How to Maximize Flow in Networks: An Oil Pipeline Case Study

1 Maximum Flow Problem

1.1 Network Flow

Network flow optimization is a fundamental problem with applications in communications, transportation, power, engineering planning, task assignment, equipment renewal, and computer‑aided design.

Flow originates at a source, travels along paths, and reaches a sink, representing the movement of people, vehicles, goods, water, cash, or information.

Each edge has a capacity limit; the flow on an edge cannot exceed its capacity.

The abstract network‑flow model has three characteristics: (1) each directed edge has a capacity; (2) total flow out of the source equals total flow into the sink; (3) for every intermediate node, inflow equals outflow.

1.2 The Maximum Flow Problem

The maximum flow problem asks, given capacities on all edges, how to route the largest possible total flow from source to sink.

2 Case Study: Oil Pipeline Network

2.1 Problem Description

The oil pipeline connects wells, storage facilities, and intermediate pump stations. The capacities between stations are shown in the following matrix:

a  b  c  d  e  f  g  h  i  j
a: 0  6  8  3  0  3  0  0  0  0
b: 0  0  0  0  0  0  0  7 10  5
c: 0  0  0  4  0  4  0  0  3  0
d: 0  0  0  0  3  0  6  4  0  0
e: 0  7  0  0  0  0  0  0  3  4
f: 0  0  0  0  0  0  0  4  0  0
g: 0  0  0  0  0  7  0  0  0  0
h: 0  0  0  0  0  0  0  1  0  3
i: 0  0  0  0  0  0  0  0  0  3
j: 0  0  0  0  0  0  0  0  0  0

Using NetworkX to solve the model yields the diagram below:

3 Summary

This article illustrated the maximum‑flow model through an oil‑pipeline transportation example.

References

youcans Python小白的数学建模课 https://www.zhihu.com/column/c_1381900867424075776