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: Dipayan@textile.iitd.ac.in

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. Natural or inherent variability at a specified time, Variability over time. Let us investigate and assess both aspects of process capability. 3

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. 4

Techniques for Process Capability Analysis

Techniques for Process Capability Analysis Histogram Probability Plot Control Charts 6

Histogram It gives an immediate visual impression of process performance. It may also immediately show the reason for poor performance. Poor process capability is due to poor process centering σ LSL μ USL Poor process capability is due to excess process variability σ μ LSL USL 7

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 14.77 15.08 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 13.42 13.14 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 14.31 13.21 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 15.58 15.79 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 15.90 13.78 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 14.62 15.65 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 14.26 15.47 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 16.43 14.41 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 13.81 15.85 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 15.16 14.84 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 14.22 12.26 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 14.31 11.93 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 13.40 14.08 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 13.21 15.32 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 15.16 14.57 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 15.22 16.80 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 15.81 14.31 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 14.18 13.69 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 16.14 15.16 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 16.11 15.12 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 16.80 17.03 16.80 17.12 16.14 15.90 16.34 16.70 16.09 17.64 15.34 8

Frequency Distribution Class Interval (cN.tex-1) Class Value xi Frequency ni (-) Relative Frequency gi Relative Frequency Density fi (cN-1.tex) 10.00-11.00 10.50 2 0.0044 11.00-12.00 11.50 8 0.0178 12.00-13.00 12.50 37 0.0822 13.00-14.00 13.50 102 0.2267 14.00-15.00 14.50 140 0.3111 15.00-16.00 15.50 104 0.2311 16.00-17.00 16.50 43 0.0956 17.00-18.00 17.50 13 0.0289 18.00-19.00 18.50 1 0.0022 TOTAL 450 1.0000 9

Histogram Mean = 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. 0 10 11 12 13 14 15 16 17 18 19 0.1 0.2 0.3 0.4 10

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. The sample data is arranged as where is the smallest observation, is the second smallest observation, and is the largest observation, and so forth. The ordered observations are then plotted again their observed cumulative frequency on the appropriate probability paper. If the hypothesized distribution adequately describes the data, the plotted points will fall approximately along a straight line. 11

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. j 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 The sample strength data can be regarded as taken from a population following normal distribution. 12

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. 3σ μ USL LSL Cp>1 Cp=1 Cp<1 14

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 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. 10.5 14.5 18.5 LSL USL That is, 75.78% of the specifications’ width is used by the process. 15

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 16

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 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. 10.5 14.5 LSL That is, 75.78% of the specifications’ width is used by the process. 17

Process Capability Ratio Versus Process Fallout [1] Assumptions: 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 18

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 Process running off-center LSL 3σ μ 3σ USL LSL 3σ μ 3σ USL 19

Interpretations When Cpk=Cp then the process is centered at the midpoint of the specifications. When Cpk<Cp then the process is running off center. When Cpk=0, the process mean is exactly equal to one of the specification limits. When Cpk<0 then the process mean lies outside the specification limit. When Cpk<-1 then the entire process lies outside the specification limits. 20

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 off-center. LSL 14.5 USL 21

Inadequacy of Cpk Let us compare the two processes, Process A and Process B. Process A B Mean 50.0 cN 57.5 cN Standard deviation 5.0 cN 2.5 cN Specification limits 35cN, 65cN Cp 1 2 Cpk Cpk interprets the processes as equally-competent. 22

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, 23

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, 24

Illustration Take the example of process A and process B. Here T=50 cN. Then, Process A B Mean 50.0 cN 57.5 cN Standard deviation 5.0 cN 2.5 cN Specification limits 35cN, 65cN Cp 1 2 Cpk Cpm 0.63 Cpmk 0.1582 25

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 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 26

Note to Non-normal Process Output 3) For non-normal data, use quantile based process capability ratio As it is known that, for normal distribution, Then, 27

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 where and are the lower and upper percentage points of the chi-square distribution with n-1 degree of freedom. 29

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 30

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 31

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. 32

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 ==0.05 Cp(high)/Cp(low) C/Cp(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 60 1.11 1.36 1.15 70 1.25 1.10 1.33 80 1.23 90 1.28 1.12 100 1.09 33

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. 34

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. 35

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 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. 36

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. 38

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. 39

References Montgomery, D. C., Introduction to Statistical Quality Control, John Wiley & Sons, Inc., Singapore, 2001. 40

Sources of Further Reading Montgomery, D. C. and Runger, G. C., Applied Statistics and Probability for Engineers, John Wiley & Sons, Inc., New Delhi, 2003. Montgomery, D. C., Introduction to Statistical Quality Control, John Wiley & Sons, Inc., Singapore, 2001. Grant, E. L. and Leavenworth, R. S., Statistical Quality Control, Tata McGraw Hill Education Private Limited, New Delhi, 2000. 41