Statistical Process Control

1. Statistical Process Control Control charts

2. Quality • Quality is an old concept • Artisan’s or craftsmen’s work characterised by qualities such as strength, beauty or finish. • In mass production, reproducibility is a big issue • Particularly dimensions of component parts • Quality was obtained through complete inspection • In 1931, Walter Shewhart published ‘Economic Control of Quality of Manufactured Product ‘ • Foundation of modern Statistical Process Control (SPC)

3. Process variability • Basketball player shoots 100 FT/day • Typical sequence 84/100, 67/100, 77/100 • All processes have this type of variability • Process variation has two components: • Natural variation, common cause or system variation (fluctuates around long run %) • Special cause or assignable cause variation (e.g. player has hand injury)

4. Common causes • Inappropriate procedures. • Poor design. • Poor maintenance of machines. • Lack of clearly defined standard operating procedures. • Poor working conditions,e.g. lighting, noise, dirt, temperature, ventilation. • Machines not suited to the job. • Substandard raw materials. • Measurement error. • Vibration in industrial processes. • Ambient temperature and humidity. • Insufficient training. • Normal wear and tear. • Variability in settings. • Computer response time • Deming – 85-94% of problems are common cause • System is the responsibility of management

5. Special causes • Operator absent. • Poor adjustment of equipment. • Operator falls asleep. • Faulty controllers. • Machine malfunction. • Computer crashes. • Poor batch of raw material. • Power surges. • Workforce responsibility • Can be addressed in the short term

6. Basic idea of control charting • Draw samples from process of interest at a sequence of time points • Choose a statistic, such as sample mean or sample proportion to describe state of process • Values of statistic plotted against time

7. Example: Monitoring thermostat production • 4 thermostats per hour sampled from assembly line and tested • Test temperature = 75. • Past experience shows thermostat response varies from test temp with s.d. = 0.5. • How should monitoring be done?

8. Control chart format • Center line = aim of process = m • Control limits = range of normal operation • Control limits decided by common cause variability (s)

9. Identifying special causes • Limits chosen to make excursions rare (e.g. 1 in 1000) • Many other ways in which process can be out of control

10. Setting control limits(m,s, known) • e.g. Monitoring hourly averages UCL = m+3s/n = 75 +3 x 0.5/  4 = 75.75 LCL = m+3s/n = 75 - 3 x 0.5/  4 = 74.25 m+3s/n m 1 4 m-3s/n 3 2

11. Estimating process parameters • Need to know common cause variability (for limits) • Special causes will disturb estimate of variability • Observations close in time are likely to differ in only common cause variation • Choose a rational subgroup of observations 1 4 3 2

12. Within subgroup variation • Common cause variation (s) = variability within subgroups • Estimate s from within subgroup standard deviation (S) • E(S) = ans n 2 3 4 5 6 7 8 9 10 an.780 .886 .921 .940 .952 .959 .965 .969 .973 • If we have k subgroups, we pool s.d.’s to get overall estimate:

13. Between subgroup variation • Variability across (between) subgroups is given by changes in subgroup means • Changes in subgroup mean are measured relative to process mean (m if known) • A point lies outside control limits when its between subgroup variation is large. • We conclude that this is due to a special cause

14. Example: Monitoring measurement process • Testing Measurement method for chemical assay (% solids in chemical) • Take a sample. Split into 3 parts. Make 3 separate measurements. • 17 different samples.All from same source. • What is being measured?

15. 1 9.0 0.40 0.21 2 0.50 3.1 1.00 3 0.96 1.80 6.5 4 1.50 0.81 6.6 5 7.6 1.00 0.55 6 7 6.3 1.07 1.90 8 0.10 0.20 7.6 9 6.7 1.11 2.20 10 0.58 7.0 1.00 11 12 0.36 5.0 0.70 13 6.3 0.42 0.80 14 0.74 1.40 7.0 15 0.66 1.30 6.4 16 17 0.97 6.0 1.90 6.5 1.60 0.90 5.6 0.56 1.10 1.20 7. 0.67 8.8 4.3 5.8 7.5 8.2 5.1 7.5 6.6 6.3 4.9 6.0 7.6 7.0 5.8 7.5 5.0 7.5 9.2 3.3 6.1 6.2 7.3 7.0 7.7 7.9 7.3 4.7 6.2 7.3 6.5 7.1 5.9 6.1 6.3 9.1 3.8 7.6 6.0 7.2 6.9 7.6 5.7 7.3 5.4 6.8 6.2 5.7 5.2 6.0 5.7 7.4 Measurement process for chemical assays Data from 17 samples Average   Standard Deviation (s) Range (R)

16. Calculation of control limits • First estimate process mean: • Then estimate process s.d.: Sometimes range in used : E(R) = bns

17. Control chart for measurement data • Solid evidence process in not in control

18. Need to take immediate action Common types of special causes • Point out of limits • Run of points above or below centerline

19. Common types of special causes • run of points going up or down • Repetitive patterns observed

20. Use of control charts • Charts can be done manually or automatically • Crucial: What happens next? • Control charts don't improve processes, people do • Incidents need to be recorded in real time • Need to track down root cause and remove it • Don’t confuse specification limits with control charts • Specifications – what is desirable • Control limits – what is currently possible • Specifications pertain to individual items • Control limits pertain to averages

21. Effectiveness of control charts • Two types of mistakes possible • Type I error: Stopping the process when it is in control. • In American or Japanese system: In British system:

22. Type II errors = 1- Power • Not stopping the process when it is out of control (e.g. process mean has shifted) • m = m0 + Ds

23. Average Run Length (ARL) • ARL = (Avg.) time needed to identify an out-of-limits signal If process is in control, p = a

24. ARL for out of control process • If process is out of control,e.g. m = m0 + Ds • p = 1- b; E(Y) = 1/p, e.g. • Chart takes very long to detect is shift is small • CUSUM charts have shorter ARL

25. Full SPC cycle • Identify special causes of variation • Bring process into statistical control • Reduce common cause effects through process improvement (narrow control limits towards centerline)

26. Control Charts for Process Variation • X bar charts look to control the central tendency of a process • Equally important to control the process variation • Many practitioners first control variation and then location • Variation measured by S (historically R)

27. Sampling distribution of s2 • What is it’s sampling distribution ? • Sums of squares of i.i.d normals are chi-squared with as many d.f. as there are terms.

29. Control limits for S chart • E(S2) = s2 (S2 is an unbiased estimator of s2) • V(S) = E(S2) – (E(S))2 = s2 – (ans)2 Normal approx:

30. S chart for Measurement data