With increased global awareness of environmental sustainability, green supply chain management (GSCM) has received considerable attention in the literature over the decade. Green supplier selection and assessment in GSCM is one of the most significant and complex challenges for supply chain managers. This paper provides a new model and approach for green supplier selections by decomposing their efficiency indicators into technical, environmental and eco-efficiency scores. We show that the models in the literature are computationally intensive and are not able to measure eco-efficiency properly. Instead of running three different models, linear goal programming is used to integrate technical, environmental and eco-efficiency objectives into a multiple objective linear programming (MOLP) data envelopment analysis (DEA) model. Therefore, the model proposed in this paper is less computationally intensive than the models in the literature. The new model provides a more valid eco-efficiency indicator of decision-making units (DMUs) by utilizing a better combination of the technical and environmental efficiency objectives compared to the conventional models. Unlike the conventional models, the new model identifies DMUs as being eco-efficient if, and only if, they are both technically and environmentally efficient. We also discuss the non-dominated weights as the solutions of the MOLP model and use them to construct technical, environmental and eco cross-efficiency matrices of the DMUs. In order to illustrate the effectiveness and applicability of the proposed model, we present the real world business case of the Hyundai Steel Company and its suppliers.