## 单纯形方法和它在线性规划问题中的应用

We first illustrate how the simplex method works on a specific example:
maximize $5 x_{1}+4 x_{2}+3 x_{3}$
subject to $2 x_{1}+3 x_{2}+x_{3} \leq 5$
$4 x_{1}+x_{2}+2 x_{3} \leq 11$
$3 x_{1}+4 x_{2}+2 x_{3} \leq 8$
$x_{1}, x_{2}, x_{3} \geq 0 .$
We start by adding so-called slack variables. For each of the less-than inequalities in (2.1) we introduce a new variable that represents the difference between the right-hand side and the left-hand side. For example, for the first inequality,
$$2 x_{1}+3 x_{2}+x_{3} \leq 5$$
we introduce the slack variable $w_{1}$ defined by
$$w_{1}=5-2 x_{1}-3 x_{2}-x_{3} .$$
It is clear then that this definition of $w_{1}$, together with the stipulation that $w_{1}$ be nonnegative, is equivalent to the original constraint. We carry out this procedure for each of the less-than constraints to get an equivalent representation of the problem:
\begin{aligned} \operatorname{maximize} & \zeta=5 x_{1}+4 x_{2}+3 x_{3} \ \text { subject to } w_{1}=5-2 x_{1}-3 x_{2}-x_{3} \ w_{2}=11-4 x_{1}-x_{2}-2 x_{3} \ w_{3}=8-3 x_{1}-4 x_{2}-2 x_{3} \ x_{1}, x_{2}, x_{3}, w_{1}, w_{2}, w_{3} \geq 0 \end{aligned}

$4 x_{1}+x_{2}+2 x_{3} \leq 11$
$3 x_{1}+4 x_{2}+2 x_{3} \leq 8$
$x_{1}, x_{2}, x_{3} \geq 0 .$

$$2 x_{1}+3 x_{2}+x_{3} \leq 5$$

$$w_{1}=5-2 x_{1}-3 x_{2}-x_{3} 。$$

\begin{aligned} \operatorname{maximize} & \zeta=5 x_{1}+4 x_{2}+3 x_{3} \ \text { subject to } w_{1}=5-2 x_{1}-3 x_{2}-x_{3} \ w_{2}=11-4 x_{1}-x_{2}-2 x_{3} \ w_{3}=8-3 x_{1}-4 x_{2}-2 x_{3} \ x_{1}, x_{2}, x_{3}, w_{1}, w_{2}, w_{3} \geq 0 \end{aligned}

## 运筹学的三个特点代写

• 优化——运筹学的目的是在给定的条件下达到某一机器或者模型的最佳性能。优化还涉及比较不同选项和缩小潜在最佳选项的范围。
• 模拟—— 这涉及构建模型，以便在应用解决方案刀具体的复杂大规模问题之前之前尝试和测试简单模型的解决方案。
• 概率和统计——这包括使用数学算法和数据挖掘来发现有用的信息和潜在的风险，做出有效的预测并测试可能的解决方法。

BS equation代写

Categories: 数学代写