A theoretical look at ordinal classification methods based on comparing actions with limiting boundaries between adjacent classes

This paper addresses the general problem of designing ordinal classification methods based on comparing actions with limiting boundaries of ordered classes (categories). The fundamental requirement of the method consists of setting a relational system (D,S), where S and D are reflexive and transitive relations, respectively, S should be compatible with the order of the set of classes, and D is a subset of S. An asymmetric preference relation P is defined from S. Other requirements are imposed on the actions which compose the limiting boundaries between adjacent classes, in such a way that each class is closed below and above. The paper proposes S-based and P-based assignment procedures. Each of them is composed of two complementary assignment procedures, which correspond through the transposition operation and should be used conjointly. The methods work under several basic conditions on the set of limiting boundaries. Under other more demanding separability requirements, each procedure fulfills the set of structural properties established for other outranking-based ordinal classification methods. Our proposal avoids the conflict between the required correspondence through the transposition operation and the assignment of limiting actions to the classes to which they belong. We thus propose very diverse S and P-based ordinal classification approaches with desirable properties, which can be designed by using decision models with the capacity to build preference relations fulfilling the basic requirements to S and D.