Search results for: Fuzzy Texture Spectrum
Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 1558

Search results for: Fuzzy Texture Spectrum

1378 Project Selection by Using Fuzzy AHP and TOPSIS Technique

Authors: S. Mahmoodzadeh, J. Shahrabi, M. Pariazar, M. S. Zaeri

Abstract:

In this article, by using fuzzy AHP and TOPSIS technique we propose a new method for project selection problem. After reviewing four common methods of comparing alternatives investment (net present value, rate of return, benefit cost analysis and payback period) we use them as criteria in AHP tree. In this methodology by utilizing improved Analytical Hierarchy Process by Fuzzy set theory, first we try to calculate weight of each criterion. Then by implementing TOPSIS algorithm, assessment of projects has been done. Obtained results have been tested in a numerical example.

Keywords: Fuzzy AHP, Project Selection, TOPSIS Technique.

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1377 Characterization of Pure Nickel Coatings Fabricated under Pulse Current Conditions

Authors: M. Sajjadnejad, H. Omidvar, M. Javanbakht, A. Mozafari

Abstract:

Pure nickel coatings have been successfully electrodeposited on copper substrates by the pulse plating technique. The influence of current density, duty cycle and pulse frequency on the surface morphology, crystal orientation, and microhardness was determined. It was found that the crystallite size of the deposit increases with increasing current density and duty cycle. The crystal orientation progressively changed from a random texture at 1 A/dm2 to (200) texture at 10 A/dm2. Increasing pulse frequency resulted in increased texture coefficient and peak intensity of (111) reflection. An increase in duty cycle resulted in considerable increase in texture coefficient and peak intensity of (311) reflection. Coatings obtained at high current densities and duty cycle present a mixed morphology of small and large grains. Maximum microhardness of 193 Hv was achieved at 4 A/dm2, 10 Hz and duty cycle of 50%. Nickel coatings with (200) texture are ductile while (111) texture improves the microhardness of the coatings.

Keywords: Current density, Duty cycle, Microstructure, Nickel, Pulse frequency.

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1376 An Application of Generalized Fuzzy Soft Sets in a Social Decision Making Problem

Authors: Nisha Singhal, Usha Chouhan

Abstract:

At present, application of the extension of soft set theory in decision making problems in day to day life is progressing rapidly. The concepts of fuzzy soft set and its properties have been evolved as an area of interest for the researchers. The generalization of the concepts recently got importance and a rapid growth in the research in this area witnessed its vital-ness. In this paper, an application of the concept of generalized fuzzy soft set to make decision in a social problem is presented. Further, this paper also highlights some of the key issues of the related areas.

Keywords: Soft set, Fuzzy Soft set, Generalized Fuzzy Soft set, Membership and Non-Membership Score.

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1375 Simplex Method for Solving Linear Programming Problems with Fuzzy Numbers

Authors: S. H. Nasseri, E. Ardil, A. Yazdani, R. Zaefarian

Abstract:

The fuzzy set theory has been applied in many fields, such as operations research, control theory, and management sciences, etc. In particular, an application of this theory in decision making problems is linear programming problems with fuzzy numbers. In this study, we present a new method for solving fuzzy number linear programming problems, by use of linear ranking function. In fact, our method is similar to simplex method that was used for solving linear programming problems in crisp environment before.

Keywords: Fuzzy number linear programming, rankingfunction, simplex method.

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1374 Performance of Nakagami Fading Channel over Energy Detection Based Spectrum Sensing

Authors: M. Ranjeeth, S. Anuradha

Abstract:

Spectrum sensing is the main feature of cognitive radio technology. Spectrum sensing gives an idea of detecting the presence of the primary users in a licensed spectrum. In this paper we compare the theoretical results of detection probability of different fading environments like Rayleigh, Rician, Nakagami-m fading channels with the simulation results using energy detection based spectrum sensing. The numerical results are plotted as Pf Vs Pd for different SNR values, fading parameters. It is observed that Nakagami fading channel performance is better than other fading channels by using energy detection in spectrum sensing. A MATLAB simulation test bench has been implemented to know the performance of energy detection in different fading channel environment.

Keywords: Spectrum sensing, Energy detection, fading channels, Probability of detection, probability of false alarm.

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1373 Application of Quality Index Method, Texture Measurements and Electronic Nose to Assess the Freshness of Atlantic Herring (Clupea harengus) Stored in Ice

Authors: Nga T.T. Mai, Emilía Martinsdóttir, Kolbrún Sveinsdóttir, Gudrun Olafsdóttir, Sigurjón Arason

Abstract:

Atlantic herring (Clupea harengus) is an important commercial fish and shows to be more and more demanded for human consumption. Therefore, it is very important to find good methods for monitoring the freshness of the fish in order to keep it in the best quality for human consumption. In this study, the fish was stored in ice up to 2 weeks. Quality changes during storage were assessed by the Quality Index Method (QIM), quantitative descriptive analysis (QDA) and Torry scheme, by texture measurements: puncture tests and Texture Profile Analysis (TPA) tests on texture analyzer TA.XT2i, and by electronic nose (e-nose) measurements using FreshSense instrument. Storage time of herring in ice could be estimated by QIM with ± 2 days using 5 herring per lot. No correlation between instrumental texture parameters and storage time or between sensory and instrumental texture variables was found. E-nose measurements could be use to detect the onset of spoilage.

Keywords: Herring, Quality Index Method (QIM), freshness, storage time.

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1372 Half-Circle Fuzzy Number Threshold Determination via Swarm Intelligence Method

Authors: P.-W. Tsai, J.-W. Chen, C.-W. Chen, C.-Y. Chen

Abstract:

In recent years, many researchers are involved in the field of fuzzy theory. However, there are still a lot of issues to be resolved. Especially on topics related to controller design such as the field of robot, artificial intelligence, and nonlinear systems etc. Besides fuzzy theory, algorithms in swarm intelligence are also a popular field for the researchers. In this paper, a concept of utilizing one of the swarm intelligence method, which is called Bacterial-GA Foraging, to find the stabilized common P matrix for the fuzzy controller system is proposed. An example is given in in the paper, as well.

Keywords: Half-circle fuzzy numbers, predictions, swarm intelligence, Lyapunov method.

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1371 Fuzzy Control of Macroeconomic Models

Authors: Andre A. Keller

Abstract:

The optimal control is one of the possible controllers for a dynamic system, having a linear quadratic regulator and using the Pontryagin-s principle or the dynamic programming method . Stochastic disturbances may affect the coefficients (multiplicative disturbances) or the equations (additive disturbances), provided that the shocks are not too great . Nevertheless, this approach encounters difficulties when uncertainties are very important or when the probability calculus is of no help with very imprecise data. The fuzzy logic contributes to a pragmatic solution of such a problem since it operates on fuzzy numbers. A fuzzy controller acts as an artificial decision maker that operates in a closed-loop system in real time. This contribution seeks to explore the tracking problem and control of dynamic macroeconomic models using a fuzzy learning algorithm. A two inputs - single output (TISO) fuzzy model is applied to the linear fluctuation model of Phillips and to the nonlinear growth model of Goodwin.

Keywords: fuzzy control, macroeconomic model, multiplier - accelerator, nonlinear accelerator, stabilization policy.

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1370 Fuzzy Logic PID Control of Automatic Voltage Regulator System

Authors: Aye Aye Mon

Abstract:

The application of a simple microcontroller to deal with a three variable input and a single output fuzzy logic controller, with Proportional – Integral – Derivative (PID) response control built-in has been tested for an automatic voltage regulator. The fuzzifiers are based on fixed range of the variables of output voltage. The control output is used to control the wiper motor of the auto transformer to adjust the voltage, using fuzzy logic principles, so that the voltage is stabilized. In this report, the author will demonstrate how fuzzy logic might provide elegant and efficient solutions in the design of multivariable control based on experimental results rather than on mathematical models.

Keywords: Fuzzy logic system, PID Controller, control systems, controlled A V R

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1369 Texture Characterization Based on a Chandrasekhar Fast Adaptive Filter

Authors: Mounir Sayadi, Farhat Fnaiech

Abstract:

In the framework of adaptive parametric modelling of images, we propose in this paper a new technique based on the Chandrasekhar fast adaptive filter for texture characterization. An Auto-Regressive (AR) linear model of texture is obtained by scanning the image row by row and modelling this data with an adaptive Chandrasekhar linear filter. The characterization efficiency of the obtained model is compared with the model adapted with the Least Mean Square (LMS) 2-D adaptive algorithm and with the cooccurrence method features. The comparison criteria is based on the computation of a characterization degree using the ratio of "betweenclass" variances with respect to "within-class" variances of the estimated coefficients. Extensive experiments show that the coefficients estimated by the use of Chandrasekhar adaptive filter give better results in texture discrimination than those estimated by other algorithms, even in a noisy context.

Keywords: Texture analysis, statistical features, adaptive filters, Chandrasekhar algorithm.

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1368 Best Co-approximation and Best Simultaneous Co-approximation in Fuzzy Normed Spaces

Authors: J. Kavikumar, N. S. Manian, M.B.K. Moorthy

Abstract:

The main purpose of this paper is to consider the t-best co-approximation and t-best simultaneous co-approximation in fuzzy normed spaces. We develop the theory of t-best co-approximation and t-best simultaneous co-approximation in quotient spaces. This new concept is employed us to improve various characterisations of t-co-proximinal and t-co-Chebyshev sets.

Keywords: Fuzzy best co-approximation, fuzzy quotient spaces, proximinality, Chebyshevity, best simultaneous co-approximation.

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1367 Robust Fuzzy Control of Nonlinear Fuzzy Impulsive Singular Perturbed Systems with Time-varying Delay

Authors: Caigen Zhou, Haibo Jiang

Abstract:

The problem of robust fuzzy control for a class of nonlinear fuzzy impulsive singular perturbed systems with time-varying delay is investigated by employing Lyapunov functions. The nonlinear delay system is built based on the well-known T–S fuzzy model. The so-called parallel distributed compensation idea is employed to design the state feedback controller. Sufficient conditions for global exponential stability of the closed-loop system are derived in terms of linear matrix inequalities (LMIs), which can be easily solved by LMI technique. Some simulations illustrate the effectiveness of the proposed method.

Keywords: T–S fuzzy model, singular perturbed systems, time-varying delay, robust control.

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1366 Adomian Method for Second-order Fuzzy Differential Equation

Authors: Lei Wang, Sizong Guo

Abstract:

In this paper, we study the numerical method for solving second-order fuzzy differential equations using Adomian method under strongly generalized differentiability. And, we present an example with initial condition having four different solutions to illustrate the efficiency of the proposed method under strongly generalized differentiability.

Keywords: Fuzzy-valued function, fuzzy initial value problem, strongly generalized differentiability, adomian decomposition method.

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1365 Processing Web-Cam Images by a Neuro-Fuzzy Approach for Vehicular Traffic Monitoring

Authors: A. Faro, D. Giordano, C. Spampinato

Abstract:

Traffic management in an urban area is highly facilitated by the knowledge of the traffic conditions in every street or highway involved in the vehicular mobility system. Aim of the paper is to propose a neuro-fuzzy approach able to compute the main parameters of a traffic system, i.e., car density, velocity and flow, by using the images collected by the web-cams located at the crossroads of the traffic network. The performances of this approach encourage its application when the traffic system is far from the saturation. A fuzzy model is also outlined to evaluate when it is suitable to use more accurate, even if more time consuming, algorithms for measuring traffic conditions near to saturation.

Keywords: Neuro-fuzzy networks, computer vision, Fuzzy systems, intelligent transportation system.

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1364 Self – Tuning Method of Fuzzy System: An Application on Greenhouse Process

Authors: M. Massour El Aoud, M. Franceschi, M. Maher

Abstract:

The approach proposed here is oriented in the direction of fuzzy system for the analysis and the synthesis of intelligent climate controllers, the simulation of the internal climate of the greenhouse is achieved by a linear model whose coefficients are obtained by identification. The use of fuzzy logic controllers for the regulation of climate variables represents a powerful way to minimize the energy cost. Strategies of reduction and optimization are adopted to facilitate the tuning and to reduce the complexity of the controller.

Keywords: Greenhouse, fuzzy logic, optimization, gradient descent.

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1363 Solving Fully Fuzzy Linear Systems by use of a Certain Decomposition of the Coefficient Matrix

Authors: S. H. Nasseri, M. Sohrabi, E. Ardil

Abstract:

In this paper, we give a certain decomposition of the coefficient matrix of the fully fuzzy linear system (FFLS) to obtain a simple algorithm for solving these systems. The new algorithm can solve FFLS in a smaller computing process. We will illustrate our method by solving some examples.

Keywords: Fully fuzzy linear system, Fuzzy number, LUdecomposition.

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1362 A Fuzzy Dynamic Load Balancing Algorithm for Homogenous Distributed Systems

Authors: Ali M. Alakeel

Abstract:

Load balancing in distributed computer systems is the process of redistributing the work load among processors in the system to improve system performance. Most of previous research in using fuzzy logic for the purpose of load balancing has only concentrated in utilizing fuzzy logic concepts in describing processors load and tasks execution length. The responsibility of the fuzzy-based load balancing process itself, however, has not been discussed and in most reported work is assumed to be performed in a distributed fashion by all nodes in the network. This paper proposes a new fuzzy dynamic load balancing algorithm for homogenous distributed systems. The proposed algorithm utilizes fuzzy logic in dealing with inaccurate load information, making load distribution decisions, and maintaining overall system stability. In terms of control, we propose a new approach that specifies how, when, and by which node the load balancing is implemented. Our approach is called Centralized-But-Distributed (CBD).

Keywords: Dynamic load balancing, fuzzy logic, distributed systems, algorithm.

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1361 Constructing a Fuzzy Net Present Value Method to Evaluating the BOT Sport Facilities

Authors: Huei-Fu Lu

Abstract:

This paper is to develop a fuzzy net present value (FNPV) method by taking vague cash flow and imprecise required rate of return into account for evaluating the value of the Build-Operate-Transfer (BOT) sport facilities. In order to clearly manifest a more realistic capital budgeting model based on the classical net present value (NPV) method, some uncertain financial elements in NPV formula will be fuzzified as triangular fuzzy numbers. Through the conscientious manipulation of fuzzy set theory, we will find that the proposed FNPV model is a more explicit extension of classical (crisp) model and could be more practicable for the financial managers to capture the essence of capital budgeting of sport facilities than non-fuzzy model.

Keywords: Fuzzy sets; Capital budgeting, Sport facility, Net present value (NPV), Build-Operate-Transfer (BOT) scheme

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1360 Spherical Spectrum Properties of Quaternionic Operators

Authors: Yiwan Guo, Fahui Zhai

Abstract:

In this paper, the similarity invariant and the upper semi-continuity of spherical spectrum, and the spherical spectrum properties for infinite direct sums of quaternionic operators are characterized, respectively. As an application of some results established, a concrete example about the computation of the spherical spectrum of a compact quaternionic operator with form of infinite direct sums of quaternionic matrices is also given.

Keywords: Spherical spectrum, Quaternionic operator, Upper semi-continuity, Direct sum of operators.

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1359 Neural Network Tuned Fuzzy Controller for MIMO System

Authors: Seema Chopra, R. Mitra, Vijay Kumar

Abstract:

In this paper, a neural network tuned fuzzy controller is proposed for controlling Multi-Input Multi-Output (MIMO) systems. For the convenience of analysis, the structure of MIMO fuzzy controller is divided into single input single-output (SISO) controllers for controlling each degree of freedom. Secondly, according to the characteristics of the system-s dynamics coupling, an appropriate coupling fuzzy controller is incorporated to improve the performance. The simulation analysis on a two-level mass–spring MIMO vibration system is carried out and results show the effectiveness of the proposed fuzzy controller. The performance though improved, the computational time and memory used is comparatively higher, because it has four fuzzy reasoning blocks and number may increase in case of other MIMO system. Then a fuzzy neural network is designed from a set of input-output training data to reduce the computing burden during implementation. This control strategy can not only simplify the implementation problem of fuzzy control, but also reduce computational time and consume less memory.

Keywords: Fuzzy Control, Neural Network, MIMO System, Optimization of Membership functions.

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1358 Speed Control of Permanent Magnet Synchronous Motor Using Evolutionary Fuzzy PID Controller

Authors: M. Umabharathi, S. Vijayabaskar

Abstract:

Evolutionary Fuzzy PID Speed Controller for Permanent Magnet Synchronous Motor (PMSM) is developed to achieve the Speed control of PMSM in Closed Loop operation and to deal with the existence of transients. Consider a Fuzzy PID control design problem, based on common control Engineering Knowledge. If the transient error is big, that Good transient performance can be obtained by increasing the P and I gains and decreasing the D gains. To autotune the control parameters of the Fuzzy PID controller, the Evolutionary Algorithms (EA) are developed. EA based Fuzzy PID controller provides better speed control and guarantees the closed loop stability. The Evolutionary Fuzzy PID controller can be implemented in real time Applications without any concern about instabilities that leads to system failure or damage.

Keywords: Evolutionary Algorithm (EA), Fuzzy system, Genetic Algorithm (GA), Membership, Permanent Magnet Synchronous Motor (PMSM).

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1357 Comparison of Mamdani and Sugeno Fuzzy Interference Systems for the Breast Cancer Risk

Authors: Alshalaa A. Shleeg, Issmail M. Ellabib

Abstract:

Breast cancer is a major health burden worldwide being a major cause of death amongst women. In this paper, Fuzzy Inference Systems (FIS) are developed for the evaluation of breast cancer risk using Mamdani-type and Sugeno-type models. The paper outlines the basic difference between Mamdani-type FIS and Sugeno-type FIS. The results demonstrated the performance comparison of the two systems and the advantages of using Sugeno- type over Mamdani-type.

Keywords: Breast cancer diagnosis, Fuzzy Inference System (FIS), Fuzzy Logic, fuzzy intelligent technique.

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1356 Minimal Spanning Tree based Fuzzy Clustering

Authors: Ágnes Vathy-Fogarassy, Balázs Feil, János Abonyi

Abstract:

Most of fuzzy clustering algorithms have some discrepancies, e.g. they are not able to detect clusters with convex shapes, the number of the clusters should be a priori known, they suffer from numerical problems, like sensitiveness to the initialization, etc. This paper studies the synergistic combination of the hierarchical and graph theoretic minimal spanning tree based clustering algorithm with the partitional Gath-Geva fuzzy clustering algorithm. The aim of this hybridization is to increase the robustness and consistency of the clustering results and to decrease the number of the heuristically defined parameters of these algorithms to decrease the influence of the user on the clustering results. For the analysis of the resulted fuzzy clusters a new fuzzy similarity measure based tool has been presented. The calculated similarities of the clusters can be used for the hierarchical clustering of the resulted fuzzy clusters, which information is useful for cluster merging and for the visualization of the clustering results. As the examples used for the illustration of the operation of the new algorithm will show, the proposed algorithm can detect clusters from data with arbitrary shape and does not suffer from the numerical problems of the classical Gath-Geva fuzzy clustering algorithm.

Keywords: Clustering, fuzzy clustering, minimal spanning tree, cluster validity, fuzzy similarity.

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1355 Target Detection with Improved Image Texture Feature Coding Method and Support Vector Machine

Authors: R. Xu, X. Zhao, X. Li, C. Kwan, C.-I Chang

Abstract:

An image texture analysis and target recognition approach of using an improved image texture feature coding method (TFCM) and Support Vector Machine (SVM) for target detection is presented. With our proposed target detection framework, targets of interest can be detected accurately. Cascade-Sliding-Window technique was also developed for automated target localization. Application to mammogram showed that over 88% of normal mammograms and 80% of abnormal mammograms can be correctly identified. The approach was also successfully applied to Synthetic Aperture Radar (SAR) and Ground Penetrating Radar (GPR) images for target detection.

Keywords: Image texture analysis, feature extraction, target detection, pattern classification.

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1354 Hybrid Artificial Bee Colony and Least Squares Method for Rule-Based Systems Learning

Authors: Ahcene Habbi, Yassine Boudouaoui

Abstract:

This paper deals with the problem of automatic rule generation for fuzzy systems design. The proposed approach is based on hybrid artificial bee colony (ABC) optimization and weighted least squares (LS) method and aims to find the structure and parameters of fuzzy systems simultaneously. More precisely, two ABC based fuzzy modeling strategies are presented and compared. The first strategy uses global optimization to learn fuzzy models, the second one hybridizes ABC and weighted least squares estimate method. The performances of the proposed ABC and ABC-LS fuzzy modeling strategies are evaluated on complex modeling problems and compared to other advanced modeling methods.

Keywords: Automatic design, learning, fuzzy rules, hybrid, swarm optimization.

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1353 Proposal of Additional Fuzzy Membership Functions in Smoothing Transition Autoregressive Models

Authors: Ε. Giovanis

Abstract:

In this paper we present, propose and examine additional membership functions for the Smoothing Transition Autoregressive (STAR) models. More specifically, we present the tangent hyperbolic, Gaussian and Generalized bell functions. Because Smoothing Transition Autoregressive (STAR) models follow fuzzy logic approach, more fuzzy membership functions should be tested. Furthermore, fuzzy rules can be incorporated or other training or computational methods can be applied as the error backpropagation or genetic algorithm instead to nonlinear squares. We examine two macroeconomic variables of US economy, the inflation rate and the 6-monthly treasury bills interest rates.

Keywords: Forecast , Fuzzy membership functions, Smoothingtransition, Time-series

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1352 Intelligent Automatic Generation Control of Two Area Interconnected Power System using Hybrid Neuro Fuzzy Controller

Authors: Sathans, A. Swarup

Abstract:

This paper presents the development and application of an adaptive neuro fuzzy inference system (ANFIS) based intelligent hybrid neuro fuzzy controller for automatic generation control (AGC) of two-area interconnected thermal power system with reheat non linearity. The dynamic response of the system has been studied for 1% step load perturbation in area-1. The performance of the proposed neuro fuzzy controller is compared against conventional proportional-integral (PI) controller, state feedback linear quadratic regulator (LQR) controller and fuzzy gain scheduled proportionalintegral (FGSPI) controller. Comparative analysis demonstrates that the proposed intelligent neuro fuzzy controller is the most effective of all in improving the transients of frequency and tie-line power deviations against small step load disturbances. Simulations have been performed using Matlab®.

Keywords: Automatic generation control, ANFIS, LQR, Hybrid neuro fuzzy controller

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1351 An Empirical Analysis of Arabic WebPages Classification using Fuzzy Operators

Authors: Ahmad T. Al-Taani, Noor Aldeen K. Al-Awad

Abstract:

In this study, a fuzzy similarity approach for Arabic web pages classification is presented. The approach uses a fuzzy term-category relation by manipulating membership degree for the training data and the degree value for a test web page. Six measures are used and compared in this study. These measures include: Einstein, Algebraic, Hamacher, MinMax, Special case fuzzy and Bounded Difference approaches. These measures are applied and compared using 50 different Arabic web pages. Einstein measure was gave best performance among the other measures. An analysis of these measures and concluding remarks are drawn in this study.

Keywords: Text classification, HTML documents, Web pages, Machine learning, Fuzzy logic, Arabic Web pages.

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1350 Ranking Fuzzy Numbers Based on Lexicographical Ordering

Authors: B. Farhadinia

Abstract:

Although so far, many methods for ranking fuzzy numbers have been discussed broadly, most of them contained some shortcomings, such as requirement of complicated calculations, inconsistency with human intuition and indiscrimination. The motivation of this study is to develop a model for ranking fuzzy numbers based on the lexicographical ordering which provides decision-makers with a simple and efficient algorithm to generate an ordering founded on a precedence. The main emphasis here is put on the ease of use and reliability. The effectiveness of the proposed method is finally demonstrated by including a comprehensive comparing different ranking methods with the present one.

Keywords: Ranking fuzzy numbers, Lexicographical ordering.

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1349 Jacobi-Based Methods in Solving Fuzzy Linear Systems

Authors: Lazim Abdullah, Nurhakimah Ab. Rahman

Abstract:

Linear systems are widely used in many fields of science and engineering. In many applications, at least some of the parameters of the system are represented by fuzzy rather than crisp numbers. Therefore it is important to perform numerical algorithms or procedures that would treat general fuzzy linear systems and solve them using iterative methods. This paper aims are to solve fuzzy linear systems using four types of Jacobi based iterative methods. Four iterative methods based on Jacobi are used for solving a general n × n fuzzy system of linear equations of the form Ax = b , where A is a crisp matrix and b an arbitrary fuzzy vector. The Jacobi, Jacobi Over-Relaxation, Refinement of Jacobi and Refinement of Jacobi Over-Relaxation methods was tested to a five by five fuzzy linear system. It is found that all the tested methods were iterated differently. Due to the effect of extrapolation parameters and the refinement, the Refinement of Jacobi Over-Relaxation method was outperformed the other three methods.

Keywords: Fuzzy linear systems, Jacobi, Jacobi Over- Relaxation, Refinement of Jacobi, Refinement of Jacobi Over- Relaxation.

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