Dynamic analysis of the proportional order-up-to policy with damped trend forecasts

We study the bullwhip behaviour in the proportional order-up-to (POUT) policy with non-stationary autoregressive integrated moving average (ARIMA) demand. We build a state-space model of the POUT policy where the damped trend forecasting method predicts ARIMA(1,1,2) demand. The POUT policy is closely related to the order-to-up (OUT) policy with the addition of a proportional feedback controller in the inventory and work-in-progress feedback loops. Our modelling approach allows us to derive and/or analyse the demand, order, and inventory variances. We also find the covariance between the demand forecast and the inventory forecast in an attempt to obtain the order variance. However, both the demand and the order variances are infinite under the non-stationary ARIMA(1,1,2) process. Thus, the traditional bullwhip measure (the ratio of the order variance divided by the demand variance) is indeterminate. Despite this difficulty, we can study the difference between the order variance and the demand variance for both the OUT and POUT policies. These differences are finite and their sign indicates whether a bullwhip effect has been generated or not. We find under non-stationary demand, the POUT policy’s bullwhip behaviour contradicts some of the existing bullwhip theory. The POUT policy sometimes generates more bullwhip than the OUT policy, revealing that existing knowledge based on stationary demand should be used with caution in non-stationary demand environments. We validate our findings with an investigation of some ARIMA(1,1,2) time series from the M4 competition.