320. Daganzo Continuous

A continuous-approximation approach to large-scale routing and logistics design (Daganzo, 1984+). Replaces discrete combinatorial optimization with closed-form formulas — sacrifices exactness for insight and fast strategic design.

320.1. The key formula: route length

For $𝑛$ customers uniformly distributed in a region of area $𝐴$, the expected length of an optimal TSP tour satisfies:

with $𝑘\approx 0.71$ (the Beardwood-Halton-Hammersley constant). For uniform Euclidean instances, this constant has been studied extensively.

For VRP: each vehicle visits $\approx 𝑄$ customers (where $𝑄$ is capacity in number of customers). Number of routes $=\frac{𝑛}{𝑄}$. Each route length:

where $𝑟$ is depot-to-zone distance and ${𝐴}_{\text{zone}}$ is the area of the zone the route covers.

320.2. Why useful

• Strategic design: how many vehicles? How big should zones be? Optimal depot location?
• Sensitivity analysis: $𝑑\frac{𝐿}{𝑑}𝑛$, etc.
• Fast benchmarking: compare network designs without solving thousands of VRPs

320.3. Square-root facility-location model

For a region with demand density $𝜌$ and warehouse fixed cost $𝑓$, the optimal number of facilities ${𝑁}^{\ast }$ minimizes:

Differentiating and solving:

— larger regions / denser demand / lower facility cost → more facilities. Square-root scaling: doubling demand doesn’t double facility count; it multiplies by $\sqrt{2}$.

320.4. Other Daganzo formulas

• Headway design in transit / dispatching: ${𝐻}^{\ast }=\sqrt{2\frac{{𝑐}_{𝑠}+{𝑐}_{ℎ}}{𝜆{𝑐}_{𝑤}}}$ for cost-of-waiting / dispatch trade-off
• Hub-spoke vs direct shipping: cost crossover conditions
• One-to-many vs many-to-many distribution

These give order-of-magnitude correct answers in seconds, useful when:

• The detailed instance isn’t known yet (strategic phase)
• Quick comparison of network designs is needed
• Sensitivity to a parameter matters more than exact cost

320.5. Limitations

• Uniform / smooth demand assumed — clustering breaks formulas
• Continuous approximation — discrete fleet sizes round
• No time windows / capacities in basic forms (extensions exist)
• Less useful for operational (day-to-day) routing than for strategic design

320.6. Where it shows up

• Logistics network design — placing DCs, sizing warehouses
• Public transit planning — bus / rail / paratransit
• Last-mile design — Amazon, UPS strategic network choices
• Reference for VRP heuristic benchmarking

320.7. See also

• VRP — operational counterpart
• Clarke-Wright — operational heuristic
• Location Pooling — strategic trade-off Daganzo formalizes