A methodology to determine the maximum value of weighted Gini–Simpson index

Weighted Gini–Simpson index is an analytical tool that promises to be widely used concerning biological and economics applications, relative to the assessment of diversity measured by compositional proportions of a system defined with a finite number of elementary states characterized by positive weights. In this paper, a current literature review on the theme is presented and the mathematical properties of the index are outlined, focusing on the location of the maximizer (maximum point) and evaluation of the maximum value, with emphasis in the role of the Lagrange multiplier critical value—closely related with the harmonic mean of the weights—which is shown to be a barrier concerning the feasibility of the solution. Sequential procedures are presented, either backward or forward, which are used to obtain the correct values of the maximum point coordinates, thus allowing for the computation of the right maximum value of the index. Also, new theoretical results are provided, such as the calculus of limits and partial derivatives related to the critical solution, used to assess of the effectiveness of the algorithms herein proposed and discussed.