data driven process optimization using real-coded genetic algorithms ~陳奇中教授演講投影片
TRANSCRIPT
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
Ref. :http://www.cas.mcmaster.ca/~cs777/presentations/3_GO_Olesya_Genetic_Algorithms.pdf
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
Ref. :http://www.cas.mcmaster.ca/~cs777/presentations/3_GO_Olesya_Genetic_Algorithms.pdf
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
θ θ θθ θ θθ θ θθ θ θ
⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥
⎣ ⎦⎣ ⎦
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:
Procedure of RCGA (Chen et al., 2008)
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
θ θ θθ θ θθ θ θθ θ θ
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
L
L
L
L
Divided intotwo group
Crossover
Procedure of RCGA (Chen et al., 2008)
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
θ θ θθ θ θθ θ θθ θ θ
⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦
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
Procedure of RCGA (Chen et al., 2008)
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
Procedure of RCGA (Chen et al., 2008)
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:
Global optimal solution
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.
RCGA parametersN=100Pc=0.1Pm=0.1
Optimal solutions
The proposed method NSGA-II
After 60 iterationsResults:
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.
RCGA parametersN=100Pc=0.1Pm=0.1
Optimal solutions
The proposed method NSGA-II
After 200 iterationsResults:
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
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
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
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
Train NN model
NN type: feed-forward
MSE index < 1e-3
Data Driven Flow Chart
Generate a group of designs 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
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.
Data Driven Flow Chart
Generate a group of design of
experiments
Train a model by neuron network
algorithm
Search optimal design parameters
by RCGA
Calculate objective function value
Reach the goal StopYes
No
Add result into the neuron
network model and count
runs=runs+1
Initialize setting runs=0
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).
Data Driven Flow Chart
Generate a group of design of
experiments
Train a model by neuron network
algorithm
Search optimal design parameters
by RCGA
Calculate objective function value
Reach the goal StopYes
No
Add result into the neuron
network model and count
runs=runs+1
Initialize setting runs=0
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 =
= −∑
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
Data Driven test 1:
Global optimal solution
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.
RCGA parametersN=100Pc=0.1Pm=0.1
( )( )
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
Growth Rate (nm/min)
Uni
form
ity in
dex
-50 -45 -40 -35 -30 -25 -20-8
-6
-4
-2
0
2
4
6
8
10
12
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
Growth Rate (nm/min)
Uni
form
ity in
dex
1 2
3 4
Data driven
-45 -40 -35 -30-20
-10
0
10
20
30
40
50
Growth Rate (nm/min)
Uni
form
ity in
dex
-50 -45 -40 -35 -30 -25 -20 -15-15
-10
-5
0
5
10
15
20
25
30
35
Growth Rate (nm/min)
Uni
form
ity in
dex
-50 -45 -40 -35 -30 -25 -20 -15-20
-10
0
10
20
30
40
50
Growth Rate (nm/min)
Uni
form
ity in
dex
-50 -45 -40 -35 -30 -25 -20-20
-10
0
10
20
30
40
50
60
Growth Rate (nm/min)
Uni
form
ity in
dex
5 6
7 8
Data driven
-50 -45 -40 -35 -30 -25-20
0
20
40
60
80
100
Growth Rate (nm/min)
Uni
form
ity in
dex
-45 -40 -35 -30 -25 -20 -15 -10 -5-10
0
10
20
30
40
50
60
Growth Rate (nm/min)
Uni
form
ity in
dex
-50 -45 -40 -35 -30 -25 -20-5
0
5
10
15
20
Growth Rate (nm/min)
Uni
form
ity in
dex
-50 -45 -40 -35 -30 -25 -20 -15 -10-20
0
20
40
60
80
100
Growth Rate (nm/min)
Uni
form
ity in
dex
9 10
11 12
Data driven
-45 -40 -35 -30 -25 -20 -15 -10 -5 0 5-10
0
10
20
30
40
50
Growth Rate (nm/min)
Uni
form
ity in
dex
-45 -40 -35 -30 -25 -20 -15 -10 -5-10
0
10
20
30
40
50
60
70
Growth Rate (nm/min)
Uni
form
ity in
dex
-45 -40 -35 -30 -25 -20 -15 -10 -5-10
0
10
20
30
40
50
60
70
Growth Rate (nm/min)
Uni
form
ity in
dex
13 14
-45 -40 -35 -30 -25 -20 -15 -10 -5-10
0
10
20
30
40
50
60
70
Growth Rate (nm/min)
Uni
form
ity in
dex
15 16
Data driven
-45 -40 -35 -30 -25 -20 -15 -10 -5-10
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 5-10
0
10
20
30
40
50
Growth Rate (nm/min)
Uni
form
ity in
dex
1817
-45 -40 -35 -30 -25 -20 -15 -10 -5 0 5-10
0
10
20
30
40
50
Growth Rate (nm/min)
Uni
form
ity in
dex
19
-45 -40 -35 -30 -25 -20 -15 -10-10
0
10
20
30
40
50
60
70
Growth Rate (nm/min)
Uni
form
ity in
dex
20
Data driven
-45 -40 -35 -30 -25 -20 -15 -10 -5 0-10
0
10
20
30
40
50
60
70
80
90
Growth Rate (nm/min)
Uni
form
ity in
dex
-45 -40 -35 -30 -25 -20 -15 -10 -5 0-10
0
10
20
30
40
50
60
70
80
90
Growth Rate (nm/min)
Uni
form
ity in
dex
-45 -40 -35 -30 -25 -20 -15 -10 -5 0-10
0
10
20
30
40
50
60
70
80
90
Growth Rate (nm/min)
Uni
form
ity in
dex
21 22
-45 -40 -35 -30 -25 -20 -15 -10 -5 0-10
0
10
20
30
40
50
60
70
80
90
Growth Rate (nm/min)
Uni
form
ity in
dex
23 24
Data driven
-45 -40 -35 -30 -25 -20 -15 -10 -5 0-10
0
10
20
30
40
50
60
70
80
90
Growth Rate (nm/min)
Uni
form
ity in
dex
-45 -40 -35 -30 -25 -20 -15 -10 -5 0-10
0
10
20
30
40
50
60
70
80
90
Growth Rate (nm/min)
Uni
form
ity in
dex
25 26
-45 -40 -35 -30 -25 -20 -15 -10 -5 0-10
0
10
20
30
40
50
60
70
80
90
Growth Rate (nm/min)
Uni
form
ity in
dex
27
-45 -40 -35 -30 -25 -20 -15 -10 -5 0-10
0
10
20
30
40
50
60
70
80
90
Growth Rate (nm/min)
Uni
form
ity in
dex
28
Data driven
-45 -40 -35 -30 -25 -20 -15 -10 -5 0-10
0
10
20
30
40
50
60
70
80
90
Growth Rate (nm/min)
Uni
form
ity in
dex
-45 -40 -35 -30 -25 -20 -15 -10 -5 0-10
0
10
20
30
40
50
60
70
80
90
Growth Rate (nm/min)
Uni
form
ity in
dex
29 30
-45 -40 -35 -30 -25 -20 -15 -10 -5 0-10
0
10
20
30
40
50
60
70
80
90
Growth Rate (nm/min)
Uni
form
ity in
dex
-45 -40 -35 -30 -25 -20 -15 -10 -5 0-10
0
10
20
30
40
50
60
70
80
90
Growth Rate (nm/min)
Uni
form
ity in
dex
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
-45 -40 -35 -30 -25 -20 -15 -10 -5 0-10
0
10
20
30
40
50
60
70
80
90
Growth Rate (nm/min)
Uni
form
ity in
dex
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. )
Operating conditions:
T= 957.9659 K
P=10 kPa
U= 20000 sccm
Performance:
44.346 (nm/ min)GR =
82.059δ =
GR−
Case C.
Operating conditions:
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.
Operating conditions:
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.