1. TM 620: Quality Management
Session Eight – 23 November 2010
Control Charts, Part II
Attributes
Special Cases

2. Shewhart’s Assumptions
The data generated by the process when it is in control:
Are normally distributed
Are independent
Have a mean and standard deviation that are fixed and unknown

3. Data on Quality Characteristics
Attribute data
Discrete
Often a count of some type
Variable data
Continuous
Often a measurement, such as length, voltage, or viscosity
We will use statistical methods for both types

4. Control Chart Concept Map
Quality Characteristic
Q-Sum Chart
n>10
n >1
Sm. shfts
X, Moving R
type of attribute
ni = n
p, np c
pvar u
X, S
X, R no
yes
variable
attribute
defective
defects

5. Names for Abnormalities
Variable data
Defective
Attribute data
Nonconforming
Does the item meet the requirements on one or more quality characteristics

6. Charts for Attributes Fraction nonconforming (p-chart)
Fixed sample size
Variable sample size
np-chart for number nonconforming
Charts for defects
c-chart
u-chart

7. Fraction Nonconforming
The ratio of the number of nonconforming items in a population to the total number of items in the population
These items may have several quality characteristics simultaneously inspected

8. The P-Chart  UCL  p  3  LCL  p  3  d p ( 1  p ) p    nm n
where m  d i p ( 1  p ) p  i  1   p nm n

11. Example; P-Chart
Operators of a sorting machine must read the zip code on a letter and diver the letter to the proper carrier route. Over one month’s time, 25 samples of 100 letters were chosen, and the number of errors was recorded. Error counts for each of the 25 days follows.

12. Example

13. Example 25 03 . 00 01 + = p
= 0.024

14. Example 25 03 . 00 01 + = p
= 0.024 p
070 .
100 )
024 1 ( 3 = - + n
UCL

15. Example 25 03 . 00 01 + = p = 0.024 p ( 1 - p ) LCL = p - = - . 022 =
070 .
100 )
024 1 ( 3 = - + n
UCL p ( 1 - p )
LCL = p - = - .
022 = 3 . p n

16. Example; P-Chart

17. Variable P-Charts Idea
Recall,
For different sample size, compute CL for each sample p ( 1 - p ) CL = p 3 n p ( 1 - p ) CL = p 3 i n i

21. Variable Size P Chart

22. Number Nonconforming np-chart
Many non-statistically trained people find the np chart easier to interpret that the p-chart

23. The np Control Chart
UCL =
CL = np
LCL =

26. C - Chart p chart shows % of parts that are defective in a lot
np chart shows # parts defective in a lot
Suppose more than one defect can occur in a particular part
C-chart shows # defects in a given lot
based on the Poisson
subgroup size constant > 25

27. Poisson
Distribution of rare events E X [ ] = l x s p e ( ) ! -

28. ® l c ® s c Poisson Distribution of rare events
Idea: let c = # defects E X [ ] = l x s p e ( ) ! - l c s c x

29. Control Limits = ± c 3 c ® l c ® s c Poisson Distribution
Idea: let c = # defects l c s c x
Control Limits = c 3 c

30. Example

31. Example

34. U - Chart
Like the c - chart, except it removes the restriction of equal sample sizes c + c + . . . + c u = 1 2 k n + n + . . . + n 1 2 k S = u / n ui i
Control Limits = u 3 u / n i i

35. Example

36. Example

39. CUSUM Chart
X-bar, R charts work well to detect significant shifts in the mean (1.5s - 2.0s)
However, suppose we wish to detect smaller shifts in the process
Consider a piston ring process with target value at 10.0 cm.
First 10 sampled at normal with m = 10
Second 10 sampled at normal with m = 11
Control limits computed at LCL = 7, UCL = 13

40. Example

41. Example

42. CUSUM Idea å å Show the cumulative effects of relatively small changes= ( x - m ) i j o j = 1 i - 1 å = ( x - m ) + ( x - m ) i o j o j = 1 = ( x - m ) + C i o i - 1

43. Example, CUSUM Table

44. Example; CUSUM Chart

45. Questions to Ask When Implementing Control Charts
Which process characteristics to control
Where the charts should be implemented in the process
What is the proper type of control chart for your process
What mechanisms are there to take action based on the analysis of the control charts
What data collection systems and computer software should be used

46. Control Chart Design Issues
Basis for sampling
Sample size
Frequency of sampling
Location of control limits

48. SPC Implementation Requirements
Top management commitment
Project champion
Initial workable project
Employee education and training
Accurate measurement system

49. Example: Catalog Company
A catalog distributer ships a variety of orders each day. The packing slips often contain errors such as wrong purchase order numbers, wrong quantities, or incorrect sizes. The data on the next slide was collected during the month of August.
What type of chart should you use to analyze this data?
Catalog Company
u bar
individual control limits will vary with sample size

52. Example: Machine Breakdowns
The number of machine breakdowns for a particular process are tracked per day over a 25 day period. The results are on the next slide.
What type of chart should you use to analyze this data?
Machine Breakdowns
C bar

55. Example: Silicon Wafer Production
The thickness of silicon wafers used in the production of semiconductors must be carefully controlled. The tolerance of one such product is specified as ± inches. In one production facility, three wafers were selected each hour and the thickness measured carefully to within one ten-thousandth of an inch.
What type of chart should you use to analyze this data?
Silicon Wafer Production
X bar and R

58. Example: Orange Juice
Frozen orange juice concentrate is packed in 6-oz. cardboard cans. These cans are formed on a machine by spinning them from cardboard stock and attaching a metal bottom panel. By inspection of a can, we may determine whether, when filled, it could possibly leak either on the side seam or around the bottom joint. Thirty samples of fifty cans each were selected at half-hour intervals over a three shift period in which the machine was in continuous operation.
What type of chart should you use to analyze this data?
Orange Juice
P or NP

62. Next Class Homework Topic Preparation Ch. 11 (12) Problems 13, 14, 21
Six Sigma, FMEA, Reliability
Preparation
Chapter 15 and Handout