Managing supply chain risks has received increased attention in recent years, aiming to shield supply chains from disruptions by predicting their occurrence and mitigating their adverse effects. At the same time, the resurgence of Artificial Intelligence (AI) has led to the investigation of machine learning techniques and their applicability in supply chain risk management. However, most works focus on prediction performance and neglect the importance of interpretability so that results can be understood by supply chain practitioners, helping them make decisions that can mitigate or prevent risks from occurring. In this work, we first propose a supply chain risk prediction framework using data-driven AI techniques and relying on the synergy between AI and supply chain experts. We then explore the trade-off between prediction performance and interpretability by implementing and applying the framework on the case of predicting delivery delays in a real-world multi-tier manufacturing supply chain. Experiment results show that prioritising interpretability over performance may require a level of compromise, especially with regard to average precision scores.