Robust Block Times Yu-Ching Lee, University of Illinois at Urbana- Champaign Diego Klabjan, Northwestern University Milind Sohoni, Indian School of Business 

The papaer • Milind Sohoni, Yu-Ching Lee, and Diego Klabjan. 2011. Robust Airline Scheduling Under Block-Time Uncertainty. Transportation Science 45, 4 (November 2011), 451-464. 

Motivation • Marketing group designs the schedule – Good idea on frequency – Solid judgment on departure times – Poor job on block times • Look at historical block times • Given percentile of historical block times – Based on the desired service level 

Motivation • Quantitative approach to block times – Compute block times (arrival times) – Robust approach with respect to various service levels – DOT service level – Passenger connection service level • Subject to – Allow minor departure time adjustments – Do not change frequency 

Modeling Framework • Given a fleeted schedule – Seat capacities are known – Schedule is known subject to allowable perturbations • O-D itinerary-based deterministic demand • Produce – Adjusted schedule – Maximize profit – Capture service level 

Modeling Framework • Profit – Operating cost – Planned revenue • Service level – Flight service level – Network service level • Standard O-D seat capacity restrictions • Departure and arrival time decision variables 

Modeling Framework • Xi,t : Random variable representing the block time of flight i departing at time t – Obtained from historical observations • We use chance/probabilistic constraints O D t Xit 

Flight Service Level • Flight service level measures the on-time performance of a single flight • A flight arriving no later than 15 min after scheduled arrival time is “on-time” on time flight service level = 0.8 block time, x P[late no more than 15 minutes] ≥ r P[block time ≤ arr-dep+15] ≥ r 

Flight Service Level • Added explicit constraints by precomputing the inverse of pdf • Block time distribution log-concave – Consider log on both sides – Now the feasible set is convex • If block time distributions stationary with respect to the departure time Log P[Xi,t ≤ arr-dep+15] ≥ Log r 

Network Service Level • Network service level measures the probability of passenger connections O O O time scheduled arrival time minimum connection time X O O time scheduled arrival time minimum connection time • NSL is a multiple of probabilities 

Network Service Level • Consider flight i P[connect from i to j1 ] · P[connect from i to j2 ] · P[connect from i to j3 ] · · · P[connect from i to jk ] ≥ q log log P[connect from i to j1 ] + log P[connect from i to j2 ] + log P[connect from i to j3 ] +· · ·+ log P[connect from i to jk ] ≥ log q 

Network Service Level • And we are again happy – Block time distribution log-concave – Does not depend on the departure time • The feasible set is convex 

Departure Time Adjustments • We penalize per minute deviation of the departure time • The new departure time must be within a “ time window”, e.g. ± 15 min, ± 30 min 

Profit Maximization Model • Objective: To maximize profit • Constraints – Planned resource (capacity, budget) – Flight service level above requirement – Network service level above requirement Min [ Operating Cost ] – [ Planned Revenue ] + [ Deviation Penalty Cost ] 

Service Level Maximization Model • Objective: To maximize service level • Constraints – Planned resource (capacity, budget) – Profit above a certain number Max FSL + weight ·NSL ＝ Max exp( log FSL ) + weight ·exp( log NSL ) ～ Max [ min log FSL ] + weight [ min log NSL ] 

Solution Methodology • The feasible set is convex • We apply standard Benders cuts – We add a cut at the point violating a service level requirement • The service level model requires an approximation to the objective function 

Computational Study • More than 1,000 flights of a legacy US carrier • Block time distributions obtained from historical data • Real cost and revenue data 

Computational Study • Model 1, approximately 1,400 flights • NSL set to 0.8 0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 Flight service level Cost 

Computational Study • Model 1 • FSL set to 0.8 0.87 0.88 0.89 0.9 0.91 0.92 0.93 0.94 Network service level cost 

Computational Study 0 0.2 0.4 0.6 0.8 1 1.2 1.4 co s t s e rvice le ve ls time window = 45 time window = 15 NSL FSL NSL * 0.7 + FSL cost Win = 45 Win = 15 Win = 45 Win = 15 Win = 45 Win = 15 $$$ 0.997 0.854 0.338 0.349 1.036 0.946 $$$$$ 0.997 0.854 0.423 0.443 1.121 1.041 $$$$$$$ 0.997 0.853 0.502 0.504 1.200 1.101 > > > > > > < < < • Model 2 

Generalized Modeling Framework • Allow the flights to be adjusted across the time intervals • Introduction of binary variables Interval 1 Interval 2 Interval 3 …… 

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