方法—数据包络法,对应急设施选址的合理性进行评估。实例验证评估指标体系的实用性和有效性。