Search results for: Boussinesq's approximation and Mean Nusselt number.

Authors: N. Mamode Khan, V. Jowaheer

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Number of breakdowns experienced by a machinery is a highly under-dispersed count random variable and its value can be attributed to the factors related to the mechanical input and output of that machinery. Analyzing such under-dispersed count observations as a function of the explanatory factors has been a challenging problem. In this paper, we aim at estimating the effects of various factors on the number of breakdowns experienced by a passenger car based on a study performed in Mauritius over a year. We remark that the number of passenger car breakdowns is highly under-dispersed. These data are therefore modelled and analyzed using Com-Poisson regression model. We use quasi-likelihood estimation approach to estimate the parameters of the model. Under-dispersion parameter is estimated to be 2.14 justifying the appropriateness of Com-Poisson distribution in modelling under-dispersed count responses recorded in this study.

Keywords: Breakdowns, under-dispersion, com-poisson, generalized linear model, quasi-likelihood estimation

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Authors: Debashree Guha, Debjani Chakraborty

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The aim of this paper is to adopt a compromise ratio (CR) methodology for fuzzy multi-attribute single-expert decision making proble. In this paper, the rating of each alternative has been described by linguistic terms, which can be expressed as triangular fuzzy numbers. The compromise ratio method for fuzzy multi-attribute single expert decision making has been considered here by taking the ranking index based on the concept that the chosen alternative should be as close as possible to the ideal solution and as far away as possible from the negative-ideal solution simultaneously. From logical point of view, the distance between two triangular fuzzy numbers also is a fuzzy number, not a crisp value. Therefore a fuzzy distance measure, which is itself a fuzzy number, has been used here to calculate the difference between two triangular fuzzy numbers. Now in this paper, with the help of this fuzzy distance measure, it has been shown that the compromise ratio is a fuzzy number and this eases the problem of the decision maker to take the decision. The computation principle and the procedure of the compromise ratio method have been described in detail in this paper. A comparative analysis of the compromise ratio method previously proposed [1] and the newly adopted method have been illustrated with two numerical examples.

Keywords: Compromise ratio method, Fuzzy multi-attributesingle-expert decision making, Fuzzy number, Linguistic variable

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Authors: Yan Ma, ShunBao Li

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A new approach is adopted in this paper based on Turk and Pentland-s eigenface method. It was found that the probability density function of the distance between the projection vector of the input face image and the average projection vector of the subject in the face database, follows Rayleigh distribution. In order to decrease the false acceptance rate and increase the recognition rate, the input face image has been recognized using two thresholds including the acceptance threshold and the rejection threshold. We also find out that the value of two thresholds will be close to each other as number of trials increases. During the training, in order to reduce the number of trials, the projection vectors for each subject has been averaged. The recognition experiments using the proposed algorithm show that the recognition rate achieves to 92.875% whilst the average number of judgment is only 2.56 times.

Keywords: Eigenface, Face Recognition, Threshold, Rayleigh Distribution, Feature Extraction

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Authors: Tugrul Oktay, Mehmet Konar, Mustafa Soylak, Firat Sal, Murat Onay, Orhan Kizilkaya

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In this paper, autonomous performance of a small manufactured unmanned helicopter is tried to be increased. For this purpose, a small unmanned helicopter is manufactured in Erciyes University, Faculty of Aeronautics and Astronautics. It is called as ZANKA-Heli-I. For performance maximization, autopilot parameters are determined via minimizing a cost function consisting of flight performance parameters such as settling time, rise time, overshoot during trajectory tracking. For this purpose, a stochastic optimization method named as simultaneous perturbation stochastic approximation is benefited. Using this approach, considerable autonomous performance increase (around %23) is obtained.

Keywords: Small helicopters, hierarchical control, stochastic optimization, autonomous performance maximization, autopilots.

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Authors: Lei Shi, Zhibin Yu, Artur J. Jaworski, Abdulrahman S. Abduljalil

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This paper investigates vortex shedding processes occurring at the end of a stack of parallel plates, due to an oscillating flow induced by an acoustic standing wave within an acoustic resonator. Here, Particle Image Velocimetry (PIV) is used to quantify the vortex shedding processes within an acoustic cycle phase-by-phase, in particular during the “ejection" of the fluid out of the stack. Standard hot-wire anemometry measurement is also applied to detect the velocity fluctuations near the end of the stack. Combination of these two measurement techniques allowed a detailed analysis of the vortex shedding phenomena. The results obtained show that, as the Reynolds number varies (by varying the plate thickness and drive ratio), different flow patterns of vortex shedding are observed by the PIV measurement. On the other hand, the time-dependent hot-wire measurements allow obtaining detailed frequency spectra of the velocity signal, used for calculating characteristic Strouhal numbers. The impact of the plate thickness and the Reynolds number on the vortex shedding pattern has been discussed. Furthermore, a detailed map of the relationship between the Strouhal number and Reynolds number has been obtained and discussed.

Keywords: Oscillatory flow, Parallel-plate thermoacoustic stack, Strouhal numbers, Vortex shedding.

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Authors: M. Babaeipour, M. Vejdanihemmat

Abstract:

Optical properties of bulk and thin film of 1H-CaAlSi for two directions (1,0,0) and (0,0,1) were studied. The calculations are carried out by Density Functional Theory (DFT) method using full potential. GGA approximation was used to calculate exchange-correlation energy. The calculations are performed by WIEN2k package. The results showed that the absorption edge is shifted backward 0.82eV in the thin film than the bulk for both directions. The static values of the real part of dielectric function for four cases were obtained. The static values of the refractive index for four cases are calculated too. The reflectivity graphs have shown an intensive difference between the reflectivity of the thin film and the bulk in the ultraviolet region.

Keywords: 1H-CaAlSi, absorption, bulk, optical, thin film.

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Authors: Chinwendu. B. Eleje, Udechukwu P. Egbuhuzor

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One of the difficulties encountered in solving nonlinear Boundary Value Problems (BVP) by many researchers is finding approximated solutions with minimum deviations from the exact solutions without so much rigor and complications. In this paper, we propose an approach to solve a two point BVP which involves a combination of Taylor series expansion method and Newton Raphson method. Furthermore, the fourth and sixth order approximated solutions are obtained and we compare their relative error and rate of convergence to the exact solution. Finally, some numerical simulations are presented to show the behavior of the solution and its derivatives.

Keywords: Newton Raphson method, non-linear boundary value problem, Taylor series approximation, Michaelis-Menten equation.

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Authors: Jianyin Lu, Masayoshi Seike, Wei Liu, Peihong Wu, Lihua Wang, Yihua Wu, Yasuhiro Naito, Hiromu Nakajima, Yasuhiro Kouchi

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To extract the important physiological factors related to diabetes from an oral glucose tolerance test (OGTT) by mathematical modeling, highly informative but convenient protocols are required. Current models require a large number of samples and extended period of testing, which is not practical for daily use. The purpose of this study is to make model assessments possible even from a reduced number of samples taken over a relatively short period. For this purpose, test values were extrapolated using a support vector machine. A good correlation was found between reference and extrapolated values in evaluated 741 OGTTs. This result indicates that a reduction in the number of clinical test is possible through a computational approach.

Keywords: SVM regression, OGTT, diabetes, mathematical model

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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: Peng Li, Chuanri Li, Tao Li

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Reliability allocation is quite important during early design and development stages for a system to apportion its specified reliability goal to subsystems. This paper improves the reliability fuzzy allocation method, and gives concrete processes on determining the factor and sub-factor sets, weight sets, judgment set, and multi-stage fuzzy evaluation. To determine the weight of factor and sub-factor sets, the modified trapezoidal numbers are proposed to reduce errors caused by subjective factors. To decrease the fuzziness in fuzzy division, an approximation method based on linear programming is employed. To compute the explicit values of fuzzy numbers, centroid method of defuzzification is considered. An example is provided to illustrate the application of the proposed reliability allocation method based on fuzzy arithmetic.

Keywords: Reliability allocation, fuzzy arithmetic, allocation weight.

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Authors: Norhashidah Hj. Mohd Ali, Teng Wai Ping

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In this paper, we present a high order group explicit method in solving the two dimensional Helmholtz equation. The presented method is derived from a nine-point fourth order finite difference approximation formula obtained from a 45-degree rotation of the standard grid which makes it possible for the construction of iterative procedure with reduced complexity. The developed method will be compared with the existing group iterative schemes available in literature in terms of computational time, iteration counts, and computational complexity. The comparative performances of the methods will be discussed and reported.

Keywords: Explicit group method, finite difference, Helmholtz equation, rotated grid, standard grid.

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Authors: Khosrow Maleknejad, Yaser Rostami

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In this paper, Semi-orthogonal B-spline scaling functions and wavelets and their dual functions are presented to approximate the solutions of integro-differential equations.The B-spline scaling functions and wavelets, their properties and the operational matrices of derivative for this function are presented to reduce the solution of integro-differential equations to the solution of algebraic equations. Here we compute B-spline scaling functions of degree 4 and their dual, then we will show that by using them we have better approximation results for the solution of integro-differential equations in comparison with less degrees of scaling functions

Keywords: Integro-differential equations, Quartic B-spline wavelet, Operational matrices.

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Authors: M. A. Bajaber, M. A. Taha

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Contractor selection in Saudi Arabia is very important due to the large construction boom and the contractor role to get over construction risks. The need for investigating contractor selection is due to the following reasons; large number of defaulted or failed projects (18%), large number of disputes attributed to contractor during the project execution stage (almost twofold), the extension of the General Agreement on Tariffs and Trade (GATT) into construction industry, and finally the few number of researches. The selection strategy is not perfect and considered as the reason behind irresponsible contractors. As a response, this research was conducted to review the contractor selection strategies as an integral part of a long advanced research to develop a good selection model. Many techniques can be used to form a selection strategy; multi criteria for optimizing decision, prequalification to discover contractor-s responsibility, bidding process for competition, third party guarantee to enhance the selection, and fuzzy techniques for ambiguities and incomplete information.

Keywords: Bidding, Construction industry, Contractor selection, Saudi Arabia.

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Authors: Zina Benouaret, Djamil Aissani

Abstract:

In this work, we introduce the qualitative and quantitative concept of the strong stability method in the risk process modeling two lines of business of the same insurance company or an insurance and re-insurance companies that divide between them both claims and premiums with a certain proportion. The approach proposed is based on the identification of the ruin probability associate to the model considered, with a stationary distribution of a Markov random process called a reversed process. Our objective, after clarifying the condition and the perturbation domain of parameters, is to obtain the stability inequality of the ruin probability which is applied to estimate the approximation error of a model with disturbance parameters by the considered model. In the stability bound obtained, all constants are explicitly written.

Keywords: Markov chain, risk models, ruin probabilities, strong stability analysis.

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Authors: Ashok Kumar Goel, Suresh Chandra Saxena, Surekha Bhanot

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In this work, a Modified Functional Link Artificial Neural Network (M-FLANN) is proposed which is simpler than a Multilayer Perceptron (MLP) and improves upon the universal approximation capability of Functional Link Artificial Neural Network (FLANN). MLP and its variants: Direct Linear Feedthrough Artificial Neural Network (DLFANN), FLANN and M-FLANN have been implemented to model a simulated Water Bath System and a Continually Stirred Tank Heater (CSTH). Their convergence speed and generalization ability have been compared. The networks have been tested for their interpolation and extrapolation capability using noise-free and noisy data. The results show that M-FLANN which is computationally cheap, performs better and has greater generalization ability than other networks considered in the work.

Keywords: DLFANN, FLANN, M-FLANN, MLP

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Authors: S. Balaji, V. Swaminathan, K. Kannan

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The Minimum Weighted Vertex Cover (MWVC) problem is a classic graph optimization NP - complete problem. Given an undirected graph G = (V, E) and weighting function defined on the vertex set, the minimum weighted vertex cover problem is to find a vertex set S V whose total weight is minimum subject to every edge of G has at least one end point in S. In this paper an effective algorithm, called Support Ratio Algorithm (SRA), is designed to find the minimum weighted vertex cover of a graph. Computational experiments are designed and conducted to study the performance of our proposed algorithm. Extensive simulation results show that the SRA can yield better solutions than other existing algorithms found in the literature for solving the minimum vertex cover problem.

Keywords: Weighted vertex cover, vertex support, approximation algorithms, NP-complete problem.

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Authors: Hassan Baghgar Bostan Abad, Ali Yazdian Varjani, Taheri Asghar

Abstract:

Variable speed drives are growing and varying. Drives expanse depend on progress in different part of science like power system, microelectronic, control methods, and so on. Artificial intelligent contains hard computation and soft computation. Artificial intelligent has found high application in most nonlinear systems same as motors drive. Because it has intelligence like human but there are no sentimental against human like angriness and.... Artificial intelligent is used for various points like approximation, control, and monitoring. Because artificial intelligent techniques can use as controller for any system without requirement to system mathematical model, it has been used in electrical drive control. With this manner, efficiency and reliability of drives increase and volume, weight and cost of them decrease.

Keywords: Artificial intelligent, electrical motor, intelligent drive and control,

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Authors: A. Jameel, G. A. Harmain, Y. Anand, J. H. Masoodi, F. A. Najar

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The present study investigates the effect of inclusions on the shape and size of crack tip plastic zones in engineering materials subjected to static loads by employing the element free Galerkin method (EFGM). The modeling of the discontinuities produced by cracks and inclusions becomes independent of the grid chosen for analysis. The standard displacement approximation is modified by adding additional enrichment functions, which introduce the effects of different discontinuities into the formulation. The level set method has been used to represent different discontinuities present in the domain. The effect of inclusions on the extent of crack tip plastic zones is investigated by solving some numerical problems by the EFGM.

Keywords: EFGM, stress intensity factors, crack tip plastic zones, inclusions.

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Authors: Hani Shafeek, Muhammed Marsudi

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The purpose of this work is examining the multiproduct multi-stage in a battery production line. To improve the performances of an assembly production line by determine the efficiency of each workstation. Data collected from every workstation. The data are throughput rate, number of operator, and number of parts that arrive and leaves during part processing. Data for the number of parts that arrives and leaves are collected at least at the amount of ten samples to make the data is possible to be analyzed by Chi-Squared Goodness Test and queuing theory. Measures of this model served as the comparison with the standard data available in the company. Validation of the task time value resulted by comparing it with the task time value based on the company database. Some performance factors for the multi-product multi-stage in a battery production line in this work are shown. The efficiency in each workstation was also shown. Total production time to produce each part can be determined by adding the total task time in each workstation. To reduce the queuing time and increase the efficiency based on the analysis any probably improvement should be done. One probably action is by increasing the number of operators how manually operate this workstation.

Keywords: Production line, manufacturing, performance measurement, queuing theory.

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Authors: S. Jani, M. Mahmoodi, M. Amini

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Magnetohydrodynamic free convection fluid flow and heat transfer in a square cavity filled with an electric conductive fluid with Prandtl number of 0.7 has been investigated numerically. The horizontal bottom wall of the cavity was kept at Th while the side and the top walls of the cavity were maintained at a constant temperature Tc with Th>Tc. The governing equations written in terms of the primitive variables were solved numerically using the finite volume method while the SIMPLER algorithm was used to couple the velocity and pressure fields. Using the developed code, a parametric study was performed, and the effects of the Rayleigh number and the Hartman number on the fluid flow and heat transfer inside the cavity were investigated. The obtained results showed that temperature distribution and flow pattern inside the cavity depended on both strength of the magnetic field and Rayleigh number. For all cases two counter rotating eddies were formed inside the cavity. The magnetic field decreased the intensity of free convection and flow velocity. Also it was found that for higher Rayleigh numbers a relatively stronger magnetic field was needed to decrease the heat transfer through free convection.

Keywords: Free Convection, Magnetic Field, Square Cavity, Numerical Simulation.

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Authors: Hsian Ming Goo

Abstract:

The paper concerns a special approximate algorithm of the square root of the specific positive integer, which is built by the use of the property of positive integer solution of the Pell’s equation, together with using some elementary theorems of matrices, and then takes it to compare with general used the Newton’s method and give a practical numerical example and error analysis; it is unexpected to find its special property: the significant figure of the approximation value of the square root of positive integer will increase one digit by one. It is well useful in some occasions.

Keywords: Special approximate algorithm, square root, Pell’s equation, Newton’s method, error analysis.

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Authors: Natarajan Vijayarangan, Prasanna S. Bidare

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Cryptography, Image watermarking and E-banking are filled with apparent oxymora and paradoxes. Random sequences are used as keys to encrypt information to be used as watermark during embedding the watermark and also to extract the watermark during detection. Also, the keys are very much utilized for 24x7x365 banking operations. Therefore a deterministic random sequence is very much useful for online applications. In order to obtain the same random sequence, we need to supply the same seed to the generator. Many researchers have used Deterministic Random Number Generators (DRNGs) for cryptographic applications and Pseudo Noise Random sequences (PNs) for watermarking. Even though, there are some weaknesses in PN due to attacks, the research community used it mostly in digital watermarking. On the other hand, DRNGs have not been widely used in online watermarking due to its computational complexity and non-robustness. Therefore, we have invented a new design of generating DRNG using Pi-series to make it useful for online Cryptographic, Digital watermarking and Banking applications.

Keywords: E-tokens, LFSR, non-linear, Pi series, pseudo random number.

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Authors: O. H. Colak, T. C. Destici, S. Ozen, H. Arman, O. Cerezci

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The wavelet transform is one of the most important method used in signal processing. In this study, we have introduced frequency-energy characteristics of local earthquakes using discrete wavelet transform. Frequency-energy characteristic was analyzed depend on difference between P and S wave arrival time and noise within records. We have found that local earthquakes have similar characteristics. If frequency-energy characteristics can be found accurately, this gives us a hint to calculate P and S wave arrival time. It can be seen that wavelet transform provides successful approximation for this. In this study, 100 earthquakes with 500 records were analyzed approximately.

Keywords: Discrete Wavelet Transform, Frequency-EnergyCharacteristics, P and S waves arrival time.

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Authors: Pan Cheng, Jin Huang, Guang Zeng

Abstract:

Elastic boundary eigensolution problems are converted into boundary integral equations by potential theory. The kernels of the boundary integral equations have both the logarithmic and Hilbert singularity simultaneously. We present the mechanical quadrature methods for solving eigensolutions of the boundary integral equations by dealing with two kinds of singularities at the same time. The methods possess high accuracy O(h3) and low computing complexity. The convergence and stability are proved based on Anselone-s collective compact theory. Bases on the asymptotic error expansion with odd powers, we can greatly improve the accuracy of the approximation, and also derive a posteriori error estimate which can be used for constructing self-adaptive algorithms. The efficiency of the algorithms are illustrated by numerical examples.

Keywords: boundary integral equation, extrapolation algorithm, aposteriori error estimate, elasticity.

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Authors: Duy Cuong Nguyen, Ali Makke, Guillaume Montay

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This work presents an improved single fiber pull-out test for fiber/matrix interface characterization. This test has been used to study the Inter-Facial Shear Strength ‘IFSS’ of hemp fibers reinforced polypropylene (PP). For this aim, the fiber diameter has been carefully measured using a tomography inspired method. The fiber section contour can then be approximated by a circle or a polygon. The results show that the IFSS is overestimated if the circular approximation is used. The Influence of the molding temperature on the IFSS has also been studied. We find that a molding temperature of 183◦C leads to better interfacial properties. Above or below this temperature the interface strength is reduced.

Keywords: Interface, pull-out, processing, temperature, hemp, polypropylene, composite.

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Authors: Almas Tayebi, Tayeb Saki Nejad, Alireza Shoukofar

Abstract:

The trial in the city, located 170 kilometers from the Iranian city of Ahvaz was Omidiyeh. The main factor in this project includes 4 levels in control (without hormones), use of hormones in the seed, vegetative and flowering stage respectively. And sub-plots included 3 varieties of vetch in three levels, with local names, was the jewel in the study of light and Auxin in the vegetative and reproductive different times in different varieties of vetch was investigated. This test has been taken in the plots in a randomized complete block with four replications. In order to study the effects of the hormone Auxin in the growth stages (seed, vegetative and flowering) to control (no hormone Auxin) on three local varieties of vetch, the essence of light and plant height, number of pods per plant, seed number The pods, seeds per plant, grain weight, grain yield, plant dry weight and protein content were measured. Among the vetch varieties for plant height, number of pods per plant, a seed per plant, grain weight, grain yield, and plant dry weight and protein levels of 1 percent of plant and seed number per pod per plant at 5% level of There was no significant difference. Interactions for grain yield per plant, grain yield and protein levels of 1 percent and the number of seeds per pod and seed weight are significant differences in levels 5 and plant height and plant dry weight of the interaction were INFLUENCE There was no significant difference in them.

Keywords: Auxin hormones, various periods of growth, production components, vetch

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Authors: Cecilia Nugraheni, Luciana Abednego

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Sudoku is a kind of logic puzzles. Each puzzle consists of a board, which is a 9×9 cells, divided into nine 3×3 subblocks and a set of numbers from 1 to 9. The aim of this puzzle is to fill in every cell of the board with a number from 1 to 9 such that in every row, every column, and every subblock contains each number exactly one. Sudoku puzzles belong to combinatorial problem (NP complete). Sudoku puzzles can be solved by using a variety of techniques/algorithms such as genetic algorithms, heuristics, integer programming, and so on. In this paper, we propose a new approach for solving Sudoku which is by modelling them as block-world problems. In block-world problems, there are a number of boxes on the table with a particular order or arrangement. The objective of this problem is to change this arrangement into the targeted arrangement with the help of two types of robots. In this paper, we present three models for Sudoku. We modellized Sudoku as parameterized multi-agent systems. A parameterized multi-agent system is a multi-agent system which consists of several uniform/similar agents and the number of the agents in the system is stated as the parameter of this system. We use Temporal Logic of Actions (TLA) for formalizing our models.

Keywords: Sudoku puzzle, block world problem, parameterized multi agent systems modelling, Temporal Logic of Actions.

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Authors: N. F. Jamaluddin, N. A. H. Aizam

Abstract:

This paper explains the educational timetabling problem, a type of scheduling problem that is considered as one of the most challenging problem in optimization and operational research. The university examination timetabling problem (UETP), which involves assigning a set number of exams into a set number of timeslots whilst fulfilling all required conditions, has been widely investigated. The limitation of available timeslots and resources with the increasing number of examinations are the main reasons in the difficulty of solving this problem. Dynamical change in the examination scheduling system adds up the complication particularly in coping up with the demand and new requirements by the communities. Our objective is to investigate these demands and requirements with subjects taken from Universiti Malaysia Terengganu (UMT), through questionnaires. Integer linear programming model which reflects the preferences obtained to produce an effective examination timetabling was formed.

Keywords: Demands, educational timetabling, integer linear programming, scheduling, university examination timetabling problem.

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Authors: San Nah Sze, Wei King Tiong

Abstract:

The multiple traveling salesman problem (mTSP) can be used to model many practical problems. The mTSP is more complicated than the traveling salesman problem (TSP) because it requires determining which cities to assign to each salesman, as well as the optimal ordering of the cities within each salesman's tour. Previous studies proposed that Genetic Algorithm (GA), Integer Programming (IP) and several neural network (NN) approaches could be used to solve mTSP. This paper compared the results for mTSP, solved with Genetic Algorithm (GA) and Nearest Neighbor Algorithm (NNA). The number of cities is clustered into a few groups using k-means clustering technique. The number of groups depends on the number of salesman. Then, each group is solved with NNA and GA as an independent TSP. It is found that k-means clustering and NNA are superior to GA in terms of performance (evaluated by fitness function) and computing time.

Keywords: Multiple Traveling Salesman Problem, GeneticAlgorithm, Nearest Neighbor Algorithm, k-Means Clustering.

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Authors: Yogi Satrya Aryadinata, Anne Laurent, Michel Sala

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Gradual patterns have been studied for many years as they contain precious information. They have been integrated in many expert systems and rule-based systems, for instance to reason on knowledge such as “the greater the number of turns, the greater the number of car crashes”. In many cases, this knowledge has been considered as a rule “the greater the number of turns → the greater the number of car crashes” Historically, works have thus been focused on the representation of such rules, studying how implication could be defined, especially fuzzy implication. These rules were defined by experts who were in charge to describe the systems they were working on in order to turn them to operate automatically. More recently, approaches have been proposed in order to mine databases for automatically discovering such knowledge. Several approaches have been studied, the main scientific topics being: how to determine what is an relevant gradual pattern, and how to discover them as efficiently as possible (in terms of both memory and CPU usage). However, in some cases, end-users are not interested in raw level knowledge, and are rather interested in trends. Moreover, it may be the case that no relevant pattern can be discovered at a low level of granularity (e.g. city), whereas some can be discovered at a higher level (e.g. county). In this paper, we thus extend gradual pattern approaches in order to consider multiple level gradual patterns. For this purpose, we consider two aggregation policies, namely horizontal and vertical.

Keywords: Gradual Pattern.

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