# Case Study Of The Newsvendor Problem

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)

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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

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The Big Data-Driven Newsvendor

A standard approach in Econometrics to model historical demand observations is to estimate the relationship between demand and any hypothetical explanatory variables via the ordinary least squares (OLS) regression. In the single variable case Demand D_i is expressed by a (linear) function of explanatory variable X_1, the slope(s) β_1, plus a y-axis intercept β_0 and an error term ϵ (15). Given the observations (X_i,D_i), the OLS estimators β_0 and β_1,are represented in formulas (16) and (17). (see, e.g. Gujarati and Porter, 2009) [look up reference]

D_i=β_0+ β_1 X_1+ϵ_i (15)

β ̂_1=(∑_(i=1)^(n )▒〖(X_i-X ̅ )(D_i-D ̅)〗)/(∑_(i=1)^n▒(X_i-X ̅ ) ) (16)

β ̂_0= D ̅-β ̂_1 X ̅ (17)

The extension to multiple variables, provides the possibility to incorporate big amount of data (features). The linear demand function becomes

D_i= β_0+ β_j X_ji+ϵ_i