## Overstock in pure Pull supply chains

I have had a couple of conversations recently that have led me to think about how much overstock we might expect in a Pull supply chain even under fairly idealistic conditions. The first was with a colleague working on a redesign of a warehouse in which a large number of products had stock outside of their reorder point/order-up-to parameters. He took this to be evidence of poor inventory control. The second was with a client whose inventory performance has improved remarkably, but whose faster-moving product showed a level of overstock to the parameters – was it realistic to try and reduce that?

## Basic Pull inventory model

Let’s set up the situation: we have a single stocking location serving customers who expect immediate availability (so we have to hold stock). Demand for each product is stable (no increasing or decreasing trends or seasonality) and follows a Normal distribution – the former is unrealistic, but the latter is a good assumption for all but slow-moving products.

Replenishment of stock from suppliers is controlled by a reorder-point/order-up-to Pull system. The reorder point is *safety stock* + *expect demand over lead time*. The order-up-to level is *reorder point* + *order quantity*. We would expect our physical stock to cycle between the *safety stock* and the *safety stock* + *order quantity*, and that our average stock holding would be *safety stock* + ½ *order quantity*.

## Demand variability

Anybody who manages supply chain inventory is familiar with the idea that – all else being equal – the level of safety stock determines the availability of a part. We hold safety stock because of the irreducible randomness in demand – the reorder point assumes an expected demand over the lead-time, and a safety stock allowance for the variability of that lead-time demand. So we accept that for some of the time, our stock will be below the planned minimum (safety stock) because demand will be higher than expected.

But this works the other way too: demand can be lower than expected. In the worst case our customers might demand none at all over the lead time. Our expected maximum stock level would then be *safety stock* + *lead-time demand* + *order quantity*, instead of our planned level of *safety stock* + *order quantity*.

## Estimating expected overstock

I am going to skip the statistical proofs – they are not too hard, but they would make this a long post and I think this is getting involved enough for most people. If anyone wants more detail, let me know.

The expected level of “under-demand” – by how much real demand fails to meet our expected lead-time demand – is a factor multiplied by the standard deviation of lead-time demand. For fairly large order quantities, this factor turns out to be about 0.4, a little lower for smaller order quantities, but let’s use 0.4 for now.

This means that the percentage of time any particular product will be over-stocked will be given by 0.4 x *standard deviation of leadtime demand* / *order quantity*.

If we have all this information for our stocked range, we could in principle run this equation for every product, and that would provide us with a “best case” overstock level to assess our performance.

## Simplified example

Suppose a product sells 100 each month, has a lead time of one month and that the ROP/OU parameters are 200 and 400. So we expect the stock to cycle between 100 and 300. Furthermore, suppose the standard deviation of monthly demand is 50.

Plugging this into the formula above, we would expect the product to have in excess of 300 in stock about 0.4 x 50 / 200 = 10% of the time. So if all products had the same sort of characteristics, we would expect about 10% of them to be in overstock at any one time.

If we change the ROP/OU parameters to be 200 and 300, so we now normally order in lots of 100 and stock cycles between 100 and 200, the equation becomes 0.4 x 50 / 100 = 20%. The smaller the order quantity, the higher the expected level of overstock. (It’s not quite as neat as inversely proportional because the 0.4 factor reduces, and in fact at a certain level the results start to flatten off.) We should note however that overall, stock levels have fallen because we have reduced our target stocks – the problem is that we hit those targets less of the time.

Now if the lead time goes up to 4 months, so the ROP/OU parameters are 500 and 600, stock will still cycle between 100 and 200, but now the standard deviation of lead-time demand goes up to 100, so we get 0.4 x 100/100 = 40%. It works the other way round too, so reducing the lead-time reduces overstocks.

## Lean lessons

For the customer-facing end of the supply chain, opportunities for smoothing demand are much more limited than they are in, for example, a manufacturing environment, where much of the demand variability is system-generated.

The results above suggest that order-size reduction on its own can have some undesirable consequences, but that should not distract us. The benefit will be lower stocks, smoother demand transition up the supply chain, and consequently more reliable supply. It should also facilitate lead-time reduction – which then mitigates the problems.

We should also bear in mind that in practice this overstock phenomenon may be massively outweighed by the effects of slow-moving stock, new product introductions, trends or seasonality, and poor management and control.

Categories: Thought Pieces, Training and Reference.

Tags: Inventory Management, Lean, Six Sigma

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