### 统计代写|运筹学作业代写operational research代考|Critical Path Analysis

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 统计代写|运筹学作业代写operational research代考|Concept of Critical Path Analysis

Critical path analysis is a network-based method designed to aid in scheduling, monitoring, and controlling large and complex projects, particularly in the construction industry. A project can be defined as a series of related activities or tasks, with each activity consuming time and resources. Finding a critical path is a major part of project management. The activities on the critical path represent tasks that will delay the entire project if they are not completed on time. Based on the critical path analysis, project managers can reschedule and reallocate labor and financial resources so that the critical tasks can be completed on time. Critical path analysis is important because it can answer a number of questions about projects, such as:

1. When will the entire project be finished?
2. What are the critical activities or tasks in the project?
3. Which activities or tasks can be delayed if necessary and by how long without delaying the entire project?
Linear programming (LP) can be used to formulate such a problem and then yield the critical path. Consider a project with $n$ activities. The objective is to minimize the time required to complete the entire project. For each activity, we are certain that before node $j$ occurs, node $i$ must occur and activity on arc $(i, j)$ must be completed. The time required by an activity on arc $(i, j)$ is denoted as $t_{i j}$. By introducing decision variables $x_{j}$ to represent the completion time of an activity on arc $(i, j)$, the mathematical model for the critical path analysis can be written as shown in Model 4.1.1.

## 统计代写|运筹学作业代写operational research代考|Example of Critical Path Analysis

Before identifying the critical path, there are four steps to follow:

1. Define the project and its activities.
2. Define the precedence relationships among the activities.
3. Assign the time requirement to each activity.
4. Draw the network connecting all of the activities.
Table $4.1$ shows a project with eight activities in which activities A and B are done first because they have no predecessor and activity $\mathrm{H}$ is the terminal point. The precedence relationships among the activities, as well as their time requirements, are listed in the table. Table $4.2$ shows the same information as in Table 4.1, except that the immediate successors of activities are shown.

After defining the precedence relationships among the activities and the time requirements to each activity, a network representing the project can be constructed (Figure 4.1).

As shown in Figure 4.1, each activity is represented by a directional arc or arrow. This type of project network is regarded as activity-on-arc (AOA) network. There are two crucial rules for the construction of AOA network: (1) each activity is represented by exactly one arc or arrow in the network, and (2) each activity must be identified by two nodes. For example, activity A, which requires 2 units of time to complete, is linked by two nodes. Node 0 is the starting point of activity $\mathrm{A}$, and node 1 is the terminal point of activity A. To prevent a violation of the rules, it is sometimes necessary to use a dummy activity with 0 task time in the network. For example, activities $A$ and $B$ are predecessors of activity D. In this case, we can add a dummy activity, shown by a dashed arrow, pointing from node 1 to 2 . By adding this dummy.

## 统计代写|运筹学作业代写operational research代考| Concept of Assembly Line–Balancing Problem

Assembly line balancing is a product-oriented layout technique in operations management. Product-oriented layouts are designed for high-volume, lowvariety products or continuous production. The problem of assembly line balancing is how to assign tasks to workstations while meeting production requirements at a minimum imbalance between labors or machines. Minimization of imbalance leads to minimization of idle time along the assembly line and maximization utilization of labors and machines.

Integer linear programming (ILP) can be used to formulate an assembly line balancing problem and then yield the optimal product layout. Consider a job with a series of tasks $(i=1,2, \ldots, m)$ to be assigned to a certain number of workstations $(j=1,2, \ldots, n)$. The objective is to minimize the number of workstations, $A_{j}$, used to complete the work. Given the precedence relationships, the earliest workstation to which task $i$ can be assigned is denoted as $E_{i}$ and the latest workstation to which task $i$ can be assigned is denoted as $L_{i}$. The time required by task $i$ is $t_{i}$, and the theoretical cycle time is $C . W_{j}$ is the subset of all tasks that can be assigned to workstation $j$, and ||$W_{j}||$ is the number of tasks in subset $W_{j} . P_{i}$ is the set of tasks that must proceed task $i$, and $S_{i}$ is the set of tasks that must succeed task $i$. By introducing decision variables $x_{i j}$ to represent the assignment of task $i$ to workstation $j$, the mathematical model for the assembly line balancing problem can be written as shown in Model 5.1.1 (Patterson and Albracht 1975; Gökçen and Erel 1998; Ağpak and Gökçen 2005).

## 统计代写|运筹学作业代写operational research代考|Concept of Critical Path Analysis

1. 整个项目什么时候完成？
2. 项目中的关键活动或任务是什么？
3. 如有必要，哪些活动或任务可以延迟，在不延迟整个项目的情况下延迟多长时间？
线性规划 (LP) 可用于制定此类问题，然后得出关键路径。考虑一个项目n活动。目标是尽量减少完成整个项目所需的时间。对于每个活动，我们确定在节点之前j发生，节点一世必须在弧上发生和活动(一世,j)必须完成。arc 上的活动所需的时间(一世,j)表示为吨一世j. 通过引入决策变量Xj表示弧上活动的完成时间(一世,j), 关键路径分析的数学模型可以写成模型 4.1.1 所示。

## 统计代写|运筹学作业代写operational research代考|Example of Critical Path Analysis

1. 定义项目及其活动。
2. 定义活动之间的优先关系。
3. 为每个活动分配时间要求。
4. 画出连接所有活动的网络。
桌子4.1显示一个包含八项活动的项目，其中活动 A 和 B 首先完成，因为它们没有前置任务和活动H是终点。活动之间的优先关系及其时间要求列于表中。桌子4.2显示与表 4.1 中相同的信息，只是显示了活动的直接后续活动。

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## MATLAB代写

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