## 4 Analysis of Single Objective Algorithms Overview
### Learner Performance-Based Behavior
As seen in the preceding section, once a neighborhood topology is created, each particle's leader for updating its position is identified while solving single-objective optimization problems. Every particle in MOO problems might have several leaders to pick from, yet just one could be chosen to update the particle's location. Usually, the leaders' group is kept in a location that is referred to as an external archive that is distinct from a swarm [12; 58]. The non-dominated solutions that are discovered are kept in this repository. The solutions that are kept in the external archive act as a reference to rearrange the particle positions within the swarm. Furthermore, the final result of the algorithm is usually supplied as the contents of the external archive [54; 59]. Nothing in the way of material is offered. The workings of a general MO-PSO algorithm are demonstrated by Algorithm 7. The distinct features that set this algorithm apart from the conventional PSO method for single-objective optimization are highlighted by the underlined procedures. First, the swarm is organized. The non-dominated particles in a swarm are used in order to form a group of leaders. As mentioned before, the group of leaders is usually stored in an external repository. After that, a quality metric is computed for every leader to choose one leader for every swarm particle. Every particle has a leader selected throughout each generation, and the flight is carried out. A mutation operator is incorporated into many MOPSO algorithms after the flight operation. The particle is then evaluated, and the corresponding personal best value is updated. In the case when the former is either dominated or both particles are incomparable, which means they are not dominated concerning one another, a new particle frequently replaces its personal best particle. The set of leaders is likewise updated when all particles are updated. This approach is usually repeated for a specific and updated quality measure of the group of leaders at the end [6; 80]. ```
1: Begin
2: Swarm Initialization
3: Initialization of leaders in external archive
4: Quality(leaders)
5: g = 0
6: While gmax > g
7: For every particle Select leader
8: Update Position (Flight)
9: Mutation
10: Evaluation
11: pbest Update
12: EndFor
13: Update leaders in external archive Quality(leaders)
14: g = g + 1;
15: EndWhile
16: The report leads to external archive
17: End
```
**Algorithm 7** MOPSO

#### 5.3.4 Multi-Objective Fitness Dependent Optimizer

In MOFDO, in order to determine the pace, it is necessary to take into consideration conditions that are outlined in Equations (13)-(15), which rely on the fitness weight (fw) value, which may be determined by applying Equation (16) to the problem's cost function values. There is importance in noting that in MOFDO, the pace denotes both domain and historical knowledge:

\[X_{i,t+1}=X_{t,i}+pace \tag{12}\]

In this case, \(x\) denotes the individual itself, \(t\) denotes current iteration, \(i\) represents current individual number, and pace is the direction and rate of movement, remembering that fw determines the pace value most of the time. On the other hand, the pace's direction (value sign) is solely determined randomly. \[\left\{ \begin{aligned} & fw=1\;or\;fw=0\;or\;fw=X\;i,t\;fitiness=0\;pace=X_{i,t}*r \;(\ref{eq:x})\\ & fw>0\;and\;fw<1\left\{\begin{aligned} & r<0,pace=(X_{i,t}-X_{i,t}^{*})*fw*-1\;(\ref{eq:x})\\ &\{r\geq 0,pace=(X_{i,t}-X_{i,t}^{*})*fw*\}(\ref{eq:x})\\ \end{aligned}\right\}\end{aligned} \right\}\]Equations (13)-(15) contain two different conditions. Initially, if fw equals 0, or, if fw equals 1, or if \(\sum_{o=1}^{n}Xi,ffitness=0\), then the pace should be estimated as Equation (12). Second, a random number r is created in the [-1, 1] range if the fw value is between zero and one. If r is negative, Equation (13) will be applied; if not, Equation (15) will be utilized in order to determine the pace fw, which is calculable with the use of Equation (16).