Set-theoretic Operations Research Articles

An algorithm for identifying reference signals under the environment of complex fuzzy sets

In this paper, we have discussed some new operations and results of set theory for complex fuzzy sets (CFSs). Moreover, we developed the basic results of CFSs under the basic operations such as complex fuzzy simple difference, bounded sum, bounded difference, dot product, bounded product, union, intersection, and Cartesian product. We explored the CFSs and discussed the related properties with examples such as complex fuzzy bounded sum over the intersection, complex fuzzy dot product over the union, etc. Identifying the reference signals under the environment of CFSs have always been a challenging. Many algorithms based on set theoretic operations and distance measures have been proposed for identifying a reference signal using any common system. But linear time invariant (LTI) system is considered easy to analyze the linear and time-varying signals. We used CFSs in signals and systems. We developed an algorithm based on convolution product and LTI system under the complex fuzzy environment. We identified a high degree of resemblance (reference signal) of the received signals to the reference signal in a linear time-invariant (LTI) system that receives an input signal and produces an output signal.

Graphical Exploration of Generalized Picture Fuzzy Hypersoft Information with Application in Human Resource Management Multiattribute Decision-Making

In this study, the graphical exploration of a novel hybrid, that is, possibility of picture fuzzy hypersoft graph ( popfhs -graph) is accomplished. The popfhs -graph is more flexible and reliable in the sense that it has the ability to tackle the limitations of picture fuzzy soft graph regarding the entitlement of multiargument approximate mapping and possibility degree-based setting. Its approximate mapping considers the Cartesian product of subclasses of parameters as domain and then maps it to the power set of universal set. The possibility degree-based setting ensures the assessment of uncertain attitude of approximate elements up to the level of acceptance. First, some of elementary notions and set theoretic operations of popfhs -graph are investigated with the support of numerical examples and pictorial representations. Second, some of its variants, types of products, and composition are also discussed. Lastly, based on aggregation of the popfhs -graph, an algorithm is proposed for multiattribute decision-making problem and validated by resolving daily-life recruitment problem for the best selection of candidate.

A user-configurable and explainable framework for ranking fuzzy numbers

Fuzzy number ranking is an important and fundamental subject matter within fuzzy set theory, just as the ranking of crisp values is important and fundamental to classical mathematics. For a user of fuzzy set theory, a fuzzy number ranking method that is user-configurable and user-explainable is highly desirable -- being user-configurable is important for ensuring the fitness of the fuzzy ranking method in relation with the assumptions of the application domain of interest, and being user-explainable is important for providing confidence to the user and the stakeholders of the appropriateness of the ranking outcomes in an expert system. In this article, a simple framework for ranking fuzzy numbers is proposed and developed. This simple framework consists of concrete constructs similar to the building blocks in a LEGO set, for the user to configure his/her own fuzzy ranking operator. The proposed framework is applied to examples taken from previous comparative studies from the fuzzy set theory literature, and to multiple-criteria-decision-making and other fuzzy ranking problems. First of all, these demonstrations will illustrate that fuzzy number ranking can indeed be done in a concrete, user-configurable, and user-explainable manner. Thus, the proposed framework represents a viable alternative for the fuzzy ranking literature, which currently consists of various ranking methods that are mathematically sophisticated yet oftentimes highly non-trivial for the user to interpret and to adapt/re-configure for his/her applications. Secondly, we note that the precursor of the proposed framework has been successfully employed to re-derive, instead of postulate, commonly used fuzzy set-theoretic operators, and to derive fuzzy arithmetic operators that do not produce anomalies plaguing other fuzzy arithmetic approaches. Thus, the applicability of such a framework to fuzzy ranking will further demonstrate its potential as a fundamental framework for tackling a wide range of issues in fuzzy set theory, rather than being an isolated approach limited to narrow aspects of fuzzy set theory.

Approximation operators via TD-matroids on two sets.

Rough set theory is an extension of set theory with two additional unary set-theoretic operators known as approximation in order to extract information and knowledge. It needs the basic, or say definable, knowledge to approximate the undefinable knowledge in a knowledge space using the pair of approximation operators. Many existed approximation operators are expressed with unary form. How to mine the knowledge which is asked by binary form with rough set has received less research attention, though there are strong needs to reveal the answer for this challenging problem. There exist many information with matroid constraints since matroid provides a platform for combinatorial algorithms especially greedy algorithm. Hence, it is necessary to consider a matroidal structure on two sets no matter the two sets are the same or not. In this paper, we investigate the construction of approximation operators expressed by binary form with matroid theory, and the constructions of matroidal structure aided by a pair of approximation operators expressed by binary form.First, we provide a kind of matroidal structure—TD-matroid defined on two sets as a generalization of Whitney classical matroid.Second, we introduce this new matroidal construction to rough set and construct a pair of approximation operators expressed with binary form.Third, using the existed pair of approximation operators expressed with binary form, we build up two concrete TD-matroids.Fourth, for TD-matroid and the approximation operators expressed by binary form on two sets, we seek out their properties with aspect of posets, respectively.Through the paper, we use some biological examples to explain and test the correct of obtained results. In summary, this paper provides a new approach to research rough set theory and matroid theory on two sets, and to study on their applications each other.

Application of the Model of Potential Containers for Packing Complex Shape Objects

The optimization problem of packing objects of arbitrary shape with generalization in dimension is considered. The purpose of this work is to develop and study algorithms that provide fast optimized placement of a large number of objects of arbitrary shape. To achieve this goal, the problem was set to improve the model of potential containers, originally developed for packing orthogonal objects in the form of rectangles or parallelepipeds. To quickly get layouts, it is proposed to switch from a polygonal representation of objects under placing to their discrete representation after voxelization. Obviously, when placing objects of complex shape are voxelized with a high degree of detail, a large number of potential containers will be formed, which will reduce the speed of solving the problem. This article presents the developed algorithms optimized by the number of potential containers processed to increase the speed of geometric design of a layout. Computational experiments have shown that the highest speed of the formation placement schemes of orthogonal polyhedrons is provided by using algorithms based on the application of the set-theoretic intersection operation. A method of removing unused potential containers is proposed, which allows several times to increase the speed of object placement. It is shown that the effectiveness of its application increases with an increase in the number of objects to be placed. A method for creating a container of complex shape form a polygonal model by applying an orthogonal polyhedron of geometric constraints to it is presented. All presented algorithms are implemented programmatically for the case of an arbitrary dimension of the problem being solved.

Some Set Theoretic Operators Preserving Ideal Hausdorff Convergence

In this article, we investigate set theoretic operators preserving the ideal Hausdorff convergence of sequences of sets. Then we showed that pointwise continuous functions are not sufficient to preserve ideal Hausdorff convergence. Finally, we proved that uniformly continuous functions preserve this type of convergence.

Amplitude interval-valued complex Pythagorean fuzzy sets with applications in signals processing

In this paper, we introduce the notion of amplitude interval-valued complex Pythagorean fuzzy sets (AIVCPFSs). The motivation for this extension is the utility of interval-valued complex fuzzy sets in membership and non-membership degree which can express the two dimensional ambiguous information as well as the interaction among any set of parameters when they are in the form of interval-valued. The principle of AIVCPFS is a mixture of the two separated theories such as interval-valued complex fuzzy set and complex Pythagorean fuzzy set which covers the truth grade (TG) and falsity grade (FG) in the form of the complex number whose real part is the sub-interval of the unit interval. We discuss some set-theoretic operations and laws of the AIVCPFSs. We study some particular examples and basic results of these operations and laws. We use AIVCPFSs in signals and systems because its behavior is similar to a Fourier transform in certain cases. Moreover, we develop a new algorithm using AIVCPFSs for applications in signals and systems by which we identify a reference signal out of the large number of signals detected by a digital receiver. We use the inverse discrete Fourier transform for the membership and non-membership functions of AIVCPFSs for incoming signals and a reference signal. Thus a method for measuring the resembling values of two signals is provided by which we can identify the reference signal.

Методы формирования ортогональных многогранников для задач раскроя и упаковки объектов произвольной геометрии

The article deals with the problem of packing objects of arbitrary geometry. Modern methods of designing irregular packing schemes use a mathematical model based on phi-functions and a hodograph vector function of dense placement. These methods make it possible to obtain exact solutions, but they are time-consuming and very sensitive to the dimension of the problem being solved and the degree of detail of the geometry of vector objects. The use of a discrete representation of placed objects in the form of orthogonal polyhedra can signifi increase the speed of construction a packing, which makes the problem of adequately transforming the shape of placed objects (vector models in the two-dimensional case and polygonal models in the three- dimensional case) relevant. The aim of the study is to systematize methods that provide the formation of orthogonal polyhedra of various dimensions for describing objects and containers of arbitrary geometry. Methods for creating orthogonal polyhedra based on set-theoretic operations (addition, subtraction and intersection), analytical modeling using a set of functions and relational operators, as well as voxelization of fl and volumetric object models are considered. The use of set-theoretic operations is best suited for the manual creation of orthogonal polyhedra with relatively simple geometry. The method of analytical modeling is intended for the formation of voxelized objects based on geometric fi es described by a set of analytically specifi functions. The application of various relational operators to obtain orthogonal polyhedra that describe the contour, internal and external regions of analytical given objects is shown. An algorithm for creating a container in the form of an orthogonal polyhedron based on a given vector model is proposed, which makes it possible to solve problems of irregular packing of objects inside containers of arbitrary shape. All the methods presented in the article are programmatically implemented with a generalization in terms of dimension and are applicable to solving any types of cutting and packing problems.

A New Optimization Approach Based on Bipolar Type-2 Fuzzy Soft Sets

The main objective of this research study is to amplify the schematic representation of human reasoning by launching the most generalized fantastic theories of bipolar type-2 fuzzy set (BT2FS) and bipolar type-2 fuzzy soft set (BT2FS f S). These incredible models are exclusively developed for the simultaneous capturing of both polarity and abstruseness inherent in equivocal interpretations. The proposed BT2FS f S theory renders an outstanding parameterized framework that skilfully wipes out the high-order uncertainty of imprecise knowledge-based systems. First, we keenly provide the formal structure of both proposed models along with the deep exploration of their elementary properties. Moreover, we explore rudimentary set-theoretical operations of developed frameworks inclusive of equality, subset, complement, union, and intersection with their noticeable results. We brilliantly formulate a highly proficient algorithm using proposed theory to disentangle the real-world multiattribute decision-making conundrums with two-sided ambiguous information. Finally, we holistically scrutinize an empirical analysis for the selection best way to cultivate the dessert city to demonstrate the remarkable accountability of the proposed methodology.

An Intelligent Multiattribute Decision-Support Framework Based on Parameterization of Neutrosophic Hypersoft Set

Hypersoft set is a novel area of interest which is able to tackle the real-world scenarios where classification of parameters into their respective sub-parametric values in the form of overlapping sets is mandatory. It employs a new approximate mapping which considers such sets in the form of sub-parametric tuples as its domain. The existing soft set-like structures are insufficient to tackle such kind of situations. This research intends to establish a novel concept of parameterization of fuzzy set under hypersoft set environment with uncertain components of intuitionistic fuzzy set and neutrosophic set. Two novel structures, i.e., fuzzy parameterized intuitionistic fuzzy hypersoft set (fpifhs-set) and fuzzy parameterized neutrosophic hypersoft set (fpnhs-set), are developed by employing algebraic techniques like theoretic, analytical, pictorial, and algorithmic techniques. After characterizing the elementary properties and set-theoretic operations of fpifhs-set and fpnhs-set, two novel algorithms are proposed to solve real-life decision-making COVID-19 problem. The results of both algorithms are compared with related already established models through certain evaluating features to judge the advantageous aspects of the proposed study. The generalization of the proposed models is discussed by describing some of their particular cases.

Hesitant T-spherical Dombi fuzzy aggregation operators and their applications in multiple criteria group decision-making

A hesitant fuzzy (HF) set is an extension of the fuzzy sets and a T-spherical fuzzy set (T-SFS) is a generalization of the spherical fuzzy set (SFS). HF set has a significant role for modelling disagreements of the decision-makers over membership degree of an element. Also, T-SFS is quite effective in the modelling of the uncertainty for decision-making (DM) problems. In this paper, we define the concept of hesitant T-spherical fuzzy (HT-SF) set (HT-SFS) by combining concepts of HF set and T-SFS, and present some set-theoretical operations of HT-SFSs. We also develop the Dombi operations on HT-SFSs. We present some aggregation operators based on Dombi operators, including hesitant T-spherical Dombi fuzzy weighted arithmetic averaging operator, hesitant T-spherical Dombi fuzzy weighted geometric averaging operator, hesitant T-spherical Dombi fuzzy ordered weighted arithmetic averaging operator, and hesitant T-spherical Dombi fuzzy ordered weighted geometric averaging operator, and investigate some properties of them. In addition, we give a multi-criteria group decision-making method and algorithm of the proposed method under the hesitant T-spherical fuzzy environment. To show the process of proposed method, we present an example related to the selection of the most suitable person for the assistant professorship position in a university. Besides this, we present a comparative analysis with existing operators to reveal the advantages and authenticity of our technique.

Complex fermatean fuzzy N-soft sets: a new hybrid model with applications

Decision-making methods play an important role in the real-life of human beings and consist of choosing the best options from a set of possible choices. This paper proposes the notion of complex Fermatean fuzzy N-soft set (hbox {CFFNS}_fS) which, by means of ranking parameters, is capable of handling two-dimensional information related to the degree of satisfaction and dissatisfaction implicit in the nature of human decisions. We define the fundamental set-theoretic operations of hbox {CFFNS}_fS and elaborate the hbox {CFFS}_fS associated with threshold. The algebraic and Yager operations on hbox {CFFNS}_f numbers are also defined. Several algorithms are proposed to demonstrate the applicability of hbox {CFFNS}_fS to multi-attribute decision making. The advanced algorithms are described and accomplished by several numerical examples. Then, a comparative study manifests the validity, feasibility, and reliability of the proposed model. This method is compared with the Fermatean fuzzy Yager weighted geometric (hbox {FFY}_wG) and the Fermatean fuzzy Yager weighted average (hbox {FFY}_wA) operators. Further, we developed a remarkable hbox {CFFNS}_f-TOPSIS approach by applying innovative hbox {CFFNS}_f weighted average operator and distance measure. The presented technique is fantastically designed for the classification of the most favorable alternative by examining the closeness of all available choices from particular ideal solutions. Afterward, we demonstrate the amenability of the initiated approach by analyzing its tremendous potential to select the best city in the USA for farming. An integrated comparative analysis with existing Fermatean fuzzy TOPSIS technique is rendered to certify the terrific capability of the established approach. Further, we decisively investigate the rationality and reliability of the presented hbox {CFFNS}_fS and hbox {CFFNS}_f-TOPSIS approach by highlighting its advantages over the existent models and TOPSIS approaches. Finally, we holistically describe the conclusion of the whole work.

Multi-Attribute Decision Support Model Based on Bijective Hypersoft Expert Set

Soft set tackles a single set of attributes whereas its extension hypersoft set is projected for dealing attribute-valued disjoint sets corresponding to distinct attributes with entitlement of multi-argument approximate function. In order to furnish soft set-like models with multi-decisive opinions of multi-experts, a new model i.e. soft expert set has been developed but this is inadequate for handling the scenario where partitioning of attributes into their respective attribute-valued sets is necessary. Hence hypersoft expert set has made its place to be developed. This article intends to develop a new type of hypersoft set called bijective hypersoft expert set which is more flexible and effective. After characterization of its essential properties and set-theoretic operations like union, relaxed and restricted AND, a decision-support system is designed which is characterized by new operations such as decision system, reduced decision system, etc. with illustrated examples. The proposed decision-support system is applied in multi-attribute decision-making process to manage a real-life application.

Refined Pythagorean Fuzzy Sets: Properties, Set-Theoretic Operations and Axiomatic Results

Refined Pythagorean fuzzy set is the generalization of fuzzy set, intuitionistic fuzzy set, Pythagorean fuzzy set, refined fuzzy set, and refined intuitionistic fuzzy set as it not only bears the characteristics of these models but also addresses their limitations for real-life scenarios with sub-grades corresponding to membership and non-memberships grades. In order to equip the existing literature with conceptual framework of refined Pythagorean fuzzy sets, essential properties, set-theoretic operations, and axiomatic results are characterized with the help of numerical examples.

Hesitant fuzzy parameterized hesitant fuzzy soft sets and their applications in decision-making

A hesitant fuzzy set is a significant tool to model disagreements of the decision-makers over a subject. Also, the notion of the soft set is another useful instrument in the modelling of some problems. In this study, the concept of hesitant fuzzy parameterized hesitant fuzzy soft (HFPHFSs) sets and set-theoretical operations of them are defined. Fundamental definitions are supported with examples to make the concepts more understandable. Based on Hamacher, Euclidean, and Hausdorff distances, some distance measure methods are proposed for HFPHFS sets. Then, two decision-making algorithms based on the proposed distance measure methods are developed under the HFPHFS-environment. Finally, some applications of the proposed methods in decision-making are presented and they are compared to the existing methods in the literature.

Intuitionistic fuzzy-valued neutrosophic multi-sets and numerical applications to classification

A single-valued neutrosophic multi-set is characterized by a sequence of truth membership degrees, a sequence of indeterminacy membership degrees and a sequence of falsity membership degrees. Nature of a single-valued neutrosophic multi-set allows us to consider multiple information in the truth, indeterminacy and falsity memberships which is pretty useful in multi-criteria group decision making. In this paper, we consider sequences of intuitionistic fuzzy values instead of numbers to define the concept of intuitionistic fuzzy-valued neutrosophic multi-set. In this manner, such a set gives more powerful information. We also present some set theoretic operations and a partial order for intuitionistic fuzzy-valued neutrosophic sets and provide some algebraic operations between intuitionistic fuzzy-valued neutrosophic values. Then, we develop two types of weighted aggregation operators with the help of intuitionistic fuzzy t-norms and t-conorms. By considering some well-known additive generators of ordinary t-norms, we give the Algebraic weighted arithmetic and geometric aggregation operators and the Einstein weighted arithmetic and geometric aggregation operators that are the particular cases of the weighted aggregation operators defined via general t-norms and t-conorms. We also define a simplified neutrosophic valued similarity measure and we use a score function for simplified neutrosophic values to rank similarities of intuitionistic fuzzy-valued neutrosophic multi-values. Finally, we give an algorithm to solve classification problems using intuitionistic fuzzy-valued neutrosophic multi-values and proposed aggregation operators and we apply the theoretical part of the paper to a real classification problem.

Decision making algorithmic techniques based on aggregation operations and similarity measures of possibility intuitionistic fuzzy hypersoft sets

<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>

Some properties of integration of real-valued function over a fuzzy interval

One of the most fundamental concepts in fuzzy set theory is the extension principle. It gives a generic way of dealing with fuzzy quantities by extending non-fuzzy mathematical concepts. There are a few examples, including the concept of fuzzy distance between fuzzy sets. The extension approach is then methodically applied to real algebra, with considerable development of fuzzy number operations. These operations are computationally appealing and generalized interval analysis. Although the set of real fuzzy numbers with extended addition or multiplication is no longer a group, it retains many structural qualities. The extension concept is demonstrated to be particularly beneficial for defining set-theoretic operations for higher fuzzy sets. We need some definitions related to our properties before we can create the properties of integration of a crisp real-valued function over a fuzzy interval. It is our goal in this article to develop and demonstrate certain characteristics of a real-valued function over a fuzzy interval in order to broaden the scope of the notion of integration of a real-valued function over a fuzzy interval. Some of these characteristics are linked to the operations of extended addition and extended subtraction, while others are not.

Towards a complete Logical Phonology model of intrasegmental changes

All changes to the internal structure of phonological segments arise from combinations of rules based on two set-theoretic operations: feature deletion by set subtraction and feature insertion by unification. Apparent cases of rules targeting underspecified segments reflect two kinds of vacuous rule application, one due to unification failure and the other due to vacuous unification. Despite this reduction of all segment-internal changes to two basic mechanisms we can account for a wide variety of patterns, including the reciprocal neutralization and apparent exceptional behavior seen in Hungarian voicing assimilation.

On union and intersection of type-2 fuzzy sets not expressible by the sup-t-norm extension principle

This paper constructs a conjunction and a disjunction operator for type-2 fuzzy sets that cannot be expressed by the sup-t-norm extension principle applied to conjunction and disjunction of type-1 fuzzy sets by means of t-norms and t-conorms on the unit interval. This result answers affirmatively an open problem posed by Hernández et al. in 2015. The duality between the set-theoretic operators for type-2 fuzzy sets is obtained from the duality of set-theoretic operations on type-1 fuzzy sets.