International institutions constitute a central focus in the field of international studies. However, unlike other actors, the operational capacities of these institutions during internal transformations are seldom highlighted, largely due to the challenges associated with quantifying this dimension. Generally, studies on this topic rely on qualitative assessments derived from the personal and intuitive perceptions of analysts. Considering the significance of cooperative groups in global governance, this study seeks to determine the precise magnitude of performance changes in the international institution under investigation  from the G5 to the G20. To address this question, an applied study was conducted using mathematical methods to derive a functional model. Relevant data were extracted from the World Bank to evaluate the results. The proposed model is applicable to all international institutions, with the current study indicating the highest efficiency for the G7 and the lowest for the G8.