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Fill rate in a periodic review order-up-to policy under auto-correlated normally distributed, possibly negative, demand

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  • Disney, Stephen M.
  • Gaalman, Gerard J.C.
  • Hedenstierna, Carl Philip T.
  • Hosoda, Takamichi

Abstract

We investigate the inventory service metric known as the fill rate—the proportion of demand that is immediately fulfilled from inventory. The task of finding analytical solutions for general cases is complicated by a range of factors including; correlation in demand, double counting of backlogs, and proper treatment of negative demand. In the literature, two approximate approaches are often proposed. Our contribution is to present a new fill rate measure for normally distributed, auto-correlated, and possibly negative demand. We treat negative demand as returns. Our approach also accounts for accumulated backlogs. The problem reduces to identifying the minimum of correlated normally distributed bivariate random variables. There exists an exact solution, but it has no closed form. However, the solution is amenable to numerical techniques, and we present a custom Microsoft Excel function for practical use. Numerical investigations reveal that the new fill rate is more robust than previous measures. Existing fill rate measures are likely to cause excessive inventory investment, especially when fill rate targets are modest, a strongly positive or negative autocorrelation in demand is present, or negative demands exist. Our fill rate calculation ensures that the target fill rate is achieved without excessive inventory investments.

Suggested Citation

  • Disney, Stephen M. & Gaalman, Gerard J.C. & Hedenstierna, Carl Philip T. & Hosoda, Takamichi, 2015. "Fill rate in a periodic review order-up-to policy under auto-correlated normally distributed, possibly negative, demand," International Journal of Production Economics, Elsevier, vol. 170(PB), pages 501-512.
    • Handle: RePEc:eee:proeco:v:170:y:2015:i:pb:p:501-512
    DOI: 10.1016/j.ijpe.2015.07.019
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    References listed on IDEAS

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    1. Cardós, Manuel & Babiloni, Eugenia, 2011. "Exact and approximate calculation of the cycle service level in periodic review inventory policies," International Journal of Production Economics, Elsevier, vol. 131(1), pages 63-68, May.
    2. Silver, Edward A. & Bischak, Diane P., 2011. "The exact fill rate in a periodic review base stock system under normally distributed demand," Omega, Elsevier, vol. 39(3), pages 346-349, June.
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    5. Guijarro, Ester & Cardós, Manuel & Babiloni, Eugenia, 2012. "On the exact calculation of the fill rate in a periodic review inventory policy under discrete demand patterns," European Journal of Operational Research, Elsevier, vol. 218(2), pages 442-447.
    6. Gaalman, Gerard & Disney, Stephen M., 2009. "On bullwhip in a family of order-up-to policies with ARMA(2,2) demand and arbitrary lead-times," International Journal of Production Economics, Elsevier, vol. 121(2), pages 454-463, October.
    7. Disney, S.M. & Farasyn, I. & Lambrecht, M. & Towill, D.R. & de Velde, W. Van, 2006. "Taming the bullwhip effect whilst watching customer service in a single supply chain echelon," European Journal of Operational Research, Elsevier, vol. 173(1), pages 151-172, August.
    8. Li, Qinyun & Disney, Stephen M. & Gaalman, Gerard, 2014. "Avoiding the bullwhip effect using Damped Trend forecasting and the Order-Up-To replenishment policy," International Journal of Production Economics, Elsevier, vol. 149(C), pages 3-16.
    9. Basu, A. P. & Ghosh, J. K., 1978. "Identifiability of the multinormal and other distributions under competing risks model," Journal of Multivariate Analysis, Elsevier, vol. 8(3), pages 413-429, September.
    10. Disney, S. M. & Towill, D. R., 2003. "On the bullwhip and inventory variance produced by an ordering policy," Omega, Elsevier, vol. 31(3), pages 157-167, June.
    11. Matthew J. Sobel, 2004. "Fill Rates of Single-Stage and Multistage Supply Systems," Manufacturing & Service Operations Management, INFORMS, vol. 6(1), pages 41-52, June.
    12. Amar Sapra & Van-Anh Truong & Rachel Q. Zhang, 2010. "How Much Demand Should Be Fulfilled?," Operations Research, INFORMS, vol. 58(3), pages 719-733, June.
    13. Larsen, C. & Thorstenson, A., 2014. "The order and volume fill rates in inventory control systems," International Journal of Production Economics, Elsevier, vol. 147(PA), pages 13-19.
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    Cited by:

    1. Veiga, Claudimar Pereira da & Veiga, Cássia Rita Pereira da & Puchalski, Weslly & Coelho, Leandro dos Santos & Tortato, Ubiratã, 2016. "Demand forecasting based on natural computing approaches applied to the foodstuff retail segment," Journal of Retailing and Consumer Services, Elsevier, vol. 31(C), pages 174-181.
    2. 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.
    3. Hedenstierna, Carl Philip T. & Disney, Stephen M., 2016. "Inventory performance under staggered deliveries and autocorrelated demand," European Journal of Operational Research, Elsevier, vol. 249(3), pages 1082-1091.
    4. Wesley Marcos Almeida & Claudimar Pereira Veiga, 2023. "Does demand forecasting matter to retailing?," Journal of Marketing Analytics, Palgrave Macmillan, vol. 11(2), pages 219-232, June.
    5. Petropoulos, Fotios & Wang, Xun & Disney, Stephen M., 2019. "The inventory performance of forecasting methods: Evidence from the M3 competition data," International Journal of Forecasting, Elsevier, vol. 35(1), pages 251-265.
    6. Bruzda, Joanna, 2020. "Demand forecasting under fill rate constraints—The case of re-order points," International Journal of Forecasting, Elsevier, vol. 36(4), pages 1342-1361.
    7. 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.
    8. Dreyfuss, Michael & Giat, Yahel, 2019. "Allocating spares to maximize the window fill rate in a periodic review inventory system," International Journal of Production Economics, Elsevier, vol. 214(C), pages 151-162.

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