Download - Statistical Quality Control in Textiles
Dr. Dipayan DasAssistant Professor
Dept. of Textile TechnologyIndian Institute of Technology Delhi
Phone: +91-11-26591402E-mail: [email protected]
Statistical Quality Control in Textiles
Module 5:
Process Capability Analysis
Introduction
Process Capability AnalysisWhen 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 LimitsThe 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 areUpper natural tolerance limit = + 3σ
Lowe natural tolerance limit = - 3σUnder 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
HistogramIt gives an immediate visual impression of process
performance. It may also immediately show the reason for poor performance.
σ
μ USLLSL
μ USLLSL
σ
Poor process capability is due to poor process centering
Poor process capability is due to excess process variability
Example: Yarn Strength (cN.tex-1) Dataset
14.11 13.09 12.52 13.40 13.94 13.40 12.72 11.09 13.28 12.34 12.72 13.84 13.78 13.35 15.15 11.44 13.82 14.59 15.71 12.32 14.99 17.97 15.76 13.56 13.31 14.03 16.01 17.71 15.67 16.69 14.08 13.06 13.60 13.51 13.17 14.53 15.35 14.31 14.99 14.7715.08 14.41 11.87 13.62 14.84 15.44 13.78 13.84 14.99 13.99 13.51 14.87 14.76 13.06 13.69 12.93 13.48 15.21 14.82 13.4213.14 12.35 14.08 13.40 13.45 13.44 12.90 14.08 14.71 13.11 12.91 14.71 14.84 15.58 14.18 13.30 14.41 12.72 13.62 14.3113.21 13.69 13.25 14.05 15.58 14.82 14.31 14.92 10.57 15.16 13.50 12.23 13.60 13.89 13.21 14.13 14.08 13.89 14.53 15.5815.79 15.58 14.67 13.62 15.90 14.43 14.53 13.81 14.92 12.23 13.26 16.32 14.58 13.87 14.31 15.03 14.67 14.41 15.26 15.9013.78 13.90 15.10 15.26 13.17 13.67 14.99 13.39 14.84 14.15 15.62 14.84 15.47 15.12 15.26 15.68 14.99 15.16 15.12 14.6215.65 16.38 15.10 14.67 16.53 15.42 15.44 17.09 15.68 15.44 15.08 14.54 14.99 15.36 14.99 14.31 16.96 14.31 14.84 14.2615.47 15.36 14.38 14.08 14.08 14.84 14.08 14.62 15.05 13.89 14.92 13.78 12.47 12.98 14.72 16.14 15.71 16.53 16.34 16.4314.41 15.21 14.04 13.44 15.85 14.18 15.44 14.94 14.84 16.19 16.53 14.67 16.08 16.19 15.49 13.85 13.85 15.16 16.11 13.8115.85 16.49 15.67 14.67 15.46 16.17 14.85 14.68 15.10 15.85 14.40 15.90 14.31 13.51 14.84 13.55 14.52 14.67 15.90 15.1614.84 13.99 15.44 14.87 14.17 15.36 13.41 15.05 15.10 13.73 14.76 14.53 14.99 14.18 15.62 15.65 13.94 14.08 15.21 14.2212.26 12.86 14.67 13.35 13.35 13.62 13.69 13.78 12.72 14.18 14.67 13.21 12.53 14.53 15.12 14.67 12.44 11.92 13.06 14.3111.93 11.82 12.93 12.72 13.41 13.62 12.72 14.48 14.09 14.31 13.14 13.06 13.25 12.19 12.91 11.97 14.09 13.56 14.04 13.4014.08 14.31 12.40 13.40 13.25 12.44 12.72 13.60 14.31 12.80 14.08 13.53 12.81 12.96 13.21 13.89 12.72 14.41 13.44 13.2115.32 15.05 15.90 13.78 15.90 15.21 14.18 16.63 15.65 15.34 16.96 17.36 16.11 15.70 15.67 14.97 14.84 15.37 15.58 15.1614.57 14.92 16.53 17.06 15.03 16.43 15.41 14.18 14.67 15.31 13.44 14.92 16.35 16.32 16.43 14.58 16.14 15.21 15.31 15.2216.80 15.65 14.43 14.53 15.56 14.97 14.87 14.41 15.94 17.17 14.31 16.34 16.48 15.90 17.12 15.68 15.94 16.35 16.96 15.8114.31 14.48 13.01 14.18 12.42 12.86 16.94 13.22 13.30 12.95 13.79 14.57 12.47 14.31 14.53 14.43 15.16 13.35 15.58 14.1813.69 14.45 14.45 11.98 14.16 14.67 14.38 13.29 12.29 14.62 13.89 13.44 14.08 14.35 14.62 13.44 15.01 14.92 14.31 16.1415.16 16.14 14.62 15.58 15.90 14.08 13.40 14.92 14.41 15.44 15.68 13.85 13.78 15.20 13.69 15.16 14.18 13.62 15.65 16.1115.12 14.62 15.77 17.51 14.58 13.73 17.89 15.62 10.84 13.32 15.86 15.94 12.60 16.19 15.68 17.49 16.70 16.80 18.02 16.8017.03 16.80 17.12 16.14 15.90 16.34 16.70 16.09 17.64 15.34
Frequency DistributionClass
Interval
(cN.tex-1)
Class Valuexi
(cN.tex-1)
Frequencyni(-)
Relative Frequency gi
(-)
Relative Frequency Density fi(cN-1.tex)
10.00-11.00 10.50 2 0.0044 0.004411.00-12.00 11.50 8 0.0178 0.017812.00-13.00 12.50 37 0.0822 0.082213.00-14.00 13.50 102 0.2267 0.226714.00-15.00 14.50 140 0.3111 0.311115.00-16.00 15.50 104 0.2311 0.231116.00-17.00 16.50 43 0.0956 0.095617.00-18.00 17.50 13 0.0289 0.028918.00-19.00 18.50 1 0.0022 0.0022
TOTAL 450 1.0000
Histogram
00 10 11 12 13 14 15 16 17 18 19
0.1
-1cN texx
-1cN texf 0.2
0.3
0.4Mean = 14.57 cN tex-1 Standard deviation = 1.30 cN tex-1
The process capability would be estimated as follows:
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.
-1 -1cN tex cN tex3 14.57 3.90x s
Probability PlotProbability 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 is arranged as where is the smallest observation, is the second smallest observation, and is the largest observation, and so forth. 2) The ordered observations are then plotted again their observed cumulative frequency on the appropriate probability paper.3) If the hypothesized distribution adequately describes the data, the plotted points will fall approximately along a straight line.
1 2, , , nx x x 1 2, , , nx x x 1x 2x nx
jx 0.5j n
Example: Yarn Strength (cN.tex-1) Dataset
Let us take that the following yarn strength data12.35, 17.17, 15.58, 10.84, 18.02, 14.05, 13.25, 14.45, 12.35, 16.19.j xj (j-0.5)/
101 10.84 0.052 11.09 0.153 12.35 0.254 13.25 0.355 14.05 0.456 14.45 0.557 15.58 0.658 16.19 0.759 17.17 0.85
10 18.02 0.95
11 12 13 14 15 16 17 18
0.05
0.10
0.25
0.50
0.75
0.90
0.95
Data
Prob
abili
ty
Normal Probability Plot
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
where USL and LSL stand for upper specification limit and
lower specification limit respectively and σ refers to the
process standard deviation.100(1/Cp) is interpreted as the
percentage of the specifications’ width used by
the process.
6pUSL LSLC
3σ μ 3σ USLLSL
Cp>1
3σ μ 3σ USLLSL
Cp=1
3σ μ 3σ USL
Cp<1
LSL
IllustrationSuppose 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
2
3.110ˆ 1.01043.078
Rd
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-1.18.5 10.5 1.3196
6 1.0104pC
That is, 75.78% of the specifications’ width is used by the process.
1cN.texx
-1cN .texf x
18.5
10.5
14.5 USLLSL
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
upper specification only3pu
USLC
lower specification only3plLSLC
IllustrationSuppose 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
2
3.110ˆ 1.01043.078
Rd
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-1.14.5 10.5 1.3196
3 1.0104plC
That is, 75.78% of the specifications’ width is used by the process.
1cN.texx
-1cN .texf x
10.5
14.5LSL
Process Capability Ratio Versus Process Fallout [1]
Assumptions:1) The quality
characteristic is normally distributed.2) The process is in
statistical control.3) The process mean is
centered between USL and LSL.
Process Capability Ratio
Process Fallout (in defective parts per million)One sided
specificationsTwo sided
specifications
0.25 226,628 453,2550.50 66,807 133,6140.60 35,931 71,8610.70 17,865 35,7290.80 8,198 16,3950.90 3,467 6,9341.00 1,350 2,7001.10 484 9671.20 159 3181.30 48 961.40 14 271.50 4 71.60 1 21.70 0.17 0.341.80 0.03 0.062.00 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
ratiomin , min ,
3 3pk pu plUSL LSLC C C
3σ μ 3σ USLLSL
3σ μ 3σ USLLSL
Process running off-center
Interpretations1) 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.
IllustrationSuppose 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 off-center.
18.5 14.5 14.5 10.0min ,3 1.0104 3 1.0104
min 1.1547,1.4846 1.1547
pkC
1cN.texx
-1cN .texf x
14.5
USLLSL
Inadequacy of Cpk
Let us compare the two processes, Process A and Process B.Process A BMean 50.0 cN 57.5 cN
Standard deviation
5.0 cN 2.5 cN
Specification limits
35cN, 65cN
35cN, 65cN
Cp 1 2Cpk 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
where τ is the square root of expected squared deviation from the target T
Then,
6pmUSL LSLC
12
T USL LSL
22 2 T
2 2 226 6 1 1pm
USL LSL USL LSL PCRCT T T
Measure of Process Capability: Cpmk
For non-centered process mean, the modified process capability ratio is
where τ is the square root of expected squared deviation from the target T
Then,
min ,3 3pmk
USL LSLC
12
T USL LSL
22 2 T
2 2min ,
3 1 3 1pmk
USL LSLCT T
IllustrationTake the example of process A and process B. Here T=50
cN. Then,Process A BMean 50.0 cN 57.5 cN
Standard deviation
5.0 cN 2.5 cN
Specification limits
35cN, 65cN
35cN, 65cN
Cp 1 2Cpk 1 1Cpm 1 0.63Cpmk 1 0.1582
Note to Non-normal Process OutputAn 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 non-normal 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
where
62
pcUSL LSLC
E x T
12
T USL LSL
Note to Non-normal Process Output3) For non-normal data, use quantile based process
capability ratio
As it is known that, for normal distribution,
Then,
0.99865 0.00135
pUSL LSLC qx x
0.00135
0.99865
33
xx
3 3 6p pUSL LSL USL LSLC q C
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
If the quality characteristic follows a normal distribution, then a 100(1-)% confidence interval on Cp is obtained as
ˆ6p
USL LSLCs
2 21 2, 1 2, 1ˆ ˆ
1 1n n
p p pC C Cn n
where and are the lower and upper percentage points of the chi-square distribution
with n-1 degree of freedom.
21 2, 1n 2
2, 1n 22
Confidence Interval on Cpk
A point estimate of Cpk is found as follows
If the quality characteristic follows a normal distribution, then a 100(1-)% confidence interval on Cpk is obtained as
ˆ min ,3 3pk
USL LSLCs s
2 22 2
1 1 1 1ˆ ˆ1 1ˆ ˆ2 1 2 19 9pk pk pkk k
C u C C un nnPCR nPCR
ExampleSuppose 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
2 2
1 1 1 11.07 1 1.96 1.07 1 1.969 450 1.07 2 450 1 9 450 1.07 2 450 1pkC
18.5 14.57 14.57 10.5ˆ min , 1.073 1.23 3 1.23pkC
0.991190 1.148810pkC
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 in such a way that H will be rejected if exceeds a critical value C.
A table of sample sizes and critical values of C to assist in testing process capability is available.
ˆpC
ˆpC
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 size
==0.10 ==0.05Cp(high)/Cp(low)
C/Cp(low)
Cp(high)/
Cp(low)
C/Cp(low)
10 1.88 1.27 2.26 1.3720 1.53 1.20 1.73 1.2630 1.41 1.16 1.55 1.2140 1.34 1.14 1.46 1.1850 1.30 1.13 1.40 1.1660 1.27 1.11 1.36 1.1570 1.25 1.10 1.33 1.1480 1.23 1.10 1.30 1.1390 1.21 1.10 1.28 1.12
100 1.20 1.09 1.26 1.11
ExampleA 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=1.33HA: 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.
Then, from table, we get, n=70 and
To demonstrate capability, the supplier must take a sample of n=70 and the sample process capability ratio Cp must
exceed C=1.46.
High 1.66 1.25Low 1.33
p
p
CC
1.10, Low 1.10 1.33 1.10 1.46Low p
p
C C CC
Note to Practical ApplicationThis example shows that in order to demonstrate that the process capability is at least equal to 1.33, the observed sample 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 if the sample based on a sample size of . Clearly, this procedure does not
account for sampling variation in the estimate of , and larger values of n and/or higher acceptable values of may
be necessary in practice.
ˆpC
1.33pC ˆpC
30 50n
ˆpC
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 & AnswersQ5: 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.
References1. Montgomery, D. C., Introduction to Statistical Quality Control, John
Wiley & Sons, Inc., Singapore, 2001.
Sources of Further Reading1. 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.