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# 数学代写|运筹学代写Operations Research代考|MAST30013 PROJECT NETWORK DIAGRAM

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## 数学代写|运筹学代写Operations Research代考|PROJECT NETWORK DIAGRAM

The best way to understand the relationships among the activities of a project is to construct a network diagram. This is explained using Illustration 14.1.

In the network given in Figure 14.1, the activities $A$ to $I$ are shown as arcs and hence this type of network is called activity on arc network. The nodes represent the events that indicate the beginning or end of an activity or set of activities.

Node 1 is the beginning of the project and node 7 is the end of the project. Node 6 represents the time where activities $E$ and $F$ are completed so that activity $I$ can start at node 6 . Node 2 represents the time of completion of activity $A$ so that activities $D$ and $E$ can begin.

The network in Figure $14.1$ is constructed as follows: Node 1 represents the start of the project. Activities $A, B$ and $C$ that have no precedence start from node 1 and finish at nodes 2 , 3 and 4 respectively. Activities $D$ and $E$ that have $A$ as precedence start at node 2 (finish of $A$ ) and end in nodes 5 and 6 respectively. Activity $F$ starts from node 3 and finishes at node 6 . Now, node 6 is the point in time where $E$ and $F$ are completed. Activity $G$ starts from node 4 and finishes at node 7 . Activity $H$ starts from node 5 and finishes at node 7 while activity $I$ starts from node 6 and finishes at node 7. Activities $H, I$ and $G$ are completed at node 7 and represents the finish of the project.

Another way of drawing the network is to represent the activities as nodes and events as arcs. This is shown in Figure 14.2.

In Figure 14.2, we have shown the activities as nodes and the precedence as arcs. This is called the activity on node network. Sometimes we add a start and finish node to indicate the beginning and the end of the project. Otherwise, we can get a disconnected network as the one shown in Figure 14.2.

Sometimes the precedence relationships may be such that when we represent the network as an activity on arc network, we require additional arcs (activities) to be created to meet the precedence. Such arcs are called dummy arcs and have zero duration. This is illustrated through an example.

## 数学代写|运筹学代写Operations Research代考|ONSIDERING BACKORDERING

In this model, we allow a backorder of $s$ units every cycle and as soon as the order quantity $Q$ arrives, we issue the backordered quantity. Figure $13.3$ shows the model. The maximum inventory held is $I_m=Q-s$. There is an inventory period of $T_1$ per cycle and a backorder period of $T_2$ per cycle.

The coefficients are:
Annual demand $=D /$ year
Order cost $=C_o$
Carrying cost $=C_c$
Shortage (backorder) $\operatorname{cost}=C_s$
Order quantity $=Q$
Backorder quantity $=s$
Maximum inventory in a cycle $=I_m$
Number of orders/year $=\frac{D}{Q}$
Annual order cost $=\frac{D C_o}{Q}$
Average inventory in the system $=\frac{I_m}{2}$
Annual inventory carrying cost $=\frac{I_m C_c}{2}$
Average shortage in the system $=\frac{s}{2}$

Annual shortage cost $=\frac{s C_s}{2}$
Total cost $T C=\frac{D C_o}{Q}+\frac{I_m C_c}{2} \times \frac{T_1}{\left(T_1+T_2\right)}+\frac{s C_s}{2} \times \frac{T_2}{\left(T_1+T_2\right)}$
From similar triangles, we get
and
Substituting, we get
$$T C=\frac{D C_o}{Q}+\frac{(Q-s)^2 C_c}{2 Q}+\frac{s^2 C_s}{2 Q}$$
The values of $Q$ and $s$ that minimize the total cost are obtained by setting the first partial derivative with respect to $Q$ and $s$ to zero. Partially differentiating with respect to $s$ and setting to zero, we get
$$s=\frac{Q C_c}{\left(C_c+C_s\right)}$$
Partially differentiating with respect to $Q$ and substituting for $s$, we get
$$Q^*=\sqrt{\frac{2 D C_o\left(C_c+C_s\right)}{C_c C_s}}$$

## 数学代写|运筹学代写Operations Research代考|ONSIDERING BACKORDERING

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$$T C=\frac{D C_o}{Q}+\frac{(Q-s)^2 C_c}{2 Q}+\frac{s^2 C_s}{2 Q}$$

$$s=\frac{Q C_c}{\left(C_c+C_s\right)}$$

$$Q^*=\sqrt{\frac{2 D C_o\left(C_c+C_s\right)}{C_c C_s}}$$

## 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。