State space investigation of the bullwhip problem with ARMA(1,1) demand processes

Using state space techniques we study a “myopic” order-up-to policy. The policy is myopic because it is optimal at minimising local inventory holding and shortage costs. In particular we study the bullwhip effect produced by the replenishment policy reacting to a stochastic ARMA(1,1) demand processes. We reveal that bullwhip is fundamentally caused by the co-variance between the inventory level and the demand forecast. We go on to highlight the impact of a simple control engineering inspired bullwhip reduction technique, a proportional controller in the inventory feedback loop. Although it can be shown this approach is always able to remove bullwhip, we expose that it is not possible to arbitrarily “tune” the proportional controller, without knowing in advance the likely structure of the demand process. We conclude with an investigation of trade-offs and optimisations in the order and inventory variance (and their sum) produced by our policy.