## 4 Problem Exploration
### Practical Case Study  
The same procedure as in Section 4.1 is implemented here for manufacturing of customizable chairs. The product consists of seven assembly input components, namely, (A) back support, (B, H) cross bars, (C) front legs, (D) back legs, (E) sledge legs, (F) seat panel, and (G) arm rest. The feature diagram describing the product structure is depicted in Figure 8. \begin{table}
\begin{tabular}{c c c c c c} \hline
**APSs** & **Qd** & **CMI** & **M(G)** & **MLT (Minutes)** & **THR (Parts/h)** \\ \hline No.1 & 0.36 & 0.1 & 0.043 & 92.3 & 65.01 \\ No.9 & 0.4 & 0.1 & 0.039 & 75.88 & 79.07 \\ No.8 & 0.471 & 0.182 & 0.055 & 75.63 & 79.33 \\ No.10 & 0.438 & 0.182 & 0.042 & 76.18 & 78.76 \\ No.4 & 0.463 & 0.182 & 0.0496 & 67.3 & 89.49 \\ No.2 & 0.47934 & 0.182 & 0.047 & 60.1 & 99.83 \\ No.7 & 0.472 & 0.25 & 0.054 & 59.68 & 100.53 \\ No.3 & 0.493 & 0.25 & 0.048 & 51.82 & 115.79 \\ No.5 & 0.486 & 0.25 & 0.0504 & 51.02 & 117.61 \\ No.6 & 0.47929 & 0.308 & 0.0494 & 43.24 & 138.25 \\ \hline \end{tabular}
\end{table} 
Table 4: Summarized results of process modularity indicators and efficiency indicators. Figure 7: Examples of Gantt charts of (**a**) APS of No.1, and (**b**) APS of No.6. The assembly process encompasses six assembly operations, which can be processed in individual workstations, or integrally. The assembly process is able to complete three types of products (P\({}_{1}\), P\({}_{2}\), and P\({}_{3}\)). The compositions of the products are as follows: P\({}_{1}\) = {A, E, F}; P\({}_{2}\) = {A, B, C, D, F, H}; P\({}_{3}\) = {A, B, C, D, F, G, H}. Original assembly process sequences for individual customized products varies, as depicted in Figure 9a, while the integral assembly process for all the product types is schematically described in Figure 9b. To optimize the integral assembly process structure, first of all, alternative process models need to be generated. Theoretically it is possible to identify 90 APS alternatives [42], which are depicted in Figure 10, where the number of all possible APS structures follows integer sequence A000669 (number of series-reduced planted trees with \(n\) leaves) [45]. As reasoned above, the MuMAL strategy was applied here to explore the influence of process modularity on assembly process efficiency, since it brings better results than MiMAL in the given application (see Section 3.1). To calculate MLT and THR, operation times for individual alternatives of products and structures are provided in Table 5.  
Figure 8: Feature product diagram. Figure 9: (a) Specific assembly process sequences; (b) whole assembly process. \begin{table}
\begin{tabular}{l c} \hline \hline
**No. of Assembled Parts** & **Assembly Time** \\ \hline
2 (e.g., A + E) & 20 \\
3 (e.g., A + E + B) & 30 \\
4 (e.g., A + D + B + F) & 40 \\
5 (e.g., AD + B + F + C + H) & 50 \\
6 (e.g., A + D + B + F + C + H) & 60 \\
7 (e.g., A + D + B + F + C + H + G) & 70 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Assembly time based on the number of assembled parts in seconds. It is worth noting that the order of APSs in Table 6 (by decreasing MLT or increasing THR) is identical, since this is a consequence of [PERSON]'s law [46], i.e., if MLT decreases, then THR increases. Based on these results, one can state that the best MLT and THR values were obtained using APS No.10, and the same structure was recognized as the optimal modular configuration according to M(G), while, according to indicator Qd, there were two most optimal modular APSs (No.4 and No.9), where the MLT and THR values were far from the optimal ones. Similarly, the CMI indicator seems to be not applicable for the given purpose, as eleven APSs were identified as optimally modular using this. Due to this reason, process modularity indicators M(G) and Qd can be considered as more relatable to measure the modularity as they are more sensitive than the CMI indicator. To express the closeness of the relationships, the Spearman correlation coefficient was employed for this purpose.