Invariance and robustness of the ordered inequality of aggregate sustainability indices by vector space theory

Sustainability assessment of manufacturing sector has been described by a single aggregate index for each process. A set of production processes give rise to an inequality ordering of such indices. This ordered hierarchy must be representative of the physical system otherwise it may yield misleading results. A recent paper has treated processes as vectors whose components are the manufacturing factors in a Euclidean space generated by a set of orthonormal vectors. Aggregate indices are defined by the Euclidean magnitude of the process vectors. A subsequent paper considered an enlarged space where the Euclidean space is a particular case. In this increased space, the robustness of the Euclidean indices was analyzed. The present work in this broader space focuses on the robustness of the ordered inequality. The central aim consists in determining the conditions by which the Euclidean indices belong to an invariant hierarchy in this space. The formalism was confirmed by providing consistent results when applied to real production processes.