## 5 Numerical Results  
This section outlines the numerical results of the experimentation. For the numerical results that appear in Tables 2-4 for the multi-objective quality indicators of hypervolume, additive epsilon, and inverted generational plus, respectively. In the column problem for the nomenclature used in the name of the instances, the first character indicates their size: S for small, M for medium, and L for large. For the hypervolume computation, the extreme points from the reference set of nondominated solutions are used as the reference point to compute the hypervolume as follows. Normalization of the objectives is applied using the extreme points, and the minimization objectives are inverted to compute the hypervolume. Table 2 and Figure 1 present the Friedman average rankings numerically and graphically, respectively, achieved by the algorithms in experimental comparison. As Table 2 shows, NSGA2 achieves outstanding performance, near 0.5 for the entire set of instances studied. NSGA2 achieves the best hypervolume median value in the 15 instances studied, followed by SMS-EMOA with seven, MOCell with five, and GWASFGA with the three best median values. The above results are over 0.48, close to the optimal hypervolume for Pareto fronts with linear shapes, since it should be approximately 0.5, half the total unit square area for a normalized two-objective space. Considering the Friedman average rankings in Table 5 best-ranked algorithm is NSGA2 with an average rank of 1.25, followed by GWASFGA, SMS-EMOA, MOCell, MOMBI, MOMBI2, AGEMOEA2, and AGEMOEA with average rankings of 2.63, 2.72, 3.40, 6.11, 6.18, 6.83, and 6.88, respectively. Regarding the additive epsilon multi-objective quality indicator, GWASGFA achieves the best median values on the 15 studied problem instances, followed by NSGA2 with 15 second-best median values achieved. The differences between the NSGA2 and GWASFGA diversity preservation mechanisms could explain the result. NSGA2 environmental selection always keeps the extreme points of the Pareto approximation, i.e., the best-found solutions in every objective, the minimum purchase cost, and the maximum cash-back/reward points. Meanwhile, GWASFGA points half of its population toward the same extreme points by referencing the slightly improved ideal point and the other half towards the opposite of the extreme points (the worst in every objective) by referencing the worsened nadir point. Moreover, as an auxiliary diversity mechanism, the direction of the GWASFGA population depends on the set of weights. Therefore, the two diversity mechanisms in GWASFGA produce Pareto approximations that require less distance to weakly dominate the reference front. Contrary to the crowding distance (NSGA2), which strongly depends on the current extreme points (ideal point), neglecting the direction of the nadir point. The above result coincides with the [PERSON] average rankings from Table 5, where the best-ranked algorithm is GWASFGA, with an average rank of 1.03, followed by NSGA2, SMS-EMOA, MOCell, MOMBI, MOMBI2, AGEMOEA, and AGEMOEA2 with average rankings of 2.05, 3.21, 3.71, 6.03, 6.53, 6.72, and 6.72, respectively. According to the inverted generational distance plus compute results, NSGA2 is the best-performing algorithm with 15 best median values. NSGA2 Pareto front approximations are notably close to the Pareto reference front; for all instances, the computed value is lower than 0.00007 (the minimum possible value is zero). The second-best performer algorithm is GWASFGA, with nine second-best median values. The Friedman average rankings confirm the above result in Table 5, where the ranked algorithm is NSGA2 with an average rank of 1.12, followed by GWASFGA, SMS-EMOA, MOCell, MOMBI, MOMBI2, AGEMOEA2, and AGEMOA with average rankings of 2.64, 2.72, 3.53, 6.05, 6.24, 6.82, and 6.89, respectively.