Expected shortage per replenishment cycle

Establish an alternative formula for expected shortage per replenishment cycle (ESC) to be evaluated using Excel. 

Given that the demand during lead time is normally distributed with a mean Di and a standard deviation o-, we have (using Equation 12.21) 

Recall that F. ) is the cumulative distribution function and/ C.) is the probability density function for the standard normal distribution with mean 0 and standard deviation 1. Using Equation 12.21 and the definition of the standard normal distribution, we have 

Substitute w = vjl into the expression for ESC. This implies that 

Using Equations 12.25 and 12.26, ESC may be evaluated using Excel as follows  

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The expected shortage per replenishment cycle (ESC) is the average units of demand that are not satisfied from inventory in stock per replenishment cycle. Given a lot size of Q (which is also the average demand in a replenishment cycle), the fraction of demand lost is thus ESC/ Q. 

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