# Expected output Player norms:[1.77552809 1.69189243 1.68448805] ```
### Experimental Analysis  
The first experiment that we present has an optimal solution. The first experiment has the settings listed in Listing 2. ```
# Skills skills=['Attack','Defense','Teamwork'] num_players=9 num_skills=3 num_teams=3
```  
```

The results shown in Figure 4 confirm that the teams formed are balanced. The metric \"Skill Distribution Heatmap\" shows the balance of skills among the three formed teams. Also, the metric \"Mean Skill Distance\" reflects how the mean skill distance varies slightly among the teams. However, the cohesion metric shows a big variation among the three formed teams. Moreover, we calculated the Gini Coef. and tabulated the results in Table 1. The Gini ratios demonstrate the homogeneity and equality among teams. The results in Table 1 indicate that all team members have different skills due to the need for different roles within the team. This could be in favor of diversity and is reflected in their high Gini Coef. of 0.89. Next, we present a second experiment with the following settings: a pool of twenty players, each with three skills. The objective is to form four teams that each contain five players. The results of the experiment are shown in Table 2.

An interpretation of the metrics used:

1. Mean Skill Distance (MSD): Lower values signify fair skill distribution. 2. Within-Team Variance (WTV): Lower values signify cohesive teams. 3. Skill Balance (SB): Lower values signify balanced teams. First, the MSD score of 0.6222 indicates that individuals are, on average, close to the average skill level of their respective teams. This suggests that the teams formed are homogeneous in terms of their individual skill levels. Second, the WTV score of 1.695 \begin{table}
\begin{tabular}{c c} \hline
**Metric** & **Value** \\ \hline Team 1 & [7, 8, 9, 16, 19] \\ Team 2 & [3, 11, 12, 15, 17] \\ Team 3 & [4, 10, 13, 14, 18] \\ Team 4 & [0, 1, 2, 5, 6] \\ Mean Skill Distance & 0.6222 \\ Within-Team Variance & 1.6956 \\ Skill Balance & 0.1052 \\ \hline \end{tabular}
\end{table} 
Table 2: Result of the second experiment. Figure 4: A sample run for which an optimal solution exists. confirms that the teams are fairly homogeneous. Finally, the SB score of 0.1052 is a favorably low value that indicates that the average skill levels among all teams are very similar. Thus, we can deduce that the proposed model successfully generates teams with a balanced distribution of skills. Next, we introduce a third experiment. In this experiment, we conducted a comparative analysis of our proposed model and two other models: the first was a genetic algorithm, and the second was a modified genetic algorithm with local search and adaptive penalties. The settings of the experiment were as follows: fifty players, each with ten listed skills. The aim was to form five balanced teams. The results are shown in Table 3.

### Interpretation of the Metrics

A proper interpretation of the results of experiment number three is necessary to understand their relevance to the models' performance. First, the mean skill distribution of 0.6222 indicates that the average variation in skill levels between the teams is moderate. Second, the within-team variance of 1.6956 suggests that there is moderate variation in skill levels within the teams. This shows that teams are diverse in terms of skill levels, with a mix of skilled and less skilled players. Third, the skill balance of 0.1052 or 10.52% suggests that the difference between the highest and lowest total skill levels between teams is relatively small compared to the average total skill level. This indicates that the teams formed are fairly balanced in terms of total skills. Next, we analyze the results of the two models used for comparison purposes, namely the genetic algorithm (GA) and the genetic algorithm with local search and adpative penalties (GA_LS_AP). First, the GA_LS_AP model outperformed the GA in all measures. However, the GA_LS_AP model could not yield teams that were as balanced as those of the GCN-MILP model.