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Certified Inventory Control of Critical Resources

Ludvig Hult, Dave Zachariah, Petre Stoica

TL;DR

The paper tackles inventory control under unknown, time-dependent demand by proposing a data-driven order policy augmented with integral action to certify a target service level $1-\alpha$ and by developing a finite-sample valid method to infer future operating costs with coverage $1-\beta$. A base predictor $\widehat{W}(\mathcal{D}_t)$ couples with a nonlinear gain $g_t(E_t)$ to produce an admissible policy $\mu^{\alpha}(\mathcal{D}_t)$ that bounds stock-outs via a count $E_t$ of critical events. For costs, nominal prediction intervals are adaptively widened with a similar gain mechanism to guarantee valid cost inference. The approach is validated on synthetic scenarios (periodic, spiking, and feedback-demand) and a real electricity-demand dataset, demonstrating high service levels and correct cost-coverage, with discussion on parameter choices and potential extensions. This yields a practically impactful framework for robust inventory control with transparent probabilistic guarantees under very weak assumptions on demand dynamics.

Abstract

Inventory control is subject to service-level requirements, in which sufficient stock levels must be maintained despite an unknown demand. We propose a data-driven order policy that certifies any prescribed service level under minimal assumptions on the unknown demand process. The policy achieves this using any online learning method along with integral action. We further propose an inference method that is valid in finite samples. The properties and theoretical guarantees of the method are illustrated using both synthetic and real-world data.

Certified Inventory Control of Critical Resources

TL;DR

The paper tackles inventory control under unknown, time-dependent demand by proposing a data-driven order policy augmented with integral action to certify a target service level and by developing a finite-sample valid method to infer future operating costs with coverage . A base predictor couples with a nonlinear gain to produce an admissible policy that bounds stock-outs via a count of critical events. For costs, nominal prediction intervals are adaptively widened with a similar gain mechanism to guarantee valid cost inference. The approach is validated on synthetic scenarios (periodic, spiking, and feedback-demand) and a real electricity-demand dataset, demonstrating high service levels and correct cost-coverage, with discussion on parameter choices and potential extensions. This yields a practically impactful framework for robust inventory control with transparent probabilistic guarantees under very weak assumptions on demand dynamics.

Abstract

Inventory control is subject to service-level requirements, in which sufficient stock levels must be maintained despite an unknown demand. We propose a data-driven order policy that certifies any prescribed service level under minimal assumptions on the unknown demand process. The policy achieves this using any online learning method along with integral action. We further propose an inference method that is valid in finite samples. The properties and theoretical guarantees of the method are illustrated using both synthetic and real-world data.
Paper Structure (14 sections, 3 theorems, 30 equations, 5 figures)

This paper contains 14 sections, 3 theorems, 30 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Method
  4. Error Bound Functions
  5. Admissible policy
  6. Valid cost inference
  7. Numerical experiments
  8. Synthetic data
  9. Periodic demand
  10. Spiking demand
  11. Feedback demand
  12. Electricity dataset
  13. Discussion
  14. CRediT authorship contribution statement

Key Result

Lemma 1

Consider any error bound function $b(t)$ with an associated gain $g_t(E_{})$. If a saturated gain $g_t(E_t) \geq g_{\circ}$ implies no error growth, i.e., $E_{t+1} =E_{t}$, then the error process is bounded as

Figures (5)

  • Figure 1: Inventory control under an unknown (periodic) demand process. Top: Stock level and orders by policy $\mu^\alpha$ that is certified to have a service level of at least $1-\alpha=95\%$. The empirical service level is larger than $96\%$. Bottom: Policy operating costs $C^{H}_{t}$ over $H=5$ steps ahead and a prediction interval $\mathcal{C}^{\beta}$ with a certified coverage level of $1-\beta=95\%$. For a full experimental description, see Section \ref{['sec:experiment noisey sine']}.
  • Figure 2: Stock, purchase and cost estimate for the SIR model. The empirical service level is $98.7\%$ exceeding the prescribed $95\% = 1-\alpha$. The cost prediction interval covers the true cost in $97\%$ of cases, exceeding the prescribed $95\% = 1-\beta$.
  • Figure 3: Stock, purchase and cost estimate for the feedback demand model. The empirical service level is $96.3\%$ exceeding the prescribed $95\% = 1-\alpha$. The cost prediction interval covers the true cost in $97.7\%$ of cases, exceeding the prescribed $95\% = 1-\beta$.
  • Figure 4: Stock, purchase and cost estimates using the demand recorded in the Elec2 dataset (and is ormalized to [0,1].) The empirical service level is $99.8\%$ exceeding the prescribed $95\% = 1-\alpha$. The cost prediction interval covers the true cost in $97\%$ of cases, exceeding the prescribed $95\% = 1-\beta$.
  • Figure 5: Error processes $E_{t}$ for experiments on the Elec2 dataset, which are provably bounded by $b(t)$ in \ref{['eq:defn:flexible error bound fun']} with parameters $T_{*}$, $b_*$ and $\alpha$. Top: $E_{t}$ is the number of critical stock events. The default $b_*=2$$T_{*}=0$ are used. Bottom: $E_{t}$ is the number of observed ($\widetilde{E}_t$) plus potential miscoverage events in the cost inference. The default $b_*=H$ and $\alpha=\beta$.

Theorems & Definitions (10)

  • definition 1: Error bound function
  • definition 2: Associated gain
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • remark 1
  • remark 2
  • proof