Process and Measurement System Capability Analysis Introduction - - PowerPoint PPT Presentation

ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Process and Measurement System Capability Analysis

Introduction “Process capability” means broadly the ability of a process to achieve satisfactory performance, in light of its inherent variability. For example, the Process Capability Ratio (PCR) Cp compares the difference between a process’s specification limits with the difference between its natural tolerance limits: Cp = USL − LSL UNTL − LNTL. But specifications are not necessary to analyze process capability.

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Process Capability Analysis Using Graphics

Histogram Histograms can be helpful in estimating process capability. Example: Bursting strength of glass containers Table 8.1 gives the bursting strengths (psi) of 100 glass containers. In R:

burst <- read.csv("Data/Table-08-01.csv"); hist(burst$Strength) # To match Figure 8.2: hist(burst$Strength, right = FALSE, breaks = seq(from = 170, to = 350, by = 20))

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

The histogram is broadly similar to the normal density, so the data may be summarized by ¯ x = 264.1, s = 32.0. The q-q plot is a better tool for comparing data with the normal distribution:

qqnorm(burst$Strength)

The q-q plot is very much what we would expect for a random sample from a normal distribution, so we estimate the natural tolerance limits µ ± 3σ by 264.1 ± 3 × 32.0 = (168.1, 360.1). We expect 99.73% of containers to have strengths in this interval.

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

The natural tolerance limits describe what to expect from this production process. Notes No specification limits are needed up to this point. However, if limits are given, they may be added to the histogram to allow a visual assessment of process capability. Probabilities like 99.73% are heavily dependent on the data being normally distributed. Small samples, even as large as 100, do not permit identification of small but important deviations from normality. For example, a sample of 100 from the t-distribution with 10 degrees

• f freedom may also look normal, but that distribution has almost 3

times the probability outside the natural tolerance limits.

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Process Capability Ratios

Recall Cp = USL − LSL UNTL − LNTL = USL − LSL 6σ . If the process is centered (µ = 1

2(USL + LSL)) then

USL − µ σ =

1 2(USL − LSL)

σ = 3Cp.

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

If the process is also normally distributed, the fraction nonconforming is P(X > USL) + P(X < LSL) = P X − µ σ > USL − µ σ

• + P

X − µ σ < LSL − µ σ

• = P(Z > 3Cp) + P(Z < −3Cp)

= 2[1 − Φ(3Cp)]. So the process capability ratio determines the fraction of nonconforming items.

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

For example, if Cp ≥ 1, then at least 99.73% of items will conform to specification (2,700 ppm nonconformant). If Cp ≥ 1.5, then at most 6.8 ppm of items will be nonconformant (“six-sigma” quality would be 3.4 ppm).

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

One-sided process capability In some contexts, there may be only one specification limit. For the example of the glass containers, LSL = 200 psi, and there is no USL. The lower process capability ratio Cpl is Cpl = µ − LSL 3σ , which is estimated by (¯ x − LSL)/(3s) = 0.67.

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Again, if the process is normally distributed, the fraction nonconforming is P(X < LSL) = P X − µ σ < LSL − µ σ

• = P(Z < −3Cpl)

= 1 − Φ(3Cpl). For the glass containers, the fraction of nonconforming containers, assuming normality, is 1 − Φ(3Cpl) = 1 − Φ(2) = 2.28%, or around 22,800 ppm.

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Off-center process capability If a process is not centered, the fraction nonconforming cannot be calculated only from Cp, and is higher than it would be for a centered process with the same σ. A modified PCR is Cpk = min(Cpu, Cpl), where Cpl is the lower process capability ratio and Cpu = USL − µ 3σ , is the corresponding upper process capability ratio.

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Clearly Cp = 1 2(Cpu + Cpl), and hence Cpk ≤ Cp. For a normally distributed process, the fraction nonconforming is 1 − Φ(3Cpk) + 1 − Φ[3(2Cp − Cpk)]. The second term is always less than or equal to the first, and much smaller for a substantially off-center process. For example, if Cp = 1.5 and Cpk = 1, the two terms are 1 − Φ(3) = 1.35 × 10−3 and 1 − Φ(6) = 9.87 × 10−10, respectively.

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Other PCRs Various other efforts have been made to define PCRs that capture process capability when the process is not centered. Statistical inference All Cps are defined in terms of population parameters, so when they are estimated from sample data, the usual issues arise: Point estimates; Interval estimates; Testing hypotheses.

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Process Capability Analysis Using a Control Chart

The process capability ratios Cp etc. do not distinguish between short-term variability, which is inherent to the process, and long-term variability, which may be attributed to assignable causes. A control chart can help to separate them, when the sample structure is known.

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

Example: bursting strengths The bursting strengths of glass containers were actually collected in m = 20 samples of size n = 5:

burst$Sample <- rep(1:20, rep(5, 20)) library(qcc) burstG <- qcc.groups(burst$Strength, burst$Sample) summary(qcc(burstG, "R")) summary(qcc(burstG, "xbar"))

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ST 435/535 Statistical Methods for Quality and Productivity Improvement / Statistical Process Control

The process appears to be in control, so the new statistics can be used to assess process capability:

process.capability(qcc(burstG, "xbar"), c(LSL = 200, USL = NA)) # C_pl = 0.6425, 95% CI is (0.5495, 0.7355)

Note that the output includes confidence intervals for the PCRs (in this case, only Cpl can be calculated, because there is no USL). We can use the more conventional pooled variance estimate of σ2, instead of the range-based calculation:

process.capability(qcc(burstG, "xbar", std.dev = "RMSDF"), spec.limits = c(LSL = 200, USL = NA))

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