The study presents a sustainable closed-loop supply chain network that integrates financial, environmental, and social objectives within a context of uncertainty. A fuzzy-based modeling approach is introduced to address uncertainty in customer demand, cost parameters, and carbon emission coefficients across the sustainable closed-loop supply chain network. Two metaheuristic methods, the non-dominated sorting genetic algorithm II (NSGA-II) and multi-objective particle swarm optimization (MOPSO), are employed to address the problem and are compared against each other. A practical case study of a battery company is employed to validate the framework. The findings indicate that MOPSO surpasses non-dominated sorting genetic algorithm II in terms of solution quality and computational efficiency, compared with NSGA-II, the proposed MOPSO achieved a 6.3% reduction in total cost and an 8.1% decrease in CO₂ emissions, while the social index reflecting recruitment and employee security increased by 12.5%. This study contributes a sustainable closed-loop supply chain network design model for the battery industry that together optimizes economic, environmental, and social objectives amid parameter uncertainty, and offers algorithmic evaluations of optimized multi-objective metaheuristics to achieve high-quality Pareto solutions.