Forward Backward Transformation - Differential Evolution: A Topic for Global Permutation-Based Combinatorial Optimization - page 49

610141920
10 16 18 23 26
13 19 23 24 29
17 23 28 31 32

410151920
10 14 19 21 24
13 17 20 25 28
17 21 24 30 31

Solution 1 = Solution 2 =

5 6 10 16 20
9 12142026
10 15 19 23 29
13 26 24 28 33

410151620
614192228
10 17 20 25 31
15 21 24 26 35

Solution 3 = Solution 4 =

5 9 10 14 20
9 15182024
10 18 21 25 28
13 22 23 30 34

1 6 10 14 20
410122024
711162327
814212731

Solution 5 = Solution 6 =

4 5 10 16 20
6 9 14 20 26
10 13 15 23 29
15 16 19 27 33

610141520
10 16 18 21 25
13 19 23 26 27
17 23 28 29 32

Solution 7 = Solution 8 =

4 5 11 16 20
10 13 17 21 23
13 16 20 22 27
17 18 24 27 32

1 5 11 15 20
4 7 15 21 25
711182426
8162228 3

Solution 9 = Solution 10 =

The fitness of each solution is given as the last right bottom entry in each solu-
tion matrix for that particular solution. The population can now be represented as in
Tab le 3 . 3.
The optimal value and its corresponding solution, for the current generation is
highlighted.
Step (2) is the forward transformation of the solution into real numbers. Using Equa-
tion 3.3, each value in the solution is transformed. An example of the first Solution 1 =
{ 1 , 2 , 3 , 4 , 5 } is given as an illustration:

1 + 1 • 500
999

1 + 2 • 500
999

x 1 = − = − 0 . 499 x 2 = − = 0 . 001

1 + 3 • 500
999

1 + 4 • 500
999

x 3 = − = 0 . 501 x 4 = − = 1 . 002

1 + 5 • 500

x 5 = −

999 = 1 . 502

Table 3.4 gives the table with values in real numbers. The results are presented in 3
d.p. format.
In Step (3), DE strategies are applied to the real population in order to find better
solutions.
An example of DE operation is shown. Strategy DE/rand/1/exp is used for this ex-
ample: u i , G +1 = x r 1 , G + F

• x r 2 , G − x r 3 , G .