The newsvendor problem with normal demand

We know from Section 7.4.4 that the optimal solution of a newsvendor problem with continuous demand is the solution of the equation

i.e., the quantile of demand distribution, corresponding to probability m/(m+c_μ). If we assume normal demand, with expected value μ and standard deviation σ, then the optimal order quantity (assuming that we want to maximize expected profit) is

Assume that items are purchased from a supplier for $10 per item and then are sold at $15, and that the salvage value of unsold items is $3. The expected value of demand over the sales window is 10,000 items, and its standard deviation is 2000 items. Then we find

Note that service level is lower than 50%, so the corresponding quantile from the standard normal distribution is negative, and we should buy less than expected demand. Indeed, statistical software yields

Note that, since the profit margin is low with respect to the cost of unsold items, we should be conservative; the larger the risk, measured by standard deviation, the less we buy.