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Optimal Solution of Scheduling Problems Using Lagrange Multipliers: Part I

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  • Marshall L. Fisher

    (University of Chicago, Chicago, Illinois)

Abstract

This paper presents an algorithm for solving resource-constrained network scheduling problems, a general class of problems that includes the classical job-shop-scheduling problem. It uses Lagrange multipliers to dualize the resource constraints, forming a Lagrangian problem in which the network constraints appear explicitly, while the resource constraints appear only in the Lagrangian function. Because the network constraints do not interact among jobs, the problem of minimizing the Lagrangian decomposes into a subproblem for each job. Algorithms are presented for solving these subproblems. Minimizing the Lagrangian with fixed multiplier values yields a lower bound on the cost of an optimal solution to the scheduling problem. The paper gives procedures for adjusting the multipliers iteratively to obtain strong bounds, and it develops a branch-and-bound algorithm that uses these bounds in the solution of the scheduling problem. Computational experience with this algorithm is discussed.

Suggested Citation

  • Marshall L. Fisher, 1973. "Optimal Solution of Scheduling Problems Using Lagrange Multipliers: Part I," Operations Research, INFORMS, vol. 21(5), pages 1114-1127, October.
    • Handle: RePEc:inm:oropre:v:21:y:1973:i:5:p:1114-1127
    DOI: 10.1287/opre.21.5.1114
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    Cited by:

    1. Claude Lemaréchal, 2007. "The omnipresence of Lagrange," Annals of Operations Research, Springer, vol. 153(1), pages 9-27, September.
    2. Nils Korber & Maximilian Rohrig & Andreas Ulbig, 2022. "A stakeholder-oriented multi-criteria optimization model for decentral multi-energy systems," Papers 2204.06545, arXiv.org.
    3. Klein, Robert & Scholl, Armin, 1999. "Computing lower bounds by destructive improvement: An application to resource-constrained project scheduling," European Journal of Operational Research, Elsevier, vol. 112(2), pages 322-346, January.
    4. Alessandro Hill & Andrea J. Brickey & Italo Cipriano & Marcos Goycoolea & Alexandra Newman, 2022. "Optimization Strategies for Resource-Constrained Project Scheduling Problems in Underground Mining," INFORMS Journal on Computing, INFORMS, vol. 34(6), pages 3042-3058, November.
    5. Marshall L. Fisher, 2004. "The Lagrangian Relaxation Method for Solving Integer Programming Problems," Management Science, INFORMS, vol. 50(12_supple), pages 1861-1871, December.
    6. Gonzalo Muñoz & Daniel Espinoza & Marcos Goycoolea & Eduardo Moreno & Maurice Queyranne & Orlando Rivera Letelier, 2018. "A study of the Bienstock–Zuckerberg algorithm: applications in mining and resource constrained project scheduling," Computational Optimization and Applications, Springer, vol. 69(2), pages 501-534, March.
    7. Alexander Abakumov & Svetlana Pak, 2023. "Role of Photosynthesis Processes in the Dynamics of the Plant Community," Mathematics, MDPI, vol. 11(13), pages 1-24, June.
    8. Norbis, Mario & MacGregor Smith, J., 1996. "An interactive decision support system for the resource Constrained Scheduling Problem," European Journal of Operational Research, Elsevier, vol. 94(1), pages 54-65, October.
    9. P N Ram Kumar & T T Narendran, 2011. "On the usage of Lagrangean Relaxation for the convoy movement problem," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 62(4), pages 722-728, April.
    10. Chatterjee A K & Mukherjee, Saral, 2006. "Unified Concept of Bottleneck," IIMA Working Papers WP2006-05-01, Indian Institute of Management Ahmedabad, Research and Publication Department.
    11. Blazewicz, Jacek & Domschke, Wolfgang & Pesch, Erwin, 1996. "The job shop scheduling problem: Conventional and new solution techniques," European Journal of Operational Research, Elsevier, vol. 93(1), pages 1-33, August.
    12. J. F. Chen & W. E. Wilhelm, 1994. "Optimizing the allocation of components to kits in small‐lot, multiechelon assembly systems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 41(2), pages 229-256, March.
    13. Mukherjee, Saral & Chatterjee, A.K., 2006. "The average shadow price for MILPs with integral resource availability and its relationship to the marginal unit shadow price," European Journal of Operational Research, Elsevier, vol. 169(1), pages 53-64, February.
    14. Jain, A. S. & Meeran, S., 1999. "Deterministic job-shop scheduling: Past, present and future," European Journal of Operational Research, Elsevier, vol. 113(2), pages 390-434, March.
    15. Aristide Mingozzi & Vittorio Maniezzo & Salvatore Ricciardelli & Lucio Bianco, 1998. "An Exact Algorithm for the Resource-Constrained Project Scheduling Problem Based on a New Mathematical Formulation," Management Science, INFORMS, vol. 44(5), pages 714-729, May.

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