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Infinite Horizon Production Planning in Time-Varying Systems with Convex Production and Inventory Costs

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  • Robert L. Smith

    (Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, Michigan 48109)

  • Rachel Q. Zhang

    (Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, Michigan 48109)

Abstract

We consider the planning of production over the infinite horizon in a system with time-varying convex production and inventory holding costs. This production lot size problem is frequently faced in industry where a forecast of future demand must be made and production is to be scheduled based on the forecast. Because forecasts of the future are costly and difficult to validate, a firm would like to minimize the number of periods into the future it needs to forecast in order to make an optimal production decision today. In this paper, we first prove that under very general conditions finite horizon versions of the problem exist that lead to an optimal production level at any decision epoch. In particular, we show it suffices for the first period infinite horizon production decision to solve for a horizon that exceeds the longest time interval over which it can prove profitable to carry inventory. We then develop a closed-form expression for computing such a horizon and provide a simple finite algorithm to recursively compute an infinite horizon optimal production schedule.

Suggested Citation

  • Robert L. Smith & Rachel Q. Zhang, 1998. "Infinite Horizon Production Planning in Time-Varying Systems with Convex Production and Inventory Costs," Management Science, INFORMS, vol. 44(9), pages 1313-1320, September.
    • Handle: RePEc:inm:ormnsc:v:44:y:1998:i:9:p:1313-1320
    DOI: 10.1287/mnsc.44.9.1313
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    References listed on IDEAS

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    1. Ford W. Harris, 1990. "How Many Parts to Make at Once," Operations Research, INFORMS, vol. 38(6), pages 947-950, December.
    2. Howard C. Kunreuther & Thomas E. Morton, 1973. "Planning Horizons for Production Smoothing with Deterministic Demands," Management Science, INFORMS, vol. 20(1), pages 110-125, September.
    3. Arthur F. Veinott, Jr., 1964. "Production Planning with Convex Costs: A Parametric Study," Management Science, INFORMS, vol. 10(3), pages 441-460, April.
    4. Chand, Suresh, 1982. "Lot sizing for products with finite demand horizon and periodic review inventory policy," European Journal of Operational Research, Elsevier, vol. 11(2), pages 145-148, October.
    5. Rolf A. Lundin & Thomas E. Morton, 1975. "Planning Horizons for the Dynamic Lot Size Model: Zabel vs. Protective Procedures and Computational Results," Operations Research, INFORMS, vol. 23(4), pages 711-734, August.
    6. Dwight R. Lee & Daniel Orr, 1977. "Further Results on Planning Horizons in the Production Smoothing Problem," Management Science, INFORMS, vol. 23(5), pages 490-498, January.
    7. James C. Bean & Robert L. Smith, 1984. "Conditions for the Existence of Planning Horizons," Mathematics of Operations Research, INFORMS, vol. 9(3), pages 391-401, August.
    8. Joseph D. Blackburn & Howard Kunreuther, 1974. "Planning Horizons for the Dynamic Lot Size Model with Backlogging," Management Science, INFORMS, vol. 21(3), pages 251-255, November.
    9. Irwin E. Schochetman & Robert L. Smith, 1989. "Infinite Horizon Optimization," Mathematics of Operations Research, INFORMS, vol. 14(3), pages 559-574, August.
    10. Harvey M. Wagner & Thomson M. Whitin, 1958. "Dynamic Version of the Economic Lot Size Model," Management Science, INFORMS, vol. 5(1), pages 89-96, October.
    11. Gerald L. Thompson & Suresh P. Sethi, 1980. "Turnpike Horizons for Production Planning," Management Science, INFORMS, vol. 26(3), pages 229-241, March.
    12. Gary D. Eppen & F. J. Gould & B. Peter Pashigian, 1969. "Extensions of the Planning Horizon Theorem in the Dynamic Lot Size Model," Management Science, INFORMS, vol. 15(5), pages 268-277, January.
    13. Howard C. Kunreuther & Thomas E. Morton, 1974. "General Planning Horizons for Production Smoothing with Deterministic Demands," Management Science, INFORMS, vol. 20(7), pages 1037-1046, March.
    14. Edward Zabel, 1964. "Some Generalizations of an Inventory Planning Horizon Theorem," Management Science, INFORMS, vol. 10(3), pages 465-471, April.
    15. Thomas E. Morton, 1978. "The Nonstationary Infinite Horizon Inventory Problem," Management Science, INFORMS, vol. 24(14), pages 1474-1482, October.
    16. Joseph Thomas, 1970. "Price-Production Decisions with Deterministic Demand," Management Science, INFORMS, vol. 16(11), pages 747-750, July.
    17. Awi Federgruen & Michal Tzur, 1991. "A Simple Forward Algorithm to Solve General Dynamic Lot Sizing Models with n Periods in 0(n log n) or 0(n) Time," Management Science, INFORMS, vol. 37(8), pages 909-925, August.
    18. C. Bes & S. P. Sethi, 1988. "Concepts of Forecast and Decision Horizons: Applications to Dynamic Stochastic Optimization Problems," Mathematics of Operations Research, INFORMS, vol. 13(2), pages 295-310, May.
    19. Suresh Chand & Suresh P. Sethi & Jean-Marie Proth, 1990. "Existence of Forecast Horizons in Undiscounted Discrete-Time Lot Size Models," Operations Research, INFORMS, vol. 38(5), pages 884-892, October.
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    Cited by:

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    2. Huang, Kai & Ahmed, Shabbir, 2010. "A stochastic programming approach for planning horizons of infinite horizon capacity planning problems," European Journal of Operational Research, Elsevier, vol. 200(1), pages 74-84, January.
    3. A Diponegoro & B R Sarker, 2007. "Operations policy for a supply chain system with fixed-interval delivery and linear demand," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 58(7), pages 901-910, July.
    4. Konur, Dinçer & Geunes, Joseph, 2016. "Supplier wholesale pricing for a retail chain: Implications of centralized vs. decentralized retailing and procurement under quantity competition," Omega, Elsevier, vol. 65(C), pages 98-110.
    5. Torpong Cheevaprawatdomrong & Robert L. Smith, 2004. "Infinite Horizon Production Scheduling in Time-Varying Systems Under Stochastic Demand," Operations Research, INFORMS, vol. 52(1), pages 105-115, February.
    6. Archis Ghate & Robert L. Smith, 2009. "Optimal Backlogging Over an Infinite Horizon Under Time-Varying Convex Production and Inventory Costs," Manufacturing & Service Operations Management, INFORMS, vol. 11(2), pages 362-368, June.
    7. Archis Ghate & Robert L. Smith, 2013. "A Linear Programming Approach to Nonstationary Infinite-Horizon Markov Decision Processes," Operations Research, INFORMS, vol. 61(2), pages 413-425, April.
    8. Erica L. Plambeck, 2013. "OM Forum —Operations Management Challenges for Some “Cleantech” Firms," Manufacturing & Service Operations Management, INFORMS, vol. 15(4), pages 527-536, October.
    9. Herbon, Avi & Khmelnitsky, Eugene & Maimon, Oded, 2004. "Effective information horizon length in measuring off-line performance of stochastic dynamic systems," European Journal of Operational Research, Elsevier, vol. 157(3), pages 688-703, September.
    10. Jing, Fuying & Chao, Xiangrui, 2022. "Forecast horizons for a two-echelon dynamic lot-sizing problem," Omega, Elsevier, vol. 110(C).
    11. Li, Xiang & Li, Yongjian & Cai, Xiaoqiang, 2013. "Double marginalization and coordination in the supply chain with uncertain supply," European Journal of Operational Research, Elsevier, vol. 226(2), pages 228-236.
    12. Suresh Chand & Vernon Ning Hsu & Suresh Sethi, 2002. "Forecast, Solution, and Rolling Horizons in Operations Management Problems: A Classified Bibliography," Manufacturing & Service Operations Management, INFORMS, vol. 4(1), pages 25-43, September.

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