This paper explores the worst-case carbon footprint of supply chain networks under uncertain link failure scenarios, with the objective of quantifying the environmental impact in extreme conditions. Supply chain networks are vulnerable to disruptions, such as link failures, which can significantly impact their carbon emissions. These failures, arising from factors like natural disasters, geopolitical risks, or logistical challenges, introduce substantial uncertainty in the network's performance. To capture the environmental impact under extreme failure conditions, we adopt a network-based approach, leveraging robust optimization to handle the uncertainty and mixed integer non-linear optimization (MINLP) to solve the resulting complex problem. The study models the supply chain as a network of nodes and edges, where nodes represent supply chain entities (e.g., suppliers, warehouses, and consumers) and edges represent transportation or logistics links. The uncertainty in the failure of these links is captured through a set of possible failure scenarios, which may grow exponentially in size. Robust optimization techniques are employed to determine the worst-case carbon footprint, considering the full range of failure scenarios, while ensuring the solution remains feasible under all possible conditions. MINLP is used to solve the optimization problem, identifying solutions that minimize carbon emissions while addressing the uncertainty in the network. The results provide key insights into managing supply chain emissions in the presence of large uncertainty, helping supply chain managers design more resilient and sustainable networks that can adapt to extreme disruptions while minimizing their environmental impact.