Competitive Facility Location Research Articles

A novel hybrid algorithm for solving continuous single-objective defensive location problem

The continuous defensive location problem (CDLP) is an NP-hard problem well investigated in the fields of competitive facility location. CDLP is a bi-level programming problem, where a decision maker locates defensive facilities with different capacities in the vertices of the network in order to avoid her/his aggressors from reaching core which is an important vertex in the network. In the present research, a hybrid method combining the imperialist competitive algorithm (ICA) and BFGS algorithm is presented to solve the CDLP. The proposed hybrid method integrates the ICA and the BFGS algorithm, providing a highly near-optimal solution. The upper-level problem solves the optimal location of defense facilities, and hybrid algorithm is applied. The lower-level problem is the shortest path problem which is solved by the Dijkstra method. The feasibility of the proposed hybrid method is demonstrated for a number of small, medium and large instances of the problem. The test results are compared with those obtained by genetic algorithm, particle swarm optimization and ICA in terms of solution accuracy and required CPU time. Simulation results reveal that the proposed hybrid method is feasible, robust and more effective in solving the CDLP than conventional metaheuristic methods.

Nash type games in competitive facilities location

In a non-cooperative game, a Nash equilibrium corresponds to a set of strategies, which ensure that none of the players are better off by unilaterally changing his strategy. This work focuses on the application of Nash Equilibrium to the competitive facility location. Its aim is to provide an up to date review of the literature concerning the discrete version of the problem.

Nash type games in competitive facilities location

In a non-cooperative game, a Nash equilibrium corresponds to a set of strategies, which ensure that none of the players are better off by unilaterally changing his strategy. This work focuses on the application of Nash Equilibrium to the competitive facility location. Its aim is to provide an up to date review of the literature concerning the discrete version of the problem.

Solution of Discrete Competitive Facility Location Problem for Firm Expansion

Solution of Discrete Competitive Facility Location Problem for Firm Expansion

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.

PARALLEL OPTIMIZATION ALGORITHM FOR COMPETITIVE FACILITY LOCATION

A stochastic search optimization algorithm is developed and applied to solve a bi-objective competitive facility location problem for firm expansion. Parallel versions of the developed algorithm for shared- and distributed-memory parallel computing systems are proposed and experimentally investigated by approximating the Pareto front of the competitive facility location problem of different scope. It is shown that the developed algorithm has advantages against its precursor in the sense of the precision of approximation. It is also shown that the proposed parallel versions of the algorithm have almost linear speed-up when solving competitive facility location problems of different scope reasonable for practical applications.

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.

A leader–follower model for discrete competitive facility location

In this paper we investigate a leader–follower (Stackelberg equilibrium) competitive location model. The competitive model is based on the concept of cover. Each facility attracts consumers within a “sphere of influence” defined by a “radius of influence.” The leader and the follower, each has a budget to be spent on the expansion of their chains either by improving their existing facilities or constructing new ones. We find the best strategy for the leader assuming that the follower, knowing the action taken by the leader, will react by investing his budget to maximize his market share. The objective of the leader is to maximize his market share following the follower׳s reaction.

A Computational approach to Convex Hull of Zero-one MIP problems with special structure

This paper presented a new approach for implementing reformulation-linearization technique (RLT) using matrix operations. It began with employing base-2 expansions of binary variables to express extended form of the main problem and generate matrices transform. The generated matrices provided a powerful tool for implementing different level of RLT. Particularly, the new representation was efficient for deriving results of the problem with special structures. Moreover, it was employed for deriving the convex hull representation of problems with generalized upper bound (GUB) constraint and packing constraints. Finally, as an application of the proposed method, a class of competitive facility location, (1∣p)-centroid, in which two competing companies successively opened their facilities, was studied. The problem was formulated as a bi-level programming problem (BLPP) and then converted into an equivalent one-level mixed-integer linear program so that it could be solved by any MIP solver. Experimental result was reported for instances with different sizes.

A hybrid genetic algorithm with solution archive for the discrete $$(r|p)$$ ( r | p ) -centroid problem

In this article we propose a hybrid genetic algorithm for the discrete $$(r|p)$$(r|p)-centroid problem. We consider the competitive facility location problem where two non-cooperating companies enter a market sequentially and compete for market share. The first decision maker, called the leader, wants to maximize his market share knowing that a follower will enter the same market. Thus, for evaluating a leader's candidate solution, a corresponding follower's subproblem needs to be solved, and the overall problem therefore is a bi-level optimization problem. This problem is $$\Sigma _2^P$$Σ2P-hard, i.e., harder than any problem in NP (if $$\hbox {P}\not =\hbox {NP}$$P?NP). A heuristic approach is employed which is based on a genetic algorithm with tabu search as local improvement procedure and a complete solution archive. The archive is used to store and convert already visited solutions in order to avoid costly unnecessary re-evaluations. Different solution evaluation methods are combined into an effective multi-level evaluation scheme. The algorithm is tested on well-known benchmark sets of both Euclidean and non-Euclidean instances as well as on larger newly created instances. Especially on the Euclidean instances our algorithm is able to exceed previous state-of-the-art heuristic approaches in solution quality and running time in most cases.

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.

Quantum binary particle swarm optimization-based algorithm for solving a class of bi-level competitive facility location problems

This paper deals with a special class of competitive facility location problems, in which two non-cooperative firms compete to capture the most of a given market, in order to maximize their profit. This paper intends to present a simple and effective nested strategy based on the quantum binary particle swarm optimization (QBPSO) method for solving the bi-level mathematical model of the problem. In solution approach, an improvement procedure is embedded into QBPSO to increase the convergence speed and generate more accurate solutions. Taguchi's method is employed to systematically determine the optimal values of QBPSO parameters. Finally, computational results on large-scale instances with up to 300 locations and 350 clients (more than 100,000 variables and 300,000 constraints at each level) confirmed the method efficiency in terms of solution quality and time.

Voronoi game on graphs

Voronoi game is a geometric model of competitive facility location problem played between two players. Users are generally modeled as points uniformly distributed on a given underlying space. Each player chooses a set of points in the underlying space to place their facilities. Each user avails service from its nearest facility. Service zone of a facility consists of the set of users which are closer to it than any other facility. Payoff of each player is defined by the quantity of users served by all of its facilities. The objective of each player is to maximize their respective payoff. In this paper we consider the two player Voronoi game where the underlying space is a road network modeled by a graph. In this framework we consider the problem of finding k optimal facility locations of Player 2 given any placement of m facilities by Player 1. Our main result is a dynamic programming based polynomial time algorithm for this problem on tree network. On the other hand, we show that the problem is strongly NP-complete for graphs. This proves that finding a winning strategy of P2 is NP-complete. Consequently, we design a 1−1e factor approximation algorithm, where e≈2.718.

Computational complexity of the discrete competitive facility location problem

We consider the discrete competitive facility location problem with finitely many consumers and finitely many locations of the facilities. Two competing companies are opening the facilities in some of these locations consequently; i.e., the first company does it first, then does second company. The goal of both is to maximize their profit obtained from serving the consumers, provided that each of the consumers chooses exactly one of the facilities opened by both companies, based on his known preferences. The complexity status is identified for two particular cases of this problem.

Parallelization of a non-linear multi-objective optimization algorithm: Application to a location problem

Real-life problems usually include conflicting objectives. Solving multi-objective problems (i.e., obtaining the complete efficient set and the corresponding Pareto-front) via exact methods is in many cases nearly intractable. In order to cope with those problems, several (meta) heuristic procedures have been developed during the last decade whose aim is to obtain a good discrete approximation of the Pareto-front. In this vein, a new multi-objective evolutionary algorithm, called FEMOEA, which can be applied to many nonlinear multi-objective optimization problems, has recently been proposed. Through a comparison with an exact interval branch-and-bound algorithm, it has been shown that FEMOEA provides very good approximations of the Pareto-front. Furthermore, it has been compared to the reference algorithms NSGA-II, SPEA2 and MOEA/D. Comprehensive computational studies have shown that, among the studied algorithms, FEMOEA was the one providing, on average, the best results for all the quality indicators analyzed. However, when the set approximating the Pareto-front must have many points (because a high precision is required), the computational time needed by FEMOEA may not be negligible at all. Furthermore, the memory requirements needed by the algorithm when solving those instances may be so high that the available memory may not be enough. In those cases, parallelizing the algorithm and running it in a parallel architecture may be the best way forward. In this work, a parallelization of FEMOEA, called FEMOEA-Paral, is presented. To show its applicability, a bi-objective competitive facility location and design problem is solved. The results show that FEMOEA-Paral is able to maintain the effectiveness of the sequential version and this by reducing the computational costs. Furthermore, the parallel version shows good scalability. The efficiency results have been analyzed by means of a profiling and tracing toolkit for performance analysis.

A review of competitive facility location in the plane

In this paper, we review competitive location models. Retail facilities operate in a competitive environment with an objective of profit and market share maximization. These facilities are different from each other in their overall attractiveness to consumers. The basic problem is the optimal location of one or more new facilities in a market where competition already exists or will exist in the future. Extensions to the models include an analysis of the optimal allocation of a budget among new facilities and their best locations, modeling location under conditions of uncertainty and future competition, incorporating the concept of a threshold in competitive location, modeling lost demand, and minimizing cannibalization.

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.

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.

On the competitive facility location problem with a free choice of suppliers

We consider a mathematical model from the class of competitive sequential facility location problems. In these problems, the competitors sequentially open their facilities, and each side aims to "capture" the consumers and maximize its profits. In the proposed model, we consider a situation of a "free" choice by each side of an open facility to service a customer. The model is formulated as a bilevel integer programming problem. We show that the problem of finding an optimal noncooperative solution can be represented as a maximization problem for a pseudo-Boolean function. We propose an algorithm for constructing an admissible noncooperative solution for fixed values of the variables in this pseudo-Boolean function. We also propose a method for constructing an upper bound on the maximal value of the pseudo-Boolean function on subsets of solutions defined by partial (0, 1)-vectors.

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.