1. TM 620: Quality Management
Session Seven – 9 November 2010
Control Charts, Part I
Variables

2. Recall: What is Quality?
Juran – Quality is fitness for use
=> we should be able to determine a set of measurable characteristics which define quality

3. Recall: What is Quality?
Taguchi – Loss from quality is proportional to the amount of variability in the system
Why?
=> if we reduce variation, we reduce loss from quality
Less rework, reduction in wasted time, effort, and money

4. Quality Improvement
The reduction of variability in processes and products
Equivalent definition:
The reduction of waste
Waste is any activity for which the customer will not pay
From Juran and Taguchi (and others);
Define the three categories of activities

5. Recall: Cost of Quality
Cost of Failure
Cost of
Control
Total Cost
Traditional View
Quality Level
Costs

6. Traditional Loss Function
LSL
USL x T
“goalpost” specifications – as long as you kick the ball between the posts, it doesn’t matter where you hit.
LSL T
USL

7. Example (Sony, 1979)
Comparing cost of two Sony television plants in Japan and San Diego. All units in San Diego fell within specifications. Japanese plant had units outside of specifications.
Loss per unit (Japan) = $0.44
Loss per unit (San Diego) = $1.33
How can this be?
Sullivan, “Reducing Variability: A New Approach to Quality,” Quality Progress, 17, no.7, 15-21, 1984.

8. Example
LSL
USL x T
U.S. Plant (2 = 8.33)
Japanese Plant (2 = 2.78)

9. Taguchi Loss Function T x x T
The Taguchi loss function is a scientific approach to tolerance design.
Taguchi suggests that no strict cut-off point divides good quality from poor quality. Rather, Taguchi assumes that losses can be approximated by a quadratic function so that larger derivations from the target correspond to increasingly larger losses. x T

10. Taguchi Loss Function T L(x) = k(x - T)2 L(x) k(x - T)2 x
Nominal is best = quality deteriorates as the actual value moves away from the target on either side x T
L(x) = k(x - T)2

11. Estimating Loss Function
Suppose we desire to make pistons with diameter D = 10 cm. Too big and they create too much friction. Too little and the engine will have lower gas mileage. Suppose tolerances are set at D = cm. Studies show that if D > 10.05, the engine will likely fail during the warranty period. Average cost of a warranty repair is $400.

12. Estimating Loss Function
L(x)
400
10.05 10
400 = k( )2
= k(.0025)

13. Estimating Loss Function
L(x)
400
10.05 10
400 = k( )2
= k(.0025)
k = 160,000

14. Example 2
Suppose we have a 1 year warranty to a watch. Suppose also that the life of the watch is exponentially distributed with a mean of 1.5 years. The warranty costs to replace the watch if it fails within one year is $25. Estimate the loss function.

15. Example 2
L(x)
f(x) 25 1
1.5
25 = k( )2
k = 100

16. Example 2
L(x)
f(x) 25 1
1.5
25 = k( )2
k = 100

17. Single Sided Loss Functions
Smaller is better
L(x) = kx2
Larger is better
L(x) = k(1/x2)

18. Example 2
L(x)
f(x) 25 1

19. Example 2
L(x)
f(x) 25 1
25 = k(1)2
k = 25

20. Expected Loss

21. Expected Loss

22. Expected Loss

23. Expected Loss

24. Expected Loss

25. Expected Loss Recall, X f(x) with finite mean  and variance 2.
E[L(x)] = E[ k(x - T)2 ]
= k E[ x2 - 2xT + T2 ]
= k E[ x2 - 2xT + T2 - 2x + 2 + 2x - 2 ]
= k E[ (x2 - 2x+ 2) - 2 + 2x - 2xT + T2 ]
= k{ E[ (x - )2 ] + E[ - 2 + 2x - 2xT + T2 ] }

26. Expected Loss
E[L(x)] = k{ E[ (x - )2 ] + E[ - 2 + 2x - 2xT + T2 ] }
Recall,
Expectation is a linear operator and
E[ (x - )2 ] = 2
E[L(x)] = k{2 - E[ 2 ] + E[ 2x - E[ 2xT ] + E[ T2 ] }

27. Expected Loss Recall, E[ax +b] = aE[x] + b = a + b
E[L(x)] = k{2 - 2 + 2 E[ x - 2T E[ x ] + T2 }
=k {2 - 2 + 22 - 2T + T2 }

28. Expected Loss Recall, E[ax +b] = aE[x] + b = a + b
E[L(x)] = k{2 - 2 + 2 E[ x - 2T E[ x ] + T2 }
=k {2 - 2 + 22 - 2T + T2 }
=k {2 + ( - T)2 }

29. Expected Loss Recall, E[ax +b] = aE[x] + b = a + b
E[L(x)] = k{2 - 2 + 2 E[ x - 2T E[ x ] + T2 }
=k {2 - 2 + 22 - 2T + T2 }
=k {2 + ( - T)2 }
= k { 2 + ( x - T)2 } = k (2 +D2 )

30. Example Since for our piston example, x = T, D2 = (x - T)2 = 0
L(x) = k2

31. Example (Piston Diam.)

32. Example (Sony) T x E[LUS(x)] = 0.16 * 8.33 = $1.33
LSL
USL x T
U.S. Plant (2 = 8.33)
Japanese Plant (2 = 2.78)
E[LUS(x)] = 0.16 * 8.33 = $1.33
E[LJ(x)] = 0.16 * 2.78 = $0.44

33. Tolerance (Pistons) Recall, 10 400 = k(10.05 - 10.00)2 = k(.0025)
L(x)
400
10.05 10
400 = k( )2
= k(.0025)
k = 160,000

34. Tolerance
L(x)
Suppose repair for an engine which will fail during warranty can be made for only $200
400
200
10.05 10
LSL
USL

35. Tolerance
L(x)
Suppose repair for an engine which will fail during warranty can be made for only $200
200 =160,000(tolerance)2
400
200
10.05 10
LSL
USL

36. Tolerance
L(x)
Suppose repair for an engine which will fail during warranty can be made for only $200
200 = 160,000(tolerance)2
tolerance = (200/160,000)1/2
= .0354
400
200
10.05 10
LSL
USL

37. Statistical Thinking
All work occurs in a system of interconnected processes
All process have variation
Understanding variation and reducing variation are important keys to success

38. Variability A certain amount of variability is inescapable
Therefore, no two products are identical
The larger the variability, the greater the probability that the customer will perceive its existence

39. Sources of Variability
Include:
Differences in materials
Differences in the performance and operation of the manufacturing equipment
Differences in the way the operators perform their tasks
Leave blank for students, make into a call-out

40. Variability and Statistics
Variability is difference from the target
Characteristics of quality must be measurable
Therefore,
Variability is described in statistical terms
We will use statistical methods in our quality improvement activities
Recall customer requirements vs. technical requirements
Use management tools as first resort, save power tools for power situations

41. Breaking down workplace barriers
Train people in the basics of the tools and methods
Deal with those who look down on or fear numerical methods
Encourage a data-driven culture of decision making
Lose the jargon
No “lying with statistics”
Why do statistics sometimes fail in the workplace?
Lack of knowledge about the tools
General disdain for all things mathematical
Cultural barriers in a company
Statistical specialists have trouble communicating
Statistics generally are poorly taught, emphasizing mathematical development rather than application
People have a poor understanding of the scientific method
Organizations lack patience in collecting data. All decisions have to be made “yesterday”
Statistics are viewed as something to buttress an already-held opinion
People fear using statistics
Most people don’t understand random variation
Statistical tools often are reactive and focus on effects rather than causes

42. Recall: Types of Errors
Type I error
Producers risk
Probability that a good product will be rejected
Type II error
Consumers risk
Probability that a nonconforming product will be available for sale
Type III error
Asking the wrong question

43. 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

44. Terms Specifications Target (or Nominal) Value
Upper Specification Limit
Lower Specification Limit
Random Variation
Non-random Variation
Process stability
Specs (manufacturing): desired measurements for the quality characteristics on the components and sub-assemblies that make up the product as well as the desired values for the quality characteristics of the final product
Specs (service): typically in terms of the max amount of time to process an order or to provide a particular service
Target: goal or desired value
USL: largest allowable value for the given quality characteristic
LSL: smallest allowable value for the given quality characteristic
Random variation
Centered around the mean
Consistent amount of dispersion
Nonrandom variation
“Special Causes”
Results from some event
Dispersion and average of the process are changing
Process that is not repeatable
Process stability
Random Variation
Not nonrandom variation
Process Charts

45. Terms Nonconforming: failure to meet one or more of the specifications
Nonconformity: a specific type of failure
Defect: a nonconformity serious enough to significantly affect the safe or effective use of the produce or completion of the service

46. Nonconforming vs. Defective
A nonconforming product is not necessarily unfit for use
A nonconforming product is considered defective if if it has one or more defects
Example: a detergent may have a concentration of an active ingredient that is below the lower spec limit but still performs acceptably

47. Classroom Exercise For a product or service in your job:
Name a quality characteristic
Give an example of a nonconformity that is not a defect
Give an example of a defect

48. Types of Inspection Receiving In Process Final None
One Hundred Percent
Acceptance Sampling
Difference between spot check and acceptance= in acceptance, inspectors that a statistically determined random sample and use a decision rule to determine acceptance or rejection of the lot based on the observed number of nonconforming items. The decision rule is called lot sentencing.
Acceptance sampling was developed after SPC

49. Quality Design & Process Variation
Lower Spec
Limit
Upper Spec
Limit
Acceptance
Sampling 60 80
100
120
140
Statistical
Process
Control 60
140
Experimental
Design
140 60

50. Variation and Control
A process that is operating with only common causes of variation is said to be in statistical control.
A process operating in the presence of special or assignable cause is said to be out of control.

51. Finding Trends and Special Causes
Inspection does not tell you about a problem until it becomes a problem
We need a mechanism to help us spot special causes when they occur
We need mechanism to help us determine when we have a trend in the data

52. Statistical Process Control
Originally developed by Walter Shewhart in 1924 at the Bell Telephone Laboratories
Late 1920s, Harold Dodge and Harry Romig developed statistically based acceptance sampling
Not recognized by industry until after World War II
Shewhart was a contemporary of Deming and Juran
Dodge and Romig also of Bell Labs
statistically based acceptance sampling is an alternative to 100% inspection

53. Definition Statistical Process Control (SPC):
“a methodology for monitoring a process to identify special causes of variation and signal the need to take corrective action when it is appropriate”
(Evans and Lindsay)

54. The quality function is not responsible for quality!

55. Statistical Process Control Tools
The magnificent seven
The tool most often associated with Statistical Process Control is Control Charts
We will spend the next few class on the different types of control charts

56. Common Causes
Special Causes

57. Histograms do not take into account changes over time.
Control charts can tell us when a process changes

58. Control Chart Applications
Establish state of statistical control
Monitor a process and signal when it goes out of control
Determine process capability
Note: Control charts will only detect the presence of assignable causes. Management, operator, and engineering action is necessary to eliminate the assignable cause.

59. Capability Versus Control
Capable
Not Capable
In Control Out of Control
IDEAL

60. Developing Control Charts
Prepare
Choose measurement
Determine how to collect data, sample size, and frequency of sampling
Set up an initial control chart
Collect Data
Record data
Calculate appropriate statistics
Plot statistics on chart

62. Types of Sampling Random Samples Systematic Samples Rational subgroups
Each piece has an equal chance of being selected for inspection
Systematic Samples
According to time or sequence
Rational subgroups
A group of data that is logically homogeneous
Computing variation between subgroups
Rational Subgroups
Subgroups or samples should be selected so that if assignable causes are present, the chance for differences between subgroups will be maximized while the chance for differences due to these assignable causes within a subgroup will be minimized.

63. Approaches to Rational Subgrouping
Each sample consists of units that were produced at the same time (or as closely together as possible)
Used when the primary purpose of the control chart is to detect process shifts
Each sample consists of units of product that are representative of all units that have been produced since the last sample was taken
Used when the purpose of the control chart is to make lot sentencing decisions

64. Next Steps Determine trial control limits
Center line (process average)
Compute UCL, LCL
Analyze and interpret results
Determine if in control
Eliminate out-of-control points
Re-compute control limits as necessary

65. Warning Limits on Control Charts
Some suggest using two sets of limits on control charts:
Action limits
Set at 3-sigma
When a point plots outside of this limit, a search for an assignable cause is made and any necessary corrective action is taken
Warning limits
Set at 2-sigma
When one or more points fall in between the warning and action limits or very close to the warning limit, be suspicious

67. Typical Out-of-Control Patterns
Point outside control limits
Hugging the center line
Hugging the control limits
Instability
Sudden shift in process average
Cycles
Trends

68. Shift in Process Average

69. Identifying Potential Shifts

70. Cycles

71. Trend

72. Western Electric Sensitizing Rules:
One point plots outside the 3-sigma control limits
Two of three consecutive points plot outside the 2-sigma warning limits
Four of five consecutive points plot beyond the 1-sigma limits
A run of eight consecutive points plot on one side of the center line
Western Electric Handbook, 1956

73. Additional sensitizing rules:
Six points in a row are steadily increasing or decreasing
Fifteen points in a row with 1-sigma limits (both above and below the center line)
Fourteen points in a row alternating up and down
Eight points in a row in both sides of the center line with none within the 1-sigma limits
An unusual or nonrandom pattern in the data
One of more points near a warning or control limit

74. Classroom Exercise
In small groups, choose two of the sensitizing rules. For each of your two rules, make up a (reasonable) situation where that rule would catch a problem and a (reasonable) situation where that rule might falsely identify a problem.

75. Final Steps Use as a problem-solving tool Compute process capability
Continue to collect and plot data
Take corrective action when necessary
Compute process capability

76. Capability vs. Stability
A process is capable if individual products consistently meet specification
A process is stable only if common variation is present in the process

77. Process Capability Calculations
PCR=(USL-LSL)/(6*sigma) where sigma=R-bar/d2

78. Process Capability
The ability of the process to perform at the required level for the given quality characteristic
Two of the methods for determining:
Find the probability that the process will produce a part below the LSL plus the probability that the process will produce a part above the USL
Process Capability Ratio
PCR = CP = (USL – LSL) / 6σ

79. Process Capability Ratio Note
There are many ways we can estimate the capability of our process
If σ is unknown, we can replace it with one of the following estimates:
The sample standard deviation S
R-bar / d2

80. PCR and One Sided Specifications
Upper specification only
CPU = (USL – μ) / 3σ
Lower specification only
CPL = (μ – LSL) / 3σ
We can use the same estimate for σ

81. PCR and an Off-Center Process
CPK = min (CPU, CPL)
Generally, if CP = CPK, then the process is centered at the midpoint of the specifications
If CP ≠ CPK, then the process is off-center
PCR=potential capability
PCR(k) = actual capability

82. Commonly Used Control Charts
Variables data
x-bar and R-charts
x-bar and s-charts
Charts for individuals (x-charts)
Attribute data
For “defectives” (p-chart, np-chart)
For “defects” (c-chart, u-chart)

83. Control Charts         x x  n
We assume that the underlying distribution is normal
with some mean  and some constant but unknown
standard deviation .
Let n x x   i n i  1

84. Distribution of x s s = n x  N (  ,  )
Recall that x is a function of random variables,
so it also is a random variable with its own
distribution. By the central limit theorem, we
know that
where, x  N (  ,  ) x s s = x x n

85. Control Charts x

      x
x 
x

86. Control Charts  3  UCL & LCL Set at
x 
x
LCL
UCL & LCL Set at
Problem: How do we estimate  &  ?  3  x

87. Control Charts m å x i x = i = 1 m m m å R R = i = 1 = f ( s ) s m

88. å Control Charts å x x = ® m m R R = = f ( s ) ® s m UCL = D R LCL = Di x = i = 1 m m m å R R = i = 1 = f ( s ) s m
UCL = x + A R
UCL = D R x 2 R 4
LCL = x - A R
LCL = D R x 2 R 3

90. Example
Suppose specialized o-rings are to be manufactured at .5 inches. Too big and they won’t provide the necessary seal. Too little and they won’t fit on the shaft. Twenty samples of 2 rings each are taken. Results follow.

92. X-Bar Control Charts X-bar charts can identify special causes of
variation, but they are only useful if the process
is stable (common cause variation).

93. Control Limits for Range
UCL = D4R = 3.268*.002 = .0065
LCL = D3 R = 0

94. Special Variables Control Charts
x-bar and s charts
x-chart for individuals

95. X-bar and S charts
Allows us to estimate the process standard deviation directly instead of indirectly through the use of the range R
S chart limits:
UCL = B6σ = B4*S-bar
Center Line = c4σ = S-bar
LCL = B5σ = B3*S-bar
X-bar chart limits
UCL = X-doublebar +A3S-bar
Center line = X-doublebar
LCL = X-doublebar -A3S-bar

97. X-chart for individuals
UCL = x-bar + 3*(MR-bar/d2)
Center line = x-bar
LCL = x-bar - 3*(MR-bar/d2)

99. Next Class Homework Topic Preparation
Ch. 11 (12) Disc. Questions 5, 6, 7
Ch. 11 (12) Problems 5
Topic
Control Charts, Part II
Preparation
Chapter 12 (13)