# Optimization of Inspection Period in Natural Stone Claddings
## 3 Results  
In this study, natural stone claddings are analyzed as a case study. To analyze the influence of the different maintenance plans, three maintenance strategies (MS) are considered: MS1--total replacement only; MS2--a combination of minor interventions and total replacement; MS3--a combination of cleaning operations, minor interventions, and total replacement. The MS1 represents the maintenance strategy most implemented by the owners/managers [45], while MS3 specifies the most complete maintenance strategy. Furthermore, as input data of the multi-objective optimization problem, a time horizon, \(t_{h}\), of 150 years is considered, and the initial severity of degradation, \(S_{w}\), of 0% (condition A) at the beginning of the analysis is adopted. In the maintenance model, it is assumed that the degradation process follows a Weibull distribution, and, in Table 2, the optimal parameters of the degradation process (degradation rate vector, \(\Theta\)) are presented in terms of the mean and standard deviation of the sojourn time in each degradation condition. Specifically, according to a Weibull distribution, for an NSC, a transition between conditions A and B takes, on average, 4.1 years to occur, with a standard deviation of 7.0 years. The same analysis can be used for other condition transitions. The choice of the best probability distribution that describes the degradation process is performed through the maximization of the logarithm of the likelihood [59]. A more detailed explanation of these assumptions is provided in [PERSON] et al. [58]. However, in relation to the data shown in Table 2, the high standard deviation of \(T_{a}\) is due to the reduced interval associated with the severity of degradation, \(S_{w}\), and the anomalies considered to quantify the severity of degradation in conditions A and B. From Table 1,the severity of degradation of condition A varies between 0 and 1%, showing that the deposit of surface dirt is sufficient to change the NSC from condition A to B. Moreover, the classification of an NSC is highly dependent on the inspector, geographic location, exposure conditions, and time of the year. The maintenance action details (costs, application zone conditions, and impacts) are presented in Table 3, and the cladding characteristics are the condition limits given in Table 1 (last column), which relates the two scales of the classification system. The information presented in Table 3 can be read in the following way: a minor intervention in an NSC has a cost of 68.80 \(\text{\text{\text{\text{\text{\text{\text{e}}}}}}}\)/m\({}^{2}\), and it is applied when the cladding reaches condition C, and improves the cladding's condition to B, with a probability of 80.4%, or maintains condition C, with a probability of 19.6%. More detailed information about the cost calculation can be found in [PERSON] et al. [58]. Regarding the design variable constraints (Equation (3)), and based on Section 2.1.2, it is assumed that the (optimal) time interval between inspection varies between 1 and 20 years. The lower limit is defined based on the literature, while the upper limit is increased in relation to the values found in the literature to determine the optimal time interval between inspection for NSC. Finally, it should be mentioned that all the solutions analyzed in this study are technically feasible.