Chapter 8
1
Chapter 8
2
Learning Objectives
Chapter 8
3
Process Capability
•
In short: The process must first be brought
into statistical control so its performance can
be predicted; then its capability to meet
specifications can be assessed.
Chapter 8
4
Process Control
Chapter 8
5
Process Capability
Chapter 8
6
Capable V.S. In-Control Process
• Capable process– A process that has aC pk
greater the 1. this means that the histogram
distribution and the normal curve are
contained within the actual specification
dimensions.
Chapter 8
7
Capable V.S. In-Control Process
• In-control process– A process that when
charted on a SPC chart and all measurements
are within the upper and lower control limits.
These measurements plots must be random in
nature, i.e., no points over or under the control
limits and do not violate the Western Electric
rules.
Chapter 8
8
Capable V.S. In-Control Process
• A process may be in-control, but not capable.
• A process may be in-control and capable.
Chapter 8
9
Process Capability
• “Natural Tolerance” (process spread) of a
process is the given range that the process
can maintain during ordinary operation
without the action of any person in making
process adjustments.
• Normally 3 Limits (6 sigma)
Chapter 8
10
How Do We Know a Process is
Capable?
CP
and CPK statistical process indexes
• These indexes tell if the process is capable of
producing features within the engineering
tolerances.
• This is essentially a grade for the process
where the higher number is better.
Chapter 8
11
The Cp Index
Cp 
Spec width
Upper spec  Lower spec

Natural tolerance
6
• Example: suppose the Spec width=10 units
and 6 =5 units
L.S.
6
Cp 
U.S.
10
 2 units
5
• This implies the spec is twice as wide as the
natural spread.
Chapter 8
12
CP and CPK
• Capability indexes are useful tools in the
analysis of capability data. The process must
be capable.
• CP is a capability index.
• Formula:
Chapter 8
Engineering Tol.
6 sigma
13
CP and CPK
• CPK is a capability index
• Formula:
USL  mean 
3 sigma
or
 mean  LSL 
3 sigma
what ever is less
USL=Upper SPEC Limit
LSL=Lower SPEC Limit
Chapter 8
14
CP and CPK
• Statistical Process Control– The use of
statistical methods or techniques such as
control charts to analyze a process or its
outputs and to take appropriate actions to
achieve and maintain a state of statistical
control and to improve the process capability.
Chapter 8
15
CP and CPK
• Characteristic– A distinguishing feature of a
process or its output on which variable data or
attribute data can be collected.
Chapter 8
16
CP and CPK
(Key)
• Dominant characteristics– Those
characteristics that greatly influence product
quality and are most important to product form,
fit, or function.
• Variable Data– Quantitative data, where
measurements are used for analysis.
Chapter 8
17
CP and CPK
• Attribute Data– Qualitative data that can be
counted for recording and analysis, such as the
presence of a required label, the installation of
all the required fasteners or the absence of
errors on an expense report. It can also be
characteristics that are inherently measurable
but where the results are recorded in a simple
yes/no fashion, such as: the acceptability of a
shaft diameter when checked on a go/no-go
gauge.
Chapter 8
18
CP and CPK
• Sigma– The standard deviation of a statistical
population.
Chapter 8
19
CP and CPK
• Capability– When the process average plus and
minus the 3 standard deviations (sigma) spread
of the distribution is contained within the
specification tolerance for variable data, or
when at least 99.73% of individuals produced
by the process meet specification for attribute
data, the process is considered to be capable.
Capability can only be determined after the
process is in statistical control.
Chapter 8
20
CP and CPK
• Capability Indices– A measure of the capability of
a process. Cp is the inherent capability of a
process and is defined as the ratio of the tolerance
to the process variation. Cpk is a measure of the
capability of a process in relation to the process
average. It is based on the distance between the
process average and the closest specification limit.
NORMAL DISTRIBUTION CURVE
68.26%
99.73%
Chapter 8
[SIGMA]
4
3
2
1
X
1
2
3
4
21
Process Fallout Table
Centered Process
Chapter 8
Process capability
ratio
0.50
0.75
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
2.00
Parts per million
defective
133,600.00
24,400.00
2,700.00
967.00
318.00
96.00
26.00
6.80
1.60
0.34
0.06
0.0018
22
Relationship of Specifications,
Control Limits, and Natural
Tolerance Limits in the Process
Chapter 8
23
Relationship of Specifications,
Control Limits, and Natural
Tolerance Limits in the Process
Chapter 8
24
Relationship of Specifications,
Control Limits, and Natural
Tolerance Limits in the Process
LSL
USL
Target
Cp
Cp
spread
Cp
+3σ
2σ
σ
-σ
-2σ
-3σ
C p is the ratio of spec. width to the natural variability present
in the process; i.e.
Chapter 8
Cp 
USL  LSL
6
25
Relationship of Specifications,
Control Limits, and Natural
Tolerance Limits in the Process
LSL
USL
Target
Center Line
(Avg)
C pk is the ratio of the distance between the process center
and the nearest spec. Limit to one half of process
3); i.e. C  min  X  LSL , USL  X 
variability
(


Chapter 8
pk


3
3


26
Chapter 8
27
Chapter 8
28
Chapter 8
29
Process Capability
When?
• Prior to taking delivery of new process
equipment
• Before approving newly-installed process
equipment for production use
• As production begins, to establish capability of
the equipment-tooling-material-operator
combination.
• On an ongoing basis to verify continuing
capability.
• When out-of-specification conditions are
found.
Chapter 8
30
Process Capability
The need for capability studies
• Products must meet specifications.
• It is more efficient for cost and timing to
produce to specification than to sort to
specification.
• Consistent performance requires inherent
capability.
• Process capability equals machine capability
plus process control.
• To verify capability, conduct capability studies.
Chapter 8
31
Process Capability Study Flowchart
決定關鍵品質特性
從製程中隨機抽30個樣
本
（短期製程能力分析）
計算平均數，標準差，
Cp/Cpk指標
是
是否通過常態性檢定
且Cpk1.23
決定SPC管制計畫中的4個
W
否
檢查並排除影響量測儀器及製程產
生變異之原因，考慮進行長程製程
能力分析
開始使用計量值或前置管制
圖
（長期製程能力分析）
使用管制圖時，每次取兩個
以上之樣本，至少收集25筆
（天）以上之資料
是
是否通過常態性檢定
且Cpk>1
檢查全距管制圖是否穩定，去掉脫
離管制界限之異常點
確認資料之正確性並使用目標值作為製程
平均，再一次估計Cp/Cpk指標
檢查平均管制圖是否穩定，
去掉異常點後再計算Cp及
Cpk之指標
決定SPC管制計畫，開始使
用 管制圖
Chapter 8
決定SPC管制計畫，開始
使用管制圖
否
是
是否通過常態性檢
定並接受Cpk>1
否
考慮以實驗設計的方法減少製程變
異，或更新設備
32
Procedures for Determining
Process Capability
1) Define the process
Is it a line, machine, operator, portion of
manufacturing, etc.?
Chapter 8
33
Procedures for Determining
Process Capability
2) Determine if Specifications are now being
met – conduct “Short Term” Process
Capability Study
a. Get current data
b. Calculate X and s (usually from grouped
data) to estimate the mean and standard
deviation of the process.
( n= 30)
Chapter 8
34
Procedures for Determining
Process Capability
c. Assume normality and estimate percent
meeting specifications. Determine from
frequency distribution if the assumption is
valid.
d. If specs are being met, generally go to
another problem. If specs are not being
met, proceed below.
Chapter 8
35
Procedures for Determining
Process Capability
3) Determine Inherent Variability, Using XR/Pre-Control Charts
a. Get consecutive samples, two (2) or more at
a time in a sample group. Get at least twenty
(20) groups over a shift or other short
production run during which operation
appears stable or without unusual problems.
Chapter 8
36
Procedures for Determining
Process Capability
b. Calculate average range R , and control limits
for R.
c. Discard any ranges outside control limits.
Assume that assignable cause would have
been identified and removed. If assumption
cannot be made, retain the range.
Chapter 8
37
Procedures for Determining
Process Capability
d. Recalculate average range and control limits.
e. Repeat steps c and d (removal of excessive
ranges and the recalculation of control limits)
until all ranges in control.
Chapter 8
38
Procedures for Determining
Process Capability
f. Estimate the standard deviation from the
last average range:
Estimate of σ= R /d2
g. Use midpoint of specification limits as the
process mean, assume normality, and
estimate percent meeting specs.
Chapter 8
39
Procedures for Determining
Process Capability
h. If specs can be met, investigate process to
determine why specs are not being met.
Chapter 8
40
Procedures for Determining
Process Capability
i. If specs cannot be met, consider
management alternatives:
Change specs
Change process
Make best of it
Drop product
j. Set up X -R charts for future control.
Chapter 8
41
Chapter 8
42
SPC Control Plan
Sheet
(1)
Supplier
(2) Supplier No.
Address
(3) Supplier Representative
City/State
(4)
Phone No.
(5)
of
.
(6) Control Characteristics
A.
F.
B.
G.
C.
H.
D.
I.
E.
J.
Chapter 8
43
SPC Control Plan
(7)
Spec
Limit
(8)
(9)
(10)
(11)
Station/
Location
Inspection
Methods
Sample
Size
Inspection
Frequency
(12)
(13)
Analysis Cpk
Method Index
(14)
Reaction to
Out of Control
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
(15)
Submitted by
.
Title
.
Chapter 8
(16)
Approved By
.
Date
.
44
DATA COLLECTION FOR CAPABILITY ANALYSIS
Char measured
Part No & Name
Operation No & Dese.
Date
SAMPLE DATA:
No
Value
No
Value
No
Value
No
Value
No
1
21
41
61
81
2
22
42
62
82
3
23
43
63
83
4
24
44
64
84
5
25
45
65
85
6
26
46
66
86
7
27
47
67
87
8
28
48
68
88
9
29
49
69
89
10
30
50
70
90
11
31
51
71
91
12
32
52
72
92
13
33
53
73
93
14
34
54
74
94
15
35
55
75
95
16
36
56
76
96
17
37
57
77
97
18
38
58
78
98
19
39
59
79
99
20
40
60
80
100
Value
Remarks
TALLY SHEET:
VALUE
TALLY
FREQUENCY
Chapter 8
45
Process Capability Studies
1) Train engineers to learn the process capability
study that is to measure the inherent
variability of the process so that the
performance potential can be detected under
normal, in control conditions.
2) Train engineers to use the control chart
method and the frequency distribution
method for measuring process capability at
99.7% confidence level.
Chapter 8
46
Process Capability Studies
3) The applications include the following:
a. Information to facilitate the design of the
product
b. Acceptance of new or reconditioned piece
of equipment.
c. Scheduling work to machine.
Chapter 8
47
Process Capability Studies
d. Setting up the machine for a production run.
e. Selection of operators.
f. Introduction of a new product or process.
Chapter 8
48
CPK
• A key characteristic will be considered
capable if the supplier can demonstrate with
90% confidence that the true Cpk exceeds
1.0. (In some cases, an alternate Cpk
requirement will be defined in the contract.)
When computing Cpk (see section 7-3.2), the
number of measurements collected must be
taken into account. Table 2.3.2 must be used to
determine the minimum calculated Cpk to
demonstrate capability
Chapter 8
49
TABLE 2.3.2 CPK
Chapter 8
50
CPK
• The values in the above table are the
calculated Cpk values required to be 90%
confident that the actual Cpk is greater than or
equal to the Cpk value at the top of the
respective column. The values listed in the
column titled “Number of measurements taken”
are the actual number of measurements, not
the number of plot points. The table assumes
that the underlying distribution of the
individual measurements are normally
distributed with a fixed mean and standard
deviation.
Chapter 8
51
CPK
• Examples:
If 30 parts are measured and the required Cpk
is 1.0, the calculated Cpk from the 30 parts
needs to be at least 1.23.
If 20 parts are measured and the calculated
Cpk is 1.93, we have sufficient confidence that
the actual Cpk is 1.50 or better.
Chapter 8
52
Cp, Cpk Index / Process Capability
Studies
(1)The test specification limits for incoming
diodes is .008 ohms (upper) and .001 (lower)
and the standard deviation for this population
is .002 ohms. What is Cp index for this
process? Does this number indicate the
process is within the specification limits?
Chapter 8
53
Cp, Cpk Index / Process Capability
Studies
Why or why not?
a. Suppose action was taken on this incoming
test process and the standard deviation is
now .001 ohms. What is the Cpk index? Is it
within the limits? Given that x =0.04
Chapter 8
54
Cp, Cpk Index / Process Capability
Studies
(2) Measurements of the solder thicknesses on
print circuit boards going through the wave
solder machine are naturally between .006
and .012. Given that
  .001
,whereas the specification limits are at
.004, .002
from the mean of .009. What is the
Cp/Cpk index for this process?
Chapter 8
55
Cp, Cpk Index / Process Capability
Studies
(3) When should a process capability study be
conducted?
Chapter 8
56
Process Capability
Natural tolerance limits are defined as follows:
Chapter 8
57
Uses of process capability data:
Chapter 8
58
Reasons for Poor Process Capability
Process may have
good potential
capability
Chapter 8
59
Chapter 8
60
Chapter 8
61
Chapter 8
62
Probability Plotting
Chapter 8
63
• The distribution may not be normal; other types of
probability plots can be useful in determining the
appropriate distribution.
Chapter 8
64
Chapter 8
65
Chapter 8
66
For the hard bake process:
Chapter 8
67
One-Sided PCR
Chapter 8
68
Interpretation of the PCR
Chapter 8
69
Assumptions for Interpretation of
Numbers in Table 8.2
• Violation of these assumptions can lead to big trouble in using the
data in Table 8.2.
Chapter 8
70
Chapter 8
71
• Cp does not take
process centering
into account
• It is a measure
of potential
capability, not
actual capability
Chapter 8
72
A Measure of Actual Capability
Chapter 8
73
Normality and Process Capability
Ratios
•
The assumption of normality is critical to the
usual interpretation of these ratios (such as
Table 8.2)
For non-normal data, options are
•
1. Transform non-normal data to normal
2. Extend the usual definitions of PCRs to handle
non-normal data
3. Modify the definitions of PCRs for general
families of distributions
Chapter 8
74
Other Types of Process Capability Ratios
•
•
•
•
First generation
Second generation
Third generation
Lots of research has been done to develop
ratios that overcome some of the problems
with the basic ones
• Not much evidence that these ratios are used to
any significant extent in practice
Chapter 8
75
Chapter 8
76
Chapter 8
77
Chapter 8
78
Chapter 8
79
Chapter 8
80
Chapter 8
81
Process Capability
Analysis using Control
Charts
Chapter 8
82
Since LSL = 200
Chapter 8
83
Chapter 8
84
Chapter 8
85
Measurement Accuracy
The closeness of measurement to the true value

Bias : Systematic Error



 Precision : Random Error

Chapter 8
86
Accuracy and Precision
We have focused
only on precision
Chapter 8
87
Chapter 8
88
Gauge R&R Studies
Chapter 8
89
Chapter 8
90
Chapter 8
91
7.8 Gauge and Measurement Systems
Capability Studies
• Determine how much of the observed
variability is due to the gauge or measurement
system
• Isolate the components of variability in the
measurement system
• Assess whether the gauge is capable (suitable
for the intended application)
Chapter 8
92
Chapter 8
93
Chapter 8
94
Chapter 8
95
The P/T ratio:
Chapter 8
96
Chapter 8
97
Estimating the Variance Components
Chapter 8
98
Chapter 8
99
The gauge is not capable by this criterion
Chapter 8
100
Discrimination Ratio
Chapter 8
101
Gauge R&R Studies Are Usually Conducted
with a Factorial Experiment
Chapter 8
102
This is a two-factor factorial experiment
ANOVA methods are used to analyze the data and yo estimate the
variance components
Chapter 8
103
Chapter 8
104
Chapter 8
105
Chapter 8
106
Chapter 8
107
• Negative estimates of a variance component
would lead to filling a reduced model, such as,
for example:
Chapter 8
108
Chapter 8
109
For this Example
Chapter 8
110
Other Topics in Gauge R&R
Studies
• Section 8.7.3 provides a description of methods to
obtain confidence intervals on the variance
components and measures of gauge R&R
• Section 8.7.4 presents a new measure of gauge
capability, the probabilities of misclassification of
parts
– Rejecting good units (producer’s risk)
– Passing bad units (consumer’s risk)
– Methods for calculating these two probabilities are given
Chapter 8
111
Statistical Tolerance
• The more realistic statistical approach is based
on the relationship between the variances of a
number of independent causes(A,B,C,…) and
the variance of the overall result. This
relationship is :
 2ass' y  A2  B2  C 2
ass' y  A2  B 2  C 2
Chapter 8
112
Statistical Tolerance
• From a process capability of 6=T and by
substitution (T=tolerance)
 Tass ' y  TA  TB  TC
2
2
2
• The approach to statistical tolerance is
illustrated by s simple mechanical
assembly of three parts as shown below :
A
.500
.005
Chapter 8
B
C
1.000
2.000
.010
.020
113
Statistical Tolerance
• Dimensions of A,B, and C determine the
overall assembly length. A conventional
specification on the assembly length would
be :
Chapter 8
A:
.500

.005
B:
1.000

.010
C:
2.000

.020
Assy :
3.500

.035
114
Statistical Tolerance
• This is illustrated by again using the
previous assembly. Assume that capability
studies indicate that if the .005 tolerance on
component a could be example to .010 , a
secondary operation could be eliminated.
What effect will this have on the total length
tolerance?
Tassy  0.0202  0.0202  0.0402
 .01 4.0  4.0  16.0
Tassy  .049  .0245
Chapter 8
115
Statistical Tolerance
• This is an increase of only .0015 for a
component increase of .005. This
illustrates an important characteristic of the
statistical combination of component
variances. The Effect of a component with
small variance is vary small; the component
with the largest variance has the greatest
effect on overall variance.
Chapter 8
116
Statistical Tolerance
• Exercise
• Compare the tolerance found using the
conventional engineering approach to the
statistically computed assembly tolerance.
• Consider the four blocks:AB, BC, CD and
DE shown below. Each of these is
independent. Determine the specification
for the total assembly AE.
A
Chapter 8
.750〞B
.002 〞
.320 〞C
.001 〞
.475 〞 D .100 〞 E
.003 〞 .002 〞
117
Statistical Tolerance
• Assume that each part component was
studied for capability, and we found that
each required process creating the
respective component linear length was
stabilized (that is, in control). Each process
had respective dimensional
characteristics as shown below:
Chapter 8
118
Component
X

AB
.750 〞
.00067 〞
BC
.320 〞
.000333
CD
.475 〞
.001 〞
DE
.100 〞
.00067 〞
Chapter 8
119
Statistical Tolerance
• From this data, it is apparent that if we
assemble these four components, that the
grand average of these parts would be
1.645 〞. This is determined by the
following calculation :
X AB
X BC
X CD
X DE
Chapter 8
Total
X AE
=.750 〞
=.320 〞
=.475 〞
= .100 〞
=.100 〞(Assembly average)120
Statistical Approach
• Assume that each part component was studied
for capability, and we found hat each required
process creating the respective component linear
length was stabilized (that is , in control ).
• Each process had respective dimension
characteristic as shown below:
Chapter 8
121
Component
AB
BC
CD
DE
Chapter 8
X
.750”
.320”
.475”
.100”

.00067”
.00033”
.001”
.00067”
122
Form this data, it is apparent that if we assemble
these four components, that the grand average of
those parts would be 1.645”. This is determined
by the following calculation:
X AB =.750”
X BC =.320”
X CD =.475”
X DE =.100”
Total
Chapter 8
X AE = 1.645” (assembly average)
123
If we could determine the standard deviation
of AE, then we could be able to find that
region
within which 99.7% of all linear lengths of AE
(the assembly) would lie. This region be
defined as:
Chapter 8
124
The standard deviation of the assembly AE can
be found by calculating the square root of the
sum of the square of each component. This can
be stated mathematically as follows:
 AE   AB 2   BC 2   CD 2   DE 2 
Therefore,
Upper tolerance  X  3  
Lower tolerance  X  3  
Total statistical tolerance 
What percent is this of the “ Worst case ”
tolerance?
Chapter 8
AE
AE
AE
AE
125
Such shafts and bearings fit together?
Is the overlap too much satisfactory assembly?
Chapter 8
126
An Analysis
The situation pictured in Figure 11-6 can be
analyzed as follows.
Let xs  a value for shaft O.D.
xb  a value for bearing I.D.
 s  0.9105, b  0.9210
Chapter 8
127
Bearing
0.9105
0.9210
Shaft
.9000
.9050
.9100
Shaft
 xs 
.9200
.9150
and bearing
 xh 
.9250
.9300
dimensions in cm
Figure 11-6 Distributions for Shaft and Bearing with Overlapping Variation
Chapter 8
128
and d  xb  xs
Hence d  b  s  0.0105
The standard deviation for the bearing I.D.,  s ,
may be approximated as
0.9195  0.9015
s 
 0.003
6
Similarly, the standard deviation for the bearing
I.D.,  b , may be approximated using
0.9335  0.9085
b 
 0.00417
6
Chapter 8
129
Hence, the standard deviation for the difference,  d
may be approximated using the equation with
a1  1 and a2  1
as below:
 d   s2   b2  
0.0032  0.004172
 0.005134
This distribution of the difference, d , will be normal,
since xb and xs are normal, with the approximate
mean value and standard deviation obtained above.
This situation is pictured in Figure 11-7.
Chapter 8
130
This area to the left of zero represents the
proportion of bearings having smaller
I.D.’s than the O.D.’s of the corresponding
shafts. It is estimated as follows:
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0  0.0105
z
 2.045
0.005134
0.0105
with  2.045  0.0204 or 2.04%
-.005
0
.005
.010
.015
.020
.025
d  xb  xs in cm
Figure 11-7 Distribution for Evaluating Mating Part Tolerances
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8.7.5 Attribute Gauge Capability
• Sometimes the output of a gauge isn’t numerical – it’s just
pass/fail
• Nominal or ordinal data is also common
• Occurs frequently in service businesses
• Common situation – do operating personnel consistently make
the same decisions regarding the units they are inspecting or
analyzing
• Example – a bank uses manual underwriting of mortgage loans
• The underwriter uses information to classify the applicant into
one of four categories; decline or category 1, 2, 3 – categories
2 & 3 are low-risk and 1 is high risk
• Compare underwriters performance relative to a “consensus”
evaluation determined by a panel of “experts”
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Thirty applicants,
three underwriters
Each underwriter
evaluates each
application twice
The applications are
“blinded” by
removing names,
SSNs, addresses,
and other identifying
information
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Attribute Gauge Capability
• Determine the proportion of time that the
underwriter agrees with him/herself – this
measures repeatability
• Determine the proportion of time that the
underwriter agrees with the correct
classification – this measures bias
• Minitab performs the analysis – using the
attribute agreement analysis routine
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8.8 Setting Specifications on Discrete Components
•
•
•
•
•
Components interact with other components
Complex assemblies
Tolerance stack-up problems
Linear combinations
Nonlinear combinations
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8.9 Estimating the Natural Tolerance Limits
of a Process
For a normal distribution with unknown mean and variance:
• Difference between tolerance limits and confidence limits
• Nonparametric tolerance limits can also be calculated
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Learning Objectives
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