A linear ordering on the class of Trapezoidal intuitionistic fuzzy numbers

Fuzzy numbers and intuitionistic fuzzy numbers are introduced in the literature to model problems involving incomplete and imprecise information in expert and intelligent systems. Ranking of TrIFNs plays an important role in an information system (Decision Making) with imprecise and inadequate information and the complete ranking on the class of trapezoidal intuitionistic fuzzy number is an open problem worldwide. Researchers from all over the world have been working in ranking of intuitionistic fuzzy numbers since 1985, but till date there is no common methodology that ranks any two arbitrary intuitionistic fuzzy numbers due to the partial ordering of TraIFNs. Different algorithms are available in the literature for solving intuitionistic fuzzy decision (or information system) problem, but each and every algorithm failed to give better result in some places due to the ranking procedure of TrIFNs. Intuitionistic fuzzy decision algorithm works better when it have a complete ranking procedure that ranks arbitrary intuitionistic fuzzy numbers. In this paper a linear (total) ordering on the class of trapezoidal intuitionistic fuzzy numbers using axiomatic set of eight different scores is introduced. The main idea of this paper is to classify and study the properties of eight different sub classes of the set of TrIFNs. Further new total order relations are defined on each of the subclasses of TrIFNs and they are extended to a complete ranking procedure on the set of TrIFNs. Finally the significance of the proposed method over existing methods is studied by illustrative examples. (C) 2016 Elsevier Ltd. All rights reserved.