The distributed permutation flow shop scheduling problem(DPFSP)has received increasing attention in recent years.The iterated greedy algorithm(IGA)serves as a powerful optimizer for addressing such a problem because o...The distributed permutation flow shop scheduling problem(DPFSP)has received increasing attention in recent years.The iterated greedy algorithm(IGA)serves as a powerful optimizer for addressing such a problem because of its straightforward,single-solution evolution framework.However,a potential draw-back of IGA is the lack of utilization of historical information,which could lead to an imbalance between exploration and exploitation,especially in large-scale DPFSPs.As a consequence,this paper develops an IGA with memory and learning mechanisms(MLIGA)to efficiently solve the DPFSP targeted at the mini-malmakespan.InMLIGA,we incorporate a memory mechanism to make a more informed selection of the initial solution at each stage of the search,by extending,reconstructing,and reinforcing the information from previous solutions.In addition,we design a twolayer cooperative reinforcement learning approach to intelligently determine the key parameters of IGA and the operations of the memory mechanism.Meanwhile,to ensure that the experience generated by each perturbation operator is fully learned and to reduce the prior parameters of MLIGA,a probability curve-based acceptance criterion is proposed by combining a cube root function with custom rules.At last,a discrete adaptive learning rate is employed to enhance the stability of the memory and learningmechanisms.Complete ablation experiments are utilized to verify the effectiveness of the memory mechanism,and the results show that this mechanism is capable of improving the performance of IGA to a large extent.Furthermore,through comparative experiments involving MLIGA and five state-of-the-art algorithms on 720 benchmarks,we have discovered that MLI-GA demonstrates significant potential for solving large-scale DPFSPs.This indicates that MLIGA is well-suited for real-world distributed flow shop scheduling.展开更多
The job shop scheduli ng problem has been studied for decades and known as an NP-hard problem. The fl exible job shop scheduling problem is a generalization of the classical job sche duling problem that allows an oper...The job shop scheduli ng problem has been studied for decades and known as an NP-hard problem. The fl exible job shop scheduling problem is a generalization of the classical job sche duling problem that allows an operation to be processed on one machine out of a set of machines. The problem is to assign each operation to a machine and find a sequence for the operations on the machine in order that the maximal completion time of all operations is minimized. A genetic algorithm is used to solve the f lexible job shop scheduling problem. A novel gene coding method aiming at job sh op problem is introduced which is intuitive and does not need repairing process to validate the gene. Computer simulations are carried out and the results show the effectiveness of the proposed algorithm.展开更多
“This trip to China has been an absolute steal,”exclaimed Keiko,a Japanese tourist purchasing cosmetics through Shanghai International Finance Center’s immediate tax refund service.Her experience mirrors a broader ...“This trip to China has been an absolute steal,”exclaimed Keiko,a Japanese tourist purchasing cosmetics through Shanghai International Finance Center’s immediate tax refund service.Her experience mirrors a broader trend:Visitors from overseas are flocking to China’s retail scene,lured by its growing shopping convenience.On April 26,six government agencies,including the Ministry of Commerce,jointly issued the refined departure tax refund policy,slashing the minimum refund threshold from 500 yuan(US$69)to 200 yuan(US$28),doubling the cash refund limit from 10,000 yuan(US$1,376)to 20,000 yuan(US$2,752),and encouraging shopping districts,tourist attractions and hotels to increase the number of tax refund stores.展开更多
THE Spring Festival,the most important traditional festival of the Chinese people,is not only an occasion for family reunion but also a time for binge shopping.Now,with digital technology enabling high penetration of ...THE Spring Festival,the most important traditional festival of the Chinese people,is not only an occasion for family reunion but also a time for binge shopping.Now,with digital technology enabling high penetration of mobile payments and e-commerce,as well as highly efficient smart logistics,infinite possibilities for shopping sprees are possible during the Spring Festival,which this year fell on January 29.展开更多
随着跨境电商的发展,TikTok Shop依靠TikTok平台的用户基数庞大在泰国市场得到了迅速扩张。首先通过对泰国市场跨境电商规模的分析,发现其呈现出不断扩大的发展趋势。进一步地,对TikTok Shop在泰国市场进行了SWOT分析,发现其海外电商体...随着跨境电商的发展,TikTok Shop依靠TikTok平台的用户基数庞大在泰国市场得到了迅速扩张。首先通过对泰国市场跨境电商规模的分析,发现其呈现出不断扩大的发展趋势。进一步地,对TikTok Shop在泰国市场进行了SWOT分析,发现其海外电商体系还不够完善、对本地市场还不够了解等等。因此,对TikTok Shop在泰国市场的发展路径提出相应建议措施并进行优化,以推动TikTok Shop在泰国市场份额的不断扩大。With the development of cross-border e-commerce, TikTok Shop has been rapidly expanding in the Thai market relying on the large user base of the TikTok platform. Firstly, by analyzing the scale of cross-border e-commerce in the Thai market, it is found that it shows an expanding development trend. Further, a SWOT analysis of TikTok Shop in the Thai market reveals that its overseas e-commerce system is not perfect enough and it does not understand the local market well enough. Therefore, the development path of TikTok Shop in Thailand market is proposed to optimize the corresponding measures to promote the continuous expansion of TikTok Shop’s market share in Thailand.展开更多
1 On a Sunday morning,Zheng Shiying drove 130 kilometers to a village on the outskirts of Beijing,not for a hike or a hot spring,but for a cup of coffee.Housed in an old stone building that looks untouched by time,by ...1 On a Sunday morning,Zheng Shiying drove 130 kilometers to a village on the outskirts of Beijing,not for a hike or a hot spring,but for a cup of coffee.Housed in an old stone building that looks untouched by time,by the time Zheng arrived,the caféwas already packed with weekend visitors,and finding a seat was a bit of a challenge.展开更多
Intelligent production is an important development direction in intelligent manufacturing,with intelligent factories playing a crucial role in promoting intelligent production.Flexible job shops,as the main form of in...Intelligent production is an important development direction in intelligent manufacturing,with intelligent factories playing a crucial role in promoting intelligent production.Flexible job shops,as the main form of intelligent factories,constantly face dynamic disturbances during the production process,including machine failures and urgent orders.This paper discusses the basic models and research methods of job shop scheduling,emphasizing the important role of dynamic job shop scheduling and its response schemes in future research.A multi-objective flexible job shop dynamic scheduling mathematical model is established,highlighting its complex and multi-constraint characteristics under different interferences.A classification discussion is conducted on the dynamic response methods and optimization objectives under machine failures,emergency orders,fuzzy completion times,and mixed dynamic events.The development process of traditional scheduling rules and intelligent methods in dynamic scheduling are also analyzed.Finally,based on the current development status of job shop scheduling and the requirements of intelligent manufacturing,the future development trends of dynamic scheduling in flexible job shops are proposed.展开更多
The flexible job shop scheduling problem(FJSP)is commonly encountered in practical manufacturing environments.A product is typically built by assembling multiple jobs during actual manufacturing.AGVs are normally used...The flexible job shop scheduling problem(FJSP)is commonly encountered in practical manufacturing environments.A product is typically built by assembling multiple jobs during actual manufacturing.AGVs are normally used to transport the jobs from the processing shop to the assembly shop,where they are assembled.Therefore,studying the integrated scheduling problem with its processing,transportation,and assembly stages is extremely beneficial and significant.This research studies the three-stage flexible job shop scheduling problem with assembly and AGV transportation(FJSP-T-A),which includes processing jobs,transporting them via AGVs,and assembling them.A mixed integer linear programming(MILP)model is established to obtain optimal solutions.As the MILP model is challenging for solving large-scale problems,a novel co-evolutionary algorithm(NCEA)with two different decoding methods is proposed.In NCEA,a restart operation is developed to improve the diversity of the population,and a multiple crossover strategy is designed to improve the quality of individuals.The validity of the MILP model is proven by analyzing its complexity.The effectiveness of the restart operator,multiple crossovers,and the proposed algorithm is demonstrated by calculating and analyzing the RPI values of each algorithm's results within the time limit and performing a paired t-test on the average values of each algorithm at the 95%confidence level.This paper studies FJSP-T-A by minimizing the makespan for the first time,and presents a MILP model and an NCEA with two different decoding methods.展开更多
The shop scheduling problem with limited buffers has broad applications in real-world production scenarios,so this research direction is of great practical significance.However,there is currently little research on th...The shop scheduling problem with limited buffers has broad applications in real-world production scenarios,so this research direction is of great practical significance.However,there is currently little research on the hybrid flow shop scheduling problem with limited buffers(LBHFSP).This paper deeply investigates the LBHFSP to optimize the goal of the total completion time.To better solve the LBHFSP,a multi-level subpopulation-based particle swarm optimization algorithm(MLPSO)is proposed,which is founded on the attributes of the LBHFSP and the shortcomings of the basic PSO(particle swarm optimization)algorithm.In MLPSO,firstly,considering the impact of the limited buffers on the process of subsequent operations,a specific circular decoding strategy is developed to accommodate the characteristics of limited buffers.Secondly,an initialization strategy based on blocking time is designed to enhance the quality and diversity of the initial population.Afterward,a multi-level subpopulation collaborative search is developed to prevent being trapped in a local optimum and improve the global exploration capability.Additionally,a local search strategy based on the first blocked job is designed to enhance the MLPSO algorithm’s exploitation capability.Lastly,numerous experiments are carried out to test the performance of the proposed MLPSO by comparing it with classical intelligent optimization and popular algorithms in recent years.The results confirm that the proposed MLPSO has an outstanding performance when compared to other algorithms when solving LBHFSP.展开更多
With the development of economic globalization,distributedmanufacturing is becomingmore andmore prevalent.Recently,integrated scheduling of distributed production and assembly has captured much concern.This research s...With the development of economic globalization,distributedmanufacturing is becomingmore andmore prevalent.Recently,integrated scheduling of distributed production and assembly has captured much concern.This research studies a distributed flexible job shop scheduling problem with assembly operations.Firstly,a mixed integer programming model is formulated to minimize the maximum completion time.Secondly,a Q-learning-assisted coevolutionary algorithmis presented to solve themodel:(1)Multiple populations are developed to seek required decisions simultaneously;(2)An encoding and decoding method based on problem features is applied to represent individuals;(3)A hybrid approach of heuristic rules and random methods is employed to acquire a high-quality population;(4)Three evolutionary strategies having crossover and mutation methods are adopted to enhance exploration capabilities;(5)Three neighborhood structures based on problem features are constructed,and a Q-learning-based iterative local search method is devised to improve exploitation abilities.The Q-learning approach is applied to intelligently select better neighborhood structures.Finally,a group of instances is constructed to perform comparison experiments.The effectiveness of the Q-learning approach is verified by comparing the developed algorithm with its variant without the Q-learning method.Three renowned meta-heuristic algorithms are used in comparison with the developed algorithm.The comparison results demonstrate that the designed method exhibits better performance in coping with the formulated problem.展开更多
As one of the most classical scheduling problems,flexible job shop scheduling problems(FJSP)find widespread applications in modern intelligent manufacturing systems.However,the majority of meta-heuristic methods for s...As one of the most classical scheduling problems,flexible job shop scheduling problems(FJSP)find widespread applications in modern intelligent manufacturing systems.However,the majority of meta-heuristic methods for solving FJSP in the literature are population-based evolutionary algorithms,which are complex and time-consuming.In this paper,we propose a fast effective singlesolution based local search algorithm with an innovative adaptive weighting-based local search(AWLS)technique for solving FJSP.The adaptive weighting technique assigns weights to each operation and adaptively updates them during the exploration.AWLS integrates a Tabu Search strategy and the adaptive weighting technique to smooth the landscape of the search space and enhance the exploration diversity.Computational experiments on 313 well-known benchmark instances demonstrate that AWLS is highly competitive with state-of-the-art algorithms in terms of both solution quality and computational efficiency,despite of its simplicity.Specifically,AWLS improves the previous best-known results in the literature on 33 instances and match the best-known results on the remaining ones except for only one under the same time limit of up to 300 s.As a strongly non-deterministic polynomia(NP)-hard problem which has been extensively studied for nearly half a century,breaking the records on these classic instances is an arduous task.Nevertheless,AWLS establishes new records on 8 challenging instances whose previous best records were established by a state-of-the-art meta-heuristic algorithm and a famous industrial solver.展开更多
A new concept of multi-shop （M ） is put forward which contains all basic shops including open shop （O）, job shop （J ）, flow shop （F ） and hybrid flow shop （H ） so that these basic shop can be scheduled toget...A new concept of multi-shop （M ） is put forward which contains all basic shops including open shop （O）, job shop （J ）, flow shop （F ） and hybrid flow shop （H ） so that these basic shop can be scheduled together. Several algorithms including ant colony optimization （ACO）, most work remaining （MWR）, least work remaining （LWR）, longest processing time （LPT） and shortest processing time （SPT） are used for scheduling the M. Numerical experiments of the M adopting data of some car and reC series benchmark instances are tested. The results show that the ACO algorithm has better performance for scheduling the M than the other algorithms, if minimizing the makespan （ Cmax^＊） is taken as the objective function. As a comparison, the separate shops contained in the M are also scheduled by the ACO algorithm for the same objective function, when the completing time of the jobs in the previous shop is taken as the ready time of these jobs in the following shop. The results show that the M has the advantage of shortening the makespan upon separate shops.展开更多
基金supported in part by the National Key Research and Development Program of China under Grant No.2021YFF0901300in part by the National Natural Science Foundation of China under Grant Nos.62173076 and 72271048.
文摘The distributed permutation flow shop scheduling problem(DPFSP)has received increasing attention in recent years.The iterated greedy algorithm(IGA)serves as a powerful optimizer for addressing such a problem because of its straightforward,single-solution evolution framework.However,a potential draw-back of IGA is the lack of utilization of historical information,which could lead to an imbalance between exploration and exploitation,especially in large-scale DPFSPs.As a consequence,this paper develops an IGA with memory and learning mechanisms(MLIGA)to efficiently solve the DPFSP targeted at the mini-malmakespan.InMLIGA,we incorporate a memory mechanism to make a more informed selection of the initial solution at each stage of the search,by extending,reconstructing,and reinforcing the information from previous solutions.In addition,we design a twolayer cooperative reinforcement learning approach to intelligently determine the key parameters of IGA and the operations of the memory mechanism.Meanwhile,to ensure that the experience generated by each perturbation operator is fully learned and to reduce the prior parameters of MLIGA,a probability curve-based acceptance criterion is proposed by combining a cube root function with custom rules.At last,a discrete adaptive learning rate is employed to enhance the stability of the memory and learningmechanisms.Complete ablation experiments are utilized to verify the effectiveness of the memory mechanism,and the results show that this mechanism is capable of improving the performance of IGA to a large extent.Furthermore,through comparative experiments involving MLIGA and five state-of-the-art algorithms on 720 benchmarks,we have discovered that MLI-GA demonstrates significant potential for solving large-scale DPFSPs.This indicates that MLIGA is well-suited for real-world distributed flow shop scheduling.
文摘The job shop scheduli ng problem has been studied for decades and known as an NP-hard problem. The fl exible job shop scheduling problem is a generalization of the classical job sche duling problem that allows an operation to be processed on one machine out of a set of machines. The problem is to assign each operation to a machine and find a sequence for the operations on the machine in order that the maximal completion time of all operations is minimized. A genetic algorithm is used to solve the f lexible job shop scheduling problem. A novel gene coding method aiming at job sh op problem is introduced which is intuitive and does not need repairing process to validate the gene. Computer simulations are carried out and the results show the effectiveness of the proposed algorithm.
文摘“This trip to China has been an absolute steal,”exclaimed Keiko,a Japanese tourist purchasing cosmetics through Shanghai International Finance Center’s immediate tax refund service.Her experience mirrors a broader trend:Visitors from overseas are flocking to China’s retail scene,lured by its growing shopping convenience.On April 26,six government agencies,including the Ministry of Commerce,jointly issued the refined departure tax refund policy,slashing the minimum refund threshold from 500 yuan(US$69)to 200 yuan(US$28),doubling the cash refund limit from 10,000 yuan(US$1,376)to 20,000 yuan(US$2,752),and encouraging shopping districts,tourist attractions and hotels to increase the number of tax refund stores.
文摘THE Spring Festival,the most important traditional festival of the Chinese people,is not only an occasion for family reunion but also a time for binge shopping.Now,with digital technology enabling high penetration of mobile payments and e-commerce,as well as highly efficient smart logistics,infinite possibilities for shopping sprees are possible during the Spring Festival,which this year fell on January 29.
文摘随着跨境电商的发展,TikTok Shop依靠TikTok平台的用户基数庞大在泰国市场得到了迅速扩张。首先通过对泰国市场跨境电商规模的分析,发现其呈现出不断扩大的发展趋势。进一步地,对TikTok Shop在泰国市场进行了SWOT分析,发现其海外电商体系还不够完善、对本地市场还不够了解等等。因此,对TikTok Shop在泰国市场的发展路径提出相应建议措施并进行优化,以推动TikTok Shop在泰国市场份额的不断扩大。With the development of cross-border e-commerce, TikTok Shop has been rapidly expanding in the Thai market relying on the large user base of the TikTok platform. Firstly, by analyzing the scale of cross-border e-commerce in the Thai market, it is found that it shows an expanding development trend. Further, a SWOT analysis of TikTok Shop in the Thai market reveals that its overseas e-commerce system is not perfect enough and it does not understand the local market well enough. Therefore, the development path of TikTok Shop in Thailand market is proposed to optimize the corresponding measures to promote the continuous expansion of TikTok Shop’s market share in Thailand.
文摘1 On a Sunday morning,Zheng Shiying drove 130 kilometers to a village on the outskirts of Beijing,not for a hike or a hot spring,but for a cup of coffee.Housed in an old stone building that looks untouched by time,by the time Zheng arrived,the caféwas already packed with weekend visitors,and finding a seat was a bit of a challenge.
基金supported by the National Key Research and Development Program Project(No.2021YFB3301300).
文摘Intelligent production is an important development direction in intelligent manufacturing,with intelligent factories playing a crucial role in promoting intelligent production.Flexible job shops,as the main form of intelligent factories,constantly face dynamic disturbances during the production process,including machine failures and urgent orders.This paper discusses the basic models and research methods of job shop scheduling,emphasizing the important role of dynamic job shop scheduling and its response schemes in future research.A multi-objective flexible job shop dynamic scheduling mathematical model is established,highlighting its complex and multi-constraint characteristics under different interferences.A classification discussion is conducted on the dynamic response methods and optimization objectives under machine failures,emergency orders,fuzzy completion times,and mixed dynamic events.The development process of traditional scheduling rules and intelligent methods in dynamic scheduling are also analyzed.Finally,based on the current development status of job shop scheduling and the requirements of intelligent manufacturing,the future development trends of dynamic scheduling in flexible job shops are proposed.
基金Supported by National Natural Science Foundation of China(Grant Nos.52205529 and 62303204)the Youth Innovation Team Program of Shandong Higher Education Institution(Grant No.2023KJ206)the Guangyue Youth Scholar Innovation Talent Program support received from Liaocheng University(Grant No.LCUGYTD2022-03)。
文摘The flexible job shop scheduling problem(FJSP)is commonly encountered in practical manufacturing environments.A product is typically built by assembling multiple jobs during actual manufacturing.AGVs are normally used to transport the jobs from the processing shop to the assembly shop,where they are assembled.Therefore,studying the integrated scheduling problem with its processing,transportation,and assembly stages is extremely beneficial and significant.This research studies the three-stage flexible job shop scheduling problem with assembly and AGV transportation(FJSP-T-A),which includes processing jobs,transporting them via AGVs,and assembling them.A mixed integer linear programming(MILP)model is established to obtain optimal solutions.As the MILP model is challenging for solving large-scale problems,a novel co-evolutionary algorithm(NCEA)with two different decoding methods is proposed.In NCEA,a restart operation is developed to improve the diversity of the population,and a multiple crossover strategy is designed to improve the quality of individuals.The validity of the MILP model is proven by analyzing its complexity.The effectiveness of the restart operator,multiple crossovers,and the proposed algorithm is demonstrated by calculating and analyzing the RPI values of each algorithm's results within the time limit and performing a paired t-test on the average values of each algorithm at the 95%confidence level.This paper studies FJSP-T-A by minimizing the makespan for the first time,and presents a MILP model and an NCEA with two different decoding methods.
基金supported in part by the National Natural Science Foundation of China under Grant No.52175490.
文摘The shop scheduling problem with limited buffers has broad applications in real-world production scenarios,so this research direction is of great practical significance.However,there is currently little research on the hybrid flow shop scheduling problem with limited buffers(LBHFSP).This paper deeply investigates the LBHFSP to optimize the goal of the total completion time.To better solve the LBHFSP,a multi-level subpopulation-based particle swarm optimization algorithm(MLPSO)is proposed,which is founded on the attributes of the LBHFSP and the shortcomings of the basic PSO(particle swarm optimization)algorithm.In MLPSO,firstly,considering the impact of the limited buffers on the process of subsequent operations,a specific circular decoding strategy is developed to accommodate the characteristics of limited buffers.Secondly,an initialization strategy based on blocking time is designed to enhance the quality and diversity of the initial population.Afterward,a multi-level subpopulation collaborative search is developed to prevent being trapped in a local optimum and improve the global exploration capability.Additionally,a local search strategy based on the first blocked job is designed to enhance the MLPSO algorithm’s exploitation capability.Lastly,numerous experiments are carried out to test the performance of the proposed MLPSO by comparing it with classical intelligent optimization and popular algorithms in recent years.The results confirm that the proposed MLPSO has an outstanding performance when compared to other algorithms when solving LBHFSP.
文摘With the development of economic globalization,distributedmanufacturing is becomingmore andmore prevalent.Recently,integrated scheduling of distributed production and assembly has captured much concern.This research studies a distributed flexible job shop scheduling problem with assembly operations.Firstly,a mixed integer programming model is formulated to minimize the maximum completion time.Secondly,a Q-learning-assisted coevolutionary algorithmis presented to solve themodel:(1)Multiple populations are developed to seek required decisions simultaneously;(2)An encoding and decoding method based on problem features is applied to represent individuals;(3)A hybrid approach of heuristic rules and random methods is employed to acquire a high-quality population;(4)Three evolutionary strategies having crossover and mutation methods are adopted to enhance exploration capabilities;(5)Three neighborhood structures based on problem features are constructed,and a Q-learning-based iterative local search method is devised to improve exploitation abilities.The Q-learning approach is applied to intelligently select better neighborhood structures.Finally,a group of instances is constructed to perform comparison experiments.The effectiveness of the Q-learning approach is verified by comparing the developed algorithm with its variant without the Q-learning method.Three renowned meta-heuristic algorithms are used in comparison with the developed algorithm.The comparison results demonstrate that the designed method exhibits better performance in coping with the formulated problem.
基金supported in part by the National Natural Science Foundation of China(NSFC)(62202192 and 72101094)the National Science Fund for Distinguished Young Scholars of China(51825502).
文摘As one of the most classical scheduling problems,flexible job shop scheduling problems(FJSP)find widespread applications in modern intelligent manufacturing systems.However,the majority of meta-heuristic methods for solving FJSP in the literature are population-based evolutionary algorithms,which are complex and time-consuming.In this paper,we propose a fast effective singlesolution based local search algorithm with an innovative adaptive weighting-based local search(AWLS)technique for solving FJSP.The adaptive weighting technique assigns weights to each operation and adaptively updates them during the exploration.AWLS integrates a Tabu Search strategy and the adaptive weighting technique to smooth the landscape of the search space and enhance the exploration diversity.Computational experiments on 313 well-known benchmark instances demonstrate that AWLS is highly competitive with state-of-the-art algorithms in terms of both solution quality and computational efficiency,despite of its simplicity.Specifically,AWLS improves the previous best-known results in the literature on 33 instances and match the best-known results on the remaining ones except for only one under the same time limit of up to 300 s.As a strongly non-deterministic polynomia(NP)-hard problem which has been extensively studied for nearly half a century,breaking the records on these classic instances is an arduous task.Nevertheless,AWLS establishes new records on 8 challenging instances whose previous best records were established by a state-of-the-art meta-heuristic algorithm and a famous industrial solver.
基金This project is supported by National Natural Science Foundation of China (No. 50575137)Provincial Natural Science Foundation of Zhejiang, China (No. Z604342)Scientific Research Fund of Zhejiang Provincial Educational Committee, China (No. 20051643).
文摘A new concept of multi-shop （M ） is put forward which contains all basic shops including open shop （O）, job shop （J ）, flow shop （F ） and hybrid flow shop （H ） so that these basic shop can be scheduled together. Several algorithms including ant colony optimization （ACO）, most work remaining （MWR）, least work remaining （LWR）, longest processing time （LPT） and shortest processing time （SPT） are used for scheduling the M. Numerical experiments of the M adopting data of some car and reC series benchmark instances are tested. The results show that the ACO algorithm has better performance for scheduling the M than the other algorithms, if minimizing the makespan （ Cmax^＊） is taken as the objective function. As a comparison, the separate shops contained in the M are also scheduled by the ACO algorithm for the same objective function, when the completing time of the jobs in the previous shop is taken as the ready time of these jobs in the following shop. The results show that the M has the advantage of shortening the makespan upon separate shops.
