Search results for: strong (weak) fuzzy domination number.

Authors: Seema Chopra, R. Mitra, Vijay Kumar

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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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Authors: Binu Thomas, Raju G., Sonam Wangmo

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In Data mining, Fuzzy clustering algorithms have demonstrated advantage over crisp clustering algorithms in dealing with the challenges posed by large collections of vague and uncertain natural data. This paper reviews concept of fuzzy logic and fuzzy clustering. The classical fuzzy c-means algorithm is presented and its limitations are highlighted. Based on the study of the fuzzy c-means algorithm and its extensions, we propose a modification to the cmeans algorithm to overcome the limitations of it in calculating the new cluster centers and in finding the membership values with natural data. The efficiency of the new modified method is demonstrated on real data collected for Bhutan-s Gross National Happiness (GNH) program.

Keywords: Adaptive fuzzy clustering, clustering, fuzzy logic, fuzzy clustering, c-means.

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Authors: Seema Chopra, R. Mitra, Vijay Kumar

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A neurofuzzy approach for a given set of input-output training data is proposed in two phases. Firstly, the data set is partitioned automatically into a set of clusters. Then a fuzzy if-then rule is extracted from each cluster to form a fuzzy rule base. Secondly, a fuzzy neural network is constructed accordingly and parameters are tuned to increase the precision of the fuzzy rule base. This network is able to learn and optimize the rule base of a Sugeno like Fuzzy inference system using Hybrid learning algorithm, which combines gradient descent, and least mean square algorithm. This proposed neurofuzzy system has the advantage of determining the number of rules automatically and also reduce the number of rules, decrease computational time, learns faster and consumes less memory. The authors also investigate that how neurofuzzy techniques can be applied in the area of control theory to design a fuzzy controller for linear and nonlinear dynamic systems modelling from a set of input/output data. The simulation analysis on a wide range of processes, to identify nonlinear components on-linely in a control system and a benchmark problem involving the prediction of a chaotic time series is carried out. Furthermore, the well-known examples of linear and nonlinear systems are also simulated under the Matlab/Simulink environment. The above combination is also illustrated in modeling the relationship between automobile trips and demographic factors.

Keywords: Fuzzy control, neuro-fuzzy techniques, fuzzy subtractive clustering, extraction of rules, and optimization of membership functions.

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Authors: D.R Prince Williams

Abstract:

The notion of S-fuzzy left h-ideals in a hemiring is introduced and it's basic properties are investigated.We also study the homomorphic image and preimage of S-fuzzy left h-ideal of hemirings.Using a collection of left h-ideals of a hemiring, S-fuzzy left h-ideal of hemirings are established.The notion of a finite-valued S-fuzzy left h-ideal is introduced,and its characterization is given.S-fuzzy relations on hemirings are discussed.The notion of direct product and S-product are introduced and some properties of the direct product and S-product of S-fuzzy left h-ideal of hemiring are also discussed.

Keywords: hemiring, left h-ideal, anti fuzzy h-ideal, S-fuzzy left hideal, t-conorm , homomorphism.

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Authors: O. Poleshchuk, E.Komarov

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This paper presents a regression model for interval type-2 fuzzy sets based on the least squares estimation technique. Unknown coefficients are assumed to be triangular fuzzy numbers. The basic idea is to determine aggregation intervals for type-1 fuzzy sets, membership functions of whose are low membership function and upper membership function of interval type-2 fuzzy set. These aggregation intervals were called weighted intervals. Low and upper membership functions of input and output interval type-2 fuzzy sets for developed regression models are considered as piecewise linear functions.

Keywords: Interval type-2 fuzzy sets, fuzzy regression, weighted interval.

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Authors: Lazim Abdullah, N orhanadiah Zam

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Increasing number of vehicles and lack of awareness among road users may lead to road accidents. However no specific literature was found to rank vehicles involved in accidents based on fuzzy variables of road users. This paper proposes a ranking of four selected motor vehicles involved in road accidents. Human and non-human factors that normally linked with road accidents are considered for ranking. The imprecision or vagueness inherent in the subjective assessment of the experts has led the application of fuzzy sets theory to deal with ranking problems. Data in form of linguistic variables were collected from three authorised personnel of three Malaysian Government agencies. The Multi Criteria Decision Making, fuzzy TOPSIS was applied in computational procedures. From the analysis, it shows that motorcycles vehicles yielded the highest closeness coefficient at 0.6225. A ranking can be drawn using the magnitude of closeness coefficient. It was indicated that the motorcycles recorded the first rank.

Keywords: Road accidents, decision making, closeness coefficient, fuzzy number

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Authors: Sara El Manar El Bouanani, Ismail Kassou

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Our objective in this paper is to propose an approach capable of clustering web messages. The clustering is carried out by assigning, with a certain probability, texts written by the same web user to the same cluster based on Stylometric features and using fuzzy clustering algorithms. Focus in the present work is on comparing the most popular algorithms in fuzzy clustering theory namely, Fuzzy C-means, Possibilistic C-means and Fuzzy Possibilistic C-Means.

Keywords: Authorship detection, fuzzy clustering, profiling, stylometric features.

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Authors: E. Koyuncu

Abstract:

The problem of Order Acceptance and Scheduling (OAS) is defined as a joint decision of which orders to accept for processing and how to schedule them. Any linear programming model representing real-world situation involves the parameters defined by the decision maker in an uncertain way or by means of language statement. Fuzzy data can be used to incorporate vagueness in the real-life situation. In this study, a fuzzy mathematical model is proposed for a single machine OAS problem, where the orders are defined by their fuzzy due dates, fuzzy processing times, and fuzzy sequence dependent setup times. The signed distance method, one of the fuzzy ranking methods, is used to handle the fuzzy constraints in the model.

Keywords: Fuzzy mathematical programming, fuzzy ranking, order acceptance, single machine scheduling.

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Authors: Chien-Hua Wang, Wei-Hsuan Lee, Chin-Tzong Pang

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In data mining, the association rules are used to find for the associations between the different items of the transactions database. As the data collected and stored, rules of value can be found through association rules, which can be applied to help managers execute marketing strategies and establish sound market frameworks. This paper aims to use Fuzzy Frequent Pattern growth (FFP-growth) to derive from fuzzy association rules. At first, we apply fuzzy partition methods and decide a membership function of quantitative value for each transaction item. Next, we implement FFP-growth to deal with the process of data mining. In addition, in order to understand the impact of Apriori algorithm and FFP-growth algorithm on the execution time and the number of generated association rules, the experiment will be performed by using different sizes of databases and thresholds. Lastly, the experiment results show FFPgrowth algorithm is more efficient than other existing methods.

Keywords: Data mining, association rule, fuzzy frequent patterngrowth.

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Authors: Hae Young Lee, Tae Ho Cho

Abstract:

In many sensor network applications, sensor nodes are deployed in open environments, and hence are vulnerable to physical attacks, potentially compromising the node's cryptographic keys. False sensing report can be injected through compromised nodes, which can lead to not only false alarms but also the depletion of limited energy resource in battery powered networks. Ye et al. proposed a statistical en-route filtering scheme (SEF) to detect such false reports during the forwarding process. In this scheme, the choice of a security threshold value is important since it trades off detection power and overhead. In this paper, we propose a fuzzy logic for determining a security threshold value in the SEF based sensor networks. The fuzzy logic determines a security threshold by considering the number of partitions in a global key pool, the number of compromised partitions, and the energy level of nodes. The fuzzy based threshold value can conserve energy, while it provides sufficient detection power.

Keywords: Fuzzy logic, security, sensor network.

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Authors: Tamanna Siddiqui, M. Afshar Alam

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Censored Production Rule is an extension of standard production rule, which is concerned with problems of reasoning with incomplete information, subject to resource constraints and problem of reasoning efficiently with exceptions. A CPR has a form: IF A (Condition) THEN B (Action) UNLESS C (Censor), Where C is the exception condition. Fuzzy CPR are obtained by augmenting ordinary fuzzy production rule “If X is A then Y is B with an exception condition and are written in the form “If X is A then Y is B Unless Z is C. Such rules are employed in situation in which the fuzzy conditional statement “If X is A then Y is B" holds frequently and the exception condition “Z is C" holds rarely. Thus “If X is A then Y is B" part of the fuzzy CPR express important information while the unless part acts only as a switch that changes the polarity of “Y is B" to “Y is not B" when the assertion “Z is C" holds. The proposed approach is an attempt to discover fuzzy censored production rules from set of discovered fuzzy if then rules in the form: A(X) ÔçÆ B(Y) || C(Z).

Keywords: Uncertainty Quantification, Fuzzy if then rules, Fuzzy Censored Production Rules, Learning algorithm.

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Authors: Hazem. A. Khorshed, Mostafa A. El Gendy, Amer. Abd El-Razik

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In this paper, The T-G-action topology on a set acted on by a fuzzy T-neighborhood (T-neighborhood, for short) group is defined as a final T-neighborhood topology with respect to a set of maps. We mainly prove that this topology is a T-regular Tneighborhood topology.

Keywords: Fuzzy set, Fuzzy topology, Triangular norm, Separation axioms.

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Authors: Mohammad Hasan Raouf, Ahmad Rouhani, Mohammad Abedini, Ebrahim Rasooli Anarmarzi

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In this paper the application of a hierarchical fuzzy system (HFS) based on MPSS and SVC in multi-machine environment is studied. Also the effect of communication lines active power variance signal between two ΔP_Tie-line regions, as one of the inputs of hierarchical fuzzy multi-input PSS and SVC (HFMPSS & SVC), on the increase of low frequency oscillation damping is examined. In the MPSS, to have better efficiency an auxiliary signal of reactive power deviation (ΔQ) is added with ΔP+ Δω input type PSS. The number of rules grows exponentially with the number of variables in a classic fuzzy system. To reduce the number of rules the HFS consists of a number of low-dimensional fuzzy systems in a hierarchical structure. Phasor model of SVC is described and used in this paper. The performances of MPSS and ΔP_Tie-line based HFMPSS and also the proposed method in damping inter-area mode of oscillation are examined in response to disturbances. The efficiency of the proposed model is examined by simulating a four-machine power system. Results show that the proposed method is performing satisfactorily within the whole range of disturbances and reduces the cost of system.

Keywords: Communication lines active power variance signal, Hierarchical fuzzy system (HFS), Multi-input power system stabilizer (MPSS), Static VAR compensator (SVC).

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Authors: Melike Şah, Konstantin Y.Degtiarev

Abstract:

This paper proposes a novel improvement of forecasting approach based on using time-invariant fuzzy time series. In contrast to traditional forecasting methods, fuzzy time series can be also applied to problems, in which historical data are linguistic values. It is shown that proposed time-invariant method improves the performance of forecasting process. Further, the effect of using different number of fuzzy sets is tested as well. As with the most of cited papers, historical enrollment of the University of Alabama is used in this study to illustrate the forecasting process. Subsequently, the performance of the proposed method is compared with existing fuzzy time series time-invariant models based on forecasting accuracy. It reveals a certain performance superiority of the proposed method over methods described in the literature.

Keywords: Forecasting, fuzzy time series, linguistic values, student enrollment, time-invariant model.

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Authors: H. Daoud Dammak, S. Bensalem, S. Zouari, M. Loulou

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This paper presents an optimized methodology to folded cascode operational transconductance amplifier (OTA) design. The design is done in different regions of operation, weak inversion, strong inversion and moderate inversion using the gm/ID methodology in order to optimize MOS transistor sizing. Using 0.35μm CMOS process, the designed folded cascode OTA achieves a DC gain of 77.5dB and a unity-gain frequency of 430MHz in strong inversion mode. In moderate inversion mode, it has a 92dB DC gain and provides a gain bandwidth product of around 69MHz. The OTA circuit has a DC gain of 75.5dB and unity-gain frequency limited to 19.14MHZ in weak inversion region.

Keywords: CMOS IC design, Folded Cascode OTA, gm/ID methodology, optimization.

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Authors: Jian Tang

Abstract:

Let S be an ordered semigroup. In this paper we first introduce the concepts of (∈,∈ ∨q)-fuzzy ideals, (∈,∈ ∨q)-fuzzy bi-ideals and (∈,∈ ∨q)-fuzzy generalized bi-ideals of an ordered semigroup S, and investigate their related properties. Furthermore, we also define the upper and lower parts of fuzzy subsets of an ordered semigroup S, and investigate the properties of (∈,∈ ∨q)-fuzzy ideals of S. Finally, characterizations of regular ordered semigroups and intra-regular ordered semigroups by means of the lower part of (∈ ,∈ ∨q)-fuzzy left ideals, (∈,∈ ∨q)-fuzzy right ideals and (∈,∈ ∨q)- fuzzy (generalized) bi-ideals are given.

Keywords: Ordered semigroup, regular ordered semigroup, intraregular ordered semigroup, (∈, ∈ ∨q)-fuzzy left (right) ideal of an ordered semigroup, (∈, ∈ ∨q)-fuzzy (generalized) bi-ideal of an ordered semigroup.

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Authors: G. M. Arun Prasath, N. Arthi

Abstract:

This work proposed a multi-objective mathematical programming approach to select the appropriate supply network elements. The multi-item multi-objective production-distribution inventory model is formulated with possible constraints under fuzzy environment. The unit cost has taken under fuzzy environment. The inventory model and warehouse location model has combined to formulate the production-distribution inventory model. Warehouse location is important in supply chain network. Particularly, if a company maintains more selling stores it cannot maintain individual secondary warehouse near to each selling store. Hence, maintaining the optimum number of secondary warehouses is important. Hence, the combined mathematical model is formulated to reduce the total expenditure of the organization by arranging the network of minimum number of secondary warehouses. Numerical example has been taken to illustrate the proposed model.

Keywords: Fuzzy inventory model, warehouse location model, triangular fuzzy number, secondary warehouse, LINGO software.

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Authors: Behnaz Zohouri, Mahdi Zowghiand, Mohsen haghighi

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In this paper, a method for decision making in fuzzy environment is presented.A new subjective and objective integrated approach is introduced that used to assign weight attributes in fuzzy multiple attribute decision making (FMADM) problems and alternatives and fmally ranked by proposed method.

Keywords: Multiple Attribute Decision Making, Triangular fuzzy numbers, ranking index, Fuzzy Entropy.

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Authors: C. Ardil

Abstract:

This article presents a multiple criteria evaluation approach to uncertainty, vagueness, and imprecision analysis for ranking alternatives with fuzzy data for decision making using the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS). The fighter aircraft evaluation and selection decision making problem is modeled in a fuzzy environment with triangular fuzzy numbers. The fuzzy decision information related to the fighter aircraft selection problem is taken into account in ordering the alternatives and selecting the best candidate. The basic fuzzy TOPSIS procedure steps transform fuzzy decision matrices into matrices of alternatives evaluated according to all decision criteria. A practical numerical example illustrates the proposed approach to the fighter aircraft selection problem.

Keywords: triangular fuzzy number (TFN), multiple criteria decision making analysis, decision making, aircraft selection, MCDMA, fuzzy TOPSIS

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Authors: Sophia Fry, Mahir Irtiza, Alexa Hoffman, Yousef Sardahi

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This study presents an intelligent control algorithm for a flexible robotic arm. Fuzzy control is used to control the motion of the arm to maintain the arm tip at the desired position while reducing vibration and increasing the system speed of response. The Fuzzy controller (FC) is based on adding the tip angular position to the arm deflection angle and using their sum as a feedback signal to the control algorithm. This reduces the complexity of the FC in terms of the input variables, number of membership functions, fuzzy rules, and control structure. Also, the design of the fuzzy controller is model-free and uses only our knowledge about the system. To show the efficacy of the FC, the control algorithm is implemented on the flexible joint manipulator (FJM) developed by Quanser. The results show that the proposed control method is effective in terms of response time, overshoot, and vibration amplitude.

Keywords: Fuzzy logic control, model-free control, flexible joint manipulators, nonlinear control.

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Authors: Dalhatu, Musa Yusha’u, Shuaibu Sidi Safiyanu, Haliru Musa Hussaini

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This study examined budget and performance of public enterprises in Nasarawa State, Nigeria in a period of 2009-2013. The study utilized secondary sources of data obtained from four selected parastatals’ budget allocation and revenue generation for the period under review. The simple correlation coefficient was used to analyze the extent of the relationship between budget allocation and revenue generation of the parastatals. Findings revealed varying results. There was positive (0.21) and weak correlation between expenditure and revenue of Nasarawa Investment and Property Development Company (NIPDC). However, the study further revealed that there was strong and weak negative relationship in the revenue and expenditure of the following parastatals over the period under review. Viz: Nasarawa State Water Board, -0.27 (weak), Nasarawa State Broadcasting Service, -0.52 (Strong) and Nasarawa State College of Agriculture, -0.36 (weak). The study therefore, recommends that government should increase its investments in NIPDC to enhance efficiency and profitability. It also recommends that government should strengthen its fiscal responsibility, accountability and transparency in public parastatals.

Keywords: Allocation, Budget, Public Enterprises, Parastatals, Performance.

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Authors: Nurhakimah Ab. Rahman, Lazim Abdullah

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According to fuzzy arithmetic, dual fuzzy polynomials cannot be replaced by fuzzy polynomials. Hence, the concept of ranking method is used to find real roots of dual fuzzy polynomial equations. Therefore, in this study we want to propose an interval type-2 dual fuzzy polynomial equation (IT2 DFPE). Then, the concept of ranking method also is used to find real roots of IT2 DFPE (if exists). We transform IT2 DFPE to system of crisp IT2 DFPE. This transformation performed with ranking method of fuzzy numbers based on three parameters namely value, ambiguity and fuzziness. At the end, we illustrate our approach by two numerical examples.

Keywords: Dual fuzzy polynomial equations, Interval type-2, Ranking method, Value.

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Authors: O. Belalia Douma, B. Boukhatem, M. Ghrici

Abstract:

Self-compacting concrete (SCC) developed in Japan in the late 80s has enabled the construction industry to reduce demand on the resources, improve the work condition and also reduce the impact of environment by elimination of the need for compaction. Fuzzy logic (FL) approaches has recently been used to model some of the human activities in many areas of civil engineering applications. Especially from these systems in the model experimental studies, very good results have been obtained. In the present study, a model for predicting compressive strength of SCC containing various proportions of fly ash, as partial replacement of cement has been developed by using Fuzzy Inference System (FIS). For the purpose of building this model, a database of experimental data were gathered from the literature and used for training and testing the model. The used data as the inputs of fuzzy logic models are arranged in a format of five parameters that cover the total binder content, fly ash replacement percentage, water content, superplasticizer and age of specimens. The training and testing results in the fuzzy logic model have shown a strong potential for predicting the compressive strength of SCC containing fly ash in the considered range.

Keywords: Self-compacting concrete, fly ash, strength prediction, fuzzy logic.

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Authors: M. A. Sharbafi, M. Shakiba Herfeh, Caro Lucas, A. Mohammadi Nejad

Abstract:

Several researchers have proposed methods about combination of Genetic Algorithm (GA) and Fuzzy Logic (the use of GA to obtain fuzzy rules and application of fuzzy logic in optimization of GA). In this paper, we suggest a new method in which fuzzy decision making is used to improve the performance of genetic algorithm. In the suggested method, we determine the alleles that enhance the fitness of chromosomes and try to insert them to the next generation. In this algorithm we try to present an innovative vaccination in the process of reproduction in genetic algorithm, with considering the trade off between exploration and exploitation.

Keywords: Genetic Algorithm, Fuzzy Decision Making.

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Authors: Chong Zhang, Mu-Xuan Tao

Abstract:

In several earthquakes, numerous reinforced concrete (RC) frames subjected to seismic excitation demonstrated a collapse pattern characterized by column hinges, though designed according to the Strong-Column-Weak-Beam (S-C-W-B) criteria. The effect of biaxial seismic excitation on the disparity between design and actual performance is carefully investigated in this article. First, a modified load contour method is proposed to derive a closed-form equation of biaxial bending moment strength, which is verified by numerical and experimental tests. Afterwards, a group of time history analyses of a simple frame modeled by fiber beam-column elements subjected to biaxial seismic excitation are conducted to verify that the current S-C-W-B criteria are not adequate to prevent the occurrence of column hinges. A biaxial over-strength factor is developed based on the proposed equation, and the reinforcement of columns is appropriately amplified with this factor to prevent the occurrence of column hinges under biaxial excitation, which is proved to be effective by another group of time history analyses.

Keywords: Biaxial bending moment strength, biaxial seismic excitation, fiber beam-column model, load contour method, strong-column-weak-beam.

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Authors: Jing Liu

Abstract:

In this paper, motivated by the ideas of dependent weighted aggregation operators, we develop some new hesitant fuzzy dependent weighted aggregation operators to aggregate the input arguments taking the form of hesitant fuzzy numbers rather than exact numbers, or intervals. In fact, we propose three hesitant fuzzy dependent weighted averaging(HFDWA) operators, and three hesitant fuzzy dependent weighted geometric(HFDWG) operators based on different weight vectors, and the most prominent characteristic of these operators is that the associated weights only depend on the aggregated hesitant fuzzy numbers and can relieve the influence of unfair hesitant fuzzy numbers on the aggregated results by assigning low weights to those “false” and “biased” ones. Some examples are given to illustrated the efficiency of the proposed operators.

Keywords: Hesitant fuzzy numbers, hesitant fuzzy dependent weighted averaging(HFDWA) operators, hesitant fuzzy dependent weighted geometric(HFDWG) operators.

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Authors: Nuanwan Soonthornphisaj, Boonserm Kijsirikul

Abstract:

This paper presents a semi-supervised learning algorithm called Iterative-Cross Training (ICT) to solve the Web pages classification problems. We apply Inductive logic programming (ILP) as a strong learner in ICT. The objective of this research is to evaluate the potential of the strong learner in order to boost the performance of the weak learner of ICT. We compare the result with the supervised Naive Bayes, which is the well-known algorithm for the text classification problem. The performance of our learning algorithm is also compare with other semi-supervised learning algorithms which are Co-Training and EM. The experimental results show that ICT algorithm outperforms those algorithms and the performance of the weak learner can be enhanced by ILP system.

Keywords: Inductive Logic Programming, Semi-supervisedLearning, Web Page Categorization

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Authors: Arshia Azam, J. Amarnath, Ch. D. V. Paradesi Rao

Abstract:

The advantage of solving the complex nonlinear problems by utilizing fuzzy logic methodologies is that the experience or expert-s knowledge described as a fuzzy rule base can be directly embedded into the systems for dealing with the problems. The current limitation of appropriate and automated designing of fuzzy controllers are focused in this paper. The structure discovery and parameter adjustment of the Branched T-S fuzzy model is addressed by a hybrid technique of type constrained sparse tree algorithms. The simulation result for different system model is evaluated and the identification error is observed to be minimum.

Keywords: Fuzzy logic, branch T-S fuzzy model, tree modeling, complex nonlinear system.

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Authors: S.K.Sardar, B.Davvaz, S.Kayal, S.K.Majumdar

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The purpose of this note is to obtain some properties of (∈,∈ ∨q)- fuzzy bi-ideals in a Γ-semigroup in order to characterize regular and intra-regular Γ-semigroups.

Keywords: Regular Γ-semigroup, belong to or quasi-coincident, (∈, ∈ ∨q)-fuzzy subsemigroup, (∈, ∈ ∨q)-fuzzy bi-ideals.

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Authors: Meredith Stevenson, John E. Porter

Abstract:

Since the pioneering work of Zadeh, fuzzy set theory has been applied to a myriad of areas. Song and Chissom introduced the concept of fuzzy time series and applied some methods to the enrollments of the University of Alabama. In recent years, a number of techniques have been proposed for forecasting based on fuzzy set theory methods. These methods have either used enrollment numbers or differences of enrollments as the universe of discourse. We propose using the year to year percentage change as the universe of discourse. In this communication, the approach of Jilani, Burney, and Ardil is modified by using the year to year percentage change as the universe of discourse. We use enrollment figures for the University of Alabama to illustrate our proposed method. The proposed method results in better forecasting accuracy than existing models.

Keywords: Fuzzy forecasting, fuzzy time series, fuzzified enrollments, time-invariant model

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