22-Mec-B4 Integrated Manufacturing Systems · May 2015

Question 3 of 7: A Control Criterion for a Purchased Watch Gear

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

Question 3: A Control Criterion for a Purchased Watch Gear (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 completed base-period study of the gear dimension — twenty-five rational subgroups of five pieces each, with the grand average and the average subgroup range already reduced.

Find. The operating rule the vendor should adopt to declare the gear process out of control, the correct way to set that rule against the horologist’s print tolerance, and the courses of action open to the vendor when the two do not match.

Approach. Build Shewhart limits for the average and the range from the base-period statistics; convert $\bar{R}$ into an estimate of the process standard deviation so that the spread of individual gears can be stated; and only then set that spread against the specification, since control limits and specification limits describe different populations and are never compared directly.

1. The criterion, part one: limits for the subgroup average. With the process standard deviation unknown, the average range supplies it through the factor $A_{2}$:$$UCL_{\bar{X}},\;LCL_{\bar{X}}=\bar{\bar{X}}\pm A_{2}\bar{R}=0.125\pm 0.577(0.002)$$$$\boxed{UCL_{\bar{X}}=0.12615\ \text{in},\qquad LCL_{\bar{X}}=0.12385\ \text{in}}$$
2. The criterion, part two: limits for the subgroup range. A chart on the average alone cannot see a change in dispersion, so the range is charted alongside it:$$UCL_{R}=D_{4}\bar{R}=2.114(0.002)=\boxed{0.00423\ \text{in}},\qquad LCL_{R}=D_{3}\bar{R}=0$$The lower range limit is zero at $n=5$, which simply means that with only five pieces a subgroup range cannot be small enough to signal.
3. State the operating rule. The vendor takes a rational subgroup of five consecutive gears at a fixed frequency, plots $\bar{X}$ and $R$, and treats the process as out of control whenever a point falls outside either pair of limits, or whenever the pattern within the limits is non-random — seven or more consecutive points on one side of the centre line, a run of seven rising or falling, or a systematic cycle. An in-control signal is a licence to leave the process alone; an out-of-control signal is an instruction to find and remove an assignable cause before the batch is submitted.
4. Convert the control information into the spread of individual pieces. The specification applies to single gears, not to averages of five, so the process standard deviation must be recovered first:$$\hat{\sigma}=\frac{\bar{R}}{d_{2}}=\frac{0.002}{2.326}=0.00086\ \text{in}$$and the natural tolerance of the process is$$6\hat{\sigma}=0.00516\ \text{in},\qquad \boxed{0.12242\ \text{in}\ \text{to}\ 0.12758\ \text{in}}$$
5. Check the internal consistency of the two results. The $\bar{X}$ limits should be exactly three standard errors wide:$$3\sigma_{\bar{X}}=\frac{3\hat{\sigma}}{\sqrt{n}}=\frac{3(0.00086)}{\sqrt{5}}=0.00115\ \text{in}$$against $A_{2}\bar{R}=0.00115$ in. The agreement confirms that $A_{2}=3/(d_{2}\sqrt{n})$ has been applied correctly and that the narrowness of the chart limits is a property of averaging, not a tighter requirement on the part.

How the criterion should compare with the specification. It should not be compared with it at all in the direct sense — and this is the substance of the second part of the question. Control limits are computed from the process and describe what the process does when only chance causes act; specification limits are set by the horologist and describe what the watch needs. Plotting the print tolerance on the $\bar{X}$ chart is a classic error, because the chart carries averages of five whose spread is $\sqrt{5}$ times smaller than that of the individual gears the customer receives. The legitimate comparison is between the natural tolerance $6\hat{\sigma}=0.00516$ in and the width of the print tolerance, expressed as the capability ratio $C_{p}=(USL-LSL)/6\hat{\sigma}$.

The paper does not print the horologist’s tolerance, so the comparison is shown for the two bands that bracket normal watch-gear practice. Against a tolerance of 0.125 ± 0.003 in the process has $C_{p}=0.006/0.00516=1.16$: the criterion and the specification are compatible, and a process held inside the control limits will deliver conforming gears. Against 0.125 ± 0.002 in the ratio falls to $C_{p}=0.004/0.00516=0.78$; the process is then incapable, and — this is the point that makes the question worth twenty marks — a perfectly in-control, perfectly centred process will still ship non-conforming gears, about 2.0 per cent of them, indefinitely.

Alternatives when the criterion is not compatible with the specification. Six courses are open, in the order a vendor should consider them. Reduce the process spread, which is the only remedy that removes the problem rather than managing it: better tooling and fixturing, a stiffer machine, tighter control of the incoming blank, temperature control of the gear-cutting operation. Re-centre the process if the incompatibility is an offset rather than a spread — free of capital cost, and often enough on its own when $C_{p}$ is adequate but $C_{pk}$ is not. Have the tolerance reviewed by the horologist’s designers, since a tolerance tighter than the function requires is a common and expensive default. Screen the output by 100 per cent inspection or automatic gauging, converting the capability shortfall into a known scrap and sorting cost; this is a holding action, not a solution, and it should be costed and time-limited. Change the process or the vendor — move the gear to a grinding or hobbing operation with an inherently smaller $\sigma$, or source it from a supplier who already has one. Finally, select and grade: if the assembly tolerates it, sort gears into classes and match them to mating parts, which trades inventory complexity for yield.