The multivariate bullwhip effect

A multivariate bullwhip expression for m products with an order-up-to inventory policy is developed. The demand models under consideration are differenced stationary vector time series with a Wold representation for which general forecasting formulas are available, resulting in a large class of possible models (including nonstationary ones). Examples are provided for common demand models and implemented on sales data. It is found that the multivariate approach gives rise to mechanisms for understanding and reducing the bullwhip effect through horizontal information sharing, particularly for the nonstationary demand case. In the stationary setting, a more nuanced approach to bullwhip reduction can be achieved by managing the relationship between cross-correlations and lead-times. A method of determining whether a multivariate or univariate approach generates a lower bullwhip effect is proposed.