Binary variable

Continuous variable

Particle position

$x^{\ast }=\left\{y\right\}$

First stage: $x=\left\{P_{hu},P_{bess},\alpha ,\beta \right\}$

Second stage: $x=\left\{\theta ,\varphi \right\}$

Velocity update

$V_i^{k+1}=\varphi \cdot w^k\cdot v_i^k+c_1\cdot r_1\cdot \left(P_{best}-x_i^k\right)+c_2\cdot r_2\cdot \left(G_{best}-x_i^k\right)$

where k is the iteration number. i is the particle number. x is the particle position. v is the particle velocity. $P_{bess}$ and $G_{best}$ are the personal and global bests, respectively. $c_1$ and $c_2$ are the acceleration coefficients. $r_1$ and $r_2$ are the random numbers between 0 and 1.

The time-varying inertia weight is $w^k=w_{max}-k\frac{w_{max}-w_{min}}{k_{max}}$ and $\varphi =\frac{2}{\left|2-C_0-\sqrt{C_0^2-4C_0}\right|}$ .

where $w_{min}$ and $w_{max}$ are the minimum and maximum values of the inertia weight, respectively. $k_{max}$ is the maximum iteration. $C_0$ is a constant coefficient, where $C_0=c_1+c_2$ and $C_0>4$ .

Particle update

$x_{ij}^k=\{\begin{array}{l}1\text{if}{u}_{ij}^k<s_{ij}^k\\ 0\text{if}{u}_{ij}^k\ge s_{ij}^k\end{array}$

where $u_{ij}^k$ is a random number between 0 and 1, and $s_{ij}^k$ is the sigmoid function ( $s_{ij}^k=\frac{1}{1+{\text{e}}^{-v_{ij}^k}}$ ).

$x_i^{k+1}=x_i^k+v_i^{k+1}$