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Exact solutions to the supply chain equations for arbitrary, time-dependent demands

Author

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  • Warburton, Roger D.H.
  • Hodgson, J.P.E.
  • Nielsen, E.H.

Abstract

We study the impact on inventory of an unexpected, non-linear, time-dependent demand and present the exact solutions over time to the supply chain equations without requiring any approximations. We begin by imposing a boundary condition of stability at infinity, from which we derive expressions for the estimated demand and the target work in progress when the demand is time-dependent. The resulting inventory equation is solved in terms of the Lambert modes with all of the demand non-linearities confined to the pre-shape function. The series solution is exact, and all terms are reasonably easy to calculate, so users can determine the inventory behavior to any desired precision. To illustrate, we solve the equations for a non-linear, quadratic time-dependence in the demand. For practical use, only a few terms in the series are required, a proposition illustrated by the For All Practical Purposes (FAPP) solution. While the paper provides a theoretical foundation, the result is decidedly practical: An accurate and reasonably easy-to-implement model that companies can use to analyze the impact of non-linear, time-dependent demands.

Suggested Citation

  • Warburton, Roger D.H. & Hodgson, J.P.E. & Nielsen, E.H., 2014. "Exact solutions to the supply chain equations for arbitrary, time-dependent demands," International Journal of Production Economics, Elsevier, vol. 151(C), pages 195-205.
    • Handle: RePEc:eee:proeco:v:151:y:2014:i:c:p:195-205
    DOI: 10.1016/j.ijpe.2013.10.015
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    1. Disney, Stephen M. & Towill, Denis R. & Warburton, Roger D.H., 2006. "On the equivalence of control theoretic, differential, and difference equation approaches to modeling supply chains," International Journal of Production Economics, Elsevier, vol. 101(1), pages 194-208, May.
    2. Yossi Aviv, 2003. "A Time-Series Framework for Supply-Chain Inventory Management," Operations Research, INFORMS, vol. 51(2), pages 210-227, April.
    3. Amann, Andreas & Schöll, Eckehard & Just, Wolfram, 2007. "Some basic remarks on eigenmode expansions of time-delay dynamics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 373(C), pages 191-202.
    4. Yossi Aviv, 2002. "Gaining Benefits from Joint Forecasting and Replenishment Processes: The Case of Auto-Correlated Demand," Manufacturing & Service Operations Management, INFORMS, vol. 4(1), pages 55-74, December.
    5. Selcuk, B. & Fransoo, J.C. & De Kok, A.G., 2006. "The effect of updating lead times on the performance of hierarchical planning systems," International Journal of Production Economics, Elsevier, vol. 104(2), pages 427-440, December.
    6. 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.
    7. Retsef Levi & Martin Pál & Robin O. Roundy & David B. Shmoys, 2007. "Approximation Algorithms for Stochastic Inventory Control Models," Mathematics of Operations Research, INFORMS, vol. 32(2), pages 284-302, May.
    8. Dejonckheere, J. & Disney, S. M. & Lambrecht, M. R. & Towill, D. R., 2004. "The impact of information enrichment on the Bullwhip effect in supply chains: A control engineering perspective," European Journal of Operational Research, Elsevier, vol. 153(3), pages 727-750, March.
    9. Warburton, R.D.H. & Disney, S.M., 2007. "Order and inventory variance amplification: The equivalence of discrete and continuous time analyses," International Journal of Production Economics, Elsevier, vol. 110(1-2), pages 128-137, October.
    10. Everette S. Gardner, Jr. & Ed. Mckenzie, 1985. "Forecasting Trends in Time Series," Management Science, INFORMS, vol. 31(10), pages 1237-1246, October.
    11. Vedran Capkun & Ari-Pekka Hameri & Lawrence A. Weiss, 2009. "On the relationship between inventory and financial performance in manufacturing companies," Post-Print hal-00493163, HAL.
    12. S. A. Roberts, 1982. "A General Class of Holt-Winters Type Forecasting Models," Management Science, INFORMS, vol. 28(7), pages 808-820, July.
    13. Lingxiu Dong & Hau L. Lee, 2003. "Optimal Policies and Approximations for a Serial Multiechelon Inventory System with Time-Correlated Demand," Operations Research, INFORMS, vol. 51(6), pages 969-980, December.
    14. Disney, Stephen M. & Lambrecht, Marc R., 2008. "On Replenishment Rules, Forecasting, and the Bullwhip Effect in Supply Chains," Foundations and Trends(R) in Technology, Information and Operations Management, now publishers, vol. 2(1), pages 1-80, April.
    15. 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.
    16. Warburton, Roger D. H., 2004. "An exact analytical solution to the production inventory control problem," International Journal of Production Economics, Elsevier, vol. 92(1), pages 81-96, November.
    17. Vedran Capkun & Ari-Pekka Hameri & Lawrence A. Weiss, 2009. "On the relationship between inventory and financial performance in manufacturing companies," Post-Print hal-00466392, HAL.
    18. A Diponegoro & B R Sarker, 2007. "Operations policy for a supply chain system with fixed-interval delivery and linear demand," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 58(7), pages 901-910, July.
    19. Frank Chen & Zvi Drezner & Jennifer K. Ryan & David Simchi-Levi, 2000. "Quantifying the Bullwhip Effect in a Simple Supply Chain: The Impact of Forecasting, Lead Times, and Information," Management Science, INFORMS, vol. 46(3), pages 436-443, March.
    20. Yossi Aviv, 2007. "On the Benefits of Collaborative Forecasting Partnerships Between Retailers and Manufacturers," Management Science, INFORMS, vol. 53(5), pages 777-794, May.
    21. Yossi Aviv & Awi Federgruen, 2001. "Capacitated Multi-Item Inventory Systems with Random and Seasonally Fluctuating Demands: Implications for Postponement Strategies," Management Science, INFORMS, vol. 47(4), pages 512-531, April.
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    Cited by:

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    2. 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.
    3. Yasser A. Davizón & César Martínez-Olvera & Rogelio Soto & Carlos Hinojosa & Piero Espino-Román, 2015. "Optimal Control Approaches to the Aggregate Production Planning Problem," Sustainability, MDPI, vol. 7(12), pages 1-16, December.

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