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Reduction criteria, upper bounds, and a dynamic programming based heuristic for the max–min $$k_i$$ k i -partitioning problem

Author

Listed:
  • Alexander Lawrinenko

    (Friedrich-Schiller-University Jena)

  • Stefan Schwerdfeger

    (Friedrich-Schiller-University Jena)

  • Rico Walter

    (Friedrich-Schiller-University Jena)

Abstract

This paper addresses the max–min $$k_i$$ k i -partitioning problem that asks for an assignment of n jobs to m parallel machines so that the minimum machine completion time is maximized and the number of jobs on each machine does not exceed a machine-dependent cardinality limit $$k_i$$ k i $$(i=1,\ldots ,m)$$ ( i = 1 , … , m ) . We propose different preprocessing as well as lifting procedures and derive several upper bound arguments. Furthermore, we introduce suited construction heuristics as well as an effective dynamic programming based improvement procedure. Results of a comprehensive computational study on a large set of randomly generated instances indicate that our algorithm quickly finds (near-)optimal solutions.

Suggested Citation

  • Alexander Lawrinenko & Stefan Schwerdfeger & Rico Walter, 2018. "Reduction criteria, upper bounds, and a dynamic programming based heuristic for the max–min $$k_i$$ k i -partitioning problem," Journal of Heuristics, Springer, vol. 24(2), pages 173-203, April.
    • Handle: RePEc:spr:joheur:v:24:y:2018:i:2:d:10.1007_s10732-017-9362-9
    DOI: 10.1007/s10732-017-9362-9
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    References listed on IDEAS

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    1. Walter, Rico & Wirth, Martin & Lawrinenko, Alexander, 2017. "Improved approaches to the exact solution of the machine covering problem," Publications of Darmstadt Technical University, Institute for Business Studies (BWL) 80530, Darmstadt Technical University, Department of Business Administration, Economics and Law, Institute for Business Studies (BWL).
    2. Rico Walter, 2013. "Comparing the minimum completion times of two longest-first scheduling-heuristics," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 21(1), pages 125-139, January.
    3. Shi Ping Chen & Yong He & Guohui Lin, 2002. "3-Partitioning Problems for Maximizing the Minimum Load," Journal of Combinatorial Optimization, Springer, vol. 6(1), pages 67-80, March.
    4. Rico Walter & Martin Wirth & Alexander Lawrinenko, 2017. "Improved approaches to the exact solution of the machine covering problem," Journal of Scheduling, Springer, vol. 20(2), pages 147-164, April.
    5. Luitpold Babel & Hans Kellerer & Vladimir Kotov, 1998. "Thek-partitioning problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 47(1), pages 59-82, February.
    6. Mauro Dell’Amico & Silvano Martello, 1995. "Optimal Scheduling of Tasks on Identical Parallel Processors," INFORMS Journal on Computing, INFORMS, vol. 7(2), pages 191-200, May.
    7. D. K. Friesen & B. L. Deuermeyer, 1981. "Analysis of Greedy Solutions for a Replacement Part Sequencing Problem," Mathematics of Operations Research, INFORMS, vol. 6(1), pages 74-87, February.
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