doe process optimization[1]
TRANSCRIPT
Ho : µ1 = µ2 ; Ha : µ1 ≠ µ2 * SSTotal = SSFactors + SSErrors * I=ABCDE ; A=BCDE ; AB = CDE * x-(zα s/√n)<µ<x+(zαs/√n)
Design of Experiments for Process Optimization
A 2 days Practical Workshop
C- - - - ++++
B - - + + - - + +
A- +- +- +- +
D- - - - - - - -
zα -z α 0
∆εσιγν οφ Εξπεριµεντσ Α 2 δαψσ Πραχτιχαλ Ωορκσηοπ
Introduction to concepts of Statistical variation, location and dispersion effects : Basic terminologies / Quality Cost Function; Process Parameters; Basic Statistical Inferences; Robust Design for a Process * Factorial Experiments / Designs : Comparison of Factorial Design to One Factor at One Time Experiment; Method to construct a Factorial Design; Estimation of main effect and interactions of Factors; Application of Yate’s Algorithm; Appropriate procedures to run a Factorial Design *Fractional Factorial Experiments : Introduction to Fractional Factorial Design; Construction of Fractional Factorial Designs; Confounding of Effects of Factors; Extracting information from Fraction Factorial Designs/experiments; Fold-over and De-confounding of Effects from Fractional Factorial Designs; Selection of Critical Process Parameters; Real life applications ~ Funneling the Factors into Critical, Trivial and Dummy; Comparison between Factorial and Fractional Factorials~the strength and weakness * Implementation : The Sequential Approach of Experimentation; Economical consideration of experiments; Randomization of Experimental Runs; Important points to take note before and during experimentation and data collection; Checklist for experimentation *Statistical Models : * Conclusion & ImplementationTaguchi Method of Experimentation*Multiresponse Experiments*Response Surface Methodology
design of experiments a 2 days practical workshop
OVERALL COURSE OBJECTIVES
At the end of this course, you will be able to:
• Apply DOE to solve process optimization problems.
• Use statistical method to evaluate the importance of each factor.
• Develop an appropriate matrix to perform the experiments.
• Apply and implement Factorial Experiments.
• Apply Fractional Factorial Experiments.
• Develop strategy to perform effective experiments to study process issues.
• Determine optimum operating condition.
• Implement DOE to meet the business requirements (cost, quality and
productivity).
i
TABLE OF CONTENTS
OVERALL COURSE OBJECTIVES........................................................................ i
TABLE OF CONTENTS......................................................................................... iii
SECTION 1: ........................................................................................................... 1
Introduction to concepts of Statistical variation, location and dispersion effects.............1 Section 1 Objectives ................................................................................................................................ 1 Definition / Basic Terminology............................................................................................................... 2 Process Parameters and Monitoring Tools .............................................................................................. 8 Basic Statistical Inferences.................................................................................................................... 13 Robust Design for a Process.................................................................................................................. 25
SECTION 2: ......................................................................................................... 29
Factorial Experiments / Designs .....................................................................................29 Section 2 Objectives .............................................................................................................................. 29 Comparison of Factorial Designs to One Factor at a Time Experiments .............................................. 30 Method to Construct a Factorial Design................................................................................................ 33 Estimation of Main Effect and Interactions of Factors.......................................................................... 35 Application of Yates’ Algorithm........................................................................................................... 42 Appropriate Procedures to Run a Factorial Design - Case Study.......................................................... 46
SECTION 3: ......................................................................................................... 53
Fractional Factorial Experiments ....................................................................................53 Section 3 Objectives:............................................................................................................................. 53 Introduction to Fractional Factorial Designs ......................................................................................... 54 Construction of Fractional Factorial Designs ........................................................................................ 55 Confounding of Effects of Factors ........................................................................................................ 56 Extracting Information from Fractional Factorial Designs / Experiments............................................. 57 Fold-Over and De-confounding of Effects from Fractional Factorial Designs ..................................... 58 Real Life Application ~ Funneling the Factors into Critical, Trivial and Dummy................................ 60
SECTION 4: ......................................................................................................... 63
Important Points for Implementation..............................................................................63 Section 4 Objectives:............................................................................................................................. 63 The Sequential Approach of Experimentation....................................................................................... 64 Economical Consideration of Experiments ........................................................................................... 65 Important Points to Take Note Before and During Experimentation and Data Collection.................... 67
iii
SECTION 1:
Introduction to concepts of Statistical
variation, location and dispersion effects
Section 1 Objectives
Section 1 Objectives
At the end of this section, you will be able to:
• Understand the importance of meeting target in Engineering
Specifications.
• Perform simple Statistical Hypothesis Testing and Parameter
Estimation.
• Understand and appreciate the importance of Robust Designs and
Optimum Parameters Settings.
1
Section 1: Introduction to concepts of Statistical variation, location and dispersion effects
Definition / Basic Terminology
Definition / Basic Terminology
Experimental units/subjects
• The basic units for which response measurement is collected.
Factors
• Distinct types of conditions that are manipulated on the
experimental units.
Quantitative factors
• Factors that can be measured based on intensity.
Qualitative factors
• Factors that do not have intensity such as different vendors,
materials, methods and others.
Factors levels
• Different modes of presence of a factor.
Treatment
• Specific combination of the levels of different factors.
Replications
• The number of experimental units on which a particular
treatment is applied.
2
Design of Experiments for Process Optimization
Response/Dependent variables
• The performance/output of certain treatment combinations.
Independent variables
• Factors / input / parameter.
Input Output Process
a. Parameter b. Factor c. Independent variable
Characteristic Response Dependent variable
Effect
• The change in response when level of certain independent
variable is changed.
Experimental error
• Variation in the observed results of an experiment when it is
repeated under situations that are as similar as possible. The
sources of experimental errors may be known or unknown.
3
Section 1: Introduction to concepts of Statistical variation, location and dispersion effects
Errors of measurements/experiments
• Small in magnitude.
• Random in nature.
• Usually normally distributed in nature.
• Affected by skills, alertness of personnel, ambient temperature,
efficiency and conditions of equipment.
• Distinct from careless mistake such as recording errors or wrong
procedures in experiment.
• Important in analysis and design of experiment.
Quality
• Classical definition :
Meeting the specifications.
• Modern definition :
Fitness to use, or
Meet or exceed customer’s requirement.
Quality
Engineering Specifications
Customer Satisfaction
Product
True Quality Substitute Quality
4
Design of Experiments for Process Optimization
Quality cost
• Loss due to poor quality products.
• Loss here refers to replacement cost, guarantee cost, repair cost
and loss of customers.
• There are 4 types of Quality cost :
Prevention
Appraisal
Internal failure
External failure
• Quality cost function is minimized when the variability between
each product is minimized.
• Quality cost traditional definition:
Target Upper Specification
Limit
Lower Specification
Limit
Loss No Loss
• Quality cost modern definition:
Upper Specification
Limit
Target Lower Specification
Limit
Loss
The amount of loss depends on how far the product deviates from target. No loss at
target
Loss = k(x-µ)²
5
Section 1: Introduction to concepts of Statistical variation, location and dispersion effects
• Quality Cost computation :
Assuming the cost of scrapping a part is $10.00, when it exceeds the target by +/- 0.50 mm. (Assuming µ= 10.00 mm)
Thus from
L = k (x - µ )²
10 = k (10.5 – 10.0 )²
k = 10/0.25 = $40.00 per mm
Hence the loss function is
L = 40 ( y – 10.0 )²
6
Design of Experiments for Process Optimization
From above example,
a. given y = 11.0 mm, what is the loss?
_______________________________________ _______________________________________ _______________________________________
b. given y = 12.0 mm, what is the loss?
_______________________________________
_______________________________________ _______________________________________ Comment on your findings.
EXERCISE 1
7
Section 1: Introduction to concepts of Statistical variation, location and dispersion effects
Process Parameters and Monitoring Tools
Process Parameters and Monitoring Tools
There is infinite number of factors affecting a process, e.g.
environmental changes, machine and human random behaviors, etc.
The 4M’s Random errors / Noises
Output
Man
Materials
Machine
Method
PROCESS
7M’s of Engineering
• Men
• Materials
• Machines
• Methods Product Quality Improvement
• Measurements
• Management
• Money
8
Design of Experiments for Process Optimization
9
Write down the process name, all input parameters that you can think of and the
required output measurements.
Relationship between input & output of a process:
X1 X2 : :
Xk
Process Product, Y
Input Output Characteristics
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
1.
2.
3.
4.
5.
EXERCISE 2
Section 1: Introduction to concepts of Statistical variation, location and dispersion effects
Some of the monitoring tools used in the production stages
Inspection / screening Low CPK
Output
Process characterization DOE / Taguchi /
change specification ??
SPC charts (Cpk , Cp , Ppk)
R&D tools (Taguchi, DOE optimization,
RSM)
Input
ShipmentProduct, Y Process
X1 X2 : :
Xk
Example of tools used in a proactive production / quality improvement approach:
Items Tools
Customers voices/inputs Survey, visits. ↓
Product design Quality Function Deployment (QFD).
↓
Process design Manufacturing Technology / Poka Yoke.
↓
Process optimization and robustness DOE / Taguchi / EVOP
Response Surface. ↓ Production control FMEA / PPA. ↓
Actual production SPC, In-Process Inspection, Statistical Techniques.
↓
Finished products Sampling inspection / 100% inspection.
↓
Con
tinuo
us Im
prov
emen
t
Ship to customer
10
Design of Experiments for Process Optimization
Data is an essential part of problem solving. As mentioned by
Sherlock Holmes
“It is a capital mistake to theorize before one has the data.”
In problem solving reality, there are two major problems to be
accomplished:
• Problem of information
• Problem of variation.
Process Control Vs Design of Experiments
Tools Nature Status
Process Control. (Control Charts, Sampling Plans.)
Data oriented. Passive.
Design of Experiments.
Information oriented. Active.
When to use Design of Experiments?
• Machine problems have been identified and eliminated.
• E.g. alignment problems, blockages, faulty parts have been
eliminated.
• Materials should be appropriate and no abnormality observed.
• People have been trained and certified to the job if necessary.
• Procedures have been standardized.
Design of Experiment should be used to solve Optimization
Problems or working towards Process Optimization.
11
Section 1: Introduction to concepts of Statistical variation, location and dispersion effects
Ip
P
Q
P
MT
3
12
EXERCISE
dentify the suitable monitoring tools that can be applied in the following rocesses:
Types of Processes
erformances Process I Process II
uality Depends on machines Depends on people
roductivity Speed of machines Skills of operators
onitoring ools usable
Design of Experiments for Process Optimization
Basic Statistical Inferences
Basic Statistical Inferences
Statistical Inference enables us to understand the general behavior
or situation through proper interpretation of a limited amount of
available information.
Deduce/Infer Samples Population
Confidence level for point estimation
• Point estimation ~ the prediction of a specific value such as
Mean or Average concerning a population.
• Confidence level ~ the degree of certainty on the accuracy or
precision of the values predicted.
Process Mean
Confidence level for Mean estimation
-z α z α
13
Section 1: Introduction to concepts of Statistical variation, location and dispersion effects
To estimate the Mean (µ) for the population, we can use the
Average ( ) and Standard Deviation (s) from the random sample
with a sample size n (n>25) and the following:
x
- (zα s/ √ n) < µ < + (zα s/ √ n ) xx zα = 1.96 for 95% confidence interval
zα = 2.58 for 99% confidence interval
Hypothesis test
• Hypothesis ~ a statement, which may or may not be true,
concerning one or more population.
• The purpose of hypothesis test is to make conclusion from
statistical analysis instead of from gut-feel.
• Decision in hypothesis test:
State of hypothesis True False
Do not reject OK Type II error
Dec
isio
n
Reject Type I error OK
• Steps :
1. State the null hypothesis ( Ho ) and alternate hypothesis ( Ha ),
e.g.
i. Ho : µ1 = µ2
ii. Ha : µ1 ≠ µ2 or Ha : µ1 > µ2
2. Calculate the appropriate statistics.
3. Test hypothesis (at 95% or 99% confidence).
14
Design of Experiments for Process Optimization
Statistical deduction for 2 distributions
1. 2 distributions can be compared for the differences in the means
as well as the differences in the variances.
2. Normally we can compare the sum of squares of 2 distributions
if they are from normal distribution family.
3. The test statistics to be used here will be F-Statistics.
Sum of Squares due to effect A Sum of Squares due to effect B
~ Fdf A, df B, α
where df A - degree of freedom for A
df B - degree of freedom for B
• Example :
Output data distribution:
Random Errors/Noises
Factors Process Output
The variation of the output can be stratified into variations due to
random errors and also variations due to factors.
This can be represented as:
SSTotal = SSFactors + SSErrors
where SS is the Sum of Squares.
Application examples will be shown in Factorial Designs.
15
Section 1: Introduction to concepts of Statistical variation, location and dispersion effects
One-Sample Hypothesis Testing Procedure
Objective: To test on the basis of sampled data whether the
population mean or population percentage differs
from specified standard or historical value.
One-Sample Hypothesis Test of Means
Case 1: n ≥ 30 and σ unknown
Example:
In the past, mean library usage per cardholder was 8.5 books during
the year. A random sample of 100 cardholders showed the
following results this year: = 9.34 books, s = 3.31 books. The
library administration would like to know whether this year’s mean
usage (µ) has changed from that for past years. Conduct the
appropriate test, controlling the α risk at .05. State the alternatives.
x
Steps in the Hypothesis Testing Procedure:
1. Ho : µ = 8.5
Sample data is used to decide whether or not Ho is rejected.
2. H1 : µ ≠ 8.5 two-sided alternative
H1 : µ > 8.5 one-sided (upper-tail) alternative
H1 : µ < 8.5 one-sided (lower-tail) alternative
Compare mean of sampled population ( = 9.34) to the
specified standard or historical value of the population mean (µo
= 8.5).
x
Select alternative hypothesis depending on the nature of
problem at hand.
16
Design of Experiments for Process Optimization
3. Level of significance of the test, α = .05.
A 5% risk of erroneously rejecting Ho when Ho is true.
In practice, α = .05 or .01.
4. Since n is greater than 30, use the z-distribution.
Test Statistic:
- µo 9.34 – 8.5 Z =
s/ √ n =
(3.31/ √100) = 2.54
x
Note: If σ is known, replace s with σ.
5. Acceptance and Critical regions with α = .05
Two-sided alternative:
Critical Region
(Reject Ho) Critical Region
Acceptance Region
(do not reject Ho)
z 0-z.025 = -1.96 z.025 = 1.96
The values 1.96 and -1.96 are included in the acceptance region.
If Ho is rejected, we conclude that the true population mean does
not equal to 8.5. It does not matter if the population mean is
more or less than 8.5.
17
Section 1: Introduction to concepts of Statistical variation, location and dispersion effects
One-sided (upper-tail) alternative:
Critical Region Acceptance Region
z
0 z.05 = 1.645
Ho will be rejected if the value of the sample mean is
significantly higher than 8.5 .
One-sided (lower-tail) alternative:
Critical Region
Acceptance Region
z
0-z.05 = -1.645
Ho will be rejected if the value of the sample mean is
significantly lower than 8.5 .
6. Conclusion: For : H1 : µ > 8.5 , reject Ho : µ = 8.5 since 2.54
is greater than 1.645 .
18
Design of Experiments for Process Optimization
Note: Acceptance and Critical Regions with α = 0.01
Two-sided alternative:
Critical Region
Critical Region
Acceptance Region
z
0-z.005 = -2.58 z.005 = 2.58
One-sided (upper-tail) alternative:
Critical Region
Acceptance Region
z
0 z.01 = 2.33
One-sided (lower-tail) alternative:
Critical Region
Acceptance Region
z 0-z.01 = -2.33
19
Section 1: Introduction to concepts of Statistical variation, location and dispersion effects
Hypothesis: Ho : µ = µo
HA : µ ≠ µo
Test statistic:
- µo - µo Z =
σ / √ n or
s / √ n
If n ≥ 30
and σ unknown
Rejection region: Reject Ho if z < -zα/2 or z > zα/2
Hypothesis: Ho : µ ≤ µo
HA : µ > µo
Test statistic:
- µo - µo Z =
σ / √ n or
s / √ n
If n ≥ 30
and σ unknown
Rejection region: Reject Ho if z > zα
-zα/2 z
Reject Reject
0 zα/2
α/2 α/2
x x
x x
α
Reject
z
0 zα
20
Design of Experiments for Process Optimization
Hypothesis: Ho : µ ≥ µo
HA : µ < µo
Test statistic:
- µo - µo Z =
σ / √ n or
s / √ n
If n ≥ 30
and σ unknown x x
Rejection region: Reject Ho if z < -zα
α
Reject
z
0-zα
21
Section 1: Introduction to concepts of Statistical variation, location and dispersion effects
One-Sample Hypothesis Test of Percentages (n ≥ 100)
Example 1: One-sided (lower-tail) alternative
The manager of Company A advertised that 90 percent of its’
customers are satisfied with the company’s services. The manager
of Company B feels that this is an exaggerated statement. In a
random sample of n = 150 of Company A’s clients, 132 (x) said
they were satisfied. What should be concluded if a test were
conducted at the .05 level of significance?
Solution:
Hypothesis Testing Procedure is essentially the same procedure
used for testing means with a large sample size except for the
following:
Test Statistic: z = ( P – πo ) / σp
p = ( x/n ) 100
πo = Hypothesized value of the population percentage
σp = Standard deviation of percentage
= πo ( 100 - πo ) / n
Ho : π = 90% ( πo = 90%)
H1 : π < 90%
α = .05
p = 132 / 150 x 100 = 88%
σp = 90 ( 100 – 90 ) / 150 = 2.4%
∴z = ( 88 – 90 ) / 2.4 = -.833
22
Design of Experiments for Process Optimization
Example 2: One-sided (upper-tail) alternative
A company anticipates that 5% of its’ employees drive to work. A
random sample of 200 employees showed that 9% (p) drive to
work. Using a significance level of .01, conduct a statistical test to
test the accuracy of the company’s assumption.
Solution:
Ho : π = 5%
H1 : π > 5%
α = .01
σp = 5 ( 100 – 5 ) / 200 = 1.5%
z = ( 9 - 5 ) / 1.5 = 2.67
23
Section 1: Introduction to concepts of Statistical variation, location and dispersion effects
Example 3:
Suppose the target property value in the previous example is 120.
How may one determine whether the production process has
deviated from this target?
Outline of solution:
We may answer this question by a test of hypothesis.
Ho : µ = 120
H1 : µ ≠ 120
Set α = 0.05
Test statistic: - µo x
z = σ / √ n
Critical region: z < -1.96, z > 1.96
Value of test statistic: z = 123.8 - 120
2 / √ 4
= 3.8
24
Design of Experiments for Process Optimization
Robust Design for a Process
Robust Design for a Process
Coding of Factors
It is very common for us to code the Process Factors. Normally we
do not need to write the actual factor in the experimental array, we
usually code them as Low (-), Medium (0) or High (+) with the
following transformation.
For example, say we have factor A, we denote
AMin – Mean of A
Mean of A = -1
AMax – Mean of A
Mean of A = +1
AMedium – Mean of A
Mean of A = 0
Effects/Types of Factors
There are three types of factors in general:
1. Critical factors - factors that have direct and big impact to
the process, usually there are only one or
two factors of this type
2. Trivial factors - factors that have little effects to the
process. There are usually many factors
of this type.
3. Dummy factors - factors that have no effect at all to the
process.
25
Section 1: Introduction to concepts of Statistical variation, location and dispersion effects
Process Models
Many models appear for production process.
There may be Linear Effect Models like those that follow Ohm’s
Law, Quadratic Models, Exponential Models or other non-linear
models.
Generally we can represent a process model by
Y = F(X) + Random Errors
The function F needs to be determined with Design of Experiments
or Regression Analysis.
In either case, data needs to be collected.
Robust Designs
A robust design will meet the following criteria:
1. The parameters combination gives the optimum performance.
2. The process is NOT sensitive to errors or noises.
3. Critical parameters have been determined and controlled
appropriately.
4. Output of the process is not varied greatly by any small
variations in the input parameters or materials.
26
Design of Experiments for Process Optimization
Selection of Factors
Factors considered for experimentation can be selected from:
1. Engineering knowledge.
2. Experiences.
3. From previous experiments.
Golden rules for experimentation:
1. Start with large number of factors at the initial experiment.
2. The range of factors should be set as close to the limit as
possible.
Funneling of factors during experimentation:
Stage of Experiments Number of Factors
Initial Experiment Many Factors
Secondary Experiment
Critical Factors (from initial experiment)
Fine Tuning Experiment
Optim
ize
27
SECTION 2:
Factorial Experiments / Designs
Section 2 Objectives
Section 2 Objectives
At the end of this section, you will be able to:
• Understand and appreciate the powers of Factorial Experiments
versus One-Factor-at a Time Experiments.
• Construct a Factorial Experiment.
• Analyze the data obtained from Factorial Experiments.
• Use Yate’s Algorithm to compute the effects of factors.
• Decide on the optimum operating condition from the Experiment
Data.
29
Section 2: Factorial Experiments / Designs
Comparison of Factorial Designs to One Factor at a Time Experiments
One Factor at a Time Experiment
Comparison of Factorial Designs to One Factor at a Time Experiments
EXERCISE 4
R A
I
V
Ohm’s Law states that:
For a given temperature,
R = V/I
The followings are some typical reading for current (I) for a
piece of 30 Ohms resistor under room temperature.
Voltage (V) Current (I) Average (I) 2.0 0.07, 0.06 2.5 0.08, 0.08 3.0 0.10, 0.10 3.5 0.11, 0.12 4.0 0.13, 0.14 4.5 0.15, 0.14 5.0 0.17, 0.17
Plot V against average I.
30
Design of Experiments for Process Optimization
Questions:
1. What is the shape of the line you obtained?
2. What is the relationship for this graph called?
3. What would happen to the current if we keep the
voltage constant but vary the temperature?
4. How many experimental runs are required to get
the effect of temperature-voltage interaction?
5. What is your opinion on the effectiveness of One-
Factor at a Time Experiment?
31
Section 2: Factorial Experiments / Designs
Classical experiment Vs Factorial Experiment
Classical experiment Factorial experiment
↓ ↓
Vary one factor at a time. Vary factors simultaneously.
↓ ↓
No information about possible
interaction between factors.
Provides information about
main and interaction effects.
↓ ↓
Set of measurements collected
useful for making inferences
about that factor alone.
Can estimate effects of
individual factors at several
levels of the other factors.
↓
Conclusions valid over a
range of experimental
condition.
↓
More efficient than Classical
Experiments.
Appropriateness of Factorial Experiments
• Most efficient experimental design for studying the effect
of two or more factors on the response.
• Under Factorial Experiment, each complete trial or
replication of the experiment will have all its’ possible
combinations of the levels of the factors investigated.
For example, if there are p levels of factor A and q levels of
factor B, then each replicate contains all the pq treatment
combinations. This design is called pxq Factorial Design.
32
Design of Experiments for Process Optimization
Method to Construct a Factorial Design
Method to Construct a Factorial Design
Experiment Set-up
A single replicate Assign pg experimental units at random,
one to each treatment combination.
Replicates Repeat experiment r times using r sets of
pq experimental units.
When we are performing a single replicate experiment, it is
impossible for us to calculate the interaction effects.
Errors have to be estimated using the general Sum of Squares
apportion. E.g. Use Sum of Squares of factors that contribute
less than 5% to Total Sum of Squares.
Recall: Effect of a factor is the change in the response
produced by a change in the level of the factor.
Frequently known as main effect because it refers to
primary factors of interest in the experiment.
Interaction Effect : Join effects of the factors are known as
interaction effects.
2 Level (2k) Factorial Design
i.e. Factors, each at only two levels (High, Low).
For a complete set of non-replicate, k factors
2 x 2 x … x 2 = 2k runs/observation
33
Section 2: Factorial Experiments / Designs
When to use 2 Level (2k) Factorial Design?
Useful in early stages of experimental work when there are
likely to be many factors investigated.
Sequentially, 2k experiments can be used to obtain optimum
operating condition.
Reasons:
1. Require relatively few runs to indicate major trends and so
determine a promising direction for further
experimentation.
2. Two levels fractional factorial designs, especially, look at a
large number of factors superficially rather than a small
number of factors thoroughly. The interpretation of the
observations can be carried out mainly by using sense and
simple arithmetic.
Note:
Since 2 levels for each factor are used, we must assume the
response is approximately linear over the range of the factor
levels chosen.
34
Design of Experiments for Process Optimization
Estimation of Main Effect and Interactions of Factors
Estimation of Main Effect and Interaction of Factors
Classical Experiment
Factor B B1 B2
A1 A1B1 A1B2 Factor A
A2 A2B1
Two observations are taken at each treatment combination.
Only the treatment means are presented for each treatment.
The effects of factors A and B are estimated using the treatment
means.
Main effect of A A2B1 - A1B1
Main effect of B A1B2 - A1B1
Interactions can cause misleading results since we have no
information from this experiment.
Factorial Experiment
Example: interaction Vs no interaction for a 2x2 Factorial
Design.
For the purposes of condensation, only the treatment means are
presented for each treatment.
35
Section 2: Factorial Experiments / Designs
Table 1 : A Factorial experiment without interaction.
Factor B B1 B2
A1 20 30 Factor A
A2 40 52
Main Effect of Factor A
(40 + 52) (20 + 30) = 2 -
2 = 21
Difference between the average response at the first level of A
and the average response at the second level of A.
Main Effect of Factor B
(30 + 52) (20 + 40) = 2 -
2 = 11
at A1 : Effect B = 30 - 20 = 10
at A2 : Effect B = 52 - 40 = 12
at B1 : Effect A = 40 - 20 = 20
at B2 : Effect A = 52 - 30 = 22
There is no interaction between A and B since the difference in
response between the levels of one factor is the same at all
levels of the other factors.
36
Design of Experiments for Process Optimization
60_ 50_ B2 Expected 40_ B1 Response 30_ B2 20_ 10_ B1 A1 A2
Figure 1 : A Factorial Experiment without significant
interaction.
B1 and B2 lines are almost parallel.
Table 2 : A Factorial experiment with interaction.
Factor B B1 B2
A1 20 40 Factor A
A2 50 12
Main Effect of Factor A
(50 + 12) (20 + 40) = 2 -
2 = 1
May conclude that there is no effect due to A.
However, at B1 : Effect A = 50 - 20 = 30
B2 : Effect A = 12 - 40 = -28
Factor A has an effect! It depends on the level of factor B.
∴There is interaction between A and B.
37
Section 2: Factorial Experiments / Designs
Main Effect of Factor B
(40 + 12) (20 + 50) = 2 -
2 = -9 at A1 : Effect B = 40 - 20 = 20
at A2 : Effect B = 12 - 50 = -38
Similarly, the effect of Factor B depends on the level of Factor
A.
60_ 50_ B1 Expected 40_ B2 Response 30_ 20_ 10_ B1 B2 A1 A2
Figure 2: A Factorial Experiment with interaction.
B1 and B2 lines are not parallel.
Note : Knowledge of the AB interaction is more useful
than knowledge of the main effects.
Summary
• When an interaction is large, the corresponding main
effects have little practical meaning!
• Examine levels of one factor, say A, with levels of the other
factor fixed to draw conclusions about the main effect of A.
• Figures 1 and 2 are useful in interpreting significant
interactions and reporting results to non-statistically trained
management.
38
Design of Experiments for Process Optimization
• However, do not utilize them as the sole technique of data
analysis since their interpretation is subjective and their
appearance is often misleading.
Example: Bonding Experiment
Objective: An experiment was designed to study and evaluate
the effect of EPI type, evaporator and alloy type at
wafer fabrication on bondability of Silicon wafers.
Factors Levels Coding Epi Type (T) Shin-Etsu - Showa-Denko + Evaporator (E) R & D - Production + Alloy (A) Rapid Thermal Alloy - Furnace +
Assignment of high and low levels.
Variable Factor : Code the low and high levels with a minus
and plus sign.
Attribute Factor : Arbitrarily code the two “Levels” with a
minus and plus sign.
Select the actual values of the – and + levels as boldly as
possible without making the experiment inoperable.
E.g.:- Two temperature extremes, two pressure extremes, two
time values, two machines; one good and one bad in
terms of performance etc.
39
Section 2: Factorial Experiments / Designs
Table of Contrast Coefficients.
Signs for calculating effects for 23 factorial example.
Test Condition AVE A T AT E AE TE ATE
1 .534 + - - + - + + - 2 .954 + + - - - - + + 3 .515 + - + - - + - + 4 .912 + + + + - - - - 5 .450 + - - + + - - + 6 .573 + + - - + + - - 7 .531 + - + - + - + - 8 .558 + + + + + + + +
Divisor * 8 4 4 4 4 4 4 4 Sum 5.027 .967 .005 -.119 -.803 -.667 .127 -.073
Contrast
Effect .628 .242 .001 -.030 -.201 -.167 .032 -.018 Sum of Squares - .116 0 .001 .081 .055 .002 0
Yi
* Obtain the divisor by counting the number of ‘+’ signs in the
column.
Note: Obtain the signs for the interactions by multiplying the signs of their respective variables.
Column Effects Calculation
3 Average + .534 + .954 + .515 + .912 + ... + .558 = .628 8
4 A -.534 + .954 - .515 + .912 - .450 + .573 - .531 + .558 = .242 4
9 AE +.534 -.954 +.515 -.912 -.450 +.573 - .531 +.558 = -.167 4
Etc Etc Etc
Note: Contrast is used to estimate all main and interaction
effects.
E.g.: CA = contrast for A
= -1(.534)+ 1(.954) - 1(.515) + 1(.912) – 1(.450) +
1(.573) – 1(.531) + 1(.558)
= .967
40
Design of Experiments for Process Optimization
The contrast coefficient is always either + 1 or - 1. The
contrast coefficients for estimating the interaction effect are just
the product of the corresponding coefficients for the two main
effects. Sum of the coefficients is equals to zero.
Definition of "Orthogonal"
Any two columns 4 to 10 are Orthogonal or balanced, since the
numbers of combinations (- -), (- +), (+ -), (+ +) in any two
columns are equal.
41
Section 2: Factorial Experiments / Designs
Application of Yates’ Algorithm
Application Of Yates’ Algorithm
Yates Algorithm for the 2k design
This is a very simple technique devised by Yates (1937) for
estimating the effects and determining the sum of squares in a 2k
Factorial Design.
Yates Algorithm, 23 Factorial example:
Test Condition
Design Matrix Variables
Run Averages Algorithm
A T E Yi (1) (2) (3) Divisor Estimate Effects
1 - - - .534 1.488 2.915 5.027 8 .628 Average2 + - - .954 1.427 2.112 .967 4 .242 A 3 - + - .515 1.023 .817 .005 4 .001 T 4 + + - .912 1.089 .150 -.119 4 -.031 AT 5 - - + .450 .420 -.061 -.803 4 -.201 E 6 + - + .573 .397 .066 -.667 4 -.167 AE 7 - + + .531 .123 -.023 .127 4 .032 TE 8 + + + .558 .027 .096 -.073 4 -.018 ATE
Procedures
1. Arrange the observations in standard order. A 2k Factorial
Design is in standard order when as in the design matrix,
• the first column of matrix consists of successive minus and
plus signs,
• the second column consists of successive pairs of minus and
plus signs,
• the third column consists of four minus signs followed by
four plus signs, and so forth.
In general, the kth column consists of 2k-1 minus signs followed by
2k-1 plus signs.
42
Design of Experiments for Process Optimization
2. Column Yi contains the corresponding average for each run. (If
the design had not been replicated, each average would be the
single observation recorded for that run).
3. Consider the averages in successive pairs.
• The first four entries in column (1) are obtained by adding
the pairs together:
1.488 = .534 + .954,
1.427 = .515 + .912 etc.
• The second four entries in column (1) are obtained by
subtracting top number from bottom number of each pair:
.420 = .954 - .534,
.397 = .912 - .515 etc.
4. Obtain column (2) from column (1) in the same way that
column (1) is obtained from column Yi.
5. Finally, column (3) is obtained from column (2) similarly. In
general, construct k columns of this type for 2k design.
6. To obtain the effects, divide column (3) by appropriate divisor.
• First divisor = 2k i.e. (23 = 8)
• Remaining divisors = 2k-1 i.e. (23-1 = 4)
7. The effects are identified by locating the plus signs in the design
matrix. The first estimate is the grand average of all
observations.
43
Section 2: Factorial Experiments / Designs
8. Obtain the sum of squares for effects in the same manner as for
table of contrast coefficients.
For cases with two or more response variables, separately determine
the significant factors (if the responses are not correlated) and their
optimum levels for each response variable.
Use engineering judgment to resolve conflict between optimum
levels suggested by the different response variables.
If the responses are correlated, we have to analyze the results based
on the value function of the variables
Example : Yate’s Algorithm, pilot plant example, single
replicate.
Design Matrix
Variables Run # T C K
Response (1) (2) (3) ID Estimate of Effect
Sum of Squares DF MS F
1 - - - 60 Average - - - - - 2 + - - 72 T 3 - + - 54 C 4 + + - 68 TC 5 - - + 52 K 6 + - + 83 TK 7 - + + 45 CK 8 + + + 80 TCK Total 34,342 - - 7 - Error
44
Design of Experiments for Process Optimization
EXERCISE 5
Find the effects of all the factors.
Run
Z Position Time Diameter Type Yield Level
1 - - - - 93.8 2 + - - 14.4 3 - + - - 37.5 - 4 4 + + - - 75.0 Z Position + 8 5 - - + - 0.6 6 + - + - 20.0 7 - + + - 0.6 Time 8 + + + - 21.3
- +
8 12
9 - - - + 74.4 10 + - - + 51.9 - 0.20 11 - + - + 84.4 Diameter + 0.25 12 + + - + 75.0 13 - - + + 92.5 14 + - + + 33.1 Type 15 - + + + 99.4
- +
Type K Type E
16 + + + + 58.8
-
45
Section 2: Factorial Experiments / Designs
Appropriate Procedures to Run a Factorial Design - Case Study
Appropriate Procedures to Run a Factorial Design - Case Study
Experimental Procedures
1. Recognition & statement of problems.
• Define objective clearly.
2. Choice of factors & levels.
• Study the feasible region.
3. Selection of a response variable.
4. Choice of experimental design.
• Proper selection of mathematical model.
5. Perform the experiment.
• Collect actual data.
• Supervise the experiments closely.
• Don’t leave too much discretion to Operators.
• Note any unusual observations.
6. Analysis of data using statistical method.
• Check data normality.
• Use software if applicable.
• Transform the data if necessary.
• Look for abnormal behavior of data.
46
Design of Experiments for Process Optimization
7. Conclusion and recommendation.
• Select the most stable (least variation) condition.
• Select the optimum conditions.
Note : Statistical application : steps 4 to 6.
Factorial Experiment Case Study
Case study: Silicon wafer saw process optimization.
Objective: To reduce the chipping of wafer at sawing process.
Initial status: Defective rate of 9000 ppm.
• Due to the machine set up, engineering suspect
the problem is caused by non-optimized
parameters performance.
• Decided to use DOE to solve the problem.
Process flow:
Mounting ⇓
Heating the wafer ⇓
Sawing Area of operation ⇓
Expansion, Probing and etching
⇓
Inspection For breakage Area of detection
⇓
100% inspection for other parameter
47
Section 2: Factorial Experiments / Designs
Planning the experiment:
Four factors were identified by engineering and
statistician to be related to the problem. They were:
Levels Factor - +
(A) Cutting speed 0.100 0.600 (B) Cutting depth 0.2 0.5 (C) Finishing mode A B (D) Spinning speed 33K 35K
Note: All settings observed the maximum
specification limits requirement.
Experiment set up:
• Using a full Factorial Design of 16 runs. • Response is breakage rate.
Factorial Design
Cutting speed
Cutting depth
Finishing mode
Spindle speed Response (%) No. of run
(A) (B) (C) (D) Breakage rate 1 + + + + 0.57 2 - + + + 0.80 3 + - + + 0.06 4 - - + + 3.47 5 + + - + 1.81 6 - + - + 4.82 7 + - - + 0.11 8 - - - + 4.05 9 + + + - 1.13
10 - + + - 0.48 11 + - + - 0.18 12 - - + - 0.98 13 + + - - 4.04 14 - + - - 6.10 15 + - - - 1.88 16 - - - - 2.33
48
Design of Experiments for Process Optimization
DOE analysis using Yate’s Algorithm:
Run Variables Run Algorithm . A B C D Ave. % (1) (2) (3) (4) Div. Est. Ide. Sum of Sq.
1 + + + + 0.57 -0.23 3.18 2.25 -0.81 8 -0.10 ABCD 0.04 2 - + + + 0.80 -3.41 0.93 3.06 0.85 8 0.11 BCD 0.05 3 + - + + 0.06 -3.01 1.45 -4.63 0.95 8 0.12 ACD 0.06 4 - - + + 3.47 -3.94 -1.61 -5.48 5.69 8 0.71 CD 2.02 5 + + - + 1.81 0.65 -2.16 3.31 4.27 8 0.53 ABD 1.14 6 - + - + 4.82 -0.80 2.47 2.36 -6.07 8 -0.76 BD 2.30 7 + - - + 0.11 -2.06 0.45 -5.89 7.93 8 -0.99 AD 3.93 8 - - - + 4.05 -0.45 5.93 -11.58 -1.43 8 -0.18 D 0.13 9 + + + - 1.13 1.37 -3.64 4.11 5.31 8 0.66 ABC 1.76
10 - + + - 0.48 3.53 -6.95 -0.16 -10.11 8 -1.26 BC 6.39 11 + - + - 0.18 6.63 -0.15 0.31 5.67 8 0.71 AC 2.01 12 - - + - 0.98 4.16 -2.51 6.38 -17.47 8 -2.18 C 19.08 13 + + - - 4.04 1.61 4.90 -10.59 3.95 8 0.49 AB 0.98 14 - + - - 6.10 1.16 10.79 -2.66 6.69 8 0.84 B 2.80 15 + - - - 1.88 10.14 2.77 15.69 -13.25 8 -1.66 A 10.97 16 - - - - 2.33 4.21 14.35 17.12 32.81 16 2.05 AVE --
Total Error 53.65
Conclusion:
Main effect and interaction:
1. An increase in cutting speed (Factor A) from
0.100 to 0.600 inch/sec. reduces the defective
rate by about 1.66%.
2. By changing the finishing mode (Factor C) from
Mode A to Mode B will reduce the rejection rate
by about 2.18%.
49
Section 2: Factorial Experiments / Designs
Two way table for process development data:
Factors A B Mean Response (% Breakage) + + 1.8875 + - 0.5575 - + 3.0500 - - 2.7075
Factor C + 0.95875 - 3.1425 5.8785 1.2225 Factor A - +
Result monitoring:
WW Qty Insp. % Breakage 1 188522 1.51 2 1797530 2.39 3 3522186 2.20 4 1998227 0.66 5 4463391 0.34 6 3998492 0.43 Before 7 5648897 0.09 Improvement 8 7067791 0.25 9 2873447 0.45 10 3625711 0.92 11 5213613 0.48 12 2164495 0.76 13 1189388 0.036 14 3517105 0.087 After 15 4424915 0.029 Improvement 16 3392512 0.018
50
Design of Experiments for Process Optimization
Results:
% defectives
Before improvement 0.867%
After improvement 0.042%
% of improvement 0.825%
Volume of production per year 360KK
Cost saving from defectives US$20K
Reduced labour cost about US$26K
Total cost saving per year about US$50K
51
SECTION 3:
Fractional Factorial Experiments
Section 3 Objectives:
Section 3 Objectives
At the end of this section, you will be able to:
• Construct the Appropriate Fractional Factorial Design from
Factorial Experiments based on your specific requirements.
• Construct the “Mirror Image” of a specific Fractional Factorial to
get the main or interaction effect.
• Analyze data from Fractional Experiment.
53
Section 3: Fractional Factorial Experiments
Introduction to Fractional Factorial Designs
Introduction to Fractional Factorial Designs
Fractional Factorial Designs
• A fraction of full Factorial Experiment.
• Produce some degree of confounding.
• Useful for screening purposes.
• Taguchi L orthogonal arrays, Plackett-Burman design are
examples of fractional factorials.
• Resolution number to determine the powers of fractional
factorial.
Defining Words
• The shortest ‘words’ to describe the confounding effect.
• E.g. I = ABCDE this is a resolution V fractional factorial.
I = ABCDE or A = BCDE means main effect A
confounded with four-factor-interaction (BCDE) effect.
AB = CDE means two factors interaction AB effect
confounded with three-factor-interaction (CDE) effect.
54
Design of Experiments for Process Optimization
Construction of Fractional Factorial Designs
Construction of Fractional Factorial Designs
General Procedures for Selection of Fractions to Perform Experiment
1. Determine the defining `words’ for these fractional factorials i.e.
by selecting the unimportant interactions to form a word.
2. Determine the sign of the word.
3. From a full factorial, select the experimental array based on the
signs.
4. If we like to have half of a fraction, then we must have one
defining word. If we like to have a quarter of a fraction, then
we must have 2 words.
5. Do not perform fractional factorial for 23 experiments.
6. We can choose any fraction of factorials from the blocks we
made.
7. If the first block does not give us satisfactory results, the next
block can be used to dealiased the confounded effects.
8. Higher resolution designs will be better as compared to lower
resolution designs.
55
Section 3: Fractional Factorial Experiments
Confounding of Effects of Factors
Confounding of Effects of Factors
Suppose, the following 23 factorial design.
E.g. we write I = ABC or resolution III, we will have the main
effects confused with 2 way interaction.
ABC = +1 (Block I)
or ABC = -1 (Block II)
Factors A B C ABC Block Run #
1 - - - - II * 2 + - - + I * 3 - + - + I 4 + + - - II * 5 - - + + I 6 + - + - II 7 - + + - II * 8 + + + + I
We can group the runs with (*) to start with block. Hence the
following will be obtained:
Design I Factors A B C Response
Run # 1 + - - Y1 2 - + - Y2 3 - - + Y3 4 + + + Y4
Hence, effect of
A + BC = (Y4 + Y1) – (Y3 + Y2) / 2 = L1
B + AC = (Y4 + Y2) – (Y3 + Y1) / 2 = L2
C + AB = (Y3 + Y4) – (Y1 + Y2) / 2 = L3
Thus, the effects are confounded. More over
µ + ABC = (Y1 + Y2 + Y3 + Y4) / 4 = L4
is also confounded.
56
Design of Experiments for Process Optimization
Extracting Information from Fractional Factorial Designs / Experiments
Extracting Information from Fractional Factorial Designs / Experiments
If interactions (some) are important, the result is confusing!
However, through some technical or engineering knowledge, if
these interactions can be ignored, then the estimate of effect A, B &
C can be obtained.
Usually, the higher resolution design will position us in a better
situation.
E.g. If, we have a defining word such as I = ABCDEF for six
factors at 2 levels each, experimental design, then main effect will
be confused with BCDEF which is not significant usually !!
57
Section 3: Fractional Factorial Experiments
Fold-Over and De-confounding of Effects from Fractional Factorial Designs
Fold-Over and De-confounding of Effects from Fractional Factorial Designs
If our fraction really doesn’t give any factorial results, we may
perform the fold over experiment like those in block II.
Hence,
Design II Factors A B C Response
Run # 1 - - - Y5 2 + + - Y6 3 + - + Y7 4 - + + Y8
Hence, effect of
A - BC = (Y6 + Y7) – (Y5 + Y8) / 2 = L1’
B - AC = (Y6 + Y8) – (Y5 + Y7) / 2 = L2’
C - AB = (Y8 + Y7) – (Y5 + Y6) / 2 = L3’
and
µ - ABC = (Y5 + Y6 + Y7 + Y8) / 4 = L4’
Hence, by applying addition or subtraction on the L1 + L1’ , we can
estimate the effects of factors we need accordingly.
Example:
Effect of A = 1/2 ( L1 + L1’ )
B = 1/2 ( L2 + L2’ )
and so on, and
Effect of BC = 1/2 ( L1 - L1’ )
AC = 1/2 ( L2 - L2’ )
and so on !!
58
Design of Experiments for Process Optimization
EXERCISE 6
Fractional Factorial Experiment
Objective: To design a new carburetor to give lower level of
unburned hydrocarbons.
Variable - - A Tension on spring Low High B Air gap Narrow Open C Size of aperture Small Large D Rate of flow of gas Slow Rapid
The following results were obtained:
A B C D Unburned hydrocarbon - + + + 8.2 - - + + 1.7 - - - + 6.2 + - - - 3.0 + - + + 6.8 + + + - 5.0 - + - - 3.8 + + - + 9.3
Analyze and comment on the results.
59
Section 3: Fractional Factorial Experiments
Real Life Application ~ Funneling the Factors into Critical, Trivial and Dummy
Real Life Application ~ Funneling the Factors into Critical, Trivial and Dummy
Case study – Fractional Factorial Application
• Mold - Bubble problem for more than a year.
• Bubble problem - 30% ~ 40% in magnitude.
• Try several engineering modification (in design) but failed to
achieve consistency and drive down the bubble contents.
• Many parameters influence the Mold such as raw material, flow
speed, pressure, heating time etc.
• Altogether 7 factors were pinpointed by engineers! Approach:
o Since there were a large number of factors, Statistician then
suggested Fractional Factorial as a screening experiment.
o Basic belief: Not all the Seven factors cited were important!
o By coding the factor as A, B, C, D, E, F, G, a 8 runs
experiment was carried out.
• Experiment Set up:
Factors Run A B C D E F G
1 - - - - - - - 2 - - - + + + + 3 - + + - - + + 4 - + + + + - - 5 + - + - + - + 6 + - + + - + - 7 + + - - + + - 8 + + - + - - +
60
Design of Experiments for Process Optimization
• Experiment Results:
Responses (% bubble) Run I II Average
1 99.3 99.3 99.3 2 6.0 2.8 4.4 3 91.2 94.2 92.7 4 0.0 1.3 0.7 5 32.7 21.7 27.2 6 70.5 66.2 68.3 7 18.7 20.2 19.4 8 69.7 90.5 80.1
Sample size = 1200 units.
• Computation of factor’s effect:
Factor Effect Sum of Square
A -0.525 0.55 B -1.575 4.96 C -3.575 25.56 D -21.275 905.25 E -72.175 10418.46 F -5.625 63.28 G +4.175 34.86
From earlier data, run #4 gives a very low % of bubble.
Controversy in experimental results Vs engineering: factor
D (Material batches) should not be very serious in
affecting the results.
Prove that interaction occurs !!
Unable to obtain the similar results during confirmation
run.
• Compile with engineering understanding and further data
analysis, we screen out 3 factors, remaining factors A, B, C, G.
61
Section 3: Fractional Factorial Experiments
• Proceed with 24-1 fractional factorial designs or resolution IV
designs as follows:
Factors Response % bubble Run A B C G I II Average
1 - - - - 2.6 3.8 3.2 2 + - - + 5.3 11.5 8.4 3 - + - + 21.8 18.8 20.3 4 + + - - 2.0 1.1 1.6 5 - - + + 12.3 12.8 12.6 6 + - + - 0.8 1.0 0.9 7 - + + - 1.8 3.3 2.6 8 + + + + 6.3 8.1 7.2
• Result analysis
Effect A + BCG = -5.15 A = -5.15
B + ACG = 1.65 B = 1.65
C + ABG = -2.55 C = -2.55
G + ABC = 10.05 G = 10.05
Assuming 3 factors interaction Not significant!
• With the above result,
Optimum parameters
Factor A High level
Factor B Low level
Factor C High level
Factor G Low level
Confirmation results for 5000 units show that with optimum
parameter average bubble 0.9% while normal parameter average
bubble 32.7%.
62
SECTION 4:
Important Points for Implementation
Section 4 Objectives:
Section 4 Objectives
At the end of this section, you will be able to:
• Perform a sequential analysis and design of experiments.
• Design experimental runs to meet Economical, Production Timing
and other specific requirements.
• Collect the right data from the experiments.
63
Section 4: Important Points For Implementation
The Sequential Approach of Experimentation
The Sequential Approach of Experimentation
Experiment Set-up
• Should be iterative or sequential.
• Should not be comprehensive in the first attempt.
• Consideration in randomization.
• Sequence of preparing the experimental units.
• Assignment of treatment to the units.
• Sequence of performing the test runs.
• Sequence of taking measurements.
First stage experiment or Screening Experiment
• Should be screening the important factors.
• Include all `thought to be' significant factors.
• Run a fractional Experiment.
• 2 levels factors are sufficient.
• Main aim - to determine feasibility region
Second stage or Fine Tuning experiment
• Only include the important or statistically significant factors.
• More levels ( > 2 levels ) and detailed experiments are needed
to obtain the response curves.
• Main Aim: To obtain the optimum conditions.
64
Design of Experiments for Process Optimization
Economical Consideration of Experiments
Economical Consideration of Experiments
Proper procedure for Design of Experiments
• The objectives of the experiment.
• The details of the physical set-up.
• The variables to be held constant and how this will be
accomplished (as well as those that are to be varied).
• The uncontrolled variables - what they are and which ones are
measurable.
• Conditions within the experimental region where the expected
outcome is known; the anticipated performance is expected to
be inferior, especially for programs where an optimum is
sought; and experimentation is impossible or unsafe.
• The budgeted size of the experiment and the deadlines that must
be met.
• The desirability and opportunities for running the experiment in
stages.
• The response variables and how they will be measured.
• The procedure for running test, including the ease with each of
the variables can be changed from one run to the next.
• Past test data and, especially, any information about different
types of repeatability.
• The anticipated complexity of the relationship between the
experimental variables and the response variables and any
anticipated interactions.
65
Section 4: Important Points For Implementation
Other special considerations
• Record down any changes in the actual experimental conditions
that differ from the planned experimental conditions.
• Collect data on other factors that might prove important.
66
Design of Experiments for Process Optimization
Important Points to Take Note Before and During Experimentation and Data Collection
Important Points to Take Note Before and During Experimentation and Data Collection
Blocking
• To reduce known source of experimental error.
• Variation among blocks does not affect differences among
experimental units since each experimental unit appears in every
block.
• To obtain a more precise experimental result.
• Treat all experimental units as uniformly as possible.
• Any changes in technique or condition should be made on this
block.
• Run experiment block by block.
• Carry out a new randomization for each block.
• If possible, blocks should also be chosen at random.
General types of Blocking Variables
• Units of test equipment.
• Machinery measuring instrument.
• Batches of raw materials.
• Day, time of processing.
• Observers / people.
• Environment.
Good blocking variables are essential for effective experimentation.
67
Section 4: Important Points For Implementation
Selection of good Blocking Variables
• Use past experience in the related subject matter.
• Analyze results of past experiments in which blocking has been
employed to determine the effectiveness of blocking variables.
Use of more than one blocking variable
We can utilize more than one variable for determining blocks if
there is no desire to study the separate effects of each of the
blocking variables.
Example:
2 blocking variables: observer and day of treatment application.
Block 1: Observer 1, day 1
Block 2: Observer 2, day 1
Block 3: Observer 1, day 2 etc
Treat the blocks as ordinary blocks and calculate usual block sum of
squares.
For two or more blocking variables, a large number of blocks is
required, use other experimental designs.
68
Design of Experiments for Process Optimization
Design of Experiments array
Table 1: Set up of Runs Versus Factors.
2? Run # Factor A Factor B Factor C Factor D Factor E
1 - - - - - 2 + - - - - 3 - + - - -
22
4 + + - - - 5 - - + - - 6 + - + - - 7 - + + - -
23
8 + + + - - 9 - - - + - 10 + - - + - 11 - + - + - 12 + + - + - 13 - - + + - 14 + - + + - 15 - + + + -
24
16 + + + + - 17 - - - - + 18 + - - - + 19 - + - - + 20 + + - - + 21 - - + - + 22 + - + - + 23 - + + - + 24 + + + - + 25 - - - + + 26 + - - + + 27 - + - + + 28 + + - + + 29 - - + + + 30 + - + + + 31 - + + + +
25
32 + + + + +
69
Section 4: Important Points For Implementation
EXERCISE 7
o Hypothesis : To determine the wire bonding parameters for the ball bond.
o 2 full factorial design, factors: ball search height, ultrasonic power, bond
force, heater block temperature, single replicate.
o The response is pull strength as shown below :
Run # Search Ht (A)
Power (B)
Force (C)
Temp (D) Response
1 300 50 50 200 10.06 2 500 50 50 200 10.62 3 300 80 50 200 10.42 4 500 80 50 200 10.60 5 300 50 90 200 5.37 6 500 50 90 200 5.21 7 300 80 90 200 10.47 8 500 80 90 200 10.57 9 300 50 50 300 10.53 10 500 50 50 300 10.45 11 300 80 50 300 10.28 12 500 80 50 300 10.13 13 300 50 90 300 2.37 14 500 50 90 300 3.09 15 300 80 90 300 9.98 16 500 80 90 300 10.36
o Analyze and comment on the results.
70
Design of Experiments for Process Optimization
EXERCISE 8
Table 2: Result from 25 full Factorial Design, reactor example.
Run # A B C D E % Reacted 1 - - - - - 61 2 + - - - - 53 3 - + - - - 63 4 + + - - - 61 5 - - + - - 53 6 + - + - - 56 7 - + + - - 54 8 + + + - - 61 9 - - - + - 69 10 + - - + - 61 11 - + - + - 94 12 + + - + - 93 13 - - + + - 66 14 + - + + - 60 15 - + + + - 95 16 + + + + - 98 17 - - - - + 56 18 + - - - + 63 19 - + - - + 70 20 + + - - + 65 21 - - + - + 59 22 + - + - + 55 23 - + + - + 67 24 + + - + 65 25 - - - + + 44 26 + - - + + 45 27 - + - + + 78 28 + + - + + 77 29 - - + + + 49 30 + - + + + 42 31 - + + + + 81 32 + + + + + 82
+
Analyse the experiment based on 25-1 Fractional Factorial Design.
71
Section 4: Important Points For Implementation
Exercise 8 :
Splus output of AOV from25 full factorial design, reactor example. > summary
Degree of freedom SS MSS
A 1 15.125 15.125 B 1 3042.000 3042.000 C 1 3.125 3.125 D 1 924.500 924.500 E 1 312.500 312.500 A:B 1 15.125 15.125 A:C 1 4.500 4.000 B:C 1 6.125 6.125 A:D 1 6.125 6.125 B:D 1 1404.500 1404.500 C:D 1 36.125 36.125 A:E 1 0.125 0.125 B:E 1 32.000 32.000 C:E 1 6.125 6.125 D:E 1 968.000 968.000 A:B:C 1 18.000 18.000 A:B:D 1 15.125 15.125 A:C:D 1 4.500 4.500 B:C:D 1 10.125 10.125 A:B:E 1 28.125 28.125 A:C:E 1 50.000 50.000 B:C:E 1 0.125 0.125 A:D:E 1 3.125 3.125 B:D:E 1 0.500 0.500 C:D:E 1 0.125 0.125 A:B:C:D 1 0.000 0.000 A:B:C:E 1 18.000 18.000 A:B:D:E 1 3.125 3.125 A:C:D:E 1 8.000 8.000 B:C:D:E 1 3.125 3.125 A:B:C:D:E 1 2.000 2.000
72
Design of Experiments for Process Optimization
73
Exercise 8 : Splus output of AOV from25-1 full factorial design, reactor example. > summary
Degree of freedom Sum of Sq Mean Sq A 1 16.00 16.00 B 1 1681.00 1681.00 C 1 0.00 0.00 D 1 600.00 600.00 A:B 1 9.00 9.00 A:C 1 1.00 1.00 B:C 1 9.00 9.00 A:D 1 2.25 2.25 B:D 1 462.25 462.25 C:D 1 0.25 0.25 A:B:C 1 361.00 361.00 A:B:D 1 20.25 20.25 A:C:D 1 6.25 6.25 B:C:D 1 6.25 6.25 A:B:C:D 1 156.25 156.25