Solving the minimum cost flow problem plays a crucial role in the design and architecture of various real-world systems. Therefore, it is essential to have fast and efficient solution methods that guarantee an optimal result. In this work, we first define the fundamental concepts and definitions that outline the characteristics of the problem. We then present in detail the cycle-canceling method and the successive shortest path method. Both methods, along with the Simplex method, are tested on randomly generated graphs, and the results are compared. Finally, we apply the methods to road network data from Piran and Ljubljana to demonstrate their effectiveness in solving real-world problems.
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