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Exploring the nonlinear dynamics of the lost-sales order-up-to policy

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  • Stephen M. Disney
  • Borja Ponte
  • Xun Wang

Abstract

With most inventory theory investigating linear models, the dynamics of nonlinear inventory systems is not well understood. We explore the dynamics of the order-up-to policy under lost-sales for the case of i.i.d. normally distributed demand and unit lead times. We consider the ideal minimum mean-squared error forecast and two alternative scenarios: partial demand observation and dynamic demand forecasting, providing a broad understanding of the operational performance of lost-sales systems. In each scenario, we obtain analytical expressions for the order, inventory, and satisfied demand distributions. This allows us to quantify the Bullwhip and inventory variance amplification ratios as well as the fill rate and the inventory cover. We show the lost sales nonlinearity induces complex behaviours in inventory systems. Interestingly, lost sales smooth supply chain dynamics, significantly affecting the trade-off between service level and average inventory holding. We also reveal the inventory downsides of demand censoring and the production damages induced by dynamic forecasts. We identify a key parameter, the relative safety margin, that characterises the performance of lost-sales systems. We finish by offering some prescriptive results for the optimal safety stock and capacity level in both a retail and a manufacturing lost-sales setting.

Suggested Citation

  • Stephen M. Disney & Borja Ponte & Xun Wang, 2021. "Exploring the nonlinear dynamics of the lost-sales order-up-to policy," International Journal of Production Research, Taylor & Francis Journals, vol. 59(19), pages 5809-5830, October.
    • Handle: RePEc:taf:tprsxx:v:59:y:2021:i:19:p:5809-5830
    DOI: 10.1080/00207543.2020.1790687
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    Cited by:

    1. Md. Iftekharul Alam Efat & Petr Hajek & Mohammad Zoynul Abedin & Rahat Uddin Azad & Md. Al Jaber & Shuvra Aditya & Mohammad Kabir Hassan, 2024. "Deep-learning model using hybrid adaptive trend estimated series for modelling and forecasting sales," Annals of Operations Research, Springer, vol. 339(1), pages 297-328, August.
    2. Li, Qinyun & Gaalman, Gerard & Disney, Stephen M., 2023. "On the equivalence of the proportional and damped trend order-up-to policies: An eigenvalue analysis," International Journal of Production Economics, Elsevier, vol. 265(C).
    3. Lin, Junyi & Huang, Hongfu & Li, Shanshan & Naim, Mohamed M., 2023. "On the dynamics of order pipeline inventory in a nonlinear order-up-to system," International Journal of Production Economics, Elsevier, vol. 266(C).
    4. Wang, Xun & Disney, Stephen M. & Ponte, Borja, 2023. "On the stationary stochastic response of an order-constrained inventory system," European Journal of Operational Research, Elsevier, vol. 304(2), pages 543-557.
    5. Eugenia Babiloni & Ester Guijarro & Juan R. Trapero, 2023. "Stock control analytics: a data-driven approach to compute the fill rate considering undershoots," Operational Research, Springer, vol. 23(1), pages 1-25, March.
    6. Yang, Liu & Liu, Kanglin & Zhang, Juan & Zelbst, Pamela J., 2024. "Inventory management with actual palletized transportation costs and lost sales," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 184(C).

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