## 3 Theory: Problem Statement and Mathematical Formulation
### Model B1 Formulation: Team Formation  
Model B1 aims to form a team of workers in each cell to ensure that part production demands can be satisfied in the worker assignment stage. Based on this idea, we proposed a new objective function for part-skill scores. The part-skill score objective consists of the summation of the total worker capabilities and the weight of each part production demands in each cell. [PERSON] (2016) motivated us to consider team capabilities to ensure that there are enough skilled workers in the cell. As forming an effective team in the cell is the main concern, two additional constraints are introduced in this model. With two additional notations to those summarised in Section 3.1, we can formulate Model B1 as follows:  
\begin{tabular}{l} Input Parameter \\ \(dem_{p}\) \\ Decision variable \\ \(z_{w}\) \\ \end{tabular}  
If model of part \(p\) is a positive integer, \(1\); otherwise, \(0\)  
If model \(w\) is assigned to cell \(c\), \(1\); otherwise, \(0\)  
The objective functions and penalty functions of Model B1 can be expressed as follows:  
\[\max\,Z_{1}=\sum_{c=1}^{C}\left(\sum_{p=1}^{P}\left(\frac{h_{pc}\cdot D_{p}}{ \sum_{p=1}^{P}\left(h_{pc}\cdot D_{p}\right)}\cdot\sum_{w=1}^{N}\left(z_{wc} \cdot s_{wt}\cdot q_{pt}\cdot g_{tc}\right)\right)\right)-\sum_{k=1}^{5}pen_{k} \tag{19}\]  
\[pen_{1}=\max\left(\sum_{p=1}^{P}\left(j\cdot\left(1-\sum_{w=1}^{W}\left(z_{wc} \cdot s_{wt}\cdot q_{pt}\cdot g_{tc}\cdot dem_{p}\right)\right)\right),0\right) \tag{20}\]  
\[pen_{2}=j\cdot\sum_{c=1}^{C}\max(L-normCOH_{c},0) \tag{21}\]  
\[pen_{3}=j\cdot\max\left(\sum_{c=1}^{C}\left(\sum_{w=1}^{W}z_{wc}-\sum_{t=1}^{ T}g_{tc}\right),0\right) \tag{22}\]  
\[pen_{4}=j\cdot\sum_{w=1}^{W}\left(\sum_{c=1}^{C}z_{wc}-1\right) \tag{23}\]  
\[pen_{5}=j\cdot\max\left(\sum_{w=1}^{W}\left(1-\sum_{c=1}^{C}z_{wc}\right),0\right) \tag{24}\]  
subject to,  
\[COH_{c}=\sum_{u,v\in W}\left(r_{uv}\cdot z_{wc}\cdot z_{vc}\right)\ \ \ \forall c \tag{25}\]  
\[maxCOH_{c}=o\cdot\sum_{u,v\in W}\left(z_{wc}\cdot z_{vc}\right)\ \ \ \forall c \tag{26}\]\[minCOH_{c}=n\cdot\sum_{u,w\in W}(z_{uc}\cdot z_{vc})\ \ \ \forall c \tag{27}\]  
\[normCOH_{c}=\frac{COH_{c}-minCOH_{c}}{maxCOH_{c}-minCOH_{c}}\ \ \ \forall c \tag{28}\]  
\[z_{uc}\in 0,1 \tag{29}\]  
\[maxCOH_{c},\ minCOH_{c},\ COH_{c}\in INT^{+} \tag{30}\]  
\[normCOH_{c}\in\ R^{+} \tag{31}\]  
Equation (19) shows the part-skill score objective function calculated using the summation of the product of part \(p\) demand weight and the total number of worker skills at part \(p\) of each cell. Equations (20)-(24) are the penalty functions of Model B1, which are considered in Equation (19). Equation (20) calculates the additional penalty when there is no skilled worker in cell \(c\) to operate the task of the demanded part \(p\). Equation (21) represents the penalty of cell \(c\) cohesion by taking Equations (25)-(28) into account, similar to Equation (6) of Model A. Equation (22) penalises the solution when the number of workers in cell \(c\) exceeds the number of total tasks in cell \(c\). Equation (23) represents the cell-transfer penalty. Equation (24) represents the additional penalty for unassigned workers. Equation (29) represents the binary decision variable in Model B1. Equation (30) represents the positive integer variable in Model B1. Finally, Equation (31) represents the positive real-number variable in Model B1.