DMAIC

Statistical methods for
EngineeringEngineering
Yun-Hsuan Yeh
Software JMP
2014/12/29 1Yun-Hsuan Yeh

Outline
TYPE OF ANALYSIS APPLICATION
Sample Plan Reduce excessive sampling
Capability Study (SPC) Cpk analysis of Machine, Process or System
Gr&R For all measuring equipment
Comparative Analysis Compare of two or more groupsComparative Analysis Compare of two or more groups
DOE For all new system & process (material, parameters)
Correlation Correlation between two objectives
Regression X & Y correlation (input and output relationship)
Reliability Risk assessments of long term
All the items linked with one another
Yun-Hsuan Yeh2014/12/29 2

DMAIC (6 sigma)
•Xs/Ys SPC selection
•Identify problem
•Customer ‘s requirement
•Select CTQ (Critical to Quality)
•Goal setting
•Time line
•Sampling Plan
•Measuring technology
•Screening DOE
•Optimization (Regression )
•Validation
•SPC control
•Documentation
•Reliability monitor
•Hypothesis testing
•ANOVA
•Contingency & Cluster
analysis

DOE (WLB RDL Short Issue)
Defect mode: RDL short (Width 15um / Spacing 10um)
Factors: PR thickness, Exposure energy, Developer time, and PR ash time.
2014/12/29 Yun-Hsuan Yeh 40
Factors: PR thickness, Exposure energy, Developer time, and PR ash time.
Faults: We should remove the trials that were anticipated. On the others side,
we should not put the depending variables in the DoE. It is waste!!

Key factor: PR thickness
Optimal solution:
Actually, we just select one or two factors for this problem
Actually, we just select one or two factors for this problem
that will save many resources.

DOE
Design of experiments are a structured development strategy for
product/process engineering in order to characterize, optimize and
control performance with minimal waste. This is accomplished by
experimenting with many factors at the same time.
Main Effects or
Linear
Interactions Powers or
Quadratics
Correlations between factor and respond.

DOE (Levels)
If unsure of levels, run a few pre-experiments to find an operating
range, then begin the experiment.
3 levels are used if the factor range is wide.
If the relationship is nonlinear, we need to add center point to the
predicting.
Yun-Hsuan Yeh
Level 1 Level 2
The real phenomenon
Predicted by linear
regression with 2
levels.
???
2014/12/29 43
Level 1 Level 2
Narrow
range
Wide
range

DOE (Full factorial designs)
A class of designs that use k factors at 2 levels.
t = nk
where
t = the number of tests
n = number of levels
k = number of factors

DOE (Fractional factorial designs)
We have a k factors and 2k-p fractional factorial designs, then the design
follows the properties:
1.Sum of each column in the sign table is 0
1allfor,0 >=∑ js
j
j
2.Sum of the product of any two columns is 0
3.Sum of the square of elements in any column is 2k-p
.,1,allfor,0
,
kjjss
kj
kj ≠>=∑
1allfor,22
>= −
∑ js pk
j
j
A B C
1 -1 -1 1
2 1 -1 -1
3 -1 1 -1
4 1 1 1
23-1 Design

DOE (Custom Design)
D or I-optimal designs are the forms of design provided by a computer
algorithm. The reasons for using D or I-optimal designs instead of
standard classical designs generally fall into two categories:
Standard factorial or fractional factorial designs require too many
runs for the amount of resources or time allowed for the experiment.
The design space is constrained (the process space contains factor
settings that are not feasible or are impossible to run).
Factors Full Fractional D-Optimal
5 32 16 16
6 64 32 28
7 128 64 35
8 256 128 43
9 512 256 52

D-Optimal (emphasize the corners) I-Optimal (emphasize the centers)
DOE (Custom Design)

Evaluation of DOE
Prediction Variance Profile is used to determine the relative
variability in the response depending on the DOE design.
Add additional samples and the variance goes down.
Modify the design by adding runs and see what happens to the
predicted variance.
We will consider DOEs for multiple linear regression using linear
and polynomials and where errors are due to noise in the data.
where is the nx1 vector, x is the nxp matrix and β is the px1 vector.
because in using to estimate the responses, we effectively estimate
parameters β . That is, we lose p degrees of freedom.
βˆˆ xy =
pn
yy
n
j
jj
−
∑ −
=
= 1
2
)(
ˆσ (MSE)
yˆ
yˆ

Prediction Variance
By definition we have,
Then , so that different x has distinct variance.
xxxx
xx
xy
1TT2
T
][ˆ
)ˆVar(
)ˆ(Var)ˆVar(
−
=
=
=
σ
β
β
xxxx
y 1TT
2
][
ˆ
]ˆVar[ −
=
σ
The Prediction Variance Profile gives a profiler of the relative variance of
prediction as a function of each factor at fixed values of the other factors.
ˆσ
The prediction variance is 0.2241 that
means the variance of experimental
error is 10, then the prediction variance
of response at X1=0 is
10x0.2241=2.241.

DOE (Correlation of Xs check)
If Xs were correlated to any great degree
(r > 0.5) then experiment was not well
designed and should be redone, this will
cause collinearity then
There is no unique solution for
regression coefficients.
Partial coefficients are estimating
something that does not occur in the
data.
This step is very important for uncontrolled
variables, engineers always ignore
uncontrolled variables usually.

Regression
Learn more about the relationship between several independent or
predictor variables and a dependent or criterion variable.
In the most general terms, least squares estimation is aimed at
minimizing the sum of squared deviations of the observed values for
the dependent variable from those predicted by the model.
2
)]([min∑ − fy θ
where is a known function of , and are
random variables, and usually assumed to have expectation of 0.
)]([min∑ −
i
i
f
fy θ
)(θf θ ii fy εθ += )( iε
0H : Individual coefficient is equal to 0.
0H : All coefficients are equal to 0.

R2
A data set has values yi each of which has an associated modeled value f
the yi are called the observed values and the modeled values f called the
predicted value.
y
2
][∑ −=
i
ireg fySS
2
][∑ −=
i
itot yySS
tot
reg
SS
SS
R −=12
y

Adjusted R2
To correct the R2 for such situations, an adjusted R2 takes into account the
degrees of freedom of an equation. When you suspect that an R2 is higher
than it should be, calculate the R2 and adjusted R2. If the R2 and the
adjusted R2 are close, then the R2 is probably accurate. If R2 is much
higher than the adjusted R2, you probably do not have enough data points
to calculate the regression accurately.
2
−−
where n is the number of data points and m is the number of independent
variables.
1
)1)(1(
1
2
2
−−
−−
−=
mn
nR
adjR

Regression (Parameter estimates)
: the coefficient is zero.0H
Therefore, the terms, Correlation, Leverage, Profit, Change of holder,
Turnover(60), and Correlation*Profit have significant influence under α=0.05.

DOE (Replication)
Replication improve the chance of detecting a statistically significant
effect in the midst of natural process variation (noise).
The variation can be determined from control charts and process
capability studies.
We can improve the power (1- β) of the DOE by adding actual
replicates where conditions are duplicated.replicates where conditions are duplicated.
Replication

DOE (Residuals)
Residuals can tell us whether our assumptions are reasonable and
our choice of model is appropriate.
Residuals can be thought of as elements of variation unexplained
by the fitted model (normality, independence, and homoscedasticity).
We expects them to be normal and independently distributed with a
Residual = Observation - Predicted response
We expects them to be normal and independently distributed with a
mean of 0 and some constant variance.

DOE (Response Surface Methods)
High-resolution fractional or full factorial to understand interactions (
add center points at this stage to test for curvature).
Response surface methods RSM) to optimize.
Contour maps (2D) and 3D surfaces guide you to the optimal solutions.
Set the limits of response
To find the safe window

Postscript
Nowadays, the problem can be solved by the experiences that will not be a
problem. Furthermore, those methods had been developed over 50 years.
The rollback or missing anticipation always occur without evaluation
and the cost is ignored usually.
There is gap between prediction and practicality certainly, so that to
evaluate the variances becomes the important objective in DOE.
Sometimes, DOE predicts the effect is 80%, but the real effect is about
40% that may be caused by poor evaluation.
Evaluation is that to reduce the risk and cost through the scientific methods.
If we have too many features may fit the training set very well, but fail to
generalize to new examples (predict the response on new examples).

6 sigma (DMAIC)
•Identify problem
•Customer ‘s requirement
•Select CTQ (Critical to Quality)
•Goal setting
•Time line
•Sampling Plan
•Measuring technology
•Screening DOE
•Optimization (Regression )
•Validation
•SPC control
•Documentation
•Reliability monitor
•Hypothesis testing
•ANOVA
•Contingency & Cluster
analysis

Population, Sample and Variable
A population includes each element from the set of observations that can
be made.
A sample consists only of observations drawn from the population.
A variable is an attribute that describes a person, place, thing, or idea.
Variables can be classified as qualitative (aka, categorical)
or quantitative (aka, numeric).

Sample Plan
Simple Random Sampling: Every member of the population has an equal
chance of being selected for your sample.
Stratified Sampling: The population is split into non-overlapping groups,
then simple random sampling is done on each group to form a sample.
Systematic Sampling: Every nth individual from the population is placed in
the sample. That cannot obtain a frame of the population you wish to study.
Cost Risk
Size
Frequency α (AQL)
β (LTPD)

Sample Plan
Alpha is the probability of mistakenly detecting a difference even when
there is no difference.
Beta is the probability of not being able to detect a significant difference.
Power (1-Beta) is the probability of correctly detecting a shift in performance.
Sample size in each
group (assumes equal
Represents the desired
power
Define the rational specifications via the pre-sampling.
group (assumes equal
sized groups)
Represents the desired
level of statistical
significance (typically
1.96).
Standard deviation of
the outcome variable Effect Size (the
difference in
means)
2
2
/2
2
difference
)Z(2 αβσ +
=
Z
n

Sample Plan (Difference)
D
The D means that we want to defect the difference under the power.
D is the accuracy of the estimate. We can determine D by experiences and
standard deviation form outcome, and power is an indicator to estimate the
sample size.

Sample Plan (K means)
D1 D3
If we fix the Di, then the sample size increases as the total variances
increase. When the Di is large, we detect the different easily with small
sample size (refer to one-way ANOVA).
D2
Total
mena

Sample Size (Traditional industrial)
Compute the UCI and LCI for the instant-in-time variation at 95%
UCL=x+1.96xS, LCI=x-1.96xS, where S is the instant-in-time variation
2
timeofperiod
2
timeininstanttotal −−−− += SSS

Sample Size (Traditional industrial)
Determine the range of the Cl at various sample sizes and compare it as a
percentage of the tolerance= (UCI-LCI)/(USL-LSL)x100 – Two side.
a percentage of the margin= (Average-LCI)/(Average-LSL)x100 – one side.
Percentage Sampling SizePercentage Sampling Size
<7% Excessive
7~15% Efficient
>15% Insufficient
The percentages can be modify to fit industrial or process form experiences.

Sample Frequency (Traditional industrial)
Use regression to correlate the run order to the data. Determine the change
in the measurement based on one unit of time (drift rate). Make sure and
examine the drift rate as the most dynamically changing segment of the
data.
Drift rate x Time/(USL-LSL) Sampling frequency
5~15% Efficient
>15% Insufficient
The percentages can be modify to fit industrial or process form experiences.

Attribute sampling (GO/NG)
Using the Poisson to calculate probabilities, the probability of exactly x
defects or defective parts in a sample n with the probability of defect
occurrence p.
!
)(
)(
x
npe
xP
xnp−
=
If the lot has 3% defective, then
it has a probability 70% of being
α
If the lot has 10% defective, then
accepted with n=40.
accepted with n=40.
β
AQL LTPD

Attribute sampling (GO/NG)
As an example, consider a wafer manufacturing process with a target of 4
defects per wafer. You want to verify that a new process meets that target
within a difference of 1 defect per wafer with a significance level of 0.05. In
the Counts per Unit window:

Postscript
Sample plan is a comparative calculation that depends on the
detection and acceptable risk .
How long should the period be decided? That must consider the
operating period, such as period of job rotation, PM period, and period
of material renewing.of material renewing.
To reduce the sample size is always a easy way to cost down,
simultaneously, we can detect the variances and risk under the
anticipations.

Gr&R
True value True value
Step 1: Set 3-10 parts and 3-5 operators for measuring 3-10 times .
Step 2: Check the tool is ready and operators is clear for operating.
Step 3: Ensure the parts will not be mixed up and operators do not know
which part he/she are measuring (random).
Step 4:Test, data collection, and analysis.
Accuracy
True value True value
Reproducibility
Operator 1
Operator 3
Operator 2
Repeatability

Gr&R
The calculation parameters, repeatability, reproducibility, and total
measurement precision ,
22
& RrrR SSS +=
where,
,
15.5
1KR
Sr
×
=
nm
R
R
m
i
n
ij
×
=
∑∑= =1 1
15.5
r
under the 99% confidence interval,
( )
15.5
2
2
1
kn
S
KR
S
r
x
R
×
−×
=
nm
R
×
=
and where, m = number of operators, n = number of parts, k = number of
repeated readings, Rij = rang of repeated readings for operator i and part j,
and = the maximum operator average – the minimum operator average.x
R

Gr&R
We have range and average to imply the K factors, since the range and
estimated standard deviation have a relationship,
2
ˆ
d
R
r =σ
Where is called the relative range,
σˆ
R






=
σˆ2
R
Ed
σˆ
and then,
2
11
2
15.5
ityRepeatabil
ˆ15.5ityRepeatabil
d
KRK
d
R
=⇒=
== σ
let,

Gr&R
Trail K1 Operations K2 Parts K3
2 4.5656 2 3.6525 2 3.6524
3 3.0419 3 2.6963 3 2.6963
4 2.5012 4 2.2991 4 2.2991
5 2.2141 5 2.0766 5 2.0766
6 2.0324 6 1.9288 6 1.9288
7 1.9046 7 1.8198 7 1.8198
Similarly, we have K2 and K3.
7 1.9046 7 1.8198 7 1.8198
8 1.8089 8 1.7398 8 1.7398
9 1.734 9 1.6721 9 1.6721
10 1.6732 10 1.6195 10 1.6195
And the precision to variation ratio is
3KRS pp ×=
Where is the range between the maximum and minimum part averages.pR

Gr&R
GRR Rating
The precision-to-tolerance ratio, GRR, shows percent of the specification
window is consumed by measurement uncertainty, and is defined as:
%100
15.5 &
×
−
×
=
UCLUSL
S
GRR rR
GRR Rating
>30% Needs improvement
10~30% Marginal
<10% Acceptable

Attribute Gauge (GO/NG)
The result of the checks made by attribute gauges is not a numerical value,
but a good/scrap status.
Agreement rate of Part
6 is 5/9 (55.56%).6 is 5/9 (55.56%).
These parts might have
been more difficult to
categorize.
We usually set the criteria of
false alarm to be <0.1.

Gauge Linearity and Bias Study
The gauge linearity and bias study is used extensively in quality control in
manufacturing but also extends beyond this practical application. It is used to
determine the accuracy of measurements through the expected measurement
range.
Criteria <5%

Degrees of Freedom
The number of degrees of freedom is the number of values in the final
calculation of a statistic that are free to vary.
In order to estimate sigma, we must first have estimated mu. Thus, mu is
replaced by x-bar in the formula for sigma. Thus, degrees of freedom are
n-1 in the equation for s below:
)( 2
−
=
xx
s
For example, imagine you have four numbers (a, b, c and d) that must add up
to a total of m; you are free to choose the first three numbers at random, but
the fourth must be chosen so that it makes the total equal to m - thus your
degree of freedom is three.
1
)(
−
−
=
n
xx
s

Central Limit Theorem
The distribution of an average tends to be Normal, even when the
distribution from which the average is computed is decidedly non-Normal.
Thus, the Central Limit theorem is the foundation for many statistical
procedures, including Quality Control Charts, because the distribution of
the phenomenon under study does not have to be Normal because its
average will be.
Furthermore, this normal distribution will have the same mean as the
parent distribution, AND, variance equal to the variance of the parent
parent distribution, AND, variance equal to the variance of the parent
divided by the sample size.
N=1 N=30

SPC (Cpk and Ppk)
The Ca, Cp, and Cpk.
where is estimated by range or sigma of subgroup.
2/)( LCLUSL
x
Ca
−
−
=
µ
σˆ6
LCLUSL
Cp
−
= CpCaCpk ×−= )1(
σˆ
The Ppk is calculated by the sigma from the raw data, that includes the
variation within subgroup (the common cause) and the variation between
subgroup (the special cause), so that Cpk ≥ Ppk.
Cpk
Ppk

On the other hand, Cpk is
we have
SPC (Cpk and PPM of OOS)





 −−
=
σσ ˆ3
,
ˆ3
min
LSLxxUSL
Cpk
3
xUSL
Cpkz
−
=×=
LSLx
z
−
=or
By standard normal distribution, the defective parts per million is
σˆ
3 CpkzU =×=
σˆ
zL =
[ ] 1000000×> UzzP
or
[ ] 1000000×< LzzP

SPC (Cpk Level)
Level Cpk PPM
A >1.67 <0.5443
Process capability measures the ability of the process to meet the design
requirements.
A >1.67 <0.5443
B 1.67~1.33 0.5443~66.07
C 1.33~1 66.07~2699.8
D <1 >2699.8

t-Test (σ is unknown)
Populations are normally distributed.
Samples are independent, The hypothesis is defined as:
If judge by F-test, then we do the t-test with the equal variation (select
the Means/Anova/Pooled t option).
210 : µµ =H
2
2
2
1 σσ =
21 XX
t
−
=
)1()1( 2
22
2
12 −+−
=
SnSn
S
Otherwise, we do the t-test with the unequal variation (select the
t-Test option ).
2
2
2
1
2
1
21
n
S
n
S
XX
t
+
−
=
21
11
21
nnpS
XX
t
+
−
=
2
)1()1(
21
2212
−+
−+−
=
nn
SnSn
Spwhere

p-value
p-value is the probability of getting an effect of the size or greater by random
chance alone (tail probability), if p-value is small then t is large and the
means of the two samples have significance (reject the hypothesis).
Yun-Hsuan Yeh
p-value
t
2
α
p-value
-t
2
α
Two tailed test
2014/12/29 28
The critical p-value is a subjective criterion that depends on your
practical problem. Generally, we set the criterion as 0.05.

Test Standard Deviations
An F-test is used to test if the variances of two populations are equal. The F
hypothesis is defined as:
F distribution is also used to test multiple means.
Test statistic:
2
2
2
10 : σσ =H
222
==
Test statistic:
Where and are the sample variances. The more this ratio deviates
from 1, the stronger the evidence for unequal population variances.
22
2
2
1 / tssF ==
2
1s 2
2s

Step 1: Unequal Variances Test
p > .05 p < .05
t-Test Stdevs are unequal
When the variation by group was not equal and therefore the t test
assuming unequal variances should select the t-test.
Step 2: Means/Anova/Pooled t t-Test
p > .05
Variances are
the same
p < .05
Variances are
different

Chi-squared test
There are basically two types of random variables and they yield two types of
data: integral and categorical. A chi square (X2) statistic is used
to investigate whether distributions of categorical variables differ from one
another. the Chi Square statistic is calculated by the formula:
∑
−
= ii
E
EO 2
2 )(
χ
where O = observed cell count and E = expected cell count.
The hypothesis is defined as:
Pearson Chi-Square results when one or more cells contain less than 5
observations cannot be trusted, an alternative to the Fisher’s Exact Test.
∑i iEall
on.distributispecifiedafollowdataThe:0H

Chi-squared test
(Lot-yield comparison)
A test for independence
Test statistic:
t.independenaretionsclassificatwoThe:0H
∑∑
−
=
j i ij
ijij
E
EO
all all
2
2
)(
χ

One way ANOVA (Means Test)
The distribution of the response variable follows a normal distribution..
All populations have a common variance.
All samples are drawn independently of each other.
An experiment with k groups and each group has n samples, j=1,…,k
.2,...: 210 ≥=== nH nµµµ
An experiment with k groups and each group has nj samples, j=1,…,k
the test statistic:
where
wb MSMSF /=
j
j
j
ji
jij
w ni
n
xx
MS ,...,1,
)1(
)(
2
,
=
−
−
=
∑
∑
1
)(
2
−
−
=
∑
k
xxn
MS
j
jj
b

Two way ANOVA
The distribution of the response variable follows a normal distribution..
All populations have a common variance.
All samples are drawn independently of each other.
The groups must have the same sample size.
0H :The means do not depend on factor A.
0H :The means do not depend on factor B.
The statistics are
0H :The means do not depend on factor B.
0H :The means do not depend on factor AxB.
wa MSMSF /=
wb MSMSF /=
wab MSMSF /=

Two way ANOVA
An experiment with a levels with factor A and b levels with factor B and
sample size is r.
1 … a
1 1…r…
1…r
b 1…r
•• jx
•ijx
b 1…r
)1(
)(
2
,,
−
−
=
∑ •
rab
xx
MS kji
ijijk
w
1
)(
2
−
−
=
∑ •••••
a
xxrb
MS i
i
a
•••x••ix
1
)(
2
−
−
=
∑ •••••
b
xxra
MS
j
j
b
)1)(1(
)(
2
−−
+−−
=
∑ ••••••••
ba
xxxxr
MS
ij
jiij
ab

Two way ANOVA
0H :The pains do not depend on gender.
0H :The pains do not depend on drug.
0H :The pains do not depend on gender x drug.
There is no interaction between the gender and drug.

Hierarchical Clustering
(Apply to tool mapping or product difference)
The basic process of hierarchical clustering is:
1. Start by assigning each item to its own cluster, so that if you have N
items, you now have N clusters, each containing just one item. Let the
distances (similarities) between the clusters equal the distances
(similarities) between the items they contain.
2. Find the closest (most similar) pair of clusters and merge them into a
single cluster, so that now you have one less cluster.
3. Compute distances (similarities) between the new cluster and each of
the old clusters.
4. Repeat steps 2 and 3 until all items are clustered into a single cluster of
size N.

Hierarchical Clustering
Single-link : is the distance between clusters and .
Complete-link:
),(),( min,
ba
ba
dCCd
ji CC
ji
∈∈
=
),( ji CCd iC jC
),(),( max badCCd =
Average-link:
where is the mean vector of .
),(),( max,
ba
ba
dCCd
ji CC
ji
∈∈
=
∑∪∈
−=
ji CC
ji aCCd
a
µ),(
µ ji
CC ∪

K-means Clustering
The process of K-means clustering is:
1. Define the initial groups' centroids. This step can be done using different
strategies. A very common one is to assign random values for the
centroids of all groups. Another approach is to use the values
of K different entities as being the centroids.
2. Assign each entity to the cluster that has the closest centroid. In order to
find the cluster with the most similar centroid, the algorithm must
calculate the distance between all the entities and each centroid.
3. Recalculate the values of the centroids. The values of the centroid's
fields are updated, taken as the average of the values of the entities'
attributes that are part of the cluster.
4. Repeat steps 2 and 3 iteratively until entities can no longer change
groups.

Reliability (Censor)
During the T hours of test we observe r failures (where r can be any number
from 0 to n). The failure times are t1, t2 ,…, tr and there are (n-r) units that
survived the entire T-hour test without failing.
r1 fail r2 fail r3 fail ………………………
∑=
i
irr
0 t1 t2 t3
n units
start test
tr
i
In the typical test scenario, we have a fixed time T to run the units to see if
they survive or fail. The data obtained are called Censored Type I data.

Reliability (Weibull)
The most general expression of the Weibull pdf is given by the two-
parameter Weibull distribution expression,
where is the shape parameter and is the scale parameter, F(T) is CDF.
When , the Weibull reduces to the Exponential model.
0T,)(
)(
1
≥=
−
−
β
αβ
β
α
β T
eTTf
β α
1=β
Yun-Hsuan Yeh
When , the Weibull reduces to the Exponential model.
1<β 0=β 1>β
)(1
)(
rateFailure
TF
Tf
−
=
1=β
2014/12/29 60
infant mortality
wear out

Reliability (Life Distribution)
The Life Distribution platform helps you discover distributional properties
of time-to-event data.
The failure rate is
0.188 at 5975 units of
time.

Reliability (Sample Size)
From historical information an
engineer knows that component's
life follows a Weibull distribution
with some and .
Under the confidence level CL, the
total number of units n can be
determined by the following
α β
determined by the following
equation with t cycle times.
We want to know
the required
sample size to
estimate the time
till probability of
units fail, with a
two-sided
absolute
precision cycle
times.
)ln()( CL
t
n βα
=

Reliability (Acceleration)
Failure may be due to mechanical fatigue, corrosion, chemical reaction,
diffusion, migration, etc. These are the same causes of failure under normal
stress; the time scale is simply different. Then, an acceleration
factor AF between stress and use means the following relationships hold:
Time-to-Fail tu = AF × ts
Time-to-Fail tu = AF × ts
Failure Probability Fu(t) = Fs(t/AF)
Reliability Ru(t) = Rs(t/AF)
PDF or Density Function fu(t) = (1/AF)fs(t/AF)
Failure Rate hu(t) = (1/AF) hs(t/AF)
u: at use/ s: at stress

Reliability (Arrhenius)
This empirically based model is known as the Arrhenius equation. The acceleration
factor AF between a higher temperature T2 and a lower temperature T1 is given by
where k is Boltzmann’s constant and is the activation energy depends on the
failure mechanism and the materials involved, and typically ranges from 0.3 to 1.5,












−
∆
=
21
11
exp
TTk
H
AF
H∆
failure mechanism and the materials involved, and typically ranges from 0.3 to 1.5,
or even higher.
Fit life by Temperature
The Hazard Profiler for failure
probability of 0.00033 at 45
degrees is 2641.5 hours.

Reliability (Thermal Cycling Test)
The AF is given by
where f is temperature-cycling frequency m=1/3, n=1.9 (for SnPb solder), and
=1414.
















−
∆






∆
∆






=
su
n
u
s
m
u
s
TTk
H
T
T
f
f
AF
max,max,
11
exp
kH /∆
=1414.kH /∆
Test conditions: 0°C ~ 100°C
(36 cycles per day).
Use conditions: 0°C ~ 35°C
(2 cycles per day)
Then AF=6.24.
By fitting Weibull, the failure
risk is 0.05136 at 400 cycles .
So that the life time is
400x6.24/(365x2)=3.41 years.

Postscript
AF (Arrhenius) can be estimated through the experiments and mode
fitting.
Acceleration experiment can reduce the testing period and to find the
impact of factors.
If there are different failure modes after reliability test, then we canIf there are different failure modes after reliability test, then we can
analyze by modes to figure out which mode influences the life time
critically.

DMAIC

Recomendados

Recomendados

Más contenido relacionado

La actualidad más candente

La actualidad más candente (20)

Similar a DMAIC

Similar a DMAIC (20)

DMAIC