Abstract
In this paper, we study the 1-maximin problem with rectilinear distance. We locate a single undesirable facility in a continuous planar region while considering the interaction between the facility and existing demand points. The distance between facility and demand points is measured in the rectilinear metric. The objective is to maximize the distance of the facility from the closest demand point. The 1-maximin problem has been formulated as an MIP model in the literature. We suggest new bounding schemes to increase the solution efficiency of the model as well as improved branch and bound strategies for implementation. Moreover, we simplify the model by eliminating some redundant integer variables. We propose an efficient solution algorithm called cut and prune method, which splits the feasible region into four equal subregions at each iteration and tries to eliminate subregions depending on the comparison of upper and lower bounds. When the sidelengths of the subregions are smaller than a predetermined value, the improved MIP model is solved to obtain the optimal solution. Computational experiments demonstrate that the solution time of the original MIP model is reduced substantially by the proposed solution approach.
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References
Appa GM and Giannikos I (1994). Is linear programming necessary for single facility location with maximin of rectilinear distance? J Opl Res Soc 45: 97–107.
Brooke A, Kendrick D and Meeraus A (1988). GAMS a User's Guide. The Scientific Press: Redwood City, CA.
Cplex (1998). Using the Cplex Callable Library, Version 6.0. ILOG, Inc.: NV, USA.
Drezner Z and Wesolowsky GO (1983). The location of an obnoxious facility with rectangular distances. J Reg Sci 23: 241–248.
Erkut E and Neuman S (1989). Analytical models for locating undesirable facilities. Eur J Oper Res 40: 275–291.
Karasakal EK and Koksalan M (2001). Generating a representative subset of the efficient frontier in multiple criteria decision making. Working paper no. 01-20, Faculty of Administration, University of Ottawa.
Mehrez A, Sinuany-Stern Z and Stulman A (1986). An enhancement of the Drezner–Wesolowsky algorithm for single-facility location with maximin of rectilinear distance. J Opl Res Soc 37: 971–977.
Melachrinoudis E (1988). An efficient computational procedure for the rectilinear maximin location problem. Trans Sci 22: 217–223.
Melachrinoudis E and Cullinane TP (1986). Locating an obnoxious facility within a polygonal region. Ann Opns Res 6: 137–145.
Morales DR, Carrizosa E and Conde E (1997). Semi-obnoxious location models: a global optimization approach. Eur J Opns Res 102: 295–301.
Plastria F (1996). Optimal location of undesirable facilities: a selective overview. Belg J Opns Res Stat Comput Sci 36: 109–127.
Sayin S (2000a). A mixed integer programming formulation for the 1-maximin problem. J Opl Res Soc 51: 371–375.
Sayin S (2000b). Measuring the quality of discrete representations of efficient sets in multiple objective mathematical programming. Math Program 87: 543–560.
Sayin S (2003). A procedure to find discrete representations of the efficient set with specified coverage errors. Oper Res 51: 427–436.
Steuer RE (1986). Multiple Criteria Optimization: Theory, Computation and Application. John Wiley: New York p. 245.
Steuer RE and Harris FW (1980). Intra-set point generation and filtering in decision and criterion space. Comput Opns Res 7: 41–53.
White DJ (1996). Rectilinear location revisited. J Opl Res Soc 47: 181–187.
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Nadirler, D., Karasakal, E. Mixed integer programming-based solution procedure for single-facility location with maximin of rectilinear distance. J Oper Res Soc 59, 563–570 (2008). https://doi.org/10.1057/palgrave.jors.2602372
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DOI: https://doi.org/10.1057/palgrave.jors.2602372