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Simple Aircraft Cost Functions
Prof Nicole AdlerUniversity of Jerusalem
Dr William SwanBoeing
2 July 2004ATRS Symposium, Istanbul
Overview
1. Cost vs.. Distance is Linear Illustration Explanation Calibration Why we care
2. Cost vs.. Airplane Size is Linear Illustration Explanation Calibration Why we care
3. Cost vs.. Distance and Size is Planar Why we care
Cost vs. Distance is Linear
• Cost for a single airplane design– Example 737-700
• Cost based on Engineering cost functions– Data from 25-year Boeing OpCost “program”– Divides cost into engineering components
• Fuel, crew, maintenance, ownership• Calibrates components from airline data
– Records of fuel burn– Knowledge of crew pay and work rules– Schedule of recurring maintenance and history of failures– Market Ownership Rents allocated to trips
Engineering Approach is Different
• Not a “black box”– We made what is inside the box
• Not a statistical calibration– Although components are calibrated against data
• Less an overall average– OpCost calibrations based on detail records
• OpCost estimates costs– For standard input cost factors: fuel, labor, capital– Ongoing function recalibration
• This report from 2001 version• 2004 version now in use
We Generate “Perfect” Data Points
• Cost for exactly the same airplane– At different distances
• Each point with identical input costs– Fuel, labor, capital
• Superb spread of data points– Costs at 1000, 1500, 2000, 3000, 4000, 5000km distances– Much larger than spreads of averages for airlines– Comparable overall average distance– Much greater sensitivity to slope
• Objective is to learn the shape of the relationship– Find appropriate algebraic form
• For ratios of costs at different distances
Cost is Linear With Distance737-700 Example
0
5
10
15
20
25
30
0 1000 2000 3000 4000 5000
Distance (km)
Tri
p C
ost
(in
dex
)
Cost DataLinear (Cost Data)
Explanation:Why is Cost Linear With Distance?
• Most costs are per hour or per cycle• Time vs. distance is linear: speed is constant
– (roughly ½ hour plus 500 mph)• Departure/arrival cycle time is about ½ hour• Some costs are allocated
– Allocation is per hour and per cycle– Ownership, for example
• Very small rise in fuel/hour for longer hours• Beyond 8 hours, crew gains 1 or 2 pilots
– Does not apply to regional distances.
Cost Formulae are Linear
Airplane cost/seat-km km/departure seats R-Squareda318 0.039 691 107 0.9998
737-600 0.038 700 110 0.9997737-700 0.035 692 126 0.9997a319 0.036 705 126 0.9997
A320-200 0.033 727 150 0.9998737-800 0.031 701 162 0.9997737-900 0.030 715 177 0.9997A321-200 0.030 725 183 0.9998757-200 0.029 782 200 0.9999757-300 0.027 815 243 0.9999
Observations
• All airplanes’ cost vs.. distance was linear
• Calibration using 6 “perfect” data points
• Least squares
• Slopes per seat-km similar
• Intercept in equivalent km cost similar
• 757s designed for longer hauls
• Otherwise comparable capabilities
Why we Care
• Costs Linear with distance means– Average cost is cost at average stage length
• We generally know these data• We can adjust and compare airlines at standard
distance
– Cost of an extra stop are separable• Stop cost independent of where in total distance
• Simplifies Network Costs– Costs are depend on total miles and departures
Costs Are Linear with Airplane Size(Example at 1500 km)
0
2
4
6
8
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100 120 140 160 180 200 220 240seats for comparable single-aisle designs
trip
co
st
at
15
00
km
(in
de
x)
DataLinear (Data)
Why we Care
• Costs Linear with Seats means– Average cost is cost at average size
• We generally know these data
• We can adjust and compare airlines at a standard size
– Cost of Frequency and Capacity are Separable• Frequency cost is independent of capacity
• Powerful Independence in Network Design– Costs and values of Frequencies
– Cost and need for capacity
Calibration for Planar Formula
• NOT
Cost = a + bSeats + c*Dist + d*Seats*Dist
• Yes:
Cost = k * (Seats + a) * (Dist + b)
= k*a*b + k*b*seats + k*a*Dist
+ k*Seats*Dist
NOTE: only 3 degrees of freedom
Why We Care
• Planar function is VERY easy to work with
• Decouples frequency, size, distance
• Vastly simplifies network design issues
• Allows comparison of airline costs after adjustment for size and stage length
• Calibration with broad ranges of size and distance means slopes are very significant
Calibration Techniques
• Calibrate each airplane vs.. distance– Two variables, k and b
• Calibrate a for least error– Unbiased– Least squared
• Compare to least % error (log form)• Compare to size-first process• Results very similar• Results also similar to 4-variable values
Calibration Formula
Cost = $0.019 * (Seats + 104) * (Dist + 722) Where Cost means total cost 2001US $ per airplane trip,
non-US cost functions.Seats means seat count in standard 2-class regional
density.Dist means airport-pair great circle distance in
kilometers.
One try at “Fair” Relative Seat CountsRegional Configurations
Airplane Nominal (all Y) 2-class (as used)
A318 117 107
737-600 122 110
737-700 140 126
A319 138 126
A320 160 150
737-800 175 162
737-900 189 177
A321 202 183
757-200 217 200
757-300 258 243
Another Try at “Fair” Relative Seat Counts
Long-haul Configurations
Airplane Nominal (all Y) 2-class (long)
767-200 238 163
767-300 280 200
767-400 315 229
A330-2 355 233
A330-3 379 268
777-200 415 308
777-300 510 385
747-400 553 429
Cost is Linear With Distance777-200 Example
0
20
40
60
80
100
120
3000 4000 5000 6000 7000 8000 9000
Distance (km)
Tri
p C
ost
Ind
ex
Data
Linear (Data)
Costs Are Linear with Airplane Size(Example at 6000 km)
0
10
20
30
40
50
60
70
80
90
100
100 150 200 250 300 350 400Long-haul seat count
Tri
p C
os
t a
t 6
00
0 k
m (
ind
ex
)
Calibration Formula
Cost = $0.0115 * (Seats + 211) * (Dist + 2200) Where Cost means total cost 2001US $ per airplane trip,
non-US International trip cost functions.Seats means seat count in standard 2-class long haul
density.Dist means airport-pair great circle distance in
kilometers.
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