## 3 Solution Method
### Diversification
Then, the top \(N\) individuals are chosen as a candidate subset containing significant solutions. ### Intensification

To enhance efficiency, an AOCBA method is developed as a more refined selection technique to identify the best solution among the significant solutions found in diversification part. In the original OCBA, it is necessary to perform all simulation replications repeatedly to calculate the statistics of overall simulation replications. The AOCBA requires only a part of extra simulation replications to determine the statistics of overall simulation replications. We proceed to determine a superb solution by the AOCBA approach from the \(N\) significant solutions. The AOCBA approach is designed to adjust the computational efforts dynamically for significant solutions. The key feature of the AOCBA is spending more computing resource on few remarkable solutions and less on most mediocre solutions. Focusing on few promising solutions does not only save computational time, but also reduces these promising estimators' variances. Through the statistics concerning objective value provided by the \(N\) significant solutions, the number of extra simulation replications is computed to provide more computational resource for promising solutions. Let \(C_{b}\) denote the permissible computing budget, and \(L_{i}\) indicate the number of simulation replications assigned to the \(i\)th solution. The initial number of simulation replications assigned to the \(N\) significant solutions is \(L_{0}\). The extra computing budget is raised by a pre-defined \(\Delta\) at each iteration. The goal of the AOCBA is to properly allocate \(C_{b}\) to \(L_{1}\), \(L_{2}\),..., \(L_{N}\) such that \(L_{1}+L_{2}+\cdots+L_{N}=C_{b}\) and maximize the probability of obtaining the optimum. The permissible computing budget \(C_{b}\) is obtained from \(C_{b}=\frac{N\times L_{b}}{s}\), where \(L_{a}=10^{4}\) indicates the simulation replications adopted in accurate evaluation and \(s\) is a time saving factor with respect to \(N\) that can be found in the optimal computing budget allocation (OCBA) procedure [25]. The step-wise process of the AOCBA (Algorithm 2) method is given below. ``` Step 0. Set the quantity of \(L_{0}\), \(l=0\), \(L_{1}^{l}=L_{0}\),..., \(L_{N}^{l}=L_{0}\), where \(l\) expresses the iteration counter. Determine the value of \(C_{b}=\frac{N\times L_{b}}{s}\). Step 1: If \(\sum\limits_{i=1}^{N}L_{i}^{l}\geq C_{b}\), stop and choose the optimum \(\mathbf{x}^{*}\) with the smallest objective value; else, go to Step 2. Step 2: Raise an extra computing budget (\(\Delta\)) to \(\sum\limits_{i=1}^{N}L_{i^{\prime}}^{l}\) and update the simulation replications by  
\[L_{j}^{l+1}=(\sum\limits_{i=1}^{N}L_{i}^{l}+\Delta)\times\theta_{j}^{l}/(\theta _{b}^{l}+\sum\limits_{i=1,i\
eq b}^{N}\theta_{i}^{l}) \tag{16}\]  
\[L_{b}^{l+1}=\frac{\theta_{b}^{l}}{\theta_{j}^{l}}\times L_{j}^{l+1} \tag{17}\]  
\[L_{i}^{l+1}=\frac{\theta_{i}^{l}}{\theta_{j}^{l}}\times L_{i}^{l+1} \tag{18}\]  
for all \(i\
eq j\
eq b\), where \(\frac{\theta_{j}^{l}}{\theta_{j}^{l}}=\left(\frac{\delta\psi(\overrightarrow{ \mathbf{r}_{j}}-\overrightarrow{\mathbf{r}_{j}})}{\delta\psi(\overrightarrow{ \mathbf{r}_{j}}-\overrightarrow{\mathbf{r}_{j}})}\right)^{2}\), \(\theta_{b}^{l}=\delta_{b}^{l}\sqrt{\sum\limits_{i=1,i\
eq b}^{N}\left(\frac{ \theta_{i}^{l}}{\theta_{i}^{l}}\right)^{2}}\), \(\overrightarrow{F_{i}}=\frac{1}{L_{i}^{l}}\sum\limits_{h=1}^{L_{i}^{l}}F_{h}( \mathbf{x}_{i})\), \(\delta_{i}^{l}=\sqrt{\sum\limits_{i=1}^{L_{i}^{l}}\left(F_{h}(\mathbf{x}_{i})- \overrightarrow{F_{i}}\right)^{2}}\), \(\mathbf{x}_{i}\) denotes the \(i\)th promising solution, \(F_{h}(\mathbf{x}_{i})\) is the objective value of \(\mathbf{x}_{i}\) for the \(h\)th simulation replication, and \(b=\text{arg}\underset{i}{\text{min}}\overrightarrow{\mathbf{r}_{i}}\).