## 4 Experimental Setup  
This section details the parameters needed to reproduce our experiments. To evaluate our proposed multi-objective problem formulation and the performance of the selection of algorithms from Section 3, we use the same initial proposed benchmark in [1] of five small instances (10 products with 25 stores), five medium instances (25 products with 50 stores), and five large instances (50 products with 100 stores). Instances are available at [[https://github.com/AASantiago/ISHOP-U-Instances](https://github.com/AASantiago/ISHOP-U-Instances)]([https://github.com/AASantiago/ISHOP-U-Instances](https://github.com/AASantiago/ISHOP-U-Instances)) (accessed on 21 April 2025). The experimentation parameters for the eight algorithms studied are the ones recommended in their original publications and appear in Table 1, except for a crossover probability \(p_{r}=1.0\), a mutation probability \(p_{m}=0.05\), a population size of 100, and 25,000 objective function evaluations as stop criterion, the above are parameters and evolutionary operators that have been already fine-tuned in [1] for the ISHOP-U.  
Following the recommendations in [18], to achieve statistical confidence, we performed 30 independent executions per problem instance; the numerical results are summarized  
\begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **AGEMOEA** & **AGEMOEA2** \\ \hline Population size: & 100 & 100 \\ Selection: & Binary Tournament & Binary Tournament \\ Recombination: & \(p_{r}=1.0\) & \(p_{r}=1.0\) \\ Mutation: & \(p_{m}=0.05\) & \(p_{m}=0.05\) \\ \hline \hline \multicolumn{3}{c}{**MOMBI**} & **MOMBI2** \\ \hline Population size: & 101 & 101 \\ Selection: & Binary Tournament & Binary Tournament \\ Recombination: & \(p_{r}=1.0\) & \(p_{r}=1.0\) \\ Mutation: & \(p_{m}=0.05\) & \(p_{m}=0.05\) \\ \hline \hline \multicolumn{3}{c}{**SMS-EMOA**} & **MOCell** \\ \hline Archive size: & - & 100 \\ Population size: & 100 & 100 \\ Selection: & Binary Tournament & Binary Tournament \\ Recombination: & \(p_{r}=1.0\) & \(p_{r}=1.0\) \\ Mutation: & \(p_{m}=0.05\) & \(p_{m}=0.05\) \\ \hline \hline \multicolumn{3}{c}{**GWASFGA**} & **NSGA2** \\ \hline Population size: & 100 & 100 \\ Selection: & Binary Tournament & Binary Tournament \\ Recombination: & \(p_{r}=1.0\) & \(p_{r}=1.0\) \\ Mutation: & \(p_{m}=0.05\) & \(p_{m}=0.05\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Parameters for the eight state-of-the-art algorithms in comparison. Parameters tuned as in the original ISHOP-U publication.  
in the result tables as the median and interquartile range (MEDIAN\({}_{IQR}\)) as measures of central tendency and dispersion, respectively. To quickly identify the best and second-best algorithms, the best and second-best results appear in gray and light gray, respectively. Due to the multi-objective nature of the problem studied, we use the same multi-objective quality indicators as in [19], the hypervolume [20] as a measure of the dominated region, the additive epsilon [21] as a measure of the necessary distance to weakly dominance the reference Pareto front, the inverted generational distance plus [22] as a measure of the solutions distance to their nearest point in the reference Pareto front. Since the real true Pareto front is unknown for this newly proposed multi-objective problem, we use the added nondominated solutions after the complete experimentation as reference fronts.