# Analytic Hierarchy Process and Multilayer Network-Based Method for Assembly Line Balancing
## 4 Case Study
\begin{table}
\begin{tabular}{l l l l l l l l l} \hline \hline  & **E1** & **E2** & **E3** & **E4** & **E5** & **E6** & **E7** & **E8** \\ \hline A1 & 1 & 1 & & & & & \\ A2 & 1 & 1 & & & & & \\ A3 & 1 & 1 & &  
The tables illustrate that the practical implementation of line balancing problems is also influenced by how much equipment is needed for the designed production line and how many skills should be learnt by the operators. All the collected information is transformed into network layers, as shown in Figure 5. The top of the figure shows the bipartite networks that represent the details of the assignments, while the bottom of the figure represents the tree layers of the network that define the activity-operator, skill-operator, and equipment-operator assignments. As can be seen, this representation is beneficial as it shows how similar operators, skills and equipment can be grouped into clusters. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline  & **S1** & **S2** & **S3** & **S4** & **S5** & **S6** \\ \hline A7 & & 1 & & & \\ A16 & & 1 & & & \\ A17 & 1 & & & & \\ A18 & & 1 & & & \\ A19 & & & 1 & & \\ A20 & & & 1 & & \\ A21 & & & 1 & & \\ A22 & & & 1 & & \\ A23 & & & 1 & & \\ A24 & & & & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: _Cont._  
Figure 5: Illustration of the skill–operator and equipment–operator assignments after line balancing. Although this is not shown in the figure, the weights of the edges represent the costs or benefits of the assignments. The final form of the network is formed based on a multi-objective optimisation of the sets of active edges. As some of these objectives are difficult to measure, we utilised the proposed AHP-based method to convert the pairwise comparisons of the experts into weights of criteria. The structure of the decision problem is represented in Figure 6. As this figure illustrates, the AHP is used to compare difficult to evaluate equipment and skill assignment costs and the importance of the objectives. The pairwise comparison was performed by a process engineer, and the resulting comparison matrices can be found in Tables 7-9. Based on the analysis of the the eigenvalues of these matrices [25], we found that the evaluations were consistent. Since the activities cannot be performed in parallel, a precedence graph defines the most crucial question, namely whether a permutation of sequence \(\pi\) is feasible. Based on the transitive closure of the adjacency matrix of the graph, the interchangeable sets  
Figure 8 presents the different time, skill and equipment-related objectives in the case of different operators. As the results show, the increase in the number of operators decreases the efficiency of the utilisation of the tools and skills (this trend is the main driving force for forming manufacturing cells). The process can be well balanced in the case of 3-5 operators; e.g., in the case of five operators, in one of the best solutions, the balancing objectives are a time cost of 94.3%, training cost of 75.0% and equipment cost of 72.8%. Figure 9 shows the different total activity times of each operator during the simulation. In this case, the station times do not differ greatly, and the result is optimal [10]. The proposed algorithm was implemented in MATLAB and is available on the website of the authors (www.abonyilab.com/about-us/software-and-data); interested readers can make further comparisons, and the proposed problem can serve as a benchmark for constrained multi-objective line balancing. Based on the simple modification of the code, the algorithm can be compared to classical simulated annealing-based line balancing; this comparison demonstrates that the main benefit of the proposed constrained handling is the acceleration of the optimisation. At the same time, the application of the inner-loop-based assignment significantly reduces the variance and increases the chance of obtaining improved line balancing results.