## 5 Empirical Results
### Box-Behnken Design (BBD)  
Following the results from the experiment of the central composite design, Box-Behnken design was performed to compare the effects of independent variables on the response variables. In this experiment, BBD with three continuous factors used the total 15-run design with one block. Compared to the central composite design, the BBD experiment did not have an embedded factorial design and extreme points, and the BBD experiment also did not have the rotatability value (\(\alpha\)) in the experimental design. The advantage of the BBD model was that the design usually had fewer design points, hence, they were often less expensive to run than the CCD design with the same number of independent variables. The BBD experiment running setup and randomized design table is presented in Figure 15 and Table 6 as follows. Figure 15 shows all set-up parameters for the experimental BBD. From the results of Table 6, TC and CSL were calculated based on considered independent variables using an excel spreadsheet. The initial model was obtained for the response variables (TC and CSL). Figure 16 shows the response surface regression of total cost versus T, Q, and SS. In the first running, the results suggest that the square terms Q*Q, SS*SS (\(p\)-values were 0.976 and 0.915 respectively), and the interaction among T*SS and Q*SS (\(p\)-values were 0.721 and 0.902, respectively) were not significant (all \(p\)-value \(>\) 0.05), while the paper considered the confidence level of the BBD at 95%. Then, these terms were eliminated from the BBD model to obtain the following final model. After excluded insignificant terms, the experimental BBD was run again, the results of the reduced model are shown in Figure 17. The results display that term SS was statistically significant while the terms T, Q, square term T*T, and the interaction of T*Q significantly affected the model (\(p\)-value \(<\) 0.05). The authors decided to keep the term SS for further analysis to see the effects of these independent variables on the objective functions. For the second objective function related to CSL (%), the BBD was performed consequentially. The response surface regression of the CSL versus T, Q, and SS is presented in Figure 18. In the first running, the results display that the term SS (\(p\)-value was 0.585) and the square terms Q*Q and SS*SS (\(p\)-values were 0.461 for both) were not significant (\(p\)-value \(>\) 0.05). In this case, the square terms Q*Q and SS*SS were eliminated from the BBD model in priority, the term SS was kept for further analysis purposes. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**StdOrder** & **RunOrder** & **PtType** & **Blocks** & **T** & **Q** & **SS** & **TC** & **CSL** \\ \hline
1 & 1 & 2 & 1 & 2 & 3250 & 1000 & 134,828.00 & 100 \\
2 & 2 & 2 & 1 & 4 & 3250 & 1000 & 65,946.84  
After excluded these terms, the BBD experiment was run again, the reduced model is shown in Figure 19. The results show that only the term Q, SS, and the square term T*T (\(p\)-values were 0.054, 0.564, and 0.067, respectively) were slightly significant because the \(p\)-value was slightly out of the confidence level at 95%. Otherwise, only the terms T was highly significant to the BBD experiment (\(p\)-value \(<\) 0.05). After reducing the BBD model into the final form, the residual plots (normal probability plot, histogram, versus fits, and versus order plot) for the response variables (TC and CSL) are displayed in Figure 20. The results show that all residual plots satisfied the normality and constant variance assumptions, which means the simulated data was random and followed normal distribution. Figure 21 shows the response optimizer of BBD to determine the feasible response variables, i.e., TC and CSL (CSL, 90%, 95%). From the result, the CSL set to target was 90% with the possible minimum total cost 70,870 of the possible settings were T = 3, Q = 3603, and SS = 1250. Otherwise,  
Figure 19: BBD—response surface regression of CSL (%) versus T, Q, and SS (reduced model).