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

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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)

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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

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Costs Are Linear with Airplane Size(Example at 6000 km)

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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.