# Optimization of Inspection Period in Natural Stone Claddings
## 3 Results
### Comparison of the Optimal Solutions for the Different Maintenance Strategies  
In Figures 4 and 5, the Pareto solutions obtained for the three maintenance strategies are compared. In Figure 4, the solutions are displayed through pairs of the four objective functions (maintenance cost, service life, efficiency index, and the number of total replacements), while, in Figure 5, the time interval between inspections (design variable), _t__insp_, is compared with the four objective functions. A total of 18 optimal solutions are obtained for each maintenance strategy. For each maintenance strategy, three optimal solutions are highlighted in Figures 4 and 5. The values of the four objective functions and the design variable are presented in Table 4. The optimal solutions, for MS1, are identified as S1 to S3; for MS2, as S4 to S6; for MS3, as S7 to S9. By comparing the three optimal solutions for MS1, solution S1, which is associated with a higher time interval between inspections, yields a lower maintenance cost, but its performance is also lower. On the other hand, solution S3, which is associated with a lower time interval between inspections, implies higher maintenance costs but ensures a higher performance of the NSC over the time horizon. Lastly, solution S2 represents a trade-off between solutions S1 and S3. A similar consideration can be made for MS2 and MS3. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameters** & \(T_{A}\) & \(T_{B}\) & \(T_{C}\) & \(T_{D}\) \\ \hline Mean (years) & 4.1 & 42.9 & 22.4 & 49.9 \\ Standard deviation (years) & 7.0 & 10.3 & 1.4 & 2.4 \\ \hline \hline \(T_{A}\)—transition between conditions A and B; & & & \\ \(T_{B}\)—transition between conditions B and C; & & & \\ \(T_{C}\)—transition between conditions C and D; & & & \\ \(T_{D}\)—transition between conditions D and E. & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Degradation rate vector, \(\Theta\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Maintenance Actions** & **Cost (\(\text{f}\)/m\({}^{2}\))** & **Application Zone Conditions** & \multicolumn{3}{c}{**Impact of the Maintenance Actions**} \\ \cline{3-5}  & & \(P_{A}\) (\%) & \(P_{B}\) (\%) & \(P_{C}\) (\%) \\ \hline Inspection & 1.03 & All & - & - & - \\ Cleaning operations & 31.37 & B & 15.0 & 85.0 & - \\ Minor interventions & 68.80 & C & 0 & 80.4 & 19.6 \\ Total replacement & 149.51 & D, E & 100.0 & - & - \\ \hline \hline \(P_{A}\)—probability of the application zone changing to condition A; & & & \\ \(P_{B}\)—probability of the application zone changing to condition B; & & & \\ \(P_{C}\)—probability of the application zone changing to condition C. & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Maintenance action details. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Maintenance Strategy** & **Optimal Solution** & \begin{tabular}{c} **Time Interval,** \\ **\(t_{\text{insp}}\) (Years)** \\ \end{tabular} & \begin{tabular}{c} **Maintenance** \\ **Costs (\(\ell\)/m\({}^{2}\))** \\ \end{tabular} & \begin{tabular}{c} **Service Life** \\ **(Years)** \\ \end{tabular} & \
\begin{tabular}{c} **Efficiency** \\ **Index** \\ \end{tabular} &
\begin{tabular}{c} **Number of** \\ **Replacements** \\ \end{tabular} \\ \hline MS1 & S1 & 18.8 & 2.54 & 70 & 0.9057 & 1.17 \\  & S2 & 10.9 & 3.61 & 71 & 0.9155 & 1.42 \\  & S3 & 1.0 & 20.47 & 70 & 0.9316 & 1.75 \\ \hline MS2 & S4 & 18.8 & 4.32 & 163 & 0.9292 & 0.14 \\  & S5 & 11.9 & 5.82 & 157 & 0.9367 & 0.35 \\  & S6 & 1.0 & 23.91 & 143 & 0.9449 & 0.57 \\ \hline MS3 & S7 & 19.0 & 14.58 & 270 & 0.9460 & 0.02 \\  & S8 & 15.4 & 18.79 & 245 & 0.9489 & 0.02 \\  & S9 & 1.0 & 74.59 & 154 & 0.9494 & 0.46 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comparison of the four objective functions and the design variable of the optimal solutions highlighted.