## 3 Methodology
### Selection of Replacement Plan Using Hybrid Chaotic Whale Optimization Algorithm
#### 3.2.3 Hybrid Chaotic Whale Optimization Algorithm
To put it in practical terms, the whales will swim randomly outside the shrinking encircling area when \(\left|\overset{\rightarrow}{A}\right|>1\), which emphasizes the global search of WOA and can be described as follows:  
\[ \begin{array}{l}\overset{\rightarrow}{D}=\left|\overset{\rightarrow}{C} \cdot\overset{\rightarrow}{X}_{rand}(t)-\overset{\rightarrow}{X}(t)\right| \\ \overset{\rightarrow}{X}(t+1)=\overset{\rightarrow}{X}_{rand}(t)- \overset{\rightarrow}{A}\cdot\overset{\rightarrow}{D}\end{array}  \tag{16}\]  
where \(\overset{\rightarrow}{X}_{rand}(t)\) is a random position vector of a search agent chosen from the current population. 2. Tent chaotic mapping strategy. The random-based optimization algorithm using chaotic variables instead of random variables are defined as the chaotic optimization algorithm, which is capable of carrying out global searches at both higher speed and accuracy than stochastic searches that depend on probabilities due to the non-repetition and ergodicity of chaos. Additionally, compared with other chaotic maps, Tent chaotic map proves efficient to enhance the performance of standard WOA when the process of population initialization is conducted. Hence, the Tent chaotic mapping strategy is introduced and expressed as follows [39]:  
\[x_{k+1,d}=\left\{ \begin{array}{cc}\frac{x_{kd}}{0.7}&x_{kd}<0.7\\ \frac{10}{3}(1-x_{kd})&x_{kd}\geq 0.7\end{array} \right. \tag{17}\]  
where \(k\) denotes the search agent \(k\), \(d\) indicates the dimension \(d\) and \(x_{kd}\in[0,1]\). Specifically, a random vector is generated as the position vector of the first search agent in the solution space of optimization problem. Then, other position vectors of the rest search agents are calculated by Equation (17), where the chaotic set is formed. Ultimately, the obtained position vectors are mapped to the solution space, which can be realized by Equation (18). \[y_{kd}=lb_{d}+x_{kd}\cdot(ub_{d}-lb_{d}) \tag{18}\]where \(lb_{d}\) and \(ub_{d}\) are the lower boundary and upper boundary of the optimization problem, and \(y\) indicates the mapped position of the search agent. 3. Levy flight strategy. Confronted with high-dimensional and multi-modal optimization problems, the performance of standard WOA remains insufficient due to the dependence on randomness, which results in the risk of local optimum. Originally, the Levy flight strategy was introduced to develop the cuckoo search (CS) algorithm, which proves efficient with both the high speed and accuracy of global convergence. Therefore, in order to prevent the optimization algorithm from trapping in the local optimum and enhance the global search performance, the Levy flight strategy is conducted to be hybridized with the chaotic WOA, which is utilized to update the position of whales again at the later stage of each iteration and can be demonstrated as follows [40]:  
\[\overset{\rightarrow}{X}(t+1)=\overset{\rightarrow}{X}_{rand}(t)-\alpha \cdot sign[rand-1/2]\oplus\textit{Levy}(s) \tag{19}\]  
where \(sign[rand-1/2]\) takes only 3 values: \(-1\), 0, or 1, the product \(\oplus\) denotes the entry wise multiplication, and \(\alpha\) is the step size related to the scales of optimization problems, which can be described as follows:  
\[\alpha=0.01\left[\overset{\rightarrow}{X}_{rand}(t)-\overset{\rightarrow}{X }(t)\right] \tag{20}\]  
The _Levy_ flight strategy proves a non-Gaussian random process, where the step lengths follow the _Levy_ distribution and can be expressed as follows:  
\[\textit{Levy}(s)\sim|s|^{-1-\beta},0<\beta\leq 2 \tag{21}\]  
where \(\beta\) is an index, \(s\) indicates the random step length of _Levy_ flight, which can be obtained by [PERSON]'s algorithm [41]:  
\[s=\frac{\mu}{|v|^{1/\beta}} \tag{22}\]  
where \(\beta\) is set to be 1.5, \(\mu\) and \(v\) obey the normal distribution and can be acquired as follows:  
\[