Secondary Membership Function Research Articles

An augmented fuzzy decision support system to analyse compatible cosmetic face masks for various complexions

AbstractBeauty face masks (BFM) are becoming increasingly popular among both men and women since they provide quick refreshment and nurture the skin. Given the wide range of skin types and the chemicals used in their formulation, it can be difficult to find a product that not only complements the skin type but is also free of potentially harmful ingredients that could endanger the consumer's health. When dealing with ambiguous situations, the multi‐attribute decision making (MADM) approach combined with fuzzy set theory is more effective. Type‐2 fuzzy sets (T2FS) provide greater flexibility in dealing with uncertainty in real‐world issues since they are characterised by a main and secondary membership function. In this research, we present the innovative idea of type‐2 linear diophantine fuzzy set (T2LDFS) as an intriguing tool for capturing expert reluctance about an issue. For analysing the discussed problem, a hybrid fuzzy VIKOR enhanced with the proposed fuzzy logic is suggested. A sensitivity and comparative analysis is carried out to establish the validity of the recommended approach.

The Enhanced Wagner–Hagras OLS–BP Hybrid Algorithm for Training IT3 NSFLS-1 for Temperature Prediction in HSM Processes

This paper presents (a) a novel hybrid learning method to train interval type-1 non-singleton type-3 fuzzy logic systems (IT3 NSFLS-1), (b) a novel method, named enhanced Wagner–Hagras (EWH) applied to IT3 NSFLS-1 fuzzy systems, which includes the level alpha 0 output to calculate the output y alpha using the average of the outputs y alpha k instead of their weighted average, and (c) the novel application of the proposed methodology to solve the problem of transfer bar surface temperature prediction in a hot strip mill. The development of the proposed methodology uses the orthogonal least square (OLS) method to train the consequent parameters and the backpropagation (BP) method to train the antecedent parameters. This methodology dynamically changes the parameters of only the level alpha 0, minimizing some criterion functions as new information becomes available to each level alpha k. The precursor sets are type-2 fuzzy sets, the consequent sets are fuzzy centroids, the inputs are type-1 non-singleton fuzzy numbers with uncertain standard deviations, and the secondary membership functions are modeled as two Gaussians with uncertain standard deviation and the same mean. Based on the firing set of the level alpha 0, the proposed methodology calculates each firing set of each level alpha k to dynamically construct and update the proposed EWH IT3 NSFLS-1 (OLS–BP) system. The proposed enhanced fuzzy system and the proposed hybrid learning algorithm were applied in a hot strip mill facility to predict the transfer bar surface temperature at the finishing mill entry zone using, as inputs, (1) the surface temperature measured by the pyrometer located at the roughing mill exit and (2) the time taken to translate the transfer bar from the exit of the roughing mill to the entry of the descale breaker of the finishing mill. Several fuzzy tools were used to make the benchmarking compositions: type-1 singleton fuzzy logic systems (T1 SFLS), type-1 adaptive network fuzzy inference systems (T1 ANFIS), type-1 radial basis function neural networks (T1 RBFNN), interval singleton type-2 fuzzy logic systems (IT2 SFLS), interval type-1 non-singleton type-2 fuzzy logic systems (IT2 NSFLS-1), type-2 ANFIS (IT2 ANFIS), IT2 RBFNN, general singleton type-2 fuzzy logic systems (GT2 SFLS), general type-1 non-singleton type-2 fuzzy logic systems (GT2 NSFLS-1), interval singleton type-3 fuzzy logic systems (IT3 SFLS), and interval type-1 non-singleton type-3 fuzzy systems (IT3 NSFLS-1). The experiments show that the proposed EWH IT3 NSFLS-1 (OLS–BP) system presented superior capability to learn the knowledge and to predict the surface temperature with the lower prediction error.

Structure Analysis of General Type-2 Fuzzy Controller and Its Application

In this paper, a general type-2 fuzzy controller is proposed. The proposed general type-2 fuzzy controller is based on α-plane representation, which converts the centroid of general type-2 fuzzy sets to centroids of several interval type-2 fuzzy sets. Also, the mathematical expression of general type-2 fuzzy controller is derived by fuzzy product operate and NT type reduction. By the mathematical expression, the relation between general type-2 fuzzy controller output with parameter of triangular secondary membership function is discussed. Finally, the control performances of proposed general type-2 fuzzy controller are compared with conventional PID controller, tradition type-1 fuzzy controller, and interval type-2 fuzzy controller. Furthermore, to show the effective and practical of the proposed controller, a nonlinear inverted pendulum system is tested. The simulation results show that general type-2 fuzzy controller achieves better control efforts than other compared controllers.

A class of general type-2 fuzzy controller based on adaptive alpha-plane for nonlinear systems

This study introduces a general type-2 Takagi–Sugeno–Kang fuzzy controller (GT2-TSKFC) for controlling uncertain systems. The proposed GT2-TSKFC uses equidistant type-2 triangular membership functions (MFs) for the antecedents, Larsen’s implication method, type-1 fuzzy sets for the consequent parameters, and a direct defuzzification method. The analytical structure of the proposed controller indicates that the alpha-plane and apex of the secondary MFs have a noticeable effect on calculating the control signal. Adaptation of the alpha-plane and apex of the secondary MFs is performed using the Lyapunov function to achieve the stability of the controlled system. The proposed controller is applied to an uncertain nonlinear inverted pendulum system. The results of the proposed control algorithm are compared with those of a general type-2 fuzzy controller with a specific number of alpha-planes, a quasi type-2 fuzzy controller, an interval type-2 fuzzy controller, and a type-1 fuzzy controller to demonstrate the robustness and effectiveness of the proposed scheme.

An Approach Towards the Design of Interval Type-3 T–S Fuzzy System

This article providesa systematic approach for the design of an interval type-3 (IT3) Takagi–Sugeno (T–S) fuzzy logic system (FLS) using <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha $</tex-math></inline-formula> - plane representation. An IT3 FLS is designed with the baseline of the general type-2 (GT2) FLS in a similar manner as an IT2 FLS was designed from the baseline of type-1 FLS. Hence, IT3 FLS evolved as a successor of GT2 FLS, where secondary membership function is an interval type-2 fuzzy set (FS), and values of tertiary membership are unity over the footprint of uncertainty of secondary membership. This extra degree of freedom in IT3 FLS provides better modeling capability as compared to GT2 FLS in the presence of a high degree of uncertainty and vagueness. The proposed system will be more appealing while dealing with uncertain information or data, which is supposed to be generated from uncertain sources; i.e., there exist uncertainties even in the presence of uncertainty. The computations needed for the design of IT3 FLS are derived using IT2 FS and GT2 FS mathematics. The design algorithms adopted for the baseline IT2 T–S fuzzy system are as per the modified interval type-2 fuzzy c-regression model clustering algorithm and hyper-plane-shaped membership function. The proposed methodology is applied to several benchmark examples and obtained results are compared with recently developed fuzzy modeling methods having a comparable number of rule bases. The proposed IT3 T–S FLS shows good performance in terms of accuracy when data is corrupted by noise and uncertainties related to missing or unvarying data exist. The computational cost is linear with design parameters and by optimum choice of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\alpha $</tex-math></inline-formula> -planes, it is still bearable considering advantages and nature of applications.

Accurate Prediction Using Triangular Type-2 Fuzzy Linear Regression: Simplifying Complex T2F Calculations

Many studies have been performed to handle the uncertainties in the data using type-1 fuzzy regression (FR). Few type-2 fuzzy (T2F) regression studies have used interval type-2 (IT2) for indeterminate modeling using type-1 fuzzy membership. The current article proposes a triangular T2F regression (TT2FR) model to ameliorate the efficiency of the model by handling the uncertainty in the data. The triangular secondary membership function is used instead of widely used interval-type models. In the proposed model, vagueness in primary and secondary fuzzy sets is minimized, and also, a specified <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> -plane of the observed value is included in the same <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">${\alpha}$</tex-math></inline-formula> -plane of the predicted value. Complex calculations of the T2F model are simplified by reducing the 3D T2F set into 2D IT2 fuzzy models.

An improved fuzzy sorting algorithm coupling bi-level programming for synergetic optimization of agricultural water resources: A case study of Fujian Province, China

The uneven allocation of water resources and the shortage of regional water resources pose great challenges to the economic development and regional development balance of the Fujian province. Optimizing the water allocation structure in different regions can effectively alleviate water pressure. In this study, a type-2 fuzzy bi-level programming (T2FBL) method is proposed to plan the agricultural water resource system in the Fujian province. This method uses an improved fuzzy sorting algorithm to deal with uncertain parameters in the system and combines the bi-level programming method to balance the trade-off between two levels of decision-makers, the uncertain information contained in the secondary membership function omitted in the interval type-2 fuzzy theory is considered in the new ranking algorithm. Multiple scenarios related to different food security needs and different risk indices are examined. The major findings are as follows: (i) With an average tolerance of 75%, the average gross agricultural output value under various scenarios increased (0.4% ∼ 7%) (average 3.89%) after optimization. (ii) The regional water allocation scheme under different food demands and different water availability scenarios is calculated, and the results show that prioritizing adjustments to the industrial water distribution structure of Fuzhou and Zhangzhou will greatly relieve the water pressure in the Fujian province. (iii) The relationship between the availability of system water resources and economic benefits is given through the calculation results of the T2FBL model. These findings can provide an in-depth understanding of the interaction between agricultural, industrial and tertiary industry water allocation and provide technical support for agricultural water resource planning issues.

An Improved General Type-2 Fuzzy Approach for Planning Energy Resources Systems with Air Quality Requirements

Due to economic development and the improvement of people’s living standards, questions that the imbalance of energy supply and demand as well as the environmental damage have become increasingly prominent. To study the contradiction between energy and environment, and considering there are a lot of uncertain factors in the actual planning, this paper improves a chance-constrained interval type-2 fuzzy method used in linear programming and extends the interval type-2 fuzzy number to the general type-2 fuzzy number by introducing the secondary membership function that makes up for the loss of uncertainty due to the fixed value of the second membership function and optimizes the accuracy of the probability in the original method. By applying the new method to the energy planning system model, an optimal planning plan with the goal of reducing air pollutant emissions as well as the lowest overall operating cost can be obtained. Compared with the interval fuzzy method, the new method proposed in this paper is able to deal with the uncertainty in the system better, and provides more accurate solutions under different satisfaction levels.

Towards Systematic Design of General Type-2 Fuzzy Logic Controllers: Analysis, Interpretation, and Tuning

This article aims to provide a new perspective on how the deployment of general type-2 (GT2) fuzzy sets affects the mapping of a class of fuzzy logic controllers (FLCs). It is shown that an α-plane represented a GT2-FLC is easily designed via baseline type-1 and interval type-2 FLCs and two design parameters (DPs). The DPs are the total number of α planes and the tuning parameter of the secondary membership function that are interpreted as sensitivity and shape DPs, respectively. We provide a clear understanding and interpretation of the sensitivity and shape DPs on controller performance through various comparative analyses. We present design approaches on how to tune the shape DP by providing a tradeoff between robustness and performance. We also propose two online scheduling mechanisms to tune the shape DP. We explore the effect of the sensitivity DP on the GT2-FLC and provide practical insights on how to tune the sensitivity DP. We present an algorithm for tuning the sensitivity DP that provides a compromise between computational time and sensitivity. We validate our analyses, interpretations, and design methods with experimental results conducted on a drone. We believe that this article provides clear explanations on the role of DPs on the performance, robustness, sensitivity, and computational time of GT2-FLCs.

Triangular Fuzzy-Rough Set Based Fuzzification of Fuzzy Rule-Based Systems

Abstract In real-world approximation problems, precise input data are economically expensive. Therefore, fuzzy methods devoted to uncertain data are in the focus of current research. Consequently, a method based on fuzzy-rough sets for fuzzification of inputs in a rule-based fuzzy system is discussed in this paper. A triangular membership function is applied to describe the nature of imprecision in data. Firstly, triangular fuzzy partitions are introduced to approximate common antecedent fuzzy rule sets. As a consequence of the proposed method, we obtain a structure of a general (non-interval) type-2 fuzzy logic system in which secondary membership functions are cropped triangular. Then, the possibility of applying so-called regular triangular norms is discussed. Finally, an experimental system constructed on precise data, which is then transformed and verified for uncertain data, is provided to demonstrate its basic properties.

Mathematical analysis of a simplified general type-2 fuzzy PID controller

For the type reduction of general type-2 fuzzy PID controller is time consuming and the mathematical expression of general type-2 fuzzy PID controller is difficult to derived. So, a simplified general type-2 fuzzy PID (SGT2-FPID) controller is studied in this article. The SGT2-FPID controller adopts triangular function as the primary and secondary membership function. Then the primary membership degree of apex for the secondary membership degree will be applied to get the output of SGT2-FPID controller, which can reduce the computation complexity of general type-2 fuzzy controller type reduction. Furthermore, the mathematical expressions of SGT2-FPID controller, type-1 fuzzy PID controller and interval type-2 fuzzy PID controller are discussed. Finally, 4 plants are applied to demonstrate the effectiveness and robustness of SGT2-FPID controller. The simulation results show that when the plants have uncertainty in model structure, measurement and external disturbance, the SGT2-FPID controller can achieve better control performances in contrast to compared controllers.

Toward a Fuzzy Logic System Based on General Forms of Interval Type-2 Fuzzy Sets

Recent years have witnessed a widespread in the use of interval type-2 fuzzy logic systems (IT2 FLSs) in real-world applications. It has been shown recently that interval type-2 fuzzy sets (IT2 FSs) are more general than interval-valued fuzzy sets (IV FSs) [1] . Hence, there is a need to explore the capabilities of the more general forms of IT2 FSs (beyond IV FSs) and the applications areas they will be more suitable for. In addition, there is a need to develop the theory of the general forms of IT2 FLSs (gfIT2 FLSs), which employ IT2 FSs that are not equivalent to IV FSs and can have nonconvex secondary membership functions (MFs). Although these systems could be considered within the scope of general type-2 FLSs (GT2 FLSs), the practical implementation of GT2 FLSs has traditionally required the secondary MFs to be convex and normal type-1 fuzzy sets (T1 FSs). In addition, the type-reduction operation still presents a challenge for GT2 FLSs because of its computational complexity. In this paper, we present a complete framework for a type-2 FLS that uses the most recent perception of IT2 FSs (the so called general forms of interval type-2 fuzzy sets, gfIT2 FSs), whose secondary grades can be nonconvex T1 FSs. This framework includes new equations for the meet and join operations of gfIT2 FSs, as well as a new type reduction procedure for the type-2 FLS involving gfIT2 FSs. In addition, we present the type-2 FLS operation for singleton and nonsingleton fuzzification. We will introduce the various operations employed within a gfIT2 FLSs, from fuzzification (including singleton and nonsingleton) to inference, type-reduction, and defuzzification. We will also present two examples in which these gfIT2 FSs arise naturally when modeling sonar sensors input noise and the antecedents/consequents from a survey including different users.

A hybrid design of shadowed type-2 fuzzy inference systems applied in diagnosis problems

Computer aided systems have been frequently used in recent times and the evolution of artificial intelligence allows the inclusion of this kind of systems in most complex problems, for example in diagnosis problems. Some of the most powerful algorithms or methods that have been applied to diagnosis problems are: Artificial Neural Networks, Support Vector Machines, Decision Threes, Fuzzy Inference Systems and hybrids of these algorithms. The present paper explores the applications of a special approximation of the Type-2 Fuzzy Inference System, which is the Shadowed Type-2 Fuzzy Inference System, and the reason for using this approach (and not another one) is because it provides a good approximation to General Type-2 Fuzzy Inference Systems, but with a computational cost reduction. Shadowed Type-2 Fuzzy Inference Systems are inspired on the Shadows Sets that simplify the traditional Type-1 Membership Function using two optimal α-cuts, which in our case are applied in the secondary membership function of the General Type-2 Fuzzy Inference System (modeled by the α-planes representation) in order to model the system with only two α-planes. Experiments were realized with eleven benchmark datasets in order to evaluate the accuracy of the proposed system. The obtained results demonstrate the advantages of using this approach over the conventional General Type-2 Fuzzy Inference Systems, and this is because we obtain better performance in most of the cases and with less computational resources.

A concrete and rational approach for building type-2 fuzzy subsethood and similarity measures via a generalized foundational model

Subsethood and similarity between fuzzy sets have always been intensely studied concepts in fuzzy set theory (FST). However, researches on subsethood and similarity for truly general type-2 fuzzy sets (T2FSs) have been comparatively scarce, because of the intrinsic difficulties of directly dealing with the secondary membership functions of very general nature. While the advent of the α-plane/z-slice representation by Mendel and his colleagues as well as by Wagner and Hagras has led to progress in confronting this challenge, there remains quite a number of limitations and unsolved issues. The contribution of this article is to utilize a generalized foundational model (introduced in Ngan, 2018) to construct T2FS subsethood and similarity measures as rationally, concretely and systematically as feasible, such that (i) these T2FS measures are applicable to truly general type-2 fuzzy sets, that (ii) the actions of these measures can be very simply understood, analyzed and even customized by the T2FS users, and that (iii) these T2FS measures can process and output results that appropriately maintain and reflect the high degree of fuzziness involved in T2FSs. Last but not least, for applications, (iv) these measures will be demonstrated on multiple criteria decision making and pattern recognition problems, and (v) in a brief sketch, we will illustrate how the generalized-foundational-model-based method of building T2FS subsethood and similarity measures can be adapted to building other T2FS measures that embrace the advantages described in (i), (ii) and (iii).

An improved general type-2 fuzzy sets type reduction and its application in general type-2 fuzzy controller design

The Karnik–Mendel (KM) or the enhanced Karnik–Mendel (EKM) algorithm is widely used for interval type-2 fuzzy sets type reduction in many applications. Compared with iterative procedures of KM/EKM, an iterative algorithm with a stop condition or an enhanced iterative algorithm with a stop condition based on the KM algorithm that converges monotonically is more efficient. In this article, a new iterative algorithm with stop condition type reduction for interval type-2 fuzzy sets is proposed, in which switch points are initialized and unidirectional search is performed based on monotone properties of the KM algorithm. Furthermore, the proposed algorithm is applied to general type-2 fuzzy sets type reduction based on α-plane representation. The experimental results of a triangular and gaussian secondary membership function show practicality and efficiency of this method. In accordance with the conventional PI, type-1 or interval type-2 fuzzy controller is difficult to achieve a desired control effect for steam temperature at collector outlet of trough solar thermal power generation system with large time delay, strong inertia and parameter time-variation, and a general type-2 fuzzy controller with more adjustable controller parameters is proposed in this article. In different working conditions, the proposed controller can reduce system overshoot and ensure system stability. Moreover, when the working condition changes, the controller can solve a model mismatch problem under same controller parameters and has faster response.

Forecasting by designing Mamdani general type-2 fuzzy logic systems optimized with quantum particle swarm optimization algorithms

Much more attention has been focused on studying and applying general type-2 fuzzy logic systems (GT2 FLSs) in recent years. The paper designs a type of Mamdani GT2 FLS for studying forecasting problems based on the data of permanent magnetic drive (PMD) loss. During the system design process, we choose the primary membership functions (MFs) of antecedent, consequent and input measurement general type-2 fuzzy sets (GT2 FSs) as Gaussian type MFs with uncertain standard deviations. The corresponding vertical slices (secondary MFs) are chosen as the triangle MFs. All the parameters of Mamdani GT2 FLSs are optimized by the quantum particle swarm optimization (QPSO) algorithms. Noisy data of PMD loss are adopted for both training and testing the proposed FLSs forecasting approaches. Simulation studies and convergence analysis are employed to show the effectiveness and feasibility of the proposed GT2 FLSs forecasting methods compared with their T1 and IT2 counterparts.

Comparing the Performance Potentials of Interval and General Type-2 Rule-Based Fuzzy Systems in Terms of Sculpting the State Space

This paper provides application-independent perspectives on why improved performance usually occurs as one goes from an interval type-2 (IT2) fuzzy system to a general type-2 (GT2) fuzzy system. This is achieved by using the horizontal-slice representation of a GT2 fuzzy set and GT2 fuzzy system and by examining first- and second-order rule partitions as well as novelty partitions for the horizontal slices. It demonstrates that, for triangle and trapezoid secondary membership functions, the numbers of first- and second-order rule partitions are exactly the same for IT2 and GT2 fuzzy systems, but that a maximum amount of change always occurs in every second-order rule partition of a GT2 fuzzy system. This does not always occur in such partitions of an IT2 fuzzy system. Furthermore, when type reduction (TR) is used in a GT2 fuzzy system, the total number of novelty partitions is directly proportional to the number of horizontal slices; consequently, there are many more such partitions in a GT2 fuzzy system that uses TR than occur in an IT2 fuzzy system that also uses TR. It is the author's conjecture that it is the maximum changes that occur in every second-order rule partition, as well as the greater number of novelty partitions when TR is used, that provide a GT2 fuzzy system with the potential to outperform an IT2 fuzzy system.

A fast analytical approximation type-reduction method for a class of spiked concave type-2 fuzzy sets

In this paper, an analytical type-reduction method called concave analytical type-reduction method with spikes (CATRS) of two kinds of spiked concave type-2 fuzzy sets is proposed. The concave type-2 fuzzy sets which are considered in the paper include triangular type-2 fuzzy set, trapezoidal type-2 fuzzy set and its spiked versions. The analytical type-reduction method is free of the following steps, such as discourse partition and discourse refinement for algorithms' discrete implementation, thus its calculation complexity is reduced. To obtain the formulae of concave triangular type-2 fuzzy set, the method proposed by Starczewski is extended from the convex case to the concave one, secondary membership function with spikes is considered as a special form of triangular fuzzy set. Centroid set is formed by all the type-1 fuzzy sets that is derived from the convex part of the primary membership with or without spikes at a given discourse partition point. Moreover, the method proposed for concave triangular type-2 fuzzy set is extended to concave trapezoidal type-2 fuzzy set with a trapezoidal fuzzy number approximation operator. The proposed analytical type-reduction methods can address a more general class of type-2 fuzzy sets, it is more convenient for type-2 fuzzy modeling and inference, and the application of type-reduction method to variable universe of discourse controller design shows that the approximation method is applicable for the adaptive fuzzy controller design.

An approach for parameterized shadowed type-2 fuzzy membership functions applied in control applications

At present time, general type-2 fuzzy logic provides better uncertainty modeling when compared to interval type-2 fuzzy logic, and this has been shown with the improvement in the performance of fuzzy logic controllers and fuzzy edge detectors. However, higher computational cost represents a limitation for many applications and implementation platforms, and it is because of this that the computational cost reduction is the main goal of this paper. The aim of this work is to model a GT2 FLS based on shadowed type-2 fuzzy sets (ST2 FS) and apply this approach to control problems. The ST2 FS consists of approximating the secondary membership function by shadowed sets and is because of this, that we present an analytic approach to obtain the shadowed set for triangular and Gaussian membership functions for any parameters. Based on this, we achieve a representation of the ST2 FS as a parameterized function, facilitating its implementation. The results are compared against the performance when a generalized type-2 fuzzy inference system is approximated using α-planes for control applications.

A New Method for Parameterization of General Type-2 Fuzzy Sets

ABSTRACTIn this paper a new method for the parameterization of general type-2 fuzzy membership functions. The proposed method describes the methodology, equations and pseudo-code for building a set of general type-2 membership functions, which are a combination of two Gaussian-type primary membership functions (Gaussian with uncertain mean, and Gaussian with uncertain standard deviation), with multiple combinations of secondary membership functions (Gaussian, double Gaussian, general bell and trapezoidal). In addition, several application cases are used to illustrate the advantages of the proposed parameterization of general type-2 fuzzy sets; where the membership functions are designed using the parameterization approach and the general type-2 inference system is approximated using the α-planes theory. Simulation results illustrate that the parameterization offers an efficient way to represent these fuzzy sets. The main idea of the approach is to facilitate the use of general type-2 fuzzy systems in real world applications. The main contribution is a proposed new form of parameterizing general type-2 fuzzy sets that simplifies the efficient design of this type of sets.