# A Hybrid Multi-Objective Optimization Method and Its Application to Electromagnetic Device Designs
## 3 A Hybrid Multi-Objective Optimization Method
### Mopso  
MOPSO is also a bio-inspired algorithm. It imitates the food hunting process of fish or a flock of birds. It is believed that in the case of a bird flying and searching randomly for food, for instance, all birds in the flock can share their discovery and help the entire flock to achieve the best hunt. Each bird or fish in the population is called a particle. In the implementation, each particle is given two variables, namely position and velocity. The critical part of the MOPSO algorithm is the update mechanism of the particle position and velocity. The updating of the particle velocity and position is governed by  
\[\mathbf{v}^{k+1}=\omega\mathbf{V}^{k}+c_{1}r_{1}\left(\mathbf{P}_{id}^{k}-\mathbf{X}^{k} \right)+c_{2}r_{2}\left(\mathbf{P}_{gd}^{k}-\mathbf{X}^{k}\right) \tag{7}\]  
\[\mathbf{X}^{k+1}=\mathbf{V}^{k+1}+\mathbf{X}^{k} \tag{8}\]  
where \(\mathbf{V}\) is the particle velocity, \(\mathbf{X}\) is the particle position, \(k\) is the index of iterations, \(\mathbf{P}_{id}^{k}\) is the individual optimal particle (Pbest) position, \(\mathbf{P}_{gd}^{k}\) is the global optimal particle (Gbest) position, \(c_{1}\) is the personal learning coefficient, \(c_{2}\) is the global learning coefficient, and \(\omega\) is the inertia weight, which represents the ability of the particle to maintain its current velocity. \(r_{1}\) and \(r_{2}\) are two random numbers between 0 and 1. The general structure of the MOPSO algorithm is shown in Algorithm 2, and an Archive is set to preserve the excellent particles in the search process. In the beginning, \(N\) particles are generated to form the population \(P\), and the velocity of each particle is initialized; \(K\) best individuals are selected to form the initial Archive \(R\); then, the velocity and position of each particle is updated according to (7)and (8) to form the new population \(P^{\prime}\). Next, the \(K\) best individuals of the mixed population are selected after the non-dominated sorting and adaptive lattice methods are applied, and Archive \(R\) is updated. The above process repeats until the termination criterion is satisfied. Here, the adaptive lattice method is used to select good individuals for the next generation, contributing to the maintenance of population diversity.  
```
Input:\(N\) (particle swarm size), \(K\) (Archive size) Output:\(R\) (Archive)
1\(P\gets Initialization(N)\);
2\(V\leftarrow\) Initialize the velocity of each particle;
3\(R\leftarrow\) Non-dominated-sort(\(P\)); // Select the better \(K\) individuals from \(P\) while termination criterion not fulfilleddo
5\(P\leftarrow\) Update the individual optimal particle according to \(P\);
6\(V\leftarrow\) Compute the velocity of each particle by (8);
7\(P^{\prime}\leftarrow\) Update \(P\) by (7) and (8);
8\(P\gets R\cup P^{\prime}\);
9\(R\leftarrow\) Select the better \(K\) individuals by Non-dominated-sort(\(P\)) and adaptive-lattice(\(P\));
10\(Gbest\leftarrow\) Update the global optimal particle according to \(R\);
11 return\(R\);
```  
**Algorithm 2**MOPSO