Γενετικοί Αλγόριθμοι καιΕφαρμογές

A E

T N E '

A EYPIN YKOANAHA T M H/Y

ATPA 2001

HPOOPIKH T N E ' A E YPIN YKOANAH A T M H/Y KPATH KATIKA T M A EPIO NIKHOPIH T I ANATAIO TPIANTH TEANO OYNTZH ... TYPORAMA / 2000 ISBN: 9605381753 K : H 31/3 Copyright 2000 & , 26222 : (0610) 314094, 314206 : (0610) 317244 . 2121/1993, .

............................................................................................................................................................. 9K 1

, , , E ................................................................................................................. 13 1.1 A , , E ................................................................................................. 16 1.1.1 .............................................................. 17 1.2 A , ................................................................... 20 1.2.1 A .................................... 25 1.2.2 ......................................................................................................... 27 ............................................................................................................................................................... 29 B................................................................................................................................................

31

K 2

A

, , , E ................................................................................................................. 32 2.1 2.2 2.3 A A .................................

35 37 48

.................................................

............................................................

............................................................................................................................................................... 55 B................................................................................................................................................

58

6

ENETIKOI AOPIMOI KAI EAPMOE

K 3

A A

, , , E ................................................................................................................. 59 3.1 3.2 A A , ................................................................... 61 , ................................................................... 79 ............................................................................................................................................................... 83 B................................................................................................................................................

85

K 4

A

, , , E ................................................................................................................. 87 4.1 4.2 A , ................................................................... 89 Y ....................................................................................... 102

............................................................................................................................................................ 106 B.............................................................................................................................................

108

K 5

A A

, , , E .............................................................................................................. 109 5.1 A , , E ............................................................................................... 111 5.1.1 K GRAY ................................................................... 111 5.2 , , E .............................................................................................. 115 5.2.1 ............................................................................................... 115

EPIEXOMENA

7

5.2.2 ........................................................... 116 5.2.3 ............................................................................................................... 117 5.2.4 ............................................................................................................ 119 5.2.5 ......................................................................................................... 122 5.2.6 ............................................................................................................ 122 ............................................................................................................................................................ 125 B.............................................................................................................................................

126

K 6

E A

, , , E .............................................................................................................. 127 6.1 6.2 T M , ................................................................ 129 A T N , ................................................................ 133 6.3 , ................................................................ 142 ............................................................................................................................................................ 152 B.............................................................................................................................................

154

A A A ...................................................................... 155 Y A .....................................................................

171 183

......................................................................................................................................................

, , , (). , , 1998. , . , , . , . . , , , , . , (Artificial Intelligence) (Machine Learning), (Computational Intelligence). , ( ). ( ) ( ) ( ) . : , . , , . .

10


. . , . , , , . . ( ), . , . . () (). , , , , . . A, . , , . , . A. , , , . -

POOO

11

, . , , . , . , , . ( , , ) , , .. , , . , , , . A . A. , . . , , . A . , . . : , . . . , . , -

12


. , , , , , , , .., . , , , . , () , . , , . , , . , , . . , , , , . , , . , . , , . .

, (A). . , . A, . , . , : A . . A. A. , A.

1

14

KEAAIO 1: EIAH TOY ENETIKOY AOPIMOY

, . , , , : (calculusbased methods): . : . . (hillclimbing). . . , . (enumerative) (random) : . ( ) , . , , . , . (iterated search): . hillclimbing ( ), . . ( ), -

E I A I K E A PAT H P H E I

15

, . (Simulated Annealing): hillclimbing. , , . (Dynamic Programming): . , . . (heuristic methods): , . , . , . , . , . , . , . , , , . , , .

16


1.1 A

, , , , , . , . , ( , , ) . , . , , ' . , . . , , , , . ,

1.1 A

17

, . , . 1950, . , , , 1970 John Holland Michigan [2].1.1.1

(Evolution of Species) , , . , , , . , . , , : ( , .. ). , . , , . , , , .

18


, , (chromosomes), . , (genes). (genotype). . , , (phenotype). (reproduction) (mutation). , (.. ), . , , , , . , . , , . , , , .. , , .. . , (.. ) , . (dominant) (recessive). ( ), (alleles). John Holland, , '70 [2]. Holland

1.1 A

19

, , . olland , . (A) . , , . . , . , . , , . , , , , . , , , . , , , . , , , . 1.1

;

;

1.2

20


1.3

;

1.1

, ; .

1.2 A

1.1.1 . , . . . A . , , A , A . , . . , , ( 46 ). A . . -

1.2 A

21

. (loci). ( ) , . , , ( ). ( ) . , . . , . A A . , , , , A . , . , A , . , A [3]. , , , . A . , -

22


, . A . , . A . . , , . (objective fitness function), . . : t, A :t t P (t ) = { x1 ,K, x n } .

xit . , ( t + 1) . (mutation) (crossover mating) . . , ( a1 , b1 , c1 , d1 , e1 ) ( a2 , b2 , c2 , d2 , e2 ) , ( crossover point = 2) ( a1 , b1 , c2 , d2 , e2 )

1.2 A

23

( a2 , b2 , c1 , d1 , e1 ) . . . , , . , , , . , , . . (mutation rate). , , . A : 1. . 2. . 3. , . 4. . 5. ( , , ..). . , [, ] , . , , .

24


1.4

, , , , .

1.5

A , : 1. E . 2. E . 3. , . 4. T 1 2. 5. T 1 3. 6. . .

1.6

() f(x) = x2 x [0, 31]. x 0 31;

1.2

. ( ), . . , , .. 0, 1, 16, 31 . , , k, . , .

1.2 A

25

1.2.1 A

A . , , : 1. . A . , , , A. , A . 2. . A , . , . , , . 3. . , A , , . , A, , . , . 4. . A , , , , A . A 5. . , ,

26


. , A , , , . 6. . , , , . A . 7. . A . . A , , . 8. . A , . O (n 3 ) , 10 1000. , , . 9. . . , . , (hillclimbing) , . . A , . 10. . A ,

1.2 A

27

, . , . 1.7

A : 1. . 2. , , . 3. . 4. , . 5. . .

1.2.2

A , . . , , ; , :1. .

, , . , ' . , A . A , , -

28


. , , . , , A , , . (.. : , , .). , A . , , . , A , .2. .

, , . , , . , , : , , ; . ' , , . , . , . , . , , , , . , ,

1.2 A

29

. , , . , , , . . , , , , . , , A . , , (. ) . , . . . , . . , . , , . , , . , . . , , -

30


. , . , A , . T, A, E . . , . A , . . , A. , A , . , . . A , . , , A. , . 1, . : A , . A. A. , A.

B I B I O PA I A

31

, () . ;

1.8

B

[1] Goldberg D.E.,GENETIC ALGORITHMS in Search, Optimization and Machine Learning, Addison Wesley Publishing Company, Inc., 1989. [2] Holland J.H., Adaptation in Natural and Artificial Systems, M.I.T. Press, 1975. [3] Michalewich Z., Genetic Algorithms + Data Structures = Evolution Programs, SpringerVerlag, 2nd ed., 1992. [4] Mitchel, Melanie, An Introduction to Genetic Algorithms, MIT Press, 1996. [5] Davis L., Handbook of Genetic Algorithms, Van Nostrand Reinhold, 1991.

A , , , . A . , , , , . , , . : (, , ) , , .

2

34

KEAAIO 2: A

, . , , , . A . A, . , . . . , . ( , , ), .

2 . 1 A

35

2.1 A

A . , . , , D. Goldberg [1], : 1. A : O A , , . , : (onoff). s f ( s ) . . , , . A, , . , . 1 on 0 off. , 11110 on off. . , , . , , A . 2. A : , , . , , , , . A ( ).

36

KEAAIO 2: A

(hillclimbing) , . , , , ( ). , A . , .. 01101, 11000, 01000 10011. , , . . 3. A : . A , . , . ' A (KnowledgeBased Genetic Algorithms). 4. A : H A, . , . , , . A , 1.2.

2.1

A ; .

2 . 1 A

37

. 1 2 16 . 3, 4 5 6, 7 8 . .

2.1

2.2 A

, A , . . , , . A . A

. , , . , () . , ( ). , . . (bits): (binary string) . , , . , -

38

KEAAIO 2: A

. A , , . . , , ( ) . : ; . , . 2.1 f(x) = x2, x [0, 31] x: . . A . , , 5, . [0, 31] 32 . .. 10010 , , 18 . , , .

A . ( ), . . . , . , .

2.2 A

39

2.1, f, . , , x, (fitness score), , f. , , A. . . , , . A, .

A. . A : 1. (Initialization) 2. (Decoding) 3. (Fitness calculation evaluation) 4. (Reproduction) I. (Selection)

II. (Crossover mating) III. (Mutation) 5. (2) A

40

KEAAIO 2: A

, A. , (, , ). , , . 2.1, 4. 32 . 20 , 4 5 . 01101, 11000, 01000 10011. , A . , . . , : . , , . . , , . . . , . , , , , . . . , , . , , , .. -

2.2 A

41

, . , A ( ): . , o.

A, . . . : . , . , , . , . , , . A . A. , , , , , , . , . , , , 1 [1]. , , 20 . , -

42

KEAAIO 2: A

( ), , 2.1: 2.1

1. 2. 3. 4.

01101 11000 01000 10011

169 576 64 361 1170

% 14.4 49.2 5.5 30.9 100.0

1170. . 2.1. A3 5%

A2 50%

A4 31% 2.1

.

A1 14%

, . , 1 169, 14.4% . 1 14,4% , 0,144. , () , . ,

2.2 A

43

, , , . [1] (mating pool).

, . , , . , , . . , . A. , . , . . A , A, , [3]. . . , . , , , , . , -

44

KEAAIO 2: A

, , (crossover probability) pc, A , , . , , . pc=1, , . , , . , p c , . X , , . , , , , . M , , .

, , , , . . . : . , , (mutation probability) pm, ( 0 1 ). ( ), A . , , . , , , . , , -

2.2 A

45

. , , P(t) t, 2.2. Procedure Genetic Algorithm begin t0 P(t) P(t) while ( not ) do begin t t+1 P(t) P(t1) P(t) P(t) end end

2.2

.

, , .

f(x) = x2 [1, 31], 2.2, ( = 1 = 0). , , . (. 2.2 ) . pm = 0.002, 20 (4 * 5) bits ;

2.2

46

KEAAIO 2: A

2.2

A

T x ( )

f ( x ) = x2

pselecti =

A/ fi f

f

fi

A

1 2 3 4

01101 11000 01000 10011

1 2 0 1

f f /4Maximum 2.2

2.2, , ;

2.3

string : k 1 l string [1, l1]. strings k+1 l . =112=3 =012 =1, strings; ) 1: 0 1 1 0 1 2: 1 1 0 0 0 ) 1: 1 1 0 0 0 2: 1 0 0 1 1

2.4

A 2.2, pc = 1 pm = 0,001. (2.3). 2.3

2.2 A

47

2.3

Z ( )

( )

N

T x ( )

f ( x ) = x2

1 2 3 4

01101 11000 11000 10011

2 1 4 3 2.3

f f /4Maximum

A . .

, , . , , .

2.4

: 5, 10, 15, 25, 50, 100. , , n = 6.

2.5

48

KEAAIO 2: A

2.6

, , ncounti : ncounti = f i / f , f i ( ) i f . , i string , . , ncounti 1,25, i 1,0 0,25. , 6 . , .

2.3

A , . 2.2 : f ( x ) = x sin(10px ) + 1.0 x [1, 2] f, x0 f ( x0 ) f ( x ) , x [1, 2]. H f . f f '( x ) = sin(10px ) + 10px cos(10px ) = 0 , tan(10px ) = -10px [3] ,

2.3

49

xi =

2i - 1 + ei , i = 1, 2, , 20

x0 = 0 , 2i + 1 - ei , i = 1, 2, , 20 ei i = 1, 2, i = 1, 2, . xi = f , xi i xi i . [1, 2], x19 = 37 / 20 + e19 = 1.85 + e19 , f(x19) p f (1.85 ) = 1.85 sin(18p + ) + 1.0 = 2.85 . 2 , A . , A 2.2, f. , .

x. , . x 3. [1, 2] 3 1000000 . , 22 , 2097152 = 221 < 3000000 222 = 4194304 x [1, 2] : :

50

KEAAIO 2: A

( < b21b20 ...b0 > )2 = (

21 i =0

bi 2i )10 = x '

x x = -1.0 + x ' 3 , 2 -122

1.0 3 . , (1000101110110101000111) 0.637107, x' = (1000101110110101000111)2 = 2288967 x = -1.0 + 2288967 , (0000000000000000000000) (1111111111111111111111) , 1 2 .

3 = 0.637197 4194303

. , 22 . 22 .

eval v f: eval ( v ) = f ( x ) , v x. , , . , v1 = (1000101110110101000111), v2 = (0000001110000000010000 ), v 3 = (1110000000111111000101),

2.3

51

x1 = 0.637197, x2 = 0.958973 x3 = 1.627888, . , : eval ( v1 ) = f ( x1 ) = 1.586345, eval ( v2 ) = f ( x2 ) = 0.078878, eval ( v 3 ) = f ( x 3 ) = 2.250650. , v3 , .

A . . , . v3 . 0 1 v3 v 3 ' = (1110100000111111000101). x 3 ' = 1.721638 f ( x 3 ' ) = -0.082257. v3. , v3 , v 3 '' = (1110000001111111000101). x 3 '' = 1.630818 f ( x 3 '' ) = 2.343555. v3, f ( x 3 ) = 2.250650. v2 v3. , , . T

52

KEAAIO 2: A

v2 = (00000 | 01110000000010000 ), v 3 = (11100 | 00000111111000101) : v2 ' = (00000 | 00000111111000101), v 3 ' = (11100 | 01110000000010000 ). f ( v2 ' ) = f ( -0.998113 ) = 0.940865, f ( v 3 ' ) = f (1.666028 ) = 2.459245. .

2.2, A: pop _ size = 50 pc = 0.25 pm = 0.01

2.4, A . 2.4

1 6 8 9 10 12

1.441942 2.250003 2.250283 2.250284 2.250363 2.328077

2.3

53

39 40 51 99 137 145

2.344251 2.345087 2.738930 2.849246 2.850217 2.850227

A, , . 150 : v max = (1111001101000100000101), x max = 1.850773. x max = 1.85 + e19 f ( x max ) 2.85. 2.7

2.4 pc 1, 2.2 p c = 0.25. ( ) ; .

1. , . , . , ; 1: 1 1 0 1 | 1 0 0 1 0 1 | 1 0 1 1 2: 0 0 0 1 | 0 1 1 0 1 1 | 1 1 0 0 2. , : . , , .

2.5

54

KEAAIO 2: A

, , , . . ; 1: 1 0 0 1 0 1 1 2: 0 1 0 1 1 0 1 : 1 1 0 1 0 0 1

2.6

, 100 ( 100), (1). ;

2.8

, . X , , , 1000 0 1. : 0.00 0.25, 0.25 0.50, 0.50 0.75, 0.75 1.00 . ; ;

2.9

10 : 0.1, 0.2, 0.05, 0.15, 0.11, 0.07, 0.04, 0.12, 0.16. , . , 10 . 1000 , .

2.3

55

, . strings, 10: 1011101011, 0000110100. : 3, 1, 6 20.

2.10

, pm. 1000 , p m = 0.001, 0.01, 0.1. .

2.11

. : 1. . 2. . 3. , . 4. , . , , . . , . , . , , .

56

KEAAIO 2: A

, . . , A . A , . . , . . , A : 1. (Initialization) 2. (Decoding) 3. (Fitness calculation evaluation) 4. (Reproduction) I. (Selection) II. (Crossover mating) III. (Mutation) 5. (2) A , , / , . , . , . , A ,

YNOH

57

. , A. . , . . , , A. . A, . , . A , . , , A. , . 2, , . , . : , , , ( , , ). , , .

58

KEAAIO 2: A

2.7

() .

B

[1] Goldberg D.E.,GENETIC ALGORITHMS in Search, Optimization and Machine Learning, Addison Wesley Publishing Company, Inc., 1989.

[2] Holland J.H., Adaptation in Natural and Artificial Systems, M.I.T. Press, 1975. [3] Michalewich Z., Genetic Algorithms + Data Structures = Evolution Programs, SpringerVerlag, 2nd ed., 1992. [4] Mitchel, Melanie, An Introduction to Genetic Algorithms, MIT Press, 1996. [5] Davis L., Handbook of Genetic Algorithms, Van Nostrand Reinhold, 1991.

A A . , A, . A. , , A. , . , . . , . , . , : , A, , /, A, , , A, , .

3

60

KEAAIO 3: A A

[1]: , . , . , A , , , , . , . , , . , . , , . , , A, . , -

3.1 A A

61

, , , (), A . , , A. . . , , . , 2. , , .

3.1 A A

. , , A. , . , . , , , . , , . , :

62

KEAAIO 3: A A

, A, , A, /, , , , A, .

, A . f(x), x = xmax, f (xmax) = max. f. f, g, g = f. , f , C, max g ( x ) = max{ f ( x ) + C } . , , k , f ( x1 ,K, x k ): R k R . xi Di = [ ai , bi ] R f ( x1 ,K, x k ) > 0, "xi Di , i = 1,K, k . f , .. q . , :

3.1 A A

63

1. . 2. . 3. . 4. . 5. . 6. . 7. () .

, . .

q , Di = [ ai , bi ] ( bi - ai ) 10 q . mi ( bi - ai ) 10 q 2mi - 1 . T, mi q i . , bin _ str , : xi = ai + decimal ( bin _ str ) bi - ai 2mi - 1 ,

(3.1)

decimal ( bin _ str ) bin _ str . 1.2. ' , m=

k i =1

mi ,

(3.2)

64

KEAAIO 3: A A

m1 x1, [ ai , bi ] , m2 x2 [ a2 , b2 ] , ... .

. , . . A . , ( ), ( ) ( ), . 3.1 2.2 2.3 2. . . , (). 3.1

/ 1 2 3

1 01111 01001 00111 x1 15 9 7 2 00111 00010 01001 x2 7 2 9 ... F(x1,x2,,xl) 225 101 123

... ... ... . . . ...

... ... ... . . . ...

. . .n

. . .00111

. . .7

. . .00101

. . .5

. . .81

3.1 A A

65

, , . , Pascal, [1]: , , pop_size m. . (type) (type declaration). , population array individual ( 1 pop_size). individual record chrom, chromosome, fitness x. , x, . , chromosome array allele ( 1 m), boolean ( bit true false). . . , , . , pop_size m . : 1. , ( f ). 2. , . 3. .

66

KEAAIO 3: A A

4. , , 3.1 . 5. , , A . ( ). T 1 2 3

T+1 1 2 3

A M

N1 N 3.1

N1 N

A.

, (slotted roulette wheel). :

1. vi , i = 1,K, pop _ size . 2. F= eval ( vi ) .

eval ( vi )

pop _ size i =1

3. pi vi , i = 1,K, pop _ size : pi = eval ( vi ) / F . 4. , (cumulative) qi vi , i = 1,K, pop _ size : qi =

i j =1

pj .

3.1 A A

67

pop_size . , : 1. r 0 1. 2. r < q1, v1, vi ( 2 i pop _ size ), qi -1 < r qi . , . , . A pc. : 1. r 0 1. 2. r < pc, . ( pc pop_size), pos [1, m1], m . pos . , : ( b1b2 K bpos bpos +1 K bm ) ( c1c2 K c pos c pos +1 K cm ) ( b1b2 K bpos c pos +1 K cm ) ( c1c2 K c pos bpos +1 K bm ) . , , . . A, pm. pm m pop_size. :

68

KEAAIO 3: A A

: 1. r 0 1. 2. r < pm, . , . , . . 3.1

pos . ;

3.2

50 33, , 0.001, 0.01 0.1;

[3], . 3.1 A . pop_size = 20, pc = 0.25 pm = 0.01. : f ( x1 , x2 ) = 21.5 + x1 sin( 4px1 ) + x2 sin( 20px2 ) , 3.0 x1 12.1 4.1 x2 5.8. . x1 15.1, [3.0, 12.1] 15.1 10000 = 151 000 -

3.1 A A

69

. 18 x1 ( ), : 217 < 151000 218 . x2 15 . , , (3.2), m = 18 + 15 = 33 . : (010001001011010000111110010100010 ) . 18 ( 010001001011010000 ) ( 3.1) x1 = -3.0 + 70352 15.1 = -3.0 + 4.05242 = 1.05242 , 262143

15 ( 11110010100010 ) x2 = 5.75533 . , (x1, x2) = (1.05242, 5.75533). f (1.05242, 5.75533) = 20.25264. : v1 = (100110100000001111111010011011111) v2 = (111000100100110111001010100011010 ) v 3 = (000010000011001000001010111011101) v 4 = (100011000101101001111000001110010 ) v5 = (000111011001010011010111111000101) v6 = (000101000010010101001010111111011) v7 = (001000100000110101111011011111011) v8 = (100001100001110100010110101100111) v9 = (010000000101100010110000001111100 ) v10 = (000001111000110000011010000111011)

70

KEAAIO 3: A A

v11 = (011001111110110101100001101111000 ) v12 = (110100010111101101000101010000000 ) v13 = (111011111010001000110000001000110 ) v14 = (010010011000001010100111100101001) v15 = (111011101101110000100011111011110 ) v16 = (110011110000011111100001101001011) v17 = (011010111111001111010001101111101) v18 = (011101000000001110100111110101101) v19 = (000101010011111111110000110001100 ) v20 = (101110010110011110011000101111110 ) . : eval(v1) = f(6.084492, 5.652242) = 26.019600 eval(v2) = f(10.348434, 4.380264) = 7.580015 eval(v3) = f(2.516603, 4.390381) = 19.526329 eval(v4) = f(5.278638, 5.593460) = 17.406725 eval(v5) = f(1.255173, 4.734458) = 25.341160 eval(v6) = f(1.811725, 4.391937) = 18.100417 eval(v7) = f(0.991471, 5.680258) = 16.020812 eval(v8) = f(4.910618, 4.703018) = 17.959701 eval(v9) = f(0.795406, 5.381472) = 16.127799 eval(v10) = f(2.554851, 4.793707) = 21.278435 eval(v11) = f(3.130078, 4.996097) = 23.410669 eval(v12) = f(9.356179, 4.239457) = 15.011619 eval(v13) = f(11.134646, 5.378671) = 27.316702 eval(v14) = f(1.335944, 5.151378) = 19.876294 eval(v15) = f(11.089025, 5.054515) = 30.060205 eval(v16) = f(9.211598, 4.993762) = 23.867227 eval(v17) = f(3.367514, 4.571343) = 13.696165

3.1 A A

71

eval(v18) = f(3.843020, 5.158226) = 15.414128 eval(v19) = f(1.746635, 5.395584) = 20.095903 eval(v20) = f(7.935998, 4.757338) = 13.666916 v15 v2 . (roulette wheel). (fitness) : F=

eval ( v ) = 387.776822i i =1

20

pi vi , i=1,,20, : p1 = eval(v1)/F = 0.067099 p2 = eval(v2)/F = 0.019547 p3 = eval(v3)/F = 0.050355 p4 = eval(v4)/F = 0.044889 p5 = eval(v5)/F = 0.065350 p6 = eval(v6)/F = 0.046677 p7 = eval(v7)/F = 0.041315 p8 = eval(v8)/F = 0.046315 p9 = eval(v9)/F = 0.041590 p11 = eval(v11)/F = 0.060372 p12 = eval(v12)/F = 0.038712 p13 = eval(v13)/F = 0.070444 p14 = eval(v14)/F = 0.051257 p15 = eval(v15)/F = 0.077519 p16 = eval(v16)/F = 0.061549 p17 = eval(v17)/F = 0.035320 p18 = eval(v18)/F = 0.039750 p19 = eval(v19)/F = 0.051823

p10 = eval(v10)/F = 0.054873 p20 = eval(v20)/F = 0.035244 (cumulative probabilities) qi vi ,i=1,,20 : q1 = 0.067099 q2 = 0.086647 q3 = 0.137001 q4 = 0.181890 q5 = 0.247240 q6 = 0.293917 q7 = 0.335232 q8 = 0.381546 q9 = 0.423137 q10 = 0.478009 q11 = 0.538381 q12 = 0.577093 q13 = 0.647537 q14 = 0.698794 q15 = 0.776314 q16 = 0.837863 q17 = 0.873182 q18 = 0.912932 q19 = 0.964756 q20 = 1.000000

20 ; . 20 [0, 1]:

72

KEAAIO 3: A A

0.513870 0.171736 0.703899 0.005398

0.175741 0.702231 0.389647 0.765682

0.308652 0.226431 0.277226 0.646473

0.534534 0.494773 0.368071 0.767139

0.947628 0.424720 0.983437 0.780237

r = 0.513870 q10 q11, v11 . r = 0.175741 q3 q4, v4 . : v1* = (011001111110110101100001101111000) (v11) v2* = (100011000101101001111000001110010) (v4) v3* = (001000100000110101111011011111011) (v7) v4* = (011001111110110101100001101111000) (v11) v5* = (000101010011111111110000110001100) (v19) v6* = (100011000101101001111000001110010) (v4) v7* = (111011101101110000100011111011110) (v15) v8* = (000111011001010011010111111000101) (v5) v9* = (011001111110110101100001101111000) (v11) v10* = (000010000011001000001010111011101) (v3) v11* = (111011101101110000100011111011110) (v15) v12* = (010000000101100010110000001111100) (v9) v13* = (000101000010010101001010111111011) (v6) v14* = (100001100001110100010110101100111) (v8) v15* = (101110010110011110011000101111110) (v20) v16* = (100110100000001111111010011011111) (v1) v17* = (000001111000110000011010000111011) (v10) v18* = (111011111010001000110000001000110) (v13) v19* = (111011101101110000100011111011110) (v15) v20* = (110011110000011111100001101001011) (v16) , , () . ,

3.1 A A

73

, . , . r 0 1. pc = 0.25, r < 0.25, o . : 0.822951 0.911720 0.031523 0.581893 0.151932 0.519760 0.869921 0.389248 0.625477 0.401154 0.166525 0.200232 0.314685 0.606758 0.674520 0.355635 0.346901 0.785402 0.758400 0.826927

v2, v11, v13 v18 ( , , ). , . (v2 v11) (v13 v18). , , pos = 9: v2 = (100011000101101001111000001110010 ) v11 = (111011101101110000100011111011110 ) . v2 = (100011000101110000100011111011110 ) v11 = (111011101101101001111000001110010 ) . , , pos = 20: v13 = (000101000010010101001010111111011) v18 = (111011111010001000110000001000110 ) . v13 = (00010100001001010100 0000001000110 ) v18 = (111011111010001000111010111111011) .

74

KEAAIO 3: A A

: v1 = (100110100000001111111010011011111) v2 = (100011000101110000100011111011110 ) v 3 = (000010000011001000001010111011101) v 4 = (100011000101101001111000001110010 ) v5 = (000111011001010011010111111000101) v6 = (000101000010010101001010111111011) v7 = (001000100000110101111011011111011) v8 = (100001100001110100010110101100111) v9 = (010000000101100010110000001111100 ) v10 = (000001111000110000011010000111011) v11 = (111011101101101001111000001110010 ) v12 = (110100010111101101000101010000000 ) v13 = (00010100001001010100 0000001000110 ) v14 = (010010011000001010100111100101001) v15 = (111011101101110000100011111011110 ) v16 = (110011110000011111100001101001011) v17 = (011010111111001111010001101111101) v18 = (111011111010001000111010111111011) v19 = (000101010011111111110000110001100 ) v20 = (101110010110011110011000101111110 ) , , pm = 0.01 , 1% . m pop_size = 33 20 = 660 , 6.6 . r [0, 1]. r < 0.01, . , 660 . , , ,

3.1 A A

75

, , : 112 349 418 429 602 4 11 13 13 19 13 19 22 33 8

, . , : v1 = (100110100000001111111010011011111) v2 = (100011000101110000100011111011110 ) v 3 = (000010000011001000001010111011101) v 4 = (011001111110010101100001101111000 ) v5 = (000111011001010011010111111000101) v6 = (000101000010010101001010111111011) v7 = (001000100000110101111011011111011) v8 = (100001100001110100010110101100111) v9 = (010000000101100010110000001111100 ) v10 = (000001111000110000011010000111011) v11 = (111011101101101001011000001110010 ) v12 = (110100010111101101000101010000000 ) v13 = (000101000010010101000100001000111) v14 = (010010011000001010100111100101001) v15 = (111011101101110000100011111011110 ) v16 = (110011110000011111100001101001011) v17 = (011010111111001111010001101111101) v18 = (111011111010001000111010111111011)

76

KEAAIO 3: A A

v19 = (111011100101110000100011111011110 ) v20 = (101110010110011110011000101111110 ) . , : eval ( v1 ) = f (3.130078, 4.996097 ) = 23.410669 eval ( v2 ) = f (5.279042, 5.054515 ) = 18.201083 eval ( v 3 ) = f ( -0.991471, 5.680258 ) = 16.020812 eval ( v 4 ) = f (3.128235, 4.996097 ) = 23.412613 eval ( v5 ) = f ( -1.746635, 5.395584 ) = 20.095903 eval ( v6 ) = f (5.278638, 5.593460 ) = 17.406725 eval ( v7 ) = f (11.089025, 5.054515 ) = 30.060205 eval ( v8 ) = f ( -1.255173, 4.734458 ) = 25.341160 eval ( v9 ) = f (3.130078, 4.996097 ) = 23.410669 eval ( v10 ) = f ( -2.516603, 4.390381) = 19.526329 eval ( v11 ) = f (11.088621, 4.743434 ) = 33.351874 eval ( v12 ) = f (0.795406, 5.381472 ) = 16.127799 eval ( v13 ) = f ( -1.811725, 4.209937 ) = 22.692462 eval ( v14 ) = f ( 4.910618, 4.703018 ) = 17.959701 eval ( v15 ) = f (7.935998, 4.757338 ) = 13.666916 eval ( v16 ) = f (6.084492, 5.652242 ) = 26.019600 eval ( v17 ) = f ( -2.554851, 4.793707 ) = 21.278435 eval ( v18 ) = f (11.134646, 5.666976 ) = 27.591064 eval ( v19 ) = f (11.059532, 5.054515 ) = 27.608441 eval ( v20 ) = f (9.211598, 4.993762 ) = 23.867227

3.1 A A

77

. F 447.049688, 387.776822. , , v11, , v15, 33.351874 30.060205. , : ; : , . , A , . , 100 1000 0.001%. : 1. , . . . 2. . ( ) . , ( ).

78

KEAAIO 3: A A

3.3

3.1 2 , , .

3.4

3.1, ;

3.5

3.1 ; 0.25 20, ; , ; .

3.6

, , ;

3.7

( ), ;

3.1

3.1, . .

3.2

79

, 1. 100 , , 1000 . 1. . 2. 50 1. 3. 1 2.

3.2

3.1 100 . 0.1 1.0, 0.1. , . .

3.3

( 2.5) 3.1. , 3.2, 100 .

3.4

3.2

A, , , . A, . ,

80

KEAAIO 3: A A

(redundant values). , 2. , . : . . , 3.1, A.

A. , A . , , . , , . . A, . A . , , , . A, . , 2. .

3.2

81

: ( ). , , (.. g(x) = 0 g(x) 0 g(x) 0. , . , , . , , ( ' ) . , , . . , , A. . A (penalty method). : ( ). , : E g(x) hi(x) 0, i = 1, 2, , n, x m. : E g ( x ) + r

F[h ( x )] , i i =1

n

r . r A

82

KEAAIO 3: A A

(redundant values). , 2. , A. .., [0 9] , 4 0000 1111. , , 1010 1111; ; , A, . , , ( ) , [3]: 1. . 2. , . 3. . , , , . : 1. (fixed remapping), , .. 10 - 15 0 - 5 . , . 2. (random remapping), . , A -

YNOH

83

.

g(x) h(x) 0 y(x) 0. ;

3.8

;

3.9

f(x,y,z) . x [20.0, 125.0], y [0, 1.2 106] z [0.1, 1.0]. ( ) 0.5, 104 0.001 . . ; , , : (20, 0, 1), 125.0, 1.26, 1.0) (50, 100000, 0.597). ;

3.5

A , , , , . , , , . , ,

84

KEAAIO 3: A A

. , , . , , A, . , , , , (), A . , . , , , . , . A, , , . A, . , (redundant values). , 2. 3.2, . , A. 3, A , Pascal. , 3.1. , , A . 3,

YNOH

85

, . , , . , , ., . , .B

[1] Goldberg D.E.,GENETIC ALGORITHMS in Search, Optimization and Machine Learning, Addison Wesley Publishing Company, Inc., 1989. [2] Michalewich Z., Genetic Algorithms + Data Structures = Evolution Programs, SpringerVerlag, 2nd ed., 1992. [3] Davis L., Handbook of Genetic Algorithms, Van Nostrand Reinhold, 1991.

A

. , , A. . , A ( ) . (building blocks) ( ), A. , . , : , , , , A.

4

88

KEAAIO 4: A

A , . A, [1, 2, 4]. . A , . [4]: A; , ( ) A; (, ) A; A ; A ; A , ; A;

4.1 A

89

( A ) A ; , . . , [6, 7, 8].4.1 A

, . , . ; . . , ( ), A. , : , , .

90

KEAAIO 4: A

, . . A, Holland [2], , , A , () . , A , . () . Holland , . ()

A , (schema) (template) . (don't care symbol) * ( = {0, 1}). ( ), *. , 10. (*111100100) : {(0111100100 ), (1111100100 )} (*1*1100100) : {(0101100100 ), (0111100100 ), (1101100100 ), (1111100100 )} . (1001110001) , (1001110001), (**********) 10. 2r , r * . , m 2m

4.1 A

91

. D. Goldberg [1, .19] . m 3m , *. , (cardinality) c (c + 1)m. , (1001110001). 210 : (1001110001) (*001110001) (1 * 01110001) (10 *1110001) M (100111000*) (** 01110001) (*0 *1110001) M (10011100 **) (***1110001) M (**********) . . * . : (order) (defining length). (Schema Theorem) . (SCHEMA THEOREM)

S ( (S)) 0 1, (fixed positions), *. , *.

92

KEAAIO 4: A

(speciality) , . , , 10, S1 = (***001 *110 ) , S2 = (**** 00 **0*) , S 3 = (11101 **001) , : o( S1 ) = 6 , o( S2 ) = 3 o( S 3 ) = 8 , S3 , , , , S1 S2 16 128 . . S ( (S)) . (compactness) . ,

d ( S1 ) = 10 - 4 = 6 , d ( S2 ) = 9 - 5 = 4 d ( S1 ) = 10 - 4 = 6 ., . . A ( ). .

. pop_size = 20 ( ) m = 33

4.1 A

93

( 3.1). , ( ) t : v1 = (100110100000001111111010011011111) v2 = (111000100100110111001010100011010 ) v 3 = (000010000011001000001010111011101) v 4 = (100011000101101001111000001110010 ) v5 = (000111011001010011010111111000101) v6 = (000101000010010101001010111111011) v7 = (001000100000110101111011011111011) v8 = (100001100001110100010110101100111) v9 = (010000000101100010110000001111100 ) v10 = (000001111000110000011010000111011) v11 = (011001111110110101100001101111000 ) v12 = (110100010111101101000101010000000 ) v13 = (111011111010001000110000001000110 ) v14 = (010010011000001010100111100101001) v15 = (111011101101110000100011111011110 ) v16 = (110011110000011111100001101001011) v17 = (011010111111001111010001101111101) v18 = (011101000000001110100111110101101) v19 = (000101010011111111110000110001100 ) v20 = (101110010110011110011000101111110 ) (S, t) t S. , S0 = (****111 **************************) , (S0, t) = 3, ( v13, v15 v16), S0. S0 o(S0) = 3 (S0) = 7 5 = 2.

94

KEAAIO 4: A

t eval (S, t). t S. p { vi1 ,K, vi p } t S. , eval ( S , t ) = (

eval ( vj =1

p

ij

)) / p .

(4.1)

, . , , . , , vi p1 = eval (vi)/F(t), F(t) . , (S, t + 1) S. 1. S, eval (S, t)/F(t), 2. S (S, t) 3. pop_size,

x ( S , t + 1) = x ( S , t ) pop _ size eval ( S , t ) / F (t )

(4.2)

F (t ) = F (t ) / pop _ size , :

x ( S , t + 1) = x ( S , t ) eval ( S , t ) / F (t ) .

(4.3)

, , .

4.1 A

95

. , . : S % ( eval ( S , t ) = F (t ) + e F (t ) ),

x ( S , t ) = x ( S , 0 ) (1 + e )t , e = (eval ( S , t ) - F (t )) / F (t )(4.4)

> 0 < 0 . . , , . S0. t S0, eval ( S0 , t ) = ( 27.316702 + 30.060205 + 23.867227 ) / 3 = 27.081378 . , F (t ) = (

eval ( v )) / pop _ size = 387.776822 / 20 = 19.388841i i =1

20

S0 eval ( S0 , t ) / F (t ) = 1.396751 . S0 . , t S0 1.396751 , t + 1 3 1.396751 = 4.19 ( 4 5), t + 2 3 1.3967512 = 5.85 ( 5 6), ... , S0 -

96

KEAAIO 4: A

, , . . t S0 . , : v1 = (011001111110110101100001101111000 ) v2 = (100011000101101001111000001110010 ) v 3 = (001000100000110101111011011111011) v 4 = (011001111110110101100001101111000 ) v5 = (000101010011111111110000110001100 ) v6 = (100011000101101001111000001110010 ) v7 = (111011101101110000100011111011110 ) v8 = (000111011001010011010111111000101) v9 = (011001111110110101100001101111000 ) v10 = (000010000011001000001010111011101) v11 = (111011101101110000100011111011110 ) v12 = (010000000101100010110000001111100 ) v13 = (000101000010010101001010111111011) v14 = (100001100001110100010110101100111) v15 = (111001100110000101000100010100001) v16 = (111001100110000101000100010100001) v17 = (111001100110000100000101010111011) v18 = (111011111010001000110000001000110 ) v19 = (111011101101110000100011111011110 ) v20 = (110011110000011111100001101001011) ( v11 ) ( v4 ) ( v7 ) ( v11 ) ( v19 ) ( v4 ) ( v15 ) ( v5 ) ( v11 ) ( v3 ) ( v15 ) ( v9 ) ( v6 ) ( v8 ) ( v20 ) ( v1 ) ( v10 ) ( v13 ) ( v15 ) ( v16 )

, S0 : v7 , v11 , v18 , v19 v20 . , ,

4.1 A

97

( ) . , .

, , . , . , . , .. v18 = (111011111010001000110000001000110 ) , 233 . , : S0 = (****111 **************************) S1 = (111 ****************************10 ) . ( 3). , , pos = 20. S0 , S0. 111 , , .. v18 = (111011111010001000110000001000110 ) v13 = (00010100001001010100 0000001000110 ) v18 = (111011111010001000111010111111011) v13 = (00010100001001010100 0000001000110 ) . , S1 , . 111 10 . , -

98

KEAAIO 4: A

. , S0 (S0) = 2, S1 (S1) = 32. , (uniformly) m 1 . S pd ( S ) =

d (S ) m -1

(4.5)

: ps ( S ) = 1 -

d (S ) . m -1

(4.6)

, S0 S1 pd ( S0 ) = 2 / 32 , p s ( S0 ) = 30 / 32 , pd ( S1 ) = 32 / 32 , p s ( S1 ) = 0 , . , pc . , : p s ( S ) = 1 - pc

d (S ) . m -1

(4.7)

, (pc = 0.25) p s ( S0 ) = 1 - 0.25 2 = 63 / 64 = 0.984375 . 32

, , . , v18 v13 111 10, S1 . , p s ( S ) 1 - pc

d (S ) . m -1

(4.8)

,

4.1 A

99

d (S ) x ( S , t + 1) x ( S , t )eval ( S , t ) / F (t )1 - pc m -1 (4.9)

, . , . S0: d ( S0 ) eval ( S0 , t ) / F (t )1 - pc = 1.396751 0.984375 = 1.374927 . m -1 , S0 . t + 1, 3 1.374927 = 4.12 t + 2, 3 1.3749272 = 5.67 . pm. . , , v19 = (111011101101110000100011111011110 ) S0 = (****111 * ** ********* **************) . , , v19 . 3.1 3, v19 : v19 = (111011100101110000100011111011110 ) , S0. 1 4 8 33, S0. 3 ( , ) : -

100

KEAAIO 4: A

. , , . pm, 1 pm. , ( ) p s ( S ) = (1 - pm ) o( S ) , , Pm 0, . < 0.

4.2

f x, l = 4, x (.. f 0011 = 3, f 1111 = 15 ). 1***, f; 0*** f;

(

)

(

)

4.3

x, x. S, k 1, l k.

4.4

102

KEAAIO 4: A

4.2

A , , . (): : , , , . , : S0 = (****111 * ** ********* **************) . , , ( ) . . , . , A, . , A . pop_size m, 2m 2pop_size . : () ( ). A. , , , . , , 11

4.2

103

S1 = ( 1 1 1 * * * * * * * * ) S2 = ( * * * * * * * * * 1 1 ) ( ), S3 = ( 1 1 1 * * * * * * 1 1) , , : S4 = ( 0 0 0 * * * * * * 0 0) S0 = ( 1 1 1 1 1 1 1 1 1 1 1) ( S3 ). A S0 ( 0 0 0 1 1 1 1 1 1 0 0 ). : ( , ) A . [3]. ( ). , , , , , . , , . , ( ) : , . : , . , s = ((1, 0), (2, 0), (3, 0) | (4, 1), (5, 1), (6, 0), (7, 1) | (8, 0), (9, 0), (10, 0), (11, 1)) , s = ((1, 0), (2, 0), (3, 0) | (7, 1), (6, 0), (5, 1), (4, 1) | (8, 0), (9, 0), (10, 0), (11, 1)) , , -

104

KEAAIO 4: A

. S3 = ( 1 1 1 * * * * * * 1 1) S3 = ((1, 1), (2, 1), (3, 1), (4, *), (5, *), (6, *), (7, *), (8, *), (9, *), (10, 1), (11, 1)), S3 = ((1, 1), (2, 1), (3, 1), (11, 1), (10, 1), (9, *), (8, *), (7, *), (6, *), (5, *), (4, *)), . . , [3]: (messy) . , [4]. . , , , 5 [3]. 4.1

A1 = 11101111, A2 = 00010100 A3 = 01000011 H1 = 1*******, H2 = 0*******, H3 = ******11, H4 = ***0*00*, H5 = 1*****1* H6 = 1110**1*. ) ; ) . ) , pm = 0.001. ) , pc = 0.85.

4.2

105

0 : A/A 1 2 3 4 10001 11100 00011 01110 20 10 5 15

4.2

pm= 0.01 pc = 1.0. ) S1=(1****) 1. ) S2=(0**1*) 1.

106

KEAAIO 4: A

, . , , , . , . () , . , . , , , . A . ( , ) , , . A. , A . , , , . , A, . , A, . . A, .

YNOH

107

, , . : , , , , A.

108

KEAAIO 4: A

B

[1] Goldberg D.E.,GENETIC ALGORITHMS in Search, Optimization and Machine Learning, Addison Wesley Publishing Company, Inc., 1989. [2] Holland J.H., Adaptation in Natural and Artificial Systems, M.I.T. Press, 1975. [3] Michalewich Z., Genetic Algorithms + Data Structures = Evolution Programs, SpringerVerlag, 2nd ed., 1992. [4] Mitchel, Melanie, An Introduction to Genetic Algorithms, MIT Press, 1996. [5] Whitley, L.D., Foundations of Genetic Algorithms 2, Ed., Morgan Kaufman, 1993. [6] Whitley, L.D. and Vose M.D., Foundations of Genetic Algorithms 3, Ed., Morgan Kaufman, 1995.

A A , . , , . () . , . , , . , , . , : , , , , .

5

110

KEAAIO 5: A A

A, . , . A , () . A , , . , 100 [500, 500] , 3000. , , 101000. A . . A. . , () . , , .

5 . 1 A

111

5.1 A

. ( ) A . , : , . , , A . , . , . , . , Gray.5.1.1 K GRAY

b = Gray g = . m .

112

KEAAIO 5: A A

M A Gray begin g1=b1 for k=2 to m do gk=bk1 XOR bk end M A Gray begin value=g1 b1=value for k=2 to m do begin if gk=1 then value=NOT value bk=value end end

1

Gray Gray

5.1 16 Gray. Gray . , . , Gray . ( m = 4), 1 1 A= 0 0 0 1 1 0 0 0 1 1 0 0 , 0 1 1 1 A -1 = 1 1 0 1 1 1 0 0 1 1 0 0 0 1

: g=Ab b=A1g, 2.

5 . 1 A

113

, , . , . . , . . 1 uk, k = 0, 1, , N1, N -1 2 2 2 J = min x N + x k + uk k =0

0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 5.1

Gray 0000 0001 0011 0010 0110 0111 0101 0100 1100 1101 1111 1110 1010 1011 1001 1000

(

)

.

x k +1 = x k + uk , k = 0, 1,..., N 1, x0 , xk R u RN . :2 J * = K0 x0 ,

Gray

Kk Riccati, k = N1, , 0 K k = 1 + K k +1 / 1 + K k +1 K N = 1. u. (200, 200) ui ( ). x0 = 100 N = 45, u = u0 ,..., u44 , J* = 16180.4.

(

)

114

KEAAIO 5: A A

5.1

Gray : . . . . . . . .

5.2

: . . . . . . . . . .

5.1

, .

5.2

115

5.2

( ) . . , . . : , , , , . , 1. . . : . , .5.2.1

.

116

KEAAIO 5: A A

, ( , ) . , . , , , . ( ) (UB LB) / (2n 1), UB LB n .5.2.2

, . . , . , , . , ( ). , , . , -

5.2

117

.5.2.3

DEC3100 [1]. 10 . 60 20000. n = 30 ( ), 30 45 = 1350 . , . . , .

( ) .

. , , . 0.25, ( 2).

118

KEAAIO 5: A A

5.2

0.6 0.00047 0.014 0.7 0.00068 0.02 0.8 0.00098 0.03 0.9 0.0015 0.045 0.95 0.0021 0.061

( ) (0.25). , , . (LB, UB). 5.3

0.6 42179 46594 0.7 46102 41806 0.8 29290 47454 0.9 52769 69624 0.95 30573 82371 31212 11275

5.3

: u1 = [0.1, 0.3, 1.2, 0.05] u2 = [1.5, 0.05, 0.8, 0.3] , .

5.2

119

5.2.4

3 . , , (16180.4). , , , . , . , . : % ( 400, 200, -200 ) ; : : d , 2d . , = 0.05 0.05, 0.1 . : . n = 30 m , m m n + log 2 d . m m = n + log 2 d = 25

m/n = 25/30 = 0.833, . , , .

, , ,

120

KEAAIO 5: A A

. . t : su = u1 ,..., um -

( t ) k t , su+1 = u1 ,..., u k ,..., um ,

u k + D t ,UB -u k ), 0, uk = u k - D t ,u k - LB ), 1, LB UB k. (t, y) [0, y] (t, y) , t . ( t ) . , , . b t 1- D t , y = y 1 - r T ,

( (

( )

r [0, 1], b ( b = 5).

, , . , u k

5.2

121

u k = u k , t , n) , n = 30 . (k, i) : k i ( 0 ) t,n 0, t , n = t,n 1, b , ( b = 1.5).

(

( )

( )

( ) ( )

[*]

3. 5.4

0.8 35265 20561 0.9 30373 26164 40256 2133

( 4). , . , , , (16205 16189). 5.3. 5.4

To .

[*]

122

KEAAIO 5: A A

5.2.5

.

, . , .

. , , . .5.2.6

. , . 5.5

0.7 23814 16248 0.8 19234 16798 0.9 27456 16198 M 6078 54

K 16188.2 16182.1

5.2

123

A . . 5.2.5. 5.6

( ) . A (N) 5 1080 184 15 3123 398 25 5137 611 35 7177 823 45 9221 1072

CPU . 30 bits . , . 5.7

( ) , =45. 5 4426 10 5355 20 7438 30 9219 40 10981 50 12734

1072 ()

124

KEAAIO 5: A A

5.2

2.097.152 . A A . , : 1. , 2. , 3. , 4. .

5.3

A A . , A , f(x) = x10, A , . 30 10 . .

5.4

.

YNOH

125

, . , , . . , . , . . , ( ) ( , ). ( ). , , , . . , . , . , . , , .

126

KEAAIO 5: A A

, . . , .B

[1] Michalewich Z., Genetic Algorithms + Data Structures = Evolution Programs, SpringerVerlag, 2nd ed., 1992. [2] Davis L., Handbook of Genetic Algorithms, Van Nostrand Reinhold, 1991.

E A , , . , , . . , . , , . . , : A , , A, A, , , .

6

128

KEAAIO 6: E A

(blueprint) (markers) , A . , , () . A (Image Processing), / (Computer Aided Desing CAD), , , T (Software Engineering), (Scheduling), (Computer Graphics) . A . A, , . .

6.1 T M

129

6.1 T M

, A , . , : A , , A, .

. , , , . . , , , . A , . , , , , .. 2%, . , , .

130

KEAAIO 6: E A

, , , , . , . , A [1], [2], [3].

(structural optimization). Goldberg [2] A . 10 , , . A : , . . 4 10 , . , , . , A . , A , . , Powel, Skolnik Tong [3], . , , : (), () A. ,

6.1 T M

131

. , , , . (Interdigitation) . , . , A . . , : , . , . , A. , , A . . , . . , , . . . , , . General Electric . General Electric, ( ), A Engineous. -

132

KEAAIO 6: E A

, , CAD. Engineous, [4]. , A. , , , , . , Engineous , (trial and error). A 100 , 10.

6.1

: . , . , . , . .

6.2

, , ;

6 . 2 A T N

133

6.2 A T N

, , . A . , : , ,

, . . , , , , , , , , . A, , , . A () , ( A) . [3].

134

KEAAIO 6: E A

, (), . , . , . , , (SigmaPi unit), , . , , . (product unit). M , , y=

X (i )i =1

N

p( i )

.

p(i) . , , , , . , , . . (Product Neural Networks) (). 6.1.

6 . 2 A T N

135

6.1:

X1

X2

X3

X4

(Error Back Propagation BP). , . , , , , . , . , A, . , Janson Frenzel [3], A, . A ' : , . . , 30 100, 32 . 37 , 1184 . : (AT)

136

KEAAIO 6: E A

1 . 1 + ATL 1. . , : 20 , 20 . . , 1000 . CAD CMOS. , CMOS, , . , , . , A . , . , A 5 20 .

( A) . ' , , A . . , A -

6 . 2 A T N

137

, . .

Harp Samad [3]. A . , . , , , . , . , , , ' , . , Harp Samad [3] , NeuroGENESYS, A . . , ( ) . , . NeuroGENESYS , , , .. A . , , ,

138

KEAAIO 6: E A

, A , , . A , , , .. , , , .. . () . , , , , (learning rate), (feedback connections). , . , A . , , . , , , .. ( blueprint ), . , ( ) ( ). : 1. () 2. (). , , , .. ,

6 . 2 A T N

139

, .. , , . , A. , , A . (markers) . , . , A. 30 100 . , . . . , , . : 1. , . 2. E . 0.01 . , , , , . -

140

KEAAIO 6: E A

( ). : 1. , , 2. , . , ( ). , , (.. , , .). F(i) i: F (i ) =

a Y ( p (i )) .j j j j =1

n

, pj , j. NeuroGENESYS j . NeuroGENESYS . , ExclusiveOR. , . . (fanout) , fanout . , , fanout. ExclusiveOR (EXOR), NeuroGENESYS (, , ).

6 . 2 A T N

141

, , 6.2.

INPUT

OUTPUT

6.2:

NeuroGENESIS

, 0.75, 0.25. , Harp Samad [3].

Jansen & Frenzel .

6.3

Jansen & Frenzel .

6.4

Harp & Samad, ;

6.5

Harp & Samad F (i ) =

6.1

a Y ( p (i ))j j j j =1

n

pj aj F.

142

KEAAIO 6: E A

6.3

A , . , . . , , . , : A , , , A, , .

(scheduling) , [5]. , , (NPcomplete) . , . ,

6.3

143

, , . . (), , . A . (System Integration Test Station) . , F14, (cockpits), (radar), ., . , , , , , . . ( ), . , , 1. , .. . 2. , .. 5 .. 3. , .. , , . Gilber Syswerda [3] (scheduling system) , . A . , -

144

KEAAIO 6: E A

. , . , (deterministic schedule builder), . FCFS (First Come First Served), , , , ... ( ). . , . A, . . , , , , ' . , , . A , . . , . , . , . -

6.3

145

. , . , . , , , ( ) . , , , A . , . , , , , , . , . , : (OrderBased Crossover). (PositionBased Crossover). (Edge Recombination Crossover). . () . . 1: , :

146

KEAAIO 6: E A

1 2

123456789 412876935 ****

, . ( ) 2, 8, 6 5. 2, 5 ,6 8. , ( 2865). , 2, 5, 6, 8: 1 (1 x 3 4 x x 7 x 9).

, 2, 8, 6, 5, 1 2 : 1 2 1 2 abcdefghij eibdfajgch **** (1 2 3 4 8 6 7 5 9 ).

: aibcfdeghj ibcdefahjg

Whitley [3], . . : (Positionased utation). (OrderBased Mutation).

6.3

147

(Scramble mutation). . , . , . . , . Whitley, Starkweather Shaner [3] GENITOR, A. (), . . , . .. ABCDEF : A AB, BC, CD, DE, EF FA. , AB BA . . . (edge map) . ( ).

(Robotics) . (robot trajectory generation) -

148

KEAAIO 6: E A

, , (rules) . , . , , , . ' , , , A, . A , , A . O Yuval Davidor [6], , A . A, . . , (endeffecter) . ( ) , , . ,

6.3

149

, . , . , , . A Davidor [6] , . . . : 64n - tuple 8 4 44 7 a1,1; a1,2 ;L; a1, n ; a2 ,1;L; a2 , n ;L; al , n 144444 2444444 4 3l n - tuples

, n n () . , l , ln. . A . . (Analogous Crossover) [6]. , . , , . 6.3.

150

KEAAIO 6: E A

6.3

M 1

M 2

, ' ( A) . , , . , . . , . , , , . , (Segregation Crossover), : 1. . 2. n . 3. n n. n . 4. 1 2 . 5. . . :

6.3

151

(Addition Mutation): n . . (Deletion Mutation): n, , . , A . 4.2 6.3. 1040. 100. A : .

. . A . A , . A Holland [7], . , Prediction Company Santa Fe, ( Prophet), A . Man Machine Interfaces Inc., A . , , 1 6.3. 6.7

152

KEAAIO 6: E A

6.2

, . .

A . , , . . , A . . , , A , . A ( A), . A , A . A , ' . , A . [5] A. A, . A . A , (highly parallel algorithms).

YNOH

153

, . A. A , . , A . , A , . , A . . . , . . A . , , . , , , . , . . A , . . , . , , -

154

KEAAIO 6: E A

. : A , , A, A , , .B

[1] Michalewich Z., Genetic Algorithms + Data Structures = Evolution Programs, SpringerVerlag, 2nd ed., 1992. [2] Goldberg D.E.,GENETIC ALGORITHMS in Search, Optimization and Machine Learning, Addison Wesley Publishing Company, Inc., 1989. [3] Davis L., Handbook of Genetic Algorithms, Van Nostrand Reinhold, 1991. [4] S. Hedberg, Emerging Genetic Algorithms, in Artificial Intelligence Expert, September 1994. [5] R. Mangano, A Genetic Algorithm White Paper, in An Introduction to Genetic Algorithm Implementation: Theory, Application, History and Future Pontential, Man Machine Interfaces Inc., 1993. [6] Y. Davidor, A Genetic Algorithm Applied to Robot Trajectory Generation, Proc. Of 3rd International Conference on Genetic Algorithms, 1989. [7] Holland J.H., Adaptation in Natural and Artificial Systems, M.I.T. Press, 1975.

1.1 , : . (chromosomes), , . (reproduction) (mutation). , , . 1.2 , , . . , 1.1.1. 1.3 , , , .. . 1.4 , , , , . , , . , -

156


. , , . . 1.5 5. , 1.1.2. 1.6 (string) l = 5, ( ) 0 (00000) 31 (11111). , . , 53 095, : 5 * 10 4 + 3 * 10 3 + 0 * 102 + 9 * 101 + 5 * 100 = 5395 , 0 1 10011 : 1 * 2 4 + 0 * 2 3 + 0 * 22 + 1 * 21 + 1 * 20 = 16 + 2 + 1 = 19 , . 1.7 1.2 . 5. 1.8 , . ( ) . , . , .

A A N T H E I A K H E N AY T O A I O O H H

157

2.1 . , , . , , . , , . , .... 2.2 1.1.3, , : 1. . A T x ( ) f ( x ) = x2 pselecti = A/ fi f 0.14 0.49 0.06 0.31 1.00 0.25 0.49 0.58 1.97 0.22 1.23 4.00 1.00 1.97 A

f13 24 8 19 169 576 64 361 1170 293 576

fi

1 2 3 4

01101 11000 01000 10011

1 2 0 1 4.0 1.0 2.0

f f /4Maximum

2. bits : 4 5 0.001 = 20 0.001 = 0.02 < 1. , bit .

158


2.3 k = 3, k + 1 = 4 2.

) 0 1 1 0 1 1 1 0 0 0 2.4

01100

) 0 1 0 0 0 1 0 0 1 1

01011

11001

10000

, , . 2.2, . , . Z ( ) ( ) N T x ( ) f ( x ) = x2

1 2 3 4

01101 11000 11000 10011

2 1 4 3

4 4 2 2

01100 11001 11011 10000

12 25 27 16

144 625 729 256 1754 439 729

f f /4Maximum


159

2.5 1. , : 1: 1 1 0 1 0 1 1 0 1 1 1 0 1 1 2: 0 0 0 1 1 0 0 1 0 1 1 1 0 0 . 2. , : 1: 1 0 0 1 1 0 1 2: 0 1 0 1 0 1 1 , , , , , . 2.6 x , f(x) = x, . 2.7 : 1. . 2. . 3. , . 4. , .

160


3.1 l , k 1 l1, pos = k [1, l]. . 3.2 pop_size m pm. : 1. 50 33 0.001 = 1.65 2. 50 33 0.01 = 16.5 3. 50 33 0.1 = 165 3.3 29 < 1510 < 210, m1 = 10 26 < 170 < 27 , m2 =7. m = m1 + m2 = 10 + 7 = 17. 3.4 6.6 . 5 , . 3.5 20 0.25 = 5. 4 . , . 3.6 , , .


161

3.7 . . 3.8 y'(x) = y(x), : y'(x) 0. , : g '( x ) = g ( x ) + r1F(h( x )) - r2 F( y ( x )) 3.9 : 1. . 2. , . 3. . , , , . , : 1. (fixed remapping), 2. (random remapping) 4.1 , , . , ( 4.1) . , 10. (*111100100) {(0111100100), (1111100100)}

162


(*1*1100100) {(0101100100), (0111100100), (1101100100), (1111100100)}. (1001110001) , (1001110001), (**********) 10. 2r , r * . , m 2m . , (1001110001). 210 : (1001110001) (*001110001) (1 * 01110001) (10 *1110001) M (100111000*) (** 01110001) (*0 *1110001) M (10011100 **) (***1110001) M (**********) . , m j = 1, , m , m j, :

m = j j =0m

(1) (1)j j =0

n

m - j m

j

,


163

( x + y) = j ( x) ( y) m j j =0

m

m

m- j

, x = y = 1,

j = 2 j =0

m

m

m

4.2 S % ( eval ( S , t ) = F (t ) + e F (t ) ), :

x ( S , t ) = x ( S , 0 ) (1 + e )t e = (eval ( S , t ) - F (t )) / F (t )(4.4)

> 0 < 0 . . , , . < 0, . 4.3 f f = b0 20 + b1 21 + b2 22 + b3 2 3 , bi = 0 1 i = 0, 1, 2, 3. S, p , F S =

( ) eval ( v ) / p ,j j =1

p

eval(vj) vj. 1*** : 1000 eval 1000 = 8

(

)

164


( ) 1010 eval (1010) = 10 1011 eval (1011) = 11 1100 eval (1100) = 12 1101 eval (1101) = 13 1110 eval (1110) = 14 1111 eval (1111) = 151001 eval 1001 = 9 , S1 = (1***) F S1 = 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 / 8 F S1 = 11.5 0*** :

( ) (

)

( )

( ) 0001 eval (0001) = 1 0010 eval (0010) = 2 0011 eval (0011) = 3 0100 eval (0100) = 4 0101 eval (0101) = 5 0110 eval (0110) = 6 0111 eval (0111) = 70000 eval 0000 = 0 , S2 = (0***) F S2 = 0 + 1 + 2 + 3 + 4 + 5 + 6 + 7 / 8 F S2 = 3.5 4.4 f : f = b0 + b1 + + bl , bi = 0 1 i = 0,1,..., l . S 2l k

( ) (

)

( )


165

k ( k l k ) l ( l ). , S : F S = ,

( ) eval ( v ) / 2j j =1

2 l -k

l -k

= k + ( k + 1) + + l / 2l - k

(

)

( ) ( l (l + 1) k ( k - 1) l (l + 1) - k ( k - 1) = =2 2 2 F S = 5.1

k + k + 1 + + l = 1 + 2 + + k + + l - 1 + 2 + + k - 1 =

(

))

()

l l +1 - k k -1 2l - k +1

( ) (

)

. , 1. 5.2 . , , , 5.5. 5.3 30. 4 30 = 120 . u1 =

166


u2 = 60 ( ). u1 = u2 = . u1 = [0.1, 0.3, 0.8, 0.3] u2 = [1.5, 0.05, 1.2, 0.05]. 5 u1 10 u2. : u1 = u2 = . u1 = [0.1, 12.8, 0.8, 0.3] u2=[1.5, 0.05, 0.8, 0.05]. : . u1 = [0.1, 0.05, 0.8, 0.3] u2 = [1.5, 0.3, 1.2, 0.05].


167

u1 u2. , , 3.4 4.9. T : u1 = [0.1, 3.4, 0.8, 0.3] u2 = [1.5, 0.3, 1.2, 4.9]. 5.4 u1 u2. u k = u k , t , n , t,n 0, t , n = t,n 1,

( )

( ) ( )

(

( ))

b t 1- D t , y = y 1 - r T ,

( )

= 20000, t = 1, b = 5 r [0, 1]. 0, 19 u1, 20 u1. 0, 19 u2, 20 u2. T u1 = [0.09925, 0.3, 1.2, 0.05] u2 = [1.5, 0.05, 0.80022, 0.3], 0 u1 = [0.10039, 0.3, 1.2, 0.05] u2 = [1.5, 0.05, 0.79936, 0.3], 1.

168


u1 u2. : u k + D t ,UB -u k ), 0, uk = u k - D t ,u k - LB ), 1,b t 1- D t , y = y 1 - r T ,

( (

( )

= 20000, t = 1, b = 5 r [0, 1]. T u1=[0.1, 0.05, 131.709, 0.3] u2=[66.991, 0.3, 1.2, 0.05], 0 u1=[0.1, 0.05, 131.453, 0.3] u2=[66.481, 0.3, 1.2, 0.05], 1. 6.1 . , A . 6.2 , . , . . , A . . 6.3 : , -


169

.., . : (AT) 1 . 1 + ATL . , . . , 1000 . 6.4 A (Error Back Propagation BP). , . , , , . , . , A, . Janson Frenzel A, . 6.5 . . , , .

170


6.6 . , , , , . ( ). 6.7 1 6.3, : 2 (3 1 2 8 7 4 6 9 5).

1.1 , (.. , ) . . , , . . , . , . , . , . , , . 1.2 , . . . , . l , 6 1.2. . , . 2.1 . [1].

172


2.2 6 7, , . 5 7, . . , . 2.3 , . , , 729, 576. , . , : , 1024, . . , . 2.4 , select , . select, , . , partsum. rand , rand : = random sumfitness, random 0 1. , sumfitness ( ), random. , repeatuntil , ,

Y O E I E I A A N T H E N PA T H P I O T H T N

173

rand. j, select. . , . , . 2.5 . 2.6 2.4, . 2.7 A . . , /, , . , . pc , pc = 1 . , . , f ( x ) = x 2 , , x. (pc = 1), x, . pc, . , , , . , , .

174


2.8 . , Matlab, , rand. 2.9 , , , 1.0. 1.0, 9 10. , . . 2.4, . , . , . 2.10 , GAlib. () (singlepoint), (). . 2.1.3. , . , . , 2.11, . [1] ( 3, . 6365). , .


175

2.11 2.10. , . , . 3.1 , : 1. . 2. . 3. . 4. . 5. . 6. . 7. () . 3.1. , . 2. , 2.1.2. 3.2 3.1, . , , . 1. , . , . 100 , 1000.

176


, ( ) . 2. ( ), . 3.3 , , , . 1, . , , . , , . , . . 3.4 , . , 2.5. . , , , . A, . ( 10 100), ( ) . , . 3.5 , (3.1). (3.2).


177

, . . . 4.1 ) 1 1, 3, 5 6. 2 2. 3 2 3. ) S, o(S), 0 1. S ( (S)) . , (1) = 1 (2) = 1 (3) = 2 (4) = 3 (5) = 2 (6) = 5 (1) = 11 = 0 (2) = 11 = 0 (3) = 87 = 1 (4) = 74 = 3 (5) = 71 = 6 (6) = 71 = 6

) S pm p s S = 1 - pm , ps(H1) = 110.001 = 0.999 ps(H2) = 110.001 = 0.999 ps(H3) = 120.001 = 0.998 ps(H4) = 130.001 = 0.997 ps(H5) = 120.001 = 0.998 ps(H6) = 150.001 = 0.995

() (

) ( ) 1 - o(S ) po S

m

178


S pc : p s S 1 - pc

()

m -1

d S)

(

m . , ps(H1) = 10.850/7 = 1 ps(H2) = 10.850/7 = 1 ps(H3) = 10.851/7 = 0.878571 ps(H4) = 10.853/7 = 0.635714 ps(H5) = 10.856/7 = 0.271429 ps(H6) = 10.856/7 = 0.271429 4.2 , , ( )

x S , t + 1) x S , t ) eval ( S , t ) / F t ) (S, t + 1) , (S, t) , eval(S, t) S F t . , (S1) = 1 1 = 0 (S2) = 4 1 = 3 F t =0 = o(S1) = 1 o(S2) = 2

(

(

(

()

(

) eval (v ) / 4 = (20 + 10 + 5 + 15) / 4 = 12.5i i =1

4

( ) ( ) eval ( S , t = 0) = (5 + 15) / 2 = 102

eval S1 , t = 0 = 20 + 10 / 2 = 15


179

, : ) x S1 , t = 1 = 2

( (

)

15 0 1 - 1 - 1 0.01 = 2.376 12.5 4 10 3 1 - 1 - 2 0.01 = 0.368 12.5 4

) x S2 , t = 1 = 2

)

, , S1 S2 1. 5.1 , . . , , . 5.2 , , 3. , , . . 1 2 3 4 5.3 A, , f(x) = x10. , , , A. 221 2097152 221

27 2097152 27 27 50 27

221 50 221

180

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