ON FUZZY ORDERINGS OF CRISP AND FUZZY INTERVALS R. FUENTES-GONZALEZ, P. BURILLO Universidad Ptiblica de Navarra. 31006 Pamplona, Spain E-mail: ($uentes, pburillo)&navarra.es G. MAYOR Universidad de las Islas Baleares. 07071 Palma de Mallorca, Spain E-mail: gmayor&ib.es First, fuzzy membership functions for the assertion '[x,y] is a positive interval' are proposed and characterized via non-decreasing real maps. Last, using those functions together with the interval- difference and the notion of average index, comparison indexes between intervals and the ones between fuzzy intervals are proposed. 1 Introduction In the literature, there are a lot of methods concerning the problem of ordering the inter- val 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 classi- fied as belonging t o two different approaches. (i) Ordering the 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), the present paper deals with ordering pro- cedures of interval numbers or fuzzy quanti- ties belonging to the aforementioned second class. Specifically, we propose a theoretical approach to the comparison of pairs of the approximate measurements [g, T] , [y - , g] , ... or pairs of fuzzy quantities A, B, ... in the fol- lowing way: (a) The crisp and fuzzy intervals are considered as points belonging t o crisp subsets. (b) Using fuzzy extensions of the characteristic function fR+ and arith- metic interval operations, comparison in- dexes R([g, o], [y,y]) € [O, 11 between inter- vals 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 the assertion '[g,T] is a positive interval' are proposed. Third, we characterize every aforementioned func- tion 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 the maps type Rpos are analysed. Finally, using average indexes (see 3), exten- sions of the previous maps Pus and RPos to a class 3J(R) of fuzzy intervals are defined. Properties and examples are included. 2 Basic notions and notations 1% this section we recall some basic arithmetic operations, results and binary relations r e lated t o the crisp and fuzzy Interval Analysis. 2.1 Some results on Interval Analysis In the field of Interval Analysis ') an in- terval number can be thought as an exten- sion of the concept of a real number. If J(R) denotes the set of interval numbers, we con- sider R as a proper subset of J(R) identify- ing x € R with [x,x] = {x). In this way, [x] = k,?], [y] = [g,g], ... denote generic in-