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“学越千山（四十）：位势法与闭回路寻找最优解”，
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Learning over a thousand mountains(40):Potential method and closed loop search for optimal solution，
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思维导图
Mind mapping
位势法
Potential Method
位势法是一种求解运输问题的简便方法，它基于“供需平衡”和“位势相等”两个基本原理。
The potential method is a convenient method for solving transportation problems, based on the two basic principles of "supply-demand balance" and "potential equality".
供需平衡：每个供应点的发出量等于其需求量，每个需求点的接收量等于其需求量。
Supply and demand balance: The output of each supply point is equal to its demand, and the reception of each demand point is equal to its demand.
位势相等：在运输问题的最优解中，从同一供应点出发的所有运输路径上的单位运价之差相等，称为该供应点的位势；同理，到达同一需求点的所有运输路径上的单位运价之差也相等，称为该需求点的位势。
Equal potential: In the optimal solution of a transportation problem, the difference in unit freight rates across all transportation paths starting from the same supply point is equal, which is called the potential of that supply point; Similarly, the difference in unit freight rates across all transportation paths that reach the same demand point is also equal, known as the potential of that demand point.
通过位势法，我们可以快速判断一个初始方案是否最优，并对其进行调整以达到最优解。
Through the potential method, we can quickly determine whether an initial solution is optimal and adjust it to achieve the optimal solution.
闭回路法
Closed Loop Method
当通过某种方法（如位势法）得到一个非最优解时，我们可以使用闭回路法来修改这个解，使其达到最优。
When a non optimal solution is obtained through a certain method, such as the potential method, we can use the Closed Loop Method to modify the solution to achieve optimal results.
闭回路是指从某个未饱和的供应点出发，依次经过若干条运输路径和节点，最后回到起点的闭合路径。在闭回路上，我们可以对运输量进行调整，以改善总运输费用。
A closed loop refers to a closed path that starts from an unsaturated supply point, passes through several transportation paths and nodes in sequence, and finally returns to the starting point. On a closed loop, we can adjust the transportation volume to improve the total transportation cost.
具体步骤如下：
1、找到一个未饱和(<0)的供应点或需求点作为起点。
2、沿着运输路径前进，直到遇到另一个未饱和的节点（供应点或需求点）。
3、形成一个闭回路，并计算闭回路上每条路径的运输费用变化。
4、选择一个能使总运输费用减少的调整方案，对闭回路上的运输量进行调整。
5、重复上述步骤，直到所有供应点和需求点都达到饱和状态，此时得到的就是最优解。
The specific steps are as follows:
1.Find an unsaturated supply or demand point as the starting point.
2.Advance along the transportation path until encountering another unsaturated node (supply or demand point).
3.Form a closed loop and calculate the transportation cost change for each path on the closed loop.
4.Choose an adjustment plan that can reduce the total transportation cost and adjust the transportation volume on the closed loop.
5.Repeat the above steps until all supply and demand points reach saturation, and the optimal solution is obtained.
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翻译：百度翻译
参考资料：百度百科，
《管理运筹学》
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