1. Statistical Process Control Operations Management Dr. Ron Lembke

2. Designed Size 10 11 12 13 14 15 16 17 18 19 20

3. Natural Variation 14.5 14.6 14.7 14.8 14.9 15.0 15.1 15.2 15.3 15.4

4. Theoretical Basis of Control Charts 95.5% of all  X fall within ± 2  Properties of normal distribution

5. Theoretical Basis of Control Charts Properties of normal distribution 99.7% of all  X fall within ± 3 

6. Skewness  Lack of symmetry  Pearson’s coefficient of skewness: Skewness = 0 Negative Skew < 0 Positive Skew > 0

7. Kurtosis  Amount of peakedness or flatness Kurtosis < 0 Kurtosis > 0 Kurtosis = 0

8. Heteroskedasticity Sub-groups with different variances

9. Design Tolerances  Design tolerance: Determined by users’ needs USL -- Upper Specification Limit LSL -- Lower Specification Limit Eg: specified size +/- 0.005 inches  No connection between tolerance and  completely unrelated to natural variation.

10. Process Capability and 6   A “capable” process has USL and LSL 3 or more standard deviations away from the mean, or 3σ.  99.7% (or more) of product is acceptable to customers LSLUSL 33 66 LSLUSL

11. Process Capability LSLUSL LSL USL CapableNot Capable LSLUSL LSLUSL

12. Process Capability  Specs: 1.5 +/- 0.01  Mean: 1.505 Std. Dev. = 0.002  Are we in trouble?

13. Process Capability  Specs: 1.5 +/- 0.01 LSL = 1.5 – 0.01 = 1.49 USL = 1.5 + 0.01 = 1.51  Mean: 1.505 Std. Dev. = 0.002 LCL = 1.505 - 3*0.002 = 1.499 UCL = 1.505 + 0.006 = 1.511 1.499 1.511.491.511 Process Specs

14. Capability Index  Capability Index (C pk ) will tell the position of the control limits relative to the design specifications.  C pk >= 1.33, process is capable  C pk < 1.33, process is not capable

15. Process Capability, C pk  Tells how well parts produced fit into specs Process Specs 33 33 LSLUSL

16. Process Capability  Tells how well parts produced fit into specs  For our example:  C pk = min[ 0.015/.006, 0.005/0.006]  C pk = min[2.5,0.833] = 0.833 < 1.33 Process not capable

17. Process Capability: Re-centered  If process were properly centered  Specs: 1.5 +/- 0.01 LTL = 1.5 – 0.01 = 1.49 UTL = 1.5 + 0.01 = 1.51  Mean: 1.5 Std. Dev. = 0.002 LCL = 1.5 - 3*0.002 = 1.494 UCL = 1.5 + 0.006 = 1.506 1.4941.511.491.506 Process Specs

18. If re-centered, it would be Capable 1.4941.511.491.506 Process Specs

19. Packaged Goods  What are the Tolerance Levels?  What we have to do to measure capability?  What are the sources of variability?

20. Production Process Make Candy PackagePut in big bags Make Candy Mix Mix % Candy irregularity Wrong wt.

21. Processes Involved  Candy Manufacturing: Are M&Ms uniform size & weight? Should be easier with plain than peanut Percentage of broken items (probably from printing)  Mixing: Is proper color mix in each bag?  Individual packages: Are same # put in each package? Is same weight put in each package?  Large bags: Are same number of packages put in each bag? Is same weight put in each bag?

22. Weighing Package and all candies  Before placing candy on scale, press “ON/TARE” button  Wait for 0.00 to appear  If it doesn’t say “g”, press Cal/Mode button a few times  Write weight down on form

23. Candy colors 1. Write Name on form 2. Write weight on form 3. Write Package # on form 4. Count # of each color and write on form 5. Count total # of candies and write on form 6. (Advanced only): Eat candies 7. Turn in forms and complete wrappers

25. Peanut Candy Weights  Avg. 2.18, stdv 0.242, c.v. = 0.111

26. Plain Candy Weights  Avg 0.858, StDev 0.035, C.V. 0.0413

27. Peanut Color Mix website  Brown 17.7%20%  Yellow 8.2%20%  Red 9.5%20%  Blue15.4%20%  Orange26.4%10%  Green22.7%10%

28. Classwebsite  Brown12.1%30%  Yellow14.7%20%  Red11.4%20%  Blue19.5%10%  Orange21.2%10%  Green21.2%10% Plain Color Mix

29. So who cares?  Dept. of Commerce  National Institutes of Standards & Technology  NIST Handbook 133  Fair Packaging and Labeling Act

30. Acceptable?

32. Package Weight  “Not Labeled for Individual Retail Sale”  If individual is 18g  MAV is 10% = 1.8g  Nothing can be below 18g – 1.8g = 16.2g

33. Goal of Control Charts  See if process is “in control” Process should show random values No trends or unlikely patterns  Visual representation much easier to interpret Tables of data – any patterns? Spot trends, unlikely patterns easily

34. NFL Control Chart?

35. Control Charts UCL LCL avg Values Sample Number

36. Definitions of Out of Control 1. No points outside control limits 2. Same number above & below center line 3. Points seem to fall randomly above and below center line 4. Most are near the center line, only a few are close to control limits 1. 8 Consecutive pts on one side of centerline 2. 2 of 3 points in outer third 3. 4 of 5 in outer two-thirds region

37. Control Charts NormalToo LowToo high 5 above, or belowRun of 5 Extreme variability

38. Control Charts UCL LCL avg 1σ1σ 2σ2σ 2σ2σ 1σ1σ

39. Control Charts 2 out of 3 in the outer third

40. Out of Control Point?  Is there an “assignable cause?” Or day-to-day variability?  If not usual variability, GET IT OUT Remove data point from data set, and recalculate control limits  If it is regular, day-to-day variability, LEAVE IT IN Include it when calculating control limits

41. Attribute Control Charts  Tell us whether points in tolerance or not p chart: percentage with given characteristic (usually whether defective or not) np chart: number of units with characteristic c chart: count # of occurrences in a fixed area of opportunity (defects per car) u chart: # of events in a changeable area of opportunity (sq. yards of paper drawn from a machine)

42. Attributes vs. Variables Attributes:  Good / bad, works / doesn’t  count % bad (P chart)  count # defects / item (C chart) Variables:  measure length, weight, temperature (x-bar chart)  measure variability in length (R chart)

43. p Chart Control Limits # Defective Items in Sample i Sample i Size

44. p Chart Control Limits # Defective Items in Sample i Sample i Size z = 2 for 95.5% limits; z = 3 for 99.7% limits # Samples

45. p Chart Control Limits # Defective Items in Sample i # Samples Sample i Size z = 2 for 95.5% limits; z = 3 for 99.7% limits

46. p Chart Example You’re manager of a 1,700 room hotel. For 7 days, you collect data on the readiness of all of the rooms that someone checked out of. Is the process in control (use z = 3)? © 1995 Corel Corp.

47. p Chart Hotel Data # RoomsNo. NotProportion DaynReady p 11,300130130/1,300 =.100 2800 90.113 340021.053 435025.071 530018.06 640012.03 760030.05

48. p Chart Control Limits

49. p Chart Solution

50. Hotel Room Readiness P-Bar

51. 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

52. 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? Hotel Example

53. Hotel Data DayDelivery Time 17.304.206.103.455.55 24.608.707.604.437.62 35.982.926.204.205.10 47.205.105.196.804.21 54.004.505.501.894.46 610.108.106.505.066.94 76.775.085.906.909.30

54. R &  X Chart Hotel Data Sample DayDelivery TimeMeanRange 17.304.206.103.455.555.32 7.30 + 4.20 + 6.10 + 3.45 + 5.55 5 Sample Mean =

55. R &  X Chart Hotel Data Sample DayDelivery TimeMeanRange 17.304.206.103.455.555.323.85 7.30 - 3.45Sample Range = LargestSmallest

56. R &  X Chart Hotel Data Sample DayDelivery TimeMeanRange 17.304.206.103.455.555.323.85 24.608.707.604.437.626.594.27 35.982.926.204.205.104.883.28 47.205.105.196.804.215.702.99 54.004.505.501.894.464.073.61 610.108.106.505.066.947.345.04 76.775.085.906.909.306.794.22

57. R Chart Control Limits Sample Range at Time i # Samples Table 10.3, p.433

58. Control Chart Limits

59. R Chart Control Limits

60. R Chart Solution UCL

61.  X Chart Control Limits Sample Range at Time i # Samples Sample Mean at Time i

62.  X Chart Control Limits A 2 from Table 10-3

63. Table 10.3 Limits

64. R &  X Chart Hotel Data Sample DayDelivery TimeMeanRange 17.304.206.103.455.555.323.85 24.608.707.604.437.626.594.27 35.982.926.204.205.104.883.28 47.205.105.196.804.215.702.99 54.004.505.501.894.464.073.61 610.108.106.505.066.947.345.04 76.775.085.906.909.306.794.22

65.  X Chart Control Limits

66.  X Chart Solution* 0 2 4 6 8 1234567  X, Minutes Day UCL LCL