317. 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.

317.1. The key formula: route length

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

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

For VRP: each vehicle visits customers (where is capacity in number of customers). Number of routes . Each route length:

where is depot-to-zone distance and is the area of the zone the route covers.

317.2. Why useful

317.3. Square-root facility-location model

For a region with demand density and warehouse fixed cost , the optimal number of facilities 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 .

317.4. Other Daganzo formulas

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

317.5. Limitations

317.6. Where it shows up

317.7. See also