## 1 Introduction  
Assembly lines are used in the mass production of complex items found in many contexts such as in the automotive sector, the field of consumer electronics, home appliances, etc. The assembly lines consist of stations that perform the tasks needed to obtain the final items. The operations to complete the tasks of the stations can be performed manually, automatically, or through a collaboration between humans and robots (defined as Cobot [1]). The assembly line design defines the assignment of operations to the stations, maximizing workload balancing [2]. Classic design methodologies assign operations exclusively to a single station for maximum productivity. The precedence constraints that characterize the product and the timing of the tasks significantly limit the maximum achievable productivity. In recent years, the increase in productivity has also been pursued with greater automation and the introduction of human-robot collaboration systems [3; 4; 5]. The increasing automation in production lines leads to high investments, so maximizing productivity is of fundamental importance for the return on investment. Some unforeseen events such as the short failures of the stations [6] reduce the profitability of assembly lines due to the throughput loss with starvation or blocking of the stations. Approaches to improve productivity and mitigate the effects of unexpected events have been proposed in the recent literature. The introduction of additional flexibility in mixed-model lines by identical stations was proposed by [PERSON] et al. [7; 8; 9]. This approach leads to layout complications and can increase costs more than a fractional task allocation. The concept of the fractional assembly line allows sharing of some tasks among the stations to reduce the cycle time with a higher throughput of the line [10]. Following this approach, two stations can share some tasks that represent additional balancing flexibility to reduce the cycle time and increase the throughput. The fractional assembly lines are characterized by some tasks split among stations; therefore, the related processing time is assigned to more stations that collaborate to perform the entire task. Some restrictions are necessary to allow the split of these tasks so that each station pair can only share one task, and each task can only be shared by two adjacent stations. The assembly line problem has to include the percentage of the task assigned to each station with the relative processing time part. The introduction of buffers with preventive maintenance policies can mitigate the effects of short failures [11; 12] increasing the capital costs and dimensions of the assembly lines. The redundancy of the operations in a flow line can reduce the effect of short failures avoiding the increment of capital costs and dimension [13; 14]. This approach concerns the possibility to move some operations to downstream stations when a failure occurs. To support this configuration, the stations need the necessary equipment to perform some operations assigned at the upstream stations. The development of digitalization in production systems supports the development of sharing and redundancy approaches. In particular, digital transformation such as \"Industry 4.0\" enables the connectivity to collect data in real-time [15] which support the rapid reaction to unforeseen events by the reconfigurations of the stations [16; 17]. The design models have to be modified to include the fractional assembly lines concept [10] and task redundancies [13; 14], and the literature proposed some approaches in recent works. The models proposed in the literature define statically the fraction of the tasks assigned to the stations. Greater benefits could be obtained by using a dynamic allocation policy. The most recent works on this topic do not address the potential improvements derived from the dynamic allocation of fractional tasks. This research proposed a fuzzy engine to support the dynamic allocation of the operations in fractional assembly lines with redundancy.