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Type-2 fuzzy sets (T2FSs) were introduced by L. A. Zadeh in 1975 as an extension of type-1 fuzzy sets (T1FSs). In this extension, the degree to which an element belongs to a set is just a label of the linguistic variable “TRUTH,” which allows to represent reality in a more appropriate way. On the other hand, negations play an essential role within fuzzy sets theory. In fact, they are necessary in order to obtain, for example, complements of fuzzy sets, dual of a t-norm or a t-conorm, entropies, implications, as well as to study the possible contradictions appearing in a fuzzy system. However, meanwhile negations on [0, 1] (set of the membership degrees of a fuzzy set) have been deeply studied throughout the literature, the same has not happened with the negations on ${\bf M}$ = $\text{Map} \;([\text{0}, \text{1}],[\text{0}, \text{1}])$ , set of functions from [0, 1] to [0, 1] (and also set of the membership degrees of a T2FS), and so many aspects of the negations on ${\bf M}$ have not yet been investigated. In a previous paper, the axioms that an operation must satisfy to be considered a negation or a strong negation on a bounded partially ordered set were established. A family of strong negations on L and set of normal and convex functions of ${\bf M}$ were also presented. Moreover, let us note that the main characteristic of fuzzy systems is just the flexibility in order to be able to represent knowledge according to each situation, offering different models among which the expert can choose the one that best suits his/her criteria. Thus, it seems useful to find broad sets of negations in ${\bf M}$ and in L, which, as far as we know, have not been done by other researchers. According to these ideas, in this paper, the authors first present new negations and strong negations on L, and then show, for the first time, some negations on ${\bf M}$ with respect to each of the two partial orders defined in this set. [ABSTRACT FROM AUTHOR]

The focus of this paper is the linguistic weighted average (LWA), where the weights are always words modeled as interval type-2 fuzzy sets (IT2 FSs), and the attributes may also (but do not have to) be words modeled as IT2 FSs; consequently, the output of the LWA is an IT2 FS. The LWA can be viewed as a generalization of the fuzzy weighted average (FWA) where the type-1 fuzzy inputs are replaced by IT2 FSs. This paper presents the theory, algorithms, and an application of the LWA. It is shown that finding the LWA can be decomposed into finding two FWAs. Since the LWA can model more uncertainties, it should have wide applications in distributed and hierarchical decision-making. [ABSTRACT FROM AUTHOR]

This paper addresses the commonly used methods for solving multiple attribute group decision-making problems under an interval-valued intuitionistic fuzzy environment. It is shown that either the most popular score function and its latterly improved versions or other related methods may lead to the undesirable results. Apart from these seemly heuristic methods, we propose a generalized p-norm knowledge-based score function for the interval-valued intuitionistic fuzzy set, which is a generalization of the score function for intuitionistic fuzzy sets as well. The proposed score function is defined as the average amount of knowledge of information depicted in terms of the interval-valued intuitionistic fuzzy sets, that regards their quantitative level of information referred to a reference level, fuzziness and qualitative significance of information. It is proved that the proposed score function has good algebraic properties, such as monotonicity under multiplication or aggregation operator, which are not preserved in the case of existing score functions. We show that the proposed score function is a simple, easily interpreted approach, which provides intuitive results in ranking the IvIFSs as well as IFSs, particularly. Several illustrative examples are performed to demonstrate the superiority of the proposed method in measuring and discriminating the IvIFSs. A comparative analysis with the preceding methods in solving decision-making problems is conducted to show the effectiveness of the proposed method in real-life applications. [ABSTRACT FROM AUTHOR]

In this paper, we propose the concept of a linguistic interval-valued Atanassov intuitionistic fuzzy set (LIVAIFS), whose membership and nonmembership degrees are represented by the interval-valued linguistic terms, for better dealing with imprecise and uncertain information during the decision-making process. In it, first some operational laws, score, and accuracy functions of LIVAIFS are defined with a brief study of related properties. Then, based on these operational laws, several aggregating operators are proposed to aggregate the linguistic interval-valued Atanassov intuitionistic fuzzy (LIVAIF) information. Some properties and inequalities are established to show the effectiveness and validity of proposed operators. Furthermore, a group decision making (GDM) approach, based on proposed operators, has been presented to solve the multiattribute GDM problems under LIVAIF environment. Finally, an illustrative example has been presented to demonstrate the approach. [ABSTRACT FROM AUTHOR]

Overlap and grouping functions, as two new classes of special binary aggregation functions, have been investigated by several authors for applications in image processing, decision making, classification, and fuzzy community detection problems. On the other hand, after Aczél studied the distributive law between two operations, the distributive laws among particular binary aggregation functions become an interesting and natural research area. In this paper, we continue to discuss this topic for uninorms and overlap and grouping functions. First, we give some basic properties for the distributive law of uninorms over overlap functions. Second, we study this distributive law when the uninorms belong to one of the usual classes (e.g., $\mathcal {U}_{\min }$ , $\mathcal {U}_{\max }$ , the family of idempotent uninorms, representable uninorms or uninorms continuous on $]0,1[^2$). Finally, we investigate the distributive law of uninorms over grouping functions by an analogous way. [ABSTRACT FROM AUTHOR]

Two processes are necessary to solve group decision making problems: A consensus process and a selection process. The consensus reaching process is necessary to obtain a final solution with a certain level of agreement between the experts; and the selection process is necessary to obtain such a final solution. In a previous paper, we present a selection process to deal with group decision making problems with incomplete fuzzy preference relations, which uses consistency measures to estimate the incomplete fuzzy preference relations. In this paper we present a consensus model. The main novelty of this consensus model is that of being guided by both consensus and consistency measures. Also, the consensus reaching process is guided automatically, without moderator, through both consensus and consistency criteria. To do that, a feedback mechanism is developed to generate advice on how experts should change or complete their preferences in order to reach a solution with high consensus and consistency degrees. In each consensus round, experts are given information on how to change their preferences, and to estimate missing values if their corresponding preference relation is incomplete. Additionally, a consensus and consistency based induced ordered weighted averaging operator to aggregate the experts' preferences is introduced, which can be used in consensus models as well as in selection processes. The main improvements of this consensus model is that it supports the management of incomplete information and it allows to achieve consistent solutions with a great level of agreement. [ABSTRACT FROM AUTHOR]

In this paper, an orness measure to reflect the or-like degree of the quasi-arithmetic mean operator is proposed. With the generating function representation method, some properties of a quasi-arithmetic mean, associated with its orness measure, are analyzed. Then, the paper focuses on two kinds of parameterized quasi-arithmetic means with exponential function and power function generators respectively. Some properties of them are examined and compared. As the aggregation value for any set always monotonically increases with the orness level, the orness level can be used as a control parameter to represent the decision maker's preference. A generic method to get the quasi-arithmetic mean operator which makes the aggregation result consistent with its orness level is also examined. Finally, the properties of the quasi-arithmetic mean and that of the exponential and power function generators are extended to the weighted quasi-arithmetic mean by considering the importance of the aggregated elements. [ABSTRACT FROM AUTHOR]

Fuzzy pattern trees have recently been introduced as a novel type of fuzzy system, specifically with regard to the modeling of classification functions in machine learning. Moreover, different algorithms for learning pattern trees from data have been proposed in the literature. While showing strong performance in terms of predictive accuracy, these algorithms exhibit a rather high computational complexity, and their runtime may become prohibitive for large datasets. In this paper, we therefore propose extensions of an existing state-of-the-art algorithm for fuzzy pattern tree induction, which are aimed at making this algorithm faster without compromising its predictive accuracy. These extensions include the use of adaptive sampling schemes, as well as heuristics for guiding the growth of pattern trees. The effectiveness of our modified algorithm is confirmed by means of several experimental studies. [ABSTRACT FROM AUTHOR]

The article presents an introduction to a special issue of "IEEE Transactions on Fuzzy Systems" on aggregation operators.

It is known that bisymmetry generalizes the simultaneous commutativity and associativity in the framework of the unit interval. In this work, we will completely characterize two classes of bisymmetric aggregation operators: one with a neutral element and the other with the vertical and horizontal sections of the idempotent elements being smooth on a finite chain, but not necessarily smooth and commutative. Thus, the previous results, based on the smoothness that is known as a very restrictive condition, are improved. For example, there is only one smooth Archimedean t-norm on a finite chain. In this paper, the discrete bisymmetric aggregation operators are explored without the limit of the smoothness. As a by-product, it is deduced that for smooth aggregation operators on a finite chain, the bisymmetry is equivalent to the commutativity and associativity, which improves the conclusion obtained by Mas et al. that associativity and bisymmetry are equivalent for commutative smooth aggregation operators on a finite chain. [ABSTRACT FROM PUBLISHER]

Fuzzy measures are monotonic set functions on a reference set; they generalize probabilities replacing the additivity condition by monotonicity. The typical application of these measures is with fuzzy integrals. Fuzzy integrals integrate a function with respect to a fuzzy measure, and they can be used to aggregate information from a set of sources (opinions from experts or criteria in a multicriteria decision-making problem). In this context, background knowledge on the sources is represented by means of the fuzzy measures. For example, interactions between criteria are represented by means of nonadditive measures. In this paper, we introduce fuzzy measures on multisets. We propose a general definition, and we then introduce a family of fuzzy measures for multisets which we show to be equivalent to distorted probabilities when the multisets are restricted to proper sets. [ABSTRACT FROM PUBLISHER]

Three ways to approximate a proximity relation R (i.e., a reflexive and symmetric fuzzy relation) by a T-transitive one where T is a continuous Archimedean t-norm are given. The first one aggregates the transitive closure R― of R with a (maximal) T-transitive relation B contained in R. The second one computes the closest homotecy of R― or B to better fit their entries with the ones of R. The third method uses nonlinear programming techniques to obtain the best approximation with respect to the Euclidean distance for T the Lukasiewicz or the product t-norm. The previous methods do not apply for the minimum t-norm. An algorithm to approximate a given proximity relation by a Mintransitive relation (a similarity) is given in the last section of the paper. [ABSTRACT FROM AUTHOR]

This paper aims at expressing MIN and MAX operations when triangular fuzzy intervals are taken as inputs, that is when Zadeh's extension principle is considered. First presented approach consists in representing fuzzy intervals by means of their profiles. In this context, a computation algorithm can be easily derived for implementing the MIN and MAX operations. Another methodology based on interval relations is then proposed for determining a general analytical expression of the MIN and MAX operations. The potential use of these expressions in the framework of uncertain aggregation operators is illustrated with the two-additive Choquet integral. [ABSTRACT FROM AUTHOR]

Aggregation operators have been used and studied for a long time. More than ten different types of means (including the arithmetic, geometric, and harmonic ones) were already studied 2000 years ago. Nevertheless, this topic has gained relevance in recent years. This is partly due to the increasing need of methods and operators for fusing information within computer programs. In this paper, we will present a personal view of the field, focusing on the application issues and on averaging aggregation operators. We will point out some research topics and open lines for future research. [ABSTRACT FROM AUTHOR]

This paper deals with a class of conjunction-like binary operations defined on a finite totally ordered set. The concept of additive generator of a discrete conjunctive aggregation operation is introduced and we obtain a description of those operations having an additive generator by means of nonstrict Archimedean t-norms. The main goal is to give a procedure for deciding whether a conjunctive aggregation operation is additively generated or not. The problem of the existence of additive generators for discrete t-norms is also discussed. [ABSTRACT FROM AUTHOR]

This paper deals with discrete copulas considered as a class of binary aggregation operators on a finite chain. A representation theorem by means of permutation matrices is given. From this characterization, we study the structure of associative discrete copulas and a theorem of decomposition of any discrete copula in terms of associative discrete copulas is obtained. Finally, some aspects concerning their extension to copulas are dealt with. [ABSTRACT FROM AUTHOR]

In this paper, it is shown that aggregation operators cannot be associative and strictly compensatory at the same time. As a way out, the rotation construction and the rotation-annihilation construction are applied to construct wide classes of associative aggregation operators that are compensatory on a subdomain only. From this point of view, the resulting operators are similar to uninorms, a class of aggregation operators that is being reported on extensively in the literature. The constructions are demonstrated by means of ample examples. [ABSTRACT FROM AUTHOR]

We propose the concept of ordered weighted averaging (OWA) generation function with some properties, illustrative examples, and usages. A number of new properties of OWA operators and relations between some well-known OWA operators are proposed and proved using OWA Generation Functions. We discuss some orness/andness adjustment methods for a predetermined OWA operator. A practical application about investment prediction problem is given with detailed illustration and analysis, which can show the advantages of the adjustment methods proposed in this paper. In particular, the concept consistent adjustment matrix is proposed and the interests and advantages of it can be shown by its special properties. [ABSTRACT FROM PUBLISHER]

Atanassov's intuitionistic fuzzy set (A-IFS) is a generalized form of Zadeh's fuzzy set, and the basic elements of an A-IFS are intuitionistic fuzzy numbers (IFNs). Recently, lots of aggregation techniques have been proposed for fusing IFNs. However, they only deal with a limited number of IFNs that take the form of discrete information. In this paper, we will first apply the definite integral to give the notion of definite integration for IFNs and investigate a lot of novel integral operators and, then, utilize these integral operators to get some new aggregation operators that can aggregate the IFNs spreading all over an area, which means that each point in a 2-D plane is an IFN that we want to aggregate. The new techniques can help us to deal with more complicated intuitionistic fuzzy information. [ABSTRACT FROM PUBLISHER]

A critical issue when selecting an ordered weighted aggregation (OWA) operator is the determination of the associated weights. For this reason, numerous weight generating methods have appeared in the literature. In this paper, a generalization of the binomial OWA operator on the basis of the Stancu polynomial is proposed and analyzed. We propose a weight function in the parametric form using the Stancu polynomial by which the weights of OWA operators can be generated easily. The proposed Stancu OWA operator provides infinitely many sets of weight vectors for a given level of the orness value. An important property of this kind of OWA operator is its orness, which remains constant, irrespective of the number of objectives aggregated and always equal to one of its parameters. This approach provides a significant advantage for generating the OWA operators’ weights over existing methods. One can choose a set of weight vectors based on his/her own preference. This class of OWA operators can utilize a prejudiced preference to determine the corresponding weight vector. The maximum entropy (Shannon) OWA operator's weights for a given level of orness is calculated by the purposed weight function and compared with the existing maximum entropy OWA operator. [ABSTRACT FROM AUTHOR]

This paper mainly investigates a particular type of multicriteria aggregation imperative called prioritized aggregations in order to handle a class of multicriteria decision making problems, in which there are prioritizations among criteria. In such problems, the satisfaction degrees of higher prioritized criteria cannot be compensated by the lower prioritized criteria. To describe the prioritizations among criteria quantitatively, Yager has introduced a prioritized measure from the viewpoint of fuzzy measure, which is validated as an effective method, except for some little imperfectness. On the basis of detailed analysis with respect to Yager's prioritized measure, we elaborate the properties of prioritized measures and then propose a generalized form of prioritized measure. Moreover, by using the generalized prioritized measure to convey the prioritized imperative, we develop a generalized prioritized measure-guided aggregation operator that is based on the Choquet integral. Additionally, the prioritized weight aggregation operator, introduced by Yager, is verified to be a special case of the generalized prioritized measure-guided aggregation operator. At length, we develop a concrete prioritized measure and the corresponding prioritized measure-guided aggregation operator under the framework of the generalized prioritized measure-guided aggregation operator and illustrate their feasibility and effectiveness through several numerical examples. [ABSTRACT FROM AUTHOR]

The ordered weighted averaging (OWA) operators are an extensively used class of aggregation operators. The weight vector that is associated with an OWA can determine the attitudinal characters of the aggregation. One of these characterizing measures is called the orness measure. The aim of this paper is to introduce orness measures in an axiomatic framework and to propose an alternate definition of orness that is based on these axioms. The proposed orness measure satisfies a more generalized set of axioms than Yager’s orness measure. We further calculate the maximum Shannon’s entropy of the OWA operator corresponding to a fixed value of orness of our proposed measure as well as of Yager’s orness. For a given level of orness, the maximum entropy corresponding to the proposed orness measure, is more than that of Yager’s. This suggests that the proposed measure is a more plausible one. [ABSTRACT FROM AUTHOR]

An intuitionistic multiplicative preference relation was recently introduced by Xia et al. to characterize the preference information that is given by a decision maker over a set of objects. All the elements of the intuitionistic multiplicative preference relation are the 2-tuples, which can simultaneously depict the degree that one object is prior to another, and the degree that the object is not prior to another. Each part of the 2-tuples takes its value from the closed interval [1/9, 9], and thus can describe the decision maker's preferences over objects more comprehensively than the traditional multiplicative preference relation. How to derive the priority weights of the objects from an intuitionistic multiplicative preference relation is an important research topic for decision making with intuitionistic multiplicative preference information. In this paper, we shall focus on solving this issue. We first define the concepts of expected intuitionistic multiplicative preference relation, left and right error matrices, etc. Then based on the geometric aggregation operator and the error propagation formula, we derive the priority weight intervals from an intuitionistic multiplicative preference relation. After that, some approaches to decision making based on intuitionistic multiplicative preference relations are developed, and furthermore, two practical examples are given to illustrate our approaches. [ABSTRACT FROM AUTHOR]

Aggregation or fusion of interval data is not a trivial task, since the necessity of arranging data arises in many aggregation functions, such as OWA operators or the Choquet integral. Some arranging procedures have been given to solve this problem, but they need certain parameters to be set. In order to solve this problem, we propose the concept of an admissible permutation of intervals. Based on this concept, which avoids any parameter selection, we propose a new approach for the interval-valued Choquet integral that takes into account every possible permutation fitting to the considered ordinal structure of data. Finally, a consensus among all the permutations is constructed. [ABSTRACT FROM AUTHOR]

The power-average (PA) operator and the power-ordered- weighted-average (POWA) operator are the two nonlinear weighted-average aggregation tools whose weighting vectors depend on the input arguments. In this paper, we develop a power-geometric (PG) operator and its weighted form, which are on the basis of the PA operator and the geometric mean, and develop a power-ordered-geometric (POG) operator and a power-ordered-weighted- geometric (POWG) operator, which are on the basis of the POWA operator and the geometric mean, and study some of their properties. We also discuss the relationship between the PA and PG operators and the relationship between the POWA and POWG operators. Then, we extend the PG and POWG operators to uncertain environments, i.e., develop an uncertain PG (UPG) operator and its weighted form, and an uncertain power-ordered-weighted- geometric (UPOWG) operator to aggregate the input arguments taking the form of interval of numerical values. Furthermore, we utilize the weighted PG and POWG operators, respectively, to develop an approach to group decision making based on multiplicative preference relations and utilize the weighted-UPG and UPOWG operators, respectively, to develop an approach to group decision making based on uncertain multiplicative preference relations. Finally, we apply both the developed approaches to broadband Internet-service selection. [ABSTRACT FROM AUTHOR]

This paper examines the practical construction of κ-Lipschitz triangular norms and conorms from empirical data. We apply a characterization of such functions based on κ-convex additive generators and translate κ-convexity of piecewise linear strictly decreasing functions into a simple set of linear inequalities on their coefficients. This is the basis of a simple linear spline-fitting algorithm, which guarantees κ-Lipschitz properly of the resulting triangular norms and conorms. [ABSTRACT FROM AUTHOR]

This paper examines disjunctive aggregation operators used in various recommender systems. A specific requirement in these systems is the property of noble reinforcement: allowing a collection of high-valued arguments to reinforce each other while avoiding reinforcement of low-valued arguments. We present a new construction of Lipschitz-continuous aggregation operators with noble reinforcement property and its refinements. [ABSTRACT FROM AUTHOR]

An intuitionistic fuzzy set, characterized by a membership function and a non-membership function, is a generalization of fuzzy set. In this paper, based on score function and accuracy function, we introduce a method for the comparison between two intuitionistic fuzzy values and then develop some aggregation operators, such as the intuitionistic fuzzy weighted averaging operator, intuitionistic fuzzy ordered weighted averaging operator, and intuitionistic fuzzy hybrid aggregation operator, for aggregating intuitionistic fuzzy values and establish various properties of these operators. [ABSTRACT FROM AUTHOR]

Since the ordered weighted averaging (OWA) operator was introduced by Yager to provide a method for aggregating multiple inputs that lie between the max and mm operators, much research that deals with uncertain information on input arguments instead of numerical values has been conducted due to the fact that information provided by human beings is in fact often vague and imprecise. In this paper, the OWA operators with uncertain weights such as interval or weak ordinal descriptions, coupled with interval input arguments, are presented in order to identify the superior course of action among a multitude of courses of action. With the help of relaxation of information representation, a burden of information specification imposed on decision maker can be, to some extent, relieved, and thus with regard to the parameters, we can obtain a less-specific expression that renders human judgments readily available. To perform the OWA aggregation, we take into account the strength of preference based on a probabilistic measure and present a way of ordering the input arguments specified by interval numbers as well as two heuristic algorithms to allocate uncertain weights to the ordered input arguments for prioritizing the courses of action. [ABSTRACT FROM AUTHOR]

The ordered weighted averaging operator attempts to provide a user-friendly class of mean type aggregation operators by allowing a user to easily and naturally go from a mental model of a desired type of aggregation to its formal realization. Our objective in this paper is to support this agenda by providing a new methodology to help a user describe the type of aggregation they desire using the idea of a stress function. With this stress function, a user can "stress" which argument values they want to give more weight in the aggregation. An important feature of this stress function is that it is only required to be nonnegative function on the unit interval. This allows a user to completely focus on the issue of where to put the stress in the aggregation without having to consider satisfaction of any other requirements. [ABSTRACT FROM AUTHOR]

One of the key operations in fuzzy logic and approximate reasoning is the fuzzy implication, which is usually performed by a binary operator I, called an implication function or, simply, an implication. Many fuzzy rule based systems do their inference processes through these operators that also take charge of the propagation of uncertainty in fuzzy reasonings. Moreover, they have proved to be useful also in other fields like composition of fuzzy relations, fuzzy relational equations, fuzzy mathematical morphology, and image processing. This paper aims to present an overview on fuzzy implication functions that usually are constructed from t-norms and t-conorms but also from other kinds of aggregation operators. The four most usual ways to define these implications are recalled and their characteristic properties stated, not only in the case of [0,1] but also in the discrete case. [ABSTRACT FROM AUTHOR]

As a result of the use of OLAP technology in new fields of knowledge and the merging of data from different sources, it has become necessary for models to support this technology. In this paper, we shall propose a new multidimensional model that can manage imprecision in both dimensions and facts and hide the complexity to the end user. The multidimensional structure is therefore able to model data imprecision resulting from the integration of data from different sources or even information from experts, which it does by means of fuzzy logic. [ABSTRACT FROM AUTHOR]

This paper is devoted to the application of aggregation operators and to the application of ordered weighting averaging (OWA) operators to data mining. In particular, we consider two application of OWA operators in this field: model building and information extraction. The latter application is oriented to the reidentification procedures. [ABSTRACT FROM AUTHOR]

Three-way decision is consistent with human cognitive processes and applied to different information systems. However, since the real world is more imprecision, hesitation, and ambiguity, experts can select interval fuzzy information to represent their evaluations. In the evaluation process, experts sometimes may not guarantee the interval fuzzy information satisfy the constraint between membership and nonmembership, which further impacts make decisions. Interval-valued $q$ -rung orthopair fuzzy set (IVq-ROFS) are proposed to improve the fault-tolerant ability and ensure the experts’ evaluation make sense. In this case, this article generalizes IVq-ROFS to three-way decision, which consists of the construction of three-way decision and the aggregation of interval continuity of IVq-ROFS. For considering the aggregation of interval continuity of IVq-ROFS, we first investigate the IVq-ROFS integral to aggregate continuous IVq-ROFSs by developing simplified interval-valued $q$ -rung orthopair fuzzy set. Particularly, in order to obtain integrable region, we design $\varepsilon$ -amendment to amend integral region of IVq-ROFS for removing unmeaning point in advance. To simplify the calculation, we further utilize Taylor series of the integral function to replace the complicated integral function. Second, a new equivalence class method is constructed to support three-way decision. Third, with the aid of the integral of IVq-ROFS and equivalence class method, we availably fuse the loss functions of the equivalence class and obtain the classifying rules of three-way decision by comparing the expected losses. Finally, we elaborate three-way decision by using a case of medical diagnosis about nephropathy and prove the validity of the investigated model. [ABSTRACT FROM AUTHOR]

Recently, frequent emergencies seriously impact people’s life and production. The timely emergency material can save more people. Hence, emergency material demand prediction is a core of emergency rescue. However, emergency is a continuous, uncertain evolution process. The emergency of different periods has various situations. The time-varying and uncertain characteristics of emergency have made this representation and prediction work much more complex. In order to address this concern, this article proposes a time-varying intuitionistic fuzzy set (TVIFS) to precisely depict the time-varying scenario and the uncertainty of emergencies. For utilizing the TVIFS of different time frames, we deeply explore the integral aggregation operator of TVIFS, which can take full account of decision hesitation and possible information. Due to the different time importance of different time frames, we investigate the integral of TVIFS with constant period weight in advance. Considering the time-varying period weight, the TVIFS integral with time-varying period weight is investigated. For the sake of calculation feasibility, inspired by the relationship between continuity and dispersion, we translate the TVIFS integral with variable period weight into the TVIFS integral with constant period weight. Furthermore, we develop time-varying intuitionistic fuzzy case-based reasoning (CBR) model to forecast the emergency materials demand by designing hybrid similarity between cases. Finally, we elaborate the application of time-varying intuitionistic fuzzy CBR model by using an example of forest fire and verify the validity of our proposed method. [ABSTRACT FROM AUTHOR]

Fuzzy sets and fuzzy rough sets are widely applied in data analysis, data mining, and decision-making. So far, the common method is to use rough approximate operators to induce aggregation functions when fuzzy rough sets are used for multicriteria decision-making (MCDM). However, they are parametric linear and the corresponding weights are additive measures. In this article, we give a novel method for MCDM based on fuzzy covering rough sets by using the nonadditive measure [i.e., fuzzy measure (FM)] and the nonlinear integral [i.e., Choquet integral (ChI)]. First, two nonadditive measures are presented by fuzzy covering lower and upper approximation operators, respectively. Moreover, both of them are FMs which are called $\beta$ -neighborhood approximation measures. Second, two types of ChIs with respect to $\beta$ -neighborhood approximation measures are constructed. A novel method, which considers the association, is presented to solve the problem of MCDM under the fuzzy covering rough set model. Third, a new approach based on $\beta$ -neighborhood approximation measures is proposed for attribute reductions in a fuzzy $\beta$ -covering information table. This approach of attribute reductions is used in MCDM. Finally, both new methods above are compared with other methods through some numerical examples and UCI datasets, respectively. [ABSTRACT FROM AUTHOR]

The fuzzy integral (FI) is an extremely flexible aggregation operator. It is used in numerous applications, such as image processing, multicriteria decision making, skeletal age-at-death estimation, and multisource (e.g., feature, algorithm, sensor, and confidence) fusion. To date, a few works have appeared on the topic of generalizing Sugeno’s original real-valued integrand and fuzzy measure (FM) for the case of higher order uncertain information (both integrand and measure). For the most part, these extensions are motivated by, and are consistent with, Zadeh’s extension principle (EP). Namely, existing extensions focus on fuzzy number (FN), i.e., convex and normal fuzzy set- (FS) valued integrands. Herein, we put forth a new definition, called the generalized FI (gFI), and efficient algorithm for calculation for FS-valued integrands. In addition, we compare the gFI, numerically and theoretically, with our non-EP-based FI extension called the nondirect FI (NDFI). Examples are investigated in the areas of skeletal age-at-death estimation in forensic anthropology and multisource fusion. These applications help demonstrate the need and benefit of the proposed work. In particular, we show there is not one supreme technique. Instead, multiple extensions are of benefit in different contexts and applications. [ABSTRACT FROM AUTHOR]

This study first defines the concept of weighted aggregation operators, and then mainly revolves around four very representative types of them, namely, weighted arithmetic mean, weighted geometrical mean, Sugeno Integral with weight vectors, and Shilkret Integral with weight vectors. This study proposes and defines the properties of weights pair consistency and values pair consistency with some propositions and counter examples for weighted aggregation operators. As the main finding and contribution of this study, we reveal and discuss a subtle property about weighted aggregation operators, called individual preference monotonicity. We prove or put forward counter examples to show that some of weighted aggregation operators have those proposed properties while some others do not. [ABSTRACT FROM AUTHOR]

Probabilistic hesitant fuzzy sets (PHFSs) add the probability value corresponding to each degree of membership on the basis of hesitant fuzzy sets, so as to express the initial decision information given by experts more accurately and comprehensively. In this article, we mainly study how to integrate large-scare probabilistic hesitant fuzzy information more efficiently. We first discuss some basic operation laws of probabilistic hesitant fuzzy numbers, based on which the concepts of continuous PHFSs and continuous probabilistic hesitant fuzzy functions (c-PHFFs) are defined. They are the main objects of our research. We further explore definite integrals of the c-PHFFs and their related properties. They have direct and powerful applications in continuous probabilistic hesitant fuzzy environments, and lay the foundation for subsequent theoretical analysis. Based on the weight density function, we finally get the weighted-integral operator of continuous probabilistic hesitant fuzzy information. Then, some important properties of this integration operator are studied, including normalization, monotonicity, boundedness, etc. We are also devoted to revealing the inner connection between the continuous probabilistic hesitant fuzzy weighted-integral operator and the probabilistic hesitant fuzzy weighted averaging operator, the latter is usually used when dealing with discrete information. At last, we state why it is necessary to introduce a novel aggregation method based on continuous probabilistic hesitant fuzzy definite integrals, and in turn provide an application of the proposed method to prove its validity and rationality. [ABSTRACT FROM AUTHOR]

Yager's ordered weighted averaging (OWA) operators are extensively employed to perform mean-type aggregations. Their success heavily depends on proper determination of the associated weights that characterize the operators. Several methods have been established to determine the weights and, thereby, to support successful OWA aggregation. In this study, the least square method, which is one of the nonlinear programs with a quadratic objective function, is reformulated by using the extreme points that correspond to its constraints and then solved by a few steps of matrix operations. Finally, we present a new weighting method called the minimizing distances from the extreme points (MDP) wherein the OWA operator weights that minimize the expected quadratic distance with respect to the set of extreme points are chosen. A different attitudinal character leads to different extreme points, and thus, the MDP seeks to find the OWA operator weights that are dynamically adjusted to be at the center of attitude-dependent polytope. [ABSTRACT FROM AUTHOR]

Aggregation operators are often needed when building preference relations in multicriteria decision making problems. Most existing approaches have limitations due to incomparability between decisions or ties due to the use of some aggregation operations that produce a ranking. The natural way of overcoming the lack of discrimination power is to refine the obtained ranking. We bring an overview of methods that enable aggregation-based rankings to be refined, generalizing concepts like discrimin (max), leximin (max), and Lorentz orderings that refine such aggregation operations like the minimum (the maximum) and the sum. [ABSTRACT FROM AUTHOR]

We establish a general framework for constructing copulas that can be regarded as a patchwork-like assembly of arbitrary copulas, with nonoverlapping rectangles as patches. We derive a family of construction methods that require the choice of a single background copula. When this background copula is the greatest copula TM, we retrieve the well-known ordinal sum construction, while in the case of the smallest copula TL, we obtain a construction method that is dual to the ordinal sum construction. Also nonsingular background copulas lead to suitable construction methods. [ABSTRACT FROM AUTHOR]

Inspired by reflection transformations and rotation transformations in plane geometry and dictionary orders in lattice theory, in this article, we explore, through exposing the possible motivation of definition of a defined linear order ≤ on R (the set of all interval numbers), how to find an easy-to-understand way to extend the ordinary linear order on R (the set of all real numbers) to a linear order on R. Theory and application aspects are also studied. For an arbitrary cardinal number κ (κ ≠ 0), we define seven operations and the point-wise order induced by ≤ on Rκ (the set of all sequences indexed by κ and consisting of interval numbers). It is found that (R, ≤), much like R (with the ordinary linear order), has some desirable properties; particularly, ≤ can be used to describe the corrected degree of possibility of an interval number is smaller than another. Generalizing the discovery process of ≤, we provide an understandable way to extend the ordinary linear order on R to a linear order on R (even on the plane R2), which meets some specific requirement, and we give a clear description on the relation between the admissible orders and those we propose. Those urge us to take a look at the possible applications of ≤ (a delegation of linear orders on R). Precisely, to propose an ordering method (based on this linear order and the matched metric and weighted averaging aggregation operator), which can be used to deal with very complicated generalized interval-valued preference hesitant relations in group decision-making problems. [ABSTRACT FROM AUTHOR]

This article focuses on distributivity and conditional distributivity for uninorms over a given continuous t-norm, closely related to the open problem recalled by Klement in the Linz2000 closing session. This problem has been solved for the well-known classes of uninorms except the class of uninorms with the continuous underlying operators. In this article, the complete characterizations of uninorms with continuous underlying operators (conditionally) distributive over a given continuous t-norm are presented. From the obtained results, we deduce that conditional distributivity is equivalent to distributivity for a pair of such operators. [ABSTRACT FROM AUTHOR]

Several real-world problems employ linguistic-based approaches to handle qualitative data. The set of linguistic terms that is utilized in the problems are mostly alleged to be symmetrically distributed. However, with the advent of time, as the complexity of the problem increases, the equidistant linguistic term set seems improper. Consequently, in such cases, experts often prefer to use the set of the unbalanced linguistic term to direct the appraisal for the problems. In this article, we tend to propose a method that is newly designed to deal with a set of unbalanced linguistic terms. In this direction, we initially propose an algorithm to represent unbalanced linguistic information via a multiplicative linguistic label set that has a global inconsistent linguistic term distribution. Furthermore, in light of the Herrera and Martínez, “2-tuple linguistic model,” we develop a novel 2-tuple approach for the unbalanced linguistic set, which is based on the notion of minimum distance measure. Finally, to validate the proposed model in the physical realm and to demonstrate the functioning of the method, a numerical example is being elucidated. The proposed methodology seeks to indicate a reduction in the computation time and also enhances the decision-makers’ evaluations. [ABSTRACT FROM AUTHOR]

Induced ordered weighted averaging (OWA) operators are important extensions of OWA operators and can suit more different decision environments. However, when practitioners perform induced OWA operators, they frequently do not consider the tied values in inducing information, therein lay the problem of possible non-uniqueness of aggregation results. From a novel perspective that can easily remove this flaw, this study uses a systematical frame to better redefine induced OWA operators, which not only covers all the spirits of the original definition of induced OWA operator but also extends it to some wider understandings and usages. One significant feature related to this frame is that we can express induced OWA operators fully in terms of weighted averaging operators, making the usages and expressions of induced OWA operators much more flexible and stricter. In detail, we propose several new definitions, such as positioning transformation, permutation-based orness, and inducing function based orness. As an example of flexibly formulating complex preferences under the proposed concepts, a novel investigators-consultants-decision maker (ICD) two-stage IOWA evaluation model is clearly proposed and illustrated. [ABSTRACT FROM AUTHOR]

Yager's q-rung orthopair fuzzy set is a generalization of fuzzy sets, whose prominent feature is that the qth power sum of the membership and the nonmembership degrees is equal to or less than one, and we call its core, an ordered pair, q-rung orthopair fuzzy number (q-ROFN). More recently, the scholars have constructed the q-rung orthopair fuzzy calculus (q-ROFC), which can effectively deal with continuous q-rung orthopair fuzzy information. Nevertheless, the q-ROFC is only based on the basic operational laws of the q-ROFNs, in fuzzy theory, Archimedean t-norms and t-conorms (ATTs) are a significant class of continuous triangular norms and conorms, which are the generalizations of the intersection and union related to fuzzy sets. Thus, in order to extend the q-ROFC to a wider area, in this article, we systematically discuss the q-rung orthopair fuzzy double integrals (q-ROFDIs) in the frame of ATTs. First, we construct the q-ROFDI in the frame of Archimedean t-conorms in detail, and then provide its concrete value. In addition, we reveal the relationships with respect to two types of q-rung orthopair fuzzy spaces. Based on which, we can easily obtain another types of q-ROFDI. After that, we investigate their fundamental properties in detail so as to comprehend these kinds of q-ROFDIs in-depth. Finally, we point out the essences of these kinds of q-ROFDIs, on the basis of which we provide a practical application to show their effectiveness and elasticity via comparing with the existing methods. [ABSTRACT FROM AUTHOR]

Determination of the weights of an ordered weighted averaging (OWA) operator is a vital constituent of the operator's aggregation procedure. Consequently, several weight generation techniques have evolved in the literature. Here, using inverse hypergeometric distribution, we use a novel parametric function to generate the weight vector of an OWA operator. The proposed inverse hypergeometric OWA (InHyp-OWA) operator generates a unique weight vector for every choice of its parameter. Also, using the proposed weight function, OWA weight vectors can be generated without solving any complicated optimization problem. An important property of InHyp-OWA operator is that for a given parameter value, its orness value remains constant, regardless of the number of objectives aggregated. Hence, InHyp-OWA operator is a new member of the family of OWA operators with constant orness. This class of OWA operators can utilize a prejudiced preference to determine the corresponding weight vector. The proposed approach provides a number of advantages over existing weight generating methods for OWA operators. One of the remarkable advantages of the InHyp-OWA operator is the fact that it generates weight vectors with strictly monotonic components. It also generalizes the well-known Borda–Kendall OWA operator. [ABSTRACT FROM AUTHOR]

Yager's q-rung orthopair fuzzy sets (q-ROFSs), which extend Zadeh's fuzzy sets, use the membership and nonmembership functions to describe things’ vague characteristics, and the sum of the qth-power for the membership and nonmembership functions is less than or equal to 1. More recently, some scholars have proposed a series of aggregation operators to fuse q-rung orthopair fuzzy discrete information. However, so far, there is no research on aggregating q-rung orthopair fuzzy continuous information. Thus, we proposed q-rung orthopair fuzzy definite integrals (q-ROFDIs) to fill this vacancy. First, we further study the operations of q-rung orthopair fuzzy numbers (q-ROFNs) that are the core of q-ROFSs. We also introduce the limit of a q-ROFN sequence. Subsequently, we construct the q-ROFDIs step-by-step, give their concrete values, and discuss their integrability criteria from two perspectives. From the perspectives of modern analysis and the operational laws of q-ROFNs, we investigate the q-ROFDIs in detail, which are concise and considerably different from the investigative techniques of the previous research on aggregating continuous information. Finally, a practical example is provided to show the effectiveness, elasticity, and superiority of the q-ROFDIs via comparing them with the existing methods. [ABSTRACT FROM AUTHOR]

We propose a version of ordered weighted averaging (OWA) operators, which are robust against inputs with outliers. Outliers may heavily bias the outputs of the standard OWA. The penalty-based method proposed here comprises both outlier detection and reallocation of weights of the OWA. At the first stage, the outliers are identified based on a robust criterion that can accommodate up to half the inputs being outliers, but at the same time not removing the inputs unnecessarily. Three numerical algorithms for calculating the optimal value of this criterion are proposed. At the second stage, the OWA weights are recalculated for a subset of clean data while preserving the overall character of the weighting vector. The method is numerically tested on simulated data and exemplified on aggregating a large number of online ratings where the outliers represent biased, missing, or erroneous evaluations. [ABSTRACT FROM AUTHOR]