I recently have come across a problem which can be categorized as a stochastic optimization. The problem seems simple, but I haven't been able to solve it yet. It has a major impact on algorithm design for real business problems.
Assume that there is a retailer who faces a stationary/stochastic demand with a known discrete distribution. Let $K$ denote the fixed cost of ordering, and $h$ and $b$ denote the holding and back-order cost of the retailer respectively. It can be shown that the optimal solution for this problem over an infinite horizon follows $(s, S)$ policy in which whenever the inventory level of the retailer falls below $s$, it brings it back to $S$ by setting an order.
What I like to know here is, if there is any upper-bound for the value of $S$ (based on $K$, $h$, and $b$)?
Extension: The problem has two stages, and in the upper stage, there is a supplier who has to respond to the orders. Assuming $K$s, $h$s and $b$s as the setup cost of the supplier, holding and back-order costs of the supplier, and the problem is solved in an integrated manner, what is the bound for the $s$ and $S$?