## 4 Problem Exploration
### Theoretical Case Study
\begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline
**Parts Assembly** & \(\mathbf{P_{1}}\) & \(\mathbf{P_{2}}\) & \(\mathbf{P_{3}}\) & \(\mathbf{P_{4}}\) & \(\mathbf{P_{5}}\) & \(\mathbf{P_{6}}\) & \(\mathbf{P_{7}}\) & \(\mathbf{P_{8}}\) & \(\mathbf{P_{9}}\) & \(\mathbf{P_{10}}\) \\ \hline A + B & 20 & 20 & 20 & - & 20 & 20 & 20 & 20 & - & - \\ A + B + V1 & 30 & 30 & 30 & - & 30 & 30 & 30 & 30 & - & - \\ AB + C + D + E + F & 50 & - & - & - & - & - & - & - & - & - \\ AB + C + D + E + F & V2 & 60 & - & - & - & - & - & - & - & - \\ ABV1 + C + D + E + F & 50 & - & - & - & - & - & - & - & - & - \\ ABV1 + C + D + E + F & V2 & 60 & - & - & - & - & - & - & - & - & - \\ AB + C + D & - & 30 & - & - & - & - & - & - & - & - \\ ABV1 + C + D & - & 30 & - & - & - & - & - & - & - & - \\ ABCD + E + F & - & 30 & - & - & - & - & 30 & - & 30 & - \\ ABCDV1 + E + F & - & 30 & - & - & - & - & 30 & - & 30 & - \\ ABCD + E + F + V2 & - & 40 & - & - & - & - & 40 & - & 40 & - \\ ABCDV1 + E + F + V2 & - & 40 & - & - & - & - & 40 & - & 40 & - \\ AB + C & - & - & 20 & - & - & - & - & - & - & - \\ ABV1 + C & - & - & 20 & - & - & - & - & - & - & - \\ ABC + D + E & - & - & 30 & - & - & - & - & - & - & - \\ ABCV1 + D + E & - & - & 30 & - & - & - & - & - & - & - \\ ABCDE + F & - & - & 20 & - & - & 20 & 20 & - & - & 20 \\ ABCDEV1 + F & - & - & 30 & - & - & - & - & - & - & - \\ ABCDE + F + V2 & - & - & 30 & - & - & 30 & 30 & - & - & 30 \\ ABCDE + V1 + F + V2 & - & - & 40 & - & - & - & - & - & - & - \\ A + B + C & - & - & - & 30 & - & - & - & - & - & - \\ A + B + C + V1 & - & - & - & 40 & - & - & - & - & - & - \\ D + E + F & - & - & - & 30 & - &Moreover, setup times for each operation were 5 s, and transport times of 8 s were assumed. The simulation was performed for 25 pieces of each product, so the total production batch was 100 products. When all the input data are provided, then it is possible to identify MLT and THR of all the APSs. Manufacturing lead time, which is the total time required to process a product through the plant, is usually calculated by the following formula [44]:  
\[MLT_{j}=\sum_{i=1}^{n_{0}}(Tsu_{ij}\ +Q_{j}*Tc_{ij}+Tt_{ij}), \tag{4}\]  
where \(MLT_{j}\) means \(MLT\) for a batch of part or product \(j\), \(Tsu_{ij}\) is setup time for operation \(i\) on part or product \(j\), \(Q_{j}\) is quantity of part or product \(j\) in the batch being processed, \(Tc_{ij}\) is cycle time for operation \(i\) on part or product \(j\), and \(Tt_{ij}\) is transport time associated with operation \(i\), where \(i\) indicates the operation sequence in the processing, \(i=1\), \(2\), \(\ldots\), \(n_{0j}\). The throughput rate indicator calculates the rate at which units move through the production process from the beginning to the end of the production process (i.e., how many units are produced per minute, hour, or day). For demonstration purposes, a model of assembly process structure No. 1 is depicted in Figure 6.  
Figure 6: The print screen from the simulation software Tecnomatix Plant simulation version 2201 showing assembly line structure with eight input components (six stable ones—A, B, C, D, E, F, and two optional ones—V1, V2).