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The most well-known type of forecast is the

QUANTILE TECHNOLOGY PART I

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QUANTILE TECHNOLOGY PART II

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Spare parts inventory management with quantiles (PDF)

In a world where most equipment manufacturers and retailers are operating in fiercely competitive markets, delivering a high service level to the existing customer base is a strategic priority for many companies. However, managing a spare parts inventory efficiently still poses a huge challenge due to size and the erratic nature of demand. This whitepaper discusses the challenges and current state of spare parts planning technology, and introduces quantile forecasting as a disruptive new approach to tackling the problem.

The terminology *quantile forecast* might sound complicated, and chances are, unless you’re deeply versed in statistics, that you’ve never heard the term before. However, **quantile forecasts – without being named that way - are routinely used in retail and manufacturing businesses**. For example, defining a reorder point for your inventory is strictly equivalent to producing a quantile forecast over the demand.

Despite radical implications of quantile forecasts for retail and manufacturing, quantiles have received little attention in the market so far. The simplest explanation is that **support for quantile forecasts was close to nonexistent in the software industry**. However, with Lokad, there is no reason to overlook anymore such a critical piece of technology.

In order to understand why *quantile* forecasts are of any use for a retailer or a manufacturer, we need to back to why forecasts are required in the first place. Demand forecasts are critical in order to ensure that the right level of resources - such as inventory, staffing or cash - is available at the right time.

However, meeting the demand with the right level of resources is typically a **very asymmetric problem**: the cost of over-allocating resources (aka over-forecasting) can vastly differ from the cost of under-allocating resources (aka under-forecasting).

- Food retailers typically seek very high service levels at 95% or more (i.e. very infrequent stock-outs). In this context, it is estimated that the marginal cost of a stock-out vastly excesses the marginal cost an extra unit of stock.
- Car manufacturers are under increasing pressure to lower their production costs. As a result, some manufacturers opt for a zero-stock strategy - and consequently a zero immediate availability - where cars can only be purchased first to be manufactured later on. In this situation, it is estimated that the marginal cost of the stock excesses the cost of the non-immediate availability.

Thus, for companies, it is typically not profitable to allocate their resources based on *raw* mean demand forecasts, as allocating too few resources 50% of the time is a poor trade-off that does not reflect the reality of the business.
Hence, companies are purposefully **introducing a bias in their resource allocations to reflect the business-specific asymmetry** that exists in their trade. Being able to better deal with this asymmetry is exactly what quantile forecasts are about.

Definition:a quantile forecast (τ, λ) where τ (tau) is the target probability and where λ (lambda) is the horizon expressed in days, represent a demand forecast over the next λ days that come with a probability of τ of being higher than the future demand (consequently a probability 1-τ of being lower than the future demand).

Quantile forecasts have been known for decades, however implementing a *native* quantile forecasting model is frequently, and rightfully, considered as being as a lot more complicated than implementing a mean forecasting model. As a result, the vast majority of forecasting software vendors (*) are only delivering mean forecasts.

*(*) As far we know, Lokad has become in March 2012, the first vendor to deliver a native industrial-grade generic quantile forecasting technology. However, among academic circles, research prototypes for quantile regression have been around for decades.*

However, as companies do require quantile forecasts, they typically leverage an **extrapolation work-around** to produce their quantile forecasts. Practically speaking, the approach consists of assuming that the demand follows a normal distribution and to add a corrective *safety* term. The classical safety stock approach follows this pattern for example.

Extrapolated quantilesare classic (mean) forecasts transformed into quantile forecasts through an extrapolation method. The term is opposed tonativequantiles where the statistical model directly produces the quantile. The extrapolation doesn't rely on input data, but rather on a distribution defineda priori. This distribution, usually thenormal distribution, tends to be the weakest link of the extrapolation process, as it differs from the reality.

Unfortunately, the extrapolation suffers from serious drawbacks in 3 frequent contexts:

- High quantiles (i.e. high service level)
- Intermittent demand
- Spiky demand (bulk orders)

In those situations, we have found that *native* quantile forecasts tend to outperform of 20% or more the best extrapolated quantile forecasts; the comparison being made by leveraging the respective quantile and classical forecasting technologies of Lokad - knowing that those already tend to outperform the competition.

The assumption that errors associated to forecasts are normally distributed is typically good for quantile targets close to the mean or the median. However, the quality of the **approximation degrades as the target percentage increases**. For high target percentages, typically all values above 90%, we have found that the extrapolation itself frequently becomes the weakest link of the forecast. In those situations, native quantiles should be favored.

The extrapolation tries to fit a **smooth curve** over the future demand in order to reflect uncertainty. However, when the demand is intermittent or sparse, there is **nothing smooth about the demand**: for each period (week, month), the number of units being sold, i.e. the observable demand, is an integer varying between 0 and 5 for example.

Historically, many *mean* forecasting models have been designed to better apprehend sparse demand; however from the quantile angle, it becomes clear that the more fundamental issue is that no mean forecast can be properly extrapolated into an accurate quantile in case of sparse demand. In contrast, native quantiles can completely fit *small-integers* patterns of the demand.

When bulk orders are present, the historical demand curve tends to have a rather spiky shape. This shape reflects that a few orders account for a significant percentage of the total demand. However, contrary to the intermittent demand case, a non-zero demand exists all the time. The fundamental issue here is not that the demand goes through *integral* values; it is that mean forecasts fail at properly projecting those spikes in the future.

Oversimplifying, there are two approaches to deal with spikes:

**Discard them**if the company decides they are not worth pre-allocating resources.- Adjust pre-allocated resources in order to
**handle them**, or a least handle a certain fraction of the spikes.

In both cases, mean forecasts behave poorly: extrapolated quantiles remain too low to capture spikes while in the same time, they are over-estimating the resources to handle non-spike demand. Native quantile forecasts address spikes in a more direct and more accurate manner.

Lokad delivers a **fully automated online service** that takes time-series as input and returns native quantile forecasts, each quantile matching its horizon and target percentage (respectively lead time and the service level in case of inventory optimization). No extrapolation is required.

The quantile forecasting process requires **zero statistical expertise**. In practice, most companies will go through Salescast, a web application also provided by Lokad, in order to get optimized reorder points; the reorder point being an inventory-specific quantile forecast.

For each time-series, the quantile forecast is just one single data point. Unlike mean forecasts, **quantile forecasts are typically not represented as a curve** that evolves over time and that extends the historical curve into the future.

Quantile forecasts behave differently statistically-speaking, however fundamental underlying demand patterns remain the same: trend, seasonality, product life-cycle, promotions ... All patterns supported by our *classic* forecasting technology are also supported by our *quantile* forecasting technology.

From a mathematical viewpoint, **quantile forecasts represent a generalization** of the classical notion of forecasts. From a practical viewpoint, quantile forecasts are typically superior (more accurate) for most business situations where risks associated to over and under estimates of the demand are not symmetric.

However, quantile forecasts are also **less readable and less intuitive**. Hence, classic forecasts remain a fundamental tool for managers to get a more intuitive grasp of the evolution of their business.

We have no plan whatsoever to deprecate classic forecasts. As a matter of fact, most of R&D efforts that we push on our forecasting technology benefit to the two types of forecasts. Quantile forecasting is a chance for us to refine our understanding of the statistical behavior of the demand. Our No1 priority remains **to deliver more accurate forecasts**.