Question 4 of 6: p- vs. u-Chart and the Demerit Chart, Positive c-Chart LCL, and a u-Chart for Paper-Roll Imperfections

Question 4: p- vs. u-Chart and the Demerit Chart, Positive c-Chart LCL, and a u-Chart for Paper-Roll Imperfections (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.

(a) p-chart vs. u-chart, and the demerit chart

The p-chart monitors the FRACTION (or percentage) NONCONFORMING — each inspected UNIT is classified as either conforming or nonconforming as a whole (pass/fail), and the chart tracks $\hat p=d/n$, the proportion of defective units in each sample; it answers "what share of items are bad?" and is used whenever a simple accept/reject classification per unit is natural (e.g. percentage of defective printed-circuit boards, percentage of late shipments). The u-chart instead monitors the AVERAGE NUMBER OF NONCONFORMITIES (individual defects/imperfections) PER INSPECTION UNIT, $\hat u=c/n$, where a single physical unit can carry MULTIPLE separate nonconformities (a unit is not simply pass/fail; it can have 0, 1, 2, ... defects) — it answers "how many defects, on average, per unit (or per roll, per vehicle, per garment) are we producing?" and is used whenever defects are COUNTED rather than units simply classified, especially when the inspection-unit size (area, length, number of items in the sample) varies from period to period, as with the paper-mill rolls in part (c). In short: p-chart = fraction of BAD UNITS; u-chart = average COUNT OF DEFECTS per unit, allowing several defects per unit and a variable inspection-unit size.

A demerit chart extends the u-chart idea by recognizing that not all nonconformities are equally serious: each defect is classified into a severity class (e.g. critical, major, minor, very minor) and assigned a demerit WEIGHT (commonly 100/50/10/1, though weights are process-specific), and the chart plots the total weighted demerit score per inspection unit (or per 100 units), $D=100c_1+50c_2+10c_3+c_4$ (for four severity classes), rather than a simple unweighted defect count. This lets a single critical defect appropriately dominate the quality signal over many trivial cosmetic defects, which a plain u-chart (treating every nonconformity as equally important) cannot distinguish.

(b) Why a positive c-chart LCL is preferred, and how to achieve it

Quality engineers prefer a strictly POSITIVE lower control limit on a c-chart (or u-chart) because a positive $LCL$ lets the chart signal an unusually GOOD result — a sample count falling below $LCL$ is itself an out-of-control signal worth investigating, since it may indicate a genuine, favourable process improvement (a beneficial assignable cause: a new inspector catching fewer real defects because the process actually improved, a raw-material change, better calibration) that should be understood and, if real, LOCKED IN as the new standard. With $LCL=0$ (the usual case when the count is low), no sample count can ever fall below the limit, so the chart can never detect and credit a genuine improvement — it becomes a one-sided (upper-limit-only) monitoring tool that only ever asks "did things get worse?", silently discarding the diagnostic value of "did things get meaningfully better?"

It is not always possible to have a positive c-chart $LCL=\bar c-3\sqrt{\bar c}$: this requires $\bar c-3\sqrt{\bar c}>0$, i.e. $\bar c>9$. Many real processes run at a low average defect count ($\bar c<9$, often the GOAL of a well-controlled process), for which $3\sigma$ limits necessarily give a negative (and therefore clipped-to-zero) lower limit. When a positive $LCL$ IS wanted and the average count is high enough (or can be raised by inspecting a larger unit), it is achieved either by increasing the size of the inspection unit $n$ (u-chart: raising $\bar u\,n$ so the expected count per inspection exceeds 9), or, more commonly, by using NARROWER control limits than the standard $3\sigma$ (a smaller multiplier $k<3$, e.g. $k$ chosen so $\bar c-k\sqrt{\bar c}>0$, accepting a slightly higher false-alarm rate $\alpha$ in exchange for the ability to detect favourable shifts) — Montgomery gives a standard table of the minimum count needed to guarantee a positive $LCL$ at various $k$ multiples of sigma.

(c) u-chart for paper-roll imperfections

Given. 15 days of inspection, rolls produced (the inspection-unit count) alternating between 18 and 20 per day.

Find. u-chart control limits (revised if necessary) and $\hat\lambda_0$, the expected nonconformities per roll.

1. Pooled estimate of imperfections per roll. Total imperfections $=17+14+7+18+15+12+11+15+12+10+11+14+9+10+14=189$; total rolls $=11(20)+4(18)=292$. $$\bar u=\frac{189}{292}=\boxed{0.6473\ \text{imperfections/roll}}.$$
2. Control limits (vary with $n_i$). $UCL_i=\bar u+3\sqrt{\bar u/n_i}$, $LCL_i=\max[0,\,\bar u-3\sqrt{\bar u/n_i}]$: $$n_i=20:\ UCL=0.6473+3\sqrt{0.6473/20}=\boxed{1.187},\quad LCL=\boxed{0.108};$$ $$n_i=18:\ UCL=0.6473+3\sqrt{0.6473/18}=\boxed{1.216},\quad LCL=\boxed{0.078}.$$
3. Check each day and revise if necessary. Daily $\hat u_i=c_i/n_i$ ranges from $7/20=0.35$ (day 3) to $18/20=0.90$ (day 4) — every value lies between its day's $LCL$ ($0.108$ or $0.078$) and $UCL$ ($1.187$ or $1.216$). No point is out of control, so no revision is needed.
4. Expected nonconformities per roll. The in-control pooled rate IS the estimate of the process's mean nonconformity rate: $$\hat\lambda_0=\bar u=\boxed{0.6473\ \text{imperfections/roll}}.$$