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Dynamics of Two Classes of Continuous-Review Inventory Systems

Author

Listed:
  • H. P. Galliher

    (Operations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts)

  • Philip M. Morse

    (Operations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts)

  • M. Simond

    (Operations Research Center, Massachusetts Institute of Technology, Cambridge, Massachusetts)

Abstract

The classes of inventory systems studied in this paper involve random, captive demand and assume continuous review of inventory and replenishment of stock in lots of size Q . The replenishment doctrine is to initiate an order for Q items whenever the sum of stock on hand and stock on order falls below a pre-determined level R . Possible examples are inventories of spare parts for maintenance, of military supplies, or of merchandising stock, so costs of purchase, of storage, and of backordenng are included in the measures of effectiveness. A general relation, between cost of back-orders and minimal-cost probability of stock-out, is derived, similar to that reported earlier [Morse, P. M. 1959. Solution of a class of discrete-time inventory problems. Opns Res. 7 67--78.] for discrete-time systems. For the first class of system studied, demands arrive at random, with a stationary probability distribution of arbitrary form, and all replenishment times are of the same length. For the second class, demand arrivals are Poisson and replenishment times are distributed exponentially. Exact solutions, together with expressions for expected values of operating cost, probability of stock-out, P out , of stock on hand, etc., are obtained for both classes of system. For the first class, specific formulas are given for the frequently-encountered cases of Poisson and stuttering Poisson demand arrivals. An asymptotic form for these solutions is obtained, valid for both classes of system over the range of parameters of practical interest. In terms of this asymptotic formulation, equations, tables, and graphs are given, from which the re-order level R and replenishment lot size Q can be determined for minimal cost and/or for a pre-set value of P out , or to satisfy other managerial requirements. For Poisson demand, comparison between the two systems shows that when Q is small, increase in variance of replenishment time makes very little difference in the requirements for R , but when Q is large an increase in variance of replenishment time requires an increase in the value of R , the re-order level, to maintain the optimality of the solution.

Suggested Citation

  • H. P. Galliher & Philip M. Morse & M. Simond, 1959. "Dynamics of Two Classes of Continuous-Review Inventory Systems," Operations Research, INFORMS, vol. 7(3), pages 362-384, June.
  • Handle: RePEc:inm:oropre:v:7:y:1959:i:3:p:362-384
    DOI: 10.1287/opre.7.3.362
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    Citations

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    Cited by:

    1. John D. C. Little, 2002. "Philip M. Morse and the Beginnings," Operations Research, INFORMS, vol. 50(1), pages 146-148, February.
    2. Michael Katehakis & Laurens Smit, 2012. "On computing optimal (Q,r) replenishment policies under quantity discounts," Annals of Operations Research, Springer, vol. 200(1), pages 279-298, November.
    3. Emre Tokgöz & Hillel Kumin, 2012. "Mixed convexity and optimization results for an (S − 1, S) inventory model under a time limit on backorders," Computational Management Science, Springer, vol. 9(4), pages 417-440, November.
    4. 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.
    5. Xin X. He & Susan H. Xu & J. Keith Ord & Jack C. Hayya, 1998. "An Inventory Model with Order Crossover," Operations Research, INFORMS, vol. 46(3-supplem), pages 112-119, June.
    6. Prak, Derk & Teunter, Rudolf & Babai, M. Z. & Syntetos, A. A. & Boylan, D, 2018. "Forecasting and Inventory Control with Compound Poisson Demand Using Periodic Demand Data," Research Report 2018010, University of Groningen, Research Institute SOM (Systems, Organisations and Management).
    7. Zhang, Huiming & Liu, Yunxiao & Li, Bo, 2014. "Notes on discrete compound Poisson model with applications to risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 59(C), pages 325-336.
    8. Kumar Muthuraman & Sridhar Seshadri & Qi Wu, 2015. "Inventory Management with Stochastic Lead Times," Mathematics of Operations Research, INFORMS, vol. 40(2), pages 302-327, February.
    9. 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.
    10. Prak, Dennis & Teunter, Ruud & Babai, Mohamed Zied & Boylan, John E. & Syntetos, Aris, 2021. "Robust compound Poisson parameter estimation for inventory control," Omega, Elsevier, vol. 104(C).
    11. 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.
    12. 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.
    13. 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.
    14. Nguyen, Duy Tan & Adulyasak, Yossiri & Landry, Sylvain, 2021. "Research manuscript: The Bullwhip Effect in rule-based supply chain planning systems–A case-based simulation at a hard goods retailer," Omega, Elsevier, vol. 98(C).
    15. 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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