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

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

  1. 1. Development of Data DrivenTechniques for ProcessOptimization Using Real-Coded Genetic Algorithms 陳奇中 Chyi-Tsong Chen ctchen@fcu.edu.tw 逢甲大學化工系 Dept. of Chem. Eng., Feng Chia Univ.
  2. 2. Outline Introduction - evolution in biology What is genetic algorithm (GA)? Optimization using RCGA (Real-coded GA) A. Single Objective (Global optimal) B. Multi-objective (Pareto front) Data Driven Techniques Using RCGA A. Single objective B. Multi-objective Application to the optimal design of MOCVD processes Conclusions
  3. 3. Introduction: Evolution in biology IMG form http://www.geo.au.dk/besoegsservice/foredrag/evolution/
  4. 4. Evolution in biology - I Organisms produce a number of offspring similar to themselves but can have variations due to: (a) Sexual reproduction Parents offspringIMG from http://www.tulane.edu/~wiser/protozoology/notes/images/ciliate.gifRef. :http://www.cas.mcmaster.ca/~cs777/presentations/3_GO_Olesya_Genetic_Algorithms.pdf
  5. 5. Evolution in biology - I Organisms produce a number of offspring similar to themselves but can have variations due to: (b) Mutations (Random changes in the DNA sequence) Before After IMG from http://offers.genetree.com/landing/images/mutation.pngIMG from http://www.tulane.edu/~wiser/protozoology/notes/images/ciliate.gifRef. :http://www.cas.mcmaster.ca/~cs777/presentations/3_GO_Olesya_Genetic_Algorithms.pdf
  6. 6. Evolution in biology - II Some offspring survive, and produce next generations, and some don’t: Ugobe Inc. Pelohttp://www.ugobe.com/Home.aspxRef. :http://www.cas.mcmaster.ca/~cs777/presentations/3_GO_Olesya_Genetic_Algorithms.pdf
  7. 7. What is genetic algorithm (GA)?GA is a particular class of evolutionary algorithm Initially 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
  8. 8. Advantages of GAGA can be regarded as a search method from multiple 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 space Have superior performance over the traditional optimal techniques, e.g., the gradient descent method.
  9. 9. Traditional 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. 1 gene 1 1 0 1 0 chromosomeIMG from http://static.howstuffworks.com/gif/cell-dna.jpg
  10. 10. Real-coded GA (RCGA) 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. 1.1 gene 1.1 0.1 15 10 0.12 chromosomeIMG from http://static.howstuffworks.com/gif/cell-dna.jpg
  11. 11. Notations of RCGA (Chen et al., 2008) Θ = [θ1 θ 2 L θ m ] is a solution set (chromosome) of the optimization problem θ i is called a gene, i∈m and m = { 1, 2 L , m }The admissible parameter space for Θ is defined as Ω Θ = { Θ ∈ ℜm | θ1,min ≤ θ1 ≤ θ1,max , θ 2,min ≤ θ 2 ≤ θ 2,max , L , θ m ,min ≤ θ m ≤ θ m ,max }
  12. 12. Procedure of RCGA (Chen et al., 2008) Reproduction (tournament selection) Discard Pr × N chromosomes with maximum values of objective Add Pr × N chromosomes with minimum values of objective Example: Pr=0.5 Sort by objective value New population⎡θ 2,1 θ 2,2 L θ 2,m ⎤ ⎡ 0.1⎤ ⎡θ 2,1 θ 2,2 L θ 2,m ⎤ ⎡ 0.1⎤⎢θ θ1,2 L θ1,m ⎥ ⎢0.2 ⎥ ⎢θ θ1,2 L θ1,m ⎥ ⎢0.2 ⎥⎢ 1,1 ⎥⎢ ⎥ add ⎢ 1,1 ⎥⎢ ⎥⎢θ 4,1 θ 4,1 L θ 4,m ⎥ ⎢ 0.3⎥ ⎢θ 2,1 θ 2,2 L θ 2,m ⎥ ⎢ 0.1⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣θ3,1 θ3,2 L θ3,m ⎦ ⎣0.4 ⎦ ⎣θ1,1 θ1,2 L θ1,m ⎦ ⎣0.2 ⎦ Discard
  13. 13. Procedure of RCGA (Chen et al., 2008) Crossover Divided chromosomes into N/2 pairs where serve as parents. Suppose that Θ1 and Θ 2 are parents of a given pair.Example: Divided into two group ⎡θ 2,1 θ 2,2 L θ 2,m ⎤ ⎡θ1,1 θ1,2 L θ1,m ⎤ ⎢θ θ1,2 L θ1,m ⎥ ⎢θ ⎣ 1,1 ⎦ ⎢ 2,1 θ 2,2 L θ 2,m ⎥ ⎥ ⎢θ3,1 θ3,2 L θ3,m ⎥ ⎢ ⎥ ⎡θ 4,1 θ 4,1 L θ 4,m ⎤ ⎣θ 4,1 θ 4,2 L θ 4,m ⎦ ⎢θ θ3,2 L θ3,m ⎥ ⎣ 3,1 ⎦
  14. 14. Procedure of RCGA (Chen et al., 2008)Crossover Θ1 if obj (Θ1 ) < obj (Θ 2 ) Θ1 ⎡θ θ 2,2 L θ 2,m ⎤ Θ1 ← Θ1 + r (Θ1 − Θ 2 ) 2,1 ⎢θ θ1,2 L θ1,m ⎥ Θ 2 ← Θ 2 + r (Θ1 − Θ 2 ) If c>Pc ⎣ 1,1 ⎦ Θ2 else random c ∈ [ 0 1] Θ 2 ⎡θ 4,1 θ 4,1 L θ 4,m ⎤ Θ1 ← Θ1 + r (Θ 2 − Θ1 ) ⎢θ ⎣ 3,1 θ3,2 L θ3,m ⎥ ⎦ Θ 2 ← Θ 2 + r (Θ 2 − Θ1 ) obj ( Θ1 ) − obj ( Θ 2 )r= max ( obj ( Θ ) ) − min ( obj ( Θ ) )
  15. 15. Procedure of RCGA (Chen et al., 2008) Mutation Randomly select Pm× N chromosomes in the current population.Example: Pm=0.5 If a generated chromosome is outside the search space Ω Θ ,then the chromosome will be bounded by Ω Θ . ⎡θ1,1 θ1,2 L θ1,m ⎤ ⎢θ θ 2,2 L θ 2,m ⎥ ⎢ 2,1 ⎥ Θ ← Θ+ s×Φ ⎢θ3,1 θ3,2 L θ3,m ⎥ ⎢ ⎥ m Φ ∈ ℜ : random vector ⎣θ 4,1 θ 4,2 L θ 4,m ⎦ s ( > 0 ) :mutation size
  16. 16. Procedure of RCGA (Chen et al., 2008)Step 1. Generate a population of N chromosomes from Ω Θ .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 max ( obj ( Θ ) ) − min ( obj ( Θ ) ) ≤ ε , 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.
  17. 17. Methods Comparison The proposed Deb et al., 2000 Chang, 2007 (Chen et al., 2008)Initial population Sobol (Pseudo Random) Random Randomreproduction tournament selection tournament selection tournament selectioncrossover •N/2 pairs by sorting •Random pair •Random pair with objective function • Simulated binary •Direction-based value crossover (SBX) •random step size •Direction-based •controlled step sizemutation Quadratic-decay Polynomial-type Random
  18. 18. Global optimization using RCGA: Single-objective min f ( x ) Single-objective function x c (x) ≤ 0 Nonlinear constraints ceq ( x ) = 0 Ax ≤ b Linear constraints A eq x = b eq x L ≤ x ≤ xU Variables constraints
  19. 19. Benchmark test 1: De Jong function, 1975 max F = 3905.93 − 100 ( x12 − x2 ) − (1 − x1 ) 2 2s.t − 3 ≤ xi ≤ 3, i = 1,2 Global optimal solution X(1,1) F=3905.93
  20. 20. Benchmark test 1: De Jong function, 1975Results: (N=100, Pr=0.2, Pc=0.3, Pm=0.3,ε=1e-4,runs=300) 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
  21. 21. Convergence of the solution
  22. 22. Benchmark test 2:min F = ( x + x2 − 11) + ( x1 + x2 − 7 ) + x1 + 3 x2 + 57 2 2 2 2 1s.t−5 ≤ xi ≤ 5, i = 1, 2Modified Himmelblau function,1993 Global optimal solution X(-3.79,-3.32) F=43.3030
  23. 23. Benchmark test 2:Results: Modified Himmelblau function, 1993 (N=100, Pr=0.2, Pc=0.3, Pm=0.3,ε=1e-4, runs=300) Methods Avg. iteration no. Avg. time (s) The proposed 16.99667 0.14383 Deb, et al., 2000 19.05667 0.16141 Chang, 2007 19.87333 0.16791
  24. 24. Convergence of the solution
  25. 25. Example 3: Gen and Cheng, 1997 min f ( x ) = 5.3578547 x32 + 0.8356891x1 x5 + 37.293239 x1 − 40792.141s.t. 0 ≤ 85.334407 + 0.0056858 x2 x5 + 0.00026 x1 x4 − 0.0022053 x3 x5 ≤ 92 90 ≤ 80.51249 + 0.0071317 x2 x5 + 0.0029955 x1 x2 + 0.0021813x32 ≤ 110 20 ≤ 9.300961 + 0.0047026 x3 x5 + 0.0012547 x1 x3 + 0.0019085 x3 x4 ≤ 25 78 ≤ x1 ≤ 102 33 ≤ x2 ≤ 45 27 ≤ x3 , x4 , x5 , ≤ 45Optimal solution (Gen and Cheng,1997) : NOT true Global Optimum x1 = 78, x2 = 33, x3 = 29.995, x4 = 45, x5 = 36.776 f = −30665.5
  26. 26. Benchmark test 3: Results: (N=100, Pr=0.2, Pc=0.3, Pm=0.3,ε=1e-4,runs=300)The proposed method Deb et al., 2000 Chang, 2007Avg. iteration no. = 38 Avg. iteration no. = 160 Avg. iteration no. = 51 x1 = 78.000 x1 = 78.000 x1 = 78.000 x2 = 33.000 x2 = 33.000 x2 = 33.000 x3 = 27.071 x3 = 27.072 x3 = 27.071 x4 = 45.000 x4 = 45.000 x4 = 45.000 x5 = 44.969 x5 = 44.966 x5 = 44.969 f = −31025.560 f = −31025.480 f = −31025.560
  27. 27. Optimization using RCGA: Multi-objective min f1 ( x ) , f 2 ( x ) ,L, f M ( x ) Multi-objective xs.t. c (x) ≤ 0 Nonlinear constraints ceq ( x ) = 0 Ax ≤ b Linear constraints A eq x = b eq x L ≤ x ≤ xU Variable constraints
  28. 28. Concept of multi-objective optimization90%40% 10 k 100 k
  29. 29. Concept of Pareto-optimal solutions : non-dominated (Goldberg, 1989) B dominate A A C dominate A B, C non-dominated B D, E non-dominated2 C E dominate A, B, C D D dominate A, B E 1
  30. 30. How does multi-objective optimization work? Non-dominated Crowding distance New Parents sorting sorting for each front Population 1 2 Front 1 Front 1 Front 1RCGA M Front 2 Front 2 Front 2 N Front 3 Front 3 Front 3 N CAT Offspring 1 2 Rejected M N
  31. 31. How to extend RCGA to multi- objective optimization problemsCrossover: Min J 1 , J 2 , L , J M J i (Θ1 ) < J i (Θ 2 ) J i (θ1 ) − J i (θ 2 )if i Θ1 ← Θ1 + r (Θ1 − Θ 2 ) ωi = M i Θ 2 ← Θ 2 + r (Θ1 − Θ 2 ) ∑ J (θ ) − J (θ ) i =1 i 1 i 2else M i Θ1 ← Θ1 + r (Θ 2 − Θ1 ) θ1 = ∑ ωiθ1i i =1 i Θ 2 ← Θ 2 + r (Θ 2 − Θ1 ) M θ 2 = ∑ ωiθ 2i i =1
  32. 32. Methods Comparison The proposed NSGA-II (Deb et al., 2000)Initial population Sobol (pseudo random) Randomreproduction Tournament selection Tournament selectioncrossover •N/2 pairs by sorting •Random pair with crowding •Simulated binary distance crossover (SBX) •Multi-direction based •controlled step sizemutation Quadratic-decay Polynomial-type
  33. 33. Benchmark test 1: FON function ⎛ 3 ⎛ 1 ⎞ ⎞ 2 Optimal solutions f1 ( x ) = 1 − exp ⎜ −∑ ⎜ xi − ⎟ ⎟ ⎜ i =1 ⎝ 3⎠ ⎟ ⎡ 1 1 ⎤ ⎝ ⎠ x1 = x2 = x3 ∈ ⎢ − , ⎥ ⎛ 3 ⎛ ⎣ 3 3⎦ 1 ⎞ ⎞ 2 f 2 ( x ) = 1 − exp ⎜ −∑ ⎜ xi + ⎟ ⎟ ⎜ i =1 ⎝ 3⎠ ⎟ ⎝ ⎠ RCGA parameterss.t. N=100 −π ≤ x ≤ π Pc=0.1 Pm=0.1
  34. 34. Results: After 20 iterations The proposed method NSGA-II After 50 iterations
  35. 35. Benchmark test 2: KUR function Optimal solutions ( ( )) n −1 f1 ( x ) = ∑ −10 exp −0.2 xi 2 + xi +12 i ( ) n f 2 ( x ) = ∑ xi + 5sin xi 3 0.8 i =1s.t. −5 ≤ xi ≤ 5 RCGA parameters N=100 Pc=0.1 Pm=0.1
  36. 36. Results: After 60 iterations The proposed method NSGA-II After 150 iterations
  37. 37. Benchmark test 3: ZTD6 function f1 ( x ) = 1 − exp ( −4 x1 ) sin 6 ( 6π x1 ) ( f 2 ( x ) = g ( x ) 1 − x1 g ( x ) ) Optimal solutions g ( x ) = 1 + 10 ( n − 1) + ∑ ( xi 2 − 10 cos ( 4π xi ) ) n i=2s.t. 0 ≤ xi ≤ 1, i = 1, 2,L ,10 RCGA parameters N=100 Pc=0.1 Pm=0.1
  38. 38. Results: After 200 iterations The proposed method NSGA-II After 500 iterations
  39. 39. Data Driven Techniques Using RCGA Single-objective process optimizationx1 y1 min f (y )x2 y2 xi M M s.t.xn ym xi ,min ≤ xi ≤ xi ,max
  40. 40. Multi-objective optimization x1 y1 x2 y2 M M xn ym min f1 ( y ) , f 2 ( y ) ,L , f M ( y ) xis.t. xi ,min_ i ≤ xi ≤ xi ,max_ i , i = 1, 2,L , n
  41. 41. Data Driven Flow Chart Initialize setting runs=0 Search optimal design parametersGenerate a group of by RCGA design of experiments Yes Reach the goal Stop Train a model by Calculate objective function value No neural network algorithm Add result into the neural network model and count runs=runs+1
  42. 42. Data Driven Flow Chart Initialize setting runs=0 Generate N chromosomes ⎡ θ1,1 θ1,2 L θ1,n ⎤ Search optimal design parameters ⎢θ θ 2,2 L θ 2,n ⎥ by RCGA Θ = ⎢ ⎥Generate a group of 2,1 design of experiments ⎢ M M the O MYes⎥ Reach goal Stop ⎢ ⎥ ⎣θ N ,1 θ N ,2 L θ N ,n ⎦ Train a model by Calculate objective Generate function value No objective function values neural network algorithm obj ( Θ ) = [ obj L objN ] Add result into T obj 1 the neural 2 network model and count runs=runs+1
  43. 43. Data Driven Flow Chart Initialize setting runs=0 Train NN model NN type: feed-forward Search optimal design parametersGenerate a group of by RCGA design of experiments Yes Reach the goal Stop Train a model by Calculate objective function value No neural network algorithm Add result into the neural MSE index < 1e-3 network model and count runs=runs+1
  44. 44. Data Driven Flow Chart Initialize setting Single-objective: runs=0 Apply direction-based RCGA to search optimal Search optimal solution according toGenerate a group of design parameters current NN model. by RCGA designs of experiments Yes Reach the goal Stop Multi-objective: Calculate objective Apply multi-direction RCGA Train a model by neural network function value to search optimal Pareto- No algorithm front according to current NN result into Add model. the neural network model and count runs=runs+1
  45. 45. Data Driven Flow Chart Initialize setting Single-objective: runs=0 Calculate the objective function value from the Search optimal solution searched by RCGA. design parametersGenerate a group of by RCGA design of experiments Yes Multi-objective: Reach the goal Stop Pick up first p points from Perato front which is Train a model by Calculate objective sorted by crowding function value No neuron network distance. algorithm Then generate the Add result into corresponding objective the neuron function value(s). network model and count runs=runs+1
  46. 46. Data Driven Flow ChartCalculate performance index: Initialize setting runs=0Single objective:Generate a group of ( y j − y j ) 1 p 2 Search optimal MSE = ∑ ˆ design parameters p by RCGA design of j =1 experiments Yes Reach the goal StopMulti-objective: ∑∑ ( ) M p Calculate objective Train a model by1 1 2 MSE = neuron network y −y ˆ function value i, j i, j No M p i =1 j =1 algorithm Add result into y Predict from NN model ˆ the neuron network model and count y Calculate from process runs=runs+1
  47. 47. Data Driven Flow Chart Initialize setting runs=0 Search optimal design parametersGenerate a group of by RCGA design of experiments Yes Reach the goal Stop Train a model by Calculate objective function value No neural network algorithm Add result into the neural network model and count runs=runs+1
  48. 48. Data Driven test 1: Single-objective 2 (min F = 3 (1 − x1 ) exp − ( x1 ) − ( x2 + 1) 2 2 MATLAB, peaks function ⎛x ⎞ − 10 ⎜ 1 − x13 − x25 ⎟ exp ( − x12 − x2 2 ) ⎝5 ⎠ 1 ( − exp − ( x1 + 1) − x2 2 ⎟ 3 2 )⎞ ⎠s.t − 3 ≤ xi ≤ 3, i = 1, 2 Global optimal solution X(0.2281, -1.6255) F= -6.55113
  49. 49. 15
  50. 50. 1015
  51. 51. 2025
  52. 52. 3035
  53. 53. 40After 20 iterationsOptimal solution: x1=0.2282 x2=-1.6255Predicted F= -6.5513cf. Global optimal solution X(0.2281,-1.6255) F= -6.5511
  54. 54. Application to the optimal designof an MOCVD reactor AIXTRON AIX200/4 The schematic of horizontal MOCVD reactor (top: 3D view; bottom: 2D side view).
  55. 55. Objective function Susceptor temperature: 600K ~ 1200K Total flow rate : 10000sccm ~ 15000sccm Pressure: 8kPa ~ 15kPa Objective function: 2 1 ⎛ 1 ⎞ 1 + β (δ n ) + (1 − GR f ) (1 − δ f ) 1 2 1 2J = α⎜ + 2 ⎟ GR 2 ⎝ GR n ⎠ 2 2 2 GR = ∫ GR dA δ =∫ GR − GR dA A GR
  56. 56. 14 15 17 12 13 16 15 18 14 19 13 20 11P (kPa) 12 11 21 8 10 9 10 5 9 7 1 6 4 8 1.5 3 1.4 1200 1.3 2 1100 4 1000 x 10 1.2 900 sccm 1.1 800 700 T (K) 1 600 Convergence of the design parameters
  57. 57. Suscuptor Temperature: 600 K ~ 1200 K Total flow rate : 10000 sccm ~ 15000 sccm Pressure: 8 kPa ~ 15 kPa Before AfterGR = ∫ GR dA = 8.65 ×10 −9 m / min GR = ∫ GR dA = 11.65 ×10 −9 m / min A A GR − GR GR − GRδ =∫ dA = 0.003504 δ =∫ dA = 0.00220 GR GR
  58. 58. Data Driven Test 2: Multi-objective Pareto-optimal solutions: CONSTR function A : 0.39 ≤ x1 ≤ 0.67 ⇒ x2 = 6 − 9 x1 f1 ( x ) = x1 B : 0.67 ≤ x1 ≤ 1 ⇒ x2 = 0 f 2 ( x ) = (1 + x2 ) x1s.t. −20 ≤ xi ≤ 20, i = 1, 2 g1 ( x ) = x2 + 9 x1 ≥ 6 g1 ( x ) = − x2 + 9 x1 ≥ 1 B RCGA parameters N=100 Pc=0.1 A Pm=0.1
  59. 59. N=30, p=5
  60. 60. 16 iterations
  61. 61. Multi-objective design ofHorizontal MOCVD process AIXTRON AIX200/4 The schematic of horizontal MOCVD reactor (top: 3D view; bottom: 2D side view).
  62. 62. Objective functions Susceptor Temperature: 833K ~ 1033K Total flow rate : 13000sccm ~ 20000sccm Pressure: 10kPa ~ 100kPa Growth of GaAs film on a 3-inch substrateObjective functions: GR = ∫ GRdA A ∫ ( GR − GR ) 2 dA δ= A
  63. 63. Design of experiments -Taguchi methodL25(56) Susceptor Temperature (K): T Total flow rate (sccm): U Pressure (kPa): P Five levels for each factor Levels Variables 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
  64. 64. Data driven 12 12 10 1 10 2 8 8 6 6Uniformity index Uniformity index 4 4 2 2 0 0 -2 -2 -4 -4 -6 -6 -8 -8 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 Growth Rate (nm/min) Growth Rate (nm/min) 12 50 10 3 40 4 8 30 6 Uniformity index 20 Uniformity index 4 10 2 0 0 -10 -2 -20 -4 -6 -30 -8 -40 -50 -45 -40 -35 -30 -25 -20 -55 -50 -45 -40 -35 -30 -25 -20 Growth Rate (nm/min) Growth Rate (nm/min)
  65. 65. Data driven 50 35 40 5 30 6 25 30 20Uniformity index Uniformity index 15 20 10 10 5 0 0 -5 -10 -10 -20 -15 -45 -40 -35 -30 -50 -45 -40 -35 -30 -25 -20 -15 Growth Rate (nm/min) Growth Rate (nm/min) 50 60 40 7 50 8 40 30Uniformity index Uniformity index 30 20 20 10 10 0 0 -10 -10 -20 -50 -45 -40 -35 -30 -25 -20 -15 -20 -50 -45 -40 -35 -30 -25 -20 Growth Rate (nm/min) Growth Rate (nm/min)
  66. 66. Data driven 100 60 80 9 50 10 40 60Uniformity index Uniformity index 30 40 20 20 10 0 0 -20 -10 -50 -45 -40 -35 -30 -25 -45 -40 -35 -30 -25 -20 -15 -10 -5 Growth Rate (nm/min) Growth Rate (nm/min) 20 100 15 11 80 12 Uniformity index 60 Uniformity index 10 40 5 20 0 0 -20 -5 -50 -45 -40 -35 -30 -25 -20 -15 -10 -50 -45 -40 -35 -30 -25 -20 Growth Rate (nm/min) Growth Rate (nm/min)
  67. 67. Data driven 70 70 60 13 60 14 50 50 Uniformity indexUniformity index 40 40 30 30 20 20 10 10 0 0 -10 -10 -45 -40 -35 -30 -25 -20 -15 -10 -5 -45 -40 -35 -30 -25 -20 -15 -10 -5 Growth Rate (nm/min) Growth Rate (nm/min) 70 50 60 15 16 40 50Uniformity index Uniformity index 30 40 30 20 20 10 10 0 0 -10 -10 -45 -40 -35 -30 -25 -20 -15 -10 -5 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 Growth Rate (nm/min) Growth Rate (nm/min)
  68. 68. Data driven 50 50 40 17 40 18 30 30Uniformity index Uniformity index 20 20 10 10 0 0 -10 -10 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 -45 -40 -35 -30 -25 -20 -15 -10 -5 Growth Rate (nm/min) Growth Rate (nm/min) 50 70 40 19 60 20 50 30Uniformity index Uniformity index 40 20 30 20 10 10 0 0 -10 -10 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 5 -45 -40 -35 -30 -25 -20 -15 -10 Growth Rate (nm/min) Growth Rate (nm/min)
  69. 69. Data driven 90 90 80 21 80 22 70 70 60 60Uniformity index Uniformity index 50 50 40 40 30 30 20 20 10 10 0 0 -10 -10 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 Growth Rate (nm/min) Growth Rate (nm/min) 90 90 80 23 80 24 70 70 60 60Uniformity index Uniformity index 50 50 40 40 30 30 20 20 10 10 0 0 -10 -10 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 Growth Rate (nm/min) Growth Rate (nm/min)
  70. 70. Data driven 90 90 80 25 80 26 70 70 60 60Uniformity index Uniformity index 50 50 40 40 30 30 20 20 10 10 0 0 -10 -10 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 Growth Rate (nm/min) Growth Rate (nm/min) 90 90 80 70 27 80 28 70 60 60Uniformity index Uniformity index 50 50 40 40 30 30 20 20 10 10 0 0 -10 -10 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 Growth Rate (nm/min) Growth Rate (nm/min)
  71. 71. Data driven 90 90 80 70 29 80 70 30 60 60Uniformity index Uniformity index 50 50 40 40 30 30 20 20 10 10 0 0 -10 -10 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 Growth Rate (nm/min) Growth Rate (nm/min) 90 90 80 70 31 80 70 32 60 60Uniformity index 50 Uniformity index 50 40 40 30 30 20 20 10 10 0 0 -10 -10 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 Growth Rate (nm/min) Growth Rate (nm/min)
  72. 72. Convergence of MSE index 3 1000 10 900 2 800 10 700 1 600 10MSE MSE 500 0 400 10 300 -1 200 10 100 -2 0 10 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 runs runs
  73. 73. Optimal Pareto-front solutions of the MOCVD 90 80 B 70 60Uniformity index 50 40 30 20 C D A 10 0 -10 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 Growth Rate (nm/min)
  74. 74. Case A. the best uniformity (min. of δ ) Operating conditions: T= 883 K P=10 kPa U= 13000 sccm Performance: GR = 3.461 (nm/ min) δ = 0.00409
  75. 75. Case B. the max. growth rate (min. − GR ) Operating conditions: T= 957.9659 K P=10 kPa U= 20000 sccm Performance: GR = 44.346 (nm/ min) δ = 82.059
  76. 76. Case C. min. J = −GR + δ Operating conditions: T= 935.82 K P=18.87 kPa U= 20000 sccm Performance: GR = 34.852 (nm/ min) δ = 3.245
  77. 77. Case D. min. J = −GR + 10δ Operating conditions: T= 917.81 K P=16.23 kPa U= 20000 sccm Performance: GR = 29.401 (nm/ min) δ = 1.024
  78. 78. ConclusionsAn efficient global optimization schemeusing a real-coded genetic algorithmshas been proposed.Effective data driven techniques forsingle objective and multi-objectiveoptimal process design have beendeveloped.The proposed schemes have been testedsuccessfully on the optimal design ofMOCVD processes.
  79. 79. Q&AThanks for your attention.

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