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An Inventory Model for Arbitrary Interval and Quantity Distributions of Demand

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  • Martin Beckmann

    (Department of Economics, Brown University)

Abstract

The inventory problem for continuous time is studied under the following assumptions about the demand process (1) an arbitrary distribution of the length of intervals between successive demands; (2) a distribution of the quantity demanded which is independent of the last quantity demanded and any previous events but may depend on the time elapsed since the last demand; (3) unfilled orders are backlogged. The delivery time is fixed. Costs considered are fixed ordering costs and proportional costs of purchase, storage and shortage. The loss function and the equations for reordering point and minimal ordering quantity are derived. Formulae are calculated for the Poisson, stuttering Poisson, geometric, negative binomial, Gamma and compounded distributions.

Suggested Citation

  • Martin Beckmann, 1961. "An Inventory Model for Arbitrary Interval and Quantity Distributions of Demand," Management Science, INFORMS, vol. 8(1), pages 35-57, October.
    • Handle: RePEc:inm:ormnsc:v:8:y:1961:i:1:p:35-57
    DOI: 10.1287/mnsc.8.1.35
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    Cited by:

    1. Girlich, Hans-Joachim & Chikan, Attila, 2001. "The origins of dynamic inventory modelling under uncertainty: (the men, their work and connection with the Stanford Studies)," International Journal of Production Economics, Elsevier, vol. 71(1-3), pages 351-363, May.
    2. Larsen, Christian & Kiesmüller, Gudrun P., 2006. "Developing a closed-form cost expression for an (R,s,nQ) policy where the demand process is compound generalized Erlang," CORAL Working Papers L-2006-09, University of Aarhus, Aarhus School of Business, Department of Business Studies.
    3. Joseph B. Mazzola & William F. McCoy & Harvey M. Wagner, 1987. "Algorithms and heuristics for variable‐yield lot sizing," Naval Research Logistics (NRL), John Wiley & Sons, vol. 34(1), pages 67-86, February.
    4. Kaj Rosling, 2002. "Inventory Cost Rate Functions with Nonlinear Shortage Costs," Operations Research, INFORMS, vol. 50(6), pages 1007-1017, December.
    5. E. P. Chew & L. A. Johnson, 1995. "Service levels in distribution systems with random customer order size," Naval Research Logistics (NRL), John Wiley & Sons, vol. 42(1), pages 39-56, February.
    6. Chew, Ek Peng & Tang, Loon Ching, 1995. "Warehouse-retailer system with stochastic demands -- Non-identical retailer case," European Journal of Operational Research, Elsevier, vol. 82(1), pages 98-110, April.
    7. Johansen, Soren Glud & Thorstenson, Anders, 1996. "Optimal (r, Q) inventory policies with Poisson demands and lost sales: discounted and undiscounted cases," International Journal of Production Economics, Elsevier, vol. 46(1), pages 359-371, December.
    8. Mekhtiev, Mirza Arif, 2013. "Analytical evaluation of lead-time demand in polytree supply chains with uncertain demand, lead-time and inter-demand time," International Journal of Production Economics, Elsevier, vol. 145(1), pages 304-317.
    9. Sandun C. Perera & Suresh P. Sethi, 2023. "A survey of stochastic inventory models with fixed costs: Optimality of (s, S) and (s, S)‐type policies—Continuous‐time case," Production and Operations Management, Production and Operations Management Society, vol. 32(1), pages 154-169, January.

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