Closed-loop supply chain network design (CLSCND) has been increasingly spotlighted over the latest decade. The focus has been given to maximize the economic performance, resource utilization and sustainability through incorporating a holistic decision-making on both forward and reverse logistics. In this paper, a new fuzzy-stochastic multi-objective mathematical model is formulated for sustainable CLSCND. The model aims at balancing the trade-off between cost effectiveness and environmental performance under different types of uncertainty. The environmental performance of CLSCND is measured by carbon emission. Moreover, the network flexibility is modeled and incorporated in the decision-making so that customer demands can be fulfilled by different means. In order to solve the complex optimization problem, the model is first defuzzilized and converted into an equivalent crisp form. Then, a sample average approximation based weighting method (SAAWM) is developed to obtain a set of Pareto optimal solutions between cost and carbon emission under different uncertain environments. The model is validated through a set of numerical experiments. The computational results show, through the incorporation with network flexibility, the proposed mathematical model and solution approach can effectively generate consistent objective values and solutions over different scenario trees and obtain robust strategic decisions on facility locations. Meanwhile, the flexibility and rationality of the decision-making on transportation management, demand allocation and facility operations can be improved as well.