Airline Fleet Assignment Cynthia Barnhart 16.75 Airline Management Outline: – – – – – Problem Definition and Objective Fleet Assignment Network Representation Fleet Assignment Models and Algorithms Extension of Fleet Assignment to Schedule Design Conclusions 3/9/2006 16.75 1 Airline Schedule Planning 3/9/2006 Schedule Design Select optimal set of flight legs in a schedule Fleet Assignment AAssign flight specifies aircraft types origin,todestination, flight legs such thatand contribution departureistime maximized Aircraft Routing Route individual aircraft honoring Contribution = Revenue - Costs maintenance restrictions Crew Scheduling Assign crew (pilots and/or flight attendants) to flight legs 16.75 2 Fleet Assignment Problem Definition • Given: ¾Flight Schedule: a set of (daily) flight legs; ¾Aircraft fleet: consisting of different fleet types; ¾Passenger demand pattern; ¾Revenue and operating cost data; • Find: A feasible fleet assignment, i.e. a mapping from flight legs to fleet types that maximizes profit = total revenue – total operating costs. 3/9/2006 16.75 3 Fleet Assignment • Class Exercise: Flight Network BOS Boston Logan ORD Chicago O’Hare 3/9/2006 CL50x (2 flights) CL33x (3 flights) CL55x (2 flights) LGA 16.75 CL30x (3 flights) New York LaGuardia 4 Fleet Assignment • Class Exercise: Flight Schedule, Fares, & Demand Flight # From To Dept Time Arr Time Fare CL301 CL302 CL303 CL331 CL332 CL333 CL501 CL502 CL551 CL552 3/9/2006 LGA LGA LGA BOS BOS BOS LGA LGA ORD ORD BOS BOS BOS LGA LGA LGA ORD ORD LGA LGA (EST) 1000 1100 1800 0700 1030 1800 1100 1500 0700 0830 16.75 (EST) 1100 1200 1900 0800 1130 1900 1400 1800 1000 1130 [$] 150 150 150 150 150 150 400 400 400 400 Demand [passengers] 250 250 100 150 300 150 150 200 200 150 5 Fleet Assignment • Class Exercise: Fleet Information Fleet type DC-9 B737 A300 3/9/2006 Number of aircraft owned 1 2 2 Capacity [seats] 120 150 250 16.75 Per flight operating cost [$000] LGA - BOS LGA – ORD 10 15 12 17 15 20 6 Fleet Assignment • Class Exercise: Find: An assignment of fleet types to the flights in this network that maximizes net profit. 3/9/2006 16.75 7 Fleet Assignment Evaluating assignment profits: cl , f := fare l × min( Dl , Cap f ) − OpCost l , f where: cl , f : profitability of assigning fleet type f to flight leg l ; farel : fare of flight leg l ; Dl : demand of flight leg l ; Cap f : capacity of fleet type f ; OpCostl , f : operating cost of assigning fleet type f to flight leg l. 3/9/2006 16.75 8 Fleet Assignment • Class Exercise: Evaluating assignment profitabilities… Profitability [$000 per day] Flight # CL301 CL302 CL303 CL331 CL332 CL333 CL501 CL502 CL551 CL552 3/9/2006 DC-9 B737 8 8 5 8 8 8 33 33 33 33 10.5 10.5 3 10.5 10.5 10.5 43 43 43 43 16.75 A300 22.5 22.5 0 7.5 22.5 7.5 40 60 60 40 9 Fleet Assignment Time-Line Network: CL332 BOS CL333 CL303 CL302 CL331 CL301 LGA CL551 CL501 CL552 CL502 ORD 0800 1000 3/9/2006 1200 1400 16.75 1600 timescale 10 Fleet Assignment Optimal solution: CL332 CL331 CL301 B737 B737 CL552 Revenue = $428,500 Cost = $148,000 3/9/2006 DC9 00 CL302 3 A DC9 A300 CL551 A300 CL333 CL303 B737 B737 A300 CL501 CL502 Leg ∆capacity CL301 CL302 CL303 CL331 CL332 CL333 CL501 CL502 CL551 CL552 -130 0 +50 0 -50 -30 0 +50 +50 0 Profit = $280,500 16.75 11 Fleet Assignment Time-Line Network: Flight arc Ground arc Airport A yv − v yv + wl,f Airport B count line 3/9/2006 16.75 12 Notations • Decision Variables – fk,i equals 1 if fleet type k is assigned to flight leg i, and 0 otherwise – yk,o,t is the number of aircraft of fleet type k, on the ground at station o, and time t • Parameters – Ck,i is the cost of assigning fleet k to flight leg i – Nk is the number of available aircraft of fleet type k – tn is the “count time” • Sets – – – – L is the set of all flight legs i K is the set of all fleet types k O is the set of all stations o CL(k) is the set of all flight arcs for fleet type k crossing the count time 3/9/2006 16.75 13 Fleet Assignment Model (FAM) Min ∑ ∑ ck,i fk,i k∈K i∈L Subject to: ∑ fk,i = 1 ∀i∈ L fk,i = 0 ∀k, o,t k∈ ∈K K yk,o,t − + ∑ i∈I (k ,o,t ) fk,i − yk,o,t + − ∑ i∈O(k ,o,t ) ∑ yk,o,tn + ∑ fk,i ≤ Nk ∀k ∈ K o∈O i∈CL(k ) fk,i ∈{0,1} yk,o,t ≥ 0 Hane et al. (1995), Abara (1989), and Jacobs, Smith and Johnson (2000) 3/9/2006 16.75 14 Constraints • Cover Constraints – Each flight must be assigned to exactly one fleet • Balance Constraints – Number of aircraft of a fleet type arriving at a station must equal the number of aircraft of that fleet type departing • Aircraft Count Constraints – Number of aircraft of a fleet type used cannot exceed the number available 3/9/2006 16.75 15 Objective Function • For each fleet - flight combination: Cost ≡ Operating cost + Spill cost –Recaptured revenue • Operating cost associated with assigning a fleet type k to a flight leg j is relatively straightforward to compute – Can capture range restrictions, noise restrictions, water restrictions, etc. by assigning “infinite” costs • Spill cost for flight leg j and fleet assignment k = average revenue per passenger on j * MAX(0, unconstrained demand for j – number of seats on k) – Unclear how to compute revenue for flight legs, given revenue is associated with itineraries • Recaptured revenue – Revenue from passengers16.75 that are recaptured back to the airline after being spilled from another flight leg 3/9/2006 16 FAM Example: Spill A B Demand = 100 Fare = $100 Fleet Type Capacity Revenue Spill Cost Op. Cost Assignment Cost Contribution i 80 $8,000 $2,000 $5,000 $7,000 $3,000 ii 100 $10,000 $0 $6,000 $6,000 $4,000 iii 120 $10,000 $0 $7,000 $7,000 $3,000 iv 150 $10,000 $0 $8,000 $8,000 $2,000 3/9/2006 16.75 17 FAM Example: Recapture 100 seats 100 seats A B ( 80, $200 ) ( 90, $250 ) ( 50, $400 ) 9AM 10AM C 20 30 X 0.3 = 9 recaptured passengers 29 ( 20, $400 ) ( Demand, Fare ) Recapture Rate 3/9/2006 16.75 18 Fleet Assignment A few observations on FAM: Nodes can be consolidated to reduce model size; ¾ Fleet-specific time-line networks are possible; ¾ Fleet assignment not aircraft assignment! ¾ Note that feasibility of FAM implies that aircraft rotations exist (takes only a little bit of thinking); ¾ However, these rotations might not be maintenance feasible… ¾ 3/9/2006 16.75 19 Fleet Assignment Solvability: FAM can be solved using standard branchand-bound software; ¾ Solution times are FAST, thanks to FAM’s small LP gaps… ¾ 3/9/2006 16.75 20 Fleet Assignment Computational Sample: 2,044 flight legs, 9 fleet types Problem size # of columns # of rows # of non-zero entries Strength of formulation Root node LP solution Best IP solution Difference Solution time [sec] 3/9/2006 16.75 18,487 7,827 50,034 21,401,658 21,401,622 36 974 21 Fleet Assignment • FAM suffers from a significant drawback in its modeling of the revenue side… • Passengers book itineraries not flight legs… • Capacity decisions on one leg will affect passenger spill on other legs… • This phenomenon is known as network effects. 3/9/2006 16.75 22 Fleet Assignment Model (FAM) Min ∑ ∑ ck,i fk,i k∈K i∈L Subject to: ∑ fk,i = 1 ∀i∈ L fk,i = 0 ∀k, o,t k∈K yk,o,t − + ∑ i∈I (k ,o,t ) fk,i − yk,o,t + − ∑ i∈O(k ,o,t ) ∑ yShortcoming: k ,o,tn + ∑ fk ,i ≤ Nk ∀k ∈ K Major o∈O i∈CL(k ) FAM assumes leg independence fk,i ∈{0,1} yk,o,t ≥ 0 Hane et al. (1995), Abara (1989), and Jacobs, Smith and Johnson (2000) 3/9/2006 16.75 23 FAM Example: Network Effects A B ( 80, $200 ) C ( 90, $250 ) ( 50, $400 ) ( Demand, Fare ) Fleet Type Capacity Spill Cost i 80 ? ii 100 ? iii 120 ? iv 150 $0 3/9/2006 16.75 Leg Interdependence Network Effects 24 Spill Cost Computation and Underlying Assumption • Given: – Spill cost for flight leg j and fleet assignment k = average revenue per passenger on j * MAX(0, unconstrained demand for j – number of seats on k) • Implication: – A passenger might be spilled from some, but not all, of the flight legs in his/ her itinerary 3/9/2006 16.75 25 An Iterative Approach Fleet Assignment Solver • FAM Solver Problem Modification – Basic Fam • Possibly with minor modifications Spill Calculation • Spill Calculation: – Simulations – Passenger mix model • Problem Modification: – Objective cost coefficient update 3/9/2006 16.75 26 Passenger Mix • Passenger Mix Model (PMIX) – Kniker (1998) – Given a fixed, fleeted schedule, unconstrained passenger demands by itinerary (requests), and recapture rates find maximum revenue for passengers on each flight leg PMIX 3/9/2006 Network Effects and Recapture 16.75 27 Problem Modification • Based on differences in expected spill from FAM and the Spill Calculator, we modify the FAM problem – Update objective cost coefficients • Cost coefficient update, many heuristics possible 3/9/2006 16.75 28 FAM Spill Calculation Heuristics • Fare Allocation – Full fare - the full fare is assigned to each leg of the itinerary – Partial fare - the fare divided by the number of legs is assigned to each leg of the itinerary – Shared fare - the fare divided by the number of capacitated legs is assigned to each capacitated leg in the itinerary • Spill Cost for each variable – Representative Fare • A “spill fare” is calculated; each passenger spilled results in a loss of revenue equal to the spill fare – Integration • Sort each itinerary by fare, spill costs are sum of x lowest fare passengers, where x = max{0, demand - capacity} 3/9/2006 16.75 29 An Illustrative Example Fleet Type A B Market X-Y Seats 100 200 Itinerary 1 flight 1 X Average Fare $200 flight 2 Y Z No. of Pax 75 Y-Z 2 $225 150 X-Z 1-2 $300 75 Partial Alloc. Full Alloc. Actual Opt. Spill Spill Spill Spilled Pax A-A $30,000 $38,125 31,875 50 X-Z, 75 Y-Z A-B $11,250 $15,625 12,500 25 X-Z, 25 X-Y B-A $22,500 $28,125 28,125 125 Y-Z B-B $3,750 $5,625 5,625 Fleet Assign. Fl. 1- Fl. 2 3/9/2006 16.75 25 Y-Z 30 Spill Calculation: Results • For a 3 fleet, 226 flights problem: – The best representative fare solution results in a gap with the optimal solution of $2,600/day – Using a shared fare scheme and integration approach, we found a solution with an $8/day gap. • By simply modifying the basic spill model, significant gains can be achieved 3/9/2006 16.75 31 Itinerary-Based Fleet Assignment • Impossible to estimate airline profit exactly using link-based costs • Enhance basic fleet assignment model to include passenger flow decision variables – Associate operating costs with fleet assignment variables – Associate revenues with passenger flow variables 3/9/2006 16.75 32 Itinerary-based Fleet Assignment Definition • Given – a fixed schedule, – number of available aircraft of different types, – unconstrained passenger demands by itinerary, and – recapture rates, Find maximum contribution ODFAM 3/9/2006 Network effects 16.75 33 Itinerary-Based FAM (IFAM) M in ∑ ∑ k ∈ K i∈ L c%k , i f k , i + ∑∑ p∈ P r ∈ P ( fare p − b pr fare r ) t rp ∑ f k ,i = 1 S u b je c t to : k∈K ∀i ∈ L Fleet Assignment FAM y k ,o ,t − + ∑ f k ,i = 0 ∀ k , o,t f k ,i ≤ N k ∀k ∈ K + ∑ ∑ δ i p t rp − ∑ ∑ δ i p b rp t rp ≥ Q i ∀i ∈ L i∈ I ( k , o , t ) f k ,i − y k ,o ,t + − i∈ O ( k , o , t ) ∑ y k ,o ,t n + o∈O ∑ f k , i SEATS ∑ ∑ i∈ CL ( k ) ConsistentPMM Spill + ∑Recapture k k r ∈ P p∈ P r ∈ P p∈ P r∈ P t rp ≥ 0 t rp ≤ D p f k , i ∈ {0 ,1} ∀p ∈ P y k , o ,t ≥ 0 Kniker (1998) 3/9/2006 16.75 34 Itinerary-Based FAM (IFAM) M in ∑ ∑ k ∈ K i∈ L c%k , i f k , i + ∑∑ p∈ P r ∈ P ( fare p − b pr fare r ) t rp 3 ∑ f k ,i = 1 ∀i ∈ L f k ,i = 0 ∀ k , o,t f k ,i ≤ N k ∀k ∈ K + ∑ ∑ δ i p t rp − ∑ ∑ δ i p b rp t rp ≥ Q i ∀i ∈ L S u b je c t to : k∈K y k ,o ,t − + ∑ i∈ I ( k , o , t ) f k ,i − y k ,o ,t + − i∈ O ( k , o , t ) ∑ y k ,o ,t n + o∈O 1 ∑ f k , i SEATS k k r ∈ P p∈ P ∑ ∑ i∈ CL ( k ) r ∈ P p∈ P 2 r ∑ tp ≤ Dp ∀p ∈ P r∈ P t rp ≥ 0 f k , i ∈ {0 ,1} y k , o ,t ≥ 0 Kniker (1998) 3/9/2006 16.75 35 Column and Constraint Generation Original RMP 1 3 5 2 7 4 6 8 3/9/2006 16.75 36 Implementation Details • Computer • RMP constraint matrix size – Workstation class computer – 2 GB RAM – CPLEX 6.5 – ~77,000 columns – ~11,000 rows • Final size • Full size schedule – – – – – ~86,000 columns – ~19,800 rows ~2,000 legs ~76,000 itineraries ~21,000 markets 9 fleet types • Solution time – LP: > 1.5 hours – IP: > 4 hours 88% Saving from Row Generation > 95% Saving from Column Generation 3/9/2006 16.75 37 Fleet Assignment Computational Sample: Problem size # of columns # of rows # of non-zero entries Strength of formulation Root node LP solution Best IP solution Difference Solution time [sec] Contribution [$/day] 3/9/2006 2,044 flight legs, 9 fleet types FAM IFAM 18,487 7,827 50,034 77,284 10,905 128,864 21,401,658 21,401,622 36 974 21,178,815 21,302,460 21,066,811 235,649 >100,000 21,066,811 16.75 38 IFAM Contributions • Annual improvements over basic FAM – Network Effects: ~$30 million – Recapture: ~$70 million • These estimates are upper bounds on achievable improvements 3/9/2006 16.75 39 Subnetwork-Based FAM • IFAM has limited opportunity for expansion to include schedule design decisions – Fractionality of solution to LP relaxation is a big issue • Need alternative fleet assignment kernel – Capture network effects Modeling – Maintains tractability Accuracy IFAM ? SFAM FAM Tractability 3/9/2006 16.75 40 Basic Concept • Isolate network effects – Spill occurs only on constrained legs 5 Potentially Constrained Flight Leg 3 Unconstrained Flight Leg Potentially Binding Itinerary 6 9 1 7 4 2 Non-Binding Itinerary 8 3/9/2006 SFAM IFAM FAM < 30% of total legs are potentially constrained < 6% of total itineraries are potentially binding 16.75 41 Modeling Challenges • Utilize composite variables (Armacost, 2000; Barnhart, Farahat and Lohatepanont, 2001) 1 1 2 3 4 5 6 7 8 9 A A A B B B C C C A B C A B C A B C 3 Fleet Types: A, B, and C Challenges Efficient 3/9/2006 column enumeration 16.75 42 SFAM Formulation m M S ηΠS Min∑∑( CΠmS ) ( fΠmS ) n m=1 n=1 n m M S ηΠS ∑∑(δ ) ( f ) Subject to: m=1 n=1 yk ,o,t− + ∑ m M S ηΠS m κ ( ∑∑ Π i∈I ( k ,o,t ) m=1 n=1 S ) k ,i n ( fΠmS ) − yk,o,t+ − n m M S ηΠS m κ ( ∑∑ Π =1 ∀i∈ L = 0 ∀k, o,t ∑ ∑∑(γ ) ( f ) ≤ Nk ∀k ∈ K S i∈O(k ,o,t ) m=1 n=1 o∈A m ΠS n ) (f ) ∑ ∑ yk,o,tn + m i ΠS n m M S ηΠS i∈CL( k ) m=1 n=1 ( f ) ∈{0,1} m ΠS n k ,i n m k ΠS n m ΠS n m ΠS n yk,o,t ≥ 0 FAM solution algorithm applies 3/9/2006 16.75 43 SFAM Results • Testing performed on full size schedules – Runtimes similar to FAM, much faster than itinerary-based approaches • Tight LP relaxations – SFAM achieve improved solutions relative to FAM and itinerary-based approach • SFAM has potential for further integration or extension – Time windows, stochastic demand, schedule design 3/9/2006 16.75 44 Caveats 2. Deterministic Demand A B ( 80, $200 ) 4. Optimal 9AM Control of Paxs 10AM C ( 70, $250 ) ( 50, $400 ) 20 3. Demand Forecast Errors 30 X 0.3 = 9 recaptured passengers 1. Recapture Rate ( 20, $400 ) Errors ( Demand, Fare ) Recapture Rate 3/9/2006 16.75 45 Recapture Rate Sensitivity Specified Recapture Rate IFAM Fleeting Decision PMix flows passengers on fleeted schedule assuming full knowledge of passenger choices Solve SolvePMix PMix with withvaried varied recapture recapturerates rates Operating Cost Estimated Revenue Fleeting Contribution 3/9/2006 16.75 46 Recapture Rate Sensitivity Improvement over Basic FAM ($/day) Recapture Rate Sensitivity 8,000 7,000 6,000 5,000 4,000 3,000 2,000 1,000 0 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 Recapture Rate Multiplier ( δ) Sensitivity of IFAM Improvement gained from network effects alone Improvement gained from network effects and recapture 3/9/2006 16.75 47 IFAM Sensitivity Analysis • Simulations FAM or IFAM Average Demand Simulate 500 realizations of demand based on Poisson distributions Fleeting Decision Demand Variations Passenger Passenger Allocation Allocation Simulation Simulation Realizations Operating Cost Estimated Revenue Fleeting Contribution 3/9/2006 16.75 48 IFAM vs. FAM Demand Variations Demand Stochasticity Realizations Demand deviation ~14% FAM Problem 1N-3A Revenue Operating Cost Contribution Problem 2N-3A Revenue Operating Cost Contribution 3/9/2006 IFAM Difference (IFAM-FAM) $ $ $ 4,858,089 2,020,959 2,837,130 $ $ $ 4,918,691 2,021,300 2,897,391 $ $ $ 60,602 341 60,261 $ $ $ 3,526,622 2,255,254 1,271,368 $ $ $ 3,513,996 2,234,172 1,279,823 $ $ $ (12,626) (21,082) 8,455 $/day 16.75 49 Demand Variations IFAM vs. FAM Demand Stochasticity Realizations Forecast Errors Data Quality Issue Demand deviation ~15% Contributions ($ million/day) Model Sensitivity to Demand Forecast Errors $1.80 $1.70 $1.60 $1.50 $1.40 $1.30 $1.20 $1.10 $1.00 -5% -4% -3% -2% -1% 0 +1% +2% +3% +4% +5% Simulated Demands (% of Forecasted Demand) FAM 3/9/2006 16.75 IFAM 50 Extending Fleet Assignment Models to Include “Incremental” Schedule Design… 3/9/2006 16.75 51 Airline Schedule Planning Schedule Design Select optimal set of flight legs in a schedule Fleet Assignment Assign aircraft types to flight legs such that contribution is maximized Aircraft Routing Crew Scheduling 3/9/2006 16.75 52 Schedule Design: Fixed Flight Network, Flexible Schedule Approach • Fleet assignment model with time windows – Allows flights to be re-timed slightly (plus/ minus 10 minutes) to allow for improved utilization of aircraft and improved capacity assignments ¾Initial step in integrating flight schedule design and fleet assignment decisions 3/9/2006 16.75 53 Example: Results Aircraft Utilization Do time windows allow a reduction in the number of required aircraft? TW = 0 P1 P2 3/9/2006 a/c req'd 365 428 TW = 20 cost 28,261,302 29,000,175 a/c req'd 363 426 16.75 cost 28,114,913 28,965,409 54 Results • Time windows can provide significant cost savings, as well as a potential for freeing aircraft – $50 million in operating costs alone for one U.S. airline 3/9/2006 16.75 55 Schedule Design: Optional Flights, Flexible Schedule Approach • Fleet assignment with “optional” flight legs – Additional flight legs representing varying flight departure times – Additional flight legs representing new flights – Option to eliminate existing flights from future flight network ¾Incremental Schedule Design 3/9/2006 16.75 56 Demand and Supply100Interactions Market Share 450 Market Share 410 Market Share 150 100 100 A 100 190 A 120 40 B 100 200 A 150 B 300 3/9/2006 B 16.75 Non-Linear Interactions 57 Formulation ∑ c~k , i f k , i + ∑ Min ∑ k ∈ K i∈ L ∑ ( fare p p∈ P r∈ P f k ,i = 1 ∀ i ∈ LF ∑ f ≤1 Flight Selection Schedule Design ∀ i ∈ LO ∑ S u b je c t to : k∈K k ,i k∈ K y k ,o ,t − + ∑ f k ,i − y k ,o ,t + − f k ,i = 0 ∀ k ,o,t f k ,i ≤ N k ∀k ∈ K p r p r r ∑ δ i t p − ∑ ∑ δ i b pt p ≥ Qi ∀i ∈ L ∑ FAM Fleet Assignment Spill +PMM Recapture i∈ I ( k , o , t ) i∈ O ( k , o , t ) ∑ y k ,o ,t n + o∈ O ∑ f k , i SEATS k − b rp fare r ) t rp k + ∑ r∈ P p∈ P ∑ i ∈ CL ( k ) r∈ P p∈ P r ∑ tp ≤ Dp ∀p ∈ P r∈ P t rp ≥ 0 f k , i ∈ {0 ,1} y k ,o ,t ≥ 0 Lohatepanont, M. and Barnhart, Cynthia, “Airline Schedule Planning: Integrated Models and Algorithms for Schedule Design and Fleet Assignment,” Transportation Science, future. 3/9/2006 16.75 58 Schedule Design: Results • Demand and supply interactions – Tractability potentially a big issue • Resulting schedules operate fewer flights – Lower operating costs – Fewer aircraft required • Order of magnitude impact: ~$100 - $350 million improvement annually for variable market demand – Rough estimates: sensitive to quality of data, spill and recapture assumptions, demand forecasts and stochasticity – Comparison to planners’ schedules – Excludes benefits from saved aircraft 3/9/2006 16.75 59