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Optimal Inventories for an (S - 1, S) System with No Backorders

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  • Stephen A. Smith

    (Xerox Palo Alto Research Center)

Abstract

This paper considers the problem of finding optimal stock levels for an inventory system with Poisson demands, arbitrary replacement time distribution, and emergency handling of shortages at a premium cost. The ordering policy is assumed to be of the (S - 1, S) type, i.e., a replacement item is ordered as soon as a unit of stock is used. A criterion for selecting the optimal stock level is formulated by minimizing the expected cost per unit time in steady state. Based on the fact that the probability of shortage for this model is given by the Erlang loss formula, it is possible to derive an approximate stocking level formula, which appears to be accurate over a wide range of parameter values. Comparisons with other related models and determination of achieved service levels are also discussed.

Suggested Citation

  • Stephen A. Smith, 1977. "Optimal Inventories for an (S - 1, S) System with No Backorders," Management Science, INFORMS, vol. 23(5), pages 522-528, January.
    • Handle: RePEc:inm:ormnsc:v:23:y:1977:i:5:p:522-528
    DOI: 10.1287/mnsc.23.5.522
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    Cited by:

    1. Kouki, Chaaben & Babai, M. Zied & Minner, Stefan, 2018. "On the benefit of dual-sourcing in managing perishable inventory," International Journal of Production Economics, Elsevier, vol. 204(C), pages 1-17.
    2. R. Dekker & R.M. Hill & M.J. Kleijn & R.H. Teunter, 2002. "On the (S − 1, S) lost sales inventory model with priority demand classes," Naval Research Logistics (NRL), John Wiley & Sons, vol. 49(6), pages 593-610, September.
    3. Johansen, Soren Glud, 2001. "Pure and modified base-stock policies for the lost sales inventory system with negligible set-up costs and constant lead times," International Journal of Production Economics, Elsevier, vol. 71(1-3), pages 391-399, May.
    4. S. Kalpakam & K. P. Sapna, 1996. "A lost sales (S — 1, S) perishable inventory system with renewal demand," Naval Research Logistics (NRL), John Wiley & Sons, vol. 43(1), pages 129-142, February.
    5. Barron, Yonit, 2022. "The continuous (S,s,Se) inventory model with dual sourcing and emergency orders," European Journal of Operational Research, Elsevier, vol. 301(1), pages 18-38.
    6. 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.
    7. Bijvank, Marco & Vis, Iris F.A., 2012. "Lost-sales inventory systems with a service level criterion," European Journal of Operational Research, Elsevier, vol. 220(3), pages 610-618.
    8. Mauricio Varas & Franco Basso & Armin Lüer-Villagra & Alejandro Mac Cawley & Sergio Maturana, 2019. "Managing premium wines using an $$(s - 1,s)$$ ( s - 1 , s ) inventory policy: a heuristic solution approach," Annals of Operations Research, Springer, vol. 280(1), pages 351-376, September.
    9. Kouki, Chaaben & Babai, M. Zied & Jemai, Zied & Minner, Stefan, 2019. "Solution procedures for lost sales base-stock inventory systems with compound Poisson demand," International Journal of Production Economics, Elsevier, vol. 209(C), pages 172-182.
    10. Bijvank, Marco & Vis, Iris F.A., 2011. "Lost-sales inventory theory: A review," European Journal of Operational Research, Elsevier, vol. 215(1), pages 1-13, November.
    11. Saha, Kushal & Bhattacharya, Subir, 2021. "‘Buy online and pick up in-store’: Implications for the store inventory," European Journal of Operational Research, Elsevier, vol. 294(3), pages 906-921.
    12. Esmail Mohebbi & Morton J.M. Posner, 1998. "A continuous‐review inventory system with lost sales and variable lead time," Naval Research Logistics (NRL), John Wiley & Sons, vol. 45(3), pages 259-278, April.
    13. Hill, R.M., 2007. "Continuous-review, lost-sales inventory models with Poisson demand, a fixed lead time and no fixed order cost," European Journal of Operational Research, Elsevier, vol. 176(2), pages 956-963, January.
    14. 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.
    15. Carl R. Schultz, 1990. "On the optimality of the (S — 1,S) policy," Naval Research Logistics (NRL), John Wiley & Sons, vol. 37(5), pages 715-723, October.
    16. Eric Logan Huggins & Tava Lennon Olsen, 2010. "Inventory Control with Generalized Expediting," Operations Research, INFORMS, vol. 58(5), pages 1414-1426, October.
    17. Kalpakam, S. & Sapna, K. P., 1995. "A two reorder level inventory system with renewal demands," European Journal of Operational Research, Elsevier, vol. 84(2), pages 402-415, July.
    18. Andersson, Jonas & Melchiors, Philip, 2001. "A two-echelon inventory model with lost sales," International Journal of Production Economics, Elsevier, vol. 69(3), pages 307-315, February.
    19. 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.
    20. Hill, Roger M., 1999. "On the suboptimality of (S-1, S) lost sales inventory policies," International Journal of Production Economics, Elsevier, vol. 59(1-3), pages 387-393, March.

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