# Confusion about how to account for leadtimes with a fixed-review period base stock(S,T) inventory policy

So we're using a fixed review period model with base stock $$S$$, and review period $$T$$ of 1 month - the period is not chosen optimally to minimise ordering cost like in an EOQ model, it's just a nice round number.

I'm trying to work out how much to order each month. My modelling assumptions are:

• In one month/ review period $$T$$, $$\mu_T$$ items are sold, and assuming demand is stochastic, I can calculate a $$\sigma_T$$
• If that's the demand & std.dev in 1 month, then by the central limit theorem, the demand & std.dev over a leadtime $$L$$ are given by
• $$\mu_{T+L}=\mu_T \cdot (T + L)$$
• $$\sigma_{T+L}=\sigma_T \cdot \sqrt{T+L}$$

Finally, I calculate the base stock $$S = \mu_{T+L} + z\cdot \sigma_{T+L}$$ where $$z$$ is some safety factor chosen based on a 95% service level. (see these notes I found, slide 21)

I'm slightly confused why this is. It looks like $$S$$ is set to cover the demand for 1 period + retrospective demand that happened over the leadtime. Isn't this double counting, I would have expected $$\mu_T$$, not $$\mu_{T+L}$$, since each order must be enough to cover the demand between now and the next order, which is $$T$$ away?

On this graph for example: It feels like accounting for $$L$$ would just shift it in time by that much, but the spacing between verticals is still $$T$$ so surely that's all the demand $$S$$ needs to account for, not $$T+L$$

The safety factor makes more sense; every review period must cover its own fluctuations in demand from $$\sigma_{T}$$ as well as any uncertainty that occurs while the next review period is waiting for its order to arrive.

Can someone provide some intuition? Why is $$S$$ set that way?

Let's say you start from zero and you order up to $$S$$ at time $$t$$. Then you know that your next inventory will come in at time $$t+T+L$$, because you order at time $$t+T$$ and it takes $$L$$ time to arrive.
An important assumption in this order strategy is that you won't inspect inventory during two consecutives $$T$$ periods.
So only at periods $$T-1$$, $$T$$, $$T+1$$, ... you will observe inventory. And at that point you will need to order up to $$S$$, knowing that you will order at $$T+1$$ at the earliest and it always takes $$L$$ time units to arrive.