Forecasting products that sell on only a few days

1The problem

Open the daily sales of almost any product in a store with a few hundred items and the pattern is the same: mostly zeros, with a sale of one to four units every few days. Forecasting literature calls this intermittent demand. It is the normal case for a small catalog, not an exception, and the habits that work for smooth series break on it.

Three habits do the most damage. The first is forecasting tomorrow from today: if today was zero, the forecast is zero, and a reorder rule fed with zero orders nothing. The second is measuring accuracy in percentages: a percentage error divides by the actual value, which is zero on most days, so the measure is undefined or infinite and is often quietly computed on the selling days only¹. The third is picking a model on that broken measure.

2A worked example

Consider one product over 28 days:

0, 0, 3, 0, 0, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 4, 0, 0, 1, 0, 0, 0, 0, 0, 2, 0

It sold 13 units on 6 of the 28 days. Four forecasts of its future demand follow.

The naive forecast repeats the last value. The mean forecast averages all 28 days. Croston's method smooths two things separately, the size of each nonzero sale and the number of days between sales, and forecasts their ratio². Syntetos and Boylan showed that this ratio is biased upward and proposed multiplying it by (1 - α/2), where α is the smoothing constant; the corrected version is usually called SBA³. We use α = 0.1 and start both smoothed values at the first observed sale.

Method	Per day	Next 7 days
Naive (last value)	0.00	0.00
Mean of 28 days	0.46	3.25
Croston, α = 0.1	0.69	4.84
SBA, α = 0.1	0.66	4.60

Source: our calculation on the example series. Croston and SBA use α = 0.1 for both sale size and interval, started at the first sale (3 units after 3 days)^{2, 3}.

The naive forecast says the product will not sell. The other three say it will sell about three to five units a week, which is what matters for a reorder. Croston and SBA sit above the mean here because the smoothed interval, 3.75 days at the end of the series, is shorter than the 28 day average of 4.67 days per sale: the method has weighted the recent gaps. Whether that is right depends on whether the recent pace will hold, which is a judgment the store has to make, but at least the forecast is in the right unit.

3Three changes

3.1Forecast over the lead time

A store does not need to know whether the product sells on Tuesday. It needs to know how many units will sell before the next delivery arrives. Summing over the lead time turns a series of mostly zeros into a quantity a reorder rule can use, and it is the quantity Croston built his method around: the stock needed to cover demand until replenishment².

3.2Default to an intermittent method

For a product that sells on a minority of days, a store should start with Croston or SBA, not with a smooth series method, because both were built for series with many zeros^{2, 3}. Every candidate should also be compared with simple benchmarks such as the mean and the naive forecast before it is trusted; the large forecasting competitions score every entry against benchmarks of this kind for that reason⁴. If a more elaborate method cannot beat the mean on held-out weeks, the mean is the forecast.

3.3Judge by a scaled error and by stock

Hyndman and Koehler proposed the mean absolute scaled error, which divides a forecast's absolute error by the in-sample error of a naive forecast, so it stays defined when actual values are zero and can be compared across products¹. That is a sound default for choosing between methods. The final test is the stock outcome: how often the product ran out, and how much was left over, when replenishment followed the forecast.

4What the competitions found

Two large public competitions tested forecasting methods at scale. The M4 competition compared 61 methods on 100,000 series and found that combinations of methods and hybrids of statistical and machine learning approaches were the most accurate, while pure machine learning methods did poorly⁴. The M5 competition used unit sales from a large US retailer, many of them intermittent at the item and store level, and there machine learning methods, notably gradient boosted trees trained across many series at once, took the top positions⁵.

These results are consistent with the position here. Machine learning gained in M5 by learning across thousands of related series with shared features such as prices, promotions and calendars. A store with a few hundred products can borrow some of that by pooling similar items, but per product, with no other inputs, a simple intermittent method and an honest error measure are the place to start.

5Limitations

The worked example is one illustrative series, chosen to show how the methods differ, not a sample from which to estimate accuracy. We do not cover products with trends, strong seasonality or promotions, where Croston's method is a poor fit, nor products new enough to have no history. We do not report results on store data of our own.

6Conclusion

Most products in a small catalog sell on few days. Forecast them over the replenishment lead time, start with an intermittent demand method, compare it with the mean, choose by a scaled error and confirm by stock outcomes. That removes most of the damage done by forecasting habits built for smooth series.