## 2 Materials and Methods
### Mathematical Formulation of the Problem  
The facility layout issue with equal spaces is formulated as a discrete assignment problem, where WS must be optimally positioned to minimize the total distance traveled by workers within the production area. The workspace is structured as a rectangular grid  
Figure 1: Facility layout and its relationship with the alphanumeric code. The diagram shows the spatial arrangement of the facility elements (RMW, MSA, FPW, WT, WS, SAR) within the production system. Each element is coded (COD), decoded (DEC), and linked to a specific workshop station (WS). Colors indicate functional categories: green for active workshop stations (WS), orange for worker stations (WT), gray for inactive or unassigned areas, and white for storage or service elements (RMW, MSA, FPW, SAR). The “Code” and “Decoding” matrices provide a numeric-alphabetic reference for identifying element positions. Cells marked with “0” indicate non-assigned zones. of 4 \(\times\) 8 equally sized cells, forming the layout matrix. The following notation is used throughout the model:  
* \(i\in\) I = {1,2,3,4}: row index of the layout matrix. * \(j\in\) J = {1,2,...8}: column index of the layout matrix. * \(k\in K\) = {1,2,...,16}: index of workstations (WS). * \(l\in L\) = {1,2,3}: index of workers. * \(N=1\)\(K\)\(\mid\) 16: total number of assignable positions in the matrix. * \(d(i,j)\),(\(i^{\prime},j^{\prime}\)): Euclidean distance between two cells: (\(i,j\)) and (\(i^{\prime},j^{\prime}\)). * \(r_{l}\): route assigned to worker \(l\), as a sequence of WS positions. * \(s_{l}\): ordered set of cells visited by worker \(l\) based on task sequence. * \(d_{l}\): total travel distance incurred by worker \(l\). * \(x_{ij}{}^{k}\in\) {0,1}: binary decision variable: 1 if workstation \(k\) is placed at cell (\(i,j\)), 0 otherwise. The objective function that minimizes the total distance of all workers is as follows: \[Min\;Z=\sum_{l\in L}d_{l}\] (1) Constraints. The total travel distance for each worker \(l\) is computed as the sum of Euclidian distances between consecutive cells in the route \(\mathbf{s}_{l}\), given the layout defined by the decision variables:  
\[\sum_{(i,j),(i^{\prime},j^{\prime})\in\;\mathbf{s}_{l}}d_{(i,j),(i^{\prime},j ^{\prime})}*\;x_{ij}^{r_{l}}*x_{ij^{\prime}}^{r_{l+1}}=d_{l}\;\;\;\forall l\in L \tag{2}\]  
Each workstation must be assigned to exactly one location:  
\[\sum_{i\in I}\sum_{j\in J}x_{ij}^{k}\geq 1\;\forall k\in K \tag{3}\]  
Each cell can host at most one workstation:  
\[\sum_{k\in K}X_{ijk}\leq 1\;\;\;\forall i\in I,\;\forall j\in J \tag{4}\]  
The total number of assigned cells equals the number of workstations:  
\[\sum_{i\in I}\sum_{j\in J}\sum_{k\in K}x_{ijk}^{k}=N \tag{5}\]  
Fixed pre-assignments. Some workstations are pre-assigned to specific locations due to physical or infrastructure constraints:  
\[x_{11}^{1}=1 \tag{6}\]  
\[x_{21}^{1}=1 \tag{7}\]  
\[x_{31}^{2}=1 \tag{8}\]  
\[x_{27}^{3}=1 \tag{9}\]  
\[Min\;Z=\sum_{l\in L}d_{l}\]0  
\[Min\;Z=\sum_{l\in L}d_{l}\]1  
\[Min\;Z=\sum_{l\in L}d_{l}\]2The proposed model seeks to optimize the facility layout by minimizing the total travel distance of all workers, as defined in objective function (1). This objective is needed for improving process efficiency and reducing idle time within the production system. Constraint (2) ensures that each worker's assigned route aligns with the spatial distribution of workstations, enabling accurate distance calculations based on Euclidean metrics between consecutive cells. Constraint (3) ensures that each workstation is assigned to exactly one location, while constraint (4) prevents multiple workstations from being placed in the same cell, preserving the validity of the layout. Constraint (5) ensures that the number of assigned workstations equals the number of assignable positions, guaranteeing full use of available space.