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Statistical Quality Control in Textiles Module 4: Shewhart Control Charts Dr. Dipayan Das Assistant Professor Dept. of Textile Technology Indian Institute of Technology Delhi Phone: +91-11-26591402 E-mail: [email protected] Introduction Why Control Charts? [1] Input Yes Manufacturing Process No Scrap or Rework No Yes Customer Is output acceptable? Is process under control? Correction Is input acceptable? Yes Output The control charts are required to know whether the manufacturing process in under control or out of control. Basis of Control Charts The basis of control charts is to checking whether the variation in the magnitude of a given characteristic of a manufactured product is arising due to random variation or assignable variation. Random variation: Natural variation or allowable variation, small magnitude Assignable variation: Non-random variation, relatively high magnitude. variation or preventable If the variation is arising due to random variation, the process is said to be under control. But, if the variation is arising due to assignable variation then the process is said to be out of control. Types of Control Charts Control Charts Other Control Charts* Shewhart Control Charts Variables x Chart R Chart s Chart Attributes p Chart np Chart c Chart * This will be discussed later on. Basics of Shewhart Control Charts Major Parts of Shewhart Control Chart Lower Control Limit (LCL): This indicates the lower limit of tolerance. If m is the underlying statistic so that E m m & Var m 2m CL = m UCL = m 3m LCL = m 3m Out of control Quality Scale Central Line (CL): This indicates the desired standard or the level of the process. Upper Control Limit (UCL): This indicates the upper limit of tolerance. UCL 3 CL 3 LCL Out of control 1 2 3 4 5 6 7 8 9 10 Sample Number Why 3? Let us assume that the probability distribution of the sample statistic m is (or tends to be) normal with mean m and standard deviation m . Then P m 3m m m 3m 0.9973 This means the probability that a random value of m falls in-between the 3- limits is 0.9973, which is very high. On the other hand, the probability that a random value of m falls outside of the 3- limits is 0.0027, which is very low. When the values of m fall in-between the 3 limits, the variations are attributed due to chance variation, then the process is considered to be statistically controlled. But, when one or many values of m fall out of the 3- limits, the variations are attributed due to assignable variation, then the process is said to be not under statistical control. Analysis of Control Chart: Process Out of Control The following one or more incidents indicate the process is said to be out of control (presence of assignable variation). A point falls outside any of the control limits. UCL Eight consecutive points fall within 3 limits. UCL 3 CL CL LCL Standard Control Action LCL 3 Analysis of Control Chart: Process Out of Control… Two out of three consecutive points fall beyond 2 limits. UCL UCL CL LCL Four out of five consecutive points fall beyond 1 limits. 2 2 CL 1 1 LCL Note: Sometimes the 2 limits are called as warning limits. Then, the 3 limits are called as action limits. Analysis of Control Chart: Process Out of Control… Presence of upward or downward trend Presence of cyclic trend UCL UCL CL CL LCL LCL Analysis of Control Chart: Process Under Control When all of the following incidents do not occur, the process is said to be under control (absence of assignable variation). 1) A point falls outside any of the control limits. 2) Eight consecutive points fall within 3 limits. 3) Two out of three consecutive points fall beyond 2 limits. 4) Four out of five consecutive points fall beyond 1 limits. 5) Presence of upward or downward trend 6) Presence of cyclic trend Shewhart Control Charts for Variables The Mean Chart (x-Chart) Let xij , j 1, 2, , n be the measurements on ith sample (i=1,2,…,k). The mean xi , range Ri , and standard deviation si for ith sample are given by xi j n 1 xij n j 1 j n j n Ri max xij min xij j 1 j 1 si j n j 1 x ij xi 2 n Then the mean x of sample means, the mean R of sample ranges, and the mean s of sample standard deviations are given by 1 ik x xi k i 1 1 ik R Ri k i 1 1 i k s si k i 1 The Mean Chart (x-Chart)… Let us now decide the control limits for xi . When the mean and standard deviation of the population from which samples are taken are given. CL = E xi 3 A n UCL= E xi +3 Var xi A 3 A n LCL = E xi 3 Var xi A The Mean Chart (x-Chart)… When the mean and standard deviation are not known. CL= x 3 x s x A1s c2 n A1 UCL= 3 x R x A R 2 d n 2 A2 3 x s x A1s c2 n A1 LCL= 3 x R x A R 2 d n 2 A2 The Range Chart (R-Chart) Let xij , j 1, 2, , n be the measurements on ith sample (i=1,2,…,k). The range Ri for ith sample is given by j n j n j 1 j 1 Ri max xij min xij Then the mean R of sample ranges is given by 1 ik R Ri k i 1 The Range Chart (R-Chart)… Let us now decide the control limits for Ri . When the standard deviation of the population from which samples are taken is known. CL = E Ri d2 UCL= E Ri +3 Var Ri d 2 3d3 d 2 3d3 D2 D2 LCL = E Ri 3 Var Ri d 2 3d3 d 2 3d3 D1 D1 The Range Chart (R-Chart)… When the standard deviation of the population is not known. CL = E Ri R 3d3 3d3 UCL= E Ri +3 Var Ri R R 1 R D4 R d2 d2 D4 3d3 3d3 LCL = E Ri 3 Var Ri R R 1 R D3 R d2 d2 D3 The Standard Deviation Chart (s-Chart) Let xij , j 1, 2, , n be the measurements on ith sample (i=1,2,…,k). The standard deviation si for ith sample is given by si j n j 1 x ij xi 2 n Then the mean s of sample standard deviations is given by 1 i k s si k i 1 The Standard Deviation Chart (s-Chart)… Let us now decide the control limits for si . When the standard deviation of the population from which samples are taken is known. CL = E si c2 UCL= E si +3 Var si c2 3c3 c2 3c3 B2 B2 LCL = E si 3 Var si c2 3c3 c2 3c3 B1 B1 The Standard Deviation Chart (s-Chart)… When the standard deviation of the population is not known. CL = E si s UCL= 3c3 c3 E si +3 Var si s 3 s 1 s B4 s c2 c2 B4 LCL = 3c3 c3 E si 3 Var si s 3 s 1 s B3 s c2 c2 B3 Table Sample size Mean Chart Standard deviation chart Range chart n A A1 A2 c2 B1 B2 B3 B4 d2 D1 D2 D3 D4 2 2.121 3.760 1.886 0.5642 0 1.843 0 3.297 1.128 0 3.686 0 3.267 3 1.232 2.394 1.023 0.7236 0 1.858 0 2.568 1.693 0 4.358 0 2.575 4 1.500 1.880 0.729 0.7979 0 1.808 0 2.266 2.059 0 4.698 0 2.282 5 1.342 1.596 0.577 0.8407 0 1.756 0 2.089 2.326 0 4.918 0 2.115 6 1.225 1.410 0.483 0.8686 0.026 1.711 0.030 1.970 2.534 0 5.078 0 2.004 7 1.134 1.277 0.419 0.8882 0.105 1.672 0.118 1.882 2.704 0.205 5.203 0.076 1.924 8 1.061 1.175 0.373 0.9027 0.167 1.638 0.185 1.815 2.847 0.387 5.307 0.136 1.864 9 1.000 1.094 0.337 0.9139 0.219 1.609 0.239 1.761 2.970 0.546 5.394 0.184 1.816 10 0.949 1.028 0.308 0.9227 0.262 1.584 0.284 1.679 3.078 0.687 5.469 0.223 1.777 11 0.905 0.973 0.285 0.9300 0.299 1.561 0.321 1.646 3.173 0.812 5.534 0.256 1.744 12 0.866 0.925 0.266 0.9359 0.331 1.541 0.354 1.618 3.258 0.924 5.592 0.284 1.716 13 0.832 0.884 0.249 0.9410 0.359 1.523 0.382 1.594 3.336 1.026 5.646 0.308 1.692 14 0.802 0.548 0.235 0.9453 0.384 1.507 0.406 1.572 3.407 1.121 5.693 0.329 1.671 15 0.775 0.816 0.223 0.9499 0.406 1.492 0.428 1.552 3.472 1.207 5.779 0.348 1.652 16 0.759 0.788 0.212 0.9523 0.427 1.478 0.448 1.534 3.532 1.285 5.817 0.364 1.636 17 0.728 0.762 0.203 0.9951 0.445 1.465 0.466 1.518 3.588 1.359 5.854 0.379 1.621 18 0.707 0.738 0.194 0.9576 0.461 1.454 0.482 1.503 3.640 1.426 5.888 0.392 1.668 19 0.688 0.717 0.187 0.9599 0.477 1.443 0.497 1.499 3.689 1.490 5.922 0.404 1.596 20 0.671 0.697 0.180 0.9619 0.491 1.433 0.510 1.477 3.735 1.548 5.950 0.414 1.586 25 0.600 0.610 0.153 0.9696 0.548 1.392 0.565 1.435 3.931 1.804 6.058 0.459 1.541 Illustration Sample No. i Yarn strength xicNtex 1 Ri cNtex 1 si cNtex 1 xij cNtex 1 1 14.11 13.09 12.52 13.40 13.94 13.40 12.72 11.09 13.28 12.34 12.99 3.02 0.83 2 14.99 17.97 15.76 13.56 13.31 14.03 16.01 17.71 15.67 16.69 15.57 4.66 1.54 3 15.08 14.41 11.87 13.62 14.84 15.44 13.78 13.84 14.99 13.99 14.19 3.57 0.97 4 13.14 12.35 14.08 13.40 13.45 13.44 12.90 14.08 14.71 13.11 13.47 2.36 0.64 5 13.21 13.69 13.25 14.05 15.58 14.82 14.31 14.92 10.57 15.16 13.96 5.01 1.36 6 15.79 15.58 14.67 13.62 15.90 14.43 14.53 13.81 14.92 12.23 14.55 3.67 1.07 7 13.78 13.90 15.10 15.26 13.17 13.67 14.99 13.39 14.84 14.15 14.23 2.09 0.72 8 15.65 16.38 15.10 14.67 16.53 15.42 15.44 17.09 15.68 15.44 15.74 2.42 0.69 9 15.47 15.36 14.38 14.08 14.08 14.84 14.08 14.62 15.05 13.89 14.59 1.58 0.54 10 14.41 15.21 14.04 13.44 14.94 14.84 16.19 14.85 2.75 0.81 14.41 3.11 0.92 15.85 14.18 15.44 Average xcNtex 1 14.41 RcNtex 1 3.11 scNtex 1 0.92 Illustration (x-chart) CL= x 14.11 cN tex 1 xicNtex 1 UCL= UCL x A2 R 14.11 0.308 3.11 cN tex 1 15.37 cN tex 1 CL LCL LCL= x A2 R 14.11 0.308 3.11 cN tex 1 13.45 cN tex 1 0 i The process average is out of control. Illustration (R-chart) CL= R 3.11cN tex 1 UCL= D4 R Ri cNtex 1 UCL 1.777 3.11cN tex 1 5.50 cN tex 1 LCL= D3 R 0.223 3.11cN tex 1 0.69cN tex 1 CL LCL i The process variability is in control. Illustration (s-chart) CL= si cNtex 1 s 0.92cN tex 1 UCL= B4 s 1 1.716 0.92cN tex UCL 1.58 cN tex 1 CL LCL= B3 s 1 0.284 0.92 cN tex 0.26 cN tex 1 LCL i The process variability is in control. Illustration (Overall Conclusion) Although the process variability is in control, the process cannot be regarded to be in statistical control since the process average is out of control. Shewhart Control Charts for Attributes Control Chart for Fraction Defective (p-Chart) The fraction defective is defined as the ratio of the number of defectives in a population to the total number of items in the population. Suppose the production process is operating in a stable manner, such that the probability that any item produced will not conform to specifications is p and that successive items produced are independent. Then each item produced is a realization of a Bernouli random variable with parameter p. If a random sample of n items of product is selected, and if D is the number of items of product that are defectives, then D has a binomial distribution with parameter n n x and p; that is The mean and P D x nCx p x 1 p , x 0,1, , n. variance of the random variable D are np and np 1 p , respectively. Control Chart for Fraction Defective (p-Chart)… The sample fraction defective is defined as the ratio of the number of p D n . defective items in the sample of size n; that is The distribution of the random variable p can be obtained from the binomial distribution. The mean and variance of p are p and p 1 p , respectively. n When the mean fraction of defectives p of the population from which samples are taken is known. CL= p p 1 p UCL= p 3 n p 1 p LCL= p 3 n Control Chart for Fraction Defective (p-Chart)… When the mean fraction of defectives p of the population is not known. Let us select m samples, each of size n. If there are Di defective items in ith sample, then the fraction defectives in the ith sample is pi Di n , i 1, 2, defectives is , m. The average of these individual sample fraction i m p CL= p i m D p i 1 mn UCL= p 3 i i 1 i m p 1 p n LCL= p 3 p 1 p n Control Chart for Number of Defectives (np-Chart) It is also possible to base a control chart on the number of defectives rather than the fraction defectives. When the mean number of defectives np of the population from which samples are taken is known. CL= np UCL= np 3 np 1 p LCL= np 3 np 1 p When the mean number of defectives np of the population is not known. CL= np UCL= np 3 np 1 p LCL= np 3 np 1 p Illustration [2] The following refers to the number of defective knitwears in samples of size 180. Sample No. No. of defectives Sample No. No. of defectives Sample No. No. of defectives 1 05 11 36 21 24 2 08 12 24 22 17 3 10 13 19 23 12 4 12 14 13 24 08 5 12 15 05 25 17 6 29 16 02 26 19 7 25 17 11 27 04 8 13 18 08 28 09 9 09 19 15 29 05 10 20 20 20 30 12 Illustration… Here, n=180 and p 423 30 180 0.0783. CL= np 14.09 UCL= np 3 np 1 p 24.90 npi UCL CL LCL= np 3 np 1 p 3.28 LCL i The process is out of control. Control Chart for Defects (c-Chart) Consider the occurrence of defects in an inspection of product(s). Suppose that defects occur in this inspection according to Poisson distribution; that is e c c x P x , x 0,1, 2, x! , Where x is the number of defects and c is known as mean and/or variance of the Poisson distribution. When the mean number of defects c in the population from which samples are taken is known. CL= c UCL= c 3 c LCL= c 3 c Note: If this calculation yields a negative value of LCL then set LCL=0. Control Chart for Defects (c-Chart)… When the mean number of defects c in the population is not known. Let us select n samples. If there are ci defects in ith sample, then the average of these defects in samples of size n is i n c CL= c i n c c i 1 n i i 1 i n UCL= c 3 c LCL= c 3 c Note: If this calculation yields a negative value of LCL then set LCL=0. Illustration [2] The following dataset refers to the number of holes (defects) in knitwears. Sample No. No. of holes Sample No. No. of holes Sample No. No. of holes 1 4 11 3 21 2 2 6 12 7 22 1 3 3 13 9 23 7 4 8 14 6 24 6 5 12 15 10 25 5 6 9 16 11 26 9 7 7 17 7 27 11 8 2 18 8 28 8 9 11 19 9 29 3 10 8 20 3 30 2 Illustration… Consider ci denote the number of holes in ith sample. CL= c 6.57 UCL ci UCL= c 3 c 14.26 CL LCL= c 3 c 1.12 0 LCL i The process is in control. Frequently Asked Questions & Answers Frequently Asked Questions & Answers Q1: Is it not possible that a process turns to be out of control because of presence of random variation? A1: Yes, it is possible, but the probability of such occurrence is very low, that is, 0.0027. Q2: Is it so that a process can be found to be out-of control even if there is no point falling out of 3-sigma limits? A2: Yes, it is possible, the presence of a run, trend, etc. can do so. Q3: If process mean is in control, but the process variability is not in control, can the process be said to be under control? A3: No. Q4: Is Shewhart control chart able to detect a small shift in process mean? A4: No. Frequently Asked Questions & Answers Q5: Name the probability distribution that the process defectives can be regarded to follow? A5: Binomial distribution. Q6: Name the probability distribution that the process defects can be regarded to follow? A6: Poission distribution. References 1. Gupta, S. C. and Kapoor, V. K., Fundamentals of Applied Statistics, Sultan Chand & Sons, New Delhi, 2007. 2. Leaf, G. A. V., Practical Statistics for the Textile Industry: Part II, The Textile Institute, UK, 1984. Sources of Further Reading 1. Leaf, G. A. V., Practical Statistics for the Textile Industry: Part I, The Textile Institute, UK, 1984. 2. Leaf, G. A. V., Practical Statistics for the Textile Industry: Part II, The Textile Institute, UK, 1984. 3. Gupta, S. C. and Kapoor, V. K., Fundamentals of Mathematical Statistics, Sultan Chand & Sons, New Delhi, 2002. 4. Gupta, S. C. and Kapoor, V. K., Fundamentals of Applied Statistics, Sultan Chand & Sons, New Delhi, 2007. 5. Montgomery, D. C., Introduction to Statistical Quality Control, John Wiley & Sons, Inc., Singapore, 2001. 6. Grant, E. L. and Leavenworth, R. S., Statistical Quality Control, Tata McGraw Hill Education Private Limited, New Delhi, 2000. 7. Montgomery, D. C. and Runger, G. C., Applied Statistics and Probability for Engineers, John Wiley & Sons, Inc., New Delhi, 2003.