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|Conversion of Primal to Dual or Dual to Primal

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|Conversion of Primal to Dual or Dual to Primal

The following steps (rules) will enable the conversion of any primal problem into its dual and vice-versa.

Step 1: (a) If in the given problem, objective function is in maximisation form proceed to step 2.
(b) Convert the objective function of the given problem to maximisation form if it is minimisation form. This conversion can be done by multiplying with -1
Ex : Min. $Z=x_1+9 x_2+x_3$ converts to Max. $Z=-2 x_1-9 x_2-x_3$

Step 2 : (a) If all constraints have “less than or equal to” (S) sign, go to step – 3
(b) If a constraint has “greater than or equal to” $(\geq)$ sign, convert it to “less than or equal to “
Ex: (i) $x_1+4 x_2+2 x_3 \geq 5$ is rewritten as $-x-4 x_2-2 x_3 \leq-5$
(ii) $3 x_1+x_2-2 x_3 \geq 4$ is rewritten as $-3 x_1-x_2+2 x_3 \leq-4$
(c) If a constraint has an “equality” sign (=), split this into two constraints in two opposite inequalities.
Ex : $2 x_1+3 x_2=5$ is split into $2 x_1+3 x_2 \leq 5$ and $2 x_1+3 x_2 \geq 5$
Then these are re-written using step – 2 (b)
$$\begin{array}{r} 2 x_1+3 x_2 \leq 5 \ \text { and }-2 x_1-3 x_2 \leq-5 \end{array}$$
Step 3: (a) If all the variables are non-negative i.e., $x_j \geq 0$, then proceed to step 4
(b) If any variable is unrestricted replace it by difference of two non-negative variables (These variables are to be replaced in the entire problem)
Ex: If $x_j$ is unrestricted, change it as $x_j^{\prime} \geq 0, x_j^{\prime \prime} \geq 0$ where
$$x_j=x_j^{\prime}-x_j^{\prime \prime}$$

## 数学代写|运筹学代写Operations Research代考|Advantages \& Applications of Duality

Duality will be more advantageous in the following cases.

Sometimes dual problem solution may be easier than primal solution particularly, when number of decision variables is considerably less than slack/surplus variables.

In the areas like economics, it is highly helpful in obtaining future decision in the activities being programmed.

In physics, it is used in parallel circuit \& series circuit theory.

In game theory, dual is employed by Column player who wishes to minimise his maximum loss while his opponent while Row player applies primal to maximise his minimum gains. However, if one problem is solved, the solution for other also can be obtained from the simplex tableau (Refer game theory : LPP method)

When a problem does not yield any solution in primal, it can be verified with dual.

Economic interpretations can be made and shadow prices can be determined enabling the managers to take further decisions.

The shadow price of a resource is the unit price that is equal to increase in profit to be realised by one additional unit of the resource. (or)

It is the change in the optimum value of the objective function per unit increase of the resource.

The shadow price can be determined from final simplex tableau using the values of $Z_j-C_j$, in primal problem.
In the illustration – 21 of this chapter, we conclude that

1. For Primal : A unit increase of resource covered by first constraint raises the profit of $x_1$ by 15 units and a unit increase in resource covered by second constraint raises profit of $x_2$ by $5 / 4$ units. Thus shadow prices of $x_1$ and $x_2$ are 15 and $5 / 4$ units separately.
2. For Dual : A unit slash in requirement of first constraint reduces the cost of $w_1$ by $5 / 16$ units while a unit slash on second constraint requirements reduces cost of $w_2$ by $75 / 8$ units. Thus shadow prices of $w_1$ and $w_2$ are $15 / 16$ and $75 / 8$ respectively.

## 数学代写|运筹学代写Operations Research代考|Conversion of Primal to Dual or Dual to Primal

(b)如果给定问题的目标函数是最小化形式，则将其转换为最大化形式。这种转换可以通过乘以-1来完成

(b)如果约束具有“大于或等于”$(\geq)$符号，将其转换为“小于或等于”

(ii) $3 x_1+x_2-2 x_3 \geq 4$被改写为$-3 x_1-x_2+2 x_3 \leq-4$
(c)如果约束有“相等”符号(=)，将其分成两个相反不等式中的两个约束。

$$\begin{array}{r} 2 x_1+3 x_2 \leq 5 \ \text { and }-2 x_1-3 x_2 \leq-5 \end{array}$$

(b)如果任意变量是不受限制的，将其替换为两个非负变量的差(这些变量在整个问题中都要替换)

$$x_j=x_j^{\prime}-x_j^{\prime \prime}$$

## 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。