# A Study on Operator Allocation in Consideration of Fatigue in Cell Manufacturing System +
## 3 The Target Model
### Formulation  
In this study, we incorporate a learning index that accounts for fatigue when updating the skill index at the end of each order. In the study of [PERSON] et al. (2019), proficiency progressed as tasks were assigned, but in this study, skill improvement was hampered by fatigue depending on the number of tasks assigned. Therefore, tasks that each operator struggles with are assigned preferentially to facilitate skill proficiency. Consequently, the objective function is set to minimize the sum of the skill index for each task allocated to  
Figure 2: Learning–fatigue curve (learning rate of 80%, fatigue index of 0.15, 0.20, 0.25).  
operators, as defined in Equation (5), where \(S_{ijk}\) is the skill index values for each operator's task, the decision variable \(x_{itc}\) represents who is assigned to which cell, and \(y_{itjk}\) represents who is assigned to which task.  
\[min\sum_{i=1}^{I}\sum_{j=1}^{J}\sum_{k=1}^{K}s_{ijk}y_{itjk}\ (\forall_{t}) \tag{5}\]  
The constraints are defined by Equations (6)-(9). Equation (6) is the constraint that operators are always placed in one cell, and Equation (7) is the constraint that the maximum number of people in one cell must not exceed \(C_{max}\).  
\[\sum_{c=1}^{C}x_{itc}=1\ (\forall_{t,i}) \tag{6}\]  
\[\sum_{i=1}^{I}x_{itc}\leq C_{max}\ (\forall_{t,c}) \tag{7}\]  
Equation (8) is a constraint that ensures that operators are assigned to adjacent tasks when they are responsible for multiple tasks.  
\[\sum_{k=F_{ijk}}^{E_{ijk}}y_{itjk}=E_{itjk}-F_{itjk}+1\leq K_{jmax}\ (\forall_{i,t,j}) \tag{8}\]  
In Equation (9), the number of operators who can be assigned to the same task within a cell is set to be one, achieving a task-dedicated system.  
\[\sum_{i=1}^{I}x_{itc}y_{itjk}=1\ (\forall_{t,c,j,k}) \tag{9}\]  
The tact time for each cell is calculated by Equation (10). Based on the tact time, the number of products to be assigned to each cell is determined by Equation (11).  
\[TactTime_{tj}=max\Bigg{\{}\sum_{k=1}^{K}\frac{ST_{jk}}{S_{1 jk}x_{1 tc}y_{1 tk}},\sum_{k=1}^{K}\frac{ST_{jk}}{S_{2 jk}x_{2 tc}y_{2 tk}},\ldots\Bigg{\}}\ (\forall_{t,cj}) \tag{10}\]  
\begin{table}
\begin{tabular}{c l} \hline \hline \multicolumn{1}{c}{**Variables**} \\ \hline \(i\) & The operator number (1, 2, \(\ldots\), 1) \\ \(c\) & The cell number (1, 2, \(\ldots\),  C) \\ \(j\) & The product number (1, 2, \(\ldots\),  J) \\ \(k\) & The task number (1, 2, \(\ldots\),  K) \\ \(t\) & The order number (1, 2, \(\ldots\),  T) \\ \(S_{ijk}\) & The skill index values for each task of each operator \\ \(x_{itc}\) & The decision variable represents who to assign to which cell \\ \(y_{itjk}\) & The decision variable represents who to assign to which task \\ \(C_{max}\) & The maximum number of operators that can be allocated to a single cell \\ \(K_{jmax}\) & The maximum number of tasks that can be allocated to a single operator \\ \(F_{itjk}\) & The first task number allocated to the operator \\ \(E_{itjk}\) & The last task number allocated to the operator \\ \(Q_{ijall}\) & The order quantity of product \(j\) in order \(t\) \\ \(Q_{tcj}\) & The order quantity in cell \(c\) of product \(j\) in order \(t\) \\ \(TaktTime_{tj}\) & The tact time in cell \(c\) of product \(j\) in order \(t\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Symbols.  
\[Q_{tej}=\ Q_{tjall}/\left(\text{{TactTime}}_{tej}\sum_{c=1}^{C}\frac{1}{\text{{TactTime }}_{tej}}\right)\ \left(\forall_{t,c,j}\right) \tag{11}\]  
The assembly time after the order completion is calculated based on the tasks assigned to the operators and the number of products. The conventional method [10] is obtained by Equation (12) and the proposed method by Equation (13).  
\[ET_{ijk}=\frac{ST_{jk}}{S_{ijk}x_{itc}y_{itjk}}/Q_{tcj}{}^{\alpha} \tag{12}\]  
\[ET_{ijk}=\frac{ST_{jk}}{S_{ijk}x_{itc}y_{itjk}}/Q_{tcj}{}^{\alpha(Q_{tcj})} \tag{13}\]  
where \(\alpha\) in Equation (12) is calculated using Equation (3) and \(\alpha\big{(}Q_{tcj}\big{)}\) in Equation (13) is calculated using Equation (4).