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The continuous grey pattern problem

Author

Listed:
  • Zvi Drezner

    (California State University-Fullerton)

  • Pawel Kalczynski

    (California State University-Fullerton)

Abstract

A new location problem is formulated and solved. It is the continuous version of the grey pattern problem which is a special case of the Quadratic Assignment Problem. The problem is a minimization of a convex function subject to non-convex constraints and has infinitely many optimal solutions. We propose several mathematical programming formulations that are suitable for a multi-start heuristic algorithm. In addition to solving these formulations by the Solver in Excel and Mathematica, a special Nelder–Mead algorithm is proposed. This special algorithm provided the best results. One suggested modification may improve the performance of the Nelder–Mead algorithm for other optimization problems as well.

Suggested Citation

  • Zvi Drezner & Pawel Kalczynski, 2017. "The continuous grey pattern problem," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 68(5), pages 469-483, May.
    • Handle: RePEc:pal:jorsoc:v:68:y:2017:i:5:d:10.1057_s41274-016-0023-4
    DOI: 10.1057/s41274-016-0023-4
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    References listed on IDEAS

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    1. Jack Elzinga & Donald W. Hearn, 1972. "Geometrical Solutions for Some Minimax Location Problems," Transportation Science, INFORMS, vol. 6(4), pages 379-394, November.
    2. Zvi Drezner & Atsuo Suzuki, 2004. "The Big Triangle Small Triangle Method for the Solution of Nonconvex Facility Location Problems," Operations Research, INFORMS, vol. 52(1), pages 128-135, February.
    3. James Vijay, 1985. "An Algorithm for the p -Center Problem in the Plane," Transportation Science, INFORMS, vol. 19(3), pages 235-245, August.
    4. Zvi Drezner, 2015. "The Quadratic Assignment Problem," Springer Books, in: Gilbert Laporte & Stefan Nickel & Francisco Saldanha da Gama (ed.), Location Science, edition 127, chapter 0, pages 345-363, Springer.
    5. Jack Brimberg & Pierre Hansen & Nenad Mladenović & Eric D. Taillard, 2000. "Improvements and Comparison of Heuristics for Solving the Uncapacitated Multisource Weber Problem," Operations Research, INFORMS, vol. 48(3), pages 444-460, June.
    6. William P. Pierskalla, 1968. "Letter to the Editor—The Multidimensional Assignment Problem," Operations Research, INFORMS, vol. 16(2), pages 422-431, April.
    7. Z Drezner & A Suzuki, 2010. "Covering continuous demand in the plane," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 61(5), pages 878-881, May.
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    Cited by:

    1. Brimberg, Jack & Kalczynski, Pawel & Drezner, Zvi, 2024. "Different formulations of the gray pattern problem and their optimal solutions," European Journal of Operational Research, Elsevier, vol. 316(1), pages 61-71.
    2. Pawel Kalczynski & Zvi Goldstein & Zvi Drezner, 2023. "An Efficient Heuristic for the k-Partitioning Problem," SN Operations Research Forum, Springer, vol. 4(4), pages 1-21, December.

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