introduction to statistical quality control, 4th edition chapter 7 process and measurement system...

Introduction to Statistical Quality Control, 4th Edition

Chapter 7

Process and Measurement System Capability Analysis


7-1. Introduction • Process capability refers to the uniformity of the process.• Variability in the process is a measure of the uniformity of

output.• Two types of variability:

– Natural or inherent variability (instantaneous)– Variability over time

• Assume that a process involves a quality characteristic that follows a normal distribution with mean , and standard deviation, . The upper and lower natural tolerance limits of the process are

UNTL = + 3LNTL = - 3


7-1. Introduction

• Process capability analysis is an engineering study to estimate process capability.

• In a product characterization study, the distribution of the quality characteristic is estimated.


7-1. Introduction Major uses of data from a process capability analysis

1. Predicting how well the process will hold the tolerances.2. Assisting product developers/designers in selecting or

modifying a process.3. Assisting in Establishing an interval between sampling

for process monitoring.4. Specifying performance requirements for new

equipment.5. Selecting between competing vendors.6. Planning the sequence of production processes when

there is an interactive effect of processes on tolerances7. Reducing the variability in a manufacturing process.


7-1. Introduction

Techniques used in process capability analysis

1. Histograms or probability plots

2. Control Charts

3. Designed Experiments


7-2. Process Capability Analysis Using a Histogram or a Probability Plot

7-2.1 Using a Histogram• The histogram along with the sample mean and

sample standard deviation provides information about process capability.

– The process capability can be estimated as– The shape of the histogram can be determined (such

as if it follows a normal distribution) – Histograms provide immediate, visual impression of

process performance.

s3x


7-2.2 Probability Plotting

• Probability plotting is useful for– Determining the shape of the distribution– Determining the center of the distribution– Determining the spread of the distribution.

• Recall normal probability plots (Chapter 2)– The mean of the distribution is given by the 50th

percentile– The standard deviation is estimated by

84th percentile – 50th percentile


7-2.2 Probability Plotting

Cautions in the use of normal probability plots• If the data do not come from the assumed

distribution, inferences about process capability drawn from the plot may be in error.

• Probability plotting is not an objective procedure (two analysts may arrive at different conclusions).


7-3. Process Capability Ratios

7-3.1 Use and Interpretation of Cp

• Recall

where LSL and USL are the lower and upper specification limits, respectively.

6

LSLUSLCp



The estimate of Cp is given by

Where the estimate can be calculated using the sample standard deviation, S, or

ˆ6

LSLUSLCp

2d/R



Piston ring diameter in Example 5-1

• The estimate of Cp is

68.1

)0099.0(6

95.7305.74Cp



One-Sided Specifications

These indices are used for upper specification and lower specification limits, respectively

3

LSLC

3

USLC

pl

pu



Assumptions

The quantities presented here (Cp, Cpu, Clu) have some very critical assumptions:

1. The quality characteristic has a normal distribution.2. The process is in statistical control3. In the case of two-sided specifications, the process mean

is centered between the lower and upper specification limits.

If any of these assumptions are violated, the resulting quantities may be in error.


7-3.2 Process Capability Ratio an Off-Center Process

• Cp does not take into account where the process mean is located relative to the specifications.

• A process capability ratio that does take into account centering is Cpk defined as

Cpk = min(Cpu, Cpl)


7-3.3 Normality and the Process Capability Ratio

• The normal distribution of the process output is an important assumption.

• If the distribution is nonnormal, Luceno (1996) introduced the index, Cpc, defined as

TXE2

6

LSLUSLCpc


7-3.3 Normality and the Process Capability Ratio

• A capability ratio involving quartiles of the process distribution is given by

• In the case of the normal distribution Cp(q) reduces to Cp

00135.099865.0p xx

LSLUSL)q(C


7-3.4 More About Process Centering

• Cpk should not be used alone as an measure of process centering.

• Cpk depends inversely on and becomes large as approaches zero. (That is, a large value of Cpk does not necessarily reveal anything about the location of the mean in the interval (LSL, USL)



• An improved capability ratio to measure process centering is Cpm.

where is the squre root of expected squared deviation from target: T =½(USL+LSL),

6

LSLUSLCpm

2222 )T(TxE



• Cpm can be rewritten another way:

where

2

p

22pm

1

C

)T(6

LSLUSLC

T



• A logical estimate of Cpm is:

where

2

ppm

V1

CC

S

xTV



Example 7-3. Consider two processes A and B.• For process A:

since process A is centered.• For process B:

0.101

0.1

1

CC

2

ppm

63.0)3(1

0.2

1

CC

22

ppm


7-3.4 More About Process Centering• A third generation process capability ratio, proposed

by Pearn et. al. (1992) is

• Cpkm has increased sensitivity to departures of the process mean from the desired target.

2

pk

2

pkpkm

1

C

T1

CC


7-3.5 Confidence Intervals and Tests on Process Capability Ratios

Cp

• Ĉp is a point estimate for the true Cp, and subject to variability. A 100(1-) percent confidence interval on Cp is

1nCC

1nC

21n,2/

pp

21n,2/1

p



Example 7-4. USL = 62, LSL = 38, n = 20,

S = 1.75, The process mean is centered. The point estimate of Cp is

95% confidence interval on Cp is

01.3C57.119

85.3229.2C

19

91.829.2

p

p

29.2)75.1(6

3862Cp



Cpk

• Ĉpk is a point estimate for the true Cpk, and subject to variability. An approximate 100(1-) percent confidence interval on Cpk is

)1n(2

1

Cn9

1Z1CC

)1n(2

1

Cn9

1Z1C

pk

2/pkpk

pk

2/pk



Example 7-5. n = 20, Ĉpk = 1.33. An approximate 95%

confidence interval on Cpk is

• The result is a very wide confidence interval ranging from below unity (bad) up to 1.67 (good). Very little has really been learned about actual process capability (small sample, n = 20.)

)19(2

1

33.1)20(9

196.1133.1C

)19(2

1

33.1)20(9

196.1133.1 pk

67 . 1 C 99 . 0pk



Cpc

• Ĉpc is a point estimate for the true Cpc, and subject to variability. An approximate 100(1-) percent confidence interval on Cpc is

where

nc

st1

CC

nc

st1

C

c

1n,2

pcpc

c

1n,2

pc

n

1ii Tx

n

1c



Example 7-5. n = 20, Ĉpk = 1.33. An approximate 95%

confidence interval on Cpk is

• The result is a very wide confidence interval ranging from below unity (bad) up to 1.67 (good). Very little has really been learned from this result, (small sample, n = 20.)

)19(2

1

33.1)20(9

196.1133.1C

)19(2

1

33.1)20(9

196.1133.1 pk

67 . 1 C 99 . 0pk



Testing Hypotheses about PCRs

• May be common practice in industry to require a supplier to demonstrate process capability.

• Demonstrate Cp meets or exceeds some particular target value, Cp0.

• This problem can be formulated using hypothesis testing procedures



Testing Hypotheses about PCRs• The hypotheses may be stated as

H0: Cp Cp0 (process is not capable)

H0: Cp Cp0 (process is capable)

• We would like to reject Ho

• Table 7-5 provides sample sizes and critical values for testing H0: Cp = Cp0



Example 7-6• H0: Cp = 1.33

H1: Cp > 1.33• High probability of detecting if process capability is

below 1.33, say 0.90. Giving Cp(Low) = 1.33• High probability of detecting if process capability

exceeds 1.66, say 0.90. Giving Cp(High) = 1.66 = = 0.10.• Determine the sample size and critical value, C, from

Table 7-5.



Example 7-6• Compute the ratio Cp(High)/Cp(Low):

• Enter Table 7-5, panel (a) (since = = 0.10). The sample size is found to be n = 70 and C/Cp(Low) = 1.10

• Calculate C:

25.133.1

66.1

)Low(C

)High(C

p

p

46.1

)10.1(33.1

)10.1)(Low(CpC



Example 7-6• Interpretation:

– To demonstrate capability, the supplier must take a sample of n = 70 parts, and the sample process capability ratio must exceed 1.46.


7-4. Process Capability Analysis Using a Control Chart

• If a process exhibits statistical control, then the process capability analysis can be conducted.

• A process can exhibit statistical control, but may not be capable.

• PCRs can be calculated using the process mean and process standard deviation estimates.


7-5. Process Capability Analysis Designed Experiments

• Systematic approach to varying the variables believed to be influential on the process. (Factors that are necessary for the development of a product).

• Designed experiments can determine the sources of variability in the process.


7-6. Gage and Measurement System Capability Studies

7-6.1 Control Charts and Tabular Methods

• Two portions of total variability:– product variability which is that variability

that is inherent to the product itself– gage variability or measurement variability

which is the variability due to measurement error

2gage

2product

2Total



and R Charts

• The variability seen on the chart can be interpreted as that due to the ability of the gage to distinguish between units of the product

• The variability seen on the R chart can be interpreted as the variability due to operator.

X

X



Precision to Tolerance (P/T) Ratio• An estimate of the standard deviation for

measurement error is

• The P/T ratio is2

gage d

Rˆ

LSLUSL

ˆ6T/P gage



• Total variability can be estimated using the sample variance. An estimate of product variability can be found using

2gage

22product

2gage

2product

2

2gage

2product

2Total

ˆSˆ

ˆˆS



Percentage of Product Characteristic Variability• A statistic for process variability that does not

depend on the specifications limits is the percentage of product characteristic variability:

100ˆ

ˆ

product

gage



Gage R&R Studies• Gage repeatability and reproducibility (R&R)

studies involve breaking the total gage variability into two portions:

– repeatability which is the basic inherent precision of the gage

– reproducibility is the variability due to different operators using the gage.



Gage R&R Studies

• Gage variability can be broken down as

• More than one operator (or different conditions) would be needed to conduct the gage R&R study.

2ityrepeatabil

2ilityreproducib

2gage

2errortmeasuremen


7-6.1 Control Charts and Tabular MethodsStatistics for Gage R&R Studies (The Tabular

Method)• Say there are p operators in the study• The standard deviation due to repeatability can be found

as

where

and d2 is based on the # of observations per part per operator.

2ityrepeatabil d

Rˆ

p

RRRR p21



Statistics for Gage R&R Studies (the Tabular Method)

• The standard deviation for reproducibility is given as

where

d2 is based on the number of operators, p

2

xilityreproducib d

Rˆ

)x,x,xmin(x

)x,x,xmax(x

xxR

p21min

p21max

minmaxx


7-6.2 Methods Based on Analysis of Variance

• The analysis of variance (Chapter 3) can be extended to analyze the data from an experiment and to estimate the appropriate components of gage variability.

• For illustration, assume there are a parts and b operators, each operator measures every part n times.



• The measurements, yijk, could be represented by the model

where i = part, j = operator, k = measurement.

n,...,2,1k

b,...,2,1j

a,...2,1i

)(y ijkijjiijk



• The variance of any observation can be given by

are the variance components.

2222ijk )y(V

2222 ,,,



• Estimating the variance components can be accomplished using the following formulas

bn

MSMSˆ

an

MSMSˆ

n

MSMSˆ

MSˆ

ABA2

ABB2

EAB2

E2

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