During future demand calculations, the safety stock concept is usually used.
One way to model the safety stock is the following:
SS = Z * sqrt(PC) * std(D)
Where, SS is the safety stock, Z is the Z-score for a given product, PC is the performance cycle (or total lead time), and std(D) is the demand standard deviation.
My question regards why the formula uses sqrt of the Lead Time instead of the pure Lead Time?
Any proof or any book/paper that shows the main idea why to use the sqrt instead of the total leadtime?
Best Answer
As Glen mentioned, additivity of variances when samples are independent: VAR(ND) = NVAR(D), with VAR = S^2 then S(N*D) = SQRT(N)*S(D)
In statistics, the standard error SE or standard deviation for a sample mean is given by the formula SE = S/SQRT(N) for the same reason as above, where N is the sample size and S is the standard deviation of the sample.
Applying this to supply chain, the standard deviation over a lead time of N days is equivalent as pulling a sample of N days and measuring the standard error of its daily demand: Average SE for N days = S(D)/SQRT(N), where S(D) is the daily demand standard deviation. Notice this is an average daily standard error.
The total SE over the length of N days lead time is N times that value = SE = N*S(D)/SQRT(N) = SQRT(N)*S(D)
The safety stock is then simply a measure of how many standard deviations away from the mean you want to be = Z*SE (standard error over the lead time).