Question 4 of 6: Zone Rules and ARL, and an Individuals/Moving-Range Chart with Detection Speed

Question 4: Zone Rules and ARL, and an Individuals/Moving-Range Chart with Detection Speed (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Part (a) — Effect of zone rules on ARL; appropriateness for an EWMA chart

Zone (sensitizing) rules divide the region between the centre line and each 3-sigma control limit into three equal-width zones (A, B, C, moving outward) and flag additional non-random patterns beyond the basic "point beyond the limits" test — e.g. 2 of 3 consecutive points in Zone A or beyond, 4 of 5 in Zone B or beyond, 8 consecutive points on one side of the centre line, or a steady run trending in one direction. Because each rule gives the chart an extra, largely independent chance to signal on every sample, applying several zone rules simultaneously makes the chart substantially more sensitive to small, sustained out-of-control shifts: the out-of-control $ARL$ falls well below what the lone 3-sigma test alone would achieve, so real disturbances are, on average, caught in far fewer samples. The cost is on the in-control side: each additional rule carries its own small false-alarm probability, and since the rules are evaluated together on every sample, their false-alarm probabilities essentially add, so the in-control $ARL$ falls too — from the basic chart's roughly 370 (at $\alpha=0.0027$) down to commonly cited combined-rule figures in the 90–150 range. Zone rules therefore trade a lower false-alarm rate for faster real-shift detection; they never improve both simultaneously.

Applying a zone rule to an EWMA chart is not appropriate, for two related reasons. First, EWMA already achieves its enhanced sensitivity to small shifts by construction — the plotted statistic $z_i=\lambda x_i+(1-\lambda)z_{i-1}$ is itself a weighted average of all past observations, so consecutive plotted points are highly autocorrelated by design, unlike the (approximately) independent points on a Shewhart $\bar X$ chart. The zone rules were derived assuming independent, identically distributed plotted points; applying them to a sequence of EWMA statistics, which are deliberately correlated, badly distorts their intended false-alarm rate (a run of EWMA points sitting in Zone B is not evidence of a shift the way it would be on an independent $\bar X$ chart — it is largely just the smoothing memory carrying over). Second, it is unnecessary: EWMA (like CUSUM) was created specifically to close the small-shift detection gap that zone rules exist to patch on a Shewhart chart, so stacking zone rules on top of EWMA duplicates functionality the chart already has, while additionally corrupting the very statistical assumption the zone-rule Type-I-error calculation depends on.

Part (b) — Individuals and moving-range charts for extrusion-sheet weight

Given. One weight measurement is recorded every half hour (a single cross-section per sampling instant, rational subgroup size $n=1$), so the appropriate chart pair is the individuals ($X$) and moving-range ($MR$) chart, not an $\bar X$/$R$ or $\bar X$/$S$ chart (those require $n\ge2$ per subgroup). The 20 recorded weights (lb) are:

Find. Trial and (if needed, revised) control limits for both charts, and the resulting in-control estimates $\hat\mu_0$ and $\hat\sigma_0$.

Approach. Compute $\bar X$ and the average moving range $\overline{MR}$ (of the 19 successive absolute differences), form trial 3-sigma limits using the individuals-chart constants ($d_2=1.128$, $D_3=0$, $D_4=3.267$ for a span-2 moving range), and revise only if a point signals out of control.

1. Centre line and moving ranges. $\bar X=\dfrac{\sum X_i}{20}=\dfrac{3512}{20}=175.6$ lb. The 19 successive moving ranges $MR_i=|X_i-X_{i-1}|$ are 5, 5, 9, 1, 4, 2, 14, 11, 5, 23, 32, 2, 14, 5, 13, 18, 34, 28, 14, giving $\overline{MR}=\dfrac{239}{19}=12.58$ lb.
2. Trial limits. $$UCL_{MR}=D_4\overline{MR}=3.267(12.58)=\boxed{41.10\ \text{lb}},\quad LCL_{MR}=D_3\overline{MR}=0.$$ $$UCL_X=\bar X+3\dfrac{\overline{MR}}{d_2}=175.6+3\left(\dfrac{12.58}{1.128}\right)=\boxed{209.05\ \text{lb}},\quad LCL_X=175.6-3\left(\dfrac{12.58}{1.128}\right)=\boxed{142.15\ \text{lb}}.$$
3. Check for out-of-control points. All 20 individual weights (min 148, max 202) fall strictly between $LCL_X=142.15$ and $UCL_X=209.05$, and all 19 moving ranges (max 34) fall below $UCL_{MR}=41.10$. No point signals on either chart, so no revision is needed — the trial limits stand as the final control limits.
4. In-control process-parameter estimates. Since both charts are in control as drawn, $$\hat\mu_0=\bar X=\boxed{175.6\ \text{lb}},\qquad \hat\sigma_0=\dfrac{\overline{MR}}{d_2}=\dfrac{12.58}{1.128}=\boxed{11.15\ \text{lb}}.$$

Part (c) — ARL and ATS in control, and after a shift to $\mu_1=185$

Given. $\hat\mu_0=175.6$ lb, $\hat\sigma_0=11.15$ lb, 3-sigma individuals-chart limits from part (b); one sample every half hour ($h=0.5$ hr); shifted mean $\mu_1=185$ lb.

Find. $ARL$ and $ATS$ ($=ARL\times h$) both in control and after the shift.

Approach. For an individuals chart ($n=1$, $k=3$), $\beta=\Phi(k-\delta)-\Phi(-k-\delta)$ with $\delta=(\mu_1-\mu_0)/\sigma$; $ARL=1/(1-\beta)$, and in control $\beta=0$ so $ARL_0=1/\alpha$.

1. In control. With exact 3-sigma limits, $\alpha=2[1-\Phi(3)]=2(0.00135)=0.0027$, so $$ARL_0=\dfrac{1}{\alpha}=\boxed{370.4},\qquad ATS_0=ARL_0\times0.5\ \text{hr}=\boxed{185.2\ \text{hr}}.$$
2. Shift size. $$\delta=\dfrac{\mu_1-\mu_0}{\hat\sigma_0}=\dfrac{185-175.6}{11.15}=\boxed{0.843\sigma}.$$
3. After the shift. $$\beta=\Phi(3-0.843)-\Phi(-3-0.843)=\Phi(2.157)-\Phi(-3.843)=0.9845-0.0001=0.9845,$$ $$ARL_1=\dfrac{1}{1-\beta}=\boxed{64.3},\qquad ATS_1=ARL_1\times0.5\ \text{hr}=\boxed{32.1\ \text{hr}}.$$

The shift is a modest 0.84$\sigma$, well short of the design point used in Question 2(c), so detection is comparatively slow: on average it takes roughly 64 samples (about 32 hours, or 4 full production days at half-hourly sampling) to signal, versus the in-control average of 370 samples (about 185 hours) — the chart still detects the shift far faster than pure chance would suggest, but nowhere near as fast as a chart specifically designed for a shift this size.