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Order-Level Inventory Systems with Independent Stochastic Leadtimes

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  • David Zalkind

    (Department of Health, Education and Welfare)

Abstract

Inventory systems where leadtimes are stochastic and independent, so that orders are not restricted to arrive in the same sequence in which they were placed, can arise if the vendor with whom orders are placed is unreliable in processing them or if orders for the same item are placed with different vendors at different times. We present a procedure for finding the optimal order-level Z (stock on hand plus on order) for a pre-determined scheduling-period t (time interval between decisions about placing orders), that is, a tZ policy. Backordering of shortages is assumed. Most results are derived when both the demand distribution and the leadtime distribution are discrete and bounded. The main concept employed in the derivation is the use of the distribution of the number of outstanding orders to find the distribution of the amount of stock on order. It is then shown that for tZ policies, the solution can be interpreted as the weighted average of constant leadtime systems. Further results that may be helpful in constructing heuristic decision rules are given. The nonoptimality of the sZ policy (i.e., order to Z whenever inventory falls to s) for independent stochastic leadtime systems is also discussed.

Suggested Citation

  • David Zalkind, 1978. "Order-Level Inventory Systems with Independent Stochastic Leadtimes," Management Science, INFORMS, vol. 24(13), pages 1384-1392, September.
  • Handle: RePEc:inm:ormnsc:v:24:y:1978:i:13:p:1384-1392
    DOI: 10.1287/mnsc.24.13.1384
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    Cited by:

    1. 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.
    2. Ryu, Si Wook & Lee, Kyung Keun, 2003. "A stochastic inventory model of dual sourced supply chain with lead-time reduction," International Journal of Production Economics, Elsevier, vol. 81(1), pages 513-524, January.
    3. Hansen, Ole & Transchel, Sandra & Friedrich, Hanno, 2023. "Replenishment strategies for lost sales inventory systems of perishables under demand and lead time uncertainty," European Journal of Operational Research, Elsevier, vol. 308(2), pages 661-675.
    4. Salal Humair & John D. Ruark & Brian Tomlin & Sean P. Willems, 2013. "Incorporating Stochastic Lead Times Into the Guaranteed Service Model of Safety Stock Optimization," Interfaces, INFORMS, vol. 43(5), pages 421-434, October.
    5. Hayya, Jack C. & Harrison, Terry P. & He, X. James, 2011. "The impact of stochastic lead time reduction on inventory cost under order crossover," European Journal of Operational Research, Elsevier, vol. 211(2), pages 274-281, June.
    6. Chatfield, Dean C. & Pritchard, Alan M., 2018. "Crossover aware base stock decisions for service-driven systems," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 114(C), pages 312-330.
    7. Lawrence W. Robinson & James R. Bradley & L. Joseph Thomas, 2001. "Consequences of Order Crossover Under Order-Up-To Inventory Policies," Manufacturing & Service Operations Management, INFORMS, vol. 3(3), pages 175-188, September.
    8. Zhou, Bin & Katehakis, Michael N. & Zhao, Yao, 2009. "Managing stochastic inventory systems with free shipping option," European Journal of Operational Research, Elsevier, vol. 196(1), pages 186-197, July.
    9. Riezebos, Jan & Zhu, Stuart X., 2020. "Inventory control with seasonality of lead times," Omega, Elsevier, vol. 92(C).
    10. Van Nieuwenhuyse, Inneke & Vandaele, Nico & Rajaram, Kumar & Karmarkar, Uday S., 2007. "Buffer sizing in multi-product multi-reactor batch processes: Impact of allocation and campaign sizing policies," European Journal of Operational Research, Elsevier, vol. 179(2), pages 424-443, June.
    11. Riezebos, Jan, 2006. "Inventory order crossovers," International Journal of Production Economics, Elsevier, vol. 104(2), pages 666-675, December.
    12. Bookbinder, James H. & Cakanyildirim, Metin, 1999. "Random lead times and expedited orders in (Q,r) inventory systems," European Journal of Operational Research, Elsevier, vol. 115(2), pages 300-313, June.
    13. Wang, Xun & Disney, Stephen M., 2017. "Mitigating variance amplification under stochastic lead-time: The proportional control approach," European Journal of Operational Research, Elsevier, vol. 256(1), pages 151-162.
    14. Alp Muharremoglu & Nan Yang, 2010. "Inventory Management with an Exogenous Supply Process," Operations Research, INFORMS, vol. 58(1), pages 111-129, February.
    15. Achin Srivastav & Sunil Agrawal, 2020. "Multi-objective optimization of mixture inventory system experiencing order crossover," Annals of Operations Research, Springer, vol. 290(1), pages 943-960, July.
    16. Disney, Stephen M. & Maltz, Arnold & Wang, Xun & Warburton, Roger D.H., 2016. "Inventory management for stochastic lead times with order crossovers," European Journal of Operational Research, Elsevier, vol. 248(2), pages 473-486.
    17. Hellemans, Tim & Boute, Robert N. & Van Houdt, Benny, 2019. "Analysis of lead time correlation under a base-stock policy," European Journal of Operational Research, Elsevier, vol. 276(2), pages 519-535.
    18. 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.
    19. James R. Bradley & Lawrence W. Robinson, 2005. "Improved Base-Stock Approximations for Independent Stochastic Lead Times with Order Crossover," Manufacturing & Service Operations Management, INFORMS, vol. 7(4), pages 319-329, November.
    20. Hayya, Jack C. & Bagchi, Uttarayan & Kim, Jeon G. & Sun, Daewon, 2008. "On static stochastic order crossover," International Journal of Production Economics, Elsevier, vol. 114(1), pages 404-413, July.
    21. Hayya, Jack C. & Bagchi, Uttarayan & Ramasesh, Ranga, 2011. "Cost relationships in stochastic inventory systems: A simulation study of the (S, S-1, t=1) model," International Journal of Production Economics, Elsevier, vol. 130(2), pages 196-202, April.
    22. 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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