Global sustainable closed-loop supply chain network considering Incoterms rules and advertisement impacts

• Introducing the globalization of operations in a sustainable closed-loop supply chain (CLSC). • Formulating and incorporating the Incoterms rules to the sustainable CLSC problem. • Considering international transportation modes for the international suppliers. • Classifying the customers based on green degrees and advertisement factors. • Developing an efficient Lagrangian-based heuristic solution for the proposed CLSC model. Industrial information integration plays a crucial role in modern supply chains by ensuring the smooth flow of data across all stages, including recovery, recycling, and disposal, which is essential for the successful implementation of a closed-loop supply chain (CLSC) model. Building on this, our paper addresses a global CLSC problem by incorporating International Commercial Terms (Incoterms) and international transportation modes, bridging global supply chain operations with sustainability criteria. This innovative approach advances the development of a globally sustainable CLSC by focusing on the integration of economic, environmental, and social factors, i.e., the triple bottom line of sustainability. Specifically, we address environmental concerns through the introduction of carbon taxation and enhance social sustainability by exploring the impact of advertising on customer satisfaction. To further refine this model, we classify customers based on their sustainability engagement and apply a fuzzy programming approach to account for uncertainty in customer demand influenced by advertising. To solve this complex global CLSC model, we conduct a thorough analysis of constraints and develop a robust Lagrangian relaxation reformulation. While the initial solution may result in infeasibility, we propose a heuristic algorithm that ensures feasible solutions. Our efficient Lagrangian-based heuristic, incorporating an adaptive strategy, is capable of solving large-scale networks with an approximate 10 % optimality gap. Ultimately, this research provides both a comprehensive framework for practitioners to improve the environmental performance and global operations of their supply chains, as well as significant theoretical contributions to the field of industrial information systems.