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Group formation is one of the key processes in collaborative learning because having adequate members in the learning groups supports good collaborative interactions among members and is fundamental to ensuring satisfactory learning performance. Several previous studies have proposed genetic algorithm-based group formation scheme that considers multiple student characteristics to optimize collaborative learning groups. However, the fitness function used in the genetic algorithm (GA) for assessing the quality of group formation may determine collaborative learning groups with unbalanced learning characteristics. Additionally, few studies considered how learning roles and interactions among peers can be used to optimize collaborative learning groups and confirmed the effects of different group formation schemes on learning performance and peer interaction. Therefore, this work proposes a novel genetic algorithm-based group formation scheme with penalty function (GAGFS-PF) that considers the heterogeneous of student's knowledge levels and learning roles, and the homogeneity of social interactions measured by social network analysis among the members in the learning group, to generate collaborative learning groups with balanced learning characteristics for improving student's learning performance and facilitate student's interactions in a collaborative problem-based learning (CPBL) environment. This work uses a quasi-experimental research method to collect quantitative data to assess the effects of three group formation schemes - the proposed GAGFS-PF, the random group formation scheme, and the self-selection group formation scheme - on the learning performance and effects of interaction in a CPBL environment and also adopts interview to enhance the results of qualitative data analysis. Namely, this study adopts a mixed study to examine the research findings. Eighty-three students from three Grade 6 classes at an elementary school in New Taipei City, Taiwan were invited to participate in the experiment. Three classes were randomly assigned to the three experimental groups that used different group formation schemes including the proposed GAGFS-PF, random group formation scheme, and self-selection group formation scheme for CPBL activities on the topic of "global warming". The results reveal that the proposed GAGFS-PF is significantly superior to the random group formation scheme in the score of a completed report assessed by two teachers during the "action 2" learning stage, among the four CPBL stages. Analytical results also show that the proposed GAGFS-PF for group formation is significantly superior to the random and self-selection group formation schemes in the effects of peer interaction, as assessed using social network measures. The interview results also support that the proposed GAGFS-PF provides benefits in determining collaborative learning groups. This work contributes a novel and useful group formation scheme for enhancing collaborative learning performance and also helps in calling for future research in this field as well.