Chapter 9A Process Capability and Statistical Quality Control

1. Chapter 9AProcess Capability and Statistical Quality Control • Process Variation • Process Capability • Process Control Procedures • Variable data • Attribute data • Acceptance Sampling • Operating Characteristic Curve

2. Basic Causes of Variation • Assignable causesare factors that can be clearly identified and possibly managed. • Common causesare inherent to the production process. In order to reduce variation due to common causes, the process must be changed. • Key: Determining which is which!

3. Types of Control Charts • Attribute (Go or no-go information) • Defectivesrefers to the acceptability of product across a range of characteristics. • p-chart application • Variable (Continuous) • Usually measured by the mean and the standard deviation. • X-bar and R chart applications

4. Types of Statistical Quality Control Statistical Statistical Quality Control Quality Control Process Acceptance Process Acceptance Control Sampling Control Sampling Variables Attributes Variables Attributes Variables Attributes Variables Attributes Charts Charts Charts Charts

5. UCL Normal Behavior LCL 1 2 3 4 5 6 Samples over time UCL Possible problem, investigate LCL 1 2 3 4 5 6 Samples over time UCL Possible problem, investigate LCL 1 2 3 4 5 6 Samples over time Statistical Process Control (SPC) Charts Look for trends! Excellent review in exhibit TN8.5.

6. x LCL UCL Control Limits We establish the Upper Control Limits (UCL) and the Lower Control Limits (LCL) with plus or minus 3 standard deviations. Based on this we can expect 99.7% of our sample observations to fall within these limits. 99.7%

7. Number of defectives Sample size Sample Example of Constructing a p-Chart: Required Data

8. Statistical Process Control Formulas:Attribute Measurements (p-Chart) Given: Compute control limits:

9. Example of Constructing a p-chart: Step 1 1. Calculate the sample proportions, p (these are what can be plotted on the p-chart) for each sample.

10. Example of Constructing a p-chart: Steps 2&3 2. Calculate the average of the sample proportions. 3. Calculate the standard deviation of the sample proportion

11. Example of Constructing a p-chart: Step 4 4. Calculate the control limits. UCL = 0.0930 LCL = -0.0197 (0)

12. Example of Constructing a p-Chart: Step 5 5. Plot the individual sample proportions, the average of the proportions, and the control limits

13. R Chart • Type of variables control chart • Interval or ratio scaled numerical data • Shows sample ranges over time • Difference between smallest & largest values in inspection sample • Monitors variability in process • Example: Weigh samples of coffee & compute ranges of samples; Plot

14. R Chart Control Limits From Table (function of sample size) Sample Range in sample i # Samples

15. R Chart Example You’re manager of a 500-room hotel. You want to analyze the time it takes to deliver luggage to the room. For 7 days, you collect data on 5 deliveries per day. Is the process in control?

16. 7.30 + 4.20 + 6.10 + 3.45 + 5.55 5 Sample Mean = R Chart Hotel Data Sample DayDelivery TimeMean Range 1 7.30 4.20 6.10 3.45 5.55 5.32

17. Sample Range = 7.30 - 3.45 R Chart Hotel Data Sample DayDelivery TimeMean Range 1 7.30 4.20 6.10 3.45 5.55 5.32 3.85 Largest Smallest

18. R Chart Hotel Data Sample DayDelivery TimeMean Range 1 7.30 4.20 6.10 3.45 5.55 5.32 3.85 2 4.60 8.70 7.60 4.43 7.62 6.59 4.27 3 5.98 2.92 6.20 4.20 5.10 4.88 3.28 4 7.20 5.10 5.19 6.80 4.21 5.70 2.99 5 4.00 4.50 5.50 1.89 4.46 4.07 3.61 6 10.10 8.10 6.50 5.06 6.94 7.34 5.04 7 6.77 5.08 5.90 6.90 9.30 6.79 4.22

19. R Chart Control Limits Solution From Table (n = 5)

20. R Chart Control Chart Solution UCL R-bar

21. X Chart • Type of variables control chart • Interval or ratio scaled numerical data • Shows sample means over time • Monitors process average • Example: Weigh samples of coffee & compute means of samples; Plot

22. X Chart Control Limits From Table Mean of sample i Range of sample i # Samples

23. X Chart Example You’re manager of a 500-room hotel. You want to analyze the time it takes to deliver luggage to the room. For 7 days, you collect data on 5 deliveries per day. Is the process in control?

24. X Chart Hotel Data Sample DayDelivery TimeMean Range 1 7.30 4.20 6.10 3.45 5.55 5.32 3.85 2 4.60 8.70 7.60 4.43 7.62 6.59 4.27 3 5.98 2.92 6.20 4.20 5.10 4.88 3.28 4 7.20 5.10 5.19 6.80 4.21 5.70 2.99 5 4.00 4.50 5.50 1.89 4.46 4.07 3.61 6 10.10 8.10 6.50 5.06 6.94 7.34 5.04 7 6.77 5.08 5.90 6.90 9.30 6.79 4.22

25. X Chart Control Limits Solution* From Table (n = 5)

26. X ChartControl Chart Solution* UCL X-bar LCL

27. X AND R CHART EXAMPLEIN-CLASS EXERCISE The following collection of data represents samples of the amount of force applied in a gluing process: Determine if the process is in control by calculating the appropriate upper and lower control limits of the X-bar and R charts.

28. X AND R CHART EXAMPLEIN-CLASS EXERCISE

29. Example of x-bar and R charts: Step 1. Calculate sample means, sample ranges, mean of means, and mean of ranges.

30. Example of x-bar and R charts: Step 2. Determine Control Limit Formulas and Necessary Tabled Values

31. Example of x-bar and R charts: Steps 3&4. Calculate x-bar Chart and Plot Values

32. Example of x-bar and R charts: Steps 5&6: Calculate R-chart and Plot Values

33. SOLUTION:Example of x-bar and R charts: 1. Is the process in Control? 2. If not, what could be the cause for the process being out of control?

34. Process Capability • Process limits - actual capabilities of process based on historical data • Tolerance limits - what process design calls for- desired performance of process

35. 1. Make bigger 2. Make smaller Bad idea Implies having greater control over process Þ Good! Process Capability • How do the limits relate to one another? You want: tolerance range > process range Two methods of accomplishing this:

36. Process Capability Measurement Cp index = Tolerance range / Process range What value(s) would you like for Cp? Þ Larger Cp indicates a more reliable and predictable process (less variability) The Cp index is based on the assumption that the process mean is centered at the midpoint of the tolerance range

38. UTL LTL

39. While the Cp index provides useful information on process variability, it does not give information on the process average relative to the tolerance limits. Note: UTL LTL

40. Refers to the LTL Refers to the UTL = process mean (Unknown but can be estimated with the grand mean) Cpk Index s= standard deviation (Unknown but can be estimated with the average range) Together, these process capability Indices show how well parts being produced conform to design specifications.

41. UTL LTL Since Cp and Cpk are different we can conclude that the process is not centered, however the Cp index tells us that the process variability is very low

42. An example of the use of process capability indices The design specifications for a machined slot is 0.5± .003 inches. Samples have been taken and the process mean is estimated to be .501. The process standard deviation is estimated to be .001. What can you say about the capability of this process to produce this dimension?

43. Process capability Machined slot (inches) 0.497 inches LTL 0.503 inches UTL  = 0.001 inches Process mean 0.501 inches

44. Sampling Distributions(The Central Limit Theorem) • Regardless of the underlying distribution, if the sample is large enough (>30), the distribution of sample means will be normally distributed around the population mean with a standard deviation of :

45. Computing Process Capability Indexes Using Control Chart Data Recall the following info from our in class exercise: Since A2 is calculated on the assumption of three sigma limits:

46. From the Central Limit Theorem: So, Therefore,

47. Suppose the Design Specs for the Gluing Process were 10.7  .2, Calculate the Cp and Cpk Indexes: Answer:

48. Note, multiplying each component of the Cpk calculation by 3 yields a Z value. You can use this to predict the % of items outside the tolerance limits: From Appendix E we would expect: .008 + .036 = .044 or 4.4% non-conforming product from this process .792 * 3 = 2.38 .597 * 3 = 1.79 .008 or .8% of the curve .036 or 3.6% of the curve

49. Capability Index – In Class Exercise • You are a manufacturer of equipment. A drive shaft is purchased from a supplier close by. The blueprint for the shaft specs indicate a tolerance of 5.5 inches ± .003 inches. Your supplier is reporting a mean of 5.501 inches. And a standard deviation of .0015 inches. • What is the Cpk index for the supplier’s process?