# A Robust Scheduling Framework for Re-Manufacturing Activities of Turbine Blades
## 5 Case Study  
The proposed robust scheduling framework has been preliminary tested in an industrial environment to support the scheduling of the re-manufacturing activities for turbine blades. In this case study, four turbine stages are considered. Each stage consists of a set of identical blades that are thus processed as a batch. The steps of the repair process are the same for all the batches, i.e., defects removal, material additive, machining, and non-destructive testing (Section 3). Within these steps, 9-12 operations in total are executed, with some differences among the different types of batches. Moreover, 2/3 operations in the process may need a successive rework. The initial state of the turbine blades is modeled in terms of their damage level, defined for each stage grounding on historical data, reported in Table 1. Starting from the damage level, a rejection rate can be defined (the probability for a blade to be too damaged to be repaired), as well as the distribution of the processing times for rework operations (Section 4.2.1). Each activity is expected to require a single renewable resource from a resource set with four different resources representing machines and human workers involved. We will show how to appropriately use the framework to provide relevant information to the decision-makers, and help them develop robust schedules in their turbine blades re-manufacturing plants. A resource constrained project scheduling model (RCPSP) is used to formalize the described scheduling problem (Section 4.1). A total of four batches are considering, resulting in 41 activities to schedule (including the start and complete dummy activity). The objective of the approach is the minimization of the maximum regret of the makespan (Section 4.3.1). The processing time of each activity is modeled with an interval derived from historical data and convolution operations (Section 4.2.1). Rework activities are described through a Bernoulli distribution modeling the probability to occur (Section 4.2.2). Thus, the integrated uncertainty is described by a set of scenarios, obtained by considering the possible occurrences of processing times and reworks for all the activities. For each scenario, a processing time vector \([p_{0},p_{1},\ldots,p_{40}]\) is used to describe the processing time for all the activities. Since only extreme scenarios need to be examined in the maximum regret minimization model [30], \(2^{(41-2)}\) scenarios in total need to be evaluated in this case study, the scenario relaxation algorithm with accelerated convergences for robust resource-constrained project scheduling problem proposed in [23] has been adopted to support the definition of robust schedules. To demonstrate the benefits of this robust scheduling framework, an alternative scheduling approach is used for comparison, namely, one only considering the expected values of processing times and a probability of 50% rework occurrence, ignoring the disturbance and volatility in the re-manufacturing process, is compared with the proposed framework. Due to the large amount of possible scenarios, it is unfeasible to test the schedule obtained through the alternative approach on all of them. To overcome this limitation, a subset of the total number of scenarios is sampled and, for each of them, the performance \begin{table}
\begin{tabular}{c c c} \hline \hline
**Damage Level** & **Rejection Rate LB (\%)** & **Rejection Rate UB (\%)** \\ \hline Heavy & 60 & 90 \\ Medium & 30 & 60 \\ Light & 0 & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Damage level table. obtained with the schedule considered (i.e., the makespan). These values are then used to assess the value of the stochastic solution (VSS), i.e., the value of exploiting stochastic information to support a robust schedule. It is calculated as the average deviation between the performance of the robust approach and the one just considering expected values (Equation (4)) over the considered set of scenarios [41].