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.     

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.     

The facility location problem with Bernoulli demands

This paper presents the facility location problem with Bernoulli demands. In this capacitated discrete location stochastic problem the goal is to define an a priori solution for the locations of the facilities and for the allocation of customers to the operating facilities that minimizes the sum of the fixed costs of the open facilities plus the expected value of the recourse function. The problem is formulated as a two-stage stochastic program and two different recourse actions are considered. For each of them, a closed form is presented for the recourse function and a deterministic equivalent formulation is obtained for the case in which the probability of demand is the same for all customers. Numerical results from computational experiments are presented and analyzed.     

DELIVERY GUARANTEES AND THE INTERDEPENDENCE OF MARKETING AND OPERATIONS

Delivery guarantees are an important element in a customer satisfaction program. When setting delivery guarantees, a firm must consider customer expectations as well as operational constraints. We develop a profit‐maximization model in which a firm's sales organization, with incomplete information on operations' status, solicits orders and quotes delivery dates. If obtained, orders are processed in a make‐to‐order facility, after which revenue is received, minus tardiness penalty if the delivery was later than quoted. We specify conditions for an optimal log‐linear decision rule and provide exact expressions for its effect on arrival rate, mean processing time, and mean cycle time.     

Strategyproof mechanism design for facility location games with weighted agents on a line

Approximation mechanism design without money was first studied in Procaccia and Tennenholtz (2009) by considering a facility location game. In general, a facility is being opened and the cost of an agent is measured by its distance to the facility. In order to achieve a good social cost, a mechanism selects the location of the facility based on the locations reported by agents. It motivates agents to strategically report their locations to get good outcomes for themselves. A mechanism is called strategyproof if no agents could manipulate to get a better outcome by telling lies regardless of any configuration of other agents. The main contribution in this paper is to explore the strategyproof mechanisms without money when agents are distinguishable. There are two main variations on the nature of agents. One is that agents prefer getting closer to the facility, while the other is that agents prefer being far away from the facility. We first consider the model that directly extends the model in Procaccia and Tennenholtz (2009). In particular, we consider the strategyproof mechanisms without money when agents are weighted. We show that the strategyproof mechanisms in the case of unweighted agents are still the best in the weighted cases. We establish tight lower and upper bounds for approximation ratios on the optimal social utility and the minimum utility when agents prefer to stay close to the facility. We then provide the lower and upper bounds on the optimal social utility and lower bound on the minimum distance per weight when agents prefer to stay far away from the facility. We also extend our study in a natural direction where two facilities must be built on a real line. Secondly, we propose an novel threshold based model to distinguish agents. In this model, we present a strategyproof mechanism that leads to optimal solutions in terms of social cost.     

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.     

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.     

Joint Initial Stocking and Transshipment—Asymptotics and Bounds

A common problem faced by many firms in their supply chains can be abstracted as follows. Periodically, or at the beginning of some selling season, the firm needs to distribute finished goods to a set of stocking locations, which, in turn, supply customer demands. Over the selling season, if and when there is a supply‐demand mismatch somewhere, a re‐distribution or transshipment will be needed. Hence, there are two decisions involved: the one‐time stocking decision at the beginning of the season and the supply/transshipment decision throughout the season. Applying a stochastic dynamic programming formulation to a two‐location model with compound Poisson demand processes, we identify the optimal supply/transshipment policy and show that the optimal initial stocking quantities can be obtained via maximizing a concave function whereas the contribution of transshipment is of order square‐root‐of T. Hence, in the context of high‐volume, fast‐moving products, the initial stocking quantity decision is a much more important contributor to the overall profit. The bounds also lead to a heuristic policy, which exhibits excellent performance in our numerical study; and we further prove both the bounds and the heuristic policy are asymptotically optimal when T approaches infinity. Extension to multiple locations is also discussed.     

The effects of seasonal demand on depot location

Many of the cost functions associated with depot location algorithms assume an overall demand for a fixed period, usually one year. This paper describes an investigation into the effect of high seasonality within such demand figures on the optimal location of storage facilities. The preliminary results indicate that a more flexible approach for certain products could be worthwhile.     

A note on the facility location problem with stochastic demands

In this research note that the single source capacitated facility location problem with general stochastic identically distributed demands is studied. The demands considered are independent and identically distributed random variables with arbitrary distribution. The unified a priori solution for the locations of facilities and for the allocation of customers to the operating facilities is found. This solution minimizes the objective function which is the sum of the fixed costs and the value of one of two different recourse functions. For each case the recourse function is given in closed form and a deterministic equivalent formulation of the model is presented. Some numerical examples are also given.     

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。     

基于阶梯型衰退效用函数的竞争选址问题

研究了连锁型企业新设施选址的2个重要因素：市场份额的划分和设施的服务半径。依照效用函数为竞争环境下设施市场份额分配的重要依据,结合消费者的空间选择行为提出了阶梯型效用函数,并应用此效用函数建立了一个竞争环境下连锁型企业新设施选址的新模型,此模型既考虑了服务设施覆盖衰退又考虑了设施的服务半径。最后,给出了求解模型的遗传算法和具体案例。结果表明,该模型和算法可以用于连锁型企业新设施选址决策中,并能够有效地得到问题的近似解。     

竞争环境下的以旧换新策略

企业通过对拥有旧产品的老消费者提供以旧换新补贴能够提升自身销量与利润。然而,面临竞争对手时企业的以旧换新决策是否会受到影响?本文求解了先后进入市场的双寡头竞争企业所面临的以旧换新与定价博弈均衡,并分析了竞争存在与否对于企业以旧换新策略产生的影响。研究结果表明,第一,面对竞争时企业的定价决策受到市场中老消费者比例、两家竞争企业各自新产品的创新提升水平、老产品的使用残值这四个因素的共同影响。第二,当老产品残值相对较低而市场中老消费者数量适中时,两企业均不提供以旧换新可能成为博弈均衡,而其他条件下,两企业均提供以旧换新为博弈均衡。第三,先进入的企业没有动机单独为消费者提供以旧换新补贴。第四,竞争对手的存在对于先进入企业自身的以旧换新决策与相应的定价策略都产生了显著的影响。     

Capacity Planning with Financial and Operational Hedging in Low‐Cost Countries

The authors of this article outline a capacity planning problem in which a risk‐averse firm reserves capacities with potential suppliers that are located in multiple low‐cost countries. While demand is uncertain, the firm also faces multi‐country foreign currency exposures. This study develops a mean‐variance model that maximizes the firm's optimal utility and derives optimal utility and optimal decisions in capacity and financial hedging size. The authors show that when demand and exchange rate risks are perfectly correlated, a risk‐averse firm, by using financial hedging, will achieve the same optimal utility as a risk‐neutral firm. In this study as well, a special case is examined regarding two suppliers in China and Vietnam. The results show that if a single supplier is contracted, financial hedging most benefits the highly risk‐averse firm when the demand and exchange rate are highly negatively related. When only one hedge is used, financial hedging dominates operational hedging only when the firm is very risk averse and the correlation between the two exchange rates have become positive. With both theoretical and numerical results, this study concludes that the two hedges are strategic tools and interact each other to maximize the optimal utility.     

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.     

Median problems with positive and negative weights on cycles and cacti

This paper deals with facility location problems on graphs with positive and negative vertex weights. We consider two different objective functions: In the first one (MWD) vertices with positive weight are assigned to the closest facility, whereas vertices with negative weight are assigned to the farthest facility. In the second one (WMD) all the vertices are assigned to the nearest facility. For the MWD model it is shown that there exists a finite set of points in the graph which contains the locations of facilities in an optimal solution. Furthermore, algorithms for both models for the 2-median problem on a cycle are developed. The algorithm for the MWD model runs in linear time, whereas the algorithm for the WMD model has a time complexity of  O(n²)\mathcal{O}(n^{2}) .     

The Use of Flexible Manufacturing Capacity in Pharmaceutical Product Introductions*

This work considers the value of the flexibility offered by production facilities that can easily be configured to produce new products. We focus on technical uncertainty as the driver of this value, while prior works focused only on demand uncertainty. Specifically, we evaluate the use of process flexibility in the context of risky new product development in the pharmaceutical industry. Flexibility has value in this setting due to the time required to build dedicated capacity, the finite duration of patent protection, and the probability that the new product will not reach the market due to technical or regulatory reasons. Having flexible capacity generates real options, which enables firms to delay the decision about constructing product‐specific capacity until the technical uncertainty is resolved. In addition, initiating production in a flexible facility can enable the firm to optimize production processes in dedicated facilities. The stochastic dynamic optimization problem is formulated to analyze the optimal capacity and allocation decisions for a flexible facility, using data from existing literature. A solution to this problem is obtained using linear programming. The result of this analysis shows both the value of flexible capacity and the optimal capacity allocation. Due to the substantial costs involved with flexibility in this context, the optimal level of flexible capacity is relatively small, suggesting products be produced for only short periods before initiating construction of dedicated facilities.     

海外市场的在线生产策略选择及其竞争策略

在参与全球化竞争的过程中,面对复杂多变的不确定海外市场,公司需要在出口(EXP)、许可证生产(LG)和海外直接投资(FDI)3类进入模式之间选择.运用在线算法研究了海外市场在线进入模式和转换时机选择问题的决策,设计了海外市场进入模式的最优在线策略.结果显示:海外市场的进入与退出成本将对进入模式的选择与转换时机产生迟滞作用,迟滞的幅度受不同模式之间进入与退出成本的差异和不同模式下单位产品成本差异的影响.在经济全球化的趋势下,模型的策略和结论对我国一些国际化经验不足却不得不进入海外市场参与竞争的企业具有一定的借鉴意义.     

Two Backorder Compensation Mechanisms in Inventory Systems with Impatient Customers

We study a compensation mechanism design problem with customer‐choice behavior in a continuous review setting where the production and demand processes are stochastic. When a stockout occurs, the firm controls backorders on the basis of certain compensation policies. Customers make decisions to maximize their utility, which is decreasing in the price, the waiting time, and the customer's impatience factor. We assume that the impatience factor is private information held by the customer only. Two compensation mechanisms are designed to control backorders, namely uniform compensation and priority auction with an admission price. Under uniform compensation, the firm offers the same discount to all customers, whereas under auction compensation, priority is granted according to the customers' bid prices. We obtain the optimal stockout price and base stock level under each mechanism, and analyze the properties of the respective optimal policies. Assuming linear waiting costs with uniformly distributed impatience factor, we find that the auction mechanism (1) maintains a lower base stock level and results in greater profit and (2) benefits customers with relatively lower or higher impatience factors, but customers with a medium impatience factor may be rendered worse off. We further show that both compensation mechanisms are suitable for products with a high unit profit, a high lost sales penalty cost, and a high holding cost.     

Location models for ceding market share and shrinking services

New location models are presented here for exploring the reduction of facilities in a region. The first of these models considers firms ceding market share to competitors under situations of financial exigency. The goal of this model is to cede the least market share, i.e., retain as much of the customer base as possible while shedding costly outlets. The second model considers a firm essentially without competition that must shrink it services for economic reasons. This firm is assumed to close outlets so that the degradation of service is limited. An example is offered within a competitive environment to demonstrate the usefulness of this modeling approach.