## 5 Statistical Analysis and Optimization  
In the rest of this section, a numerical case study of the system under investigation was conducted, where the data used in the analysis satisfied the feasibility condition (Equation (9)) and such data were based on previous studies of [PERSON] et al. [17]. Furthermore, by taking into consideration the validation of the simulation model from the previous section, the data used ensured that the obtained results were representative of the productive system. In this numerical case study, three independent variables were considered \((Z,A_{sub},A_{o})\) and the dependent variable was defined by the total incurred cost. Several simulation runs were conducted based on a factorial \(3^{3}\) design. For each combination of the independent variables \((Z,A_{sub},A_{o})\), the factorial design \(3^{3}\) was replicated two times; this meant that a total of \(3^{3}\times 2\) = 54 simulation runs were needed for the analysis. The simulation time used in the analysis was defined in 100,000 time units to guarantee steady-state conditions. The input data for the parameters of the simulation model are presented in Table 2.  
Table 3 shows the values of several cost categories used in the simulation. To simplify the calculations, it should be noted that to ensure that \(A_{sub}<A_{o}\), this age was defined as a fraction of age \(A_{o}\), i.e., \(A_{sub}=k\cdot A_{o}\), where \(k\in[0,\,1]\). The values of the independent variables \((Z,\,k,A_{o})\) shown in Table 4 were selected via the observation of several off-line simulations. In the following, taking into account the information in Tables 2-4, the simulation model generated the necessary output data to determine a second-order regression model for the total cost. Such a regression model was analyzed through the use of statistical software called STATGRAPHICS, where an ANOVA was performed. Then, the regression model for the total cost was the following:  
\[ \begin{split} TC(Z,\,k,A_{o})=\\ 809.977+4.74576\cdot Z-3.70433\cdot A_{o}-1705.16\cdot k+0.043567 \cdot Z^{2}\\ -0.0161327\cdot Z\cdot A_{o}\,\,-7.5515\cdot Z\cdot k+0.00580246 \cdot{A_{o}}^{2}\\ +3.67873\cdot A_{o}\cdot k+1036.04\cdot k^{2}\end{split}  \tag{17}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Factor** & **Low Level** & **High Level** & **Description** \\ \hline \(Z\) & 5 & 50 & Inventory level \\ \(k\) & 0.5 & 0.99 & Critical parameter to trigger subcontracting \\ \(A_{o}\) & 50 & 300 & Critical age to trigger major maintenance \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Domains of the independent variables. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline Parameter: & \(q_{12}\) & \(q_{21}\) & \(q_{13}\) & \(q_{31}\) & \(q_{12}^{sub}\) & \(q_{21}^{sub}\) \\ Value: & 0.1 & 1.5 & 0.15 & 5 & 0.25 & 0.3 \\ \hline Parameter: & \(u_{max}\) & \(d\) & \(k_{1}\) & \(\eta_{o}\) & \(b_{0}\) & \(b_{1}\) \\ Value: & 9 & 5 & 0.5 & 0.1 & 0.01 & 0.49 \\ \hline Parameter: & \(\eta_{1}\) & \(\eta_{2}\) & \(k_{1}\) & & & \\ Value: & \(15\times 10^{-6.2}\) & 2.4 & 0.5 & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Data for the numerical case study. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline Parameter: & \(C^{+}\) & \(C^{-}\) & \(C_{R}\) & \(C_{M}\) & \(C_{def}\) & \(C_{pro}\) & \(C_{sub}\) \\ Value: & 1 & 40 & 100 & 3000 & 20 & 5 & 15 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Cost parameters. The analysis of variance of the regression model is presented in Table 5, where it is evident that all the main factors, most of the quadratic effects, and most of the interactions were significant since the \(p\)-value indicators were less than 5%. The contour plots of the total cost reported by the regression model are presented in Figure 5 in two-dimensional space.