1. Statistical Process Control
Operations Management
Dr. Ron Lembke

2. Designed Size

3. Natural Variation

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

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:
Positive Skew > 0
Skewness = 0
Negative Skew < 0

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

8. Design Tolerances Design tolerance:
Determined by users’ needs
UTL -- Upper Tolerance Limit
LTL -- Lower Tolerance Limit
Eg: specified size +/ inches
No connection between tolerance and 
completely unrelated to natural variation.

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

10. Process Capability
Capable Not Capable
LTL
UTL
LTL
UTL
LTL
UTL
LTL
UTL

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

12. Process Capability Specs: 1.5 +/- 0.01 Mean: 1.505 Std. Dev. = 0.002
LTL = 1.5 – 0.01 = 1.49
UTL = = 1.51
Mean: Std. Dev. = 0.002
LCL = *0.002 = 1.499
UCL = = 1.511
Process
Specs
1.49
1.499
1.51
1.511

13. Capability Index
Capability Index (Cpk) will tell the position of the control limits relative to the design specifications.
Cpk>= 1.0, process is capable
Cpk< 1.0, process is not capable

14. Process Capability, Cpk
Tells how well parts produced fit into specs
Process
Specs 3 3
LTL
UTL

15. Process Capability Tells how well parts produced fit into specs
For our example:
Cpk= min[ 0.015/.006, 0.005/0.006]
Cpk= min[2.5,0.833] = < 1 Process not capable

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

17. If re-centered, it would be Capable
Process
Specs
1.49
1.494
1.506
1.51

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

19. Production Process Mix % Wrong wt. Wrong wt. Candy irregularity
Make Candy
Make Candy
Make Candy
Mix
Package
Put in big bags
Make Candy
Mix %
Wrong wt.
Wrong wt.
Make Candy
Make Candy
Candy irregularity

20. Processes Involved Candy Manufacturing: Mixing: Individual packages:
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?

21. 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

22. Candy colors Write Name on form Write weight on form
Write Package # on form
Count # of each color and write on form
Count total # of candies and write on form
(Advanced only): Eat candies
Turn in forms and complete wrappers

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

25. Plain Candy Weights
Avg 0.858, StDev 0.035, C.V

26. Peanut Color Mix website Brown 17.7% 20% Yellow 8.2% 20% Red 9.5% 20%
Blue 15.4% 20%
Orange 26.4% 10%
Green 22.7% 10%

27. Plain Color Mix Class website Brown 12.1% 30% Yellow 14.7% 20%
Red 11.4% 20%
Blue 19.5% 10%
Orange 21.2% 10%
Green 21.2% 10%

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

29. Acceptable?

31. 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

32. Goal of Control Charts collect and present data visually
allow us to see when trend appears
see when “out of control” point occurs

33. Process Control Charts
Graph of sample data plotted over time X
UCL
Process Average ± 3
LCL
Time

34. Process Control Charts
Graph of sample data plotted over time X
UCL
Assignable Cause Variation
LCL
Natural Variation
Time

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

36. Attributes vs. Variables
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)

37. 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)

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

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

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

41. p Chart Example
You’re manager of a 500-room hotel. You want to achieve the highest level of service. For 7 days, you collect data on the readiness of 200 rooms. Is the process in control (use z = 3)?
© 1995 Corel Corp.

42. p Chart Hotel Data No. No. Not Day Rooms Ready Proportion
/200 =

43. p Chart Control Limits

44. p Chart Control Limits

45. p Chart Solution

46. p Chart Solution

47. p Chart
UCL
LCL

48. R Chart Type of variables control chart Shows sample ranges over time
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

49. Hotel 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?

50. Hotel Data Day Delivery Time

51. R &X Chart Hotel Data Sample Day Delivery Time Mean Range
Sample Mean =

52. R &X Chart Hotel Data Sample Day Delivery Time Mean Range
Largest
Smallest
Sample Range =

53. R &X Chart Hotel Data Sample Day Delivery Time Mean Range

54. R Chart Control Limits From Exhibit 6.13 Sample Range at Time i
# Samples

55. Control Chart Limits

56.  R Chart Control Limits R 3 . 85  4 . 27    4 . 22 R    3 .k  R i 3 . 85  4 . 27    4 . 22 R  i  1   3 .
894 k 7

57.  R Chart Solution R 3 . 85  4 . 27    4 . 22 R    3 . 894 k 71   3 .
894 k 7
UCL  D  R 
(2.11)
(3.894)  8 .
232 R 4
From (n = 5)
LCL  D  R 
(0)
(3.894)  R 3

58. R Chart Solution
UCL

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

60. X Chart Control Limits
From Table 6-13

61. X Chart Control Limits
From 6.13
Sample Mean at Time i
Sample Range at Time i
# Samples

62. Exhibit 6.13 Limits

63. R &X Chart Hotel Data Sample Day Delivery Time Mean Range

64. X Chart Control Limitsk  X i 5 . 32  6 . 59    6 . 79 X  i  1   5 .
813 k 7 k  R i 3 . 85  4 . 27    4 . 22 R  i  1   3 .
894 k 7

65. X Chart Control Limitsk  X i 5 . 32  6 . 59    6 . 79 X  i  1   5 .
813 k 7 k  R i 3 . 85  4 . 27    4 . 22
From 6.13 (n = 5) R  i  1   3 .
894 k 7
UCL  X  A  R  5 .
813  .
58 * 3 .
894  8 .
060 X 2

66. X Chart Solution   X 5 . 32  6 . 59    6 . 79 X    5 . 813 k
From 6.13 (n = 5)  R i 3 . 85  4 . 27    4 . 22 R  i  1   3 .
894 k 7
UCL  X  A  R  5 .
813  (0 .
58)
(3.894) = 8.060 X 2
LCL  X  A  R  5 .
813  (0 .
58)
(3.894) = 3.566 X 2

67. X Chart Solution* `
X, Minutes 8
UCL 6 4 2
LCL 1 2 3 4 5 6 7
Day

68. Thinking Challenge
You’re manager of a 500-room hotel. The hotel owner tells you that it takes too long to deliver luggage to the room (even if the process may be in control). What do you do? N
© 1995 Corel Corp.

69. Solution Redesign the luggage delivery process Use TQM tools
Cause & effect diagrams
Process flow charts
Pareto charts
Method
People
Too Long
Material
Equipment