  Motivated by how forecast errors exacerbate order fluctuations in supply
chains, we use tools from robust control theory to characterize and compute the
worst-case order fluctuation experienced by an individual supply chain vendor
under bounded forecast errors and demand fluctuations. Building on existing
discrete time, linear time-invariant (LTI) models of supply chains, we
separately model forecast error and demand fluctuations as inputs to the
inventory dynamics. We then define a transient Bullwhip measure to evaluate the
vendor's worst-case order fluctuation and show that for bounded forecast errors
and demand fluctuations, this measure is equivalent to the disturbance to
control peak gain. To compute the controller that minimizes the worst-case peak
gain, we formulate an optimization problem with bilinear matrix inequalities
and show that solving this problem is equivalent to minimizing a quasi-convex
function on a bounded domain. In contrast to the existing Bullwhip measure in
literature, the transient Bullwhip measure has an explicit dependency on the
forecast error and does not need the forecast to be a deterministic function of
the demand history. This explicit dependency enables us to separately quantify
the transient Bullwhip measure's sensitivity to forecast error and demand
fluctuations. We empirically verify our model for vendors with non-zero
perishable rates and order backlogging rates.
