## 5 Simulation Results  
### Simulation Examples and Test Results  
Two examples presented in [29] are used to illustrate the application of the EHOO method. The first one is a small problem with 12 agent groups and five call types, and the size of the search space is \(20^{12}\). The second example is a large problem with 15 agent groups and 20 call types, and the size of the search space is \(30^{15}\). Table 1 shows the arrival rates in calls per hour and the agent skill groups of the small problem. The overall service level requirements are \(b_{0}=0.8\), and \(b_{j}=0.5\) for \(j=1,\,\ldots,\,5\). Table 2 shows the arrival rates in calls per hour and the agent skill groups of the large problem. The overall service level requirements are \(b_{0}=0.8\), and \(b_{j}=0.5\) for \(j=1,\,\ldots,\,20\). The cost in group \(k\) is \(c_{k}=1+0.1\times s_{k}\), where \(s_{k}\) is the number of skills in group \(k\). The service times are lognormal-distributed with mean \(\mu_{k,j}=5\) minutes and standard deviation \(\sigma_{k,j}=3\) minutes for all \(j\) and \(k\). The waiting times are exponentially distributed with mean \(\rho_{j}=6\) minutes. Calls will abandon before \(\tau=20\) seconds. Every simulation replication has a length with \(T+T/20\). Each simulation replication is divided into 21 batches, and the first batch is discarded. The choice for \(T\) is \(T=50\) h. \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline
**Call Type** & \(\lambda_{j}\) & & & & & **Agent Groups** & & & & \\ \hline
1 & 440 & 1 & 3 & 4 & 5 & 7 & 8 & 9 & 11 & 12 \\
2 & 540 & & 3 & & 6 & 7 & 8 & & 11 & 12 \\
3 & 440 & 2 & & 4 & 6 & 7 & & 9 & 10 & 11 & 12 \\
4 & 540 & & & & 5 & & & 10 & & 12 \\
5 & 440 & & & & & & 8 & 9 & 10 & 11 & 12 \\ \hline \end{tabular}
\end{table} 
Table 1: Skill groups of the small problem. In both problems, the RMTS was trained by arbitrarily choosing \(M=9604\) samples. The quantity \(M=9604\) was provided by the sample size calculation in a confidence interval for 1% and a confidence level for 95% [30]. In the choice of parameters for modeling RMTS, \(\delta=0.5\) and \(\xi=10^{-14}\) were obtained by the analytic approach. The objective value of each sample was obtained by accurate evaluation. The integrated factor \(\lambda\) was 0.9 in the small problem. After running numerous hand-tuning experiments, the values of parameters used in IEHO were \(\alpha_{\max}=0.9\), \(\alpha_{\min}=0.3\), \(\beta_{\max}=0.3\), \(\beta_{\min}=0.1\), \(I=5\), \(c_{i}=10\), \(i=1,\ldots,5\), \(\Phi=\sum\limits_{i=1}^{I}c_{i}=50\) and \(t_{\max}=500\). Tentative experiments have shown that IEHO with the above values has performed better over 30 runs. The IEHO beneficially explored the solution space at the beginning and tended to exploit excellent solutions closer to the end. When the iterative processes proceeded, the parameters \(\alpha\) and \(\beta\) were dynamically modified to emphasize diversity in early searches and intensity in later searches. Figure 5 presents the curves of \(\alpha\) and \(\beta\) along with iterations. The value of \(\alpha\) decreases exponentially. A large \(\alpha\) is in favor of promoting diversity at the beginning, then a small \(\alpha\) enhances fine-tuning of local search toward the end. The value of \(\beta\) is exponentially increased to keep a balance between explorative and exploitative processes. At first, \(\beta\) has a small value which can achieve the exploration goal more effectively. When the iterative process proceeds, \(\beta\) is exponentially increased and the search process progressively concentrates around the optimum to accomplish local exploitation. The number of significant solutions was \(N=5\). The following parameters were used in the AOCBA, \(L_{0}=20\), \(\Delta=10\) and \(L_{a}=10^{4}\). The time saving factor \(s\) corresponding to \(N=5\) is 2.08 [25]. Accordingly, the permissible computing budget \(C_{b}\) is 24,038. Table 3 demonstrates the superb solution \(\mathbf{x}^{\star}\), cost, service-levels and the CPU times in the small problem.