SummaryThe infinite period stationary inventory model is considered. There is a constant lead time, a nonnegative set‐up cost, a linear purchase cost, a holding and shortage cost function, a fixed discount factor β, 0<β<1, and total backlogging of unfilled demand. Both the total discounted cost (β<1) and the average cost (β= 1) criteria are considered. Under the assumption that the negatives of the one period holding and shortage costs are unimodal, a unified proof of the existence of an optimal (s.S) policy is given. As a by‐product of the proof upper and lower bounds on the optimal values ofsandSare found. New results simplify the algorithm of Veinottand Wagnerfor finding an optimal (s, S) policy for the case β<1. Further it is shown that the conditions imposed on the one period holding and shortage costs can be weakened