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# 数学代写|运筹学代写Operations Research代考|ADAPTING TO OTHER MODEL FORMS

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## 数学代写|运筹学代写Operations Research代考|ADAPTING TO OTHER MODEL FORMS

Thus far we have presented the details of the simplex method under the assumptions that the problem is in our standard form (maximize $Z$ subject to functional constraints in $\leq$ form and nonnegativity constraints on all variables) and that $b_i \geq 0$ for all $i=1,2, \ldots, m$. In this section we point out how to make the adjustments required for other legitimate forms of the linear programming model. You will see that all these adjustments can be made during the initialization, so the rest of the simplex method can then be applied just as you have learned it already.

The only serious problem introduced by the other forms for functional constraints (the $=$ or $\geq$ forms, or having a negative right-hand side) lies in identifying an initial $B F$ solution. Before, this initial solution was found very conveniently by letting the slack variables be the initial basic variables, so that each one just equals the nonnegative right-hand side of its equation. Now, something else must be done. The standard approach that is used for all these cases is the artificial-variable technique. This technique constructs a more convenient artificial problem by introducing a dummy variable (called an artificial variable) into each constraint that needs one. This new variable is introduced just for the purpose of being the initial basic variable for that equation. The usual nonnegativity constraints are placed on these variables, and the objective function also is modified to impose an exorbitant penalty on their having values larger than zero. The iterations of the simplex method then automatically force the artificial variables to disappear (become zero), one at a time, until they are all gone, after which the real problem is solved.

To illustrate the artificial-variable technique, first we consider the case where the only nonstandard form in the problem is the presence of one or more equality constraints.
Equality Constraints
Any equality constraint
$$a_{i 1} x_1+a_{i 2} x_2+\cdots+a_{i n} x_n=b_i$$
actually is equivalent to a pair of inequality constraints:
\begin{aligned} & a_{i 1} x_1+a_{i 2} x_2+\cdots+a_{i n} x_n \leq b_i \ & a_{i 1} x_1+a_{i 2} x_2+\cdots+a_{i n} x_n \geq b_i . \end{aligned}

## 数学代写|运筹学代写Operations Research代考|Negative Right-Hand Sides

The technique mentioned in the preceding sentence for dealing with an equality constraint with a negative right-hand side (namely, multiply through both sides by -1 ) also works for any inequality constraint with a negative right-hand side. Multiplying through both sides of an inequality by -1 also reverses the direction of the inequality; i.e., $\leq$ changes to $\geq$ or vice versa. For example, doing this to the constraint
$$\left.x_1-x_2 \leq-1 \quad \text { (that is, } x_1 \leq x_2-1\right)$$
gives the equivalent constraint
$$-x_1+x_2 \geq 1 \quad \text { (that is, } x_2-1 \geq x_1 \text { ) }$$
but now the right-hand side is positive. Having nonnegative right-hand sides for all the functional constraints enables the simplex method to begin, because (after augmenting) these right-hand sides become the respective values of the initial basic variables, which must satisfy nonnegativity constraints.

## 数学代写|运筹学代写Operations Research代考|ADAPTING TO OTHER MODEL FORMS

$$a_{i 1} x_1+a_{i 2} x_2+\cdots+a_{i n} x_n=b_i$$

\begin{aligned} & a_{i 1} x_1+a_{i 2} x_2+\cdots+a_{i n} x_n \leq b_i \ & a_{i 1} x_1+a_{i 2} x_2+\cdots+a_{i n} x_n \geq b_i . \end{aligned}

## 数学代写|运筹学代写Operations Research代考|Negative Right-Hand Sides

$$\left.x_1-x_2 \leq-1 \quad \text { (that is, } x_1 \leq x_2-1\right)$$

$$-x_1+x_2 \geq 1 \quad \text { (that is, } x_2-1 \geq x_1 \text { ) }$$

## 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。