The plant location problem with demand-dependent setup costs and centralized allocation

Suppose that customers are situated at the nodes of a transportation network, and a service company plans to locate a number of facilities that will serve the customers. The objective is to minimize the sum of the total setup cost and the total transportation cost. The setup cost of a facility is demand-dependent, that is, it depends on the number of customers that are served by the facility. Centralized allocation of customers to facilities is assumed, that is, the service company makes a decision about allocation of customers to facilities. In the case of a general network, the model can be formulated as a mixed integer programming problem. For the case of a tree network, we develop a polynomial-time dynamic programming algorithm.     

Optimal pricing for service facilities with self-optimizing customers

Many service industries (e.g., walk-in clinics, vehicle inspection facilities, and data-processing centers) have customers who choose among congested facilities, and select the facility with the lowest combination of travel cost plus congestion cost at the facility. In general, customers over-utilize attractive facilities, causing higher costs than if customers were assigned to facilities to minimize total costs. Optimal facility prices induce customers to select facilities that minimize total cost. We find optimal facility prices and show they equal charging customers for the impact (net costs and benefits) they cause for others. We explore a rich flexibility that allows a range of optimal prices, useful when negotiating the implementation of facility fees. Facility prices can be positive or negative (price discounts), and can be adjusted to be all positive, or to provide net subsidy or net revenue. We contribute to unifying and generalizing several disparate streams of research.     

带惩罚的容错设施布局问题的近似算法

在带惩罚的容错设施布局问题中, 给定顾客集合、地址集合、以及每个顾客和各个地址之间的连接费用, 这里假设连接费用是可度量的. 每位顾客有各自的服务需求, 每个地址可以开设任意多个设施, 顾客可以被安排连接到某些地址的一些开设的设施上以满足其需求, 也可以被拒绝, 但这时要支付拒绝该顾客所带来的惩罚费用. 目标是确定哪些顾客的服务需求被拒绝并开设一些设施, 将未被拒绝的顾客连接到不同的开设设施上, 使得开设费用、连接费用和惩罚费用总和最小. 给出了带惩罚的容错设施布局问题的线性整数规划及其对偶规划, 进一步, 给出了基于其线性规划和对偶规划舍入的4-近似算法.     

Location-allocation on a line with demand-dependent costs

This paper extends the location-allocation formulation by making the cost charged to users by a facility a function of the total number of users patronizing the facility. Users select their facility based on facility charges and transportation costs. We explore equilibria where each customer selects the least expensive facility (cost and transportation) and where the facility is at a point that minimizes travel costs for its customers. The problem in its general form is quite complex. An interesting special case is studied: facilities and customers are located on a finite line segment and demand is distributed on the line by a given density function.     

考虑应急设施中断风险与防御的可靠性选址模型研究

为提高应急设施运行的可靠性和抵御中断风险的能力, 研究中断情境下的应急设施选址-分配决策问题。扩展传统无容量限制的固定费用选址模型, 从抵御设施中断的视角和提高服务质量的视角建立选址布局网络的双目标优化模型, 以应急设施的建立成本和抵御设施中断的加固成本最小为目标, 以最大化覆盖服务质量水平为目标, 在加固预算有限及最大最小容量限制约束下, 构建中断情境下应急设施的可靠性选址决策优化模型。针对所构建模型的特性利用非支配排序多目标遗传算法(NSGA-Ⅱ)求解该模型, 得到多目标的Pareto前沿解集。以不同的算例分析和验证模型和算法的可行性。在获得Pareto前沿的同时对不同中断概率进行灵敏度分析, 给出Pareto最优解集的分布及应急设施选址布局网络的拓扑结构。     

A note on the allocation of queuing facilities using a minisum criterion

This paper suggests a formulation and a solution procedure for resource allocation problems which consider a central planner, m static queuing facilities providing a homogeneous service at their locations, and a known set of demand points or customers. It is assumed that upon a request for service the customer is routed to a facility by a probabilistic assignment. The objective is to determine how to allocate a limited number of servers to the facilities, and to specify demand rates from customers to facilities in order to minimize a weighted sum of response times. This sum measures the total time lost in the system due to two sources: travel time from customer to facility locations and waiting time for service at the facilities. The setting does not allow for cooperation between the facilities.     

Optimizing pricing and location decisions for competitive service facilities charging uniform price

In this paper, we present the problem of optimizing the location and pricing for a set of new service facilities entering a competitive marketplace. We assume that the new facilities must charge the same (uniform) price and the objective is to optimize the overall profit for the new facilities. Demand for service is assumed to be concentrated at discrete demand points (customer markets); customers in each market patronize the facility providing the highest utility. Customer demand function is assumed to be elastic; the demand is affected by the price, facility attractiveness, and the travel cost for the highest-utility facility. We provide both structural and algorithmic results, as well as some managerial insights for this problem. We show that the optimal price can be selected from a certain finite set of values that can be computed in advance; this fact is used to develop an efficient mathematical programming formulation for our model.     

Cooperative facility location games

The location of facilities in order to provide service for customers is a well-studied problem in the operations research literature. In the basic model, there is a predefined cost for opening a facility and also for connecting a customer to a facility, the goal being to minimize the total cost. Often, both in the case of public facilities (such as libraries, municipal swimming pools, fire stations, … ) and private facilities (such as distribution centers, switching stations, … ), we may want to find a ‘fair’ allocation of the total cost to the customers—this is known as the cost allocation problem. A central question in cooperative game theory is whether the total cost can be allocated to the customers such that no coalition of customers has any incentive to build their own facility or to ask a competitor to service them. We establish strong connections between fair cost allocations and linear programming relaxations for several variants of the facility location problem. In particular, we show that a fair cost allocation exists if and only if there is no integrality gap for a corresponding linear programming relaxation; this was only known for the simplest unconstrained variant of the facility location problem. Moreover, we introduce a subtle variant of randomized rounding and derive new proofs for the existence of fair cost allocations for several classes of instances. We also show that it is in general NP-complete to decide whether a fair cost allocation exists and whether a given allocation is fair.     

Inventory sharing in integrated network design and inventory optimization with low-demand parts

Service Parts Logistics (SPL) problems induce strong interaction between network design and inventory stocking due to high costs and low demands of parts and response time based service requirements. These pressures motivate the inventory sharing practice among stocking facilities. We incorporate inventory sharing effects within a simplified version of the integrated SPL problem, capturing the sharing fill rates in 2-facility inventory sharing pools. The problem decides which facilities in which pools should be stocked and how the demand should be allocated to stocked facilities, given full inventory sharing between the facilities within each pool so as to minimize the total facility, inventory and transportation costs subject to a time-based service level constraint. Our analysis for the single pool problem leads us to model this otherwise non-linear integer optimization problem as a modified version of the binary knapsack problem. Our numerical results show that a greedy heuristic for a network of 100 facilities is on average within 0.12% of the optimal solution. Furthermore, we observe that a greater degree of sharing occurs when a large amount of customer demands are located in the area overlapping the time windows of both facilities in 2-facility pools.     

Locating semi-obnoxious facilities with expropriation: minisum criterion

This paper considers the problem of locating semi-obnoxious facilities assuming that demand points within a certain distance from an open facility are expropriated at a given price. The objective is to locate the facilities so as to minimize the total weighted transportation cost and expropriation cost. Models are developed for both single and multiple facilities. For the case of locating a single facility, finite dominating sets are determined for the problems on a plane and on a network. An efficient algorithm is developed for the problem on a network. For the case of locating multiple facilities, a branch-and-bound procedure using Lagrangian relaxation is proposed and its efficiency is tested with computational experiments.     

The \alpha -cost minimization model for capacitated facility location-allocation problem with uncertain demands

Facility location-allocation problem aims at determining the locations of some facilities to serve a set of spatially distributed customers and the allocation of each customer to the facilities such that the total transportation cost is minimized. In real life, the facility location-allocation problem often comes with uncertainty for lack of the information about the customers’ demands. Within the framework of uncertainty theory, this paper proposes an uncertain facility location-allocation model by means of chance-constraints, in which the customers’ demands are assumed to be uncertain variables. An equivalent crisp model is obtained via the \(\alpha \) -optimistic criterion of the total transportation cost. Besides, a hybrid intelligent algorithm is designed to solve the uncertain facility location-allocation problem, and its viability and effectiveness are illustrated by a numerical example.     

Optimizing capacity,pricing and location decisions on a congested network with balking

In this paper, we consider the problem of making simultaneous decisions on the location, service rate (capacity) and the price of providing service for facilities on a network. We assume that the demand for service from each node of the network follows a Poisson process. The demand is assumed to depend on both price and distance. All facilities are assumed to charge the same price and customers wishing to obtain service choose a facility according to a Multinomial Logit function. Upon arrival to a facility, customers may join the system after observing the number of people in the queue. Service time at each facility is assumed to be exponentially distributed. We first present several structural results. Then, we propose an algorithm to obtain the optimal service rate and an approximate optimal price at each facility. We also develop a heuristic algorithm to find the locations of the facilities based on the tabu search method. We demonstrate the efficiency of the algorithms numerically.     

Optimal admission and pricing control problems in service industries with multiple servers and sideline profit

This paper deals with the problem of selecting profitable customer orders sequentially arriving at a company operating in service industries with multiple servers in which two classes of services are provided. The first class of service is designed to meet the particular needs of customers; and the company (1) makes a decision on whether to accept or to reject the order for this service (admission control) and (2) decides a price of the order and offers it to an arriving customer (pricing control). The second class of service is provided as a sideline, which prevents servers from being idle when the number of customer orders for the first class is less than the number of servers. This yields the sideline profit. A cost is paid to search for customer orders, which is called the search cost. In the context of search cost, the company has an option whether to conduct the search or not. In this paper, we discuss both admission control and pricing control problems within an identical framework as well as examine the structure of the optimal policies to maximize the total expected net profit gained over an infinite planning horizon. We show that when the sideline profit is large, the optimal policies may not be monotone in the number of customer orders in the system. Copyright © 2008 John Wiley & Sons, Ltd.     

Lower bounds for the two-stage uncapacitated facility location problem

In the two-stage uncapacitated facility location problem, a set of customers is served from a set of depots which receives the product from a set of plants. If a plant or depot serves a product, a fixed cost must be paid, and there are different transportation costs between plants and depots, and depots and customers. The objective is to locate plants and depots, given both sets of potential locations, such that each customer is served and the total cost is as minimal as possible. In this paper, we present a mixed integer formulation based on twice-indexed transportation variables, and perform an analysis of several Lagrangian relaxations which are obtained from it, trying to determine good lower bounds on its optimal value. Computational results are also presented which support the theoretical potential of one of the relaxations.     

需求导向的容量设施竞争选址问题研究

客户意愿与容量限制是竞争设施选址问题中两个重要的影响因素,在考虑客户意愿与设施容量共同作用条件下,建立了最小化企业总成本以及每个客户费用为目标的竞争设施选址问题优化模型,通过设计需求导向服务分配机制解决设施与客户之间服务关系分配问题,结合模拟退火思想提出了求解模型的算法。最后利用数值例子分析了需求导向服务分配机制以及目标权重、预算限额等参数对于选址决策的影响,其中考虑需求导向因素会适当增加企业的总成本,但可以减少客户所付出的费用从而增强对客户的吸引力;另外企业的预算限额对于企业的设施选址决策有着重要的影响,企业所能获取的市场份额与其选址预算限额呈正相关的关系;而客户所需付出的总费用与企业提供服务的总成本两者之间则呈负相关的关系,因此需要通过服务质量与成本之间的权衡实现最理想的选址决策。     

带有线性惩罚的鲁棒k-种产品设施选址问题的近似算法

$k$-种产品设施选址问题是指存在一组客户和一组可以建设设施的地址。现有$k$种不同的产品,每一客户均需要$k$种不同的产品,且每一设施最多只能生产一种产品。问题的要求是从若干地址中选择一组地址来建立设施,对所要建立的设施指定其生产的产品,并为每一个客户提供一组指派确保每一客户都有$k$个设施来为其提供$k$种不同的产品,使得设施建设费用与运输费用之和最小。对于$k$-种产品设施选址问题,我们通常简写为$k$-PUFLP,其中,当所有设施建设费用为0时,记为$k$-PUFLPN。本文对$k$-PUFLPN进行线性舍入,通过分析最优分数解特殊结构,当$k\geq 3$时分析算法将$k$-PUFLPN的近似比从$\frac{3k}{2}-1$提升到了$\frac{3k}{2}-\frac{3}{2}$。鲁棒$k$-种产品设施选址问题是指在该问题中,最多有$q$个客户可以不被服务。我们首次对无容量限制下建设费用为0时的鲁棒$k$-种产品选址问题建立模型,当$k\geq 3$,得到了$\frac{3k}{2}-\frac{3}{2}$近似算法。对顾客伴有线性惩罚的鲁棒$k$-种产品设施选址问题,本文同时考虑异常值与惩罚性,利用$k$-PUFLPN中最优整数解与最优分数解的关系,得到了$\frac{3k}{2}-\frac{3}{2}$近似算法。     

p-Median theorems for competitive locations

In competitive location theory, one wishes to optimally choose the locations ofr facilities to compete againstp existing facilities for providing service (or goods) to the customers who are at given discrete points (or nodes). One normally assumes that: (a) the level of demand of each customer is fixed (i.e. this demand is not a function of how far a customer is from a facility), and (b) the customer always uses the closest available facility. In this paper we study competitive locations when one or both of the above assumptions have been relaxed. In particular, we show that for each case and under certain assumptions, there exists a set of optimal locations which consists entirely of nodes.This work was supported by a National Science Foundation Grant ECS-8121741.     

A facility location model with safety stock costs: analysis of the cost of single-sourcing requirements

We consider a supply chain setting where multiple uncapacitated facilities serve a set of customers with a single product. The majority of literature on such problems requires assigning all of any given customer??s demand to a single facility. While this single-sourcing strategy is optimal under linear (or concave) cost structures, it will often be suboptimal under the nonlinear costs that arise in the presence of safety stock costs. Our primary goal is to characterize the incremental costs that result from a single-sourcing strategy. We propose a general model that uses a cardinality constraint on the number of supply facilities that may serve a customer. The result is a complex mixed-integer nonlinear programming problem. We provide a generalized Benders decomposition algorithm for the case in which a customer??s demand may be split among an arbitrary number of supply facilities. The Benders subproblem takes the form of an uncapacitated, nonlinear transportation problem, a relevant and interesting problem in its own right. We provide analysis and insight on this subproblem, which allows us to devise a hybrid algorithm based on an outer approximation of this subproblem to accelerate the generalized Benders decomposition algorithm. We also provide computational results for the general model that permit characterizing the costs that arise from a single-sourcing strategy.     

Efficient heuristics for the rectilinear distance capacitated multi-facility Weber problem

In this paper, we consider the capacitated multi-facility Weber problem with rectilinear distance. This problem is concerned with locating m capacitated facilities in the Euclidean plane to satisfy the demand of n customers with the minimum total transportation cost. The demand and location of each customer are known a priori and the transportation cost between customers and facilities is proportional to the rectilinear distance separating them. We first give a new mixed integer linear programming formulation of the problem by making use of a well-known necessary condition for the optimal facility locations. We then propose new heuristic solution methods based on this formulation. Computational results on benchmark instances indicate that the new methods can provide very good solutions within a reasonable amount of computation time.     

Filtering Policies in Loss Queuing Network Location Problems

In this paper we deal with two stationary loss queuing network location models. We analyze the influence of filtering policies on the locational aspect of the problems. We assume that requests for service are placed at nodes of a transportation network and they arrive in time as independent homogeneous Poisson processes with different input rates. The considered policies only cover a given proportion of requests even if there are idle service units. This proportion is stationary and fixed in advance and only depends on the node where the request is originated. The objective is to find the location of the facilities together with the filtering policy to be applied that minimize the expected total cost per unit time with respect to a given cost structure. Properties and computational results are presented enabling the resolution of these problems efficiently and showing the good performance of filtering policies in terms of both the overall operating costs, and the demand that is served.