<p>Sustainable supplier selection and uncertainty in decision-making are two critical issues in supply chain management. In this paper, we address multi-period, multi-product, and capacity-constrained supplier selection and order allocation problems under a carbon cap-and-trade regulation. We develop a mixed-integer programming model to determine the selection of suppliers and their order quantities for each time period, aiming to minimize the overall cost of the supply chain. Furthermore, we explore the robust version of supplier selection and order allocation by constructing an adjustable uncertainty set to handle uncertain market demand. Subsequently, we develop robust counterparts to ensure computational tractability, resulting in a mixed-integer second-order cone programming model. A heuristic method is proposed to yield a recipe for solving large-scale problems efficiently. Finally, using both in-sample and out-of-sample data, we compare the proposed adjustable robust optimization model with other models, including the robust optimization model under an ellipsoidal uncertainty set, the robust optimization model under a box uncertainty set, the deterministic model, and the stochastic optimization model, to verify its effectiveness and feasibility. Unlike robust methods using the box and ellipsoidal uncertainty sets, the adjustable robust method can provide decision-makers with a scheme that has an adjustable degree of decision conservatism under the premise of probability guarantees. The proposed heuristic approach shows superiority to other existing methods and off-the-shelf solver Gurobi in terms of solution quality.</p>