The rapid depletion of fossil fuels and the adverse environmental effects of combustion has growing concerns about the sustainability of fossil fuel and the world turns its focus on cleaner energy sources. Animal fat-based biodiesel is a sensible alternative to non-renewable energy sources as it's produced from inedible, affordable, and easily available biomass. The effective design of a biodiesel supply chain is pivotal for making it more advantageous than fossil fuels from a social, economic, and environmental standpoint. This model reduces the biodiesel supply chain cost while minimizing its environmental impact and maximizing its social impact. It is planned to have multi-supply sites for providing beef, chicken, and mutton fat, and numerous biorefineries, rendering factories, and distribution depots. The augmented ϵ-constraint approach combined with the lexicographic optimization technique is used to solve the multi-objective optimization model. A numerical example is analyzed, and the results show that the environmental impact of establishing biodiesel production facilities is 99.5% of the environmental impact of the entire biodiesel supply chain and that the cost of installing biodiesel production plants is 99% of the cost of the entire biodiesel supply chain. A heterogeneous fleet of trucks is incorporated to distribute the material throughout the supply chain whereby the impact of transportation on the environment is 0.0007% of the total environmental impact of the biodiesel supply chain and transportation cost amount to 0.0014% of the total costs of the biodiesel supply chain. The model demonstrates the biodiesel supply chain network under a sustainability framework and addresses the environmental and social impact based on the life cycle assessment which accelerates the transition to a renewable energy future.