## 2 Integration of Ordinal Optimization with Ant Lion Optimizer
### Informed Ant Lion Optimization
The number of stages, denoted as \(n_{s}\), is obtained by

\[n_{s}=\arg\biggl{\{}\min_{n_{s}}(L_{0}\times e^{n_{s}-1}\leq L_{a}<L_{0}\times e ^{n_{s}},1\leq N/e^{n_{s}-1}<N_{\text{min}})\biggr{\}} \tag{14}\]

where \(L_{a}\) denotes the amount of replications in precise estimation, and \(N_{\text{min}}\) denotes the specified minimum size of selected subset. Equation (14) determines \(n_{s}\) by satisfying at least one of the following two conditions: (i) the value of \(L_{n_{s}}\) in the last stage exceeds the value of precise estimation, i.e., \(L_{n_{s}}>L_{a}\), and (ii) the size of selected subset in the last stage is smaller than the specified minimum size, i.e., \(N_{n_{s}}<N_{\text{min}}\). When the value of \(n_{s}\) is obtained, a simulation with \(L_{i}=e^{i}\times L_{0}\) replications is adopted to calculate \(\overline{f}(\mathbf{x})\) in the \(i\)-th stage. Next, these \(N/e^{i-1}\) critical designs are ranked based on the value of \(\overline{f}(\mathbf{x})\) and the former \(N/e^{i}\) critical designs are selected into the selected subset for the (\(i+1\))-th stage. A simulation with \(L_{a}\) replications is adopted to calculate \(\overline{f}_{a}(\mathbf{x})\) of all \(N_{n_{s}}\) critical designs in the last stage. The critical design with the smallest \(\overline{f}_{a}(\mathbf{x})\) in the last stage is the quasi-optimal design. ### The OALO Algorithm

The OALO algorithm can be expressed by the flow diagram as shown in Figure 2. Described below are the step-wise procedures of the OALO Algorithm 3. ```
Step 0: Set the values of \(M,\overline{V},w_{\text{min}},w_{\text{max}},\;a_{\text{min}},\;a_{\text{max}}, \;k_{\text{max}},\;N,\;L_{0},\;L_{a}\) and \(N_{\text{min}}\) Step 1: Randomly choose \(M\)\(\mathbf{x}^{\prime}\)s from solution space and calculate \(f_{a}(\mathbf{x})\), then train the MARS off-line using these \(M\) designs. Step 2: Randomly yield \(\mathbf{x}\) ants \(\mathbf{a}^{\prime}\)s and antlions \(\mathbf{x}^{\prime}\)s as the initial population and adopt Algorithm 2. After Algorithm 2 terminates, rank all the final \(\mathbf{x}\)\(\mathbf{x}^{\prime}\)s based on their approximate fitness from the lowest to the highest and choose the prior \(N\)\(\mathbf{x}\)'s to be the \(N\) exceptional designs. Step 3. Decide the number of stages, \(n_{s}\), of the R&S procedure. Step 4. For \(i=1\) to \(n_{s}-1\), perform the simulation with replications \(L_{i}=e^{i}L_{0}\) to estimate \(\overline{f}(\mathbf{x})\) of the \(N_{i}=N/e^{i-1}\) designs in the \(i\)-th stage. Rank these \(N/e^{i-1}\) designs based on their \(\overline{f}(\mathbf{x})\) and select the former \(N/e^{i}\) designs as the critical designs for the (\(i+1\))-th stage. Step 5. Perform the simulation with replications \(L_{a}\) to calculate \(\overline{f}_{a}(\mathbf{x})\) of the \(N/e^{n_{s}-1}\) designs. The design with the smallest \(\overline{f}_{a}(\mathbf{x})\) is the quasi-optimal design. ```

**Algorithm 3**The OALO. ## 3 Optimization of a Communication System

### Problem Statement

Queueing designs play an important role in managing communication system between different components of a large-scale distributed system [38]. The optimal routing optimization in design of queueing networks is a challenging problem that arises in many real-life situations. Such questions are arising in management of computer systems and data networks. In the last few decades, there has been an upsurge of interest in the system modeling methods, such as discrete-event simulation and queueing theory. In general, the discrete-event simulation can enable the modeling of system operation control [39]. Unlike discrete-event simulation, queueing theory requires very little data and obtains the relatively simple formula to predict various performance measures. In designing queueing systems, it is important to have a good balance between service to messages and economic considerations. Queues happen when there are limited resources for providing a service. In

Figure 2: Flow diagram of the proposed OALO algorithm.