## 5 Experimental Results
### Performance of MILP and CP Models
#### 5.1.2 Experimental Results  
In this section, we intend to evaluate the proposed models. We set the stopping criteria parameters as follows: the time limit CPU is equal to 30 min and the maximum iterations equal to 1000. The performances of the models are evaluated thanks to real data of the workshop. Test results are discussed below. Several set instance sets have been created with \(N\) ranging from  
* {5, 8, 10, 13, 15 to 20} for small-sized instances. * {30, 40, 50, 65, 70, 75, 80 to 100} for large-sized instances  
For each set, at least two test instances were generated by varying the number of operations. For each problem class, an effort measurement is completed by calculating the associated total tardiness and the required CPU time. The optimal values that are obtained for tardiness have been distinguished with bold numbers. When applying both formulations to the test instances, a total of 160 experiments were carried out. Instances without Resource Calendar Constraints  
We now present some results on the solution quality obtained with the different models that do not take into account resource calendar constraints. 1. Small-Sized Instances  
In this subsection, the general performance of the MILP and CP models is evaluated by a set of small-sized instances. Several instance sets have been created with \(n\) ranging from 5 to 20. Table 5 provides an overview of the obtained results. For each problem, the name, the number of jobs, the number of operations and machines, the total tardiness T in minutes, and the solution time in seconds are shown for both models (\(\mathrm{MILP_{wav}}\) and \(\mathrm{CP_{wav}}\)). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{**Instance Characteristics**} & \multicolumn{2}{c}{\(\mathrm{MILP_{wav}}\)} & \multicolumn{2}{c}{\(\mathrm{CP_{wav}}\)} \\ \hline
**Instance** & **N** & **O** & \(T_{\mathrm{MILP}}\) & \(\mathrm{CPU_{MILP}}\) & \(T_{\mathrm{CP}}\) & \(\mathrm{CPU_{CP}}\) \\ \hline WOS1 & 5 & 7 & **0** & 5 & **0** & 4 \\ WOS2 & 5 & 12 & **0** & 8 & **0** & 4 \\ WOS3 & 5 & 20 & **0** & 12 & **0** & 4 \\ WOS4 & 8 & 10 & **0** & 11 & **0** & 4 \\ WOS5 & 8 & 25 & **0** & 50 & **0** & 5 \\ WOS6 & 8 & 30 & **0** & 69 & **0** & 5 \\ WOS7 & 10 & 17 & **0** & 58 & **0** & 5 \\ WOS8 & 10 & 29 & **0** & 110 & **0** & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Main characteristics of the considered small-sized instances and comparison of \(MILP_{wav}\) and \(\mathrm{CP_{wav}}\) models. Large-Sized Problems  
To further validate the performance of the proposed models, larger-sized instances are evaluated. Table 6 summarizes the corresponding computational results. As can be observed, up to \(\mathrm{N}=40\) and \(\mathrm{O}=88\), the MIP model is unable to find a solution within 1800 s, whereas the CP model still finds an optimal solution for all instances in 38 s on average. The first conclusion that can be drawn is that CP is much faster than MILP. This experimentation confirms CP's outstanding performance for the problem under study. Instances with Resource Calendar Constraints  
This subsection shows the results of the instances on the models that incorporate resource calendar constraints. 1. Small-Sized Instances  
The results of the computational comparison for each combination of n and m are presented in Table 7.