Upper bounds for objective functions of discrete competitive facility location problems

Under study is the problem of locating facilities when two competing companies successively open their facilities. Each client chooses an open facility according to his own preferences and return interests to the leader firm or to the follower firm. The problem is to locate the leader firm so as to realize the maximum profit (gain) subject to the responses of the follower company and the available preferences of clients. We give some formulations of the problems under consideration in the form of two-level integer linear programming problems and, equivalently, as pseudo-Boolean two-level programming problems. We suggest a method of constructing some upper bounds for the objective functions of the competitive facility location problems. Our algorithm consists in constructing an auxiliary pseudo-Boolean function, which we call an estimation function, and finding the minimum value of this function. For the special case of the competitive facility location problems on paths, we give polynomial-time algorithms for finding optimal solutions. Some results of computational experiments allow us to estimate the accuracy of calculating the upper bounds for the competitive location problems on paths.     

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.     

A mathematical model of market competition

We consider a mathematical model of decision making by a company attempting to win a market share. We assume that the company releases its products to the market under the competitive conditions that another company is making similar products. Both companies can vary the kinds of their products on the market as well as the prices in accordance with consumer preferences. Each company aims to maximize its profit. A mathematical statement of the decision-making problem for the market players is a bilevel mathematical programming problem that reduces to a competitive facility location problem. As regards the latter, we propose a method for finding an upper bound for the optimal value of the objective function and an algorithm for constructing an approximate solution. The algorithm amounts to local ascent search in a neighborhood of a particular form, which starts with an initial approximate solution obtained simultaneously with an upper bound. We give a computational example of the problem under study which demonstrates the output of the algorithm.     

A capacitated competitive facility location problem

We consider a mathematical model similar in a sense to competitive location problems. There are two competing parties that sequentially open their facilities aiming to “capture” customers and maximize profit. In our model, we assume that facilities’ capacities are bounded. The model is formulated as a bilevel integer mathematical program, and we study the problem of obtaining its optimal (cooperative) solution. It is shown that the problem can be reformulated as that of maximization of a pseudo-Boolean function with the number of arguments equal to the number of places available for facility opening. We propose an algorithm for calculating an upper bound for values that the function takes on subsets which are specified by partial (0, 1)-vectors.     

Bilevel “Defender–Attacker” Model with Multiple Attack Scenarios

We consider a bilevel “defender-attacker” model built on the basis of the Stackelberg game. In this model, given is a set of the objects providing social services for a known set of customers and presenting potential targets for a possible attack. At the first step, the Leader (defender) makes a decision on the protection of some of the objects on the basis of his/her limited resources. Some Follower (attacker), who is also limited in resources, decides then to attack unprotected objects, knowing the decision of the Leader. It is assumed that the Follower can evaluate the importance of each object and makes a rational decision trying to maximize the total importance of the objects attacked. The Leader does not know the attack scenario (the Follower’s priorities for selecting targets for the attack). But, the Leader can consider several possible scenarios that cover the Follower’s plans. The Leader’s problem is then to select the set of objects for protection so that, given the set of possible attack scenarios and assuming the rational behavior of the Follower, to minimize the total costs of protecting the objects and eliminating the consequences of the attack associated with the reassignment of the facilities for customer service. The proposed model may be presented as a bilevelmixed-integer programming problem that includes an upper-level problem (the Leader problem) and a lower-level problem (the Follower problem). The main efforts in this article are aimed at reformulation of the problem as some one-level mathematical programming problems. These formulations are constructed using the properties of the optimal solution of the Follower’s problem, which makes it possible to formulate necessary and sufficient optimality conditions in the form of linear relations.     

Algorithms for a Facility Location Problem with Stochastic Customer Demand and Immobile Servers

This paper studies a facility location problem with stochastic customer demand and immobile servers. Motivated by applications to locating bank automated teller machines (ATMs) or Internet mirror sites, these models are developed for situations in which immobile service facilities are congested by stochastic demand originating from nearby customer locations. Customers are assumed to visit the closest open facility. The objective of this problem is to minimize customers' total traveling cost and waiting cost. In addition, there is a restriction on the number of facilities that may be opened and an upper bound on the allowable expected waiting time at a facility. Three heuristic algorithms are developed, including a greedy-dropping procedure, a tabu search approach and an -optimal branch-and-bound method. These methods are compared computationally on a bank location data set from Amherst, New York.     

An Improved Branch & Bound Method for the Uncapacitated Competitive Location Problem

In this paper, the problem of locating new facilities in a competitive environment is considered. The problem is formulated as the firm expected profit maximization and a set of nodes is selected in a graph representing the geographical zone. Profit depends on fixed and deterministic location costs and, since customers are independent decision-makers, on the expected market share. The problem is an instance of nonlinear integer programming, because the objective function is concave and submodular. Due to this complexity a branch & bound method is developed for solving small size problems (that is, when the number of nodes is less than 50), while a heuristic is necessary for larger problems. The branch & bound is called data-correcting method, while the approximate solutions are obtained using the heuristic-concentration method.     

Competitive facility location models

Two classes of competitive facility location models are considered, in which several persons (players) sequentially or simultaneously open facilities for serving clients. The first class consists of discrete two-level programming models. The second class consists of game models with several independent players pursuing selfish goals. For the first class, its relationship with pseudo-Boolean functions is established and a novel method for constructing a family of upper and lower bounds on the optimum is proposed. For the second class, the tight PLS-completeness of the problem of finding Nash equilibriums is proved.     

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.     

Algorithms for the generalized quadratic assignment problem combining Lagrangean decomposition and the Reformulation-Linearization Technique

In this paper, we propose two exact algorithms for the GQAP (generalized quadratic assignment problem). In this problem, given M facilities and N locations, the facility space requirements, the location available space, the facility installation costs, the flows between facilities, and the distance costs between locations, one must assign each facility to exactly one location so that each location has sufficient space for all facilities assigned to it and the sum of the products of the facility flows by the corresponding distance costs plus the sum of the installation costs is minimized. This problem generalizes the well-known quadratic assignment problem (QAP). Both exact algorithms combine a previously proposed branch-and-bound scheme with a new Lagrangean relaxation procedure over a known RLT (Reformulation-Linearization Technique) formulation. We also apply transformational lower bounding techniques to improve the performance of the new procedure. We report detailed experimental results where 19 out of 21 instances with up to 35 facilities are solved in up to a few days of running time. Six of these instances were open.     

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.     

Online facility location with facility movements

In the online facility location problem demand points arrive one at a time and the goal is to decide where and when to open a facility. In this paper we consider a new version of the online facility location problem, where the algorithm is allowed to move the opened facilities in the metric space. We consider the uniform case where each facility has the same constant cost. We present an algorithm which is 2-competitive for the general case and we prove that it is 3/2-competitive if the metric space is the line. We also prove that no algorithm with smaller competitive ratio than \({(\sqrt{13}+1)/4\approx 1.1514}\) exists. We also present an empirical analysis which shows that the algorithm gives very good results in the average case.     

Optimizing bandwidth allocation in elastic optical networks with application to scheduling

We study a problem of optimal bandwidth allocation in the elastic optical networks technology, where usable frequency intervals are of variable width. In this setting, each lightpath has a lower and upper bound on the width of its frequency interval, as well as an associated profit, and we seek a bandwidth assignment that maximizes the total profit. This problem is known to be NP-complete. We strengthen this result by showing that, in fact, the problem is inapproximable within any constant ratio even on a path network. We further derive NP-hardness results and present approximation algorithms for several special cases of the path and ring networks, which are of practical interest. Finally, while in general our problem is hard to approximate, we show that an optimal solution can be obtained by allowing resource augmentation. Some of our results resolve open problems posed by Shalom et al. (2013) [28]. Our study has applications also in real-time scheduling.     

A Lagrangian heuristic for concave cost facility location problems: the plant location and technology acquisition problem

We propose a Lagrangian heuristic for facility location problems with concave cost functions and apply it to solve the plant location and technology acquisition problem. The problem is decomposed into a mixed integer subproblem and a set of trivial single-variable concave minimization subproblems. We are able to give a closed-form expression for the optimal Lagrangian multipliers such that the Lagrangian bound is obtained in a single iteration. Since the solution of the first subproblem is feasible to the original problem, a feasible solution and an upper bound are readily available. The Lagrangian heuristic can be embedded in a branch-and-bound scheme to close the optimality gap. Computational results show that the approach is capable of reaching high quality solutions efficiently. The proposed approach can be tailored to solve many concave-cost facility location problems.     

A discrete long-term location–price problem under the assumption of discriminatory pricing: Formulations and parametric analysis

We consider a competitive location problem in which a new firm has to make decisions on the locations of several new facilities as well as on its price setting in order to maximise profit. Under the assumption of discriminatory prices, competing firms set a specific price for each market area. The customers buy one unit of a single homogeneous price-inelastic product from the facility that offers the lowest price in the area the consumers belong to. Three customer choice rules are considered in order to break ties in the offered prices. We prove that, considering long-term competition on price, this problem can be reduced to a problem with decisions on location only. For each one of the choice rules the location problem is formulated as an integer programming model and a parametric analysis of these models is given. To conclude, an application with real data is presented.     

Local search with a generalized neighborhood in the optimization problem for pseudo-Boolean functions

In the optimization problem for pseudo-Boolean functions we consider a local search algorithm with a generalized neighborhood. This neighborhood is constructed for a locally optimal solution and includes nearby locally optimal solutions. We present some results of simulations for pseudo-Boolean functions whose optimization is equivalent to the problems of facility location, set covering, and competitive facility location. The goal of these experiments is to obtain a comparative estimate for the locally optimal solutions found by the standard local search algorithm and the local search algorithm using a generalized neighborhood.     

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 Bilevel Stochastic Programming Problem with Random Parameters in the Follower’s Objective Function

Under study is a bilevel stochastic linear programming problem with quantile criterion. Bilevel programming problems can be considered as formalization of the process of interaction between two parties. The first party is a Leader making a decision first; the second is a Follower making a decision knowing the Leader’s strategy and the realization of the random parameters. It is assumed that the Follower’s problem is linear if the realization of the random parameters and the Leader’s strategy are given. The aim of the Leader is the minimization of the quantile function of a loss function that depends on his own strategy and the optimal Follower’s strategy. It is shown that the Follower’s problem has a unique solution with probability 1 if the distribution of the random parameters is absolutely continuous. The lower-semicontinuity of the loss function is proved and some conditions are obtained of the solvability of the problem under consideration. Some example shows that the continuity of the quantile function cannot be provided. The sample average approximation of the problem is formulated. The conditions are given to provide that, as the sample size increases, the sample average approximation converges to the original problem with respect to the strategy and the objective value. It is shown that the convergence conditions hold for almost all values of the reliability level. A model example is given of determining the tax rate, and the numerical experiments are executed for this example.     

LP-based approximation algorithms for capacitated facility location

In the capacitated facility location problem with hard capacities, we are given a set of facilities, ${\mathcal{F}}$ , and a set of clients ${\mathcal{D}}$ in a common metric space. Each facility i has a facility opening cost f _i and capacity u _i that specifies the maximum number of clients that may be assigned to this facility. We want to open some facilities from the set ${\mathcal{F}}$ and assign each client to an open facility so that at most u _i clients are assigned to any open facility i. The cost of assigning client j to facility i is given by the distance c _ij , and our goal is to minimize the sum of the facility opening costs and the client assignment costs. The only known approximation algorithms that deliver solutions within a constant factor of optimal for this NP-hard problem are based on local search techniques. It is an open problem to devise an approximation algorithm for this problem based on a linear programming lower bound (or indeed, to prove a constant integrality gap for any LP relaxation). We make progress on this question by giving a 5-approximation algorithm for the special case in which all of the facility costs are equal, by rounding the optimal solution to the standard LP relaxation. One notable aspect of our algorithm is that it relies on partitioning the input into a collection of single-demand capacitated facility location problems, approximately solving them, and then combining these solutions in a natural way.