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

Abstract

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 behavior for slow-moving integer demand. We forecast this 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 incoherent with the integer demand. However, the conditional median forecasts are integer-valued and produce coherent integer order and inventory levels. The conditional median forecasts minimize the expected absolute forecasting error, but it is not possible to obtain closed forms for the variances. 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. Conditional median bullwhip can be both lower and higher than the conditional mean bullwhip; 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 autoregressive, 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.