A function that takes
n numbers as input and outputs one number is said to be homogeneous whenever the result of multiplying each input by a certain factor
λ yields the original output multiplied by ...that same factor. This concept has been extended by the notion of homogeneity, which generalizes the product in the expression of homogeneity by a general function
g and the effect of the factor
λ by an automorphism. However, the effect of parameter
λ remains unchanged for all the input values. In this study, we generalize further the condition of homogeneity by introducing
ℱ‐homogeneity, which is defined with respect to a family of functions, enabling a different behavior for each of the inputs. Next, we study the properties that are satisfied by this family of functions and, moreover, we link this concept with the condition of directional monotonicity, which is a trendy property in the framework of aggregation functions. To achieve that, we generalize directional monotonicity by
ℱ directional monotonicity, which is defined with respect to a family of functions
ℱ and a family of vectors
V. Finally, we show how the introduced concepts could be applied in two different problems of computer vision: a snow detection problem and image thresholding improvement.
A new concept of equivalence between intervals and the induced indistinguishability between interval‐valued (IV) fuzzy sets are proposed and considered. A new notion of the degree of IV equivalence ...is presented where partial or linear orders and the width of intervals are involved reflecting uncertainty. Furthermore, construction methods of the considered equivalences are provided and the relation between them and other notions of IV equivalences are studied. Finally, a methodology to apply the proposed equivalences in the field of image processing is shown, along with an illustrative example.
The notion of directional monotonicity emerged as a relaxation of the monotonicity condition of aggregation functions. As the extension of aggregation functions to fuse more complex information than ...numeric data, directional monotonicity was extended to the framework of multidimensional data, with respect to the product order, which is a partial order. In this work, we present the notion of admissible order for multidimensional data and we define the concept of directional monotonicity for multidimensional fusion functions with respect to an admissible order. Moreover, we study the main properties of directionally monotone functions in this new context. We conclude that, while some of the properties are still valid (e.g. the set of directions of increasingness is still closed under convex combinations), some of the main ones no longer hold (e.g. there does not exist a finite set of directions that characterize standard monotonicity in terms of directional monotonicity).
In the theory of aggregation, there is a trend towards the relaxation of the axiom of monotonicity and also towards the extension of the definition to other domains besides real numbers. In this ...work, we join both approaches by defining the concept of directional monotonicity for functions that take values in Riesz spaces. Additionally, we adapt this notion in order to work in certain convex sublattices of a Riesz space, which makes it possible to define the concept of directional monotonicity for functions whose purpose is to fuse uncertain data coming from type-2 fuzzy sets, fuzzy multisets, n-dimensional fuzzy sets, Atanassov intuitionistic fuzzy sets and interval-valued fuzzy sets, among others. Focusing on the latter, we characterize directional monotonicity of interval-valued representable functions in terms of standard directional monotonicity.
•A new cohesion measure for Hesitant Fuzzy Linguistic Term Set is introduced.•A flexible cohesion-driven Consensus Reaching Process for Large Groups is defined.•An adaptive feedback process that ...guides the consensus process is presented.•A simulation of several scenarios to evaluate the performance of the model is shown.
Large-scale group decision-making (LSGDM) under uncertainty modelled by comparative linguistic expressions based on a hesitant fuzzy linguistic term set (HFLTS) has recently attracted the interest of many researchers and research, due to the necessity of its function in LSGDM, and the challenges it faces such as the managing of the scalability problem, uncertainty of experts’ opinions and dealing with polarized conflicting opinions. To smooth out such discrepancies and obtain agreed solutions Consensus Reaching Processes (CRPs) for LSGDM have been applied, in which experts are grouped into sub-groups according to the closeness of their opinions to deal with scalability. However, most CRPs for LSGDM are driven by a majority rule, in which larger sub-groups, where there might be internal disagreements, lead the consensus. In such processes, the internal disagreements can produce unsatisfactory solutions. Consequently, the majority view should be complemented by additional mechanisms that also measure the strength of the sub-groups’ opinions. A good measurement of such strength is the cohesion among the sub-group members. Therefore, in this paper, a new cohesion measure for HFLTS based on restricted equivalence functions for measuring the experts’ sub-group cohesiveness is introduced to drive the consensus process together the majority and thus reduce the impact of internal disagreements risen in majority driven CRPs. It is then integrated in a new cohesion-driven CRP approach based on LSGDM to deal with comparative linguistic expressions based on HFLTS. An experimental analysis on different large scale scenarios will show the performance and importance of cohesion in consensus based LSGDM.
In this paper, the problem of measuring the degree of subsethood in the interval-valued fuzzy setting is addressed. Taking into account the widths of the intervals, two types of interval subsethood ...measures are proposed. Additionally, their relation and main properties are studied. These developments are made both with respect to the regular partial order of intervals and with respect to admissible orders. Finally, some construction methods of the introduced interval subsethood measures with the use interval-valued aggregation functions are examined.
Interval-valued fuzzy sets are a generalization of classical fuzzy sets where the membership values are intervals. The epistemic interpretation of interval-valued fuzzy sets assumes that there is one ...real-valued membership degree of an element within the membership interval of possible membership degrees. Considering this epistemic interpretation, we propose a new measure, called IV-embedding, to compare the precision of two interval-valued fuzzy sets. An axiomatic definition for this concept as well as a construction method are provided. The construction method is based on aggregation operators and the concept of interval embedding, which is also introduced and deeply studied.
Overlap and grouping functions are special kinds of non necessarily associative aggregation operators proposed for many applications, mainly when the associativity property is not strongly required. ...The classes of overlap and grouping functions are richer than the classes of t-norms and t-conorms, respectively, concerning some properties like idempotency, homogeneity, and, mainly, the self-closedness feature with respect to the convex sum and the aggregation by generalized composition of overlap/grouping functions. In previous works, we introduced some classes of fuzzy implications derived by overlap and/or grouping functions, namely, the residual implications RO-implications, the strong implications (G,N)-implications and the Quantum Logic implications QL-implications, for overlap functions O, grouping functions G and fuzzy negations N. Such implications do not necessarily satisfy certain properties, but only weaker versions of these properties, e.g., the exchange principle. However, in general, such properties are not demanded for many applications. In this paper, we analyze the so-called law of O-Conditionality, O(x,I(x,y))≤y, for any fuzzy implication I and overlap function O, and, in particular, for RO-implications, (G,N)-implications, QL-implications and D-implications derived from tuples (O,G,N), the latter also introduced in this paper. We also study the conditional antecedent boundary condition for such fuzzy implications, since we prove that this property, associated to the left ordering property, is important for the analysis of the O-Conditionality. We show that the use of overlap functions to implement de generalized Modus Ponens, as the scheme enabled by the law of O-Conditionality, provides more generality than the laws of T-conditionality and U-conditionality, for t-norms T and uninorms U, respectively.
In this work, we propose a new notion of monotonicity: strengthened ordered directional monotonicity. This generalization of monotonicity is based on directional monotonicity and ordered directional ...monotonicity, two recent weaker forms of monotonicity. We discuss the relation between those different notions of monotonicity from a theoretical point of view. Additionally, along with the introduction of two families of functions and a study of their connection to the considered monotonicity notions, we define an operation between functions that generalizes the Choquet integral and the Łukasiewicz implication.
In this paper, we introduce the notion of ordered directionally monotone function as a type of function which allows monotonicity along different directions in different points. In particular, these ...functions take into account the ordinal size of the coordinates of the inputs in order to fuse them. We show several examples of these functions and we study their properties. Finally, we present an illustrative example of an application which justifies the introduction and the study of the concept of ordered directional monotonicity.
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