Discretization results for the Huff and Pareto-Huff competitive location models on networks

In this paper we prove that there always exists a finite set that includes an optimal solution for the Huff and the Pareto-Huff competitive models on networks with the assumption of a concave function of the distance. In the Huff model, there is always a vertex of the network that belongs to the solution set. For the Pareto-Huff model, we prove that there is always an optimal solution at, or an ε-optimal solution close to, a vertex or an isodistant point, a new concept introduced in this paper.     

Efficient solution approaches for a discrete multi-facility competitive interaction model

In this paper, we present efficient solution approaches for discrete multi-facility competitive interaction model. Applying the concept of “Tangent Line Approximation” presented by the authors in their previous work, we develop efficient computational approaches—both exact and approximate (with controllable error bound α). Computational experiments show that the approximate approach (with small α) performs extremely well solving large scale problems while the exact approach performs very well for small to medium-sized problems.     

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.     

Recent insights in Huff-like competitive facility location and design

Recently, several articles appeared on the location–design problem that firms face when entering a competing market. All use a Huff-like attraction model. We discuss the formulation of the base model, the different settings studied in the papers and summarise their findings.     

Sequential competitive location on networks

We present a survey of recent developments in the field of sequential competitive location problems, including the closely related class of voting location problems, i.e. problems of locating resources as the result of a collective election. Our focus is on models where possible locations are not a priori restricted to a finite set of points. Furthermore, we restrict our attention to problems defined on networks. Since a line, i.e. an interval of one-dimensional real space, may be interpreted as a special type of network and because models defined on lines might contain ideas worth adopting in more general network models, we include these models as well, yet without describing them in detail for the sake of brevity.     

Relaxed voting and competitive location under monotonous gain functions on trees

We examine competitive location problems where two competitors serve a good to users located in a network. Users decide for one of the competitors based on the distance induced by an underlying tree graph. The competitors place their server sequentially into the network. The goal of each competitor is to maximize his benefit which depends on the total user demand served. Typical competitive location problems include the (1,X₁)-medianoid, the (1,1)-centroid, and the Stackelberg location problem.An additional relaxation parameter introduces a robustness of the model against small changes in distance. We introduce monotonous gain functions as a general framework to describe the above competitive location problems as well as several problems from the area of voting location such as Simpson, Condorcet, security, and plurality.In this paper we provide a linear running time algorithm for determining an absolute solution in a tree where competitors are allowed to place on nodes or on inner points. Furthermore we discuss the application of our approach to the discrete case.     

Competitive facility location model with concave demand

We consider a spatial interaction model for locating a set of new facilities that compete for customer demand with each other, as well as with some pre-existing facilities to capture the “market expansion” and the “market cannibalization” effects. Customer demand is assumed to be a concave non-decreasing function of the total utility derived by each customer from the service offered by the facilities. The problem is formulated as a non-linear Knapsack problem, for which we develop a novel solution approach based on constructing an efficient piecewise linear approximation scheme for the objective function. This allows us to develop exact and α-optimal solution approaches capable of dealing with relatively large-scale instances of the model. We also develop a fast Heuristic Algorithm for which a tight worst-case error bound is established.     

A competitive hub location and pricing problem

We formulate and solve a new hub location and pricing problem, describing a situation in which an existing transportation company operates a hub and spoke network, and a new company wants to enter into the same market, using an incomplete hub and spoke network. The entrant maximizes its profit by choosing the best hub locations and network topology and applying optimal pricing, considering that the existing company applies mill pricing. Customers’ behavior is modeled using a logit discrete choice model. We solve instances derived from the CAB dataset using a genetic algorithm and a closed expression for the optimal pricing. Our model confirms that, in competitive settings, seeking the largest market share is dominated by profit maximization. We also describe some conditions under which it is not convenient for the entrant to enter the market.     

Optimal location and design of a competitive facility

A single facility has to be located in competition with fixed existing facilities of similar type. Demand is supposed to be concentrated at a finite number of points, and consumers patronise the facility to which they are attracted most. Attraction is expressed by some function of the quality of the facility and its distance to demand. For existing facilities quality is fixed, while quality of the new facility may be freely chosen at known costs. The total demand captured by the new facility generates income. The question is to find that location and quality for the new facility which maximises the resulting profits.It is shown that this problem is well posed as soon as consumers are novelty oriented, i.e. attraction ties are resolved in favour of the new facility. Solution of the problem then may be reduced to a bicriterion maxcovering-minquantile problem for which solution methods are known. In the planar case with Euclidean distances and a variety of attraction functions this leads to a finite algorithm polynomial in the number of consumers, whereas, for more general instances, the search of a maximal profit solution is reduced to solving a series of small-scale nonlinear optimisation problems. Alternative tie-resolution rules are finally shown to result in problems in which optimal solutions might not exist.Mathematics Subject Classification (2000):90B85, 90C30, 90C29, 91B42Partially supported by Grant PB96-1416-C02-02 of the D.G.E.S. and Grant BFM2002-04525-C02-02 of Ministerio de Ciencia y Tecnología, Spain     

Facility location problems in the plane based on reverse nearest neighbor queries

For a finite set of points S, the (monochromatic) reverse nearest neighbor (RNN) rule associates with any query point q the subset of points in S that have q as its nearest neighbor. In the bichromatic reverse nearest neighbor (BRNN) rule, sets of red and blue points are given and any blue query is associated with the subset of red points that have it as its nearest blue neighbor. In this paper we introduce and study new optimization problems in the plane based on the bichromatic reverse nearest neighbor (BRNN) rule. We provide efficient algorithms to compute a new blue point under criteria such as: (1) the number of associated red points is maximum (MAXCOV criterion); (2) the maximum distance to the associated red points is minimum (MINMAX criterion); (3) the minimum distance to the associated red points is maximum (MAXMIN criterion). These problems arise in the competitive location area where competing facilities are established. Our solutions use techniques from computational geometry, such as the concept of depth of an arrangement of disks or upper envelope of surface patches in three dimensions.     

一类多商品设施选址问题的基于线性松弛解的启发式方法

多商品设施选址问题是众多设施选址问题中一类重要而困难的问题.在这一问题中,顾客的需求可能包含不止一种商品.对于大规模问题,成熟的商业求解器往往不能在满意的时间内找到高质量的可行解.研究了无容量限制的单货源多商品设施选址问题的一般形式,并给出了应用于此类问题的两个启发式方法.这两个方法基于原选址问题的线性规划松弛问题的最优解,分别通过求解紧问题和邻域搜索的方式给出了原问题的一个可行上界.理论分析指出所提方法可以实施于任意可行问题的实例.数值结果表明所提方法可以显著地提高求解器求解此类设施选址问题的求解效率.     

Competitive facility location and design with reactions of competitors already in the market

A new retail facility is to locate and its service quality is to determine where similar facilities of competitors offering the same goods are already present. The market share captured by each facility depends on its distance to customers and its quality, which is described by a probabilistic Huff-like model. In order to maximize the profit of the new facility, a two-stage method is developed, which takes into account the reactions of the competitors. In the quality decision stage, the competitive decision process occurring among facilities is modelled as a game, whose solution is given by its Nash equilibrium. The solution, which can be represented as functions of the location of the new facility, is obtained by analytical resolution of a system of equations in the case of one facility in the market or by polynomial approximation in the case of multiple facilities. In the location decision stage, an interval based global optimization method is used to determine the best location of the new facility. Numerical experiments on randomly generated instances demonstrate the effectiveness of the method.     

The danger model: application to a competitive market

The behavior of the firm in a competitive market based on the idea of the human system, i.e., using a danger activator and a defence system, is modelled. The proposed model uses three variables: market share ratio, danger index and the ratio of relative investment between the firm and the total investments including the competition. The danger activator, the defence and the market reaction functions, which explain how the danger index becomes activated, how the firm reacts to a danger signal, and the market reaction to the firm’s actions, respectively, are carefully constructed. This leads to a parametric dynamic system that governs the behavior of the competitive market. The following five classical behaviors of a firm result: monopoly, below aimed market share, aimed market share, above aimed market share and out of market. Formulas for a sensitivity analysis are derived to determine how and how much the equilibrium points of the dynamic system change when the parameters change. All the concepts are illustrated by graphs that show the equilibrium points and the trajectories of the system.     

Competitive facility location on decentralized supply chains

This paper addresses a novel competitive facility location problem about a firm that intends to enter an existing decentralized supply chain comprised of three tiers of players with competition: manufacturers, retailers and consumers. It first proposes a variational inequality for the supply chain network equilibrium model with production capacity constraints, and then employs the logarithmic-quadratic proximal prediction–correction method as a solution algorithm. Based on this model, this paper develops a generic mathematical program with equilibrium constraints for the competitive facility location problem, which can simultaneously determine facility locations of the entering firm and the production levels of these facilities so as to optimize an objective. Subsequently, a hybrid genetic algorithm that incorporates with the logarithmic-quadratic proximal prediction–correction method is developed for solving the proposed mathematical program with an equilibrium constraint. Finally, this paper carries out some numerical examples to evaluate proposed models and solution algorithms.     

A D.C. biobjective location model

In this paper we address the biobjective problem of locating a semiobnoxious facility, that must provide service to a given set of demand points and, at the same time, has some negative effect on given regions in the plane. In the model considered, the location of the new facility is selected in such a way that it gives answer to these contradicting aims: minimize the service cost (given by a quite general function of the distances to the demand points) and maximize the distance to the nearest affected region, in order to reduce the negative impact. Instead of addressing the problem following the traditional trend in the literature (i.e., by aggregation of the two objectives into a single one), we will focus our attention in the construction of a finite -dominating set, that is, a finite feasible subset that approximates the Pareto-optimal outcome for the biobjective problem. This approach involves the resolution of univariate d.c. optimization problems, for each of which we show that a d.c. decomposition of its objective can be obtained, allowing us to use standard d.c. optimization techniques.     

Competitive facility location and design problem

We develop a spatial interaction model that seeks to simultaneously optimize location and design decisions for a set of new facilities. The facilities compete for customer demand with pre-existing competitive facilities and with each other. The customer demand is assumed to be elastic, expanding as the utility of the service offered by the facilities increases. Increases in the utility can be achieved by increasing the number of facilities, design improvements, or locating facilities closer to the customer.     

The k-level facility location game

We propose a cost-sharing scheme for the k-level facility location game that is cross-monotonic, competitive, and 6-approximate cost recovery. This extends the recent result for the 1-level facility location game of Pál and Tardos.     

On the convergence of a modified algorithm for the spherical facility location problem

We study the spherical facility location problem which is a more realistic model than the Euclidean facilities location. We present a modified algorithm for this problem, which has the following good properties: (a) It is very easy to initialize the algorithm with an arbitrary point as its starting point; (b) Under suitable assumptions, it is proved that the algorithm globally converges to a global minimizer of the problem.