## 2 Integration of Ordinal Optimization with Ant Lion Optimizer
### Informed Ant Lion Optimization  
With the aid of the MARS approximation model, existing search approaches can be adopted to determine \(N\) exceptional designs from the solution space. ALO is a biologically approach inspired by the hunting behavior of antlions and ants getting trapped in the trap set by the antlion [7]. The ALO has a high exploration capability with the help of random walk and roulette wheel to build traps. The time-varying boundary shrinking mechanism and elitism are used to increase exploitation efficiency of the ALO. There are many advantages of ALO, such as avoidance of local optima, ease of implementation, reduced need for parameter adjustment, and high precision. ALO has been successfully employed to solve the multi-robot path planning problem with obstacles [34], prediction of soil shear strength [35], and structural damage assessment [36]. [PERSON] et al. presented a comprehensive literature review on well-established researches of ALO from 2015 to 2018 [37]. The comparative results revealed the dominance of ALO over other SI approaches including artificial bee colony (ABC), firefly algorithm (FA), ant colony optimization (ACO), cuckoo search (CS), bat algorithm (BA), and biogeography-based optimization (BBO). With the help of random walks and roulette wheel for building traps, ALO has a high exploration capability. The shrinking of trap boundaries and elitism provide the ALO with a high exploitation efficiency. These merits make ALO to avoid immature convergence shortcomings. Accordingly, it is particularly well suited to meet the requirements in global exploration. However, there are some issues of the original ALO, such as the local optima stagnation and occurrence of premature convergence for some problems. Therefore, the RALO is designed to enhance the convergent speed of the original ALO. The RALO has two self-adaptive control parameters, which are sliding factor and composition factor. The sliding factor determines the ants which are shifted toward the antlion using a given rate of slippage. The RALO uses a small value of sliding factor to increase diversification. When the sliding factor is small, the RALO intends to perform the global search. On the contrary, the RALO uses a large value of sliding factor to improve intensification. As the sliding factor is increased, the RALO intends to carry out the local search around the local optimum. The composition factor determines the degree of the elite antlion which affects the movements of all ants. The balance between exploitation and exploration of the RALO is mainly controlled by the composition factor. Large value of composition factor generates new positions of ants far from the elite antlion which will result in high exploration ability. Therefore, a large value of composition factor intends to perform exploration, while a small value leads to perform exploitation. There are six steps of hunting prey in ALO: (i) using roulette wheel to construct antlion's traps, (ii) random walk of ants, (iii) enter the ants to traps, (iv) adaptive shrinking boundaries of antlion's trap, (v) catching ants and re-building traps, and (vi) performing elitism. The notations shown below are used in RALO. \(\Psi\) indicates the amount of ants and antlions, \(k_{\text{max}}\) is the maximum number of iterations. The position of the \(i\)th ant at iteration \(k\) is denoted by \(\mathbf{x}_{i}^{k}=[a_{i,1}^{k},\dots,a_{i,m}^{k}]\). The position of the \(i\)th antlion at iteration \(k\) is denoted by \(\mathbf{x}_{i}^{k}=[x_{i,1}^{k},\dots,x_{i,m}^{k}]\). The position of the elite antlion is denoted by \(\mathbf{x}^{*}=[x_{1}^{*},\dots,x_{m}^{*}]\). \(w^{k}\) is a sliding factor at iteration \(k\), \(w^{k}\in[w_{\text{min}},w_{\text{max}}]\), where \(w_{\text{min}}\) and \(w_{\text{max}}\) indicate the minimum and maximum value, respectively. \(\alpha^{k}\) is a composition factor at iteration \(k\), \(\alpha^{k}\in[\alpha_{\min},\alpha_{\max}]\), where \(\alpha_{\min}\) and \(\alpha_{\max}\) represent the minimum and maximum value, respectively.