Interval-valued Sets Research Articles

Complex Pythagorean neutrosophic normal interval-valued set with an aggregation operators using score values

The complex Pythagorean neutrosophic normal interval-valued set approach solves the multiple-attribute decision-making problem. We introduce the new concepts such as complex Pythagorean neutrosophic normal interval-valued weighted averaging, complex Pythagorean neutrosophic normal interval-valued weighted geometric, complex generalized Pythagorean neutrosophic normal interval-valued weighted averaging and complex generalized Pythagorean neutrosophic normal interval-valued weighted geometric operator. We demonstrate that complex Pythagorean neutrosophic normal interval-valued set satisfy algebraic structures such as associative, idempotent, bounded, commutative and monotonic properties. Additionally, we develop algorithm and flowchart that solve problems using these operators. Examples of using enhanced score values and accuracy values in real-world environments are provided in this paper. Artificial intelligence refers to the simulation or approximation of human intelligence in machines. Its goals include computer enhanced learning, reasoning and perception. Artificial intelligence is being used today across different industries, from finance to healthcare. Agricultural robots have been described as being highly dependent on computer and machine tool technology. Four factors can be used to evaluate the quality of a robotics system: the controller’s sophistication, the software efficiency, the maximum moment of inertia, and the manufacturer’s reliability. The best alternative can be determined by comparing expert opinions to the criteria. Therefore, the parameter ∇ has a very significant impact on the results of the model. This comparison aims to prove that the models under consideration are valid and valuable by comparing them with the available and proposed models. In conclusion, the value of ∇ significantly impacts the model performance. Based on the comparison and sensitivity analysis, we conclude that the proposed aggregation operation is superior and more reliable than the existing one. The criteria were compared to the most appropriate options based on expert assessments.

A novel group decision-making method for interval-valued q-rung dual hesitant fuzzy information using extended power average operator and Frank operations

This paper advances the field of multi-attribute group decision making (MAGDM) by proposing a novel framework based on interval-valued q-rung dual hesitant fuzzy sets (IVq-RDHFSs). IVq-RDHFSs, which surpass most existing fuzzy sets, effectively represent complex fuzzy information by describing membership and non-membership degrees through interval value sets. However, prior MAGDM methods based on IVq-RDHFSs have been limited by the functions of operation rules and aggregation operators (AOs). This limitation is addressed through the construction of a new MAGDM framework, leveraging the robust Frank t-norm and t-conorm (FTT) operation and the extended power average (EPA) operator. The proposed framework features the interval-valued q-rung dual hesitant fuzzy Frank weighted extended power average (IVq-RDHFFWEPA) operator to obtain comprehensive evaluation values. The paper also introduces novel techniques for determining the weights of decision-makers and attributes. Practical applications of the proposed method are demonstrated through the assessment of desalination technology selection and rural green eco-tourism projects. Sensitivity and comparison analyses validate the superior functionality, accuracy, and flexibility of this method compared to many state-of-the-art methods. The contributions of this paper are two-fold: it develops efficient measurement techniques for IVq-RDHFSs, such as distance and weight calculation, and it introduces a comprehensive MAGDM method by integrating FTT and EPA under IVq-RDHFSs, which improves the efficiency of solving decision-making problems.

Some novel Kulisch-Miranker type inclusions for a generalized class of Godunova-Levin stochastic processes

<abstract><p>Mathematical inequalities supporting interval-valued stochastic processes are rarely addressed. Recently, Afzal et al. introduced the notion of $ \mathtt{h} $-Godunova-Levin stochastic processes and developed Hermite-Hadamard and Jensen type inequalities in the setting of interval-valued functions. This note introduces a more generalized class of Godunova-Levin stochastic process that unifies several previously published results through the use of Kulisch-Miranker type order relations that are rarely discussed in relation to stochastic processes. Further, it is the first time that fractional version of Hermite-Hadamard inequality has been developed by using interval-valued stochastic processes in conjunction with a classical operator. Moreover, we give new modified forms for Ostrowski type results and present a new way to treat Jensen type inclusions under interval stochastic processes by using a discrete sequential form. We end with an open problem regarding Milne type results and discuss the importance of different types of order relations related to inequality terms in interval-valued settings.</p></abstract>

Cosine trigonometric rules applied to average and geometric aggregating operators using extension q-rung interval-valued neutrosophic set approach.

We introduce the concept of cosine trigonometric q-rung Diophantine neutrosophic interval-valued set (CosTq-rung DioNSIVS). The fact that CosTq-rung DioNSIVS combines q-rung neutrosophic interval-valued set, q-rung neutrosophic set and neutrosophic interval-valued set is one of its distinguishing characteristics. A new idea of CosTq-rung DioNSIVWA, CosTq-rung DioNSIVWG, GCosTq-rung DioNSIVWA and GCosTq-rung DioNSIVWG is proposed in this study. We also look at the idempotency, boundedness, commutativity, and monotonicity of the CosTq-rung DioNSIVS based on algebraic operations. We considered new kinds of two distances in the proposed models, besides Euclidean and Hamming distances. The CosTq-rung DioNSIVS method was used to analyze the cosine trigonometric aggregation procedures. The study's concluding results include several fascinating and captivating discoveries.

Multi-granulation rough approximations under normal distribution

Multi-granulation decision-theoretic rough set effectively combines Bayesian decision approaches with multi-granulation rough set theory, and provides an important theoretical framework for studying rough set. In this paper, we explore several extensional models of multi-granulation decision-theoretic rough sets under the normal distribution of the decision loss function. Using the 3σ rule of normal distribution, we transform the decision loss of the multi-granulation decision-theoretic rough set into a set of interval values. We construct the upper and lower approximations from the optimistic, weakly optimistic, pessimistic, weakly pessimistic, optimistic-pessimistic, weakly optimistic-pessimistic, pessimistic-optimistic, and weakly pessimistic-optimistic viewpoints, and provide the decision rules of the proposed rough set models. The work in this paper brings the decision behavior based on a multi-granulation decision-theoretic rough set closer to the actual situation.

Multiple Attribute Trigonometric Decision-Making and Its Application to the Selection of Engineers

A novel method is presented for solving MADM under a sine trigonometric Pythagorean neutrosophic normal interval-valued set (ST-PyNSNIVS). An identifying feature of ST-PyNSNIVS is that it is a combination of PyNSIVS, PyNSS, and IVNSS. This article proposes a novel concept of ST-PyNSNIVWA, ST-PyNSNIVWG, ST-GPyNSNIVWA, and ST-GPyNSNIVWG. In addition, we acquired a flowchart and an algorithm that interact with MADM and are called ST-PyNSNIVWA, ST-PyNSNIVWG, ST-GPyNSNIVWA, and ST-GPyNSNIVWG, respectively. In addition to Euclidean and Hamming distances, we addressed new types of two distances in the suggested models, which are future expansions of real-life instances. The sine trigonometric aggregation operations were examined using the PyNSNIV set technique. They are more straightforward and practical, and you can arrive at the best option quickly. Consequently, the conclusions of the defined models are more accurate and closely correlated with Σ . Our analysis shows that the investigated models are valid and useful by comparing them to some of the current models. As a final result of the study, some intriguing and enthralling findings are presented.

Some New Versions of Fractional Inequalities for Exponential Trigonometric Convex Mappings via Ordered Relation on Interval-Valued Settings

This paper’s main goal is to introduce left and right exponential trigonometric convex interval-valued mappings and to go over some of their important characteristics. Additionally, we demonstrate the Hermite–Hadamard inequality for interval-valued functions by utilizing fractional integrals with exponential kernels. Moreover, we use the idea of left and right exponential trigonometric convex interval-valued mappings to show various findings for midpoint- and Pachpatte-type inequalities. Additionally, we show that the results provided in this paper are expansions of several of the results already demonstrated in prior publications The suggested research generates variants that are applicable for conducting in-depth analyses of fractal theory, optimization, and research challenges in several practical domains, such as computer science, quantum mechanics, and quantum physics.

Medical robotic engineering selection based on square root neutrosophic normal interval-valued sets and their aggregated operators

<abstract><p>We introduce the concepts of multiple attribute decision-making (MADM) using square root neutrosophic normal interval-valued sets (SRNSNIVS). The square root neutrosophic (SRNS), interval-valued NS, and neutrosophic normal interval-valued (NSNIV) sets are extensions of SRNSNIVS. A historical analysis of several aggregating operations is presented in this article. In this article, we discuss a novel idea for the square root NSNIV weighted averaging (SRNSNIVWA), NSNIV weighted geometric (SRNSNIVWG), generalized SRNSNIV weighted averaging (GSRNSNIVWA), and generalized SRNSNIV weighted geometric (GSRNSNIVWG). Examples are provided for the use of Euclidean distances and Hamming distances. Various algebraic operations will be applied to these sets in this communication. This results in more accurate models and is closed to an integer $ \Delta $. A medical robotics system is described as combining computer science and machine tool technology. There are five types of robotics such as Pharma robotics, Robotic-assisted biopsy, Antibacterial nano-materials, AI diagnostics, and AI epidemiology. A robotics system should be selected based on four criteria, including robot controller features, affordable off-line programming software, safety codes, and the manufacturer's experience and reputation. Using expert judgments and criteria, we will be able to decide which options are the most appropriate. Several of the proposed and current models are also compared in order to demonstrate the reliability and usefulness of the models under study. Additionally, the findings of the study are fascinating and intriguing.</p></abstract>

An inclusive study on the fundamentals of interval-valued fuzzy hypersoft set

When compared to its extension, hypersoft set, a soft set only deals with a single set of attributes, while a hypersoft set deals with several attribute-valued disjoint sets that correspond to various attributes. Several researchers have developed models based on soft sets, but the majority of these models suffer from limitations since they are inappropriate for interval-type data or uncertain data. In order to address these issues, a novel model interval-valued fuzzy hypersoft set (IV F HS -set) is presented in this research article. This model not only resolves the inadequacy of soft set for distinct attributes for non-overlapping attribute-valued sets, but also addresses the limitations of soft set-like models with having data in interval environment. This work modifies the current fuzzy hypersoft set concept and introduces certain fundamental ideas, such as subset, not set, whole set, and absolute relative null set, relative absolute set and aggregation operations e.g. intersection, union, extended intersection, restricted union, complement, OR, AND, difference, restricted difference are discussed under IV F HS -set environment with illustrated examples. Some new hybrids of fuzzy hypersoft set under interval-valued settings are also discussed. Moreover, some extensions of IV F HS -set are presented along with different operations.

Interval-Valued Fuzzy Normal Subnearrings of a Near Rings

The theory of interval-valued set is to concentrate the single membership functions. The manuscript aims to develop the concept of interval-valued Fuzzy normal subnearrings of a near ring (i-v FNSN). Based on these concepts, we analyzed some properties and results for the development of the theorem. We also initiate the study of homomorphism, union, and some algebraic properties. The concept of intersection is also proved. The results are beneficial to researchers.

Some New Integral Inequalities for Generalized Preinvex Functions in Interval-Valued Settings

In recent years, there has been a significant amount of research on the extension of convex functions which are known as preinvex functions. In this paper, we have used this approach to generalize the preinvex interval-valued function in terms of (£1, £2)-preinvex interval-valued functions because of its extraordinary applications in both pure and applied mathematics. The idea of (£1, £2)-preinvex interval-valued functions is explained in this work. By using the Riemann integral operator, we obtain Hermite-Hadamard and Fejér-type inequalities for (£1, £2)-preinvex interval-valued functions. To discuss the validity of our main results, we provide non-trivial examples. Some exceptional cases have been discussed that can be seen as applications of main outcomes.

Review: a generalized belief interval-valued soft set with applications in decision making

Review: a generalized belief interval-valued soft set with applications in decision making

New Riemann–Liouville Fractional-Order Inclusions for Convex Functions via Interval-Valued Settings Associated with Pseudo-Order Relations

In this study, we focus on the newly introduced concept of LR-convex interval-valued functions to establish new variants of the Hermite–Hadamard (H-H) type and Pachpatte type inequalities for Riemann–Liouville fractional integrals. By presenting some numerical examples, we also verify the correctness of the results that we have derived in this paper. Because the results, which are related to the differintegral of the ϱ1+ϱ22 type, are novel in the context of the LR-convex interval-valued functions, we believe that this will be a useful contribution for motivating future research in this area.

A Ranking Method Based on Relative Score under the Interval-Valued Intuitionistic Z-Linguistic Environment and Its Application in Rural Revitalization Strategy

Rural Revitalization Strategy (RRS) is a crucial measure for China’s development. However, owing to the uncertainty of information and the neglect of its credibility, decision-makers (DMs) make inaccurate expression and unreasonable processing of decision information. Therefore, in order to better express decision information, the concept of interval-valued intuitionistic Z-linguistic set (IVIZLS) considering the uncertainty and credibility is proposed. And the corresponding operation rules, score function, accuracy function, and aggregation operators are introduced. Then, a ranking method based on relative score (RMRS) under the interval-valued intuitionistic Z-linguistic (IVIZL) environment for multi-attribute group decision-making (MAGDM) problem is introduced. Finally, a numerical example of RRS in China is used to validate the rationality and validity of the RMRS.

Hermite-Hadamard Inequalities in Fractional Calculus for Left and Right Harmonically Convex Functions via Interval-Valued Settings

The purpose of this study is to define a new class of harmonically convex functions, which is known as left and right harmonically convex interval-valued function (LR-𝓗-convex IV-F), and to establish novel inclusions for a newly defined class of interval-valued functions (IV-Fs) linked to Hermite–Hadamard (H-H) and Hermite–Hadamard–Fejér (H-H-Fejér) type inequalities via interval-valued Riemann–Liouville fractional integrals (IV-RL-fractional integrals). We also attain some related inequalities for the product of two LR-𝓗-convex IV-Fs. These findings enable us to identify a new class of inclusions that may be seen as significant generalizations of results proved by Iscan and Chen. Some examples are included in our findings that may be used to determine the validity of the results. The findings in this work can be seen as a considerable advance over previously published findings.

A novel kind of generalized belief interval-valued soft set and its application in decision- making

A novel kind of generalized belief interval-valued soft set and its application in decision- making

Some New Versions of Integral Inequalities for Left and Right Preinvex Functions in the Interval-Valued Settings

The principles of convexity and symmetry are inextricably linked. Because of the considerable association that has emerged between the two in recent years, we may apply what we learn from one to the other. In this paper, our aim is to establish the relation between integral inequalities and interval-valued functions (IV-Fs) based upon the pseudo-order relation. Firstly, we discuss the properties of left and right preinvex interval-valued functions (left and right preinvex IV-Fs). Then, we obtain Hermite–Hadamard (𝓗-𝓗) and Hermite–Hadamard–Fejér (𝓗-𝓗-Fejér) type inequality and some related integral inequalities with the support of left and right preinvex IV-Fs via pseudo-order relation and interval Riemann integral. Moreover, some exceptional special cases are also discussed. Some useful examples are also given to prove the validity of our main results.

Fractional Calculus for Convex Functions in Interval-Valued Settings and Inequalities

In this paper, we discuss the Riemann–Liouville fractional integral operator for left and right convex interval-valued functions (left and right convex I∙V-F), as well as various related notions and concepts. First, the authors used the Riemann–Liouville fractional integral to prove Hermite–Hadamard type (𝓗–𝓗 type) inequality. Furthermore, 𝓗–𝓗 type inequalities for the product of two left and right convex I∙V-Fs have been established. Finally, for left and right convex I∙V-Fs, we found the Riemann–Liouville fractional integral Hermite–Hadamard type inequality (𝓗–𝓗 Fejér type inequality). The findings of this research show that this methodology may be applied directly and is computationally simple and precise.

Riemann–Liouville Fractional Integral Inequalities for Generalized Pre-Invex Functions of Interval-Valued Settings Based upon Pseudo Order Relation

The concepts of convex and non-convex functions play a key role in the study of optimization. So, with the help of these ideas, some inequalities can also be established. Moreover, the principles of convexity and symmetry are inextricably linked. In the last two years, convexity and symmetry have emerged as a new field due to considerable association. In this paper, we study a new version of interval-valued functions (I-V·Fs), known as left and right χ-pre-invex interval-valued functions (LR-χ-pre-invex I-V·Fs). For this class of non-convex I-V·Fs, we derive numerous new dynamic inequalities interval Riemann–Liouville fractional integral operators. The applications of these repercussions are taken into account in a unique way. In addition, instructive instances are provided to aid our conclusions. Meanwhile, we’ll discuss a few specific examples that may be extrapolated from our primary findings.

The α-interval valued fuzzy sets defined on α-interval valued set

In this study, ?-interval valued set is defined whose elements are closed sub-intervals including ? of unit interval that is I = [0, 1]. With different order relation on this set, the properties of ?-interval valued set are examined. By the help of this order relation, it is shown that ?-interval valued set is complete lattice. Negation function on ?-interval valued set is given in order to study the theoretical properties of this set. By means of discussions on ?-interval valued set, the fundamental features of ?-interval valued set are studied. By the help of ?-interval valued set, ?-interval valued fuzzy sets are defined. The fundamental algebraic properties of these sets are examined. The level subsets of ?-interval valued fuzzy sets are defined to give the relations between ?-interval valued sets and crisp sets. With the help of this definition, some propositions and examples are given.