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数学代写|运筹学代写Operations Research代考|MATH3202 Shortest Path in a Manhattan Network

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数学代写|运筹学代写Operations Research代考|Shortest Path in a Manhattan Network

An efficient algorithm for the shortest-path problem in Figure $5.2$ is the recursive approach of dynamic programming. The basic principle of this approach is to divide the original problem into a series of related and easily solvable subproblems. The main observation of the recursive approach is that a shortest path from the starting point $A=(0,0)$ to the endpoint $B$ would be easy to calculate if a shortest path from each of the points $(1,0)$ and $(0,1)$ to $B$ were known. In general, one can observe that a shortest path from point $(x, y)$ to the endpoint $B$ could easily be calculated if the shortest path to $B$ from each of the points $(x+1, y)$ and $(x, y+1)$ were known. The original problem can therefore be divided into a series of nested subproblems of decreasing size. The smallest subproblem is the problem that calculates the shortest path to the endpoint $B$ from each of the points $(n-1, m)$ and $(n, m-1)$. The solution to this subproblem is trivial. To concretize the ideas, we define
$f(x, y)=$ minimum travel distance from $(x, y)$ to the endpoint $B$.
This function is called the value function and is crucial in dynamic programming. Note that this function is defined for every point $(x, y)$ even though the goal is to find $f(0,0)$. However, by defining $f(x, y)$ for every point $(x, y)$, it is possible to create a recursive algorithm for $f(x, y)$ that will eventually lead to the desired value $f(0,0)$ for the starting point $A=(0,0)$. The data of the problem are
\begin{aligned} & R(x, y)=\text { travel distance from point }(x, y) \text { to point }(x+1, y) \ & U(x, y)=\text { travel distance from point }(x, y) \text { to point }(x, y+1) . \end{aligned}
The algorithm is initiated with
$$f(n-1, m)=R(n-1, m) \quad \text { and } \quad f(n, m-1)=U(n, m-1) .$$

数学代写|运筹学代写Operations Research代考|Flexibility of Dynamic Programming

In this section, we first describe the general structure of dynamic programming problems. Then, we show how flexible the dynamic programming approach is by considering the Manhattan network for two other optimization criteria. First, we consider the determination of the safest path from $A$ to $B$ when risks are associated with passing through edges in the network. Then, we consider the determination of the path from $A$ to $B$ for which the greatest distance covered in one step is the least.

Every dynamic programming problem consists of several key components. The problem can be divided into stages $n$, with a decision required at each stage. Stages are also called decision epochs: the moments at which a decision must be made. Each stage has a number of states associated with it. The state space $S_n$ is the set of possible states $i$ which can occur at stage $n$. The state contains all the information that is needed to make an optimal decision. Decisions are also called actions. The decision space $D_n(i)$ is the set of decisions $d$ which are feasible in state $i$ at stage $n$. As a consequence of a decision, two things happen: the decision maker receives an immediate reward, and there is a transition to another state in the next stage. We define $r_n(i, d)$ as the immediate reward during stage $n$ as a consequence of decision $d$ in state $i$. Naturally, these are rewards in a maximization setting and costs in a minimization setting. Next to the immediate reward, the decision $d$ in state $i$ at stage $n$ causes a transition to state $j$ in stage $n+1$. In deterministic dynamic programming problems, which we are considering right now, the decision chosen at any stage fully characterizes how the state at the current stage is transformed into the state at the next stage. The fact that a decision causes an immediate reward as well as a transition to another state is at the heart of optimization in dynamic programming problems: a decision is optimal if it achieves the maximum value of the sum of the immediate reward and the rewards that can be earned from the next stage onward.

数学代写|运筹学代写Operations Research代考|Shortest Path in a Manhattan Network

$f(x, y)=$ 最小行驶距离 $(x, y)$ 到終点 $B$.

$R(x, y)=$ travel distance from point $(x, y)$ to point $(x+1, y) \quad U(x, y)=$ travel distance from point $(x, y)$ to point $(x, y+1)$.

$$f(n-1, m)=R(n-1, m) \quad \text { and } \quad f(n, m-1)=U(n, m-1) .$$

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。