## 3 Development of Hybrid RF Models for Predicting the Rock EM
### Metaheuristic Optimization Algorithms
#### 3.1.2 Multi-Verse Optimizer (MVO)  
The multi-verse optimizer (MVO) is a MOA-based physics algorithm proposed by [PERSON] et al. [54], which is inspired by the idea that the universe moves from white holes to black holes through wormholes to achieve a stable situation. The white holes are believed to be an important part of the original universe, the black holes have an irresistible pull on everything including light beams, and the wormhole is a bridge or passage connecting different universes. In the MVO algorithm, the birth of the universe is always related to the objects transfer. The universe with a low inflation rate is more likely to take in more objects. The optimization process of MVO can be described as follows:  
1. Population--the initial population of the universes in the searching space is defined using the Equation (6). \[U=\begin{bmatrix}u_{1}^{1}&u_{1}^{2}&\cdots&u_{1}^{d}\\ u_{2}^{2}&u_{2}^{2}&\cdots&u_{2}^{d}\\ \vdots&\vdots&\vdots&\vdots\\ u_{n}^{1}&u_{n}^{2}&\cdots&u_{n}^{d}\end{bmatrix}\] (6) where \(u_{n}^{d}\) indicates the parameter of the \(n\)-th universe in the \(d\)-dimension searching space.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Variables**} & \multicolumn{6}{c}{**Statistical Information**} \\ \cline{2-7}  & **Sign** & **Unit** & **Min** & **Max** & **Mean** & **St. D** \\ \hline Point load index & PLI & MPa & 0.890 & 12.530 & 4.365 & 2.839 \\ Porosity & Pn & \% & 0.100 & 10.270 & 1.957 & 3.047 \\ P-wave velocity & V\({}_{\rm p}\) & km/s & 2.823 & 7.943 & 5.575 & 0.892 \\ Schmidt hammer rebound number & SHRN & / & 25.630 & 72.000 & 47.093 & 13.795 \\ Elasticity modulus & EM & GPa & 3.050 & 183.300 & 60.139 & 44.832 \\ \hline \hline \end{tabular} Note: Min—minimum value; Max—maximum value; St. D—standard deviation value.  
\end{table}
Table 1: Detailed description of the input and output variables.  
2. Exploration and exploitation--the function of wormholes is to help objects move from one universe to another (see Figure 2). Thus, this mechanism by which objects are exchanged between universes through wormholes can be described as: \[u_{j}=\left\{\begin{array}{ll}u_{j}+TDR\cdot((ub-lb)\cdot r_{3}+lb),&r_{2}<0.5 \text{ and }r_{1}<WEP\\ u_{j}+TDR\cdot((ub-lb)\cdot r_{3}+lb),&r_{2}>0.5\text{ and }r_{1}<WEP\\ u_{j},&r_{1}>WEP\end{array}\right.\] (7) where \(u_{j}\) represents the \(j\)-th parameter of the best universe. \(ub\) and \(lb\) are the lower and upper bounds of the multi-universes space, respectively. \(r_{1}\), \(r_{2}\) and \(r_{3}\) indicate three random numbers within the range of [0, 1]. \(TDR\) and \(WEP\) represent two coefficients, the former is the wormhole existence probability, and the latter is named the travelling-distance rate. These coefficients can be calculated using the Equations (8) and (9). \[WEP=\min+t\cdot(\frac{\max-\min}{T})\] (8) \[TDR=1-\frac{t^{1/\varepsilon}}{T^{1/\varepsilon}}\] (9) where \(\min\) and \(\max\) represent the minimum maximum values, respectively. \(t\) and \(T\) indicate the current iteration and the maximum iteration, respectively. \(\varepsilon\) is the exploitation accuracy.