## 6 Industrial Case Study
### Calibration of the Simulated Annealing Parameters  
The customized SA procedure proposed in Section 5 is applied to find the optimal synchronization solution for the MALSP suggested in the case study. It is developed in common coding software environment and experiments are carried out on a standard PC with a 2.30 GHz intel 7-4650U CPU and 8 GB RAM. With reference to the case under study, the proposed procedure has been carefully tuned to define the most suitable values for the parameters adopted both for the control of the annealing process and for the building of new feasible solutions. The control parameters needed are the initial and the final temperature (\(T_{i}\) and \(T_{f}\) respectively), the cooling ratio (\(\lambda\)) and the so-called epoch length (K) (i.e., the number of neighborhood searches at each temperature level). Their ranges are determined based \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Task** & **Duration** & **Mounting** & **Equipment** & **Number of Simultaneous** \\
**[\#]** & **[S]** & **Position** & **Required** & **Workers [\#]** \\ \hline
1 & 67 & MED\_6 & - & 1 \\
59 & 114 & LOW\_1 & D & 1 \\
348 & 276 & MED\_13 & M & 2 \\
608 & 1037 & TOP\_8 & R & 1 \\ \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Example of input data for some assembly tasks. Figure 8: Car mounting positions identification. on the previous and similar literature contributions and their specifications. To evaluate the effects of each parameter on the procedure results, some preliminary experiments have been conducted following the design of experiment principles. As a result of this process, \(T_{i}\) and \(T_{f}\). are set to 21 and 1.5, respectively. In this way, at a specific iteration, the probability of accepting solutions that are worse than the current ones is initially high and progressively drops to lower values while the procedure is running, in accordance with the adopted objective function. \(\lambda\) and K are set at 0.90 and 50, respectively, to ensure a proper cooling rate and an appropriate neighborhood search at each temperature level. The parameters employed to build feasible solutions are \(\delta\) and \(XP\) (see paragraph 5.2, step 6). They are set at 40 and 15, correspondingly. Moreover, the smoothness index target value (SI\({}^{\text{min}}\)) needs to be configured to give proportional weights to all the contributions of the objective function used to evaluate solutions (Equation (5)). This parameter depends on factor S as previously shown in Equation (8), which has been fixed at 3% of the cycle time for the examined instance. Table 2 summarizes the parameter values selected for the considered case study. Setting parameters deeply influences the execution of the procedure and the number of iterations (i.e., the number of feasible solutions generated during the annealing process) needed to achieve the optimal solution. Figure 9 shows the objective function values in relation to the number of iterations performed. Initially, even very poor solutions are easily accepted, but as the number of iterations increases, only smaller worsening can be embraced. Moreover, the objective function improvements obtained through the iterations are highlighted in the figure. Good values are gained from the first procedure iteration (objective function equal to 3.50) due to the initial priority list selection according to the RPW method. The best result (objective function equal to 1.74) is achieved at the 449 th feasible synchronization solution generation with a computational time of 4086 s by using the abovementioned PC. This optimal solution for the case study is illustrated in detail in the following section, discussing its main features.