## 6 Proposed Approach
### Improved Adaptive Particle Swarm Optimization  
The particle swarm optimization (PSO) is an evolutionary optimization technology by choosing a number of particles to search for best solution [41]. In PSO, each particle represents a random solution for a specific problem, and it can calculate the fitness function value. The position and velocity of the particle is given by a directional vector respectively. During the process of iteration, each particle search for a better solution dynamically by personal best value \(P_{best}\) and global best value \(G_{best}\). Let there are \(m\) particles in \(h\)-dimensional space, the position of each particle \(i\left(1\leq i\leq n\right)\) during \(t\)th iteration is \(X_{ih}\left(t\right)=\left(x_{i1},x_{2},...,x_{ih}\right)\), and the velocity is \(V_{ih}\left(t\right)=\left(v_{i1},v_{i2},...,v_{ih}\right)\). The best position a particle has experienced is \(P_{best}=\left(p_{i1},p_{i2},...,p_{ih}\right)\), and the best position all the particles has experienced is \(G_{best}=\left(g_{i1},g_{i2},...,g_{ih}\right)\). During each iteration, each particle updates its velocity and position according to the following equations.  
\[V_{ih}\left(t+1\right)=\omega\cdot V_{ih}\left(t\right)+c_{1}\cdot rand_{1} \cdot\left(P_{best}-X_{ih}\left(t\right)\right)+c_{2}\cdot rand_{2}\cdot \left(G_{best}-X_{ih}\left(t\right)\right) \tag{18}\]  
\[X_{ih}\left(t+1\right)=X_{ih}\left(t\right)+V_{ih}\left(t+1\right),1\leq i\leq n,1\leq h\leq H \tag{19}\]  
where \(\omega,0<\omega<1\) is the inertia weight, \(c_{1},c_{2},0\leq c_{1},c_{2}\leq 2\) are the acceleration coefficients and \(rand_{1},rand_{2},0\leq rand_{1},rand_{2}\leq 2\) are random values. To achieve better optimization, this paper make an improvement for inertia weight.  
In PSO, the inertia weight \(\omega\) play a vital role in adjusting velocity by considering how the former velocity affect current velocity. To adapt to actual situations, it is necessary to adjust the pace of searching for best solution by improving inertia weight. The bigger inertia weight \(\omega\) benefits a global search, and the smaller one benefits a local search. The traditional linear decrement inertia weight is as follows.  
\[\omega=\omega_{\max}-\frac{\omega_{\max}-\omega_{\min}}{T_{\max}}\times t \tag{20}\]  
where \(T_{\max}\) is the maximal number of iteration, and \(t\) is the current number of iteration. \(\omega_{\max}\) is the maximal inertia weight, and \(\omega_{\min}\) is the minimal inertia weight. Although the inertia weight can effectively avoid the fluctuation of global best position, it cannot be guaranteed to avoid being caught in local optima with the number of iteration increasing. To solve the problem of partial best, it is necessary to balance the global and partial search. The inertia weight in this paper is defined by the following equation.  
\[\omega\left(k\right)=\begin{cases}\omega_{\max}-\frac{\omega_{\max}-\frac{1}{k }\omega_{\max}}{\sharp{it_{\max}}}\times k,k\leq\frac{2}{\sharp{it_{\max}}}\\ \left(\frac{1}{2}\omega_{\max}-\omega_{\min}\right)\times e^{-\beta\cdot\left( k-\frac{2}{\sharp{it_{\max}}}\right)}+\omega_{\min},k>\frac{2}{\sharp{it_{\max}}} \end{cases} \tag{21}\]  
where the inertia weight is a non-linear dynamic adaptive value. \({it_{\max}}\) is the maximal number of iteration, and \(k\) is the current number of iteration. \(\omega_{\max}\) is the maximal interation weight, and \(\omega_{\min}\) is the minimal weight. When the number of iteration \(k\) is in the interval \(\left[0,\frac{2}{\sharp{it_{\max}}}\right]\), the linear decreasing of the inertia weight benefit to getting close to global optima and avoid local optima. After \(k\) is over \(\left[0,\frac{2}{\sharp{it_{\max}}}\right]\), the index decrease of inertia weight benifit to accurate search and getting close to optimal area quickly.