## What This Article Covers

- What is the Croston Forecasting Model?
- What is Intermittent Demand?
- How Does Croston Work?
- Smoothing and Moving Average Forecasting
- Croston Versus Moving Average Forecasting
- The Implications for Supply Planning.
- What are the Benefits of Croston?
- The Importance of Being Cognizant of the Effort Versus Return Placed into Forecasting.

## Introduction

Croston is a method that has been specialized for dealing with intermittent demand. The literature is inconclusive as to whether Croston is much more beneficial over exponential smoothing with a very small alpha and a high beta value (more emphasis placed on earlier periods). Croston is also presented as more useful than using moving average forecasting. In this article, we will use moving average forecasting without zeros for intermittent demand.

Regardless of the state of the academic literature, the decision has been made by both companies looking to improve the forecast accuracy of intermittent demand items and (most) software vendors. They have concluded that Croston shows benefits significantly higher than alternative solutions like exponential smoothing and moving average.

This post will investigate whether this assumption bears out with the facts, particularly with the events of real life implementations. It will also bring up the question, often not discussed, as to how much better Croston’s performs that exponential smoothing or moving average forecasting.

## How Does Croston Work?

Croston has a complicated formula. However, its output is very simple.

The spreadsheet screenshot below emulates what Croston does. Croston is very rarely directly compared to a moving average in the format shown below. In most circumstances, it is represented by a formula. The approach below does not get into the math of Croston but makes its output transparent and comparable.

The math behind this is complicated, but the output is remarkably similar to performing an exponential smoothing which includes more periods and does not weight the most recent periods very highly. So is this all there is to Croston? No. There is a timing element which also must be covered to appreciate what the formula is doing fully.

## Timing of Occurrence

One of the commonly listed benefits by those technically knowledgeable in how Croston works are its timing benefits.

Below are two Croston forecasts based upon slightly demanding histories.

Below the Croston method, we have moving average forecasting that does not use any zeros. With more non-zero data points, Croston begins to show benefits.

Here is why:

- Firstly it tries to “detect” the cyclic/periodicity of demand pattern. In this case, it suggested an order could occur possibly after 3.5 (4 after roundup) zero period.
- Secondly, if the recent periods are zero periods, it further adjusts the next occurrence from last non-zero period.

So the objects of the forecast are predicting the consumption at the right moment with right quantity. Croston does try to predict the “right moment,” which is more sophisticated than the moving average. But, how relevant is this in “real life?” To answer this question, the following factors should be considered, but most often are not when Croston is proposed on forecast improvement projects.

## 1. The Demand Over Risk Period and the Lead Time Duration

The benefit of Croston is very much dependent upon the lead-time of the product being forecasted. Eventually, it is an average demand across the periods. If the lead-time is short, then timing can be seen to have some benefits. The actual translation of this benefit is open to debate.

## 2. The Variance of the Risk Period

One common experience about Croston is that many people have over time become frustrated with forecast inaccuracy. This is actually due to the fact the forecast measurement is not only measuring the amount but also the timing.

For example:

Forecast: 0, 0, 5, 0, 0, 5

Actual: 0, 0, 5, 0, 0, 5

The forecast error in this example would be zero 0. But if the actual lags by one month and is…

0, 5, 0, 0, 5, 0

…then the forecast error would be quite large. (This further exaggerates the problem of bullwhip effect.) The Higher variance, of course, drives up the required safety stock. In this situation, using a straight average is the safer bet. The following matrix can help break down when Croston might be useful.

## 3. The Implications of Supply Planning

What much research misses out on Croston is the impact on supply planning, or more precisely how supply planning already adjusts to intermittent demand. This is not a criticism restricted to how Croston is analyzed. Many demand planning methods are measured without consideration given to supply planning. Demand planning methods do not exist in a vacuum, and eventually, the demand plan will be released to supply planning.

The following are factors that must be considered in supply planning for a complete analysis of Croston.

- The safety stock (contributing to the final inventory position) calculation in supply planning is based on the demand over risk period and the variance associated the demand within the risk period.
- The effect on (S,s-1) ordering/replenishment policy. The standard ordering/replenishing policies are (r,q), (S,s) and (S,s-1). (r,q) And (r, S) can be discarded for this analysis as they are primarily used products with higher volume demand histories. Therefore, the one to focus on here is (S,s-1).

(To read about (S,s-1) see this **post.**)

## Croston Benefits to (S, s-1)

(S,s-1) Environments can see benefits from the use of Croston. Let us move to the next example to see how.

In the earlier example, the forecast is 0, 0, 5.5, 0, 0 while the average is 1.33 over the five periods. Under an (S,s-1) inventory policy, the supply planning system will suggest an order of 1.33 units per month… There would be 2.66 units sitting in inventory prior the demand for 5.5 occurring. The lower the average monthly forecast, the larger the problem. There are two factors to consider when analyzing the benefits of Croston on supply planning.

- Supply planning is always integer planning; it can’t order a percentage of an item, it must order 0 or 1 as part of a setup function. (This is a common problem in service parts planning which due to low demand optimal ordering quantities are often are percentages of a unit, which must then be rounded up or rounded down.)
- In many cases, items can not be orders in single quantities but must be purchased in multiple units due to either economic order quantity considerations or pack sizes (the minimum packaged quantity sold by the supplier)

For more detail on the (S,s-1) inventory policy see the **post.**

## The Outcome of Croston when Supply Planning is Considered

Much of the benefit of Croston is eliminated when supply planning is taken into account. Croston may be able to create a slightly better “timed” forecast than a long duration moving average. But if orders are naturally rounded by supply planning, the benefit is lost. Companies that should be focusing on Croston are those that use (S,s-1) (with low demand). This is because it is the only area Croston can provide value over and above a long duration moving average or another smoothing method.

As explained at the beginning of this article, this is not the general interpretation of Croston. Instead, Croston is viewed as not only marginally beneficial. It is considered highly advantageous to all products with low and or intermittent demand. This is primarily because many proponents of Croston are not considering supply planning. That is how the relatively minor benefits are already accounted for in supply planning.

This is not the only factor contributing to the continued promotion of Croston. Elements of wishful thinking element in the hopes that are pinned upon Croston. Frustrated by an inability to reduce the forecast error of difficult to forecast items, Croston is seen by decision makers in companies as a magic bullet because “it’s so good at intermittent demand.”

## Being Cognizant of the Effort Put Into Forecasting

Sophisticated forecasting approaches are often recommended without consideration for how much effort they are to implement and maintain versus their pay off. Michael Gilliland is one specialist in this are who makes this a focal point and writes very convincingly on the topic:

Sometimes the best way to deal with forecasting problems is to just get rid of them. When a product has very low sales it may be hard to sense any patterns in the data. There may be long periods of no sales at all, with occasional spikes in sales. There are a number of techniques to deal with such intermittent demand, with Croston’s method and its variations being perhaps the best known. But there is no easy or sure answer, and trying to forecast these low volume items ignores a very important question: Is it worth our efforts to even bother?

## Conclusion

The output of Croston is not nearly as unique as thought and as generally presented. Croston is a complicated formula. Difficult enough such that it is not understandable to most people without a mathematics background. The black box aspect of Croston tends to convince people who it must lead to significantly improved forecasts.

Croston can be easily emulated with exponential smoothing or move average forecasting. While for intermittent demand, Croston will provide higher accuracy than exponential smoothing or moving average forecasting, the relevance of forecast accuracy must be taken in concert with supply planning.

Any timing benefit is usually adjusted by order lot sizing, and or safety stock in supply planning. Demand history must not only be lumpy but must also be **very low for Croston to be of value**. That is the accuracy benefit is in most cases minimal compared to exponential smoothing or moving average forecasting. On the other hand, if there are no lot sizes at play, which may be the case for expensive items, then Croston may provide benefit over exponential smoothing and move average forecasting.

Even in these environments, both authors are skeptical as to its real life benefits. Croston can be seen as a specialty forecasting method that provides value in certain limited circumstance.

## Our Work on Crostons

Crostons is now commonly found in forecasting applications. We use Crostons quite a bit on projects because we often run into products with intermittent demand. When we run best-fit forecast procedures there is always some portion of the product database that is assigned to the Crostons forecasting model. Other portions of the database go out on moving average forecasting, regression, seasonal trend, etc.

**Would you like to find out how much your company could benefit from Crostons in forecast accuracy?** We offer forecasting consulting.

- We can calculate the which product locations will be allocated to Crostons.
- We can calculate the forecast accuracy improvement percentage from Crostons.
- We can show you how to account for problematic, intermittent and high forecast error items far more effectively than the traditional methods of dealing with these items.
- We can do all of this remotely and at a reasonable cost. All we need is clean data history from you.

For more information fill out the form at the end of this page.

## References

Co author thanks, to Wayne Fu who helped with all the math for Crostons.

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