Mutual Conversion Between Preference Maps And Cook-Seiford Vectors

In group decision making, the preference map and Cook-Seiford vector are two concepts as ways of describing ties-permitted ordinal rankings. This paper shows that they are equivalent for representing ties-permitted ordinal rankings. Transformation formulas from one to the other are given and the inherent consistency of the mutual conversion is discussed. The proposed methods are illustrated by some examples. Some possible future applications of the proposed formulas are also pointed out.Comment: 7 pages, 1 figure, 1 table, 1 theorem, some formula

Measuring Knowledge for Recognition and Knowledge Entropy

People employ their knowledge to recognize things. This paper is concerned with how to measure people's knowledge for recognition and how it changes. The discussion is based on three assumptions. Firstly, we construct two evolution process equations, of which one is for uncertainty and knowledge, and the other for uncertainty and ignorance. Secondly, by solving the equations, formulas for measuring the levels of knowledge and the levels of ignorance are obtained in two particular cases. Thirdly, a new concept of knowledge entropy is introduced. Its similarity with Boltzmann's entropy and its difference with Shannon's Entropy are examined. Finally, it is pointed out that the obtained formulas of knowledge and knowledge entropy reflect two fundamental principles: (1) The knowledge level of a group is not necessarily a simple sum of the individuals' knowledge levels; and (2) An individual's knowledge entropy never increases if the individual's thirst for knowledge never decreases.Comment: Measuring knowledge for recognition and knowledge entrop