On the equivalence of the proportional and damped trend order-up-to policies: An eigenvalue analysis

We investigate the equivalence of the order-up-to (OUT) replenishment policy with damped trend forecasting (OUT-DT) to the proportional OUT (POUT) policy via an eigenvalue (zero-pole) analysis. We show when the damped trend forecasting parameters are selected from a specific region of the parameter space, the Bullwhip Avoidance(BA) region, the OUT-DT policy potentially possesses some desirable characteristics. Under an independent and identically distributed (i.i.d.) random demand (the simplest random demand) we show the OUT-DT policy: (a) can eliminate the bullwhip effect, (b) has the same dynamic response as the POUT policy (because of the superposition principle, this holds true for all demands, not just i.i.d. demands), (c) forecasting parameters can be set so as to minimize the sum of the inventory and order variances, and (d) has a similar order and inventory variance as the POUT policy when non-linear constraints are present. We investigate the case when the forecasting parameters are selected from the BA region and correlated demand with one auto-regressive term, one integrated term, and two moving average terms, ARIMA(1,1,2), is present. We reveal the effect of finite lead times on the bullwhip effect (order variance) using an approach based on the order of the eigenvalues (the zeros and poles). We reveal either: (a) the bullwhip effect is always present and always increasing in the lead time or (b) a smoothing effect can be present with short lead times (and the order variance may even be decreasing in the lead time) but the bullwhip effect may return with long lead times.