## 2 Literature Review
The simulations are conducted by introducing allocation policies for the shared tasks between the stations that work in real-time. These policies allocate the shared tasks completely to one or consecutive stations evaluating the level of the buffers. Table 1 classifies the main characteristics of the literature review considering the following issues: the dynamic assembly line balancing (D-ALB) problem addressed, the cross-trained workers (C-TW), the objective of throughput maximization (T-max) or cycle time minimization (CT-min), the application of heuristic rules, the possibility to apply work sharing (WS) for the fractional assembly line, the development of linear programming or mixed linear programming (LP), the main focus on the design of the flow line, the introduction of demand uncertain, the introduction of failures (Fail), the introduction of the redundancy of the tasks (red), the dynamic allocation (D-Al) of the percentage of the fractional tasks between two stations, the use of the fuzzy logic, and the introduction of the processing time uncertain (PT-un). \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Refs. **} & \multicolumn{2}{c}{**D-**} & \multirow{2}{*}{**C-TW**} & \multicolumn{2}{c}{**T-Max/CT-**} & \multicolumn{2}{c}{**Heuristic**} & \multirow{2}{*}{**WS**} & \multirow{2}{*}{**LP**} & \multirow{2}{*}{**Design**} & \multicolumn{2}{c}{**Demand**} & \multirow{2}{*}{**Fail**} & \multirow{2}{*}{**Red**} & \multirow{2}{*}{**D-Al**} & \multirow{2}{*}{**Fuzzy**} & \multirow{2}{*}{**PT-un**} \\  & **ALB** & & & & & & & & & & & \\ \hline
[19] & X & X & X & X & & & & & & & \\ \hline
[20] & X & & X & & & X & X & X & & & \\ \hline
[21] & X & X & X & & X & & & & & & \\ \hline
[22] & X & X & X & & & & & X & X & & \\ \hline
[23] & & & & X & & X & & & & & \\ \hline \hline \As highlighted in Table 1, the discussion of the works proposed on the fractional and redundancy assembly lines highlights how the model proposed is focused only on the design of the line, while the real-time allocation of the shared tasks is investigated only in one work. Few works studied the integration of the fractional and redundancy tasks together. The research proposed in this paper overcomes two of the principal limits of the literature which are the control of the percentage of the shared tasks between two adjacent stations in real-time (dynamic allocation) considering some states information of the flow line and the integration of the fractional tasks with the redundancy possibility. The model developed is based on a fuzzy inference engine that combines the information of the upstream and downstream buffers of each station to assign the percentage of the shared tasks between two adjacent stations. The focus of the dynamic allocation of the shared tasks is to improve the performance of the flow line under uncertain conditions. The fuzzy logic model developed allows for obtaining the dynamic distribution of the fractional tasks with more gradual changes under dynamic scenarios. Simulation models test the proposed fuzzy policy under different dynamic conditions of failures, processing time variations, and the inter-arrival time of the raw items. Then, the first research question of this paper is the following:  
RQ1: _Can a fuzzy policy support the dynamic allocation of the shared task in a fractional and redundancy assembly line optimizing the main performance measures_? The studies proposed in the literature do not evaluate the different causes of variations such as the parameters of the failures (mean time between failures and mean time to repair), several standard deviation levels of the processing time fluctuations, and the level of the utilization of the assembly line; then, the second research question is the following:  
RQ2: _What are the operating conditions of the assembly line in which the dynamic allocation of the shared tasks leads to better results_?