# A Robust Scheduling Framework for Re-Manufacturing Activities of Turbine Blades
## 4 Robust Scheduling Framework
### Robustness Measures
#### 4.3.2 Minimizing the Value-at-Risk/Conditional Value-at-Risk of the Objective Function  
The minimization of the maximum regret has the advantage of only requiring knowledge about the extreme scenarios and, consequently, the simple knowledge of the extreme values of the domains for uncertain variable is enough. At the same time, focusing the optimization on worst-case scenarios, that may be unlikely to occur, tends to be too conservative. To cope with this limitation, different risk measures can be used to guide the optimization, namely, the value-at-risk (VaR) and the conditional value-at-risk (CVaR) [32]. These indicators take into consideration the whole distribution of uncertain variables and, thus, are able to consider the impact of uncertain events both in terms of their effect and occurrence probability. The use of both the VaR and CVaR was initiated in the financial area [32] but their popularity in robust scheduling area is rapidly increasing [21; 33; 34]. In the case of the deterministic RCPSP, a solution to the RCPSP is a schedule \(s\), i.e., a vector of starting times \((s_{0},s_{1},\ldots,s_{n},s_{n+1})\), with each activity duration is a constant. While for the scheduling problem under study, the decision maker does not know which exact information of activity duration, and yet a number of sequencing decisions need to be made. Hence, the execution of the project with uncertain activity durations is a dynamic decision process, and a schedule solution which denoted as a vector \(x\), is a policy, which defines actions at the start of the project and at the completion times of activities. A vector of random variables \(y=\{p_{1},\ldots,p_{n}\}\) models the random processing times, governing by a probability measure \(P\) on \(Y\) and independent of scheduling decision \(x\). The probability distribution of the makespan, \(f_{C_{max}}(x,y)\), depends on the values of \(x\) and \(y\). For a given schedule \(x\), the resulting cumulative density function (cdf) for the makespan is defined as:  
\[F_{C_{max}}(x,\zeta)=P(f_{C_{max}}(x,y)\leq\zeta|x) \tag{1}\]  
Then, the value-at-risk \(\alpha\) (\(VaR_{\alpha}\)) of \(C_{max}\), associated with a schedule decision \(x\), denoted as \(\zeta_{\alpha}(x)\), is defined according to the following:  
\[\zeta_{\alpha}(x)=min\{\zeta|F_{C_{max}}(x,\zeta)\geq\alpha\} \tag{2}\]  
Further, the \(\alpha-CVaR\) of (1) associated to a schedule \(x\) is the mean of the \(\alpha-\)tail distribution (3) [35]. \[F_{X}^{\alpha}(z)=\left\{ \begin{array}{ll}0,&\text{when }z<\zeta_{\alpha}(X)\\ \frac{F_{X}(z)-\alpha}{1-\alpha},&\text{when }z\geq\zeta_{\alpha}(X)\end{array} \right. \tag{3}\]  
By considering the proposed Equation (2), VaR can be specified as the risk measure on the random \(f(x,y)\), and minimizing \(\zeta\) corresponds to seeking the schedule solution with the smallest possible VaR measure for a specified \(\alpha\) value, i.e., for confidence level \(\alpha\), \(VaR_{\alpha}\) is the \((1-\alpha)\)- quantile of the makespan distribution which is the largest value that ensures that the probability of obtaining a makespan less than this value is lower than \(1-\alpha\)[33]. CVaR at confidence level \(\alpha\), i.e., \(CVaR_{\alpha}\), is defined as the expected value of makespan smaller than the \((1-\alpha)\)-quantile of the probability distribution of makespan, i.e., \(VaR_{\alpha}\)[36]. CVaR is the expected value of the makespan for the worst \(\alpha\)% cases with a value greater than the VaR. The main difficulty in pursuing robustness through the minimization of the VaR/CVaR lies in the need to calculate the distribution of the objective function, e.g., the makespan. In scheduling problems affected by uncertainty, thus entail the capability of dealing with the correlation among all the possible paths in the network of activities (Section 4.1) [37; 38]. To overcome this difficulty, a Markovian Activity Network(MAN) approach can be used to support the analytical estimation of this distribution.