Branch-and-bound algorithm for a competitive facility location problem

We study a mathematical model generalizing the well-known facility location problem. In this model we consider two competing sides successively placing their facilities and aiming to “capture” consumers, in order to make maximal profit. We state the problem as a bilevel integer programming problem, regarding optimal noncooperative solutions as optimal solutions. We propose a branch-and-bound algorithm for finding the optimal noncooperative solution. While constructing the algorithm, we represent our problem as the problem of maximizing a pseudo-Boolean function. An important ingredient of the algorithm is a method for calculating an upper bound for the values of the pseudo-Boolean function on subsets of solutions. We present the results of a simulation demonstrating the computational capabilities of the proposed algorithm.     

A cross decomposition procedure for the facility location problem with a choice of facility type

Consider a capacitated facility location problem in which each customer is assumed to have a unit demand, and each facility capacity has to be chosen from the given set of admissible levels. Under the restriction that each customer's unit demand be met by exactly one facility, the objective is to select a set of facilities to open, along with their capacities, and to assign customer's demand to them so as to minimize the total cost which includes fixed costs of opening facilities as well as variable assignment costs. The problem is modelled as a pure 0–1 program which extends the scope of applicability significantly over that by conventional location models. Based on Cross Decomposition recently developed by Van Roy, a solution procedure is proposed, when exploits the special structure of the problem. Computational results with a set of test problems shows the superiority of our solution procedure to other related ones.     

Randomized local search for the discrete competitive facility location problem

Consider a finite set of consumers that two competing companies are willing to service. The companies open facilities one by one. The set of locations available to open facilities is finite. The problem is to find a facility location for the first company that maximizes its profit given that the second company also makes its decision by maximizing the profit. We propose a randomized local search scheme that employs an internal local search procedure to estimate the solutions being enumerated. Numerical experiments with random input data show that the scheme is able to find high quality approximate solutions for examples with dimension that has not been amenable to previously known algorithms.     

Approximating the Pareto-front of a planar bi-objective competitive facility location and design problem

A bi-objective competitive facility location and design problem is considered. The problem of obtaining a complete representation of the efficient set and its corresponding Pareto-front has been previously tackled through exact general methods, but they require high computational effort. In this work, we propose a new evolutionary multi-objective optimization algorithm, named FEMOEA, which deals with the problem at hand in a fast and efficient way. It combines ideas from different multi-objective and single-objective optimization evolutionary algorithms, although it also incorporates new devices which help to reduce the computational requirements, and also to improve the quality of the provided solutions. The performance of the algorithm is analyzed by comparing it to other (meta)heuristics previously proposed in the literature. In particular, the reference algorithms MOEA/D, SPEA2 and NSGA-II have been considered. A comprehensive computational study shows that the new heuristic method outperforms, on average, the three heuristic algorithms. Additionally, it reduces, on average, the computing time of the exact methods by approximately 99%, and this offering high-quality discrete approximations of the true Pareto-front.     

Location equilibria for a continuous competitive facility location problem under delivered pricing

The problem of finding location equilibria of a location-price game where firms first select their locations and then set delivered prices in order to maximise their profits is investigated. Assuming that firms set the equilibrium prices in the second stage, the game can be reduced to a location game for which a global minimiser of the social cost is a location equilibrium, provided that the demand is completely inelastic and the marginal production cost is constant. When the set of feasible locations is a region of the plane the minimisation of the social cost becomes a hard-to-solve global optimisation problem. We propose an exact interval branch-and-bound algorithm suitable for small and medium size problems and an alternating Weiszfeld-like heuristic for larger instances. The latter approach is based on a new iterative formula for which the validity of the descent property is proved. The proposed heuristic performs extremely well against the exact method when tested on small to medium size instances while requiring a tiny fraction of its computational time.     

基于竞争环境的截流设施选址与车辆路径问题

研究竞争环境下截流设施选址与带时间窗的多中心车辆路径问题。首先,在考虑设施覆盖范围衰退的情况下,利用阶梯型效用函数和偏离距离描述消费者的选择行为,并确定截流设施的需求量;然后,采用基于聚集度的启发式算法对门店进行分类,借助双层规划法,建立门店选址与车辆路径安排的多目标整数规划模型;最后,采用改进的蚁群算法进行求解。通过分析对比实验结果,验证了模型的有效性和可行性。     

A planar single-facility competitive location and design problem under the multi-deterministic choice rule

A new customer choice rule, which may model in some cases the actual patronising behaviour of customers towards the facilities closer to reality than other existing rules, is proposed. According to the new rule, customers split their demand among the firms in the market by patronising only one facility from each firm, the one with the highest utility, and the demand is split among those facilities proportionally to their attraction. The influence of the choice rule in the location of facilities is investigated. In particular, a new continuous competitive single-facility location and design problem using this new rule is proposed. Both exact and heuristic methods are proposed to solve it. A comparison with the classical proportional (or Huff) choice rule when solving the location model reveals that both the location and the quality of the new facility to be located may be quite different depending on the patronising behaviour of customers. Most importantly, the profit that the locating chain may lose if a wrong choice is made can be quite high in some instances.     

The congested facility location problem

Consider a network facility location problem where congestion arises at facilities, and is represented by delay functions that approximate the queueing process. We strive to minimize the sum of customers' transportation and waiting times, and facilities' fixed and variable costs. The problem is solved using a column generation technique within a branch-and-bound scheme. Numerical results are reported and a bilevel (user-optimized) formulation considered, among other extensions.     

Bilevel competitive facility location and pricing problems

We propose new models for competitive facility location and pricing as bilevel Boolean linear programming problems. We obtain results that characterize the complexity of the problem where a monopolist’s profit on each of the markets is defined with a monotone nonincreasing function of the servicing cost. For this problem, we also propose two approximate algorithms based on the ideas of alternating heuristics and local search. We give results of a computational experiment that show a possibility for fast computation of approximate solutions.     

A leader-follower game in competitive facility location

We address the problem of locating new facilities of a firm or franchise that enters a market where a competitor operates existing facilities. The goal of the new entrant firm is to decide the location and attractiveness of its new facilities that maximize its profit. The competitor can react by opening new facilities, closing existing ones, and adjusting the attractiveness levels of its existing facilities, with the aim of maximizing its own profit. The demand is assumed to be aggregated at certain points in the plane and the new facilities of both the firm and the competitor can be located at predetermined candidate sites. We employ the gravity-based rule in modeling the behavior of the customers where the probability that a customer visits a certain facility is proportional to the facility attractiveness and inversely proportional to the distance between the facility site and demand point. We formulate a bilevel mixed-integer nonlinear programming model where the firm entering the market is the leader and the competitor is the follower. We propose heuristics that combine tabu search with exact solution methods.     

Capacitated facility location problem with general setup cost

This paper presents an extension of the capacitated facility location problem (CFLP), in which the general setup cost functions and multiple facilities in one site are considered. The setup costs consist of a fixed term (site setup cost) plus a second term (facility setup costs). The facility setup cost functions are generally non-linear functions of the size of the facility in the same site. Two equivalent mixed integer linear programming (MIP) models are formulated for the problem and solved by general MIP solver. A Lagrangian heuristic algorithm (LHA) is also developed to find approximate solutions for this NP-hard problem. Extensive computational experiments are taken on randomly generated data and also well-known existing data (with some necessary modifications). The detailed results are provided and the heuristic algorithm is shown to be efficient.     

A single facility location problem with a weighted maximin-minimax rectilinear distance

This paper provides an algorithm for locating a single facility in a region, where the objective function is composed of the weighted maximin and minimax rectilinear distances from a set of given demand points. This weighted objective function is applicable when the facility to be located is somewhat desirable but it should not be too close to the demand points, since it also has some undesirable effects. It has been proven in this paper, that it is enough to test for optimality all the intersection points of any two lines forming the equirectilinear distances between any pair of demand points or boundary lines of the region. The algorithm developed here tests these intersection points. The efficient set of points and their optimality range are found. This parametric form of the solution provides an optimal solution for any desired weight.     

The single facility location problem with minimum distance constraints

We consider the problem of locating a single facility (server) in the plane, where the location of the facility is restricted to be outside a specified forbidden region (neighborhood) around each demand point. Two models are discussed. In the restricted 1-median model, the objective is to minimize the sum of the weighted rectilinear distances from the n customers to the facility. We present an O(n log n) algorithm for this model, improving upon the O(n³) complexity bound of the algorithm by Brimberg and Wesolowsky (1995). In the restricted 1-center model the objective is to minimize the maximum of the weighted rectilinear distances between the customers and the serving facility. We present an O(n log n) algorithm for finding an optimal 1-center. We also discuss some related models, involving the Euclidean norm.     

RAMP algorithms for the capacitated facility location problem

Annals of Mathematics and Artificial Intelligence - In this paper, we address the Capacitated Facility Location Problem (CFLP) in which the assignment of facilities to customers must ensure enough...     

有约束竞争选址问题的降阶回溯算法

有约束竞争选址问题是组合优化中一个经典的NP-hard问题,现有算法研究该问题时或是无法求得最优解或是求解速度慢.针对现有算法的缺点,首先在这个经典问题的基础上进行修改,构建了一个新的数学模型;接着对该模型的数学性质进行研究,并在数学性质的基础上提出了上下界算法和降阶子算法对问题进行降阶,达到了缩减问题搜索解空间的目的,降阶的过程中既有单个的降阶,也有成批的降阶;然后在前面的基础上设计了一个回溯子算法来求解问题的最优解;最后通过两个示例分析更清楚地阐述该算法的原理,结果证明该算法可以较快求得最优解.     

A computational study of a nonlinear minsum facility location problem

A discrete location problem with nonlinear objective is addressed. A set of p plants is to be open to serve a given set of clients. Together with the locations, the number p of facilities is also a decision variable. The objective is to minimize the total cost, represented as the transportation cost between clients and plants, plus an increasing nonlinear function of p.     

The facility and transfer points location problem

In this paper, we investigate the location of a facility and several transfer points to serve as collector points for customers who need the services of the facility. For example, demand for emergency services is generated at a set of demand points that need the services of a central facility (such as a hospital). Patients are transferred to a helicopter pad (transfer point) at normal speed, and from there they are transferred to the facility at increased speed. The model involves the location of multiple transfer points and one facility. Locating one transfer point when the set of demand points and the location of the facility are known was investigated in Berman et al. (2004a). Location of several transfer points when the location of the facility is given is investigated in Berman et al. (2004b). In this paper, we propose heuristic approaches for the solution of this problem and report computational experiments on a test set of 40 problems.     

The multi-period incremental service facility location problem

In this paper we introduce the multi-period incremental service facility location problem where the goal is to set a number of new facilities over a finite time horizon so as to cover dynamically the demand of a given set of customers. We prove that the coefficient matrix of the allocation subproblem that results when fixing the set of facilities to open is totally unimodular. This allows to solve efficiently the Lagrangean problem that relaxes constraints requiring customers to be assigned to open facilities. We propose a solution approach that provides both lower and upper bounds by combining subgradient optimization to solve a Lagrangean dual with an ad hoc heuristic that uses information from the Lagrangean subproblem to generate feasible solutions. Numerical results obtained in the computational experiments show that the obtained solutions are very good. In general, we get very small percent gaps between upper and lower bounds with little computation effort.     

Models and algorithms for competitive facility location problems with different customer behavior

Competitive facility location problems arise in the context of two non-cooperating companies, a leader and a follower, competing for market share from a given set of customers. We assume that the firms place a given number of facilities on locations taken from a discrete set of possible points. For this bi-level optimization problem we consider six different customer behavior scenarios from the literature: binary, proportional and partially binary, each combined with essential and unessential demand. The decision making for the leader and the follower depends on these scenarios. In this work we present mixed integer linear programming models for the follower problem of each scenario and use them in combination with an evolutionary algorithm to optimize the location selection for the leader. A complete solution archive is used to detect already visited candidate solutions and convert them efficiently into similar, not yet considered ones. We present numerical results of our algorithm and compare them to so far state-of-the-art approaches from the literature. Our method shows good performance in all customer behavior scenarios and is able to outperform previous solution procedures on many occasions.