## 2 Integration of Ordinal Optimization with Ant Lion Optimizer
### Informed Ant Lion Optimization
 \right.,\ j=1,\ldots,m \tag{9}\]

where \(w^{k}\) denotes the sliding factor at iteration \(k\), \(u_{j}^{k}\) and \(v_{j}^{k}\) express the minimum and maximum of the \(j\)-th variable at iteration \(k\), \(x_{j}^{k}\) is the \(j\)-th antilon position at iteration \(k\), which can be either antilon \(xs_{j}^{k}\) selected by the roulette wheel or the elite antilon \(x_{j}^{*}\) determined by the following equation. \[x_{j}^{k}=\left\{ \begin{array}{l}xs_{j}^{k},rand[0,1]>0.5\\ x_{j}^{k},rand[0,1]\leq 0.5\end{array} \right.,\ j=1,\ldots,m \tag{10}\]

(b) Update positions of all ants. \[\mathbf{a}_{i}^{k+1}=a^{k}\times\mathbf{R}_{S}^{k+1}+\left(1-a^{k}\right) \times\mathbf{R}_{E}^{k+1},\ i=1,\ldots,\Psi \tag{11}\]

where \(a^{k}\) denotes the composition factor at iteration \(k\), \(\mathbf{R}_{S}^{k+1}\) denotes the random walk around an antilon which is selected by the roulette wheel at iteration \(k+1\), \(\mathbf{R}_{E}^{k+1}\) denotes the random walk around the elite antilon at iteration \(k+1\). **Step 6**: Calculate the fitness Calculate the fitness \(F(\mathbf{a}_{i}^{k+1})\) of ant \(\mathbf{a}_{i}^{k+1}\) and the fitness \(F(\mathbf{x}_{i}^{k+1})\) of antilon \(\mathbf{x}_{i}^{k+1}\) assisted by MARS, \(i=1,\ldots,\Psi\). **Step 7**: Replace an antilon with it corresponding ant Apply the greedy selection between \(\mathbf{a}_{i}^{k+1}\) and \(\mathbf{x}_{i}^{k+1}\). If \(F(\mathbf{a}_{i}^{k+1})<F(\mathbf{x}_{i}^{k+1})\), then set \(\mathbf{x}_{i}^{k+1}=\mathbf{a}_{i}^{k+1}\), \(i=1,\ldots,\Psi\). **Step 8**: Update sliding factor and composition factor. \[w^{k+1}=w_{\min}+(w_{\max}-w_{\min})\times\left(1-\exp\!\left(-\frac{w_{\max }}{w_{\min}}\times\frac{k+1}{k_{\max}}\right)\right) \tag{12}\]

\[a^{k+1}=a_{\min}+(a_{\max}-a_{\min})\times\exp\!\left(\ln\!\left(\frac{a_{\min }}{a_{\max}}\right)^{2}\times\frac{k+1}{k_{\max}}\right) \tag{13}\]
**Step 9**: Elitism Apply the greedy selection between \(\mathbf{x}_{i}^{k+1}\) and \(\mathbf{x}^{*}\). If \(F(\mathbf{x}_{i}^{k+1})<F(\mathbf{x}^{*})\), then set \(\mathbf{x}^{*}=\mathbf{x}_{i}^{k+1}\), \(i=1,\ldots,\Psi\). **Step 10**: Stop criteria If \(k\geq k_{\max}\), then stop; else, let \(k=k+1\) and return to Step 4. **Algorithm 2**_cont._

The RALO terminates when it reaches a specified maximum number of iterations \(k_{\max}\). When the RALO has stopped, the final \(\Psi\) antlions are ranked according to the fitness. Then, the former \(N\) antlions are selected as the exceptional designs. ### Ranking and Selection

We continue to determine the quasi-optimal design from the \(N\) exceptional designs using the R&S procedure. The main idea of the R&S procedure is spending more computing efforts on few critical designs and less on most non-critical designs. The proposed R&S procedure composes of multiple stages, which select and allocate the most computational budget to critical designs that has a high probability to be the quasi-optimal design. The number of critical designs in each stage is decreased gradually. Remaining designs are continuing to perform simulation and some of them are eliminated in each stage, andthe best one obtained in the last stage is the quasi-optimal design. The computational complexity can be gradually reduced, because the number of the critical designs had been greatly decreased when the evaluations are more refined. The more refined evaluations used in those stages are simulation with various numbers of replications ranging from tiny to large. First, we choose an initial amount of replications as \(L_{0}\). The amount of replications and the amount of critical designs in the \(i\)th stage are denoted as \(L_{i}\) and \(N_{i}\), respectively. We set \(L_{i}=eL_{i-1}\) (or \(L_{i}=e^{i}L_{0}\)) for \(i=1,2,\ldots\), and \(N_{i}=N_{i-1}/e\) (or \(N_{i}=N_{1}/e^{i-1}\)) for \(i=2,3,\ldots\), where \(N_{1}=N\). The Monte Carlo simulation with exponential rate provides a substantial computational speed-up without noticeable overshoot of the Probability of Correct Selection (PCS). The value of the \(N_{i}\) is rounded to the nearest integer.