As a fundamental problem in Operations Research, sparse process flexibility design (SPFD) aims to design a manufacturing network across industries that achieves a trade-off between the efficiency and robustness of supply chains. In this study, we propose a novel solution to this problem with the help of computational optimal transport techniques. Given a set of supply-demand pairs, we formulate the SPFD task approximately as a group sparse optimal transport (GSOT) problem, in which a group of couplings between the supplies and demands is optimized with a group sparse regularizer. We solve this optimization problem via an algorithmic framework of alternating direction method of multipliers (ADMM), in which the target network topology is updated by soft-thresholding shrinkage, and the couplings of the OT problems are updated via a smooth OT algorithm in parallel. This optimization algorithm has guaranteed convergence and provides a generalized framework for the SPFD task, which is applicable regardless of whether the supplies and demands are balanced. Experiments show that our GSOT-based method can outperform representative heuristic methods in various SPFD tasks. Additionally, when implementing the GSOT method, the proposed ADMM-based optimization algorithm is comparable or superior to the commercial software Gurobi. The code is available at https://github.com/Dixin-s-Lab/GSOT.