Abstract
In this paper we investigate planar location models with equity objectives. Two objectives are analyzed: (1) Minimizing the variance of the distances to the facility, and (2) minimizing the range of the distances. The problems are solved using the global optimization technique “Big Triangle Small Triangle”. Computational experiments provided excellent results. Solving a problem with 10,000 demand points required less than 5 s of computer time for finding the minimum variance, and less than half that time for finding the minimum range.
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References
Al-Khayyal F, Tuy H, Zhou F (2002) Large-scale single facility continuous location by D.C. optimization.” Optimization 51:271–292
Carrizosa E (2001) An optimal bound for d.c. programs with convex constraints. Math Methods Oper Res 54:47–51
Carrizosa E, Plastria F (1998) Locating an undesirable facility by generalized cutting planes. Math Oper Res 23:680–694
Drezner Z (1998) Finding whether a point is inside a polygon and its application to forbidden regions. J Manage Sci & Reg Dev 1:41–48
Drezner T (2004) Location of casualty collection points. Environ Plan C 22:899–912
Drezner T, Drezner Z (2004) Finding the optimal solution to the huff competitive location model. Comput Manage Sci 1:193–208
Drezner T, Drezner Z, Salhi S (2002) Solving the multiple competitive facilities location problem. Euro J Oper Res 142:138–151
Drezner T, Drezner Z, Salhi S (2006) A multi-objective heuristic approach for the casualty points location problem. J Oper Res Soc 58:727–734
Drezner Z, Klamroth K, Schöbel A, Wesolowsky GO (2002). The weber problem. In: Drezner Z, Hamacher H (eds). Facility location: applications and theory. Springer, Berlin Heidelberg New York
Drezner Z, Suzuki A (2004) The big triangle small triangle method for the solution of non-convex facility location problems. Oper Res 52:128–135
Drezner Z, Thisse J-F, Wesolowsky GO (1986) The minimax-min location problem. J Reg Sci 26:87–101
Eiselt HA, Laporte G (1995). Objectives in locations problems. Ch. 9. In: Facility location: a survey of applications and methods, Drezner Z (ed), Springer, Berlin Heidelberg, New York, pp 151–180
Elzinga DJ, Hearn DW (1972) Geometrical solutions for some minimax location problems. Transportation Sci 6:379–394
Hakimi SL (1964) Optimal location of switching centers and the absolute centers and medians of a graph. Oper Res 12:450–459
Hakimi SL (1965) Optimum distribution of switching centers in a communication network and some related graph theoretic problems. Oper Res 13:462–475
Kucherenko S, Sytsko Y (2004) Applications of deterministic low-discrepency sequences to nonlinear global optimization problems. Comput Optim Appl 30:297–318
Love RF, Morris JG, Wesolowsky GO (1988) Facilities location: models and methods. North Holland Amsterdam
Maimon O (1986) The variance equity measure in locational decision theory. Ann Oper Res 6:147–160
Maimon O (1988) An algorithm for the lorenz measure in locational decisions on trees. J Algorithms 9:583–596
Maranas CD, Floudas CA (1994) A global optimization method for weber’s problem with attraction and repulsion. In: Large scale optimization: state of the art. Hager WW, Hearn DW, Pardalos PM (eds) Kluwer, Dordrecht
Marsh M, Schilling D (1994) Equity measurement in facility location analysis: a review and framework. Euro J Oper Res 74:1–17
Minieka E (1970) The m-Center problem. SIAM Rev 12:138–139
Mladenovic N, Petrovic J, Kovacevic-Vojcic V, Cangalovic M (2003) Solving a spread-spectrum radar polyphase code designproblem by tabu search and variable neighborhood search. Eur J Oper Res 151:389–399
Okabe A, Boots B, Sugihara K, Chin S-N (2000) Spatial tessellations: concepts and applications of voronoi diagrams. 2nd edn. Wiley, Chichester
Schöber A (1999) Locating lines and hyperplanes: theory and algorithms. Kluwer, Dordrecht
Strekalovsky AS (1998) Global optimality conditions for nonconvex optimization. J Global Optim 12:415–434
Strekalovsky AS, Yakovleva TV (2004) On a local and global search involved in nonconvex optimization problems. Autom Remote Control 65:375–387
Sylvester JJ (1857) A question in the geometry of situation. Q J Pure Appl Math 1:79
Tuy H, Al-Khayyal F, Zhou F (1995) A D.C. optimization method for single facility location problems. J Global Optim 7:209–227
Weber A (1909) ÜBer Den Standort Der Industrien, 1. Teil: Reine Theorie Des Standortes, Tübingen, Germany. (English Translation by C. J. Friedeich (1957), Theory of the Location of Industries, Chicago University Press, Chicago.)
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Drezner, T., Drezner, Z. Equity Models in Planar Location. CMS 4, 1–16 (2007). https://doi.org/10.1007/s10287-006-0021-0
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DOI: https://doi.org/10.1007/s10287-006-0021-0