  This paper studies a continuous-review backlogged inventory model considered
by Helmes et al. (2015) 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 $(s,S)$-type policy obtained by Helmes
et al. (2015) 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 can not 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 $(s,\{S(x):x\leq
s\})$ policy for the other initial levels. Under $(s,\{S(x):x\leq s\})$ policy,
it orders nothing for $x>s$ and orders up to level $S(x)$ for $x\leq s$, where
$S(x)$ is a non-constant function of $x$. We further prove the optimality of
such $(s,\{S(x):x\leq s\})$ 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 common lower bound approach.
