Posted on Categories:Operations Research, 数学代写, 运筹学

# 数学代写|运筹学代写Operations Research代考|THE UPPER BOUND TECHNIQUE

avatest™

## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

•最快12小时交付

•200+ 英语母语导师

•70分以下全额退款

## 数学代写|运筹学代写Operations Research代考|THE UPPER BOUND TECHNIQUE

It is fairly common in linear programming problems for some of or all the individual $x_j$ variables to have upper bound constraints
$$x_j \leq u_j$$
where $u_j$ is a positive constant representing the maximum feasible value of $x_j$. We pointed out in Sec. 4.8 that the most important determinant of computation time for the simplex method is the number of functional constraints, whereas the number of nonnegativity constraints is relatively unimportant. Therefore, having a large number of upper bound constraints among the functional constraints greatly increases the computational effort required.

The upper bound technique avoids this increased effort by removing the upper bound constraints from the functional constraints and treating them separately, essentially like nonnegativity constraints. Removing the upper bound constraints in this way causes no problems as long as none of the variables gets increased over its upper bound. The only time the simplex method increases some of the variables is when the entering basic variable is increased to obtain a new BF solution. Therefore, the upper bound technique simply applies the simplex method in the usual way to the remainder of the problem (i.e., without the upper bound constraints) but with the one additional restriction that each new BF solution must satisfy the upper bound constraints in addition to the usual lower bound (nonnegativity) constraints.

To implement this idea, note that a decision variable $x_j$ with an upper bound constraint $x_j \leq u_j$ can always be replaced by
$$x_j=u_j-y_j$$
where $y_j$ would then be the decision variable. In other words, you have a choice between letting the decision variable be the amount above zero $\left(x_j\right)$ or the amount below $u_j\left(y_j=u_j-x_j\right)$. (We shall refer to $x_j$ and $y_j$ as complementary decision variables.) Because
$$0 \leq x_j \leq u_j$$
it also follows that
$$0 \leq y_j \leq u_j .$$
Thus, at any point during the simplex method, you can either
Use $x_j$, where $0 \leq x_j \leq u_j$,
or 2. Replace $x_j$ by $u_j-y_j$, where $0 \leq y_j \leq u_j$.

## 数学代写|运筹学代写Operations Research代考|AN INTERIOR-POINT ALGORITHM

In Sec. 4.9 we discussed a dramatic development in linear programming that occurred in 1984, the invention by Narendra Karmarkar of AT\&T Bell Laboratories of a powerful algorithm for solving huge linear programming problems with an approach very different from the simplex method. We now introduce the nature of Karmarkar’s approach by describing a relatively elementary variant (the “affine” or “affine-scaling” variant) of his algorithm. ${ }^1$ (Your OR Courseware also includes this variant under the title, Solve Automatically by the Interior-Point Algorithm.)

Throughout this section we shall focus on Karmarkar’s main ideas on an intuitive level while avoiding mathematical details. In particular, we shall bypass certain details that are needed for the full implementation of the algorithm (e.g., how to find an initial feasible trial solution) but are not central to a basic conceptual understanding. The ideas to be described can be summarized as follows:

Concept 1: Shoot through the interior of the feasible region toward an optimal solution. Concept 2: Move in a direction that improves the objective function value at the fastest possible rate.
Concept 3: Transform the feasible region to place the current trial solution near its center, thereby enabling a large improvement when concept 2 is implemented.
To illustrate these ideas throughout the section, we shall use the following example:
Maximize $Z=x_1+2 x_2$,
subject to
$$x_1+x_2 \leq 8$$
and
$$x_1 \geq 0, \quad x_2 \geq 0 .$$
This problem is depicted graphically in Fig. 7.3, where the optimal solution is seen to be $\left(x_1, x_2\right)=(0,8)$ with $Z=16$.

## 数学代写|运筹学代写Operations Research代考|THE UPPER BOUND TECHNIQUE

$$x_j \leq u_j$$

## 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。