Due to their numerous applications such as in decision making, information fusion, game theory , and data mining , Choquet integrals have recently attracted much attention. In this study, two generalization types of Choquet integrals are presented. First, a generalized Choquet type integral of a single-valued function is introduced with respect to a set-function and measure. Several of its properties, such as convergence theorems and Jensen's inequality, are proved. Second, in the spirit of the single-valued Choquet integral, a generalized Choquet type set-valued integral for a single-valued function with respect to a set-multifunction and measure is introduced using Aumann integrals as well as various properties, including convergence theorems.

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 abstract 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 abstract 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  .Finally, we show how the introduced concepts could be applied

In the creation of better multiple attribute decision-making (MADM) patterns to address the ambiguity in the expanding sophisticated of expert systems, the hypothesis of interval-valued intuitionistic fuzzy sets has proven to be an effective and advantageous technique. We employ Aczel–Alsina operations to remedy the MADM issue, wherein all data supplied by decision-makers is conveyed as interval-valued intuitionistic fuzzy (IVIF) decision matrices with all components described by an IVIF number (IVIFN). This allows us to satisfy much more demands from fuzzy decision-making concerns (IVIFN). In the framework of IVIFNs, we primarily describe several novel Aczel–Alsina operations. On the basis of these operations, we construct several novel IVIF aggregation operators, such as the IVIF Aczel–Alsina weighted averaging operator, the IVIF Aczel–Alsina order weighted averaging operator, and IVIF Aczel–Alsina hybrid averaging operator. We built up several features of such operators. We recommend an MADM technique dependent on the advanced IVIF aggregation operators. To demonstrate the effectiveness of the developed technique, we present an overview of research scientist selection. The experimental results show the viability and benefits of the created strategy by contrasting it with the different strategies. This paper reveals that some existing IVIF aggregation operators are particular instances of the operators induced in this paper.

This study proposes some standard and general forms of induced ordered weighted averaging (GnIOWA) operators where the inductive information is ordered weighted averaging (OWA) weight vectors instead of real numbers. It shows the usefulness of such type of generalized induced OWA in decision-making and evaluation and many other applications. We propose three weights allocation methods that are specifically designed for the proposed GnIOWA operators. For each of the proposed weights allocation methods, a numerical example is also attached accordingly. With the use of convex/concave Regular Increasing Monotone quantifiers, we further discuss some mathematical properties of these weights allocation methods.

The partitioned Bonferroni mean (PBM) operator, which was oriented as an elementary attempt to outstretch the Bonferroni mean (BM) operator, has enlarged the class of BM-type aggregation operators for information accumulation by modeling interrelationship among pairwise disjoint partition sets with the presupposition that the criteria of intra-partition are homogeneously related to each other, while no relationship exists among criteria of inter-partition. Although PBM has encountered a lot of attraction from the researchers due to its versatility in information aggregation technique, the principal disadvantage of the existing PBM definitions evolution is that they do not provide any specification regarding the relationship among criteria of partition structure during design, development, and applications of PBM over unalike situations of information fusion. This consideration propels us to focus on the systematic investigation of different variations of PBM operators based on various mandatory requisites to be imposed on information retrieved from the partition sets. In this regard, we propose the construction of novel generalized partitioned Bonferroni mean (GPBM) operator by befitting its suitable components to provide a descriptive configuration, which is quite interpretable, understandable and thus facilitates the ability to model specific mandatory prerequisites in a single operator. To enrich the capacity for modeling real-life decision situations, the PBM operator is customized to propose optional partitioned Bonferroni mean (OPBM) operator that captures partition-wise interrelationship among attributes while taking into consideration optional conditions jumbled in each partition set. Furthermore, we demonstrate the construction methodology of generalized OPBM operator that amalgamate the concept of GPBM and OPBM operator to enhance and model-specific requirements along with optional requirements as per the desires of decision makers.

This study first revamps Yager prioritized ordered weighted averaging operators, and condenses them into a conceptual frame with putting aside one realization from Yager's original proposal. Then, based on elicited conceptual frame called Yager prioritized preference conceptual frame, this article proposes three distinct realizations to the conceptual frame with corresponding different instances, in which some evaluation models with weights determination methods are provided. Numerical examples are also presented immediately after every instance. Lastly, this study proposes the concepts of outer monotonic, inner monotonic, and global monotonic weights functions, and discusses some related properties, which are often embodied in preferences of decision makers.

In this work, we discuss the law of α-migrativity as applied to fuzzy implication functions in a meaningful way. A generalisation of this law leads us to Pexider-type functional equations connected with the law of importation, viz., the generalised law of importation I(C(x,α),y)=I(x,J(α,y)) (GLI) and the generalised cross-law of importation I(C(x,α),y)=J(x,I(α,y)) (CLI), where C is a generalised conjunction. In this article we investigate only (GLI). We begin by showing that the satisfaction of law of importation by the pairs (C, I) and/or (C, J) does not necessarily lead to the satisfaction of (GLI). Hence, we study the conditions under which these three laws are related.

In this paper, a numerical Monte Carlo integration for Choquet integrals is proposed by using a generalized version of mean value theorem based on Choquet integral. In special cases, this generalization can help us to have the classical Monte Carlo integration and the mean value theorem over some unbounded regions.

Inspired by the Zadeh approach to fuzzy connectives in fuzzy set theory and by some applications, we introduce and study set-based extended functions, and in particular, set-based extended aggregation functions. These functions reflect neither reordering nor repetition of input values, and, linking different arities, they introduce serious constraints for extended functions. A complete characterization of set-based extended (aggregation) functions is given, and some constructions of such functions are also proposed, including several examples.

Existing extensions to Yager's ordered weighted averaging (OWA) operators enlarge the application range and to encompass more principles and properties related to OWA aggregation. However, these extensions do not provide a strict and convenient way to model evaluation scenarios with complex or grouped preferences. Based on earlier studies and recent evolutionary changes in OWA operators, we propose formulation paradigms for induced OWA aggregation and a related weight function with self-contained properties that make it possible to model such complex preference-involved evaluation problems in a systematic way. The new formulations have some recursive forms that provide more ways to apply OWA aggregation and deserve further study from a mathematical perspective. In addition, the new proposal generalizes almost all of the well-known extensions to the original OWA operators. We provide an example showing the representative use of such paradigms in decision-making and evaluation problems.

The hesitant fuzzy sets are a new efficient mathematical approach to study imprecise, uncertain, or incomplete knowledge. This paper focuses to extend this approach on a lattice and shows that two large application concepts of the fuzzy set theory namely resolution identity and representation theorem are true under this extended definition.

Aggregation functions are mostly used in decision-making situations that require information fusion in a meaningful manner. The main purpose of aggregation is to turn a group of input data into a single and comprehensive one. However, in real decision-making and system evaluation problems, the decision maker may exhibit only some amount of certainty in her decision inputs. In this study, we show how to aggregate these certainty degrees assigned to a group of inputs in an intuitive and reasonable manner. One of the interesting aspects of the problem is that the value aggregation is independent of their certainty degrees while the certainty aggregation essentially depends on both the input values and the value aggregation function. The construction of the aggregation function gives rise to a fuzzy measure that satisfies some very interesting properties. The technique presented here has wide range of applications.

Regular Increasing Monotone (RIM) quantifier and Ordered Weighted Function are important counterparts of discrete ordered weighted averaging operators. Some important characteristics such as entropy, Moment, and Step/Hurwicz degree have already been proposed and studied by several researchers. The main propose of this paper is to put the concepts of entropy, Moment, and Step/Hurwicz degree for RIM quantifier into a continuous environment. Some well-defined representative families of RIM quantifiers are also presented. The metric spaces of RIM quantifiers are discussed.

We study the properties of OWA multiplication monoid. By introducing and-accumulation vectors of OWA operators, we consider the set of all n-dimensional OWA operators as a lattice. Then, we analyze some monotonicity properties of OWA operators based on an ordering induced by and-accumulation vectors. We also show that the lattice-theoretical operations are a kind of counterpart of the OWA multiplication monoid (and its dual, additive monoid). An example of using the OWA multiplication monoid and the lattice-theoretical structure in decision making problem is provided.

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 operators and then propose a standard metric space for the set of all n-dimension OWA operators. We analyze and discuss some properties of the introduced OWA metric and further propose a metric space of Choquet integrals represented by the underlying fuzzy measures. Some applications in decision making of OWA distances are also presented in this study.

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 concept is illustrated. Furthermore, the algebraic structure of L-fuzzy bags is studied and the concept of α-cuts, L-fuzzy bag relations, and related theorems are given.

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, interval-valued fuzzy sets, intuitionistic fuzzy sets, etc. Some illustrative examples are added, considering the interval lattice L([0, 1]), the intuitionistic lattice LI, and the diamond lattice.

The Choquet integral and the Sugeno integral provide a useful tool in many problems in engineering and social choice where the aggregation of data is required. In this paper, previous results of Hong (Nonlinear Analysis 2011 74:7296–7303) are improved by relaxing some of their requirements. Carlson's, Sandor's, Bushell–Okrasinski's type inequalities and Fatou's lemma for universal integral are studied in a rather general form, thus generalizing some recent results. © 2012 Wiley Periodicals, Inc.

We study the cross-migrativity of triangular norms. The classes of continuous triangular norms, which are cross-migrative with respect to some strict or nilpotent triangular norm, respectively, are completely characterized, as well as those which are cross-migrative with respect to the greatest and smallest triangular norms, respectively. As a by-product, parametric systems of equivalence relations on the classes of strict and nilpotent triangular norms are found. © 2012 Wiley Periodicals, Inc.

We bring an overview of fuzzy integrals, including historical remarks. These integrals can be viewed as an average membership value of fuzzy sets, and they are related to fuzzy measures. The Choquet integral can be traced back to 1925. The Sugeno integral has a predecessor in the Shilkret integral from 1971. Some other fuzzy integrals and the corresponding discrete integrals are introduced too. A closer look to the geometric interpretation of fuzzy integrals is also given, resulting among others the weakest and the strongest regular fuzzy integral. An application of the Choquet integral to additive impreciseness measuring of fuzzy quantities with interesting consequences for fuzzy measures is presented. Finally, recent development and streaming of the fuzzy integral theory is discussed. © 2008 Wiley Periodicals, Inc.