Accurate preference-based method to obtain the deterministically optimal and satisfactory fairness-efficiency trade-off

The resource allocation problem is a classic multi-objective challenge, particularly when balancing the fairness-efficiency trade-off. To achieve a deterministically optimal and satisfactory solution, researchers frequently employ preference-based methods, including selecting among Pareto solutions based on the decision-maker's a posteriori preference and using deterministic models incorporating a priori preferences. In this study, we address two main challenges—specifically, (1) the limitations in measuring the abstract concepts of fairness and efficiency and (2) finding a deterministically optimal and satisfactory balance between fairness and efficiency. We apply a Gini impurity index derived from the classification and regression tree to calculate fairness, ensuring the Gini index function's differentiability. Additionally, we unify the scales of fairness and efficiency to facilitate calculation. Using accurate preference information, we employ the extended interval goal programming method to solve the model and achieve a deterministically optimal and satisfactory solution. The comparative analysis results demonstrate that our model (1) efficiently addresses the real-world water resource allocation problem concerning the fairness-efficiency trade-off; and (2) generates fewer penalties, with an average improvement ratio of 8% in the case study, using more refined penalty functions that align closer to the decision-maker's real and nonlinear preferences.