Fuzzy Calculus Research Articles

Continuities, Derivatives, and Differentials of $q$-Rung Orthopair Fuzzy Functions

Yager's $q$ -rung orthopair fuzzy set ( $q$ -ROFS) is a powerful tool to handle uncertainty and vagueness in real life. It expands the spatial scope of membership and nonmembership, and therefore has a wider range of constraints and stronger modeling capabilities. However, to date, there is no investigation for $q$ -rung orthopair fuzzy derivatives and differentials, which are very important for further developing $q$ -rung orthopair fuzzy calculus ( $q$ -ROFC). The basic elements of a $q$ -ROFS are $q$ -rung orthopair fuzzy numbers ( $q$ -ROFNs), based on which we propose the $q$ -rung orthopair fuzzy functions ( $q$ -ROFFs) and discuss their continuities in detail. Subsequently, we study the derivative of the $q$ -ROFF, which reveals an accurate description on rate of change for continuous $q$ -ROFF. Next, the differential operation of $q$ -ROFF is established; thereby providing an effective approximation on nonlinear problem in the $q$ -ROFF environment. Finally, we present numerical examples as explicit applications of $q$ -ROFC.

Additive Integrals of q -Rung Orthopair Fuzzy Functions.

The q -rung orthopair fuzzy set ( q -ROFS) is a powerful tool to deal with uncertainty and ambiguity in real life. The theoretical basis for processing the continuous q -rung orthopair fuzzy information is q -rung orthopair fuzzy calculus ( q -ROFC) and the main object is q -rung orthopair fuzzy functions ( q -ROFFs). Recently, the authors proposed derivatives and differentials of q -ROFFs in the framework of q -ROFC. In this paper, we aim to further study the q -rung orthopair fuzzy integral ( q -ROFI). It is the most important and fundamental part of the q -ROFC theoretical system with direct and powerful applications. Our contribution is the indefinite and definite integrals, and bridges the fuzzy calculus theoretical gap of the nonlinear q -ROFFs. In particular, we begin with the indefinite integral of q -ROFFs, which can be regarded as the anti-derivatives operations of our previous work. Some of their basic properties are discussed. Next, we give the accurate concept of definite integrals of q -ROFFs under additive operations, and obtain the explicit integral formula. Some properties of q -ROFIs, such as comparison, algebraic operations, and mean value theorem are analyzed. Finally, we generalize the q -ROFI to the case when membership and nonmembership functions are allowed to be correlated. After the theoretical results have been established, we present some numerical examples to demonstrate the rationality and effectiveness of integrating continuous q -rung orthopair fuzzy data with the q -ROFIs.

Bornological Coderivative and Subdifferential Calculus in Smooth Banach Spaces

In this paper, we study bornological generalized differential properties of sets with nonsmooth boundaries, nonsmooth functions, and set-valued mappings in smooth Banach spaces. We establish a fuzzy intersection rule for bornological normal cones and develop fuzzy calculus for bornological generalized differential constructions as well as exact calculus for the limiting counterparts of these constructions.

Simplified Interval-Valued Intuitionistic Fuzzy Line Integrals and Their Application

In the past few years, the intuitionistic fuzzy calculus (IFC) has been proposed, which is based on some basic operational laws of intuitionistic fuzzy numbers (IFNs). The interval-valued intuitionistic fuzzy sets have received great attention of scholars since they can describe the characters of things comprehensively. However, so far, the interval-valued IFC is in the original stage. In this paper, we focus on investigating the simplified interval-valued intuitionistic fuzzy line integrals (SIVIFLIs). First, we construct the SIVIFLIs step by step, and then provide their concrete values. In order to further describe the characteristic of the SIVIFLIs, we discuss their basic properties, and particularly, we investigate a kind of the order in intuitionistic fuzzy circumstance in detail. Finally, we offer an application and address it by the proposed SIVIFLIs to demonstrate their effectiveness and practicability.

Foundation of Interval‐Valued Intuitionistic Fuzzy Limit and Differential Theory and an Application to Continuous Data

The intuitionistic fuzzy calculus (IFC), based on the basic operational laws of intuitionistic fuzzy numbers (IFNs), has been put forward. However, the interval‐valued IFC (IVIFC), based on the basic operational laws of interval‐valued IFNs (IVIFNs), is only in the original stage. To further develop the theory of the IVIFC and make it be rigorous, the primary task is to systematically investigate the characteristics of the limits and differentials, which is a foundation of the IVIFC. Moreover, there is quite a lot of literature on IVIFNs; however, the scholars did not reveal the relationships between IFNs and the IVIFNs. To do that, we first investigate the limit of interval‐valued intuitionistic fuzzy sequences, and then, we focus on investigating the limit, the continuity, and the differential of IVIFFs in detail and reveal their relationships. After that, due to the fact that the IFC and the IVIFC are based on the basic operational laws of IFNs and IVIFNs, respectively, we reveal the relationships between the IFNs and the IVIFNs via some homomorphic mappings. Finally, a case study about continuous data of IVIFNs is provided to illustrate the advantages of continuous data.

Television Rating Control in the Multichannel
 Environment Using Trend Fuzzy Knowledge Bases
 and Monitoring Results †

The purpose of this study is to control the ratio of programs of different genres whenforming the broadcast grid in order to increase and maintain the rating of a channel. In themultichannel environment, television rating controls consist of selecting content, the ratings ofwhich are completely restored after advertising. The hybrid approach to rule set refinement basedon fuzzy relational calculus simplifies the process of expert recommendation systems construction.By analogy with the problem of the inverted pendulum control, the managerial actions aim to retainthe balance between the fuzzy demand and supply. The increase or decrease trends of the demandand supply are described by primary fuzzy relations. The rule-based solutions of fuzzy relationalequations connect significance measures of the primary fuzzy terms. Program set refinement bysolving fuzzy relational equations allows avoiding procedures of content-based selective filtering.The solution set generation corresponds to the granulation of television time, where each solutionrepresents the time slot and the granulated rating of the content. In automated media planning,generation of the weekly TV program in the form of the granular solution provides the decrease oftime needed for the programming of the channel broadcast grid.

Intuitionistic Fuzzy Double Integrals and Their Fundamental Properties

Intuitionistic fuzzy set (IFS), as a significant generalization of the fuzzy set (FS), takes the membership function and the non-membership function into consideration simultaneously. The core components of an IFS are the ordered pairs which are called intuitionistic fuzzy numbers (IFNs). More recently, the intuitionistic fuzzy calculus (IFC) has been established, which is based on the basic operational laws of IFNs. As far as the IFC with several variables is concerned, its theory is still in the initial stage. The purpose of this paper is to systematically establish the theory of the intuitionistic fuzzy double integrals (IFDIs). To accomplish this, we first construct the IFDIs, and then assign them concrete values. To understand the IFDIs in depth, we investigate their fundamental properties in detail and offer simple proofs for them. Next, we give a counterexample to verify that in the IFC the first and second mean value theorems do not hold, and interpret the reason from the perspective of the geometric figure. In addition, we compare the results in the IFC with those in the classical calculus to demonstrate that they are thoroughly different despite their forms being sometimes similar. Finally, we offer an application to show the utility of the proposed IFDIs.

SOME FUNDAMENTAL RESULTS ON FUZZY CALCULUS

In this paper, we study fuzzy calculus in two main branches differential and integral. Some rules for finding limit and $gH$-derivative of $gH$-difference, constant multiple of two fuzzy-valued functions are obtained and we also present fuzzy chain rule for calculating $gH$-derivative of a composite function. Two techniques namely, Leibniz's rule and integration by parts are introduced for fuzzy integrals. Furthermore, we prove three essential theorems such as a fuzzy intermediate value theorem, fuzzy mean value theorem for integral and mean value theorem for $gH$-derivative. We derive a Bolzano's theorem, Rolle's theorem and some properties for $gH$-differentiable functions. To illustrate and explain these rules and theorems, we have provided several examples in details.

Fuzzy Calculus Theory and Its Applications

Fuzzy Calculus Theory and Its Applications

Line Integrals of Intuitionistic Fuzzy Calculus and Their Properties

Atanassov's intuitionistic fuzzy set (A-IFS) is a powerful tool to handle uncertainty and vagueness in real life. The basic elements of an A-IFS are intuitionistic fuzzy values, based on which the intuitionistic fuzzy calculus (IFC) has been proposed recently. However, to date, there is no investigation for intuitionistic fuzzy line integrals (IFLIs), which are very important for further developing IFC. In this paper, we propose the IFLIs and give their concrete values. After that, we investigate a series of basic properties of the IFLIs in detail, moreover, in order to show the utility of the proposed IFLIs, we offer an example, and finally, we discuss the relationships among the additive IFLI, the multiplicative IFLI, and the intuitionistic fuzzy aggregation operators.

Multiple Definite Integrals of Intuitionistic Fuzzy Calculus and Isomorphic Mappings

Intuitionistic fuzzy set (A-IFS) introduced by Atanassov, a generalized form of fuzzy sets, is a useful tool to deal with vagueness and ambiguity. Each element of an A-IFS is called an intuitionistic fuzzy value, based on which the intuitionistic fuzzy calculus (IFC) has been proposed recently. However, the existing study only discusses the one-variable IFC, in order to enrich the theory of the IFC; in this paper, we put forward the intuitionistic fuzzy multiple definite integrals (IFMDI). We first define some concepts related to the IFMDI, based on which we reveal the relationships between the IFMDI and the intuitionistic fuzzy definite integrals. After that we investigate the basic properties of the IFMDI in detail, and finally, we apply the knowledge of abstract algebra to construct some isomorphic mappings, based on which, we interpret the reason why there are some parallel properties between the additive IFMDI and the multiplicative IFMDI.

Uncertainty Analysis of the Estimated Risk in Formal Safety Assessment

An uncertainty analysis is required to be carried out in formal safety assessment (FSA) by the International Maritime Organization. The purpose of this article is to introduce the uncertainty analysis technique into the FSA process. Based on the uncertainty identification of input parameters, probability and possibility distributions are used to model the aleatory and epistemic uncertainties, respectively. An approach which combines the Monte Carlo random sampling of probability distribution functions with the a-cuts for fuzzy calculus is proposed to propagate the uncertainties. One output of the FSA process is societal risk (SR), which can be evaluated in the two-dimensional frequency–fatality (FN) diagram. Thus, the confidence-level-based SR is presented to represent the uncertainty of SR in two dimensions. In addition, a method for time window selection is proposed to estimate the magnitude of uncertainties, which is an important aspect of modeling uncertainties. Finally, a case study is carried out on an FSA study on cruise ships. The results show that the uncertainty analysis of SR generates a two-dimensional area for a certain degree of confidence in the FN diagram rather than a single FN curve, which provides more information to authorities to produce effective risk control measures.

Fundamental Properties With Respect to the Completeness of Intuitionistic Fuzzy Partially Ordered Set

Intuitionistic fuzzy set (A-IFS), originally introduced by Atanassov in 1983, is a generalization of a fuzzy set. The basic elements of an A-IFS are intuitionistic fuzzy values (IFVs), based on which the intuitionistic fuzzy calculus (IFC) has been proposed recently. To avoid relying too much upon the classical calculus and make the IFC to be an independent subject, it is necessary to develop the limit theory of the IFC. In this paper, we first define the concepts of the supremum and the infimum with respect to IFVs, and investigate their properties in detail. Then, we reveal the relationships among them and four types of limits, and finally, we give a series of fundamental theorems with respect to the completeness of intuitionistic fuzzy partially ordered set.

A Unification of Intuitionistic Fuzzy Calculus Theories Based on Subtraction Derivatives and Division Derivatives

Intuitionistic fuzzy calculus replaces real numbers in classical calculus with intuitionistic fuzzy numbers that are the basic elements of Atanassov's intuitionistic fuzzy sets. Intuitionistic fuzzy calculus consists of two parallel parts, which are respectively developed based on the subtraction derivatives and the division derivatives. This paper focuses on building the relationships between the two independent intuitionistic fuzzy calculus theories, and unifying them into a whole theory.

The chain and substitution rules of interval-valued intuitionistic fuzzy calculus

The interval-valued intuitionistic fuzzy set proposed by Atanassov is the extension of intuitionistic fuzzy set. It extends the membership degree and non-membership to interval values instead of a single value. So it contains more possible values and maybe more considerate. Among all the researches, the exploration on the calculus of interval-valued intuitionistic fuzzy set is entirely new. Recently, Zhao et al. (Int J Comput Intell Syst 9:36–56, 2016) proposed the concept of interval-valued intuitionistic fuzzy function (IVIFF) and gave a calculation method of derivative and differential of IVIFF. Based on this work, in this paper, firstly, we utilize a new and easier method to express the derivative and differential of IVIFF. Secondly, we propose the chain rules of derivative and the form invariance of differential in the interval-valued intuitionistic fuzzy environment. In addition, some properties of the substation rules for interval-valued intuitionistic fuzzy indefinite integrals and definite integrals are also developed.

Infinite Intuitionistic Fuzzy Series and Product

Since the Atanassov's intuitionistic fuzzy set theory was introduced, many intuitionistic fuzzy aggregation operators have been proposed for the integration of the intuitionistic fuzzy information. It is necessary to extend them to accommodate the infinite situations so that we can deal with the large amount of intuitionistic fuzzy data, and, at the same time, we must solve an important issue that whether the sum of an infinite number of intuitionistic fuzzy numbers is convergent or not. Thus, in this paper, we first put forward the infinite intuitionistic fuzzy series and product. Then, we investigate the properties and the convergences of the infinite intuitionistic fuzzy series and product, respectively. These results greatly enrich the intuitionistic fuzzy calculus theory.

Relationships Between Two Types of Intuitionistic Fuzzy Definite Integrals

Intuitionistic fuzzy numbers (IFNs) are the basic components of Atanassov's intuitionistic fuzzy sets, which are very convenient and comprehensive in depicting the fuzzy characters of things in real life. In the existing literature, there are mainly two different types of definite integrals with respect to IFNs, which were developed from completely different aspects. The main purpose of this paper is to demonstrate their relationships in detail by deriving the definite integrals of two novel intuitionistic fuzzy functions. It is worth pointing out that one of the two types of definite integrals is shown to be a special case of another one, and finally, we also derive some other useful results for intuitionistic fuzzy calculus based on the division derivative.

Chain and Substitution Rules of Intuitionistic Fuzzy Calculus

The intuitionistic fuzzy set, originally given by Atanassov, can be considered as a set that consists of intuitionistic fuzzy numbers (IFNs), each of which is described by a membership degree, a nonmembership degree, and a hesitancy degree. Recently, the calculus of IFNs has been proposed, and the derivatives, differentials, indefinite integrals, and definite integrals with respect to IFNs have also been developed in previous research studies. In this paper, some novel results of intuitionistic fuzzy calculus are derived, which include mainly the chain rule of derivatives, the form invariance of differentials, and the substitution rules for indefinite integrals and definite integrals in intuitionistic fuzzy environment.

Fuzzy Logic-Based Formalisms for Gynecology Disease Diagnosis

AbstractThe very basis of the present article is the fact that the medical knowledge consisting of clinical presentation, diagnosis, and treatment of a disease is with imprecision and uncertainty. The overall approach in gynecological disease diagnosis could be divided into three distinct stages, and this was confirmed by seven experienced gynecologists. Stage 1 refers to an initial screening process in order to arrive at a single disease diagnosis for the patients, which is based only on the subjective information provided by patients to the physician. In stage 2, the patient who has not received a single diagnostic label in stage 1 is further investigated for a single disease diagnosis using past history criteria. If stage 2 fails to arrive at a single disease diagnosis for a patient, then physical examination and various tests like imaging tests, blood tests, etc., are conducted, and the test results are processed in stage 3. In stage 1, we have revisited fuzzy relational calculus and mathematically evaluated the perceptions of the domain experts (gynecologists) with respect to 31 gynecological diseases. The paper also presents the research findings with a case study focused on stage 2 using a type 1 fuzzy inference system. Out of 226 patients, 50 are correctly diagnosed for a single disease and 147 for multiple diseases in stage 1. The paper concludes that fuzzy relational calculus is an effective method as an “initial screening” process to arrive at a single disease diagnosis. We have identified 29 out of 226 patients satisfying past history criteria to achieve a single disease diagnosis by stage 2. Investigations for stage 3 are in progress.

Limit properties and derivative operations in the metric space of intuitionistic fuzzy numbers

Intuitionistic fuzzy numbers (IFNs) are a useful tool to depict the uncertain information in real life. Based on IFNs, the intuitionistic fuzzy calculus (IFC) has been put forward recently. To further develop the IFC theory, in this paper, we investigate the limit properties of IFCs, and study the intuitionistic fuzzy infinitesimals and their orders. We also discuss the continuity, the derivatives and the differentials of intuitionistic fuzzy functions in detail, and reveal their relationships. Additionally, we define the metric space of the IFNs, based on which, a series of desirable results are obtained. These results are similar to the ones in the classical calculus.