# Influence of Manufacturing Process Modularity on Lead Time Performances and Complexity
## 6 Testing and Analysis of the Indicators through Theoretical Process Models  
For this purpose, process models from work by [PERSON] [56] who explored the complexity of six [PERSON]'s process graphs [57] were used for the theoretical experiment. It is assumed that 100 parts are produced on 22 machines (see Figure 6). In addition, set-up times, processing cycle times, and transport times were assigned. The allocated set-up time was two seconds, and processing cycle time was one minute for all 22 machines. Transport times were assigned based on the distance between the machines at five seconds between two machines located next to each other (e.g., M2 to M3); ten seconds between machines, e.g., M2 and M9; and 15 s between machines that are located further away from each other (e.g., M2 and M15). Figure 4: (a) Probabilities (\(p_{\text{iik}}\)) and related complexity values; (b) partial complexities from the machine view and part view, and cumulated complexities of machines and parts production. Figure 5: Print screen of the selected process model from Tecnomatix Plant Simulation. Process modularity and complexity values were enumerated using Equations (4) and (8), while MLT values were obtained using the simulation tool--Tecnomatix Plant Simulation. The obtained values are depicted using graphs in Figure 7.  
Following the obtained results, one can see from Figure 7a when comparing process modularity and MLT, that process modularity in Graph 1 is the highest while its MLT is the shortest. Process modularity of the remaining graphs has the same tendency in relation to MLT values. Then, it is possible to state from these summarized results, that modularity values are in relation with MLT values: as the modularity increases, MLT decreases, and vice versa. When comparing process modularity in relation to operational complexity from Figure 7b, it can be seen that Graph 1 has the lowest complexity, while its process modularity is the highest. The rest of the graphs report the same tendency, and it can be said that \(M(G)\) positively impacts the operational complexity of manufacturing systems. To answer the research questions mentioned in the introduction section and to confirm the positive impact of process modularity on MLT and process complexity, the two methods were further employed to find to what extent the \(M(G)\) effects MLT, and how modularity influences the process complexity of manufacturing systems. Figure 6: Theoretical process models. Figure 7: (a) Relationship between modularity and MLT; (b) relationship between modularity and process complexity for theoretical process models. At first, the Spearman correlation analysis was used to test these relationships. Spearman correlation coefficient rated using a known scale [58] between \(M(G)\) and manufacturing lead time, and between \(M(G)\) and process complexity equals (0.91 and 0.98, respectively). Based on these results, it means that there are very strong relationships. Then, the sensitivity index \"\(I\)\" is used to assess sensitivity between the \(M(G)\) and MLT values, as well as between \(M(G)\) and \(PCI\) values. For this purpose, the following formula was applied [59]:  
\[I=\frac{(y_{2}-y_{1})/y_{0}}{2\cdot\Delta x/x_{0}} \tag{10}\]  
where \(y_{0}\) is the model output calculated with an initial value \(x_{0}\) of the parameter \(x\); \(x_{1}=x_{0}-\Delta x\) and \(x_{2}=x_{0}+\Delta x\) with corresponding values \(y_{1}\) and \(y_{2}\).