This study proposes a novel framework for evaluating the efficiency of two-stage network decision-making units by integrating the hyperbolic distance function (HDF) with conic optimization. Building on the direct HDF model of Hassanasab et al. (2019), the problem is reformulated within a conic programming structure that explicitly captures internal flows and inter-stage dependencies. The proposed model measures overall efficiency by simultaneously contracting inputs and expanding both intermediate products and final outputs. Unlike conventional approaches, it introduces two stage-specific hyperbolic efficiency parameters, enabling input contraction and output expansion at each stage while implicitly regulating intermediate products and preserving the sequential production structure. The resulting primal and dual linear programming formulations provide additional economic insights, including stage-level shadow prices. Numerical experiments using synthetic and real-world data demonstrate improved discriminatory power and clearer managerial guidance. These results underscore the effectiveness of the proposed conic two-stage HDF framework for evaluating complex multi-stage systems.
