Search results for: multiobjective programming

Authors: Egor E. Nuzhin, Nikolay V. Brilliantov

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We propose an approach for addressing Reinforcement Learning (RL) problems. It combines the ideas of umbrella sampling, borrowed from Monte Carlo technique of computational physics and chemistry, with optimal control methods, and is realized on the base of neural networks. This results in a powerful algorithm, designed to solve hard RL problems – the problems, with long-time delayed reward, state-traps sticking and a lack of terminal states. It outperforms the prominent algorithms, such as PPO, RND, iLQR and VI, which are among the most efficient for the hard problems. The new algorithm deals with a continuous ensemble of agents and expected return, that includes the ensemble entropy. This results in a quick and efficient search of the optimal policy in terms of ”exploration-exploitation trade-off” in the state-action space.

Keywords: umbrella sampling, reinforcement learning, policy gradient, dynamic programming

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Authors: Sagar Hedaoo, Fazle Baki, Ahmed Azab

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In supply chain management, network design for scalable manufacturing facilities is an emerging field of research. Facility location allocation assigns facilities to customers to optimize the overall cost of the supply chain. To further optimize the costs, capacities of these facilities can be changed in accordance with customer demands. A mathematical model is formulated to fully express the problem at hand and to solve small-to-mid range instances. A dedicated constraint has been developed to restrict emissions in line with the Kyoto protocol. This problem is NP-Hard; hence, a simulated annealing metaheuristic has been developed to solve larger instances. A case study on the USA-Canada cross border crossing is used.

Keywords: emission, mixed integer linear programming, metaheuristic, simulated annealing

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Authors: Kainat Affrin, P. Reshma, G. Narendra Kumar

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Optimization of timetable is the need of the day for the rescheduling and routing of trains in real time. Trains are scheduled in parallel with the road transport vehicles to the same destination. As the number of trains is restricted due to single track, customers usually opt for road transport to use frequently. The air pollution increases as the density of vehicles on road transport is increased. Use of an alternate mode of transport like train helps in reducing air-pollution. This paper mainly aims at attracting the passengers to Train transport by proper rescheduling of trains using hybrid of stop-skip algorithm and iterative convex programming algorithm. Rescheduling of train bi-directionally is achieved on a single track with dynamic dual time and varying stops. Introduction of more trains attract customers to use rail transport frequently, thereby decreasing the pollution. The results are simulated using Network Simulator (NS-2).

Keywords: air pollution, AODV, re-scheduling, WSNs

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Authors: Abdelhak-Djamel Seriai, Selim Kebir, Allaoua Chaoui

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During the last decades, object oriented program- ming has been massively used to build large-scale systems. However, evolution and maintenance of such systems become a laborious task because of the lack of object oriented programming to offer a precise view of the functional building blocks of the system. This lack is caused by the fine granularity of classes and objects. In this paper, we use a post object-oriented technology namely software components, to propose an approach based on the identification of the functional building blocks of an object oriented system by analyzing its source code. These functional blocks are specified as software components and the result is a multi-layer component based software architecture.

Keywords: software comprehension, software component, object oriented, software architecture, reverse engineering

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Authors: Lisa Tribuzio

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This paper explores the complexity of working with Muslim communities in Australia. It will draw upon the notions of belonging, social inclusion and effective community programming to engage Muslim communities in Western environments given the current global political climate. Factors taken into consideration for effective engagement include: family engagement, considering key practices such as Ramadan, fasting and prayer and food requirements, gender relations, core values around faith and spirituality, considering attitudes towards self disclosure in a counseling setting and the notion of Us and Them in the media and systems and its effect on minority communities. It will explore recent research in the field from Australian researchers as well as recommendations from United Nations in working with Muslim communities. It will also explore current practice models applied in Australia in engaging effectively with diverse communities and addressing racism and discrimination in innovative ways.

Keywords: Muslim, cultural diversity, social inclusion, racism

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Authors: Z. Jannoo, B. W. Yap, N. Auchoybur, M. A. Lazim

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The two common approaches to Structural Equation Modeling (SEM) are the Covariance-Based SEM (CB-SEM) and Partial Least Squares SEM (PLS-SEM). There is much debate on the performance of CB-SEM and PLS-SEM for small sample size and when distributions are non-normal. This study evaluates the performance of CB-SEM and PLS-SEM under normality and non-normality conditions via a simulation. Monte Carlo Simulation in R programming language was employed to generate data based on the theoretical model with one endogenous and four exogenous variables. Each latent variable has three indicators. For normal distributions, CB-SEM estimates were found to be inaccurate for small sample size while PLS-SEM could produce the path estimates. Meanwhile, for a larger sample size, CB-SEM estimates have lower variability compared to PLS-SEM. Under non-normality, CB-SEM path estimates were inaccurate for small sample size. However, CB-SEM estimates are more accurate than those of PLS-SEM for sample size of 50 and above. The PLS-SEM estimates are not accurate unless sample size is very large.

Keywords: CB-SEM, Monte Carlo simulation, normality conditions, non-normality, PLS-SEM

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Authors: Shah Muhammad Butt, Ahmed Masaud Ansari

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Present Financial crisis in Higher Educational Institutes (HEIs) facing lots of problems considerable budget cuts, make difficult to meet the ever growing IT-based research and learning needs, institutions are rapidly planning and promoting cloud-based approaches for their academic and research needs. A cost effective Hybrid Cloud framework for HEI’s will provide educational services for campus or intercampus communication. Hybrid Cloud Framework comprises Private and Public Cloud approaches. This paper will propose the framework based on the Open Source Cloud (OpenNebula for Virtualization, Eucalyptus for Infrastructure, and Aneka for programming development environment) combined with CSP’s services which are delivered to the end-user via the Internet from public clouds.

Keywords: educational services, hybrid campus cloud, open source, electrical and systems sciences

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Authors: Noriki Amano

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The final goal of this study is to create another way in which people enjoy music through the performance of 'Wearable Music'. Concretely speaking, we generate colorful costumes in real- time from music and to realize their dressing by projecting them to a person. For this purpose, we propose three methods in this study. First, a method of giving color to music in a three-dimensionally way. Second, a method of generating images of costumes from music. Third, a method of wearing the images of music. In particular, this study stands out from other related work in that we generate images of unique costumes from music and realize to wear them. In this study, we use the technique of generative arts to generate images of unique costumes and project the images to the fog generated around a person from 3-way using projectors. From this study, we can get how to enjoy music as 'wearable'. Furthermore, we are also able to have the prospect of unconventional entertainment based on the fusion between music and costumes.

Keywords: entertainment computing, costumes, music, generative programming

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Authors: Çağrı Özgün Kibiroğlu, Zeynep Turgut

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This paper considers a hub location problem where the network service area partitioned into predetermined zones (represented by node clusters is given) and potential hub nodes capacity levels are determined a priori as a selection criteria of hub to investigate congestion effect on network. The objective is to design hub network by determining all required hub locations in the node clusters and also allocate non-hub nodes to hubs such that the total cost including transportation cost, opening cost of hubs and penalty cost for exceed of capacity level at hubs is minimized. A mixed integer linear programming model is developed introducing additional constraints to the traditional model of capacitated multiple allocation hub location problem and empirically tested.

Keywords: hub location problem, p-hub median problem, clustering, congestion

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Authors: Anubhav Shrivastava Shivarudraswamy, Bhat Lakshya

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Distributed generation has indeed become a major area of interest in recent years. Distributed generation can address a large number of loads in a power line and hence has better efficiency over the conventional methods. However, there are certain drawbacks associated with it, an increase in voltage being the major one. This paper addresses the voltage control at the buses for an IEEE 30 bus system by regulating reactive power. For carrying out the analysis, the suitable location for placing distributed generators (DG) is identified through load flow analysis and seeing where the voltage profile is dipping. MATLAB programming is used to regulate the voltage at all buses within +/- 5% of the base value even after the introduction of DGs. Three methods for regulation of voltage are discussed. A sensitivity based analysis is then carried out to determine the priority among the various methods listed in the paper.

Keywords: distributed generators, distributed system, reactive power, voltage control, sensitivity analysis

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Authors: Stephen Kirkup

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This paper discusses the implementation of the boundary element method (BEM) on an Excel spreadsheet and how it can be used in teaching vector calculus and simulation. There are two separate spreadheets, within which Laplace equation is solved by the BEM in two dimensions (LIBEM2) and axisymmetric three dimensions (LBEMA). The main algorithms are implemented in the associated programming language within Excel, Visual Basic for Applications (VBA). The BEM only requires a boundary mesh and hence it is a relatively accessible method. The BEM in the open spreadsheet environment is demonstrated as being useful as an aid to teaching and learning. The application of the BEM implemented on a spreadsheet for educational purposes in introductory vector calculus and simulation is explored. The development of assignment work is discussed, and sample results from student work are given. The spreadsheets were found to be useful tools in developing the students’ understanding of vector calculus and in simulating heat conduction.

Keywords: boundary element method, Laplace’s equation, vector calculus, simulation, education

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Authors: Nagmani Lnu

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Often, Quality Engineering refers to testing the applications that either have a User Interface (UI) or an Application Programming Interface (API). We often find mature test practices, standards, and automation regarding UI or API testing. However, another kind is present in almost all types of industries that deal with data in bulk and often get handled through something called a Batch Application. This is primarily an offline application companies develop to process large data sets that often deal with multiple business rules. The challenge gets more prominent when we try to automate batch testing. This paper describes the approaches taken to test a Batch application from a Financial Industry to test the payment settlement process (a critical use case in all kinds of FinTech companies), resulting in 100% test automation in Test Creation and Test execution. One can follow this approach for any other batch use cases to achieve a higher efficiency in their testing process.

Keywords: batch testing, batch test automation, batch test strategy, payments testing, payments settlement testing

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Authors: Almagul Assainova , Dariya Abykenova, Liudmila Goncharenko, Sergey Sybachin, Saule Rakhimova, Abay Aman

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The recognition of a handwritten Kazakh text is a relevant objective today for the digitization of materials. The study presents a model of a hybrid neural network for handwriting recognition, which includes a convolutional neural network and a multi-layer perceptron. Each network includes 1024 input neurons and 42 output neurons. The model is implemented in the program, written in the Python programming language using the EMNIST database, NumPy, Keras, and Tensorflow modules. The neural network training of such specific letters of the Kazakh alphabet as ә, ғ, қ, ң, ө, ұ, ү, h, і was conducted. The neural network model and the program created on its basis can be used in electronic document management systems to digitize the Kazakh text.

Keywords: handwriting recognition system, image recognition, Kazakh font, machine learning, neural networks

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Authors: Zhenqiu Liu

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Visualizing the heterogeneity within cell-populations for single-cell RNA-seq data is crucial for studying the functional diversity of a cell. However, because of the high level of noises, outlier, and dropouts, it is very challenging to measure the cell-to-cell similarity (distance), visualize and cluster the data in a low-dimension. Minimum volume embedding (MVE) projects the data into a lower-dimensional space and is a promising tool for data visualization. However, it is computationally inefficient to solve a semi-definite programming (SDP) when the sample size is large. Therefore, it is not applicable to single-cell RNA-seq data with thousands of samples. In this paper, we develop an efficient algorithm with an accelerated proximal gradient method and visualize the single-cell RNA-seq data efficiently. We demonstrate that the proposed approach separates known subpopulations more accurately in single-cell data sets than other existing dimension reduction methods.

Keywords: single-cell RNA-seq, minimum volume embedding, visualization, accelerated proximal gradient method

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Authors: Tianyu Gao

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Digital architectures use computer-aided design, programming, simulation, and imaging to create virtual forms and physical structures. Today's customers want to know more about their buildings. They want an automatic thermostat to learn their behavior and contact them, such as the doors and windows they want to open with a mobile app. Therefore, the architectural display form is more closely related to the customer's experience. Based on the purpose of building informationization, this paper studies the steel structure design based on BIM. Taking the Zigan office building in Hangzhou as an example, it is divided into four parts, namely, the digital design modulus of the steel structure, the node analysis of the steel structure, the digital production and construction of the steel structure. Through the application of BIM software, the architectural design can be synergized, and the building components can be informationized. Not only can the architectural design be feedback in the early stage, but also the stability of the construction can be guaranteed. In this way, the monitoring of the entire life cycle of the building and the meeting of customer needs can be realized.

Keywords: digital architectures, BIM, steel structure, architectural design

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Authors: Jonathon Smith, Alba Agostino

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Autism, also referred to as autism spectrum disorder (ASD), is commonly associated with difficulties in social and communication skills. Other characteristics common to autistic individuals include repetitive behaviours, difficulties adhering to routine, as well as paying attention. Recent findings indicate that autism is the fastest-growing neurodevelopmental disorder in North America, yet programming aimed at improving the quality of autistic individual’s real-world social interactions is limited. Although there are social skills programs for autistic youth, participation appears to improve social knowledge, but that knowledge does not improve social competence or transfer to the participant’s daily social interactions. Peers are less likely to interact with autistic people based thin slice judgements, meaning that even when an autistic youth has successfully completed a social skills program, they most likely will still be rejected by peers and not have a social group to participate in. Recently, many researchers are exploring therapeutic interventions using Dungeon and Dragons (D&D) for conditions such as social anxiety, loneliness, and identity exploration. D&D is a table-top role-playing game (TTRPG) based on social play experience where the players must communicate, plan, negotiate, and compromise with other players to achieve a shared goal. The game encourages players to assume the role of their character and act out their play within the rules of the game with the guidance of the games dungeon master. The popularity Dungeons and Dragons has increased at a rapid rate, and many suggest that there social-emotional benefits of joining and participating in these types of gaming experiences, however this is an under researched topic and studies examining the benefits of such games is lacking in the field. The main purpose of this exploratory study is to examine the autistic youth’s experiences of participating in a D&D club. Participants of this study were four high functioning autistic youth between the ages of 14-18 (average age – 16) enrolled in a D&D Club that was specifically designed for neurodiverse youth. The youth participation with the club ranged from 4 months to 8 months. All participants completed a 30–40-minute semi-structured interview where they were able to express their perceptions as participants of the D&D club. Preliminary findings suggest that the game provided a place for the youth to engage in authentic social interactions. Additionally, preliminary results suggest that the youth report being in a positive space with other neurodivergent youth created an atmosphere where they felt confident and could connect with others. The findings from this study will aid clinicians, researchers, and educators in developing programming aimed at improving social interactions and connections for autistic youth.

Keywords: autism, social connection, dungeons and dragons, neurodivergent affirming space

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Authors: T. Martini, J. M. Martínez

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An algorithmic acceleration strategy based on quasi-Newton (or secant) methods is displayed for address the practical problem of accelerating the convergence of the Newton-Lagrange method in the case of convergence to critical multipliers. Since the Newton-Lagrange iteration converges locally at a linear rate, it is natural to conjecture that quasi-Newton methods based on the so called secant equation and some minimal variation principle, could converge superlinearly, thus restoring the convergence properties of Newton's method. This strategy can also be applied to accelerate the convergence of algorithms applied to fixed-points problems. Computational experience is reported illustrating the efficiency of this strategy to solve fixed-point problems with linear convergence rate.

Keywords: algorithmic acceleration, fixed-point problems, nonlinear programming, quasi-newton method

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Authors: Miklos Taliga

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In the present case study we examined the development and testing methods of systems that contain safety-critical elements in different industrial fields. Consequentially, we observed the classical object-oriented development and testing environment, as both medical technology and automobile industry approaches the development of safety critical elements that way. Subsequently, we examined model-based development. We introduce the quality parameters that define development and testing. While taking modern agile methodology (scrum) into consideration, we examined whether and to what extent the methodologies we found fit into this environment.

Keywords: safety-critical elements, quality managent, unit verification, model base testing, agile methods, scrum, metamodel, object-oriented programming, field specific modelling, sprint, user story, UML Standard

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Authors: Ali Vaezi, Jyotirmoy Dalal, Manish Verma

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The railroad is one of the primary transportation modes for hazardous materials (hazmat) shipments in North America. Installing an emergency response network capable of providing a commensurate response is one of the primary levers to contain (or mitigate) the adverse consequences from rail hazmat incidents. To this end, we propose a two-stage stochastic program to determine the location of and equipment packages to be stockpiled at each response facility. The raw input data collected from publicly available reports were processed, fed into the proposed optimization program, and then tested on a realistic railroad network in Ontario (Canada). From the resulting analyses, we conclude that the decisions based only on empirical datasets would undermine the effectiveness of the resulting network; coverage can be improved by redistributing equipment in the network, purchasing equipment with higher containment capacity, and making use of a disutility multiplier factor.

Keywords: hazmat, rail network, stochastic programming, emergency response

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Authors: Artur Matuck, Guilherme F. Nobre

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Machines can be either tool, media, or social agents. Advances in technology have been delivering machines capable of autonomous expression, both through communication and art. This paper deals with models (theoretical approach) and experiments (applied approach) related to artificial agents. On one hand it traces how social sciences' scholars have worked with topics such as text automatization, man-machine writing cooperation, and communication. On the other hand it covers how computer sciences' scholars have built communicative and artistic machines, including the programming of creativity. The aim is to present a brief survey on artificially intelligent communicators and artificially creative writers, and provide the basis to understand the meta-authorship and also to new and further man-machine co-authorship.

Keywords: artificial communication, artificial creativity, artificial writers, meta-authorship, robotic art

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Authors: Filipe Cardoso de Oliveira, Lino Marcos da Silva, Ademar Nogueira do Nascimento, Cristiano Hora de Oliveira Fontes

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This article presents a mathematical formulation for the production Break-Even Point problem in a non-homogeneous system. The optimization problem aims to obtain the composition of the best product mix in a non-homogeneous industrial plant, with the lowest cost until the breakeven point is reached. The problem constraints represent real limitations of a generic non-homogeneous industrial plant for n different products. The proposed model is able to solve the equilibrium point problem simultaneously for all products, unlike the existing approaches that propose a resolution in a sequential way, considering each product in isolation and providing a sub-optimal solution to the problem. The results indicate that the product mix found through the proposed model has economical advantages over the traditional approach used.

Keywords: branch and bound, break-even point, non-homogeneous production system, integer linear programming, management accounting

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Authors: H. C. Nguyen

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This contribution considers seismic effects on the stability of slope and footing resting on a slope. The seismic force is simply treated as static inertial force through the values of acceleration factor. All domains are assumed to be plasticity deformations approximated using node-based smoothed finite element method (NS-FEM). The failure mechanism and safety factor were then explored using numerical procedure based on upper bound approach in which optimization problem was formed as second order cone programming (SOCP). The data obtained confirm that upper bound procedure using NS-FEM and SOCP can give stable and rapid convergence results of seismic stability factors.

Keywords: upper bound analysis, safety factor, slope stability, footing resting on slope

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Authors: Asma’a Mahfoud, Abu Bakar Md. Sultan, Abdul Azim Abd, Norhayati Mohd Ali, Novia Admodisastro

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Obfuscation is a practice to make something difficult and complicated. Programming code is ordinarily obfuscated to protect the intellectual property (IP) and prevent the attacker from reverse engineering (RE) a copyrighted software program. Obfuscation may involve encrypting some or all the code, transforming out potentially revealing data, renaming useful classes and variables (identifiers) names to meaningless labels, or adding unused or meaningless code to an application binary. Obfuscation techniques were not performing effectively recently as the reversing tools are able to break the obfuscated code. We propose in this paper a hybrid obfuscation technique that contains three approaches of renaming. Experimentation was conducted to test the effectiveness of the proposed technique. The experimentation has presented a promising result, where the reversing tools were not able to read the code.

Keywords: intellectual property, obfuscation, software security, reverse engineering

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Authors: Nurul Liyana Abdul Aziz, Nur Aidya Hanum Aizam

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Course timetabling problems occur every semester in a university which includes the allocation of resources (subjects, lecturers and students) to a number of fixed rooms and timeslots. The assignment is carried out in a way such that there are no conflicts within rooms, students and lecturers, as well as fulfilling a range of constraints. The constraints consist of rules and policies set up by the universities as well as lecturers’ and students’ preferences of courses to be allocated in specific timeslots. This paper specifically focuses on the preferences of the course timetabling problem in one of the public universities in Malaysia. The demands will be considered into our existing mathematical model to make it more generalized and can be used widely. We have distributed questionnaires to a number of lecturers and students of the university to investigate their demands and preferences for their desired course timetable. We classify the preferences thus converting them to construct one mathematical model that can produce such timetable.

Keywords: university course timetabling problem, integer programming, preferences, constraints

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Authors: Qing K. Zhu

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Very Large Scale Integration (VLSI) design becomes very complex due to the continuous integration of millions of gates in one chip based on Moore’s law. Designers have encountered numerous report files during design iterations using timing and noise analysis tools. This paper presented our work using data mining techniques combined with HTML tables to extract and represent critical timing/noise data. When we apply this data-mining tool in real applications, the running speed is important. The software employs table look-up techniques in the programming for the reasonable running speed based on performance testing results. We added several advanced features for the application in one industry chip design.

Keywords: VLSI design, data mining, big data, HTML forms, web, VLSI, EDA, timing, noise

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Authors: Duygu Kan, Mehmet Cayoren

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Sommerfeld-integrals (SIs) are commonly encountered in electromagnetics problems involving analysis of antennas and scatterers embedded in planar multilayered media. Generally speaking, the analytical solution of SIs is unavailable, and it is well known that numerical evaluation of SIs is very time consuming and computationally expensive due to the highly oscillating and slowly decaying nature of the integrands. Therefore, fast computation of SIs has a paramount importance. In this paper, a parallel code has been developed to speed up the computation of SI in the framework of calculation of dyadic Green’s function in multilayered media. OpenMP shared memory approach is used to parallelize the SI algorithm and resulted in significant time savings. Moreover accelerating the computation of dyadic Green’s function is discussed based on the parallel SI algorithm developed.

Keywords: Sommerfeld-integrals, multilayer dyadic Green’s function, OpenMP, shared memory parallel programming

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Authors: A. Yacob, N. Hj. Ali, M. H. Yusoff, M. Y. MohdSaman, W. M. A. F. W. Hamzah

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Personalized learning becomes increasingly popular which not is restricted by time, place or any other barriers. This study proposes an analysis of Personalized Learning using Item Response Theory which considers course material difficulty and learner ability. The study investigates twenty undergraduate students at TATI University College, who are taking programming subject. By using the IRT, it was found that, finding the most appropriate problem levels to each student include high and low level test items together is not a problem. Thus, the student abilities can be asses more accurately and fairly. Learners who experience more anxiety will affect a heavier cognitive load and receive lower test scores. Instructors are encouraged to provide a supportive learning environment to enhance learning effectiveness because Cognitive Load Theory concerns the limited capacity of the brain to absorb new information.

Keywords: assessment, item response theory, cognitive load theory, learning, motivation, performance

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Authors: Mohsen Ziaee

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Almost all existing researches on the flowshop scheduling problems focus on the permutation schedules and there is insufficient study dedicated to the general flowshop scheduling problems in the literature, since the modeling and solving of the general flowshop scheduling problems are more difficult than the permutation ones, especially for the large-size problem instances. This paper considers the general flowshop scheduling problem with the objective function of the makespan (F//Cmax). We first find the optimal solution of the problem by solving a mixed integer linear programming model. An efficient heuristic method is then presented to solve the problem. An ant colony optimization algorithm is also proposed for the problem. In order to evaluate the performance of the methods, computational experiments are designed and performed. Numerical results show that the heuristic algorithm can result in reasonable solutions with low computational effort and even achieve optimal solutions in some cases.

Keywords: scheduling, general flow shop scheduling problem, makespan, heuristic

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Authors: M. H. Hussain, I. Musirin, A. F. Abidin, S. R. A. Rahim

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This paper presents the implementation of Artificial Bees Colony (ABC) algorithm in solving Directional OverCurrent Relays (DOCRs) coordination problem for near-end faults occurring in fixed network topology. The coordination optimization of DOCRs is formulated as linear programming (LP) problem. The objective function is introduced to minimize the operating time of the associated relay which depends on the time multiplier setting. The proposed technique is to taken as a technique for comparison purpose in order to highlight its superiority. The proposed algorithms have been tested successfully on 8 bus test system. The simulation results demonstrated that the ABC algorithm which has been proved to have good search ability is capable in dealing with constraint optimization problems.

Keywords: artificial bees colony, directional overcurrent relay coordination problem, relay settings, time multiplier setting

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Authors: H. Samadiyeh, R. Khajavi

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A new FD-based procedure, modal finite difference method (MFDM), is proposed for seismic wave propagation modeling, in which simulation is dealt with in the modal space. The method employs eigenvalues of a characteristic matrix formed by appropriate time-space FD stencils. Since MFD runs for different modes are totally independent of each other, MFDM can easily be parallelized while considerable simplicity in parallel-algorithm is also achieved. There is no requirement to any domain-decomposition procedure and inter-core data exchange. More important is the possibility to skip processing of less-significant modes, which enables one to adjust the procedure up to the level of accuracy needed. Thus, in addition to considerable ease of parallel programming, computation and storage costs are significantly reduced. The method is qualified for its efficiency by some numerical examples.

Keywords: Finite Difference Method, Graphics Processing Unit (GPU), Message Passing Interface (MPI), Modal, Wave propagation

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