The cross-border e-commerce supply chain network (CBESCN) has extensive geographical coverage, trade barriers and complexity of cross-border logistics issues, which makes its construction and development face many challenges. This article focuses on solving the operation optimisation problem of CBESCN under the background of the Internet of Things. A genetic algorithm constructed and solved the resource scheduling model of the supply chain of e-commerce enterprises in international trade. In addition, the mobile edge computing (MEC) optimisation scheme based on partial computation unloading is involved. The initial offload ratio is set and supply chain resources are allocated, then the remaining computing resources are distributed according to the server’s computing power. Finally, the offload is optimised according to the resource allocation. The experimental results show that time delay and cost adjustment strategies can improve the CBE supply chain’s comprehensive ability. The supply chain optimisation scheme proposed in this article can effectively use supply chain resources according to the requirements of computing tasks to reduce the total delay of task execution and the consumption of node computing.