# Calcualation of the Standardized Loss Function

```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 &micro; and standard deviation σ, then it can be shown that
&micro;
&para;
r−&micro;
n(r) = σL
= σL(z).
σ
The standardized variate z is equal to (r − &micro;)/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
```