Interval-valued Sets Research Articles

Square root Diophantine neutrosophic normal interval-valued sets and their aggregated operators in application to multiple attribute decision making

We discuss innovative square root Diophantine neutrosophic normal interval-valued set (SRDioNSNIVS)- based approaches to multiple attribute decision-making (MADM) problems. Square root neutrosophic sets, interval-valued Diophantine neutrosophic sets and neutrosophic normal interval-valued (NSNIV) sets are both extensions of square root Diophantine neutrosophic sets. In this section, we will look over several aggregating operations and how those interpretations have evolved over time. The article is focused on a novel idea known as square root NSNIV weighted averaging (SRDioNSNIVWA), square root NSNIV weighted geometric (SRDioNSNIVWG), generalized square root NSNIV weighted averaging (GSRDioNSNIVWA), and generalized square root NSNIV weighted geometric (GSRDioNSNIVWG). In order to solve MADM problems, we also begin an algorithm based on the aforementioned operators. The use of the euclidean and hamming distances is described, and examples from real-world situations are given. The main characteristics of these sets under various algebraic operations will be discussed in this communication. They are more practical and straightforward, and the ideal choice may be determined quickly. As a result, the defined models are more accurate and closely tied to Φ. In order to show the reliability and usefulness of the models under examination, we also compare a few of the proposed and current models. The study’s results are also fascinating and intriguing.

Multiple attribute decision making for square root diophantine neutrosophic interval-valued sets and their aggregated operators

Square root Diophantine neutrosophic interval-valued set (SRDioNIVS) approaches to multiple attribute decisionmaking (MADM) problems. The square root neutrosophic sets, interval-valued Diophantine neutrosophic sets are both extensions of square root Diophantine neutrosophic sets. In this section, we discuss aggregating operations and how those interprtautions have evolved over time. The paper is focused on a novel idea known as square root neutrosophic interval-valued weighted averaging (SRDioNIVWA), square root neutrosophic interval-valued weighted geometric (SRDioNIVWG), generalized square root neutrosophic interval-valued weighted averaging (GSRDioNIVWA), and generalized square root neutrosophic interval-valued weighted geometric (GSRDioNIVWG). We also begin an algorithm using these operators. The use of the euclidean and hamming distances is described, and examples from real-world problems are inserted. As a result, the defined models are more accurate and closely tied to Ξ. In order to show the reliability and usefulness of the models under examination, we also compare a few of the proposed and current models. The study’s results are also fascinating and intriguing.

Multi-Attribute Decision-Support System Based on Aggregations of Interval-Valued Complex Neutrosophic Hypersoft Set

Hypersoft set is an emerging field of study that is meant to address the insufficiency and the limitation of existing soft-set-like models regarding the consideration and the entitlement of multi-argument approximate function. This type of function maps the multi-subparametric tuples to the power set of the universe. It focuses on the partitioning of each attribute into its attribute-valued set that is missing in existing soft-set-like structures. This study aims to introduce novel concepts of complex intuitionistic fuzzy set and complex neutrosophic set under the hypersoft set environment with interval-valued settings. Two novel structures, that is, interval-valued complex intuitionistic hypersoft set (IV-CIFHS-set) and interval-valued complex neutrosophic hypersoft set (IV-CNHS-set), are developed via employing theoretic, axiomatic, graphical, and algorithmic approaches. After conceptual characterization of essential elementary notions of these structures, decision-support systems are presented with the proposal of algorithms to assist the decision-making process. The proposed algorithms are validated with the help of real-world applications. A comprehensive inter-cum-intra comparison of proposed structures is discussed with the existing relevant models, and their generalization is elaborated under certain evaluating features.

Multi-granularity belief interval-valued soft set

A new model named multi-granularity belief interval-valued soft set is introduced in this paper. Some basic properties about it are presented and illustrated. The improved concepts of the soft belief value and soft belief degree are proposed, which provided an easier and better compared horizontally and vertically method among the different objects and different parameters. An algorithm for decision-making problems on multi-granularity belief interval-valued soft set is put forward and its validity is proved by the application of an example. Moreover, the newly proposed algorithm is compared with existing method to indicate its extensive application.

Multi-granularity distance measure for interval-valued intuitionistic fuzzy concepts

The interval-valued intuitionistic fuzzy (IvIF) set, a combination of intuitionistic fuzzy sets and interval-valued sets, has been widely employed for multi-attribute decision-making and group decision-making. However, two urgent problems remain to be solved. The first is to determine the coarseness/fineness relation in the IvIF granular space, and the other is to determine the difference between two IvIF granular structures with the same uncertainty. In this paper, a pair of IvIF granular structure distances is proposed to address the aforementioned issues. Then, a more generalized coarseness/fineness relation is defined for the IvIF granular space. Furthermore, two complementary uncertainty measures were formulated based on the coarseness/fineness relation. Finally, we investigate the IvIF approximate granular structure of a target concept and propose a new algorithm for attribute reduction from the perspective of distance. The experiments show that our algorithm possesses a shorter reduct than the other algorithms, ensuring relatively high classification accuracy. Furthermore, a case study is presented to demonstrate the feasibility of using the proposed method to solve a large linguistic group decision-making problem.

The Riemann-Lebesgue Integral of Interval-Valued Multifunctions

We study Riemann-Lebesgue integrability for interval-valued multifunctions relative to an interval-valued set multifunction. Some classic properties of the RL integral, such as monotonicity, order continuity, bounded variation, convergence are obtained. An application of interval-valued multifunctions to image processing is given for the purpose of illustration; an example is given in case of fractal image coding for image compression, and for edge detection algorithm. In these contexts, the image modelization as an interval valued multifunction is crucial since allows to take into account the presence of quantization errors (such as the so-called round-off error) in the discretization process of a real world analogue visual signal into a digital discrete one.

Long-term learning for type-2 neural-fuzzy systems

The development of a new long-term learning framework for interval-valued neural-fuzzy systems is presented for the first time in this article. The need for such a framework is twofold: to address continuous batch learning of data sets, and to take advantage the extra degree of freedom that type-2 Fuzzy Logic systems offer for better model predictive ability. The presented long-term learning framework uses principles of granular computing (GrC) to capture information/knowledge from raw data in the form of interval-valued sets in order to build a computational mechanism that has the ability to adapt to new information in an additive and long-term learning fashion. The latter, is to accommodate new input–output mappings and new classes of data without significantly disturbing existing input–output mappings, therefore maintaining existing performance while creating and integrating new knowledge (rules). This is achieved via an iterative algorithmic process, which involves a two-step operation: iterative rule-base growth (capturing new knowledge) and iterative rule-base pruning (removing redundant knowledge) for type-2 rules. The two-step operation helps create a growing, but sustainable model structure. The performance of the proposed system is demonstrated using a number of well-known non-linear benchmark functions as well as a highly nonlinear multivariate real industrial case study. Simulation results show that the performance of the original model structure is maintained and it is comparable to the updated model's performance following the incremental learning routine. The study is concluded by evaluating the performance of the proposed framework in frequent and consecutive model updates where the balance between model accuracy and complexity is further assessed.

Width-Based Interval-Valued Distances and Fuzzy Entropies

In this paper, we study distance measures between interval-valued fuzzy sets and entropies of interval-valued fuzzy sets. These are well-known and widely used notions in the fuzzy sets theory. The novelty of our approach is twofold: on one hand, it considers the width of intervals in order to connect the uncertainty of the output with the uncertainty of the inputs. On the other hand, it makes use of total orders between intervals, instead of partial ones, so that the usefulness of the notions related with some kind of monotonicity is fully recovered in the interval-valued setting. The construction of distance measures and entropies is done by aggregating interval-valued restricted dissimilarity functions and interval-valued normal E N -functions. For this reason, we first study these functions, both in line with the two above stated considerations. Finally, we present an illustrative example in image thresholding using an expression of the proposed interval-valued entropy to show the validity of our approach.

INTEGRABILITY OF AN INTERVAL-VALUED MULTIFUNCTION WITH RESPECT TO AN INTERVAL-VALUED SET MULTIFUNCTION

Intervals are related to the representation of uncertainty. In this sense, we introduce an integral of Gould type for an interval-valued multifunction relative to an interval-valued set multifunction, with respect to Guo and Zhang order relation. Classicaland specific properties of this new type of integral are established and several examples and applications from multicriteria decision making problems are provided.

An Efficient Signature Verification Method Based on an Interval Symbolic Representation and a Fuzzy Similarity Measure

In this paper, an efficient offline signature verification method based on an interval symbolic representation and a fuzzy similarity measure is proposed. In the feature extraction step, a set of local binary pattern-based features is computed from both the signature image and its under-sampled bitmap. Interval-valued symbolic data is then created for each feature in every signature class. As a result, a signature model composed of a set of interval values (corresponding to the number of features) is obtained for each individual’s handwritten signature class. A novel fuzzy similarity measure is further proposed to compute the similarity between a test sample signature and the corresponding interval-valued symbolic model for the verification of the test sample. To evaluate the proposed verification approach, a benchmark offline English signature data set (GPDS-300) and a large data set (BHSig260) composed of Bangla and Hindi offline signatures were used. A comparison of our results with some recent signature verification methods available in the literature was provided in terms of average error rate and we noted that the proposed method always outperforms when the number of training samples is eight or more.

Unbalanced interval-valued OWA operators

In this work, we introduce a new class of functions defined on the interval-valued setting. These functions extend classical OWA operators but allow for different weighting vectors to handle the lower bounds and the upper bounds of the considered intervals. As a consequence, the resulting functions need not be an interval-valued aggregation function, so we study, in the case of the lexicographical order, when these operators give an interval as output and are monotone. We also discuss an illustrative example on a decision making problem in order to show the usefulness of our developments.

An interval type-2 fuzzy logic PSS with the optimal H∞ tracking control for multi-machine power system

In this paper, the interval type-2 fuzzy logic controller IT2FLC as a power system stabiliser PSS combined with the optimal H∞ tracking control is proposed in this study. The multi-machine power system is considered for evaluation and implementation of the robust proposed controllers. The type-2 fuzzy logic based on interval value sets is capable for modelling the uncertainty and imprecision and to overcome the drawbacks of the conventional PSS. The input signals to the IT2FLC are considered as speed deviation and acceleration. The optimal H&infin tracking control guarantees the convergence of the errors to the neighbourhood of zero. The objective of the proposed method is to enhance the stability and the dynamic response of the multi-machine power system in different operating conditions. The performance with IT2FL-PSS and H∞ is compared to the performance of the power system with IT1FL-PSS and H∞ and with conventional PSS. In order to test the effectiveness of the proposed method, the simulation results show the damping of the oscillations of the angle and angular speed with reduced overshoots and quick settling time.

An interval type-2 fuzzy logic PSS with the optimal H∞ tracking control for multi-machine power system

In this paper, the interval type-2 fuzzy logic controller (IT2FLC) as a power system stabiliser (PSS) combined with the optimal H∞ tracking control is proposed in this study. The multi-machine power system is considered for evaluation and implementation of the robust proposed controllers. The type-2 fuzzy logic based on interval value sets is capable for modelling the uncertainty and imprecision and to overcome the drawbacks of the conventional PSS. The input signals to the IT2FLC are considered as speed deviation and acceleration. The optimal H&infin tracking control guarantees the convergence of the errors to the neighbourhood of zero. The objective of the proposed method is to enhance the stability and the dynamic response of the multi-machine power system in different operating conditions. The performance with (IT2FL-PSS and H∞) is compared to the performance of the power system with (IT1FL-PSS and H∞) and with conventional PSS. In order to test the effectiveness of the proposed method, the simulation results show the damping of the oscillations of the angle and angular speed with reduced overshoots and quick settling time.

An interval extension of homogeneous and pseudo-homogeneous t-norms and t-conorms

In this paper, the notion of homogeneity is extended to the interval-valued setting. In particular, we focus on the case of interval-valued t-norms and t-conorms and we characterize interval-valued homogeneous t-norms and t-conorms of interval-valued order. We also introduce the notions of interval-valued pseudo-homogeneous t-norm and interval pseudo-homogeneous t-conorm are introduced and characterized. We illustrate our results with two examples, one in image processing and the other one in decision making.

Study of Fuzzy Evaluation Method on Ecological Civilization Construction Based on Grey Interval Number

Ecological civilization evaluation is an important part of the sustainable development strategy. However, because of subjective and objective reasons, it is becoming increasing clear that much data or information is of poor quality, which has serious effects on the research results. With the analysis on the cause and process of the low quality data, this article presents a fuzzy evaluation method based on gray interval number, which would assess the ecological civilization performance. The application of data-gray-correction model could change the original statistics into gray interval number to reduce the influence of statistical error. A modified FAHP is used to capture and convert experts’ subjective judgment. A modified TOPSIS on three parameters interval-valued sets is applied to rank the objects. The research show that: corrected gray interval number could contain much more information than original crisp data; compared to traditional assessment algorithms, results by fuzzy evaluation method based on interval number would match more reality.

A practical resource-searching method for manufacturing grid

Manufacturing grid has become more and more popular because it can realize the resource share and collaboration within the worldwide scope. In most cases, the resource and task information in manufacturing grid is fuzzy and uncertain, but the resource-searching methods now are focused on deterministic value and have some limitations for manufacturing grid. This paper proposes a unified information model in which the information parameters are described as a set of interval values, and the conversion rules from the given capability parameters into intervals are discussed. On these bases, a new optimized resource-searching method is designed to find the best required manufacturing resource for the task by calculating the interval distance of the task and the resources. The method is verified to be effective by an instance. Furthermore, it is practical by comparing it with other resource-searching methods.

Groups of negations on the unit square.

The main results are about the groups of the negations on the unit square, which is considered as a bilattice. It is proven that all the automorphisms on it form a group; the set, containing the monotonic isomorphisms and the strict negations of the first (or the second or the third) kind, with the operator “composition,” is a group G 2 (or G 3 or G 4, correspondingly). All these four kinds of mappings form a group G 5. And all the groups G i, i = 2,3, 4 are normal subgroups of G 5. Moreover, for G 5, a generator set is given, which consists of all the involutive negations of the second kind and the standard negation of the first kind. As a subset of the unit square, the interval-valued set is also studied. Two groups are found: one group consists of all the isomorphisms on L I, and the other group contains all the isomorphisms and all the strict negations on L I, which keep the diagonal. Moreover, the former is a normal subgroup of the latter. And all the involutive negations on the interval-valued set form a generator set of the latter group.

Remarks on monotone interval-valued set multifunctions

In this paper, motivated by the representation of uncertainty, we discuss the problem of interval-valued set multifunctions which are monotone with respect to Guo and Zhang order relation [7]. We prove that a set multifunction μ:C→Pkc(R+) is a multisubmeasure in this order relation if and only if for every A∈C,μ(A)=[m1(A),m2(A)],m1,m2:C→R+ being submeasures in the sense of Drewnowski [5]. As application, related results concerning variation and continuity properties are established.

Classifier fusion with interval-valued weights

The article presents a new approach of calculating the weight of base classifiers from a committee of classifiers. The obtained weights are interpreted in the context of the interval-valued sets. The work proposes four different ways of calculating weights which consider both the correctness and incorrectness of the classification. The proposed weights have been used in the algorithms which combine the outputs of base classifiers. In this work we use both the outputs, represented by rank and measure level. Research experiments have involved several bases available in the UCI repository and two data sets that have generated distributions. The performed experiments compare algorithms which are based on calculating the weights according to the resubstitution and algorithms proposed in the work. The ensemble of classifiers has also been compared with the base classifiers entering the committee.

Some Characterizations of the Choquet Integral with Respect to a Monotone Interval-Valued Set Function

Intervals can be used in the representation of uncertainty. In this regard, we consider monotone interval-valued set functions and the Choquet integral. This paper investigates characterizations of monotone interval-valued set functions and provides applications of the Choquet integral with respect to monotone interval-valued set functions, on the space of measurable functions with the Hausdorff metric.