Smallest Position Value Approach - Differential Evolution: A Topic for Global Permutation-Based Combinatorial Optimization - page 129

R
k =27 into the first position of partial solution for Case A

Ta b l e 5 . 5 . Insertion of node π

j 1 2 3 4 5 6 7 8 9 10 11

j

π 27 1 22 20 50 10 33 44 41 25 24

d π j π j +1 d 27 , 1 d 1 , 22 d 22 , 20 d 20 , 50 d 50 , 10 d 10 , 33 d 33 , 44 d 44 , 41 d 41 , 25 d 25 , 27 d 24 , 27

174 8 7 15 21 17 12 17 20 21 14 22

F ( )= d 1 , 22 + d 22 , 20 + d 20 , 50 + d 50 , 10 + d 10 , 33 + d 33 , 44 + d 44 , 41 + d 41 , 25 + d 25 , 24 +

d 24 , 1 + d 27 , 1 + d 24 , 27 −

π

d 24 , 1

F ( )= d 1 , 22 + d 22 , 20 + d 20 , 50 + d 50 , 10 + d 10 , 33 + d 33 , 44 + d 44 , 41 + d 41 , 25 + d 25 , 24 +

d 27 , 1 + d 24 , 27

π

k in the first position of the partial tour D

B Insertion of node π π

a Remove = d

1

π

m π

b Add = d

k + d m π k π 1

π π

D are fit-
ness function values of the tour after insertion and the partial tour,
respectively.

)= F π

D + Add

) and F π

b F ( π − Remove ,where F ( π

Example B:

Remove = d

1

π

m π

Remove = d 10 π 1

π

Remove = d 24 , 1
Add = d + d R

k

R
k π

1

π

m π

π

Add = d 1 + d

10 π 1 π 1

π π

Add = d 24 , 27 + d 27 , 1
F (

)= F π

D + Add

π − Remove

F ( )= d 1 , 22 + d 22 , 20 + d 20 , 50 + d 50 , 10 + d 10 , 33 + d 33 , 44 + d 44 , 41 + d 41 , 25 +

d 25 , 24 + d 24 , 1 + d 24 , 27 + d 27 , 1 −

π

d 24 , 1

F ( )= d 1 , 22 + d 22 , 20 + d 20 , 50 + d 50 , 10 + d 10 , 33 + d 33 , 44 + d 44 , 41 + d 41 , 25 + d 25 , 24 +

d 24 , 27 + d 27 , 1

π

k between an edge π v

u ,

C Insertion of node π π

a Remove = d π u v

π

b Add = d k + d

u k π v

π π π

D are fit-
ness function values of the tour after insertion and the partial tour,
respectively.

)= F π

D + Add

) and F π

b F ( π − Remove ,where F ( π