Genetic Algorithm for Multi-Objective Optimization 2001 Shinya Watanabe
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1 Genetic Algorithm for Multi-Objective Optimization 2001 Shinya Watanabe
2 i VEGA... 17
3 ii WBGA MOGA NSGA NPGA SPEA NSGA-II SPEA (error : I error ) (cover rate: I cover ) ,, (Max and Min and Average: I MMA ) Ratio of Non-dominated Individuals: I RNI ) (Sampling of the Pareto Frontier Lines of Intersection: I LI GA GA GA... 65
4 iii GA GA
5 iv A 157
6 v 2.1 The concept of Pareto optimal solution Schematic of GA Flowchart of GA procedure Schematic of GA search Schematic of VEGA Distribution of Pareto Solutions in VEGA The concept of Pareto-ranking The concept of Non-dominated Sorting method The SPEA fitness assignment scheme Schematic of the NSGA-II procedure The crowding distance calculation Comparison of fitness assignement schemes in SPEA and SPEA Archive truncation method used in SPEA Schematic of I cover An example of I MMA Schematic of I RNI(X,Y ) Schematic of I LI Schematic of master slave model Schematic of distributed population model Schematic of neighborhood model Schematic of the Total Sharing procedure Pareto optimum individuals (T2) Pareto optimum individuals (T4) I error of T2 and T
7 vi 5.8 I cover of T2 and T Schematic of DRMOGA Pareto optimum individuals(ndp) Pareto optimum individuals(vb3) Pareto optimum individuals (f2 f3, VB3) I error of NDP I cover of NDP and VB Schematic of DC-Scheme Pareto optimum individuals (ZDT4) Pareto optimum individuals (KUR) I error of ZDT I cover of ZDT4 and KUR Pareto-optimal front of F discon I cover and I error of F discon (N = 10) I MMA of F discon (N = 10) I RNI and I LI of F discon (N = 10) Pareto optimum individuals(f discon (N = 10)) I cover and I error of F discon (N = 100) I MMA of F discon (N = 100) I RNI and I LI of F discon (N = 100) Pareto optimum individuals(f discon (N = 100)) I cover and I error of ZDT I MMA of ZDT I RNI and I LI of ZDT Pareto optimum individuals(zdt4) I cover and I error of ZDT I MMA of ZDT I RNI and I LI of ZDT Pareto optimum individuals(zdt6) I cover of KUR I MMA of KUR I RNI and I LI of KUR Pareto optimum individuals(kur)
8 vii 6.22 I cover of KP I MMA of KP I RNI and I LI of KP Pareto optimum individuals(kp-2) I cover of p I error of p I RNI and I LI of F discon (N = 10) Pareto optimum individuals(f discon (N = 10)) I RNI and I LI of F discon (N = 100) Pareto optimum individuals(f discon (N = 100)) I RNI and I LI of ZDT Pareto optimum individuals(zdt4) I RNI and I LI of ZDT Pareto optimum individuals(zdt6) I RNI and I LI of KUR Pareto optimum individuals(kur) Optimization System Coding method Derived non-dominated solutions Derived non-dominated solutions(sfc,nox) Derived non-dominated solutions(sfc,soot) Derived non-dominated solutions(nox,soot) Concept of sequence-pair Coding example of sequence-pair Horizontal/Vertical Constrain graphs Placement-based Partially Exchanging Crossover(PPEX) Results of I LI (33 blocks) I MMA of 33 blocks Derived non-dominated solutions(33 blocks) Results of I LI (50 modules) Results of I LI (100 modules) I MMA of 50 modules I MMA of 100 modules
9 viii 7.18 Derived non-dominated solutions(50 modules) Derived non-dominated solutions(100 modules) Results of I LI (500 blocks) I MMA of 500 blocks Derived non-dominated solutions(500 blocks) The placement of the modules(33 modules) The placement of the modules(100 modules)
10 ix 3.1 A brief history of EMO A example of weight vector Table GA parameters Specification of the target diesel engine GA parameters Cluster System Calculation time GA Parameter
11 ( ) (Evolutionary Multi-Objective Optimization:
12 2 1 EMO) 2 9 EMO 1 (Genetic Algorithm: GA) GA 2 9 GA GA GA 1 GA 2 9 Deb NSGA-II 8 Zitzler SPEA2 9 GA GA 2 GA GA GA GA GA GA (Neighborhood Cultivation GA :) 2 GA
13 i) NOx 3 1 NOx 3 ii) , GA 2 2 4
14 4 1 3 GA GA GA 3 Schaffer VEGA Hajela WBGA Fonseca MOGA Horn NPGA Srinivas NSGA Deb NSGA-II Zitzler SPEA SPEA2 4 GA GA GA GA GA 5 GA GA 3 6 GA 6 NSGA-II, SPEA2 GA, NSGA-II SPEA
15 ( )., ( ) 1 Pareto-optimal solution 1 Multiobjective Optimization Problems MOPs
16 n k f i (x 1,x 2,...,x n ) (i =1, 2,...,k) (2.1) m g j (x 1,x 2,...,x n ) 0 (j =1, 2,...,m) (2.2) ( ) 1 f i (x) (Pareto-optimal solution) Pareto x 1 x 2 I(x =(x 1,x 2,...,x n )) a) f i (x 1 ) f i (x 2 )( i =1,...,k) x 1 x 2 b) f i (x 1 ) <f i (x 2 )( i =1,...,k) x 1 x 2 x 1 x 2 x 1 x 2 2 x 0 I a) x 0 x I x 0 (Weak Pareto-optimal solution) 2
17 2.3 7 f2(x) Feasible region Weak Pareto-optimal solutions o Pareto-optimal solutions on f1(x) 2.1 The concept of Pareto optimal solution b) x 0 x I x 0 (Pareto-optimal solution) ( 2.1 ) (weight method) w i wf(x) 1 w =(w 1,w 2,...,w k ) 3
18 8 2 k min wf(x) = w i f i (x) (2.3) x X w i 0 i=1 k w k = 1 (2.4) w ( ε )(constraint method) 1 f j (x) (k 1) ε i (i =1, 2,...,k,i k) ε i=1 min x X f j (x) (2.5) subject to w i f i (x) ε i,i=1, 2,...,k,i j (2.6) i ε k f 1 f 1 f 1 f 2 f
19 (Genetic Algorithm: GA) GA 1
20
21 (Evolutionary Computation: EC) (Evolutionary Multi- Criterion Optimization: EMO) 2, 6 9 EMO 1 (Genetic Algorithm: GA) GA GA 2 GA 4, 10 GA, 1 GA 1 ( ) 2 1
22 12 3 GA GA 3.2 (Genetic Algorithm: GA) GA Michigan Holland 1975 Adaptation in Natural and Artificial System Goldberg Genetic Algorithms in Search, Optimization, and Machine Learning 4 4, GA GA (Individual) Chromosome (Geno Type) (Pheno Type) (Population) GA (Selection) (Crossover) (Mutation) 3.1 GA GA 3.2 Initialization : (Population Size)
23 Objective Function GA Operators Selection Crossover Mutation etc.. Chromosome Decoding Pheno Type Encoding Geno Type Gene Individual 3.1 Schematic of GA Evaluation : (Evaluation Value) Selection : Crossover : (Crossover Rate)
24 14 3 Start Initialization Evaluation Selection Crossover Mutation Evaluation No Terminate Check Yes End 3.2 Flowchart of GA procedure Mutation : (Gene) (locus) DNA (Mutation Rate) Terminate Check : GA Goldberg GA 4 4
25 GA GA 1 12, 13, GA No.6 No.1 Optimum Solutions A : global B,C : local A Schematic of GA search GA
26 Schaffer VEGA 3 EMO 01(Conference on Evolutionary Multi-Criterion Optimization) 18 (Genetic Algorithm: GA) GA 18 GA GA GA GA GA 1 GA 2 GA 1 ( ) 3 ( ) GA
27 ( ) GA Schaffer VEGA(Vector Evaluated Genetic Algorithm) Goldberg Fonseca MOGA(Multi-Objective Genetic Algorithm) GA A brief history of EMO Year Name Proposer(s) Characteristic 1985 VEGA Schaffer The first multi-objective GA 1993 WBGA Hajela and Lin Weighted function 1993 MOGA Fonseca and Fleming Pareto-based selection 1993 NPGA Horn and Nafpiliotis Niching 1994 NSGA Srinivas and Deb Non-dominated Sorting 1999 SPEA Zitzler and Thiele Archiving + elitism 2000 NSGA-II Deb Crowding distance 2001 SPEA2 Zitzler and etc Archive truncation + improved fitness assignment shceme VEGA Schaffer 1985 GA (Vector Evaluated Genetic Algorithm: VEGA) 3 VEGA ( ) VEGA GA VEGA VEGA
28 18 3 t generation The objrctive function f1 The partial population 1 t +1 generation popration fp The partial population P popration Selection Crossover & Mutation 3.4 Schematic of VEGA M N VEGA Step 1 i =1 q = N/M Step 2 j =1+(i 1) q j = i q F (x j )=f i (x j ) (3.1) Step 3 q P i Step 4 i = M Step 5 i = i +1 Step 2 Step 5 P i P (P = M i=1 P i). P VEGA GA VEGA
29 VEGA 1 VEGA ( 3.5) Dominated Solution Non-dominated Solution f2 Pareto-optimal solution f1 3.5 Distribution of Pareto Solutions in VEGA WBGA Hajela Lin 1993 WBGA( Weight-Based Genetic Algorithm ) 2 (2.3.1 ) GA 1 GA, 1 WBGA 2 a) b) ( )
30 20 3 a) WBGA F (x) = k i=1 w if i (x) w i x 1 F (x 1 )= M j=1 f w xi w j (x 1 ) fj min j f max j f min j (3.2) (3.2) (3.2) x i w A example of weight vector Table x w Weight vector (0.1,0.9) (0.2,0.8) (0.3,0.7)... (0.9,0.1) Sh(d) 2 i j d i,j d i,j = x i w x j w (3.3) d i,j Sh(d i,j ) (niche count) nc i nc i Sh(d i,j ) F i = F (xi )/nc i Sh(d i,j ) i (σ share ) j 0 1 (3.4) { Sh(d i,j )= 1 d i,j σ share, ifd i,j σ share ; 0, otherwise. (3.4) (3.4) Sh(d i,j ) (d i,j ) 1 σ share 0
31 Sh(d i,j ) i nc i (3.5) N nc i = Sh(d i,k ) (3.5) k=1 Step 1 f j fj max fj min Step 2 i =1, 2,...,N d i,k = x i w xk w (3.4) Sh(d i,k ) i nc i (3.5) Step 3 i =1, 2,...,N F i = F (xi )/nc i WBGA GA GA WBGA x w GA WBGA WBGA 2
32 22 3 b) VEGA K w (k) (k =1, 2,...,K) N (3.2) w (k) N w (k) N/K w (k) K 1 1 ( ) K w (k) (k = 1, 2,...,K) 1 K M Step 1 k=1 Step 2 w (k) x (i) F j F (x (i) )= M j=1 f w x(i) w j (x (i) ) fj min j f max j f min j (3.6) F N/K. P k Step 3 Step 4 P k. N/K k<k k 1 ( k = k +1) Step2 P = K k=1 P k. P <N N
33 Step 2 N K K N K N O(N ) 2 GA MOGA 1993 Fonseca MOGA(Multi-Objective Genetic Algorithm) 19 MOGA X i n i X i r(x i ) r(x i )=1+n i (3.7) ( 1 ) Fonseca Fleming ( 3.4) i j
34 24 3 f2(x) f1(x) The concept of Pareto-ranking (3.8) f max k ( d ij = M f i k f j k k=1 f max k f min k ) 2 (3.8),fk min k (3.4) Sh(d ij ) nc i = j=1 µ ri Sh(d i,k ) (3.9) (3.9) µ ri r i MOGA Step 1 (µ(j) =0(j =1,...,N):i =1) µ(j) j Step 2 i n i i r i r i =1+n i r i µ(r i ) (µ(r i )=µ(r i )+1) Step 3 i<n i = i +1 Step 2 Step 4 Step 4 µ(r i ) > 0 r i r (3.10)
35 r i 1 F i = N µ(k) 0.5(µ(r i ) 1) (3.10) k=1 r i =1 i (3.10) F i = N 0.5(µ(1) 1) F i µ(1) N N µ(1) + 1 r c =1 Step 5 r c i (3.9) F j = F j/nc j F j F jµ(r c ) µ(rc) k=1 F F k j (3.11) Step 6 r c <r r c = r c +1 Step 5 r i 1 ( ) r i ( ) MOGA MOGA MOGA ( 1 ) 2 MOGA
36 NSGA GA(Non-dominated Sorting Genetic Algorithm: NSGA) 1994 Deb, Srinivas Goldberg 4 1 (nondominated sorting) MOGA NSGA NSGA (3.12) ( d i,j = P 1 x i k x j k k=1 x max k x min k ) 2 (3.12) NSGA NSGA Step 1 r =1 Step 2 (P ) r Step 3 P r = r +1 Step 4 ( P ) Step 2 Step 3 MOGA 3.7 NSGA
37 f2(x) f2(x) f1(x) f1(x) (a) Pareto Ranking method(moga) (b) Non-dominated Sorting method(nsga) 3.7 The concept of Non-dominated Sorting method Step 1 σ share ɛ F min = N + ɛ, j =1 Step 2 P : (P 1,P 2,...,P ρ ) = Sort(P, ) Step 3 P j q Step 3a F (q) j = F min ɛ Step 3b (3.4) P j nc p Step 3c F q j = F q j nc q Step 4 P j F min = min( F q j : q P j) Step 5 j = j +1 P j P ρ (j ρ) Step 3 NSGA NSGA
38 28 3 σ share NSGA σ share NPGA NPGA(Niched Pareto Genetic Algorithm) 1994 Horn Nafpliotis MOGA NSGA 21 NPGA 11 NPGA NPGA NPGA N P 2 i j t dom ( N) 2 i j 1 i j i j 2 i j Q, NPGA winner = NPGA-tournament(i, j, Q) Step T1 P t dom T ij Step T2 i T ij α i j T ij α j
39 Step T3 α i =0 α j > 0 i Step T4 α i > 0 α j =0 j Step T5 Q < 2 i j 5 Q 2 Q i j i σ share (k Q) i j d ik ( ) d ik = M f i m fm k 2 fm max fm min (3.13) (3.13) f max m f min m m=1 m Step T6 nc i nc j i j NPGA 2 GA NPGA 2 NPGA NPGA Step 1 P i =1,Q = Step 2 1 p 1 =NPGA-tournament(i, i+ 1,Q). Step 3 i = i +2 2 p 2 =NPGA-tournament(i, i +1,Q). Step 4 p 1 p 2 c 1 c 2 c 1 c 2 Step 5 Q Q = Q {c 1,c 2 }.
40 30 3 Step 6 i = i +1 i<n Step 2 Q = N/2 P i =1 Step 2 NPGA VEGA NSGA MOGA GA NPGA NPGA t dom N NPGA M NPGA O(Mt dom ), O(N 2 ) NPGA NPGA σ share t dom 2 NPGA GA NSGA MOGA σ share NPGA i σ share NSGA MOGA (3.4) i Sh(d) NPGA NPGA σ share t dom SPEA 1999 Zitzler,Thiele SPEA 6 SPEA a) ( ) GA 2, 9, 21
41 b) c) SPEA Step 1 P P Step 2 P P Step 3 P 1 Step 4 P N P Step 5 P P Step 6 P + P, P SPEA Step 7 P Step 8 Step2 SPEA SPEA
42 32 3 SPEA 2 P ( )P Step 1 i P s i [0, 1) 4 s i P N i j P n (s i = n N +1 ) i s i (f i = s i ) Step 2 j P j i P s i : f j =1+ i,i j s i where f j [1,N) (3.14) (3.14) 1 P P SPEA 3.8 SPEA 3.8 σ share SPEA P N N P N N (N > N) P 5 N cluster distance 4 SPEA Strength Strength 5
43 f2(x) 1/5 4/5 6/5 4/5 13/5 10/5 16/5 12/5 population P extermal population P 3/5 f1(x) 3.8 The SPEA fitness assignment scheme C 1 C 2 d 12 (i C 1,j C 2 ) d = 1 C 1 C 2 i C 1,j C 2 d(i, j) (3.15) SPEA d(i, j) (3.15) SPEA N N Step 1 C N i P C = {C 1,C 2,...,CN } Step 2 C N Step 5 Step 3 Step 3 ( C 2 ) Step 4 d C 1 C 2 ; C 1 Step 2 Step 5
44 34 3 SPEA SPEA SPEA SPEA 2 SPEA N N N SPEA N N SPEA N N SPEA 1:4 SPEA NSGA-II NSGA-II(Elitist Non-Dominated Sorting Genetic Algorithm) NSGA Deb, Agrawal NSGA-II NSGA ( ) (Crowded Tournament Selection Operator) (Crowding Distance) NSGA-II NSGA
45 NSGA-II NSGA NSGA-II NSGA-II P t Q t 2 NSGA-II P t Q t t P t Q t Q t Q t Q t P t R t = P t Q t 2N R t N P t+1 NSGA-II Step 1 R t = P t Q t R t ( ) F i,i=1, 2,..., etc. Step 2 P t+1 = i =1 P t+1 + F i <N P t+1 = P t+1 F i i = i +1 Step 3 (Crowding-sort) N P t+1 P t+1 Step 4 P t+1 Q t+1 NSGA-II P t Q t R t N P t+1 ( )Q t P t P t
46 36 3 P t Q t NSGA-II P t 3.9 P t Non-dominated sorting F 1 F 2 Crowding distance sorting P t+1 F 3 Q t R t F 4 Rejected 3.9 Schematic of the NSGA-II procedure (Crowded Tournament Selection Operator) (r) (F r ) 2 i 2 1) (i rank ) 2) (i distance ) i j 2 i j 1) i j :(i rank <j rank ) 2) i j i j :(i rank = j rank )and(i distance >j distance )
47 (Crowding Distance) i ( ) 3.10 front F 1 f2(x) i-1 front F 2 d i2 i i+1 d i f1(x) The crowding distance calculation Step C1 F l :l = F i :d i =0 Step C2 m =1, 2,...,M :I m = sort(f m,>). Step C3 m =1, 2,...,M, ( m ) : d I m 1 = d I m l = 6 (j =2,...,l 1) 6 I j j d I m j = d I m j + f Im j+1 m f Im j 1 m f max m f min m (3.16)
48 38 3 NSGA-II σ share P t Q t R t = P t Q t N N SPEA2 SPEA2(Strength Pareto Evolutionary Algorithm 2) SPEA Zitzler SPEA NSGA-II SPEA SPEA2 SPEA SPEA2 SPEA2 N
49 NSGA-II SPEA2 SPEA2 NSGA-II 2 ( P P ) Step 1 : P 0 :P 0 =0.Set t =0. Step 2 : P t P t (3.3.9 ) Step 3 : P t P t P t+1 P t+1 > N P t+1 P t+1 < N P t N P t+1 P t+1 P t+1 N Step 4 : t T P t+1 Step 5 Step 5 : P t+1 N P t+1 Step 6 : P t+1 (Environmental Selection) (Mating selection) P t P t P t P t P t+1 (Mate) SPEA2 NSGA-II (binary tournament selection)
50 40 3 SPEA2 i s(i) f(i) s(i) i 0 SPEA SPEA f2(x) 1/5 4/5 6/5 13/5 4/5 10/5 16/5 12/5 f2(x) /5 0 f1(x) (a) SPEA fitness assiignment f1(x) (a) SPEA2 fitness assiignment 3.11 Comparison of fitness assignement schemes in SPEA and SPEA SPEA2 s(i) SPEA SPEA2 SPEA SPEA2 SPEA 2 (i) (ii) (archive truncation method) SPEA2 P t P t 0 P t+1
51 P t+1 = i i P t + P t F (i) < 1 (3.17) ( P t+1 = N ) N P t+1 P t + P t P t + P t F (i) 1 N P t+1 i P t+1 1 P t+1 = N P t+1 N Step A1 i j k =2 Step A2 i j k (σi k,σk j )7 Step A3 σi k >σj k σk i <σj k σi k = σk j k = k +1 Step A SPEA2 SPEA 9 NSGA-II NSGA-II 7 σi k i k
52 42 3 f 2 f f 1 f 1 (a) before (b) after (N=5) 3.12 Archive truncation method used in SPEA2 SPEA SPEA σ k O(M 3 )(M = N + N) 2 3 O(M 2 log M) 3.4 GA NSGA-II SPEA2 2, 9 GA
53 i) ii) iii) NSGA-II SPEA2 GA GA GA GA 23 22, 24 GA GA GA GA PC GA GA 2 1. GA 2. GA
54 44 3 GA
55 GA 3 3 6, 22, 25,
56 46 4 i) ii) iii) iv) 2 v) (error : I error ) (I error ) 3 k x k = (x k i,...,xk m) f(x 1,...,x m )=C e(x k ) (4.1) e(x k )=C f(x k i,...,x k m) (4.1) I error e(x) E 1 N e(x k ) (4.2) N (4.2) N k=1
57 Min Max Max f2 Min f1 4.1 Schematic of I cover (cover rate: I cover ) (I cover ) 22 f k I coverk I coverk N k N (4.3) (4.3) N N k I cover I coverk I cover 1 N I coverk (4.4) N k=
58 48 4 Maximum f(x) Avrage Minimum 4.2 An example of I MMA 4.2.3,, (Max and Min and Average: I MMA ),., I MMA. I MMA 4.2., I MMA,,, Ratio of Non-dominated Individuals: I RNI ) I RNI 2 Tan X Y S U., S U, S P., S P I RNI(X,Y ) 4.3 I RNI(X,Y ) 100
59 Method X Method Y 67% 33% Method X f2 Method X = 4/6 = Method Y = 2/6 = f 1 f2 f2 Method Y f 1 f Schematic of I RNI(X,Y ) (Sampling of the Pareto Frontier Lines of Intersection: I LI ), Knowels Corne (X, Y ) 4.4. ( 4.4 ).,,,. 2 (X, Y ), I LI (X, Y ), X Y., X I LI (X, Y )=1, I LI (Y,X)=0. I LI (X, Y ) 3
60 50 4 f2 X Y f Schematic of I LI i) ii) iii) X Y I LI (X, Y )=1,I LI (Y,X)=0
61 (I error ) (I cover ) (I MMA ) 3 2 (I RNI ) (I LI ) 2 3 1) (I error ) (I RNI ) 2) (I cover ) (I MMA ) 3) (I LI )
62
63 GA 2 GA GA GA GA GA 30 GA GA 23 22, 24 GA
64 54 5 GA GA 1 PC 31 GA GA 3 GA(Total Sharing Genetic Algorithm: TSGA) GA(Divided Range Multi-Objective Genetic Algorithm: DRMOGA) (Distributed Cooperation Scheme: DC-Scheme) GA 5.2 GA GA 3
65 5.2 GA 55 Population Shared Memory PE 5 PE 4 PE 1 PE 2 PE 3 Individual PE # Evaluation Processor 5.1 Schematic of master slave model (master slave model) (distributed population model) (subpopulation)
66 56 5 Sub population Individual Migration 5.2 Schematic of distributed population model GA 27, 28 1 GA (Migration) (Migration interval) (Migration rate) PC (neighborhood model) GA
67 Processor Neighborhood Individual Exchange 5.3 Schematic of neighborhood model GA 3 30 GA GA 1) 2)
68 58 5 (Total Sharing GA: TSGA) 1) GA TSGA ( ) TSGA GA TSGA TSGA i) 33 ii) TSGA N P N 2N N 1 TSGA N N S N (N S <N) Fonseca 19 N N S N
69 TSGA M, N q = N/M Step 1 t S t. Step 2 S t >N Step 3 (t = t +1) Step 3 (Pi t) 1 (P t ) (P t = M i=1 P i t). Step 4 P P N Step 5 P q t = t +1 Step 4 Step TS1 Fonseca 19 P 2 P P P N> P N P P,P P N < P Step TS2 Step TS2 P N > P P N N s P,P P N < P Step TS3 Step TS N P 3 N 5.4 TSGA 2 1
70 60 5 if all population>p size ( ) 1 island 2 island P - 1 island Sharing with all populations 1 island 2 island P - 1 island P island P island 5.4 Schematic of the Total Sharing procedure GA(Single population GA: SGA) GA(Distributed GA:DGA) 27, 28 GA(TSGA) 2 34 T2, T4 T2 min f 1 (x)=x 2 1 x 2 min f 2 (x)= 1 2 x 1 x 2 1 s.t. g 1 (x)= 1 T2 : 2 x 1 + x g 2 (x)= 1 2 x 1 + x g 3 (x)= 1 2 x 1 + x g 4 (x)=x 1 0 g 5 (x)=x 2 0 (5.1) T4
71 GA parameters SGA DGA TSGA Terminal generation 250 Total population size 100 Crossover rate 1.0 Mutation rate 0.0 Number of islands - 5 Migration interval - 5 (sort interval) Migration rate min f 1 (x)= 2x 1 + x 2 min f 2 (x)=x 2 s.t. T4 : g 1 (x)=x 2 1 x 2 0 g 2 (x)=x 1 0 g 3 (x)=x (5.2) GA 35 GA SGA DGA TSGA 3 GA T2 T ,
72 T2 T DGA SGA TSGA TSGA SGA T4 T4 T2 5.6 DGA 5.8 DGA 5.7 T2 DGA SGA SGA f TSGA f f2 f2 f DGA f Pareto optimum individuals (T2)
73 f2 f SGA TSGA f1 f1 5.6 f DGA Pareto optimum individuals (T4) f SGA DGA TSGA (a) T I error of T2 and T4 SGA DGA TSGA (b) T4
74 SGA DGA TSGA (a) T I cover of T2 and T4 SGA DGA TSGA (b) T4 TSGA DGA ( ) DGA T2 T4 TSGA DGA SGA
75 DGA SGA TSGA DGA SGA TSGA TSGA DGA DGA SGA DGA SGA 2 GA 5.4 GA GA 1 TSGA GA TSGA (Divided Range Multi-Objective Genetic Algorithm: DRMOGA) GA DRMOGA GA N K f 1 f M M Step 1 N Step 2
76 66 5 division 1 division 2 division 3 f2 (x) Pareto-optimal solution Min f 1 (x) Max 5.9 Schematic of DRMOGA Step 3 f i f i f 1 f M N/K N/K K Step 4 GA GA Step 5 Step 5 GA s Step 3 K s (M =2) f 1 3 (K =3) 5.9 DRMOGA DGA SGA
77 DRMOGA DGA SGA GA Fonseca MOGA 19 DRMOGA NDP VB3 2 NDP min f 1 = x 1 min f 2 = x 2. min f n 1 = x n 1 NDP : min f n = x n s.t. g j = x j (j =1, 2,,n) g n+k = x k 6(k =1, 2,,n) g 2n+1 =1 x 1 x 2 x n (5.3) (5.3) NDP n (5.3) g 2n+1 VB3 min f 1 (x)=0.5(x x2 2 ) + sin(x2 1 + x2 2 ) min f 2 (x)= (3x 1 2x 2 +4) 2 + (x 1 x 2 +1) VB3 : min f 3 (x)= x 2 1 +x exp( x2 1 x2 2 ) s.t. g 1 (x)=x 1 3 g 2 (x)=x 2 3 (5.4) VB3 (5.4) Veldhuizen (x 1,x 2 )=(0.0, 0.0) f 3 (x) = 0.1 (x 1,x 2 )=(, ) f 3 (x) =0.0
78 f 2 (x) -2-3 SGA f 2 (x) -2-3 DGA f 1 (x) f 1 (x) 0-1 f 2 (x) -2-3 DRMOGA f 1 (x) 5.10 Pareto optimum individuals(ndp) 1 2 f 2 (x) f 2 (x) [15, 17.5] f 3 (x) =15 f 2 (x) [15, 17.5], 5.1 DRMOGA s VB3 f 2 (x) f 2 (x),f NDP NDP VB VB3.
79 f 3 (x) f 3 (x) f 2 (x) 15.5 f 3 (x) SGA f 1 (x) f (x) 15 0 DGA f 1 (x) f 2 (x) DRMOGA f 1 (x) 8 Pareto optimum individuals(vb3) f 3 (x) f 3 (x) f 2 (x) SGA f 2 (x) DGA f 3 (x) f 2 (x) DRMOGA Pareto optimum individuals (f2 f3, VB3)
80 SGA 5.13 DGA DRMOGA I error of NDP SGA DGA DRMOGA (a) NDP SGA I cover of NDP and VB3 DGA DRMOGA (b) VB3 NDP 5.10 SGA DRMOGA DGA DRMOGA SGA DGA DRMOGA 5.13 DRMOGA SGA DGA 2
81 VB3 (5.4) 3 3 f 2 (x) =15 f 2 (x) = DRMOGA SGA DGA 5.14 f 2 f SGA DRMOGA SGA DGA f 2 (x) =15 DRMOGA 2 DRMOGA VB3 f 2 f 3 DRMOGA SGA DRMOGA DGA DGA SGA DRMOGA DGA 5.3 DGA SGA DRMOGA SGA
82 72 5 DGA DRMOGA DRMOGA DGA GA GA(Divided Range Multi- Objective Genetic Algorithm: DRMOGA) DRMOGA DRMOGA 1. GA DGA SGA DRMOGA 2. DRMOGA 5.5 TSGA DRMOGA GA (Distributed Cooperation Scheme: DC- Scheme) GA GA
83 DC-Scheme GA GA(Single Objective Genetic Algorithm: SOGA) DC-Scheme M SOGA M GA(Multi Objective Genetic Algorithm: MOGA) 1 M +1 DC-Scheme GA SOGA MOGA DC-Scheme DC-Scheme DC-Scheme GA GA DC-Scheme MOGA SOGA MOGA DC-Scheme MOGA M+1 DC-Scheme MOGA SOGA GA M M SOGA GA MOGA M M +1 M SOGA 1 MOGA
84 74 5 DC-Scheme GA GA 2 SOGA MOGA F i (x) i =1, 2...M MOGA M B Mi N SOGA B Si MOGA B Mi SOGA B Si B Mi >B Si MOGA i SOGA B Si >B Mi SOGA MOGA GA 6, 8, 9. DC-Scheme N DC-Scheme Step 1
85 MOGA Group Multi-objective GAs Single-objective GAs SOGA(F1) Group Single-objective GAs SOGA(F2) Group 5.15 Schematic of DC-Scheme Step 2 MOGA F i M SOGA Step 3 GA Step 4 Step 5 MOGA B Mi F i SOGA F i SOGA B Si MOGA Step 6 B Mi B Si B Si >B Mi SOGA MOGA MOGA B Mi B Si MOGA SOGA Step 7 3. DC-Scheme
86 DC-Scheme DC-Scheme GA Fonseca 5 MOGA Zitler SPEA2 9 Deb NSGA-II 8 DC-Scheme MOGA ( ) MOGA Fonseca MOGA 5 GA GA DGA 27, Zitzler Deb ZDT4 40 Kursawe KUR 37 ZDT4 min f 1 (x)=x 1 x 1 min f 2 (x)=g(x) 1 g(x) ZDT4 : s.t. g(x) = i=2 [x2 i 10 cos(4πx i)] x 1 [0, 1], x i [ 5, 5], i=2,...,10 (5.5) 10 2 x 1 x i =0.0(i =2,...,10) KUR KUR : min min s.t. f 1 = n i=1 x ( 10 exp( i + x2 i+1 )) f 2 = n i=1 ( x i sin(x i ) 3 ) x i [ 5, 5], i=1,...,n, n= 100 (5.6) f 1 (x) 2 f 2 (x) 100
87 f 2 (x) f 2 (x) f 2 (x) MOGA Pareto-optimal solutions f 1 (x) SPEA2 Pareto-optimal solutions f 1 (x) NSGA-II Pareto-optimal solutions f 1 (x) f 2 (x) f 2 (x) f 2 (x) MOGA with DC Pareto-optimal solutions f 1 (x) SPEA2 with DC Pareto-optimal solutions f 1 (x) NSGA-II with DC Pareto-optimal solutions f 1 (x) 5.16 Pareto optimum individuals (ZDT4) ZDT ZDT4 KUR 5.16, 5.17 ZDT ZDT4 KUR KUR.
88 78 5 f 2 (x) f 2 (x) f 2 (x) MOGA f 1 (x) SPEA f 1 (x) NSGA-II f 1 (x) f 2 (x) f 2 (x) f 2 (x) MOGA with DC f 1 (x) SPEA2 with DC f 1 (x) NSGA-II with DC f 1 (x) 5.17 Pareto optimum individuals (KUR) without DC without DC without DC with DC with DC with DC MOGA SPEA2 NSGA-II 5.18 I error of ZDT4
89 without DC without DC without DC with DC with DC with DC MOGA SPEA2 NSGA-II (a) zdt without DC without DC without DC with DC with DC with DC MOGA SPEA2 NSGA-II I cover of ZDT4 and KUR (b) kur ZDT4 ZDT4 f 1 x 1 DC-Scheme DC-Scheme f 1 f 1 DC-Scheme DC-Scheme 5.18 DC-Scheme ZDT4 DC-Scheme KUR 100 DC-Scheme DC-Scheme DC-Scheme
90 80 5 DC-Scheme KUR DC-Scheme DC-Scheme DC-Scheme GA GA ZDT4 DC-Scheme (Distributed Cooperation Scheme: DC-Scheme) DC-Scheme GA GA GA DC-Scheme GA DC-Scheme MOGA 5 SPEA2 9 NSGA-II 8 3 GA GA DC-Scheme 5.6 GA 3 GA(Total Sharing Genetic Algorithm :TSGA) GA(Divided Range Multi-Objective Genetic Algorithm :DRMOGA) (Distributed Cooperation Scheme :DC-Scheme) TSGA DRMOGA GA GA 1)
91 ) TSGA DGA DRMOGA DRMOGA DRMOGA GA 1 GA DRMOGA DC-Scheme GA GA GA GA GA
92
93 GA Deb NSGA-II 8 Zitzler SPEA2 9 Erickson NPGA2 7 Fonseca MOGA GA DRMOGA 1 GA GA GA GA GA(Neighborhood Cultivation GA: ) Deb NSGA-II 8 Zitzler SPEA2 9
94 GA 1999 SPEA 6 NPGA2 7 NSGA-II 8 SPEA2 9 GA GA 6 9 a) (elitism) 8, 9 NSGA-II SPEA2 8, 9 b) NSGA-II SPEA2 8, 9 (mating selection) c)
95 SPEA2 (truncation method) 9 NSGA-II (crowding distance calculation) 8 d) GA MOGA SPEA2 NSGA-II 7 9 NPGA 2 21
96 86 6 NSGA NSGA-II SPEA2 e) NPGA2 7 ( ) O i = O i O i,min O i,max O i,min (6.1) (6.1) O i i O i O i O i,max O i,min i 6.3 (Neighborhood Cultivation GA: ) 6.2
97 Step1 t =1 (A t ). Step2 (A t ) (P t ) P t 1 f j (x) t M (t j (mod M)) Step 3 4 i 0 Step3 Step2 (P t ) i i +1 2 Step4 2 Step3 2 i 2 i Step3 Step4 (P t+1 ) Step5 (P t+1 ) (A t ) (A t+1 ) SPEA2 (environmental selection) 1 SPEA2 NSGA-II N Step6 t = t +1 Step
98 f i (x) t M (t i (mod M)) (A t ) SPEA2 9 (3.3.9 ) SPEA2 Step1 (P t+1 ) (A t ) R t Step2 R t SPEA2 (3.3.9 ) Step3 R t A t+1 SPEA2 NSGA-II (A t ) (P t ) (mating selection) 2 (A t ) (P t ) 6.4 Zitzler SPEA2 Deb NSGA-II (non-). 2
99 non-. non , 9 2 Deb (F discon ) 39 Zitzler Deb ZDT4 ZDT6 40 Kursawe KUR 37 4 (F discon ) ZDT6 ZDT4 KUR ( (5.5) (5.6) ) (F discon ) min f 1 = x 1, min f 2 (x) = g(x) h(f 1,g) F discon : s.t. g(x 2,...,x N )=1+10 h(f 1,g) = 1 N i=2 x i N 1 ) 0.25 f 1 g g sin(10πf 1) ( f1 x i [0, 1], i=1,...,n (6.2) [0,1] x i = 0 for i =2,...,N [0,1] x 1
100 90 6 f 2 (x) f 1 (x) Pareto-optimal front of F discon 6.1 f 1 (x 1 )=x 1 ([0, 1]) f 2 (x i )=1 x x 1 sin(10πx 1 ) ZDT6 min f 1 =1 exp( 4x 1 ) sin 6 (6πx 1 ) ( ) 2 f1 min f 2 = g(x) 1 g ZDT6 : s.t. (6.3) ( N i=2 g(x)=1+9 x ) 0.25 i 0 N 1 x i [0, 1], i=1,...,10 f 1 (x) x 1 f 1 (x) [0.28, 0.65] x 1 ([0.05, 0.125]) x 1 ([0.05, 0.125]) f 1 (x) [0.28, 0.65]
101 KP750 2 max f i (x)= 750 j=1 x j p (i,j) s.t. KP750 2 : g i (x)= 750 j=1 x j w (i,j) W i 1 i k, k =2 (6.4) p (i,j) w (i,j) i j W i i ( ) KP750 2 Zitzler 6, GA GA UNDX 41 1 Zitzler Deb 8, I MMA. I error. 2 I RNI I LI.
102 ( SPEA2 NSGA-II non-) F discon ZDT4, ZDT6 x 1 0 x 1 x 1 0 ZDT4 KUR F discon F discon 10 (N = 10) 100 (N = 100) 2 N =10 F discon (N = 10) 10 (N = 10) I cover I error 6.2 I MMA 6.3 I RNI I LI SPEA2 NSGA-II non- (a) Icover SPEA2 NSGA-II non- (b) Ierror 6.2 I cover and I error of F discon (N = 10)
103 ( ) non- SPEA2 NSGA-II F discon (N = 100) 100 (N = 100) I cover I error 6.6 I MMA 6.7 I RNI I LI N =10 f 1 (x) f 2 (x) -0.1 SPEA2 NSGA-II non- 6.3 I MMA of F discon (N = 10) SPEA2 NSGA-II non- SPEA2 50% 50% NSGA-II algorithm A a% b% algorithm B SPEA2 50% 50% NSGA-II 49% 51% 49% 51% non- 46% 54% 46% 54% non- 49% 51% 49% 51% 50% 50% 46% 54% 46% 54% 50% 50% (a) I RNI (b) I LI 6.4 I RNI and I LI of F discon (N = 10)
104 94 6 f 2(x) f 2(x) f1(x) SPEA f1(x) f 2(x) f 2(x) n f1(x) NSGA-II f1(x) Pareto optimum individuals(f discon (N = 10)) SPEA2 NSGA-II non- (a) Icover 0.0 SPEA2 NSGA-II non- (b) Ierror 6.6 I cover and I error of F discon (N = 100)
105 (b) non- SPEA2 NSGA 6.6(a) I cover 6.9 x 1 x 1 ZDT4 ZDT6 6.9 f 1 (x) SPEA2 NSGA-II non- SPEA2 NSGA-II non- f 2 (x) 6.7 I MMA of F discon (N = 100) SPEA2 SPEA2 40% 60% NSGA-II 39% 61% NSGA-II 14% 86% 22% 78% non- 13% 87% 22% 78% non- 8% 92% 14% 86% 38% 62% 7% 93% 14% 86% 37% 63% (a) I RNI (b) I LI 6.8 I RNI and I LI of F discon (N = 100)
106 96 6 f 2(x) f1(x) f 2(x) n f1(x) f 2(x) SPEA f1(x) f 2(x) NSGA-II f1(x) 6.9 Pareto optimum individuals(f discon (N = 100)) 250 ZDT4 I cover I error 6.10 I MMA 6.11 I RNI I LI 6.12?? ZDT4 6.10(b) 6.11 non- SPEA2 NSGA-II 6.12 non- SPEA2 NSGA-II non- SPEA2 NSGA-II
107 SPEA2 NSGA-II non- (a) Icover 0.0 SPEA2 NSGA-II non- (b) Ierror 6.10 I cover and I error of ZDT4 f 1 (x) SPEA2 NSGA-II non- f 2 (x) SPEA2 NSGA-II non I MMA of ZDT4 SPEA2 SPEA2 49% 51% NSGA-II 49% 51% NSGA-II 32% 68% 34% 66% non- 32% 68% 34% 66% non- 25% 75% 28% 72% 48% 52% 25% 75% 28% 72% 48% 52% (a) I RNI (b) I LI 6.12 I RNI and I LI of ZDT4
108 SPEA2 NSGA-II 6 6 f 2(x) 4 f 2(x) Pareto-optimal front Pareto-optimal front f1(x) f1(x) 6 6 non- f 2(x) 4 f 2(x) Pareto-optimal front Pareto-optimal front f1(x) f1(x) 6.13 Pareto optimum individuals(zdt4) non- SPEA2 NSGA-II SPEA2 NSGA-II ZDT4 non- 6.10(a) ZDT6
109 SPEA2 NSGA-II non- (a) I RNI 0.0 SPEA2 NSGA-II non- (b) I LI 6.14 I cover and I error of ZDT6 f 1 (x) x 1 f 1 (x) I cover I error 6.14 I MMA 6.15 I RNI I LI ZDT6 ZDT4 f 1 (x) x f 1 (x) 6.16 non- SPEA2 NSGA-II SPEA2 NSGA-II f 1 (x) SPEA2 NSGA-II non- f 2 (x) SPEA2 NSGA-II non I MMA of ZDT6
110 100 6 SPEA2 SPEA2 38% 61% NSGA-II 38% 61% NSGA-II 80% 20% 90% 10% non- 80% 20% 90% 10% non- 21% 79% 12% 88% 49% 51% 22% 78% 12% 88% 49% 51% (a) I RNI (b) I LI 6.16 I RNI and I LI of ZDT6 2 2 SPEA2 NSGA-II f 2(x) 1 f 2(x) Pareto-optimal front 0.5 Pareto-optimal front f 1 (x) f 1 (x) non- f 2(x) 1 f 2(x) Pareto-optimal front 0.5 Pareto-optimal front f 1 (x) f 1 (x) 6.17 Pareto optimum individuals(zdt6)
111 ZDT6 ZDT4 SPEA2 NSGA-II ZDT4 ZDT4 ZDT6 KUR I cover I error 6.18 I MMA 6.19 I RNI I LI f 1 (x) f 2 (x) Cover rate SPEA2 NSGA-II non I cover of KUR f 1 (x) SPEA2 NSGA-II non- f 2 (x) SPEA2 NSGA-II non I MMA of KUR
112 102 6 SPEA2 SPEA2 41% 59% NSGA-II 40% 60% NSGA-II 38% 62% 47% 53% non- 36% 64% 46% 54% non- 6% 94% 8% 92% 9% 91% 5% 95% 7% 93% 8% 92% (a) I RNI (b) I LI 6.20 I RNI and I LI of KUR SPEA2 100 NSGA-II f 2(x) f 2(x) f1(x) f1(x) n f 2(x) f 2(x) f1(x) f1(x) 6.21 Pareto optimum individuals(kur)
113 ZDT4 ZDT i) ii) non- SPEA2,NSGA-II SPEA2,NSGA-II KUR KUR f 1 (x) SPEA2,NSGA-II, non- non- KUR x 1 ZDT4, ZDT6 KP I cover 6.22 I MMA 6.23 I RNI I LI KP750 2
114 Cover rate SPEA2 NSGA-II non I cover of KP750 2 f 1 (x) SPEA2 NSGA-II non- f 2 (x) SPEA2 NSGA-II non I MMA of KP750 2 SPEA2 SPEA2 51% 49% NSGA-II 49% 51% NSGA-II 64% 36% 66% 34% non- 56% 44% 60% 40% non- 3% 3% 25% 75% 21% 79% 97% 21% 79% 19% 81% 97% (a) I RNI (b) I LI 6.24 I RNI and I LI of KP750 2
115 SPEA NSGA-II f 2(x) f 2(x) f1(x) f1(x) n f 2(x) f 2(x) f1(x) f1(x) 6.25 Pareto optimum individuals(kp-2) KUR non- SPEA2,NSGA-II 6.24 NSGA-II SPEA2 non- NSGA-II SPEA2 non- KP750 2 ( ) 4 5 NSGA-II, SPEA2, non- 4
116 ZDT4 KUR non- SPEA2 NSGA-II ZDT6 SPEA2,NSGA-II non- SPEA2 NSGA-II KUR KP750 2 non- I RNI I LI F discon ZDT4 ZDT6 x 1 non- KUR KP750 2 F discon (N=100) KUR non- SPEA2 NSGA-II x j t P (t j (mod P ))
117 p- (plan sorting) (object sorting) (a) discontinuous problem ( N=10 ) p- (plan sorting) (object sorting) (b) discontinuous problem ( N=100 ) p- (plan sorting) (c) ZDT4 (object sorting) 0.0 p- (plan sorting) (d) ZDT6 (object sorting) 0.00 p- (plan sorting) (e) KUR (object sorting) 6.26 I cover of p- ( ) ( p- ) I cover I error F discon (N = 10) F discon 10 (N = 10) I RNI I LI KUR I error 6.27
118 p- (plan sorting) (object sorting) (a) discontinuous problem ( N=10 ) p- (plan sorting) (object sorting) (b) discontinuous problem ( N=100 ) p- (plan sorting) (c) ZDT4 (object sorting) 0.0 p- (plan sorting) (d) ZDT6 (object sorting) 6.27 I error of p-
119 p p p SPEA SPEA2 p SPEA SPEA2 p NSGA-II NSGA-II p NSGA-II NSGA-II p non non- p non non- (a) I RNI (b) I LI 6.28 I RNI and I LI of F discon (N = 10) p f 2(x) f1(x) 6.29 Pareto optimum individuals(f discon (N = 10)) p I cover 6.27 I error F discon (N = 100) F discon 100 (N = 100) I RNI I LI N 100 p- I error p- p-
120 p F discon (N = 100) ZDT4 ZDT6 x 1 (= f 1 ) 0 f 2 0 p p p- 70 p SPEA SPEA2 p SPEA SPEA2 p NSGA-II NSGA-II p NSGA-II NSGA-II p non non- p non non-ncg (a) I RNI (b) I LI 6.30 I RNI and I LI of F discon (N = 100) f 2(x) p f1(x) f 2(x) f1(x) 6.31 Pareto optimum individuals(f discon (N = 100))
121 p- p- non- ZDT4 ZDT4 I RNI I LI ZDT4 p- I cover I error F discon (N = 100) ZDT4 p- p p p SPEA SPEA2 p SPEA SPEA2 p NSGA-II NSGA-II p NSGA-II NSGA-II p non non- p non non- (a) I RNI (b) I LI 6.32 I RNI and I LI of ZDT p- 6 f 2(x) 4 f 2(x) Pareto-optimal front Pareto-optimal front f1(x) f1(x) 6.33 Pareto optimum individuals(zdt4)
122 112 6 p p p SPEA SPEA2 p SPEA SPEA2 p NSGA-II NSGA-II p NSGA-II NSGA-II p non non- p non non- (a) I RNI (b) I LI 6.34 I RNI and I LI of ZDT p- 1.5 f 2(x) 1 f 2(x) Pareto-optimal front 0.5 Pareto-optimal front f 1 (x) f 1 (x) 6.35 Pareto optimum individuals(zdt6) p- non- F discon (N = 100) ZDT4 x 1 (= f 1 ) 0 ZDT6 ZDT6 I RNI I LI ZDT6 F discon (N = 100) ZDT4 p p- ZDT6 ZDT4 F discon (N = 100) F discon ZDT4 ZDT6 p- ZDT4
123 p- ZDT6 ZDT4 p- KUR KUR I RNI I LI p- N 100 F discon (N = 100) p- KUR 1 16 p p- 85 p SPEA SPEA2 p SPEA SPEA2 p NSGA-II 92 8 NSGA-II p NSGA-II 93 7 NSGA-II p non non- p non non- (a) I RNI (b) I LI 6.36 I RNI and I LI of KUR p- 100 f 2(x) f 2(x) f1(x) f1(x) 6.37 Pareto optimum individuals(kur)
124 114 6 p- non- (p-) () 2 F discon (N = 10) F discon (N = 100) ZDT4 ZDT6 KUR 5 i) p- F discon (N = 100) ii) KUR p- p- F discon ZDT4 ZDT6 p- non- x 1 0 p- 1 1
125 GA GA(Neighborhood Cultivation GA: ) Deb NSGA-II Zitzler SPEA2 ZDT6 SPEA2 NSGA-II non- GA 1 GA
126
127 GA(Neighborhood Cultivation GA: ) i) ii) 2 NOx
128 (NOx) ) 2) 3) NOx NOx 3 NOx 3 NOx NOx
129 PC CO 2 CO HC CO HC CO 2 NOx PM(Particulate Matter)
130 120 7 NOx 3 PM NOx PM NOx PM NOx PM NOx NO NO 2 N 2 O N 2 O 2 PM Particulate Matter Soot PM PM NOx PM ) (Thermodynamic Model) 2) (Phenomenological Model) : 3) (Detailed Multidimensional Model): 3 NOx
131 wrapper input file HIDECS output file optimizer analyzer 7.1 Optimization System 1 HIDECS 46 HIDECS KIVA code 47 1 HIDECS Injection Rate Atmosphere Condition Specific Fuel Consumption SFC NOx Soot HIDECS 7.1 HIDECS 4 18 /36 = 0.5
132 Specification of the target diesel engine Injected Fuel Volume 40mg/st Injection Timing ATDC 5.0 Injection Duration ATDC HIDECS Specific Fuel Consumption g/kwh Indicated Mean Effective Pressure kpa Indicated Power kw NOx g/kwh Soot g/kwh Specific Fuel Consumption SFC NOx Soot 36 HIDECS ( a l) 5 ATDC After Top Dead Center. ATDC 5.0 5
133 b a a-l : design variable d f h c e g j i l k Injection Rate(%) Time 7.2 Coding method bit gray GA parameters population size 100 crossover rate 1.0 crossover method 1X totale length of the chromosome bit mutation rate 1/120 total evaluation number PC FastEthernet Switching Hub
134 Cluster System CPU Memory OS Network Communication library Pentium III (1GHz)* Mb Linux2.4.4 FastEthernet TCP/IP LAM Soot (g/h Wh) SFC (g/w Wh) NOx (g/k Wh) 7.3 Derived non-dominated solutions
135 NOx (g/k Wh) Injection Rate (%) SFC (g/k Wh) Injection Timing Injection Timing Injection Timing Injection Timing 35 1 SFC:183.7, NOx:1.74, Soot: SFC:184.4, NOx:1.5, Soot: SFC:196.1, NOx:0.78, Soot: SFC299.6, NOx:0.43, Soot:0.15 Injection Rate (%) Injection Rate (%) Injection Rate (%) 7.4 Derived non-dominated solutions(sfc,nox) Soot (g/k Wh) Injection Rate (%) Injection Timing 1 SFC:183.7, NOx:1.74, Soot:0.26 Injection Rate (%) SFC (g/k Wh) Injection Rate (%) Injection Timing Injection Timing 2 SFC:202.0, NOx:1.6, Soot: SFC:229.9, NOx:1.2, Soot: Injection Rate (%) Injection Timing 4 SFC:267.8, NOx:1.05, Soot: Derived non-dominated solutions(sfc,soot)
136 SMOKE (g/k Wh) Injection Rate (%) Injection Rate (%) NOx (g/k Wh) Injection Timing Injection Timing Injection Timing 1 SFC299.6, NOx:0.43, Soot: SFC:291.9,NOx:0.64,Soot: SFC:278.1,NOx:0.87,Soot:0.10 Injection Rate (%) Injection Rate (%) Injection Timing 4 SFC:267.8, NOx:1.05, Soot: Derived non-dominated solutions(nox,soot) (Specific Fuel Consumption: SFC) NOx Soot 7.4 -NOx NOx 1 NOx Pilot 1 4 NOx NOx NOx ( )
137 Calculation time Total culculation time(sec) (About 3 hours) The average execution time of one trial of HIDECS(sec) The execution time of only master node running(sec) NOx NOx NOx HIDECS % 95% 3 HIDECS
138 128 7 GA HIDECS HIDECS NOx NOx NOx NOx GA HIDECS 7.3, ,, 2.,. 50 PPEX 51
139 ,, GA(Neighborhood Cultivation Genetic Algorithm :) 52, SPEA2, NSGA-II , (LSI), 48, ,,,.,.,.,.,,,., 49, 50, 53 LSI (Sequence-Pair) 50 BSG 53., BSG.,. 2 (Γ,Γ + ). (Γ,Γ + ) (a), 2 X,Y Γ,Γ + X,Y Γ,Γ + =(XY,XY) Γ Γ + X Y X Y Γ
140 130 7 Γ + Γ Γ + 7.7(b) (a)., Γ,Γ (c) (Γ, Γ + ) θ 7.8(c) (Γ,Γ + ) 7.8(b) 7.8(b) 7.7 X is the left of Y X is the right of Y X is the upper of Y X is the lower of Y (Γ_,Γ+ ) (XY,XY) (YX,YX) (YX, XY) (XY, YX) (a) relative position of each block Y Y X Y (Γ_,Γ + ) = (YX, YX) (Γ_,Γ + ) = (XY, YX) Y (Γ_,Γ + ) = (YX, XY) (Γ_,Γ + ) = (XY, XY) (b) concept figure 7.7 Concept of sequence-pair e a b c f d (a) floor plan e a 3 1 b 4 0 c 5 d f (b) relative position f c d e a b e c a b f d a b c d e f (c) coding of solution direction of block 7.8 Coding example of sequence-pair
141 end e a b e a b start c end c f d f d width 7.9 height Horizontal/Vertical Constrain graphs start (a), 2. min f 1 (x) =width (7.1) min f 2 (x) =height (7.2), width, height. 2,,., 7.9
142 ,., Zitler SPEA2, Deb NSGA-II, (non-). GA, ( Placement-based Partially Exchanging Crossover PPEX). 51,,. PPEX. Step 1: 1. Step 2: M c M nc M c Γ,Γ + Step 3: M nc, 6., PPEX 4., 1 a, b (M c )., (Γ,Γ + )=(da, da), 2 (Γ,Γ + )=(ad, ad). (M nc ) 1 1.,., 4 (33, 50, 100, 500 ), 6 (oblique grid), 7.8(b).
143 c 3 4 e 5 0 d 1 2 a c 3 4 e 5 0 a 1 2 d c 2 3 b f Parent f a e 3 d c 2 3 b f Child e a f 3 d 4 5 b b Parent 2 Child Placement-based Partially Exchanging Crossover(PPEX), Zitzler SPEA2 9 Deb NSGA- II 2 3 non-(non-) 4. GA GA GA Parameter number of blocks 33, 50, 100, 500 population size 200 crossover rate 1.0 mutation rate 1/bit length terminal generation 400 number of trial 30
144 134 7 SPEA2 46% 54% NSGA-II 54% 46% 56% 44% non- 36% 64% 39% 61% 31% 69% 7.11 Results of I LI (33 blocks), 33,50,100, VLSI CAD MCNC ami ,,SPEA2,NSGA-2,non non , I LI 7.11, I MMA 7.12., 7.13., ,,., 7.12 non-., non-., 7.11 I LI, non-
145 f1(x) SPEA2 NSGA-II non- f2(x) SPEA2 NSGA-II non I MMA of 33 blocks f 2(x) f 2(x) non f1(x) f1(x) SPEA f1(x) f 2(x) f 2(x) NSGA-II f1(x) 7.13 Derived non-dominated solutions(33 blocks)
146 136 7 SPEA2 50% 50% NSGA-II 56% 44% 57% 43% non- 37% 63% 37% 63% 28% 72% 7.14 Results of I LI (50 modules).,. 50, , I LI , I MMA 7.16, , , , 33.,,., I LI, I MMA., ,,., 33, 100.
147 SPEA2 42% 58% NSGA-II 59% 41% 61% 39% non- 37% 63% 42% 58% 25% 75% 7.15 Results of I LI (100 modules) f1(x) non- SPEA2 NSGA-II non- SPEA2 NSGA-II f 2(x) 7.16 I MMA of 50 modules f1(x) f 2(x) SPEA2 NSGA-II non- SPEA2 NSGA-II non I MMA of 100 modules
148 138 7 f 2(x) f 2(x) f1(x) SPEA f 2(x) f 2(x) f1(x) non f1(x) NSGA-II f1(x) 7.18 Derived non-dominated solutions(50 modules) , I LI 7.20,I MMA 7.21., I LI (, SPEA2) I LI (, NSGA-II) (50%, 50%). I LI,,.,., 7.22,. 1) SPEA2 NSGA-II. 2), SPEA2 NSGA-II. 500, ,
149 f 2(x) f 2(x) f1(x) SPEA f1(x) f 2(x) f 2(x) f1(x) non- NSGA-II f1(x) 7.19 Derived non-dominated solutions(100 modules) SPEA2 50% 50% NSGA-II 65% 35% 65% 35% non- 47% 53% 47% 53% 23% 77% 7.20 Results of I LI (500 blocks)
150 f1(x) SPEA2 NSGA-II non- SPEA2 NSGA-II non- f2(x) 7.21 I MMA of 500 blocks f 2(x) f1(x) f 2(x) SPEA2 f 2(x) f1(x) f1(x) f 2(x) non- NSGA-II f1(x) 7.22 Derived non-dominated solutions(500 blocks)
151 SPEA2 NSGA-II, non-,.,,, non- SPEA2,NSGA-II,.,.,. 33, , , A E ( ) A E A E,,. A E,., , 33.,., ,, 100! 100! (> ).
152 142 7 area A B C D aspect ratio (width/length) E A[0.21, ] B[0.39, ] C[0.57, ] D[0.767, ] E[0.991, ] 7.23 The placement of the modules(33 modules), 8000 (200, 400 ),.,,,.,,., 2,, , LSI, GA SPEA2, NSGA-II, non-.,.
I, II 1, A = A 4 : 6 = max{ A, } A A 10 10%
1 2006.4.17. A 3-312 tel: 092-726-4774, e-mail: hara@math.kyushu-u.ac.jp, http://www.math.kyushu-u.ac.jp/ hara/lectures/lectures-j.html Office hours: B A I ɛ-δ ɛ-δ 1. 2. A 1. 1. 2. 3. 4. 5. 2. ɛ-δ 1. ɛ-n
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最新耐震構造解析 ( 第 3 版 ) サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/052093 このサンプルページの内容は, 第 3 版 1 刷発行時のものです. i 3 10 3 2000 2007 26 8 2 SI SI 20 1996 2000 SI 15 3 ii 1 56 6
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Signal Detection Based on Belief Propagation in a Massive MIMO System Takeo Ohgane Hokkaido University, Japan 28 October 2013 Background (1) 2 Massive MIMO An order of 100 antenna elements channel capacity
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微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. このサンプルページの内容は, 初版 1 刷発行時のものです.
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