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# 数学代考| The Transportation Simplex Algorithm 运筹学代写

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## 运筹学代写

It is now possible to put together the components developed to this point in the form of a complete revised simplex procedure for the transportation problem. The steps are:
Step 1. Compute an initial basic feasible solution using the Northwest Corner Rule or some other method.
Step 2. Compute the simplex multipliers and the reduced cost coefficients. If all relative cost coefficients are nonnegative, stop; the solution is optimal. Otherwise, go to Step 3 .
Step 3. Select a nonbasic variable corresponding to a negative cost coefficient to enter the basis (usually the one corresponding to the most negative cost basic variable with a minus assigned to it. Update the solution. Go to Step $2 .$

Example 7 We can now completely solve the problem that was introduced in Example 5 of the first section. The requirements and a first basic feasible solution indicated on the array should be ignored at this point, since they cannot be computed until the next step is completed.
\begin{tabular}{|l|l|l|l|l|l|}
\hline 10 & 20 & & & & 30 \
\hline & 30 & $20^{-}$ & $30^{+}$ & & 80 \
\hline & & & $10^{0}$ & & 10 \
\hline & $+$ & $40^{-}$ & $20^{0}$ & 60 \
\hline 10 & 50 & 20 & 80 & 20 & \
\hline
\end{tabular}
The cost coefficients of the problem are shown in the array below, with the circled cells corresponding to the current basic variables. The simplex multipliers, computed by row and column scanning, are shown as well.
$$\begin{array}{|ccccc|c} \hline \text { (3) } & 4 & 6 & 8 & 9 & 5 \ 2 & (2 & 4 & (5) & 5 & 3 \ 2 & 2 & 2 & (3) & 2 & 1 \ 3 & 3 & 2 & 4 & (2) & 2 \ \hline-2 & -1 & 1 & 2 & 0 & \end{array}$$ The reduced cost coefficients are found by subtracting $u_{i}+v_{j}$ from $c_{i j}$. In this case Thus $\mathrm{a}+$ is entered into this cell in the original array, and the cycle of zeros and plus scanning once a complete cycle is determined.)
scanning once a complete cycle is determined.) $4.5$ The Simplex Method for Transportation Problems
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The smallest basic variable with a minus sign is 20 and, accordingly, 20 is added or subtracted from elements of the cycle as indicated by the signs. This leads to the new basic feasible solution shown in the array below:

The new simplex multipliers corresponding to the new basis are computed, and the cost array is revised as shown below. In this case all reduced cost coefficients are positive, indicating that the current solution is optimal.
$$\begin{array}{|ccccc|c} \hline \text { (3) } & 4 & 6 & 8 & 9 & 5 \ 2 & (2) & 4 & 5 & 5 & 3 \ 2 & 2 & 2 & (3) & 2 & 1 \ 3 & 3 & (2) & 4 & (2) & 2 \ \hline-2 & -1 & 0 & 2 & 0 & \ \hline \end{array}$$
As in all linear programming problems, degeneracy, corresponding to a basic variable having the value zero, can occur in the transportation problem. If degeneracy is encountered in the simplex procedure, it can be handled quite easily by introduction of the standard perturbation method (see Exercise 15, Chap. 4). In this method a zero-valued basic variable is assigned the value $\varepsilon$ and is then treated in the usual way. If it later leaves the basis, then the $\varepsilon$ can be dropped.
Example 8 To illustrate the method of dealing with degeneracy, consider a modification of Example 7, with the fourth row sum changed from 60 to 20 and the fourth column sum changed from 80 to 40 . Then the initial basic feasible solution found by the Northwest Corner Rule is degenerate. An $\varepsilon$ is placed in the array for the zero-valued basic variable as shown below:

\begin{表格}{|l|l|l|l|l|l|}
\hline 10 & 20 & & & & 30 \
\hline & 30 & $20^{-}$ & $30^{+}$ & & 80 \
\hline & & & $10^{0}$ & & 10 \
\hline & $+$ & $40^{-}$ & $20^{0}$ & 60 \
\hline 10 & 50 & 20 & 80 & 20 & \
\hline
\end{表格}

$$\begin{数组}{|ccccc|c} \hline \text { (3) } & 4 & 6 & 8 & 9 & 5 \ 2 & (2 & 4 & (5) & 5 & 3 \ 2 & 2 & 2 & (3) & 2 & 1 \ 3 & 3 & 2 & 4 & (2) & 2 \ \hline-2 & -1 & 1 & 2 & 0 & \结束{数组}$$ 通过从 $c_{i j}$ 中减去 $u_{i}+v_{j}$ 找到降低的成本系数。在这种情况下，因此 $\mathrm{a}+$ 被输入到原始数组中的这个单元格中，并且确定了一个完整的周期后零和加扫描的周期。）

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$$\begin{数组}{|ccccc|c} \hline \text { (3) } & 4 & 6 & 8 & 9 & 5 \ 2 & (2) & 4 & 5 & 5 & 3 \ 2 & 2 & 2 & (3) & 2 & 1 \ 3 & 3 & (2) & 4 & (2) & 2 \ \hline-2 & -1 & 0 & 2 & 0 & \ \hline \结束{数组}$$

## 什么是运筹学代写

• 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。

## 运筹学代写的三个特点

• 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。