## 2 Integration of Ordinal Optimization with Ant Lion Optimizer
### Informed Ant Lion Optimization
First, we define the values of \(L_{0}\), \(L_{a}\), \(N_{\min}\) and calculate the value of \(n_{s}\) by Equation (14). For \(i=1\) to \(n_{s}-1\), a stochastic simulation with \(L_{i}=e^{i}\times L_{0}\) replications is performed to compute \(\overline{f}(\mathbf{x})\) of the \(N_{i}=N/e^{i-1}\) critical designs. Then, we rank these \(N/e^{i-1}\) critical designs based on their \(\overline{f}(\mathbf{x})\) and select the former \(N/e^{i}\) critical designs as the selected subset for the (\(i+1\))-th stage. Finally, a stochastic simulation with \(L_{a}\) replications is performed to calculate \(\overline{f}_{a}(\mathbf{x})\) of all \(N/e^{n_{s}-1}\) critical designs in the last stage. The critical design with the smallest \(\overline{f}_{a}(\mathbf{x})\) is the quasi-optimal design. ## 4 Experimental Results

### Test Examples and Simulation Results

Two problems presented in [41] are utilized to test the performance of the OALO algorithm. The first problem is a small example with 3 networks as shown in Figure 5. The number of messages is \(n=1000\). The cost per unit time in system is \(K=\$0.005\). The processing costs per service are $0.03 for network 1, $0.01 for network 2 and $0.005 for network 3. The message interarrival times have an exponential distribution with mean \(1/\lambda=1\) time unit. The means of transit time are \(\mu_{j}=1\), 2, and 3 for networks 1, 2, and 3, respectively. The second problem is a large example with 10 networks. The number of messages is \(n=1000\). The cost per unit time in system is \(K=\$0.005\). The processing cost per service for network \(j\) is \(c_{j}=1/j,j=1,\ldots,10\). The interarrival rate of messages is \(\lambda=1\). The mean of transit time through network \(j\) is \(\mu_{j}=j,j=1,\ldots,10\). Figure 5: Queueing network design of 3 networks. In small example, the MARS was trained by randomly sampling \(M=384\) designs. The amount \(M=384\) was produced using the sample size formula for a 95% confidence level and a 5% confidence interval [42]. The number of sampling designs depends on the parameters and complexity of the trained MARS model. For MARS, the performance is expected to reach a maximum threshold no matter how much more data is included in the training. In other words, there is a point where more data may not improve the model. Accordingly, the additional samples become redundant when the value is larger than 384. The objective value of each design was estimated using precise estimation. Due to the random nature of stochastic simulation process, 30 trials (complete repetitions of the entire experiment) were carried out to check the consistency. After running multiple experimental trials with different settings, the parameter settings in RALO for small example were \(\alpha_{\text{max}}=0.8\), \(\alpha_{\text{min}}=0.2\), \(w_{\text{max}}=6\)\(w_{\text{min}}\)\(=1.5\), \(\Psi=20\) and \(k_{\text{max}}=100\). There is no existing systematic way to determine the preferable ranges of the two control parameters, \([\alpha_{\text{min}},\alpha_{\text{max}}]\), \([w_{\text{min}},w_{\text{max}}]\) and the value of \(N\). Although a proper parameter setting can obtain a good performance, however such setting is problem dependent. Thus, the preferable ranges of \([\alpha_{\text{min}},\alpha_{\text{max}}]\), \([w_{\text{min}},w_{\text{max}}]\) and the value of \(N\) were determined using a series of hand-tuning experiments after analyzing the influence on solution accuracy and convergence rate. The values of \(\alpha\) and \(w\) were dynamically adjusted to strengthen exploration in early iterations and exploitation in later iterations. Figure 6 displays the variations of \(\alpha\) and \(w\) over iterations. The function of \(\alpha\) is an exponentially decreasing function. A large value of \(\alpha\) tends to favor and promote exploration at the beginning, however, a small value of \(\alpha\) increases the local search tendency which leads to fast convergence speed. The function of \(w\) is an exponentially increasing function.