## 2 Integration of Ordinal Optimization with Ant Lion Optimizer
### Informed Ant Lion Optimization
The pseudo-code of RALO is represented in Algorithm 1. ```
Input: Amount of ants and antlions, range of two control parameters and maximum number of iterations (\(k_{\max}\)). Output: The best antilon. Initialize the positions of all ants \(a_{i}^{0}\) and antlions \(\mathbf{x}_{i}^{0}\) inside \(\mathbf{V}\) and \(\mathbf{U}\) bounds. Evaluate the fitness of all antlions by MARS. Determine the elite antlion \(\mathbf{x}^{*}\). while\(k\leq k_{\max}\)do for an ant \(\mathbf{a}_{i}^{k}\)do  Choose an antlon \(\mathbf{x}_{i}^{k}\) using the roulette wheel. Random walk of \(\mathbf{a}_{i}^{k}\) nearby antlon \(\mathbf{x}_{i}^{k}\) to generate \(\mathbf{R}_{S}^{k+1}\)  Random walk of \(\mathbf{a}_{i}^{k}\) nearby elite antlon \(\mathbf{x}^{*}\) to generate \(\mathbf{R}_{E}^{k+1}\)  Update the minimum and maximum of each design variable. Update the position of \(\mathbf{a}_{i}^{k+1}\) based on \(\mathbf{R}_{S}^{k+1}\) and \(\mathbf{R}_{E}^{k+1}\) End for  Evaluate the fitness of all ants and antlions by MARS. Replace an antlon with it corresponding ant. Update sliding factor and composition factor. Update the elite antlon. End while
```  
**Algorithm 1**Pseudo-code of the R ALO algorithm. The steps involved in the RALO Algorithm 2 are briefly explained below. ```
Step 1: Configuring basic parameters (a)  Setting the values of \(\Psi\), \(w_{\min}w_{\max}\), \(\alpha_{\min}\), \(\alpha_{\max}\) and \(k_{\max}\). Set \(u_{j}^{0}=U_{j}\), \(v_{j}^{0}=V_{j}\), \(j=1,\ldots,m\). Let \(k=0\), where \(k\) indicates the iteration index. Step 2: Initializing a population (a)  A population containing \(\Psi\) ants and \(\Psi\) antlions are generated. \[a_{i,j}^{0}=V_{j}+\left\lfloor rand[0,1]\times(U_{j}-V_{j})\right\rfloor,\ i=1,\ldots,\Psi,\ j=1,\ldots,m\] (5) \[x_{i,j}^{0}=V_{j}+\left\lfloor rand[0,1]\times(U_{j}-V_{j})\right\rfloor,\ i=1,\ldots,\Psi,\ j=1,\ldots,m\] (6) where \(rand[0,1]\) indicates a random number in range from \(0\) to \(1\), and \(V_{j}\) and \(U_{j}\) express the lower and upper bound, respectively. Calculate the fitness \(F(\mathbf{x}_{i}^{0})\) of antlion assisted by MARS, \(i=1,\ldots,\Psi\). Step 3: Ranking Rank the \(\Psi\) antlions based on the fitness from the smallest to the largest, and determine the elite antlon \(\mathbf{x}^{*}=[x_{i}^{*},\ldots,x_{m}^{*}]\). Step 4: Random walk of ants Generate new position of each ant. \[a_{i,j}^{k+1}=\frac{\left(a_{i,j}^{k}-\min\text{R}_{i,j}\right)\times(u_{j}^{ k}-v_{j}^{k})}{\max\text{R}_{i,j}-\min\text{R}_{i,j}}+v_{j}^{k},\ i=1,\ldots,\Psi,\ j=1,\ldots,m\] (7) where \(\text{R}_{i,j}\)\(=[0,cusum(bintrad^{1}),\ldots,cusum(bintrad^{k}),\ldots,cusum(bintrad^{k_{\max}})]\), \(cusum\) refers to the cumulative sum, \(bintrad^{k}\) denotes the random number at iteration \(k\), either \(1\) or \(-1\), \(\max\text{R}_{i,j}\) and \(\min\text{R}_{i,j}\) denote the minimum and maximum random walk of the \(j\)-th variable for the \(i\)-th ant, respectively, \(u_{j}^{k}\) and \(v_{j}^{k}\) express the minimum and maximum of the \(j\)-th variable at iteration \(k\). ```  
**Algorithm 2**The RALO  
The steps involved in the RALO Algorithm 2 are briefly explained below. ```
Step 2: Initializing a population (a)  A population containing \(\Psi\) ants and \(\Psi\) antlions \(\mathbf{x}_{i



**Step 5**: Slide ants in a trap

(a) Update the minimum and maximum of the \(j\)-th variable. \[v_{j}^{k+1}=\left\{ \begin{array}{l}x_{j}^{k}+\frac{v_{j}^{k}}{10^{k+1}(k_{ \max})},rand[0,1]>0.5\\ x_{j}^{k}-\frac{v_{j}^{k}}{10^{k+1}(k_{\max})},rand[0,1]\leq 0.5\end{array}  \right.,\ j=1,\ldots,m \tag{8}\]

\[u_{j}^{k+1}=\left\{ \begin{array}{l}x_{j}^{k}+\frac{u_{j}^{k}}{10^{k+1}(k_{ \max})},rand[0,1]>0.5\\ x_{j}^{k}-\frac{u_{j}^{k}}{10^{k+1}(k_{\max})},rand[0,1]\leq 0.5\end{array}