## 4 Staffing a Multi-Model Assembly Line
### Balancing the Multi-Model Line  
The last step to design a line is the assignment of tasks to workstations [46]. Each model (\(k\)) can be considered as a separate SMALBP because the time devoted to each product (\(TP_{k}\)) is now known  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Task (\(i\))** & **Alpha** & **Beta** & **Gamma** \\ \hline
1 & 0.51042 & 0.44989 & 0.61362 \\
2 & 0.59549 & 0.98975 & 0.99713 \\
3 & 0.34028 & 0.53986 & 0 \\
4 & 0.42535 & 0.44989 & 0.38351 \\
5 & 0.42535 & 0.26993 & 0.15340 \\
6 & 0.17014 & 0.08998 & 0.30681 \\
7 & 0.34028 & 0.08998 & 0.23011 \\
8 & 0.59549 & 0.35991 & 0.99713 \\
9 & 0.25521 & 0.35991 & 0.07670 \\
10 & 0.08507 & 0.08998 & 0.07670 \\
11 & 0.85069 & 0.98975 & 0.61362 \\
12 & 0.08507 & 0 & 0.23011 \\ \hline \hline \multicolumn{3}{c}{\(TWu_{k}=\Sigma Wu^{*}_{ik}\)} & 4.679 & 4.679 & 4.679 \\ \hline \hline \end{tabular}
\end{table}
Table 7: New unit workload (\(Wu^{*}_{ik}\)) per task and model.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Model** & **Unit Load** & **Production** & **Production Rate** & **Daily Output** \\ **(\(k\))** & \(TWu_{k}\)** & **(min/Day)** & **(Units/Day)** & **(Units)** \\  & & \(TP_{k}\)** & \(P_{ik}\)** & \(P_{k}\)** \\ \hline Alpha & 2.674 & 274.0 & 2450.00 & 1400 \\ Beta & 1.264 & 130.0 & 2591.35 & 700 \\ Gamma & 0.741 & 76.0 & 2209.02 & 350 \\ \hline Total & 4.679 & 480.0 & - & 2450 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Expected performance of the multi-model line.  
(as well as the minimum number of workstations). Therefore, each model can be solved using _gammazumi_ charts, heuristics or exact methods. In our example, in order not to miss the optimal solution, ILP was used, model by model. Expression (23) is the multi-model objective function and Equations (24)-(27) show linear constraints. Each variable \(x_{ij}\) (or \(x_{hi}\) ) is 1 if task \(i\) is assigned to workstation \(j\). Otherwise, \(x_{ij}\) is zero. Twelve tasks (\(|I|=12\)) and up to six (\(|I|=6\)) possible workstations are considered in the model because the required number of workstations might not be the theoretical minimum value. If we compare Equation (25) with [PERSON] model [44], workload values (\(Wu^{*}_{ik}\)) found in previous sections are used instead of task times and the capacity of each workstation (\(E_{j}\)) replaces the cycle time of the line. \(E_{j}\) may be either zero or one (one operator), depending on whether the station is necessary or not in that line. Table 9 displays the optimal solution for each model. The result shows that, for every model, all work elements are assigned to five workstations or operators.  
\[\text{Objective function}:\ Min\ \sum_{j}E_{j} \tag{23}\]  
Subject to  
\[\sum_{j}x_{ij}=1\ \forall i \tag{24}\]  
\[\sum_{i}Wu^{*}_{ik}x_{ij}\leq E_{j}\ \forall j \tag{25}\]  
\[\sum_{j}\ j\!:\!x_{ij}\leq\sum_{j}\ j\!:\!x_{hi}\ \forall i\ \prec h \tag{26}\]  
\[E_{j+1}\leq E_{j}\ \forall j \tag{27}\]  
To measure the efficiency of a multi-model line, we develop Equation (28) in terms of load and capacity [34]. In our example, since the total multi-model unit workload of the system (\(TWu\)) is 4.68 and the capacity (\(\omega\)) is 5, the efficiency is 0.936 or 93.6 percent.  
\[Efficiency=\frac{TWu}{w}\cdot 100 \tag{28}\]