## 3 Theory: Problem Statement and Mathematical Formulation
### Model A Formulation  
In addition to the notation summarised in Section 3.1, Model A can be formulated using two additional variables, as follows:  
\begin{tabular}{l l} Variable & \\ \(y_{wt}\) & \\ Decision variable & \\ \(x_{wt}\) & \\ \end{tabular}  
Figure 1: Illustration of TFWAP.  
The objective functions and penalty functions of Model A can be expressed as follows:  
\[\min Z_{1}=\sum_{p=1}^{p}\Bigg{(}\sum_{w=1}^{W}\frac{3600\cdot q_{pt}\cdot x_{wp }\cdot\alpha_{wt}}{ST_{p}}-D_{p}\Bigg{)}+\sum_{k=1}^{5}pen_{k} \tag{1}\]  
\[\min Z_{2}=\frac{1}{C}\sum_{c=1}^{C}\text{IVR}_{c}+\sum_{k=1}^{5}pen_{k} \tag{2}\]  
\[pen_{1}=\sum_{p=1}^{p}\max\Bigg{(}j\cdot\Bigg{(}D_{p}-\sum_{w=1}^{W}\frac{3600 \cdot q_{pt}\cdot x_{wp}\cdot\alpha_{wt}}{ST_{p}}\Bigg{)},0\Bigg{)} \tag{3}\]  
\[pen_{2}=\max\Bigg{(}\sum_{w=1}^{W}\Bigg{(}j\cdot\Bigg{(}\sum_{p=1}^{p}x_{wp}-A \Bigg{)}\Bigg{)},0\Bigg{)} \tag{4}\]  
\[pen_{3}=j\cdot\sum_{w=1}^{W}\Bigg{(}\sum_{c=1}^{C}y_{wc}-1\Bigg{)} \tag{5}\]  
\[pen_{4}=j\cdot\sum_{c=1}^{C}\max(L-normCOH_{c},0) \tag{6}\]  
\[pen_{5}=j\cdot\sum_{c=1}^{C}\max\Bigg{(}\Bigg{(}\sum_{w=1}^{W}y_{wc}-\sum_{t= 1}^{T}g_{tc}\Bigg{)},0\Bigg{)} \tag{7}\]  
subject to,  
\[IVR_{c}=\frac{1}{T_{c}}\sqrt{\frac{\sum_{w=1}^{W}\left(I_{wc}-\overline{I_{c} }\right)^{2}}{\sum_{w=1}^{W}y_{wc}}}\quad\forall c \tag{8}\]  
\[\overline{I}_{c}=\frac{1}{\sum_{w=1}^{W}y_{wc}}\sum_{w=1}^{W}I_{wc}\quad\forall c \tag{9}\]  
\[I_{wc}=A-\sum_{P=1}^{p}x_{wp}\cdot h_{pc}\cdot s_{wt}\cdot q_{pt}\quad\forall c,t,w \tag{10}\]  
\[y_{wc}=x_{wp}\cdot h_{pc}\quad\forall w,c,p \tag{11}\]  
\[COH_{c}=\sum_{u,vcW}(r_{uv}\cdot y_{wc}\cdot y_{vc})\quad\forall c,p \tag{12}\]  
\[maxCOH_{c}=o\cdot\sum_{u,vcW}(y_{wc}\cdot y_{vc})\quad\forall c,p \tag{13}\]  
\[minCOH_{c}=n\cdot\sum_{u,vcW}(y_{wc}\cdot y_{vc})\quad\forall c,p \tag{14}\]  
\[normCOH_{c}=\frac{COH_{c}-minCOH_{c}}{maxCOH_{c}-minCOH_{c}}\quad\forall c,p \tag{15}\]  
\[x_{wp},y_{wc}\in 0,1 \tag{16}\]  
\[I_{wc},COH_{c},\ maxCOH_{c},\ minCOH_{c}\ \in INT^{+} \tag{17}\]  
\[IVR_{c}\,\overline{I}_{c},normCOH_{c}\ \in\ R^{+} \tag{18}\]  
Equations (1) and (2) represent the two objective functions: the total inventory level and the average cell worker's idle time variation of all cells. Equations (1) and (2) can be calculated by considering the summation of the penalty functions of Equations (3)-(7). Equation (1) can be calculated using the sum of the parts produced by all workers for each part and its demand. Equation (2) can be calculated using the cell worker's idle time variation of cell \(c\) from Equation (8). Equation (9) shows the average worker idle time in cell \(c\). Equation (10) calculates the idle time of worker \(w\) in cell \(c\). Equation (3) represents the unsatisfied part of the demand penalty function. If the solution cannot satisfy the demand at part \(p\), then the penalty value of part p becomes positive. Otherwise, the penalty value for part \(p\) is zero. Equation (4) represents the working overtime penalty. The summation of the worker's working hours must not exceed the available working hours, \(A\). If the working hours of worker \(w\) exceeds \(A\), the penalty value of worker \(w\) can be calculated. Equation (5) addresses the cell-transfer penalty by considering Equation (11), which ensures that worker \(w\) works in cell \(c\) if worker \(w\) operates part \(p\) which belongs to cell \(c\). Equation (6) represents the summation of the cohesion penalties using Equations (12)-(15). Equation (12) represents the total cohesion value of cell \(c\) and calculates the possible maximum total cohesion score of cell \(c\). Equation (13) calculates the possible minimum cohesion score of cell \(c\). Equation (15) calculates the normalised cohesion score of cell \(c\). Equation (7) penalises the solution when the total number of workers in cell c exceeds the number of total tasks in cell \(c\). Equation (16) defines the binary variables. Equation (17) defines the positive integer variables. Finally, Equation (18) defines the positive real-number variables.