YOU ARE DOWNLOADING DOCUMENT

Please tick the box to continue:

Transcript
Page 1: DOE Process Optimization[1]

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

Page 2: DOE Process Optimization[1]
Page 3: DOE Process Optimization[1]

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

Page 4: DOE Process Optimization[1]
Page 5: DOE Process Optimization[1]

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

Page 6: DOE Process Optimization[1]
Page 7: DOE Process Optimization[1]

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

Page 8: DOE Process Optimization[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

Page 9: DOE Process Optimization[1]

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

Page 10: DOE Process Optimization[1]

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

Page 11: DOE Process Optimization[1]

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

Page 12: DOE Process Optimization[1]

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

Page 13: DOE Process Optimization[1]

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

Page 14: DOE Process Optimization[1]

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

Page 15: DOE Process Optimization[1]

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

Page 16: DOE Process Optimization[1]

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

Page 17: DOE Process Optimization[1]

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

Page 18: DOE Process Optimization[1]

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

Page 19: DOE Process Optimization[1]

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

Page 20: DOE Process Optimization[1]

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

Page 21: DOE Process Optimization[1]

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

Page 22: DOE Process Optimization[1]

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

Page 23: DOE Process Optimization[1]

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

Page 24: DOE Process Optimization[1]

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

Page 25: DOE Process Optimization[1]

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

Page 26: DOE Process Optimization[1]

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

Page 27: DOE Process Optimization[1]

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

Page 28: DOE Process Optimization[1]

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

Page 29: DOE Process Optimization[1]

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

Page 30: DOE Process Optimization[1]

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

Page 31: DOE Process Optimization[1]

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

Page 32: DOE Process Optimization[1]

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

Page 33: DOE Process Optimization[1]

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

Page 34: DOE Process Optimization[1]
Page 35: DOE Process Optimization[1]

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

Page 36: DOE Process Optimization[1]

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

Page 37: DOE Process Optimization[1]

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

Page 38: DOE Process Optimization[1]

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

Page 39: DOE Process Optimization[1]

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

Page 40: DOE Process Optimization[1]

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

Page 41: DOE Process Optimization[1]

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

Page 42: DOE Process Optimization[1]

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

Page 43: DOE Process Optimization[1]

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

Page 44: DOE Process Optimization[1]

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

Page 45: DOE Process Optimization[1]

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

Page 46: DOE Process Optimization[1]

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

Page 47: DOE Process Optimization[1]

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

Page 48: DOE Process Optimization[1]

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

Page 49: DOE Process Optimization[1]

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

Page 50: DOE Process Optimization[1]

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

Page 51: DOE Process Optimization[1]

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

Page 52: DOE Process Optimization[1]

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

Page 53: DOE Process Optimization[1]

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

Page 54: DOE Process Optimization[1]

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

Page 55: DOE Process Optimization[1]

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

Page 56: DOE Process Optimization[1]

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

Page 57: DOE Process Optimization[1]

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

Page 58: DOE Process Optimization[1]
Page 59: DOE Process Optimization[1]

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

Page 60: DOE Process Optimization[1]

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

Page 61: DOE Process Optimization[1]

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

Page 62: DOE Process Optimization[1]

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

Page 63: DOE Process Optimization[1]

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

Page 64: DOE Process Optimization[1]

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

Page 65: DOE Process Optimization[1]

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

Page 66: DOE Process Optimization[1]

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

Page 67: DOE Process Optimization[1]

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

Page 68: DOE Process Optimization[1]

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

Page 69: DOE Process Optimization[1]

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

Page 70: DOE Process Optimization[1]

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

Page 71: DOE Process Optimization[1]

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

Page 72: DOE Process Optimization[1]

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

Page 73: DOE Process Optimization[1]

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

Page 74: DOE Process Optimization[1]

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

Page 75: DOE Process Optimization[1]

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

Page 76: DOE Process Optimization[1]

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

Page 77: DOE Process Optimization[1]

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

Page 78: DOE Process Optimization[1]

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

Page 79: DOE Process Optimization[1]

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


Related Documents