IMPULSE CONTROL WITH DISCONTINUOUS SETUP COSTS: DISCOUNTED COST CRITERION

This paper studies a continuous-review backlogged inventory model considered by [K. L. Helmes, R. H. Stockbridge, and C. Zhu, SIAM J. Control. Optim., 53 (2015), pp. 2100-2140] but with discontinuous quantity-dependent setup cost for each order. In particular, the setup cost is characterized by a two-step function and a higher cost would be charged once the order quantity exceeds a threshold Q. Unlike the optimality of the (s, S)-type policy obtained by Helmes, Stockbridge, and Zhu for continuous setup cost with the discounted cost criterion, we find that, in our model, although some (s, S)-type policy is indeed optimal in some cases, the (s, S)-type policy cannot always be optimal. In particular, we show that there exist cases in which an (s, S) policy is optimal for some initial levels but it is strictly worse than a generalized (s, {S(x) : x <= s}) policy for the other initial levels. Under (s, {S(x) : x <= s}) policy, it orders nothing for x > s and orders up to level S(x) for x <= s, where S(x) is a nonconstant function of x. We further prove the optimality of such (s, {S(x) : x <= s}) policy in a large subset of admissible policies for those initial levels. Moreover, the optimality is obtained through establishing a more general lower bound theorem which will also be applicable in solving some other optimization problems by the lower bound approach.