# Metaheuristic with Cooperative Processes for the University Course Timetabling Problem
## 4 Computational Results
### Statistical Analysis  
The test data are the values obtained by each algorithm (SACP, ACO, GA, ILS, TS, SA) with respect to its objective function (Table 5). For small instances, all algorithms find the global optimum solution, and for the statistical analysis purposes, the worst value obtained by each algorithm is used in order to know if there are differences in the effects of each type of algorithm in the worst of the results obtained. For the other medium and large instances, the statistical analysis uses the best value obtained in each algorithm. The null hypothesis, \(H_{0}\) in Equation (14), indicates that the means of the results are equal. The alternative hypothesis, Equation \(H_{1}\) in (15), points out that the means are not equals or at least one is different. Statistical analysis tests whether the null hypothesis is true. If the null hypothesis is true, then there will be no difference in the effects of the analyzed algorithms. \[H_{0}:\overline{X_{1}}=\overline{X_{2}}=\ldots=\overline{X_{r}} \tag{14}\]  
\[H_{1}:Not\text{ {all are the same}} \tag{15}\]  
For conducting the statistical analysis, both data normality and homoscedasticity are first verified. Figure 10 presents the normality of the data of each algorithm. The results show that there is no normality in the data of any algorithm since the points are not located on the diagonal of the graphs. The homoscedasticity analysis for SACP, ILS and TS for the same number of data is performed graphically with the boxplot. Figure 11 shows that the boxes are not the same, since there is a greater dispersion of data in the TS box, and ILS follows and, finally, the least dispersion has SACP. TS presents the greatest variance and SACP is the one that has the smallest variance of the three algorithms. Therefore, a common variance cannot be admitted. It indicates that there is no homoscedasticity between the three algorithms. Applying the Bartlett test with the Bartlett-test function of R [44], the \(p\)-value = 0.001169 is found (0.001169 \(<\) 0.05), this shows that there are different variances; therefore, it is confirmed that there is no homoscedasticity when having a small \(p\)-value. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Problem**} & \multicolumn{6}{c}{**Metaheuristics**} \\ \cline{2-7}  & **SACP** & **ACO** & **GA** & **ILS** & **TS** & **SA** \\ \hline Easy01 & 0–0(1) & 0–7(4) & 0–19(5) & 0–5(3) & 0–34(6) & 0–3(2) \\ \hline Easy02 & 0–0(1) & 0–15(4) & 1–30(6) & 0–10(3) & 0–27(5) & 0–6(2) \\ \hline Easy03 & 0–0(1) & 0–12(4) & 0–22(5) & 0–7(3) & 0–29(6) & 0–4(2) \\ \hline Easy04 & 0–0(1) & 0–7(4) & 0–21(5) & 0–6(3) & 0–24(6) & 0–5(2) \\ \hline Easy05 & 0–0(1) & 0–5(3) & 0–13(5) & 0–6(4) & 0–21(6) & 0–1(2) \\ \hline Medium01 & 48(1) & 182(6) & 181(5) & 120(3) & 172(4) & 71(2) \\ \hline Medium02 & 49(1) & 194(6) & 170(4) & 121(3) & 183(5) & 70(2) \\ \hline Medium03 & 73(1) & 252(6) & 246(5) & 171(3) & 226(4) & 111(2) \\ \hline Medium04 & 49(1) & 168(6) & 148(4) & 107(3) & 163(5) & 65(2) \\ \hline Medium05 & 36(1) & 176(4) & 208(5) & 142(3) & 238(6) & 55(2) \\ \hline Large01 & 321(1) & \({}^{\star}\) (6) & \({}^{\star}\) (6) & 900(2) & 1170(3) & \({}^{\star}\) (6) \\ \hline Large02 & 285(1) & 705(3) & 1010(5) & 690(2) & 1000(4) & \({}^{\star}\) (6) \\ \hline Average ranking & 1 & 4.7 & 5 & 2.9 & 5 & 1.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of results obtained by the SACP metaheuristic with other results obtained by other metaheuristics at a time limit of 9000 s for large problems, 900 s for medium problems and 90 s for small problems. Figure 10: Normality graphs for SACP, ACO, GA, ILS, TS and SA algorithms. The homoscedasticity analysis for SACP, ACO and GA for the same number of data is performed graphically with the boxplot. Figure 12 shows that the boxes are not the same, since a greater dispersion of data is observed in the ACO and GA boxes and the lower dispersion is presented in SACP. SACP is the algorithm that has the least variance.