Enhanced Karnik-Mendel Algorithm 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>

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.

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.

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>

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.

Discrete Non-iterative Centroid Type-Reduction Algorithms on General Type-2 Fuzzy Logic Systems

Since the alpha-planes expressions of general type-2 fuzzy sets (GT2 FSs) have been proposed, general type-2 fuzzy logic systems (GT2 FLSs) that are dependent on GT2 FSs are becoming quite popular to fuzzy logic researchers. Usually enhanced Karnik–Mendel (EKM) algorithms are adopted for performing the kernel block of type-reduction (TR). However, the essence of EKM-based TR process probably hinders the GT2 FLSs from real-world applications. It is an intriguing as well as unsolved problem for comparing EKM algorithms with other non-iterative algorithms. This paper provides a framework encompassing fuzzy reasoning, defuzzification, as well as type-reduction. Furthermore, the continuous of NT (CNT) algorithm is shown to be a precise approach when it is used to execute the centroid TR of GT2 fuzzy logic systems. Four computer tests display the characteristics of discrete Nagar-Bardini (NB) and Nie-Tan (NT) non-iterative algorithms. Compared with EKM approach, the developed one exhibits some superiorities in terms of guaranteeing high computation accuracy and low computation burdens, which broadens the application ranges for the proposed method.

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.

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.

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.

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.

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.

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.

Advantages of the Enhanced Opposite Direction Searching Algorithm for Computing the Centroid of An Interval Type‐2 Fuzzy Set

AbstractComputing the centroid of an interval type‐2 fuzzy set (IT2 FS) is an important operation in a type‐2 fuzzy logic system (where it is called type‐reduction), but it is also a potentially time‐consuming operation. In this paper, an enhanced opposite direction searching (EODS) algorithm is presented for doing this. The EODS comes from an early version of the IT2 FS type‐reduction method called the opposite direction searching (ODS) algorithm, which has been proven faster than the most commonly used Enhanced Karnik‐Mendel (EKM) method. The EODS differs from the ODS in two high speed formulas for calculating the centroid endings. Quantitative analysis on the mathematical operations and comparisons performed by EODS, ODS, and EKM algorithms shows that EODS could save about 50% of the calculations and comparisons in relation to ODS. Compared with EKM, it could save about 67% to 80% of the calculations and comparisons. Simulation experiments have been performed to compare EODS with the ODS and EKM methods in terms of average CPU time. The experimental results validate the quantitative analysis.

Study on enhanced Karnik–Mendel algorithms: Initialization explanations and computation improvements

Computing the centroid of an interval type-2 fuzzy set is an important operation in a type-2 fuzzy logic system, and is usually implemented by Karnik–Mendel (KM) iterative algorithms. By connecting KM algorithms and continuous KM algorithms together, this paper gives theoretical explanations on the initialization methods of KM and Enhanced Karnik–Mendel (EKM) algorithms, proposes exact methods for centroid computation of an interval type-2 fuzzy set, and extends the Enhanced Karnik–Mendel (EKM) algorithms to three different forms of weighted EKM (WEKM) algorithms. It shows that EKM algorithms become a special case of the WEKM algorithms when the weights of the latter are constant value. It also shows that, in general, the weighted EKM algorithms have smaller absolute error and faster convergence speed than the EKM algorithms which make them very attractive for real-time applications of fuzzy logic system. Four numerical examples are used to illustrate and analyze the performance of WEKM algorithms.

Computing the centroid of a general type-2 fuzzy set by means of the centroid-flow algorithm

Previous studies have shown that the centroid of a general type-2 fuzzy set (T2 FS) A can be obtained by taking the union of the centroids of all the α-planes (each raised to level α) of A. Karnik-Mendel (KM) or the enhanced KM (EKM) algorithms are used to compute the centroid of each α-plane. The iterative features in KM/EKM algorithms can be time-consuming, especially when the algorithms have to be repeated for many α-planes. This paper proposes a new method named centroid-flow (CF) algorithm to compute the centroid of A without having to apply KM/EKM algorithms for every α-plane. Extensive simulations have shown that the CF algorithm can reduce the computation time by 75%-80 % and 50% -75%, compared with KM and EKM algorithms, respectively, and still maintains satisfactory computation accuracy for various T2 FSs when the primary variable x and α -plane are discretized finely enough.

Enhanced Karnik--Mendel Algorithms

The Karnik-Mendel (KM) algorithms are iterative procedures widely used in fuzzy logic theory. They are known to converge monotonically and superexponentially fast; however, several (usually two to six) iterations are still needed before convergence occurs. Methods to reduce their computational cost are proposed in this paper. Extensive simulations show that, on average, the enhanced KM algorithms can save about two iterations, which corresponds to more than a 39% reduction in computation time. An additional (at least) 23% computational cost can be saved if no sorting of the inputs is needed.