A decomposition approach for facility location and relocation problem with uncertain number of future facilities

In this paper, we discuss two challenges of long term facility location problem that occur simultaneously; future demand change and uncertain number of future facilities. We introduce a mathematical model that minimizes the initial and expected future weighted travel distance of customers. Our model allows relocation for the future instances by closing some of the facilities that were located initially and opening new ones, without exceeding a given budget. We present an integer programming formulation of the problem and develop a decomposition algorithm that can produce near optimal solutions in a fast manner. We compare the performance of our mathematical model against another method adapted from the literature and perform sensitivity analysis. We present numerical results that compare the performance of the proposed decomposition algorithm against the exact algorithm for the problem.     

An upper bound for the competitive location and capacity choice problem with multiple demand scenarios

A new mathematical model is considered related to competitive location problems where two competing parties, the Leader and the Follower, successively open their facilities and try to win customers. In the model, we consider a situation of several alternative demand scenarios which differ by the composition of customers and their preferences.We assume that the costs of opening a facility depend on its capacity; therefore, the Leader, making decisions on the placement of facilities, must determine their capacities taking into account all possible demand scenarios and the response of the Follower. For the bilevel model suggested, a problem of finding an optimistic optimal solution is formulated. We show that this problem can be represented as a problem of maximizing a pseudo- Boolean function with the number of variables equal to the number of possible locations of the Leader’s facilities.We propose a novel systemof estimating the subsets that allows us to supplement the estimating problems, used to calculate the upper bounds for the constructed pseudo-Boolean function, with additional constraints which improve the upper bounds.     

Lagrangian heuristics for the two-echelon,single-source,capacitated facility location problem

Facility location problems form an important class of integer programming problems, with application in the distribution and transportation industries. In this paper we are concerned with a particular type of facility location problem in which there exist two echelons of facilities. Each facility in the second echelon has limited capacity and can be supplied by only one facility (or depot) in the first echelon. Each customer is serviced by only one facility in the second echelon. The number and location of facilities in both echelons together with the allocation of customers to the second-echelon facilities are to be determined simultaneously. We propose a mathematical model for this problem and consider six heuristics based on Lagrangian relaxation for its solution. To solve the dual problem we make use of a subgradient optimization procedure. We present numerical results for a large suite of test problems. These indicate that the lower-bounds obtained from some relaxations have a duality gap which frequently is one third of the one obtained from traditional linear programming relaxation. Furthermore, the overall solution time for the heuristics are less than the time to solve the LP relaxation.     

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.     

Incorporating congestion in preventive healthcare facility network design

Preventive healthcare aims at reducing the likelihood and severity of potentially life-threatening illnesses by protection and early detection. The level of participation to preventive healthcare programs is a crucial factor in terms of their effectiveness and efficiency. This paper provides a methodology for designing a network of preventive healthcare facilities so as to maximize participation. The number of facilities to be established and the location of each facility are the main determinants of the configuration of a healthcare facility network. We use the total (travel, waiting and service) time required for receiving the preventive service as a proxy for accessibility of a healthcare facility, and assume that each client would seek the services of the facility with minimum expected total time. At each facility, which we model as an M/M/1 queue so as to capture the level of congestion, the expected number of participants from each population zone decreases with the expected total time. In order to ensure service quality, the facilities cannot be operated unless their level of activity exceeds a minimum workload requirement. The arising mathematical formulation is highly nonlinear, and hence we provide a heuristic solution framework for this problem. Four heuristics are compared in terms of accuracy and computational requirements. The most efficient heuristic is utilized in solving a real life problem that involves the breast cancer screening center network in Montreal. In the context of this case, we found out that centralizing the total system capacity at the locations preferred by clients is a more effective strategy than decentralization by the use of a larger number of smaller facilities. We also show that the proposed methodology can be used in making the investment trade-off between expanding the total system capacity and changing the behavior of potential clients toward preventive healthcare programs by advertisement and education.     

Location of terror response facilities: A game between state and terrorist

We study a leader follower game with two players: a terrorist and a state where the later one installs facilities that provide support in case of a terrorist attack. While the Terrorist attacks one of the metropolitan areas to maximize his utility, the State, which acts as a leader, installs the facilities such that the metropolitan area attacked is the one that minimizes her disutility (i.e., minimizes ‘loss’). We solve the problem efficiently for one facility and we formulate it as a mathematical programming problem for a general number of facilities. We demonstrate the problem via a case study of the 20 largest metropolitan areas in the United States.     

p-Hub approach for the optimal park-and-ride facility location problem

Park and Ride facilities (P&R) are car parks at which users can transfer to public transportation to reach their final destination. We propose a mixed linear programming formulation to determine the location of a fixed number of P&R facilities so that their usage is maximized. The facilities are modeled as hubs. Commuters can use one of the P&R facilities or choose to travel by car to their destinations, and their behavior follows a logit model. We apply a p-hub approach considering that users incur in a known generalized cost of using each P&R facility as input for the logit model. For small instances of the problem, we propose a novel linearization of the logit model, which allows transforming the binary nonlinear programming problem into a mixed linear programming formulation. A modification of the Heuristic Concentration Integer (HCI) procedure is applied to solve larger instances of the problem. Numerical experiments are performed, including a case in Queens, NY. Further research is proposed.     

Variable neighbourhood search heuristics for the probabilistic multi-source Weber problem

The Multi-source Weber Problem (MWP) is concerned with locating m facilities in the Euclidean plane, and allocating these facilities to n customers at minimum total cost. The deterministic version of the problem, which assumes that customer locations and demands are known with certainty, is a non-convex optimization problem and difficult to solve. In this work, we focus on a probabilistic extension and consider the situation where customer locations are randomly distributed according to a bivariate distribution. We first present a mathematical programming formulation for the probabilistic MWP called the PMWP. For its solution, we propose two heuristics based on variable neighbourhood search (VNS). Computational results obtained on a number of test instances show that the VNS heuristics improve the performance of a probabilistic alternate location-allocation heuristic referred to as PALA. In its original form, the applicability of the new heuristics depends on the existence of a closed-form expression for the expected distances between facilities and customers. Unfortunately, such an expression exists only for a few distance function and probability distribution combinations. We therefore use two approximation methods for the expected distances, which make the VNS heuristics applicable for any distance function and bivariate distribution of customer locations.     

The design of multiactivity multifacility systems

We consider a service/distribution system in which each of N activities is to be carried out at one or several facility locations. Each activity is to be assigned to one out of a specified set of configurations; each configuration is a specific subset of the set of L facilities being considered, along with a specific strategy for their use. We call such a system a multiactivity multifacility system and present a mathematical formulation for its optimal design that includes capacity restrictions at the facilities and the treatment of multiple criteria. The design problem is simply to choose an appropriate configuration for each of the N activities. We discuss various criteria, and we show that the multiactivity multifacility design problem includes many familiar discrete location problems as special cases. We introduce a 0–1 linear optimization model called the Team Generalized Assignment Problem (T-GAP) and show that parametric solution of a T-GAP will yield all efficient solutions of the multiactivity multifacility design problem with multiple criteria. Rather than attempting to find all efficient solutions, however, we advocate an interactive approach and describe an interactive branch-and-bound algorithm that solves the design problem as a finite sequence of T-GAP's.     

Multi-Item, Multi-Facility Supply Chain Planning: Models, Complexities, and Algorithms

We propose a planning model for products manufactured across multiple manufacturing facilities sharing similar production capabilities. The need for cross-facility capacity management is most evident in high-tech industries that have capital-intensive equipment and a short technology life cycle. We propose a multicommodity flow network model where each commodity represents a product and the network structure represents manufacturing facilities in the supply chain capable of producing the products. We analyze in depth the product-level (single-commodity, multi-facility) subproblem when the capacity constraints are relaxed. We prove that even the general-cost version of this uncapacitated subproblem is NP-complete. We show that there exists an optimization algorithm that is polynomial in the number of facilities, but exponential in the number of periods. We further show that under special cost structures the shortest-path algorithm could achieve optimality. We analyze cases when the optimal solution does not correspond to a source-to-sink path, thus the shortest path algorithm would fail. To solve the overall (multicommodity) planning problem we develop a Lagrangean decomposition scheme, which separates the planning decisions into a resource subproblem, and a number of product-level subproblems. The Lagrangean multipliers are updated iteratively using a subgradient search algorithm. Through extensive computational testing, we show that the shortest path algorithm serves as an effective heuristic for the product-level subproblem (a mixed integer program), yielding high quality solutions with only a fraction (roughly 2%) of the computer time.     

Parking Capacity and Pricing in Park'n Ride Trips: A Continuous Equilibrium Network Design Problem

In this paper we consider the problem of designing parking facilities for park'n ride trips. We present a new continuous equilibrium network design problem to decide the capacity and fare of these parking lots at a tactical level. We assume that the parking facilities have already been located and other topological decisions have already been taken.The modeling approach proposed is mathematical programming with equilibrium constraints. In the outer optimization problem, a central Authority evaluates the performance of the transport network for each network design decision. In the inner problem a multimodal traffic assignment with combined modes, formulated as a variational inequality problem, generates the share demand for modes of transportation, and for parking facilities as a function of the design variables of the parking lots. The objective is to make optimal parking investment and pricing decisions in order to minimize the total travel cost in a subnetwork of the multimodal transportation system.We present a new development in model formulation based on the use of generalized parking link cost as a design variable.The bilevel model is solved by a simulated annealing algorithm applied to the continuous and non-negative design decision variables. Numerical tests are reported in order to illustrate the use of the model, and the ability of the approach to solve applications of moderate size.     

随机容错设施布局问题的近似算法

在确定性的容错设施布局问题中, 给定顾客的集合和地址的集合. 在每个地址上可以开设任意数目的不同设施. 每个顾客j有连接需求r_j. 允许将顾客j连到同一地址的不同设施上. 目标是开设一些设施并将每个顾客j连到r_j个不同的设施上, 使得总开设费用和连接费用最小. 研究两阶段随机容错设施布局问题(SFTFP), 顾客的集合事先不知道, 但是具有有限多个场景并知道其概率分布. 每个场景指定需要服务的顾客的子集. 并且每个设施有两种类型的开设费用. 在第一阶段根据顾客的随机信息确定性地开设一些设施, 在第二阶段根据顾客的真实信息再增加开设一些设施．给出随机容错布局问题的线性整数规划和基于线性规划舍入的5-近似算法.     

Location of an undesirable facility in a polygonal region with forbidden zones

In this paper, we develop the problem of locating an undesirable facility in a bounded polygonal region (with forbidden polygonal zones), using Euclidean distances, under an objective function that generalizes the maximin and maxisum criteria, and includes other criteria such as the linear combinations of these criterions. We identify a finite dominating set (finite set of points to which an optimal solution must belong) for this problem and show that an optimum solution can be found in polynomial time in the number of vertices of the polygons in the model and the number of existing facilities.     

Planar maximal covering location problem under block norm distance measure

This paper introduces a new model for the planar maximal covering location problem (PMCLP) under different block norms. The problem involves locating g facilities anywhere on the plane in order to cover the maximum number of n given demand points. The generalization, in this paper, is that the distance measures assigned to facilities are block norms of different types and different proximity measures. First, the PMCLP under different block norms is modelled as a maximum clique partition problem on an equivalent multi-interval graph. Then, the equivalent graph problem is modelled as an unconstrained binary quadratic problem (UQP). Both the maximum clique partition problem and the UQP are NP-hard problems; therefore, we solve the UQP format through a genetic algorithm heuristic. Computational examples are given.     

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.     

The branch-and-bound algorithm for a competitive facility location problem with the prescribed choice of suppliers

In the mathematical model under study, the two competing sides consecutively place their facilities aiming to capture consumers and maximize profits. The model amounts to a bilevel integer programming problem. We take the optimal noncooperative solutions as optimal to this problem. To find approximate and optimal solutions, we propose a branch-and-bound algorithm. Simulations show that the algorithm can be applied to solve the individual problems of low and medium dimension.     

The allocation of shared fixed costs

We consider the problem of sharing the fixed costs of facilities among a number of users. Typically the users have a benefit or revenue from the use of the facilities. Although the problem can be formulated and solved as an integer programme this provides limited accounting information. Such information is often needed in order to (i) decide on which facilities are viable and (ii) to charge the users. It is shown that it is impossible to meet both these needs in a satisfactory way. We examine different ways of partially meeting them. In addition, we consider the issue of fairness among different possible cost allocations and how such ‘fair’ costs may be derived.     

A new lot sizing and scheduling heuristic for multi-site biopharmaceutical production

Biopharmaceutical manufacturing requires high investments and long-term production planning. For large biopharmaceutical companies, planning typically involves multiple products and several production facilities. Production is usually done in batches with a substantial set-up cost and time for switching between products. The goal is to satisfy demand while minimising manufacturing, set-up and inventory costs. The resulting production planning problem is thus a variant of the capacitated lot-sizing and scheduling problem, and a complex combinatorial optimisation problem. Inspired by genetic algorithm approaches to job shop scheduling, this paper proposes a tailored construction heuristic that schedules demands of multiple products sequentially across several facilities to build a multi-year production plan (solution). The sequence in which the construction heuristic schedules the different demands is optimised by a genetic algorithm. We demonstrate the effectiveness of the approach on a biopharmaceutical lot sizing problem and compare it with a mathematical programming model from the literature. We show that the genetic algorithm can outperform the mathematical programming model for certain scenarios because the discretisation of time in mathematical programming artificially restricts the solution space.     

Locating and capacity planning for retailers of a new supply chain to compete on the plane

This paper investigates the network design problem of a two-level supply chain (SC), which is applicable for industries such as automotive, fuel and tyre manufacturing. Models presented in this paper aim at locating retail facilities of an SC and identifying their required capacities in the presence of existing competing retailers of a rival SC. We consider feasible locating space of the retail facilities on the continuous plane with bounded constraints and static competition among the rivals of the markets with deterministic demands. The problem is used for both essential and luxury product cases; hence, we consider elastic and inelastic demands, both. The models discussed in this paper are non-linear and non-convex which are difficult to solve. We use interval branch-and-bound as optimization algorithm for small size single-retailer problems, but for large-scale, multi-retailer problems we need to have more efficient methods. Therefore, we apply a heuristic algorithm (H1), a simulated annealing (SA) algorithm, an interior point (IP) algorithm, a genetic algorithm (GA) and a pattern search algorithm for solving multi-retailer problem with elastic and inelastic demands. Computational results obtained from performing different solution approaches for both elastic and inelastic show that mostly IP, PS, and H1 methods outperform the other approaches. The computational results on a real-life case are also promising. Several extended mathematical models and an example of a typical case with details are presented in the appendices of the paper.     

Location and sizing of facilities on a line

The location-allocation problem in its basic form assumes that the number of new facilities to be located is known and the capacities are unlimited. When the locations of the facilities and demand points (or customers) are restricted to the real line, the basic model may be solved efficiently by dynamic programming. In this note, we show that when the number of facilities and their capacities are included in the decision process, the problem may actually be easier to solve.