## 4 Materials and Methods
### New Model after Throwing the Same Permutation Assumption
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{\begin{tabular}{c} **Interval** \\ **Variables** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Boolean** \\ **Variables** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **DPFSP Model** \\ **No_Overlap** \\ **Constraints** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **DFSP Model** \\ **Interval** \\ **Variables** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **DFSP Model** \\ **No_Overlap** \\ **Constraints** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Other Constraints** \\ **Variables** \\ \end{tabular} } & \multirow{2}{*}{\begin{tabular}{c} **Interval** \\ **Constraints** \\ \end{tabular} } & \multirow{2}{*}{\
\begin{tabular}{c} **No_Overlap** \\ **Constraints** \\ \end{tabular} } & \multirow{2}{*}{
\begin{tabular}{c} **Other Constraints** \\ **Constraints** \\ \end{tabular} } \\ \hline \(nmF\) & \(n(n-1)F\) & \(F\) & \(n(n-1)F(m+1)+nF(m-1)\) & \(nmF\) & \(mF\) & \(n+nF(m-1)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Formulas for the number of decision variables and constraints of the DPFSP model and the DFSP model (\(n\) = numbers of jobs, \(m\) = number of machines, and \(F\) = number of factories). CP approach degrades and solutions are relatively far from the best-known ones. For the rest of the problem instances (i.e., places 61 to 120), the model is so big that the CP solver has trouble creating the model in the first place, let alone reaching good solutions. Figures 3 and 4 show the optimal schedules that result from the DPFSP model and the DFSP model, respectively, for the fourth problem instance of the group of problems with two factories, six jobs, and four machines of the small dataset. The Y-axis of the figure displays the factories, and for each factory, each machine using the notation f<factory_id>_m<machine_id>. The schedule produced by the DFSP model has a makespan of 436 time units, which is smaller than the schedule created using the DPFSP model, which is 440 time units. The more favorable result of the schedule by the DFSP model is attributed to the fact that each machine can have a different permutation of jobs processed in it. So, in Figure 4 at factory 1, machines 0 and 1 process jobs in the sequence j4, j1, j0, j2, but machines 2 and 3 process jobs using the sequence j1, j4, j0, j2. On the other hand, as shown in Figure 3, all four machines of factory 1 peruse the same permutation of jobs, which is j4, j0, j3, j1. The greater degree of freedom given to the DFSP model allows the CP solver to find a better solution than the DPFSP for this problem instance. In Section 5, it will be seen that this occurs in many other problem instances of the small dataset and the large dataset. If the \"permutation\" component of the problem is added to it not as a physical requirement (e.g., conveyor belt) but as a means of making the problem easier for various solvers to search for near-optimal solutions, then dropping it does not change the gist of the problem. The DPFSP model is able to find solutions in the larger search space that allows different job permutations for the machines of the same factory, but it satisfies the vital constraint of processing each job using the same sequence of machines (i.e., workflow), which is \(0,1,2,...,\left|\mathbb{M}\right|-1\) in this case, and can be any sequence of machines by naming the machines appropriately. The \"good\" behavior of the DFSP model can be attributed to the efficient implementations of the global constraint no_overlap and the interval variables that the CP solver employs.