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# 数学代写|运筹学代写Operations Research代考|THE SHORTEST-PATH PROBLEM

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## 数学代写|运筹学代写Operations Research代考|THE SHORTEST-PATH PROBLEM

Although several other versions of the shortest-path problem (including some for directed networks) are mentioned at the end of the section, we shall focus on the following simple version. Consider an undirected and connected network with two special nodes called the origin and the destination. Associated with each of the links (undirected arcs) is a nonnegative distance. The objective is to find the shortest path (the path with the minimum total distance) from the origin to the destination.

A relatively straightforward algorithm is available for this problem. The essence of this procedure is that it fans out from the origin, successively identifying the shortest path to each of the nodes of the network in the ascending order of their (shortest) distances from the origin, thereby solving the problem when the destination node is reached. We shall first outline the method and then illustrate it by solving the shortest-path problem encountered by the Seervada Park management in Sec. 9.1.
Algorithm for the Shortest-Path Problem.
Objective of $\mathrm{n}$ th iteration: Find the $n$th nearest node to the origin (to be repeated for $n=1,2, \ldots$ until the $n$th nearest node is the destination.
Input for $\mathrm{n}$ th iteration: $n-1$ nearest nodes to the origin (solved for at the previous iterations), including their shortest path and distance from the origin. (These nodes, plus the origin, will be called solved nodes; the others are unsolved nodes.)

Candidates for $\mathrm{n}$ th nearest node: Each solved node that is directly connected by a link to one or more unsolved nodes provides one candidate – the unsolved node with the shortest connecting link. (Ties provide additional candidates.)
Calculation of $\mathrm{n}$ th nearest node: For each such solved node and its candidate, add the distance between them and the distance of the shortest path from the origin to this solved node. The candidate with the smallest such total distance is the $n$th nearest node (ties provide additional solved nodes), and its shortest path is the one generating this distance.

## 数学代写|运筹学代写Operations Research代考|Applying This Algorithm to the Seervada Park Shortest-Path Problem

The Seervada Park management needs to find the shortest path from the park entrance (node $O$ ) to the scenic wonder (node $T$ ) through the road system shown in Fig. 9.1. Applying the above algorithm to this problem yields the results shown in Table 9.2 (where the tie for the second nearest node allows skipping directly to seeking the fourth nearest node next). The first column $(n)$ indicates the iteration count. The second column simply lists the solved nodes for beginning the current iteration after deleting the irrelevant ones (those not connected directly to any unsolved node). The third column then gives the candidates for the $n$th nearest node (the unsolved nodes with the shortest connecting link to a solved node). The fourth column calculates the distance of the shortest path from the origin to each of these candidates (namely, the distance to the solved node plus the link distance to the candidate). The candidate with the smallest such distance is the $n$th nearest node to the origin, as listed in the fifth column. The last two columns summarize the information for this newest solved node that is needed to proceed to subsequent iterations (namely, the distance of the shortest path from the origin to this node and the last link on this shortest path).

Now let us relate these columns directly to the outline given for the algorithm. The input for $\mathrm{n}$ th iteration is provided by the fifth and sixth columns for the preceding iterations, where the solved nodes in the fifth column are then listed in the second column for the current iteration after deleting those that are no longer directly connected to unsolved nodes. The candidates for $\mathrm{n}$ th nearest node next are listed in the third column for the current iteration. The calculation of $\mathrm{n}$ th nearest node is performed in the fourth column, and the results are recorded in the last three columns for the current iteration.

After the work shown in Table 9.2 is completed, the shortest path from the destination to the origin can be traced back through the last column of Table 9.2 as either $T \rightarrow D \rightarrow E \rightarrow B \rightarrow A \rightarrow O$ or $T \rightarrow D \rightarrow B \rightarrow A \rightarrow O$. Therefore, the two alternates for the shortest path from the origin to the destination have been identified as $O \rightarrow A \rightarrow B \rightarrow$ $E \rightarrow D \rightarrow T$ and $O \rightarrow A \rightarrow B \rightarrow D \rightarrow T$, with a total distance of 13 miles on either path.

## 数学代写|运筹学代写Operations Research代考|THE SHORTEST-PATH PROBLEM

$\ maththrm {n}$第n次迭代的目标:找到离原点最近的$n$节点(对于$n=1,2， \ldots$重复，直到$n$最近的节点是目的地)。
$\mathrm{n}$ th迭代的输入:$n-1$离原点最近的节点(在前一次迭代中求解)，包括它们的最短路径和到原点的距离。(这些节点加上原点，称为已解节点;其他是未解决的节点。)

$\ mathm {n}$第th最近节点的候选节点:每个通过链路直接连接到一个或多个未解决节点的已解决节点提供一个候选节点-具有最短连接链路的未解决节点。(领带提供了更多的候选人。)
$\mathrm{n}$ th最近节点的计算:对于每个已解节点及其候选节点，将它们之间的距离和从原点到该已解节点的最短路径的距离相加。总距离最小的候选节点是第n个最近的节点(领带提供了额外的已解节点)，它的最短路径是产生这个距离的节点。

## 数学代写|运筹学代写Operations Research代考|Applying This Algorithm to the Seervada Park Shortest-Path Problem

Seervada公园管理需要通过图9.1所示的道路系统，找到从公园入口(节点$O$)到景区奇观(节点$T$)的最短路径。将上述算法应用于此问题会产生如表9.2所示的结果(其中第二个最近节点的平局允许直接跳转到接下来寻找第四个最近节点)。第一列$(n)$表示迭代计数。第二列只是列出了在删除不相关节点(没有直接连接到任何未解决节点的节点)之后开始当前迭代的已解决节点。然后，第三列给出了第n个最近节点的候选节点(到已解节点的连接链路最短的未解节点)。第四列计算从原点到每个候选节点的最短路径的距离(即到已解节点的距离加上到候选节点的链接距离)。距离最小的候选节点是距离原点第n个最近的节点，如第五列所示。最后两列总结了这个最新解决的节点的信息，这些信息用于后续迭代(即，从原点到该节点的最短路径的距离以及该最短路径上的最后一个链接)。

## 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。