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)
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
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Let c be the procurement cost, p the price, and g a salvage value. The decision variables are realized sales quantity s_i for all i demand scenarios and target inventory B. Historical demand observation of are represented by D_i. In the objective (5) maximizes the profit, consisting of the sample average revenue from sold and salvaged products minus the purchasing cost. The subsequent constraints ensure that for each scenario sales cannot be larger than the Demand (6) and the inventory (7). Non-negativity constraints for sales and inventory in (8) and