On Planar medianoid Competitive Location Problems with Manhattan Distance

A medianoid problem is a competitive location problem that determines the locations of a number of new service facilities that are competing with existing facilities for service to customers. This paper studies the medianoid problem on the plane with Manhattan distance. For the medianoid problem with binary customer preferences, i.e., a case where customers choose the closest facility to satisfy their entire demand, we show that the general problem is NP-hard and present solution methods to solve various special cases in polynomial time. We also show that the problem with partially binary customer preferences can be solved with a similar approach we develop for the model with binary customer preferences.

[1]  Hans-Christoph Wirth,et al.  An O(n(logn)2/loglogn) algorithm for the single maximum coverage location or the (1, Xp)-medianoid problem on trees , 2009, Inf. Process. Lett..

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

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

[4]  Andreas Drexl,et al.  Facility location models for distribution system design , 2005, Eur. J. Oper. Res..

[5]  Nimrod Megiddo,et al.  On the Complexity of Some Common Geometric Location Problems , 1984, SIAM J. Comput..

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

[7]  Robert J. Fowler,et al.  Optimal Packing and Covering in the Plane are NP-Complete , 1981, Inf. Process. Lett..

[8]  Gilbert Laporte,et al.  Competitive Location Models: A Framework and Bibliography , 1993, Transp. Sci..

[9]  Blas Pelegrín,et al.  On Nash equilibria of a competitive location-design problem , 2011, Eur. J. Oper. Res..

[10]  David R. Wood,et al.  An algorithm for finding a maximum clique in a graph , 1997, Oper. Res. Lett..

[11]  Charles ReVelle,et al.  THE MAXIMUM CAPTURE OR “SPHERE OF INFLUENCE” LOCATION PROBLEM: HOTELLING REVISITED ON A NETWORK* , 1986 .

[12]  Pierre Hansen,et al.  The p-median problem: A survey of metaheuristic approaches , 2005, Eur. J. Oper. Res..

[13]  R. Suárez-Vega,et al.  Competitive Multifacility Location on Networks: The (R|Xp)-Medianoid Problem , 2004 .

[14]  Z. Drezner.  Note-On a Modified One-Center Model , 1981 .

[15]  Tammy Drezner,et al.  Locating a single new facility among existing unequally attractive facilities , 1994 .

[16]  S. Hakimi.  On locating new facilities in a competitive environment , 1983 .

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

[18]  Gilbert Laporte,et al.  Demand allocation functions , 1998 .

[19]  Takao Asano,et al.  Finding the Connected Components and a Maximum Clique of an Intersection Graph of Rectangles in the Plane , 1983, J. Algorithms.

[20]  Timothy J. Lowe,et al.  Tool design problems in a punch press flexible manufacturing system. , 1998 .

[21]  José Muñoz-Pérez,et al.  Competitive location with rectilinear distances , 1995 .