## 7 Computational Experiments  
The BR-DES algorithm was implemented in Python 3.7 and executed using the CPython interpreter. The testing was conducted on a standard personal computer equipped with an Intel QuadCore i7 CPU running at 2.4 GHz and 8 GB RAM with the Ubuntu 18.04 operating system. When constructing the experimental data sets, the quality level of each bundle was randomly generated using a uniform probability distribution between 1 (minimum level) and 10 (maximum level). Then, the quality of a solution was computed as the average of the quality of the retrieved bundles. The input quantity was randomly generated using a uniform probability distribution between 700 and 1300, expressed in kilograms. We assume that two bundles enter together as input, each with an average weight of 500 kg. Finally, since a customer might require many bundles, the output quantity was randomly generated using a uniform probability distribution between 700 and 10,000, expressed in kilograms. Table 3 displays the data corresponding to Instance 1, where type 0 represents an input point and type 1 refers to an output point. Both the desired quality and quantity are provided as reference values. These reference values will be compared with those associated with the solution provided by each algorithm to calculate the mismatch. Similar data for the remaining instances, as well as the algorithm code, are available at [[https://github.com/mattianeroni/Shuttle-Lift-Crane-AS-RS](https://github.com/mattianeroni/Shuttle-Lift-Crane-AS-RS)]([https://github.com/mattianeroni/Shuttle-Lift-Crane-AS-RS](https://github.com/mattianeroni/Shuttle-Lift-Crane-AS-RS)) (accessed on 15 January 2024). As can be seen in Tables 4 and 5, our algorithm demonstrates an efficient performance, since it is capable of providing competitive solutions in just a few seconds, while other approaches require noticeably longer times without reaching the same solution quality. To validate the proposed methodology, a comparison with three distinct approaches is presented. The selected benchmark approaches are as follows: (i) a random algorithm utilizing a uniform probability distribution to select from different possibilities at each step of the solution-construction process; (ii) a greedy heuristic that consistently chooses the top-listed element in the candidates' list, sorted based on a minimum time criterion; and (iii) the simulated annealing method proposed by [PERSON] et al. (2016) for a similar problem, adapted for the specific problem considered in this study by incorporating new constraints. The random approach serves as a lower-bound benchmark and was obtained by implementing the proposed BR-DES while substituting the geometric distribution with a uniform distribution. This adjustment was achieved by setting the beta value to 0 in the BR-DES algorithm. However, when employing such low beta values, the algorithm often failed to find a feasible solution. To address this limitation, we utilized a geometric probability distribution with a slightly higher beta value of 0.3, allowing the algorithm to discover feasible solutions in most test scenarios, although not all. On the other hand, the greedy algorithm presents a practical solution for warehouse system managers and was acquired by implementing the proposed BR-DES with a beta value of 1. The SA algorithm by [PERSON] et al. (2016) is currently the only algorithm explicitly tailored for this type of automated warehouse system (i.e., an SLC-AS/RS). While theoretically serving as a solid benchmark, the original SA algorithm did not consider any constraints regarding the sequence of request processing. To ensure a fair comparison, we integrated these constraints into the SA algorithm. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Type** & **Bay** & **Desired Quality** & **Desired Quantity** & **Length (in Shelves)** \\ \hline  
\hline \hline \end{tabular}
\end{table} 
Table 3: Example of an instance data set—Instance 1.