As a complicated optimization problem,parallel batch processing machines scheduling problem(PBPMSP)exists in many real-life manufacturing industries such as textiles and semiconductors.Machine eligibility means that a...As a complicated optimization problem,parallel batch processing machines scheduling problem(PBPMSP)exists in many real-life manufacturing industries such as textiles and semiconductors.Machine eligibility means that at least one machine is not eligible for at least one job.PBPMSP and scheduling problems with machine eligibility are frequently considered;however,PBPMSP with machine eligibility is seldom explored.This study investigates PBPMSP with machine eligibility in fabric dyeing and presents a novel shuffled frog-leaping algorithm with competition(CSFLA)to minimize makespan.In CSFLA,the initial population is produced in a heuristic and random way,and the competitive search of memeplexes comprises two phases.Competition between any two memeplexes is done in the first phase,then iteration times are adjusted based on competition,and search strategies are adjusted adaptively based on the evolution quality of memeplexes in the second phase.An adaptive population shuffling is given.Computational experiments are conducted on 100 instances.The computational results showed that the new strategies of CSFLA are effective and that CSFLA has promising advantages in solving the considered PBPMSP.展开更多
Fabric dyeing is a critical production process in the clothing industry and heavily relies on batch processing machines(BPM).In this study,the parallel BPM scheduling problem with machine eligibility in fabric dyeing ...Fabric dyeing is a critical production process in the clothing industry and heavily relies on batch processing machines(BPM).In this study,the parallel BPM scheduling problem with machine eligibility in fabric dyeing is considered,and an adaptive cooperated shuffled frog-leaping algorithm(ACSFLA)is proposed to minimize makespan and total tardiness simultaneously.ACSFLA determines the search times for each memeplex based on its quality,with more searches in high-quality memeplexes.An adaptive cooperated and diversified search mechanism is applied,dynamically adjusting search strategies for each memeplex based on their dominance relationships and quality.During the cooperated search,ACSFLA uses a segmented and dynamic targeted search approach,while in non-cooperated scenarios,the search focuses on local search around superior solutions to improve efficiency.Furthermore,ACSFLA employs adaptive population division and partial population shuffling strategies.Through these strategies,memeplexes with low evolutionary potential are selected for reconstruction in the next generation,while thosewithhighevolutionarypotential are retained to continue their evolution.Toevaluate the performance of ACSFLA,comparative experiments were conducted using ACSFLA,SFLA,ASFLA,MOABC,and NSGA-CC in 90 instances.The computational results reveal that ACSFLA outperforms the other algorithms in 78 of the 90 test cases,highlighting its advantages in solving the parallel BPM scheduling problem with machine eligibility.展开更多
This study focuses on the scheduling problem of unrelated parallel batch processing machines(BPM)with release times,a scenario derived from the moulding process in a foundry.In this process,a batch is initially formed...This study focuses on the scheduling problem of unrelated parallel batch processing machines(BPM)with release times,a scenario derived from the moulding process in a foundry.In this process,a batch is initially formed,placed in a sandbox,and then the sandbox is positioned on a BPM formoulding.The complexity of the scheduling problem increases due to the consideration of BPM capacity and sandbox volume.To minimize the makespan,a new cooperated imperialist competitive algorithm(CICA)is introduced.In CICA,the number of empires is not a parameter,and four empires aremaintained throughout the search process.Two types of assimilations are achieved:The strongest and weakest empires cooperate in their assimilation,while the remaining two empires,having a close normalization total cost,combine in their assimilation.A new form of imperialist competition is proposed to prevent insufficient competition,and the unique features of the problem are effectively utilized.Computational experiments are conducted across several instances,and a significant amount of experimental results show that the newstrategies of CICAare effective,indicating promising advantages for the considered BPMscheduling problems.展开更多
To enhance the rationality of the layout of electric vehicle charging stations,meet the actual needs of users,and optimise the service range and coverage efficiency of charging stations,this paper proposes an optimisa...To enhance the rationality of the layout of electric vehicle charging stations,meet the actual needs of users,and optimise the service range and coverage efficiency of charging stations,this paper proposes an optimisation strategy for the layout of electric vehicle charging stations that integrates Mini Batch K-Means and simulated annealing algorithms.By constructing a circle-like service area model with the charging station as the centre and a certain distance as the radius,the maximum coverage of electric vehicle charging stations in the region and the influence of different regional environments on charging demand are considered.Based on the real data of electric vehicle charging stations in Nanjing,Jiangsu Province,this paper uses the model proposed in this paper to optimise the layout of charging stations in the study area.The results show that the optimisation strategy incorporating Mini Batch K-Means and simulated annealing algorithms outperforms the existing charging station layouts in terms of coverage and the number of stations served,and compared to the original charging station layouts,the optimised charging station layouts have flatter Lorentzian curves and are closer to the average distribution.The proposed optimisation strategy not only improves the service efficiency and user satisfaction of EV(Electric Vehicle)charging stations but also provides a reference for the layout optimisation of EV charging stations in other cities,which has important practical value and promotion potential.展开更多
This paper presents a new highly parallel algorithm for computing the minimum-norm least-squares solution of inconsistent linear equations Ax = b(A∈Rm×n,b∈R (A)). By this algorithm the solution x = A + b is obt...This paper presents a new highly parallel algorithm for computing the minimum-norm least-squares solution of inconsistent linear equations Ax = b(A∈Rm×n,b∈R (A)). By this algorithm the solution x = A + b is obtained in T = n(log2m + log2(n - r + 1) + 5) + log2m + 1 steps with P=mn processors when m × 2(n - 1) and with P = 2n(n - 1) processors otherwise.展开更多
Order-recursive least-squares(ORLS)algorithms are applied to the prob-lems of estimation and identification of FIR or ARMA system parameters where a fixedset of input signal samples is available and the desired order ...Order-recursive least-squares(ORLS)algorithms are applied to the prob-lems of estimation and identification of FIR or ARMA system parameters where a fixedset of input signal samples is available and the desired order of the underlying model isunknown.On the basis of several universal formulae for updating nonsymmetric projec-tion operators,this paper presents three kinds of LS algorithms,called nonsymmetric,symmetric and square root normalized fast ORLS algorithms,respectively.As to the au-thors’ knowledge,the first and the third have not been so far provided,and the second isone of those which have the lowest computational requirement.Several simplified versionsof the algorithms are also considered.展开更多
The optimal scheduling of multi-product batch process is studied and a new mathematics model targeting the maximum profit is proposed, which can be solved by the modified genetic algorithm (MGA) with mixed coding (seq...The optimal scheduling of multi-product batch process is studied and a new mathematics model targeting the maximum profit is proposed, which can be solved by the modified genetic algorithm (MGA) with mixed coding (sequence coding and decimal coding) developed by us. In which, the partially matched cross over (PMX) and reverse mutation are used for the sequence coding, whereas the arithmetic crossover and heteropic mutation are used for the decimal coding. In addition, the relationship between production scale and production cost is analyzed and the maximum profit is always a trade-off of the production scale and production cost. Two examples are solved to demonstrate the effectiveness of the method.展开更多
In order to solve the flexible job shop scheduling problem with variable batches,we propose an improved multiobjective optimization algorithm,which combines the idea of inverse scheduling.First,a flexible job shop pro...In order to solve the flexible job shop scheduling problem with variable batches,we propose an improved multiobjective optimization algorithm,which combines the idea of inverse scheduling.First,a flexible job shop problem with the variable batches scheduling model is formulated.Second,we propose a batch optimization algorithm with inverse scheduling in which the batch size is adjusted by the dynamic feedback batch adjusting method.Moreover,in order to increase the diversity of the population,two methods are developed.One is the threshold to control the neighborhood updating,and the other is the dynamic clustering algorithm to update the population.Finally,a group of experiments are carried out.The results show that the improved multi-objective optimization algorithm can ensure the diversity of Pareto solutions effectively,and has effective performance in solving the flexible job shop scheduling problem with variable batches.展开更多
The secure socket layer/ transport layer security（SSL/TLS） handshake protocol uses public key cryptographic algorithms such as RSA for key establishment. Typically, public key cryptographic algorithm is computationa...The secure socket layer/ transport layer security（SSL/TLS） handshake protocol uses public key cryptographic algorithms such as RSA for key establishment. Typically, public key cryptographic algorithm is computational intensive due to the modular multiplications. Therefore, SSL/TLS servers often become swamped while performing public key decryptions when the simultaneous requests increase quickly. A batch RSA decryption algorithm was proposed. The novel algorithm provides the reasonable response time and optimizes server performance significantly. The decryption speedup is proportional to the batch size b, for instance, the speedup factor is 4, while in Shacham＇s scheme the acceleration rate is only 2.5 when b = 4.展开更多
In this contribution we present an online scheduling algorithm for a real world multiproduct batch plant. The overall mixed integer nonlinear programming (MINLP) problem is hierarchically structured into a mixed integ...In this contribution we present an online scheduling algorithm for a real world multiproduct batch plant. The overall mixed integer nonlinear programming (MINLP) problem is hierarchically structured into a mixed integer linear programming (MILP) problem first and then a reduced dimensional MINLP problem, which are optimized by mathematical programming (MP) and genetic algorithm (GA) respectively. The basis idea relies on combining MP with GA to exploit their complementary capacity. The key features of the hierarchical model are explained and illustrated with some real world cases from the multiproduct batch plants.展开更多
This paper considers a scheduling problem in industrial make-and-pack batch production process. This process equips with sequence-dependent changeover time, multipurpose storage units with limited capacity, storage ti...This paper considers a scheduling problem in industrial make-and-pack batch production process. This process equips with sequence-dependent changeover time, multipurpose storage units with limited capacity, storage time, batch splitting, partial equipment connectivity and transfer time. The objective is to make a production plan to satisfy all constraints while meeting demand requirement of packed products from various product families. This problem is NP-hard and the problem size is exponentially large for a realistic-sized problem. Therefore,we propose a genetic algorithm to handle this problem. Solutions to the problems are represented by chromosomes of product family sequences. These sequences are decoded to assign the resource for producing packed products according to forward assignment strategy and resource selection rules. These techniques greatly reduce unnecessary search space and improve search speed. In addition, design of experiment is carefully utilized to determine appropriate parameter settings. Ant colony optimization and Tabu search are also implemented for comparison. At the end of each heuristics, local search is applied for the packed product sequence to improve makespan. In an experimental analysis, all heuristics show the capability to solve large instances within reasonable computational time. In all problem instances, genetic algorithm averagely outperforms ant colony optimization and Tabu search with slightly longer computational time.展开更多
In parallel-batching machine scheduling, all jobs in a batch start and complete at the same time, and the processing time of the batch is the maximum processing time of any job in it. For the unbounded parallel-batchi...In parallel-batching machine scheduling, all jobs in a batch start and complete at the same time, and the processing time of the batch is the maximum processing time of any job in it. For the unbounded parallel-batching machine scheduling problem of minimizing the maximum lateness, denoted 1|p-batch|L_(max), a dynamic programming algorithm with time complexity O(n^2) is well known in the literature.Later, this algorithm is improved to be an O(n log n) algorithm. In this note, we present another O(n log n) algorithm with simplifications on data structure and implementation details.展开更多
Levenberg-Marquardt(LM)algorithm is applied for the optimization of the heat transfer of a batch reactor.The validity of the approach is verified through comparison with experimental results.It is found that the mathe...Levenberg-Marquardt(LM)algorithm is applied for the optimization of the heat transfer of a batch reactor.The validity of the approach is verified through comparison with experimental results.It is found that the mathematical model can properly describe the heat transfer relationships characterizing the considered system,with the error being kept within±2℃.Indeed,the difference between the actual measured values and the model calculated value curve is within±1.5℃,which is in agreement with the model assumptions and demonstrates the reliability and effectiveness of the algorithm applied to the batch reactor heat transfer model.Therefore,the present work provides a theoretical reference for the conversion of practical problems in the field of chemical production into mathematical models.展开更多
To improve the productivity,the resource utilization and reduce the production cost of flexible job shops,this paper designs an improved two-layer optimization algorithm for the dual-resource scheduling optimization p...To improve the productivity,the resource utilization and reduce the production cost of flexible job shops,this paper designs an improved two-layer optimization algorithm for the dual-resource scheduling optimization problem of flexible job shop considering workpiece batching.Firstly,a mathematical model is established to minimize the maximum completion time.Secondly,an improved two-layer optimization algorithm is designed:the outer layer algorithm uses an improved PSO(Particle Swarm Optimization)to solve the workpiece batching problem,and the inner layer algorithm uses an improved GA(Genetic Algorithm)to solve the dual-resource scheduling problem.Then,a rescheduling method is designed to solve the task disturbance problem,represented by machine failures,occurring in the workshop production process.Finally,the superiority and effectiveness of the improved two-layer optimization algorithm are verified by two typical cases.The case results show that the improved two-layer optimization algorithm increases the average productivity by 7.44% compared to the ordinary two-layer optimization algorithm.By setting the different numbers of AGVs(Automated Guided Vehicles)and analyzing the impact on the production cycle of the whole order,this paper uses two indicators,the maximum completion time decreasing rate and the average AGV load time,to obtain the optimal number of AGVs,which saves the cost of production while ensuring the production efficiency.This research combines the solved problem with the real production process,which improves the productivity and reduces the production cost of the flexible job shop,and provides new ideas for the subsequent research.展开更多
基金supported by the National Natural Science Foundation of China(Grant Number 61573264).
文摘As a complicated optimization problem,parallel batch processing machines scheduling problem(PBPMSP)exists in many real-life manufacturing industries such as textiles and semiconductors.Machine eligibility means that at least one machine is not eligible for at least one job.PBPMSP and scheduling problems with machine eligibility are frequently considered;however,PBPMSP with machine eligibility is seldom explored.This study investigates PBPMSP with machine eligibility in fabric dyeing and presents a novel shuffled frog-leaping algorithm with competition(CSFLA)to minimize makespan.In CSFLA,the initial population is produced in a heuristic and random way,and the competitive search of memeplexes comprises two phases.Competition between any two memeplexes is done in the first phase,then iteration times are adjusted based on competition,and search strategies are adjusted adaptively based on the evolution quality of memeplexes in the second phase.An adaptive population shuffling is given.Computational experiments are conducted on 100 instances.The computational results showed that the new strategies of CSFLA are effective and that CSFLA has promising advantages in solving the considered PBPMSP.
文摘Fabric dyeing is a critical production process in the clothing industry and heavily relies on batch processing machines(BPM).In this study,the parallel BPM scheduling problem with machine eligibility in fabric dyeing is considered,and an adaptive cooperated shuffled frog-leaping algorithm(ACSFLA)is proposed to minimize makespan and total tardiness simultaneously.ACSFLA determines the search times for each memeplex based on its quality,with more searches in high-quality memeplexes.An adaptive cooperated and diversified search mechanism is applied,dynamically adjusting search strategies for each memeplex based on their dominance relationships and quality.During the cooperated search,ACSFLA uses a segmented and dynamic targeted search approach,while in non-cooperated scenarios,the search focuses on local search around superior solutions to improve efficiency.Furthermore,ACSFLA employs adaptive population division and partial population shuffling strategies.Through these strategies,memeplexes with low evolutionary potential are selected for reconstruction in the next generation,while thosewithhighevolutionarypotential are retained to continue their evolution.Toevaluate the performance of ACSFLA,comparative experiments were conducted using ACSFLA,SFLA,ASFLA,MOABC,and NSGA-CC in 90 instances.The computational results reveal that ACSFLA outperforms the other algorithms in 78 of the 90 test cases,highlighting its advantages in solving the parallel BPM scheduling problem with machine eligibility.
基金the National Natural Science Foundation of China(Grant Number 61573264).
文摘This study focuses on the scheduling problem of unrelated parallel batch processing machines(BPM)with release times,a scenario derived from the moulding process in a foundry.In this process,a batch is initially formed,placed in a sandbox,and then the sandbox is positioned on a BPM formoulding.The complexity of the scheduling problem increases due to the consideration of BPM capacity and sandbox volume.To minimize the makespan,a new cooperated imperialist competitive algorithm(CICA)is introduced.In CICA,the number of empires is not a parameter,and four empires aremaintained throughout the search process.Two types of assimilations are achieved:The strongest and weakest empires cooperate in their assimilation,while the remaining two empires,having a close normalization total cost,combine in their assimilation.A new form of imperialist competition is proposed to prevent insufficient competition,and the unique features of the problem are effectively utilized.Computational experiments are conducted across several instances,and a significant amount of experimental results show that the newstrategies of CICAare effective,indicating promising advantages for the considered BPMscheduling problems.
基金supported by the Jiangsu Provincial College Students Innovation andEntrepreneurship Training Plan Project(grant number 202411276037Z)the Nanjing Institute ofTechnology Fund for Research Startup Projects of Introduced Talents(grant number TB202406012).
文摘To enhance the rationality of the layout of electric vehicle charging stations,meet the actual needs of users,and optimise the service range and coverage efficiency of charging stations,this paper proposes an optimisation strategy for the layout of electric vehicle charging stations that integrates Mini Batch K-Means and simulated annealing algorithms.By constructing a circle-like service area model with the charging station as the centre and a certain distance as the radius,the maximum coverage of electric vehicle charging stations in the region and the influence of different regional environments on charging demand are considered.Based on the real data of electric vehicle charging stations in Nanjing,Jiangsu Province,this paper uses the model proposed in this paper to optimise the layout of charging stations in the study area.The results show that the optimisation strategy incorporating Mini Batch K-Means and simulated annealing algorithms outperforms the existing charging station layouts in terms of coverage and the number of stations served,and compared to the original charging station layouts,the optimised charging station layouts have flatter Lorentzian curves and are closer to the average distribution.The proposed optimisation strategy not only improves the service efficiency and user satisfaction of EV(Electric Vehicle)charging stations but also provides a reference for the layout optimisation of EV charging stations in other cities,which has important practical value and promotion potential.
基金This project is supported by the National Natural Science Foundation of China
文摘This paper presents a new highly parallel algorithm for computing the minimum-norm least-squares solution of inconsistent linear equations Ax = b(A∈Rm×n,b∈R (A)). By this algorithm the solution x = A + b is obtained in T = n(log2m + log2(n - r + 1) + 5) + log2m + 1 steps with P=mn processors when m × 2(n - 1) and with P = 2n(n - 1) processors otherwise.
文摘Order-recursive least-squares(ORLS)algorithms are applied to the prob-lems of estimation and identification of FIR or ARMA system parameters where a fixedset of input signal samples is available and the desired order of the underlying model isunknown.On the basis of several universal formulae for updating nonsymmetric projec-tion operators,this paper presents three kinds of LS algorithms,called nonsymmetric,symmetric and square root normalized fast ORLS algorithms,respectively.As to the au-thors’ knowledge,the first and the third have not been so far provided,and the second isone of those which have the lowest computational requirement.Several simplified versionsof the algorithms are also considered.
文摘The optimal scheduling of multi-product batch process is studied and a new mathematics model targeting the maximum profit is proposed, which can be solved by the modified genetic algorithm (MGA) with mixed coding (sequence coding and decimal coding) developed by us. In which, the partially matched cross over (PMX) and reverse mutation are used for the sequence coding, whereas the arithmetic crossover and heteropic mutation are used for the decimal coding. In addition, the relationship between production scale and production cost is analyzed and the maximum profit is always a trade-off of the production scale and production cost. Two examples are solved to demonstrate the effectiveness of the method.
基金supported by the National Key R&D Plan(2020YFB1712902)the National Natural Science Foundation of China(52075036).
文摘In order to solve the flexible job shop scheduling problem with variable batches,we propose an improved multiobjective optimization algorithm,which combines the idea of inverse scheduling.First,a flexible job shop problem with the variable batches scheduling model is formulated.Second,we propose a batch optimization algorithm with inverse scheduling in which the batch size is adjusted by the dynamic feedback batch adjusting method.Moreover,in order to increase the diversity of the population,two methods are developed.One is the threshold to control the neighborhood updating,and the other is the dynamic clustering algorithm to update the population.Finally,a group of experiments are carried out.The results show that the improved multi-objective optimization algorithm can ensure the diversity of Pareto solutions effectively,and has effective performance in solving the flexible job shop scheduling problem with variable batches.
基金The National Natural Science Foundation of China (No. 60273049, 60303026, 60473020)
文摘The secure socket layer/ transport layer security（SSL/TLS） handshake protocol uses public key cryptographic algorithms such as RSA for key establishment. Typically, public key cryptographic algorithm is computational intensive due to the modular multiplications. Therefore, SSL/TLS servers often become swamped while performing public key decryptions when the simultaneous requests increase quickly. A batch RSA decryption algorithm was proposed. The novel algorithm provides the reasonable response time and optimizes server performance significantly. The decryption speedup is proportional to the batch size b, for instance, the speedup factor is 4, while in Shacham＇s scheme the acceleration rate is only 2.5 when b = 4.
基金Supported by the National 973 Program of China (No. G2000263).
文摘In this contribution we present an online scheduling algorithm for a real world multiproduct batch plant. The overall mixed integer nonlinear programming (MINLP) problem is hierarchically structured into a mixed integer linear programming (MILP) problem first and then a reduced dimensional MINLP problem, which are optimized by mathematical programming (MP) and genetic algorithm (GA) respectively. The basis idea relies on combining MP with GA to exploit their complementary capacity. The key features of the hierarchical model are explained and illustrated with some real world cases from the multiproduct batch plants.
基金Thailand Research Fund (Grant #MRG5480176)National Research University Project of Thailand Office of Higher Education Commission
文摘This paper considers a scheduling problem in industrial make-and-pack batch production process. This process equips with sequence-dependent changeover time, multipurpose storage units with limited capacity, storage time, batch splitting, partial equipment connectivity and transfer time. The objective is to make a production plan to satisfy all constraints while meeting demand requirement of packed products from various product families. This problem is NP-hard and the problem size is exponentially large for a realistic-sized problem. Therefore,we propose a genetic algorithm to handle this problem. Solutions to the problems are represented by chromosomes of product family sequences. These sequences are decoded to assign the resource for producing packed products according to forward assignment strategy and resource selection rules. These techniques greatly reduce unnecessary search space and improve search speed. In addition, design of experiment is carefully utilized to determine appropriate parameter settings. Ant colony optimization and Tabu search are also implemented for comparison. At the end of each heuristics, local search is applied for the packed product sequence to improve makespan. In an experimental analysis, all heuristics show the capability to solve large instances within reasonable computational time. In all problem instances, genetic algorithm averagely outperforms ant colony optimization and Tabu search with slightly longer computational time.
基金Supported by NSFC(11571323 11201121)+1 种基金NSFSTDOHN(162300410221)NSFEDOHN(2013GGJS-079)
文摘In parallel-batching machine scheduling, all jobs in a batch start and complete at the same time, and the processing time of the batch is the maximum processing time of any job in it. For the unbounded parallel-batching machine scheduling problem of minimizing the maximum lateness, denoted 1|p-batch|L_(max), a dynamic programming algorithm with time complexity O(n^2) is well known in the literature.Later, this algorithm is improved to be an O(n log n) algorithm. In this note, we present another O(n log n) algorithm with simplifications on data structure and implementation details.
文摘Levenberg-Marquardt(LM)algorithm is applied for the optimization of the heat transfer of a batch reactor.The validity of the approach is verified through comparison with experimental results.It is found that the mathematical model can properly describe the heat transfer relationships characterizing the considered system,with the error being kept within±2℃.Indeed,the difference between the actual measured values and the model calculated value curve is within±1.5℃,which is in agreement with the model assumptions and demonstrates the reliability and effectiveness of the algorithm applied to the batch reactor heat transfer model.Therefore,the present work provides a theoretical reference for the conversion of practical problems in the field of chemical production into mathematical models.
文摘To improve the productivity,the resource utilization and reduce the production cost of flexible job shops,this paper designs an improved two-layer optimization algorithm for the dual-resource scheduling optimization problem of flexible job shop considering workpiece batching.Firstly,a mathematical model is established to minimize the maximum completion time.Secondly,an improved two-layer optimization algorithm is designed:the outer layer algorithm uses an improved PSO(Particle Swarm Optimization)to solve the workpiece batching problem,and the inner layer algorithm uses an improved GA(Genetic Algorithm)to solve the dual-resource scheduling problem.Then,a rescheduling method is designed to solve the task disturbance problem,represented by machine failures,occurring in the workshop production process.Finally,the superiority and effectiveness of the improved two-layer optimization algorithm are verified by two typical cases.The case results show that the improved two-layer optimization algorithm increases the average productivity by 7.44% compared to the ordinary two-layer optimization algorithm.By setting the different numbers of AGVs(Automated Guided Vehicles)and analyzing the impact on the production cycle of the whole order,this paper uses two indicators,the maximum completion time decreasing rate and the average AGV load time,to obtain the optimal number of AGVs,which saves the cost of production while ensuring the production efficiency.This research combines the solved problem with the real production process,which improves the productivity and reduces the production cost of the flexible job shop,and provides new ideas for the subsequent research.
