Comprehensive Guide to Minimum Cost Flow and Network Optimization

Minimum Cost Flow Problem

The problem of finding the cheapest way to send flow from supply nodes to demand nodes while respecting capacities.

Steps to Solve Min Cost Flow

Identify node types → Set flow variables → Add balance constraints → Add capacity constraints → Minimize total cost.

Supply Node

A node with positive net flow; sends units.

Demand Node

A node with negative net flow; requires units.

Transshipment Node

A node with zero net flow; only passes flow through.

Negative Net Flow

Indicates a demand node.

Five Types of Network Problems

Shortest Path, Maximum Flow, Minimum Cost Flow, Assignment, Transportation.

When to Use Min Cost Flow

If you have supplies, demands, and shipping costs.

When to Use Max Flow

If the goal is to maximize flow from one source to a sink.

Transportation Problem

Sends goods from one source to many destinations at minimum cost.

Counting Supply Nodes

Count all nodes with positive net flow.

Counting Arcs in Network

Count each listed connection from one node to another.

Assumptions of Min Cost Flow

Known costs/capacities, linear costs, balanced network, flows follow arc direction.

Discontinuity

A break, gap, or jump in a graph.

Piecewise Linear Function

Graph with multiple straight segments connected with different slopes.

Decreasing Marginal Returns

Curve that increases at a decreasing rate; flattens as x increases.

Proportional Relationship

Straight line through the origin (y = mx).

Non-Proportional Relationship

Line not through origin or any curved relationship.

Truth About Nonlinear Models

Marginal effects change; outputs vary at different rates.

Marketing vs Sales Relationship

Nonlinear due to diminishing returns.

Example of Nonlinear Function

Overtime cost, marketing diminishing returns, quadratic or exponential functions.

False Statement About Marketing Graph

"Sales increase proportionally with spending."

Triple Bottom Line

People, Planet, Profit.

People + Profit = ?

Equitable.

Sustainability

Meeting today's needs without harming future generations.

Defensive CSR

Doing minimum required to avoid trouble.

Reactive CSR

Responding only when challenged.

Accommodative CSR

Going above minimum compliance when asked.

Proactive CSR

Actively leading in CSR initiatives.

Economic Dimension of TBL

Profitability, ROI, cost savings.

Environmental Dimension of TBL

Emissions, waste, resource use.

Social Dimension of TBL

Employee welfare, community impact, fairness.

Written Report Components

Title, Exec Summary, Intro, Methods, Results, Conclusion, Appendix.

Where Do Extra Tables Go?

Appendix.

Where Do Research Limitations Go?

End of methods section.

Oral Report Consideration

Use visuals that support key points and avoid text-heavy slides.

Bar/Column Graph

Used to compare categories.

Line Graph

Used for trends over time.

Pie Chart

Used to show parts of a whole.

False Visualization Statement

"Use as many colors as possible."

Research Participant Rights

Confidentiality, consent, quality research, safety, voluntary participation.

Good Report Title

Clear, concise, includes topic + purpose.

Support Materials for Written/Oral

Use charts, tables, visuals, appendices, slides, etc.