Chapter 9A
Process Capability and SPC
9A-2
OBJECTIVES
• Process Variation
• Process Capability
• Process Control Procedures
–
–
Variable data
Attribute data
9A-3
Basic Forms of Variation
Assignable variation
is caused by
factors that can be
clearly identified
and possibly
managed
Common variation is
inherent in the
production process
Example: A poorly trained
employee that creates
variation in finished
product output.
Example: A molding
process that always leaves
“burrs” or flaws on a
molded item.
9A-4
Taguchi’s View of Variation
Traditional view is that quality within the LS and US is good
and that the cost of quality outside this range is constant, where
Taguchi views costs as increasing as variability increases, so seek
to achieve zero defects and that will truly minimize quality costs.
High
High
Incremental
Cost of
Variability
Incremental
Cost of
Variability
Zero
Zero
Lower Target
Spec
Spec
Upper
Spec
Traditional View
Lower
Spec
Target
Spec
Upper
Spec
Taguchi’s View
Process Capability
• Process limits
• Specification limits
• How do the limits relate to one
another?
9A-6
Process Capability Index, Cpk
Capability Index shows
how well parts being
produced fit into design
limit specifications.
 X  LTL
UTL - X 

C pk = min 
or
3 
 3
As a production process
produces items small
shifts in equipment or
systems can cause
differences in
production
performance from
differing samples.
Shifts in Process Mean
9A-7
Process Capability – A Standard Measure of How
Good a Process Is.
A simple ratio:
Specification Width
_________________________________________________________
Actual “Process Width”
Generally, the bigger the better.
9A-8
Process Capability
C pk
 X  LTL UTL  X 
 Min 
;

3 
 3
This is a “one-sided” Capability Index
Concentration on the side which is closest to
the specification - closest to being “bad”
9A-9
The Cereal Box Example
• We are the maker of this cereal. Consumer reports has
just published an article that shows that we frequently
have less than 16 ounces of cereal in a box.
• Let’s assume that the government says that we must be
within ± 5 percent of the weight advertised on the box.
• Upper Tolerance Limit = 16 + .05(16) = 16.8 ounces
• Lower Tolerance Limit = 16 – .05(16) = 15.2 ounces
• We go out and buy 1,000 boxes of cereal and find that
they weight an average of 15.875 ounces with a standard
deviation of .529 ounces.
9A-10
Cereal Box Process Capability
• Specification or Tolerance
Limits
– Upper Spec = 16.8 oz
– Lower Spec = 15.2 oz
• Observed Weight
– Mean = 15.875 oz
– Std Dev = .529 oz
C pk
C pk
 X  LTL UTL  X 
 Min 
;

3 
 3
15.875  15.2 16.8  15.875 
 Min 
;

3
(.
529
)
3
(.
529
)


C pk  Min.4253; .5829
C pk  .4253
9A-11
What does a Cpk of .4253 mean?
• An index that shows how well the units
being produced fit within the
specification limits.
• This is a process that will produce a
relatively high number of defects.
• Many companies look for a Cpk of 1.3
or better… 6-Sigma company wants
2.0!
9A-12
Types of Statistical Sampling
• Attribute (Go or no-go
information)
–
–
–
Defectives refers to the
acceptability of product across a
range of characteristics.
Defects refers to the number of
defects per unit which may be
higher than the number of
defectives.
p-chart application
• Variable (Continuous)
–
–
Usually measured by the mean and
the standard deviation.
X-bar and R chart applications
Statistical
Process Normal Behavior
Control
(SPC) Charts
9A-13
UCL
LCL
1
2
3
4
5
6
Samples
over time
UCL
Possible problem, investigate
LCL
1
2
3
4
5
6
Samples
over time
UCL
Possible problem, investigate
LCL
1
2
3
4
5
6
Samples
over time
9A-14
Control Limits are based on the Normal Curve
x
m
-3
-2
-1
Standard
deviation
units or “z”
units.
0
1
2
3
z
9A-15
Control Limits
We establish the Upper Control Limits
(UCL) and the Lower Control Limits
(LCL) with plus or minus 3 standard
deviations from some x-bar or mean
value. Based on this we can expect
99.7% of our sample observations to
fall within these limits.
99.7%
LCL
UCL
x
9A-16
Example of Constructing a p-Chart:
Required Data
Sample
No. of
No.
Samples
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
100
100
100
100
100
100
100
100
100
100
100
100
100
100
100
Number of
defects found
in each sample
4
2
5
3
6
4
3
7
1
2
3
2
2
8
3
9A-17
Statistical Process Control Formulas:
Attribute Measurements (p-Chart)
T o ta l N u m b e r o f D e fe c tiv e s
Given: p =
T o ta l N u m b e r o f O b s e rv a tio n s
sp =
p (1 - p)
n
Compute control limits:
UCL = p + z sp
LCL = p - z sp
9A-18
Example of Constructing a p-chart: Step 1
1. Calculate the
sample proportions,
p (these are what
can be plotted on the
p-chart) for each
sample
Sample
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
n Defectives
100
4
100
2
100
5
100
3
100
6
100
4
100
3
100
7
100
1
100
2
100
3
100
2
100
2
100
8
100
3
p
0.04
0.02
0.05
0.03
0.06
0.04
0.03
0.07
0.01
0.02
0.03
0.02
0.02
0.08
0.03
9A-19
Example of Constructing a p-chart: Steps 2&3
2. Calculate the average of the sample proportions
55
p =
1500
= 0.036
3. Calculate the standard deviation of the
sample proportion
sp =
p (1 - p)
=
n
.036(1- .036)
= .0188
100
9A-20
Example of Constructing a p-chart: Step 4
4. Calculate the control limits
UCL = p + z sp
LCL = p - z sp
.036  3(.0188)
UCL = 0.0924
LCL = -0.0204 (or 0)
9A-21
Example of Constructing a p-Chart: Step 5
5. Plot the individual sample proportions, the average
of the proportions, and the control limits
0.16
0.14
0.12
UCL
0.1
p 0.08
0.06
0.04
0.02
LCL
0
1
2
3
4
5
6
7
8
9
O b servation
10
11
12
13
14
15
9A-22
Example of x-bar and R Charts:
Required Data
Sample
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Obs 1
10.68
10.79
10.78
10.59
10.69
10.75
10.79
10.74
10.77
10.72
10.79
10.62
10.66
10.81
10.66
Obs 2
10.689
10.86
10.667
10.727
10.708
10.714
10.713
10.779
10.773
10.671
10.821
10.802
10.822
10.749
10.681
Obs 3
10.776
10.601
10.838
10.812
10.79
10.738
10.689
10.11
10.641
10.708
10.764
10.818
10.893
10.859
10.644
Obs 4
10.798
10.746
10.785
10.775
10.758
10.719
10.877
10.737
10.644
10.85
10.658
10.872
10.544
10.801
10.747
Obs 5
10.714
10.779
10.723
10.73
10.671
10.606
10.603
10.75
10.725
10.712
10.708
10.727
10.75
10.701
10.728
9A-23
Example of x-bar and R charts: Step 1. Calculate sample means, sample
ranges, mean of means, and mean of ranges.
Sample
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Obs 1
10.68
10.79
10.78
10.59
10.69
10.75
10.79
10.74
10.77
10.72
10.79
10.62
10.66
10.81
10.66
Obs 2
10.689
10.86
10.667
10.727
10.708
10.714
10.713
10.779
10.773
10.671
10.821
10.802
10.822
10.749
10.681
Obs 3
10.776
10.601
10.838
10.812
10.79
10.738
10.689
10.11
10.641
10.708
10.764
10.818
10.893
10.859
10.644
Obs 4
10.798
10.746
10.785
10.775
10.758
10.719
10.877
10.737
10.644
10.85
10.658
10.872
10.544
10.801
10.747
Obs 5
10.714
10.779
10.723
10.73
10.671
10.606
10.603
10.75
10.725
10.712
10.708
10.727
10.75
10.701
10.728
Averages
Avg
10.732
10.755
10.759
10.727
10.724
10.705
10.735
10.624
10.710
10.732
10.748
10.768
10.733
10.783
10.692
Range
0.116
0.259
0.171
0.221
0.119
0.143
0.274
0.669
0.132
0.179
0.163
0.250
0.349
0.158
0.103
10.728 0.220400
9A-24
Example of x-bar and R charts: Step 2. Determine Control Limit Formulas
and Necessary Tabled Values
x Chart Control Limits
UCL = x + A 2 R
LCL = x - A 2 R
R Chart Control Limits
UCL = D 4 R
LCL = D 3 R
From Exhibit 9A.6
n
2
3
4
5
6
7
8
9
10
11
A2
1.88
1.02
0.73
0.58
0.48
0.42
0.37
0.34
0.31
0.29
D3
0
0
0
0
0
0.08
0.14
0.18
0.22
0.26
D4
3.27
2.57
2.28
2.11
2.00
1.92
1.86
1.82
1.78
1.74
9A-26
Example of x-bar and R charts: Steps 3&4. Calculate x-bar Chart and Plot
Values
UCL = x + A 2 R  10.728 - .58(0.2204 ) = 10.856
LCL = x - A 2 R  10.728 - .58(0.2204 ) = 10.601
10.900
UCL
10.850
M ean s
10.800
10.750
10.700
10.650
10.600
LCL
10.550
1
2
3
4
5
6
7
8
Sam ple
9
10
11
12
13
14
15
9A-27
Example of x-bar and R charts: Steps 5&6. Calculate R-chart and Plot
Values
UCL = D 4 R  ( 2.11)(0.2204)  0.46504
LCL = D3 R  (0)(0.2204)  0
0 .8 0 0
0 .7 0 0
0 .6 0 0
0 .5 0 0
R
UCL
0 .4 0 0
0 .3 0 0
0 .2 0 0
0 .1 0 0
LCL
0 .0 0 0
1
2
3
4
5
6
7
8
S a m p le
9
10
11
12
13
14
15
9A-29
Question Bowl
A methodology that is used to
show how well parts being
produced fit into a range
specified by design limits is
which of the following?
a. Capability index
b. Producer’s risk
c. Consumer’s risk
d. AQL
e. None of the above
Answer: a. Capability index
9A-30
Question Bowl
You want to prepare a p chart
and you observe 200
samples with 10 in each,
and find 5 defective units.
What is the resulting
“fraction defective”?
a. 25
b. 2.5
c. 0.0025
d. 0.00025
e. Can not be computed on
data above
Answer: c. 0.0025 (5/(2000x10)=0.0025)
9A-31
Question Bowl
You want to prepare an x-bar chart.
If the number of observations in
a “subgroup” is 10, what is the
appropriate “factor” used in the
computation of the UCL and
LCL?
a. 1.88
b. 0.31
c. 0.22
d. 1.78
e. None of the above
Answer: b. 0.31
9A-32
Question Bowl
You want to prepare an R
chart. If the number of
observations in a
“subgroup” is 5, what is the
appropriate “factor” used in
the computation of the
LCL?
a. 0
b. 0.88
Answer: a. 0
c. 1.88
d. 2.11
e. None of the above
9A-33
Question Bowl
You want to prepare an R chart.
If the number of
observations in a
“subgroup” is 3, what is the
appropriate “factor” used in
the computation of the UCL?
a. 0.87
b. 1.00
c. 1.88
d. 2.11
e. None of the above
Answer: e. None of the above
9A-34
End of Chapter 9A
1-34