On the order-up-to policy with intermittent integer demand and logically consistent forecasts

We measure the impact of a first-order integer auto-regressive, INAR(1), demand process on order-up-to (OUT) replenishment policy dynamics. We obtain a unique understanding of the bullwhip behaviour for slow moving integer demand. We forecast the integer demand in two ways; with a conditional mean and a conditional median. We investigate the impact of the two forecasting methods on the bullwhip effect and inventory variance generated by the OUT replenishment policy. While the conditional mean forecasts result in the tightest inventory control, they result in real-valued orders and inventory levels which is inconsistent with the integer demand. However, the conditional median forecasts are integer-valued and produce logically consistent integer order and inventory levels. The conditional median forecasts minimise the expected absolute forecasting error, but it is not possible to obtain closed form variance expressions. Numerical experiments reveal existing results remain valid with high volume correlated demand. However, for low volume demand, the impact of the integer demand on the bullwhip effect is often significant. Bullwhip with conditional median forecasts can be both lower and higher than with conditional mean forecasts; indeed it can even be higher than a known conditional mean upper bound (that is valid for all lead times under real-valued, first-order auto-regressive, AR(1), demand), depending on the auto-regressive parameter. Numerical experiments confirm the conditional mean inventory variance is a lower bound for the conditional median inventory variance.