Application of design of experiments
in computer simulation study
Shu YAMADA
[email protected]
and
Hiroe TSUBAKI
[email protected]
Supported by
Grant-in-Aid for Scientific Research 16200021 (Representative: Hiroe
Tsubaki), Ministry of Education, Culture, Science and Technology
University of Tsukuba
MBA in International Business
July 11, 2006
1
July 11, 2006
Introduction
Application of computer simulation:
- Computer Aided Engineering in manufacturing
- R & D stage in pharmaceutical industry
Advantages of computer simulation:
- reduction of time
- better solution by examining many possibilities,
- sharing knowledge by describing a model
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MBA in International Business
Shu YAMADA and iroe TSUBAKI
2
July 11, 2006
Computer simulation
Simulation study of a phenomena by computer calculation
in stead of physical experiments such as finite
element method
Sometimes called
Digital engineering
CAE (computer aided engineering)
Simulation study
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MBA in International Business
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July 11, 2006
1. Interpretation of requirements
Output
Define
Input
Application of DOE in computer
simulation study
y
x1 , x2 ,  , x p
2. Developing a computer
simulation model
DOE helps well
Specialist
knowledge
Validation

y   x1 , x2 ,, x p

3. Application of the
developed simulation model
Validation:
Comparison of simulation
results to reality
Stage
Screening
x1 , x2 , x3 , x4 ,  , x p
Specialist
knowledge
Approximation
Design
Fractional
factorial design
Screening
Supersaturated
design
Central
Approximation composite
Spece filling
design
y  ˆ x1 , x3 
 ˆ0  ˆ1 x1  ˆ2 x3  ˆ13 x1 x3  ˆ11x12  ˆ33 x33
Application of DOE
depending on the
situation
Analysis
Stepwise selection
by F statisitc
Second order
model
Various models
Optimization in terms of various viewpoints
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MBA in International Business
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July 11, 2006
Outline of this talk
1. Validation of the developing model
Example: Forging of an automobile parts
Technique: Sequential experiments
2. Screening of many factors
Example: Cantilever
Technique: Supersaturated design and F statistic
3. Approximation of the response
Example: Wire bonding
Technique: Non-linear model and uniform design
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MBA in International Business
Shu YAMADA and iroe TSUBAKI
5
July 11, 2006
1. Validation of the developing model
knowledge
in the field
Validation

y   x1 , x2 ,, x p

Validation of the developing model by comparing with the
reality
compare
Physical
experiments
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MBA in International Business
Simulation
result
Shu YAMADA and iroe TSUBAKI
6
July 11, 2006
Forging example
x2
x1
protrusion
depth
height (response)
x1: punch depth
x2: punch width
x3: shape of corner (w)
x4: shape of corner (h)
y: height
chamfer
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Mr. Taomoto (Aisin Seiki Co.)
Shu YAMADA and iroe TSUBAKI
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July 11, 2006
What should be done?
Simulation
y
(a) Judgment:
Appropriate or not?
Physical
x1
Simulation experiments
(b) Adjustable by changing
computational parameters?
(Young ratio, mechanical
property,… )
y
Physical experiment
y
punch depth (d)
x1
Physical
(c) Revise the simulation model
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Simulationx1
Shu YAMADA and iroe TSUBAKI
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July 11, 2006
Simulation experiments
(a) Judgment: Appropriate or not?
Application of statistical tests
Physical experiment
(b) Adjustable by changing computational parameters?
How to determine the level of the computational
parameters systematically
(c) Revise the simulation model
Not a statistical problem
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MBA in International Business
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July 11, 2006
Validation of the developing computer simulation model
protrusion height
Simulation experiments
Physical experiment
Adjusted simulation
results
Find appropriate levels of
computational parameters to fit
the simulation results to physical
experimental results
punch depth (x1)
Computational parameters
Yong ratio, poison ratio, etc.
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MBA in International Business
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July 11, 2006
Problem formulation of adjusting
computational parameters
Requirement: Small run number is better
Aim:
To determine the level of computation parameters z1, z2, …, zq
to minimize the difference between the physical experiment
results and computer simulation results over the interested
region of x1, x2, …, xp.
Simulation results
Physical experiments
Minimization of
y   x1 , x 2 , , x p , z1 , z 2 ,, z q 
Y x1 , x 2 ,, x p 
  x ,, x
xR
1
, z1 ,, z q   Y x1 ,, x p  dx
2
p
by computational parameters z1 ,  , z q
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July 11, 2006
An approach by “easy to change” factor
Punch depth (x1) : Easy to change factor levels because
it does not require remaking of mold. Physical
experiments can be performed by using the mold by
several levels.
At each combination of computational parameters, the
discrepancy between physical and experimental
experiments are calculated by
D
5
  x
i 1
1
 xi1 , x2  a 2 ,..., x4  a 4 , z1 ,, z 4   Y x1  xi1 , x2  a2 ,..., x4  a 4 
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Shu YAMADA and iroe TSUBAKI
12
D
July 11, 2006
5
  x
i 1
1
 xi1 , x2  a 2 ,..., x4  a 4 , z1 ,, z 4   Y x1  xi1 , x2  a2 ,..., x4  a 4 
2
Computational parameters
No.
z1
z2
z3
z4
Discrepancy
1
0
0
0
0
21
2
1
0
0
0
11
3
0
1
0
0
19
Sequential
4
0
experiments
0
1
0
20
5
0
0
0
1
13
6
1
1
0
0
11
7
1
0
1
0
10
1
0
0
1
11
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The optimum level of the
computational parameters z1,
…, z4 are found by analyzing
the relation between
“discrepancy” and z1, …, z4
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13
July 11, 2006
protrusion height
Original simulation results
Physical experiment
Adjusted simulation
results
punch depth (x1)
University of Tsukuba
MBA in International Business
Shu YAMADA and iroe TSUBAKI
14
July 11, 2006
Future problems at validation stage
(1) Statistical tests to judge the appropriateness of the
simulation model
(2) Identification of the trend
(3) Design to examine the simulation model efficiently
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MBA in International Business
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July 11, 2006
2. Screening problem
Example: cantilever
Ex. FEM measures the maximum stress
Factors x1, x2,
…, xp
Theoretical equations (x1, x2, …, xp) are applied to calculate the
response variables under given factor level
Design a beam in which one side is fixed to the wall
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July 11, 2006
Design a beam (Iwata and Yamada (2004))
Stepping cantilever
Factors
Hight x1～x15
Levels
No.1 30(mm)
No.2 35(mm)
No.3 40(mm)
Response
Maximum stress
Variation of the maximum stress
Weight of the beam
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MBA in International Business
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July 11, 2006
An application of three-level
supersaturated design
Requirements
The relation between the responses and their factors are
complicated
It takes several hours to calculate the response in a design
There are many factor effects
Linear effects
Interaction effects
Quadratic effects
15
105
15
Impossible to estimate all factor effects simultaneously.
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July 11, 2006
Application of three-level supersaturated design
(Yamada and Lin (1999))
No.
[1]
[2]
1 1
2 1
3 1
4 1
5 1
6 1
7 1
8 1
9 1
10 2
11 2
12 2
13 2
14 2
15 2
16 2
17 2
18 2
19 3
20 3
21 3
22 3
23 3
24 3
25 3
26 3
27 3
University
1
1
1
2
2
2
3
3
3
1
1
1
2
2
2
3
3
3
1
1
1
2
2
2
3
3
of3
[3]
[4]
[5]
[6]
1 1 1
2 2 2
3 3 3
1 2 3
2 3 1
3 1 2
1 3 2
2 1 3
3 2 1
1 1 1
2 2 2
3 3 3
1 2 3
2 3 1
3 1 2
1 3 2
2 1 3
3 2 1
1 1 1
2 2 2
3 3 3
1 2 3
2 3 1
3 1 2
1 3 2
2 1 3
3 2 1
Tsukuba
MBA in International Business
[7]
1
3
3
3
2
2
2
1
1
1
3
3
3
2
2
2
1
1
1
3
3
3
2
2
2
1
1
[8]
1
1
3
2
3
2
1
3
2
1
1
3
2
3
2
1
3
2
1
1
3
2
3
2
1
3
2
[9]
1
3
2
1
1
3
2
3
2
1
3
2
1
1
3
2
3
2
1
3
2
1
1
3
2
3
2
[10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
1
2
1
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2
1
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1
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1
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1
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3
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1
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1
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1
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1
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1
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1
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1
1
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1
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1
1
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1
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1
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2
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1
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1
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1
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1
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1
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2
1
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Shu YAMADA and iroe TSUBAKI
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19
July 11, 2006
Screening procedure
More than There are many factor effects
Linear effects
15
Interaction effects
105
Quadratic effects
15
(0) Impossible to assess all possibilities 2135  4.05648  10 31
(1) Stepwise selection of F value
(2) Stepwise selection of F value with order principle and
effect heredity
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July 11, 2006
Analysis strategy
Order principle
Lower order terms are more important higher order terms
(Linear effect, interaction and quadratic,...)
Effect heredity
When two-factor interaction is detected,
(i) at least one factor effect of the two factors
(ii) both of the linear effects
should be included in the model. The strategy (ii) is implemented.
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Procedure to ensure effect heredity
and order principle (EO)
Step 1 Candidate set x1 ,...,x15 
Step 2
Quadratic term of the selected effect is added to the
candidate set
2
Ex 1 x1 ,..., x15 x2 
Interaction term of the selected two factors is added
Ex 2 x1 ,..., x15 x1 x2 , x22 
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Applied design
YAMADA, S. and LIN, D. K. J., (1999), Threelevel supersaturated design, Statistics and
Probability Letters, 45, 31-39. etc
n=9
(i)
Ordinal
F
F
Effects
Effects
x4×x10
21.111 x4×x10
21.111
x6^2
13.192
x6^2
13.192
x3×x12
9.776
x3×x7
8.933
x9×x13
28.830 x6×x13
4.185
x5×x7
643.876
x7×x8
5.143
x1×x11
288.850
x3×x8
14.766
x3
66.380 x6×x15
517.325
n=18
Yamada, S., Ikebe, Y., Hashiguchi, H. and Niki,
N., (1999), Construction of three-level
supersaturated design, Journal of
Statistical Planning and Inference, 81,
183-193.
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Ordinal
F
Effects
x6×x15
7.813
x2
9.433
x1
12.97
x1×x3
7.459
x1^2
9.279
x6×x12
18.255
x1×x14
7.606
x15^2
47.734
x3×x14
23.437
x10×x13
10.4
x9×x12
21.208
x12×x13
24.418
x2×x14
17.5
x1×x9
16.78
x1×x12
124.909
x9×x13 2075.621
(i)
Effects
x１
x2
x1×x3
x1^2
x6×x12
x3×x10
x1×x6
x2×x12
x12×x4
x2×x8
x12×x5
x7×x9
x3
x1×x8
x2×x11
x7×x10
F
7.771
29.683
8.462
0.627
14.027
14.88
6.538
22.981
9.372
3.866
17.868
9.8
8.601
8.158
17.898
44345.04
23
July 11, 2006
What is a right choice
Consistency with the knowledge in the mechanical engineering
Physical property
1．The factors closing to the wall are important
such as x1,x2, x3
2. Sometimes, the edge side are important
such as x14, x15
3. Interaction can be considered at a connected two factors
such as x1×x2，x2×x3
x1
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x2
x3 x4
x5
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x15
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July 11, 2006
n=18
EO select a reasonable
selection in terms of
the physical property of
the cantilever
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Ordinal
Effects
F
x6×x15
7.813
x2
9.433
x1
12.97
x1×x3
7.459
x1^2
9.279
x6×x12
18.255
x1×x14
7.606
x15^2
47.734
x3×x14
23.437
x10×x13
10.4
x9×x12
21.208
x12×x13
24.418
x2×x14
17.5
x1×x9
16.78
x1×x12
124.909
x9×x13 2075.621
EO
Effects
x１
x2
x1×x3
x1^2
x6×x12
x3×x10
x1×x6
x2×x12
x12×x4
x2×x8
x12×x5
x7×x9
x3
x1×x8
x2×x11
x7×x10
Shu YAMADA and iroe TSUBAKI
F
Effects
7.771
x1
29.683
x2
8.462
x1^2
0.627
x3
14.027 x1×x3
14.88
x2^2
6.538 x×12
22.981
x×7
9.372 x7×x12
3.866
x3^2
17.868
x8
9.8 x1×x2
8.601 x8×x12
8.158
x4
17.898
x4^2
44345.04
x7^2
F
7.711
29.683
6.066
5.852
10.466
28.566
3.758
5.631
9.498
3.614
4.785
8.77
5.445
2.8
831.322
156.486
25
July 11, 2006
3. Approximation and optimization
Example: wire bonding in IC
Yamazaki, Masuda and Yoshino,(2005), Analysis of wire-loop resonance
during al wire bonding, 11th Symposium on Microjoining in Electrics,
February 3-4, 2005, Yokohama
Outline:
Recent years AL wire bonding by
microjoining is widely applied in many
types of IC.
Finite Element Method obtained that it
sometimes occurs resonance problem at
the mircojoining.
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Background
(1)The shape of the wire is shown in the right
(2)Material: AL
x2: Height
x1: Width 2mm - 4mm
x2: Height 0.5mm-1.5mm
x3: Diameter 0.03 - 0.04 mm
Fix
Vibration
(3) FEM obtains the moment along with the
x1: Width
f: frequency at the joining
(4) The connected point will be broken when the moment is higher than
certain level.
(5) The amplitude and its frequency is determined by x1, x2, x3．
(6) Given the level of x1, x2 and x3, FEM analysis requires time for
calculation to analyze the frequency and response analysis．
(7) The response, moment, is a multi-modulus because of the resonance at
several frequencies.
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Output of FEM software
x1: 3.14
x2: 1.28
x3: 0.03
Complex function
x1: 3.00
x2: 0.78
x3: 0.03
FEMAP v8.2.1+ CAFEM v8.0
The peaks will be determined by x1: width, x2: height and x3: diameter
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Requirements
(1)Outline of the factors in process
x1: Width Some restrictions because of the location with other parts
x2: Height controllable in the range (0.5-1.5mm)
x3: Diameter
Specified in the priori process
f: Frequency
Controllable by selecting the bonder
(2)The tentative levels of x1, x3 are determined in the priori process. Based
on the tentative levels, optimum levels of x2 and f is explored. There is a
need to consider the robustness against the difference of f from the
specified value.
(3) It takes a long time to evaluate the frequency under a set of levels of x1,
x2 and x3. Re-calculation is inefficient when the levels are slightly
revised.
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Strategy
(1) The final goal is to find a good approximated function of M:
moment at the fixed point by x1: width, x2: height, x3: diameter
and f: frequency such that
M=g(x1, x2, x3, f)
(2) In the future, various types of wires are applied in the IC design.
Thus, the above approximation is helpful.
(3) To find a smooth function, 210-level design is utilized.
(4) Because of the complex relation, uniform design will be beneficial.
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MBA in International Business
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30
July 11, 2006
x1: width
x2: height x3: diameter
2.000
0.929
0.03
2.000
0.929
0.04
2.143
1.357
0.03
2.143
1.357
0.04
2.286
0.571
0.03
2.286
0.571
0.04
2.429
1.143
0.03
2.429
1.143
0.04
2.571
0.714
0.03
2.571
0.714
0.04
2.714
1.500
0.03
2.714
1.500
0.04
2.857
1.000
0.03
2.857
1.000
0.04
3.000
0.786
0.03
3.000
0.786
0.04
3.143
1.286
0.03
3.143
1.286
0.04
3.286
0.500
0.03
3.286
0.500
0.04
3.429
1.214
0.03
3.429
1.214
0.04
3.571
0.857
0.03
3.571
0.857
0.04
3.714
1.429
0.03
3.714
1.429
0.04
3.857
0.643
0.03
3.857
0.643
0.04
4.000
1.071
0.03
4.000
1.071
0.04
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Uniform design
x2: 高さ
1.500
1.000
0.500
2.000
3.000
4.000
Three dimensional uniform design may be the best
choice. However U15 15 2  2 is applied because of
computational restriction,
 
Shu YAMADA and iroe TSUBAKI
31
July 11, 2006
http://www.math.hkbu.edu.hk/~ktfang/
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32
July 11, 2006
x1: widthx2: height
x3:
2.000 0.929
2.000 0.929
2.143 1.357
2.143 1.357
2.286 0.571
2.286 0.571
2.429 1.143
2.429 1.143
2.571 0.714
2.571 0.714
2.714 1.500
2.714 1.500
2.857 1.000
2.857 1.000
3.000 0.786
3.000 0.786
3.143 1.286
3.143 1.286
3.286 0.500
3.286 0.500
3.429 1.214
3.429 1.214
3.571 0.857
3.571 0.857
3.714 1.429
3.714 1.429
3.857 0.643
3.857 0.643
4.000 1.071
4.000 1.071
diameter
0.03
0.04
0.03
0.04
0.03
0.04
0.03
0.04
0.03
0.04
0.03
0.04
0.03
0.04
0.03
0.04
0.03
0.04
0.03
0.04
0.03
0.04
0.03
0.04
0.03
0.04
0.03
0.04
0.03
0.04
University of Tsukuba
MBA in International Business
frequency (210 levels)
x1: 3.00
x2: 0.78
x3: 0.03
x1: 3.14
x2:1.28
x3: 003
Shu YAMADA and iroe TSUBAKI
33
July 11, 2006
Approximation
(1) It is beneficial to find levels of x1, x2, x3 and f with resonance
precisely rather than an well fitted model uniformly in the space
of x1, x2, x3 and f.
(2) The main aim is to find levels x1, x2, x3 and f with resonance. The
estimation of the moment is not the major aim.
(3) A RBF (radial basis function) like model is applied for the
approximation.
 1  f  m 2 
 1  f  m 2 
 1  f  m 2 
b0  b1 f  a1 exp   
 2

1
s1
   a2 exp   
 2
 

2
s2
   a3 exp   
 2
 

3
s3
   ...
 
＊RBF is sometimes applied to describe the impulse response in neural network.
University of Tsukuba
MBA in International Business
Shu YAMADA and iroe TSUBAKI
34
July 11, 2006
Model
60
80
 1  f  m 2 
 1  f  m 2 
 1  f  m 2 
3
1
2
   ...
   a2 exp   
   a3 exp   
b0  b1 f  a1 exp   
 2  s3  
 2  s1  
 2  s2  






a3
40
x2=0.78
predict.c
y
x1=3.00,
a1 s
1
a2
s3
s2
b0  b1 f
0
20
X3=0.03
m1
5
University of Tsukuba
MBA in International Business
m2
10
fr
Shu YAMADA and iroe TSUBAKI
15
m3
20
f
35
July 11, 2006
100
0
50
predict.c
y
150
x1: 3.14
x2:1.28
x3: 003
University of Tsukuba
MBA in International Business
5
10
Shu YAMADA and iroe TSUBAKI
15
fr
20
36
July 11, 2006
(4) Fit a model for each run (No. 1~30), i.e. estimate the
following parameters
b0 , b1 , a1 , m1 , s1 , a2 , m2 , s2 , a3 , m3 , s3 ,...
(5) The estimates are treated as response variables
whose factors are x1, x2, x3 and f. An approximation
of M=g(x1, x2, x3, f) is derived as follows:
 1  f  m 2 
 1  f  m 2 
 1  f  m 2 
3
1
2




   ...
  a2 exp  
  a3 exp   
b0  b1 f  a1 exp  
 2  s3  
 2  s1  
 2  s2  






University of Tsukuba
MBA in International Business
Shu YAMADA and iroe TSUBAKI
37
July 11, 2006
(4) Fit a model for each run (No. 1~30), i.e.
estimate the parameters
x1
2.000
2.000
2.143
2.143
2.286
2.286
2.429
2.429
2.571
2.571
2.714
2.714
2.857
2.857
3.000
3.000
3.143
3.143
3.286
3.286
3.429
3.429
3.571
3.571
3.714
3.714
3.857
3.857
4.000
4.000
x2
0.929
0.929
1.357
1.357
0.571
0.571
1.143
1.143
0.714
0.714
1.500
1.500
1.000
1.000
0.786
0.786
1.286
1.286
0.500
0.500
1.214
1.214
0.857
0.857
1.429
1.429
0.643
0.643
1.071
1.071
b0
x3
0.03 17.670
0.04 52.260
0.03 21.540
0.04 37.930
0.03 8.959
0.04 31.385
0.03 12.330
0.04 29.725
0.03 7.583
0.04 23.157
0.03 18.510
0.04 33.480
0.03 10.875
0.04 17.040
0.03 0.586
0.04 17.831
0.03 16.776
0.04 25.530
0.03 -11.743
0.04 22.721
0.03 9.249
0.04 21.301
0.03
0.04
0.03 11.540
0.04 20.490
0.03 -6.811
0.04 -31.606
0.03 0.000
0.04 0.000
b1
a1
m1
s1
s2
a3
m3
s3
a4
m4
s4
0.623
-1.634
1.824
4.022
-0.090
-0.272
1.322
1.670
0.261
-0.468
1.341
2.874
-0.063
0.419
0.586
-0.626
-0.217
0.505
2.958
0.754
0.031
-0.345
1.061
282.100
671.628
302.600
763.700
40.913
70.654
218.100
531.040
111.000
227.231
191.600
525.700
113.908
272.300
44.980
84.585
157.526
316.400
62.348
216.967
109.600
252.741
29.487
6.343
8.396
3.842
5.101
7.422
9.843
4.241
5.625
5.699
7.523
2.842
3.781
3.951
5.232
4.216
5.573
2.893
3.829
4.222
5.538
2.710
3.604
3.089
0.205
0.284
0.128
0.160
0.194
0.050
0.141
0.182
0.178
0.239
0.104
0.119
0.132
0.174
0.157
0.150
0.074
0.125
0.178
0.161
0.076
0.099
0.074
198.000
13.930
0.391
1.000
1.000
1.000
1.000
1.000
1.000
292.900
597.900
15.110
8.181
10.820
18.500
0.234
0.301
0.363
302.900
645.300
14.860
19.640
0.393
0.493
178.700
330.353
95.320
155.645
207.000
417.200
80.169
116.000
28.530
8.239
112.730
206.000
34.831
118.535
75.270
127.970
21.239
9.313
12.305
13.670
18.090
6.130
8.120
9.102
12.040
10.330
13.995
6.481
8.584
11.114
14.752
6.254
8.289
7.688
0.258
0.331
0.395
0.570
0.174
0.226
0.258
0.299
0.245
0.353
0.180
0.234
0.154
0.309
0.180
0.227
0.138
205.600
16.940
0.439
220.800
473.400
73.502
11.150
14.780
16.331
0.301
0.391
0.454
204.000
17.460
0.422
59.780
18.110
0.500
124.995
248.100
288.200
11.758
15.560
17.869
0.347
0.433
0.495
46.940
17.826
0.386
20.338
0.485
2.186
2.894
3.022
3.992
2.348
3.108
0.070
0.076
0.134
0.208
0.111
0.072
87.177
161.700
20.396
32.853
31.679
35.098
4.933
6.544
7.821
10.362
5.714
7.585
0.142
0.182
0.112
0.092
0.148
0.115
0.315
7.658
0.407
0.309 140.716
0.8923
0.261
30.218
0.329
30.060
0.3548 105.416
0.455
0.050
71.379
0.362 401.240
0.207
98.762
281.000
30.954
108.136
31.305
75.876
11.250
14.870
13.309
17.6878
8.929
11.830
12.8338
16.780
10.000
13.345
17.250
-0.095
0.322
2.330
7.409
0.871
19.129
72.680
117.079
59.114
255.737
92.258
179.700
164.662
597.156
6.000
100.634
13.523
18.210
20.275
0.265
0.313
0.389
15.301
20.136
0.400
0.587
University of Tsukuba
MBA in International Business
a2
m2
Shu YAMADA and iroe TSUBAKI
38
July 11, 2006
(5) The estimates are treated as response
variables whose factors are x1, x2, x3 and f.
b0
Constant
x1
x2
z3
x1*x2
x1*x3
x2*x3
x1^2
x2^2
b1
a1
m1
s1
a2
m2
s2
a3
m3
s3
-96.1
53.0
-743.0
14.6
0.1
-346.0
35.7
0.2
-1779.1
37.1
-2.0
15.6
-32.0
12.5
-6.4
0.0
-68.9
-12.6
-0.2
-98.9
-11.4
0.8
-37.3
16.6
252.9
-10.1
0.1
-131.4
-27.8
-0.5
-400.3
-21.7
0.7
5640.0
-1125.3
32652.3
391.5
1.9
30519.3
746.6
44.5
126459.5
909.8
53.4
19.1
-5.5
-261.3
2.8
-0.1
-187.0
6.3
0.3
-89.9
6.1
-1465.8
433.7
-12456.0
-59.3
-10519.6
-88.2
-7.9
-17220.4
-98.4
22767.2
-86.5
10586.4
-196.6
-12.5
-42641.2
-175.3
-30.5
87.0
0.6
81.2
0.9
98.2
0.4
-0.2
0.8
269.9
3.4
934.8
-0.9
4.1
 1  f  m 2 
 1  f  m 2 
 1  f  m 2 
3
1
2




   ...
  a exp  
  a exp   
b0  b1 f  a1 exp  
 2  s3  
 2  s1   2
 2  s 2   3






b0  96.1  15.6 x1 - 37.3 x 2  5640.0 x3  19.1x1 x 2 - 1465.8 x1 x3
b1  53.0 - 32.0 x1  16.6 x 2 - 1125.3 x3 - 5.5 x1 x 2  433.7 x1 x3  4.1x12
a 1  -743.0  12.5 x1  252.9 x 2  32652.3 x3 - 261.3 x1 x 2 - 12456.0 x1 x3  22767.2 x 2 x3  87.0 x12
m1  14.6 - 6.4 x1 - 10.1x 2  391.5 x3  2.8 x1 x 2 - 59.3 x1 x3 - 86.5 x 2 x3  0.6 x12  0.8 x 22
s1  0.1 0 - 0.0 x1  0.1x 2  1.9 x3  0.1x1 x 2

University of Tsukuba
MBA in International Business
Shu YAMADA and iroe TSUBAKI
39
July 11, 2006
0
200
400
predict.c
y
600
800
1000
Using the approximation
(x1=2.143, x2=1.357, x3=0.03)
5
10
15
20
fr
University of Tsukuba
MBA in International Business
Shu YAMADA and iroe TSUBAKI
40
July 11, 2006
40
0
20
predict.c
y
60
80
(x1=3.000, x2=0.786, x3=0.03)
5
10
15
20
fr
University of Tsukuba
MBA in International Business
Shu YAMADA and iroe TSUBAKI
41
July 11, 2006
Comments
(1) The essence of the case study is exploring an approximation of
multi-modulus function by uniform design and RBF.
(2) Fitting by RBF brings a good fitting. It is suggested that RBF is
beneficial to fit response to frequency.
(3) It is concerned the over fitting in the case study. The fitness
should be validated.
(4) The parameters a1, m1, a2, m2,… are estimated precisely, for
example the adjusted R^2 is more than 90%. On the other hand,
there is a need to estimate s1, s2, … more precisely.
University of Tsukuba
MBA in International Business
Shu YAMADA and iroe TSUBAKI
42
July 11, 2006
Application of the approximation
x1: width 3mm, x3 diameter 0.3
1
ｘ2
height
0.9
0.8
15
1
10
0.9
5
0.7
0
0.8
0.7
5
10
f
0.6
ｘ2 height
0.6
15
20
0.5
0.5
5
Good
choice
10
15
20
frequency
University of Tsukuba
MBA in International Business
Shu YAMADA and iroe TSUBAKI
43
July 11, 2006
4. Grammar of DOE in computer simulation
study
1. Interpreting requirements
Output
Define
2. Developing a simulator
Input
y
x1 , x2 ,  , x p
knowledge
in the field
Validation

y   x1 , x2 ,, x p

Validation:
Comparison of simulation
results to reality
3. Applying the simulator
Stage
x1 , x2 , x3 , x4 ,  , x p
y  ˆ x1 , x3 
 ˆ0  ˆ1 x1  ˆ2 x3  ˆ13 x1 x3  ˆ11x12  ˆ33 x33
University of Tsukuba
MBA in International Business
Design
Fractional
factorial design
Screening
Supersaturated
design
Central
Approximation composite
Spece filling
design
Application of DOE
depending on the
situation
Analysis
Stepwise selection
by F statisitc
Second order
model
Various models
Optimization from various viewpoints
Shu YAMADA and iroe TSUBAKI
44