Interval valued strict preference with Zadeh triples

Preference modelling and choice theory are common to many different areas including operational research, economics, artificial intelligence and social choice theory. We consider ''vague preferences'' and introduce a new technique to model this vagueness with the aim of making a choice at the final stage. Our basic tools of modelling will be fuzzy relations and interval valued fuzzy sets. Specifically, we propose that the initial vagueness in the weak preferences of a decision maker is represented by a fuzzy relation and further constructs from this concept introduce a higher-order vagueness which is represented by interval valued fuzzy sets. We derive necessary and sufficient conditions on the representation of this initial vagueness such that a complete ranking of the alternatives is possible. It is shown that conditions weaker than min-transitivity on the representation of initial vagueness are necessary and sufficient for the alternatives to be partially ranked. Furthermore, two linearity conditions are shown to make the ordering of the alternatives a complete order. Conditions for the existence of unfuzzy non-dominated alternatives are also explored.