chapter 81. 2 3 learning objectives chapter 84 process capability in short: the process must first...
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Chapter 8 1
Chapter 8 2
Chapter 8 3
Learning Objectives
Chapter 8 4
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 5
Process Control
Chapter 8 6
Process Capability
Chapter 8 7
Capable V.S. In-Control Process
• Capable process– A process that has a greater the 1. this means that the histogram distribution and the normal curve are contained within the actual specification dimensions.
pkC
Chapter 8 8
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 9
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 10
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 Limits (6 sigma)3
Chapter 8 11
How Do We Know a Process is Capable?
• 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.
CP and CPK statistical process indexes
Chapter 8 12
The Cp Index
• Example: suppose the Spec width=10 units and =5 units
• This implies the spec is twice as wide as the natural spread.
6
Spec width Upper spec Lower spec
CpNatural tolerance
6
L.S. U.S.610
25pC units
Chapter 8 13
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:.
6
Engineering Tol
sigma
Chapter 8 14
CP and CPK
• CPK is a capability index
• Formula:
what ever is less
USL=Upper SPEC Limit
LSL=Lower SPEC Limit
3 3
USL mean mean LSLor
sigma sigma
Chapter 8 15
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 16
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 17
CP and CPK
• 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.
(Key)
Chapter 8 18
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 19
CP and CPK
• Sigma– The standard deviation of a statistical population.
Chapter 8 20
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 21
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.
1234 1 2 3 4
68.26%
99.73%
NORMAL DISTRIBUTION CURVE
[SIGMA] X
Chapter 8 22
Process Fallout Table
Process capability Parts per million ratio defective 0.50 133,600.00
0.75 24,400.00 1.00 2,700.00 1.10 967.00
1.20 318.00 1.30 96.00 1.40 26.00 1.50 6.80 1.60 1.60 1.70 0.34 1.80 0.06 2.00 0.0018
Centered 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
Chapter 8 25
Relationship of Specifications, Control Limits, and Natural
Tolerance Limits in the Process
pC pC
pC
σ -3σ-2σ-σ+3σ 2σ
spread
TargetLSL USL
is the ratio of spec. width to the natural variability present in the process; i.e.
6p
USL LSLC
pC
Chapter 8 26
Relationship of Specifications, Control Limits, and Natural
Tolerance Limits in the Process
pkC
(Avg)
TargetLSL USL
is the ratio of the distance between the process center and the nearest spec. Limit to one half of process variability ( ); i.e. 3
min ,3 3pk
X LSL USL XC
Center Line
Chapter 8 27
Chapter 8 28
Chapter 8 29
Chapter 8 30
Process Capability
• 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.
When?
Chapter 8 31
Process Capability
• 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.
The need for capability studies
Chapter 8 32
Process Capability Study Flowchart
決定關鍵品質特性
從製程中隨機抽 30 個樣本
計算平均數，標準差， Cp/Cpk 指標
是否通過常態性檢定且 Cpk1.23
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
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
檢查全距管制圖是否穩定，去掉脫離管制界限之異常點
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
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
是否通過常態性檢定並接受 Cpk>1
決定 SPC 管制計畫，開始使用 管制圖
決定 SPC 管制計畫，開始使用管制圖
考慮以實驗設計的方法減少製程變異，或更新設備
是
決定 SPC 管制計畫中的 4 個W
開始使用計量值或前置管制圖
否
否
否是
（短期製程能力分析）
（長期製程能力分析）
是否通過常態性檢定且 Cpk>1
是
Chapter 8 33
Procedures for Determining Process Capability
1) Define the process
Is it a line, machine, operator, portion of manufacturing, etc.?
Chapter 8 34
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 and s (usually from grouped data) to estimate the mean and standard deviation of the process.
( n= 30)
X
Chapter 8 35
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 36
Procedures for Determining Process Capability
3) Determine Inherent Variability, Using -R/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.
X
Chapter 8 37
Procedures for Determining Process Capability
b. Calculate average range , 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.
R
Chapter 8 38
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 39
Procedures for Determining Process Capability
f. Estimate the standard deviation from the last average range:
Estimate of σ= /d2
g. Use midpoint of specification limits as the process mean, assume normality, and estimate percent meeting specs.
R
Chapter 8 40
Procedures for Determining Process Capability
h. If specs can be met, investigate process to determine why specs are not being met.
Chapter 8 41
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 -R charts for future control.X
Chapter 8 42
Chapter 8 43
SPC Control Plan
Supplier (2) Supplier No.
Address (3) Supplier Representative
City/State (4)
Phone No. (5)
A.
F.
B.
G.
C.
H.
D.
I.
E.
J.
(1)
Control Characteristics
Sheet of .
(6)
Chapter 8 44
SPC Control Plan
Spec
Limit
Station/
Location
Inspection
Methods
Sample
Size
Inspection
Frequency
Analysis
Method
Cpk
Index
Reaction to
Out of Control
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
(7) (8) (9) (10) (11) (12) (13) (14)
(15)
Submitted by .
Title .
(16)
Approved By .
Date .
Chapter 8 45
DATA COLLECTION FOR CAPABILITY ANALYSIS
Part No & NameChar measured
Operation No & Dese. Date
SAMPLE DATA:
NoValue No Value No Value No Value No Value
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
Remarks
TALLY SHEET:
VALUE
TALLY
FREQUENCY
Chapter 8 46
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 47
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 48
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 49
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 50
TABLE 2.3.2 CPK
Chapter 8 51
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 52
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 53
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 54
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 =0.04x
Chapter 8 55
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
,whereas the specification limits are at from the mean of .009. What is the
Cp/Cpk index for this process?
.001
.004, .002
Chapter 8 56
Cp, Cpk Index / Process Capability Studies
(3) When should a process capability study be conducted?
Chapter 8 57
Process Capability
Natural tolerance limits are defined as follows:Natural tolerance limits are defined as follows:
Chapter 8 58
Uses of process capability data:
Chapter 8 59
Process may have good potential
capability
Reasons for Poor Process Capability
Chapter 8 60
Chapter 8 61
Chapter 8 62
Chapter 8 63
Probability Plotting
Chapter 8 64
• The distribution may not be normal; other types of probability plots can be useful in determining the appropriate distribution.
Chapter 8 65
Chapter 8 66
Chapter 8 67
For the hard bake process:
Chapter 8 68
One-Sided PCR
Chapter 8 69
Interpretation of the PCR
Chapter 8 70
• Violation of these assumptions can lead to big trouble in using the data in Table 8.2.
Assumptions for Interpretation of Numbers in Table 8.2
Chapter 8 71
Chapter 8 72
• Cp does not take process centering into account
• It is a measure of potential capability, not actual capability
Chapter 8 73
A Measure of Actual Capability
Chapter 8 74
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 are1. Transform non-normal data to normal2. Extend the usual definitions of PCRs to handle
non-normal data3. Modify the definitions of PCRs for general
families of distributions
Chapter 8 75
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 76
Chapter 8 77
Chapter 8 78
Chapter 8 79
Chapter 8 80
Chapter 8 81
Chapter 8 82
Process Capability Analysis using Control Charts
Chapter 8 83
Since LSL = 200
Chapter 8 84
Chapter 8 85
Chapter 8 86
:
:
Bias Systematic Error
Precision Random Error
Measurement Accuracy
The closeness of measurement to the true value
Chapter 8 87
Accuracy and Precision
We have focused only on precision
Chapter 8 88
Chapter 8 89
Gauge R&R Studies
Chapter 8 90
Chapter 8 91
Chapter 8 92
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 93
Chapter 8 94
Chapter 8 95
Chapter 8 96
The P/T ratio:
Chapter 8 97
Chapter 8 98
Estimating the Variance Components
Chapter 8 99
Chapter 8 100
The gauge is not capable by this criterion
Chapter 8 101
Discrimination Ratio
Chapter 8 102
Gauge R&R Studies Are Usually Conducted with a Factorial Experiment
Chapter 8 103
This is a two-factor factorial experiment
ANOVA methods are used to analyze the data and yo estimate the variance components
Chapter 8 104
Chapter 8 105
Chapter 8 106
Chapter 8 107
Chapter 8 108
• Negative estimates of a variance component would lead to filling a reduced model, such as, for example:
Chapter 8 109
Chapter 8 110
For this Example
Chapter 8 111
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 8 112
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 :
2222 ' CBAyass
222' CBAyass
Chapter 8 113
Statistical Tolerance
• From a process capability of 6=T and by substitution (T=tolerance)
• The approach to statistical tolerance is illustrated by s simple mechanical assembly of three parts as shown below :
222' CBA
TTTyTass
.005 .010
A B C
.500 1.000 2.000
.020
Chapter 8 114
Statistical Tolerance
• Dimensions of A,B, and C determine the overall assembly length. A conventional specification on the assembly length would be :
A: .500 .005
B: 1.000 .010
C: 2.000 .020
Assy : 3.500 .035
Chapter 8 115
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?
0245.049.
0.160.40.401.
040.0020.0020.0 222
Tassy
Tassy
Chapter 8 116
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 8 117
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.
.002 〞 .001 〞
.750 〞 .320 〞 .475 〞.003 〞
.100 〞.002 〞
A B C D E
Chapter 8 118
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 8 119
Component X
AB .750 〞 .00067 〞
BC .320 〞 .000333
CD .475 〞 .001 〞
DE .100 〞 .00067 〞
Chapter 8 120
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 :
=.750 〞
=.320 〞
=.475 〞
= .100 〞
Total =.100 〞 (Assembly average)
ABX
BCX
CDX
DEX
AEX
Chapter 8 121
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 8 122
Component X
AB .750” .00067”
BC .320” .00033”
CD .475” .001”
DE .100” .00067”
Chapter 8 123
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:
=.750”
=.320”
=.475”
=.100”
Total = 1.645” (assembly average)
ABX
BCX
CDX
DEX
AEX
Chapter 8 124
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 8 125
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:
Therefore, Upper tolerance Lower tolerance Total statistical tolerance What percent is this of the “ Worst case ” tolerance?
2 2 2 2AE AB BC CD DE
3AE AEX
3AE AEX
Chapter 8 126
Such shafts and bearings fit together? Is the overlap too much satisfactory assembly?
Chapter 8 127
The situation pictured in Figure 11-6 can be analyzed as follows.
Let a value for shaft O.D.
a value for bearing I.D.
0.9105, 0.9210
s
b
s b
x
x
An Analysis
Chapter 8 128
.9000 .9050 .9100 .9150 .9200 .9250 .9300
Shaft sx and bearing hx dimensions in cm
Figure 11-6 Distributions for Shaft and Bearing with Overlapping Variation
0.91
05
0.92
10
Shaft Bearing
Chapter 8 129
and
Hence
The standard deviation for the bearing I.D., ,
may be approximated as
Similarly, the standard deviation for the bearing
I.D., , may be approximated using
sb xxd
0105.0 sbd
s
003.06
9015.09195.0
s
b
00417.06
9085.09335.0
b
Chapter 8 130
Hence, the standard deviation for the difference, may be approximated using the equation with and
as below:
This distribution of the difference, , will be normal, since and are normal, with the approximate mean value and standard deviation obtained above. This situation is pictured in Figure 11-7.
d
11 a 12 a
005134.0
00417.0003.0 2222
bsd
d
sxbx
Chapter 8 131
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:
Chapter 8 132
045.2005134.0
0105.00
z
%04.20204.0045.2 orwith
-.005 0 .005 .010 .015 .020 .025
Figure 11-7 Distribution for Evaluating Mating Part Tolerances
b sd x x in cm
0.01
05
Chapter 8 133
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”
Chapter 8 134
Thirty applicants, three underwriters
Each underwriter evaluates each application twice
The applications are “blinded” by removing names, SSNs, addresses, and other identifying information
Chapter 8 135
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
Chapter 8 136
Chapter 8 137
Chapter 8 138
8.8 Setting Specifications on Discrete Components
• Components interact with other components
• Complex assemblies
• Tolerance stack-up problems
• Linear combinations
• Nonlinear combinations
Chapter 8 139
Chapter 8 140
Chapter 8 141
Chapter 8 142
Chapter 8 143
Chapter 8 144
• Difference between tolerance limits and confidence limits
• Nonparametric tolerance limits can also be calculated
8.9 Estimating the Natural Tolerance Limits of a Process
For a normal distribution with unknown mean and variance:
Chapter 8 145
Chapter 8 146
Chapter 8 147
Learning Objectives