Search results for: coefficient inverse problem

Authors: Noratiqah Mohd Ariff, Abdul Aziz Jemain, Mohd Aftar Abu Bakar

Abstract:

The construction of Intensity-Duration-Frequency (IDF) curves is one of the most common and useful tools in order to design hydraulic structures and to provide a mathematical relationship between rainfall characteristics. IDF curves, especially those in Peninsular Malaysia, are often built using moving windows of rainfalls. However, these windows do not represent the actual rainfall events since the duration of rainfalls is usually prefixed. Hence, instead of using moving windows, this study aims to find regionalized distributions for IDF curves of extreme rainfalls based on storm events. Homogeneity test is performed on annual maximum of storm intensities to identify homogeneous regions of storms in Peninsular Malaysia. The L-moment method is then used to regionalized Generalized Extreme Value (GEV) distribution of these annual maximums and subsequently. IDF curves are constructed using the regional distributions. The differences between the IDF curves obtained and IDF curves found using at-site GEV distributions are observed through the computation of the coefficient of variation of root mean square error, mean percentage difference and the coefficient of determination. The small differences implied that the construction of IDF curves could be simplified by finding a general probability distribution of each region. This will also help in constructing IDF curves for sites with no rainfall station.

Keywords: IDF curves, L-moments, regionalization, storm events.

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Authors: Wahyudin P. Syam, Ibrahim M. Al-Harkan

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This study compares three meta heuristics to minimize makespan (Cmax) for Hybrid Flow Shop (HFS) Scheduling Problem with Parallel Machines. This problem is known to be NP-Hard. This study proposes three algorithms among improvement heuristic searches which are: Genetic Algorithm (GA), Simulated Annealing (SA), and Tabu Search (TS). SA and TS are known as deterministic improvement heuristic search. GA is known as stochastic improvement heuristic search. A comprehensive comparison from these three improvement heuristic searches is presented. The results for the experiments conducted show that TS is effective and efficient to solve HFS scheduling problems.

Keywords: Flow shop, genetic algorithm, simulated annealing, tabu search.

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Authors: Numan Unaldi, Samil Temel, Süleyman Demirci

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A novel undecimated wavelet transform based contrast enhancement algorithmis proposed to for both gray scale andcolor images. Contrast enhancement is realized by tuning the magnitude of approximation coefficients at each level with respect to the approximation coefficients of one higher level during the inverse transform phase in a center/surround  enhancement sense.The performance of the proposed algorithm is evaluated using a statistical visual contrast measure (VCM). Experimental results on the proposed algorithm show improvement in terms of the VCM.

Keywords: Image enhancement, local contrast enhancement, visual contrast measure.

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Authors: A. M. El-Zonkoly

Abstract:

In this paper, based on steady-state models of Flexible AC Transmission System (FACTS) devices, the sizing of static synchronous series compensator (SSSC) controllers in transmission network is formed as an optimization problem. The objective of this problem is to reduce the transmission losses in the network. The optimization problem is solved using particle swarm optimization (PSO) technique. The Newton-Raphson load flow algorithm is modified to consider the insertion of the SSSC devices in the network. A numerical example, illustrating the effectiveness of the proposed algorithm, is introduced. In addition, a novel model of a 3- phase voltage source converter (VSC) that is suitable for series connected FACTS a controller is introduced. The model is verified by simulation using Power System Blockset (PSB) and Simulink software.

Keywords: FACTS, Modeling, PSO, SSSC, Transmission lossreduction.

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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: Udeme O. Ini, Edikan E. Akpanibah

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This paper studies the optimal investment strategies for a plan member (PM) in a defined contribution (DC) pension scheme with transaction cost, taxes on invested funds and couple risky assets (stocks) under the Ornstein-Uhlenbeck (O-U) process. The PM’s portfolio is assumed to consist of a risk-free asset and two risky assets where the two risky assets are driven by the O-U process. The Legendre transformation and dual theory is use to transform the resultant optimal control problem which is a nonlinear partial differential equation (PDE) into linear PDE and the resultant linear PDE is then solved for the explicit solutions of the optimal investment strategies for PM exhibiting constant absolute risk aversion (CARA) using change of variable technique. Furthermore, theoretical analysis is used to study the influences of some sensitive parameters on the optimal investment strategies with observations that the optimal investment strategies for the two risky assets increase with increase in the dividend and decreases with increase in tax on the invested funds, risk averse coefficient, initial fund size and the transaction cost.

Keywords: Ornstein-Uhlenbeck process, portfolio management, Legendre transforms, CARA utility.

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Authors: María S. Avila-García, John N. Carter, Robert I. Damper

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An important problem in speech research is the automatic extraction of information about the shape and dimensions of the vocal tract during real-time speech production. We have previously developed Southampton dynamic magnetic resonance imaging (SDMRI) as an approach to the solution of this problem.However, the SDMRI images are very noisy so that shape extraction is a major challenge. In this paper, we address the problem of tongue shape extraction, which poses difficulties because this is a highly deforming non-parametric shape. We show that combining active shape models with the dynamic Hough transform allows the tongue shape to be reliably tracked in the image sequence.

Keywords: Vocal tract imaging, speech production, active shapemodels, dynamic Hough transform, object tracking.

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Authors: Nancy, Ling Sze Leung

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Hong Kong is one of the regions in the world where Total Fertility Rate (TFR) is very low. In 2001, the TFR dropped until 0.931, which means 1 woman even cannot give birth to one child on average. However, after the reformation of the 'Right of Abode of Hong Kong' in 2001 and the Chinese Central Government loosened the disembarkation procedure of mainland Chinese (mainlander) to enter Hong Kong in 2003; mainlander couples started to cross the border for giving births in Hong Kong. This action raises Hong Kong-s TFR quickly from 0.931 (2001) to 1.094 (2010). Usually, an increasing trend of TFR means a sign of rejuvenation in low-fertility, but in the case of Hong Kong, the increase of TFR does not, rather it generates other population problems. This paper is going to discuss do mainlanders- births help to solve the low-fertility problem in Hong Kong.

Keywords: Cross Border Birth, Low-fertility, Mainlander, Total Fertility Rate

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Authors: Paola Lecca, Lorenzo Dematte, Corrado Priami

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The present models and simulation algorithms of intracellular stochastic kinetics are usually based on the premise that diffusion is so fast that the concentrations of all the involved species are homogeneous in space. However, recents experimental measurements of intracellular diffusion constants indicate that the assumption of a homogeneous well-stirred cytosol is not necessarily valid even for small prokaryotic cells. In this work a mathematical treatment of diffusion that can be incorporated in a stochastic algorithm simulating the dynamics of a reaction-diffusion system is presented. The movement of a molecule A from a region i to a region j of the space is represented as a first order reaction Ai k- ! Aj , where the rate constant k depends on the diffusion coefficient. The diffusion coefficients are modeled as function of the local concentration of the solutes, their intrinsic viscosities, their frictional coefficients and the temperature of the system. The stochastic time evolution of the system is given by the occurrence of diffusion events and chemical reaction events. At each time step an event (reaction or diffusion) is selected from a probability distribution of waiting times determined by the intrinsic reaction kinetics and diffusion dynamics. To demonstrate the method the simulation results of the reaction-diffusion system of chaperoneassisted protein folding in cytoplasm are shown.

Keywords: Reaction-diffusion systems, diffusion coefficient, stochastic simulation algorithm.

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Authors: Zoe Chang, Max Williams, Gautham Das

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The purpose of this study is to evaluate the incorporation of hemp fibers (HF) in concrete. Hemp fiber reinforced concrete (HFRC) is becoming more popular as an alternative for regular mix designs. This study was done to evaluate the compressive strength of HFRC regarding mix procedure. HF were obtained from the manufacturer and hand processed to ensure uniformity in width and length. The fibers were added to concrete as both wet and dry mix to investigate and optimize the mix design process. Results indicated that the dry mix had a compressive strength of 1157 psi compared to the wet mix of 985 psi. This dry mix compressive strength was within range of the standard mix compressive strength of 1533 psi. The statistical analysis revealed that the mix design process needs further optimization and uniformity concerning the addition of HF. Regression analysis revealed that the standard mix design had a coefficient of 0.9 as compared to the dry mix of 0.375 indicating a variation in the mixing process. While completing the dry mix, the addition of plain HF caused them to intertwine creating lumps and inconsistency. However, during the wet mixing process, combining water and HF before incorporation allows the fibers to uniformly disperse within the mix hence the regression analysis indicated a better coefficient of 0.55. This study concludes that HRFC is a viable alternative to regular mixes however more research surrounding its characteristics needs to be conducted.

Keywords: hemp fibers, hemp reinforced concrete, wet and dry, freeze thaw testing, compressive strength

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Authors: Harshavardhan Reddy, Balaji Subramanian

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A systematic parametric study to find the optimum Bellmouth profile by relating geometric and performance parameters to satisfy a set of specifications is reported. A careful aerodynamic design of Bellmouth intake is critical to properly direct the flow with minimal losses and maximal flow uniformity into the honeycomb located inside the settling chamber of an indraft wind tunnel, thus improving the efficiency of the entire unit. Design charts for elliptically profiled Bellmouth's with two different contraction ratios (9 and 18) and three different test section speeds (25 m/s, 50 m/s, and 75 m/s) were presented. A significant performance improvement - especially in the coefficient of discharge and in the flow angularity and boundary layer thickness at the honeycomb inlet - was observed when an entry corner radius (r/D = 0.08) was added to the Bellmouth profile. The nonuniformity at the honeycomb inlet drops by about three times (~1% to 0.3%) when moving from square to regular octagonal cross-section. An octagonal cross-sectioned Bellmouth intake with L/d = 0.55, D/d = 1.625, and r/D = 0.08 met all the four target performance specifications and is proposed as the best choice for a low-speed wind tunnel.

Keywords: Bellmouth intake, low-speed wind tunnel, coefficient of discharge, nonuniformity, flow angularity, boundary layer thickness, CFD, aerodynamics.

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Authors: K. Krishna Veni, S. Raja Balachandar

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This paper presents a heuristic to solve large size 0-1 Multi constrained Knapsack problem (01MKP) which is NP-hard. Many researchers are used heuristic operator to identify the redundant constraints of Linear Programming Problem before applying the regular procedure to solve it. We use the intercept matrix to identify the zero valued variables of 01MKP which is known as redundant variables. In this heuristic, first the dominance property of the intercept matrix of constraints is exploited to reduce the search space to find the optimal or near optimal solutions of 01MKP, second, we improve the solution by using the pseudo-utility ratio based on surrogate constraint of 01MKP. This heuristic is tested for benchmark problems of sizes upto 2500, taken from literature and the results are compared with optimum solutions. Space and computational complexity of solving 01MKP using this approach are also presented. The encouraging results especially for relatively large size test problems indicate that this heuristic can successfully be used for finding good solutions for highly constrained NP-hard problems.

Keywords: 0-1 Multi constrained Knapsack problem, heuristic, computational complexity, NP-Hard problems.

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Authors: Mario Mastriani, Marcelo Naiouf

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The Gram-Schmidt Process (GSP) is used to convert a non-orthogonal basis (a set of linearly independent vectors) into an orthonormal basis (a set of orthogonal, unit-length vectors). The process consists of taking each vector and then subtracting the elements in common with the previous vectors. This paper introduces an Enhanced version of the Gram-Schmidt Process (EGSP) with inverse, which is useful for signal and image processing applications.

Keywords: Digital filters, digital signal and image processing, Gram-Schmidt Process, orthonormalization.

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Authors: U. C. Amadi, N. A. Udoh

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One of the major challenges faced in solving initial and boundary problems is how to find approximate solutions with minimal deviation from the exact solution without so much rigor and complications. The Taylor series method provides a simple way of obtaining an infinite series which converges to the exact solution for initial value problems and this method of solution is somewhat limited for a two point boundary problem since the infinite series has to be truncated to include the boundary conditions. In this paper, the Ying Buzu Shu algorithm is used to solve a two point boundary nonlinear diffusion problem for the fourth and sixth order solution and compare their relative error and rate of convergence to the exact solution.

Keywords: Ying Buzu Shu, nonlinear boundary problem, Taylor series algorithm, infinite series.

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Authors: Hany Osman, M. F. Baki

Abstract:

We address the balancing problem of transfer lines in this paper to find the optimal line balancing that minimizes the nonproductive time. We focus on the tool change time and face orientation change time both of which influence the makespane. We consider machine capacity limitations and technological constraints associated with the manufacturing process of auto cylinder heads. The problem is represented by a mixed integer programming model that aims at distributing the design features to workstations and sequencing the machining processes at a minimum non-productive time. The proposed model is solved by an algorithm established using linearization schemes and Benders- decomposition approach. The experiments show the efficiency of the algorithm in reaching the exact solution of small and medium problem instances at reasonable time.

Keywords: Transfer line balancing, Benders' decomposition, Linearization.

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Authors: Alireza Dehghani Pilehvarani, Mojtaba Hakimzadeh, Mohammad Jafari Far, Reza Sedaghati

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Distributed Generation (DG) can help in reducing the cost of electricity to the costumer, relieve network congestion and provide environmentally friendly energy close to load centers. Its capacity is also scalable and it provides voltage support at distribution level. Hence, DG placement and penetration level is an important problem for both the utility and DG owner. DG allocation and capacity determination is a nonlinear optimization problem. The objective function of this problem is the minimization of the total loss of the distribution system. Also high levels of penetration of DG are a new challenge for traditional electric power systems. This paper presents a new methodology for the optimal placement of DG and penetration level of DG in distribution system based on General Algebraic Modeling System (GAMS) and Genetic Algorithm (GA).

Keywords: Distributed Generation, Location, Loss Reduction, Distribution Network, GA, GAMS.

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Authors: A. GuhaRay, C. V. S. P. Kiranmayi, S. Rudraraju

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The present study considers the effect of variation of different geotechnical random variables in the design of stone column-foundation systems for assessing the bearing capacity and consolidation settlement of highly compressible soil. The soil and stone column properties, spacing, diameter and arrangement of stone columns are considered as the random variables. Probability of failure (P_f) is computed for a target degree of consolidation and a target safe load by Monte Carlo Simulation (MCS). The study shows that the variation in coefficient of radial consolidation (c_r) and cohesion of soil (c_s) are two most important factors influencing Pf. If the coefficient of variation (COV) of c_r exceeds 20%, P_f exceeds 0.001, which is unsafe following the guidelines of US Army Corps of Engineers. The bearing capacity also exceeds its safe value for COV of c_s > 30%. It is also observed that as the spacing between the stone column increases, the probability of reaching a target degree of consolidation decreases. Accordingly, design guidelines, considering both consolidation and bearing capacity of improved ground, are proposed for different spacing and diameter of stone columns and geotechnical random variables.

Keywords: Bearing capacity, consolidation, geotechnical random variables, probability of failure, stone columns.

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Authors: Hanieh Tarbiat Khosrowshahi, Mojtaba Shakeri

Abstract:

Wireless Sensor Networks (WSNs) consist of a set of sensor nodes with limited capability. WSNs may suffer from multiple node failures when they are exposed to harsh environments such as military zones or disaster locations and lose connectivity by getting partitioned into disjoint segments. Relay nodes (RNs) are alternatively introduced to restore connectivity. They cost more than sensors as they benefit from mobility, more power and more transmission range, enforcing a minimum number of them to be used. This paper addresses the problem of RN placement in a multiple disjoint network by developing a genetic algorithm (GA). The problem is reintroduced as the Steiner tree problem (which is known to be an NP-hard problem) by the aim of finding the minimum number of Steiner points where RNs are to be placed for restoring connectivity. An upper bound to the number of RNs is first computed to set up the length of initial chromosomes. The GA algorithm then iteratively reduces the number of RNs and determines their location at the same time. Experimental results indicate that the proposed GA is capable of establishing network connectivity using a reasonable number of RNs compared to the best existing work.

Keywords: Connectivity restoration, genetic algorithms, multiple-node failure, relay nodes, wireless sensor networks.

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Authors: Anatoli Torokhti, Stan Miklavcic

Abstract:

We study a new technique for optimal data compression subject to conditions of causality and different types of memory. The technique is based on the assumption that some information about compressed data can be obtained from a solution of the associated problem without constraints of causality and memory. This allows us to consider two separate problem related to compression and decompression subject to those constraints. Their solutions are given and the analysis of the associated errors is provided.

Keywords: stochastic signals, optimization problems in signal processing.

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Authors: Sylvia Encheva

Abstract:

The importance of hints in an intelligent tutoring system is well understood. The problems however related to their delivering are quite a few. In this paper we propose delivering of hints to be based on considering their usefulness. By this we mean that a hint is regarded as useful to a student if the student has succeeded to solve a problem after the hint was suggested to her/him. Methods from the theory of partial orderings are further applied facilitating an automated process of offering individualized advises on how to proceed in order to solve a particular problem.

Keywords: Decision support services, uncertainty management, partial orderings.

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Authors: Mohamad R. Khaldi, Christine S. Khoury, Guy M. Naim

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The steady-state operation of maintaining voltage stability is done by switching various controllers scattered all over the power network. When a contingency occurs, whether forced or unforced, the dispatcher is to alleviate the problem in a minimum time, cost, and effort. Persistent problem may lead to blackout. The dispatcher is to have the appropriate switching of controllers in terms of type, location, and size to remove the contingency and maintain voltage stability. Wrong switching may worsen the problem and that may lead to blackout. This work proposed and used a Fuzzy CMeans Clustering (FCMC) to assist the dispatcher in the decision making. The FCMC is used in the static voltage stability to map instantaneously a contingency to a set of controllers where the types, locations, and amount of switching are induced.

Keywords: Fuzzy logic, Power system control, Reactive power control, Voltage control

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Authors: Vahid Hajipour, Samira V. Noshafagh, Reza Tavakkoli-Moghaddam

Abstract:

This article proposes a novel Pareto-based multiobjective meta-heuristic algorithm named non-dominated ranking genetic algorithm (NRGA) to solve multi-facility location-allocation problem. In NRGA, a fitness value representing rank is assigned to each individual of the population. Moreover, two features ranked based roulette wheel selection including select the fronts and choose solutions from the fronts, are utilized. The proposed solving methodology is validated using several examples taken from the specialized literature. The performance of our approach shows that NRGA algorithm is able to generate true and well distributed Pareto optimal solutions.

Keywords: Non-dominated ranking genetic algorithm, Pareto solutions, Multi-facility location-allocation problem.

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Authors: Chang-Lin Hu, Yao-You Cheng, Meng-Lin Li

Abstract:

The need of high frame-rate imaging has been triggered by the new applications of ultrasound imaging to transient elastography and real-time 3D ultrasound. Using plane wave excitation (PWE) is one of the methods to achieve very high frame-rate imaging since an image can be formed with a single insonification. However, due to the lack of transmit focusing, the image quality with PWE is lower compared with those using conventional focused transmission. To solve this problem, we propose a filter-retrieved transmit focusing (FRF) technique combined with cross-correlation weighting (FRF+CC weighting) for high frame-rate imaging with PWE. A restrospective focusing filter is designed to simultaneously minimize the predefined sidelobe energy associated with single PWE and the filter energy related to the signal-to-noise-ratio (SNR). This filter attempts to maintain the mainlobe signals and to reduce the sidelobe ones, which gives similar mainlobe signals and different sidelobes between the original PWE and the FRF baseband data. Normalized cross-correlation coefficient at zero lag is calculated to quantify the degree of similarity at each imaging point and used as a weighting matrix to the FRF baseband data to further suppress sidelobes, thus improving the filter-retrieved focusing quality.

Keywords: retrospective synthetic focusing, high frame rate, correlation weighting.

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Authors: G.Mehdiyeva, M.Imanova, V.Ibrahimov

Abstract:

Taking into account that many problems of natural sciences and engineering are reduced to solving initial-value problem for ordinary differential equations, beginning from Newton, the scientists investigate approximate solution of ordinary differential equations. There are papers of different authors devoted to the solution of initial value problem for ODE. The Euler-s known method that was developed under the guidance of the famous scientists Adams, Runge and Kutta is the most popular one among these methods. Recently the scientists began to construct the methods preserving some properties of Adams and Runge-Kutta methods and called them hybrid methods. The constructions of such methods are investigated from the middle of the XX century. Here we investigate one generalization of multistep and hybrid methods and on their base we construct specific methods of accuracy order p = 5 and p = 6 for k = 1 ( k is the order of the difference method).

Keywords: Multistep and hybrid methods, initial value problem, degree and stability of hybrid methods

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Authors: L. Della Ragione, G. Meccariello

Abstract:

In recent years in Italy the progress of the automobile industry, in the field of reduction of emissions values, is very remarkable. Nevertheless their evaluation and reduction is a key problem, especially in the cities, that account for more than 50% of world population. In this paper we dealt with the problem of describing a quantitatively approach for the reconstruction of GPS coordinates and altitude, in the context of correlation study between driving cycles / emission / geographical location, during an experimental campaign realized with some instrumented cars.

Keywords: Air pollution, Driving cycles, GPS signal.

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Authors: Abhishek Swaroop, Awadhesh Kumar Singh

Abstract:

The group mutual exclusion (GME) problem is a variant of the mutual exclusion problem. In the present paper a token-based group mutual exclusion algorithm, capable of handling transient faults, is proposed. The algorithm uses the concept of dynamic request sets. A time out mechanism is used to detect the token loss; also, a distributed scheme is used to regenerate the token. The worst case message complexity of the algorithm is n+1. The maximum concurrency and forum switch complexity of the algorithm are n and min (n, m) respectively, where n is the number of processes and m is the number of groups. The algorithm also satisfies another desirable property called smooth admission. The scheme can also be adapted to handle the extended group mutual exclusion problem.

Keywords: Dynamic request sets, Fault tolerance, Smoothadmission, Transient faults.

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Authors: J. Kluabwang

Abstract:

An economic operation scheduling problem of a hydro-thermal power generation system has been properly solved by the proposed multipath adaptive tabu search algorithm (MATS). Four reservoirs with their own hydro plants and another one thermal plant are integrated to be a studied system used to formulate the objective function under complicated constraints, eg water managements, power balance and thermal generator limits. MATS with four subsearch units (ATSs) and two stages of discarding mechanism (DM), has been setting and trying to solve the problem through 25 trials under function evaluation criterion. It is shown that MATS can provide superior results with respect to single ATS and other previous methods, genetic algorithms (GA) and differential evolution (DE).

Keywords: Hydro-thermal scheduling problem, economic dispatch, adaptive tabu search, multipath adaptive tabu search

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Authors: W. K. Wong, Z. X. Guo, P.Y. Mok

Abstract:

This paper presents a multi-objective order allocation planning problem with the consideration of various real-world production features. A novel hybrid intelligent optimization model, integrating a multi-objective memetic optimization process, a Monte Carlo simulation technique and a heuristic pruning technique, is proposed to handle this problem. Experiments based on industrial data are conducted to validate the proposed model. Results show that (1) the proposed model can effectively solve the investigated problem by providing effective production decision-making solutions, which outperformsan NSGA-II-based optimization process and an industrial method.

Keywords: Multi-objective order allocation planning, Pareto optimization, Memetic algorithm, Mento Carlo simulation

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Authors: Xin Luo, Jin Huang, Pan Cheng

Abstract:

By means of Sidi-Israeli’s quadrature rules, mechanical quadrature methods (MQMs) for solving the first kind boundary integral equations (BIEs) of steady state Stokes problem are presented. The convergence of numerical solutions by MQMs is proved based on Anselone’s collective compact and asymptotical compact theory, and the asymptotic expansions with the odd powers of the errors are provided, which implies that the accuracy of the approximations by MQMs possesses high accuracy order O (h3). Finally, the numerical examples show the efficiency of our methods.

Keywords: Stokes problem, boundary integral equation, mechanical quadrature methods, asymptotic expansions.

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Authors: Shyam S.N. Perera

Abstract:

An optimal control problem for a mathematical model of efficiency of antiviral therapy in hepatitis B virus infections is considered. The aim of the study is to control the new viral production, block the new infection cells and maintain the number of uninfected cells in the given range. The optimal controls represent the efficiency of antiviral therapy in inhibiting viral production and preventing new infections. Defining the cost functional, the optimal control problem is converted into the constrained optimization problem and the first order optimality system is derived. For the numerical simulation, we propose the steepest descent algorithm based on the adjoint variable method. A computer program in MATLAB is developed for the numerical simulations.

Keywords: Virus infection model, Optimal control, Adjoint system, Steepest descent

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