8 Process Capability Analysis 8.1 Introduction • a Process Capability

8 Process Capability Analysis 8.1 Introduction • a Process Capability

8 Process Capability Analysis 8.1 Introduction • A process capability analysis relates the inherent variability in a process to specifications or requirements for the product produced by that process. • There are many ways of analyzing the capability of a process. The most common being: (1) Histograms and probability plots (3) Process capability ratios. (2) The control chart (4) Designed experiments. • Process capability measures the uniformity of a process. Process variability (variance) and systematic deviations from a target value (bias) are the primary sources of nonuniformity. • We will study the two major components of process variability: { Short-term variability which reflects the inherent random variability at a point in time. { Long-term variability which reflects the variability over time. • It is common to take a 6σ spread as a measure of process capability (where σ comes from the distribution of the product quality characteristic of interest). • When the distribution is assumed to be normal N(µ, σ), we define the natural tolerance limits to be µ ± 3σ. In this case, 99.73% of process output will be within the tolerance limits. • One way to estimate of process capability is to find a probability distribution that best de- scribes data from that process (e.g. normal, weibull, gamma, lognormal, etc.). Once an acceptable distribution has been found a process capability analysis is performed by compar- ing the properties of fitted distribution to specification limits. • When the researcher observes the process directly and can control or monitor the data- collection procedure, the study is a true process capability study because by controlling data collection and knowing the time sequence of the data, inferences can be made about the stability of the process over time. • Major applications of data from a process capability analysis are: 1. Predicting how well the process will meet tolerances. 2. Assisting, when necessary, in adjusting a process. 3. Reducing the variability in a manufacturing process. 4. Specifying performance requirements for new equipment. 5. Selecting between competing suppliers. 8.2 Using a Histogram or Probability Plots • One advantage of using a histogram is the immediate visual impression of process performance and that it could possibly indicate a reason for poor performance (off-target, outliers, skewness, bimodality, etc.). • For a histogram to be moderately stable so that it can reliably estimate process capability, Montgomery recommends that at least 100 observations be taken from the process. 114 • The histogram along with the mean x and standard deviation s enable us to assess process capability by looking first at the shape of the histogram. If it reasonably approximates a normal distribution, then x ± 3s can be used when assessing process capability. • A normal probability plot with a test for normality (such as a Kolmogorov-Smirnov test) are commonly used as supplementary checks of normality. Example 1: I used SAS to generate two data sets of 250 values from two distributions having µ = 20. • The first data set contains 250 random values from a normal N(20; 1) distribution. The variable is denoted NORMAL. • The second data set contains 250 random values from a gamma (.5,40) distribution. The variable is denoted GAMMA. • Suppose the lower and upper specification limits are LSL=17 and USL=23, respectively. • Histograms (1) and (2) have a normal pdf superimposed on the normal and gamma data histograms, respectively. • Histograms (3) and (4) have a gamma pdf superimposed on the normal and true gamma data histograms, respectively. • The estimated parameters shown below each plot are the maximum likelihood estimates (MLEs). • The quality of the fitted distribution to the hypothesized distribution can be assessed with goodness-of-fit tests. • SAS can output the results for the (i) Anderson-Darling Test, (ii) Cramer Von-Mises Test, (3) Kolmogorov-Smirnov Test, and (4) the (not-recommended) Chi-Square Goodness-of-Fit Test. SAS Summary Statistics for the Normal(20,1) sample data: -------------------------------------------------------- The CAPABILITY Procedure Variable: _normal Moments N 250 Sum Weights 250 Mean 20.0544549 Sum Observations 5013.61374 Std Deviation 0.98469082 Variance 0.96961602 Skewness 0.12451955 Kurtosis 0.28268622 Uncorrected SS 100786.725 Corrected SS 241.434388 Coeff Variation 4.91008519 Std Error Mean 0.06227732 Basic Statistical Measures Location Variability Mean 20.05445 Std Deviation 0.98469 Median 19.98213 Variance 0.96962 Mode . Range 5.90262 Interquartile Range 1.36525 115 Tests for Normality Test --Statistic--- -----p Value------ Shapiro-Wilk W 0.994503 Pr < W 0.5028 Kolmogorov-Smirnov D 0.037547 Pr > D >0.1500 Cramer-von Mises W-Sq 0.059499 Pr > W-Sq >0.2500 Anderson-Darling A-Sq 0.378384 Pr > A-Sq >0.2500 Quantiles Extreme Observations Quantile Estimate -------Lowest------- -------Highest------ 100% Max 23.3518731 Value Obs Value Obs 99% 22.4471940 95% 21.7529221 17.4492496 239 22.0698526 33 90% 21.2904965 17.5315695 29 22.1224929 213 75% Q3 20.7315978 17.8064801 234 22.4471940 70 50% Median 19.9821281 17.8153951 214 22.9921756 56 25% Q1 19.3663452 17.8168153 5 23.3518731 61 10% 18.9203281 5% 18.4856329 1% 17.8064801 0% Min 17.4492496 Specification Limits --------Limit-------- ------Percent------- Lower (LSL) 17.00000 % < LSL 0.00000 Target 20.00000 % Between 99.60000 Upper (USL) 23.00000 % > USL 0.40000 Process Capability Indices Index Value 95% Confidence Limits Cp 1.015547 0.926361 1.104631 CPL 1.033981 0.934039 1.133481 CPU 0.997113 0.900105 1.093675 Cpk 0.997113 0.900280 1.093946 Cpm 1.013998 0.926976 1.104973 SAS Summary Statistics for the Gamma(.5,40) sample data: -------------------------------------------------------- The CAPABILITY Procedure Variable: _gamma Moments N 250 Sum Weights 250 Mean 19.834919 Sum Observations 4958.72976 Std Deviation 3.20121968 Variance 10.2478074 116 Skewness 0.300444 Kurtosis -0.1739938 Uncorrected SS 100907.707 Corrected SS 2551.70405 Coeff Variation 16.1393131 Std Error Mean 0.20246291 Basic Statistical Measures Location Variability Mean 19.83492 Std Deviation 3.20122 Median 19.73126 Variance 10.24781 Mode . Range 18.15474 Interquartile Range 4.66168 Tests for Normality Test --Statistic--- -----p Value------ Shapiro-Wilk W 0.990459 Pr < W 0.1010 Kolmogorov-Smirnov D 0.048292 Pr > D >0.1500 Cramer-von Mises W-Sq 0.109171 Pr > W-Sq 0.0878 Anderson-Darling A-Sq 0.650947 Pr > A-Sq 0.0909 Quantiles Extreme Observations Quantile Estimate -------Lowest------- -------Highest------ 100% Max 30.1185447 Value Obs Value Obs 99% 27.1717936 95% 25.3220567 11.9638075 218 26.9835419 173 90% 23.9723333 13.2888758 174 27.1096947 28 75% Q3 22.2090666 13.5740292 234 27.1717936 96 50% Median 19.7312570 13.6244628 1 27.2287022 176 25% Q1 17.5473890 13.6678656 17 30.1185447 5 10% 15.7160413 5% 14.9488884 1% 13.5740292 0% Min 11.9638075 Specification Limits --------Limit-------- ------Percent------- Lower (LSL) 17.00000 % < LSL 18.40000 Target 20.00000 % Between 64.00000 Upper (USL) 23.00000 % > USL 17.60000 Process Capability Indices Index Value 95% Confidence Limits Cp 0.312381 0.284947 0.339783 CPL 0.295192 0.246202 0.343748 CPU 0.329570 0.278902 0.379784 Cpk 0.295192 0.246412 0.343971 Cpm 0.311966 0.285194 0.339956 117 PROCESS CAPABILITY COMPARISON OF NORMAL AND GAMMA DATA Distribution of _normal 25 N 250 Summary Statistics Cp 1.02 Mean 20.05 Cpk 1.00 Std Dev 0.985 Cpm 1.01 Skewness 0.125 20 Kurtosis 0.283 15 t n e c r e P 10 5 0 17.1 17.7 18.3 18.9 19.5 20.1 20.7 21.3 21.9 22.5 23.1 _normal Specifications and Curve Lower=17 Target=20 Upper=23 Normal(Mu=20.054 Sigma=0.9847) PROCESS CAPABILITY COMPARISON OF NORMAL AND GAMMA DATA Distribution of _gamma 25 N 250 Summary Statistics Cp 0.31 Mean 19.83 Cpk 0.30 Std Dev 3.201 Cpm 0.31 Skewness 0.300 20 Kurtosis -.174 15 t n e c r e P 10 5 0 10 12 14 16 18 20 22 24 26 28 30 _gamma Specifications and Curve Lower=17 Target=20 Upper=23 Normal(Mu=19.835 Sigma=3.2012) 118 PROCESS CAPABILITY COMPARISON OF NORMAL AND GAMMA DATA Distribution of _normal 25 N 250 Summary Statistics Cp 1.02 Mean 20.05 Cpk 1.00 Std Dev 0.985 Cpm 1.01 Skewness 0.125 20 Kurtosis 0.283 15 t n e c r e P 10 5 0 17.1 17.7 18.3 18.9 19.5 20.1 20.7 21.3 21.9 22.5 23.1 _normal Specifications and Curve Lower=17 Target=20 Upper=23 Gamma(Theta=0 Alpha=417 Sigma=0.05) PROCESS CAPABILITY COMPARISON OF NORMAL AND GAMMA DATA Distribution of _gamma 30 N 250 Summary Statistics Cp 0.31 Mean 19.83 Cpk 0.30 Std Dev 3.201 25 Cpm 0.31 Skewness 0.300 Kurtosis -.174 20 t n e c r 15 e P 10 5 0 10 12 14 16 18 20 22 24 26 28 30 _gamma Specifications and Curve Lower=17 Target=20 Upper=23 Gamma(Theta=0 Alpha=38.6 Sigma=0.51) 119 SAS Code for Process Capability Example with Normal and Gamma Data DM 'LOG; CLEAR; OUT; CLEAR;'; * ODS PRINTER PDF file='C:\COURSES\ST528\SAS\cp1.pdf'; ODS LISTING; OPTIONS LS=78 PS=500 NONUMBER NODATE; *******************************************************************; *** NORMAL AND GAMMA VARIATES FROM DISTRIBUTIONS WITH MEAN = 20 ***; *******************************************************************; DATA in; DO N = 1 TO 250; _normal = 20 + RANNOR(5510); ** NORMAL(20,1) **; _gamma = .5*RANGAM(20921,40); ** GAMMA(.5,40) **; OUTPUT; END; SYMBOL1 VALUE=dot WIDTH=3 L=1; TITLE 'PROCESS CAPABILITY COMPARISON OF NORMAL AND GAMMA DATA'; PROC CAPABILITY DATA=in; VAR _normal _gamma; ** Specify responses ; SPEC LSL=17 USL=23 TARGET=20 ; ** Enter specifications; *** Make histograms of the normal and gamma data ; *** with the MLE normal pdf and statistics

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