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数学代写|运筹学代写Operations Research代考|IMSE560 Parallel Machine Scheduling Models

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数学代写|运筹学代写Operations Research代考|Parallel Machine Scheduling Models

All scheduling models in this section have in common that the tasks can now be processed on more than one machine. In general, we assume that a given number $m$ of machines are available. We further assume that these machines are identical in the sense that not only can all machines process each of the tasks, but it takes the same amount of time to process a task, regardless of the machine it is processed on.
First consider the objective of minimizing makespan $C_{\max }$. It can be demonstrated that this problem is very difficult from a computational point of view, even for just two machines. This means that we typically have to resort to heuristics to solve the problem (except in cases, in which there is ample time to find exact solutions). The most popular heuristic method for this type of problem is the longest processing time first (or $L P T$ ) algorithm. This heuristic method belongs to the class of list scheduling methods. All list scheduling methods first produce a priority list of tasks, starting with the job that is assigned the highest priority. Using this list and starting with the task that has the highest priority, jobs are then assigned one at a time to the first available machine. In particular, the longest processing time first algorithm can be described by the following rule:

LPT Algorithm: Put the tasks in order of nonincreasing processing times. Starting at the top, assign the first available task to the first available machine. Repeat the procedure until all tasks have been scheduled.

数学代写|运筹学代写Operations Research代考|Dedicated Machine Scheduling Models

This section deals with different types of dedicated machine scheduling models. The first such model includes an open shop. Recall that in an open shop, each task must be processed on each of a number of different machines, performing different operations. The sequence of machines, in which the jobs are processed, is immaterial. Here, we will deal only with the case of two machines, which happens to be easy, while problems with three or more machines are difficult. Minimizing the schedule length (makespan) $C_{\text {max }}$ is easy. Optimal schedules can be found by means of the Longest Alternate Processing Time (LAPT) algorithm. It can be described as follows:

LAPT Algorithm: Whenever a machine becomes idle, schedule the task on it that has the longest processing time on the other machine, provided the task has not yet been processed on that machine and is available at that time. If a task is not available, the task with the next longest processing time on the other machine is scheduled. Ties are broken arbitrarily.

数学代写|运筹学代写Operations Research代考|Parallel Machine Scheduling Models

LPT 算法: 将任务按处理时间不增加的顺序排列。从顺部㰢始，将第一个可用任务分配给第一台可用机器。重复此过程，直到所有 任务都已安排好。

数学代写|运筹学代写Operations Research代考|Dedicated Machine Scheduling Models

LAPT算法: 每当一台机器空闲时，安排在另一台机器上处理时间最长的烀，前是是该仼务尚末在该机器上处理并且当时可用。 如果某个仼务不可用，则安排另一台机器上处理时间第二长的任务。关系被任意打破。

MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。