## 4 Optimization Method
### Improved Scheduling Strategy
#### 4.1.5 Improved Artificial Bee Colony Algorithm
\begin{array}{cc}1/(1+f_{i})&f_{i}>0\\ 1+\big{|}f_{i}\big{|}&\textit{otherwise}\end{array} \right. \tag{28}\]where \(fit_{i}\) represents the fitness value of the \(i\)th nectar source; \(f_{i}\) represents the value of the objective function of the \(i\)th solution. The greedy algorithm (29) is used to compare the fitnessvalues of new and old honey sources and update the honey sources. \[x_{i}=\left\{ \begin{array}{ll}u_{i}&fit(x_{i})<fit(u_{i})\\ x_{i}&otherwise\end{array} \right. \tag{29}\]  
(c) The Observer Bee Stage of the Dynamic Adjustment of \(P_{i}\)  
\(P_{i}\) is the selection probability of each honey source in the typical ABC algorithm. After the hired bees update the honey source, they fly back to the information exchange area to share the honey source.The observer bees calculated the \(P_{i}\) according to the honey source information shared by the hired bees.The calculated \(P_{i}\) is used to perform roulette to select the honey source. According to Equation (24), the observer bees update the nectar source near the selected nectar source and determine the nectar amount of the new nectar source.If the \(fit\) of the new solution is better than the previous one, the observer bee will remember the new solution instead of the old one. \[P_{i}=fit_{i}/\sum_{i=1}^{N}fit_{i} \tag{30}\]  
\[P_{i}^{d}=\left[(P_{max}-P_{0})\frac{t^{2}}{t_{max}^{2}}+P_{0}\right]P_{i} \tag{31}\]  
The \(P_{i}\) formula of the typical ABC algorithm is shown in Equation (30). The higher the fitness value of a honey source, the greater the probability that a better honey source will be selected. However, in the later stage of the algorithm, there are already many dominant individuals in the population. If the population is selected with fixed \(P_{i}\), the dominant individuals may be destroyed and the convergence rate will be slowed down. In this paper, \(P_{i}\) is improved into a dynamic function with the number of iterations as the independent variable, as shown in Equation (31), where \(P_{i}^{d}\) is the selection probability of each honey source in the IABC algorithm. \(P_{0}\) and \(P_{max}\) are the minimum and maximum selection probabilities, respectively. In the initial stage of the algorithm, \(P_{i}^{d}\) is close to \(P_{i}\). With the progress of the algorithm, \(P_{i}^{d}\) increases slightly, and in the later stage, dominant individuals can be better highlighted. The convergence rate of the algorithm is thus accelerated. (d) Detection Bee Stage  
When the mining times of the honey source reaches the _limit_, the fitness value of a certain honey source remains unchanged. This means that the honey source mining is trapped into a local optimum. Then, the honey source should be abandoned, and the hired bee changes to the detection bee. A new honey source location is generated according to Equation (23). The specific steps of the improved artificial bee colony algorithm are as follows:  
1. Initialize population: initializing all parameters, e.g., the total number of bees \(NP\), the maximum number of iterations \(t_{max}\), the control parameter _limit_, lower (\(l_{d}\)) and upper (\(u_{d}\)) bounds of the search space; and randomly generating initial solution {\(x_{i}\)\(i\) = 1, 2..., \(NP\)};
2. Calculate the adaptive value of each bee in the population;
3. Set the parameters of the whale search strategy: \(a\), \(b\), \(l\), \(p\). The hired bee generates a new solution \(u_{i,d}\), according to Equation (27), and calculate the fitness value;
4. The hired bee selects the nectar source according to the greedy strategy;
5. Calculate the selection probability \(P_{i}\) according to Equations (30) and (31);
6. The observer bees select the honey source according to the probability \(P_{i}\), and generate a new one near the honey source according to Equation (24). Meanwhile, the fitness value of the new honey source is calculated. Finally, the source of honey is selected using the greedy algorithm;* Determine whether detection bees exist.