## 3 MVO Algorithm and Its Application to Solve the RNP-WMN Problem  
### MVO Algorithm  
[PERSON] et al. proposed the Multi-Verse Optimizer Algorithm (MVO), a nature-inspired algorithm [24]. The MVO method is founded on three cosmological ideas: white holes, black holes, and worm holes. It employs exploration, exploitation, and local search to find the best answer among a large number of candidate solutions. The multi-verse theory, which emerged after the Big Bang theory, inspired MVO. The Big Bang theory states that there was a massive explosion that resulted in the existence of the universe we live in. According to the multi-verse theory, there were multiple big bangs, each of which resulted in the existence of a different universe. As a result, multi-verse believes that there are other universes besides the one we live in. Furthermore, according to the multi-verse theory, these universes can interact and collide with one another, and each universe has its own set of properties. MVO was primarily inspired by three concepts from multi-verse theory: white holes, black holes, and wormholes. White holes have never been observed, but physicists believe they can be explained by the big bang or collisions between parallel universes. Black holes attract everything due to their strong gravitational pull. Finally, wormholes connect all parts of the universe and serve as a time/space tunnel through which objects can travel. The original MVO paper mentioned that every universe has an inflation rate that causes space to expand. MVO is a population-based algorithm that searches in two stages: exploration and exploitation. The MVO algorithm considers solutions to be universes and variables within a solution to be variables within a universe, where \(d\) represents the number of objects, \(n\) represents the number of universes, \(X_{j}^{best}\) is the \(j\)-th parameter of the best universe formed so far, the TDR factor is calculated by Equation (10), the \(WEP\) factor is calculated by Equation (9), \(lb_{j}\) shows the lower bound of the \(j\)-th variable, \(ub_{j}\) is the upper bound of the \(j\)-th variable, \(x_{i}^{j}\) indicates the \(j\)-th parameter of the \(i\)-th universe, and \(rand_{2},rand_{3},rand_{4}\) are random numbers in \([0,1]\). Equations (11) and (12) are used to update the universes. In addition, each universe is given an inflation rate (\(U_{i}\)) based on the fitness function value for that universe, \(NI(U_{i})\) is the normalized inflation rate of the ith universe. The MVO algorithm is demonstrated in Algorithm 1. \[WEP=min+t\times(\frac{max-min}{Max\_iter}) \tag{9}\]  
\[TDR=1-(\frac{t}{Max\_iter})^{\frac{1}{p}} \tag{10}\]  
\[x_{i}^{j}(t+1)=\left\{ \begin{array}{ll}x_{k}^{j}(t),&rand_{1}<NI(U_{i})\\ x_{i}^{j}(t),&rand_{1}\geq NI(U_{i})\end{array} \right. \tag{11}\]  
\[x_{i}^{j}=\left\{ \begin{array}{ll}X_{j}^{best}+TDR\times((ub_{j}-lb_{j}) \times rand_{4}+lb_{j}),&rand_{3}<0.5\\ X_{j}^{best}-TDR\times((ub_{j}-lb_{j})\times rand_{4}+lb_{j}),&otherwise\end{array} \right. \tag{12}\]```
Input: Initial parameters Output: Optimal solution Initialize universe population considering upper bound and lower bound \(t=0\) while\(t<=Max\_iter\)do  Calculate the fitness of each universe in the population \(Sorted\_U\) is assigned by sorted universes  Update best universe \(X^{best}\) forach universe \(i\)do  Update \(WEP\) and \(TDR\) by Equations (9) and (10), respectively.