The Network Flow Identity (NFI) is a macroscopic equation that balances physical road constraints with user routing behavior. The fundamental identity states: Total Demand = Total Capacity × Efficiency × Max Congestion.

The Merit of the Proposed Method

Traditional transport models require exhaustive, highly detailed Origin-Destination (O-D) matrices and complex microsimulations. The NFI framework acts as a high-speed, strategic "sanity check." By utilizing continuous convex geometry and linear programming, NFI Sign Analysis allows urban planners to instantly screen political or engineering proposals to see if they are mathematically possible before spending time and money on detailed simulations.

1. NFI States

The core identity defines the physical relationship between the network variables: Total Demand ($\kappa$) = Total Capacity ($c$) × Efficiency ($\varpi$) × Max Congestion ($\xi$). Leave one input blank in the State Calculator to instantly deduce the missing network metric.

2. NFI Ratios (Policy Changes)

To analyze trade-offs between two scenarios, we use the NFI Ratio formula: $r_\kappa = r_c \cdot r_\varpi \cdot r_\xi$. This is incredibly useful because you do not need to know the actual demand or capacity of the city—you only need to know the percentage change you want to enforce. Leave one field blank to find the required percentage shift.

3. Mixed States & Ratios

This calculator bridges the gap between current reality and future planning. It takes your Current States ($X_0$) and proposed Policy Changes ($\Delta\%$), calculates the NFI ratios, and projects the Future States ($X_1$). Most importantly, it verifies strict physical limits—such as ensuring a policy doesn't require an impossible efficiency greater than 100%.

4. Sign Analysis (Feasibility Screening)

A strategic policy screening tool. Before calculating specific numbers, Sign Analysis uses the direction of change (+, -, 0) to flag structurally impossible policies. For example, increasing demand (+), decreasing capacity (-), and decreasing congestion (-) without changing efficiency (0) is flagged as an INFEASIBLE pipe dream, saving simulation resources.

How to Use the Sign Analysis Calculator

Define your policy goal by setting the desired direction of change for the four NFI variables:

• Demand (+, 0, -): The total volume of traffic entering the network.
• Capacity (+, 0, -): The total physical road space (e.g., adding or removing lanes).
• Efficiency (+, 0, -): How evenly traffic is distributed. An increase (+) requires changing driver habits to utilize empty alternative routes.
• Max Congestion (+, 0, -): The severity of the worst bottleneck in the system.

The tool calculates the geometric boundaries of these combinations. If it returns INFEASIBLE, your policy goals contradict the laws of network physics and cannot be achieved simultaneously.

Applications & Examples

• Strategic Road Diets (The "Political Magic" Scenario): A city wants to remove a lane of traffic (Capacity -) and accommodate 10% more daily trips (Demand +), while promising voters less severe rush-hour gridlock (Max Congestion -). Testing the Pareto Apex Tuple (+, -, +, -) proves this is structurally feasible only if the city can drastically improve routing efficiency (+), such as by implementing aggressive congestion pricing or adaptive signals.
• Debunking Flawed Proposals: A council member proposes accommodating a demand increase (Demand +) by executing a road diet (Capacity -) while keeping congestion from worsening (Max Congestion -), but assumes driver behavior will remain exactly the same (Efficiency 0). Checking the tuple (+, -, 0, -) reveals this policy is mathematically impossible.