This study focuses on improving Pareto front approximation methods for solving three-objective mixed-integer optimization problems under realistic computational constraints. In particular, a sustainable facility location problem with a mixed transportation fleet is considered, incorporating economic, environmental, and social objectives. To efficiently generate high-quality Pareto solutions, a hybrid solution framework is proposed that integrates a nested ε-constraint method with several enhancements, including an AUGMECON-based scalarization, adaptive grid generation, and a refinement mechanism. The adaptive strategy dynamically updates ε-levels based on intermediate solutions to better explore the feasible region, while the refinement stage identifies sparse regions of the Pareto front using both distance- and hypervolume-based criteria and intensifies the search locally. A comprehensive computational study is conducted on benchmark instances, where multiple solution variants are evaluated under a fixed runtime budget. In addition, a one-at-a-time sensitivity analysis is performed to assess the impact of key algorithmic parameters, including grid density, ε penalty coefficient, and refinement depth. The results show that the proposed hybrid framework improves Pareto front quality in terms of hypervolume and solution distribution compared to the standard ε-constraint method. In particular, the combination of adaptive grid and refinement mechanisms produces more evenly distributed and representative solution sets without increasing computational time.

Keywords

Facility Location, Multi-Objective Optimization, Sustainable Supply Chain