Search results for: quadratic programming

Authors: Xia Cui, Guang-wei Yuan, Jing-yan Yue

Abstract:

A Picard-Newton iteration method is studied to accelerate the numerical solution procedure of a class of two-dimensional nonlinear coupled parabolic-hyperbolic system. The Picard-Newton iteration is designed by adding higher-order terms of small quantity to an existing Picard iteration. The discrete functional analysis and inductive hypothesis reasoning techniques are used to overcome difficulties coming from nonlinearity and coupling, and theoretical analysis is made for the convergence and approximation properties of the iteration scheme. The Picard-Newton iteration has a quadratic convergent ratio, and its solution has second order spatial approximation and first order temporal approximation to the exact solution of the original problem. Numerical tests verify the results of the theoretical analysis, and show the Picard-Newton iteration is more efficient than the Picard iteration.

Keywords: Nonlinearity, iterative acceleration, coupled parabolic hyperbolic system, quadratic convergence, numerical analysis.

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Authors: Tunjo Perić, Marin Fatović

Abstract:

In this paper a new methodology for vendor selection and supply quotas determination (VSSQD) is proposed. The problem of VSSQD is solved by the model that combines revised weighting method for determining the objective function coefficients, and a multiple objective linear programming (MOLP) method based on the cooperative game theory for VSSQD. The criteria used for VSSQD are: (1) purchase costs and (2) product quality supplied by individual vendors. The proposed methodology has been tested on the example of flour purchase for a bakery with two decision makers.

Keywords: Cooperative game theory, multiple objective linear programming, revised weighting method, vendor selection.

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Authors: Marc C. Robini, Pierre-Jean Viverge, Yuemin Zhu, Jianhua Luo

Abstract:

The standard approach to image reconstruction is to stabilize the problem by including an edge-preserving roughness penalty in addition to faithfulness to the data. However, this methodology produces noisy object boundaries and creates a staircase effect. The existing attempts to favor the formation of smooth contour lines take the edge field explicitly into account; they either are computationally expensive or produce disappointing results. In this paper, we propose to incorporate the smoothness of the edge field in an implicit way by means of an additional penalty term defined in the wavelet domain. We also derive an efficient half-quadratic algorithm to solve the resulting optimization problem, including the case when the data fidelity term is non-quadratic and the cost function is nonconvex. Numerical experiments show that our technique preserves edge sharpness while smoothing contour lines; it produces visually pleasing reconstructions which are quantitatively better than those obtained without wavelet-domain constraints.

Keywords:

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Authors: Taehan Lee

Abstract:

We consider a network design problem which has shortest routing restriction based on the values determined by the installed facilities on each arc. In conventional multicommodity network design problem, a commodity can be routed through any possible path when the capacity is available. But, we consider a problem in which the commodity between two nodes must be routed on a path which has shortest metric value and the link metric value is determined by the installed facilities on the link. By this routing restriction, the problem has a distinct characteristic. We present an integer programming formulation containing the primal-dual optimality conditions to the shortest path routing. We give some computational results for the model.

Keywords: Integer programming, multicommodity network design, routing, shortest path.

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Authors: A. Udomsakdigool, S. Bangsaranthip

Abstract:

This paper presents an algorithm which combining ant colony optimization in the dynamic programming for solving a dynamic facility layout problem. The problem is separated into 2 phases, static and dynamic phase. In static phase, ant colony optimization is used to find the best ranked of layouts for each period. Then the dynamic programming (DP) procedure is performed in the dynamic phase to evaluate the layout set during multi-period planning horizon. The proposed algorithm is tested over many problems with size ranging from 9 to 49 departments, 2 and 4 periods. The experimental results show that the proposed method is an alternative way for the plant layout designer to determine the layouts during multi-period planning horizon.

Keywords: Ant colony optimization, Dynamicprogramming, Dynamic facility layout planning, Metaheuristic

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Authors: C. Ardil

Abstract:

A multiple-criteria decision support system is proposed for the best aircraft selection decision. Various strategic, economic, environmental, and risk-related factors can directly or indirectly influence this choice, and they should be taken into account in the decision-making process. The paper suggests a multiple-criteria analysis to aid in the airline management's decision-making process when choosing an appropriate aircraft. In terms of the suggested approach, an integrated entropic preference optimization programming (POP) for fleet modeling risk analysis is applied. The findings of the study of multiple criteria analysis indicate that the A321(neo) aircraft type is the best alternative in this particular optimization instance. The proposed methodology can be applied to other complex engineering problems involving multiple criteria analysis.

Keywords: Aircraft selection, decision making, multiple criteria decision making, preference optimization programming, POP, entropic weight method, TOPSIS, WSM, WPM

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Authors: G. Shojatalab

Abstract:

In this paper, all variables are supposed to be integer and positive. In this modern method, objective function is assumed to be maximized or minimized but constraints are always explained like less or equal to. In this method, choosing a dual combination of ideal nonequivalent and omitting one of variables. With continuing this act, finally, having one nonequivalent with (n-m+1) unknown quantities in which final nonequivalent, m is counter for constraints, n is counter for variables of decision.

Keywords: Integer, Programming, Operation Research, Variables of decision.

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Authors: Walid S. Abd El-hamid, Sherif S. El-Etriby, Mohiy M. Hadhoud

Abstract:

Regression testing is a maintenance activity applied to modified software to provide confidence that the changed parts are correct and that the unchanged parts have not been adversely affected by the modifications. Regression test selection techniques reduce the cost of regression testing, by selecting a subset of an existing test suite to use in retesting modified programs. This paper presents the first general regression-test-selection technique, which based on code and allows selecting test cases for any programs written in any programming language. Then it handles incomplete program. We also describe RTSDiff, a regression-test-selection system that implements the proposed technique. The results of the empirical studied that performed in four programming languages java, C#, Cµ and Visual basic show that the efficiency and effective in reducing the size of test suit.

Keywords: Regression testing, testing, test selection, softwareevolution, software maintenance.

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Authors: V.Sekar, T.G.Palanivelu, B.Revathi

Abstract:

Electromagnetic interference (EMI) is one of the serious problems in most electrical and electronic appliances including fluorescent lamps. The electronic ballast used to regulate the power flow through the lamp is the major cause for EMI. The interference is because of the high frequency switching operation of the ballast. Formerly, some EMI mitigation techniques were in practice, but they were not satisfactory because of the hardware complexity in the circuit design, increased parasitic components and power consumption and so on. The majority of the researchers have their spotlight only on EMI mitigation without considering the other constraints such as cost, effective operation of the equipment etc. In this paper, we propose a technique for EMI mitigation in fluorescent lamps by integrating Frequency Modulation and Evolutionary Programming. By the Frequency Modulation technique, the switching at a single central frequency is extended to a range of frequencies, and so, the power is distributed throughout the range of frequencies leading to EMI mitigation. But in order to meet the operating frequency of the ballast and the operating power of the fluorescent lamps, an optimal modulation index is necessary for Frequency Modulation. The optimal modulation index is determined using Evolutionary Programming. Thereby, the proposed technique mitigates the EMI to a satisfactory level without disturbing the operation of the fluorescent lamp.

Keywords: Ballast, Electromagnetic interference (EMI), EMImitigation, Evolutionary programming (EP), Fluorescent lamp, Frequency Modulation (FM), Modulation index.

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Authors: C. Ardil

Abstract:

In this research, the Balkan peninsula countries' developmental integration into European Union represents the strategic economic development objectives of the countries in the region. In order to objectively analyze the level of economic development competition of Balkan Peninsula countries, the mathematical compromise programming technique of multicriteria evaluation is used in this ranking problem. The primary aim of this research is to explain the role and significance of the multicriteria method evaluation using a real example of compromise solutions. Using the mathematical compromise programming technique, twelve countries of the Balkan Peninsula are economically evaluated and mutually compared. The economic development evaluation of the countries is performed according to five evaluation criteria forming the basis for economic development evaluation. The multiattribute model is solved using the mathematical compromise programming technique for producing different Pareto solutions. The results obtained by the multicriteria evaluation gives the possibility of identification and evaluation of the most eminent economic development indicators for each country separately. Finally, in this way, the proposed method has proved to be a successful model for the evaluation of the Balkan peninsula countries' economic development competition.

Keywords: Balkan peninsula countries, standard deviation, multicriteria decision making, mathematical compromise programming, multicriteria decision making, multicriteria analysis, multicriteria decision analysis.

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Authors: Mohsen Ziaee

Abstract:

In this paper, a mixed integer linear programming (MILP) model is presented to solve the flexible job shop scheduling problem (FJSP). This problem is one of the hardest combinatorial problems. The objective considered is the minimization of the makespan. The computational results of the proposed MILP model were compared with those of the best known mathematical model in the literature in terms of the computational time. The results show that our model has better performance with respect to all the considered performance measures including relative percentage deviation (RPD) value, number of constraints, and total number of variables. By this improved mathematical model, larger FJS problems can be optimally solved in reasonable time, and therefore, the model would be a better tool for the performance evaluation of the approximation algorithms developed for the problem.

Keywords: Scheduling, flexible job shop, makespan, mixed integer linear programming.

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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: Labiba Gilani, Asifullah Khan, Anwar M. Mirza

Abstract:

This paper introduces a technique of distortion estimation in image watermarking using Genetic Programming (GP). The distortion is estimated by considering the problem of obtaining a distorted watermarked signal from the original watermarked signal as a function regression problem. This function regression problem is solved using GP, where the original watermarked signal is considered as an independent variable. GP-based distortion estimation scheme is checked for Gaussian attack and Jpeg compression attack. We have used Gaussian attacks of different strengths by changing the standard deviation. JPEG compression attack is also varied by adding various distortions. Experimental results demonstrate that the proposed technique is able to detect the watermark even in the case of strong distortions and is more robust against attacks.

Keywords: Blind Watermarking, Genetic Programming (GP), Fitness Function, Discrete Cosine Transform (DCT).

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Authors: Sun Li-ping, Zhu Jian-xun, Liu Sheng-nan

Abstract:

In order to study the performance of dynamic positioning system during S-lay operations, dynamic positioning system is simulated with the hull-stinger-pipe coupling effect. The roller of stinger is simulated by the generalized elastic contact theory. The stinger is composed of Morrison members. Force on pipe is calculated by lumped mass method. Time domain of fully coupled barge model is analyzed combining with PID controller, Kalman filter and allocation of thrust using Sequential Quadratic Programming method. It is also analyzed that the effect of hull wave frequency motion on pipe-stinger coupling force and dynamic positioning system. Besides, it is studied that how S-lay operations affect the dynamic positioning accuracy. The simulation results are proved to be available by checking pipe stress with API criterion. The effect of heave and yaw motion cannot be ignored on hull-stinger-pipe coupling force and dynamic positioning system. It is important to decrease the barge’s pitch motion and lay pipe in head sea in order to improve safety of the S-lay installation and dynamic positioning.

Keywords: S-lay operation, dynamic positioning, coupling motion; time domain, allocation of thrust.

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Authors: Zahra Khalid, Gul Muhammad Khan, Arbab Masood Ahmad

Abstract:

Cartesian Genetic Programming (CGP) is explored to design an optimal circuit capable of early stage breast cancer detection. CGP is used to evolve simple multiplexer circuits for detection of malignancy in the Fine Needle Aspiration (FNA) samples of breast. The data set used is extracted from Wisconsins Breast Cancer Database (WBCD). A range of experiments were performed, each with different set of network parameters. The best evolved network detected malignancy with an accuracy of 99.14%, which is higher than that produced with most of the contemporary non-linear techniques that are computational expensive than the proposed system. The evolved network comprises of simple multiplexers and can be implemented easily in hardware without any further complications or inaccuracy, being the digital circuit.

Keywords: Breast cancer detection, cartesian genetic programming, evolvable hardware, fine needle aspiration (FNA).

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Authors: Chompunut Jantarasorn, Chutima Prommak

Abstract:

The important issue considered in the widespread deployment of Wireless Sensor Networks (WSNs) is an efficiency of the energy consumption. In this paper, we present a study of the optimal relay station planning problems using Binary Integer Linear Programming (BILP) model to minimize the energy consumption in WSNs. Our key contribution is that the proposed model not only ensures the required network lifetime but also guarantees the radio connectivity at high level of communication quality. Specially, we take into account effects of noise, signal quality limitation and bit error rate characteristics. Numerical experiments were conducted in various network scenarios. We analyzed the effects of different sensor node densities and distribution on the energy consumption.

Keywords: Binary Integer Linear Programming, BILP, Energy consumption, Optimal node placement and Wireless sensor networks.

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Authors: Chuan Li, Ming Liang, Qibing Yu

Abstract:

To improve the dynamics response of the vehicle passive suspension, a two-terminal mass is suggested to connect in parallel with the suspension strut. Three performance criteria, tire grip, ride comfort and suspension deflection, are taken into consideration to optimize the suspension parameters. However, the three criteria are conflicting and non-commensurable. For this reason, the Chebyshev goal programming method is applied to find the best tradeoff among the three objectives. A simulation case is presented to describe the multi-objective optimization procedure. For comparison, the Chebyshev method is also employed to optimize the design of a conventional passive suspension. The effectiveness of the proposed design method has been clearly demonstrated by the result. It is also shown that the suspension with a two-terminal mass in parallel has better performance in terms of the three objectives.

Keywords: Vehicle, passive suspension, two-terminal mass, optimization, Chebyshev goal programming

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Authors: Nittaya Kerdprasop, Kittisak Kerdprasop

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Frequent patterns are patterns such as sets of features or items that appear in data frequently. Finding such frequent patterns has become an important data mining task because it reveals associations, correlations, and many other interesting relationships hidden in a dataset. Most of the proposed frequent pattern mining algorithms have been implemented with imperative programming languages such as C, Cµ, Java. The imperative paradigm is significantly inefficient when itemset is large and the frequent pattern is long. We suggest a high-level declarative style of programming using a functional language. Our supposition is that the problem of frequent pattern discovery can be efficiently and concisely implemented via a functional paradigm since pattern matching is a fundamental feature supported by most functional languages. Our frequent pattern mining implementation using the Haskell language confirms our hypothesis about conciseness of the program. The performance studies on speed and memory usage support our intuition on efficiency of functional language.

Keywords: Association, frequent pattern mining, functionalprogramming, pattern matching.

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Authors: Erik Delarue, David Bekaert, Ronnie Belmans, William D'haeseleer

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This paper presents the development of an electricity simulation model taking into account electrical network constraints, applied on the Belgian power system. The base of the model is optimizing an extensive Unit Commitment (UC) problem through the use of Mixed Integer Linear Programming (MILP). Electrical constraints are incorporated through the implementation of a DC load flow. The model encloses the Belgian power system in a 220 – 380 kV high voltage network (i.e., 93 power plants and 106 nodes). The model features the use of pumping storage facilities as well as the inclusion of spinning reserves in a single optimization process. Solution times of the model stay below reasonable values.

Keywords: Electricity generation modeling, Unit Commitment(UC), Mixed Integer Linear Programming (MILP), DC load flow.

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Authors: Janya Sainui

Abstract:

As a rapid growth of digital videos and data communications, video summarization that provides a shorter version of the video for fast video browsing and retrieval is necessary. Key frame extraction is one of the mechanisms to generate video summary. In general, the extracted key frames should both represent the entire video content and contain minimum redundancy. However, most of the existing approaches heuristically select key frames; hence, the selected key frames may not be the most different frames and/or not cover the entire content of a video. In this paper, we propose a method of video summarization which provides the reasonable objective functions for selecting key frames. In particular, we apply a statistical dependency measure called quadratic mutual informaion as our objective functions for maximizing the coverage of the entire video content as well as minimizing the redundancy among selected key frames. The proposed key frame extraction algorithm finds key frames as an optimization problem. Through experiments, we demonstrate the success of the proposed video summarization approach that produces video summary with better coverage of the entire video content while less redundancy among key frames comparing to the state-of-the-art approaches.

Keywords: Video summarization, key frame extraction, dependency measure, quadratic mutual information, optimization.

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Authors: Dinesh Rao, M.G.M. Khan, Sabiha Khan

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Calibration estimation is a method of adjusting the original design weights to improve the survey estimates by using auxiliary information such as the known population total (or mean) of the auxiliary variables. A calibration estimator uses calibrated weights that are determined to minimize a given distance measure to the original design weights while satisfying a set of constraints related to the auxiliary information. In this paper, we propose a new multivariate calibration estimator for the population mean in the stratified sampling design, which incorporates information available for more than one auxiliary variable. The problem of determining the optimum calibrated weights is formulated as a Mathematical Programming Problem (MPP) that is solved using the Lagrange multiplier technique.

Keywords: Calibration estimation, Stratified sampling, Multivariate auxiliary information, Mathematical programming problem, Lagrange multiplier technique.

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Authors: A. Maddi, A. Guessoum, D. Berkani

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The purpose of this paper is to provide a practical example to the Linear Quadratic Gaussian (LQG) controller. This method includes a description and some discussion of the discrete Kalman state estimator. One aspect of this optimality is that the estimator incorporates all information that can be provided to it. It processes all available measurements, regardless of their precision, to estimate the current value of the variables of interest, with use of knowledge of the system and measurement device dynamics, the statistical description of the system noises, measurement errors, and uncertainty in the dynamics models. Since the time of its introduction, the Kalman filter has been the subject of extensive research and application, particularly in the area of autonomous or assisted navigation. For example, to determine the velocity of an aircraft or sideslip angle, one could use a Doppler radar, the velocity indications of an inertial navigation system, or the relative wind information in the air data system. Rather than ignore any of these outputs, a Kalman filter could be built to combine all of this data and knowledge of the various systems- dynamics to generate an overall best estimate of velocity and sideslip angle.

Keywords: Aircraft motion, Kalman filter, LQG control, Lateral stability, State estimator.

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Authors: J. Fernandez de Canete, P. Del Saz Orozco, S. Gonzalez-Perez

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LabVIEW and SIMULINK are two most widely used graphical programming environments for designing digital signal processing and control systems. Unlike conventional text-based programming languages such as C, Cµ and MATLAB, graphical programming involves block-based code developments, allowing a more efficient mechanism to build and analyze control systems. In this paper a LabVIEW environment has been employed as a graphical user interface for monitoring the operation of a controlled distillation column, by visualizing both the closed loop performance and the user selected control conditions, while the column dynamics has been modeled under the SIMULINK environment. This tool has been applied to the PID based decoupled control of a binary distillation column. By means of such integrated environments the control designer is able to monitor and control the plant behavior and optimize the response when both, the quality improvement of distillation products and the operation efficiency tasks, are considered.

Keywords: Distillation control, software tools, SIMULINKLabVIEWinterface.

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Authors: J. A. López, C.I. López, J.E. Olguín, C. Camargo, J. M. López

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The institutions seek to improve their performance and quality of service, so that their patients are satisfied. This research project aims, conduct a time study program in the area of gynecological surgery, to determine the current level of capacity and optimize the programming time in order to adequately respond to demand. The system is analyzed by waiting lines and uses the simulation using ARENA to evaluate proposals for improvement and optimization programming time each of the surgeries.

Keywords: Time study, waiting lines, reducing time, simulation.

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Authors: Kaushalendra K. Khadanga, Lee Hee Hyol

Abstract:

Passenger comfort has been paramount in the design of suspension systems of high speed cars. To analyze the effect of vibration on vehicle ride quality, a vertical model of a six degree of freedom railway passenger vehicle, with front and rear suspension, is built. It includes car body flexible effects and vertical rigid modes. A second order linear shaping filter is constructed to model Gaussian white noise into random rail excitation. The temporal correlation between the front and rear wheels is given by a second order Pade approximation. The complete track and the vehicle model are then designed. An active secondary suspension system based on a Linear Quadratic Gaussian (LQG) optimal control method is designed. The results show that the LQG control method reduces the vertical acceleration, pitching acceleration and vertical bending vibration of the car body as compared to the passive system.

Keywords: Active suspension, bending vibration, railway vehicle, vibration control.

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Authors: Chaimongkol Limpianchob

Abstract:

This work addresses the problem of production planning that arises in the production of aromatic coconuts from Samudsakhorn province in Thailand. The planning involves the forwarding of aromatic coconuts from the harvest areas to the factory, which is classified into two groups; self-owned areas and contracted areas, the decisions of aromatic coconuts flow in the plant, and addressing a question of which warehouse will be in use. The problem is formulated as a mixed-integer linear programming model within supply chain management framework. The objective function seeks to minimize the total cost including the harvesting, labor and inventory costs. Constraints on the system include the production activities in the company and demand requirements. Numerical results are presented to demonstrate the feasibility of coconuts supply chain model compared with base case.

Keywords: Aromatic coconut, supply chain management, production planning, mixed-integer linear programming.

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Authors: M. Imran Akram, Ahmed Pasha, Nabeel Iqbal

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In this paper a real-time trajectory generation algorithm for computing 2-D optimal paths for autonomous aerial vehicles has been discussed. A dynamic programming approach is adopted to compute k-best paths by minimizing a cost function. Collision detection is implemented to detect intersection of the paths with obstacles. Our contribution is a novel approach to the problem of trajectory generation that is computationally efficient and offers considerable gain over existing techniques.

Keywords: dynamic programming, graph search, path planning.

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Authors: Hajar Sadeghi, Payman Moallem, S. Amirhassn Monadjemi

Abstract:

This paper presents a new feature based dense stereo matching algorithm to obtain the dense disparity map via dynamic programming. After extraction of some proper features, we use some matching constraints such as epipolar line, disparity limit, ordering and limit of directional derivative of disparity as well. Also, a coarseto- fine multiresolution strategy is used to decrease the search space and therefore increase the accuracy and processing speed. The proposed method links the detected feature points into the chains and compares some of the feature points from different chains, to increase the matching speed. We also employ color stereo matching to increase the accuracy of the algorithm. Then after feature matching, we use the dynamic programming to obtain the dense disparity map. It differs from the classical DP methods in the stereo vision, since it employs sparse disparity map obtained from the feature based matching stage. The DP is also performed further on a scan line, between any matched two feature points on that scan line. Thus our algorithm is truly an optimization method. Our algorithm offers a good trade off in terms of accuracy and computational efficiency. Regarding the results of our experiments, the proposed algorithm increases the accuracy from 20 to 70%, and reduces the running time of the algorithm almost 70%.

Keywords: Chain Correspondence, Color Stereo Matching, Dynamic Programming, Epipolar Line, Stereo Vision.

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Authors: Vahid Nademi

Abstract:

In response to widely used wearable medical devices equipped with a continuous glucose monitor (CGM) and insulin pump, the advanced control methods are still demanding to get the full benefit of these devices. Unlike costly clinical trials, implementing effective insulin-glucose control strategies can provide significant contributions to the patients suffering from chronic diseases such as diabetes. This study deals with a key role of two-layer insulin-glucose regulator based on model-predictive-control (MPC) scheme so that the patient’s predicted glucose profile is in compliance with the insulin level injected through insulin pump automatically. It is achieved by iterative optimization algorithm which is called an integrated perturbation analysis and sequential quadratic programming (IPA-SQP) solver for handling uncertainties due to unexpected variations in glucose-insulin values and body’s characteristics. The feasibility evaluation of the discussed control approach is also studied by means of numerical simulations of two case scenarios via measured data. The obtained results are presented to verify the superior and reliable performance of the proposed control scheme with no negative impact on patient safety.

Keywords: Blood glucose monitoring, insulin pump, optimization, predictive control, diabetes disease.

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Authors: Wuthichai Wongthatsanekorn

Abstract:

Plastic waste is a big issue in Thailand, but the amount of recycled plastic in Thailand is still low due to the high investment and operating cost. Hence, the rest of plastic waste are burnt to destroy or sent to the landfills. In order to be financial viable, an effective reverse logistics infrastructure is required to support the product recovery activities. However, there is a conflict between reducing the cost and raising environmental protection level. The purpose of this study is to build a goal programming (GP) so that it can be used to help analyze the proper planning of the Thailand-s plastic recycling system that involves multiple objectives. This study considers three objectives; reducing total cost, increasing the amount of plastic recovery, and raising the desired plastic materials in recycling process. The results from two priority structures show that it is necessary to raise the total cost budget in order to achieve targets on amount of recycled plastic and desired plastic materials.

Keywords: Goal Programming, Plastic Recycling, Thailand.

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