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Planar Maximum Coverage Location Problem with Partial Coverage and Rectangular Demand and Service Zones

Author

Listed:
  • Manish Bansal

    (Grado Department of Industrial and Systems Engineering, Virginia Tech, Blacksburg, Virginia 24061)

  • Kiavash Kianfar

    (Department of Industrial and Systems Engineering, Texas A&M University, College Station, Texas 77840)

Abstract

We study the planar maximum coverage location problem (MCLP) with rectilinear distance and rectangular demand zones in the case where “partial coverage” is allowed in its true sense, i.e., when covering part of a demand zone is allowed and the coverage accrued as a result of this is proportional to the demand of the covered part only. We pose the problem in a slightly more general form by allowing service zones to be rectangular instead of squares, thereby addressing applications in camera view-frame selection as well. More specifically, our problem, referred to as PMCLP-PCR (planar MCLP with partial coverage and rectangular demand and service zones), is to position a given number of rectangular service zones (SZs) on the two-dimensional plane to (partially) cover a set of existing (possibly overlapping) rectangular demand zones (DZs) such that the total covered demand is maximized. Previous studies on (planar) MCLP have assumed binary coverage, even when nonpoint objects such as lines or polygons have been used to represent demand. Under the binary coverage assumption, the problem can be readily formulated and solved as a binary linear program; whereas, partial coverage, although much more realistic, cannot be efficiently handled by binary linear programming, making PMCLP-PCR much more challenging to solve. In this paper, we first prove that PMCLP-PCR is NP-hard if the number of SZs is part of the input. We then present an improved algorithm for the single-SZ PMCLP-PCR, which is at least two times faster than the existing exact plateau vertex traversal algorithm. Next, we study multi-SZ PMCLP-PCR for the first time and prove theoretical properties that significantly reduce the search space for solving this problem, and we present a customized branch-and-bound exact algorithm to solve it. Our computational experiments show that this algorithm can solve relatively large instances of multi-SZ PMCLP-PCR in a short time. We also propose a fast polynomial time heuristic algorithm. Having optimal solutions from our exact algorithm, we benchmark the quality of solutions obtained from our heuristic algorithm. Our results show that for all the random instances solved to optimality by our exact algorithm, our heuristic algorithm finds a solution in a fraction of a second, where its objective value is at least 91% of the optimal objective in 90% of the instances (and at least 69% of the optimal objective in all the instances).

Suggested Citation

  • Manish Bansal & Kiavash Kianfar, 2017. "Planar Maximum Coverage Location Problem with Partial Coverage and Rectangular Demand and Service Zones," INFORMS Journal on Computing, INFORMS, vol. 29(1), pages 152-169, February.
    • Handle: RePEc:inm:orijoc:v:29:y:2017:i:1:p:152-169
    DOI: 10.1287/ijoc.2016.0722
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    References listed on IDEAS

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    1. Alan T. Murray & Morton E. O'Kelly, 2002. "Assessing representation error in point-based coverage modeling," Journal of Geographical Systems, Springer, vol. 4(2), pages 171-191, June.
    2. Daoqin Tong & Alan T. Murray, 2009. "Maximising coverage of spatial demand for service," Papers in Regional Science, Wiley Blackwell, vol. 88(1), pages 85-97, March.
    3. Watson-Gandy, C. D. T., 1982. "Heuristic procedures for the m-partial cover problem on a plane," European Journal of Operational Research, Elsevier, vol. 11(2), pages 149-157, October.
    4. Drezner, Zvi, 1986. "The p-cover problem," European Journal of Operational Research, Elsevier, vol. 26(2), pages 312-313, August.
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    4. Marianov, Vladimir & Eiselt, H.A., 2024. "Fifty Years of Location Theory - A Selective Review," European Journal of Operational Research, Elsevier, vol. 318(3), pages 701-718.
    5. Baldomero-Naranjo, Marta & Kalcsics, Jörg & Marín, Alfredo & Rodríguez-Chía, Antonio M., 2022. "Upgrading edges in the maximal covering location problem," European Journal of Operational Research, Elsevier, vol. 303(1), pages 14-36.
    6. Chen, Liang & Chen, Sheng-Jie & Chen, Wei-Kun & Dai, Yu-Hong & Quan, Tao & Chen, Juan, 2023. "Efficient presolving methods for solving maximal covering and partial set covering location problems," European Journal of Operational Research, Elsevier, vol. 311(1), pages 73-87.
    7. Hu, Xiaoxuan & Zhu, Waiming & Ma, Huawei & An, Bo & Zhi, Yanling & Wu, Yi, 2021. "Orientational variable-length strip covering problem: A branch-and-price-based algorithm," European Journal of Operational Research, Elsevier, vol. 289(1), pages 254-269.

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