Competitive facility location with random attractiveness

Customers’ perception of a particular facility’s attractiveness is likely to be heterogeneous. However, existing competitive facility location models assume that facilities’ attractiveness levels are fixed. We extend the gravity model assuming randomly distributed facilities’ attractiveness. We propose two effective solution methods. One is based on discretizing the attractiveness level distribution. The second is based on the concept of an “effective” attractiveness. Effective attractiveness is the level of fixed attractiveness whose calculated optimal market share approximately equals the expected optimal market share under random attractiveness. We show how effective attractiveness is calculated.     

A discrete competitive facility location model with variable attractiveness

We consider the discrete version of the competitive facility location problem in which new facilities have to be located by a new market entrant firm to compete against already existing facilities that may belong to one or more competitors. The demand is assumed to be aggregated at certain points in the plane and the new facilities can be located at predetermined candidate sites. We employ Huff's gravity-based rule in modelling the behaviour of the customers where the probability that customers at a demand point patronize a certain facility is proportional to the facility attractiveness and inversely proportional to the distance between the facility site and demand point. The objective of the firm is to determine the locations of the new facilities and their attractiveness levels so as to maximize the profit, which is calculated as the revenue from the customers less the fixed cost of opening the facilities and variable cost of setting their attractiveness levels. We formulate a mixed-integer nonlinear programming model for this problem and propose three methods for its solution: a Lagrangean heuristic, a branch-and-bound method with Lagrangean relaxation, and another branch-and-bound method with nonlinear programming relaxation. Computational results obtained on a set of randomly generated instances show that the last method outperforms the others in terms of accuracy and efficiency and can provide an optimal solution in a reasonable amount of time.     

A note on the m-center problem with rectilinear distances

Given n demand points on a plane, the problem we consider is to locate a given number, m, of facilities on the plane so that the maximum of the set of rectilinear distances of each demand point to its nearest facility is minimized. This problem is known as the m-center problem on the plane. A related problem seeks to determine, for a given r, the minimum number of facilities and their locations so as to ensure that every point is within r units of rectilinear distance from its nearest facility. We formulate the latter problem as a problem of covering nodes by cliques of an intersection graph. Certain bounds are established on the size of the problem. An efficient algorithm is provided to generate this set-covering problem. Computational results with this approach are summarized.     

The minimum equitable radius location problem with continuous demand

We analyze the location of p facilities satisfying continuous area demand. Three objectives are considered: (i) the p-center objective (to minimize the maximum distance between all points in the area and their closest facility), (ii) equalizing the load service by the facilities, and (iii) the minimum equitable radius – minimizing the maximum radius from each point to its closest facility subject to the constraint that each facility services the same load. The paper offers three contributions: (i) a new problem – the minimum equitable radius is presented and solved by an efficient algorithm, (ii) an improved and efficient algorithm is developed for the solution of the p-center problem, and (iii) an improved algorithm for the equitable load problem is developed. Extensive computational experiments demonstrated the superiority of the new solution algorithms.     

The gravity p-median model

In this paper we propose a new model for the p-median problem. In the standard p-median problem it is assumed that each demand point is served by the closest facility. In many situations (for example, when demand points are communities of customers and each customer makes his own selection of the facility) demand is divided among the facilities. Each customer selects a facility which is not necessarily the closest one. In the gravity p-median problem it is assumed that customers divide their patronage among the facilities with the probability that a customer patronizes a facility being proportional to the attractiveness of that facility and to a decreasing utility function of the distance to the facility.     

Competitive facility location problem with attractiveness adjustment of the follower: A bilevel programming model and its solution

We are concerned with a problem in which a firm or franchise enters a market by locating new facilities where there are existing facilities belonging to a competitor. The firm aims at finding the location and attractiveness of each facility to be opened so as to maximize its profit. The competitor, on the other hand, can react by adjusting the attractiveness of its existing facilities with the objective of maximizing its own profit. The demand is assumed to be aggregated at certain points in the plane and the facilities of the firm can be located at predetermined candidate sites. We employ Huff’s gravity-based rule in modeling the behavior of the customers where the fraction of customers at a demand point that visit 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. In order to find the optimal solution of this model, we convert it into an equivalent one-level mixed-integer nonlinear program so that it can be solved by global optimization methods. Apart from reporting computational results obtained on a set of randomly generated instances, we also compute the benefit the leader firm derives from anticipating the competitor’s reaction of adjusting the attractiveness levels of its facilities. The results on the test instances indicate that the benefit is 58.33% on the average.     

A multi-objective genetic algorithm for a bi-objective facility location problem with partial coverage

In this study, we present a bi-objective facility location model that considers both partial coverage and service to uncovered demands. Due to limited number of facilities to be opened, some of the demand nodes may not be within full or partial coverage distance of a facility. However, a demand node that is not within the coverage distance of a facility should get service from the nearest facility within the shortest possible time. In this model, it is assumed that demand nodes within the predefined distance of opened facilities are fully covered, and after that distance the coverage level decreases linearly. The objectives are defined as the maximization of full and partial coverage, and the minimization of the maximum distance between uncovered demand nodes and their nearest facilities. We develop a new multi-objective genetic algorithm (MOGA) called modified SPEA-II (mSPEA-II). In this method, the fitness function of SPEA-II is modified and the crowding distance of NSGA-II is used. The performance of mSPEA-II is tested on randomly generated problems of different sizes. The results are compared with the solutions of the most well-known MOGAs, NSGA-II and SPEA-II. Computational experiments show that mSPEA-II outperforms both NSGA-II and SPEA-II.     

The equitable location problem on the plane

This paper considers the problem of locating M facilities on the unit square so as to minimize the maximal demand faced by each facility subject to closest assignments and coverage constraints. Focusing on uniform demand over the unit square, we develop upper and lower bounds on feasibility of the problem for a given number of facilities and coverage radius. Based on these bounds and numerical experiments we suggest a heuristic to solve the problem. Our computational results show that the heuristic is very efficient, as the average gap between its solutions and the lower bound is 4.34%.     

基于改进NSGA-II算法的多级服务设施备用覆盖选址决策模型

为了应对跨区域突发事件过程中受灾点服务差异化需求的问题,建立了应急储备设施点的多级备用覆盖选址决策模型,即一个需求点由多个应急设施提供不同质量水平的服务,并考虑设施繁忙状态下由其他设施点提供服务的状况,使模型更加符合实际应用。首次通过设计分段的染色体编码方式改进NSGA-II算法提升运算效率以更好地解决多目标选址决策问题,将改进方法下得到的Pareto解分布与NSGA-II算法下的仿真结果进行对比分析,结合设施点的部署策略得到不同的空间布局方案。证明了模型的可行性及改进NSGA-II算法在解决设施点多目标选址决策问题时的有效性。     

A multi-objective integrated facility location-hardening model: Analyzing the pre- and post-disruption tradeoff

Two methods of reducing the risk of disruptions to distribution systems are (1) strategically locating facilities to mitigate against disruptions and (2) hardening facilities. These two activities have been treated separately in most of the academic literature. This article integrates facility location and facility hardening decisions by studying the minimax facility location and hardening problem (MFLHP), which seeks to minimize the maximum distance from a demand point to its closest located facility after facility disruptions. The formulation assumes that the decision maker is risk averse and thus interested in mitigating against the facility disruption scenario with the largest consequence, an objective that is appropriate for modeling facility interdiction. By taking advantage of the MFLHP’s structure, a natural three-stage formulation is reformulated as a single-stage mixed-integer program (MIP). Rather than solving the MIP directly, the MFLHP can be decomposed into sub-problems and solved using a binary search algorithm. This binary search algorithm is the basis for a multi-objective algorithm, which computes the Pareto-efficient set for the pre- and post-disruption maximum distance. The multi-objective algorithm is illustrated in a numerical example, and experimental results are presented that analyze the tradeoff between objectives.     

Validating the Gravity-Based Competitive Location Model Using Inferred Attractiveness

The attractiveness of retail facilities is an essential component of models analyzing competition among retail facilities. In this paper we introduce an innovative method for inferring retail facility attractiveness. Readily available data from secondary sources about customers' buying power and sales volumes obtained by competing retail facilities are used. The gravity-based competitive facility location model is used to predict sales. The attractiveness of the retail facilities are inferred from these data.The procedure is used to confirm the gravity competitive facility location model. Inferred attractiveness results based on empirical data from Orange County, California, were compared with an independent survey with excellent match.     

Discrete cooperative covering problems

A family of discrete cooperative covering problems is analysed in this paper. Each facility emits a signal that decays by the distance and each demand point observes the total signal emitted by all facilities. A demand point is covered if its cumulative signal exceeds a given threshold. We wish to maximize coverage by selecting locations for p facilities from a given set of potential sites. Two other problems that can be solved by the max-cover approach are the equivalents to set covering and p-centre problems. The problems are formulated, analysed and solved on networks. Optimal and heuristic algorithms are proposed and extensive computational experiments reported.     

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.     

Improving solution of discrete competitive facility location problems

We consider discrete competitive facility location problems in this paper. Such problems could be viewed as a search of nodes in a network, composed of candidate and customer demand nodes, which connections correspond to attractiveness between customers and facilities located at the candidate nodes. The number of customers is usually very large. For some models of customer behavior exact solution approaches could be used. However, for other models and/or when the size of problem is too high to solve exactly, heuristic algorithms may be used. The solution of discrete competitive facility location problems using genetic algorithms is considered in this paper. The new strategies for dynamic adjustment of some parameters of genetic algorithm, such as probabilities for the crossover and mutation operations are proposed and applied to improve the canonical genetic algorithm. The algorithm is also specially adopted to solve discrete competitive facility location problems by proposing a strategy for selection of the most promising values of the variables in the mutation procedure. The developed genetic algorithm is demonstrated by solving instances of competitive facility location problems for an entering firm.     

竞争环境下截流设施选址问题

研究企业新建设施时,市场上已有设施存在的情况下,使本企业总体利润最大的截流设施选址问题。在一般截留设施选址模型的基础上引入引力模型,消费者到某个设施接受服务的概率与偏离距离及设施的吸引力相关,同时设施的建设费用与设施吸引力正相关,建立非线性整数规划模型并使用贪婪算法进行求解。数值分析表明,该算法求解速度快,模型计算精度较高。     

Covering continuous demand in the plane

Continuous demand is generated in a convex polygon. A facility located in the area covers demand within a given radius. The objective is to find the locations for p facilities that cover the maximum demand in the area. A procedure that calculates the total area covered by a set of facilities is developed. A multi start heuristic approach for solving this problem is proposed by applying a gradient search from a randomly generated set of p locations for the facilities. Computational experiments for covering a square area illustrate the effectiveness of the proposed algorithm.     

基于选址效益的联合覆盖模型研究

传统的覆盖模型含有“全有全无”和“单一覆盖”两个假设,即假设需求点在设施的服务半径内才被覆盖,否则不被覆盖;需求点只能被最近的设施覆盖。这两条假设在实际应用中均存在不合理之处。松弛了这两条假设,研究逐渐覆盖和联合覆盖。在保证每个需求点都享受到最低服务水平的情况下,提出了选址效益最大化的联合覆盖模型。由于目标函数中含有分式,通过引入辅助变量的方法,将含有分式目标函数的非线性规划转化成等价的线性规划。最后,通过数值算例分析了最低服务水平限制对最佳选址方案的影响,并得到选址成本、总服务水平和单位成本服务水平随最低服务水平限制的变化,同时对影响模型的重要参数做了敏感性分析。     

Finding a representative nondominated set for multi-objective mixed integer programs

In this paper, we develop algorithms to find small representative sets of nondominated points that are well spread over the nondominated frontiers for multi-objective mixed integer programs. We evaluate the quality of representations of the sets by a Tchebycheff distance-based coverage gap measure. The first algorithm aims to substantially improve the computational efficiency of an existing algorithm that is designed to continue generating new points until the decision maker (DM) finds the generated set satisfactory. The algorithm improves the coverage gap value in each iteration by including the worst represented point into the set. The second algorithm, on the other hand, guarantees to achieve a desired coverage gap value imposed by the DM at the outset. In generating a new point, the algorithm constructs territories around the previously generated points that are inadmissible for the new point based on the desired coverage gap value. The third algorithm brings a holistic approach considering the solution space and the number of representative points that will be generated together. The algorithm first approximates the nondominated set by a hypersurface and uses it to plan the locations of the representative points. We conduct computational experiments on randomly generated instances of multi-objective knapsack, assignment, and mixed integer knapsack problems and show that the algorithms work well.     

Optimal algorithms for the α-neighbor p-center problem

Assigning multiple service facilities to demand points is important when demand points are required to withstand service facility failures. Such failures may result from a multitude of causes, ranging from technical difficulties to natural disasters. The α-neighbor p-center problem deals with locating p service facilities. Each demand point is assigned to its nearest α service facilities, thus it is able to withstand up to α − 1 service facility failures. The objective is to minimize the maximum distance between a demand point and its αth nearest service facility. We present two optimal algorithms for both the continuous and discrete α-neighbor p-center problem. We present experimental results comparing the performance of the two optimal algorithms for α = 2. We also present experimental results showing the performance of the relaxation algorithm for α = 1, 2, 3.     

The variable radius covering problem

In this paper we propose a covering problem where the covering radius of a facility is controlled by the decision-maker; the cost of achieving a certain covering distance is assumed to be a monotonically increasing function of the distance (i.e., it costs more to establish a facility with a greater covering radius). The problem is to cover all demand points at a minimum cost by finding optimal number, locations and coverage radii for the facilities. Both, the planar and discrete versions of the model are considered. Heuristic approaches are suggested for solving large problems in the plane. These methods were tested on a set of planar problems. Mathematical programming formulations are proposed for the discrete problem, and a solution approach is suggested and tested.