Examples of Network Flow Problem Examples

Introduction to Network Flow Problems

Network flow problems involve optimizing the flow of resources through a network. These problems are common in logistics, telecommunications, and transportation. By modeling the flow, we can determine the most efficient way to allocate resources or transport goods through a network of nodes and edges.

Example 1: Optimizing Delivery Routes

In a city, a delivery company needs to transport packages from a central warehouse to several locations. The goal is to minimize travel time and costs while ensuring all packages are delivered.

The company represents the city as a directed graph where:

• Nodes are locations (warehouse and delivery points).
• Edges are possible routes with associated costs (e.g., distance, traffic).

Context:
The company has the following delivery points: A, B, C, and D. The warehouse is at point W. Each route has a specific cost:

• W to A: $5
• W to B: $10
• W to C: $8
• W to D: $7

To solve the problem, the company can use an algorithm like the Ford-Fulkerson method to find the minimum cost flow that satisfies the delivery requirements, ensuring all packages reach their destinations while minimizing costs.

Notes:
Variations include considering time windows for deliveries or multiple warehouses.

Example 2: Water Distribution System

A municipal water supply system must distribute water from a treatment plant to various neighborhoods. The challenge is to ensure an adequate supply while minimizing waste.

In this model:

• Nodes represent the treatment plant and neighborhoods.
• Edges represent pipelines with capacities (maximum flow rates).

Context:
Consider the treatment plant (TP) supplying neighborhoods N1, N2, and N3. The capacities of the pipelines are as follows:

• TP to N1: 100 liters/hour
• TP to N2: 150 liters/hour
• TP to N3: 200 liters/hour

Using the maximum flow algorithm, the city can determine how to allocate water to each neighborhood without exceeding pipeline capacities, ensuring all areas receive their required water supply.

Notes:
This example can be modified to include different demand levels for each neighborhood or additional sources of water.

Example 3: Traffic Flow Management

City planners aim to optimize traffic flow at an intersection to reduce congestion and improve travel times. This scenario can be modeled as a network flow problem.

In this network:

• Nodes represent intersections.
• Edges represent roads with traffic capacities (maximum number of vehicles).

Context:
At a busy intersection, the following roads and their capacities exist:

• Road A to B: 200 vehicles/hour
• Road A to C: 150 vehicles/hour
• Road B to C: 100 vehicles/hour

City planners can apply the network flow algorithms to analyze traffic patterns and determine the optimal traffic light timings or road expansions needed to manage flow effectively, thereby improving overall traffic conditions.

Notes:
This can be expanded to include factors like pedestrian crossings, emergency vehicle access, or public transport routes.