Sustainability in a supply chain is a demand on the one hand and a challenge on the other. It is necessary to balance between these dimensions in order to fulfill the purpose of the supply chain. Therefore, in the first phase—i.e., in procurement—it is necessary to take into account its sustainability. In this paper, a sustainable supplier was selected respecting all three aspects of sustainability. The evaluation was carried out on the basis of a total of 21 criteria arranged into two levels and three groups. A new Interval Rough SAW (simple additive weighting) method, which represents a contribution to the treatment of problems belonging to the multi-criteria decision-making (MCDM), was developed. The integration of interval rough numbers with the SAW method was completed. In addition, the full consistency method (FUCOM) was applied to determine the weights of the criteria. The integrated FUCOM-Interval Rough SAW model enables treatment of multi-criteria problems while reducing subjectivity to the lowest possible level and eliminating uncertainties and ambiguities. The results obtained were determined throughout a sensitivity analysis consisting of a change in the weight of the criteria and the influence of dynamic matrices on the change in ranks. In addition, Spearman’s rank correlation coefficient (SCC) was calculated to confirm the stability of the previously obtained results.