Abstract
Consider a region on a plane with a set of points with positive weights and rectangles that have to be place in that region without intersections. Either the maximal sum of weights of the points in rectangles or the total sum must be minimal. We consider the case of two rectangles. The original continuous problem is reduced to a discrete one by introducing equivalence classes. We propose polynomial combinatorial algorithms for solving the problem. We conduct a computational experiment to compare the efficiency of developed algorithms with the IBM ILOG CPLEX suite with an integer programming model.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Davydov, I.A., Kochetov, Yu.A., Mladenovich, N., and D. Urosevic, Fast Metaheuristics for the Discrete (r|p)-Centroid Problem, Autom. Remote Control, 2014, vol. 75, no. 4, pp. 677–687.
Zabudskii, G.G., Model Building and Location Problem Solving in a Plane with Forbidden Gaps, Autom. Remote Control, 2006, vol. 67, no. 12, pp. 1986–1990.
Zabudskii, G.G. and Amzin, I.V., Search Region Contraction of the Weber Problem Solution on the Plane with Rectangular Forbidden Zones, Autom. Remote Control, 2012, vol. 73, no. 5, pp. 821–830.
Zabudskii, G.G. and Koval’, A.A., Solving a Maximin Location Problem on the Plane with Given Accuracy, Autom. Remote Control, 2014, vol. 75, no. 7, pp. 1221–1230.
Kolokolov, A.A., Levanova, T.V., and Pozdnyakov, Yu.S., Algorithms of Artificial Immune System for a Variant Placement Problem of Telecommunication Centers, Izv. Irkut. Gos. Univ., Mat., 2013, no. 1, pp. 35–44.
Kochetov, Yu.A., Pashchenko, M.G., and Plyasunov, A.V., On the Complexity of Local Search in the p-Median Problem, Diskret. Anal. Issled. Oper., 2005, ser. 2, vol. 12, no. 2, pp. 44–71.
Panyukov, A.V. and Shangin, R.E., An Exact Algorithm for the Discrete Weber’s Problem for a k-Tree, Diskret. Anal. Issled. Oper., 2014, no. 3, pp. 64–75.
Kartak, V.M., Mesyagutov, M.A., Mukhacheva, E.A., and A. S. Filippova, Local Search of Orthogonal Packings Using the Lower Bounds, Autom. Remote Control, 2009, vol. 70, no. 6, pp. 1054–1066.
Farahani, R.Z. and Hekmatfar, M., Facility Location: Concepts, Models, Algorithms and Case Studies, Heidelberg: Physica-Verlag, 2009.
Tamir, A., Locating Two Obnoxious Facilities Using the Weighted Maximin Criteriom, Oper. Res. Lett., 2006, vol. 34, no. 1. pp. 97–105.
Astrakov, S.N., Erzin, A.I., and Zalyubovskii, V.V., Sensor Networks and Covering a Plane with Circles, Diskret. Anal. Issled. Oper., 2009, vol. 16, no. 3, pp. 3–19.
Drezner, Z. and Wesolowsky, G.O., Finding the Circle or Rectangle Containing the Minimum Weight of Points, Location Sci., 1994, vol. 2, no. 2, pp. 83–90.
Zabudskii, G.G. and Burlakov, Yu.A., Optimal Placement of a Dangerous Object on a Plane with Regard to Various Influence Zones, Omsk. Nauch. Vest., 2011, no. 3 (103), pp. 18–22.
Zabudskii, G.G., Keiner, T.I., and Koval’, A.A., Minimax and Maximin Placement Problems for Dangerous Objects on a Plane, Proc. Russ. Conf. Statistics, Modeling, Optimization, Chelyabinks: Yuzhn. Ural. Gos. Univ., 2011, pp. 117–120.
Author information
Authors and Affiliations
Corresponding author
Additional information
Original Russian Text © G.G. Zabudskii, T.I. Keiner, 2017, published in Avtomatika i Telemekhanika, 2017, No. 9, pp. 131–144.
Rights and permissions
About this article
Cite this article
Zabudskii, G.G., Keiner, T.I. Optimal placement of rectangles on a plane with fixed objects. Autom Remote Control 78, 1651–1661 (2017). https://doi.org/10.1134/S0005117917090090
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117917090090