## 4 Methodology and Mathematical Model  
This section presents the methodology adopted to develop a new MIP model for the version of MALSP addressed in this paper that leverages and further develops some aspects proposed in [PERSON] et al. [38]. First, the proposed methodology suggests defining the indices of the model to develop, which represents the entities considered in the MALSP. Then, it is necessary to identify what are the decision variables. This step is very delicate since inadequate variable modeling would result in a problem difficult to be solved or with missing relevant aspects. Furthermore, the objective function has to be defined considering all the different performances of the assembly line that would be maximized, resulting in a probable trade-off among them. Finally, it is necessary to identify the set of the different constraints that should adequately model all the peculiarities of the considered problem.  
In particular, to solve this problem, the required decisions concern the total number of workstations, the number of workers, and the equipment types allocated to them, the subset of tasks assigned to each worker, and the scheduling of these tasks during the cycle time. The objective proposed is to determine the optimal compromise between minimizing the total number of workers on the line, the workstation number, and the workload variation between operators. In the presented model, the following assumptions are considered. Mass production of one homogeneous product on a serial line layout is taken into account with a fixed and constant cycle time. Product assembly has \(J\) tasks to be processed. Each task has a processing time \(t_{j}\), which is deterministic and independent from the assigned workstation.  
The following paragraph presents the model indices, decision variables, and parameters. Indices are adopted to represent the sets of tasks (\(j\), \(h\) = 1,.., ), workers (\(w\), \(v\) = 1,.., W), workstations (\(k\) = 1,.., K), mounting positions (\(l\), \(f\) = 1,.., L) and equipment types  
Figure 5: The framework of the multi-manned assembly line synchronization problem (MALSP).  
(\(e=1,..,E\)) of the multi-manned assembly. The MALSP requires some decision variables in addition to the traditional task to worker assignment \(x_{j\mu}\) and the workstation opening \(r_{k}\), such as the workstation equipment \(y_{ek}\), the worker to workstation belonging \(z_{\mu\text{sk}}\), the task starting time \(s_{i}\), and the tasks sequencing \(\text{u}_{\text{h}}\). The latter two variables are instrumental in managing the scheduling problem arising in the synchronization of tasks. Moreover, each task \(j\) is distinguished by the predecessor tasks \(p_{hj}\), in addition to other parameters specific to the problem under study that include the equipment needed \(b_{j\mu}\), the number of workers required \(g_{j}\), and its mounting position \(d_{jl}\). The maximum number of workers per workstation \(M\) and the maximum number of pieces of equipment for each type \(e\) on the line \(N_{e}\) are two key criteria to be set by the assembly line designer depending on the considered case study. Indeed, they have to fulfill certain preconditions, such as the compatibility with the product size and structure, the workstation layout, and the equipment costs. Similarly, the incompatibility between mounting positions \(a_{j\mu}\) determines whether different workers can simultaneously occupy positions \(i\) and \(j\) during the performance of their tasks.