Showing papers on "Membership function published in 2023"

Generalized Frame for Orthopair Fuzzy Sets: (m,n)-Fuzzy Sets and Their Applications to Multi-Criteria Decision-Making Methods

Abstract: Orthopairs (pairs of disjoint sets) have points in common with many approaches to managing vaguness/uncertainty such as fuzzy sets, rough sets, soft sets, etc. Indeed, they are successfully employed to address partial knowledge, consensus, and borderline cases. One of the generalized versions of orthopairs is intuitionistic fuzzy sets which is a well-known theory for researchers interested in fuzzy set theory. To extend the area of application of fuzzy set theory and address more empirical situations, the limitation that the grades of membership and non-membership must be calibrated with the same power should be canceled. To this end, we dedicate this manuscript to introducing a generalized frame for orthopair fuzzy sets called “(m,n)-Fuzzy sets”, which will be an efficient tool to deal with issues that require different importances for the degrees of membership and non-membership and cannot be addressed by the fuzzification tools existing in the published literature. We first establish its fundamental set of operations and investigate its abstract properties that can then be transmitted to the various models they are in connection with. Then, to rank (m,n)-Fuzzy sets, we define the functions of score and accuracy, and formulate aggregation operators to be used with (m,n)-Fuzzy sets. Ultimately, we develop the successful technique “aggregation operators” to handle multi-criteria decision-making problems in the environment of (m,n)-Fuzzy sets. The proposed technique has been illustrated and analyzed via a numerical example.

A new adaptive membership function with CUB uncertainty with application to cluster analysis of Likert-type data

Abstract: Likert-type scales are commonly used in both academia and industry to capture human feelings since they are user-friendly, easy-to-develop and easy-to administer. This kind of scales generate ordinal variables made up of a set of rank ordered items. Since the distance between two consecutive items cannot be either defined or presumed equal, this kind of variable cannot be analysed by either statistical methods defined on a metric space or parametric tests. Therefore, Likert-type variables cannot be used as segmentation variables of a traditional cluster analysis unless pre-transformed. In such context, fuzzy numbers have been suggested as a way to recode Likert-type variables. Fuzzy numbers are defined by a membership function whose form is usually determined by an expert. In practice, researchers usually define one membership function for each Likert-type scale, not considering the peculiar characteristics of neither questions nor respondents. In this way, the individual uncertainty against each question is considered equal and constant. To overcome this limitation and to reduce the expert’s subjectivity, in this study an adaptive membership function based on CUB model is suggested to pre-transform Likert-type variables into fuzzy numbers before the adoption of a clustering algorithm. After a theoretical presentation of the method, an application using real data will be presented to demonstrate how the method works.

Evaluation of Fuzzy Measures using Dempster-Shafer Belief Structure: A Classifier Fusion Framework

Abstract: This paper studies the high complexity of the calculation of fuzzy measures which can be used in fuzzy integrals to combine the decisions of different learning algorithms. To this end, this paper proposes an alternative low complexity method for the calculation of fuzzy measures that have been applied to Choquet integral for the fusion of deep learning models across different application domains for increasing the accuracy of the overall model. The paper shows that the Dempster-Shafer (DS) belief structure provides partial information about the fuzzy measures associated with a variable, and the paper devises a method to use this partial information for the calculation of fuzzy measures. An infinite number of fuzzy measures is associated with the DS belief structure. This paper proposes a theorem to calculate the general form of a specific set of fuzzy measures associated with the DS belief structure. This specific set of fuzzy measures can be expressed as a weighted summation of the basic assignment function of the DS belief structure. The main advantage of expressing the fuzzy measures in this format is that the monotonic condition which needs to be maintained during the calculation of the fuzzy measure can be avoided and only the basic assignment function needs to be evaluated. The calculation of the basic assignment function is formulated using a method inspired by the Monte Carlo approach used to calculate Value Functions in Markov Decision Process.

Multi-objective optimization to the transportation problem considering non-linear fuzzy membership functions

Abstract: Considering the uncertainty of transporting goods from numerous origins to diverse destinations is a critical task for the decision-maker (DM). The ultimate goal of the DM is to make the right decisions that optimize the profit or loss of the organization under the vagueness of the uncontrollable effects. In this paper, mathematical models are proposed using fuzzy non-linear membership functions for the transportation problem considering the parameters' uncertainty that can help the DM to optimize the multi-objective transportation problems (MOTP) and to achieve the desired goals by choosing a confidence level of the uncertain parameters. Based on DM's selection of the confidence level, a compromise solution of the uncertain multi-objective transportation (UMOTP) is obtained along with the satisfaction level in percent for the DM. Two non-linear fuzzy membership functions are considered: the exponential and the hyperbolic functions. Using both membership functions, the sensitivity analysis was implemented by considering different confidence levels. According to the experimental results, the hyperbolic membership function gives 100% DM's satisfaction in many instances. Moreover, it shows stability against the exponential and linear functions.

Membership score of an Interval-Valued Pythagorean Fuzzy Numbers and its applications

Abstract: The main aim of the paper is to define a new Membership score function on the class of interval-valued Pythagorean fuzzy numbers which can overcome the drawbacks of existing familiar ranking methods. In this paper, firstly, we show the limitations of various ranking functions in ordering/ comparing any two arbitrary interval-valued Pythagorean fuzzy numbers in detail. Secondly, we define a new Membership score function on the class of interval-valued Pythagorean fuzzy numbers and study their properties. Then we compare our proposed method with many other different existing methods for showing the efficacy of the proposed method. Finally, we show the applicability of the proposed Membership score function in solving interval-valued Pythagorean fuzzy multi-criteria decision-making problems using a numerical example.

Type-2 Fuzzy Relations: An Approach towards Representing Uncertainty in Associative Medical Relationships

Abstract: The acquisition of precise values such as symptoms, signs, laboratory test results, and diseases/diagnoses for expressing meaningful associative relationships between medical entities has always been regarded as a critical part of developing medical knowledge-based systems. After the introduction of fuzzy sets, researchers became aware of the fact that a central problem in the use of fuzzy sets is constructing the membership function values. The complication arises from the uncertainty associated with assigning an exact membership grade for each element within the considered fuzzy set. Type-2 fuzzy set handles this problem by allocating a different fuzzy set to each element. This paper addresses the subject of medical knowledge acquisition and representation by proposing consistent interval type-2 fuzzy relations in the context of fuzzy inclusion as a measure of representing the degrees of association between medical entities. The concept of interval type-2 fuzzy relation will be introduced to represent the uncertainty and vagueness between medical entities.

Design and Performance Analysis of a Multi-level Fuzzy-Based Stabilizer to Dampen Low-Frequency Oscillation in Single-Machine Infinite Bus Systems

Abstract: Power systems are frequently viewed as complex, nonlinear, and dynamic systems. This system is constantly subjected to small disturbances that can result in synchronization loss and system failure. To fix this issue, power system stabilizers are applied to generate extra excitation control signals. Conventional power system stabilizer (CPSS) is difficult to track the dynamic nature of the load since stabilizer gains are determined under specific working conditions. In this paper, a multi-level fuzzy-based stabilizer uses the variation of rotor speed and acceleration as an input to mitigate low-frequency oscillations (LFOs) in single-machine infinite bus systems. The system is represented mathematically by the Heffron Philips K-coefficients model. The controller’s performance was investigated for disturbances exposed to inputs of various membership functions, such as a triangular, gaussian, generalized bell, and trapezoidal. Each membership function is compared. For instance, a multi-level fuzzy-based stabilizer with a triangular membership function settled the rotor angle, rotor speed, and electrical torque deviations 29.5%, 5.9%, and 39.7% faster than the gaussian membership function fuzzy-based PSS, respectively. The study’s findings revealed that the triangular membership function performed better than other membership functions.

New Relaxed Static Output Feedback Stabilization of T–S Fuzzy Systems with Time-Varying Delays

Abstract: This paper investigates the imperfect premise matching-based (IPMB) static output feedback (SOF) controller for T–S fuzzy systems with time-varying delays. Firstly, by employing integral inequality techniques, the membership-function-independent fuzzy SOF design methods in terms of LMIs are presented based on imperfect premise matching (IPM) strategy. The obtained stabilization conditions do not contain equality constraints, and the output matrices do not have rank constraints. Secondly, some suitable relaxation variables are employed, and more relaxed membership-function-dependent (MFD) stabilization conditions are obtained by considering the local boundary information of membership functions (MFs). Finally, two simulation examples are given to show the progressiveness of the proposed methods in this paper.

Bilateral fuzzy sets and their three-way decisions: a new perspective of fuzzy logic

Abstract: Fuzzy sets provide an effective method for dealing with uncertain and imprecise problems. For data of intermediate fuzzy distribution, membership degrees of objects whose attribute values are larger or smaller than the normal value would be the same and carried out the same decision. However, objects with different values mean that the information they contain is different for the decision-making problem. The decision process of calculating membership degrees in fuzzy set will lose the information of data itself. Therefore, bilateral fuzzy sets and their three-way decisions are proposed. First, the deviation degree is proposed in order to distinguish these objects. Compared with the membership degree, the deviation degree extends the mapping range from [0, 1] to [- 1, 1]. For six typical membership functions, their corresponding deviation functions are discussed and deduced. Second, the concept of bilateral fuzzy sets is proposed and the corresponding operation rules are analyzed and proved. Then, three-way decisions and approximations based on bilateral fuzzy sets are constructed. Next, for the optimization of threshold, principle of least cost is extended to the three-way decisions model based on bilateral fuzzy sets, and theoretical derivation is carried out. Finally, based on probability statistics, the principle based on confidence interval is proposed, which provides a new perspective for threshold calculation.

Fuzzy Bayesian inference for mapping vague and place-based regions: a case study of sectarian territory

Abstract: Abstract The problem of mapping regions with socially-derived boundaries has been a topic of discussion in the GIS literature for many years. Fuzzy approaches have frequently been suggested as solutions, but none have been adopted. This is likely due to difficulties associated with determining suitable membership functions, which are often as arbitrary as the crisp boundaries that they seek to replace. This paper presents a novel approach to fuzzy geographical modelling that replaces the membership function with a possibility distribution that is estimated using Bayesian inference. In this method, data from multiple sources are combined to estimate the degree to which a given location is a member of a given set and the level of uncertainty associated with that estimate. The Fuzzy Bayesian Inference approach is demonstrated through a case study in which census data are combined with perceptual and behavioural evidence to model the territory of two segregated groups (Catholics and Protestants) in Belfast, Northern Ireland, UK. This novel method provides a robust empirical basis for the use of fuzzy models in GIS, and therefore has applications for mapping a range of socially-derived and otherwise vague boundaries.

The measurement and evaluation of efficiency performance in manufacturing: a fuzzy data envelopment analysis (dea) approach

Abstract: The measurement and evaluation of technical efficiency of manufacturing environments is constrained in part by the unavailability of precise production data. A fuzzy mathematical programming approach is used to accommodate the imprecision associated with some of the input and output production variables. This approach requires that the analyst know how the fuzzy input and output variables can be expressed in terms of their risk-free and impossible bounds and a membership function. The membership function represents the degree to which a production scenario is realistically implementable. Technical efficiency scores are computed for different values of the membership function so as to identify uniquely different production scenarios which can subsequently lead to efficiency improvement strategies. The approach is illustrated in the context of a preprint and packaging manufacturing line which inserts commercial pamphlets into newspapers.

Application of artificial intelligence technologies to assess water salinity

Abstract: Topical issues of developing theoretical and methodological tools for constructing a fuzzy logical model for assessing water salinity are considered. When constructing a Sugeno fuzzy logical model for estimating water salinity, a rational number of rules and effective values of their membership functions were chosen. Initially, the membership function parameters were obtained from water industry experts. In the future, it is necessary to adjust the parameters of the membership function using neural networks to obtain the minimum number of fuzzy rules.

Which Are the Correct Membership Functions? Correct “And”- and “Or”- Operations? Correct Defuzzification Procedure?

Abstract: Even in the 1990s, when many successful examples of fuzzy control appeared all the time, many users were somewhat reluctant to use fuzzy control. One of the main reasons for this reluctance was the perceived subjective character of fuzzy techniques—for the same natural-language rules, different experts may select somewhat different membership functions and thus get somewhat different control/recommendation strategies. In this paper, we promote the idea that this selection does not have to be subjective. We can always select the “correct” membership functions, i.e., functions for which, on previously tested case, we got the best possible control. Similarly, we can select the “correct” and- and or-operations, the correct defuzzification procedure, etc.

A new on-line self-adapting fuzzy controller design using unidimensional input-output with dynamical membership functions

Abstract: Here, we develop a fuzzy controller using a new online self-adapting design. The objective of this work is to control a nonlinear process by using a one-dimensional input rule variable, instead of error and error variation. The initial limits of the fuzzy logic membership functions are mostly depend on experiments and previous knowledge of the dynamic process behaviors. Generally, the membership function parameters have a significant impact on control signal amplitude and, consequently on the convergence and stability of the controller-plant system. The proposed technique determines the limits of the antecedent membership functions online using the k th and k - 1 th outputs of the controlled plant and reference model, respectively. Meanwhile, the limits of the consequent membership functions are calculated using error and error variation. This approach ensures: (i) that the input/output variables have the required fuzzy space, (ii) the controlled plant follows the desired reference model, and (iii) the control signal amplitude is within acceptable limits. Additionally, (iiii) it takes into account the dynamic variability of the process and the existence of an overshoot. The membership function parameters are updated continuously through a self-adapting procedure, ensuring improved control performance. Ultimately, the proposed approach is improved using two nonlinear systems.

A interval valued intuitionistic fuzzy multi criteria decision making

Abstract: The limitations and disadvantages of the available intuitionistic fuzzy set scoring functions are investigated. Two improved methods for handling multi-criteria fuzzy decision-making problems are provided. They are based on the two theories of intuitionistic fuzzy set and cross entropy, with the adoption of cross entropy of the degree of membership from the degree of non-membership handling the effect of hesitancy degree. Score function method and weighted score function method are their names. This study presents and investigates a novel strategy for ranking interval-valued intuitionistic fuzzy sets. Examples using numbers are used to demonstrate the technique.

Solution of Transportation Problem Using Interval-Valued Pythagorean Fuzzy Approach

Abstract: In the implementation of real-world problems, there is increased demand on administration to figure out how to get things from point A to point B as cheaply as possible. Transportation problem (TP) models are particularly effective in reducing these difficulties. In a traditional transportation problem, the decision-maker is confident in the product’s transportation cost, availability, and demand. But in a real-life scenario, these variables are uncontrollable due to an imprecision of data. To deal with the ambiguous data and haziness of real-world situations, the concept of a fuzzy set (FS) and its extensions have been defined. It can be difficult to determine the value of membership and non-membership at a given point in some cases. As a result, we have used the membership as well as non-membership values as an interval in the proposed work to find the solution to the transportation problem using an interval-valued Pythagorean fuzzy set (IVPFS). We used the proposed score function (SF) to solve three different types of interval-valued Pythagorean fuzzy transportation problems (IVPFTP) and compared the results with existing score function in the literature. Also we have discussed the algorithm for IVPFTP in the proposed work. Numerical problems for the effectiveness of various types of models are providing to support our work. Finally, we discussed the work's outcome and conclusion.

Solution of a Pollution‐Based Economic Order Quantity Model Under Triangular Dense Fuzzy Environment

Abstract: In this chapter, the main issue is environmental pollution, which considers pollution-sensitive economic order quantity (EOQ) model for several items under so many styles of organized via triangular dense fuzzy environment. Here, we constructed a new pollution function via this model, then we have discussed about a substantive case study in a sponge iron industry. We have proposed an EOQ model where we applied the cost minimization technique depending on environmental pollution. The fuzzy model parameters develop a triangular dense fuzzy mathematical model by stirring the demand rate and all cost items of the inventory management system as triangular dense fuzzy numbers. We reorganized the introduced model and analyzed it with an old methodology. This model has been investigated using crisp and triangular dense fuzzy. Finally, we get numerical results when the inventory cost function reaches its minimum. LINGO software and MATLAB software have been used to draw various graphs and numerical illustrations. At last, we support our model using sensitivity analysis with graphical presentations.

A new ranking method for trapezoidal intuitionistic fuzzy numbers and its application to multi-criteria decision making

Abstract: The ranking of intuitionistic fuzzy numbers is paramount in the decision making process in a fuzzy and uncertain environment. In this paper, a new ranking function is defined, which is based on Robust’s ranking index of the membership function and the non-membership function of trapezoidal intuitionistic fuzzy numbers. The mentioned function also incorporates a parameter for the attitude of the decision factors. The given method is illustrated through several numerical examples and is studied in comparison to other already-existent methods. Starting from the new classification method, an algorithm for solving fuzzy multi-criteria decision-making (MCDM) problems is proposed. The application of said algorithm implies accepting the subjectivity of the deciding factors, and offers a clear perspective on the way in which the subjective attitude influences the decision-making process. Finally, a MCDM problem is solved to outline the advantages of the algorithm proposed in this paper.

Interval Type-2 Fuzzy Neural Networks for Multi-Label Classification

Abstract: Prediction of multi-dimensional labels plays an important role in machine learning problems. We found that the classical binary labels could not reflect the contents and their relationships in an instance. Hence, we propose a multi-label classification model based on interval type-2 fuzzy logic. In the proposed model, we use a deep neural network to predict the type-1 fuzzy membership of an instance and another one to predict the fuzzifiers of the membership to generate interval type-2 fuzzy memberships. We also propose a loss function to measure the similarities between binary labels in datasets and interval type-2 fuzzy memberships generated by our model. The experiments validate that our approach outperforms baselines on multi-label classification benchmarks.

A Fuzzy Set Extension of Schelling's Spatial Segregation Model

Abstract: This study explores a possible segregation mechanism assuming fuzzy group membership. We construct a fuzzy set extension of Schelling's spatial segregation model. In the fuzzy Schelling model, each agent is assumed to have fuzzy membership in groups, which is typically assumed to represent the strength of the agent's group identity. The degree of membership is represented by the value of the membership function. The model assumes that agents want to be with agents with the same or stronger (less fuzzy) group identity than themselves. Agents decide whether to stay or move depending on whether their neighborhood satisfies their desires. Analyzing a series of simulations reveals that: First, the fuzzy Schelling model can reproduce segregation at the macro level; here, segregation is formed by the accumulation of agents' modest desires and actions. This is the most important property of the Schelling model. Second, agents' behavior and situation differ depending on the fuzziness of their membership. Notably, agents with less fuzzy membership play an important role in the system's equilibrium. Third, the tendency to reach equilibrium differs depending on the density of the space, required similarity level in the neighborhood, and initial distribution of membership values. Finally, we discuss the implications of the results.

Finite Difference Method for Intuitionistic Fuzzy Partial Differential Equations

Abstract: In this paper, we investigate an intuitionistic fuzzy Poisson equation with uncertain parameters, considering the parameters as intuitionistic fuzzy numbers. We apply a finite difference method to solve the ‘Intuitionistic fuzzy Poisson equation’. The continuity of the membership and non-membership functions (which imply the continuity of the hesitancy function) is used to obtain qualitative properties on regular α-cut and β-cut of the intuitionistic fuzzy solution. The fuzzification of the deterministic α-cut and β-cut solutions obtained lead to the intuitionistic fuzzy solution. Finally, an example is presented to illustrate the proposed methodology, as well as to show a graphical representation of its corresponding intuitionistic fuzzy solution.

Type-2 Fuzzy Set Approach to Image Analysis

Abstract: Segmentation is an essential task in image analysis process. Due to non-homogeneous intensities, blurred boundaries, noise and minimum of contrast it is a challenging task for image analysists. It has wide range of applications in all fields exclusively in the field medical imaging for disease deionization and early detection. The root cause for non-homogeneous intensities is uncertainty. Various tools have been introduced to handle uncertainty. We have introduced type-2 fuzzy based image segmentation process for edge detection in blurred areas of an image. When compared with classical fuzzy set, it has upper and lower membership values. Since it has more membership values it can handle higher level of uncertainty. In this chapter we have proposed equivalence function associated with strong negation relation which will address each intensity $${\mathcal{I}}_{t}$$ of an image $$\mathcal{I}$$ through the membership values. This method is verified with thermographic breast cancer image data set and the results were satisfactory.

Intuitionistic Trapezoidal Fuzzy based Aggregation Operator: Applications in Medical Diagnosis

Abstract: Since the inception of fuzzy sets given by Zadeh, uncertainty arises due to partial information or imprecise information has been measured. The generalized version of fuzzy sets has been introduced by Atanassov, known as intuitionistic fuzzy sets (IFSs), which have wide applications in decision making processes and consider both membership and non-membership functions. A generalized fuzzy based decision making method with aggregation operator has been discussed with application in medical diagnosis. In this paper, IF based method have been discussed for the application of the diagnosis of the type of child cancer.

Evaluation Model of Students’ Employability Based on Fuzzy Theory Algorithm

Abstract: The concept of “College Students’ employability” has been put forward with the increasing attention of all sectors of society. In recent years, with the enrollment expansion of major colleges and universities year after year, the number of graduates has also increased year after year. Graduates can’t find jobs after graduation, but enterprises can’t recruit people under the banner of high salary. This is the so-called employment gap. Therefore, this paper studies the evaluation model of students’ employability based on fuzzy theory algorithm. Fuzzy theory is a model used to evaluate the performance of students’ employability. It was first proposed by m.s.guttman and r.l.schafer, who developed it as a tool to evaluate the effectiveness of training programs (Guttman and Schafer, 1969). The fuzzy evaluation model is based on two elements: “fuzzy set” and “fuzzy rule”. Fuzzy sets are represented by a mathematical function called membership function, which can be applied to any data set under some conditions; Fuzzy rules are logical expressions that represent the judgment of membership.