Transcript
Page 1: 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

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Introduction

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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.

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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.

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Techniques for Process Capability Analysis

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Techniques for Process Capability Analysis

Histogram Probability Plot Control Charts

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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

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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

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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

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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

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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

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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.

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Measures of Process Capability Analysis

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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

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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

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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

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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

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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

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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

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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.

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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

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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.

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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

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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

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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

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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

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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

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Inferential Properties of Process Capability Ratios

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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

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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

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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

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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

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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

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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.

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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

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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

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Frequently Asked Questions & Answers

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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.

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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.

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References1. Montgomery, D. C., Introduction to Statistical Quality Control, John

Wiley & Sons, Inc., Singapore, 2001.

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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.


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