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A theoretical validation of the DDMRP reorder policy

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Abstract

A recent heuristic called Demand Driven MRP, widely implemented using modern ERP systems, proposes reorder policy based on buffers. Buffers are amounts of inventory positioned and set to control the net flow position, responding to stochastic demand and lead time. Our primary goal is to propose a theoretical foundation for such a heuristic approach. To this aim, we develop an optimization model inspired by the main principles behind the heuristic algorithm. Specifically, optimal policies are of the type (s(t), S(t)) with time-varying thresholds that react to short-run real orders. We introduce constraints related to the service levels, that are written as tail risk measures to ensure fulfillment of realized demand with a predetermined probability. Interestingly, it turns out that such constraints allow to analytically justify an empirical rule that the DDMRP employs to set the risk parameters used in the heuristic. Finally, we use our model as a benchmark to theoretically validate and contextualize the aforementioned heuristic.

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Notes

  1. An explicit formulation for \({\tilde{d}}_s\) will be provided when running simulations in Sect. 4.

  2. According to the DDMRP methodology, the green zone can also take into account the presence of minimum order quantities or order cycles. For more details on these aspects, see Slack et al. (2010) or Jacobs et al. (2011).

  3. Time-varying standard deviations could also be considered, but this would make notations and derivations more cumbersome.

  4. Formally, it is defined a filtered probability space \((\Omega , \mathcal F, (\mathcal F_t)_{0\le t\le T} \mathbb P)\) such that \(L,\,D,\,\Xi \) are \(\mathcal F\)-measurable. \(h_t\) and \(ADU_t\) are, instead, \(\mathcal F_t\) measurable; in this respect, they are “known”at time t.

  5. Concerning constraints (15)–(16), \(\ell \) fictitious values of \(u_s\), \(s=-\ell +1, \dots , 0\) are set to zero.

  6. \(VaR_{\varepsilon }\) is defined as the (smallest) quantity \(x\in \mathbb R\) such that \(\mathbb P(D\ge x)\le \varepsilon \). The parameter \(\varepsilon \) can be fixed considering the decision-maker’s risk aversion; for example, if \(\varepsilon =0.1\), then \(\Phi ^{-1}(1-\varepsilon )\approx 1.28\); if \(\varepsilon =0.3\), \(\Phi ^{-1}(1-\varepsilon )\approx 0.52\) (McNeil et al. 2015). \(\varepsilon = 0.1\) means that an out-of-stock event occurs with probability 10%.

  7. In detail, \(\sqrt{1+z^2}\approx 1+\frac{1}{2} z^2\) and \(e^z=1+z+\frac{1}{2}z^2\), if z is small.

  8. Concerning constraints (23)–(24), \(\ell \) fictitious values of \(u_s\), \(s=-\ell +1, \dots , 0\) are set to zero.

  9. Having set \(q_2=0\), the computation of the peak involves only one period into the future (see Equation (21)).

  10. The seed in the numerical simulation is fixed to ensure the comparability of all instances of the different experiments.

  11. Depending on the lead time, each simulation is run over a time period defined as \(T+\ell \). Ex post, we neglect the first \(\ell \) periods to reach a stationary behavior of inventory, ensuring the comparability of the three experiments. Concerning the computation time, the more demanding case is the one with a small lead time. The software takes approximately 1654s to solve such instance.

  12. As an example, consider the production of small components of great precision (such as lenses for personal devices), requiring ultra precision and micro machining, or high expedition costs (Jáuregui et al. 2010).

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Authors and Affiliations

  1. Department of Management, Università Ca’ Foscari Venezia, Fondamenta San Giobbe 873, Cannaregio, 30121, Venice, Italy

    Daniela Favaretto, Alessandro Marin & Marco Tolotti

  2. Qantica S.r.l., Padua, Italy

    Alessandro Marin

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  1. Daniela Favaretto
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Correspondence to Marco Tolotti.

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The authors acknowledge financial support by Qantica S.r.l.; they thank Raffaele Pesenti for insipring discussions and three anonymous reviewers for their valuable comments on an earlier version of the paper.

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Favaretto, D., Marin, A. & Tolotti, M. A theoretical validation of the DDMRP reorder policy. Comput Manag Sci 20, 8 (2023). https://doi.org/10.1007/s10287-023-00443-5

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