1. 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 Statistical Quality Control in Textiles Module 4: Shewhart Control Charts

2. Introduction

3. Why Control Charts? [1] Is input acceptable? Is output acceptable? Manufacturing Process Output Input Customer Is process under control? Yes No Scrap or Rework Correction Yes The control charts are required to know whether the manufacturing process in under control or out of control.

4. 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 or preventable variation, relatively high magnitude. 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.

5. Types of Control Charts Control Charts x Chart R Chart s Chart c Chart np Chart p Chart Shewhart Control Charts Other Control Charts* VariablesAttributes * This will be discussed later on.

6. Basics of Shewhart Control Charts

7. Major Parts of Shewhart Control Chart 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. Lower Control Limit (LCL) : This indicates the lower limit of tolerance. If m is the underlying statistic so that & CL = UCL = LCL = CL UCL LCL 33 33 Out of control 10912345678 Sample Number Quality Scale

8. 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 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.

9. 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. Eight consecutive points fall within 3  limits. CL UCL LCL CL UCL LCL 33 33 Standard Control Action

10. Two out of three consecutive points fall beyond 2  limits. Four out of five consecutive points fall beyond 1  limits. CL UCL LCL 22 22 11 11 CL UCL 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…

11. Presence of upward or downward trend CL UCL LCL Presence of cyclic trend CL UCL LCL Analysis of Control Chart: Process Out of Control…

12. 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

13. Shewhart Control Charts for Variables

14. The Mean Chart ( x -Chart) Let be the measurements on i th sample ( i =1,2,…, k ). The mean, range, and standard deviation for i th sample are given by Then the mean of sample means, the mean of sample ranges, and the mean of sample standard deviations are given by

15. The Mean Chart ( x -Chart)… Let us now decide the control limits for When the mean  and standard deviation  of the population from which samples are taken are given. CL = UCL= LCL =

16. The Mean Chart ( x -Chart)… When the mean  and standard deviation  are not known. UCL= LCL= CL=

17. The Range Chart ( R -Chart) Let be the measurements on i th sample ( i =1,2,…, k ). The range for i th sample is given by Then the mean of sample ranges is given by

18. The Range Chart ( R -Chart)… When the standard deviation  of the population from which samples are taken is known. CL = UCL= LCL = Let us now decide the control limits for

19. The Range Chart ( R -Chart)… When the standard deviation  of the population is not known. CL = UCL= LCL =

20. The Standard Deviation Chart ( s -Chart) Let be the measurements on i th sample ( i =1,2,…, k ). The standard deviation for i th sample is given by Then the mean of sample standard deviations is given by

21. The Standard Deviation Chart ( s -Chart)… Let us now decide the control limits for When the standard deviation  of the population from which samples are taken is known. CL = UCL= LCL =

22. The Standard Deviation Chart ( s -Chart)… When the standard deviation  of the population is not known. CL = UCL= LCL =

23. Table Sample size Mean ChartStandard deviation chartRange chart nAA1A1 A2A2 c2c2 B1B1 B2B2 B3B3 B4B4 d2d2 D1D1 D2D2 D3D3 D4D4 22.1213.7601.8860.564201.84303.2971.12803.68603.267 31.2322.3941.0230.723601.85802.5681.69304.35802.575 41.5001.8800.7290.797901.80802.2662.05904.69802.282 51.3421.5960.5770.840701.75602.0892.32604.91802.115 61.2251.4100.4830.86860.0261.7110.0301.9702.53405.07802.004 71.1341.2770.4190.88820.1051.6720.1181.8822.7040.2055.2030.0761.924 81.0611.1750.3730.90270.1671.6380.1851.8152.8470.3875.3070.1361.864 91.0001.0940.3370.91390.2191.6090.2391.7612.9700.5465.3940.1841.816 100.9491.0280.3080.92270.2621.5840.2841.6793.0780.6875.4690.2231.777 110.9050.9730.2850.93000.2991.5610.3211.6463.1730.8125.5340.2561.744 120.8660.9250.2660.93590.3311.5410.3541.6183.2580.9245.5920.2841.716 130.8320.8840.2490.94100.3591.5230.3821.5943.3361.0265.6460.3081.692 140.8020.5480.2350.94530.3841.5070.4061.5723.4071.1215.6930.3291.671 150.7750.8160.2230.94990.4061.4920.4281.5523.4721.2075.7790.3481.652 160.7590.7880.2120.95230.4271.4780.4481.5343.5321.2855.8170.3641.636 170.7280.7620.2030.99510.4451.4650.4661.5183.5881.3595.8540.3791.621 180.7070.7380.1940.95760.4611.4540.4821.5033.6401.4265.8880.3921.668 190.6880.7170.1870.95990.4771.4430.4971.4993.6891.4905.9220.4041.596 200.6710.6970.1800.96190.4911.4330.5101.4773.7351.5485.9500.4141.586 250.6000.6100.1530.96960.5481.3920.5651.4353.9311.8046.0580.4591.541

24. Illustration Sample No. Yarn strength 114.1113.0912.5213.4013.9413.4012.7211.0913.2812.34 12.993.020.83 214.9917.9715.7613.5613.3114.0316.0117.7115.6716.69 15.574.661.54 315.0814.4111.8713.6214.8415.4413.7813.8414.9913.99 14.193.570.97 413.1412.3514.0813.4013.4513.4412.9014.0814.7113.11 13.472.360.64 513.2113.6913.2514.0515.5814.8214.3114.9210.5715.16 13.965.011.36 615.7915.5814.6713.6215.9014.4314.5313.8114.9212.23 14.553.671.07 713.7813.9015.1015.2613.1713.6714.9913.3914.8414.15 14.232.090.72 815.6516.3815.1014.6716.5315.4215.4417.0915.6815.44 15.742.420.69 915.4715.3614.3814.08 14.8414.0814.6215.0513.89 14.591.580.54 1014.4115.2114.0413.4415.8514.1815.4414.9414.8416.19 14.852.750.81 Average 14.413.110.92

25. Illustration ( x -chart) CL 0 UCL LCL CL= UCL= LCL= The process average is out of control.

26. Illustration ( R -chart) CL= UCL= LCL= CL UCL LCL The process variability is in control.

27. Illustration ( s -chart) CL= UCL= LCL= CL UCL LCL The process variability is in control.

28. 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.

29. Shewhart Control Charts for Attributes

30. 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 and p ; that is The mean and variance of the random variable D are and respectively.

31. Control Chart for Fraction Defective ( p -Chart)… The sample fraction defective is defined as the ratio of the number of defective items in the sample of size n ; that is The distribution of the random variable can be obtained from the binomial distribution. The mean and variance of are and respectively. When the mean fraction of defectives p of the population from which samples are taken is known. UCL=LCL=CL=

32. 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 D i defective items in i th sample, then the fraction defectives in the i th sample is The average of these individual sample fraction defectives is UCL=LCL= CL=

33. 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. When the mean number of defectives np of the population is not known. UCL=LCL= CL= UCL=LCL= CL=

34. Illustration [2] Sample No. No. of defectives Sample No. No. of defectives Sample No. No. of defectives 10511362124 20812242217 31013192312 4 14132408 51215052517 62916022619 72517112704 81318082809 9 19152905 1020 3012 The following refers to the number of defective knitwears in samples of size 180.

35. Illustration… Here, n =180 and UCL= LCL= CL= CL UCL LCL The process is out of control.

36. 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 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. Note: If this calculation yields a negative value of LCL then set LCL=0. UCL=LCL=CL=

37. 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 c i defects in i th sample, then the average of these defects in samples of size n is Note: If this calculation yields a negative value of LCL then set LCL=0. UCL=LCL=CL=

38. 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 14113212 26127221 33139237 48146246 5121510255 691611269 771772711 82188288 911199293 108203302

39. Illustration… Consider denote the number of holes in i th sample. UCL= LCL= CL= LCL CL UCL The process is in control.

40. Frequently Asked Questions & Answers

41. 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.

42. 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.

43. 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.

44. 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.