## 3 Methodology
### Selection of Replacement Plan Using Hybrid Chaotic Whale Optimization Algorithm
#### 3.2.3 Hybrid Chaotic Whale Optimization Algorithm  
In accordance with the established mathematical model, the hybrid chaotic whale optimization algorithm is introduced to solve the objective function \(\min L\), where both the modified comparison matrix and hierarchical weights will be output automatically. 1. Standard whale optimization algorithm. Derived from the bubble-net hunting behavior of humpback whales (Figure 5), the standard whale optimization algorithm was proposed as a meta-heuristic algorithm to handle optimization problems in 2016, which contains 3 principal operations as follows:  
* Encircling prey. The humpback whales are capable of recognizing the location of prey, which indicates the solution space in WOA, and encircle it. Additionally, the current best candidate solution is assumed to be the target prey or close to the optimum, which motivates other search agents to update their positions towards the best one. This behavior is simulated by Equation (12) as follows:  
\[ \begin{array}{ll}&\overset{\rightarrow}{D}=\left|\overset{\rightarrow}{ C}\cdot\overset{\rightarrow}{X^{*}}(t)-\overset{\rightarrow}{X}(t)\right|\\ &\overset{\rightarrow}{X}(t+1)=\overset{\rightarrow}{X^{*}}(t)-\overset{ \rightarrow}{A}\cdot\overset{\rightarrow}{D}\end{array}  \tag{12}\]  
where \(t\) denotes the current iteration, and \(\overset{\rightarrow}{D}\) and \(\overset{\rightarrow}{X}\) are position vectors. In particular, \(\overset{\rightarrow}{X^{*}}\) is the position vector of the best solution acquired so far which should be updated in each iteration if a better solution occurs. \(\overset{\rightarrow}{A}\) and \(\overset{\rightarrow}{C}\) are the coefficient vectors, which can be calculated as follows:  
\[ \begin{array}{ll}&\overset{\rightarrow}{A}=2\overset{\rightarrow}{a} \cdot\overset{\rightarrow}{r}-\overset{\rightarrow}{a}\\ &\overset{\rightarrow}{C}=2\cdot\overset{\rightarrow}{r}\end{array}  \tag{13}\]  
where \(\overset{\rightarrow}{a}\) is decreased from 2 to 0 linearly over the course of iterations in both exploration and exploitation phases, and \(\overset{\rightarrow}{r}\) is a random vector in \([0,1]\). * Bubble-net attacking (exploitation phase). In order to model the bubble-net attacking behavior of humpback whales mathematically, the shrinking encircling mechanism and spiral updating position strategy are introduced, where the mechanism is realized by setting random values for \(\overset{\rightarrow}{A}\) in \([-1,1]\), and the spiral updating position strategy is described, as Equation (15) demonstrates:  
\[ \begin{array}{l}\overset{\rightarrow}{X}(t+1)=\overset{\rightarrow}{D^{ \prime}}.\ e^{bl}\cdot\cos(2\pi l)+\overset{\rightarrow}{X^{*}}(t)\\ \overset{\rightarrow}{D^{\prime}}=\left|\overset{\rightarrow}{X^{*}}(t)- \overset{\rightarrow}{X}(t)\right|\end{array}  \tag{14}\]  
where \(\overset{\rightarrow}{D^{\prime}}\) indicates the distance of the whale to the prey, and \(b\) is a constant for defining the logarithmic spiral shapes, random number \(l\in[-1,1]\). It is assumed that there is a probability of 50% to switch between either the encircling prey mode or the spiral model to update the position of whales, which can be mathematically expressed as follows:  
\[ \begin{array}{l}\overset{\rightarrow}{X}(t+1)=\overset{\rightarrow}{X^{*} }(t)-\overset{\rightarrow}{A}\cdot\overset{\rightarrow}{D}&ifp<0.5\\ \overset{\rightarrow}{X}(t+1)=\overset{\rightarrow}{D^{\prime}}.\ e^{bl} \cdot\cos(2\pi l)+\overset{\rightarrow}{X^{*}}(t)&ifp\geq 0.5\end{array}  \tag{15}\]  
where random number \(p\in[0,1]\). * Search for prey (exploration phase). The humpback whales search randomly according to the position of each other, which is simulated through the similar approach based on the variation of \(\overset{\rightarrow}{A}\). The exploration phase and the exploitation phase differ in that the position of search agent is updated by a randomly chosen search agent rather than by the best search agent at present.