Mathematical Problems in Engineering The latest articles from Hindawi Publishing Corporation © 2015 , Hindawi Publishing Corporation . All rights reserved. Robust Object Tracking Based on Simplified Codebook Masked Camshift Algorithm Thu, 30 Jul 2015 14:13:38 +0000 Moving targets detection and tracking is an important and basic issue in the field of intelligent video surveillance. The classical Codebook algorithm is simplified in this paper by introducing the average intensity into the Codebook model instead of the original minimal and maximal intensities. And a hierarchical matching method between the current pixel and codeword is also proposed according to the average intensity in the high and low intensity areas, respectively. Based on the simplified Codebook algorithm, this paper then proposes a robust object tracking algorithm called Simplified Codebook Masked Camshift algorithm (SCMC algorithm), which combines the simplified Codebook algorithm and Camshift algorithm together. It is designed to overcome the sensitiveness of traditional Camshift algorithm to background color interference. It uses simplified Codebook to detect moving objects, whose result is employed to mask color probability distribution image, based on which we then use Camshift to predict the centroid and size of these objects. Experiment results show that the proposed simplified Codebook algorithm simultaneously improves the detection accuracy and computational efficiency. And they also show that the SCMC algorithm can significantly reduce the possibility of false convergence and result in a higher correct tracking rate, as compared with the traditional Camshift algorithm. Yuanyuan Zhang, Xiaomei Zhao, Fengjiao Li, Jiande Sun, Shuming Jiang, and Changying Chen Copyright © 2015 Yuanyuan Zhang et al. All rights reserved. Optimization of Vacuum-Microwave Radiation Pretreatment on Extraction of Ganoderma Polysaccharides Thu, 30 Jul 2015 13:31:33 +0000 A new process of vacuum-microwave (VM) radiation pretreatment for extracting polysaccharides from the Ganoderma lucidum was proposed, and the parameters were optimized by response surface methodology (RSM). The orthogonal-central composite design scheme was used and the responsive surfaces methodology of three factors and five levels was adopted, and the factors influencing the technological parameters and its interaction terms were analyzed and regressed. The optimal parameters were obtained as follows: the infiltration time of 70 min, microwave power density of 11.2 W/g, and VM irradiation time of 180 s. In consequence, the extraction yield was up to 1.775% when VM radiation was conducted in advance. Compared to the traditional hot-water extraction method, VM pretreatment can shorten the extraction time by more than a half, and the polysaccharide extraction yield was increased by 48.1%. It holds significant potential for further investigation, development, and application. C. F. Song, S. G. Wang, J. Yang, Z. W. Cui, and Y. H. Gu Copyright © 2015 C. F. Song et al. All rights reserved. A Generalized Stability Theorem for Discrete-Time Nonautonomous Chaos System with Applications Thu, 30 Jul 2015 11:36:04 +0000 Firstly, this study introduces a definition of generalized stability (GST) in discrete-time nonautonomous chaos system (DNCS), which is an extension for chaos generalized synchronization. Secondly, a constructive theorem of DNCS has been proposed. As an example, a GST DNCS is constructed based on a novel 4-dimensional discrete chaotic map. Numerical simulations show that the dynamic behaviors of this map have chaotic attractor characteristics. As one application, we design a chaotic pseudorandom number generator (CPRNG) based on the GST DNCS. We use the SP800-22 test suite to test the randomness of four 100-key streams consisting of 1,000,000 bits generated by the CPRNG, the RC4 algorithm, the ZUC algorithm, and a 6-dimensional CGS-based CPRNG, respectively. The numerical results show that the randomness performances of the two CPRNGs are promising. In addition, theoretically the key space of the CPRNG is larger than 21116. As another application, this study designs a stream avalanche encryption scheme (SAES) in RGB image encryption. The results show that the GST DNCS is able to generate the avalanche effects which are similar to those generated via ideal CPRNGs. Mei Zhang, Danling Wang, Lequan Min, and Xue Wang Copyright © 2015 Mei Zhang et al. All rights reserved. Patch Similarity Modulus and Difference Curvature Based Fourth-Order Partial Differential Equation for Image Denoising Thu, 30 Jul 2015 08:38:44 +0000 The traditional fourth-order nonlinear diffusion denoising model suffers the isolated speckles and the loss of fine details in the processed image. For this reason, a new fourth-order partial differential equation based on the patch similarity modulus and the difference curvature is proposed for image denoising. First, based on the intensity similarity of neighbor pixels, this paper presents a new edge indicator called patch similarity modulus, which is strongly robust to noise. Furthermore, the difference curvature which can effectively distinguish between edges and noise is incorporated into the denoising algorithm to determine the diffusion process by adaptively adjusting the size of the diffusion coefficient. The experimental results show that the proposed algorithm can not only preserve edges and texture details, but also avoid isolated speckles and staircase effect while filtering out noise. And the proposed algorithm has a better performance for the images with abundant details. Additionally, the subjective visual quality and objective evaluation index of the denoised image obtained by the proposed algorithm are higher than the ones from the related methods. Yunjiao Bai, Quan Zhang, Hong Shangguan, Zhiguo Gui, Yi Liu, and Yanli Liu Copyright © 2015 Yunjiao Bai et al. All rights reserved. Downstream-Conditioned Maximum Entropy Method for Exit Boundary Conditions in the Lattice Boltzmann Method Thu, 30 Jul 2015 06:37:33 +0000 A method for modeling outflow boundary conditions in the lattice Boltzmann method (LBM) based on the maximization of the local entropy is presented. The maximization procedure is constrained by macroscopic values and downstream components. The method is applied to fully developed boundary conditions of the Navier-Stokes equations in rectangular channels. Comparisons are made with other alternative methods. In addition, the new downstream-conditioned entropy is studied and it was found that there is a correlation with the velocity gradient during the flow development. Javier A. Dottori, Gustavo A. Boroni, and Alejandro Clausse Copyright © 2015 Javier A. Dottori et al. All rights reserved. Two General Extension Algorithms of Latin Hypercube Sampling Wed, 29 Jul 2015 14:12:18 +0000 For reserving original sampling points to reduce the simulation runs, two general extension algorithms of Latin Hypercube Sampling (LHS) are proposed. The extension algorithms start with an original LHS of size and construct a new LHS of size that contains the original points as many as possible. In order to get a strict LHS of larger size, some original points might be deleted. The relationship of original sampling points in the new LHS structure is shown by a simple undirected acyclic graph. The basic general extension algorithm is proposed to reserve the most original points, but it costs too much time. Therefore, a general extension algorithm based on greedy algorithm is proposed to reduce the extension time, which cannot guarantee to contain the most original points. These algorithms are illustrated by an example and applied to evaluating the sample means to demonstrate the effectiveness. Zhi-zhao Liu, Wei Li, and Ming Yang Copyright © 2015 Zhi-zhao Liu et al. All rights reserved. Prediction of Protein-Protein Interaction By Metasample-Based Sparse Representation Wed, 29 Jul 2015 11:57:58 +0000 Protein-protein interactions (PPIs) play key roles in many cellular processes such as transcription regulation, cell metabolism, and endocrine function. Understanding these interactions takes a great promotion to the pathogenesis and treatment of various diseases. A large amount of data has been generated by experimental techniques; however, most of these data are usually incomplete or noisy, and the current biological experimental techniques are always very time-consuming and expensive. In this paper, we proposed a novel method (metasample-based sparse representation classification, MSRC) for PPIs prediction. A group of metasamples are extracted from the original training samples and then use the -regularized least square method to express a new testing sample as the linear combination of these metasamples. PPIs prediction is achieved by using a discrimination function defined in the representation coefficients. The MSRC is applied to PPIs dataset; it achieves 84.9% sensitivity, and 94.55% specificity, which is slightly lower than support vector machine (SVM) and much higher than naive Bayes (NB), neural networks (NN), and k-nearest neighbor (KNN). The result shows that the MSRC is efficient for PPIs prediction. Xiuquan Du, Xinrui Li, Hanqian Zhang, and Yanping Zhang Copyright © 2015 Xiuquan Du et al. All rights reserved. New Developments in Sliding Mode Control and Its Applications 2014 Wed, 29 Jul 2015 11:54:10 +0000 Ligang Wu, Rongni Yang, Guanghui Sun, Xudong Zhao, and Peng Shi Copyright © 2015 Ligang Wu et al. All rights reserved. Exact Static Analysis of In-Plane Curved Timoshenko Beams with Strong Nonlinear Boundary Conditions Wed, 29 Jul 2015 11:26:58 +0000 Analytical solutions have been developed for nonlinear boundary problems. In this paper, the shifting function method is applied to develop the static deflection of in-plane curved Timoshenko beams with nonlinear boundary conditions. Three coupled governing differential equations are derived via the Hamilton’s principle. The mathematical modeling of the curved beam system can be decomposed into a complete sixth-order ordinary differential characteristic equation and the associated boundary conditions. It is shown that the proposed method is valid and performs well for problems with strong nonlinearity. Sen-Yung Lee and Qian-Zhi Yan Copyright © 2015 Sen-Yung Lee and Qian-Zhi Yan. All rights reserved. Complexity Uncertainty Analysis of Dynamic in a Dual-Channel Energy Supply Chain Model with Heterogeneous Retailers Wed, 29 Jul 2015 11:26:01 +0000 This paper analyses the dynamics of dual-channel energy supply chain model with heterogeneous retailers (as regards the type of expectations’ formation). On the basis of analyzing the stabilities of four fixed points in the three-dimensional dynamic system, local stable regions of Nash equilibrium are obtained. Effects of S on the stable regions and profit are studied. Simulation results show that the adjustment of price speed has an obvious impact on the complexity of competition. The performances of the model in different period are measured by using the index of average profit. The results show that unstable behavior in economic system is often an unfavorable outcome. So this paper discusses the application of parameters control method when the model is in chaos and then allows the oligarchs to eliminate the negative effects. Ting Li, Junhai Ma, and Lijian Sun Copyright © 2015 Ting Li et al. All rights reserved. Remarks on the Regularity Criterion of the Navier-Stokes Equations with Nonlinear Damping Wed, 29 Jul 2015 11:25:20 +0000 This paper is concerned with the regularity criterion of weak solutions to the three-dimensional Navier-Stokes equations with nonlinear damping in critical weak spaces. It is proved that if the weak solution satisfies ,  , then the weak solution of Navier-Stokes equations with nonlinear damping is regular on Weihua Wang and Guopeng Zhou Copyright © 2015 Weihua Wang and Guopeng Zhou. All rights reserved. Physical Manipulation of Calcium Oscillation in Shen-Larter Cell System Wed, 29 Jul 2015 10:25:06 +0000 Experimental observations show that calcium signals in response to external stimuli encode information via frequency modulation or alternatively via amplitude modulation. Here, by arguments of bifurcation theory and bifurcation control theorem, the amplitude and frequency manipulation of limit cycle in Shen-Larter cell system are investigated. A nonlinear feedback controller is then applied to the system, in order to assign the peak frequency to a prescribed value, such that bifurcation is minimized or eliminated in the closed-loop system. And numerical simulations are presented in order to illustrate the theoretical analysis. Hongkun Zuo Copyright © 2015 Hongkun Zuo. All rights reserved. Recent Advances on the Theory and Applications of Networked Control Systems Wed, 29 Jul 2015 10:24:23 +0000 Yun-Bo Zhao, Xi-Ming Sun, Jinhui Zhang, and Peng Shi Copyright © 2015 Yun-Bo Zhao et al. All rights reserved. Asymptotic Behavior of Weak Solutions to the Generalized Nonlinear Partial Differential Equation Model Wed, 29 Jul 2015 10:01:17 +0000 This paper investigates the asymptotic behavior of weak solutions to the generalized nonlinear partial differential equation model. It is proved that every perturbed weak solution of the perturbed generalized nonlinear partial differential equations asymptotically converges to the solution of the original system under the large perturbation. Min Wu and Yousheng Wu Copyright © 2015 Min Wu and Yousheng Wu. All rights reserved. Adaptive Second Order Sliding Mode Control of a Fuel Cell Hybrid System for Electric Vehicle Applications Wed, 29 Jul 2015 08:11:28 +0000 We present an adaptive-gain second order sliding mode (SOSM) control applied to a hybrid power system for electric vehicle applications. The main advantage of the adaptive SOSM is that it does not require the upper bound of the uncertainty. The proposed hybrid system consists of a polymer electrolyte membrane fuel cell (PEMFC) with a unidirectional DC/DC converter and a Li-ion battery stack with a bidirectional DC/DC converter, where the PEMFC is employed as the primary energy source and the battery is employed as the second energy source. One of the main limitations of the FC is its slow dynamics mainly due to the air-feed system and fuel-delivery system. Fuel starvation phenomenon will occur during fast load demand. Therefore, the second energy source is required to assist the main source to improve system perofrmance. The proposed energy management system contains two cascade control structures, which are used to regulate the fuel cell and battery currents to track the given reference currents and stabilize the DC bus voltage while satisfying the physical limitations. The proposed control strategy is evaluated for two real driving cycles, that is, Urban Dynamometer Driving Schedule (UDDS) and Highway Fuel Economy Driving Schedule (HWFET). Jianxing Liu, Yue Zhao, Bo Geng, and Bing Xiao Copyright © 2015 Jianxing Liu et al. All rights reserved. Subdivision Error Analysis and Compensation for Photoelectric Angle Encoder in a Telescope Control System Wed, 29 Jul 2015 08:02:46 +0000 As the position sensor, photoelectric angle encoder affects the accuracy and stability of telescope control system (TCS). A TCS-based subdivision error compensation method for encoder is proposed. Six types of subdivision error sources are extracted through mathematical expressions of subdivision signals first. Then the period length relationships between subdivision signals and subdivision errors are deduced. And the error compensation algorithm only utilizing the shaft position of TCS is put forward, along with two control models; Model I is that the algorithm applies only to the speed loop of TCS and Model II is applied to both speed loop and position loop. Combined with actual project, elevation jittering phenomenon of the telescope is discussed to decide the necessity of DC-type subdivision error compensation. Low-speed elevation performance before and after error compensation is compared to help decide that Model II is preferred. In contrast to original performance, the maximum position error of the elevation with DC subdivision error compensation is reduced by approximately 47.9% from 1.42″ to 0.74″. The elevation gets a huge decrease in jitters. This method can compensate the encoder subdivision errors effectively and improve the stability of TCS. Yanrui Su, Qiang Wang, Fabao Yan, Xiang Liu, and Yongmei Huang Copyright © 2015 Yanrui Su et al. All rights reserved. Predicting Component Failures Using Latent Dirichlet Allocation Wed, 29 Jul 2015 07:03:33 +0000 Latent Dirichlet Allocation (LDA) is a statistical topic model that has been widely used to abstract semantic information from software source code. Failure refers to an observable error in the program behavior. This work investigates whether semantic information and failures recorded in the history can be used to predict component failures. We use LDA to abstract topics from source code and a new metric (topic failure density) is proposed by mapping failures to these topics. Exploring the basic information of topics from neighboring versions of a system, we obtain a similarity matrix. Multiply the Topic Failure Density (TFD) by the similarity matrix to get the TFD of the next version. The prediction results achieve an average 77.8% agreement with the real failures by considering the top 3 and last 3 components descending ordered by the number of failures. We use the Spearman coefficient to measure the statistical correlation between the actual and estimated failure rate. The validation results range from 0.5342 to 0.8337 which beats the similar method. It suggests that our predictor based on similarity of topics does a fine job of component failure prediction. Hailin Liu, Ling Xu, Mengning Yang, Meng Yan, and Xiaohong Zhang Copyright © 2015 Hailin Liu et al. All rights reserved. Output-Feedback Tracking Control for Networked Control Systems Wed, 29 Jul 2015 06:41:07 +0000 This paper investigates the observer-based tracking problem of networked output-feedback control systems with consideration of data transmission delays, data-packet dropouts, and sampling effects. Different from other approaches, this paper offers a single-step procedure to handle nonconvex terms that appear in the process of designing observer-based output-feedback control, and then establishes a set of linear matrix inequality conditions for the solvability of the tracking problem. Finally, two numerical examples are given to illustrate the effectiveness of our result. Sung Hyun Kim Copyright © 2015 Sung Hyun Kim. All rights reserved. Output Feedback and Single-Phase Sliding Mode Control for Complex Interconnected Systems Wed, 29 Jul 2015 06:20:17 +0000 This paper generalized a new sliding mode control (SMC) without reaching phase to solve two important problems in the stability of complex interconnected systems: (1) a decentralized controller that uses only output variables directly and (2) the stability of complex interconnected systems ensured for all time. A new sliding surface is firstly designed to construct a single-phase SMC in which the desired motion is determined from the initial time instant. A new lemma is secondly established for the controller design using only output variables. The proposed single-phase SMC and the decentralized output feedback controller ensure the robust stability of complex interconnected systems from the beginning to the end. One of the key features of the single phase SMC scheme is that reaching time, which is required in most of the existing two phases of SMC approaches to stabilize the interconnected systems, is removed. Finally, a numerical example is used to demonstrate the efficacy of the method. Yao-Wen Tsai and Van Van Huynh Copyright © 2015 Yao-Wen Tsai and Van Van Huynh. All rights reserved. Hierarchic Analysis Method to Evaluate Rock Burst Risk Tue, 28 Jul 2015 13:16:40 +0000 In order to reasonably evaluate the risk of rock bursts in mines, the factors impacting rock bursts and the existing grading criterion on the risk of rock bursts were studied. By building a model of hierarchic analysis method, the natural factors, technology factors, and management factors that influence rock bursts were analyzed and researched, which determined the degree of each factor’s influence (i.e., weight) and comprehensive index. Then the grade of rock burst risk was assessed. The results showed that the assessment level generated by the model accurately reflected the actual risk degree of rock bursts in mines. The model improved the maneuverability and practicability of existing evaluation criteria and also enhanced the accuracy and science of rock burst risk assessment. Ming Ji, Hong-jun Guo, Yi-dong Zhang, Tao Li, and Lin-sheng Gao Copyright © 2015 Ming Ji et al. All rights reserved. Fundamental Theories and Practice in Service Robotics Tue, 28 Jul 2015 11:59:59 +0000 ShaoPing Bai, DeZhen Song, JiZhong Xiao, Trung Dung Ngo, and YongSheng Ou Copyright © 2015 ShaoPing Bai et al. All rights reserved. A Compensated Numerical Method for Solving Stochastic Differential Equations with Variable Delays and Random Jump Magnitudes Tue, 28 Jul 2015 11:29:15 +0000 Stochastic differential equations with jumps are of a wide application area especially in mathematical finance. In general, it is hard to obtain their analytical solutions and the construction of some numerical solutions with good performance is therefore an important task in practice. In this study, a compensated split-step method is proposed to numerically solve the stochastic differential equations with variable delays and random jump magnitudes. It is proved that the numerical solutions converge to the analytical solutions in mean-square with the approximate rate of 1/2. Furthermore, the mean-square stability of the exact solutions and the numerical solutions are investigated via a linear test equation and the results show that the proposed numerical method shares both the mean-square stability and the so-called A-stability. Ying Du and Changlin Mei Copyright © 2015 Ying Du and Changlin Mei. All rights reserved. Parallel-Machine Scheduling with Time-Dependent and Machine Availability Constraints Tue, 28 Jul 2015 11:27:20 +0000 We consider the parallel-machine scheduling problem in which the machines have availability constraints and the processing time of each job is simple linear increasing function of its starting times. For the makespan minimization problem, which is NP-hard in the strong sense, we discuss the Longest Deteriorating Rate algorithm and List Scheduling algorithm; we also provide a lower bound of any optimal schedule. For the total completion time minimization problem, we analyze the strong NP-hardness, and we present a dynamic programming algorithm and a fully polynomial time approximation scheme for the two-machine problem. Furthermore, we extended the dynamic programming algorithm to the total weighted completion time minimization problem. Cuixia Miao and Juan Zou Copyright © 2015 Cuixia Miao and Juan Zou. All rights reserved. A Novel Numerical Method of Two-Dimensional Fredholm Integral Equations of the Second Kind Tue, 28 Jul 2015 11:10:36 +0000 A novel numerical method is developed for solving two-dimensional linear Fredholm integral equations of the second kind by integral mean value theorem. In the proposed algorithm, each element of the generated discrete matrix is not required to calculate integrals, and the approximate integral operator is convergent according to collectively compact theory. Convergence and error analyses of the approximate solution are provided. In addition, an algorithm is given. The reliability and efficiency of the proposed method will be illustrated by comparison with some numerical results. Yanying Ma, Jin Huang, and Hu Li Copyright © 2015 Yanying Ma et al. All rights reserved. A Novel Model for Lattice-Based Authorized Searchable Encryption with Special Keyword Tue, 28 Jul 2015 09:36:20 +0000 Data stored in the cloud servers, keyword search, and access controls are two important capabilities which should be supported. Public-keyword encryption with keyword search (PEKS) and attribute based encryption (ABE) are corresponding solutions. Meanwhile, as we step into postquantum era, pairing related assumption is fragile. Lattice is an ideal choice for building secure encryption scheme against quantum attack. Based on this, we propose the first mathematical model for lattice-based authorized searchable encryption. Data owners can sort the ciphertext by specific keywords such as time; data users satisfying the access control hand the trapdoor generated with the keyword to the cloud sever; the cloud sever sends back the corresponding ciphertext. The security of our schemes is based on the worst-case hardness on lattices, called learning with errors (LWE) assumption. In addition, our scheme achieves attribute-hiding, which could protect the sensitive information of data user. Fugeng Zeng and Chunxiang Xu Copyright © 2015 Fugeng Zeng and Chunxiang Xu. All rights reserved. Study on Individual Traffic Police On-Duty Behavior Analysis Method with Time Series Scheduling Tue, 28 Jul 2015 08:31:24 +0000 In the traditional traffic police scheduling and dispatching, the applications of the position information are restricted. This paper presented a model of the traffic police on-duty behavior analysis based on time series, in order to improve the efficiency of traffic police scheduling and dispatching system. Firstly, it proposes the steps for the behavior analysis of individual traffic police on-duty. Secondly, it elaborates division method of individual traffic police on-duty behavior from background element definition and semantic concept description. Thirdly the paper builds a model concerning individual traffic police on duty behavior by applying state automaton. Finally it describes the implementation methods of key technologies on individual traffic police on duty behavior. Huai-jun Peng, Yong Qin, Yan-fang Yang, Zundong Zhang, and Abderrahim Chariete Copyright © 2015 Huai-jun Peng et al. All rights reserved. Single Machine Group Scheduling with Position Dependent Processing Times and Ready Times Tue, 28 Jul 2015 07:55:45 +0000 We investigate a single machine group scheduling problem with position dependent processing times and ready times. The actual processing time of a job is a function of positive group-dependent job-independent positional factors. The actual setup time of the group is a linear function of the total completion time of the former group. Each job has a release time. The decision should be taken regarding possible sequences of jobs in each group and group sequence to minimize the makespan. We show that jobs in each group are scheduled in nondecreasing order of its release time and the groups are arranged in nondecreasing order of some certain conditions. We also present a polynomial time solution procedure for the special case of the proposed problem. Xingong Zhang and Qiulian Xie Copyright © 2015 Xingong Zhang and Qiulian Xie. All rights reserved. Landing Impact Analysis of a Bioinspired Intermittent Hopping Robot with Consideration of Friction Tue, 28 Jul 2015 07:38:18 +0000 Multibody dynamics for the flying and landing oblique impact processes of a bioinspired intermittent hopping robot is derived in this paper by using the impulse-momentum principle. The dynamics model that involves the multibody configuration, mass distribution of the robot, and friction is solved by the linear complementarity conditions in terms of different impact types. The computational and experimental data is compared. And the influence factors of landing impact are analysed as well. Based on the influence rules for landing impact, a technical design of solution is proposed for adjusting the robot’s attitude during the jumping and for absorbing the impact energy during the landing. Lessons learned from the theoretical and experimental results have general applicability to the motion prediction, performance analysis, and landing stability study for intermittent hopping robots or other legged robots. Long Bai, Wenjie Ge, Xiaohong Chen, Qian Tang, and Rong Xiang Copyright © 2015 Long Bai et al. All rights reserved. Advanced Modeling and Services Based Mathematics for Ubiquitous Computing Tue, 28 Jul 2015 06:52:55 +0000 Jong Hyuk Park, Hong Shen, Jian-nong Cao, Fatos Xhafa, and Young-Sik Jeong Copyright © 2015 Jong Hyuk Park et al. All rights reserved. Single-Machine Scheduling with Workload-Dependent Maintenance Duration to Minimize Maximum Lateness Tue, 28 Jul 2015 06:21:09 +0000 A single-machine scheduling problem with a mandatory maintenance whose duration is workload-dependent is considered. The start time of the maintenance is restricted to a time window. The objective is to determine the start time of the maintenance and schedule all the jobs to the machine such that the maximum lateness is minimized. An approximation algorithm based on the classical Earliest Due Date first rule is proposed. It is showed that the proposed algorithm is optimal for some special cases and that it has a tight bound for the scheduling problem under consideration. Ting Wang and Dehua Xu Copyright © 2015 Ting Wang and Dehua Xu. All rights reserved.