Search results for: Analytical model

Authors: Amir Alani, Mojtaba Mahmoodian, Anna Romanova, Asaad Faramarzi

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Pipelines are extensively used engineering structures which convey fluid from one place to another. Most of the time, pipelines are placed underground and are encumbered by soil weight and traffic loads. Corrosion of pipe material is the most common form of pipeline deterioration and should be considered in both the strength and serviceability analysis of pipes. The study in this research focuses on concrete pipes in sewage systems (concrete sewers). This research firstly investigates how to involve the effect of corrosion as a time dependent process of deterioration in the structural and failure analysis of this type of pipe. Then three probabilistic time dependent reliability analysis methods including the first passage probability theory, the gamma distributed degradation model and the Monte Carlo simulation technique are discussed and developed. Sensitivity analysis indexes which can be used to identify the most important parameters that affect pipe failure are also discussed. The reliability analysis methods developed in this paper contribute as rational tools for decision makers with regard to the strengthening and rehabilitation of existing pipelines. The results can be used to obtain a cost-effective strategy for the management of the sewer system.

Keywords: Reliability analysis, service life prediction, Monte Carlo simulation method, first passage probability theory, gamma distributed degradation model.

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

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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: J. Y. Lee, H. S. Lim, S. H. Yoon

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This paper presents an analysis of the diagonal crack widths of RC members with various types of materials by simulating a compatibility-aided truss model. The analytical results indicated that the diagonal crack width was influenced by not only the shear reinforcement ratio but also the yield strength of shear reinforcement and the compressive strength of concrete. The yield strength of shear reinforcement and the compressive strength of concrete decreased the diagonal shear crack width of RC members for the same shear force because of the change of shear failure modes. However, regarding the maximum shear crack width at shear failure, the shear crack width of the beam with high strength materials was greater than that of the beam with normal strength materials.

Keywords: Diagonal crack width, high strength stirrups, high strength concrete, RC members, shear behavior.

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Authors: Michel Soto Chalhoub

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Building code-related literature provides recommendations on normalizing approaches to the calculation of the dynamic properties of structures. Most building codes make a distinction among types of structural systems, construction material, and configuration through a numerical coefficient in the expression for the fundamental period. The period is then used in normalized response spectra to compute base shear. The typical parameter used in simplified code formulas for the fundamental period is overall building height raised to a power determined from analytical and experimental results. However, reinforced concrete buildings which constitute the majority of built space in less developed countries pose additional challenges to the ones built with homogeneous material such as steel, or with concrete under stricter quality control. In the present paper, the particularities of reinforced concrete buildings are explored and related to current methods of equivalent static analysis. A comparative study is presented between the Uniform Building Code, commonly used for buildings within and outside the USA, and data from the Middle East used to model 151 reinforced concrete buildings of varying number of bays, number of floors, overall building height, and individual story height. The fundamental period was calculated using eigenvalue matrix computation. The results were also used in a separate regression analysis where the computed period serves as dependent variable, while five building properties serve as independent variables. The statistical analysis shed light on important parameters that simplified code formulas need to account for including individual story height, overall building height, floor plan, number of bays, and concrete properties. Such inclusions are important for reinforced concrete buildings of special conditions due to the level of concrete damage, aging, or materials quality control during construction. Overall results of the present analysis show that simplified code formulas for fundamental period and base shear may be applied but they require revisions to account for multiple parameters. The conclusion above is confirmed by the analytical model where fundamental periods were computed using numerical techniques and eigenvalue solutions. This recommendation is particularly relevant to code upgrades in less developed countries where it is customary to adopt, and mildly adapt international codes. We also note the necessity of further research using empirical data from buildings in Lebanon that were subjected to severe damage due to impulse loading or accelerated aging. However, we excluded this study from the present paper and left it for future research as it has its own peculiarities and requires a different type of analysis.

Keywords: Seismic behavior, reinforced concrete, simplified code formulas, equivalent static analysis, base shear, response spectra.

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Authors: Hui Hsin Huang

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The downstream manufactures will order their materials from different upstream suppliers to maintain a certain level of the demand. This paper proposes a bivariate model to portray this phenomenon of material demand. We use empirical data to estimate the parameters of model and evaluate the RMSD of model calibration. The results show that the model has better fitness.

Keywords: Farlie-Gumbel-Morgenstern family of bivariate distributions, multi-source ordering, materials demand quantity, recency, ordering time.

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Authors: Adibah Shuib, Zati Aqmar Zaharudin

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Our study is concerned with the development of an Emergency Medical Services (EMS) ambulance location and allocation model called the Time-based Ambulance Zoning Optimization Model (TAZ_OPT). This paper presents the framework of the study. The model is formulated using the goal programming (GP), where the goals are to determine the satellite locations of ambulances and the number of ambulances to be allocated at these locations. The model aims at maximizing the expected demand coverage based on probability of reaching the emergency location within targetted time, and minimizing the ambulance busyness likelihood value. Among the benefits of the model is the increased accessibility and availability of ambulances, thus, enhanced quality of the EMS ambulance services.

Keywords: Optimization, Ambulance Location, Location facilities.

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Authors: A. Lakrim, D. Tahri

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This paper sets out a behavioral macro-model of a Merged PiN and Schottky (MPS) diode based on silicon carbide (SiC). This model holds good for both static and dynamic electrothermal simulations for industrial applications. Its parameters have been worked out from datasheets curves by drawing on the optimization method: Simulated Annealing (SA) for the SiC MPS diodes made available in the industry. The model also adopts the Analog Behavioral Model (ABM) of PSPICE in which it has been implemented. The thermal behavior of the devices was also taken into consideration by making use of Foster’ canonical network as figured out from electro-thermal measurement provided by the manufacturer of the device.

Keywords: SiC MPS Diode, electro-thermal, SPICE Model.

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Authors: X. Gumara Rigol, I. Martínez de Apellaniz Anzuola, A. Garcia Serrano, A. Franzi Cros, O. Vidal Calbet, A. Al Maruf

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Traditional federated architectures for data warehousing work well when corporations have existing regional data warehouses and there is a need to aggregate data at a global level. Schibsted Media Group has been maturing from a decentralised organisation into a more globalised one and needed to build both some of the regional data warehouses for some brands at the same time as the global one. In this paper, we present the architectural alternatives studied and why a custom federated approach was the notable recommendation to go further with the implementation. Although the data warehouses are logically federated, the implementation uses a single database system which presented many advantages like: cost reduction and improved data access to global users allowing consumers of the data to have a common data model for detailed analysis across different geographies and a flexible layer for local specific needs in the same place.

Keywords: Data integration, data warehousing, federated architecture, online analytical processing.

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Authors: H. M. Mobarak, Helen Wu, Chunhui Yang

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In this study, an analytical model is developed to study crack breathing behavior under the effect of crack location and unbalance force. Crack breathing behavior is determined using effectual bending angle by studying the transient change in closed area of the crack. The status of the crack of a balanced shaft is symmetrical about shaft rotational angle and the duration of each crack status remains unchanged. The global stiffness of the balanced shaft is independent of crack location. Different crack breathing behavior for the unbalanced shaft has been observed. The influence of crack location on the unbalanced shaft stiffness can be divided into three regions. When the crack is located between 0.3L and 0.8335L, where L is the total length of the shaft, the unbalanced shaft is less stiff and when located outside this region it is stiffer than the balanced shaft. It was also found that unbalanced shaft stiffness has a maximum value with a crack at 0.1946L, a minimum value at 0.8053L and same value as balanced shaft at 0.3L and 0.8335L.

Keywords: Cracked shaft, crack location, shaft stiffness, unbalanced force.

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Authors: Chantawan Noisri, Robert Harold Buchanan Exell

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A simple model for studying convectional lifting processes in the tropics is described in this paper with some tests of the model in dry air. The model consists of the density equation, the wind equation, the vertical velocity equation, and the temperature equation. The model domain is two-dimensional with length 100 km and height 17.5 km. Plan for experiments to investigate the effects of the heating surface, the deep convection approximation and the treatment of velocities at the boundaries are discussed. Equations for the simplified treatment of moisture in the atmosphere in future numerical experiments are also given.

Keywords: Numerical weather prediction, Finite differences, Convection lifting.

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Authors: Keny Ordaz-Hernandez, Xavier Fischer, Fouad Bennis

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In a particular case of behavioural model reduction by ANNs, a validity domain shortening has been found. In mechanics, as in other domains, the notion of validity domain allows the engineer to choose a valid model for a particular analysis or simulation. In the study of mechanical behaviour for a cantilever beam (using linear and non-linear models), Multi-Layer Perceptron (MLP) Backpropagation (BP) networks have been applied as model reduction technique. This reduced model is constructed to be more efficient than the non-reduced model. Within a less extended domain, the ANN reduced model estimates correctly the non-linear response, with a lower computational cost. It has been found that the neural network model is not able to approximate the linear behaviour while it does approximate the non-linear behaviour very well. The details of the case are provided with an example of the cantilever beam behaviour modelling.

Keywords: artificial neural network, validity domain, cantileverbeam, non-linear behaviour, model reduction.

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Authors: Irtaza M. Syed, Kaamran Raahemifar

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This paper presents a model predictive control (MPC) of a utility interactive (UI) single phase inverter (SPI) for a photovoltaic (PV) system at residential/distribution level. The proposed model uses single-phase phase locked loop (PLL) to synchronize SPI with the grid and performs MPC control in a dq reference frame. SPI model consists of boost converter (BC), maximum power point tracking (MPPT) control, and a full bridge (FB) voltage source inverter (VSI). No PI regulators to tune and carrier and modulating waves are required to produce switching sequence. Instead, the operational model of VSI is used to synthesize sinusoidal current and track the reference. Model is validated using a three kW PV system at the input of UI-SPI in Matlab/Simulink. Implementation and results demonstrate simplicity and accuracy, as well as reliability of the model.

Keywords: Matlab/Simulink, Model Predictive Control, Phase Locked Loop, Single Phase Inverter, Voltage Source Inverter.

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Authors: Marco Raciti Castelli, Giulia Simioni, Ernesto Benini

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This paper presents a model for the evaluation of energy performance and aerodynamic forces acting on a three-bladed small vertical axis Darrieus wind turbine depending on blade chord curvature with respect to rotor axis. The adopted survey methodology is based on an analytical code coupled to a solid modeling software, capable of generating the desired blade geometry depending on the blade design geometric parameters, which is linked to a finite volume CFD code for the calculation of rotor performance. After describing and validating the model with experimental data, the results of numerical simulations are proposed on the bases of two different blade profile architectures, which are respectively characterized by a straight chord and by a curved one, having a chord radius equal to rotor external circumference. A CFD campaign of analysis is completed for three blade-candidate airfoil sections, that is the recently-developed DU 06-W-200 cambered blade profile, a classical symmetrical NACA 0021 and its derived cambered airfoil, characterized by a curved chord, having a chord radius equal to rotor external circumference. The effects of blade chord curvature on angle of attack, blade tangential and normal forces are first investigated and then the overall rotor torque and power are analyzed as a function of blade azimuthal position, achieving a numerical quantification of the influence of blade camber on overall rotor performance.

Keywords: VAWT, NACA 0021, DU 06-W-200, cambered airfoil

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Authors: Jin Y. Park, Jeong Wan Lee

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An experimental and analytical research on shear buckling of a comparably large polymer composite I-section is presented. It is known that shear buckling load of a large span composite beam is difficult to determine experimentally. In order to sensitively detect shear buckling of the tested I-section, twenty strain rosettes and eight displacement sensors were applied and attached on the web and flange surfaces. The tested specimen was a pultruded composite beam made of vinylester resin, E-glass, carbon fibers and micro-fillers. Various coupon tests were performed before the shear buckling test to obtain fundamental material properties of the Isection. An asymmetric four-point bending loading scheme was utilized for the shear test. The loading scheme resulted in a high shear and almost zero moment condition at the center of the web panel. The shear buckling load was successfully determined after analyzing the obtained test data from strain rosettes and displacement sensors. An analytical approach was also performed to verify the experimental results and to support the discussed experimental program.

Keywords: Strain sensor, displacement sensor, shear buckling, polymer composite I-section, asymmetric loading.

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Authors: Lertlop, W

Abstract:

This research aims to study the appropriate model of Science teaching for students, academic achievement and to survey students- attitudes toward using appropriate for students in Mathayomsuksa 3 in Ang-Thong province. The research results were as follows: 1. The appropriate model of Science teaching for Mathayomsuksa 3 students in Ang-Thong province including these following five steps: Step 1. The introduction of the lessons. Step 2. Setting the hypothesis. Step 3. Doing the experiment /survey. Step 4. Making conclusion. Step 5. Applying to daily life or other subjects. 2. There is no significant difference between using appropriate model teaching and regular teaching at 0.05 level significant difference. 3. There is a significant difference between before and after teaching using appropriate model of Science teaching at 0.05 level. 4. The satisfaction of students- attitudes to using the appropriate model of Science teaching for students was in intermediate level.

Keywords: Pedagogy, science teaching model, Ang-Thong province.

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Authors: Víctor Hugo Medina García, Darío Alejandro Segura Torres

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This paper presents the adaptation of the knowledge management model and intellectual capital measurement NOVA to the needs of work or research project must be developed when conducting a program of graduate-level master. Brackets are added in each of the blocks which is represented in the original model NOVA and which allows to represent those involved in each of these.

Keywords: Knowledge management, masters programs, Nova model, research projects

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Authors: Saadia Drissi, Siham Benhadou, Hicham Medromi

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The cloud computing is an innovative paradigm that introduces several changes in technology that have resulted a new ways for cloud providers to deliver their services to cloud consumers mainly in term of security risk assessment, thus, adapting a current risk assessment tools to cloud computing is a very difficult task due to its several characteristics that challenge the effectiveness of risk assessment approaches. As consequence, there is a need of risk assessment model adapted to cloud computing. This paper requires a new risk assessment model based on multi-agent system and AHP model as fundamental steps towards the development of flexible risk assessment approach regarding cloud consumers.

Keywords: Cloud computing, risk assessment model, multi-agent system, AHP model, cloud consumer.

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Authors: Ali A. Al-Jazaairry, Tahsin T. Sabbagh

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One of the important concerns within the field of geotechnical engineering is the presence of cavities in soils. This present work is an attempt to understand the behaviour of strip footing subjected to inclined load and constructed on cavitied soil. The failure mechanism of strip footing located above such soils was studied analytically. The capability of analytical model to correctly expect the system behaviour is assessed by carrying out verification analysis on available studies. The study was prepared by finite element software (PLAXIS) in which an elastic-perfectly plastic soil model was used. It was indicated, from the results of the study, that the load carrying capacity of foundation constructed on cavity can be analysed well using such analysis. The research covered many foundation cases, and in each foundation case, there occurs a critical depth under which the presence of cavities has shown minimum impact on the foundation performance. When cavities are found above this critical depth, the load carrying capacity of the foundation differs with many influences, such as the location and size of the cavity and footing depth. Figures involving the load carrying capacity with the affecting factors studied are presented. These figures offer information beneficial for the design of strip footings rested on underground cavities. Moreover, the results might be used to design a shallow foundation constructed on cavitied soil, whereas the obtained failure mechanisms may be employed to improve numerical solutions for this kind of problems.

Keywords: Axial load, cavity, inclined load, strip footing.

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Authors: Z. Veselý, M. Honner, J. Mach

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The aim of the performed work is to establish the 2D and 3D model of direct unsteady task of sample heat treatment by moving source employing computer model on the basis of finite element method. Complex boundary condition on heat loaded sample surface is the essential feature of the task. Computer model describes heat treatment of the sample during heat source movement over the sample surface. It is started from 2D task of sample cross section as a basic model. Possibilities of extension from 2D to 3D task are discussed. The effect of the addition of third model dimension on temperature distribution in the sample is showed. Comparison of various model parameters on the sample temperatures is observed. Influence of heat source motion on the depth of material heat treatment is shown for several velocities of the movement. Presented computer model is prepared for the utilization in laser treatment of machine parts.

Keywords: Computer simulation, unsteady model, heat treatment, complex boundary condition, moving heat source.

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Authors: Zeinab Sayed Abdel-Rehim

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A computational fluid dynamic (CFD-Fluent 6.2) for two-dimensional fluid flow is applied to predict the pressure drop and heat transfer characteristics of laminar and turbulent flow past staggered flat-tube bank. Effect of aspect ratio ((H/D)/(L/D)) on pressure drop, temperature, and velocity contour for laminar and turbulent flow over staggered flat-tube bank is studied. The theoretical results of the present models are compared with previously published experimental data of different authors. Satisfactory agreement is demonstrated. Also, the comparison between the present study and others analytical methods for the Re number with Nu number is done. The results show as the Reynolds number increases the maximum velocity in the passage between the upper and lower tubes increases. The comparisons show a fair agreement especially in the turbulent flow region. The good agreement of the data of this work with these recommended analytical methods validates the current study.

Keywords: Aspect ratio ((H/D)/(L/D)), CFD, fluid flow, heat transfer, staggered arrangement, tube bank, and turbulent flow.

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Authors: Reza Hassanpour, Asadollah Shahbahrami, Stephan Wong

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Skin color based tracking techniques often assume a static skin color model obtained either from an offline set of library images or the first few frames of a video stream. These models can show a weak performance in presence of changing lighting or imaging conditions. We propose an adaptive skin color model based on the Gaussian mixture model to handle the changing conditions. Initial estimation of the number and weights of skin color clusters are obtained using a modified form of the general Expectation maximization algorithm, The model adapts to changes in imaging conditions and refines the model parameters dynamically using spatial and temporal constraints. Experimental results show that the method can be used in effectively tracking of hand and face regions.

Keywords: Face detection, Segmentation, Tracking, Gaussian Mixture Model, Adaptation.

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Authors: Shih-Ching Lo, Yu-Ping Liao

Abstract:

Social-economic variables influence transportation demand largely. Analyses of discrete choice model consider social-economic variables to study traveler-s mode choice and demand. However, to calibrate the discrete choice model needs to have plenty of questionnaire survey. Also, an aggregative model is proposed. The historical data of passenger volumes for high speed rail and domestic civil aviation are employed to calibrate and validate the model. In this study, models with different social-economic variables, which are oil price, GDP per capita, CPI and economic growth rate, are compared. From the results, the model with the oil price is better than models with the other social-economic variables.

Keywords: forecasting, passenger volume, dynamic competitive model, social-economic variables, oil price.

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Authors: Anastasia Rita Widiarti, Phalita Nari Wastu

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Hidden Markov Model (HMM) is a stochastic method which has been used in various signal processing and character recognition. This study proposes to use HMM to recognize Javanese characters from a number of different handwritings, whereby HMM is used to optimize the number of state and feature extraction. An 85.7 % accuracy is obtained as the best result in 16-stated vertical model using pure HMM. This initial result is satisfactory for prompting further research.

Keywords: Character recognition, off-line handwritingrecognition, Hidden Markov Model.

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Authors: Imen Sfaihi, Noureddine Hamdi

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In this paper, we investigate the study of techniques for scheduling users for resource allocation in the case of multiple input and multiple output (MIMO) packet transmission systems. In these systems, transmit antennas are assigned to one user or dynamically to different users using spatial multiplexing. The allocation of all transmit antennas to one user cannot take full advantages of multi-user diversity. Therefore, we developed the case when resources are allocated dynamically. At each time slot users have to feed back their channel information on an uplink feedback channel. Channel information considered available in the schedulers is the zero forcing (ZF) post detection signal to interference plus noise ratio. Our analysis study concerns the round robin and the opportunistic schemes. In this paper, we present an overview and a complete capacity analysis of these schemes. The main results in our study are to give an analytical form of system capacity using the ZF receiver at the user terminal. Simulations have been carried out to validate all proposed analytical solutions and to compare the performance of these schemes.

Keywords: MIMO, scheduling, ZF receiver, spatial multiplexing, round robin scheduling, opportunistic.

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Authors: Brian A. Smith, Ronald W. McClendon, Gerrit Hoogenboom

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The mitigation of crop loss due to damaging freezes requires accurate air temperature prediction models. An improved model for temperature prediction in Georgia was developed by including information on seasonality and modifying parameters of an existing artificial neural network model. Alternative models were compared by instantiating and training multiple networks for each model. The inclusion of up to 24 hours of prior weather information and inputs reflecting the day of year were among improvements that reduced average four-hour prediction error by 0.18°C compared to the prior model. Results strongly suggest model developers should instantiate and train multiple networks with different initial weights to establish appropriate model parameters.

Keywords: Time-series forecasting, weather modeling.

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Authors: Georgi I. Petkov, Ivan I. Vankov, Yolina A. Petrova

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A structure-based model of category learning and categorization at different levels of abstraction is presented. The model compares different structures and expresses their similarity implicitly in the forms of mappings. Based on this similarity, the model can categorize different targets either as members of categories that it already has or creates new categories. The model is novel using two threshold parameters to evaluate the structural correspondence. If the similarity between two structures exceeds the higher threshold, a new sub-ordinate category is created. Vice versa, if the similarity does not exceed the higher threshold but does the lower one, the model creates a new category on higher level of abstraction.

Keywords: Analogy-making, categorization, learning of categories, abstraction, hierarchical structure.

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Authors: R. Ramli, M. I. Ghazali, H. Ibrahim, M. M. Kasim, F. M. Kamal, S.Vikneswari

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Teachers form the backbone of any educational system, hence selecting qualified candidates is very crucial. In Malaysia, the decision making in the selection process involves a few stages: Initial filtering through academic achievement, taking entry examination and going through an interview session. The last stage is the most challenging since it highly depends on human judgment. Therefore, this study sought to identify the selection criteria for teacher candidates that form the basis for an efficient multi-criteria teacher-candidate selection model for that last stage. The relevant criteria were determined from the literature and also based on expert input that is those who were involved in interviewing teacher candidates from a public university offering the formal training program. There are three main competency criteria that were identified which are content of knowledge, communication skills and personality. Further, each main criterion was divided into a few subcriteria. The Analytical Hierarchy Process (AHP) technique was employed to allocate weights for the criteria and later, integrated a Simple Weighted Average (SWA) scoring approach to develop the selection model. Subsequently, a web-based Decision Support System was developed to assist in the process of selecting the qualified teacher candidates. The Teacher-Candidate Selection (TeCaS) system is able to assist the panel of interviewers during the selection process which involves a large amount of complex qualitative judgments.

Keywords: Analytic Hierarchy Process, Simple Weighted Average, Decision Support System, Multi-criteria decision making problem.

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Authors: Kostas Metaxiotis

Abstract:

Portfolio optimization is one of the most important topics in finance. This paper proposes a mean–variance–skewness (MVS) portfolio optimization model. Traditionally, the portfolio optimization problem is solved by using the mean–variance (MV) framework. In this study, we formulate the proposed model as a three-objective optimization problem, where the portfolio's expected return and skewness are maximized whereas the portfolio risk is minimized. For solving the proposed three-objective portfolio optimization model we apply an adapted version of the non-dominated sorting genetic algorithm (NSGAII). Finally, we use a real dataset from FTSE-100 for validating the proposed model.

Keywords: Evolutionary algorithms, portfolio optimization, skewness, stock selection.

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Authors: C. Patrascioiu, V. Matei, N. Nicolae

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The paper describes the experiments and the kinetic parameters calculus of the gasoil hydrofining. They are presented experimental results of gasoil hidrofining using Mo and promoted with Ni on aluminum support catalyst. The authors have adapted a kinetic model gasoil hydrofining. Using this proposed kinetic model and the experimental data they have calculated the parameters of the model. The numerical calculus is based on minimizing the difference between the experimental sulf concentration and kinetic model estimation.

Keywords: Hydrofining, kinetic, modeling, optimization.

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Authors: Da-kuan Wei, Xian-zhong Zhou, Dong-jun Xin, Zhi-wei Chen

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The information systems with incomplete attribute values and fuzzy decisions commonly exist in practical problems. On the base of the notion of variable precision rough set model for incomplete information system and the rough set model for incomplete and fuzzy decision information system, the variable rough set model for incomplete and fuzzy decision information system is constructed, which is the generalization of the variable precision rough set model for incomplete information system and that of rough set model for incomplete and fuzzy decision information system. The knowledge reduction and heuristic algorithm, built on the method and theory of precision reduction, are proposed.

Keywords: Rough set, Incomplete and fuzzy decision information system, Limited valued tolerance relation, Knowledge reduction, Variable rough set model

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