The extended Bonferroni mean (EBM) recently proposed differs from the classical Bonferroni mean, as it aims to capture the heterogeneous interrelationship among the attributes instead of presupposing a homogeneous relation among them. In this study, we generalize the EBM to explicitly and profoundly understand its aggregation mechanism by defining a composite aggregation function. We adopt the approach of optimizing the choice of weighting vectors for the generalized EBM (GEBM) with respect to the least absolute deviation of residuals. We also investigate several desirable properties of the GEBM. Our special interest in this study is to investigate the ability of the GEBM to model mandatory requirements. Finally, the influence of replacing the conjunctive of the GEBM is analyzed to show how the change of the conjunctive affects the global andness and orness of the GEBM. Meanwhile, the aggregation mechanism of the EBM is specified and provided with quite intuitive interpretations for application.