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Calcualation of the Standardized Loss Function The expected number of shortages that occur in one cycle is n(r) = E(max(D− r, 0)). When the demand is normally distributed, n(r) is computed by using the standardized loss function. The standardized loss function L(z) is defined as Z ∞ L(z) = (t − z)φ(t)dt z where φ(t) is the standardized normal density. If lead time demand is normal with mean µ and standard deviation σ, then it can be shown that µ ¶ r−µ n(r) = σL = σL(z). σ The standardized variate z is equal to (r − µ)/σ. Table 1 gives several values of L(z). z 0.00 0.50 1.00 1.70 1.75 2.00 L(z) 0.3989 0.1978 0.0833 0.0183 0.0162 0.0085 Table 1: The Standardized Loss Function The standardized loss function, L(z), can be computed in Excel (or any other program). To do so, write L(z) in the following way: Z ∞ Z ∞ L(z) = (t − z)φ(t)dt = tφ(t)dt − z(1 − Φ(z)) = φ(z) − z(1 − Φ(z)). z z To program this formula into Excel, use the definition of the standard normal density function, which is 1 exp(−0.5z 2 ), φ(z) = p (2π) and the built-in Excel function normdist(), which returns the value of Φ(). In this way one could compute (Q, r) policies within the spreadsheet. Alternatively, one could build a table of L(z), which can then be embedded into a search routine. 1