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Two Combinatorial Algorithms for the Constrained Assignment Problem with Bounds and Penalties

Author

Listed:
  • Guojun Hu

    (School of Mathematics and Statistics, Yunnan University, Kunming 650504, China)

  • Junran Lichen

    (School of Mathematics and Physics, Beijing University of Chemical Technology, No. 15, North Third Ring East Road, Beijing 100190, China)

  • Pengxiang Pan

    (School of Mathematics and Statistics, Yunnan University, Kunming 650504, China)

Abstract

In the paper, we consider a generalization of the classical assignment problem, which is called the constrained assignment problem with bounds and penalties (CA-BP). Specifically, given a set of machines and a set of independent jobs, each machine has a lower and upper bound on the number of jobs that can be executed, and each job must be either executed on some machine with a given processing time or rejected with a penalty that we must pay for. No job can be executed on more than one machine. We aim to find an assignment scheme for these jobs that satisfies the constraints mentioned above. The objective is to minimize the total processing time of executed jobs as well as the penalties from rejected jobs. The CA-BP is related to some practical applications such as edge computing , which involves selecting tasks and processing them on the edge servers of an internet network. As a result, a motivation of this study is to improve the efficiency of internet networks by limiting the lower bound of the number of objects processed by each edge server. Our main contribution is modifying the previous network flow algorithms to satisfy the lower capacity constraints, for which we design two exact combinatorial algorithms to solve the CA-BP. Our methodologies and results bring novel perspectives into other research areas related to the assignment problem.

Suggested Citation

  • Guojun Hu & Junran Lichen & Pengxiang Pan, 2023. "Two Combinatorial Algorithms for the Constrained Assignment Problem with Bounds and Penalties," Mathematics, MDPI, vol. 11(24), pages 1-12, December.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:24:p:4883-:d:1294753
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    References listed on IDEAS

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    1. Yongwen Hu & Qunpo Liu, 2021. "A Network Flow Algorithm for Solving Generalized Assignment Problem," Mathematical Problems in Engineering, Hindawi, vol. 2021, pages 1-8, January.
    2. Hongyu He & Yanzhi Zhao & Xiaojun Ma & Yuan-Yuan Lu & Na Ren & Ji-Bo Wang, 2023. "Study on Scheduling Problems with Learning Effects and Past Sequence Delivery Times," Mathematics, MDPI, vol. 11(19), pages 1-19, September.
    3. Zong-Jun Wei & Li-Yan Wang & Lei Zhang & Ji-Bo Wang & Ershen Wang, 2023. "Single-Machine Maintenance Activity Scheduling with Convex Resource Constraints and Learning Effects," Mathematics, MDPI, vol. 11(16), pages 1-21, August.
    4. H. W. Kuhn, 1955. "The Hungarian method for the assignment problem," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 2(1‐2), pages 83-97, March.
    5. Pentico, David W., 2007. "Assignment problems: A golden anniversary survey," European Journal of Operational Research, Elsevier, vol. 176(2), pages 774-793, January.
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