## 4 Simulation Results and Comparisons  
### Test Example and Results  
Two test examples were provided by [PERSON] [26]. The first one is a small problem with three products and six nodes, as shown in Figure 3. The second one is a large problem with four products and 12 nodes, as shown in Figure 5. In both problems, the interarrival times of orders follow a normal distribution with a mean of 30 and a standard deviation of 5. The risk is \(\alpha=10\%\). The computational experiment was coded in MATLAB 9.1 (R2016b) (MathWorks: Natick, MA, USA) and conducted using an Intel Core i7, 3.2 GHz CPU, 4 GB RAM computer. In the small problem, each order has a batch of 10, and the probabilities for Products 1, 2, and 3 are 0.5, 0.35, and 0.15, respectively. The time horizon H is 600, so 200 sets of raw material are ordered. The processing times for 1-2, 1-3, 2-4, 2-5, 3-5, and 3-6 are normally distributed with parameters (4,1), (3,1), (5,2), (4,1), (4,1) and (3,1). In the large problem, each order has a batch of 10, and the probabilities for Products 1, 2, 3, and 4 are 0.5, 0.25, 0.1, and 0.15, respectively. The time horizon H is 1200, so 400 sets of raw material are ordered. The processing times for 1-2, 1-3, 1-4, 2-5, 2-6, 2-7, 3-5, 3-6, 4-7, 4-8,  
Figure 5: Four product production system. 5-9, 5-10, 6-9, 6-10, 6-11, 7-11, 7-12, 8-11 and 8-12 are normally distributed with parameters (4,1), (3,1), (5,2), (4,1), (4,1), (5,2), (4,2), (4,1), (5,1), (4,2), (3,1), (5,2), (5,1), (4,1), (5,1), (3,1), (4,2), (5,2) and (4,1). In the two problems, the RMTS was trained off-line by arbitrarily selecting \(M=9604\) x from the search space, and their \(F_{\rm s}({\bf x})\) was estimate by precise evaluation. The sampling value \(M=9604\) was calculated by the sample size formula using a confidence level of 95% and a confidence interval of 1% [27]. In the choice of parameters for modeling RMTS, \(\sigma=0.5\) and \(\rho=10^{-14}\) were obtained by the analytic approach. For the small problem, the penalty weight \(\lambda\) was 0.9. The number of outstanding solutions was \(N\) = 5. The parameters used in Algorithm 1 were obtained by a series of hand-tuning experiments as: \(\Psi=10\), \(ST_{\rm min}=0.1\), \(ST_{\rm max}=0.5\), \(\gamma_{\rm min}=0.1\), \(\gamma_{\rm max}=0.3\), and \(k_{\rm max}=1000\). Elementary experiments demonstrated that ITSA with these parameters performed uniformly better over 30 independent runs. The developed ITSA favorably explored the entire search space at the early iterations and likely exploited excellent solutions at the later iterations. As the search evolved, the values of \(ST\) and \(\gamma\) were dynamically changed to strengthen diversification at first and intensification toward the end. Figure 6 presents the trends of \(ST\) and \(\gamma\) over iterations. The value of \(ST\) is exponentially increased. A small value of \(ST\) favors diversity in the beginning, while a large value makes fine tuning ability near the end. The parameter \(\gamma\) exponentially decreased to achieve the balance between exploration and exploitation. At the beginning, most of the search utilized randomly selected pitches to accomplish global exploration. As the searching process progressed, \(\gamma\) was exponentially decreased and the search progressively revolved around the best tree in memory to realize local exploitation. The parameters of Algorithm 2 were \(L_{0}\) = 20, \(\Delta\) = 10 and \(L_{s}=10^{4}\). The reduction factor \(s\) corresponded to \(N\) = 5 is 2.08 [22]. Thus, the available computing budget \(C_{b}\) was 24038. Table 1 shows the excellent solution \({\bf x}^{*}\), the corresponding \(E[f({\bf x})]\), \({\rm P}(g({\bf x})\geq b)\), \(F({\bf x})\) and the consumed CPU times. For the large problem, the penalty weight \(\lambda\) was also 0.9. The parameters in Algorithm 1 were: \(\Psi=50\), \(ST_{\rm min}=0.1\), \(ST_{\rm max}=0.5\), \(\gamma_{\rm min}=0.1\), \(\gamma_{\rm max}=0.3\), and \(k_{\rm max}=2000\). The parameters in Algorithm 2 were: \(L_{0}\) = 20, \(\Delta\) = 10 and \(L_{s}=10^{4}\).