Tool Wear is Gradually Increasing, but the Control Chart Remains Silent? —— Five Steps for Monitoring Small Drifts with EWMA and CUSUM

An incident occurred at a machining company: during a quarterly inventory check, it was discovered that the scrap rate of a critical dimension had quietly tripled. However, after reviewing three months of control charts, not a single alarm point was found. The engineers only realized upon re-examining the data that the process center had gradually shifted by nearly one standard deviation over the three months, but each chart appeared "normal." The charts did not report incorrectly; the wrong type of chart was selected.

Shewhart control charts (individuals chart, mean and range chart) are adept at detecting sudden jumps but are naturally insensitive to "gradual changes." Common failure modes such as tool wear, heater tube aging, catalyst degradation, roller wear, nozzle clogging, and environmental temperature rise are all gradual. To intercept these issues, two additional charts that accumulate historical information are needed: EWMA and CUSUM.

1. Why Shewhart Charts Cannot Detect "Gradual Changes"

The stability logic of conventional control charts is straightforward: compare the current point with the control limits, and at most, check if the last few points form a chain or trend. Since they do not accumulate historical data, the slower the drift, the harder it is to detect.

To illustrate this with simulated data: when an individuals chart uses a 3σ control limit, the process is stable with an average of one false alarm every 370 points. Once the center shifts by 1σ, it takes an average of about 44 points to trigger an alarm; if the shift is only 0.5σ, it takes an average of 155 points. At a sampling frequency of one part every two hours, a 1σ drift would take three to four days to trigger an alarm, and a 0.5σ drift would take about half a month—during which time, the produced parts might already be installed in customer products.

Therefore, the correct approach is not to abandon Shewhart charts (they are still the best for detecting sudden jumps), but to add a chart specifically designed to monitor small drifts.

2. EWMA Chart: Assigning a "Forgetting Weight" to Historical Data

EWMA (Exponentially Weighted Moving Average) averages historical observations with exponentially decaying weights: the weight of the current point is λ, and the cumulative value of the previous point is (1−λ), i.e., Z = λx + (1−λ)Z (previous value). λ is the weight of new data, typically ranging from 0.05 to 0.3, starting from 0.2.

The control limits for EWMA are not ±3σ, but rather ±L·σ·√(λ/(2−λ)). When λ = 0.2 and L = 3, the steady-state limits are approximately ±1σ—much tighter than Shewhart charts, which is why it can detect small drifts.

Similarly, using simulated data for testing: when λ = 0.2 and L = 3, the process is stable with an average of one false alarm every 560 points. After a 1σ shift, it takes an average of about 11 points to trigger an alarm, and for a 0.5σ shift, it takes about 44 points. In other words, the ability to detect 0.5σ to 1σ drifts is three to four times faster than an individuals chart, with fewer false alarms.

The trade-off must also be clear: the smaller the λ, the more sensitive it is, but the slower it reacts to sudden jumps. A single point out of bounds does not necessarily mean the process is out of control; it must be re-measured to confirm, and then check if the cumulative value continues to be one-sided.

3. CUSUM Chart: Accumulating Deviations

The concept of CUSUM (Cumulative Sum) is more straightforward—accumulate the deviations of each point relative to the target value. It maintains two accumulators: C⁺ for upward deviations and C⁻ for downward deviations. For each point, the part exceeding μ₀±k is added to the accumulator, and if it becomes negative, it is reset to zero (indicating that the accumulation in that direction has been cleared).

There are only two parameters: k is generally half of the desired shift, so for a 1σ shift, k = 0.5σ; the decision limit h is typically 4 to 5σ. Simulation results show that when k = 0.5σ and h = 5σ, the process is stable with an average of one false alarm every 460 points. For a 1σ shift, it takes about 10 points to trigger an alarm, and for a 0.5σ shift, it takes about 38 points—sensitivity is comparable to EWMA.

CUSUM has an additional advantage over EWMA: when it alarms, it can trace back to the point where the shift began, directly correlating the time with "that day the material batch was changed/that day the tool was changed/that day the parameters were adjusted." The trade-off is that if h is tightened from 5σ to 4σ, sensitivity increases, but the average interval between false alarms in the stable period drops to about 166 points. The field must determine if this is acceptable.

4. An Example

In a certain injection molding workshop, the critical dimension tolerance is 0.10mm, and one part is sampled every two hours, accumulating 80 points. The individuals chart (3σ limit) did not trigger any alarms throughout, but nonconforming products began to appear after the 60th point, increasing the monthly defect rate from 0.5% to 3%.

Replaying the data: using the first 40 points (a period confirmed to be stable) to calculate the baseline μ₀ and σ, where σ is estimated by the moving range. The EWMA (λ = 0.2, L = 3) went out of bounds at the 52nd point; the CUSUM (k = 0.5σ, h = 5σ) triggered an alarm at the 49th point, indicating that the shift started at the 33rd point—exactly when the material batch was changed and the mold cooling water circuit began to scale.

Both charts triggered alarms approximately 25 points before the nonconforming products appeared, which, at the sampling frequency, is about two days in advance. After cleaning the scale, changing the material batch, and re-establishing the baseline, both charts returned to normal.

5. Five Steps for Implementation

Step 1: Define Target Shift and Sampling Frequency

First, answer "what size of shift needs to be detected, and within what time frame?" For small shifts of 0.5σ to 1σ, choose λ = 0.1 to 0.2 or k = 0.5σ. If only large changes of 2σ or more are of concern, conventional control charts are sufficient, and these two charts are not necessary.

Step 2: Establish the Baseline

Use at least 20 to 25 points of data from a confirmed stable period to estimate μ₀ and σ (for individuals data, use the moving range method). Before this step, perform MSA: the measurement system variation should typically be controlled within 10% to 30% to avoid drowning out 1σ signals with measurement noise.

Step 3: Set Parameters and Backtest

For EWMA, set λ and L; for CUSUM, set k and h. Then, "replay" real data from the past six months to one year: did the alarms occur before the actual anomalies? Is the false alarm frequency acceptable in the field? Parameters that do not pass the backtest should not be implemented.

Step 4: Use Both Charts and Clearly Define Response Rules

Shewhart charts handle sudden jumps, while EWMA or CUSUM handle gradual changes, using the same batch of data. Clearly define what actions should be taken by whom and within what time frame after an alarm: first, re-measure to rule out measurement issues, then check the material batch, parameters, and tooling, and finally decide whether to stop the machine. Charts without response rules will not be acted upon even if they alarm.

Step 5: Recalculate and Version Management

After changing materials, tooling, repairing molds, or adjusting parameters, re-establish the baseline and freeze the old chart. Record the parameters, baseline date, and version number to avoid using a three-month-old baseline to judge today's status.

6. Three Common Pitfalls

Pitfall 1: Copying Parameters

λ, L, k, and h determine the balance between sensitivity and false alarms. Copying parameters from others means using their field standards to judge your process, leading to frequent false alarms and the eventual abandonment of the charts.

Pitfall 2: Discarding Shewhart Charts

EWMA and CUSUM are less responsive to sudden jumps compared to conventional charts. Using only one of these charts increases the risk of missing sudden anomalies.

Pitfall 3: Including Abnormal Segments in the Baseline

Using contaminated data to estimate μ₀ and σ will widen the control limits, causing both charts to become "silent." Similarly, if the measurement system resolution is insufficient or the data is autocorrelated, these charts will generate many false signals.

Gradual drifts are not without signals; they are just naturally invisible to conventional control charts. Adding a chart that accumulates historical data can intercept drifts before they become nonconforming, which is the most economical way to handle issues like tool wear and equipment aging.

Gradual drifts are the most expensive—use a chart that accumulates historical data to intercept them before they become nonconforming.

Knowledge code: 6.3.1

Version: v20260916

Author: QTank QTank is dedicated to providing systematic professional knowledge, methodologies, and practical tools for quality management practitioners, helping enterprises continuously improve their quality capabilities.