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Stochastic leadtimes in continuous‐time inventory models

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  • Paul Zipkin

Abstract

This paper shows that one of the fundamental results of inventory theory is valid under conditions much broader than those treated previously. The result characterizes the distributions of inventory level and inventory position in the standard, continuous‐time model with backorders, and leads to the relatively easy calculation of key performance measures. We treat both fixed and random leadtimes, and we examine both stationary and limiting distributions under different assumptions. We consider demand processes described by several general classes of compound‐counting processes and a variety of order policies. For the stochastic‐leadtime case we provide the first explicit proof of the result, assuming the leadtimes are generated according to a specific, but plausible, scenario.

Suggested Citation

  • Paul Zipkin, 1986. "Stochastic leadtimes in continuous‐time inventory models," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 33(4), pages 763-774, November.
    • Handle: RePEc:wly:navlog:v:33:y:1986:i:4:p:763-774
    DOI: 10.1002/nav.3800330419
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    Cited by:

    1. Tao Lu & Jan C. Fransoo & Chung-Yee Lee, 2017. "Carrier Portfolio Management for Shipping Seasonal Products," Operations Research, INFORMS, vol. 65(5), pages 1250-1266, October.
    2. Ruud Heuts & Jan de Klein, 1995. "An (s, q) inventory model with stochastic and interrelated lead times," Naval Research Logistics (NRL), John Wiley & Sons, vol. 42(5), pages 839-859, August.
    3. Alain Bensoussan & Lama Moussawi-Haidar & Metin Çakanyıldırım, 2010. "Inventory control with an order-time constraint: optimality, uniqueness and significance," Annals of Operations Research, Springer, vol. 181(1), pages 603-640, December.
    4. Daniela Favaretto & Alessandro Marin & Marco Tolotti, 2023. "A theoretical validation of the DDMRP reorder policy," Computational Management Science, Springer, vol. 20(1), pages 1-28, December.
    5. Vipul Agrawal & Sridhar Seshadri, 2000. "Distribution free bounds for service constrained (Q, r) inventory systems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 47(8), pages 635-656, December.
    6. Achin Srivastav & Sunil Agrawal, 2020. "On a single item single stage mixture inventory models with independent stochastic lead times," Operational Research, Springer, vol. 20(4), pages 2189-2227, December.
    7. Ben-Ammar, Oussama & Bettayeb, Belgacem & Dolgui, Alexandre, 2019. "Optimization of multi-period supply planning under stochastic lead times and a dynamic demand," International Journal of Production Economics, Elsevier, vol. 218(C), pages 106-117.
    8. Johansen, Søren Glud, 2021. "The Markov model for base-stock control of an inventory system with Poisson demand, non-crossing lead times and lost sales," International Journal of Production Economics, Elsevier, vol. 231(C).
    9. Y. Barron, 2019. "A state-dependent perishability (s, S) inventory model with random batch demands," Annals of Operations Research, Springer, vol. 280(1), pages 65-98, September.
    10. Michna, Zbigniew & Disney, Stephen M. & Nielsen, Peter, 2020. "The impact of stochastic lead times on the bullwhip effect under correlated demand and moving average forecasts," Omega, Elsevier, vol. 93(C).
    11. Paul Zipkin, 1988. "The use of phase‐type distributions in inventory‐control models," Naval Research Logistics (NRL), John Wiley & Sons, vol. 35(2), pages 247-257, April.
    12. Paul Zipkin, 1991. "Evaluation of base‐stock policies in multiechelon inventory systems with compound‐poisson demands," Naval Research Logistics (NRL), John Wiley & Sons, vol. 38(3), pages 397-412, June.
    13. Yu‐Sheng Zheng & Fangruo Chen, 1992. "Inventory policies with quantized ordering," Naval Research Logistics (NRL), John Wiley & Sons, vol. 39(3), pages 285-305, April.
    14. Riezebos, Jan & Zhu, Stuart X., 2020. "Inventory control with seasonality of lead times," Omega, Elsevier, vol. 92(C).
    15. Tamer Boyacı & Guillermo Gallego, 2002. "Managing waiting times of backordered demands in single‐stage (Q, r) inventory systems," Naval Research Logistics (NRL), John Wiley & Sons, vol. 49(6), pages 557-573, September.
    16. Thomas Wensing & Heinrich Kuhn, 2015. "Analysis of production and inventory systems when orders may cross over," Annals of Operations Research, Springer, vol. 231(1), pages 265-281, August.
    17. Jing‐Sheng Song & Paul H. Zipkin, 1992. "Evaluation of base‐stock policies in multiechelon inventory systems with state‐dependent demands part I: State‐independent policies," Naval Research Logistics (NRL), John Wiley & Sons, vol. 39(5), pages 715-728, August.
    18. F. G. Badía & C. Sangüesa, 2015. "Inventory models with nonlinear shortage costs and stochastic lead times; applications of shape properties of randomly stopped counting processes," Naval Research Logistics (NRL), John Wiley & Sons, vol. 62(5), pages 345-356, August.
    19. Yonit Barron & Opher Baron, 2020. "The residual time approach for (Q, r) model under perishability, general lead times, and lost sales," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(3), pages 601-648, December.
    20. Marcus Ang & Karl Sigman & Jing-Sheng Song & Hanqin Zhang, 2017. "Closed-Form Approximations for Optimal ( r , q ) and ( S , T ) Policies in a Parallel Processing Environment," Operations Research, INFORMS, vol. 65(5), pages 1414-1428, October.

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