Geometric Optimal Transport for Sustainable Closed-Loop Supply Chain: A Fused Gromov–Wasserstein Framework for Structural and Attribute Inefficiency Diagnosis

Designing sustainable closed-loop supply chain (CLSC) networks requires jointly assessing node-level operational attributes (recovery efficiency, processing capacity, unit cost) and inter-node spatial structure. Existing methods, including mixed-integer programming, multi-objective metaheuristics, and graph-matching, typically optimize a single cost dimension and do not decompose structural connectivity from attribute-level inefficiency. We propose a Fused Gromov–Wasserstein (FGW) diagnostic framework that combines the Wasserstein distance (attribute similarity) and the Gromov–Wasserstein distance (structural alignment) via a convex trade-off parameter α, solved using the conditional gradient algorithm. Supply–capacity imbalances are resolved by marginal rescaling, with residual unabsorbed mass reported as a diagnostic indicator of infrastructure shortfall. The framework is applied to an eight-echelon PET bottle recovery and filament manufacturing network across 24 synthetic benchmark instances at three scale classes. The FGW cost decomposes exactly into feature and structural components, allowing bottleneck arcs to be diagnosed as attribute-driven or structure-driven. Under this benchmark, bottleneck cost decreases with network size, the most frequent bottleneck arc shifts from the collection interface in small networks to the mid-chain processing handoff in large networks, and attribute heterogeneity accounts for the majority of FGW cost (57.9%, conditional on the normalization and weighting scheme used) across all 144 arc–instance combinations. These results position FGW as a tractable, interpretable diagnostic layer for circular supply chain analysis, complementing rather than replacing classical CLSC design models.