Radko Mesiar's research while affiliated with Palacký University Olomouc and other places
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Publications (164)
In this article, we study calculus for gH-subdifferential of convex interval-valued functions (IVFs) and apply it in a nonconvex composite model of an interval optimization problem (IOP). Towards this, we define convexity, convex hull, closedness, and boundedness of a set of interval vectors. In identifying the closedness of the convex hull of a se...
It is well known that the usual point-wise ordering over the set T of t-norms makes it a poset but not a lattice, i.e., the point-wise maximum or minimum of two t-norms need not always be a t-norm again. In this work, we propose, two binary operations on the set TCA of continuous Archimedean t-norms and obtain, via these binary operations, a partia...
A useful expansion of the intuitionistic fuzzy set (IFS) for dealing with ambiguities in information is the Pythagorean fuzzy set (PFS), which is one of the most frequently used fuzzy sets in data science. Due to these circumstances, the Aczel-Alsina operations are used in this study to formulate several Pythagorean fuzzy (PF) Aczel-Alsina aggregat...
In this article, the concepts of gH-subgradient and gH-subdifferential of interval-valued functions are illustrated. Several important characteristics of the gH-subdifferential of a convex interval-valued function, e.g., closeness, boundedness, chain rule, etc. are studied. Alongside, we prove that gH-subdifferential of a gH-differentiable convex i...
In a recent paper by the authors, Jensen's inequality for Choquet integral was given, and a wrong assertion—“Jensen's inequality does not hold for asymmetric Choquet integral” was made. This paper can be viewed as a continuation of the previous one, Jensen's inequality for asymmetric Choquet integral is proved, the error is corrected. As its genera...
The history of integration stretches back to the nineteenth century BC where first attempts to determine areas and volumes appeared. Reflecting the genuine properties of the related objects, the classical integral theory is based on the‐additivity of the measure. Coming from the statistical mechanics and the potential theory, Gustave Choquet has in...
The structure of ordinal sums of conjunctive and disjunctive functions is convenient for classification into the classes Yes, No and Maybe containing a tendency to the classes Yes and No. It especially holds when task is expressed by short vague requirements. The averaging part is covered by any averaging function. Concerning this part (class Maybe...
In this study, a \emph{$gH$-subgradient technique} is developed to obtain efficient solutions to the optimization problems with nonsmooth nonlinear convex interval-valued functions. The algorithmic implementation of the developed $gH$-subgradient technique is illustrated. As an application of the proposed \emph{$gH$-subgradient technique}, an $\ell...
In this article, we study $gH$-subdifferential calculus of convex interval-valued functions (IVFs) and apply it in a nonconvex composite model of interval optimization problems (IOPs). It is found that the $gH$-directional derivative of maximum of finitely many comparable IVFs is the maximum of their $gH$-directional derivative. Proposed concepts o...
In this paper, it is shown that Jensen's inequality for Choquet integral given by R. Wang ten years ago is incorrect, then it is revisited, the modified Jensen's and reverse Jensen's inequalities for Choquet integral are proved. Then Jensen's inequality for generalized Choquet integral, obtained by the authors in a recent paper, is modified accordi...
In this article, the concepts of gH-subgradients and gH-subdifferentials of interval-valued functions are illustrated. Several important characteristics of the gH-subdifferential of a convex interval-valued function, e.g., closeness, boundedness, chain rule, etc. are studied. Alongside, we prove that gH-subdifferential of a gH-differentiable convex...
We propose a novel classification according to aggregation functions of mixed behavior by variability in ordinal sums of conjunctive and disjunctive functions. Consequently, domain experts are empowered to assign only the most important observations regarding the considered attributes. This has the advantage that the variability of the functions pr...
Due to many applications, the Choquet integral as a powerful tool for modeling non-deterministic problems needs to be further extended. Therefore the paper is devoted to a generalization of the Choquet integral. As a basis, the pseudo-integral for bounded integrand is extended to the case for nonnegative integrands at first, and then the generalize...
The main property of disjunction is substitutability, i.e., the fully satisfied predicate substitutes the rejected one. But, in many real–world cases disjunction is expressed as the fusion of full and optional alternatives, which is expressed as OR ELSE connective. Generally, this logical connective provides a solution lower than or equal to the MA...
Based on the random vector (X+Z,Y+Z) we study the perturbation CX+Z,Y+Z of the copula CX,Y of the random vector (X,Y) when the random noise Z is independent of both X and Y. We also propose a bivariate copula-based modification of the Irwin-Hall distribution and use it to extend the results of Mesiar et al. (2019) and present several examples for i...
This paper considers some information measures such as normalized divergence measure, similarity measure, dissimilarity measure and normalized distance measure in an intuitionistic fuzzy environment (IFE), which measure the uncertainty and hesitancy, and which can be applied to the selection of alternatives in group decision problems. We introduce...
In many real–world cases, disjunction is expressed as the fusion of full alternatives and less relevant ones, which leads to an OR ELSE connective. Obviously, this connective, so-called intensified disjunction, should provide a solution lower than or equal to the MAX operator, and higher than or equal to the projection of the full alternative. Furt...
In this paper we study the smallest and the greatest M-Lipschitz continuous n-ary aggregation functions with a given diagonal section. We show that several properties that were studied for the smallest and the greatest 1-Lipschitz continuous binary aggregation functions with a given diagonal section extend naturally to higher dimensions while consi...
This book collects the abstracts of the contributions presented at AGOP 2017, the 9th International Summer School on Aggregation Operators. The conference took place in Skövde (Sweden) in June 2017. Contributions include works from theory and fundamentals of aggregation functions to their use in applications. Aggregation functions are usually defin...
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...
This contribution first addresses axiomatic definitions of some important information measures of Atanassov's intuitionistic fuzzy sets (AIFSs), such as, distance measure, similarity measure, entropy measure and recently developed knowledge measure. After that, we study the relationship between similarity measures and knowledge measures, and that b...
Connection between the theory of aggregation
functions and formal concept analysis is discussed and studied, thus filling
a gap in the literature by building
a bridge between these two theories, one of them living in the world of data
fusion, the second one in the area of data mining. We show how Galois
connections can be used to describe an impor...
We show that as the dimensionality increases, more and more interesting classes of operations can be identified between the class of n-quasi-copulas and the class of n-copulas. One such class is the class of supermodular n-quasi-copulas. We observe that some properties of 2-copulas that cannot be generalized to higher-dimensional copulas, hold true...
Two new generalizations of the relation of comonotonicity of lattice-valued
vectors are introduced and discussed. These new relations coincide on
distributive lattices and they share several properties with the
comonotonicity for the real-valued
vectors (which need not hold for $L$-valued vectors
comonotonicity, in general). Based on these newly i...
This chapter studies L-fuzzy bags and some of its applications in which L is a complete lattice. Furthermore, the concepts of \(\alpha \)-cuts, (L-fuzzy) bag relations and related theorems are given. The chapter ends with the characterization of the algebraic structure of bags and L-fuzzy bags.
The famous Hirsch index has been introduced just ca. ten years ago. Despite that, it is already widely used in many decision-making tasks, like in evaluation of individual scientists, research grant allocation, or even production planning. It is known that the h-index is related to the discrete Sugeno integral and the Ky Fan metric introduced in th...
Ordered weighted average (OWA) operators with their weighting vectors are very important in many applications. We show that directly taking Minkowski distances (including Manhattan distance and Euclidean distance) as the distances for any two OWA operator is not reasonable. In this study, we propose the standard distance measures for any two OWA op...
Fuzzy relation inequalities based on max-F composition are discussed, where F is a binary aggregation on [0,1]. For a fixed fuzzy relation inequalities system A ◦F x ≤ b, we characterize all matrices A′ For which the solution set of the system A′ ◦F x ≤ b is the same as the original solution set. Similarly, for a fixed matrix A, the possible pertur...
The Bonferroni mean (BM) operator, originally introduced by Bonferroni, assumes homogeneous relationship among the input arguments. Recently, extended BM (EBM) operator is developed to capture heterogeneous relationship among real arguments and linguistic 2-tuple data. In this paper, we investigate the EBM operator with Atanassov's intuitionistic f...
A definition of L-fuzzy bags is introduced and studied. In this
approach, according to the concept given by M. Delgado et al. (2009), each bag
has two parts: function and summary information. Then, the definition of L-fuzzy
bag expected value is introduced. In the case L = [0, 1], several integral-based
fuzzy bag expected values are prepared. By so...
We study compatible aggregation functions on a general bounded distributive lattice L,
where the compatibility is related to the congruences on L. As a by-product, a new proof
of an earlier result of G. Grätzer is obtained. Moreover, our results yield a new characterization
of discrete Sugeno integrals on bounded distributive lattices.
We study and propose some new construction methods to obtain uninorms on bounded lattices. Considering an arbitrary bounded lattice L, we show the existence of idempotent uninorms on L for any element e ∈ L\{0, 1} playing the role of a neutral element. By our construction method, we obtain the smallest idempotent uninorm and the greatest idempotent...
Two construction methods for aggregation functions based on a restricted a priori known decomposition set and decomposition weighing function are introduced and studied. The outgoing aggregation functions are either superadditive or subadditive. Several examples, including illustrative figures, show the potential of the introduced construction meth...
We show that the class of all aggregation functions on $[0,1]$ can be generated
as a composition of infinitary sup-operation acting on sets with cardinality
not exceeding $\mathfrak{c}$, $b$-medians $\mathsf{Med}_b$, $b\in[0,1[$, and unary aggregation functions $1_{]0,1]}$ and $1_{[a,1]}$, $a\in ]0,1]$.
Moreover, we show that we cannot relax the ca...
This study introduces a revised definition for fuzzy bags. It is based on the definition of bags given
by Delgado et al. (Int J Intell Syst 2009;24:706–721) in which each bag has two parts: function
and summary information. The new definition is given as a special case of L-fuzzy bags, where L
is a complete lattice. By some examples, the new concep...
In this paper, we point out several problems in the different definitions (and related results) of penalty functions found in the literature. Then, we propose a new standard definition of penalty functions that overcomes such problems. Some results related to averaging aggregation functions, in terms of penalty functions, are presented, as the char...
This paper has three specific aims. First, some probability inequalities, including Hölder’s inequality, Lyapunov’s inequality, Minkowski’s inequality, concentration inequalities and Fatou’s lemma for Choquet-like expectation based on a monotone measure are shown, extending previous work of many researchers. Second, we generalize some theorems abou...
Classical Bonferroni mean (BM), defined by Bonferroni in 1950, assumes homogeneous relation among the attributes, i.e., each attribute Ai is related to the rest of the attributes
A \ Ai , where A = {A1 A2 ...An} denotes the attribute set. In this paper, we emphasize the importance of having an aggregation operator, which we will refer to as the ext...
Integrals on finite spaces (e.g., sets of criteria in multicriteria decision support) based on capacities are discussed, axiomatized and examplified. We introduce first the universal integrals, covering the Choquet, Shilkret and Sugeno integrals. Based on optimization approach, we discuss decomposition and superdecomposition integrals. We introduce...
In this study we merge the concepts of Choquet-like integrals and the Choquet integral with respect to level dependent capacities. For finite spaces and piece-wise constant level-dependent capacities our approach can be represented as a ℓ-ordinal sum of Choquet-like integrals acting on subdomains of the considered scale, and thus it can be regarded...
We introduce a new property of the discrete Sugeno integrals which can be seen as their characterization, too. This property, compatibility with respect to congruences on [0, 1], stresses the importance of the Sugeno integrals in multicriteria decision support as well.
In this paper we deal with fusion functions, i.e., mappings from [0, 1]n into [0, 1]. As a generalization of the standard monotonicity and recently introduced weak monotonicity, we introduce and study the directional monotonicity of fusion functions. For distinguished fusion functions the sets of all directions in which they are increasing are dete...
Inspired by the notion of a conic semi-copula, we introduce upper conic, lower conic and biconic semi-copulas with a given section. Such semi-copulas are constructed by linear interpolation on segments connecting the graph of a strict negation operator to the points and/or . The important subclasses of upper conic, lower conic and biconic (quasi-)c...
Classically, unsupervised machine learning techniques are applied on data sets with fixed number of attributes (variables). However, many problems encountered in the field of informetrics face us with the need to extend these kinds of methods in a way such that they may be computed over a set of nonincreasingly ordered vectors of unequal lengths. T...
Having in mind that ordinal sum construction of triangular norms (triangular conorms) may not work on bounded lattices, in general, we propose a modification of ordinal sums of t-norms (t-conorms) resulting to a t-norm (t-conorm) on an arbitrary bounded lattice. In particular, our method can be applied to define connectives for fuzzy sets type 2, i...
The most applied class of copulas for fitting purposes is undoubtedly the class of Archimedean copulas due to their representation by means of single functions of one variable, i.e., by means of additive generators or the corresponding pseudo-inverses. In this paper, we characterize aggregation functions preserving additive generators (pseudo-inver...
In this study we merge the concepts of Choquet-like integrals and the Choquet integral with respect to level dependent capacities. For finite spaces and piece-wise constant level-dependent capacities our approach can be represented as a phi-ordinal sum of Choquet-like integrals acting on subdomains of the considered scale, and thus it can be regard...
The theory of classical measures
and integral reflects the genuine property of several quantities in standard physics and/or geometry, namely the σ-additivity. Though monotone measure
not assuming σ-additivity appeared naturally in models extending the classical ones (for example, inner
and outer measures,
where the related integral was considered...
After a brief presentation of the history of aggregation, we recall the concept of aggregation functions on [0; 1] and on a general interval /⊆ [—∞, ∞]. We give a list of basic examples as well as some peculiar examples of aggregation functions. After discussing the classification of aggregation functions on [0; 1] and presenting the prototypical e...
In this study, we discuss additive generators of copulas with a fixed dimension n≥2n≥2 and additive generators that yield copulas for any dimension n≥2n≥2. We review the reported methods used to construct additive generators of copulas, and we introduce and exemplify some new construction methods.
In this paper, a new class of bi-cooperative games with fuzzy bi-coalitions is proposed in multilinear extension form. The extension is shown to be unique. The solution concept discussed in [3] is investigated and characterized for this class of games.
In this paper, we introduce the notion of a fuzzy Bi-cooperative game in multilinear extension form. An LG value as a possible solution concept is obtained using standard fuzzy game theoretic axioms.
Copulas are nothing else but supermodular functions with absorbing element 0 and neutral element 1. Although a copula C cannot be modular on the unit square itself, it is effectively so on every rectangle with zero C-volume. In this paper, we show that an appropriate modular function on the unit square can be transformed into a copula by performing...
Aggregation functions on [0,1] with annihilator 0 can be seen as a generalized product on [0,1]. We study the generalized product on the bipolar scale [-1,1], stressing the axiomatic point of view. Based on newly introduced bipolar properties, such as the bipolar increasingness, bipolar unit element, bipolar idempotent element, several kinds of gen...
We recall first graded classes of copula - based integrals and their specific form when a finite universe X is considered. Subsequently, copula - based generalizations of OWA operators are introduced, as copula - based integrals with respect to symmetric capacities. As a particular class of our new operators, recently introduced OMA operators are o...
We study uninorms on bounded lattices. We show the existence of uninorms with neutral element which is different from the top element and the bottom element using the fact that the t-norms and t-conorms on arbitrary lattice L always exist. As a by-product we give a method of constructing uninorms. Using this method, we obtain also the smallest unin...
In this paper, the concept of the Reidemeister closure condition is adopted in order to characterize associativity of uninorms with a special attention paid to the class of representable uninorms. Thus, conditions replacing the associativity requirement for such uninorms are given. Further, the attention is focused on the uninorms where the underly...
This paper deals with the Lipschitz property of triangular subnorms. Unlike the case of triangular norms, for these operations the problem is still open and presents an interesting variety of situations. We provide some characterization results by weakening the notion of convexity, introducing two generalized versions of convexity for real function...
This work presents a method of extending t-norms, t-conorms and fuzzy negations to a lattice-valued setting by preserving the largest possible number of properties of these fuzzy connectives which are invariants under homomorphisms. Further, we also apply this method to extend De Morgan triples, automorphisms and n-dimensional t-norms.
The Choquet, Sugeno, and Shilkret integrals with respect to monotone measures, as well as their generalization – the universal integral, stand for a useful tool in decision support systems. In this paper we propose a general construction method for aggregation operators that may be used in assessing output of scientists. We show that the most often...
We are interested in developing a framework for determining the degree of satisfaction of some imprecisely stated conditions by some relevant variables whose value itself may be imprecise or uncertain. We particularly focus on the use of fuzzy measures as they can provide a unified representation of various types of uncertainty and imprecision. Sta...
In this work we present the definition of strong fuzzy subsethood measure as a unifiying concept for the different notions of fuzzy subsethood that can be found in the literature. We analyze the relations of our new concept with the definitions by Kitainik ( [20]), Young ( [26]) and Sinha and Dougherty ( [23]) and we prove that the most relevant pr...
We discuss a new approach to integration introduced recently by Even and Lehrer and its relationship to several integrals known from the literature. Decomposition integrals are based on integral sums related to some (possibly constraint) systems of set systems, such as finite chains or finite partitions. A special stress is put on the integrals whi...
We consider the problem of choosing a total order between intervals in multiexpert decision making problems. To do so, we first start researching the additivity of interval-valued aggregation functions. Next, we briefly treat the problem of preserving admissible orders by linear transformations. We study the construction of interval-valued ordered...
We introduce a new class of dependence functions and method for constructing tail and Pickands dependence functions. Finally, we conjecture the structure of d-dimensional Archimax copulas and discuss some classes of d-dimensional Archimax copulas.
Decomposition integrals recently proposed by Even and Lehrer are deeply studied and discussed. Characterization of integrals recognizing and distinguishing the underlying measures is given. As a by-product, a graded class of integrals varying from Shilkret integral to Choquet integral is proposed.
We survey the recent developments in the studies of cooperative games under fuzzy environment. The basic problems of a cooperative game in both crisp and fuzzy contexts are to find how the coalitions form vis-á-vis how the coalitions distribute the worth. One of the fuzzification processes assumes that the coalitions thus formed are fuzzy in nature...
In our contribution we discuss the power stability of 1-Lipschitz binary aggregation functions. The main result is the characterization of power stable quasi-copulas by means of their dependence functions. The notions are illustrated by examples.
The mean defined by Bonferroni in 1950 (known by the same name) averages all non-identical product pairs of the inputs. Its generalizations to date have been able to capture unique behavior that may be desired in some decision-making contexts such as the ability to model mandatory requirements. In this paper, we propose a composition that averages...
Integration of simple functions is a corner stone of general integration theory and it covers integration over finite spaces discussed in this paper. Different kinds of decomposition and subdecomposition of simple functions into basic functions sums, as well as different kinds of pseudo-operations exploited for integration and sumation result into...
In this contribution, the well-known ordinal sum technique of posets is generalized by allowing for a lattice ordered index set instead of a linearly ordered index set, and we argue for the merits of this generalization. We
will call such a proposed sum-type construction a lattice-based sum. Our new approach of lattice-based sum extends also the ho...
This paper deals with the concept of the “size” or “extent” of the information in the sense of measuring the improvement of our knowledge after obtaining a message. Standard approaches are based on the probabilistic parameters of the considered information source. Here we deal with situations when the unknown probabilities are subjectively or vague...
In 2004, Rodríguez-Lallena and Úbeda-Flores have introduced a class of bivariate copulas which generalizes some known families such as the Farlie-Gumbel-Morgenstern distributions. In 2006, Dolati and Úbeda-Flores presented multivariate generalizations of this class, also they investigated symmetry, dependence concepts and measuring the dependence a...
After a short history of integration on real line, some examples of optimization tasks are given to illustrate the philosophy behind some types of integrals with respect to monotone measures and related to the standard arithmetics on real line. Basic integrals are then described both in discrete case and general case. A general approach to integrat...
The process of assessing individual authors should rely upon a proper aggregation of reliable and valid papers’ quality metrics. Citations are merely one possible way to measure appreciation of publications. In this study we propose some new, SJR- and SNIP-based indicators, which not only take into account the broadly conceived popularity of a pape...
Fuzzy integrals can be seen as expressions for the expectation of fuzzy events. Based on the general concept of universal integrals, we present universal fuzzy integrals on abstract spaces. In particular, we discuss their discrete versions linked to finite universes. An axiomatic approach to discrete universal fuzzy integrals is also given.
In this note, the relations between weak null-additivity and pseudometric generating property of monotone measures are discussed. We show that on finite continuous monotone measure spaces (X,ℱ,μ), if the measurable space (X,ℱ) is S-compact (especially, if X is countable), then the weak null-additivity is equivalent to the pseudometric generating pr...
This paper studies the relation between associativity of uninorms and geometry of their level sets which is enabled by adopting the concepts of web geometry, a branch of differential geometry, and the Reidemeister closure condition. Based on this result, the structure of some special classes of uninorms is described. Namely, it is the class of unin...
This paper brings a generalization of the migrativity property of aggregation functions, suggested in earlier work of some of the present authors by imposing the α-migrativity property of Durante and Sarkoci for all values of α instead of a single one. Replacing the algebraic product by an arbitrary aggregation function B naturally leads to the pro...
In this paper, a kind of regularity of monotone measures is shown by using the pseudometric generating property of set function (for short (p.g.p.)) and an equivalence condition for Egoroff's theorem. Lusin's theorem, which is well-known in classical measure theory, is generalized to monotone measure spaces. The continuity from above and below of s...
Citations
... The Hermite-Hadamard inequality, the Jensen's and Jensen-Mercer inequality and Steffensen's inequality are a few notable ones among the many interesting inequalities that have been examined, (see [6][7][8][9][10] and references therein). Mathematical inequality researchers continue to be interested in many versions of these inequalities involving certain families of functions [11][12][13][14][15]. Among other techniques, some significantly used tools to prove integral inequalities are interpolating polynomials. ...
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... Chalco-Cano et al. [4] presented a Newton method for the search of non-dominant solutions of FOPs by using gH-differentiability. For more articles on fuzzy optimization, interested readers can refer to [18,13,20,27,1,8,10,12,11] and the reference therein. ...
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... Aggregation operators Application/Problem Senapati et al. (2022a) IFAAWA, IFAAOWA, and IFAAHA MCDM human resource selection Senapati et al. (2022b) Pythagorean fuzzy AA-weighted average, order weighted average and hybrid average operators A group strategy on monetary system of a multinational company in China. Senapati et al. (2023b) IFAAWG, IFAAOWG, and IFAAHG To choice best health-care waste (HCW) disposal method Ali and Naeem (2023) , -ROFAAWA; , -ROFAAOWA; , -ROFAAHA; , -ROFAAWG; , -ROFAAOWG; , -ROFAAHG A tool for determining the impact of this social hazard based on its causes Hussain et al. (2023) IFAAHM, IFAAWHM, IFAAGHM, and IFAAWGHM To choose best solar panels Jabeen et al. (2023) -ROFAAPBM and -ROFAAWPBM Disease is diagnosed with the MADM Senapati et al. (2023c) IF AA power weighted geometric and arithmetic operators A case study on sustainable transportation of Novi Sad Karabacak (2023) Interval neutrosophic AA-weighted arithmetic, ordered weighted average, weighted geometric, and hybrid weighted average operators To solve an emerging technology selection problem Farid and Riaz (2023) -ROFAAWA, -ROFAAOWA, -ROFAAHA, -ROFAAWG, -ROFAAOWG, and -ROFAAHG operators ...
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... In the context of nonsmooth calculus for nondifferentiable convex IVFs, Ghosh et al. [12] has recently proposed the idea of gH-subgradient and gH-subdifferential. The same article [12] found that gH-directional derivative is the maximum of all the products of the direction and gH-subgradients. ...
... In 1965, Zadeh [18] introduced the concept of fuzzy theory, which has since undergone extensive research and various applications, including Choquet integrals of set-valued functions [5,6,[20][21][22], fuzzy set-valued measures [9,10,16], fuzzy random variable applications [1,4,17], theory for general quantum systems interacting with linear dissipative systems [3], and more. The relationship between fuzzy theory and probability theory has been a subject of much discussion [1,12,14], as both frameworks aim to capture the concept of uncertainty using membership functions and probability density functions (PDFs) whose values lie within the interval [0, 1]. ...
... This also enables to integrate of human experience, conceptual knowledge, and contextual understanding into machine learning architectures, which is a notable advantage. This "human-inthe-loop" or "expert-in-the-loop" approach can, in some cases, lead to more robust, reliable, and interpretable results [22,23,25]. ...
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... An interesting research topic for many researchers has been the related theory and applications of the Choquet integral on the discrete data sets (see (Faigle and Grabisch 2011;Grabisch and Labreuche 2016) and references therein). Many researchers have worked on different generalizations of the Choquet integral (Labreuche and Grabisch 2007;Grabisch 1996;Imaoka 1997;Horanská and Šipošová 2018;Karczmarek et al. 2018;Klement et al. 2009;Narukawa and Torra 2005;Zhang et al. 2022). In the continuous case, working on this concept can be complicated. ...
... [pert] θ , coinciding with C (θ ) for negative parameters θ and with C [θ ] for θ ≥ 0, which makes sure that we obtain a copula at least for each θ ∈ [−1, 1]. Note that the family of copulas ( 1] was considered in [69,Example 4.3] in the context of perturbations of a copula C ∈ C , further developing some ideas and suggestions in [66,67], see also [95][96][97]. ...
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... It is well known that measures are important for real-life applications in information processing (Żywica and Baczyński, 2022). Thus, information measures (IMs) play a key role not only in fuzzy theory but also in neutrosophic theory, where they are viewed as useful mathematical tools for tackling uncertainty (Singh et al., 2020). There are three basic types of measures that are often mentioned in recent neutrophic and fuzzy set studies: similarity (Chai et al., 2021;Dong et al., 2021;Cherif et al., 2022), entropy (Gou et al., 2017;Zeng et al., 2020;Quek et al., 2020) and cross-entropy (Kumar et al., 2020a,b;Tian et al., 2021). ...
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... Furthermore, Stefanini et al. [28] depicted Karush-Kuhn-Tucker (KKT) conditions for NIVOPs based on the directional gH-derivative. Utilizing the gH-Gâteaux derivative [29], Ghosh has developed the optimality conditions for unconstrained NIVOPs. Later, Liu [30] proposed a subgradient-based neurodynamic algorithm to explore the NIVOPs. ...