When bullwhip increases in the lead time: An eigenvalue analysis of ARMA demand

Problem definition. The impact of lead times on the bullwhip effect produced by the order-up-to (OUT) replenishment policy is studied. Practical relevance. Under general auto-regressive moving average (ARMA) demand, we investigate when the OUT policy possesses an always-increasing-in-the-lead-time bullwhip effect and when it does not. Methodology. A bullwhip measure based on the difference between the demand and order variance is combined with a novel analysis based on the eigenvalues and impulse response of the ARMA demand process. Contribution. We show a positive demand impulse response is a necessary and sufficient condition for an increasing in the lead time bullwhip effect. The ordering of zeros and poles (the eigenvalue ordering) of the z-transform transfer function of the demand process reveals when the demand impulse is positive. To provide further insight, we study ARMA(2,2) demand, which contains six different eigenvalue orderings. Two of these orderings satisfy a sufficient condition (positive demand eigenvalues in a particular order) for a positive impulse response. Two orderings satisfy the inverse of this sufficient condition and do not possess a positive impulse response. The final two orderings do not satisfy the sufficient condition, nor its inverse, but do contain positive impulse responses. Managerial implications. Our findings are important as reducing lead-times is often advocated as an improvement action to reduce the bullwhip effect. By identifying the demand characteristics that lead to a bullwhip effect that increases in the lead time we offer prescriptive advice on when, and when not, to invest in lead time reduction.