Modeling uncertainty from bidimensional histograms by intuitionistic fuzzy sets

In this paper we introduce the concentration and dilation operators on interval valued intuitionistic fuzzy soft sets of root type which is a generalization of fuzzy set, intuitionistic fuzzy set, interval valued intuitionistic fuzzy set and intuitionistic fuzzy set of root type and establish some properties of these operators. We define Hamming distance between two interval valued intuitionistic fuzzy soft sets of root type and it is shown that it is a metric. A similarity measure based on this Hamming distance is defined and some properties are established. We also develop a decision making method based on the similarity measure of Hamming distance between interval valued intuitionistic fuzzy soft sets of root type. We develop an algorithm for the decision making problem and illustrate its working by means of examples.

<abstract><p>The study on N-soft sets (NSSs) has been significantly developed recently. Hybrid models such as fuzzy N-soft sets, Intuitionistic fuzzy N-soft sets, and hesitant fuzzy N-soft sets were introduced to combine fuzzy sets, intuitionistic fuzzy sets and hesitant fuzzy sets with NSSs. Related to the hybrid models, it was also constructed some complements, operations and related properties. This article aims to construct a new hybrid model called hesitant intuitionistic fuzzy N-soft sets (HIFNSSs) to combine intuitionistic fuzzy N-soft sets and hesitant fuzzy N-soft sets. Moreover, we generalise HIFNSSs to generalized hesitant intuitionistic fuzzy N-soft sets (GHIFNSSs) as a hybrid model between generalized hesitant intuitionistic fuzzy sets and N-soft sets. It was also defined some complements of GHIFNSSs, intersection and union operations between GHIFNSSs, and proved that the operations between some particular complements hold De Morgan Law. In applying a GHIFNSS, we provide an algorithm for decision-making problems and its numerical illustration.</p></abstract>

<abstract><p>Soft set has limitation for the consideration of disjoint attribute-valued sets corresponding to distinct attributes whereas hypersoft set, an extension of soft set, fully addresses this scarcity by replacing the approximate function of soft sets with multi-argument approximate function. Some structures (i.e., possibility fuzzy soft set, possibility intuitionistic fuzzy soft set) exist in literature in which a possibility of each element in the universe is attached with the parameterization of fuzzy sets and intuitionistic fuzzy sets while defining fuzzy soft set and intuitionistic fuzzy soft set respectively. This study aims to generalize the existing structure (i.e., possibility intuitionistic fuzzy soft set) and to make it adequate for multi-argument approximate function. Therefore, firstly, the elementary notion of possibility intuitionistic fuzzy hypersoft set is developed and some of its elementary properties i.e., subset, null set, absolute set and complement, are discussed with numerical examples. Secondly, its set-theoretic operations i.e., union, intersection, AND, OR and relevant laws are investigated with the help of numerical examples, matrix and graphical representations. Moreover, algorithms based on AND/OR operations are proposed and are elaborated with illustrative examples. Lastly, similarity measure between two possibility intuitionistic fuzzy hypersoft sets is characterized with the help of example. This concept of similarity measure is successfully applied in decision making to judge the eligibility of a candidate for an appropriate job. The proposed similarity formulation is compared with the relevant existing models and validity of the generalization of the proposed structure is discussed.</p></abstract>

The basic notions in rough set theory are lower and upper approximation operators defined by a fixed binary relation. This paper proposes an intuitionistic fuzzy rough set (IFRS) model which is a combination of intuitionistic fuzzy set (IFS) and rough set. We approximate an IFS by using soft binary relations instead of fixed binary relations. By using this technique, we get two pairs of intuitionistic fuzzy (IF) soft sets, called the upper approximation and lower approximation with respect to foresets and aftersets. Properties of newly defined rough set model (IFRS) are studied. Similarity relations between IFS with respect to this rough set model (IFRS) are also studied. Finally, an algorithm is constructed depending on these approximations of IFSs and score function for decision-making problems, although a method of decision-making algorithm has been introduced for fuzzy sets already. But, this new IFRS model is more accurate to solve the problem because IFS has degree of nonmembership and degree of hesitant.

Aggregation operators (AOs) are utilized to overcome the effects of attributes under some specific degree of weight in the decision-making (DM) process. The AOs have a large capacity to deal with uncertain and unpredictable information in multi-attribute decision-making (MADM) problems. The Hamy mean (HM) aggregation tools are well-known aggregation models, which are utilized to define correlation among different input arguments adequately. The intuitionistic fuzzy (IF) sets (IFS) can express unpredictable and vague information. The Aczel Alsina aggregation expressions are extensions of triangular norms. Recently, Aczel Alsina aggregation tools attained a lot of attentions from numerous research scholars. By inspiring the robustness and reliability of Aczel Alsina aggregation tools, we expose some appropriate operations of Aczel Alsina expressions under consideration of IF information. In this manuscript, we developed an intuitionistic fuzzy Aczel Alsina HM (IFAAHM) and an intuitionistic fuzzy Aczel Alsina weighted HM (IFAAWHM) operator. We also expressed the theory of Dual HM (DHM) tools and established a series of new approaches including intuitionistic fuzzy Aczel Alsina Dual HM (IFAADHM) and intuitionistic fuzzy Aczel Alsina weighted Dual HM (IFAAWDHM) operators. Some reliable characteristics and special cases of our derived approaches are also presented. The authors solved an application of a MADM technique under consideration of our derived approaches. To check the reliability and dependency of our derived mythologies, we gave an experimental case study to evaluate a desirable construction material based on some specific criteria of different Alternatives. To see the advantages and compatibility of our derived approaches, by comparing the results of existing approaches with the results of currently discussed AOs.

Cubic intuitionistic fuzzy (CIF) set (CIFS) is one of the newly developed extension of the intuitionistic fuzzy set (IFS) in which data are represented in terms of their interval numbers membership and non-membership degrees and further the degree of agreeness, as well as disagreeness corresponding to these intervals, are given in the form of an IFS. Its fundamental characteristic lies in the fact that it is a combined version of both interval-valued IFS and IFS rather than being confined to any single fuzzy environment. Under this environment, the present work focused on exploring the structural characteristics of the CIFS by defining operational laws between them. Further, based on these operational laws, we propose some new generalized CIF averaging aggregation operators and group decision-making methods. Finally, an illustrative example is provided to discuss the reliability of the proposed operators.

Multi-attribute group decision-making based on q-rung orthopair fuzzy Aczel–Alsina power aggregation operators

We introduce the notion of intuitionistic interval-valued fuzzy sets as the another generalization of interval-valued fuzzy sets and intuitionistic fuzzy sets and hence fuzzy sets. Also we introduce some operations over intuitionistic interval-valued fuzzy sets. And we study some fundamental properties of intuitionistic interval-valued fuzzy sets and operations.

This paper researched and constructed residuation implication operator of intuitionistic fuzzy T-norms.Based on this,the paper deducted the intuitionistic fuzzy rough sets model,and proved that the Pawlak rough sets,intuitionistic fuzzy sets,fuzzy rough sets,rough fuzzy sets and fuzzy T-norms rough sets were special cases of intuitionistic fuzzy rough sets.At last,some properties of intuitionistic fuzzy sets were presented and proved.

The dual hesitant fuzzy sets (DHFSs) were proposed by Zhu et al. (2012), which encompass fuzzy sets, intuitionistic fuzzy sets, hesitant fuzzy sets, and fuzzy multisets as special cases. Correlation measures analysis is an important research topic. In this paper, we define the correlation measures for dual hesitant fuzzy information and then discuss their properties in detail. One numerical example is provided to illustrate these correlation measures. Then we present a direct transfer algorithm with respect to the problem of complex operation of matrix synthesis when reconstructing an equivalent correlation matrix for clustering DHFSs. Furthermore, we prove that the direct transfer algorithm is equivalent to transfer closure algorithm, but its asymptotic time complexity and space complexity are superior to the latter. Another real world example, that is, diamond evaluation and classification, is employed to show the effectiveness of the association coefficient and the algorithm for clustering DHFSs.

Similarity measures of intuitionistic fuzzy sets based on Lp metric

Rough set theory has been combined with intuitionistic fuzzy sets in dealing with uncertainty decision making. This paper proposes a general decision-making framework based on the intuitionistic fuzzy rough set model over two universes. We first present the intuitionistic fuzzy rough set model over two universes with a constructive approach and discuss the basic properties of this model. We then give a new approach of decision making in uncertainty environment by using the intuitionistic fuzzy rough sets over two universes. Further, the principal steps of the decision method established in this paper are presented in detail. Finally, an example of handling medical diagnosis problem illustrates this approach.

In this paper, we merge the concepts of soft set theory and a bipolar intuitionistic fuzzy set. A bipolar intuitionistic fuzzy soft ideal in a BCK-algebra is defined as a soft set over the set of elements in the BCK-algebra, with each element associated with an intuitionistic fuzzy set. This relationship captures degrees of uncertainty, hesitancy, and non-membership degrees within the context of BCK-algebras. We investigate basic operations on bipolar intuitionistic fuzzy soft ideals such as union, intersection, AND, and OR. The intersection, union, AND, and OR of two bipolar intuitionistic fuzzy soft ideals is a bipolar intuitionistic fuzzy soft ideal. We also demonstrate how to use a bipolar intuitionistic fuzzy soft set to solve a problem involving decision making. Finally, we provide a general approach for handling decision-making problems using a bipolar intuitionistic fuzzy soft set.

This article represents a colonogram enhancement approach using intuitionistic fuzzy set (IFS) and fuzzy soft set (FSS) to improve the visual quality and highlight the local details in enhanced images without artifacts. The proposed method has the advantages of IFSs and fuzzy soft sets (FSSs) which can deal with gray‐level ambiguity in colonoscopy images. The combination of IFS and FSS works on intricate color variations with an adaptive parametric maps. First, the S‐membership function has been applied on the images to map into intuitionistic fuzzy space. We obtain intuitionistic fuzzy image from source image followed by decomposition of several blocks. Thereafter, the FSS is employed to get subsequent intuitionistic fuzzy soft matrix from individual image block. The FSS utilizes a class of parametric coefficients to obtain the hesitant score of individual pixels in each block through the soft‐score measure (SSM). Finally, the proposed method intensifies each membership degree in all blocks in the fuzzy plane using SSM and achieves an enhanced colonogram via defuzzification. Extensive experiment involving large data sets shows that the designed method exhibits better performance in contrast enhancement and visual quality improvement of colonograms for malformation detection (such as a polyps, malignant precancerous cells) in comparison with the state‐of‐art methods.

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Molodtsov (Comput Math Appl 37:19–31, 1999 [6]) initiated a novel concept called soft sets, a new mathematical tool for dealing with uncertainties. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts (Aktas and Cagman in Inf Sci 1(77):2726–2735, 2007 [1]). Research works on soft sets are very active and progressing rapidly in these years. In 2001, Maji et al. (J Fuzzy Math 9(3):589–602, 2001 [5]) proposed the idea of intuitionistic fuzzy soft set theory and established some results on them. Based on an equivalence relation on the universe of discourse, Dubois and Prade (Int J Gen Syst 17:191–209, 1990 [3]) introduced the lower and upper approximation of fuzzy sets in a Pawlak approximation space and obtained a new notion called rough fuzzy sets. Feng et al. (Soft Compt 14:899–911, 2009 [4]) introduced lower and upper soft rough approximation of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy sets which is the extension of Dubois and Prade’s rough fuzzy sets. The aim of this chapter is to consider lower and upper soft rough intuitionistic fuzzy approximation of intuitionistic fuzzy sets in intuitionistic fuzzy soft approximation space (IF soft approximation space) and obtain a new hybrid model called soft rough intuitionistic fuzzy sets which can be seen as extension of both the previous work by Dubois and Prade and Feng et al.