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# 数学代写|The Primal Simplex Method 运筹学代考

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## 运筹学代写

In the last section we showed how it is possible to transform from one basic feasible solution to another (or determine that the solution set is unbounded) by arbitrarily selecting an incoming column. The idea of the simplex method is to select the column so that the resulting new basic feasible solution will yield a lower value to the objective function than the previous one. This then provides the final link in the simplex procedure. By an elementary calculation, which is derived below, it is possible to determine which nonbasic column should enter the basis so that the objective value is reduced, and by another simple calculation, derived in the previous section, it is possible to then determine which current basic column should leave in order to maintain feasibility.
Determining an Optimal Feasible Solution
As usual, let us assume that $\mathbf{B}$ consists of the first $m$ columns of $\mathbf{A}$. Then by partitioning $\mathbf{A}, \mathbf{x}$, and $\mathbf{c}^{T}$ as
$$\begin{gathered} \mathbf{A}=[\mathbf{B}, \mathbf{D}] \ \mathbf{x}{=}\left(\mathbf{x}{\mathbf{B}} ; \mathbf{x}{\mathbf{D}}\right), \quad \mathbf{c}^{T}=\left[\mathbf{c}{\mathbf{B}}^{T}, \mathbf{c}{\mathbf{D}}^{T}\right] . \end{gathered}$$ 82 4 The Simplex Method Suppose we have a basic feasible solution $$\mathbf{x}{\mathbf{B}}=\overline{\mathbf{a}}{0}:=\mathbf{B}^{-1} \mathbf{b} \geq \mathbf{0} \text { and } \mathbf{x}{\mathbf{D}}=\mathbf{0} .$$
The value of the objective function corresponding to any solution $\mathbf{x}$ is
$$z=c_{1} x_{1}+c_{2} x_{2}+\cdots+c_{n} x_{n}=\mathbf{c}{\mathbf{B}}^{T} \mathbf{x}{\mathbf{B}}+\mathbf{c}{\mathbf{D}}^{T} \mathbf{x}{\mathbf{D}}$$
and hence for the current basic solution, the corresponding value is
$$z_{0}=\mathbf{c}{\mathbf{B}}^{T} \mathbf{B}^{-1} \mathbf{b},$$ where $\mathbf{c}{\mathbf{B}}^{T}=\left(c_{1}, c_{2}, \ldots, c_{m}\right)$ and $\mathbf{c}{\mathbf{D}}^{T}=\left(c{m+1}, c_{m+2}, \ldots, c_{n}\right)$. However, for any value of $\mathbf{x}{\mathbf{D}}$ the necessary value of $\mathbf{x}{\mathbf{B}}$ is determined from $m$ equality constraints of the linear program, that is, from $\mathbf{A x}=\mathbf{b}$
and this general expression when substituted in the cost function \begin{aligned} z &=\mathbf{c}{\mathbf{B}}^{T}\left(\mathbf{B}^{-1} \mathbf{b}-\mathbf{B}^{-1} \mathbf{D} \mathbf{x}{\mathbf{D}}\right)+\mathbf{c}{\mathbf{D}}^{T} \mathbf{x}{\mathbf{D}} \ &=\mathbf{c}{\mathbf{B}}^{T} \mathbf{B}^{-1} \mathbf{b}+\left(\mathbf{c}{\mathbf{D}}^{T}-\mathbf{c}{\mathbf{B}}^{T} \mathbf{B}^{-1} \mathbf{D}\right) \mathbf{x}{\mathbf{D}} \ &=z_{0}+\left(\mathbf{c}{\mathbf{D}}^{T}-\mathbf{y}^{T} \mathbf{D}\right) \mathbf{x}{\mathbf{D}} \end{aligned}
which expresses the cost of any feasible solution to (4.1) in terms of independent variable in $\mathbf{x}{\mathbf{D}}$. Here, $\mathbf{y}^{T}=\mathbf{c}{\mathbf{B}}^{T} \mathbf{B}^{-1}$ is the simplex multipliers or shadow prices correspond Let

$$\开始{聚集} \mathbf{A}=[\mathbf{B}, \mathbf{D}] \ \mathbf{x}{=}\left(\mathbf{x}{\mathbf{B}} ; \mathbf{x}{\mathbf{D}}\right), \quad \mathbf{c} ^{T}=\left[\mathbf{c}{\mathbf{B}}^{T}, \mathbf{c}{\mathbf{D}}^{T}\right] 。 \结束{聚集}$$ 82 4 单纯形法 假设我们有一个基本可行的解决方案 $$\mathbf{x}{\mathbf{B}}=\overline{\mathbf{a}}{0}:=\mathbf{B}^{-1} \mathbf{b} \geq \mathbf{0 } \text { 和 } \mathbf{x}{\mathbf{D}}=\mathbf{0} 。$$

$$z=c_{1} x_{1}+c_{2} x_{2}+\cdots+c_{n} x_{n}=\mathbf{c}{\mathbf{B}}^{T} \ mathbf{x}{\mathbf{B}}+\mathbf{c}{\mathbf{D}}^{T} \mathbf{x}{\mathbf{D}}$$

$$z_{0}=\mathbf{c}{\mathbf{B}}^{T} \mathbf{B}^{-1} \mathbf{b},$$ 其中 $\mathbf{c}{\mathbf{B}}^{T}=\left(c_{1}, c_{2}, \ldots, c_{m}\right)$ 和 $\mathbf{c }{\mathbf{D}}^{T}=\left(c{m+1}, c_{m+2}, \ldots, c_{n}\right)$。然而，对于 $\mathbf{x}{\mathbf{D}}$ 的任何值，$\mathbf{x}{\mathbf{B}}$ 的必要值由线性规划，即从 $\mathbf{A x}=\mathbf{b}$

## 什么是运筹学代考

• 确定需要解决的问题。
• 围绕问题构建一个类似于现实世界和变量的模型。
• 使用模型得出问题的解决方案。
• 在模型上测试每个解决方案并分析其成功。
• 实施解决实际问题的方法。

## 运筹学代考的三个特点

• 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。