What the Count Leaves Out · Book XV

The algebra is exact. The weights are symbols: this chapter uses no emission, fuel or noise figure, and it does not say which weight is larger. That is the point: the comparison turns on a number the count cannot supply.

1 · The cost term

Keep the model of chapter 1: n cities, one hub, d passengers per ordered pair per day, s seats per flight, whole aircraft, at least one flight per direction per route per day. Let F be flights per day and S passenger-segments per day. Add a cost with two weights, in any one unit:

C = αF + βS α = cost of one flight, β = cost of one passenger flying one segment

α could stand for fuel burned in take-off and climb, or the noise of a movement; β for the extra weight carried or the extra distance flown by a passenger. The chapter does not assign either. It asks only how the comparison depends on ρ = β/α, flights per passenger-segment.

2 · The threshold

From chapter 1, S_hub − S_direct = 2(n−1)²d − n(n−1)d = (n−1)(n−2)d. The hub is cheaper when β·(n−1)(n−2)d < α·(F_direct − F_hub).

StatementIf F_direct > F_hub, the hub is cheaper exactly when ρ < ρ*(d) = (F_direct − F_hub) / ((n−1)(n−2)d). If F_direct ≤ F_hub, the hub is never cheaper for any ρ ≥ 0.

The script checks this against a direct cost comparison on 4,283 cases (every d from 1 to 150 at n = 50 and s = 150, with exact rational ρ on both sides of the threshold) and finds no disagreement.

3 · Numbers for 50 cities and 150 seats

d	F direct	F hub	S direct	S hub	hub cheaper when ρ <
1	2,450	98	2,450	4,802	1.0000
4	2,450	196	9,800	19,208	0.2396 (= 2,254 / 9,408)
20	2,450	686	49,000	96,040	0.0375
50	2,450	1,666	122,500	240,100	0.0067
73	2,450	2,352	178,850	350,546	0.0006 (= 98 / 171,696)
74 and up	2,450	2,450 or more	hub flies more		never

The threshold falls fast. At 4 passengers per pair per day the hub wins unless a passenger-segment weighs more than about a quarter of a flight. At 73 it wins only if a passenger-segment weighs less than about 6 parts in 10,000 of a flight. From 74 passengers per pair per day the hub never wins, whatever the weights, because it has no fewer flights and always more passenger-segments.

What the table says about the countChapter 1 declared the hub the winner for d up to 73 on flights alone. That is the case ρ = 0, where passengers cost nothing. Any positive weight on passenger-segments shrinks the range where the hub wins, and the range depends on a ratio that flight counts cannot give.

4 · Where the flights land

A cost can also be counted by place. Every hub flight has the hub at one end, so the hub airport sees F_hub movements a day. In the direct network each airport sees 2(n−1)⌈d/s⌉ = 98 movements a day at these values. At d = 4 the hub airport sees 196, twice as many as any airport in the direct network. At d = 20 it sees 686, and at d = 73, 2,352. The hub network has fewer flights in total but concentrates them on one place. Whether that is better or worse depends on how the cost of a movement grows with the number of movements at one airport, and this chapter does not model that.

What this does not show It does not show which weight is larger, or that any carrier's costs are linear in flights and passenger-segments. It uses no emission, fuel, noise or exposure data. It leaves out freight, connection time, aircraft size and the cost of the hub's ground transport. The threshold in section 2 is exact for the stated model; the numbers in section 3 depend on n = 50, s = 150 and the minimum-frequency rule.

5 · Run it

python3 book15/ch02-verify.py uses only the standard library, prints every number on this page and exits non-zero on any failure. The Lean file for these counts is not written.