## 6 Analysis of Results
### Large-Sized Instances with Fixed Profit Margins

Table 4: Comparison of results between the non-linear solvers and our BR-GRASP approach (relaxed version of the problem). \begin{table}
\begin{tabular}{c|c|c c c c c c c c c c c c c c c c c c c c c c c c c c c} \multicolumn{13}{c}{} & \multicolumn{13}{c}{_Bot Selection (attractiveness)_} & \multicolumn{13}{c}{_AVG Architectures (SD)_} & \multicolumn{13}{c}{_AVG Architectures (SD)_} & \multicolumn{13}{c}{_AVG Time (sec. )_} \\ \hline \multicolumn{  
From Table 5, we can notice that the BR-GRASP and the BR-ILS (in that order) outperform the other methodologies. In particular, the BR-GRASP provides the BKS in 37 out of 40 tested instances. Although the BR-GRASP was not able to find the BKS in some cases, it was able to find solutions with a minor deviation (less than 0.10%) from it. According to our experiments, the greedy methodology runs extremely fast, since this methodology does not incorporate a local search mechanism and it is not embedded inside a multi-start process. Comparing the remaining methodologies, all of them require similar computing times, although the BR-GRASP reaches high-quality solutions faster than other approaches. Figure 2 shows, for each methodology, a boxplot of the percentage gaps with respect to the BKS. From this Figure, one can conclude that the greedy methodology--which could be assimilated to a human behavior--represents by far the worst approach. In effect, it shows an average gap of about 3% with respect to the BKS, a gap that can grow up to 5% in some instances. It can be seen that biased-randomized techniques enhance the ILS and GRASP approaches. The main reason why our BR-GRASP performs slightly better than our BR-ILS might is the existence of inter-period dependencies. These inter-period dependencies might sometimes penalize partial-scope destruction-reconstruction processes (ILS) versus full-scope destruction-reconstruction ones (GRASP), since the former might get trapped more easily in a local minimum. ANOVA and Fisher tests were run in order to analyze if the performance differences were statistically significant. As expected, the ANOVA test resulted in the existence of significant differences among the algorithms (p-value = 0.000). Actually, according to the Fisher pairwise test, except for the comparison between algorithms BR-GRASP and BR-ILS, all other differences in performance are statistically significant. Figure 3 illustrates the convergence behavior with time of the BR-GRASP algorithm for an instance composed of 500 articles, 5 tables, and 5 periods. The convergence is analyzed by period \(P_{i}\), \(i\in\{1,2,\ldots,5\}\), which runs for 1.25 seconds (\(0.5\times 500\times 5\) milliseconds per period) for this problem. Notice that all periods present similar convergence behavior with time. Additionally, the range value of attractiveness is similar across the periods. This is explained by the small increment in attractiveness (3%) associated with new items, as well as by the large reduction in attractiveness (10%) associated with items that have lost their novelty effect. Figure 2: Comparison of percentage gaps w.r.t. the best-known solution (BKS) achieved by each methodology.