# Relation \"Greater than or Equal to\" between Ordered Fuzzy Numbers
## 1 Introduction  
The concept of ordered fuzzy number (OFN) was intuitively introduced by [PERSON] [1; 2; 3; 4] as an extension of the notion of fuzzy number (FN) which is widely interpreted as imprecise approximation of real number. OFNs' usefulness follows from the fact that it is interpreted as FN with additional information about the location of the approximated number. [PERSON] [1; 2; 3; 4] has determined arithmetic for OFNs as an extension of results obtained by [PERSON] and [PERSON] [5] for FNs. For formal reasons, the [PERSON]' theory was revised [6] in such a way that revised OFN definition fully corresponds to the intuitive [PERSON]'s definition of OFN. OFNs are always defined without use of any ordering relation between FNs. Knowing this fact makes it easier to read the section on ordering relationship between OFNs. This paper is linked to the revised OFNs' theory. In decision analysis, economics and finance, OFNs are frequently employed to evaluate the alternatives in modelling a real-world problem [7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19]. On the other hand, the OFN theory has an important disadvantage. This disadvantage is due to the lack of formal mathematical models associated with OFNs. Therefore, an important goal of further formal research should be to fill these theoretical gaps. If any alternatives are evaluated by OFNs then their ranking leads to OFNs' arrangement which is pre-given as an ordering relation \"greater than or equal to\" between OFNs. Since the notion of OFN is interpreted as an extension of the notion of FN, any formal model of order between OFNs should be consistent with the fixed ordering relation between FNs. Unlike in the case of real numbers, FNs have no natural order. A straightforward approach to the ordering of FNs is to convert each compared FN into a real number. Any procedure of this conversion is called a \"defuzzification method\" [20]. Representative examples of FNs' arrangement using different defuzzification methods are presented in [20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51; 52; 53; 54; 55; 56]. Each individual defuzzification method, however, pays attention to a special aspect of an FN. As a consequence, each approach suffers from some defectsthat only one real number is associated with each FN. [PERSON] [57] pointed out that \"by reducing the whole of our analysis to a single number, we are losing much of the information. We have purposely been keeping throughout our calculations\". [PERSON] and [PERSON] [58] introduced defuzzification methods for OFNs. Some applications of OFN arrangement using defuzzification methods are presented in [8; 16; 18; 19]. On the other side, in [17], it is shown that the use of defuzzification methods has a significant impact on the ordering of OFNs. In an extreme case, the use of defuzzification procedures can totally blur the true picture of arrangement of OFNs. It can lead to results deviating from real ranking of decision alternatives, which will increase the hazard of making a wrong decision. For this reasons, OFNs arrangement should be described by a fuzzy relation which compares OFNs pairwise. In this way, we can compare OFNs without losing information about the imprecision and orientation of evaluated OFNs. This approach is more realistic. For FNs, fuzzy order relations can be defined in two ways. First of all, fuzzy order of FNs can be determined using \(\alpha\)-cuts. Representative examples of FNs' arrangement using \(\alpha\)-cuts are presented in [59; 60; 61]. At present, the \(\alpha\)-cuts theory dedicated to OFNs is unknown. Therefore, in the current moment, any formal models of ordering with use of \(\alpha\)-cuts cannot be extended to the case of OFNs. Moreover, [PERSON] [62] defined fuzzy order of FN applying the [PERSON]'s Extension Principle [63; 64; 65]. This method does not raise any objections.