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:

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


  • $Z$ = 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?

  • $\begingroup$ 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? $\endgroup$ – LarrySnyder610 Aug 16 '20 at 20:35
  • $\begingroup$ 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. $\endgroup$ – cmp Aug 17 '20 at 6:55
  • $\begingroup$ "We get an idea of how much stock we can sell" -- how is this different from demand uncertainty? $\endgroup$ – LarrySnyder610 Aug 18 '20 at 1:12
  • $\begingroup$ 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. $\endgroup$ – cmp Aug 18 '20 at 7:01

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.


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