The ordering methods of interval-valued fuzzy cardinal numbers with application in an uncertain decision making Krzysztof Dyczkowski Faculty of Mathematics and Computer Science Adam Mickiewicz University Pozna´ n, Poland chris@amu.edu.pl Barbara P˛ ekala Institute of Computer Sciences University of Rzeszów Rzeszów, Poland bpekala@ur.edu.pl Michal Baczy´ nski Faculty of Science and Technology University of Silesia Katowice, Poland michal.baczynski@us.edu.pl Jaroslaw Szkola Institute of Computer Sciences University of Rzeszów Rzeszów, Poland jszkola@ur.edu.pl Tomasz Pilka Faculty of Mathematics and Computer Science Adam Mickiewicz University Pozna´ n, Poland pilka@amu.edu.pl Abstract—In this contribution we propose new methodology to compare interval-valued fuzzy cardinal numbers (IVFCN). The new methods are based on interval subsethood measures which take into account widths of the intervals. An application of introduced methodology is presented on an example of decision algorithm for medical diagnosis support. Index Terms—interval-valued cardinal number, interval-valued aggregation function, selection of interval-valued cardinal num- bers I. I NTRODUCTION Many new methods and theories take into account impreci- sion and uncertainty since Zadeh introduced fuzzy sets [1] in 1965. Especially, in the literature, many authors have proposed different approaches to the definition of different types of dis- tance measures, similarity measures and subsethood, inclusion or equivalence measures between fuzzy sets (e.g. [2], [3], [4], [5], [6]). We focus on subsethood measures, many applications of them have been proposed and they have been adapted and applied in different settings [7], [8]. As extensions of classical fuzzy set theory, intuitionistic fuzzy sets [9] and interval- valued fuzzy sets [10], [11] they are very useful in dealing with imprecision and uncertainty (see [12] for more details). In this setting, different proposals for subsethood measures between interval-valued fuzzy sets have been proposed [5], [13]. The motivation of the present paper is to propose a more natural tool for estimating the degree of subsethood between interval-valued fuzzy sets taking into account the widths of the intervals and we assume that the precise membership degree of an element in a given set is a number included in the membership interval. For such interpretation, the width of the membership interval of an element reflects the lack of precise membership degree of that element. Hence, the fact that two elements have the same membership intervals does not necessarily mean that their corresponding membership values are the same. This is why we have taken into account the importance of the notion of width of intervals while defining new types of subsethood measures. This approach reflects the meaning of the interval values and is better adapted to real applications. That allowed to construct an effective method for comparing and ordering interval valued fuzzy cardinal numbers. Such numbers are of great importance in solving decision problems in which uncertainty occurs (see [14], [15], [16], [17], [18], [19]). The paper is organized as follows. In Section 2, basic infor- mation of interval-valued fuzzy setting are recalled. Next, in Section 3, an interval subsethood measure for interval-valued fuzzy values by using partial or linear orders is presented. Especially, some construction methods are proposed. Finally, we propose the methodology to compare of interval-valued fuzzy cardinal numbers and its application in decision model of medical diagnosis support. II. PRELIMINARIES A. Interval-valued fuzzy set theory. Orders in the interval- valued fuzzy settings We use the following notation for the set of intervals L I = {[x , x]: x , x ∈ [0, 1] and x ≤ x}, which are the basis of interval-valued fuzzy sets introduced by L. A. Zadeh [10] and R. Sambuc [11]. Definition 1 (cf. [11], [10]). An interval-valued fuzzy set IVFS  A in X is a mapping  A : X → L I such that  A(x)=[A (x), A(x)] ∈ L I for x ∈ X, where  A ∩  B =  〈x,  min{A (x),B (x)}, min{ A(x), B(x)}  〉 : x ∈ X  , 978-1-7281-6932-3/20/$31.00 ©2020 IEEE