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A Queuing Model for an Inventory Problem

Author

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  • William Karush

    (The Ramo-Wooldridge Corporation, Los Angeles, California)

Abstract

The relation between lost sales and inventory level is an important problem in inventory control. An explicit mathematical solution is obtained by methods of general interest for a probabilistic model that arose in connection with consulting work for an industrial client. Customer demand for a given commodity is a Poisson process with mean rate (lambda), and replenishment time for restocking is random. At any moment, the constant inventory n is divided between in-stock amount n 0 , and inreplenishment process amount n - n 0 . Customer arrival when n 0 > 0 results in a unit sale and the initiation of replenishment of that unit. Successive replenishment times are independent. Customer arrival when n 0 = 0, results in a lost sale. The unique stationary probabilities p ( n 0 | n ) of the states n 0 (fixed n ), are obtained, they are given by the Erlang congestion formula, and depend upon the replenishment time only to the extent of its mean value. A generalization is obtained where (lambda) may be a function of the state of the system. The ratio of lost sales to total demand, given by p (0| n ), is shown to be convex decreasing in n . The problem of allocation of inventory dollars among various competing commodities, so as to minimize over-all lost sales dollars, is treated.

Suggested Citation

  • William Karush, 1957. "A Queuing Model for an Inventory Problem," Operations Research, INFORMS, vol. 5(5), pages 693-703, October.
    • Handle: RePEc:inm:oropre:v:5:y:1957:i:5:p:693-703
    DOI: 10.1287/opre.5.5.693
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    Cited by:

    1. Li, Xiaopeng, 2013. "An integrated modeling framework for design of logistics networks with expedited shipment services," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 56(C), pages 46-63.
    2. Rahimi-Ghahroodi, S. & Al Hanbali, A. & Zijm, W.H.M. & Timmer, J.B., 2019. "Emergency supply contracts for a service provider with limited local resources," Reliability Engineering and System Safety, Elsevier, vol. 189(C), pages 445-460.
    3. 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).
    4. van Wingerden, E. & Tan, T. & Van Houtum, G.J., 2019. "The impact of an emergency warehouse in a two-echelon spare parts network," European Journal of Operational Research, Elsevier, vol. 276(3), pages 983-997.
    5. Woonghee Tim Huh & Ganesh Janakiraman, 2010. "Base‐stock policies in capacitated assembly systems: Convexity properties," Naval Research Logistics (NRL), John Wiley & Sons, vol. 57(2), pages 109-118, March.
    6. Woonghee Tim Huh & Ganesh Janakiraman & John A. Muckstadt & Paat Rusmevichientong, 2009. "Asymptotic Optimality of Order-Up-To Policies in Lost Sales Inventory Systems," Management Science, INFORMS, vol. 55(3), pages 404-420, March.
    7. Zhe Yuan & Haoxuan Xu & Yeming (Yale) Gong & Chengbin Chu & Jinlong Zhang, 2017. "Designing public storage warehouses with high demand for revenue maximisation," International Journal of Production Research, Taylor & Francis Journals, vol. 55(13), pages 3686-3700, July.
    8. Erhun Özkan & Geert-Jan Houtum & Yasemin Serin, 2015. "A new approximate evaluation method for two-echelon inventory systems with emergency shipments," Annals of Operations Research, Springer, vol. 224(1), pages 147-169, January.
    9. Johansen, Soren Glud, 2005. "Base-stock policies for the lost sales inventory system with Poisson demand and Erlangian lead times," International Journal of Production Economics, Elsevier, vol. 93(1), pages 429-437, January.
    10. Cui, Jianxun & Zhao, Meng & Li, Xiaopeng & Parsafard, Mohsen & An, Shi, 2016. "Reliable design of an integrated supply chain with expedited shipments under disruption risks," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 95(C), pages 143-163.
    11. Kurtuluş Barış Öner & Alan Scheller-Wolf & Geert-Jan van Houtum, 2013. "Redundancy Optimization for Critical Components in High-Availability Technical Systems," Operations Research, INFORMS, vol. 61(1), pages 244-264, February.
    12. Ganesh Janakiraman & Robin O. Roundy, 2004. "Lost-Sales Problems with Stochastic Lead Times: Convexity Results for Base-Stock Policies," Operations Research, INFORMS, vol. 52(5), pages 795-803, October.
    13. Christian Howard & Johan Marklund & Tarkan Tan & Ingrid Reijnen, 2015. "Inventory Control in a Spare Parts Distribution System with Emergency Stocks and Pipeline Information," Manufacturing & Service Operations Management, INFORMS, vol. 17(2), pages 142-156, May.
    14. Zümbül Atan & Lawrence V. Snyder & George R. Wilson, 2018. "Transshipment policies for systems with multiple retailers and two demand classes," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 40(1), pages 159-186, January.
    15. Kranenburg, A.A. & Van Houtum, G.J., 2007. "Effect of commonality on spare parts provisioning costs for capital goods," International Journal of Production Economics, Elsevier, vol. 108(1-2), pages 221-227, July.

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