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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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    Cited by:

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    2. 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.
    3. Chan, Chi Kin & Zhou, Yan & Wong, Kar Hung, 2018. "A dynamic equilibrium model of the oligopolistic closed-loop supply chain network under uncertain and time-dependent demands," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 118(C), pages 325-354.

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