## 5 Numerical Results  
The simulation results are compared to the first benchmark which is the model without redundancy. Table 6 reports the legend used for the graphs of the results; the first letter \"X\" denotes the number of the case study reported in Table 3. The second letter can be \"\(Q\)\" for the benchmark proposed in [27] and \"\(F\)\" for the fuzzy model. The fuzzy models are tested in four cases:  
* \"a\", it has been considered that only the processing time of the fractional tasks assigned to the upstream station can be allocated to the downstream station (first case Table 4). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Station 1** & **Station 2** & **Station 3** & **Station 4** \\ \hline \multicolumn{5}{c}{**First case**} \\ \hline Processing time assigned only to the station & 10 & 10 & 10 & 10 \\ \hline Fractional processing time & 2 & 2 & 2 & 2 \\ \hline \multicolumn{5}{c}{**Second case**} \\ \hline Processing time assigned only to the station & 8 & 8 & 8 & 8 \\ \hline Fractional processing time & 4 & 4 & 4 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Processing time of the stations. \"b\", it has been considered that the total processing time of the fractional tasks of the couple stations can be allocated between the upstream and downstream station (first case Table 4). * \"c\", it has been considered that only the processing time of the fractional tasks assigned to the upstream station can be allocated to the downstream station (second case Table 4). * \"d\", it has been considered that the total processing time of the fractional tasks of the couple stations can be allocated between the upstream and downstream station (second case Table 5). The following figures report the performance measures studied for each inter-arrival condition for all models to provide a complete overview of the main effects as the percentage difference from the benchmark case. Figure 4 shows the percentage difference of the average time in system measures compared to the benchmark case without redundancy. The model with higher productivity where the raw items are always available highlights how the fuzzy models are always better than the queue models. The improvements are higher from fuzzy \(a\) to fuzzy \(d\) for all simulations. This means that the increment of the fractional processing time leads to improving this performance. The introduction of the inter-arrival time of the raw items drastically changes this performance. If the inter-arrival time follows an exponential distribution, which is characterized by a relevant variation, the queue models lead to better results. Generally, the improvements are lower than the case with raw items always available. If the inter-arrival time follows a normal distribution with a standard deviation of 10% or 20%, the fuzzy models lead to better results than the queue models. The coefficient of variation of the inter-arrival time has a relevant effect on this performance. The effect of the failures (cases 1 to 10) is well absorbed by the fuzzy models as shown in Figure 5.  
Figure 5 shows the standard deviation of the average time in system as the percentage difference compared to the benchmark. The fuzzy models have the same behavior of the average time in system for the case in which the raw items are always available. When the inter-arrival time of the raw items is introduced (both exponential and normal), the queue models reduce the standard deviation of the average time in system. Then, the average time in system and standard deviation of this performance measure have the same behavior studying the different models. Figure 6 reports the work in process (WIP) of the flow line as the percentage difference compared to the benchmark. The fuzzy models allow a relevant reduction of the work in process to be obtained in the case of maximum productivity. Additionally, for this performance, when the inter-arrival of the raw items follows an exponential distribution, the queue models is the better approach.