In an earlier article on Forecasting with Intermittent Demand, a reader asked me whether my (Structured Inference Base) SIB approach for intermittent demand modeling was applicable to daily and weekly data, as well as the monthly example I gave. It is, and could in fact work for any intermittent time series.

The point of this article is that the Croston method, while widely implemented for forecasting intermittent demand, may not be applicable in real-world data applications. The reason is that the underlying assumptions about the independence of intervals and demand volumes may not be valid in practice. The method and its variants essentially take interval durations and nonzero demand volumes as separate data series for forecast modeling purposes.

When you start looking at real data, you may find that the demand volumes can depend on the interval durations, as I will now demonstrate with a series of 52 intermittent weekly demand (ID) data.

Intermittent data  (also known as sporadic demand) comes about when a product experiences several periods of zero demand. This has become more common during the pandemic supply chain disruptions, especially in the retail industry. Forecasting intermittent demand occurs in practice, when

1. modeling product mix based on future patterns of demand at the item and ship-to location level
2.  selecting the most appropriate inventory control policy based on type of stocked item
3. setting safety stock levels for SKUs at multiple locations

Often in these situations, when demand occurs, it is small, and sometimes highly variable in size. For example, the level of safety stock depends on the service level you select, on the replenishment lead time and on the reliability of the forecast.

Here, I will show that a mutual information measure I(ID, LZI) gives the error in assuming the independence of nonzero intermittent demand volumes (ID) and the zero interval durations (LZI). This formula is widely used in information theory and climatology applications and can be applied in intermittent demand applications, as well. In the mutual information formula (and the Kullback-Leibler divergence measure KL), P represents a discrete probability distribution; that is, like positive weights, whose sum is one.

Let us try to validate this with some real data. You can follow the procedure with the spreadsheet below. The intermittent demand series is depicted in Column A. There are 38 nonzero demand volumes and 18 zeroes of duration 1 and 2, giving a total of 52 weeks.

In using the mutual information formula, P(ID) is the Intermittent Demand volume alphabet profile (Col D); P(LZI) is the Lagtime Zero Interval alphabet profile (Col C); P(ID, LZI) is the Joint alphabet profile (Col B), denoted by JOINT in the spreadsheet. P(.) is a marginal alphabet profile in the formula.

In our use of the formula, P defines an alphabet profile, which is coded from the original data into numbers between 0 and 1, summing up to 1. Thus, to obtain the alphabet profile P(ID) in column D, we divide the sum of the ID history (=73) into each nonzero demand value in Col A. These 38 ratios sum to one. You can see on the graphs below that the original history profile and the corresponding alphabet profile have the same pattern and differ only by the scale on the y-axis. 

The zeros for LZI in the dataset are the same number, so the alphabet profile P(LZI) in column C is a set of 14 constant numbers (=1/14), also summing to one.

In the mutual information formula, there is a joint alphabet profile P(ID, LZI) that needs to be created (Col B). In our formulation of the relationship between intervals and demand volumes, we postulate that each interval is followed by a nonzero demand. We call the intervals LZI, for Lagtime Zero Interval. In this data set, there are 3 two-zero intervals, 8 one-zero intervals, and 27 no-zero intervals (27= 38-11). In this way, there is an LZI value associated with each demand volume. This process creates a conditional distribution of demand volumes dependent on the LZI variable.

The joint alphabet profile is obtained by multiplying each value in Col D by its corresponding weight in the LZI_0, LZI_1, LZI_2 distribution (= 0.78082, 0.16438, 0.05479). The results are added up and subtracted from 1. That difference is apportioned to the remaining 14 zeros in Col B ( 0.03201 = 0.44814/14).

The Accuracy of the Independence Assumption

It the demand volumes and interval durations are independent the mutual information measure I(ID, LZI) = 0. If I(ID, LZI) = 0, it implies independence. It can be shown that the measure I(ID, LZI) ≥ 0, so that a positive value of I(ID, LZI) indicates a lack of independence or association. In this example, I(ID, LZI) = 0.56, suggesting a lack of independence. (row 56: = Col E – Col F – Col G). You will have to run many of your own examples to establish a range away from zero to test the assumption.