Computational Intelligence - February 2017 - 36

Algorithm 2 Allocation of individuals to Subpopulations.

Objective Space
V1

1.2

V2

1.0

f2

0.6

V3

0.4
0.2

V4
0

0.2

0.4

0.6
f1

0.8

1.0

Output: P1, f, PK .
1. for k ! 1 to K do
2.
Initialize P k as the solutions in Q whose objective values are in X k ;
3.
if | P k | 1 S then
4.
randomly select S -| Pk | solutions from Q and add

0.8

0

input: Q : a set of individual solutions.

Individual
Boundary

1.2

Figure 6 An illustration of the M2M decomposition strategy.

5.
6.
7.

8.
9. end

them to Pk .
end
if | Pk | 2 S then
rank the solutions in Pk using the selection method
based on the feasibility rules [34] and remove from
Pk the | Pk | - S lowest ranked solutions.
end

Algorithm 1 M2M-Based eMO Algorithm for the Multi-objective
TA Planning Problem.

input:
    *  the maximum number of function evaluations;
    *  K: the number of subproblems;
    *  K unit direction vectors: v 1, f, v K ;
    *  S: the subpopulation size.
Output: a set of nondominated solutions in , Kk = 1 Pk .
1 initialization: Initialize S * K solutions, calculate their
objective values and constraint violation value, and then
use them to set P1, f, PK .
2 while the maximum number of function evaluations is
not reached do
3
Set r = Q ;
4
for k ! 1 to K do
5
foreach x ! Pk do
6
Choose y and apply genetic operators on
x and y to generate a new solution z ;
If z violates constraint (6), repair it;
calculate the objective values and constraint
violation value of z ;
R: = R , {z} ;

7
8
9
10
11

end

12

Q: = R , (, Kk = 1 P k) ;
use Q to set P1, f, P K .

13
14

end
Output the solutions in , Kk = 1 P k .

15 end

independent, which means that each individual needs to only
compete with its counterparts located in the same subpopulation in the process of selection. Therefore, infeasible solutions
are more likely to survive than they would be in a

36

IEEE ComputatIonal IntEllIgEnCE magazInE | FEbruary 2017

single-population algorithm because of less selection pressure.
In this way, a certain number of good infeasible solutions will
be kept to make the population search more effective. What is
more, M2M has a strong capability to maintain the population
diversity, which is desirable in multi-objective optimization.
Considering the complex constraints of the proposed
model, we design an EMO algorithm based on the M2M
decomposition. First, K unit vectors v 1, f, v K in R m+ are generated in the first quadrant. The R m+ is then divided into K
subregions Ω 1, f, Ω K , where Ω k ( k = 1, f, K ) is:
Ω k = {u ! R m+ | G u, v k H # G u, v j H for any j = 1, f, K}, (11)
where G u, v j H is the acute angle between u and v j . Accordingly, we obtain K subpopulations, and those subpopulations
are used to optimize the problem collaboratively.
Fig. 6 shows an illustration of the M2M decomposition
strategy. There are four subregions (K = 4) in the two-dimensional objective space (m = 2), and the four direction vectors
v 1, v 2, v 3, v 4 are evenly distributed in the first quadrant. Each
vector represents a center of a subregion and the dash-dots
lines represent the boundaries of these subregions. Then the
population is divided into four subpopulations according to the
acute angle with the direction vectors.
G. Main Framework of the M2M-based EMO Algorithm
for Multi-objective TA Planning

In this section, the main framework of the proposed EMO
algorithm based on the M2M decomposition for multi-objective TA planning is given by Algorithm 1 and Algorithm 2
in detail.
V. Computational Experiments and Analysis

In this section, three different test networks are generated for
computational simulation. Although realistic instances are not



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