Why a single distance multiplier has to fail

Detour is not a fixed ratio. Short hauls average 2.62 times the straight line, long hauls 1.59, because the first-mile penalty is fixed and gets amortised over distance.

Logistics models routinely convert straight-line distance to road distance by multiplying by a constant. Pick the constant well and the estimate is decent on average. The problem is that the error is not spread evenly around that average. It is concentrated at short distances, where the multiplier is off by the largest margin, and short distances are where most hauls happen.

The problem

A single circuity factor treats detour as a fixed proportion of the trip. Double the straight-line distance and the road distance doubles too. That is the assumption embedded in every uniform multiplier, and it is worth stating explicitly, because it is doing more work than it looks like.

Across 435,848 measured truck routes, stratified into distance bands:

Circuity falls monotonically as distance rises. Short hauls average 2.62, long hauls 1.59, a 65% difference in the same direction across every band.

The instinct is that this is backwards. A longer trip has more opportunity to deviate, more terrain to cross, more chances to be forced off course. It should accumulate more detour, not less.

Why the ratio falls

Road networks are hierarchical. Local roads follow terrain because they were built cheaply along whatever contour was available. Highways cut through terrain because someone paid to make them straight.

A trip has three parts: getting out of wherever it starts, running along the trunk network, and getting into wherever it ends. The first and last parts are expensive in detour terms and roughly fixed in length, because they are determined by how far it is to the nearest highway on-ramp rather than by how far you are going. The middle part is nearly straight and scales with the trip.

So the total detour is close to a constant plus a proportional term. Expressed as a ratio, that constant gets divided by a growing denominator.

road ≈ (fixed first-mile and last-mile overhead) + (near-straight middle)

CF = road / straight  →  falls as straight grows

For a 15 km haul that never reaches a highway, the whole trip is the expensive part, and the ratio stays high. For a 120 km haul, the same fixed overhead is amortised across a long straight middle, and the ratio approaches the highway's own circuity, which is close to 1.

Travel speed confirms the mechanism independently. Short routes average 35.9 km/h and long routes 57.5 km/h. If short routes were merely unlucky in their geometry, speed would not track distance this cleanly. It tracks because the two measurements are reporting the same fact: short hauls stay on slow winding roads, long hauls do not.

What this breaks

Any uniform multiplier, in both directions at once. Fit a constant to the whole sample and it underestimates short hauls and overestimates long ones. There is no value that is right for both, so calibrating more carefully does not help. The functional form is wrong.

Zone-based factors, including the ones in the previous post. Cutting the state into five terrain zones and assigning each a mean circuity captures spatial variation and still applies one number per zone regardless of haul length. That is an improvement on a single statewide value and it inherits the same defect at a smaller scale. In deployment this shows up exactly where the mechanism predicts: transport cost error runs about 5.5% for procurement radii under 30 km and rises sharply beyond 40 km, because long hauls use highways that are far straighter than their zone average.

Cost models that scale linearly with straight-line distance. If cost per kilometre is fixed and distance is estimated by a constant multiplier, the model has compounded two errors that point the same way.

The variance, not just the mean. Standard deviation falls from 2.479 in the shortest band to 0.394 in the longest. Short-haul circuity is not only higher, it is far less predictable, because whether a specific origin happens to sit near a decent road dominates the outcome. Any confidence interval on a short-haul estimate should be much wider, and a single multiplier communicates no uncertainty at all.

What it means for siting

The practical reading inverts a common assumption.

A facility sited close to abundant biomass draws on short hauls at a circuity of roughly 2.6, travelling at 36 km/h. A facility sited a little further out but on a highway corridor draws on longer hauls at 1.6, travelling at 58 km/h. The second facility's trucks cover more ground per hour and waste less of their distance.

Proximity to biomass does not imply cheap transport. The relevant question is not how far the wood is but what network the trucks are on to get it, and those come apart in mountainous terrain more often than the intuition suggests.

Where this analysis is limited

The longest band is thin. 1,646 routes in the 100 to 150 km band against 203,727 in the 25 to 50 band. The trend across the first three bands is on solid ground. The fourth is directionally consistent and undersampled, because few California forest origins have many cluster destinations beyond 100 km, which is the same terrain fact the analysis is about.

Distance bands are a crude instrument. Circuity is being treated as a step function of distance when the mechanism implies something smooth and monotone. A continuous model, or a factor stratified by both zone and distance, would fit better. Neither has been built, and the distance-stratified lookup is the obvious next thing.

The bands conflate distance with terrain. Long hauls tend to originate in places with highway access, and highway access correlates with gentler terrain. Some of the 65% gap may be terrain composition rather than the amortisation mechanism. Separating the two needs the origin held fixed while distance varies, which the current sampling design does not isolate cleanly.

Speed and circuity are not independent evidence. Both derive from the same routing engine and the same truck profile. They agree, which is reassuring, and they are not two separate confirmations.

Method details at /docs/transport-method.

References

1. Ballou, R.H., Rahardja, H. and Sakai, N. (2002). Selected Country Circuity Factors for Road Travel Distance Estimation. Transportation Research Part A, 36, 843–848.

2. Levinson, D. and El-Geneidy, A. (2009). The Minimum Circuity Frontier and the Journey to Work. Regional Science and Urban Economics.

3. Contreras, M. and Chung, W. (2016). Analysis of Transportation Costs in Forest Biomass Supply Chains.

4. OpenStreetMap contributors. Open Source Routing Machine.