Transcript SQC_Module5_DD-done

Statistical Quality Control in Textiles
Module 5:
Process Capability Analysis
Dr. Dipayan Das
Assistant Professor
Dept. of Textile Technology
Indian Institute of Technology Delhi
Phone: +91-11-26591402
E-mail: [email protected]
Introduction
Process Capability Analysis
When the process is operating under control, we are often required
to obtain some information about the performance or capability of
the process.
Process capability refers to the uniformity of the process. The
variability in the process is a measure of uniformity of the output.
There are two ways to think about this variability.
1) Natural or inherent variability at a specified time,
2) Variability over time.
Let us investigate and assess both aspects of process capability.
Natural Tolerance Limits
The six-sigma spread in the distribution of product quality
characteristic is customarily taken as a measure of process
capability. Then the upper and lower natural tolerance limits are
Upper natural tolerance limit =  + 3σ
Lowe natural tolerance limit =  - 3σ
Under the assumption of normal distribution, the natural tolerance
limits include 99.73% of the process output falls inside the natural
tolerance limits, that is, 0.27% (2700 parts per million) falls outside
the natural tolerance limits.
Techniques for Process Capability Analysis
Techniques for Process Capability Analysis
 Histogram
 Probability Plot
 Control Charts
Histogram
It gives an immediate visual impression of process performance. It
may also immediately show the reason for poor performance.
σ
μ
LSL
μ
USL
Poor process capability
is due to excess process
variability
σ
LSL
Poor process capability
is due to poor process
centering
USL
Example: Yarn Strength (cN.tex-1) Dataset
14.11
14.99
15.08
13.14
13.21
15.79
13.78
15.65
15.47
14.41
15.85
14.84
12.26
11.93
14.08
15.32
14.57
16.80
14.31
13.69
15.16
15.12
17.03
13.09
17.97
14.41
12.35
13.69
15.58
13.90
16.38
15.36
15.21
16.49
13.99
12.86
11.82
14.31
15.05
14.92
15.65
14.48
14.45
16.14
14.62
16.80
12.52
15.76
11.87
14.08
13.25
14.67
15.10
15.10
14.38
14.04
15.67
15.44
14.67
12.93
12.40
15.90
16.53
14.43
13.01
14.45
14.62
15.77
17.12
13.40
13.56
13.62
13.40
14.05
13.62
15.26
14.67
14.08
13.44
14.67
14.87
13.35
12.72
13.40
13.78
17.06
14.53
14.18
11.98
15.58
17.51
16.14
13.94
13.31
14.84
13.45
15.58
15.90
13.17
16.53
14.08
15.85
15.46
14.17
13.35
13.41
13.25
15.90
15.03
15.56
12.42
14.16
15.90
14.58
15.90
13.40
14.03
15.44
13.44
14.82
14.43
13.67
15.42
14.84
14.18
16.17
15.36
13.62
13.62
12.44
15.21
16.43
14.97
12.86
14.67
14.08
13.73
16.34
12.72
16.01
13.78
12.90
14.31
14.53
14.99
15.44
14.08
15.44
14.85
13.41
13.69
12.72
12.72
14.18
15.41
14.87
16.94
14.38
13.40
17.89
16.70
11.09
17.71
13.84
14.08
14.92
13.81
13.39
17.09
14.62
14.94
14.68
15.05
13.78
14.48
13.60
16.63
14.18
14.41
13.22
13.29
14.92
15.62
16.09
13.28
15.67
14.99
14.71
10.57
14.92
14.84
15.68
15.05
14.84
15.10
15.10
12.72
14.09
14.31
15.65
14.67
15.94
13.30
12.29
14.41
10.84
17.64
12.34
16.69
13.99
13.11
15.16
12.23
14.15
15.44
13.89
16.19
15.85
13.73
14.18
14.31
12.80
15.34
15.31
17.17
12.95
14.62
15.44
13.32
15.34
12.72
14.08
13.51
12.91
13.50
13.26
15.62
15.08
14.92
16.53
14.40
14.76
14.67
13.14
14.08
16.96
13.44
14.31
13.79
13.89
15.68
15.86
13.84
13.06
14.87
14.71
12.23
16.32
14.84
14.54
13.78
14.67
15.90
14.53
13.21
13.06
13.53
17.36
14.92
16.34
14.57
13.44
13.85
15.94
13.78
13.60
14.76
14.84
13.60
14.58
15.47
14.99
12.47
16.08
14.31
14.99
12.53
13.25
12.81
16.11
16.35
16.48
12.47
14.08
13.78
12.60
13.35
13.51
13.06
15.58
13.89
13.87
15.12
15.36
12.98
16.19
13.51
14.18
14.53
12.19
12.96
15.70
16.32
15.90
14.31
14.35
15.20
16.19
15.15
13.17
13.69
14.18
13.21
14.31
15.26
14.99
14.72
15.49
14.84
15.62
15.12
12.91
13.21
15.67
16.43
17.12
14.53
14.62
13.69
15.68
11.44
14.53
12.93
13.30
14.13
15.03
15.68
14.31
16.14
13.85
13.55
15.65
14.67
11.97
13.89
14.97
14.58
15.68
14.43
13.44
15.16
17.49
13.82
15.35
13.48
14.41
14.08
14.67
14.99
16.96
15.71
13.85
14.52
13.94
12.44
14.09
12.72
14.84
16.14
15.94
15.16
15.01
14.18
16.70
14.59
14.31
15.21
12.72
13.89
14.41
15.16
14.31
16.53
15.16
14.67
14.08
11.92
13.56
14.41
15.37
15.21
16.35
13.35
14.92
13.62
16.80
15.71
14.99
14.82
13.62
14.53
15.26
15.12
14.84
16.34
16.11
15.90
15.21
13.06
14.04
13.44
15.58
15.31
16.96
15.58
14.31
15.65
18.02
12.32
14.77
13.42
14.31
15.58
15.90
14.62
14.26
16.43
13.81
15.16
14.22
14.31
13.40
13.21
15.16
15.22
15.81
14.18
16.14
16.11
16.80
Frequency Distribution
Class Interval
(cN.tex-1)
Class Value
xi
(cN.tex-1)
Frequency
ni
(-)
Relative
Frequency gi
(-)
Relative Frequency
Density fi
(cN-1.tex)
10.00-11.00
10.50
2
0.0044
0.0044
11.00-12.00
11.50
8
0.0178
0.0178
12.00-13.00
12.50
37
0.0822
0.0822
13.00-14.00
13.50
102
0.2267
0.2267
14.00-15.00
14.50
140
0.3111
0.3111
15.00-16.00
15.50
104
0.2311
0.2311
16.00-17.00
16.50
43
0.0956
0.0956
17.00-18.00
17.50
13
0.0289
0.0289
18.00-19.00
18.50
1
0.0022
0.0022
450
1.0000
TOTAL
Histogram
Mean = 14.57 cN tex-1
0.4
Standard deviation = 1.30 cN tex-1
0.3
The process capability would be
estimated as follows:
xcNtex-1   3scNtex-1   14.57  3.90

f cN-1 tex  0.2


0.1
0
0 10 11 12 13 14 15 16 17 18 19
xcNtex-1 





If we assume that yarn strength
follows normal distribution then it
can be said that 99.73% of the
yarns manufactured by this
process will break between 10.67
cN tex-1 to 18.47 cN tex-1.
Note that process capability can
be estimated independent of the
specifications on strength of yarn.
Probability Plot
Probability plot can determine the shape, center, and spread of the distribution. It
often produces reasonable results for moderately small samples (which the
histogram will not).
Generally, a probability plot is a graph of the ordered data (ascending order) versus
the sample cumulative frequency on special paper with a vertical scale chosen so
that the cumulative frequency distribution of the assumed type (say normal
distribution) is a straight line.
The procedure to obtain a probability plot is as follows.
1) The sample data x1 , x2 , , xn is arranged as x1 , x 2 , , x n where x1 is the
smallest observation, x 2 is the second smallest observation, and x n is the largest
observation, and so forth.
2) The ordered observations x j  are then plotted again their observed cumulative
frequency  j  0.5 n on the appropriate probability paper.
3) If the hypothesized distribution adequately describes the data, the plotted points
will fall approximately along a straight line.
Example: Yarn Strength (cN.tex-1) Dataset
Let us take that the following yarn strength data
12.35, 17.17, 15.58, 10.84, 18.02, 14.05, 13.25, 14.45, 12.35, 16.19.
xj
(j-0.5)/10
1
10.84
0.05
2
11.09
0.15
3
12.35
0.25
4
13.25
0.35
5
14.05
0.45
6
14.45
0.55
7
15.58
0.65
8
16.19
0.75
9
17.17
0.85
10
18.02
0.95
Normal Probability Plot
0.95
0.90
0.75
Probability
j
0.50
0.25
0.10
0.05
11
12
13
14
15
Data
16
17
18
The sample strength data can be regarded as taken from a
population following normal distribution.
Measures of Process Capability Analysis
Measure of Process Capability: Cp
Process capability ratio (Cp), when
the process is centered at nominal
dimension, is defined below
Cp 
USL  LSL
6
where USL and LSL stand for
upper specification limit and lower
specification limit respectively and
σ refers to the process standard
deviation.
Cp>1
LSL
3σ
μ
3σ
USL
Cp=1
LSL
3σ
μ
3σ USL
100(1/Cp) is interpreted as the
percentage of the specifications’
width used by the process.
Cp<1
LSL
3σ
μ
3σ
USL
Illustration
Suppose the specifications of yarn strength are given as 14.50±4
cN.tex-1. As the process standard deviation σ is not given, we need to
estimate this
R 3.110
ˆ 

 1.0104
d2 3.078
f  x  cN-1 .tex



We assume that the yarn
strength
follows
normal
distribution with mean at
14.50 cN.tex-1 and standard
deviation at 1.0104 cN.tex-1.
Cp 
18.5  10.5
 1.3196
6 1.0104
10.5
LSL
14.5
xcN.tex 1  18.5


USL
That is, 75.78% of the specifications’ width is used by the process.
Measure of Process Capability: Cpu and Cpl
The earlier expression of Cp assumes that the process has both upper
and lower specification limits. However, many practical situations
can give only one specification limit. In that case, the one-sided Cp is
defined by
C pu
USL  

3
 upper specification only 
  LSL
3
 lower specification only 
C pl 
Illustration
Suppose the low specification limit of yarn strength are given as
14.50 - 4 cN.tex-1. As the process standard deviation σ is not given,
we need to estimate this
R 3.110
ˆ 

 1.0104
d2 3.078
f  x  cN-1 .tex



We assume that the yarn
strength
follows
normal
distribution with mean at
14.50 cN.tex-1 and standard
deviation at 1.0104 cN.tex-1.
C pl 
14.5  10.5
 1.3196
3 1.0104
10.5
LSL
14.5
xcN.tex 1 


That is, 75.78% of the specifications’ width is used by the process.
Process Capability Ratio Versus Process Fallout [1]
Assumptions:
1)
2)
3)
The quality characteristic
is normally distributed.
The process is in statistical
control.
The process mean is
centered between USL and
LSL.
Process
Capability
Ratio
Process Fallout (in defective parts per million)
One sided
specifications
Two sided specifications
0.25
226,628
453,255
0.50
66,807
133,614
0.60
35,931
71,861
0.70
17,865
35,729
0.80
8,198
16,395
0.90
3,467
6,934
1.00
1,350
2,700
1.10
484
967
1.20
159
318
1.30
48
96
1.40
14
27
1.50
4
7
1.60
1
2
1.70
0.17
0.34
1.80
0.03
0.06
2.00
0.0009
0.0018
Measure of Process Capability: Cpk
We observed that Cp measures the
capability of a centered process.
But, all process are not necessarily
be always centered at the nominal
dimension, that is, processes may
also run off-center, then the actual
capability
of
non-centered
processes will be less than that
indicated by Cp. In the case when
the process is running off-center,
the capability of a process is
measured by the following ratio
C pk
Process running off-center
LSL
LSL
USL     LSL 
 min C pu , C pl   min 
,
3 
 3
μ
3σ
3σ
3σ
μ
USL
3σ
USL
Interpretations
1)
When Cpk=Cp then the process is centered at the midpoint of the
specifications.
2)
When Cpk<Cp then the process is running off center.
3)
When Cpk=0, the process mean is exactly equal to one of the
specification limits.
4)
When Cpk<0 then the process mean lies outside the specification
limit.
5)
When Cpk<-1 then the entire process lies outside the specification
limits.
Illustration
Suppose the specifications of yarn strength are given as 14±4 cN.tex-1.
We assume that the
yarn strength follows
normal
distribution
with mean at 14.50
cN.tex-1 and standard
deviation
at
1.0104
cN.tex-1. Clearly, the
process is running offcenter.
f  x  cN-1 .tex
 18.5  14.5 14.5  10.0 
C pk  min 
,

3

1.0104
3

1.0104


 min 1.1547,1.4846   1.1547



LSL
USL
14.5
xcN.tex 1 


Inadequacy of Cpk
Let us compare the two processes, Process A and Process B.
Process
Mean
Standard
deviation
Specification
limits
A
50.0 cN
5.0 cN
B
57.5 cN
2.5 cN
35cN, 65cN 35cN, 65cN
Cp
1
2
Cpk
1
1
Cpk interprets the processes as equally-competent.
Measure of Process Capability: Cpm
One way to address this difficulty is to use a process capability ratio
that is a better indicator of centering. One such modified ratio is
C pm 
USL  LSL
6
where τ is the square root of expected squared deviation from the
target T
T
Then, 2  2     T 
C pm 
1
USL  LSL 
2
2
USL  LSL
6 2     T 
2

USL  LSL
  T 
6 1  

  
2

PCR
  T 
1 

  
2
Measure of Process Capability: Cpmk
For non-centered process mean, the modified process capability ratio
is
 USL     LSL 
C pmk  min 
,

3 
 3
where τ is the square root of expected squared deviation from the
target T
T
Then, 2  2     T 
2
C pmk
1
USL  LSL 
2




USL  
  LSL


 min 
,
2
2 
 3 1     T  3 1     T  



 






 

Illustration
Take the example of process A and process B. Here T=50 cN. Then,
Process
Mean
Standard
deviation
Specification
limits
A
50.0 cN
5.0 cN
B
57.5 cN
2.5 cN
35cN, 65cN 35cN, 65cN
Cp
1
2
Cpk
1
1
Cpm
1
0.63
Cpmk
1
0.1582
Note to Non-normal Process Output
An important assumption underlying the earlier expressions and
interpretations of process capability ratio are based on a normal
distribution of process output. If the underlying distribution is nonnormal then
1) Use suitable transformation to see if the data can be reasonably
regarded as taken from a population following normal distribution.
2) For non-normal data, find out the standard capability index
C pc 
where T 
1
USL  LSL
2
USL  LSL

6
E x T
2
Note to Non-normal Process Output
3) For non-normal data, use quantile based process capability ratio
Cp q 
USL  LSL
x0.99865  x0.00135
As it is known that, for normal distribution,
x0.00135    3
x0.99865    3
Then,
USL  LSL
USL  LSL
Cp  q 

 Cp
6
   3     3 
Inferential Properties of Process Capability Ratios
Confidence Interval on Cp
In practice, the point estimate of Cp is found by replacing  by
sample standard deviation s. Thus, a point estimate of Cp is found as
follows
USL  LSL
ˆ
Cp 
6s
If the quality characteristic follows a normal distribution, then a
100(1-)% confidence interval on Cp is obtained as
Cˆ p
12 2,n1
n 1
 C p  Cˆ p
2 2,n1
n 1
2
where 1 2,n1
and 2 2,n1
are the lower  2 and upper
 2 percentage points of the chi-square distribution with n-1
degree of freedom.
Confidence Interval on Cpk
A point estimate of Cpk is found as follows
 USL     LSL 
ˆ
C pk  min 
,

3
s
3
s


If the quality characteristic follows a normal distribution, then a
100(1-)% confidence interval on Cpk is obtained as

1
1
ˆ
C pk 1  u 2

2
ˆ
2  n  1
9nPCRk




1
1
ˆ
   C pk  C pk 1  u 2


2
ˆ
2  n  1 
9nPCRk


Example
Suppose the specifications of yarn strength are given as 14.50±4
cN.tex-1. A random sample of 450 yarn specimens exhibits mean yarn
strength as 14.57 cN.tex-1 and standard deviation of yarn strength as
1.23 cN tex-1. Then, the 95% confidence interval on process capability
ratio is found as follows
 18.5  14.57 14.57  10.5 
Cˆ pk  min 
,
  1.07
3 1.23 
 3 1.23

1
1
1.07 1  1.96

9  450 1.07 2 2  450  1




1
1
   C pk  1.07 1  1.96


2
9

450

1.07
2
450

1

 


0.991190  C pk  1.148810
Test of Hypothesis about Cp
Many a times the suppliers are required to demonstrate the process
capability as a part of contractual agreement. It is then necessary
that Cp exceeds a particular target value say Cp0. Then the statements
of hypotheses are formulated as follows.
H: Cp=Cpo (The process is not capable.)
HA: Cp>Cpo (The process is capable.)
The supplier would like to reject H thereby demonstrating that the
process is capable. The test can be formulated in terms of Cˆ p in
such a way that H will be rejected if Cˆ p exceeds a critical value C.
A table of sample sizes and critical values of C to assist in testing
process capability is available.
Test of Hypothesis about Cp (Continued)
The Cp(high) is
defined as a
process
capability that is
accepted with
probability 1-
and Cp(low) is
defined as a
process
capability that is
likely
to
be
rejected
with
probability 1-.
Sample
==0.10
==0.05
size
Cp(high)/Cp C/Cp(low) Cp(high)/ C/Cp(low)
Cp(low)
(low)
10
1.88
1.27
2.26
1.37
20
1.53
1.20
1.73
1.26
30
1.41
1.16
1.55
1.21
40
1.34
1.14
1.46
1.18
50
1.30
1.13
1.40
1.16
60
1.27
1.11
1.36
1.15
70
1.25
1.10
1.33
1.14
80
1.23
1.10
1.30
1.13
90
1.21
1.10
1.28
1.12
100
1.20
1.09
1.26
1.11
Example
A fabric producer has instructed a yarn supplier that, in order to
qualify for business with his company, the supplier must
demonstrate that his process capability exceeds Cp=1.33. Thus, the
supplier is interested in establishing a procedure to test the
hypothesis
H: Cp=1.33
HA: Cp>1.33
The supplier wants to be sure that if the process capability is below
1.33 there will be a high probability of detecting this (say, 0.90),
whereas if the process capability exceeds 1.66 there will be a high
probability of judging the process capable (again, say 0.90).
Then, Cp(low)=1.33, Cp(high)=1.66, and ==0.10.
Example (Continued)
Let us first find out the sample size n and the critical value C.
C p  High 
C p  Low 

1.66
 1.25
1.33
Then, from table, we get, n=70 and
C
 1.10, C  C p  Low  1.10  1.33 1.10  1.46
C p  Low 
To demonstrate capability, the supplier must take a sample of n=70
and the sample process capability ratio Cp must exceed C=1.46.
Note to Practical Application
This example shows that in order to demonstrate that the process
capability is at least equal to 1.33, the observed sample Cˆ p will have
to exceed 1.33 by a considerable amount. This illustrates that some
common industrial practices may be questionable statistically. For
example, it is a fairly common practice in industry to accept the
process as capable at the level C p  1.33 if the sample Cˆ p based on a
sample size of 30  n  50 . Clearly, this procedure does not account
for sampling variation in the estimate of 
, and larger values of
n and/or higher acceptable values of Cˆ p may
be
necessary
in
practice.
Frequently Asked Questions & Answers
Frequently Asked Questions & Answers
Q1: Does process capability refer to the uniformity of the process?
A1: Yes.
Q2: State the two reasons for poor process capability.
A2: The two reasons for poor process capability are poor process centering and
excess process variability.
Q3: What is the advantage of probability plot over histogram in assessing process
capability?
A3: The probability plot requires relatively small data, while the histogram
requires relatively large data to assess process capability.
Q4: What are the measures of process capability?
A4: The measures of process capability are Cp, Cpu, Cpl, Cpk, Cpm, Cpmk.
Frequently Asked Questions & Answers
Q5: is it so that the higher is the process capability ratio the lower is the process
fall out?
A5: Yes
Q6: Can Cp and Cpk be negative?
A6: Cp cannot be negative, but Cpk can be negative.
Q7: What is the merit of Cpm over Cp or Cpmk over Cpk?
A7: Cp and Cpk are not the adequate measures of process centering, whereas Cpm
and Cpmk are known to be the adequate measures of process centering.
Q8: Is it required to check the normality character of a process before finding the
process capability?
A8: Yes, otherwise the capability of the process may be misinterpreted.
References
1.
Montgomery, D. C., Introduction to Statistical Quality Control, John Wiley &
Sons, Inc., Singapore, 2001.
Sources of Further Reading
1.
Montgomery, D. C. and Runger, G. C., Applied Statistics and Probability for
Engineers, John Wiley & Sons, Inc., New Delhi, 2003.
2.
Montgomery, D. C., Introduction to Statistical Quality Control, John Wiley &
Sons, Inc., Singapore, 2001.
3.
Grant, E. L. and Leavenworth, R. S., Statistical Quality Control, Tata McGraw
Hill Education Private Limited, New Delhi, 2000.