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Mitigating Transient Bullwhip Effects Under Imperfect Demand Forecasts

Sarah H. Q. Li, Florian Dörfler

TL;DR

The work addresses transient demand amplification (Bullwhip) due to forecast errors in supply chains by casting a discrete-time, single-vendor inventory model into a robust control framework. It defines the transient Bullwhip as the worst-case $\ell_\infty$ gain from disturbances to orders, and derives a computable upper bound using inescapable ellipsoids, with the bound scaling as $\hat{\epsilon}\sqrt{\epsilon_f^2+(\epsilon_f+\epsilon_d)^2}$. A forecast-driven affine controller is shown to minimize this bound; the nonconvex BMI is handled via a quasi-convex function $f(\lambda)$ enabling a bisection search to obtain the peak-gain-minimizing policy. Simulations reveal a monotone dependence on the parameter $\lambda$ and indicate forecast quality primarily affects inventory fluctuations, while order fluctuations remain tightly controlled under the proposed controller. The approach is extensible to multi-vendor settings and offers a practical, robust-control-based method to mitigate transient Bullwhip effects.

Abstract

Motivated by how forecast errors exacerbate order fluctuations in supply chains, we leverage robust feedback controller synthesis to characterize, compute, and minimize the worst-case order fluctuation experienced by an individual supply chain vendor. Assuming bounded forecast errors and demand fluctuations, we model forecast error and demand fluctuations as inputs to linear inventory dynamics, and use the $\ell_\infty$ gain to define a transient Bullwhip measure. In contrast to the existing Bullwhip measure, the transient Bullwhip measure explicitly depends on the forecast error. This enables us to separately quantify the transient Bullwhip measure's sensitivity to forecast error and demand fluctuations. To compute the controller that minimizes the worst-case peak gain, we formulate an optimization problem with bilinear matrix inequalities and show that it is equivalent to minimizing a quasi-convex function on a bounded domain. We simulate our model for vendors with non-zero perishable rates and order backlogging rates, and prove that the transient Bullwhip measure can be bounded by a monotonic quasi-convex function whose dependency on the product backlog rate and perishing rate is verified in simulation.

Mitigating Transient Bullwhip Effects Under Imperfect Demand Forecasts

TL;DR

The work addresses transient demand amplification (Bullwhip) due to forecast errors in supply chains by casting a discrete-time, single-vendor inventory model into a robust control framework. It defines the transient Bullwhip as the worst-case gain from disturbances to orders, and derives a computable upper bound using inescapable ellipsoids, with the bound scaling as . A forecast-driven affine controller is shown to minimize this bound; the nonconvex BMI is handled via a quasi-convex function enabling a bisection search to obtain the peak-gain-minimizing policy. Simulations reveal a monotone dependence on the parameter and indicate forecast quality primarily affects inventory fluctuations, while order fluctuations remain tightly controlled under the proposed controller. The approach is extensible to multi-vendor settings and offers a practical, robust-control-based method to mitigate transient Bullwhip effects.

Abstract

Motivated by how forecast errors exacerbate order fluctuations in supply chains, we leverage robust feedback controller synthesis to characterize, compute, and minimize the worst-case order fluctuation experienced by an individual supply chain vendor. Assuming bounded forecast errors and demand fluctuations, we model forecast error and demand fluctuations as inputs to linear inventory dynamics, and use the gain to define a transient Bullwhip measure. In contrast to the existing Bullwhip measure, the transient Bullwhip measure explicitly depends on the forecast error. This enables us to separately quantify the transient Bullwhip measure's sensitivity to forecast error and demand fluctuations. To compute the controller that minimizes the worst-case peak gain, we formulate an optimization problem with bilinear matrix inequalities and show that it is equivalent to minimizing a quasi-convex function on a bounded domain. We simulate our model for vendors with non-zero perishable rates and order backlogging rates, and prove that the transient Bullwhip measure can be bounded by a monotonic quasi-convex function whose dependency on the product backlog rate and perishing rate is verified in simulation.
Paper Structure (16 sections, 5 theorems, 35 equations, 4 figures)

This paper contains 16 sections, 5 theorems, 35 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Supply chain Inventory Model
  3. Steady-state: perfect forecast and stationary demand
  4. Imperfect forecast and noisy demand
  5. Transient Bullwhip as peak-to-peak gain
  6. Bounding the transient Bullwhip measure
  7. Bullwhip-minimizing feedback control design
  8. Simulated controller performance
  9. Impact of backlog and perishing rates
  10. Controller performance under different forecast oracles
  11. Conclusion
  12. Appendix
  13. Proof of Lemma \ref{['lem:normalized_gain']}
  14. Proof of Theorem \ref{['thm:ellipsoid_bound']}
  15. Proof of Theorem \ref{['thm:output_gain_bmi']}
  16. ...and 1 more sections

Key Result

Lemma 1

The inventory dynamics eqn:stationary_lti is stable if and only if $|\lambda_{+}| < 1$ and $|\lambda_-| < 1$, where $\lambda_{\pm}$ are given by where $a = \gamma_P - \alpha$ and $b = \beta - 1$.

Figures (4)

  • Figure 1: Comparing the transient behavior of two ordering schemes.
  • Figure 2: Peak gains $\sqrt{f(\lambda)}$ as a function of $\lambda$ for perishing rate $\beta = 0.5$ (top) and backlogging rate $\alpha = 0.5$ (bottom) and varying backlog rates.
  • Figure 3: Maximum order fluctuations over $1000$ time steps under increasing forecast error, $\epsilon_f \in [0, \epsilon_d]$. The order fluctuates below $0.04$ and the theoretical bound $\gamma^\star \sqrt{\epsilon_f^2 + (\epsilon_f + \epsilon_d)^2}$ monotonically increases between $0.1 - 0.2$.
  • Figure 4: Maximum inventory fluctuations over $1000$ time steps as a function of increasing forecast error, $\epsilon_f \in [0, \epsilon_d]$.

Theorems & Definitions (13)

  • Lemma 1: Steady-state stability
  • Example 1: Non-perishable goods
  • Example 2
  • Definition 1: Transient Bullwhip Measure
  • Definition 2: Inescapable ellipsoid
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • ...and 3 more