Karnik-Mendel Algorithms Research Articles

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.

Study on Centroid Type‐Reduction of Interval Type‐2 Fuzzy Logic Systems Based on Noniterative Algorithms

Interval type‐2 fuzzy logic systems have favorable abilities to cope with uncertainties in many applications. While the block type‐reduction under the guidance of inference plays the central role in the systems, Karnik‐Mendel (KM) iterative algorithms are standard algorithms to perform the type‐reduction; however, the high computational cost of type‐reduction process may hinder them from real applications. The comparison between the KM algorithms and other alternative algorithms is still an open problem. This paper introduces the related theory of interval type‐2 fuzzy sets and discusses the blocks of fuzzy reasoning, type‐reduction, and defuzzification of interval type‐2 fuzzy logic systems by combining the Nagar‐Bardini (NB) and Nie‐Tan (NT) noniterative algorithms for solving the centroids of output interval type‐2 fuzzy sets. Moreover, the continuous version of NT (CNT) algorithms is proved to be accurate algorithms for performing the type‐reduction. Four computer simulation examples are provided to illustrate and analyze the performances of two kinds of noniterative algorithms. The NB and NT algorithms are superior to the KM algorithms on both calculation accuracy and time, which afford the potential application value for designer and adopters of type‐2 fuzzy logic systems.

A Comment on “A Direct Approach for Determining the Switch Points in the Karnik–Mendel Algorithm”

This letter is a supplement to the previous paper “A Direct Approach for Determining the Switch Points in the Karnik–Mendel Algorithm”. In the previous paper, the enhanced iterative algorithm with stop condition (EIASC) was shown to be the most inefficient in R. Such outcome is apparently different from the results in another paper in which EIASC was illustrated to be the most efficient in MATLAB. An investigation has been made into this apparent inconsistency and it can be confirmed that both the results in R and MATLAB are valid for the EIASC algorithm. The main reason for such phenomenon is the efficiency difference of loop operations in R and MATLAB. It should be noted that the efficiency of an algorithm is closely related to its implementation in practice. In this letter, we update the comparisons of the three algorithms in the previous paper, based on optimized implementations under five programming languages (MATLAB, R, Python, C, and Java). From this, we conclude that results in one programming language cannot be simply extended to all languages.

Study on centroid type-reduction of general type-2 fuzzy logic systems with weighted Nie–Tan algorithms

In recent years, researching on general type-2 fuzzy logic systems (GT2 FLSs) has become a hot topic as the development of alpha-planes representation of general type-2 fuzzy sets. The iterative Karnik–Mendel (KM) algorithms are used to perform the key block of type-reduction (TR) of GT2 FLSs. However, the KM algorithms are computationally intensive and time-consuming, which is not adapted to real-time applications. In the enhanced types of algorithms, the noniterative Nie–Tan (NT) algorithms decrease the computational cost greatly. Moreover, the closed-form Nie–Tan algorithms which calculate the outputs by averaging the lower and upper bounds of the membership functions have been proved to be actually an accurate algorithm for performing TR. The paper expands the NT algorithms to three different forms of weighted NT (WNT) algorithms according to the Newton–Cotes quadrature formulas of numerical integration techniques. Four computer simulation examples are adopted to analyze the performances of WNT algorithms when solving the type-reduction of general type-2 fuzzy logic systems. The proposed WNT algorithms have smaller absolute errors and faster convergence speed compared with NT algorithms, which provide the potential value for designers and adopters of GT2 FLSs.

General Type-2 Radial Basis Function Neural Network: A Data-Driven Fuzzy Model

This paper proposes a new General Type-2 Radial Basis Function Neural Network (GT2-RBFNN) that is functionally equivalent to a GT2 Fuzzy Logic System (FLS) of either Takagi-Sugeno-Kang (TSK) or Mamdani type. The neural structure of the GT2-RBFNN is based on the α-planes representation, in which the antecedent and consequent part of each fuzzy rule uses GT2 Fuzzy Sets (FSs). To reduce the iterative nature of the Karnik-Mendel algorithm, the Enhaned-Karnik-Mendel (EKM) type-reduction and three popular direct-defuzzification methods, namely the 1) Nie-Tan approach (NT), the 2) Wu-Mendel uncertain bounds method (WU) and the 3) Biglarbegian-Melek-Mendel algorithm (BMM) are used. Hence, this paper provides four different architectures of the GT2-RBFNN and their parametric optimisation. Such optimisation is a two-stage methodology that first implements an Iterative Information Granulation (IIG) approach to estimate the antecedent parameters of each fuzzy rule. Secondly, each consequent part and the fuzzy rule base of the GT2-RBFNN is optimised using an Adaptive Gradient Descent method (AGD) respectively. A number of popular benchmark data sets, the identification of a nonlinear system and the prediction of chaotic time series are considered. The reported comparative analysis of experimental results is used to evaluate the performance of the suggested GT2 RBFNN with respect to other popular methodologies.

Conjunctive water management under multiple uncertainties: A centroid-based type-2 fuzzy-probabilistic programming approach

In this study, a centroid-based type-2 fuzzy-probabilistic programming (CT2FP) approach is developed for supporting conjunctive use of surface water and groundwater under multiple uncertainties. CT2FP can not only tackle uncertainties expressed as type-2 fuzzy sets (in both objective function and constraints) but also address complexity with characteristic of randomness and two-layer fuzziness (i.e., type-2 fuzzy random variables). Solution method based on α-plane theory, enhanced Karnik–Mendel algorithm (EKM) and interactive algorithm are proposed to transform type-2 fuzzy-probabilistic constraints into their deterministic equivalents. A case study in Zhangweinan River Basin (China) is used to demonstrate the applicability of the proposed approach. Scenarios associated with different constraint-violation risk levels are examined to generate applicable cropping patterns and water-allocation schemes. The amount of groundwater used for irrigation can be determined (i.e., more than [462.84, 495.78] × 106 m3 in dry season and no more than [470.83, 537.19] × 106 m3 in wet season, respectively) to address the conflict between food security and ecological protection. The relationship among crop area, water allocation, and economic benefit can be reflected to enhance the agricultural sustainable development for the study basin.

A Direct Approach for Determining the Switch Points in the Karnik–Mendel Algorithm

The Karnik–Mendel algorithm is used to compute the centroid of interval type-2 fuzzy sets, determining the switch points needed for the lower and upper bounds of the centroid, through an iterative process. It is commonly acknowledged that there is no closed-form solution for determining such switch points. Many enhanced algorithms have been proposed to improve the computational efficiency of the Karnik–Mendel algorithm. However, all of these algorithms are still based on iterative procedures. In this paper, a direct approach based on derivatives for determining the switch points without multiple iterations has been proposed, together with mathematical proof that these switch points are correctly determining the lower and upper bounds of the centroid. Experimental simulations show that the direct approach obtains the same switch points, but is more computationally efficient than any of the existing (iterative) algorithms. Thus, we propose that this algorithm should be used in any application of interval type-2 fuzzy sets in which the centroid is required.

An Empirical Study on Design Partner Selection in Green Product Collaboration Design

Green production has become an important issue in PCD (Product Collaboration Design) for almost every enterprise, and will determine the sustainability of enterprises in the long term. The choice of design partner is a necessary condition in order to achieve green production. For the uncertain, fuzzy, and dynamic information such as unknown indices and weights, fuzzy semantics, and dynamic time factors in GPCD (Green Product Collaboration Design), a two-stage dynamic hybrid MADM (Multi-Attribute Decision Making) approach based on fuzzy DEMATEL (Decision-Making and Trial Evaluation Laboratory), fuzzy KMA (Karnik–Mendel Algorithm), and fuzzy VIKOR (VlseKriterjumska Optimizacija I Kompromisno Resenje) was proposed. In the first stage, fuzzy DEMATEL was used to determine the evaluation indices. Then, in the second stage, to accurately depict the dynamic information generated by the different phases of a product design, the dynamic evaluation method based on fuzzy theories was employed, and the weights of the indices were calculated by fuzzy KMA, then sorted by fuzzy VIKOR. Finally, a case study and a comparative analysis wre provided to illustrate the effectiveness of the proposed approach.

Study on centroid type-reduction of general type-2 fuzzy logic systems with weighted enhanced Karnik–Mendel algorithms

With the development of $$\alpha $$ -planes representation of general type-2 fuzzy sets (GT2 FSs), general type-2 fuzzy logic systems (GT2 FLSs) based on GT2 FSs have become a hot topic in academic field. While type-reduction (TR) is most critical block for a T2 FLS, generally speaking, the most popular Karnik–Mendel (KM) or enhanced KM (EKM) algorithms are used to perform the TR. The paper connects the EKM and the continuous version of EKM algorithms together and expands the EKM algorithms to three different forms of weighted EKM (WEKM) algorithms resort to the Newton–Cotes quadrature formulas of numerical integration techniques, while the EKM algorithms just become a special case of the WEKM algorithms. Four computer simulation examples are used to illustrate the performances of the WEKM algorithms. Compared with the EKM algorithms, the WEKM algorithms have smaller absolute error and faster convergence speed to compute the centroid defuzzified value of GT2 FLSs in general, which make them potentially applicable for T2 FLSs designers and adopters.

Improving the Speed of Center of Sets Type Reduction in Interval Type-2 Fuzzy Systems by Eliminating the Need for Sorting

In the deployment of interval type-2 fuzzy systems, one of the most important steps is the type reduction. The commonly used center of sets type reducer requires the solution of two nonlinear constrained optimization problems. Frequently used approaches to solve them are the Karnik–Mendel algorithms and their variants. However, these algorithms suffer from the need for sorting, which is known to be computationally very expensive. Using the reformulations proposed in this paper for center of sets type reducer, it is possible to eliminate the need for sorting. This makes interval type-2 fuzzy systems more appropriate for cost-sensitive real-time applications. Extensive simulations are presented to illustrate the faster convergence speed of the proposed method over six other enhanced variants of the Karnik–Mendel algorithm as applied to center of sets type reduction of interval type-2 fuzzy systems.

New Methodology to Approximate Type-Reduction Based on a Continuous Root-Finding Karnik Mendel Algorithm

Interval Type-2 fuzzy systems allow the possibility of considering uncertainty in models based on fuzzy systems, and enable an increase of robustness in solutions to applications, but also increase the complexity of the fuzzy system design. Several attempts have been previously proposed to reduce the computational cost of the type-reduction stage, as this process requires a lot of computing time because it is basically a numerical approximation based on sampling, and the computational cost is proportional to the number of samples, but also the error is inversely proportional to the number of samples. Several works have focused on reducing the computational cost of type-reduction by developing strategies to reduce the number of operations. The first type-reduction method was proposed by Karnik and Mendel (KM), and then was followed by its enhanced version called EKM. Then continuous versions were called CKM and CEKM, and there were variations of this and also other types of variations that eliminate the type-reduction process reducing the computational cost to a Type-1 defuzzification, such as the Nie-Tan versions and similar enhancements. In this work we analyzed and proposed a variant of CEKM by viewing this process as solving a root-finding problem, in this way taking advantage of existing numerical methods to solve the type-reduction problem, the main objective being eliminating the type-reduction process and also providing a continuous solution of the defuzzification.

Autocratic decision making using group recommendations based on interval type-2 fuzzy sets, enhanced Karnik–Mendel algorithms, and the ordered weighted aggregation operator

Autocratic decision making using group recommendations (ADMUGRs) is a very important research topic in the research area of multiple attributes group decision making (MAGDM), which automatically modifies the weights of decision makers (DMs) to get MAGDM results. In this paper, a new method for ADMUGRs is proposed based on interval type-2 fuzzy sets (IT2FSs), enhanced Karnik-Mendel (EKM) algorithms, and the ordered weighted aggregation (OWA) operator. Each DM uses linguistic terms represented by IT2FSs to evaluate the attributes of alternatives, where different attributes can have different weights in [0, 1]. The evaluating matrices (EMs) and the weighting vectors (WVs) of the attributes given by the DMs are assessed by linguistic terms represented by IT2FSs. The proposed method can overcome the shortcomings of Chen and Hong's method (2014), Chen and Lee's method (2010) and Cheng et al.’s method (2016) for ADMUGRs in IT2FSs environments.

Voltage sag assessment using type‐2 fuzzy system considering uncertainties in distribution system

Power quality (PQ) is the key concern in every corner of the world today. Regarding PQ voltage sag is one of the most relevant issues. The causes of faults in distribution system resulting voltage sag are impact of wind speed on conductors and attachments, faults caused by cattle, animal, mice, rat, birds, bat, snakes and vagaries of weather seems to be definite but found unpredictable in quantum. Considering the intensity and the impacts of these uncertainties as an input interval type-2 fuzzy system is applied to ascertain fault rates of individual lines in distribution systems. The fault rate of each line is calculated using type reduction Karnik–Mendel algorithm. Selection of the location of fault is done by Monte Carlo simulation to assess voltage sag in each bus of the distribution system. This approach is applied to IEEE 30-bus system and Barak Valley 37-bus distribution system for validation.

Using genetic algorithms to evolve type-2 fuzzy logic systems for predicting bankruptcy

PurposeType-2 fuzzy sets became attractive in practice because of their footprint of uncertainty that gives them more degrees of freedom. This paper aims to use genetic algorithms (GAs) to design an interval Type-2 fuzzy logic system (IT2FLS) for the purpose of predicting bankruptcy.Design/methodology/approachThe shape of type-2 membership functions, the parameters giving their spread and location in the fuzzy partitions and the set of fuzzy rules are evolved at the same time by encoding all together into the chromosome representation. The enhanced Karnik–Mendel algorithms are used for the centroid type-reduction and defuzzification stage. The performance in predicting bankruptcy is evaluated by benchmarking IT2FLSs against type-1 FLSs. The experimental setup consists of evolving 100 configurations for both the T1FLS and IT2FLS and comparing their in-sample and out-of-sample average accuracy.FindingsThe experiments confirm that representing and capturing uncertainty with more degrees of freedom is an important advantage. It is this extra potential of IT2FLSs that allows them to outperform T1FLS, especially in terms of generalization capability.Originality/valueThe strategy followed in this paper is to train an IT2FLS from scratch rather than tuning the parameters of an existing T1FLS. Because this leads to solving a mixed integer optimization problem, the GA-based approach is specifically designed and uses genetic operators that are most suited for such a case: tournament selection, extended Laplace crossover and power mutation. Finally, the trained IT2FLS is applied to bankruptcy prediction, and its generalization capability is compared with related techniques.

Revisiting Karnik–Mendel Algorithms in the framework of Linear Fractional Programming

For Interval Type-2 (IT2) fuzzy sets and systems, calculating the centroid and performing type reduction are operations that must be taken into consideration. The Karnik–Mendel Algorithm (KMA) and its enhancements have been usually employed to perform these operations. In KMAs, these IT2 fuzzy operations are defined as nonlinear optimization problems which are solved iteratively by finding the optimal Switching Points (SPs). In this study, we will examine these optimization problems and will reveal the nature of KMAs in the framework of Linear Fractional Programming (LFP). In this context, we will firstly transform IT2 fuzzy operations into 0-1 LFP problems and then we will show that there exists a direct relationship between the SPs of the KMAs and the solution vectors of the defined LFP problems. Hence, the meaning of the SPs will be revealed in the framework of LFP theory. Thus, it will be concluded that the KMAs can be seen as LFP methods. Then, by taking account of the LFP representations, we will show that it is possible to perform the IT2 fuzzy operations via well-known LFP methods, namely the Charnes Cooper Transformation and the Dinkelbach's Algorithm (DA). It is believed that the presented LFP approaches will be very helpful in employing IT2 fuzzy sets and systems in different programming languages since they only use and employ built-in Linear Programming (LP) routines. However, since LP solvers are used, it will be concluded that the LFP based approaches are computationally intensive and thus might not be feasible for real-time applications. Therefore, we will present a computationally efficient implementation of presented the DA based method, namely the KMK Algorithm (KMKA). Then, through analytical investigations on the solution vectors and optimality conditions of the KMKA, it will be revealed that the KMA is algorithmic equivalent to the presented implementation of the DA based KMKA. Thus, it will be revealed that the commonly employed KMA and its enhancements can be seen as efficient implementations of the DA in the framework of LFP. Based on this connection, since the DA is an adaptation of the Newton method to a non-differentiable context, it will be also concluded that the discrete version of KMA can be also seen equivalent to a root-finding problem. We will present comparative numerical results to validate the analytical derivations and observations.

Modification on enhanced Karnik–Mendel algorithm

Karnik–Mendel(KM) algorithm and its enhancements are among the most popular type reduction algorithms in the literature. Enhanced KM (EKM) algorithm is computationally fast and can quickly locate left and right switch points in just a few iteration. This paper proposes a subtle yet very effective modification to EKM algorithm to further improve its computational requirement. The modification relates to how the initial right switch point is determined. Comprehensive simulation results for different cases and scenarios provide the statistical proof for the validity of conclusions drawn on the superiority of the proposed initialization compared to initialization used in the original EKM algorithm. The superiority is quantitatively measured in the number of saved iterations and convergence speed.

Design of a qualitative classification model through fuzzy support vector machine with type‐2 fuzzy expected regression classifier preset

Methods of qualitative analysis, such as qualitative classification, have gained importance as an essential complement of existing quantitative analysis in numerous fields. Only a few models have been developed to deal with qualitative inputs in the form of type‐2 fuzzy(T2F) sets properly, given that traditional defuzzification method like the Karnik–Mendel algorithm performs dimensionality reduction at the cost of loss of information. To improve the situation, we define the expected value and variance of T2F set in this paper. By using a combination of them, we transfer the vertical three‐dimensional uncertainty of T2F set to horizontal range uncertainty without much distortion of information. Additionally, current classification models are unsuitable to the partial classification problem if an output is not fully assigned to a single class. We build a comprehensive qualitative classification model based on fuzzy support vector machine (FSVM) combined with type‐2 fuzzy expected regression (FER) to solve the partial classification problem as mentioned. This classifier (i.e. FER‐FSVM) makes it possible to achieve the discrimination of output while characterizing membership for each class in terms of multidimensional qualitative inputs (attributes) in the form of T2F sets. FER‐FSVM also can self‐learn the data structure and shift between FER or FSVM for classification automatically, thus largely improving the efficiency of the classification process. The new model is almost 7 times more efficient than FSVM, as shown by our empirical experiments. © 2016 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc.

An algorithmic study of relative cardinalities for interval-valued fuzzy sets

The main topic of this paper is the notion of relative cardinality for interval-valued fuzzy sets – its definition, properties and computation. First we define relative cardinality for interval-valued fuzzy sets following the concept of uncertainty modelling given by Mendel's Wavy-Slice Representation Theorem. We expand on previous approaches by considering relative cardinality based on different t-norms and scalar cardinalities and we initiate an investigation of its properties and possible applications. Drawing on the Nguyen–Kreinovich and Karnik–Mendel algorithms, we propose efficient algorithms to compute relative cardinality depending on a chosen t-norm. This seems to be the first such broad and consistent analysis to have been made of relative cardinality for interval-valued fuzzy sets. As a promising application we consider using interval-valued relative cardinality to construct the family of parameterised subsethood measures.

Forecasting studies by designing Mamdani interval type-2 fuzzy logic systems: With the combination of BP algorithms and KM algorithms

A type of Mamdani interval type-2 fuzzy logic systems is designed for historical data based forecasting problem in the paper. In the Mamdani interval type-2 fuzzy logic systems design, the antecedent, consequent, and input measurement primary membership functions of type-2 fuzzy sets are chosen as Gaussian type-2 membership functions with uncertain standard deviation. Some excellent elementary vectors and partitioned matrices are used to combine Karnik–Mendel (KM) algorithms with back propagation (BP) algorithms by matrix transformation, and the challenging mission of computing derivatives in such systems can be solved. The parameters of the proposed type-2 fuzzy logic systems are also tuned. Two examples, including the historical competition data of European Network on Intelligent Technologies (EUNITE) (three o’clock from January 1, 1997 to December 9, 1998) and the price data of West Texas Intermediate (WTI) crude oil (from January 3, 2011 to December 30, 2011) are used to test traditional linear time series forecasting methods and more advanced fuzzy logic systems forecasting methods. Monte Carlo simulation studies and convergence analysis are employed to illustrate the effectiveness of the proposed type-2 fuzzy logic systems methods compared with their type-1 counterparts methods for forecasting.

Comparative Study of Type-1 Fuzzy Logic and Type-2 Fuzzy Logic

Medical diagnosis is a complex process which can be attributed to the complexities, uncertainties and vagueness of the symptoms involved, and sometimes also because of their complex relationship with the final diagnosis output. Traditional systems for diagnosis very often incorporate certain inabilities that eventually lead to the vagueness in the result. Besides this, imprecise and incomplete knowledge are difficult for these traditional disease diagnosis expert systems to analyze. The fuzzy logic has carved a niche in medical diagnosis, for its ability to handle the dynamic nature of the disease diagnosis and medication. Various approaches of Fuzzy Logic, namely, Type-1 Fuzzy Logic, Interval Type-2 Fuzzy Logic, and General Type-2 Fuzzy Logic are being used for decision making in medical diagnosis. In this paper, a comparative study of the various parameters of Type-1 Fuzzy Logic and Interval type-2 Fuzzy Logic is conducted to understand their respective advantages in the medical diagnosis. Former, being a standard fuzzy logic methodology has been used widely for diagnosis of almost every disease, and the latter, which is also known as ' Layered Type-1 Fuzzy Logic', is being widely used for the diagnosis of a few diseases only. Type-1 Fuzzy Logic is rather a simple approach and results in the fast generation of outputs, but Type-2 Fuzzy Logic can provide better results in many cases. A study is conducted on type-2 diabetes and heart related diseases, to understand the disease-specific nature of the two approaches. Type-2 Fuzzy Logic uses Karnik-Mendel (K-M) algorithm for type reduction. The comparison is drawn on the basis of accuracy, rule base and the differences of their outputs. In this way, this analysis helps to understand the advantages and disadvantages of both the approaches in the medical diagnosis. General Terms Type-1 Fuzzy Logic (T1FL), Interval Type-2 Fuzzy Logic (T2FL), Type-2 diabetes, Probability.