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Consecutive Optimizers for a Partitioning Problem with Applications to Optimal Inventory Groupings for Joint Replenishment

Author

Listed:
  • A. K. Chakravarty

    (University of Wisconsin, Milwaukee, Wisconsin)

  • J. B. Orlin

    (Massachusetts Institute of Technology, Cambridge, Massachusetts)

  • U. G. Rothblum

    (Technion, Israel Institute of Technology, Haifa, Israel)

Abstract

We consider several subclasses of the problem of grouping n items (indexed 1, 2, …, n ) into m subsets so as to minimize the function g ( S 1 , …, S m ). In general, these problems are very difficult to solve to optimality, even for the case m = 2. We provide several sufficient conditions on g (·) that guarantee that there is an optimum partition in which each subset consists of consecutive integers (or else the partition S 1 , …, S m satisfies a more general condition called “semiconsecutiveness”). Moreover, by restricting attention to “consecutive” (or “semiconsecutive”) partitions, we can solve the partition problem in polynomial time for small values of m . If, in addition, g is symmetric, then the partition problem is solvable in purely polynomial time. We apply these results to generalizations of a problem in inventory groupings considered by the authors in a previous paper. We also relate the results to the Neyman-Pearson lemma in statistical hypothesis testing and to a graph partitioning problem of Barnes and Hoffman.

Suggested Citation

  • A. K. Chakravarty & J. B. Orlin & U. G. Rothblum, 1985. "Consecutive Optimizers for a Partitioning Problem with Applications to Optimal Inventory Groupings for Joint Replenishment," Operations Research, INFORMS, vol. 33(4), pages 820-834, August.
    • Handle: RePEc:inm:oropre:v:33:y:1985:i:4:p:820-834
    DOI: 10.1287/opre.33.4.820
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    Citations

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    Cited by:

    1. Jia Shu & Chung-Piaw Teo & Zuo-Jun Max Shen, 2005. "Stochastic Transportation-Inventory Network Design Problem," Operations Research, INFORMS, vol. 53(1), pages 48-60, February.
    2. Max Shen, Zuo-Jun & Qi, Lian, 2007. "Incorporating inventory and routing costs in strategic location models," European Journal of Operational Research, Elsevier, vol. 179(2), pages 372-389, June.
    3. Braouezec, Yann, 2016. "On the welfare effects of regulating the number of discriminatory prices," Research in Economics, Elsevier, vol. 70(4), pages 588-607.
    4. Frank K. Hwang & Shmuel Onn & Uriel G. Rothblum, 2000. "Explicit solution of partitioning problems over a 1‐dimensional parameter space," Naval Research Logistics (NRL), John Wiley & Sons, vol. 47(6), pages 531-540, September.
    5. Chung‐Lun Li & Zhi‐Long Chen, 2006. "Bin‐packing problem with concave costs of bin utilization," Naval Research Logistics (NRL), John Wiley & Sons, vol. 53(4), pages 298-308, June.
    6. Amiya K. Chakravarty & G. E. Martin, 1989. "Discount pricing policies for inventories subject to declining demand," Naval Research Logistics (NRL), John Wiley & Sons, vol. 36(1), pages 89-102, February.
    7. Gerard J. Chang & Fu-Loong Chen & Lingling Huang & Frank K. Hwang & Su-Tzu Nuan & Uriel G. Rothblum & I-Fan Sun & Jan-Wen Wang & Hong-Gwa Yeh, 1998. "Sortabilities of Partition Properties," Journal of Combinatorial Optimization, Springer, vol. 2(4), pages 413-427, December.
    8. Yann Braouezec, 2013. "The Welfare Effects of Regulating the Number of Market Segments," Working Papers 2013-ECO-11, IESEG School of Management.
    9. Frank K. Hwang & Uriel G. Rothblum, 2011. "On the number of separable partitions," Journal of Combinatorial Optimization, Springer, vol. 21(4), pages 423-433, May.

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