## 4 Methodology and Mathematical Model
### Constraints  
Concerning the model constraints, those stated by the Equations (9)-(12) characterize the majority of ALB problems. Constraint (9) ensures that each worker belongs exclusively to one station. Constraint (10) fastens the tasks to be assigned only to the workers that belong to any workstation. Constraints (11) and (12) are the workstation opening decision constraints. Constraint (12) ensures that workstations are loaded in an increasing manner noting the sequence of the workstation indices. Constraints (13) and (14) limit the task processing period. The former ensures that any assembly operation has to be finished within the cycle time; the latter guarantees the precedence relations among the tasks.  
In the case of large product assembly, it is possible that some tasks require cooperation between two or more workers for their execution. Thus, constraint (15) forces each task to be assigned uniquely to the number of workers needed to perform the task, ensuring that these workers belong to the same workstation.  
Constraints (16) and (17) are about equipment requirements. Constraint (16) allows a task to be executed in a workstation only if it is equipped properly. Constraint (17) manages the feasibility of the equipment sharing since it enables the installation of a maximum of one unit of each equipment type in every individual workstation.  
Constraints (18) and (19) consider the layout restrictions in which the line has to be installed, and the assembly system designer preferences. They guarantee that the maximum number of workers and the maximum number of each equipment type are not exceeded in any workstation.  
Constraints (20) and (21) ensure the feasibility of the synchronization, considering that different workers at the same station can execute several tasks simultaneously on a product and each of these tasks has specific technological predecessors. Constraint (20) guarantees that two tasks cannot precede each other simultaneously. Constraint (21) limits the decision variable \(u_{jh}\) to 0 if there is no parallelism between tasks \(i\) and \(h\) in the examined workstation. Constraint (22) ensures that each worker can handle one task at a time. For the duration of any task in a specific mounting position, constraint (23) disables the execution of all the tasks that involve incompatible mounting positions.  
Finally, constraints (24)-(28) show the integer variables. Constraint (29) indicates that the start time of every task should be non-negative.  
\[\sum_{k}z_{\text{z}uk}\leq 1 \forall w \tag{9}\] \[x_{jw}\leq\sum_{k}z_{\text{z}uk} \forall w,\forall j\] (10) \[x_{jw}\cdot z_{\text{z}uk}\leq r_{k} \forall k,\;\forall j,\forall w\] (11) \[r_{(k+1)}\;\leq\;r_{k} \forall k\] (12) \[s_{j}+t_{j}\leq\;c \forall j\] (13) \[p_{jh}\cdot(s_{j}+t_{j})\;\leq\;s_{h} \forall h,\forall j\] (14) \[\sum_{k}\left[\sum_{w}(x_{jw}-g_{j}+1)\cdot z_{\text{z}uk}\right] =1 \forall\;j\] (15) \[x_{jw}\cdot z_{\text{z}uk}\cdot b_{jx}\leq y_{ek} \forall k,\;\forall\;w,\forall j,\forall e\] (16) \[\sum_{e}y_{ek}\leq\;1 \forall k\] (17) \[\sum_{w}z_{\text{z}uk}\leq M \forall k\] (18) \[\sum_{k}y_{ek}\leq N_{e} \forall e\] (19) \[u_{jh}+u_{jj}\leq 1 \forall h,\forall j\] (20) \[u_{jh}\cdot(s_{j}+t_{j})\leq s_{h} \forall h,\forall j\] (21) \[x_{jw}\cdot x_{\text{z}uw}\leq u_{jh}+u_{hj} \forall j,\forall w,\forall h\] (22) \[\left(1-a_{lf}\right)\cdot\left(d_{jl}\cdot x_{jw}\cdot z_{\text{ z}uk}\right)+\left(d_{hf}\cdot x_{hw}\cdot z_{\text{z}uk}\right)\leq\left(u_{jh}+u_{ hj}\right) \forall l,\forall f,\forall j,\forall w,\forall k,\forall h,\forall v\] (23) \[x_{jw}\in\{0,1\} \forall j,\forall w\] (24) \[z_{\text{z}uk}\in\{0,1\} \forall w,\forall k\] (25) \[v_{ek}\in\{0,1\} \forall e,\forall k\] (26) \[r_{k}\in\{0,1\} \forall k\] (27) \[u_{jh}\in\{0,1\} \forall j,\forall h\] (28) \[s_{j}\in R^{+} \forall j \tag{29}\]