Search results for: scheduling scheme

Authors: Harun Aydilek, Asiye Aydilek, Ali Allahverdi

Abstract:

The total tardiness is a widely used performance measure in the scheduling literature. This performance measure is particularly important in situations where there is a cost to complete a job beyond its due date. The cost of scheduling increases as the gap between a job's due date and its completion time increases. Such costs may also be penalty costs in contracts, loss of goodwill. This performance measure is important as the fulfillment of due dates of customers has to be taken into account while making scheduling decisions. The problem is addressed in the literature, however, it has been assumed zero setup times. Even though this assumption may be valid for some environments, it is not valid for some other scheduling environments. When setup times are treated as separate from processing times, it is possible to increase machine utilization and to reduce total tardiness. Therefore, non-zero setup times need to be considered as separate. A dominance relation is developed and several algorithms are proposed. The developed dominance relation is utilized in the proposed algorithms. Extensive computational experiments are conducted for the evaluation of the algorithms. The experiments indicated that the developed algorithms perform much better than the existing algorithms in the literature. More specifically, one of the newly proposed algorithms reduces the error of the best existing algorithm in the literature by 40 percent.

Keywords: algorithm, assembly flowshop, dominance relation, total tardiness

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Authors: Aydin Teymourifar, Gurkan Ozturk

Abstract:

In this paper, a hybrid distributed algorithm has been suggested for the multi-objective job shop scheduling problem. Many new approaches are used at design steps of the distributed algorithm. Co-evolutionary structure of the algorithm and competition between different communicated hybrid algorithms, which are executed simultaneously, causes to efficient search. Using several machines for distributing the algorithms, at the iteration and solution levels, increases computational speed. The proposed algorithm is able to find the Pareto solutions of the big problems in shorter time than other algorithm in the literature. Apache Spark and Hadoop platforms have been used for the distribution of the algorithm. The suggested algorithm and implementations have been compared with results of the successful algorithms in the literature. Results prove the efficiency and high speed of the algorithm.

Keywords: distributed algorithms, Apache Spark, Hadoop, job shop scheduling, multi-objective optimization

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Authors: Abdollah Kavousi Fard

Abstract:

This paper addresses the optimal scheduling of hybrid reconfigurable microgrids considering hybrid electric vehicle charging demands. A stochastic framework based on unscented transform to model the high uncertainties of renewable energy sources including wind turbine and photovoltaic panels, as well as the hybrid electric vehicles’ charging demand. In order to get to the optimal scheduling, the network reconfiguration is employed as an effective tool for changing the power supply path and avoiding possible congestions. The simulation results are analyzed and discussed in three different scenarios including coordinated, uncoordinated and smart charging demand of hybrid electric vehicles. A typical grid-connected microgrid is employed to show the satisfying performance of the proposed method.

Keywords: microgrid, renewable energy sources, reconfiguration, optimization

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Authors: Myoung-Jin Kim, Yeong-Seop Ahn, Hyun-Jee Yang, Hyoung-Kyu Song

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This paper proposes the new cooperative communication scheme for the wireless communication system. When the receiver is located in the edge of coverage, the signal from the transmitter is distorted by the inter-cell interference (ICI) and power reduction by distance. In order to improve communication performance, the proposed scheme adds the relay. By using the relay, the receiver receives the signal from the transmitter and relay at the same time. Therefore, the new cooperative communication scheme obtains diversity gain and is improved by the relay.

Keywords: cooperative communication, diversity gain, OFDM, MIMO

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Authors: Tarek Aboueldahab, Hanan Farag

Abstract:

Parallel Job Shop Scheduling Problem (JSP) is a multi-objective and multi constrains NP- optimization problem. Traditional Artificial Intelligence techniques have been widely used; however, they could be trapped into the local minimum without reaching the optimum solution, so we propose a hybrid Artificial Intelligence model (AI) with Discrete Breeding Swarm (DBS) added to traditional Artificial Intelligence to avoid this trapping. This model is applied in the cost minimization of the Car Sequencing and Operator Allocation (CSOA) problem. The practical experiment shows that our model outperforms other techniques in cost minimization.

Keywords: parallel job shop scheduling problem, artificial intelligence, discrete breeding swarm, car sequencing and operator allocation, cost minimization

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Authors: Vijay Kumar Kukreja, Ravneet Kaur

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In this paper, one dimensional advection diffusion model is analyzed using finite difference method based on Crank-Nicolson scheme. A practical problem of filter cake washing of chemical engineering is analyzed. The model is converted into dimensionless form. For the grid Ω × ω = [0, 1] × [0, T], the Crank-Nicolson spatial derivative scheme is used in space domain and forward difference scheme is used in time domain. The scheme is found to be unconditionally convergent, stable, first order accurate in time and second order accurate in space domain. For a test problem, numerical results are compared with the analytical ones for different values of parameter.

Keywords: Crank-Nicolson scheme, Lax-Richtmyer theorem, stability, consistency, Peclet number, Greschgorin circle

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Authors: Hyun-Jun Shin, Jae-Jeong Kim, Hyoung-Kyu Song

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In broadcasting and cellular system, a cooperative scheme is proposed for the improvement of performance of bit error rate. Up to date, the coverage of broadcasting system coexists with the coverage of cellular system. Therefore each user in a cellular coverage is frequently involved in a broadcasting coverage. The proposed cooperative scheme is derived from the shared areas. The users receive signals from both broadcasting base station and cellular base station. The proposed scheme selects a cellular base station of a worse channel to achieve better performance of bit error rate in cooperation. The performance of the proposed scheme is evaluated in fading channel.

Keywords: cooperative communication, diversity, STBC, CDD, channel condition, broadcasting system, cellular system

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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: Shivam Dwivedi, Sumit Prakash Gupta, Durga Toshniwal

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It is a challenging task in today’s date to keep defence forces in the highest state of combat readiness with budgetary constraints. A huge amount of time and money is squandered in the unnecessary and expensive traditional maintenance activities. To overcome this limitation Defence Intelligent Mission Readiness and Maintenance Scheduling System has been proposed, which ameliorates the maintenance system by diagnosing the condition and predicting the maintenance requirements. Based on new data mining algorithms, this system intelligently optimises mission readiness for imminent operations and maintenance scheduling in repair echelons. With modified data mining algorithms such as Weighted Feature Ranking Genetic Algorithm and SVM-Random Forest Linear ensemble, it improves the reliability, availability and safety, alongside reducing maintenance cost and Equipment Out of Action (EOA) time. The results clearly conclude that the introduced algorithms have an edge over the conventional data mining algorithms. The system utilizing the intelligent condition-based maintenance approach improves the operational and maintenance decision strategy of the defence force.

Keywords: condition based maintenance, data mining, defence maintenance, ensemble, genetic algorithms, maintenance scheduling, mission capability

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Authors: AmirHossein Mohajerzadeh, Mohammad Alishahi, Saeed Aslishahi, Mohsen Zabihi

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In this paper, we consider sensor nodes with the capability of measuring the bearings (relative angle to the target). We use geometric methods to select a set of observer nodes which are responsible for collecting data from the target. Considering the characteristics of target tracking applications, it is clear that significant numbers of sensor nodes are usually inactive. Therefore, in order to minimize the total network energy consumption, a set of sensor nodes, called sentinel, is periodically selected for monitoring, controlling the environment and transmitting data through the network. The other nodes are inactive. Furthermore, the proposed algorithm provides a joint scheduling and routing algorithm to transmit data between network nodes and the fusion center (FC) in which not only provides an efficient way to estimate the target position but also provides an efficient target tracking. Performance evaluation confirms the superiority of the proposed algorithm.

Keywords: coverage, routing, scheduling, target tracking, wireless sensor networks

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Authors: Quoc Khanh Dang, Thomas Bourdeaud’huy, Khaled Mesghouni, Armand Toguy´eni

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This paper studies a train routing and scheduling problem for busy railway stations. Our objective is to allow trains to be routed in dense areas that are reaching saturation. Unlike traditional methods that allocate all resources to setup a route for a train and until the route is freed, our work focuses on the use of resources as trains progress through the railway node. This technique allows a larger number of trains to be routed simultaneously in a railway node and thus reduces their current saturation. To deal with this problem, this study proposes an abstract model and a mixed-integer linear programming formulation to solve it. The applicability of our method is illustrated on a didactic example.

Keywords: busy railway stations, mixed-integer linear programming, offline railway station management, train platforming, train routing, train scheduling

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Authors: Aydin Teymourifar, Gurkan Ozturk

Abstract:

In this paper, a hybrid distributed algorithm has been suggested for multi-objective dynamic flexible job shop scheduling problem. The proposed algorithm is high level, in which several algorithms search the space on different machines simultaneously also it is a hybrid algorithm that takes advantages of the artificial intelligence, evolutionary and optimization methods. Distribution is done at different levels and new approaches are used for design of the algorithm. Apache spark and Hadoop frameworks have been used for the distribution of the algorithm. The Pareto optimality approach is used for solving the multi-objective benchmarks. The suggested algorithm that is able to solve large-size problems in short times has been compared with the successful algorithms of the literature. The results prove high speed and efficiency of the algorithm.

Keywords: distributed algorithms, apache-spark, Hadoop, flexible dynamic job shop scheduling, multi-objective optimization

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Authors: Yang-Kuei Lin, Yun-Xi Zhang

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This research focuses on the bicriteria correlated parallel machine scheduling problem. The two objective functions considered in this problem are to minimize maximum lateness and total weighted completion time. We first present a mixed integer programming (MIP) model that can find the entire efficient frontier for the studied problem. Next, we have proposed a bicriteria heuristic that can find non-dominated solutions for the studied problem. The performance of the proposed bicriteria heuristic is compared with the efficient frontier generated by solving the MIP model. Computational results indicate that the proposed bicriteria heuristic can solve the problem efficiently and find a set of diverse solutions that are uniformly distributed along the efficient frontier.

Keywords: bicriteria, correlated parallel machines, heuristic, scheduling

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Authors: Yong-Sung Kim, Je-Seok Shin, Ho-Jun Jo, Jin-O Kim

Abstract:

Home Energy Management System (HEMS) which makes the residential consumers contribute to the demand response is attracting attention in recent years. An aim of HEMS is to minimize their electricity cost by controlling the use of their appliances according to electricity price. The use of appliances in HEMS may be affected by some conditions such as external temperature and electricity price. Therefore, the user’s usage pattern of appliances should be modeled according to the external conditions, and the resultant usage pattern is related to the user’s comfortability on use of each appliances. This paper proposes a methodology to model the usage pattern based on the historical data with the copula function. Through copula function, the usage range of each appliance can be obtained and is able to satisfy the appropriate user’s comfort according to the external conditions for next day. Within the usage range, an optimal scheduling for appliances would be conducted so as to minimize an electricity cost with considering user’s comfort. Among the home appliance, electric heater (EH) is a representative appliance which is affected by the external temperature. In this paper, an optimal scheduling algorithm for an electric heater (EH) is addressed based on the method of branch and bound. As a result, scenarios for the EH usage are obtained according to user’s comfort levels and then the residential consumer would select the best scenario. The case study shows the effects of the proposed algorithm compared with the traditional operation of the EH, and it also represents impacts of the comfort level on the scheduling result.

Keywords: load scheduling, usage pattern, user’s comfort, copula function, branch and bound, electric heater

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

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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 (UETP)

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Authors: Liu Yi, Bao Liyong

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Aiming at the problem of low efficiency of pipelined scheduling in periodic query-qualified service, this paper proposes a system service resource scheduling strategy with parallel optimized qualified service polling control. The paper constructs the polling queuing system and its mathematical model; firstly, the first-order and second-order characteristic parameter equations are obtained by partial derivation of the probability mother function of the system state variables, and the complete mathematical, analytical expressions of each system parameter are deduced after the joint solution. The simulation experimental results are consistent with the theoretical calculated values. The system performance analysis shows that the average captain and average period of the system have been greatly improved, which can better adapt to the service demand of delay-sensitive data in the dense data environment.

Keywords: polling, parallel scheduling, mean queue length, average cycle time

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Authors: Xu Jie

Abstract:

As a key network security method, cryptographic services must fully cope with problems such as the wide variety of cryptographic algorithms, high concurrency requirements, random job crossovers, and instantaneous surges in workloads. Its complexity and dynamics also make it difficult for traditional static security policies to cope with the ever-changing situation. Cyber Threats and Environment. Traditional resource scheduling algorithms are inadequate when facing complex decisionmaking problems in dynamic environments. A network cryptographic resource allocation algorithm based on reinforcement learning is proposed, aiming to optimize task energy consumption, migration cost, and fitness of differentiated services (including user, data, and task security). By modeling the multi-job collaborative cryptographic service scheduling problem as a multiobjective optimized job flow scheduling problem, and using a multi-agent reinforcement learning method, efficient scheduling and optimal configuration of cryptographic service resources are achieved. By introducing reinforcement learning, resource allocation strategies can be adjusted in real time in a dynamic environment, improving resource utilization and achieving load balancing. Experimental results show that this algorithm has significant advantages in path planning length, system delay and network load balancing, and effectively solves the problem of complex resource scheduling in cryptographic services.

Keywords: multi-agent reinforcement learning, non-stationary dynamics, multi-agent systems, cooperative and competitive agents

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Authors: Pius W. Molo Chin

Abstract:

Real life problems such as the Huxley equation are always modeled as nonlinear differential equations. These problems need accurate and reliable methods for their solutions. In this paper, we propose a nonstandard finite difference method in time and the Galerkin combined with the compactness method in the space variables. This coupled method, is used to analyze a two dimensional Huxley equation for the existence and uniqueness of the continuous solution of the problem in appropriate spaces to be defined. We proceed to design a numerical scheme consisting of the aforementioned method and show that the scheme is stable. We further show that the stable scheme converges with the rate which is optimal in both the L2 as well as the H1-norms. Furthermore, we show that the scheme replicates the decaying qualities of the exact solution. Numerical experiments are presented with the help of an example to justify the validity of the designed scheme.

Keywords: Huxley equations, non-standard finite difference method, Galerkin method, optimal rate of convergence

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Authors: Wipada Chaiwchan, Patcharee Klinhom

Abstract:

This research is aimed to develop the online-class scheduling management system and improve as a complex problem solution, this must take into consideration in various conditions and factors. In addition to the number of courses, the number of students and a timetable to study, the physical characteristics of each class room and regulations used in the class scheduling must also be taken into consideration. This system is developed to assist management in the class scheduling for convenience and efficiency. It can provide several instructors to schedule simultaneously. Both lecturers and students can check and publish a timetable and other documents associated with the system online immediately. It is developed in a web-based application. PHP is used as a developing tool. The database management system was MySQL. The tool that is used for efficiency testing of the system is questionnaire. The system was evaluated by using a Black-Box testing. The sample was composed of 2 groups: 5 experts and 100 general users. The average and the standard deviation of results from the experts were 3.50 and 0.67. The average and the standard deviation of results from the general users were 3.54 and 0.54. In summary, the results from the research indicated that the satisfaction of users was in a good level. Therefore, this system could be implemented in an actual workplace and satisfy the users’ requirement effectively

Keywords: timetable, schedule, management system, online

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Authors: Ibrahim Özbek, Erdoğan Mehmet Özkan

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In this study, we generalize (k,n) threshold secret sharing scheme on the study Ozbek and Siap to the codes over the ring Fq+ αFq. In this way, it is mentioned that the method obtained in that article can also be used on codes over rings, and new advantages to be obtained. The method of securely sharing the key in cryptography, which Shamir first systematized and Massey carried over to codes, became usable for all error-correcting codes. The firewall of this scheme is based on the hardness of the syndrome decoding problem. Also, an open study area is left for those working for other rings and code classes. All codes that correct errors with this method have been the working area of this method.

Keywords: secret sharing scheme, linear codes, algebra, finite rings

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Authors: Aydin Teymourifar, Gurkan Ozturk, Ozan Bahadir

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NSGA-II is one of the most well-known and most widely used evolutionary algorithms. In addition to its new versions, such as NSGA-III, there are several modified types of this algorithm in the literature. In this paper, a hybrid NSGA-II algorithm has been suggested for solving the multi-objective flexible job shop scheduling problem. For a better search, new neighborhood-based crossover and mutation operators are defined. To create new generations, the neighbors of the selected individuals by the tournament selection are constructed. Also, at the end of each iteration, before sorting, neighbors of a certain number of good solutions are derived, except for solutions protected by elitism. The neighbors are generated using a constraint-based neural network that uses various constructs. The non-dominated sorting and crowding distance operators are same as the classic NSGA-II. A comparison based on some multi-objective benchmarks from the literature shows the efficiency of the algorithm.

Keywords: flexible job shop scheduling problem, multi-objective optimization, NSGA-II algorithm, neighborhood structures

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Authors: Hanife Merve Öztürk, Sıdıka Dalgan

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In this project, a capacity analysis study is done at TAT A. Ş. Maret Plant. Production capacity of products which generate 80% of sales amount are determined. Obtained data entered the LEKIN Scheduling Program and we get production schedules by using heuristic methods. Besides heuristic methods, as mathematical model, disjunctive programming formulation is adapted to flexible job shop problems by adding a new constraint to find optimal schedule solution.

Keywords: scheduling, flexible job shop problem, shifting bottleneck heuristic, mathematical modelling

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Authors: F. J. Adeyeye, D. Eni, K. M. Okedoye

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In this paper we present a Hybrid of Adomian decomposition method (ADM). This is the substitution of a One-step method of Taylor’s series approximation of orders I and II, into the nonlinear part of Adomian decomposition method resulting in a convergent series scheme. This scheme is applied to solve some Logistic problems represented as Abelian differential equation and the results are compared with the actual solution and Runge-kutta of order IV in order to ascertain the accuracy and efficiency of the scheme. The findings shows that the scheme is efficient enough to solve logistic problems considered in this paper.

Keywords: Adomian decomposition method, nonlinear part, one-step method, Taylor series approximation, hybrid of Adomian polynomial, logistic problem, Malthusian parameter, Verhulst Model

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Authors: Elmabruk Laias, Abdalla M. Hanashi, Mohammed Alnas

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Worldwide Interoperability for Microwave Access (WiMAX) became one of the most challenging issues, since it was responsible for distributing available resources of the network among all users this leaded to the demand of constructing and designing high efficient scheduling algorithms in order to improve the network utilization, to increase the network throughput, and to minimize the end-to-end delay. In this study, the proposed algorithm focuses on an efficient mechanism to serve non-real time traffic in congested networks by considering channel status.

Keywords: WiMAX, Quality of Services (QoS), OPNE, Diff-Serv (DS).

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Authors: Wenqian Shi, Xie He, Ziyin Huang, Zi Yu

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The dramatic changes in the global economic situation have produced dramatic changes to companies’ supply chain systems. A variety of opportunities and challenges make the traditional manufacturing industry feel pressured, and the manufacturing industry must seek a new way out as soon as possible. This paper presents a case study of the advanced planning and scheduling technology problem encountered by an electrical and electronics manufacturer. The objective is to seek the minimum cost of production planning and order management. Digitalization is applied to the problem, and the results demonstrate that significant production performances can be achieved in the face of the existing production of each link and order management systems to analyze and optimize. This paper can also provide some practical implications in various manufacturing industries. Finally, future research directions are discussed.

Keywords: advanced planning and scheduling, case study, production planning, supply chain optimization

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Authors: Khaled Moaddy

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In this paper, we present a fast and accurate numerical scheme for the solution of a Laplace equation with Dirichlet boundary conditions. The non-standard finite difference scheme (NSFD) is applied to construct the numerical solutions of a Laplace equation with two different Dirichlet boundary conditions. The solutions obtained using NSFD are compared with the solutions obtained using the standard finite difference scheme (SFD). The NSFD scheme is demonstrated to be reliable and efficient.

Keywords: standard finite difference schemes, non-standard schemes, Laplace equation, Dirichlet boundary conditions

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Authors: Hesham A. Khalek, Shafik S. Khoury, Remon F. Aziz, Mohamed A. Hakam

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Slipforming operation’s linearity is a source of planning complications, and operation is usually subjected to bottlenecks at any point, so careful planning is required in order to achieve success. On the other hand, Discrete-event simulation concepts can be applied to simulate and analyze construction operations and to efficiently support construction scheduling. Nevertheless, preparation of input data for construction simulation is very challenging, time-consuming and human prone-error source. Therefore, to enhance the benefits of using DES in construction scheduling, this study proposes an integrated module to establish a framework for automating the generation of time schedules and decision support for Slipform construction projects, particularly through the project feasibility study phase by using data exchange between project data stored in an Intermediate database, DES and Scheduling software. Using the stored information, proposed system creates construction tasks attribute [e.g. activities durations, material quantities and resources amount], then DES uses all the given information to create a proposal for the construction schedule automatically. This research is considered a demonstration of a flexible Slipform project modeling, rapid scenario-based planning and schedule generation approach that may be of interest to both practitioners and researchers.

Keywords: discrete-event simulation, modeling, construction planning, data exchange, scheduling generation, EZstrobe

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Authors: Jose Francisco Ferreira Ribeiro, Alexandre Bevilacqua Leoneti, Andre Lucirton Costa

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This paper presents a mathematical model in binary variables 0/1 to make the assignment of surgical procedures to the operating rooms in a hospital. The proposed mathematical model is based on the generalized assignment problem, which maximizes the sum of preferences for the use of the operating rooms by doctors, respecting the time available in each room. The corresponding program was written in Visual Basic of Microsoft Excel, and tested to schedule surgeries at St. Lydia Hospital in Ribeirao Preto, Brazil.

Keywords: generalized assignment problem, logistics, optimization, scheduling

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Authors: Nhat-To Huynh, Chen-Fu Chien

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Textile batch dyeing scheduling problem is complicated which includes batch formation, batch assignment on machines, batch sequencing with sequence-dependent setup time. Most manufacturers schedule their orders manually that are time consuming and inefficient. More power methods are needed to improve the solution. Motivated by the real needs, this study aims to propose approaches in which genetic algorithm is developed with multi-subpopulation and hybridised with estimation of distribution algorithm to solve the constructed problem for minimising the makespan. A heuristic algorithm is designed and embedded into the proposed algorithms to improve the ability to get out of the local optima. In addition, an empirical study is conducted in a textile company in Taiwan to validate the proposed approaches. The results have showed that proposed approaches are more efficient than simulated annealing algorithm.

Keywords: estimation of distribution algorithm, genetic algorithm, multi-subpopulation, scheduling, textile dyeing

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Authors: H. M. Soroush

Abstract:

The problem of scheduling products and services for on-time deliveries is of paramount importance in today’s competitive environments. It arises in many manufacturing and service organizations where it is desirable to complete jobs (products or services) with different weights (penalties) on or before their due dates. In such environments, schedules should frequently decide whether to schedule a job based on its processing time, due-date, and the penalty for tardy delivery to improve the system performance. For example, it is common to measure the weighted number of late jobs or the percentage of on-time shipments to evaluate the performance of a semiconductor production facility or an automobile assembly line. In this paper, we address the problem of scheduling a set of jobs on a server where processing times or due-dates of jobs are random variables and fixed weights (penalties) are imposed on the jobs’ late deliveries. The goal is to find the schedule that minimizes the expected weighted number of tardy jobs. The problem is NP-hard to solve; however, we explore three scenarios of the problem wherein: (i) both processing times and due-dates are stochastic; (ii) processing times are stochastic and due-dates are deterministic; and (iii) processing times are deterministic and due-dates are stochastic. We prove that special cases of these scenarios are solvable optimally in polynomial time, and introduce efficient heuristic methods for the general cases. Our computational results show that the heuristics perform well in yielding either optimal or near optimal sequences. The results also demonstrate that the stochasticity of processing times or due-dates can affect scheduling decisions. Moreover, the proposed problem is general in the sense that its special cases reduce to some new and some classical stochastic single machine models.

Keywords: number of late jobs, scheduling, single server, stochastic

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