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数学代写|运筹学代写Operations Research代考|OPR561 Another Formulation for Subtour Elimination

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数学代写|运筹学代写Operations Research代考|Another Formulation for Subtour Elimination

Let us consider another type of subtour elimination constraint of the form:
\begin{aligned} U_i-U_j+n X_{i j} \leq n-1 & & \text { for } i=1,2, \ldots, n-1 \text { and } j=2,3, \ldots, n \ U_j \geq 0 & & \text { (Bellmore and Nemhauser, 1968) } \end{aligned}

For our 5 -city example we will have $(n-1)^2$ constraints. Let us consider an infeasible solution having a subtour (not involving City 1) $X_{45}=X_{54}=1$. The two relevant constraints are:
and
\begin{aligned} & U_4-U_5+5 X_{45} \leq 4 \ & U_5-U_4+5 X_{54} \leq 4 \end{aligned}
This is clearly infeasible because the two constraints when added together gives
$$5 X_{45}+5 X_{54} \leq 8$$
which is infeasible for $X_{45}=X_{54}=1$. Therefore, every subtour not involving City 1 will violate the relevant set of constraints.
Let us consider a subtour involving City 1 given by
$$X_{12}=X_{23}=X_{31}=1$$
We have two constraints:
\begin{aligned} & U_1-U_2+5 X_{12} \leq 4 \ & U_2-U_3+5 X_{23} \leq 4 \end{aligned}
The constraint $U_3-U_1+5 X_{31} \leq 4$ does not exist for $U_j=1$.
Adding the two constraints, we get
$$U_1-U_3+10 \leq 8 \quad \text { for } X_{12}=X_{23}=1$$
It is possible to have values $U_1$ and $U_3$ that satisfy the constraints and, therefore, the constraint $U_i-U_j+n X_{i j} \leq n-1$ is unable to prevent subtours involving City 1 from occurring. However, we realize that for every subtour involving City 1 there has to be a subtour that does not involve City 1 (the subtour $X_{45}=X_{54}=1$ in our example) and the constraints are able to prevent them from happenning. Therefore, the constraints eliminate all subtours. The only requirement is that we define $d_{i j}=\infty$ (or $M$ ) so that singleton subtours are indirectly eliminated.

数学代写|运筹学代写Operations Research代考|The TSP and the Theory of NP-Completeness

Any algorithm that solves a problem carries out various steps that can be reduced to additions, subtraction, multiplication and division and other basic operations. Assuming that each of these operations take unit processing time, it is possible to represent the time taken to implement an algorithm in terms parameters such as problem, size, etc. This function can be a polynomial function or an exponential function. If it is a polynomial function, we say that the algorithm has a complexity of $O\left(N^k\right)$ where $N$ is a problem parameter (say, size) and $k$ is the order of the polynomial. The order is the power corresponding to the highest degree in the polynomial and is used because the rate of increase of the polynomial depends on the order of the polynomial. If the function is exponential, we say that the algorithm is exponential. Examples of exponential functions could be $n !, e^n$, etc.

An algorithm is polynomial if the order of complexity is of the form $O\left(N^k\right)$ and the problem is in the category of “easy” problems. Examples of easy problems are matrix inversion, solving linear equations, assignment problem, etc.

A decision problem is one that has a YES/NO answer. NP is a class of problems with the property that for any instance for which the answer is YES, there is a polynomial proof of YES. If two problems are in NP and if an instance of one can be converted in polynomial time to an instance of another, the problems are reducible. $P$ is a class of problems in NP where there exists a polynomial algorithm. $P$ is polynomially reducible to $Q$. NP complete is a subset of $P$, where the problems are reducible to each other. An optimization problem where the decision problem lies in NP complete is called NP hard. The TSP is an important problem in the class of NP complete problems.

A given problem is NP complete if it can be transformed into zero-one integer programming problem in polynomial time and if zero-one integer programming problem can be transformed to it in polynomial time (Bertsimas and Tsitsiklis, 1997). To show that a given problem is NP complete, it is customary to reduce it to a known NP complete problem. There are several instances where a given problem has been shown to be NP complete by showing that it is reducible to the TSP. (For further reading on the theory of NP-completeness, the reader is referred to Garey and Johnson, 1979, Papadimitriou and Steiglitz, 1982 and Wilf, 1975).

数学代写|运筹学代写Operations Research代考|Another Formulation for Subtour Elimination

\Begin{aligned} U_i-U_j+n X_{i j} \leq n-1 & & \text { for } i=1,2, \ldots, n-1 \text { and } j=2,3, \ldots, n \j U_j \geq 0 & & \text { （Bellmore and Nemhauser, 1968）}。 \end{aligned}

\begin{aligned} & u_4-u_5+5 x_{45} \leq 4\leq & u_5-u_4+5 x_{54} \x_{54}。 \end{aligned}

$$5 x_{45}+5 x_{54}。\leq 8$$

$$X_{12}=X_{23}=X_{31}=1$$

begin{aligned} & u_1-u_2+5 x_{12} \leq 4\leq & u_2-u_3+5 x_{23} \u_2-u_3+5 x_{23} leq 4 \end{aligned} 约束$U_3-U_1+5 X_{31} \leq 4$ 对于$U_j=1$不存在。 将这两个约束相加，我们得到 $$U_1-U_3+10leq 8 quad`text { for }. x_{12}=x_{23}=1$$

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。