Data Driven Process Optimization Using Real-Coded Genetic Algorithms ~陳奇中教授演講投影片

Development of Data Driven Techniques for Process Optimization Using Real-Coded Genetic Algorithms

陳奇中Chyi-Tsong Chen

[email protected]逢甲大學化工系

Dept. of Chem. Eng., Feng Chia Univ.

Outline

Introduction - evolution in biologyWhat is genetic algorithm (GA)?Optimization using RCGA (Real-coded GA)

A. Single Objective (Global optimal)B. Multi-objective (Pareto front)

Data Driven Techniques Using RCGAA. Single objectiveB. Multi-objective

Application to the optimal design of MOCVD processesConclusions

Introduction: Evolution in biology

IMG form http://www.geo.au.dk/besoegsservice/foredrag/evolution/

Organisms produce a number of offspring similar to themselves but can have variations due to:(a) Sexual reproduction

Evolution in biology - I

Ref. :http://www.cas.mcmaster.ca/~cs777/presentations/3_GO_Olesya_Genetic_Algorithms.pdf

Parents offspring

IMG from http://www.tulane.edu/~wiser/protozoology/notes/images/ciliate.gif

Organisms produce a number of offspring similar to themselves but can have variations due to:(b) Mutations (Random changes in the DNA sequence)

Evolution in biology - I


Before After

IMG from http://www.tulane.edu/~wiser/protozoology/notes/images/ciliate.gifIMG from http://offers.genetree.com/landing/images/mutation.png

Some offspring survive, and produce next generations, and some don’t:

Evolution in biology - II


http://www.ugobe.com/Home.aspx

Ugobe Inc. Pelo

What is genetic algorithm (GA)?

GA is a particular class of evolutionary algorithmInitially developed by Prof. John Holland

"Adaptation in natural and artificial systems“, University of Michigan press, 1975

Based on Darwin’s theory of evolution“Natural Selection” & “Survival of the fittest”

物競天擇適者生存不適者淘汰

Imitate the mechanism of biological evolution

- Reprodution- Crossover - Mutation

GA can be regarded as a search method frommultiple directions – reproduction, crossover, mutation

Provide efficient techniques to search optimal solutions for optimization problems having

- Discontinuous- Highly nonlinear- Stochastic- Has unreliable or undefined derivatives

Provide solutions for highly complex search spaceHave superior performance over the traditional optimal techniques, e.g., the gradient descent method.

Advantages of GA

All variables of interest must be encoded as binary digits (genes) forming a string (chromosome).

Gene – a single encoding of part of the solution space.

Chromosome – a string of genes that represent a solution.

Traditional GA

IMG from http://static.howstuffworks.com/gif/cell-dna.jpg

1

1 1 0 1 0

gene

chromosome

All genes in chromosome are real numbers- suitable for most systems.- genes are directly real values during genetic

operations. - the length of chromosomes is shorter than that in

binary-coded, so it can be easily performed.

Real-coded GA (RCGA)

1.1

1.1 0.1 15 10 0.12

gene

chromosome

IMG from http://static.howstuffworks.com/gif/cell-dna.jpg

Notations of RCGA (Chen et al., 2008)

is a solution set (chromosome) of the optimization problem

is called a gene, and

The admissible parameter space for is defined as

[ ]1 2 mθ θ θ=Θ L

iθ i m∈ 1, 2 ,m m= L

Θ

1,min 1 1,max 2,min 2 2,max

,min ,max

| , ,,

m

m m m

θ θ θ θ θ θ

θ θ θΘ = Θ∈ℜ ≤ ≤ ≤ ≤

≤ ≤

ΩL

Reproduction (tournament selection)Discard Pr × N chromosomes with maximum values of objectiveAdd Pr × N chromosomes with minimum values of objective

Example: Pr=0.5

Procedure of RCGA (Chen et al., 2008)

2,1 2,2 2,

1,1 1,2 1,

4,1 4,1 4,

3,1 3,2 3,

0.10.20.30.4

m

m

m

m

θ θ θθ θ θθ θ θθ θ θ

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦

L

L

L

L

Sort by objective value

Discard

2,1 2,2 2,

1,1 1,2 1,

2,1 2,2 2,

1,1 1,2 1,

0.10.20.10.2

m

m

m

m


⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎣ ⎦⎣ ⎦

L

L

L

L

add

New population

CrossoverDivided chromosomes into N/2 pairs where serve as parents. Suppose that and are parents of a given pair.

Example:


1Θ 2Θ

2,1 2,2 2,

1,1 1,2 1,

m

m

θ θ θθ θ θ⎡ ⎤⎢ ⎥⎣ ⎦

L

L

4,1 4,1 4,

3,1 3,2 3,

m

m

θ θ θθ θ θ⎡ ⎤⎢ ⎥⎣ ⎦

L

L

1,1 1,2 1,

2,1 2,2 2,

3,1 3,2 3,

4,1 4,2 4,

m

m

m

m


⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

L

L

L

L

Divided intotwo group

Crossover


2,1 2,2 2,

1,1 1,2 1,

m

m

θ θ θθ θ θ⎡ ⎤⎢ ⎥⎣ ⎦

L

L

4,1 4,1 4,

3,1 3,2 3,

m

m

θ θ θθ θ θ⎡ ⎤⎢ ⎥⎣ ⎦

L

L

1Θ

2Θ

[ ]0 1c∈

1 2

1 1 1 2

2 2 1 2

1 1 2 1

2 2 2 1

( ) ( )

( )

( )

( )

( )

if obj obj

r

r

else

r

r

Θ < Θ

Θ ← Θ + Θ −Θ

Θ ← Θ + Θ −Θ

Θ ← Θ + Θ −Θ

Θ ← Θ + Θ −Θ

If c>Pc

1Θ

2Θrandom

( ) ( )( )( ) ( )( )

1 2

max minobj obj

robj obj

Θ − Θ=

−Θ Θ

MutationRandomly select Pm× N chromosomes in the current population.

Example: Pm=0.5

1,1 1,2 1,

2,1 2,2 2,

3,1 3,2 3,

4,1 4,2 4,

m

m

m

m


⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

L

L

L

L

sΘ ← Θ + ×Φ

( ): random vector

0 :mutation size

m

sΦ∈ℜ

>

If a generated chromosome is outside the search space ,then the chromosome will be bounded by .

ΘΩ

ΘΩ


Step 1. Generate a population of N chromosomes from .

Step 2. Evaluate the corresponding objective function value for each chromosome in the population.

Step 3. If the pre-specified number of generations, G , is reached, or , then stop.

Step 4. Perform operations of reproduction, crossover, and mutation.Notice that if the objective function value of offspring chromosome is bigger than the objective function value of parent chromosome, then the parent chromosome will be retained in this generation.

Step 5. Go back to Step 2.

ΘΩ

( )( ) ( )( )max minobj obj ε− ≤Θ Θ


Methods ComparisonThe proposed(Chen et al., 2008)

Deb et al., 2000 Chang, 2007

Initial population Sobol (Pseudo Random) Random Random

reproduction tournament selection tournament selection tournament selection

crossover •N/2 pairs by sorting with objective function value

•Direction-based•controlled step size

•Random pair • Simulated binary

crossover (SBX)

•Random pair•Direction-based•random step size

mutation Quadratic-decay Polynomial-type Random

Global optimization using RCGA: Single-objective

( )min fx

x

( )( )

q eq

q

e

e

0

0

L U

≤

=

≤=

≤ ≤

Ax bA

c x

c x

b

x x x

x

Nonlinear constraints

Linear constraints

Variables constraints

Single-objective function

Benchmark test 1:

( ) ( )2 221 2 1max 3905.93 100 1

.3 3, 1,2i

F x x x

s tx i

= − − − −

− ≤ ≤ =

Global optimal solution

X(1,1)

F=3905.93

De Jong function, 1975

Methods Avg. iteration no. Avg. time (s)

The proposed 13.7633 0.11814

Deb, et al., 2000 15.4733 0.13213

Chang, 2007 15.31333 0.13130

Results:

(N=100, Pr=0.2, Pc=0.3, Pm=0.3,ε=1e-4,runs=300)

Benchmark test 1: De Jong function, 1975

Convergence of the solution

Benchmark test 2:


X(-3.79,-3.32)

F=43.3030

( ) ( )2 22 21 2 1 2 1 2min 11 7 3 57

.5 5, 1, 2i

F x x x x x x

s tx i

= + − + + − + + +

− ≤ ≤ =

Modified Himmelblau function, 1993

Methods Avg. iteration no. Avg. time (s)The proposed 16.99667 0.14383Deb, et al., 2000 19.05667 0.16141Chang, 2007 19.87333 0.16791

Benchmark test 2:

Results:

(N=100, Pr=0.2, Pc=0.3, Pm=0.3,ε=1e-4, runs=300)

Modified Himmelblau function, 1993

Convergence of the solution

Example 3: Gen and Cheng, 1997

( ) 23 1 5 1

2 5 1 4 3 52

2 5 1 2 3

3 5

min 5.3578547 0.8356891 37.293239 40792.141. .

0 85.334407 0.0056858 0.00026 0.0022053 92

90 80.51249 0.0071317 0.0029955 0.0021813 11020 9.300961 0.0047026 0.0012

f x x x x xs t

x x x x x x

x x x x xx x

= + + −

≤ + + − ≤

≤ + + + ≤≤ + + 1 3 3 4

1

2

3 4 5

547 0.0019085 2578 10233 4527 , , , 45

x x x xxxx x x

+ ≤≤ ≤≤ ≤≤ ≤

Optimal solution (Gen and Cheng,1997) :

1 2 3 4 578, 33, 29.995, 45, 36.77630665.5

x x x x xf= = = = == −

NOT true Global Optimum

Benchmark test 3:

Results: (N=100, Pr=0.2, Pc=0.3, Pm=0.3,ε=1e-4,runs=300)

Avg. iteration no. = 38The proposed method

1

2

3

4

5

78.00033.00027.07145.00044.96931025.560

xxxxxf

===

==

= −

Deb et al., 2000Avg. iteration no. = 160

1

2

3

4

5

78.00033.00027.07245.00044.96631025.480

xxxxxf

===

==

= −

Chang, 2007

Avg. iteration no. = 51

1

2

3

4

5

78.00033.00027.07145.00044.96931025.560

xxxxxf

===

==

= −

Optimization using RCGA:Multi-objective

( ) ( ) ( )1 2min , , , Mf f fx

x x xL Multi-objective

s.t.

( )( )

q eq

q

e

e

0

0

L U

≤

=

≤=

≤ ≤

Ax bA

c x

c x

b

x x x

x

Nonlinear constraints

Linear constraints

Variable constraints

Concept of multi-objective optimization

10 k 100 k

40%

90%

2

1

Concept of Pareto-optimal solutions : non-dominated

A

B

CD

B dominate A

C dominate A

B, C non-dominated

D, E non-dominated

E dominate A, B, C

D dominate A, B

E

(Goldberg, 1989)

Parents

Offspring

M

M

1

1

N

N

2

2

Non-dominatedsorting

Front 1

Front 2 N

Rejected

Crowding distance sorting for each front

Front 1

Front 2

Front 3

New Population

RCGA

Front 3 Front 3

Front 1

Front 2

How does multi-objective optimization work?

CAT

How to extend RCGA to multi-objective optimization problems

1 2, , , MMin J J JL

1 2

1 1 1 2

2 2 1 2

1 1 2 1

2 2 2 1

( ) ( )

( )

( )

( )

( )

i i

i

i

i

i

if J J

r

r

else

r

r

Θ < Θ

Θ ← Θ + Θ −Θ

Θ ← Θ + Θ −Θ

Θ ← Θ + Θ −Θ

Θ ← Θ + Θ −Θ

( ) ( )

( ) ( )1 2

1 21

i ii M

i ii

J J

J J

θ θω

θ θ=

−=

−∑

1 11

2 21

Mi

iiM

ii

i

θ ωθ

θ ωθ

=

=

=

=

∑

∑

Crossover:

Methods Comparison

The proposed NSGA-II(Deb et al., 2000)

Initial population Sobol (pseudo random) Random

reproduction Tournament selection Tournament selection

crossover •N/2 pairs by sorting with crowding distance

•Multi-direction based•controlled step size

•Random pair•Simulated binary crossover (SBX)

mutation Quadratic-decay Polynomial-type

FON function

Benchmark test 1:

( )

( )

23

11

23

21

11 exp3

11 exp3

ii

ii

f x x

f x x

xπ π

=

=

⎛ ⎞⎛ ⎞= − − −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠⎛ ⎞⎛ ⎞= − − +⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

− ≤ ≤

∑

∑s.t.

Optimal solutions

1 2 31 1,3 3

x x x ⎡ ⎤= = ∈ −⎢ ⎥⎣ ⎦

RCGA parametersN=100Pc=0.1Pm=0.1

The proposed method NSGA-II

After 20 iterationsResults:

After 50 iterations

KUR function

Benchmark test 2:

( ) ( )( )( ) ( )

12 2

1 1

0.8 32

1

10exp 0.2

5sin

5 5

n

i iin

i ii

i

f x x x

f x x x

x

−

+

=

= − − +

= +

− ≤ ≤

∑

∑s.t.


Optimal solutions



After 150 iterations

ZTD6 function

Benchmark test 3:

( ) ( ) ( )( ) ( ) ( )( )( ) ( ) ( )( )

61 1 1

2 1

2

2

1 exp 4 sin 6

1

1 10 1 10cos 4

0 1, 1, 2, ,10

n

i ii

i

f x x x

f x g x x g x

g x n x x

x i

π

π=

= − −

= −

= + − + −

≤ ≤ =

∑

L

s.t.


Optimal solutions



After 500 iterations

Data Driven Techniques Using RCGA

Single-objective process optimization

min ( )ix

f y

s.t.

,min ,maxi i ix x x≤ ≤

1x

2x

nxM

my

2y1y

M

Multi-objective optimization

1x

2x

nxM

my

2y1y

M

( ) ( ) ( )1 2min , , ,i

Mxf f fy y yL

s.t.

,min_ ,max_ , 1, 2, ,i i i i ix x x i n≤ ≤ = L

Data Driven Flow Chart

Generate a group of design of

experiments

Train a model by neural network

algorithm

Search optimal design parameters

by RCGA

Calculate objective function value

Reach the goal StopYes

No

Add result into the neural

network model and count

runs=runs+1

Initialize setting runs=0



experiments


algorithm


by RCGA



No



runs=runs+1


1,1 1,2 1,

2,1 2,2 2,

,1 ,2 ,

n

n

N N N n

θ θ θθ θ θ

θ θ θ

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

Θ

L

L

M M O M

L

Generate N chromosomes

Generate objective function values

( ) [ ]1 2T

Nobj obj obj=obj Θ L



experiments


algorithm


by RCGA



No



runs=runs+1


Train NN model

NN type: feed-forward

MSE index < 1e-3


Generate a group of designs of

experiments


algorithm


by RCGA



No



runs=runs+1


Single-objective:Apply direction-based RCGA to search optimal solution according to current NN model.

Multi-objective:Apply multi-direction RCGA to search optimal Pareto-front according to current NN model.



experiments

Train a model by neuron network

algorithm


by RCGA



No

Add result into the neuron


runs=runs+1


Single-objective:Calculate the objective function value from the solution searched by RCGA.

Multi-objective:Pick up first p points from Perato front which is sorted by crowding distance.

Then generate the corresponding objective function value(s).



experiments

Train a model by neuron network

algorithm


by RCGA



No

Add result into the neuron


runs=runs+1


Calculate performance index:

Single objective:

Multi-objective:

( )2

, ,1 1

1 1 ˆpM

i j i ji j

MSE y yM p = =

= −∑∑

y

y

Predict from NN model

Calculate from process

( )2

1

1 ˆp

j jj

MSE y yp =

= −∑



experiments


algorithm


by RCGA



No



runs=runs+1


Data Driven test 1:


X(0.2281, -1.6255)

F= -6.55113

( ) ( ) ( )(( )

( )( )

2 2 21 1 2

3 5 2 211 2 1 2

2 21 2

min 3 1 exp 1

10 exp5

1 exp 13

.3 3, 1, 2i

F x x x

x x x x x

x x

s tx i

= − − − +

⎛ ⎞− − − − −⎜ ⎟⎝ ⎠

⎞− − + − ⎟⎠

− ≤ ≤ =

MATLAB, peaks function

Single-objective

1

5

10

15

20

25

30

35

After 20 iterations

Optimal solution: x1=0.2282

x2=-1.6255

Predicted F= -6.5513

40

cf. Global optimal solution

X(0.2281,-1.6255)

F= -6.5511

AIXTRON AIX200/4

The schematic of horizontal MOCVD reactor (top: 3D view; bottom: 2D side view).

Application to the optimal design of an MOCVD reactor

Susceptor temperature: 600K ~ 1200K Total flow rate : 10000sccm ~ 15000sccm Pressure: 8kPa ~ 15kPa

( ) ( ) ( )2

2 221 1 1 1 11 12 2 2 2n f f

nJ GR

GRα β δ δ⎛ ⎞= + + − + −⎜ ⎟⎝ ⎠

GR

Objective function:

GR dAGR

A= ∫ GR GR dA

GRδ −= ∫

Objective function

600700

800900

10001100

1200

11.1

1.21.3

1.41.5

x 104

8

9

10

11

12

13

14

15

T (K)sccm

P (k

Pa)

Convergence of the design parameters

1

2

3

45

67

8 910

11

12 13

1415

1617

18

19

20

21

600700

800900

10001100

1200

11.1

1.21.3

1.41.5

x 104

8

9

10

11

12

13

14

15

T (K)sccm

P (k

Pa)

1

2

3

45

67

8 910

11

12 13

1415

1617

18

19

20

21

Before After

Suscuptor Temperature: 600 K ~ 1200 K Total flow rate : 10000 sccm ~ 15000 sccm Pressure: 8 kPa ~ 15 kPa

98.65 10 / minGR dA

GR mA

−= = ×∫

0.003504GR GR dAGR

δ −= =∫

911.65 10 / minGR dA

GR mA

−= = ×∫

0.00220GR GR dAGR

δ −= =∫

CONSTR function

Data Driven Test 2: Multi-objective

( )( ) ( )

1 1

2 2 11

20 20, 1,2i

f x x

f x x x

x i

=

= +

− ≤ ≤ =s.t.


( )( )

1 2 1

1 2 1

9 6

9 1

g x x x

g x x x

= + ≥

= − + ≥

Pareto-optimal solutions：

1 2 1

1 2

0.39 0.67 6 90.67 1 0

x x xx x

≤ ≤ ⇒ = −≤ ≤ ⇒ =

A :

B :

A

B

N=30, p=5

16 iterations

AIXTRON AIX200/4

The schematic of horizontal MOCVD reactor (top: 3D view; bottom: 2D side view).

Multi-objective design of Horizontal MOCVD process

Susceptor Temperature: 833K ~ 1033K Total flow rate : 13000sccm ~ 20000sccm Pressure: 10kPa ~ 100kPa

Growth of GaAs film on a 3-inch substrate

Objective functions:

GRdAGR

A= ∫

( )2AGR GR d

Aδ

−= ∫

Objective functions

Design of experiments -Taguchi methodL25(56)

Susceptor Temperature (K): TTotal flow rate (sccm): UPressure (kPa): P

VariablesLevels

Level 1 Level 2 Level 3 Level 4 Level 5

T 833 883 933 983 1033

P 10 25 50 75 100

U 13000 14000 16000 18000 20000

Five levels for each factor

-55 -50 -45 -40 -35 -30 -25 -20-40

-30

-20

-10

0

10

20

30

40

50

Growth Rate (nm/min)

Uni

form

ity in

dex

-45 -40 -35 -30 -25 -20 -15 -10 -5 0-8

-6

-4

-2

0

2

4

6

8

10

12


Uni

form

ity in

dex

-50 -45 -40 -35 -30 -25 -20-8

-6

-4

-2

0

2

4

6

8

10

12


Uni

form

ity in

dex

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-6

-4

-2

0

2

4

6

8

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12


Uni

form

ity in

dex

1 2

3 4

Data driven

-45 -40 -35 -30-20

-10

0

10

20

30

40

50


Uni

form

ity in

dex

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-10

-5

0

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35


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0

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-10

0

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Data driven

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0

20

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100


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0

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dex

9 10

11 12

Data driven

-45 -40 -35 -30 -25 -20 -15 -10 -5 0 5-10

0

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50


Uni

form

ity in

dex

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0

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Data driven

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0

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1817

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ity in

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0

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20

Data driven

-45 -40 -35 -30 -25 -20 -15 -10 -5 0-10

0

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Uni

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ity in

dex

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0

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21 22

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23 24

Data driven

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0

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0

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25 26

-45 -40 -35 -30 -25 -20 -15 -10 -5 0-10

0

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Uni

form

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27

-45 -40 -35 -30 -25 -20 -15 -10 -5 0-10

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Uni

form

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28

Data driven

-45 -40 -35 -30 -25 -20 -15 -10 -5 0-10

0

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Uni

form

ity in

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Uni

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29 30

-45 -40 -35 -30 -25 -20 -15 -10 -5 0-10

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Uni

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31 32

Data driven

Convergence of MSE index

0 5 10 15 20 25 30 350

100

200

300

400

500

600

700

800

900

1000

runs

MSE

0 5 10 15 20 25 30 3510-2

10-1

100

101

102

103

runs

MSE

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Optimal Pareto-front solutions of the MOCVD

AD

C

B

Case A. the best uniformity(min. of )

Operating conditions:

T= 883 K

P=10 kPa

U= 13000 sccm

Performance:

3.461 (nm/ min)GR =

0.00409δ =

δ

Case B. the max. growth rate (min. )


T= 957.9659 K

P=10 kPa

U= 20000 sccm

Performance:

44.346 (nm/ min)GR =

82.059δ =

GR−

Case C.


T= 935.82 K

P=18.87 kPa

U= 20000 sccm

Performance:

34.852 (nm/ min)GR =

3.245δ =

min. J GR δ= − +

Case D.


T= 917.81 K

P=16.23 kPa

U= 20000 sccm

Performance:

29.401 (nm/ min)GR =

1.024δ =

10min. J GR δ= − +

Conclusions

An efficient global optimization scheme using a real-coded genetic algorithms has been proposed. Effective data driven techniques for single objective and multi-objective optimal process design have been developed.The proposed schemes have been tested successfully on the optimal design of MOCVD processes.

Q & A

Thanks for your attention.

Technology

Data Driven Process Optimization Using Real-Coded Genetic Algorithms ~陳奇中教授演講投影片