## 3 Methodology
### Methodological Framework
#### 3.1.1 Problem Statement  
The present study focuses on the runway infrastructure which, in accordance with ASTM D-5340-20, could represent a single branch, divided into S homogeneous sections and N = Ks-S sample units, whose extension has to be equal to 450 \(\pm\) 180 m\({}^{2}\)[43]. The S homogeneous sections consider the differentiation of the pavement deterioration and the dissimilar stresses depending on the load given by the passage of the aircraft landing gear. Let Q be the number of the performance indicators q used for evaluating the pavement condition. In this regard, let p_k^q (0) the performance indicator value of the k-th sample unit at time zero. Hence, the approach of this study involves the identification of specific work-zones. Specifically, they consist in the grouping of sample units belonging to the same section and featured by a similar pavement deterioration condition at time zero. To this end, on the basis of pre-established performance indicators, q and J cluster levels, in this paper, a well-established K-means method is used, adequately adapted to ensure the grouping of only adjacent sample units, as detailed further in Section 3.2. Figure 1 illustrates the airport pavement schematization, including sample units and work-zone limits, explained above. At time zero, each performance indicator \(q\) of the \(w\)-th work-zone is equal to the worst value among those evaluated in the \(U_{w}\) sample units \(u_{w}\) (\(u_{w}=1,\,\ldots,\,U_{w}\)) belonging to it:  
\[p_{w}^{q}(0)=\min_{u_{w}=1,\,u_{w},U_{w}}p_{u_{w}}^{q}(0) \tag{1}\]Each performance indicator \(q\) is strictly connected to a deterioration rate from which to estimate the future pavement conditions, i.e., time-dependent [43]. In this regard, this study fixes a planning period, hereinafter denoted as \(T\), as the number of years \(t\) in which to conduct the M&R activities in the work-zones subject to performance deterioration. Consequently, assuming a limit threshold \(Th(q)\) for each performance indicator \(q\), the \(W\) critical work-zones are defined as those characterized by at least one indicator \(q\) below the respective threshold at time \(t\in T\). Therefore, the \(W\) critical work-zones need at least one M&R activity within the planning period \(T\), to ensure the quality requirements of the pavement. Moreover, to ensure that the pavement maintenance problem has effects on the execution of the M&R activities carried out even in the last year of \(T\), the approach used in this research work considers a control time horizon. It is equal to \(T+2\) years and indicates the number of years \(t\) in which the work-zones have to be the performance indicators \(q\) above the relative thresholds \(Th_{q}\). However, the solution of the proposed problem is a matrix where \(W\) is the set of critical work-zones, \(T\) is the planning time horizon and \(x_{wt}\in I\) is the activity at time \(t\) on the critical work-zone \(w\), being \(I=\{0,1,\ldots,N\}\) the set of available M&R actions. Specifically, it can be configured as follows:  
\[X= \begin{bmatrix}x_{11}&\ldots&x_{1T}\\ \ldots&x_{wt}&\ldots\\ x_{W1}&\ldots&x_{WT}\end{bmatrix}  \tag{2}\]  
If \(x_{wt}=0\), no action is conducted for the critical work-zone \(w\) at time \(t\) and the related cost is zero. Let \(C(x_{wt})\) be the maintenance cost associated with maintenance action \(x_{wt}\) on work-zone \(w\) at time \(t\), then the optimization problem can be formulated as follows:  
\[\min C_{t} \tag{3}\]  
s.t. \[P^{q}(x_{wt})\geq Th_{q},\ \forall q\in 1,\ldots,Q;\ w=1,\ldots,W;\ t=1,\ldots,T \tag{4}\]  
\[C_{t}=\sum_{w=1}^{W}\sum_{t=1}^{T}C(x_{wt}) \tag{5}\]  
with:  
\(C_{t}\) total maintenance costs [**6**]. \(C(x_{wt})\) maintenance cost associated to the activity \(x_{wt}\) on work-zone \(w\) at time \(t\) [**6**]. \(P^{q}(x_{wt})\) the \(q\)-th performance measure of work-zone \(w\) at time \(t\) after the maintenance intervention \(x_{w,t}\). \(q=1,\ldots,Q\) index of performance indicator. Figure 1: Airport runway schematization.