## 4 Swarm Hyper-heuristic Algorithm
### Swarm Hyper-heuristic Algorithm
#### 4.1.2 Heuristic Selection  
Heuristic selection is the procedure of using one high-level strategy (HLS) to manipulate or manage a set of low-level heuristics (LLH). It is noted that the performance of the swarm hyper-heuristic algorithm depends mainly on how to balance the two basic search mechanisms of intensification and diversification. There are two issues for this problem, including how to evaluate the intensification and diversification performance after one iteration and what strategy is used to select the operator according to the intensification and diversification performance. The without learning SHH uses the strategy of giving equal probability to select each operator in each iteration, whereas the online learning SHH has the ability to evaluate probabilities of choosing which operators before or after an iteration. In this paper, online learning SSH was chosen to improve the adaptability. The intensification and diversification of the algorithm are measured by a certain, and then the selection probabilities of the centralized and diversified operations are adjusted so that the intensification and diversification are more balanced. The degree of intensification is determined by the value of the best cost function before and after one iteration. For a minimization problem, the algorithm is more intensified in this iteration if \(f(x_{b}^{t-1})>f(x_{b}^{t})\). The degree of diversification can be measured by the sum of the distances between solutions and the central solution, as shown in the following equation:  
\[d=\sum_{i=1}^{N}\|x_{i}-x_{c}\| \tag{12}\]  
where \(x_{c}=\sum_{i=1}^{N}x_{i}/N\) is the central solution. For a minimization problem, the algorithm is less diversified in this iteration if \(d^{t-1}>d^{t}\). Therefore, for a minimization problem, the relationship between intensification and diversification after one iteration is shown in Figure 5. In the iterative process, the degree of diversification and intensification of the algorithm should be maintained at a considerable level. If the trend of algorithm diversification is detected to decrease, the selection probability of diversified operator in the algorithm should be appropriately increased. \begin{table}
\begin{tabular}{c c c} \hline
**Operator Name** & **Operator Formula** & **Note to Explain** \\ \hline Random move in the neighborhood & \(x_{i}:=x_{i}+\lambda_{\min}\cdot rand\cdot u\) & \(\lambda_{\min}\) is an intensification step length \\ Following better solutions & \(x_{i}:=x_{i}+\lambda\cdot rand\cdot\left(x_{j}-x_{i}\right)\) & \(\forall x_{j},f\big{(}x_{j}\big{)}\leq f(x_{i})\) \\ Following the best solution & \(x_{i}:=x_{i}+\lambda\cdot rand\cdot\left(x_{b}-x_{i}\right)\) & \(x_{b}\) is the best solution in population \\ Following own best & \(x_{i}:=x_{i}+\lambda\cdot rand\cdot\left(x_{i}^{best}-x_{i}\right)\) & \(x_{i}^{best}\) is the best performance of the \\ Random long jump & \(x_{i}:=x_{i}+\lambda_{\max}\cdot u\) & \(\lambda_{\max}\) is a diversification step length \\ Mutation & \(x_{i}:=m(x_{i})\) & \(m\) is a variation function \\ Run away from the worst & \(x_{i}:=x_{i}+\lambda\cdot rand\cdot\left(x_{i}-x_{w}\right)\) & \(x_{w}\) is the worst solution in population \\ Run away from worse solution & \(x_{i}:=x_{i}+\lambda\cdot rand\cdot\left(x_{i}-x_{j}\right)\) & \(\forall x_{j},f\big{(}x_{j}\big{)}\geq f(x_{i})\) \\ Improving local search & \(x_{i}:=x_{i}+\lambda_{\min}\cdot rand\cdot u\) & \(u=\text{argmin}\{f(x_{i}+\lambda_{\min}\cdot rand\cdot u)\) \\  & & \(:u\in\left\{u_{1},u_{2},\ldots,u_{i},\ldots,u_{m},\overset{\rightarrow}{0} \right\}\}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Some widely and recently used operators. The change of the degree of intensification or diversification can be achieved by modifying the selected probability of each operator. Consider an increase of the degree of diversification of search behavior, and the probability modifiers for all diversified operations by \(P^{t}(o_{i})=\gamma P^{t-1}(o_{i})\).