## 2 Particle Swarm Optimization with Sigmoid Increasing Inertia Weight Algorithm
### Theory of SIPSO  
The PSO algorithm shows strong global optimization ability and poor performance for local optimization due to the fact that inertia weight \(\omega\) directly affects the velocity and the position of the particles. Adjusting \(\omega\) can effectively adjust the optimization ability of the PSO algorithm; therefore, it is necessary to select the appropriate value or function for the calculation of the inertia weight according to the situation. Two kinds of inertia weight can be selected according to the whole optimization process. The first one is the fixed inertia weight, which keeps the exploration and development ability of the flight particles consistent. The other is the time-varying inertia weight, which changes within a certain range, which means that the particles have different exploration and development abilities at different times.  
The PSO algorithm is able to search globally first and then locally. Due to the global search in the early stage being random and essentially non-directional, the local search in the later stage is based on the found global solution, which makes it easy to form the common local optimal solution by using the strategy of the increasing curve and taking advantage of the parallel characteristics of individual and global extremum of the particle swarm. In essence, the slow increase in inertia weight in the early stage can ensure sufficient local search ability of the particles of the algorithm. With the increase in inertia weight in the later stage, the global search ability of the particles will increase. At the moment, there will be some beneficial perturbations due to the existence of individual and global extremum--this helps the particles escape from the trap of the local optimum in the later stage of the algorithm. For this reason, an improved algorithm is proposed based on the following two aspects:  
1. Each particle still maintains its extremum at the end of the algorithm. In other words, the new algorithm not only finds the optimal solution but also obtains several better sub-optimal solutions.
2. The unique memory is insensitive to the population size. Even if the population number decreases, the performance will not decrease obviously.  
In this study, the improved sigmoid increasing inertia weight can be defined as,  
\[\omega=\omega_{min}+\frac{\omega_{max}-\omega_{min}}{1+e^{a-b\times\frac{iter }{iter_{max}}}} \tag{5}\]  
where \(a\) and \(b\) are parameters for adjustment, which are carefully chosen through numerical simulations. The improved algorithm results in the inertia weight showing an increasing sigmoid trend in the whole iterative process.  
To verify the new algorithm, two PSO-based methods are introduced. The first one is the sigmoid-function-based new particle swarm optimization algorithm (SNPSO) with increasing inertia weight proposed by [PERSON] et al. (2018), and the second one is the sigmoid function-based adaptive particle swarm optimization algorithm (SAPSO) with decreasing inertia weight proposed by [PERSON] et al. [51] The inertia weight formula of the SNPSO algorithm is,  
\[\omega=\omega_{min}+\frac{\omega_{max}-\omega_{min}}{1+e^{u\times(iter-0.75\times iter _{max})}} \tag{6}\]  
where \(u=10^{(\log{(iter_{max})-2)}}\).  
The inertia weight formula of the SAPSO algorithm is,  
\[\omega=\omega_{max}-\frac{\omega_{max}-\omega_{min}}{1+e^{3.4-0.7 iter}} \tag{7}\]