P.lappas Genetics TSP

. , [email protected]

(Travelling Salesman Problem, TSP)

;

3

Leonhard Euler (1736)

Konigsberg

puzzle

( Kaliningrad ) Pregel

, A B, , C D 7

4

, ;

5

Sir William Rowan Hamilton (1856)

,

(Around the World)

John Thomas Graves (1806-1870)

20

30

6

, , , , , , , , , , , , , , , , , ,

/ , ,

7

Asymmetric TSP (aTSP)

1

2

5

3

4

6

i j (i,j)

i j

cij

cji

= ,

= 1,2,3,4,5,6

= 1,2 , 2,3 , 2,4 , 2,5 , 3,6 , 4,3 , 4,5 , 5,1 , 5,3 , 6,1 , 6,4

, =

8

Symmetric TSP (sTSP)

1 2

4

6

3

7

i j {i,j}

i j cij = cij

= ,

= 1,2,3,4,5,6,7

= 1,2 , 1,4 , 1,5 , 2,3 , 2,4 , 2,6 , 3,4 , 3,6 , 3,7 , 4,5

= , =

+

{j,i}

5

9

aTSP

10

1

2

5

3

4

6

1-2-3-6-4-5-1: Hamiltonian Cycle Tour

2-5-3: Path (Directed)

1-2-5-1: Cycle Subtour

TSP Hamiltonian Cycle

aTSP

11

min ( , )

(1)

s.t.

{:( , )}

= 1, = 1, , (2)

{ :( , )}

= 1, = 1, , (3)

= 0 1 , = , : , , (4)

1, = 2,3, , 1 5

TSP

12

Xcord Ycord

1 X1 Y1

2 X2 Y2

...

n Xn Yn

= ( )2 + ( )2

Ycord

Xcord

1

4 2

5 6

3

n

TSP

13

n-1

n-2

n-3

2

,

(n-1)!

TSP

14

(n-1)!/2 Hamiltonian

Hamiltonian

Hamiltonian

5 12

10 181440

20

100

1000

. .

NP-hard

TSP

15

1954 Dantzing, Fulkerson, Johnson 49

1971 Held, Karp 57

1971 Held, Karp 64

1975 Camerini, Fratta, Maffioli 67

1975 Miliotis 80

1977 Grotchel 120

1980 Padberg, Hong 318

1987 Padberg, Rinaldi 532

1987 Grotchel, Holland 666

1992 Applegate, Bixby, Cook, Chvatal 3038





(Metaheuristics)

17

(Genetic Algorithm, GA)

(Ant Colony Optimization, ACO)

(Tabu Search, TS)

(Simulated Annealing, SA)

(Greedy Randomized Adaptive Search Procedure, GRASP)

(Metaheuristics)

18

(GA)

(MA)

(ACO)

(SA)

(TS)

(GRASP)

(ILS)

(GLS)

(VNS)

Heuristics vs. Metaheuristics

19

Heuristics

Metaheuristics

(Genetic Algorithms)

20

Holland (60)

(survival of the fittest)


21

(Individuals)

(Generation)

( (parents)) (crossover)

(mutation)

( (offsprings))


22

(fitness function)

(Chromosome) (Genome)

( )

(Gene)

( )


23

(Encoding)

(Path Representation)

x = [x1 x2 xn]

x1 x2 xn

xi: ( )

Nearest-Neighbor Initialization

Nearest Neighbor ()

24

=

3293

358

11

2546

98410

3116

10

Tour 1 (Solution 1):

1 3 4 2 5 1

Total Cost = 2 + 4 + 8 + 11 + 3 = 28

Tour 2 (Solution 2):

213452

Total Cost = 3 + 2 + 4 + 10 + 11 = 30


25

(Roulette-wheel Selection)

Individual 1

Fitness = 10

Individual 2

Fitness = 20

Individual 3

Fitness = 30

Individual 4

Fitness = 40

20

10

30

40


26

() 2

4

Individual 1: 10%

Individual 2: 20%

Individual 3: 30%

Individual 4: 40%


27

{P1 P2}, {P3 P4} {C1 C2}, {C3 C4}

,


28

(Crossover ) Partially Matched Crossover (PMX; Goldberg and Lingle, 1985)

Order Crossover (OX; Davis, 1985)

Cycle Crossover (CX; Oliver et al., 1987)

Order-based Crossover (OBX; Syswerda, 1991)

Inver-over Crossover (Tao and Michalewicz, 1998)

(Mutation) Inversion (Beyer and Schwefel, 2002)

Insertion (Beyer and Schwefel, 2002)

Displacement (Michalewitz, 1996)

Reciprocal Exchange (Banzhaf, 1990)

Partially Matched Crossover (PMX)

29

(single-point crossover)

P1 = [ 2 3 4 5 6 1 ]

P2 = [ 3 2 6 1 4 5 ]

c1 = [ 2 3 4 1 4 5 ]

c2 = [ 3 2 6 5 6 1 ]

c1 4, c2 6,

c1 = [ 2 3 6 1 4 5 ]

c2 = [ 3 2 6 5 4 1 ]

Order Crossover (OX)

30

P1 = [ 9 2 3 8 4 5 6 1 7 ]

P2 = [ 4 5 2 1 8 7 6 9 3 ]

subtour P1

[ 8 4 5 6 ]

c1 = [ - - - 8 4 5 6 - - ]

c1 1,2,3,7,9 P2 {2,1,7,9,3}

c1 = [ 2 1 7 8 4 5 6 9 3 ]

subtour P2

[ 1 8 7 6 ]

c2 = [ - - - 1 8 7 6 - - ]

c2 2,3,4,5,9 P1 {9,2,3,4,5}

c2 = [ 9 2 3 1 8 7 6 4 5 ]

TSP Mutation

31

Inversion (randomly)

x = 1 5 4 7 6 2 3

xm = 1 6 7 4 5 2 3

Insertion (randomly)

i k

i=4, k=2

x = 1 5 4 7 6 2 3

xm = 1 7 5 4 6 2 3

32

()

TSP -

33

TSP DATA Best Known Solution GA (runs)

berlin52.tsp 7542 7809, 7954, 7862, 7895, 7667, 7937 8095, 7913, 8009, 8439, 8088, 8063 8063, 7950, 7938, 8189, 8173, 7989 8374, 7862

gr96.tsp 55209 60884, 60403, 61370, 63110, 59017

lin105.tsp 14379 15472, 14918, 16384

ch130.tsp 6110 7926, 7977, 7303

ulysses22.tsp 7013 7181, 7013, 7195, 7100, 7013, 7114

ulysses16.tsp 6859 6870, 6859, 6870, 6870, 6859

P.lappas Genetics TSP

Documents

Transcript of P.lappas Genetics TSP