Transcript Document

ENGM 620: Quality Management
Session 8 – 23 October 2012
• Control Charts, Part I
– Variables
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
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
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
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: 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
Types of Errors
Truth
HO
HO
HA
No
Error
Type II
b
Type I
a
No
Error
Accept
HA
A Parable
Where should we put the additional armor?
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
Terms
•
•
•
•
•
•
•
Specifications
Target (or Nominal) Value
Upper Specification Limit
Lower Specification Limit
Random Variation
Non-random Variation
Process stability
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
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
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
Types of Inspection
• Receiving
• In Process
• Final
• None
• One Hundred Percent
• Acceptance Sampling
Quality Design & Process Variation
Lower Spec
Limit
Upper Spec
Limit
Acceptance
Sampling
60
80
100
120
140
Statistical
Process
Control
60
140
Experimental
Design
60
140
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.
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
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
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)
Statistical Process Control
Tools
• The magnificent seven
• The tool most often associated with
Statistical Process Control is Control
Charts
Common
Causes
Special
Causes
Histograms do
not take into
account
changes over
time.
Control charts
can tell us when a
process changes
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.
Capability Versus Control
Control
Capability
Capable
Not Capable
In Control
IDEAL
Out of Control
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)
Control Charts
  




We assume that the underlying distribution is normal
with some mean  and some constant but unknown
standard deviation .
Let
n
xi
x
i 1 n
Distribution of x
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
x  N (  , x )
where,
x 
x
n
Control Charts
x
  




x
x

x
Control Charts
x
UCL

x
LCL
UCL & LCL Set at 3 x
Problem: How do we estimate  &  ?
Control Charts
x
m
x
i 1
i
m

m
R
R
i 1
m
 f ( )  
Control Charts
x
m
x
i 1
i
m

m
R
R
i 1
m
 f ( )  
UCLx  x + A2 R
UCLR  D4 R
LCLx  x  A2 R
LCLR  D3 R
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.
Part
No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Measurements
1
2
0.502 0.504
0.495 0.497
0.492 0.496
0.501 0.498
0.507 0.508
0.504 0.504
0.497 0.496
0.493 0.496
0.502 0.501
0.498 0.500
0.505 0.507
0.502 0.499
0.495 0.497
0.499 0.496
0.503 0.507
0.507 0.509
0.503 0.501
0.497 0.493
0.504 0.508
0.505 0.503
Avg =
Std. =
x
0.503
0.496
0.494
0.500
0.508
0.504
0.497
0.495
0.502
0.499
0.506
0.501
0.496
0.498
0.505
0.508
0.502
0.495
0.506
0.504
0.501
x
0.0047
R
0.002
0.002
0.004
0.003
0.001
0.000
0.001
0.003
0.001
0.002
0.002
0.003
0.002
0.003
0.004
0.002
0.002
0.004
0.004
0.002
0.002
R
R Chart
UCL
R
0.0077 0.002
0.0077 0.002
0.0077 0.004
0.0077 0.003
0.0077 0.001
0.0077 0.000
0.0077 0.001
0.0077 0.003
0.0077 0.001
0.0077 0.002
0.0077 0.002
0.0077 0.003
0.0077 0.002
0.0077 0.003
0.0077 0.004
0.0077 0.002
0.0077 0.002
0.0077 0.004
0.0077 0.004
0.0077 0.002
X Chart
UCL
0.5052
0.5052
0.5052
0.5052
0.5052
0.5052
0.5052
0.5052
0.5052
0.5052
0.5052
0.5052
0.5052
0.5052
0.5052
0.5052
0.5052
0.5052
0.5052
0.5052
LCL
0.4964
0.4964
0.4964
0.4964
0.4964
0.4964
0.4964
0.4964
0.4964
0.4964
0.4964
0.4964
0.4964
0.4964
0.4964
0.4964
0.4964
0.4964
0.4964
0.4964
Xbar
0.503
0.496
0.494
0.500
0.508
0.504
0.497
0.495
0.502
0.499
0.506
0.501
0.496
0.498
0.505
0.508
0.502
0.495
0.506
0.504
X-Bar Control Charts
X-Bar Chart
0.510
x
0.505
0.500
0.495
0.490
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
X-bar charts can identify special causes of
variation, but they are only useful if the process
is stable (common cause variation).
Control Limits for Range
UCL = D4R = 3.268*.002 = .0065
LCL = D3 R = 0
R Chart
0.010
Range
0.008
0.006
0.004
0.002
0.000
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19 20
Observation
Why Monitor Both Process Mean and Process
Variability?
Process Over Time
Process Doing OK
Lower
Specification
Limit
Control Charts
Upper
Specification
Limit
X-bar
R
X-bar
R
X-bar
R
Mean shift in process
Increase in process variance
37
Teminology
• Causes of Variation:
– Assignable Causes
• Keep the process from
operating predictably
• Things that we can do
something about
– Common / Chance
Causes
• Meaning of Control:
– In Specification
• Meets customer
constraints on product
– In Statistical Control
• No Assignable
Causes of variation
present in the process
• Random, inherent
variation in the process
38
Shift in Process Average
Identifying Potential Shifts
Cycles
Trend
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
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
Special Variables Control
Charts
• x-bar and s charts
• x-chart for individuals
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
X-chart for individuals
• UCL = x-bar + 3*(MR-bar/d2)
• Center line = x-bar
• LCL = x-bar - 3*(MR-bar/d2)
Next Class
• Homework
– Ch. 11 Disc. Questions 5, 7
– Ch. 11 Problems 6, 11
• Preparation
– Chapter 11, Process Capability