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# 数学代写|The Simplex Method for Transportation Problems 运筹学代考

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## 运筹学代写

The transportation problem was stated briefly in Chap. 2. We restate it here. There are $m$ origins that contain various amounts of a commodity that must be shipped to $n$ destinations to meet demand requirements. Specifically, origin $i$ contains an amount $a_{i}$, and destination $j$ has a requirement of amount $b_{j}$. It is assumed that the system is balanced in the sense that total supply equals total demand. That is,
$$\sum_{i=1}^{m} a_{i}=\sum_{j=1}^{n} b_{j} .$$
The numbers $a_{i}$ and $b_{j}, i=1,2, \ldots, m ; j=1,2, \ldots, n$, are assumed to be nonnegative, and in many applications they are in fact nonnegative integers. There is a unit cost $c_{i j}$ associated with the shipping of the commodity from origin
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4 The Simplex Method $i$ to destination $j$. The problem is to find the shipping pattern between origins and destinations that satisfies all the requirements and minimizes the total shipping cost. In mathematical terms the above problem can be expressed as finding a set of $x_{i j}$ ‘ $\mathrm{s}, i=1,2, \ldots, m ; j=1,2, \ldots, n$, to
$$\begin{gathered} \text { minimize } \sum_{i=1}^{m} \sum_{j=1}^{n} c_{i j} x_{i j} \ \text { subject to } \sum_{j=1}^{n} x_{i j}=a_{i} \quad \text { for } i=1,2, \ldots, m \ \sum_{i=1}^{m} x_{i j}=b_{j} \quad \text { for } \quad j=1,2, \ldots, n \ x_{i j} \geqslant 0 \quad \text { for all } \quad i \text { and } j \end{gathered}$$
This mathematical problem, together with the assumption (4.28), is the general transportation problem. In the shipping context, the variables $x_{i j}$ represent the amounts of the commodity shipped from origin $i$ to destination $j$.
The structure of the problem can be seen more clearly by writing the constraint equations in standard form:
(4.30)
$x_{1 n}+x_{2 n} \quad+x_{m n}=b_{n}$
The structure is perhaps even more evident when the coefficient matrix $\mathbf{A}$ of the system of equations above is expressed in vector-matrix notation as
$$\mathbf{A}=\left[\begin{array}{cccc} \mathbf{1}^{T} & & & \ & \mathbf{1}^{T} & & \ & & \ddots & \ & & & \mathbf{1}^{T} \ \mathbf{I} & \mathbf{I} & \cdots & \mathbf{I} \end{array}\right]$$
4.5 The Simplex Method for Transportation Problems
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where $\mathbf{1}=(1,1, \ldots, 1)$ is $n$-dimensional, and where each $\mathbf{I}$ is an $n \times n$ identity matrix.

In practice it is usually unnecessary to write out the constraint equations of the transportation problem in the explicit form (4.30). A specific transportation problem is generally defined by simply presenting the data in compact form, such as:
$$\mathbf{a}=\left(a_{1}, a_{2}, \ldots, a_{m}\right)$$

$$\sum_{i=1}^{m} a_{i}=\sum_{j=1}^{n} b_{j} 。$$

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4 单纯形法$i$ 到目的地$j$。问题是找到满足所有要求并最小化总运输成本的起点和目的地之间的运输模式。用数学术语，上述问题可以表示为找到一组 $x_{i j}$ ‘ $\mathrm{s}, i=1,2, \ldots, m ; j=1,2, \ldots, n$, 到
$$\开始{聚集} \text { 最小化 } \sum_{i=1}^{m} \sum_{j=1}^{n} c_{i j} x_{i j} \ \text { 服从 } \sum_{j=1}^{n} x_{i j}=a_{i} \quad \text { for } i=1,2, \ldots, m \ \sum_{i=1}^{m} x_{i j}=b_{j} \quad \text { for } \quad j=1,2, \ldots, n \ x_{i j} \geqslant 0 \quad \text { 对于所有 } \quad i \text { 和 } j \结束{聚集}$$

(4.30)
$x_{1 n}+x_{2 n} \quad+x_{m n}=b_{n}$

$$\mathbf{A}=\left[\begin{数组}{cccc} \mathbf{1}^{T} & & & \ & \mathbf{1}^{T} & & \ & & \ddots & \ & & & \mathbf{1}^{T} \ \mathbf{I} & \mathbf{I} & \cdots & \mathbf{I} \end{数组}\right]$$
4.5 交通问题的单纯形法
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$$\mathbf{a}=\left(a_{1}, a_{2}, \ldots, a_{m}\right)$$

## 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。