By increasing world food demand, the agricultural food supply chain has become a key element in the global business competitive environment. Also, the growing population causes increased demand for edible vegetable oils. Edible oil is one of the main products in the home food basket, which shows the importance and attention to the supply chain network of edible vegetable oils. As a result, the present research aims to design a sustainable supply chain network of edible vegetable oils under uncertainty. A mixed-integer linear programming model has developed with three objective functions, including minimizing costs, minimizing environmental impacts, and maximizing social impact. The multi-objective mathematical model has been solved by the preemptive fuzzy goal programming method. In this study, the demand for final edible oil and meals is considered uncertain. A case study has been considered in the supply chain network design of Iranian edible vegetable oils to validate the proposed model. In order to collect the dataset of the input parameters, we considered the three aspects of big data, namely variety, velocity, and volume. Furthermore, some sensitivity analyses have been provided for the main parameters of the present research. The obtained results show that the most important component of the cost objective function is production costs. Furthermore, import tariff and its value have a significant impact on the objective functions and decision variables, including area under cultivation, water consumption, and number of workers. Finally, some suggestions are presented for managers and experts in this industry.