In a previous post, I discussed one of the thornier problems demand planners sometimes face: working with product demand data characterized by what statisticians call skewness — a situation that can necessitate costly inventory investments. This sort of problematic data is found in several different scenarios. In at least one, the combination of intermittent demand and very effective sales promotions, the problem lends itself to an effective solution.

Reviewing terms, recall that “service level” is the probability of not stocking out while waiting for a replenishment order to arrive, while “fill rate” is the percentage of demand that is satisfied immediately from stock. In my previous post, “The Scourge of Skewness”, I pointed out that a certain type of demand distribution, having a “long right tail”, will lead to fill rates that can be much lower than service levels. I also pointed out that sometimes the only way to improve the fill rate is to increase the target service level to an unusually high level, which can be expensive.

In this post, I’ll look at solving the problem in one special case: skewness resulting from effective sales promotions mixed with “intermittent demand”. Intermittent demand has a large proportion of zero values, with nonzero values mixed in at random. Successful sales promotions, obviously positive, have a downside: they can confuse the “demand signal” with spikes in your demand history, and can undermine forecasts and bias safety stock calculations. When intermittent demand and effective sales promotions are the source of your data’s skewness, methods exist to work around the problem to achieve both higher fill rates and more accurate demand forecasts.

How Promotions Increase Skewness

Successful promotions abruptly increase item demand. This creates anomalies, or “outliers”, which contribute to forming a skewed distribution. Knowing when promotions occurred in the past, we can adjust an item’s record of past demand. We produce an alternate demand history as if there had been no promotions, by replacing the outliers with values more representative of the “natural” level of demand. These adjustments reduce demand skewness. Reduced skewness can lead to significant reductions in both expected forecasts and safety stocks, which add together to form reorder points.

Successful promotions are likely to be repeated. When that happens, the promotion effects can be added in to demand forecasts to increase their accuracy. The effect of future promotions on inventory management will be to increase the risk of stockouts, so a sensible response is to work at the operational level to build up temporary supply, in a quantity keyed to the estimated impact of prior promotions on the effected items.

Using Event Modeling to Improve Demand Forecasting

It’s possible to model the impact of like events, and apply this to planned events in the future. Doing so can improve your forecast in two big ways: by projecting the demand jolt you expect from a planned event; and rationalizing the spikes in the past that were caused by events, making your baseline activity more visible and more accurately forecastable. We do a lot of this in SmartForecasts, so allow me to use our experience there to show you what I mean.

Event Modeling entails the following steps:
• Automatically estimating the impact of previous promotions (which is a useful result in itself).
• Adjusting historical demand to statistically remove the effect of promotions.
• Creating promotion-free forecasts.
• Revising the forecasts for any future time periods in which promotions are planned.

We call this this type of analysis “Promo forecasting”. We use the word “promotions” to describe what you do yourself to improve your results. We use “events” to describe what the world does to you, usually to your detriment; examples include strikes, power outages, warehouse fires and other unlucky happenings.

To understand how Event Modeling can help you cope with skewness when doing demand forecasting for high-volume items, consider Figures 1-3.

Figure 1 shows that this item’s demand pattern is clearly seasonal, and the forecast is both seasonal and “tight”, meaning that the forecast uncertainty interval (“margin of error”, shown in cyan lines) is very narrow.

Figure 2 shows an alternative history in which a promotion in June 2014 reversed the usual seasonal low associated with June sales. This demand pattern was forecasted using the Automatic forecasting tournament in SmartForecasts, as in Figure 1. This time, the promotion scrambled the seasonal pattern enough to create an inappropriate non-seasonal forecast, and one that has a much larger margin of error.

Finally, Figure 3 shows how Promo forecasting handles the same promoted scenario, retaining a seasonal forecast and building into the forecast an estimate of the effect of a planned repeat promotion in 2015.

The Case of Intermittent Demand

In Figure 1, the item was a high-volume finished good and the task was demand forecasting. Promo modeling is also useful when dealing with the task of setting safety stocks and reorder points for items with intermittent demand, whether the items are finished goods, components or spare parts. Intermittent demand very often has a skewed distribution that makes it difficult to achieve high item availability with a small investment in inventory.

Figure 4 illustrates the problem that a successful promotion can accidentally create for inventory management. If such a spike arises from the natural, un-promoted demand, then the only way to maintain high fill rates is to provide safety stocks large enough to cope with these random surges. In this case, the big spike in demand of 500 units in February 2013 was the result of a one-time promotion.

Taking Account of Promotions to Improve Inventory Management

Unwittingly treating the spike in the example above as part of the natural demand variability results in a poor fill rate. To achieve a target service level of, say, 95% with a lead time of one month would require a reorder point of 38 units, computed as the sum of an expected forecast over the one month replenishment lead time of 21 units supplemented by a safety stock of 17 units. This investment would result in a disappointing fill rate of only 36%.

However, recognizing that the spike is a one-time promotion and replacing the 500 units with 0 obviously would make a big difference. The reorder point would drop from 38 units to 31 (the sum of an expected demand of 7 units and a safety stock of 24 units) and the fill rate would increase to 94%.

Of course, it is not ok to just throw out inconvenient demand spikes whenever they make life uncomfortable; there has to be a valid “business story” behind the adjustment of historical demand. If the spike is the result of a data processing error, then by all means, fix it. If the spike coincides with a promotion, then replacing the spike with, say, the median demand (often zero, as in this example) will result in a much more sustainable inventory investment that still meets aggressive performance targets. Future promotions of the same type on the same item will require some extra effort to prepare for the temporary surge in demand, but the recommended reorder point will be correct in the long run.

Thomas Willemain, PhD, co-founded Smart Software and currently serves as Senior Vice President for Research. Dr. Willemain also serves as Professor Emeritus of Industrial and Systems Engineering at Rensselear Polytechnic Institute and as a member of the research staff at the Center for Computing Sciences, Institute for Defense Analyses.

Demand planners have to cope with multiple problems to get their job done. One is the Irritation of Intermittency. The “now you see it, now you don’t” character of intermittent demand, with its heavy mix of zero values, forces the use of advanced statistical methods, such as Smart Software’s patented Markov Bootstrap algorithm. But even within the dark realm of intermittent demand, there are degrees of difficulty: planners must further cope with the potentially costly Scourge of Skewness.

Skewness is a statistical term describing the degree to which a demand distribution is not symmetrical. The classic (and largely mythic) “bell-shaped” curve is symmetric, with equal chances of demand in any time period falling below or above the average. In contrast, a skewed distribution is lopsided, with most values falling either above or below the average. In most cases, demand data are positively skewed, with a long tail of values extending toward the higher end of the demand scale.

Bar graphs of two time series
Figure 1: Two intermittent demand series with different levels of skewness
Figure 1 shows two time series of 60 months of intermittent demand. Both are positively skewed, but the data in the bottom panel are more skewed. Both series have nearly the same average demand, but the one on top is a mix of 0’s, 1’s and 2’s, while the one on the bottom is a mix of 0’s, 1’s and 4’s.

What makes positive skewness a problem is that it reduces an item’s fill rate. Fill rate is an important inventory management performance metric. It measures the percentage of demand that is satisfied immediately from on-hand inventory. Any backorders or lost sales reduce the fill rate (besides squandering customer good will).

Fill rate is a companion to the other key performance metric: Service level. Service level measures the chance that an item will stock out during the replenishment lead time. Lead time is measured from the moment when inventory drops to or below an item’s reorder point, triggering a replenishment order, until the arrival of the replacement inventory.

Inventory management software, such as Smart Software’s SmartForecasts, can analyze demand patterns to calculate the reorder point required to achieve a specified service level target. To hit a 95% service level for the item in the top panel of Figure 1, assuming a lead time of 1 month, the required reorder point is 3; for the bottom item, the reorder point is 1. (The first reorder point is 3 to allow for the distinct possibility that future demand values will exceed the largest values, 2, observed so far. In fact, values as large as 8 are possible.) See Figure 2.

Histograms of two time series
Figure 2: Distributions of total demand during a replenishment lead time of 1 month
(Figure 2 plots the predicted distribution of demand over the lead time. The green bars represent the probability that any particular level of demand will materialize.)

Using the required reorder point of 3 units, the fill rate for the less skewed item is a healthy 93%. However, the fill rate for the more skewed item is a troubling 44%, even though this item too achieves a service level of 95%. This is the scourge of skewness.

The explanation for the difference in fill rates is the degree of skewness. The reorder point for the more skewed item is 1 unit. Having 1 unit on hand at the start of the lead time will be sufficient to handle 95% of the demands arriving during a 1 month lead time. However, the monthly demand could reach above 15 units, so when the more skewed unit stocks out, it will “stock out big time”, losing a much larger number of units.

Most demand planners would be proud to achieve a 95% service level and a 93% fill rate. Most would be troubled, and puzzled, by achieving the 95% service level but only a 44% fill rate. This partial failure would not be their fault: it can be traced directly to the nasty skewness in the distribution of monthly demand values.

There is no painless fix to this problem. The only way to boost the fill rate in this situation is to raise the service level target, which will in turn boost the reorder point, which finally will reduce both the frequency of stockouts and their size whenever they occur. In this example, raising the reorder point from 1 unit to 3 units will achieve a 99% service level and boost fill rate to a respectable, but not outstanding, 84%. This improvement would come at the cost of essentially tripling the dollars tied up in managing this more skewed item.

Thomas Willemain, PhD, co-founded Smart Software and currently serves as Senior Vice President for Research. Dr. Willemain also serves as Professor Emeritus of Industrial and Systems Engineering at Rensselear Polytechnic Institute and as a member of the research staff at the Center for Computing Sciences, Institute for Defense Analyses.