# A Survey on Search Strategy of Evolutionary Multi-Objective Optimization Algorithms
## 3 Classification of Multi-Objective Evolutionary Algorithms from the View of Search Strategy
### Evaluation Index-Based MOEA Algorithms
#### 3.3.2 Indicator-Based Evolution Algorithm  
[PERSON] and [PERSON] proposed a generalized evolutionary model in 2004, which first proposed IBEA by embedding the evaluation indicator function as an adaptation evaluation method in MOEA. The rating indicator function is formally defined as a function that maps a group of vectors to a group of real numbers. In other words, it is a mathematical function that allocates a numerical value to a vector I, F \(\to R\); using the all-order relationship on the set of real numbers, it is possible to compare the quality between different sets of vectors in F space. In IBEA, a simpler way of assigning fitness is specified as follows.  
\[F\Big{(}x^{1}\Big{)}=\sum_{x^{2}\in P\big{\{}x^{1}\big{\}}}-e^{-I\big{(}\{x^{2} \},\{x^{1}\}\big{)}/k}. \tag{6}\]  
where \(k\) is a scaling factor greater than 0 and \(I\) is a real number function that maps the group of Pareto approximation solutions in the objective space to a group of real numbers in Equation (6). In this context, it is defined in Equation (7):  
\[I_{HD}(A,B)=\begin{cases}HV(B)-HV(A),&\text{ if }\forall x^{2}\in B\exists x ^{2}\in A:x^{1}>x^{2}\\ HV(A+B)-HV(A),&\text{ else }\end{cases}. \tag{7}\]  
As shown in Figure 10, A and B are two solution sets containing only one individual, respectively. In (a), when the individuals in set A and the individuals in the group B are not dominated by each other, the left yellow part indicates the area of the independently dominated region of set A, then there is \(I_{HD}(B,A)=HV(A+B)-HV(B)>0\). The red part indicates the area of the independently dominated region of set B, and there is \(I_{HD}(A,B)=HV(A+B)-HV(A)>0\). In (b), the gray part of the figure indicates the area of the independently dominated region of set B, then there is \(I_{HD}(A,B)=HV(A+B)-HV(A)>0\), \(I_{HD}(B,A)=HV(A)-HV(B)<0\), and \(I_{HD}(A,B)=-I_{HD}(B,A)\).  
Figure 10: The illustrations of \(I_{HD}\) indicator. In (**a**), when the individuals in the set A and the individuals in the set B are not dominated by each other, the yellow and red parts on the left indicate the areas of the independently dominated regions of A and B, respectively. In (**b**), when A is dominated by B, the gray part of the figure indicates the area of the independently dominated region of B. This figure is adopted from [91].  
By iteratively eliminating the most poorly performing individuals in the population and adjusting the fitness values of the remainder, Algorithm 3 performs mating selection and environmental selection.  
```
1:Initialize the population \(P\) with evolutionary iteration number \(t=0\), termination condition \(T\), and initialization parameter \(k\)
2:Calculate the fitness value of individuals in \(P\)
3:While \(|P|>N\) do
4:Select the least adapted from population \(P\) and remove it from population \(P\);
5:Recalculate the fitness values of individuals in population \(P\)
6:End if \(t>T\) or other termination conditions are met.
7:Select individuals from \(P\) into the mating pool \(Q\) using binary pennant selection and crossover and mutation operations are performed on \(Q\) to produce a new population \(Q^{\prime}\).
8:\(P=P\cup R\)
9:\(t=t+1\), turn back to step 2
```  
**Algorithm 3** IBEA