On fuzzy orderings of crisp and fuzzy intervals

In the literature, there are a lot of methods concerning the problem of ordering the interval numbers k , ~ ] , [y,g], ... (', ') or the fuzzy ones A, B , ... (see 4, and for a n overview of methods). All of these methods can be classified a s belonging t o two different approaches. (i) Ordering t h e crisp or the fuzzy intervals using binary order relations: [ c , ~ ] 4 [ y , ~ ] , A 5 B, .... (ii) Giving comparison indexes R([c,%], [ y , ~ ] ) or R(A,B) in [O, 11. In relation with previous works (6, 2 ) , t h e present paper deals with ordering procedures of interval numbers or fuzzy quantities belonging t o the aforementioned second class. Specifically, we propose a theoretical approach t o t he comparison of pairs of t he approximate measurements [g, T] , [y -, g] , ... or pairs of fuzzy quantities A, B , ... in the following way: (a) The crisp and fuzzy intervals are considered as points belonging t o crisp subsets. (b) Using fuzzy extensions of t he characteristic function fR+ and arithmetic interval operations, comparison indexes R([g, o], [y, y]) € [O, 11 between intervals and the on& R(A, B ) € [O, 11 between fuzzy intervals are defined. The paper is structured as follows. First, some necessary basic results on the crisp and fuzzy Interval Analysis fields are p r e sented. Second, fuzzy membership functions (denoted by Pos) for t he assertion '[g,T] is a positive interval' are proposed. Third, we characterize every aforementioned function Pos by a real non-decreasing map gp,, : [O, I] + [O, i]. Fourth, using a map Pos and the interval-difference [y] [x] = [y --?, jj-4, the associated comparison index Rpos as a fuzzy relation in J (R) is defined. Some p r o p erties of t he maps type Rpos are analysed. Finally, using average indexes (see 3), extensions of t he previous maps Pus and RPos t o a class 3 J ( R ) of fuzzy intervals are defined. Properties and examples are included.