Johnson Algorithm for Minimum Completion Time

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Questions and Answers

在两机流水车间过程中，约翰逊算法通过什么方式划分作业？

按照作业在第一台机器上的加工时间 (correct)

按照作业的先后顺序

按照作业的编号

按照作业在第二台机器上的加工时间

在S1组中，作业是如何排序的？

按照它们的编号

按照它们在第一台机器上的加工时间 (correct)

基于它们在第二台机器上的加工时间

按照它们所需的总加工时间

S2组中的作业是如何排序的？

按照它们在第二台机器上的加工时间 (correct)

基于它们在第一台机器上的加工时间

按照作业的编号

按照它们所需的总加工时间

约翰逊算法主要用于哪种情况中？

两台机器处理多个作业 Signup and view all the answers

约翰逊算法是哪一类最优化方法？

分治算法 Signup and view all the answers

约翰逊算法是谁在1954年开发的？

Frank W. Johnson Signup and view all the answers

Johnson算法在生产环境中优化调度的目标是什么？

最小化加工时间 Signup and view all the answers

在两台机器流水车间中，Johnson算法如何帮助最小化流水作业的最大完工时间？

通过优化作业顺序 Signup and view all the answers

在研究中，Johnson算法与什么调度规则结合使用以进一步优化调度过程？

最后忙碌机器规则 Signup and view all the answers

Johnson算法如何对流水作业进行分组以优化调度？

根据作业的加工时间 Signup and view all the answers

Johnson算法在哪种流水车间问题中展现了重要性？

带有不可用周期问题 Signup and view all the answers

在两台机器流水车间问题中，什么情况下Johnson算法提供了最优解？

S1的作业先于S2的作业 Signup and view all the answers

根据约翰逊算法，作业i应该在作业j之前进行处理的条件是什么？

min(ai, bij) Signup and view all the answers

约翰逊算法的核心目标是什么？

最小化总体完成时间 Signup and view all the answers

约翰逊算法在两机流水车间中如何帮助优化调度？

根据作业优先级确定处理顺序 Signup and view all the answers

在约翰逊算法中，最小化makespan指的是什么？

所有作业完成时间的总和 Signup and view all the answers

约翰逊算法主要通过什么方式帮助解决两台机器流水车间问题？

根据作业处理顺序最小化完成时间 Signup and view all the answers

约翰逊算法的规则基于两个机器上作业的什么属性来排序？

工期与先期的属性 Signup and view all the answers

Study Notes

Johnson Algorithm for Minimum Completion Time

The Johnson Algorithm is a branch and bound method used in combinatorial optimization to find the best possible schedules for multiple jobs, particularly in manufacturing systems. It was developed in 1954 by Frank W. Johnson and has become a cornerstone in scheduling theory, especially in the context of two-machine flow shops, which is a system consisting of two machines and n jobs that need to be processed through these machines. Here, we focus on how the Johnson Algorithm helps minimize completion time in such scenarios.

Understanding the Johnson Algorithm

The Johnson Algorithm operates under specific rules to determine the best order in which jobs should be processed during the two-machine flow shop process. These rules involve dividing jobs into two groups, S1 and S2, based on their processing times:

S1 contains jobs with shorter processing times on the first machine than on the second machine (i.e., p1 >= p2), while
S2 contains jobs with longer processing times on the first machine than on the second machine (i.e., p1 < p2)

Once the jobs are grouped into these categories, the algorithm applies the following sequential steps:

Order the jobs in S1 based on their processing times on the first machine in increasing order (e.g., in ascending order of p1).
Order the jobs in S2 based on their processing times on the second machine in decreasing order (e.g., in descending order of p2).
Sequence jobs from S1 first, followed by jobs from S2.

This sequence is the optimal solution to the two-machine flow shop problem when the objective is to minimize the makespan, given all ready times are equal to zero and non-preemption.

Johnson Algorithm in Practice

The Johnson Algorithm has been applied in various research contexts to optimize scheduling in production environments. For example, in a research study, the algorithm was used to effectively schedule jobs in a production company to minimize the makespan. Another paper demonstrated the importance of the Johnson Algorithm for flow shop scheduling problems with unavailability periods and proposed a modification to the algorithm for the same problem.

Additionally, the Johnson Algorithm has been combined with other scheduling rules, such as the Last Busy Machine (LBM) rule, to create a heuristic that further optimizes the scheduling process.

Conclusion

The Johnson Algorithm is a fundamental algorithm in scheduling theory, particularly in minimizing completion time in two-machine flow shops. By grouping jobs into S1 and S2 based on their processing times and applying specific ordering and sequencing rules, the algorithm provides the optimal solution to minimize the makespan in such systems. Its application has been shown to be effective in various research settings, further highlighting its importance in the field of flow shop scheduling.

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Description

Explore the Johnson Algorithm, a branch and bound method used in combinatorial optimization to minimize completion time in two-machine flow shops. Understand how jobs are grouped into S1 and S2 based on processing times and the sequential steps involved in determining the optimal order for job processing. Discover the practical applications of the Johnson Algorithm in production scheduling and its significance in scheduling theory.