# Estimate of mean lead time demand

I am trying to estimate the order-up-to level of inventory, $$y_t$$, according to ref [1]

$$y_t = \hat{D}_t^L + z \hat{\sigma}^L_{et}$$

where $$\hat{D}_t^L$$ is an estimate of the mean lead-time demand, $$\hat{\sigma}^L_{et}$$ is n estimate of the standard deviation of the $$L$$ period forecast error, and $$z$$ is a constant chosen to meet a desired service level. Assume that the retailer uses moving average to estimate $$\hat{D}_t^L$$ based on the demand observations from the previous $$p$$ periods, then

$$\hat{D}_{t}^{L}=L\left(\frac{\sum_{i=1}^{p} D_{t-i}}{p}\right)$$.

According to this equation, the estimated mean lead-time demand seems to only cover the demand during the lead time.

However, I checked out the definition from Lokad Quantitative Supply Chain[2], "The lead demand (also called lead time demand) is the total demand between now and the anticipated time for the delivery after the next one if a reorder is made now to replenish the inventory." This seems to mean that the mean lead-time demand should cover the demand during the lead time in this period plus the demand during the next period. This definition makes sense because assuming the safety stock factor is zero, the estimated order-up-to level is equal to the estimated lead-time demand, which should cover the demand from the current reorder time to the delivery time after the next one.

Can anyone help to clarify the definition of lead-time demand?

[1] Chen, F., Drezner, Z., Ryan, J. K., & Simchi-Levi, D. (2000). Quantifying the bullwhip effect in a simple supply chain: The impact of forecasting, lead times, and information. Management science, 46(3), 436-443.

The lead-time demand is the cumulative demand in $$L+R$$ consecutive periods, where $$L$$ is the lead time and $$R$$ is the reorder interval (the number of periods between orders). (At least, that's how we define it in our book, and I think that definition is consistent with other usages elsewhere.) Often $$R=1$$ and the lead-time demand is the demand over $$L+1$$ periods.

In the Chen et al. paper, they use a somewhat non-standard sequence of events, in which the demand is observed before the order is placed. A lead time of $$L$$ in the Chen sequence of events is effectively like a lead time of $$L-1$$ in the "usual" sequence of events (in which you observe the demand after the order is placed).

So, for Chen et al., we have a reorder interval of $$R$$ and an "effective" lead time of $$L-1$$, so the lead-time demand is the demand over $$L$$ periods.

I don't know why the website you linked to says "the delivery after the next one". Maybe they mean "the delivery after the next order", i.e., "the next delivery"?

I ran this past a colleague in supply chain management who confirmed my understanding of "demand during lead time" to be the demand that occurs between when a replenishment order is placed and when it arrives. I don't recall seeing it defined to include the demand occurring between orders, and in particular I think that in the (unusual? purely hypothetical?) case of immediate replenishment, the lead time demand is considered to be zero.

Regarding "the delivery after the next one", there are cases where orders overlap (you place a replenishment order and, before it shows up, place another one). As my colleague reminded me, this is usually handled by basing your reorder point on "inventory position", which includes both on-hand inventory and replenishment orders already made, rather than just on on-hand inventory.

• Thanks! I also think lead-time demand is the cumulative demand during the lead time. Per LarrySnyder610's answer, the estimated mean lead-time demand is used to determine the order-up-to level, so it should cover the cumulative demand during the lead time and the next review period/reorder interval. As such, the estimated mean lead-time demand is different from the demand during lead time. Commented Oct 23, 2021 at 14:21