# Safety stock calculation with production forecast variance

I am trying to find an safety stock calculation where the expression incorporates:

• Lead time variance
• Sales demand variance &
• Production forecast variance

My calculation so far is based on the first two:

$$\text{Safety stock}=Z\sqrt{\left( \frac{PC}{T} \times \sigma_D^2 \right) + (\sigma_{LT} × \mu_D)}$$

where

• $$Z$$ = Z score
• $$PC$$ = Lead time
• $$\sigma_D$$ = Std of sales demand
• $$\sigma_{LT}$$ = Std of lead time
• $$\mu_D$$ = Mean sales demand

I would be very grateful if anyone could sense check this and let me know how historical production forecast error could be included please?

• What do you mean by “production forecast”? Do you mean that the production quantity is random? I.e., if you order Q, you might receive something other than Q? – LarrySnyder610 Aug 16 '20 at 20:35
• We work on a push system, so stock is replenished every day i.e. no ordering/reordering. We get an idea of how much stock we can sell for contractual and discretionary sales by way of a forecast. This is of daily granularity but is updated weekly. It also possesses considerable error, which this uncertainty of which, I would like to embed in the SS calculation. – cmp Aug 17 '20 at 6:55
• "We get an idea of how much stock we can sell" -- how is this different from demand uncertainty? – LarrySnyder610 Aug 18 '20 at 1:12
• Perhaps I should reword that to the amount of stock 'available' to sell. We get an idea of our production one week in advance: i.e. in Week 1 you will have 10 tonnes to sell, in week 2 you will have 20 tonnes. Hopefully this is clear. – cmp Aug 18 '20 at 7:01

## 1 Answer

It sounds like you have a problem that involves three types of uncertainty:

1. Demand uncertainty
2. Lead time uncertainty
3. Yield uncertainty

Yield uncertainty is a somewhat general term that refers to uncertainty in the amount of supply available. (Related terms include capacity uncertainty and supply disruptions.)

With uncertainty sources #1 and 2, you can use the formula you listed. I am not aware of models that combine all 3 sources, but you might search the literature for those three terms simultaneously.

It might also be possible to merge #1 and 3: If the demand is $$D$$ and the supply is $$S$$, then the random variable of interest is really $$D-S$$, and you might be able to formulate the problem using that r.v. in place of the demand.