Transcript QualityControl_basicconcepts

Department of Engineering Management, Information and Systems
EMIS 7370/5370 STAT 5340
Probability and Statistics for Scientists and Engineers
An Application of Probability & Statistics
Statistical Quality Control
UPDATED 11/20/06
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering
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Statistical Quality Control
Statistical Quality Control is an application of
probabilitistic and statistical techniques to quality
control
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Statistical Quality Control - Elements
Analysis
of process
capability
Process
improvement
Statistical
process
control
Acceptance
sampling
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Statistical Quality Control - Basic Concepts
• Quality begins with customer requirements
• Quality must be designed in. It cannot be inspected
in!
• Quality depends on:
Parts selection and procurement
Material
Manufacturing/production processes
Logistic processes
.
.
.
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Statistical Quality Control - What is quality?
• Quality is meeting the customer’s needs over the
life cycle of the product at the best value to the
customer
• Quality has many dimensions
Reliability
Maintainability
Performance
Durability
Conformance (to requirements and
expectations)
.
.
.
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Obstacles to Quality Improvement
• Making it happen
- 99% agree that management is the problem
not the workers
- 35% of the problem is ‘not invented here’
syndrome
getting their attention and education
resistance to change
- 15% gaining management commitment
- 14% communication
getting the word out within the company
• Failure of management to understand ‘variation’
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Statistical Process Control - Definition
The application of statistical techniques
is to understand and analyze the variation in a
process.
- Joseph Juran
Quality Control Handbook
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The Situation
In many situations, our knowledge is limited to the
information that can be obtained from data that has
been obtained or that will be obtained
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The Problem
The challenge is to obtain the maximum information
from the data and to arrive at the most accurate
conclusions
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Nature of Data
Most data are characterized by variation, as opposed
to deterministic, due to variation in
• Processes and materials
• Product/Manufacturing
• Inspection & Measurement
• Operation
• Environment
• etc
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Need
Methods and techniques are needed for analysis of
data that account for
• Variation in the data
• Uncertainty in conclusion
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Statistics
• Statistics is the science of analyzing data and
drawing conclusions
• Statistical methods and techniques that provide
tools for:
- experimental design
- analysis of data
- making inferences
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Statistical Process Control (SPC)
• SPC is a powerful collection of problem-solving
tools useful in achieving process stability and
improving capability through the reduction of
variability.
• SPC can be applied to any process
• Seven major tools
1. Histogram or stem and leaf display
2. Check sheet
3. Pareto chart
4. Cause and effect diagram
5. Defect concentration diagram
6. Scatter diagram
7. Control chart
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Statistical Process Control
Causes of Variation
Assignable (special) - Intermittent sources
of variation that are unpredictable. Signaled
by violation of Western Electric rules
Common (natural) - Sources of variation
always present affecting all output from a
process
Only management can affect common causes of
variation
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Statistical Process Control - Histograms
Histograms - Questions to ask
• What is the shape of distribution?
• What would you expect shape to be?
• If computer generated, is data really normal?
• Is variation acceptable?
• Is the centering acceptable?
• Did you generate a histogram with and without
outlier points?
• Did you include specification limits and process
limits on the histogram?
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Statistical Process Control - Histograms
LSL
USL
• The shape shows the nature of the distribution of the
data
• The central tendency (average) and variability are
easily seen
• Specification limits can be used to display the
capability of the process
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Process Capability
Refers to the uniformity of the process.
Variability in the process is a measure of the
uniformity of the output.
- Instantaneous variability is the natural or
inherent variability at a specified time
- Variability over time
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Process Capability
A critical performance measure that addresses
process results relative to process/product
specifications.
A capable process is one for which the process
outputs meet or exceed expectation.
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Measures of Process Capability
Customary to use the six sigma spread in the
distribution of the product quality characteristic
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Key Points
The proportion of the process output that will fall
outside the natural tolerance limits.
• Is 0.27% (or 2700 nonconforming parts per million)
if the distribution is normal
• May differ considerably from 0.27% if the
distribution is not normal
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Process Capability Measures or Indices
Process capability indices are used to measure the
process variability due to common causes present
in the process
• The Cp index
Inherent or potential measure of capability
Cp =
specification spread
process spread
• The CpK index
Realized or actual measure of capability
• Other indices
CpM, CpMK
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Measure of Potential Process Capability, Cp
• Cp measures potential or inherent capability of the
process, given that the process is stable
• Cp is defined as
, for two-sided
USL  LSL specifications
Cp 
6σ
and
C pL 
C pU
  LSL
, for lower
specifications only
3σ
USL  

3σ
, for upper
specifications only
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Measure of Potential Process Capability, CpK
• CpK measures realized process capability relative to
actual production, given a stable process
• CpK is defined as
C pK
   LSL USL   
 min 
,
3σ 
 3σ
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Statistical Process Control
ppm = parts per million
Interpretation
CpK < 1
= process not capable
1  CpK < 1.5
= process capable, monitor
frequently
CpK  1.5
= process capable, monitor
infrequently
Pareto CpK’s to attack worst problems
Can only convert CpK, Cp to ppm if distribution normal
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Interpretation of Cp
 1 
P    100%
C 
 p
is the percentage of the specification band used up
by the process
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Statistical Process Control
Impact of special causes on process capability
process
stable
process
unstable
time
time
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Statistical Process Flow Diagram
• Expresses detailed knowledge of the process
• Identifies process flow and interaction among the
process steps
• Identifies potential control points
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20
100
16
80
12
60
8
40
4
20
Cumulative percent
Number of occurrences
Statistical Process Control - Pareto Diagram
0
0
• Identifies the most significant problems to be worked
first
• Historically 80% of the problems are due to 20% of the
factors
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• Shows the vital few
Statistical Process Control - Scatter Plot
x
x
Temp.
x
x
x
x
x
x
x
x
x
x
x
x
Pressure
• Identifies the relationship between two variables
• A positive, negative, or no relationship can be easily
detected
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Background of Six Sigma
• Six Sigma is a business initiative first espoused by
Motorola in the early 1990’s.
• Six Sigma strategy involves the use of statistical
tools within a structured methodology for gaining
the knowledge needed to achieve better, faster, and less
expensive products and services than the competition.
• A Six Sigma initiative in a company is designed
to change the culture through breakthrough
improvement by focusing on out-of-the-box
thinking in order to achieve aggressive, stretch goals.
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Motorola’s Six Sigma Ten Steps
1.
2.
3.
4.
5.
6.
7.
Prioritize opportunities for improvement
Select the appropriate team
Describe the total process
Perform measurement system analysis
Identify and describe the potential critical process
Isolate and verify the critical processes
Perform process and measurement system
capability studies
8. Implement optimum operating conditions and
control methodology
9. Monitor processes over time/continuous
improvement
10. Reduce common cause variation toward
achieving six sigma
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Product Specification
Lower
Specification
Limit
Nominal
Specification
Upper
Specification
Limit
x
Target
(Ideal level for use in product)
Tolerance
(Product
characteristic)
(Maximum range of variation of the product
characteristic that will still work in the product.)
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Traditional US Approach to Quality
(Make it to specifications)
No-Good
Loss ($)
No-Good
Good
LSL
T
USL
x
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Setting Specification Limits on Discrete Components
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Variability Reduction
Variability reduction is a modern concept of design
and manufacturing excellence
• Reducing variability around the target value leads
to better performing, more uniform, defect-free
product
• Virtually eliminates rework and waste
• Consistent with continuous improvement concept
Don’t just conform to specifications
reject
accept
Reduce variability
around the target
reject
target
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True Impact of Product Variability
• Sources of loss
- scrap
- rework
- warranty obligations
- decline of reputation
- forfeiture of market share
• Loss function - dollar loss due to deviation of
product from ideal characteristic
• Loss characteristic is continuous - not a step
function.
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Representative Loss Function Characteristics
Loss
$
Loss
$
T
Loss
$
x
x
x
X nominal is best
X smaller is better
X larger is better
L = k (x - T)2
L = k (x2)
L = k (1/x2)
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Variability-Loss Relationship
LSL
USL
Target
Loss
Maximum
$ loss
per item
$ savings
due to
reduced
variability
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Loss Computation for Total Product Population
X nominal is best
L = k (x - T)2
1
f(x) 
e
σ 2π
 (x  T) 2
2σ 2
Loss
$
Loss
$
T
x
T
x
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Statistical Tolerancing - Convention
Normal Probability
Distribution
0.00135
LTL
-3
0.9973
Nominal

0.00135
UTL
+3
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Statistical Tolerancing - Concept
LTL
Nominal
UTL
x
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Caution
For a normal distribution, the natural tolerance
limits include 99.73% of the variable, or put
another way, only 0.27% of the process output will
fall outside the natural tolerance limits. Two points
should be remembered:
1. 0.27% outside the natural tolerances sounds
small, but this corresponds to 2700 nonconforming
parts per million.
2. If the distribution of process output is non
normal, then the percentage of output falling
outside   3 may differ considerably from 0.27%.
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Normal Distribution - Example
The diameter of a metal shaft used in a disk-drive unit
is normally distributed with mean 0.2508 inches and
standard deviation 0.0005 inches. The specifications
on the shaft have been established as 0.2500  0.0015
inches. We wish to determine what fraction of
the shafts produced conform to specifications.
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Normal Distribution - Example Solution
Pmeeting spec   P0.2485  x  0.2515
 Px  0.2515  P0.2485  x 
 0.2515 - 0.2508 
 0.2485 - 0.2508 
 
  

0.0005
0.0005




 1.40   4.60
 0.91924  0.0000
 0.91924
f(x)
0.2500
0.2485
LSL
nominal
0.2508
0.2515
USL
x
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Normal Distribution - Example Solution
Thus, we would expect the process yield to be
approximately 91.92%; that is, about 91.92% of
the shafts produced conform to specifications. Note
that almost all of the nonconforming shafts are too
large, because the process mean is located very
near to the upper specification limit. Suppose we
can recenter the manufacturing process, perhaps
by adjusting the machine, so that the process mean
is exactly equal to the nominal value of 0.2500.
Then we have
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Normal Distribution - Example Solution
P0.2485  x  0.2515  Px  0.2515  P0.2485  x 
 0.2515 - 0.2500 
 0.2485 - 0.2500 
 
  

0.0005
0.0005




 3.00   3.00
 0.99865  0.00135
 0.9973
f(x)
0.2485
LSL
nominal
0.2500
x
0.2515
USL
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Normal Distribution - Example
Using a normal probability distribution as a model
for a quality characteristic with the specification
limits at three standard deviations on either side of
the mean. Now it turns out that in this situation the
probability of producing a product within these
specifications is 0.9973, which corresponds to 2700
parts per million (ppm) defective. This is referred to
as three-sigma quality performance, and it actually
sounds pretty good. However, suppose we have a
product that consists of an assembly of 100
components or parts and all 100 parts must be
non-defective for the product to function
satisfactorily.
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Normal Distribution - Example
The probability that any specific unit of product is
non-defective is
0.9973 x 0.9973 x . . . x 0.9973
= (0.9973)100
= 0.7631
That is, about 23.7% of the products produced under
three sigma quality will be defective. This is not an
acceptable situation, because many high technology
products are made up of thousands of components.
An automobile has about 200,000 components and
an airplane has several million!
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Weibull Distribution - Example
The random variable X can modeled by a Weibull
distribution with  = ½ and  = 1000. The spec time limit is
set at x = 4000. What is the proportion of items not
meeting spec?
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Weibull Distribution - Example
The fraction of items not meeting spec is
PX  4000  1  P( X  4000)
 1  F(4000)
1/2
e
 4000 


1000


 e 2
 0.1353
That is, all but about 13.53% of the items will not meet
spec.
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Statistical Process Control - Control Charts
Interpretation based on Western Electric rules
1. Analyze the chart by separating it into equal zones
above and below the centerline
A
B
C
C
B
A
UCL
Centerline
LCL
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Statistical Process Control - Control Charts
2. A process is out of statistical control if:
(a) any point is above or below the control limits
(b) two out of three points in a row in zone A
or above
(c) four out of five points in a row in zone B
or above
(d) eight in a row in zone C or above
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Statistical Process Control - Control Charts
• In general specification limits should not be on
control charts
• Data must be displayed in time sequence
• Management controls the natural variation between
the control limits
• Do not tweak the process
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Statistical Process Control - Control Charts
x
UCL
x
x
x
x
x
x
x
x
x
x
CL
x
x
x
LCL
•
•
•
•
Helps reduce variability
Monitors performance over time
Allows process corrections to prevent rejections
Trends and out-of-control conditions are immediately
detected
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The Normal Distribution and the Control Charts
+3
Upper Control Limit
+2
+1

Center Line
Process Average
1
2
3
Lower Control Limit
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General Model for the Shewhart Control Chart
UCL = W + KW
Center Line = W
LCL = W - KW
where
W is a statistic that measures a quality
characteristic
W is the mean of W
W is the standard deviation of W
K is the distance of the control limits from the
center line, in multiples of W
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Types of Error that Can Occur When Using Control Charts
Actual State of Process
Control Chart Indicates
Only Common Causes
Out
of
Control
Control
Special Causes
A
B
False Alarm
Correct Decision
C
D
Correct Decision
Failure to Detect
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Use of Control Chart
• Control charts are a proven technique for improving
productivity
• Control charts are effective in defect prevention
• Control charts prevent unnecessary process adjustment
• Control charts provide diagnostic information.
• Control charts provide information about process
capability
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Types of Control Chart
Measurement
(variables)
Data
Counts
(attributes)
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Types of Control Chart
X
One
Moving
Range
Measurement
(variables)
X
Multiple
R
S
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Types of Control Chart
p
Defectives
np
Counts
(attributes)
c
Defects

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