Sugeno Integral Research Articles

Novel Fuzzy Ostrowski Integral Inequalities for Convex Fuzzy-Valued Mappings over a Harmonic Convex Set: Extending Real-Valued Intervals Without the Sugeno Integrals

This study presents new fuzzy adaptations of Ostrowski’s integral inequalities through a novel class of convex fuzzy-valued mappings defined over a harmonic convex set, avoiding the use of the Sugeno integral. These innovative inequalities generalize the recently developed interval forms of real-valued Ostrowski inequalities. Their formulations incorporate integrability concepts for fuzzy-valued mappings (FVMs), applying the Kaleva integral and a Kulisch–Miranker fuzzy order relation. The fuzzy order relation is constructed via a level-wise approach based on the Kulisch–Miranker order within the fuzzy number space. Additionally, numerical examples illustrate the effectiveness and significance of the proposed theoretical model. Various applications are explored using different means, and some complex cases are derived.

Application of the Sugeno integral in Fuzzy Rule-Based Classification

Fuzzy Rule-Based Classification System (FRBCS) is a well-known technique to deal with classification problems. Recent studies have considered the usage of the Choquet integral and its generalizations (e.g.: CT-integral, CF-Integral and CC-integral) to enhance the performance of such systems. Such fuzzy integrals were applied to the Fuzzy Reasoning Method (FRM) to aggregate the fired fuzzy rules when classifying new data. However, the Sugeno integral, another well-known aggregation operator, obtained good results in other applications, such as brain–computer interfaces. These facts led to the present study, in which we consider the Sugeno integral in classification problems. That is, the Sugeno integral is applied in the FRM of a widely used FRBCS, and its performance is analyzed over 33 different datasets from the literature, also considering different fuzzy measures. To show the efficiency of this new approach, the results obtained are also compared with previous studies that involved the application of different aggregation functions. Finally, we perform a statistical analysis of the application.

Stochastically ordered aggregation operators

In aggregation theory, there exists a large number of aggregation functions that are defined in terms of rearrangements in increasing order of the arguments. Prominent examples are the Ordered Weighted Operator and the Choquet and Sugeno integrals. Following a probability approach, ordering random variables by means of stochastic orders can be also a way to define aggregations of random variables. However, stochastic orders are not total orders, thus pairs of incomparable distributions can appear. This paper is focused on the definition of aggregations of random variables that take into account the stochastic ordination of the components of the input random vectors. Three alternatives are presented, the first one by using expected values and admissible permutations, then a modification for multivariate Gaussian random vectors and a third one that involves a transformation of the initial random vectors in new ones whose components are ordered with respect to the usual stochastic order. A deep theoretical study of the properties of all the proposals is made. A practical example regarding temperature prediction is provided

Ostrowski and Čebyšev type inequalities for interval-valued functions and applications.

As an essential part of classical analysis, Ostrowski and Čebyšev type inequalities have recently attracted considerable attention. Due to its universality, the non-additive integral inequality takes several forms, including Sugeno integrals, Choquet integrals, and pseudo-integrals. Set-valued analysis, a well-known generalization of classical analysis, is frequently employed in studying mathematical economics, control theory, etc. Inspired by pioneering work on interval-valued inequalities, this paper establishes specific Ostrowski and Čebyšev type inequalities for interval-valued functions. Moreover, the error estimation to quadrature rules is presented as some applications for illustrating our results. In addition, illustrative examples are offered to demonstrate the applicability of the mathematical methods presented.

Rights Systems and Aggregation Functions on Property Spaces

The notion of decisive coalitions of voters with different grades of decisiveness is a part of the mathematical framework for many models in social choice theory. More generally, we study aggregation problems in which a subgroup of decision makers have the right to determine the properties of the aggregate. Then, we introduce property spaces and rights to properties and characterize aggregation operators that are consistent with rights to properties. Moreover, we define congruences in property spaces, and we propose a generalization of the Sugeno integral in our framework.

An extended TOPSIS and entropy measure based on Sugeno integral in Pythagorean fuzzy set setting

As an extension of the concepts of fuzzy set and intuitionistic fuzzy set, the concept of Pythagorean fuzzy set better models some real life problems. Distance, entropy, and similarity measures between Pythagorean fuzzy sets play important roles in decision making. In this paper, we give a new entropy measure for Pythagorean fuzzy sets via the Sugeno integral that uses fuzzy measures to model the interaction between criteria. Moreover, we provide a theoretical approach to construct a similarity measure based on entropies. Combining this theoretical approach with the proposed entropy, we define a distance measure that considers the interaction between criteria. Finally, using the proposed distance measure, we provide an extended Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) for multi-criteria decision making and apply the proposed technique to a real life problem from the literature. Finally, a comparative analysis is conducted to compare the results of this paper with those of previous studies in the literature.

Sugeno Integral Based on Overlap Function and Its Application to Fuzzy Quantifiers and Multi-Attribute Decision-Making

The overlap function is an important class of aggregation function that is closely related to the continuous triangular norm. It has important applications in information fusion, image processing, information classification, intelligent decision-making, etc. The usual multi-attribute decision-making (MADM) is to select the decision object that performs well on all attributes (indicators), which is quite demanding. The MADM based on fuzzy quantifiers is to select the decision object that performs well on a certain proportion or quantification (such as most, many, more than half, etc.) of attributes. Therefore, it is necessary to study how to express and calculate fuzzy quantifiers such as most, many, etc. In this paper, the Sugeno integral based on the overlap function (called the O-Sugeno integral) is used as a new information fusion tool, and some related properties are studied. Then, the truth value of a linguistic quantified proposition can be estimated by using the O-Sugeno integral, and the O-Sugeno integral semantics of fuzzy quantifiers is proposed. Finally, the MADM method based on the O-Sugeno integral semantics of fuzzy quantifiers is proposed and the feasibility of our method is verified by several illustrative examples such as the logistics park location problem.

Sugeno-Like Operators in Preference and Uncertain Environments

Sugeno-like operators are binary operations based generalization of Sugeno integral and are still defined on real valued fuzzy measures. This work discusses the aggregation methods in the situation where both inputs and fuzzy measures are attached with numerical uncertainties. That is, when an input vector and a fuzzy measure are given, each of the entries of the vector and the measure value on each subset can be attached with numerical uncertainty degrees. To fulfill this meaningful aim of performing Sugeno-like aggregation with numerical uncertainties, it needs two parts of work. We firstly formally define basic uncertain vector and basic uncertain fuzzy measure. Then we discuss the methods to construct or adjust basic uncertain vector and basic uncertain fuzzy measure in two situations, the general fuzzy measure situation and the probability measure only situation, respectively. With uncertain input and uncertain fuzzy measure, we then accordingly propose the corresponding restricted Sugeno-like operator, for which some different logic restrictions are analyzed. All the proposals also have full consistency for they can immediately degenerates into Sugeno-like operator when the attached uncertainties disappear.

Sugeno Integral for Hermite–Hadamard Inequality and Quasi-Arithmetic Means

Abstract In this paper, we present the Sugeno integral of Hermite–Hadamard inequality for the case of quasi-arithmetically convex (q-ac) functions which acts as a generator for all quasi-arithmetic means in the frame work of Sugeno integral.

Correction to: Sugeno Integral of Set-Valued Functions with Respect to Multi-submeasures and Its Application in MADM

Correction to: Sugeno Integral of Set-Valued Functions with Respect to Multi-submeasures and Its Application in MADM

Computing Sugeno integrals

The main goal of the present paper is to furnish practical procedures for the approximate computation (with preassigned precision) of the Sugeno integral of a general positive measurable function with respect to a general monotone measure. To this end, we use our two-steps methods (the simple two-steps method and the topologic method using the iterated convergence theorem). The aforementioned methods reduce the general computation to the well-known computation in the finite discrete case, which can be done with computer-aided methods.

Towards a Flexible Assessment of Compliance with Clinical Protocols Using Fuzzy Aggregation Techniques

In healthcare settings, compliance with clinical protocols and medical guidelines is important to ensure high-quality, safe and effective treatment of patients. How to measure compliance and how to represent compliance information in an interpretable and actionable way is still an open challenge. In this paper, we propose new metrics for compliance assessments. For this purpose, we use two fuzzy aggregation techniques, namely the OWA operator and the Sugeno integral. The proposed measures take into consideration three factors: (i) the degree of compliance with a single activity, (ii) the degree of compliance of a patient, and (iii) the importance of the activities. The proposed measures are applied to two clinical protocols used in practice. We demonstrate that the proposed measures for compliance can further aid clinicians in assessing the aspect of protocol compliance when evaluating the effectiveness of implemented clinical protocols.

A new characterization of congruences and the discrete Sugeno integral on LB-valued general fuzzy automata

The present study aimed at exploring the idempotent and compatible aggregation functions on a complete infinitely distributive lattice B of propositions about the LB-valued general fuzzy automata (for simplicity, LB-valued GFA), where the compatibility is related to the congruences on B. Further, we have shown that all aggregation functions on B can be obtained as usual composition of lattice operations ∨,∧ and certain unary and binary aggregation functions. Moreover, our results yield a new characterization of discrete Sugeno integrals on B.

The decision-making process in learners’ failure using fuzzy scale

This study aims to determine how to use an ambiguous scale to assess student performance and proficiency, whether students perform well or poorly in subjects such as reading comprehension, vocabulary, and punctuation. In this study, Sugeno Integral is a good choice for decision-making. Although other scales may be better than the one used, Sugeno Integral is easier to use than different scales, as evidenced by the results obtained.

On Join-Dense Subsets of Certain Families of Aggregation Functions

Several important classes of aggregation functions defined on a bounded lattice form a lattice with respect to the pointwise operations of join and meet, respectively. The lattice structure of such classes is usually very complex; thus, it is very useful to characterize them by some appropriate sets of functions. In this paper, we focus on the three important classes of aggregation functions, namely the lattice of all aggregation functions, the lattice of idempotent aggregation functions, and the lattice of Sugeno integrals (defined on distributive lattices) and characterize their lattices by means of join-dense subsets. Moreover, the minimality of these sets is discussed.

Generalized upper Sugeno integrals: The transformation theorem and related results

In this paper, we deal with the transformation theorem for the generalized upper Sugeno integral I∘. We formulate the sufficient and necessary conditions under which the transformation holds, thus providing a complete solution to this problem. We also provide some representations of the I∘ integral via left-continuous (right-continuous) modifications of level measures. We characterize the class of binary operations for which the I∘ integral can be represented via the quantile function. As a consequence, we get complete solutions to these problems for the smallest semicopula-based universal integral. We thus solve several open problems related to this class of integrals in the literature.

On comonotone k-maxitive aggregation functions

The aim of this paper is to study comonotone k-maxitive aggregation functions, i.e., aggregation functions fulfilling k-maxitivity for comonotone inputs. We characterize all comonotone k-maxitive aggregation functions and clarify the relation between comonotone k-maxitive aggregation functions and Sugeno integrals, level-dependent Sugeno integrals and Sugeno integrals w.r.t. k-maxitive fuzzy measure. The subset of the unit hypercube fully determining comonotone k-maxitive aggregation function is found and a construction method based on this set is proposed.

Godunova Type Inequality for Sugeno Integral

Godunova Type Inequality for Sugeno Integral

(Max,⊕)-transforms and genetic algorithms for fuzzy measure identification

Fuzzy measures generalize additive measures and probabilities. Their advantage with respect to additive ones is that they permit to model interactions between objects. Mesiar introduced in 1999 k-order Pan-additive fuzzy measures that generalize k-order additive and k-maxitive ones. They are related to the Möbius transform and related generalizations. In this paper we introduce some other transforms that we call (Max,+) and (Max,⊕) that permit to represent fuzzy measures in a convenient way when we use genetic algorithms in fuzzy measure identification problems. We illustrate its use identifying a measure for a subjective evaluation problem using a Choquet integral and a Sugeno integral.

The Alternative Representation of the Bipolar Sugeno Integral

In the context of decision support systems, bi-capacities were introduced as an extension of classical capacities. Many bipolar fuzzy integrals related to the bi-capacities have been presented in recent years. One of these integrals is the Sugeno integral concerning aggregation on bipolar scales. The paper aims to build an equivalent representation of the bipolar Sugeno integral. Therefore, we first employ in this paper the framework based on a ternary-criterion set for proposing an alternative formula of the bipolar Sugeno integral to be suitable for bipolar scales. Then, we discuss some basic properties and give an illustrative example of this representation. This representation is consistent as an extension of the representation concerning the classical capacities and aggregation on the Sugeno integral unipolar scales.