VSOP fuzzy numbers and their fuzzy ordering

In this paper, we extend the conventional fuzzy numbers and introduce a new fuzzy number system named vector space of ordered pairs (VSOP). Then, we axiomatically define the fuzzy comparison relations (>, greater than or equal to and similar or equal to) on VSOP starting from the requirements of fuzzy ordering and vector ordering. A concept of grading functions is introduced to prove some interesting properties of the fuzzy comparison relations. We also introduce a concept of grading component operators which construct a new grading function from given grading functions, and prove the necessary and sufficient condition of the grading component operators. The fuzzy comparison relations constructed here are not only for VSOP fuzzy numbers, but are also directly applicable to conventional fuzzy numbers. Examples of fuzzy comparison relations which are applicable to conventional fuzzy numbers are also described. (C) 1998 Elsevier Science B.V.