Karnik-Mendel Algorithms Research Articles

Study on sensible beginning divided-search enhanced Karnik-Mendel algorithms for centroid type-reduction of general type-2 fuzzy logic systems

<abstract> <p>General type-2 fuzzy logic systems (GT2 FLSs) on the basis of alpha-plane representation of GT2 fuzzy sets (FSs) have attracted considerable attention in recent years. For the kernel type-reduction (TR) block of GT2 FLSs, the enhanced Karnik-Mendel (EKM) algorithm is the most popular approach. This paper proposes the sensible beginning divided-search EKM (SBDEKM) algorithms for completing the centroid TR of GT2 FLSs. Computer simulations are provided to show the performances of the SBDEKM algorithms. Compared with EKM algorithms and sensible beginning EKM (SBEKM) algorithms, the SBDEKM algorithms have almost the same accuracies and better computational efficiency.</p> </abstract>

Anti-lock braking control using an interval type-2 fuzzy neural network scheme for electric vehicles

Control precision and robustness can be regarded as essential challenges of slip ratio control because of the external uncertainties of road conditions and the internal delay response of braking actuators. In addition, it is difficult to obtain a trade-off between braking energy recovery efficiency and braking safety under some extreme braking conditions. To synchronously improve both slip ratio control and braking energy recovery efficiency for different road conditions, an anti-lock braking system (ABS) using a novel interval type-2 fuzzy neural network (IT2FNN) control scheme is proposed for electrohydraulic braking systems. The novel IT2FNN control scheme with five layers is designed to calculate the commanded braking torque. The membership function layer utilizes type-2 fuzzy sets to describe the slip ratio error degree, which enhances the scheme’s anti-interference ability, and the enhanced Karnik-Mendel (EKM) algorithm is used in the type-reduction layer to accelerate computation. Additionally, the network adjusts the parameters of the membership function and rules by decreasing the performance function value corresponding to the braking torque error to improve slip ratio control and enhance the self-adaptation capacity. Simulations showed that IT2FNN control can not only improve slip ratio control performance and decrease the braking torque error but also enhance the energy recovery efficiency of regenerative braking on different road surfaces.

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

General type-2 fuzzy logic systems have received wide concerns in current academic subject. Type-reduction is the kernel module for the systems. This paper shows the interpretations for the beginning of Karnik–Mendel (KM) algorithms. According to the famous numerical integration technique, the weighting approaches of enhanced Karnik–Mendel (EKM) algorithms are put forward. Then, the sensible beginning weighted enhanced Karnik–Mendel (SBWEKM) algorithms are put forward to perform the centroid type-reduction. Compared with the EKM algorithms, WEKM algorithms and SBEKM algorithms, this approach helps to improve both the absolute errors and convergence speeds as shown in four computer simulation experiments.

Study on non-iterative algorithms for center-of-sets type-reduction of Takagi–Sugeno–Kang type general type-2 fuzzy logic systems

The paper performs the center-of-sets (COS) type-reduction (TR) and de-fuzzification for Takagi–Sugeno–Kang (TSK) type general type-2 fuzzy logic systems (GT2 FLSs) on the basis of the alpha-planes expression of general type-2 fuzzy sets. Actually, comparing the popular Karnik–Mendel (KM) algorithms with other non-iterative algorithms is an important question in T2 society. Here the modules of fuzzy inference, COS TR, and de-fuzzification for TSK type GT2 FLSs are discussed by means of non-iterative Nagar–Bardini (NB) algorithms, Nie–Tan (NT) algorithms, and Begian–Melek–Mendel (BMM) algorithms. Simulation instances are constructed to illustrate the performances of three types of non-iterative algorithms compared with the KM algorithms. It is proved that, the proposed non-iterative algorithms can enhance the computational efficiencies significantly, which afford the potential application value for designers of GT2 FLSs.

Multi-objective green 4-dimensional transportation problems for damageable items through type-2 fuzzy random goal programming

As per ECOFYS’s11https://ingmarschumacher.files.wordpress.com/2013/05/asn-ecofys-2013-world-ghg-emissions-flow-chart-2010.pdf. report, road transportation contributes 10.5% of the GHGE, which is the cause of global warming. With the infrastructural developments, there are several routes between different cities, even in developing countries. In a business, objectives’ goals are always uncertain. With these facts, in this investigation, a multi-objective green four-dimensional transportation problem for damageable items is formulated and solved. Here four objectives — minimization of operational cost, transportation time, carbon emission amount and maximization of retailers’ satisfaction, whose goals are type-2 fuzzy random are considered. There are several connecting paths between sources and destinations with varied traffic congestion and maximum possible velocities for different conveyances. To get a better compromise solution for multi-objectives, the type-2 fuzzy random goal programming method (T2FRGPM) is developed using the Enhanced Karnik-Mendel algorithm for type reduction of type-2 fuzzy set The model is solved by the proposed T2FRGPM and illustrated with some numerical data using Lingo 11.0. A real-life example and several particular models are presented. The effectiveness of the proposed method, the importance of the four-dimensional transportation problem, and the objectives are established. A trade-off between transportation time and carbon emission amount is presented.

A Novel Approach to Type-Reduction and Design of Interval Type-2 Fuzzy Logic Systems

Abstract Fuzzy logic systems, unlike black-box models, are known as transparent artificial intelligence systems that have explainable rules of reasoning. Type 2 fuzzy systems extend the field of application to tasks that require the introduction of uncertainty in the rules, e.g. for handling corrupted data. Most practical implementations use interval type-2 sets and process interval membership grades. The key role in the design of type-2 interval fuzzy logic systems is played by the type-2 inference defuzzification method. In type-2 systems this generally takes place in two steps: type-reduction first, then standard defuzzification. The only precise type-reduction method is the iterative method known as Karnik-Mendel (KM) algorithm with its enhancement modifications. The known non-iterative methods deliver only an approximation of the boundaries of a type-reduced set and, in special cases, they diminish the profits that result from the use of type-2 fuzzy logic systems. In this paper, we propose a novel type-reduction method based on a smooth approximation of maximum/minimum, and we call this method a smooth type-reduction. Replacing the iterative KM algorithm by the smooth type-reduction, we obtain a structure of an adaptive interval type-2 fuzzy logic which is non-iterative and as close to an approximation of the KM algorithm as we like.

Design of sampling-based noniterative algorithms for centroid type-reduction of general type-2 fuzzy logic systems

General type-2 (GT2) fuzzy logic systems (FLSs) become a popular research topic for the past few years. Usually the Karnik–Mendel algorithms are the most prevalent approach to complete the type-reduction. Nonetheless, the iterative quality of these types of computational intensive algorithms might impede applying them. For the improved types of algorithms, some noniterative algorithms can enhance the calculation efficiencies greatly, while it is still an open problem for comparing the relation between the discrete TR algorithms and corresponding continuous TR algorithms. First, the sum and integral operations in discrete and continuous noniterative algorithms are compared. Then, three kinds of noniterative algorithms originate from the type-reduction of interval type-2 FLSs are extended to complete the centroid type-reduction of general T2 FLSs. Four computer simulations prove that while changing the number of samples suitably, the calculational results of discrete types of algorithms may accurately gain on the related continuous types of algorithms, and the calculational times of discrete types of algorithms are obviously less than the continuous types of algorithms, this may offer the possible meaning for designing and applying T2 FLSs.

Design of reasonable initialization weighted enhanced Karnik-Mendel algorithms for centroid type-reduction of interval type-2 fuzzy logic systems

<abstract> <p>Interval type-2 fuzzy logic systems (IT2 FLSs) already become an emerging technology in recent years. As the most popular type-reduction (TR) algorithms, Karnik-Mendel (KM) algorithms own the advantage of maintaining the uncertainties flow in systems. This paper analyzes the initialization for KM types of algorithms. Furthermore, the weighting approaches of them are also given by means of the Newton-Cotes quadrature formulas. Importantly, the reasonable initialization weighted enhanced Karnik-Mendel (RIWEKM) algorithms are provided to complete the centroid type-reduction of IT2 FLSs. Three computer simulation experiments illustrate that, the proposed RIWEKM algorithms own both smaller absolute errors and faster convergence speeds in contrast to the EKM and RIEKM algorithms.</p> </abstract>

Stochastic type-2 fuzzy modelling on GHG emission prediction for gas-fired power plant

This paper demonstrates the implementation of type-2 fuzzy logic model in prediction of greenhouse gas emission for gas fuelled power plant. The data collection and analysis on the GHG emission had been conducted with emphasis on the industrial scale with many uncertainty factors. The developed model will be used as a prediction tools to evaluate potential GHG emission for future study and mitigation planning. In this work, 100 sample sizes had been used as training dataset in the model framework which serve as the input information while 14 sample sizes are used for model validation to achieve model generalization. A novel Karnik-Mendel (KM) algorithm had been proposed with the Genetic Algorithm (GA) model optimization to achieve high performance and accuracy for the respective predictive model. In conclusion, the result showed that the model able to give an outstanding accuracy. On the validation analysis, the results showed developed model able to give satisfactory prediction with R2 of 0.978 and 0.84968 for training and validation, respectively.

An efficient non-iterative method for computing the centroid of an interval type-2 fuzzy set

Interval type-2 fuzzy logic systems (IT2FLSs) have better abilities to cope with uncertainties in many applications. One major drawback of IT2FLSs is the high computational cost of the iterative Karnik-Mendel (KM) algorithms in type-reduction (TR). From the practical point of view, this prevents using IT2FLS in real-world applications. To address this issue, a novel non-iterative method called Moradi-Zirkohi-Lin (MZL) TR method is proposed for computing the centroid of an IT2FLS. This makes the practical implementation of the IT2FLSs simpler. Comparative simulation results show that the proposed method outperforms the KM TR method in terms of computational burden. Besides, closer results, in terms of accuracy, to the KM TR method among the existing non-iterative TR methods are also achieved by the proposed TR method.

Comparison of modified Karnik-Mendel algorithm-based interval type-2 ANFIS and type-1 ANFIS

Purpose This study aims to propose, as the first time, the interval type-2 adaptive network-fuzzy inference system (ANFIS) structure, which is given better results compared to previously presented in the open literature. So, the ANFIS can be used effectively for training of interval type-2 fuzzy logic system (IT2FLS) parameters. Design/methodology/approach Karnik–Mendel algorithm (KMA) is modified to use in interval type-2 ANFIS. The modified Karnik–Mendel algorithm (M-KMA) is implemented to change the uncertain ANFIS parameters into known ones. In this way, the interval type-2 ANFIS removes uncertainties of IT2FLS. Therefore, the interval type-2 ANFIS is reduced to a simple one, i.e. less mathematical operation required. Only consequent parameters are trained, and the consequent parameters are chosen in the form of crisp. Findings By applying the mentioned procedure, it can be shown that interval type-2 ANFIS has generally better results compared to type-1 ANFIS. However, it was noticed that the worst results obtained in the case of interval type-2 ANFIS are equal to the best result obtained in the case of type-1 ANFIS. Therefore, users in this field can use this approach in solving nonlinear problems. Practical implications The interval type-2 ANFIS can be used as controller for highly nonlinear systems such as air vehicles. Originality/value As stated in the open literature, it is ineffective to use ANFIS for IT2FLS. In this study, the KMA is modified for IT2FLS, and it is seen that the ANFIS can be used effectively for IT2FLS.

Comparative Study between Type I and Type II Fuzzy Controller for Semi Active Suspension System

In this article, a type-2 fuzzy interval controller is proposed to solve the nonlinear control problems of semi-active suspension system. A suspension model with two degrees of freedom and A fuzzy approach for controller synthesis were proposed. The performance of the IT2FLC-based semi-active vehicle suspension system in terms of sprung mass displacement, suspension deflection and tire deflection are compared to the homologous fuzzy type-1 controller (T1FLC), and to the passive suspension system conventional using MATLAB / SIMULINK software for simulation and controller design. The vehicle parameters, called suspension deflection and speed of suspended mass are given as inputs for both controllers. The Csemi control signal is the variable damping coefficient. Inputs and outputs are presented by triangular membership functions. Mamdani inference system is used, along with a Karnik-Mendel algorithm to locate the center of gravity in reduction type for IT2FLC controller. Simulation results show that IT2FLC-based semi-active suspension system outperforms T1FLC and passive suspension system. Thus, they show a major improvement in control signal i.e. IT2FLC controller generates a lower damping coefficient than T1FLC controller. In addition, a remarkable reduction in signal energy by IT2FLC compared to same semi-active suspension system with T1FLC.

New efficient algorithms for the centroid of an interval type-2 fuzzy set

In a type-2 fuzzy logic system , one of the important operations is to calculate the centroid of an interval type-2 fuzzy set (IT2 FS). In this paper, two novel algorithms called binary algorithms are proposed to calculate the centroid of IT2 FSs. Then, the outputs of the proposed binary algorithms are proven to be the optimal values. After analyzing the computational complexities of Karnik-Mendel (KM) algorithms, enhanced Karnik-Mendel (EKM) algorithms, enhanced iterative algorithms based on stopping condition (EIASC) algorithms and the proposed binary algorithms, it is found that the proposed binary algorithms are superior to the KM algorithms, EKM algorithms and EIASC algorithms. Finally, two extended binary algorithms are proposed to compute the centroid of an IT2 FS. The efficiencies of the proposed binary algorithms and extended binary algorithms are demonstrated by extensive simulations.

Improvement of Enhanced Opposite Direction Searching Algorithm

Type-reduction is an important part of the interval type-2 fuzzy logic system (IT2 FLS), which is also the main factor that influences the effectiveness of IT2 FLS. The enhanced opposite direction searching (EODS) algorithm is currently the fastest form of the Karnik–Mendel algorithm, but it requires a large number of iterations. In this article, an improvement of the enhanced opposite direction searching (IEODS) algorithm is presented to obtain the type reduced set more efficiently. The IEODS algorithm improves the iterative process of the original EODS algorithm through the width of the upper and lower member functions of the IT2 FS. In this way, the IEODS algorithm uses a single-sided iteration process. It has a better initial point to greatly reduce the number of iterations. Quantitative analysis and simulation based experiments show that the IEODS algorithm can save more than 90% of the iterations required in the EODS algorithm. It makes the IEODS algorithm more suitable for practical applications, especially to various development platforms with an explanatory programming language as a tool, greatly expanding the application range of the IT2 FLS.

Study on center-of-sets type-reduction of interval type-2 fuzzy logic systems with noniterative algorithms

In recent years, interval type-2 fuzzy logic systems (IT2 FLSs) have become a hot topic for the capability of coping with uncertainties. Compared with the centroid type-reduction (TR), investigating the center-of-sets (COS) TR of IT2 FLSs is more favorable for applying IT2 FLSs. Actually, it is still an open question for comparing Karnik-Mendel (KM) types of algorithms and other types of alternative algorithms for COS TR. This paper gives the block of fuzzy reasoning, COS TR, and defuzzification of IT2 FLSs based on Nagar-Bardini (NB), Nie-Tan (NT) and Begian-Melek-Mendel (BMM) noniterative algorithms. Six simulation experiments are used to show the performances of three types of noniterative algorithms. The proposed noniterative algorithms can obtain much higher computational efficiencies compared with the KM algorithms, which give the potential value for designing T2 FLSs.

Green 4D transportation problems with breakable incompatible items under type-2 fuzzy-random environment

As the carbon emission (CE) from a vehicle depends on the factors such as the surface condition of the traveled route, conveyance type, load, speed, curb weight of the vehicles, etc., nowadays, consideration of the appropriate route, conveyance and corresponding CE are given maximum importance in eco-friendly transportation systems. The present study investigates a four-dimensional transportation problem (4DTP) for breakable/damageable items, some of which are pairwise incompatible, with carbon tax policy under crisp and type-2 fuzzy/fuzzy random environments. Here, unit transportation costs are taken as type-2 fuzzy and the products’ availabilities, demands, conveyance capacities are type-2 fuzzy-random. Type-2 fuzzy and fuzzy-random data are respectively transformed to the corresponding crisp and random data using the centroid-based type-2 fuzzy-probabilistic programming with Enhanced Karnik-Mendel (EKM) algorithm. Then a modified interactive algorithm is proposed to convert the above transportation problem into two crisp sub-constrained optimization problems which are solved by Generalized Reduced Gradient method using Lingo 11.0. The models are illustrated with some numerical data. As particular cases, several models are deduced. A real life example is illustrated. Effects of consideration of different paths, conveyances with different capacities, breakability, carbon emission costs, and incompatibilty of items are demonstrated. It is shown that the routes of shortest distance and minimum unit transportation cost are not the optimum paths for the model’s minimum cost. Here, road’s specific constants also play an important role. It is observed that for minimum total cost, there is a trade off between the total carbon emission and total transportation cost. Some managerial decisions are also derived.

FUZZY INFERENCE SYSTEMS BASE ON POLYNOMIALCONSEQUENTS OF FUZZY RULES

Various fuzzy inference systems that operate on the basis of polynomial consequents of fuzzy rules. As well as inference methods for such systems, in particular, Takagi-Sugeno fuzzy inference systems, their differences from other popular fuzzy systems, such as Mamdani systems, etc., are considered. The attention is focused on the features of the functioning of such systems both in the construction of elementary fuzzy systems. The Systems for which the calculation of the general logical conclusion involves intermediate levels of logical inference with many hierarchically interconnected blocks of fuzzy rules. Fuzzy sets of type 2 are considered, the membership index of which is a fuzzy term of the first type. This allows you to take into account the secondary fuzziness of linguistic concepts in the design of intelligent systems based on fuzzy inference. Fuzzy systems of the second type based on Takagi-Sugeno systems and the iterative Karnik-Mendel algorithm are considered to obtain a logical conclusion for fuzzy systems with the interval membership functions of the second type in the antecedents of fuzzy rules. The proposed procedure for lowering the order of fuzzy rules for higher-order Takagi-Sugeno fuzzy systems is described and justified. A fuzzy inference method for higher-order fuzzy systems based on the partition of a set of input variables is proposed. It is proposed to build a separate block of fuzzy rules for each of the input subspaces in the presence of a common polynomial. Which is a higher-order consequent, that reduces the total number of fuzzy rules in blocks.

Study on reasonable initialization enhanced Karnik-Mendel algorithms for centroid type-reduction of interval type-2 fuzzy logic systems

Type-reduction (TR) is a key block for interval type-2 fuzzy logic systems (IT2 FLSs). In general, Karnik-Mendel (KM) (or enhanced Karnik-Mendel (EKM)) algorithms are used to perform the TR. These two types of algorithms have the advantage of preserving the uncertainties of membership functions (MFs) flow in IT2 FLSs. This paper gives the initialization explanations of KM and EKM algorithms, and proposes reasonable initialization enhanced Karnik-Mendel (RIEKM) algorithms for centroid TR of IT2 FLSs. By considering the accurate continuous Nie-Tan (CNT) algorithms as the benchmark, four computer simulation examples are adopted to illustrate and analyze the performances of RIEKM algorithms for solving the centroid TR and defuzzification of IT2 FLSs. Compared with the EKM algorithms, the proposed RIEKM algorithms have smaller absolute errors and faster convergence speeds, which afford the potential value for designing and applying IT2 FLSs.

Combinatorial Iterative Algorithms for Computing the Centroid of an Interval Type-2 Fuzzy Set

Computing the centroid of an interval type-2 fuzzy set (IT2 FS) is an important type-reduction method. The aim of this paper is to develop a new method to calculate the centroid of an IT2 FS when the problems of centroid computation of an IT2 FS are continuous. For the continuous centroid computation problems, the structures of optimal solutions are strictly proven from mathematics for the first time in this paper. Furthermore, we also prove that the structures of the optimal solutions are unique in the sense of almost everywhere equal, i.e., if there are two optimal solutions $f_1(x)$ and $f_2(x)$ , the Lebesgue measure of $\{x|f_1(x)\neq f_2(x)\}$ is equal to 0. Subsequently, a combinatorial iterative (CI) method is proposed to solve the roots of the sufficiently differentiable objective functions. It is proven that the convergence of the proposed iterative method is at least sixth order. Based on the proposed iterative method, two algorithms, called CI algorithms, are devised to compute the centroid of an IT2 FS. The efficiencies of CI algorithms are demonstrated by comparing the continuous Karnik–Mendel algorithms and the Hallye's methods with the CI algorithms through three numerical examples.

Analytical structures of interval type-2 fuzzy controllers using product AND operations

Interval type-2 fuzzy controllers have received attention because of their potential in handling uncertainties better than type-1 (T1) fuzzy controllers. However, directly deriving the analytical structure of interval type-2 fuzzy controllers is a great challenge due to the iterative computation of Karnik–Mendel algorithm and similar ones. Analytical structure is necessary for the analysis and design of control systems (e.g. stability analysis). In this study, a novel technique is proposed for deriving the analytical structure of interval type-2 fuzzy controllers based on the configurations of the triangle interval type-2 input fuzzy sets, T1 output fuzzy singletons, product AND operators, Karnik–Mendel center-of-set type reducer, and centroid defuzzifier. Input space is partitioned into a group of input combinations (ICs), which are produced by superposing input ICs and case ICs, to obtain an analytical structure. Input ICs are evaluated by the upper/lower membership functions and case ICs using the equation of the terminating condition of Karnik–Mendel algorithm. An explicit analytical expression is obtained on any IC because each IC has the same membership functions and switch points. Finally, two examples are used to verify the validity of our method.