Hybridizing Infeasibility Driven and Constrained- Domination Principle with MOEA/D for Constrained Multiobjective Evolutionary Optimization - Technologies and Applications of Artificial Intelligence - page 251

2.1 Constraint Violation

Constraint violation (CV) [2] is a scalar value derived through the summation of total
violations of the inequality and equality constraints. If CV equals to zero, the solution
is feasible. The degree of constraint violation of an individual on the jth constraint is
defined as:

G

max{0, gx ( )}, 1 ≤≤ j l

⊧

G

j

Gx () =

G

(4)

⊨

j

max{0,| hx ( ) | − ʴ }, l + 1 ≤ j m ≤

⊩

j

In the proposed approach, the degree of constraint violation of an individual is de-
fined as:

G G G

∑

m

Gx () = Gx ()/max( ()) Gx

(5)

j j

j = 1

2.2 Constrained-Domination Principle

Constrained-domination principle (CDP) [3] defines a domination relation of two
solutions in constraint handling problem. A solution A is said to constrained-dominate
a solution B, if any of the following conditions is true.
1) Solutions A and B are both feasible and solution A dominates solution B.
2) Solutions A and B are not both feasible, but solution A has a smaller overall
constraint violation.
In the proposed approach, CDP is used as a basic comparison mechanism to
conduct population update in the MOEA/D. the pseudo code of CDP is shown in
algorithm 1.

Algorithm 1. CDP in population update in MOEA/D

1: if

Gx () = Gx + ( ) 0 =

then

j j 1

2: if

fx f >+ then
3: swap

() ( )

j

(, xx + ) ;

j j 1

4: end if
5: else
6: if

Gx () > Gx + ( ) then

j j 1

7: swap (, xx + ) ;

j j 1

8: end if
9: end if