# Optimization of Inspection Period in Natural Stone Claddings
## Materials and Methods
### Formulation of the Optimization Problem  
Multi-objective optimization problems involve two or more objective functions that are in conflict, meaning that the improvement of one goal implies the sacrifice of another goal [52]. In the same way, as in the single-objective optimization problems, multi-objective optimization problems may contain different constraints that any feasible solution must satisfy, i.e., a multi-objective optimization problem can be expressed as follows [53]:  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Degradation** & **Anomalies** & \(k_{a,n}\) & \begin{tabular}{c} **\% Area** \\ **Affected** \\ \end{tabular}  & \begin{tabular}{c} **Severity of** \\ **Degradation (\%)** \\ \end{tabular}  \\ \hline \multirow{7}{*}{B (\(k_{n}=1\))} & \multirow{7}{*}{ \begin{tabular}{c} Visual or \\ surface \\ degradation \\ anomalies \\ \end{tabular}  } & \begin{tabular}{c} Surface dirt \\ Moisture stains/localized \\ stains/color change \\ Flatness deficiencies \\ Material degradation \({}^{*}\leq 1\%\) \\ plate thickness \\ Material degradation \({}^{*}\leq 10\%\) \\ plate thickness \\ Material degradation \({}^{*}\leq 10\%\) \\ plate thickness \\ Cracking width \(\leq 1\) mm \\ \end{tabular}  & \begin{tabular}{c} 0.13 \\ \(>\)10 \\ 0.13 \\ 0.13 \\ 0.13 \\ 0.13 \\ 0.13 \\ 1.00 \\ \end{tabular}  & \begin{tabular}{c} \(>\)10 \\ \(\leq\)15 \\ \(\leq\)10 \\ - \\ - \\ - \\ - \\ - \\ - \\ - \\ - \\ - \\ - \\ - \\ - \\ - \\ - \\ - \\ - \\ - \\ - \\ - \\ \end{tabular}  \\ \hline \multirow{7}{*}{C (\(k_{n}=2\))} & \multirow{7}{*}{\ \begin{tabular}{c} Visual or \\ surface \\ degradation \\ anomalies \\ \end{tabular}  } &
\begin{tabular}{c} Moisture stains/localized \\ stains/color change \\ Moss, lichen, algae \\ growth/parasitic \\ vegetation/efflorescence \\ Flatness deficiencies \\ Joint material degradation \({}^{*}\leq 30\%\) \\ Material loss—open joint \\ Scaling of stone near the edges \\ Partial loss of stone material \\ Material degradation \({}^{*}\leq 10\%\) \\ plate thickness \\ - \\  
\[ \begin{array}{llll}\textbf{Minimize/maximize:}&f_{m}(x)&m=1,\;2,\;\ldots,\;M\\ \textbf{Subject to:}&g_{j}(x)\geq 0&j=1,\;2,\;\ldots,\;J\\ &h_{k}(x)=0&k=1,\;2,\;\ldots,\;K\\ &x_{i,L}\leq x_{i}\leq x_{i,U}&i=1,\;2,\;\ldots,\;I\end{array}  \tag{2}\]  
where \(f_{m}(x)\) is the objective function, and \(g_{j}(x)\), \(h_{k}(x)\), \(x_{i,L}\), \(x_{i,U}\) the constraints. However, in multi-objective optimization problems, the concept of an optimal solution is replaced by the concept of non-dominated solutions. By definition, a feasible solution \(x^{(1)}\) dominates the feasible solution \(x^{(2)}\) if both the following conditions are true [53]:  
1. The solution \(x^{(1)}\) is no worse than \(x^{(2)}\) in any objective;
2. The solution \(x^{(1)}\) is strictly better than \(x^{(2)}\) in at least one objective. All solutions that are not dominated by any other solution of the set are called a non-dominated solution. In Figure 2, an example of a multi-objective optimization problem is shown. From the set of six solutions shown in Figure 2, the solutions linked by the dashed line are non-dominated solutions. The improvement in one objective, \(f_{1}\), requires a sacrifice in the other objective, \(f_{2}\). These solutions arrange a non-dominated front when viewed together on the objective space [53]. In this study, the optimization problem considers four objectives. The multi-objective algorithm is used to optimize the time interval between inspections. Therefore, in this optimization problem, the time interval between inspections is considered the design variable, while the objective functions are the: (i) minimization of the maintenance costs; (ii) maximization of the service life; (iii) maximization of the efficiency index; (iv) minimization of the number of replacements. The design variable is subjected to a constraint, i.e., the (optimal) time interval between inspections should vary between a maximum and a minimum value. The analytical formulation of this optimization problem is given by:  
\[