## 2 Integration of Ordinal Optimization with Ant Lion Optimizer
### Informed Ant Lion Optimization
A small value of \(w\) achieves diversity more efficiently or effectively. When the optimization processes proceed, the value of \(w\) is exponentially increased to gradually improve local exploitation. Figure 7 presents the sliding ratio between original ALO and RALO. The value of exceptional designs was \(N=10\). The parameter settings in R&S for small example were \(L_{0}=50\), \(N_{\text{min}}=2\), \(L_{a}=10^{3}\), and \(n_{\text{s}}=3\). Table 1 shows the quasi-optimal design \(\mathbf{x}^{*}\), cost and the central processing unit (CPU) times of the best run among the 30 trials in small example. Figure 6: Cuves of \(\alpha\) and \(w\) along with iterations. In large example, the MARS was trained by randomly sampling \(M=9604\) designs. The amount \(M=9604\) was produced using the sample size formula for a 95% confidence level and a 1% confidence interval. The sample mean of each design was estimated using precise estimation. After running multiple experimental trials with different settings, the parameter settings in RALO for large example were \(\alpha_{\max}=0.9\), \(\alpha_{\min}=0.1\), \(w_{\max}=6\)\(w_{\min}\)\(=1\), \(\Psi=200\) and \(k_{\max}=1000\). The value of exceptional designs was \(N=100\). The parameter settings in R&S for large example were \(L_{0}=10\), \(N_{\min}=2\), \(L_{a}=10^{3}\), and \(n_{s}=5\). Table 2 demonstrates the amount of critical designs and replications in each stage. Since the parameter settings in R&S are irrelevant to the random nature of stochastic simulation process, the values listed in Table 2 are applied to the 30 trials. Table 3 shows the quasi-optimal design \(\mathbf{x}^{*}\), cost and the CPU times of the best run among the 30 trials in large example. Since the MARS is trained off-line, its training time is not included in the CPU time consumed by the OALO in Tables 1 and 3. The CPU time contains the computing effort consumed by the RALO and R&S procedure. The CPU time was less than 100 s for all trials which were fast enough to meet the specification for real time applications. The stochastic simulation process can be used in optimal queuing design of a communication system with any distribution of arrival rates, even for high-dimension problems. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\mathbf{x}^{*}\) & **Cost** & **CPU Times (s)** \\ \hline \([54,64]^{\mathrm{T}}\) & 32.59 & 18.52 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The quasi-optimal design \(\mathbf{x}^{*}\), cost and the CPU times of the best run in small example. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{Stage\;i}\) & \(\mathbf{1}\) & \(\mathbf{2}\) & \(\mathbf{3}\) & \(\mathbf{4}\) & \(\mathbf{5}\) \\ \hline \(N_{i}\) & 100 & 37 & 14 & 5 & 2 \\ \(L_{i}\) & 27 & 74 & 201 & 546 & 1000 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Number of critical designs and replications in each stage. Figure 7: Comparison of sliding ratio between original ALO and RALO. Based the large deviation theory, the OALO can achieve an exponential convergence rate in stochastic simulation models. Detail convergence analysis can refer to Chap 2.4 in [24] for detailed proof. The proposed OALO method can be applied to more complex simulation-based optimization problems, such as optimal reinsurance-investment problems, multiskill call center with preference lists, flow lines with multiple-products, and portfolio optimization problems with real-world constraints. The benefit of the OALO method is to reduce the required simulation time significantly by determining quasi-optimal design instead of finding the best design. Nevertheless, yielding a candidate subset of designs may not be very satisfactory in some cases. The drawback of the OALO method is that it does not provide an absolute guarantee of the global optimality. ### Comparisons

To test the quality and efficiency of the OALO algorithm, three competing approaches, GA, ES and particle swarm optimization (PSO), were adopted to solve the large example. In the employed GA [43], the size of population was 200.