On optimal scale combinations in generalized multi-scale set-valued ordered information systems

As a computing paradigm inspired by human cognition, granular computing has demonstrated remarkable effectiveness in processing large data sets. Multi-scale rough set analysis, a prominent framework within multi-granular computing, requires optimal scale selection as a critical prerequisite for knowledge extraction from multi-scale data. This study investigates optimal scale selection in generalized multi-scale set-valued ordered information systems (GMSOISs) using Dempster-Shafer evidence theory and information quantification. We first formalize GMSOISs by defining granular information transformations based on inclusion criteria. We then establish dominance relations over object sets induced by attribute subsets under different scale combinations, along with their associated information granules. Building on these constructs, we further derive lower/upper approximations and quantify belief/plausibility degrees of decision dominance classes in generalized multi-scale set-valued ordered decision systems (GMSODSs). Finally, six types of optimal scale combinations are rigorously defined for GMSOISs, consistent GMSODSs, and inconsistent GMSODSs, and their relationships are systematically elucidated. Case studies also validate the proposed theoretical framework with concrete examples.