On the one hand, supply chain management of agri-food products under uncertain conditions has a significant impact on food security and, on the other hand, increases the profits of supply chain components. Moreover, considering the sustainability concepts leads to more social and environmental benefits. The present study investigates the canned food supply chain under uncertain conditions and sustainability concepts by considering strategic and operational decisions and different characteristics. The proposed model is a multi-echelon, multi-period, multi-product, multi-objective location-inventory-routing problem (LIRP) in which the vehicle fleet is considered heterogeneously. The objectives of this model are to (1) minimize costs, (2) minimize customer dissatisfaction, (3) maximize production throughput, and (4) maximize job opportunities. In this study, carbon cap and trade mechanism are used to minimize environmental damage. Robust fuzzy stochastic programming (RFSP) is employed to cope and control uncertainties. The multi-objective optimization problem is implemented on a real case and solved using the Torabi and Hassini (TH) method. The results of this study showed that with increasing confidence levels, the severity of the problem increased and the values of the objective functions worsened. Also, using the relative value of stochastic solution (RVSS) criterion demonstrated that the effect of utilizing the RFSP approach on the first and second objective functions was higher than that the nominal approach showed itself. Finally, sensitivity analysis is performed on two parameters: the selling price of products to foreign customers and the cost of purchasing products from farms. The results of this study showed that changing these two parameters had a significant effect on the first and second objective functions.