# A Decomposition Algorithm for Dynamic Car Sequencing Problems with Buffers
## 5 Experiments and Results
### Experimental and Computational Results  
As described in Section 4.3, for PGDA, the value of the parameter \(M\) needs to be obtained by evaluating the quality of the solution for different experimental arithmetic cases. First the number of violations \(VioP\) for the sequence of cars in the paint shop are counted before reordering, and the number of violations \(VioM\) and the time cost \(tm\) of the sorting are obtained by the statistical algorithm when \(M\) takes different values on different test arithmetic cases, until the values of \(VioM\) and \(tm\) values do not vary with \(M\). The detailed results are shown in Figure 6, and it can be seen that when \(M=2\), \(VioM\) tends to achieve the minimum value, \(tm\) is within an acceptable range, and the value of the objective function Equation (1) is optimal. There is usually a range of values for each parameter of the genetic algorithm and different combinations of these parameter values greatly affect the performance of the algorithm. Here, the solution quality of the algorithm is evaluated by different experiments with different combinations of parameter values, and better values of the genetic algorithm parameters are obtained. In all experimental arithmetic cases using the genetic algorithm, the population size is 50, \(p_{c}^{max}\) is 1, \(p_{c}^{min}\) is 0.6, \(k_{c}\) is 3, \(p_{m}^{max}\) is 0.1, \(p_{m}^{min}\) is 0.01, \(k_{m}\) is 2, and the maximum number of iterations is 100. Based on the proposed simple rule-based heuristic entry algorithm and four different heuristic exit algorithms, i.e., GDA, PGDA, GA, and DGA, experiments were conducted on DCSPwB of different sizes, and the number of violations \(VioP\) for the sequence of cars in the paint shop before reordering, \(Vio\_GDA/Vio\_PGDA/Vio\_GA/Vio\_DGA\) for the sequence of cars after reordering by each algorithm, and the time cost \(t\_GDA/t\_PGDA/t\_GA/t\_DGA\) were counted. The experimental results are shown in Table 5 and Figure 7. In Table 5, the first column identifies the test arithmetic, the next column shows the number of violations of the car sequence in the paint shop, and the next eight columns show the results of GDA, PGDA, GA, and DGA. Due to the random nature of the genetic algorithm, the results of the genetic algorithm are the average of five runs. According to Table 5, we can see that PGDA, GA, and DGA all have a substantial reduction in the number of violations for the reordered car sequence, while GDA does not work well in solving the dynamic car sequencing problem. In Figure 7, we see the comparison between these four algorithms in different test cases. PGDA outperforms the other algorithms in terms of solution quality as well as time cost of sorting, and the objective function value \(F\) reaches the optimum. However, the choice of its parameter \(M\) often requires several experiments based on the buffer capacity and upstream workshop car sequences to evaluate the solution quality obtained with different parameters. The \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** & **Unit** \\ \hline \(L\) & 6 & \\ \(V\) & 10 & \\ \(TL\) & 9 & \(s\) \\ \(t_{el}\) & [18, 12, 6, 0, 12, 18] & \(s\) \\ \(t_{fl}\) & [18, 12, 6, 0, 12, 18] & \(s\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Buffer parameters. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Order of Entering the Car** & **Car Type** & \(o_{1}\) & \(o_{2}\) & \(o_{3}\) \\ \hline
1 & K1 & & ✓ & \\
2 & K2 & ✓ & & ✓ \\
3 & K2 & ✓ & & ✓ \\
4 & K1 & & ✓ & \\ \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) \\
22 & K3 & & ✓ & \\
23 & K1 & & ✓ & \\ \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) \\ \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Dataset introduction. solution quality of DGA and GA is comparable, but the convergence of DGA is better than GA. Figure 8 shows the convergence curves of several groups of genetic algorithms in the solution process.