Zero-Inflated Intermittent Demand Distorts MAPE Toward Infinity, Mandating Scaled Errors for Spare Parts Inventory

In high-stakes industries such as aerospace maintenance, automotive aftermarket, and industrial manufacturing, the most capital-intensive inventory is often the item that rarely sells. For the vast majority of specialized spare parts, demand manifests as a sparse series of zeros punctuated by unpredictable, sporadic spikes.[8]

When supply chain planners attempt to forecast this intermittent demand, they typically rely on the Mean Absolute Percentage Error to evaluate their models. Because this metric is simple to understand and built into almost all legacy enterprise software, it remains the default standard globally.[7][8]

However, the percentage error calculation contains a fatal mathematical flaw when applied to zero-heavy data. Because the formula divides the absolute error by the actual demand for each period, a month with zero sales renders the metric mathematically undefined.[6][8]

The Mathematical Trap of Percentage Errors

This mathematical breakdown forces forecasting software into a dangerous set of hidden compromises. To avoid division by zero, systems often drop zero-demand periods from the evaluation entirely, or they add a small constant to the denominator to force a calculation.[7]

Neither workaround solves the underlying statistical distortion. By dropping zero-demand periods, the model ignores the most common state of a spare part, evaluating accuracy only on the rare days when a transaction actually occurs. By adding a constant, the system artificially bounds the error but retains a massive penalty for over-forecasting.[7][8]

Even when demand is non-zero but very low, the metric produces extreme distortions. If a planner forecasts two units to maintain a minimum safety stock, but actual demand is only one unit, the calculation registers a 100 percent error.[7]

These massive percentage swings for tiny absolute differences distort how performance is judged across the supply chain. Planners see a high error rate, assume the forecast is entirely broken, and begin manually overriding the system, which ultimately degrades inventory performance.[7]

The Asymmetry of Percentage Errors

The most destructive feature of this metric in spare parts forecasting is its fundamental asymmetry. The calculation treats over-forecasting and under-forecasting entirely differently, creating a mathematical incentive that actively harms inventory readiness and corporate capital allocation.[7][8]

If a system forecasts zero units but actual demand is one, the absolute error is one, and the percentage error is capped at 100 percent. Conversely, if the system forecasts one unit but actual demand is zero, the division by zero drives the percentage error toward infinity.[8]

An optimization algorithm trained to minimize this metric will quickly learn the asymmetry. It realizes that predicting zero is the safest mathematical strategy, as it strictly bounds the maximum possible error. Consequently, the metric unintentionally encourages planners to under-stock.[8]

This phenomenon, known as the point-forecast paradox, triggers a severe bullwhip effect in spare parts management. It causes catastrophic stockouts during sudden demand spikes, which then induce panicked, just-in-case overstocking that unnecessarily tethers corporate capital for years.[8]

Why Spare Parts Break Traditional Models

Traditional forecasting models are architected for high-volume, smooth demand where historical continuity serves as a reliable proxy for future requirements. They assume that demand follows a normal distribution around a moving average, with errors distributed evenly across the timeline.[2][8]

Intermittent demand violently violates these assumptions. To quantify this, statisticians use the Average Inter-Demand Interval and the squared Coefficient of Variation of non-zero demand sizes. Items with an average interval above 1.32 are formally classified as intermittent or lumpy.[5]

The classification scheme developed by forecasting researchers divides demand into four distinct cells based on these two diagnostics. When the interval is below 1.32 and the variation is low, demand is considered smooth, and standard exponential smoothing applies perfectly.[5]

However, when the interval exceeds 1.32 but the variation remains low, the demand is intermittent. If both thresholds are exceeded, the demand is classified as lumpy, representing the most difficult forecasting challenge in modern supply chain management.[5]

For these specific items, the timing of the next sale is highly uncertain, and the size of the order varies wildly when it finally arrives. Standard exponential smoothing methods fail completely in this environment, yielding a flat, fractional forecast.[5][6]

A fractional prediction, such as 0.2 units per month, never matches reality. When evaluated using percentage errors, these fractional forecasts generate massive penalties during both zero-demand months and sudden spikes, leaving supply chain leaders blind to the actual probability of a stockout.[7][8]

The Mean Absolute Scaled Error Solution

To resolve the division-by-zero problem, statisticians Rob Hyndman and Anne Koehler introduced the Mean Absolute Scaled Error in 2006. This metric abandons percentage-based calculations entirely, providing a robust alternative that does not exhibit asymptotic behavior near zero.[1][3]

Instead of dividing the error by the actual demand of a single period, the scaled error compares the forecast against the in-sample mean absolute error of a naive method. The naive method simply assumes that the next period will exactly repeat the previous period's demand.[3][8]

Because the denominator is based on the average variation across the entire historical series, it never equals zero unless the historical data is perfectly flat. This makes the scaled error mathematically stable and highly effective for zero-inflated arrays.[3]

A scaled error value less than 1.0 indicates that the forecasting model provides more predictive value than the naive baseline. Crucially, it treats over-forecasting and under-forecasting symmetrically, removing the mathematical incentive to artificially predict zero and preserving safety stock.[3][8]

Croston's Method and the SBA Correction

While scaled errors provide a valid evaluation metric, generating the forecast itself requires specialized algorithms. Croston's method, developed in 1972, addresses intermittency by separately smoothing the size of non-zero demands and the intervals between them.[6]

Both the demand size and the demand interval are updated using exponential smoothing, but only in periods when actual demand occurs. The final forecast per period is calculated by dividing the smoothed size by the smoothed interval.[6]

However, Croston's method contains a systematic positive bias. The ratio of two separately smoothed estimators is not itself unbiased, meaning the method overestimates the mean demand rate, especially when the smoothing constant is large.[4]

To correct this, the Syntetos-Boylan Approximation applies a deflating correction factor to the Croston forecast. By multiplying the forecast by a factor of one minus half the smoothing constant, the approximation removes the upward bias while preserving the intuitive decomposition.[4][5]

This approximation has become the de facto standard for forecasting intermittent demand. It is widely implemented in production planning software across the aerospace, defense, and pharmaceutical sectors to deliver more accurate mean forecasts at no extra computational cost.[5]

Transitioning to Inventory-Utility Metrics

While scaled errors and bias corrections fix the statistical flaws of traditional methods, leading supply chain teams are pushing beyond statistical accuracy entirely. Modern organizations are transitioning toward inventory-utility metrics that evaluate models based on the total financial cost of the error.[8]

An inventory-utility metric calculates the sum of holding costs incurred by over-forecasting and the stockout costs caused by under-forecasting. In sporadic environments, a model with higher statistical variance but lower stockout costs is often the superior business choice.[8]

For high-margin components, the cost of an operational shutdown vastly exceeds the cost of holding one extra spare part on the shelf. By aligning the forecasting metric with profit-and-loss goals, organizations can manage uncertainty rather than attempting to eliminate it.[8]

Furthermore, modern forecasting teams are increasingly deploying global machine learning models to handle these lumpy distributions. Rather than training an isolated model for a single spare part, global models learn across thousands of items simultaneously to identify hidden demand triggers.[7][8]

This cross-learning allows the algorithm to detect patterns that a single-item time series could never reveal. When combined with utility-based loss functions, these models optimize directly for service levels rather than arbitrary statistical perfection.[7]

By replacing percentage errors with scaled metrics, implementing bias corrections, and aligning the final evaluation with financial utility, organizations escape the point-forecast paradox. This paradigm shift moves the objective from predicting an exact number to ensuring operational readiness.[8]