# Reorder point with stochastic lead time and demand

I'm trying to determine the optimal reorder point for some products. The reorder point must be greater than the demand during lead time a $\%$ of the times that I should determine, let's say $95\%$.

Problem is demand is of course stochastic. I have made forecasts using Holt/HW Methods, so I have a mean value $\mathbf{D}$ and then I can check the daily error of that demand on passed events. Giving a distribution of that mean's error of that as this:

Then I have lead time, distributed for example as this:

There's a well-known formula that assumes that both distributions are normal, and thus the reorder point can be easily calculated by using the inverse cumulative distribution:

$$ROP = D\cdot LT + Z(CS)\cdot \sqrt{LT\cdot \sigma_{\text{demand}}^2 + D^2\cdot \sigma_{\text{lead time}}^2}$$ Being $CS$ that $95\%$, $Z$ is the inverse cumulative distribution and $\sigma$ the standard deviation.

But these distributions doesn't seem normal at all, so given that I have the historic data, how could I estimate an aproppiate reorder point for that goal of $95\%$ of times being greater than demand during lead time.

• did you get an answer, i have the same issue – tim Jan 10 '16 at 1:22
• No, I did not... – Pinx0 Jan 10 '16 at 1:43
• can you please share with me where you got this formula your using... my issue is similar to yours I am using the Holts Winter Forecasting Model to predict 1 period, and all the results from testing various items have U-Stat Scores below 1. I already know average and variability for supplier lead time for each item. What formula can I use to include the supplier lead time average, sd, and also the projected sales forecast Known Variables Supplier Lead Time + Sd 1 period out for projected sales Current On Hand Inventory Thank you very much . – tim Jan 10 '16 at 18:21
• The square root formula for the standard deviation of demand over lead time above is widely known and used in inventory control. It's used in commercial software like SAP and Oracle. The derivation is pretty straightforward, but I'm not going to put it into a comment. – jbowman Jan 10 '16 at 20:11