1. BMM 3633 Industrial Engineering

2. Learning Objectives: Define the concept and application of SPC chart.
Construct a control chart for
variable and attribute data.
Calculate process capability
ratio (Cp) and process
capability index (Cpk).
Learning Objectives:

3. Contents: Statistical Process Control Control Charts
Development of Control
Chart
Control Chart Patterns
Process Capability
Contents:

4. Statistical Process Control
Take periodic samples from process
Plot sample points on control chart
Determine if process is within limits
Prevent quality problems
UCL
LCL

5. Statistical Process Control (cont..)
Variation
Common Causes
Variation inherent in a process
Can be eliminated only through improvements in the system
Special Causes
Variation due to identifiable factors
Can be modified through operator or management action

6. Statistical Process Control (cont..)
Types of Data
Attribute data
Product characteristic evaluated with a discrete choice
Good/bad, yes/no
Variable data
Product characteristic that can be measured
Length, size, weight, height, time, velocity

7. Control Charts Graph establishing process control limits
Charts for variables
Mean (x-bar), Range (R)
Charts for attributes
p and c

8. Control Charts (cont..) Process Control Chart 1 2 3 4 5 6 7 8 9 10
Sample number
Upper
control
limit
Process
average
Lower
Out of control

9. A Process is In Control if
Control Charts (cont..)
A Process is In Control if
No sample points outside limits
Most points near process average
About equal number of points above & below centerline
Points appear randomly distributed

10. Development of Control Chart
Based on in-control data
If non-random causes present discard data
Correct control chart limits

11. Development of Control Chart (cont..)
Control Charts for Attributes
p Charts
Calculate percent defectives in sample
c Charts
Count number of defects in item

12. Control Charts for Variables
Development of Control Chart (cont..)
Control Charts for Variables
Mean chart ( x-Chart )
Uses average of a sample (control the central tendency of the process)
Range chart ( R-Chart )
Uses amount of dispersion in a sample (control the dispersion of the sample)

13. Setting Chart Limits For x-Charts when we know s
Upper control limit (UCL) = x + zsx
Lower control limit (LCL) = x - zsx
where x = mean of the sample means or a target value set for the process
z = number of normal standard deviations
sx = standard deviation of the sample means
= s/ n
s = population standard deviation
n = sample size

14. Example Hour 1 Hour Mean Hour Mean Sample Weight of Number Oat Flakes
1 17
2 13
3 16
4 18
5 17
6 16
7 15
8 17
9 16
Mean 16.1
s = 1
Hour Mean Hour Mean
n = 9
For 99.73% control limits, z = 3
UCLx = x + zsx = (1/3) = 17 ozs
LCLx = x - zsx = (1/3) = 15 ozs

15. Variation due to natural causes
Example (cont..)
Control Chart for sample of 9 boxes
Variation due to assignable causes
Out of control
Sample number
| | | | | | | | | | | |
17 = UCL
15 = LCL
16 = Mean
Variation due to natural causes
Out of control

16. Setting Chart Limits (cont..)
For x-Charts when we don’t know s
Upper control limit (UCL) = x + A2R
Lower control limit (LCL) = x - A2R
where R = average range of the samples
A2 = control chart factor found in the table
x = mean of the sample means

17. Setting Chart Limits (cont..)
Control Chart Factors
Sample Size Mean Factor Upper Range Lower Range
n A2 D4 D3

18. Example Process average x = 12 ounces Average range R = .25
Sample size n = 5

19. Example (cont..) Process average x = 12 ounces Average range R = .25
Sample size n = 5
UCLx = x + A2R
= 12 + (.577)(.25) =
= ounces
From Table

20. Example (cont..) Process average x = 12 ounces Average range R = .25
Sample size n = 5
UCL =
Mean = 12
LCL =
UCLx = x + A2R
= 12 + (.577)(.25) =
= ounces
LCLx = x - A2R =
= ounces

21. Exercise OBSERVATIONS (SLIP-RING DIAMETER, CM) SAMPLE k 1 2 3 4 5 x R

22. Restaurant Control Limits
For salmon filets at Darden Restaurants
Sample Mean
x Bar Chart
UCL =
x –
LCL –
| | | | | | | | |
11.5 –
11.0 –
10.5 –
Sample Range
Range Chart
UCL =
R =
LCL = 0
| | | | | | | | |
0.8 –
0.4 –
0.0 –

23. Restaurant Control Limits (cont..)
Capability Histogram
LSL
USL
10.2
10.5
10.8
11.1
11.4
11.7
12.0
Specifications
LSL
USL 12
Capability
Mean =
Std.dev = 1.88
Cp = 1.77
Cpk = 1.7

24. R – Chart Type of variables control chart
Shows sample ranges over time
Difference between smallest and largest values in sample
Monitors process variability
Independent from process mean

25. Setting Chart Limits For R-Charts Upper control limit (UCLR) = D4R
Lower control limit (LCLR) = D3R
where
R = average range of the samples
D3 and D4 = control chart factors from Table

26. Example Average range R = 5.3 pounds Sample size n = 5
From Table, D4 = 2.115, D3 = 0
UCL = 11.2
Mean = 5.3
LCL = 0
UCLR = D4R
= (2.115)(5.3)
= 11.2 pounds
LCLR = D3R
= (0)(5.3)
= 0 pounds

27. Exercise OBSERVATIONS (SLIP-RING DIAMETER, CM) SAMPLE k 1 2 3 4 5 x R

28. Mean and Range Charts (a)
These sampling distributions result in the charts below
(Sampling mean is shifting upward but range is consistent)
x-chart
(x-chart detects shift in central tendency)
UCL
LCL
R-chart
(R-chart does not detect change in mean)
UCL
LCL

29. Mean and Range Charts (cont..)
(b)
These sampling distributions result in the charts below
(Sampling mean is constant but dispersion is increasing)
x-chart
(x-chart does not detect the increase in dispersion)
UCL
LCL
R-chart
(R-chart detects increase in dispersion)
UCL
LCL

30. Using x- and R-Charts Together
Mean and Range Charts (cont..)
Using x- and R-Charts Together
Each measures the process differently
Both process average and variability must be in control
CHAPTER 9

31. The Normal Distribution
=0 1 2 3
-1
-2
-3
95%
99.74%

32. Steps In Creating Control Charts
Take samples from the population and compute the appropriate sample statistic
Use the sample statistic to calculate control limits and draw the control chart
Plot sample results on the control chart and determine the state of the process (in or out of control)
Investigate possible assignable causes and take any indicated actions
Continue sampling from the process and reset the control limits when necessary

33. Control Charts for Attributes
For variables that are categorical
Good/bad, yes/no, acceptable/unacceptable
Measurement is typically counting defectives
Charts may measure
Percent defective (p-chart)
Number of defects (c-chart)

34. Control Limits for p-Charts
Population will be a binomial distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics
UCLp = p + zsp ^
p(1 - p) n
ðp = ^
LCLp = p - zsp ^
where p = mean fraction defective in the sample
z = number of standard deviations
ðp = standard deviation of the sampling distribution
n = sample size ^

35. Control Limits for p-Charts (cont..)
Control Chart Z Values
Smaller Z values make more sensitive charts
Z = 3.00 is standard
Compromise between sensitivity and errors
CHAPTER 9

36. Example Samples of work of 20 clerks; each clerk entered 100 records
Sample Number Fraction Sample Number Fraction
Number of Errors Defective Number of Errors Defective
Total = 80
p = = 0.04 80
(100)(20)
(0.04)( )
100
sp = = 0.02 ^

37. Example (cont..) UCLp = p + zsp = 0.04 + 3(0.02) = 0.10^
LCLp = p - zsp = (0.02) = 0 ^
.11 –
.10 –
.09 –
.08 –
.07 –
.06 –
.05 –
.04 –
.03 –
.02 –
.01 –
.00 –
Sample number
Fraction defective
| | | | | | | | | |
UCLp = 0.10
LCLp = 0.00
p = 0.04

38. Possible assignable causes present
Example (cont..)
UCLp = p + zsp = (0.02) = 0.10 ^
Possible assignable causes present
LCLp = p - zsp = (0.02) = 0 ^
.11 –
.10 –
.09 –
.08 –
.07 –
.06 –
.05 –
.04 –
.03 –
.02 –
.01 –
.00 –
Sample number
Fraction defective
| | | | | | | | | |
UCLp = 0.10
LCLp = 0.00
p = 0.04

39. Exercise 20 samples of 100 pairs of jeans 1 6 .06 11 9 .09
Sample Number Fraction Sample Number Fraction
Number of Errors Defective Number of Errors Defective
Total = 200

40. Control Limits for c-Charts
Population will be a Poisson distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics
UCLc = c + 3 c
LCLc = c - 3 c
where c = mean number defective in the sample

41. Example c = 54 complaints/9 days = 6 complaints/day UCLc = c + 3 c
Number of complaints: 3, 0, 8, 9, 6, 7, 4, 9, 8 over 9-day period
c = 54 complaints/9 days = 6 complaints/day
UCLc = c + 3 c =
= 13.35 | 1 2 3 4 5 6 7 8 9
Day
Number defective
14 –
12 –
10 –
8 –
6 –
4 –
2 –
0 –
UCLc = 13.35
LCLc = 0
c = 6
LCLc = c - 3 c =
= 0

42. Exercise The number of defects in 15 sample rooms
Sample Number Sample Number Sample Number
Number of Defects Number of Defects Number of Defects
Total = 190

43. Which Control Chart to Use
Variables Data
Using an x-Chart and R-Chart
Observations are variables
Collect samples of n = 4, or n = 5, or more, each from a stable process and compute the mean for the x-chart and range for the R-chart
Track samples of n observations each.

44. Which Control Chart to Use (cont..)
Attribute Data
Using the p-Chart
Observations are attributes that can be categorized as good or bad (or pass–fail, or functional–broken), that is, in two states.
We deal with fraction, proportion, or percent defectives.
There are several samples, with many observations in each. For example, 20 samples of n = 100 observations in each.

45. Which Control Chart to Use (cont..)
Attribute Data
Using a c-Chart
Observations are attributes whose defects per unit of output can be counted.
We deal with the number counted, which is a small part of the possible occurrences.
Defects may be: number of blemishes on a desk; complaints in a day; crimes in a year; broken seats in a stadium; typos in a chapter of this text; or flaws in a bolt of cloth.

46. Patterns in Control Charts
Upper control limit
Target
Lower control limit
Normal behavior. Process is “in control.”

47. Patterns in Control Charts (cont..)
Upper control limit
Target
Lower control limit
Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated.
One plot out above (or below). Investigate for cause. Process is “out of control.”

48. Patterns in Control Charts (cont..)
Upper control limit
Target
Lower control limit
Trends in either direction, 5 plots. Investigate for cause of progressive change.

49. Patterns in Control Charts (cont..)
Upper control limit
Target
Lower control limit
Two plots very near lower (or upper) control. Investigate for cause.

50. Patterns in Control Charts (cont..)
Upper control limit
Target
Lower control limit
Run of 5 above (or below) central line. Investigate for cause.

51. Patterns in Control Charts (cont..)
Upper control limit
Target
Lower control limit
Erratic behavior. Investigate.

52. Sample Size Determination
Attribute control charts
50 to 100 parts in a sample
Variable control charts
2 to 10 parts in a sample
CHAPTER 9

53. Process Capability
The natural variation of a process should be small enough to produce products that meet the standards required
A process in statistical control does not necessarily meet the design specifications
Process capability is a measure of the relationship between the natural variation of the process and the design specifications

54. Process Capability Ratio
Cp =
Upper Specification - Lower Specification 6σ
A capable process must have a Cp of at least 1.0
Does not look at how well the process is centered in the specification range
Often a target value of Cp = 1.33 is used to allow for off-center processes
Six Sigma quality requires a Cp = 2.0

55. Upper Specification - Lower Specification
Example
Insurance claims process
Process mean x = minutes
Process standard deviation s = minutes
Design specification = 210 ± 3 minutes
Cp =
Upper Specification - Lower Specification 6σ

56. Upper Specification - Lower Specification
Example (cont..)
Insurance claims process
Process mean x = minutes
Process standard deviation s = minutes
Design specification = 210 ± 3 minutes
Cp =
Upper Specification - Lower Specification 6σ
= = 1.938
6(0.516)

57. Upper Specification - Lower Specification
Example (cont..)
Insurance claims process
Process mean x = minutes
Process standard deviation s = minutes
Design specification = 210 ± 3 minutes
Cp =
Upper Specification - Lower Specification 6σ
= = 1.938
6(0.516)
Process is capable

58. Exercise Net weight specification = 9.0 oz  0.5 oz
Process mean = 8.80 oz
Process standard deviation = 0.12 oz
CHAPTER 9

59. Exercise
In a GE insurance claims process, x = minutes, and  = minutes. The design specification to meet customer expectations is 210 ± 3 minutes. Determine the process capability ratio. Is the process capable?
CHAPTER 9

60. Process Capability Index
Cpk = minimum of ,
Upper Specification - x Limit 3σ
Lower x - Specification Limit
A capable process must have a Cpk of at least 1.0
A capable process is not necessarily in the center of the specification, but it falls within the specification limit at both extremes

61. Example New Cutting Machine New process mean x = 0.250 inches
Process standard deviation s = inches
Upper Specification Limit = inches
Lower Specification Limit = inches

62. Example (cont..) New Cutting Machine New process mean x = 0.250 inches
Process standard deviation s = inches
Upper Specification Limit = inches
Lower Specification Limit = inches
Cpk = minimum of ,
(0.251)
(3)0.0005

63. New machine is NOT capable
Example (cont..)
New Cutting Machine
New process mean x = inches
Process standard deviation s = inches
Upper Specification Limit = inches
Lower Specification Limit = inches
Cpk = minimum of ,
(0.251)
(3)0.0005
(0.249)
Both calculations result in
New machine is NOT capable
Cpk = = 0.67
0.001
0.0015

64. Exercise Net weight specification = 9.0 oz  0.5 oz
Process mean = 8.80 oz
Process standard deviation = 0.12 oz
CHAPTER 9

65. Interpreting Cpk Cpk = negative number Cpk = zero
Cpk = between 0 and 1
Cpk = 1
Cpk > 1

66. Any Questions???

67. Thank You