On (s, S) Policies - Johnson, Ellis L. (Management Science, 1968)
For the infinite horizon, single item inventory problem, assume an ordering cost M(i) + K(j) for changing the stock level from i to j (for example, M(i) + K(j) = c · (j − i) + K), a holding plus shortage cost l(j), and a probability φ (j, k) of demand j − k when the stock level is j. New conditions for optimality of (s, S) policies are given, and a computational method is given. Both the total discounted cost and the average cost criteria are treated. The method is simplified when the demand distribution is Bernoulli or geometric. When, in addition, the costs K(j) = K, and l(x) is −Lx for x > 0, zero for 0 ≦ x ≦ I, for some I ≧ 0, and H(x − I) for x > I, then conditions are given for an optimal policy (s*, S*) to have s* = −1 or S* = I. Further, for the same assumptions but without the integrality restriction on s and S, specific formulas are given for s* and S* when the criterion is average cost; and equations, which can be solved for s* and S*, are given when the criterion is total discounted cost.