I am currently an intern working on an inventory/stock policy and there is this particular modeling task on which I am blocked (and I have no statistical supervisor). I have a background mostly in economics with some applied econometrics, and I am out of my depths here.
AS IS: They use a safety stock formula (the type 1, see wiki) to calculate their service level. This formula gives them a Z value (K in the formulate) that they plug into a standard normal distribution in order to get a theoretical service level (the minimum being 50% when safety stocks are 0 (K = 0)). https://en.wikipedia.org/wiki/Safety_stock They apply this for a calculated level of demand per leadtime and they want to apply this formula for demand that is assumed to follow a poisson or negative binomial distribution (low demand (few points), lots of zero demand periods).
So far, I have different ideas.
For example, apply a Johnson transformation on my data to normalise them, but it doesn't always work (using R packages like suppdist or jtrans). And then add the mean of the demand to K (to reach the minimum of 50%). A boxcox transformation could also be considered.
Another idea was to use the data to find the parameters, simulate the distribution (still problematic with small data) and shift the distributions to center at 0. The idea would be to reuse the K above. I did not find how to do such a shift to be honest or how to modify K so that it reaches the right values (min 50%).
Side note: Oracle Inventory Optimization and other APS seem to have a way to calculate the right negative binomial and poisson models, but I can't seem to find their technique...
I kindly ask for criticisms and hopefully improvement ideas. Ideally, I would like to know how to use the same principle used on the normal distribution (formula with K) and apply it to the Negative binomial and Poisson distributions.
EDIT: Problem solved. As suggested by jbowman, applying the negative binomial on lead time demand works, I had to use the parameters calculated using the variance to mean ratio and the mean of lead time demand. The program (APS) whose results I had to reproduce used a maximum VMR in its parameters. Finally, for the poisson distribution I went with a Hermite distribution (assuming poisson demand and normal lead time).