Simple Aircraft Cost Functions Prof Nicole Adler University of Jerusalem Dr William Swan Boeing 2 July 2004 ATRS 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 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 Distance Example

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

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)

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 + b  Seats + 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 Counts Regional Configurations AirplaneNominal (all Y)2-class (as used) A A A A

Another Try at “Fair” Relative Seat Counts Long-haul Configurations AirplaneNominal (all Y)2-class (long) A A

Cost is Linear With Distance Example

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

Calibration Formula Cost = $ * (Seats + 211) * (Dist ) 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.