22-Mec-B4 Integrated Manufacturing Systems · May 2015

Question 4 of 7: Process Aim and Capability for Dustless Chalk

Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)

Notes on this paper

Paper format. National Exams, May 2015 — 07-Mec-B4 Integrated Manufacturing Systems. Three hours, open book, any non-communicating calculator permitted. Seven questions are printed; any five constitute a complete paper and only the first five appearing in the answer book are marked, each of equal value (20 marks). All seven are solved here, because the complete set is the study resource. Questions 5, 6 and 7 are explicitly essay questions, in which the examiners award marks for clarity and organisation as well as content.

Reference texts. E. S. Buffa and R. K. Sarin, Modern Production / Operations Management, 8th ed. (requirements schedules, economic lot size, economic order interval, part-period balancing, production planning); R. B. Chase and F. R. Jacobs, Operations and Supply Chain Management, 16th ed. (demand components, adaptive forecasting, aggregate planning, statistical quality control); B. W. Niebel and A. Freivalds, Methods, Standards, and Work Design, 13th ed. (time study, performance rating, allowances, wage incentive plans); D. C. Montgomery, Introduction to Statistical Quality Control, 8th ed. (Shewhart charts, process capability); M. P. Groover, Automation, Production Systems, and Computer-Integrated Manufacturing, 5th ed. (process planning, CAPP, machinability data systems, maintenance); S. Nahmias and T. L. Olsen, Production and Operations Analysis, 7th ed. (forecasting, aggregate planning).

Question 4: Process Aim and Capability for Dustless Chalk (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.

Given. A two-sided specification on chalk density and a single large sample summarising the process as it currently runs.

Given data
Quantity	Symbol	Value
Lower specification limit	LSL	4.4 gm/cc
Upper specification limit	USL	5.0 gm/cc
Sample size	n	100 pieces
Sample average density	x̄	4.8 gm/cc
Sample standard deviation	s	0.2 gm/cc

Find. Whether the process is aimed at the correct density; if it is not, the aim it should be set to; and whether the process spread is small enough to meet the density requirement at all.

Figure 4.1 — the process distribution against the specification. The distribution is both off-target (current aim 4.8 against a midpoint of 4.7) and far too wide: its six-sigma spread of 1.2 gm/cc is twice the 0.6 gm/cc tolerance, so the shaded non-conforming tails cannot be removed by re-aiming alone.

Approach. Treat the two questions separately, because they have different answers and different remedies: aim is a question about the location of the distribution and is tested against the specification midpoint using the standard error of the mean; capability is a question about its width and is settled by comparing $6s$ with the tolerance.

Establish the target the process ought to be aimed at. For a two-sided specification with no stated preference, the aim that minimises total non-conformance is the midpoint:$$\mu_{0}=\frac{LSL+USL}{2}=\frac{4.4+5.0}{2}=4.700\ \text{gm/cc}$$The observed aim is 4.800 gm/cc, an offset of $+0.100$ gm/cc, or half a standard deviation.
Test whether that offset is real or sampling noise. With $n=100$ the standard error of the mean is small:$$\sigma_{\bar{x}}=\frac{s}{\sqrt{n}}=\frac{0.2}{\sqrt{100}}=0.020\ \text{gm/cc},\qquad z=\frac{\bar{x}-\mu_{0}}{\sigma_{\bar{x}}}=\frac{4.800-4.700}{0.020}=5.00$$A 95 per cent confidence interval for the true mean is $4.800\pm1.96(0.020)=4.761$ to $4.839$ gm/cc, which excludes 4.700 comfortably. The offset is real.
Answer the first two parts. $$\boxed{\text{The process is NOT aimed correctly; the aim should be moved from } 4.800 \text{ to } 4.700\ \text{gm/cc}}$$This is a settings change — binder ratio, compaction pressure, mould fill — and it costs the manufacturer nothing but the adjustment.
Now test capability, which is a separate question. The tolerance band is $USL-LSL=0.6$ gm/cc while the process spread is$$6s=6(0.2)=1.2\ \text{gm/cc}$$so the potential capability ratio is$$C_{p}=\frac{USL-LSL}{6s}=\frac{0.6}{1.2}=\boxed{0.50}$$and, accounting for the present offset, $C_{pk}=\min(USL-\bar{x},\;\bar{x}-LSL)/3s=0.2/0.6=0.33$. Both are far below 1.00.
Quantify what that means in scrap. As the process currently runs, the two tails are$$z_{U}=\frac{5.0-4.8}{0.2}=1.00\;\Rightarrow\;15.87\ \text{per cent too dense},\qquad z_{L}=\frac{4.4-4.8}{0.2}=-2.00\;\Rightarrow\;2.28\ \text{per cent too light}$$for a total of $\boxed{18.14\ \text{per cent non-conforming}}$. Re-aiming to 4.700 gm/cc puts both limits at $z=\pm1.50$ and reduces this to $2(6.68)=13.36$ per cent — a worthwhile and free improvement, but still roughly one stick of chalk in seven.
State the variability reduction the specification actually demands. Setting $C_{p}=1$ and solving for the standard deviation:$$\sigma_{\text{req}}=\frac{USL-LSL}{6}=\frac{0.6}{6}=\boxed{0.100\ \text{gm/cc}}$$which is a 50 per cent reduction from the present 0.2 gm/cc and would leave 0.27 per cent non-conforming on a centred process. A modern capability requirement of $C_{p}=1.33$ would need $\sigma\le0.075$ gm/cc, a 62 per cent reduction.

The two findings must be reported together, because acting on the first alone would be a mistake. Re-aiming is the immediate action and it is free, but it takes the reject rate only from 18.1 per cent to 13.4 per cent. The process is not capable of meeting the density requirement: the density distribution is twice as wide as the tolerance it must live in, and no amount of centring can fix a width problem. The manufacturer must either halve the density variability — tighter control of raw-material bulk density, mixing time, moisture and compaction — or return to the designers and ask whether a band of 4.4 to 5.0 gm/cc is genuinely what the product needs, and screen the output in the meantime.

Final results — Question 4
Question asked	Result
Specification midpoint (proper aim)	4.700 gm/cc
Standard error of the mean	0.020 gm/cc
Test of the aim	z = 5.00; 95 per cent CI 4.761 to 4.839 excludes 4.700
Is the process aimed properly?	No — it runs 0.100 gm/cc too dense
What should the aim be?	4.700 gm/cc
Process spread, 6s	1.200 gm/cc against a 0.600 gm/cc tolerance
C_p / C_pk	0.50 / 0.33
Is the process capable?	No — C_p = 0.50
Non-conforming, current aim	18.14 per cent (15.87 high + 2.28 low)
Non-conforming, re-aimed to 4.700	13.36 per cent
σ required for C_p = 1.00 / 1.33	0.100 / 0.075 gm/cc

Check: the sample statistics are treated as the process parameters. With $n=100$ the sample standard deviation is a close estimate of $\sigma$ (its own relative standard error is about 7 per cent), so $s$ is used directly in the capability ratios rather than carrying a chi-square interval through the answer. Normality of the density distribution is also assumed, which is what licenses the tail percentages; the capability verdict does not depend on it, since $6s=2(USL-LSL)$ regardless of shape. A lower 95 per cent confidence bound on $C_{p}$ is 0.43, so the conclusion “not capable” is safe.

Concept & Theory

Topic. Process centring versus process capability — two independent properties of a distribution measured against a specification.

Key relations. Target $\mu_{0}=(LSL+USL)/2$; standard error $\sigma_{\bar{x}}=s/\sqrt{n}$; $C_{p}=(USL-LSL)/6s$; $C_{pk}=\min(USL-\bar{x},\bar{x}-LSL)/3s$; fraction non-conforming $=\Phi\!\left((LSL-\mu)/\sigma\right)+1-\Phi\!\left((USL-\mu)/\sigma\right)$.

Why this works. A distribution sitting inside a tolerance band can fail in two independent ways: it can be in the wrong place, or it can be too wide. $C_{p}$ isolates width by asking what fraction of the tolerance the process spread consumes, deliberately ignoring where the mean sits; $C_{pk}$ then penalises the offset. Separating them is what makes the answer actionable, because the two failures have completely different remedies and completely different costs. Re-centring is an adjustment; narrowing the distribution is an engineering project. Reporting only $C_{pk}$ — 0.33 here — would hide the fact that even a perfectly centred process would still be half as good as it needs to be.

Common pitfall. Concluding that the process is capable because the sample mean lies inside the specification. The mean of a distribution twice as wide as the tolerance will almost always lie inside it; what matters is the spread. The second trap is dividing by $\sigma_{\bar{x}}=s/\sqrt{n}$ when computing the fraction non-conforming — the specification applies to individual sticks of chalk, so the tail areas use $s$, not $s/\sqrt{n}$; only the test of the aim uses the standard error.

Source. Montgomery, Introduction to Statistical Quality Control, 8th ed., ch. 8 (process capability analysis, C_p and C_pk, capability confidence bounds); Chase & Jacobs, Operations and Supply Chain Management, 16th ed., ch. 12 (process capability index and capability versus centring).

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