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# 数学代写|最优化作业代写optimization theory代考|APPLICATION OF THE PRINCIPLE OF OPTIMALITY TO DECISION-MAKING

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## 数学代写|最优化作业代写optimization theory代考APPLICATION OF THE PRINCIPLE OF OPTIMALITY TO DECISION-MAKING

The following example illustrates the procedure for making a single optimal decision with the aid of the principle of optimality.

Consider a process whose current state is $b$. The paths resulting from all allowable decisions at $b$ are shown in Fig. 3-2(a). The optimal paths from $c, d$, and $e$ to the terminal point $f$ are shown in Fig. 3-2(b). The principle of

optimality implies that if $b-c$ is the initial segment of the optimal path from $b$ to $f$, then $c-f$ is the terminal segment of this optimal path. The same reasoning applied to initial segments $b-d$ and $b-e$ indicates that the paths in Fig. 3-2(c) are the only candidates for the optimal trajectory from $b$ to $f$. The optimal trajectory that starts at $b$ is found by comparing
\begin{aligned} & C_{b c f}^=J_{b c}+J_{c f}^ \ & C_{b d f}^=J_{b d}+J_{d f}^ \ & C_{b e f}^=J_{b e}+J_{e f}^ . \end{aligned}
The minimum of these costs must be the one associated with the optimal decision at point $b$.

Dynamic programming is a computational technique which extends the above decision-making concept to sequences of decisions which together define an optimal policy and trajectory. The optimal routing problem in the next section illustrates the procedure.

## 数学代写|最优化作业代写optimization theory代考|DYNAMIC PROGRAMMING APPLIED TO A ROUTING PROBLEM

A motorist wishes to know how to minimize the cost of reaching some destination $h$ from his current location. He can only travel (one-way as indicated) on the streets shown on his map (Fig. 3-3), and at the intersectionto-intersection costs given.

Instead of trying all allowable paths leading from each intersection to $h$ and selecting the one with lowest cost (an exhaustive search), consider the application of the principle of optimality. In this problem, “state” refers to the intersection and a “decision” is the choice of heading (control) elected by the driver when he leaves an intersection.

Suppose the motorist is at $c$; from there he can go, only to $d$ or $f$, and then on to $h$. Let $J_{c d}$ denote the cost of moving from $c$ to $d$ and $J_{c f}$ the cost from $c$ to $f$. Assume that the motorist already knows the minimum costs, $J_{d h}^$ and $J_{f h}^$, to reach the final destination $h$ from $d$ and $f$. (In this example, $J_{d h}^=10$ and $J_{f h}^=5$.) Then the minimum cost $J_{c h}^$ to reach $h$ from $c$ is the smaller of $$C_{c d h}^=J_{c d}+J_{d h}^=\text { minimum cost to reach } h \text { from } c \text { via } d$$ and $$C_{c f h}^=J_{c f}+J_{f h}^=\text { minimum cost to reach } h \text { from } c \text { via } f .$$ Thus, \begin{aligned} J_{c h}^ & =\min \left{C_{c d h}^, C_{c f h}^\right} \ & =\min {15,8} \ & =8 \end{aligned}
and the optimal decision at $c$ is to go to $f$.

## 数学代写|最优化作业代写optimization theory代考APPLICATION OF THE PRINCIPLE OF OPTIMALITY TO DECISION-MAKING

\begin{aligned} & C_{b c f}^=J_{b c}+J_{c f}^ \ & C_{b d f}^=J_{b d}+J_{d f}^ \ & C_{b e f}^=J_{b e}+J_{e f}^ . \end{aligned}

## 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。