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Figure 16-13 Histogram for vane opening.
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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
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Process Capability Ratio
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The process capability ratio (PCR) is PCR USL 6 LSL (16-20)
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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
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Therefore, the PCR is estimated to be PCR USL 6 LSL 40 20 612.152 1.55
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The PCR has a natural interpretation: (1 PCR)100 is just the percentage of the speci cations width used by the process. Thus, the vane-opening process uses approximately (1 1.55)100 64.5% of the speci cations width. Figure 16-14(a) shows a process for which the PCR exceeds unity. Since the process natural tolerance limits lie inside the specifications, very few defective or nonconforming units will be produced. If PCR 1, as shown in Fig. 16-14(b), more nonconforming units result. In fact, for a normally distributed process, if PCR 1, the fraction
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PCR > 1
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Figure 16-14 Process fallout and the process capability ratio (PCR).
nonconforming is 0.27%, or 2700 parts per million. Finally, when the PCR is less than unity, as in Fig. 16-14(c), the process is very yield-sensitive and a large number of nonconforming units will be produced. The definition of the PCR given in Equation 16-19 implicitly assumes that the process is centered at the nominal dimension. If the process is running off-center, its actual capability will be less than indicated by the PCR. It is convenient to think of PCR as a measure of potential capability, that is, capability with a centered process. If the process is not centered, a measure of actual capability is often used. This ratio, called PCRk , is defined below.
PCRk PCRk min c USL 3 , LSL 3 d (16-21)
In effect, PCRk is a one-sided process capability ratio that is calculated relative to the speci cation limit nearest to the process mean. For the vane-opening process, we nd that the
estimate of the process capability ratio PCRk is ^ PCR k min c min c USL 3 x x , LSL d 3 1.06, 33.19 20 312.152 2.04 d 1.06
40 33.19 312.152
Note that if PCR PCRk, the process is centered at the nominal dimension. Since ^ 1.06 for the vane-opening process and ^ 1.55, the process is obviously runPCR PCR k ning off-center, as was rst noted in Figs. 16-14 and 16-17. This off-center operation was ultimately traced to an oversized wax tool. Changing the tooling resulted in a substantial improvement in the process (Montgomery, 2001). The fractions of nonconforming output (or fallout) below the lower speci cation limit and above the upper speci cation limit are often of interest. Suppose that the output from a normally distributed process in statistical control is denoted as X. The fractions are determined from P1X LSL2 P1Z 1LSL 2 2 P1X USL2 P1Z 1USL 2 2
For an electronic manufacturing process a current has speci cations of 100 10 milliamperes. The process mean and standard deviation are 107.0 and 1.5, respectively. The process mean is nearer to the USL. Consequently, PCR 1110 902 16 1.52 2.22 and PCRk 1110 1072 13 1.52 0.67
The small PCRk indicates that the process is likely to produce currents outside of the speci cation limits. From the normal distribution in Appendix Table II P1X P1X LSL2 USL2 P1Z P1Z 1110 190 1072 1.52 1072 1.52 P1Z P1Z 11.332 22 0
For this example, the relatively large probability of exceeding the USL is a warning of potential problems with this criterion even if none of the measured observations in a preliminary sample exceed this limit. We emphasize that the fraction-nonconforming calculation assumes that the observations are normally distributed and the process is in control. Departures from normality can seriously affect the results. The calculation should be interpreted as an approximate guideline for process performance. To make matters worse, and need to be estimated from the data available and a small sample size can result in poor estimates that further degrade the calculation. Montgomery (2001) provides guidelines on appropriate values of the PCR and a table relating fallout for a normally distributed process in statistical control to the value of PCR. Many U.S. companies use PCR 1.33 as a minimum acceptable target and PCR 1.66 as a minimum target for strength, safety, or critical characteristics. Some companies require that internal processes and those at suppliers achieve a PCRk 2.0. Figure 16-15 illustrates a process with PCR PCRk 2.0. Assuming a normal distribution, the calculated fallout for this process is 0.0018 parts per million. A process with PCRk 2.0 is referred to as a sixsigma process because the distance from the process mean to the nearest speci cation is six standard deviations. The reason that such a large process capability is often required is that it