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(a) Using all the data, compute trial control limits for individual observations and moving-range charts. Determine whether the process is in statistical control. If not, assume assignable causes can be found to eliminate these samples and revise the control limits. (b) Estimate the process mean and standard deviation for the in-control process.

16-7

An important consequence of this path independence are the Cauchy-Riemann equations. Ifwe make dz approach zero along the x-axis, df /dz = ou/ox + iOv/oy. If we make dz approach zero along the y-axis, df /dz = Ov/oy - iou/oy. Thus,





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It is usually necessary to obtain some information about the process capability, that is, the performance of the process when it is operating in control. Two graphical tools, the tolerance chart (or tier chart) and the histogram, are helpful in assessing process capability.

(C.20)





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The tolerance chart for all 20 samples from the vane-manufacturing process is shown in Fig. 16-12. The speci cations on vane opening are 0.5030 0.0010 in. In terms of the coded data, the upper speci cation limit is USL 40 and the lower speci cation limit is LSL 20, and these limits are shown on the chart in Fig. 16-12. Each measurement is plotted on the tolerance chart. Measurements from the same subgroup are connected with lines. The tolerance chart is useful in revealing patterns over time in the individual measurements, or it may show that a particular value of x or r was produced by one or two unusual observations in the sample. For example, note the two unusual observations in sample 9 and the single unusual observation in sample 8. Note also that it is appropriate to plot the speci cation limits on the tolerance chart, since it is a chart of individual measurements. It is never appropriate to plot speci cation limits on a control chart or to use the speci cations in determining the control limits. Speci cation limits and control limits are unrelated. Finally, note from Fig. 16-12 that the process is running off-center from the nominal dimension of 30 (or 0.5030 inch). The histogram for the vane-opening measurements is shown in Fig. 16-13. The observations from samples 6, 8, 9, 11, and 19 (corresponding to out of-control points on either the X or R chart) have been deleted from this histogram. The general impression from examining this histogram is that the process is capable of meeting the speci cation but that it is running off-center. Another way to express process capability is in terms of an index that is defined as follows.

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18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 LSL Nominal dimension Vane opening USL

The Cauchy-Riemann equations (C.20) have a powerful application in physics for finding solutions to the two-dimensional Laplace equation. Differentiating the first Cauchy-Riemann equation in (C.20) by x, the second by y and adding, we find that

The process capability ratio (PCR) is PCR USL 6 LSL (16-20)

The numerator of PCR is the width of the speci cations. The limits 3 on either side of the process mean are sometimes called natural tolerance limits, for these represent limits that an in-control process should meet with most of the units produced. Consequently, 6 is often referred to as the width of the process. For the vane opening, where our sample size is 5, we could estimate as

(C.21)

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