## 2 Mathematical Modeling
### Particle Swarm Optimization  
The particles in PSO traverse the search space by adhering to the present optimal particles. The system is initially filled with random particles (solutions) that update their positions in each iteration to seek for optima. Each particle's position is updated based  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**S.No** & **NSGA-II Parameter** & **Value** \\ \hline
1. & Scheduling Objectives & 2 (Reflection Loss and Total Thickness) \\ \hline
2. & Population Size & 100 \\ \hline
3. & Number of generations & 150 \\ \hline
4. & Crossover type & Simulated Binary crossover \\ \hline
5. & Crossover distribution index & 20 \\ \hline
6. & Crossover probability & 0.75 \\ \hline
7. & Mutation type & Polynomial mutation \\ \hline
8. & Mutation distribution index & 20 \\ \hline
9. & Mutation probability & 0.25 \\ \hline
10. & Selection Strategy & Binary tournament \\ \hline
11. & Tournament size & 2 \\ \hline
12. & Lower bounds & [1 1 0 0] \\ \hline
13. & Upper bounds & [21 21 10 10] \\ \hline \hline \end{tabular}
\end{table}
Table 1: NSGA-II parameters.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**S.No** & **Material** & **Code** \\ \hline
1. & PP/Epoxy & 1. \\ \hline  
\hline \end{tabular}
\end{table}
Table 2: Material coding for GA.  
on two optimal values: the best solution (fitness) achieved by the particle itself, known as \(p_{best}\), and the best value obtained by any particle in the swarm, known as \(g_{best}\). The fundamental steps of this algorithm are outlined below, along with relevant texts and equations presented in Figure 3.  
Upon determining both best and best, the particle velocity is updated for each dimension using Equation (7).  
\[u_{n}=w\cdot u_{n}+c_{1}rand(\ )\cdot(pbest_{n}-x_{n})+c_{2}rand(\ )\cdot(gbest_{n}-x_{n}) \tag{7}\]  
where \(u_{n}\) denotes particle velocity in the \(u\)th dimension, and \(x_{n}\) is the particle nth dimension position. A random number \(rand()\) is generated between 0 and 1. The inertia weight is represented by w, then learning factors are \(c1\) and \(c2\), respectively. Using Equation (8), the \(u\)th position will be updated following every iteration.  
\[x_{n}=x_{n}+\Delta t\cdot u_{n} \tag{8}\]  
where \(\Delta t\) is a given time step.  
Figure 3: PSO flowchart.  
The parameter w represents the particle's motion without external influences, and an increased value of w reduces its susceptibility to _pbest_ and _gbest_ effects. The Particle Swarm Optimization (PSO) algorithm determines the best position of the particle and colony using parameters \(c1\) and \(c2\), the inertia weight w, and the maximum number of iterations. The swarm size, typically 100 or less, is crucial for the algorithm. However, it can be modified to address discrete problems, as introduced by [PERSON] and [PERSON] in a binary PSO modification.  
There are a lot of parallels between PSO and other evolutionary computing methods, like Genetic Algorithms (GAs). PSO, on the other hand, does not usually have evolutionary operators like mutation and crossing. When compared to GAs, PSO is easier to use in any computer language and does not need as many factors to be changed. On top of that, PSO uses less computing power than GAs for the same population size. GAs are used in PSO, which has been successfully used in many engineering fields, including teaching artificial neural networks, controlling complex systems, and optimizing functions.