The bullwhip effect with correlated lead times and autocorrelated demand

We quantify the bullwhip effect in a two-echelon supply chain when demand follows a first-order autoregressive random process and the lead times form a correlated stationary sequence of random variables. We assume future demands are predicted with the minimum mean squared error method; the random lead times can be estimated using any method. We analyse the impact of the autocorrelated demands and autocorrelated lead times on the bullwhip effect. We consider several cases of mutual lead time dependence, such as a first-order autoregressive random process and a first-order integer autoregressive random process. We explore the use of naïve, moving average, minimum mean squared error, and exponential smoothing forecasting methods for predicting lead times. We show how the bullwhip is influenced by demand correlation, lead time autocorrelation, and the parameters of the lead time forecasts. We reveal that minima and maxima exist in the bullwhip effect. With moving average forecasts of negatively correlated lead times, we observe an even-odd phenomenon in the bullwhip measure. Theoretical results are confirmed by a Monte Carlo simulation and a practical approach. We also derive a formula for the variance of the lead time demand forecast error.