## 4 Experiment Details and Results
### Final Solution  
As already mentioned, the solution for coverage is used as a starting point in order to find a feasible solution that also fulfills the diversity constraints. As a next step, the heuristic approach of Algorithm 1 has been followed. The result of the proposed solution is shown in Figure 3:  
As observed in the figure above, for every value of coverage, as the diversity threshold increases, the average rating of recommended items is similar or drops. Furthermore, for greater coverage thresholds, the problem becomes more difficult to solve and the average rating drops more rapidly with the increase of diversity threshold. More specifically for coverage threshold \(Kc=2.5\), the problem has no solution for a Diversity threshold greater than 0.0005. In Table 2, the total and average rating of the recommended items are presented for different values of Diversity threshold and for Coverage threshold \(Kc=1\) and \(Kc=2.5\). As observed from the table and the figure above, the total rating of all the recommended items for all users is very high and, in most cases, it is the same as the coverage solution. This shows that the heuristic solution chosen may not be the optimal one but produces a result that is very close to the optimal. Figure 3: Average rating for different values of diversity and coverage. Figure 2: Coverage per provider for \(K=2\) and \(K=2.5\). In order to provide a better view on how diversity influences the total rating, in the following plots the total rating for coverage threshold \(K=1.5\) and \(K=2\) for different thresholds of diversity is presented (Figures 4 and 5):  
From the plots above (Figures 4 and 5), it is observed that, as the diversity threshold increases, the total rating decreases, while the decrease of the total rating is more important for greater diversity thresholds and not proportional. Concerning the relation between diversity and coverage, it is observed that, for a bigger coverage threshold, the total rating decreases faster with the increase of diversity. This is also what was expected, because, for a higher coverage threshold, on the one hand a stricter constraint is introduced, and on the other, less exchanges with items from other categories are allowed. Moreover, as observed in Table 2, the results in terms of total or average rating are very close to the ones of the unconstrained solution. The total rating of the latter serves as the upper bound for the optimal solution, because the constrained problem cannot provide more relevant recommendations than the unconstrained one. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Diversity** & **Total Rating** & **Average Rating** & **Total Rating** & **Average Rating** \\
**Threshold D** & **(\(Kc=1\))** & **(\(Kc=1\))** & **(\(Kc=2.5\))** & **(\(Kc=2.5\))** \\ \hline
0.0001 & 30,124.411 & 4.9384 & 30,124.411 & 4.9384 \\ \hline
0.0002 & 30,124.411 & 4.9384 & 30,124.411 & 4.9384 \\ \hline
0.00025 & 30,124.411 & 4.9384 & 30,124.411 & 4.9384 \\ \hline
0.0005 & 30,124.411 & 4.9384 & 30,110.984 & 4.9362 \\ \hline
0.00075 & 30,124.411 & 4.9384 & No solution & No solution \\ \hline
0.001 & 30,124.120 & 4.9383 & No solution & No solution \\ \hline
0.002 & 30,119.159 & 4.9375 & No solution & No solution \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Average and total rating for different values of diversity and coverage. Figure 4: Total rating for different category Diversity thresholds for \(K=1.5\). Figure 5: Total rating for different category Diversity thresholds for \(K=2\). As stated before, the approach that has been followed is heuristic and does not always provide the optimal solution to the optimization problem. Instead, it always provides a solution that is close to the optimal one. Specifically, the algorithm that has been proposed follows a greedy approach and avoids backtracking to find a better solution, in case a good enough solution has been found at a previous step.