Mastering the Minimum Cost Flow Problem: Concepts and Solution Algorithms
This article explains the minimum cost flow problem—delivering a specified amount of flow from supply nodes to demand nodes at the lowest possible cost—covers its linear programming formulation, highlights key solution methods such as successive shortest path, cycle canceling, primal‑dual, network simplex, and outlines its primary applications in distribution network optimization.
1 Minimum Cost Flow Problem
In practice we often want to accomplish transportation tasks while minimizing the transportation cost. The minimum cost flow problem seeks to transport a certain amount of flow from source (supply) nodes to sink (demand) nodes at the lowest possible cost. The numbers of supply and demand nodes may be one or many, but each supply node’s capacity and each demand node’s requirement are fixed.
The minimum cost flow problem can be described by the following linear programming formulation:
2 Solution Algorithms
Many methods exist for solving the minimum cost flow problem, including the successive shortest path algorithm, the cycle‑canceling algorithm, the primal‑dual algorithm, the network simplex algorithm, and the out‑of‑kilter (non‑balanced network flow) method.
The most important application of the minimum cost flow problem is the optimization of distribution networks, such as determining how to ship goods from origins to intermediate hubs and then to customers. Transportation, assignment, transshipment, maximum flow, and shortest path problems are all special cases of the minimum cost flow problem.
3 Summary
This article introduced the concept of the minimum cost flow model.
References
youcans Python小白的数学建模课 https://www.zhihu.com/column/c_1381900867424075776
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