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The single facility location problem with average-distances. (English) Zbl 1162.90500

Summary: This paper considers a location problem in \(\mathbb R^n\) , where the demand is not necessarily concentrated at points but it is distributed in hypercubes following a Uniform probability distribution. The goal is to locate a service facility minimizing the weighted sum of average distances (measured with \(\ell_p\) norms) to these demand hypercubes. In order to do that, we present an iterative scheme that provides a sequence converging to an optimal solution of the problem for \(p \in [1,2]\). For the planar case, analytical expressions of this iterative procedure are obtained for \(p=2\) and \(p=1\), where two different approaches are proposed. The paper ends with a computational analysis of the proposed methodology, comparing its efficiency with a standard minimizer.

MSC:

90B85 Continuous location
90B15 Stochastic network models in operations research
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[1] Beckenbach EF, Bellman R (1965) Inequalities, 2nd edn. Springer, Berlin · Zbl 0128.27401
[2] Brimberg J, Love RF (1992) A new distance function for modeling travel distances in transportation network. Transp Sci Oper Res Soc Am 26(2):129–137 · Zbl 0766.90054
[3] Brimberg J, Love RF (1993) Global convergence of a generalized iterative procedure for the minisum location problem with l p distances. Oper Res Soc Am 41(6):1153–1163 · Zbl 0795.90037 · doi:10.1287/opre.41.6.1153
[4] Brimberg J, Wesolowsky GO (2000) Facility location with closest rectangular distances. Nav Res Logist 47:77–84 · Zbl 0953.90033 · doi:10.1002/(SICI)1520-6750(200002)47:1<77::AID-NAV5>3.0.CO;2-#
[5] Brimberg J, Wesolowsky GO (2002) Locating facilities by minimax relative to closest points of demand areas. Comput Oper Res 29:625–636 · Zbl 1001.90042 · doi:10.1016/S0305-0548(00)00106-4
[6] Carrizosa E, Conde E, Muñoz M, Puerto J (1995) The generalized Weber problem with expected distances. RAIRO Rech Oper 29:35–57 · Zbl 0835.90040
[7] Carrizosa E, Muñoz M, Puerto J (1998a) The Weber problem with regional demand. Eur J Oper Res 104:358–365 · Zbl 0955.90063 · doi:10.1016/S0377-2217(97)00190-2
[8] Carrizosa E, Muñoz M, Puerto J (1998b) Location and shape of a rectangular facility in \(\mathbb{R}\) n . Convexity properties. Math Program 83:277–290 · Zbl 0920.90089
[9] Chen R (2001) Optimal location of a single facility with circular demand areas. Comput Math Appl 41:1049–1061 · Zbl 0980.90047 · doi:10.1016/S0898-1221(00)00339-4
[10] Drezner Z (1986) Location of regional facilities. Nav Res Logist Q 33:523–529 · Zbl 0593.90029 · doi:10.1002/nav.3800330316
[11] Drezner Z, Wesolowsky G (1980) Optimal location of a demand facility relative to area demand. Nav Res Logist Q 27:199–206 · Zbl 0443.90028 · doi:10.1002/nav.3800270204
[12] Ioffe AD, Levin VL (1972) Subdifferential of convex functions. Trans Mosc Math Soc 26:1–72
[13] Love RF, Morris JG (1979) Mathematical models of road travel distances. Manag Sci 25(2):130–139 · Zbl 0419.90053 · doi:10.1287/mnsc.25.2.130
[14] Garkavi A, Smatkov V (1974) On the Lame point and its generalizations in a normed space. Mat Sb 95:267–286 · Zbl 0318.42029 · doi:10.1070/SM1974v024n02ABEH002187
[15] Nickel S, Puerto J, Rodríguez-Chía AM (2003) An approach to location models involving sets as existing facilities. Math Oper Res 28(4):693–715 · Zbl 1082.90059 · doi:10.1287/moor.28.4.693.20521
[16] Puerto J, Rodríguez-Chía AM (1999) Location of a moving service facility. Math Methods Oper Res 49:373–393 · Zbl 0941.90047 · doi:10.1007/s001860050055
[17] Puerto J, Rodríguez-Chía AM (2006) New models for locating a moving service facility. Math Methods Oper Res 63(1):31–51 · Zbl 1103.90063 · doi:10.1007/s00186-005-0011-y
[18] Weiszfeld E (1937) Sur le point lequel la somme des distances de n points donnés est minimum. Tohoku Math J 43:355–386 · JFM 63.0583.01
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