## 3 PSO Based on ABC Search (F-WAPSO)
### WAPSO Based on ABC  
WAPSO improves the ability of searching global optimal solutions and accelerates the search speed. However, at the end of iterations, the randomness of the algorithm will be weakened and it will fall into the local optimum due to the increase of its dependence on the global optimal solution. To strengthen the optimization ability of the algorithm, ABC was introduced to search WAPSO and enhance its development ability. ABC deals with the function optimization problem by simulating the actual honey collecting mechanism of bees. It regards the nectar source as a feasible solution in the solution space of the optimization problem. The position of the nectar source represents the value of the feasible solution, and its concentration represents the quality of the feasible solution. The higher the concentration of the nectar source, the better the quality of the feasible solution. ABC keeps the good feasible solution and eliminates the bad feasible solution in the continuous iterative calculation, and gradually approaches the global optimal solution in the solution space. In a standard ABC, there are three types of individual bees: the employed forager, onlooker bee, and scout bee. Scout bees do a random global search at the beginning. Employed foragers guide the colony in the optimal direction and transmit nectar source information to onlooker bees. Onlooker bees compare the information provided by multiple employed foragers and choose the place with the highest nectar concentration, so as to jump out of the local optimum [21; 22]. The operation flow of ABC follows. First step: initialization. Each scout bee randomly selects a position in the solution space. At the beginning of the problem-solving process, half of the artificial bees in the population will be initialized as scout bees in the solution space. The detailed formula is expressed in Equation (15):  
\[x_{ij}=x_{j}^{L}+r\times\left(x_{j}^{UI}-x_{j}^{L}\right) \tag{15}\]  
where \(x_{ij}\) represents the \(i\)th initial solution in the \(j\)th dimension; \(x_{j}^{L}\) and \(x_{j}^{UI}\) stand for the lower and upper bounds of the \(j\)th dimension, respectively, and \(r{\in}(0,1)\) is a random number. Second step: search phase of employed foragers. The employed foragers search for a new nectar source around the one that they have found and select a nectar source with a higher concentration than the old one. The specific calculation formula is shown in Equation (16):  
\[x_{ij}(\tau+1)=x_{ij}(\tau)+(2r-1)\times rang_{j} \tag{16}\]  
where \(x_{ij}(\tau)\) and \(x_{ij}(\tau+1)\) represent the solution of the \(i\)th employed forager at time \(\tau\) and \(\tau+1\) in the \(j\)th dimension; \(rang_{j}\) is the search range of the \(j\)th dimension. Third step: search phase of onlooker bees. The higher the nectar concentration, the more likely is the source to be selected by onlooker bees. The probability density formula is shown in Equation (17):  
\[p_{i}=\frac{fit_{i}}{\sum\limits_{i=1}^{N/2}fit_{i}} \tag{17}\]where \(p_{i}\) is the probability that the \(i\)th nectar source is selected by onlooker bees, \(N\) is the population size, and \(\hat{f}t_{i}\) is the fitness of the \(i\)th nectar source. WAPSO particles in the hybrid algorithm are regarded as employed foragers in ABC, and onlooker bees conduct a greedy search around them. Onlooker bees are set to search around each particle as shown in Equation (18):  
\[rang_{j}=\frac{x_{j}^{U}-x_{j}^{L}}{10},j=1,\ 2,\ \ldots,\ D \tag{18}\]  
where \(x_{j}^{L}\) and \(x_{j}^{U}\) represent the lower and upper bounds of the decision variable, respectively, in the \(j\)th dimension, and \(D\) represents the dimension [23; 24]. In the continuous searching process of ABC, particles always develop toward a better fitness function value, and the original solution is replaced only when a better solution is found. The hybrid algorithm first conducts a WAPSO search, then introduces ABC search and the particle with the best fitness function value is selected as the global optimal solution.