## 4 Simulation Environment  
The simulations have been conducted considering three factors: the inter-arrival of the raw items, the failures, and the processing time variability. The inter-arrival studied are three cases: the first case considers the raw items always available to simulate the case with higher throughput; then, the inter-arrival follows an exponential distribution (with two possible values) that have a higher coefficient of variation (CV = 1); finally, the inter-arrival follows a normal distribution with means and 20% of the variation (CV = 0.2). The unforeseen events concern the failures and the variations in processing time. The failures are characterized by a Mean Time To Repair (MTTR fixed to 10) and a Mean Time Between Failures (MTBF) with two cases that lead to 96 % and about 92% of efficiency. The variations in the processing time follow a normal distribution of 20 % and 40 % of the standard deviation of the mean (see Table 4). The fuzzy sets of the buffer levels (Figure 2) are the following:  
* a\({}_{1}=-3\); b\({}_{1}=0\); c\({}_{1}=3\);
* a\({}_{m}=0\); b\({}_{m}=3\); c\({}_{m}=6\);
* a\({}_{h}=3\); b\({}_{h}=6\); c\({}_{h}=9\). To investigate only the effect of the redundancy models, the simulation models consider a perfectly balanced line. Two different parts of the fractional tasks' processing time are considered with the same cycle time. Table 5: the first case concerns a two-unit time of fractional processing time, while the second case considers an increment to four-unit time of the fractional processing time. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Exp. No.** & **MTBF** & **MTTR** & **Inter-Arrival** &
\begin{tabular}{c} **Standard Deviation of the** \\ **Processing Time** \\ \end{tabular} \\ \hline  
ORM (13 10\%) & 40\% \\ \hline
20 & - & - & NORM (14 10\%) & 40\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Experimental classes. The models tested are the following: the first benchmark is a model without the possibility of task redundancy, while the second benchmark is a model proposed in the literature [27] that considers the buffer level to decide the allocation of the fractional tasks between two consecutive stations. This model allocates the entire fraction task to the first or second station of the couple. The better values for the buffer thresholds are obtained by simulating several cases to determine the best performance. The proposed fuzzy approach is compared to these two benchmarks to highlight the real added value by a fuzzy method. The simulation models are developed using the software package SIMUL8. The simulation length is 28,800 min, enough to assure the stability of the simulations. Simulations are conducted following the terminating simulation approach. For each experimental class, a number of replicates were conducted, capably ensuring a confidence interval of 5% and 95% of the confidence level for each performance measure. The performance measures to evaluate the proposed method are the following:  
* The throughput rate of the assembly line as the number of final items completed over the simulation horizon. In this study, this performance does not depend on the cycle time because the design of the flow line is provided. * The average time in the assembly line of the items: this average time includes the production time in the stations and the time blocked in the queue when the stations are working on other items or are in a failure state. This measure determines the lead time which is a relevant metric to evaluate how quickly the flow line can respond to changing market needs. * The standard deviation of the average time in the assembly line to evaluate the stability of the average time: this metric evaluates the stability of the average time in the system and, therefore, of the lead time. * The work in process of the assembly line as the sum of the items in queues: this metric evaluates the costs due to the temporary inventory of the items and space needed for the buffer.