The order-up-to policy sweet spot: Using proportional controllers to eliminate the bullwhip problem
Abstract
We develop a discrete control theory model of a stochastic demand pattern with both Auto Regressive and Moving Average (ARMA) components. We show that the bullwhip effect arises when the myopic Order-Up-To (OUT) policy is used. This policy is optimal when the ordering cost is linear. We then derive a set of z-transform transfer functions of a modified policy that allows us to avoid the bullwhip problem by incorporating a proportional controller into the inventory position feedback loop. With this technique, the order variation can be reduced to the same level as the demand variation. However, bullwhip-effect avoidance in our policy always comes at the costs of holding extra inventory. When the ordering cost is piece-wise linear and increasing, we compare the total cost per period under the two types of ordering policies: with and without bullwhip effect reduction. Numerical examples reveal that the cost saving can be substantial if order variance is reduced using the proportional controller.