  Modeling equity in the allocation of scarce resources is a fast-growing
concern in the humanitarian logistics field. The Gini coefficient is one of the
most widely recognized measures of inequity and it was originally characterized
by means of the Lorenz curve, which is a mathematical function that links the
cumulative share of income to rank-ordered groups in a population. So far,
humanitarian logistics models that have approached equity using the Gini
coefficient do not actually optimize its original formulation, but use
alternative definitions that do not necessarily replicate that original Gini
measure. In this paper, we derive the original Gini coefficient via the Lorenz
curve to optimize the effectiveness-equity trade-off in a humanitarian
location-allocation problem. We also propose new valid inequalities based on an
upper-bounding Lorenz curve to tighten the linear relaxation of our model and
develop a clustering-based construction of the Lorenz curve that requires fewer
additional constraints and variables than the original one. The computational
study, based on the floods and landslides in Rio de Janeiro state, Brazil,
reveals that while alternative Gini definitions have interesting properties,
they can generate vastly different decisions compared to the original Gini
coefficient. In addition, viewed from the perspective of the original Gini
coefficient, these decisions can be significantly less equitable.
