# Metaheuristic with Cooperative Processes for the University Course Timetabling Problem
## 4 Computational Results
### Statistical Analysis
Therefore, a common variance cannot be admitted, which indicates that there is no homoscedasticity. Applying the Bartlett test with the Bartlett-test function of R [44], the \(p\)-value = 0.002409 is found (0.002409 < 0.05), this shows that there are different variances; therefore, there is no homoscedasticity when having a small \(p\)-value. The homoscedasticity analysis for SACP and SA for the same number of data is performed graphically with the boxplot. Figure 13 shows that the boxes are not the same since a greater dispersion of data is observed in the SA box and the lower dispersion is presented by SACP. The greatest variance is observed in SA, while SACP has the smallest variance. So, a common variance cannot be admitted so there is no homoscedasticity. Homoscedasticity is analyzed with the Bartlett test, applying the Bartlett-test function of R [44]. The \(p\)-value 0.3233 > 0.05 indicates that there are very simi  
Figure 11: Boxplot of the SACP, ILS and TS algorithms. Figure 12: Boxplot of the SACP, ACO and GA algorithms. the Bartlett test indicates that there is homoscedasticity when having a \(p\)-value that exceeds the significance level of 0.05. In order to know if there are significant differences in the behavior of the objetive function values obtained by the algorithms, it is required to analyze the results obtained by the six algorithms. As there are not both assumptions of normality and homoscedasticity of the data provided by the six algorithms, and also because the number of data analyzed by each algorithm is small, Kruskal-Wallis non-parametric method, proposed by [PERSON] and [PERSON] (1975), will be applied. The Kruskal-Wallis test requires that the observations of each algorithm, which must be different, be presented in ranges. The Equation (16) presents the Kruskal-Wallis test. If there are observations with equal values, then Equation (16) is divided by the relation (17), where \(H\) is the value of the contrast statistic, C is the number of the algorithms to analyze (in this case there are six algorithms), \(n_{i}\) is the number of observations in the \(ith\) algorithm (number of results of each algorithm), \(N=\sum n_{i}\) is the total number of results of all algorithms and \(R_{i}\) is the sum of the ranges in the \(ith\) algorithm. \[H=\frac{12}{N(N+1)}\sum_{i=1}^{C}\frac{R_{i}^{2}}{n_{i}}-3(N+1) \tag{16}\]  
In relation 17, the sum is over all groups of repeated data and \(T=t^{3}-t\) is for each group of repeated data. Where \(t\) is the number of repeated results in the group. \[1-\frac{\sum T}{N^{3}-N} \tag{17}\]  
From Table 5, the first ten data, which are the results for small and medium problems, are taken from each algorithm in order to make a comparison of all the algorithms. The Kruskal-Wallis test is applied with the Kruskal-test function of R ([PERSON], 1975) to obtain the p-value. The contrast statistic H is 26.304 and the \(p\)-value is 7.79 \(\times\) 10\({}^{-5}\) < 0.05; therefore, the null hypothesis of equality of the effects of algorithms is rejected. This indicates that there are differences in the effects of algorithm types by having a \(p\)-value lower than their significance level. Figure 13: Boxplot of the SACP and SA algorithms. Post Hoc Comparisons  
Since the Kruskal-Wallis test is significant, it implies that at least two groups of algorithms from among those compared are significantly different, but does not indicate which ones. To find out, it is necessary to make a comparison between them. Comparison is made with the results obtained from SACP metaheuristic vs. results obtained by each metaheuristic presented in Table 5 (metaheuristic 1 vs. metaheuristic 2) as follows SACP vs. ACO, SACP vs. GA, SACP vs. ILS, SACP vs. TS and SACP vs. SA, with the Mann Whitney Wilcoxon method [46] represented by Equations (18)-(20), where \(n_{1}\) is the number of results obtained by metaheuristic 1, \(n_{2}\) is the number of results obtained of metaheuristic 2, \(R_{1}\) is the sum of the ranges in metaheuristic 2.