## 6 Analysis of Results
### Large-Sized Instances with Fixed Profit Margins  
Due to the satisfactory performance of our biased-randomized approach in small-sized instances, the BR-GRASP and the BR-ILS were also tested in solving more realistic (large-scale) instances of the complete (non-smooth) version of the problem. In this case, no variations in the selling price of the products were considered. A lower bound on the attractiveness value any item can achieve is imposed. In our experiments, we set \(a=0\), while \(b\) takes a sufficiently large value (i.e., in practice the attractiveness value of an item grows after each period in which the item has not been displayed). For each of the 40 generated instances, 10 runs were performed (each run using a different seed for the pseudo-random number generator). To evaluate the performance of each methodology, we consider the percentage gap between the best-found solution using that methodology (i.e., the one with the highest attractiveness value) and the BKS obtained with any solution methodology. Thus, the lower the gap, the better the performance of the methodology. The solutions generated by our BR-GRASP and BR-ILS algorithms are compared with those generated by a greedy strategy, a'standard' GRASP, and a'standard' ILS (i.e., without considering BR techniques). The greedy methodology approximates human behavior (e.g., an expert manager) when selecting articles to be displayed. In the standard GRASP algorithm, and after some preliminary tests, the parameter \(\alpha\) which controls the size of the RCL was set to 0.1. The standard ILS algorithm uses a greedy selection technique to build solutions. Regarding the perturbation stage, the destruction rate was defined by the set \(k\in\{0.1,0.3,0.4,0.55,0.7,0.8,1\}\). Table 5 provides the summarized results of the solutions generated by the greedy (G), BR-GRASP (BR-GR), GRASP (GR), ILS and BR-ILS methodologies. For each instance and methodology, the following data is provided: the attractiveness value of the best-found solution, the average (AVG) attractiveness value obtained after 10 runs, as well as its corresponding standard error (SE), the gap with respect to the BKS, and the average CPU time employed to find the corresponding solutions. The best-found solution for each instance is presented in bold. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Instance Details**} & \multicolumn{4}{c}{**GAP (\%) w.r.t. BKS**} & \multicolumn{4}{c}{**Time (s)**} \\ \cline{2-13}
**Instance** & \(|\mathbf{I}|\) & \(|\mathbf{H}|\) & \(|\mathbf{T}|\) & **bin var constraints** & **BKS** & **COIN-OR** & **NEOS** & **BR-GR** & **COIN-OR** & **NEOS** & **BR-GR** \\ \hline
25i-5c-2p-2t-6 it & 25 & 2 & 2 & 100 & \(>50\) & 311.4 & 4.82 & 3.21 & 0.00 & 19 & 23 & 1 \\
25i-5c-3p-2t-6 it & 25 & 3 & 2 & 150 & \(>75\) & 440.4 & 2.04 & 2.68 & 0.00 & 10 & 23 & 1 \\
25i-5c-4p-2t-6 it & 25 & 4 & 2 & 200 & \(>100\) & 562.9 & 0.00 & 1.03 & 1.49 & 37 & 69 & 4 \\
50i-5c-4p-2t-6 it & 50 & 4 & 2 & 400 & \(>200\) & 768.7 & 0.91 & 0.00 & 0.85 & 196 & 46 & 5 \\
50i-5c-4p-4t-6 it & 50 & 4 & 4 & 800 & \(>200\) & 1250.3 & 0.00 & 2.62 & 2.58 & \(>700\) & \(>300\) & 3 \\
75i-5c-2p-2t-6 it & 75 & 2 & 2 & 300 & \(>150\) & 392.4 & 3.52 & 0.00 & 2.47 & 50 & 79 & 5 \\
75i-5c-2p-3t-6 it & 75 & 2 & 3 & 450 & \(>150\) & 557.7 & 3.39 & 1.27 & 0.00 & 148 & 165 & 5 \\
75i-5c-3p-3t-6 it & 75 & 3 & 3 & 675 & \(>225\) & 1033.9 & N/A & N/A & 0.00 & \(>3600\) & \(>1200\) & 64 \\
75i-5c-3p-4t-6 it & 75 & 3 & 4 & 900 & \(>225\) & 1352.1 & N/A & N/A & 0.00 & \(>3600\) & \(>1200\) & 83 \\
100i-5c-2p-3t-6 it & 100 & 2 & 3 & 600 & \(>200\) & 583.1 & N/A & 1.73 & 0.00 & \(>3600\) & \(>120\) & 34 \\
100i-5c-3p-2t-6 it & 100 & 3 & 2 & 600 & \(>300\) & 631.6 & 0.24 & 0.00 & 1.35 & \(>700\) & \(>300\) & 4 \\
100i-5c-3p-4t-6 it & 100 & 3 & 4 & 1200 & \(>300\) & 1122.7 & N/A & N/A & 0.00 & \(>3600\) & \(>1200\) & 64 \\
100i-5c-4p-4t-6 it & 100 & 4 & 4 & 1600 & \(>400\) & 1496.9 & N/A & N/A & 0.00 & \(>3600\) & \(>1200\) & 61 \\ \hline \hline \end{tabular}
\end{table}