Incomplete generalized multi-scale ordered information systems and optimal scale combination selection

In response to the situation involving ordered attribute values, missing attribute values, and varying scales across attributes within multi-scale information systems, this paper proposes incomplete generalized multi-scale ordered information systems and defines the upper approximations and lower approximations with dominance relations. We study the problem of selecting the optimal scale combination based on knowledge distance. A new knowledge distance is proposed in incomplete generalized multi-scale ordered decision systems to reflect the difference between one knowledge space and another knowledge space, as to study its monotonicity, and so on. We respectively give the concept of distance-optimal scale combination in information systems and decision systems, prove its equivalence to the traditional optimal scale combination through derivation, and then use knowledge distance to characterize the numerical features of the optimal scale combination. The proposed method's intuitiveness and effectiveness are ultimately validated through numerical examples.