 # Newsvendor Problem Case Study

The newsvendor problem is a mathematical model which is used to determine the optimal stock under uncertainty. In the following, the newsvendor context under cost minimization will be introduced.
Let h be the unit holding cost respectively the unit overage cost (as we regard the pure cost context) and b the unit penalty of not serving demand (or unit backorder cost) respectively the unit underage cost. Then, the target inventory B is equal to the mean demand µ plus safety stock SS. The safety stock consists of k times the standard deviation σ. In the basic newsvendor case the optimal k is defined by the inverse of the assumed demand distribution of the critical fractile cf (underage cost divided by underage plus overage cost).

B= μ+SS (1)

B= μ+kσ (2)

k=F^(-1) (cf) (3)

cf=b/(b+h) (4)

With only sample historical demand observations at hand, in addition to distributional assumptions about f, estimations for the mean and the standard deviation have to be made in order to determine the order amount according to the formulas (1) - (4). This can be done by estimating mean and standard deviation directly out of the sample with the method of moments or as a function of hypothetical explanatory variables using ordinary least squares (OLS) regression.
The Small Data-Driven Newsvendor
In practice, the true underlying