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Logistics scheduling: Analysis of two-stage problems


This paper studies the coordination effects between stages for scheduling problems where decision-making is a two-stage process. Two stages are considered as one system. The system can be a supply chain that links two stages, one stage representing a manufacturer; and the other, a distributor. It also can represent a single manufacturer, while each stage represents a different department responsible for a part of operations. A problem that jointly considers both stages in order to achieve ideal overall system performance is defined as a system problem. In practice, at times, it might not be feasible for the two stages to make coordinated decisions due to (i) the lack of channels that allow decision makers at the two stages to cooperate, and/or (ii) the optimal solution to the system problem is too difficult (or costly) to achieve.

Two practical approaches are applied to solve a variant of two-stage logistic scheduling problems. The Forward Approach is defined as a solution procedure by which the first stage of the system problem is solved first, followed by the second stage. Similarly, the Backward Approach is defined as a solution procedure by which the second stage of the system problem is solved prior to solving the first stage. In each approach, two stages are solved sequentially and the solution generated is treated as a heuristic solution with respect to the corresponding system problem. When decision makers at two stages make decisions locally without considering consequences to the entire system, ineffectiveness may result — even when each stage optimally solves its own problem. The trade-off between the time complexity and the solution quality is the main concern. This paper provides the worst-case performance analysis for each approach.

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  1. Ahmadi, J. H., R. H. Ahmadi, S. Dasu, C. S. Tang, “Batching and scheduling jobs on batch and discrete processors”, Operations Research, Vol.39, pp750–763, 1992.

    MathSciNet  Google Scholar 

  2. Chang, Y.-C., C.-Y. Lee, “Machine scheduling with job delivery coordination”, To appear in European Journal of Operational Research, 2002.

  3. Chen, Z.-L., “Scheduling and common due date assignment with earliness-tardiness penalties and batch delivery costs”, European Journal of Operational Research, Vol. 93, pp49–60, 1996.

    Article  MATH  Google Scholar 

  4. Cheng, T.C.E., V. S. Gordon, M. Y. Kovalyov, “Single machine scheduling with batch deliveries”, European Journal of Operational Research, Vol. 94, pp277–283, 1996.

    Article  MATH  Google Scholar 

  5. Conway, R. W., W. L. Maxwell, L. W. Miller, Theory of Scheduling, Addison-Wesley, Reading, MA, 1967.

    MATH  Google Scholar 

  6. Garey, M.R., D. S. Johnson, and R. Sethi, “The complexity of flowshop and jobshop scheduling”, Mathematics of Operations Research, Vol. 1, pp117–129, 1976.

    MathSciNet  MATH  Google Scholar 

  7. Gonzalez, T., S. Sahni, “Flowshop and jobshop schedules: complexity and approximation”, Operations Research, Vol. 26, pp36–52, 1978.

    MathSciNet  MATH  Article  Google Scholar 

  8. Graham, R. L., E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, “Optimization and approximation in deterministic sequencing and scheduling: a survey”, Annals of Discrete Mathematics, Vol. 5, pp287–326, 1979.

    MathSciNet  Article  MATH  Google Scholar 

  9. Hall, N. G., M. A. Lesaoana, C. N. Potts, “Scheduling with fixed delivery dates”, Operations Research, Vol. 49, pp134–144, 2001.

    Article  MathSciNet  Google Scholar 

  10. Hall, N. G., C. N. Potts, “Supply chain scheduling: batching and delivery”, Operations Research, Vol. 51, pp566–584, 2003.

    Article  MathSciNet  Google Scholar 

  11. Herrmann, J. W., C.-Y. Lee, “On scheduling to minimize earliness-tardiness and batch delivery costs with a common due date”, European Journal of Operational Research, Vol. 70, pp272–288, 1993.

    Article  MATH  Google Scholar 

  12. Hoogeveen, J. A., T. Kawaguchi, “Minimizing total completion time in a two-machine flowshop: analysis of special cases”, Mathematics of Operations Research, Vol. 24, pp887–910, 1999.

    MathSciNet  MATH  Google Scholar 

  13. Hurink, J., S. Knust, “Makespan minimization for flow-shop problems with transportation times”, Discrete Applied Mathematics, Vol. 112, pp199–216, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  14. Ignall, E., L. Schrage, “Application of the branch and bound technique for some flow-shop scheduling problems”, Operations Research, Vol. 13, pp400–412, 1965.

    MathSciNet  Google Scholar 

  15. Jackson, J. R, “Scheduling a production line to minimize maximum tardiness”, Research report 43. Management Science Research Project, University of California, Los Angeles, CA, 1955.

    Google Scholar 

  16. Johnson, S. M., “Optimal two-and three-stage production schedules with setup times included”, Naval Research Logistic Quarterly, Vol. 1, pp61–68, 1954.

    Google Scholar 

  17. Lee, C.-Y., Z.-L. Chen, “Machine scheduling with transportation considerations”, Journal of Scheduling, Vol. 4, pp3–24, 2001.

    Article  MathSciNet  MATH  Google Scholar 

  18. Lenstra, J. K., A. H. G. Rinnooy Kan, P. Brucker, “Complexity of machine scheduling problems”, Annals of Discrete Mathematics, Vol. 1, pp343–362, 1977.

    MathSciNet  Article  Google Scholar 

  19. Potts, C. N., M. Y. Kovalyov, “Scheduling with batching: a review”, European Journal of Operational Research, Vol. 120, pp228–249, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  20. Potts, C. N., L. N. Van Wassenhove, “Integrating scheduling with batching and lot-sizing: a review of algorithms and complexity”, Journal of the Operational Research Society, Vol. 43, pp395–406, 1992.

    MATH  Google Scholar 

  21. Sarmiento, A. M., R. Nagi, “A review of integrated analysis of production-distribution systems”, IIE Transactions, Vol. 31, pp1061–1074, 1999.

    Article  Google Scholar 

  22. Schrage, L. E., “A proof of the optimality of the shortest remaining processing time discipline”, Operations Research, Vol. 16, pp687–690, 1968.

    MATH  Google Scholar 

  23. Smith, W. E., “Various Optimizers for Single-stage Production”, Naval Research Logistics Quarterly, Vol. 3, pp59–66, 1956.

    MathSciNet  Google Scholar 

  24. Thomas, D. J., P. M. Griffin, “Coordinated supply chain management”, European Journal of Operational Research, Vol. 94, pp1–15, 1996.

    Article  MATH  Google Scholar 

  25. Webster, S., K. R. Baker, “Scheduling groups of jobs on a single machine”, Operations Research, Vol. 43, pp692–703, 1995.

    MathSciNet  MATH  Google Scholar 

  26. Van de Velde, S. L., “Minimizing the sum of the job completion times in the two-machine flow shop by Lagrangian relaxation”, Annals of Operations Research, Vol. 26, pp257–268, 1990.

    MATH  MathSciNet  Google Scholar 

  27. Yang, X., “Scheduling with generalized batch delivery dates and earliness penalties”, IIE Transactions, Vol. 32, pp735–741, 2000.

    Article  Google Scholar 

  28. Yuan, J., “A note on the complexity of single-machine scheduling with a common due date, earliness-tardiness, and batch delivery costs”, European Journal of Operational Research, Vol. 94, pp203–205, 1996.

    Article  MATH  Google Scholar 

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This research is supported in part by Hong Kong RGC Grant HKUST 6010/02E.

Yung-Chia Chang received her Ph. D. in Industrial Engineering from Texas A&M University in 2001. She is currently the President of Davicom America Corp., the US division of Davicom Semiconductor, Inc., a fabless IC design house headquartered in Taiwan. She continues to be interested in the area of production scheduling.

Chung-Yee Lee is Head and Professor of the Industrial Engineering and Engineering Management Department at the Hong Kong University of Science & Technology (HKUST). He is also the Founding Director of Logistics and Supply Chain Management Institute at HKUST. He was Rockwell Professor in the Department of Industrial Engineering at Texas A&M University from 1996 to 2001. Before joining Texas A&M University he was a faculty member in the Department of Industrial and Systems Engineering at the University of Florida. He worked as a plant manager and also had few years consulting experience in Taiwan. During the 20 years in the U.S. he has engaged in numerous research projects sponsored by NSF, IBM, Motorola, AT&T Paradyne, Harris Semiconductor, Northern Telecom, and Martin Marietta. His research areas are Logistics and Supply Chain Management, and Production Scheduling. He was the Editor for IIE Transactions on Scheduling and Logistics in 1997–2000. Currently, he is serving in several editorial board. He has published more than 90 papers in refereed journals. He received his Bachelor degree in Electronic Engineering (1972) and a Master degree in Management Sciences (MBA Program) (1976) from National Chiao-Tung University in Taiwan. He also received a Master degree in Industrial Engineering from Northwestern University in 1980 and his Ph.D. degree in Operations Research from Yale University in 1984.

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Chang, YC., Lee, CY. Logistics scheduling: Analysis of two-stage problems. J. Syst. Sci. Syst. Eng. 12, 385–407 (2003).

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  • Logistics scheduling
  • worst case analysis
  • dynamic programming