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# 数学代写|运筹学代写Operations Research代考|IMSE560 Minimum Spanning Trees

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## 数学代写|运筹学代写Operations Research代考|Minimum Spanning Trees

Suppose that we have a given set of nodes and that for each pair of nodes, the cost of establishing an edge between the two nodes is known. How can we construct, at minimum cost, a network of edges such that any two nodes are connected by a path? This is the minimum spanning tree problem for an undirected network.
The minimum spanning tree problem arises in the construction of telephone and pipeline networks to connect a given number of places. As an illustration, consider the example of an oil company that wants to construct a network of pipelines to connect five oil extraction sites to a receiving terminal. Every extraction site must be connected to the terminal, either directly or indirectly. Table $3.5$ shows the distances between the extraction sites and the terminal. The five extraction sites are denoted by $N_1, \ldots, N_5$, and the terminal is denoted by $N_0$. How should the oil company connect the extraction sites and the terminal to minimize the overall length of the pipelines?

We need the concept of a spanning tree to solve the problem. A spanning tree is a network in which any two nodes are connected by a path and there are no cycles (a cycle is a path in which the starting point and endpoint coincide). In a spanning tree, there is a unique path between any two nodes (verify this result using a proof by contradiction: assume that there are two nodes between which two different paths exist and then deduce a contradiction). Also verify that a spanning tree has $n-1$ arcs if there are $n$ nodes. Figure $3.10$ gives two spanning trees for the oil example. These spanning trees have respective lengths 171 and 143.

## 数学代写|运筹学代写Operations Research代考|Minimum-Cost Flow Problems

Many optimization problems in networks can be formulated as minimum-cost flow problems. This problem in fact comes down to the determination of a distribution plan to send a certain product through a network at minimum cost in order to satisfy demand for the product in certain nodes using the supply in other nodes. For the general formulation of the minimum-cost flow problem, we assume that we have a directed network $G=(X, A)$, where $X=\left{N_1, \ldots, N_m\right}$ is the set of nodes and $A$ is the set of $\operatorname{arcs}\left(N_i, N_j\right)$ between the nodes. The cost $c_{i j}$ is given for every $\operatorname{arc}\left(N_i, N_j\right)$, where
$c_{i j}=$ cost per unit of product passing through arc $\left(N_i, N_j\right)$.

Integers $l_{i j} \geq 0$ and $u_{i j} \geq 0$ are associated with every arc $\left(N_i, N_j\right)$, where
$l_{i j}=$ minimum quantity of product to be transported through $\operatorname{arc}\left(N_i, N_j\right)$,
$u_{i j}=$ maximum quantity of product that can be transported through $\operatorname{arc}\left(N_i, N_j\right)$.

## 数学代写|运筹学代写Operations Research代考|Minimum-Cost Flow Problems

$c_{i j}=$每单位产品通过$operatorname{arc}\left(N_i, N_j\right)$的成本。

$l_{i j}=$通过$operatorname{arc}\left(N_i, N_j\right)$运输的最小产品数量。
$u_{i j}=$可通过$operatorname{arc}\left(N_i, N_j\right)$运输的最大产品数量。

## 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。