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Constrained location of competitive facilities in the plane. (English) Zbl 1061.90074
Summary: This paper examines a competitive facility location problem occurring in the plane. A new gravity-based utility model is developed, in which the capacity of a facility serves as its measure of attractiveness. A new problem formulation is given, having elastic gravity-based demand, along with capacity, forbidden region, and budget constraints. Two solution algorithms are presented, one based on the big square small square method, and the second based on a penalty function formulation using fixed-point iteration. Computational testing is presented, comparing these two algorithms along with a general-purpose nonlinear solver.

90B80 Discrete location and assignment
90C26 Nonconvex programming, global optimization
Full Text: DOI
[1] Hakimi, S.L., On locating new facilities in a competitive environment, European journal of operational research, 12, 29-35, (1983) · Zbl 0499.90026
[2] Emir H, Francis RL. Aggregation of demand points for the covering location model. Presentation, INFORMS 2001 Annual Meeting, Miami, FL.
[3] Drezner, T.; Drezner, Z., Replacing continuous demand with discrete demand in a competitive location model, Naval research logistics, 44, 81-95, (1997) · Zbl 0882.90083
[4] Hotelling, H., Stability in competition, Economic journal, 39, 41-57, (1929)
[5] Drezner, Z., Competitive location strategies for two facilities, Regional science and economics, 12, 485-493, (1982)
[6] Huff, D.L., Defining and estimating a trade area, Journal of marketing, 28, 34-38, (1964)
[7] Huff, D.L., A programmed solution for approximating an optimum retail location, Land economics, 42, 293-303, (1966)
[8] Drezner, T., Optimal continuous location of a retail facility, facility attractiveness, and market Sharean interactive model, Journal of retailing, 70, 49-64, (1994)
[9] Weiszfeld, E., Sur le point pour lequel la somme des distances de n points donnes est minimum, Tohoku mathematical journal, 43, 355-386, (1936) · Zbl 0017.18007
[10] Drezner, T., Location of multiple retail facilities with limited budget constraints—in continuous space, Journal of retailing and consumer services, 5, 3, 173-184, (1998)
[11] Drezner, T.; Drezner, Z.; Salhi, S., Solving the multiple competitive facilities location problem, European journal of operational research, 142, 138-151, (2002) · Zbl 1081.90575
[12] Penn M, Kariv O. Competitive location in trees: parts I and II. Working paper.
[13] Yang, H.; Wong, S.-.C., A continuous equilibrium model for estimating market areas of competitive facilities with elastic demand and market externality, Transportation science, 34, 216-217, (2000) · Zbl 1008.91507
[14] Berman, O.; Krass, D., Locating multiple competitive facilitiesspatial interaction models with variable expenditures, Annals of operations research, 111, 1, 197-225, (2002) · Zbl 1013.90086
[15] Hansen, P.; Peeters, D.; Thisse, J.-.F., On the location of an obnoxious facility, Sistemi urbani, 3, 299-317, (1981)
[16] Nakanishi, M.; Cooper, L.G., Parameter estimate for multiplicative interactive choice modelleast squares approach, Journal of marketing research, 11, 303-311, (1974)
[17] Larson, R.C.; Sadiq, G., Facility locations with the Manhattan metric in the presence of barriers to travel, Operations research, 31, 4, 652-669, (1983) · Zbl 0521.90045
[18] Hamacher, H.W.; Nickel, S., Restricted planar location problems and applications, Naval research logistics, 42, 967-992, (1995) · Zbl 0845.90082
[19] Butt, S.E.; Cavalier, T.M., An efficient algorithm for facility location in the presence of forbidden regions, European journal of operational research, 90, 1, 56-70, (1996) · Zbl 0916.90177
[20] McGarvey, R.G.; Cavalier, T.M., A global optimal approach to facility location in the presence of forbidden regions, Computers and industrial engineering, 45, 1, 1-15, (2003)
[21] Savas, S.; Batta, R.; Nagi, R., Finite-size facility placement in the presence of barriers to rectilinear travel, Operations research, 50, 6, 1018-1031, (2002) · Zbl 1163.90623
[22] Hansen, P.; Peeters, D.; Richard, D.; Thisse, J.-.F., The minisum and minimax location problems revisited, Operations research, 33, 1, 1251-1265, (1985) · Zbl 0582.90027
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