This paper proposes a novel Binary Grey Wolf Optimizer (BGWO), which innovatively incorporates chaotic mapping and a sigmoid transfer function to adapt the original Grey Wolf Optimizer (GWO) to binary search spaces. The proposed BGWO is applied to combinatorial optimization problems such as feature selection and mobile edge computing offloading. Experimental results demonstrate that BGWO outperforms the Binary Whale Optimization Algorithm (BWOA) in feature selection across multiple datasets and in mobile edge computing offloading tasks, achieving higher classification accuracy and resource allocation efficiency. Additionally, BGWO exhibits superior performance in resource allocation under competitive conditions in complex dynamic environments, validating its ability to balance global exploration and local exploitation. The outstanding performance of this algorithm highlights its broad applicability to classification problems and other optimization tasks. Future research could further explore parameter tuning strategies for BGWO and its scalability in large-scale and real-time data scenarios.