## 2 The Measurement of the Risk  
In this study we assume that \(f(x,y)\) represents a loss related with an array of decision variables \(x\), which in turn is selected from a specific subset \(X\) of \(R^{n}\) and selected from the random variable \(y\) in \(R^{m}\). In this case, the \(x\) vector could be understood as an array of existing selections, whereas the uncertainty set is specified by \(y\) vector. Thus, the objective is to achieve the optimal solution for the decision vector \(x\) by minimizing the cost function of the model which, in vector \(y\), is bound to the uncertainty. As seen in the introduction, _VaR_ is one of the most frequently utilized risk measurements, meaning that it specifically fit for cases in which loss distribution functions with a fat tail behavior are employed [48]. By employing a probabilistic approach where the probability density function of \(y\) is represented by \(P\) associated with \(f(x,y)\) and given a confidence level \(\beta\), the \(\beta\)-_VaR_ is known as the minimum cost \(\alpha\) for all cases in which the loss probability above \(\beta\) never exceeds \(1-\beta\):  
\[\beta-VaR=\min\{\alpha\in R:P\{f(x,y)\leq\alpha\}\geq\beta\}for0\leq\beta\leq 1 \tag{1}\]  
_VaR_ is a popular risk measure mostly known and generally employed in the financial sector. However, in cases of practical optimization problems, this risk measure is noncoherent and is undermined by undesirable mathematical properties such as the absence of convexity and subadditivity, thus making it quite unappealing for optimization purposes. For this reason, the conditional value at risk (_CVaR_) is employed in ref. [48] as an improved alternative to _VaR_, which is also known as mean shortfall or average value at risk. This makes it mostly suitable for practical optimization problems which require a real-time solution. Thus, given the confidence level \(\beta\), _CVaR_ is given by the following equation.  
\[\beta-CVaR=E_{y}(f(x,y)|f(x,y)\geq\beta-VaR) \tag{2}\]  
The above equation specifies the conditional estimated value of the cost, limited to its value surpassing the \(\beta\)-percentile. When compared to typical robust optimization models, minimizing _CVaR_ guarantees a higher flexibility in choosing the objective and could also enhance the performance by utilizing the distributional information on the uncertain parameter \(y\). As a matter of fact, if the _CVaR_ of the cost is minimized it is also minimized the risk of the system being vulnerable to great lossesinstead of minimizing the worst-case cost scenario. Furthermore, in cases of functions of linear cost, the minimization of \(CVaR\) can be expressed as a modest linear optimization problem, making it more suitable for practical applications [49].  
By utilizing a sample originating from the distribution of the uncertain parameter \(y\), \(CVaR\) is expressed by the following equation.  
\[CVaR_{\beta}=\min(\alpha+\frac{1}{M(1-\beta)}\sum_{i=1}^{M}\left[f(x,y_{i})- \alpha\right]^{+}) \tag{3}\]  
where, the positive components of \(z\) are given by \(z^{+}\), \(\beta\)-\(VaR\) is given by \(\alpha\), the number of Monte Carlo routes is represented by \(M\) in which the estimated value of \(\beta\)-\(CVaR\) in the cost function is assessed, and finally \(y_{i}\) specifies the \(i\)th uncertain variable's created path. In order to reach a resolution for this problem, it is usually recommended to substitute \(0^{+}\) with an array of constraints. Thus, the expression of the \(CVaR\) minimization is given by the following equation.  
\[CVaR_{\beta}=\min(\alpha+\frac{1}{M(1-\beta)}\sum_{i=1}^{M}z_{i})\text{ subject to }:z_{i}\geq 0,z_{i}\geq f(x,y_{i})-\alpha \tag{4}\]