Classical supply chain networks primarily focus on economic optimization by determining the optimal supply chain configuration in a chaotic business environment. Owing to rapidly increasing greenhouse gas emissions, there has been immense pressure from society to design eco-efficient supply chain networks to reduce the carbon emissions generated from supply chain activities at a reasonable cost. This study addresses the eco-efficient supply chain network design problem considering carbon trading and trade credit from suppliers to maximize the total supply chain profit in both the physical and carbon markets. The problem determines the optimal number, location, and capacity of facilities (e.g., production and distribution centers), optimal product flow among entities in the network, optimal product-selling price, and optimal economic order quantity for suppliers under different trade credit schemes. A robust fuzzy optimization model based on the integration of robust optimization and fuzzy programming was applied to address the uncertainties in demand and relevant costs. A case study of a Taiwanese steel firm was conducted to demonstrate the efficacy and efficiency of the proposed model. The results show that the proposed model improves the total supply chain profit, including the profits from physical and carbon markets, by around 3%, and reduces the computation time by approximately 72.44 %, compared to scenario-based robust stochastic programming. Our findings also show that the optimal configuration of the supply chain network is sensitive to different scenarios of carbon trade, and the selection of suppliers is affected by the trade credit policy.