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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: enter image description here 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?

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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.

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