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On the equivalence of the proportional and damped trend order-up-to policies: An eigenvalue analysis

Author

Listed:
  • Li, Qinyun
  • Gaalman, Gerard
  • Disney, Stephen M.

Abstract

We investigate the equivalence of the order-up-to (OUT) replenishment policy with damped trend forecasting (OUT-DT) to the proportional OUT (POUT) policy via an eigenvalue (zero-pole) analysis. We show when the damped trend forecasting parameters are selected from a specific region of the parameter space, the Bullwhip Avoidance(BA) region, the OUT-DT policy potentially possesses some desirable characteristics. Under an independent and identically distributed (i.i.d.) random demand (the simplest random demand) we show the OUT-DT policy: (a) can eliminate the bullwhip effect, (b) has the same dynamic response as the POUT policy (because of the superposition principle, this holds true for all demands, not just i.i.d. demands), (c) forecasting parameters can be set so as to minimize the sum of the inventory and order variances, and (d) has a similar order and inventory variance as the POUT policy when non-linear constraints are present. We investigate the case when the forecasting parameters are selected from the BA region and correlated demand with one auto-regressive term, one integrated term, and two moving average terms, ARIMA(1,1,2), is present. We reveal the effect of finite lead times on the bullwhip effect (order variance) using an approach based on the order of the eigenvalues (the zeros and poles). We reveal either: (a) the bullwhip effect is always present and always increasing in the lead time or (b) a smoothing effect can be present with short lead times (and the order variance may even be decreasing in the lead time) but the bullwhip effect may return with long lead times.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:proeco:v:265:y:2023:i:c:s0925527323002372
    DOI: 10.1016/j.ijpe.2023.109005
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    References listed on IDEAS

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    1. Peter R. Winters, 1960. "Forecasting Sales by Exponentially Weighted Moving Averages," Management Science, INFORMS, vol. 6(3), pages 324-342, April.
    2. Boute, Robert N. & Disney, Stephen M. & Lambrecht, Marc R. & Van Houdt, Benny, 2009. "Designing replenishment rules in a two-echelon supply chain with a flexible or an inflexible capacity strategy," International Journal of Production Economics, Elsevier, vol. 119(1), pages 187-198, May.
    3. E S Gardner & E McKenzie, 2011. "Why the damped trend works," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 62(6), pages 1177-1180, June.
    4. Chiang, Chung-Yean & Lin, Winston T. & Suresh, Nallan C., 2016. "An empirically-simulated investigation of the impact of demand forecasting on the bullwhip effect: Evidence from U.S. auto industry," International Journal of Production Economics, Elsevier, vol. 177(C), pages 53-65.
    5. Hosoda, Takamichi & Disney, Stephen M., 2009. "Impact of market demand mis-specification on a two-level supply chain," International Journal of Production Economics, Elsevier, vol. 121(2), pages 739-751, October.
    6. Everette S. Gardner, 1990. "Evaluating Forecast Performance in an Inventory Control System," Management Science, INFORMS, vol. 36(4), pages 490-499, April.
    7. Qinyun Li & Stephen M. Disney, 2017. "Revisiting rescheduling: MRP nervousness and the bullwhip effect," International Journal of Production Research, Taylor & Francis Journals, vol. 55(7), pages 1992-2012, April.
    8. 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.
    9. 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.
    10. Disney, S. M. & Towill, D. R. & van de Velde, W., 2004. "Variance amplification and the golden ratio in production and inventory control," International Journal of Production Economics, Elsevier, vol. 90(3), pages 295-309, August.
    11. Hau L. Lee & V. Padmanabhan & Seungjin Whang, 1997. "Information Distortion in a Supply Chain: The Bullwhip Effect," Management Science, INFORMS, vol. 43(4), pages 546-558, April.
    12. Rostami-Tabar, Bahman & Disney, Stephen M., 2023. "On the order-up-to policy with intermittent integer demand and logically consistent forecasts," International Journal of Production Economics, Elsevier, vol. 257(C).
    13. 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.
    14. Gaalman, Gerard & Disney, Stephen M. & Wang, Xun, 2022. "When bullwhip increases in the lead time: An eigenvalue analysis of ARMA demand," International Journal of Production Economics, Elsevier, vol. 250(C).
    15. Lin, J. & Naim, M.M. & Purvis, L. & Gosling, J., 2017. "The extension and exploitation of the inventory and order based production control system archetype from 1982 to 2015," International Journal of Production Economics, Elsevier, vol. 194(C), pages 135-152.
    16. Borja Ponte & Xun Wang & David de la Fuente & Stephen M. Disney, 2017. "Exploring nonlinear supply chains: the dynamics of capacity constraints," International Journal of Production Research, Taylor & Francis Journals, vol. 55(14), pages 4053-4067, July.
    17. S. A. Roberts, 1982. "A General Class of Holt-Winters Type Forecasting Models," Management Science, INFORMS, vol. 28(7), pages 808-820, July.
    18. 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.
    19. 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.
    20. Makridakis, Spyros & Hibon, Michele, 2000. "The M3-Competition: results, conclusions and implications," International Journal of Forecasting, Elsevier, vol. 16(4), pages 451-476.
    21. Huang, Shupeng & Potter, Andrew & Eyers, Daniel & Li, Qinyun, 2021. "The influence of online review adoption on the profitability of capacitated supply chains," Omega, Elsevier, vol. 105(C).
    22. Charles C. Holt & Franco Modigliani & Herbert A. Simon, 1955. "A Linear Decision Rule for Production and Employment Scheduling," Management Science, INFORMS, vol. 2(1), pages 1-30, October.
    23. Dejonckheere, J. & Disney, S. M. & Lambrecht, M. R. & Towill, D. R., 2003. "Measuring and avoiding the bullwhip effect: A control theoretic approach," European Journal of Operational Research, Elsevier, vol. 147(3), pages 567-590, June.
    24. Rob Hyndman & Muhammad Akram & Blyth Archibald, 2008. "The admissible parameter space for exponential smoothing models," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(2), pages 407-426, June.
    25. 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.
    26. Paolo Priore & Borja Ponte & Rafael Rosillo & David de la Fuente, 2019. "Applying machine learning to the dynamic selection of replenishment policies in fast-changing supply chain environments," International Journal of Production Research, Taylor & Francis Journals, vol. 57(11), pages 3663-3677, June.
    27. Ponte, Borja & Dominguez, Roberto & Cannella, Salvatore & Framinan, Jose M., 2022. "The implications of batching in the bullwhip effect and customer service of closed-loop supply chains," International Journal of Production Economics, Elsevier, vol. 244(C).
    28. Robert N. Boute & Stephen M. Disney & Joren Gijsbrechts & Jan A. Van Mieghem, 2022. "Dual Sourcing and Smoothing Under Nonstationary Demand Time Series: Reshoring with SpeedFactories," Management Science, INFORMS, vol. 68(2), pages 1039-1057, February.
    29. Salvatore Cannella & Borja Ponte & Roberto Dominguez & Jose M. Framinan, 2021. "Proportional order-up-to policies for closed-loop supply chains: the dynamic effects of inventory controllers," International Journal of Production Research, Taylor & Francis Journals, vol. 59(11), pages 3323-3337, June.
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