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.

[1]  Blas Pelegrín,et al.  All Stackelberg Location Equilibria in the Hotelling's Duopoly Model on a Tree with Parametric Prices , 2003, Ann. Oper. Res..

[2]  Jonathan F. Bard,et al.  Practical Bilevel Optimization: Algorithms and Applications (Nonconvex Optimization and Its Applications) , 2006 .

[3]  Lee G. Cooper,et al.  Parameter Estimation for a Multiplicative Competitive Interaction Model—Least Squares Approach , 1974 .

[4]  Winfried Steiner,et al.  A Stackelberg-Nash model for new product design , 2010, OR Spectr..

[5]  Jonathan F. Bard,et al.  Practical Bilevel Optimization: Algorithms and Applications , 1998 .

[6]  C. Adjiman,et al.  Global optimization of mixed‐integer nonlinear problems , 2000 .

[7]  Richard A. Waltz,et al.  KNITRO User's Manual Version 6.0 , 2009 .

[8]  D. Huff.  A Programmed Solution for Approximating an Optimum Retail Location , 1966 .

[9]  Blas Pelegrín,et al.  On a branch-and-bound approach for a Huff-like Stackelberg location problem , 2009, OR Spectr..

[10]  Frank Plastria,et al.  Static competitive facility location: An overview of optimisation approaches , 2001, Eur. J. Oper. Res..

[11]  Charles ReVelle,et al.  Competitive Location and Pricing on Networks , 1999 .

[12]  I. K. Altinel,et al.  Competitive facility location problem with attractiveness adjustment of the follower: A bilevel programming model and its solution , 2011, Eur. J. Oper. Res..

[13]  David L. Huff,et al.  Defining and Estimating a Trading Area , 1964 .

[14]  Z. Drezner.  Competitive location strategies for two facilities , 1982 .

[15]  W. Reilly.  The law of retail gravitation , 1931 .

[16]  Jonathan F. Bard,et al.  The Mixed Integer Linear Bilevel Programming Problem , 1990, Oper. Res..

[17]  Frank Plastria,et al.  Discrete models for competitive location with foresight , 2008, Comput. Oper. Res..

[18]  Christodoulos A. Floudas,et al.  αBB: A global optimization method for general constrained nonconvex problems , 1995, J. Glob. Optim..

[19]  Fred W. Glover,et al.  Principles of Tabu Search , 2007, Handbook of Approximation Algorithms and Metaheuristics.

[20]  C. Adjiman,et al.  A global optimization method, αBB, for general twice-differentiable constrained NLPs—II. Implementation and computational results , 1998 .

[21]  C. Floudas,et al.  Active constraint strategy for flexibility analysis in chemical processes , 1987 .

[22]  Z. Drezner,et al.  FACILITY LOCATION IN ANTICIPATION OF FUTURE COMPETITION. , 1998 .

[23]  A. Neumaier,et al.  A global optimization method, αBB, for general twice-differentiable constrained NLPs — I. Theoretical advances , 1998 .

[24]  Christodoulos A. Floudas,et al.  Deterministic Global Optimization: Theory, Methods and (NONCONVEX OPTIMIZATION AND ITS APPLICATIONS Volume 37) (Nonconvex Optimization and Its Applications) , 2005 .

[25]  Kathrin Fischer,et al.  Sequential Discrete p-Facility Models for Competitive Location Planning , 2002, Ann. Oper. Res..

[26]  Horst A. Eiselt,et al.  An alternating heuristic for medianoid and centroid problems in the plane , 2003, Comput. Oper. Res..

[27]  Christodoulos A. Floudas,et al.  Deterministic global optimization - theory, methods and applications , 2010, Nonconvex optimization and its applications.

[28]  Dimitri P. Bertsekas,et al.  Nonlinear Programming , 1997 .

[29]  Christodoulos A. Floudas,et al.  Global optimization of MINLP problems in Process Synthesis and Design , 1997 .

[30]  D. Serra,et al.  Market capture by two competitors: The pre-emptive location problem , 1994 .

[31]  I. K. Altinel,et al.  A Hybrid Tabu Search Heuristic for a Bilevel Competitive Facility Location Model , 2010, Hybrid Metaheuristics.

[32]  W. Press,et al.  Numerical Recipes: The Art of Scientific Computing , 1987 .

[33]  Gilbert Laporte,et al.  Sequential location problems , 1997 .