Models and algorithms for integrated airline schedule planning and revenue management

Transcription

1 Models and algorithms for integrated airline schedule planning and revenue management Bilge Atasoy, Matteo Salani, Michel Bierlaire TRISTAN VIII June 14, / 23

2 Motivation Flexibility in decision support tools, demand responsive transportation systems... through... a better understanding of demand behavior, integration of explicit supply-demand interactions, endogenous demand variables that can be controlled by the optimization models, considering demand early in the planning phase. 2/ 23

3 Related Literature Supply-demand interactions in air transport planning Lohatepanont and Barnhart (2004) Wang, Shebalov and Klabjan (2012) Exogenous demand models; iterative supply-demand models Jacobs, Smith and Johnson (2008) Dumas, Aithnard and Soumis (2009) Endogenous demand models - explicit integration Airlines: Schön (2008) Railways: Cordone and Redaelli (2011) Revenue management: Talluri and van Ryzin (2004) 3/ 23

4 Itinerary choice model Market segments, s, defined by the class and each OD pair Itinerary choice among the set of alternatives, I s, for each segment s For each itinerary i I s the utility is defined by: V i = ASC i + β p ln(p i ) + β time time i + β morning morning i V i = V i (p i,z i,β) - ASC i : alternative specific constant - p is the only policy variable and included as log - p and time are interacted with non-stop/stop - morning is 1 if the itinerary is a morning itinerary 4/ 23

5 Estimation Revealed preferences (RP) data: Booking data from a major European airline Lack of variability Price inelastic demand RP data is combined with a stated preferences (SP) data Time, cost and morning parameters are fixed to be the same for the two datasets. A scale parameter is introduced for SP to capture the differences in variance. 5/ 23

6 Market shares Market share and demand for itinerary i in market segment s: ms i = exp(v i(p i,z i,β)) exp(v j (p j,z j,β)) D sms i j I s Consider a new variable υ s = 1 j I s exp(v j ) ms i = υ s exp(βln(p i ) + c i ) i I s ms i = 1 υ s 0 6/ 23

7 Integrated airline scheduling, fleeting and pricing Decision variables: x k,f : binary, assignment of aircraft k to flight f π h k,f : allocated seats for class h on flight f aircraft k p i : price of itinerary i ms i : market share of itinerary i No-revenue itineraries I s I s for segment s, no control of airline. 7/ 23

8 Integrated model - Scheduling & fleeting max h H D s s S h i (I s \I s ) ms i p i C k,f x k,f : revenue - cost (1) k K f F s.t. x k,f = 1: mandatory flights f F M (2) k K x k,f 1: optional flights f F O (3) k K y k,a,t + x k,f = y k,a,t + + x k,f : flow conservation [k,a,t] N (4) f In(k,a,t) f Out(k,a,t) y k,a,mine a + x k,f R k : fleet size k K (5) a A f CT y k,a,mine a = y k,a,maxe + a : cyclic schedule k K,a A (6) π h k,f Q kx k,f : seat capacity f F,k K (7) h H x k,f {0,1} k K,f F (8) y k,a,t 0 [k,a,t] N (9) 8/ 23

9 Integrated model - Revenue management - Pricing s S h D s i (I s \I s ) δ i,f ms i π h k,f : demand - capacity h H,f F (10) k K i I s ms i = 1: market coverage h H,s S h (11) ms i υ s exp(v i (p i,z i ;β)): market share h H,s S h,i (I s \ I s) (12) ms j = υ s exp(v j (p j,z j ;β)): market share - competitors h H,s S h,j I s (13) π h k,f 0 h H,k K,f F (14) LB i p i UB i : bounds on price h H,s S h,i (I s \ I s) (15) ms i 0 h H,s S h,i I s (16) υ s 0 h H,s S h (17) 9/ 23

10 Heuristic method Mixed Integer Non-convex Problem A heuristic procedure based on two subproblems: FAM LS : price-inelastic schedule planning model MILP Prices fixed Optimizes the schedule design and fleet assignment REV LS : Revenue management with fixed capacity NLP Schedule design and fleet assignment fixed Solves pricing, seat allocation Local search based on spill (lost passengers) Price sampling Fixing a subset of FAs & VNS 10/ 23

11 Data and results 25 data instances are generated from ROADEF 2009 dataset. Integrated model is solved... with BONMIN solver as a sequential approach - 1 st iteration of the heuristic with the heuristic Up to around 35 flights 3 aircraft types BONMIN works quite fine. Integrated model improves the sequential approach by 2% on the average The average demand and capacity of the aircraft types at hand are key factors Heuristic finds the solutions at all instances 11/ 23

12 Data and results no airports flights flights per route demand per flight fleet composition BONMIN Sequential Local search heuristic Integrated model approach (SA) Average over 5 replications max 24 hours max 2 hours Profit Time % deviation Time %deviation %impr. Time Profit Profit (sec) from BONMIN (sec) from BONMIN over SA (sec) ,772 1, , % 5 155, % 0.94% ,726 84, , % , % 1.22% 1, ,197 18, , % , % 0.26% , , % , % 1.41% 1, ,457 79, , % , % 4.84% 2, ,496 85, , % 4, , % 0.12% 6,832 12/ 23

13 Sensitivity Analysis Leg-based FAM IFAM choicebased recapture IFAM choicebased recapture & pricing Fleeting & Scheduling Decisions RMM choice based recapture / pricing Resulting Profit Joint work with Prof. Cynthia Barnhart 13/ 23

14 Sensitivity to demand fluctuations Total market segment demand is assumed to be known Fluctuations in reality Average demand is perturbed in a range [-30%, +30%] For each average demand 500 simulations with Poisson 14/ 23

15 Sensitivity to demand fluctuations Profit FAM IFAM' IFAM-PR' 0-30% -25% -20% -15% -10% -5% 0% 5% 10% 15% 20% 25% 30% Perturbation on average market demand (D s ) 77 flights 4 aircraft types - heuristic solution

16 Non-convexity How to deal with non-convexity?... In the literature: inverse-demand function piecewise linear approximation A general utility specification... 16/ 23

17 Transformation of the logit model ms i = exp(v i) exp(v j ), j I s A logarithmic transformation: V i = βln(p i ) + c i ms i = υ s exp(βln(p i ) + c i ) ms i = υ s + βp i + c i ms i ln(ms i), υ s ln(υ s ), p i ln(p i). 17/ 23

18 Transformation of the logit model ms i = exp(v i) exp(v j ), j I s A logarithmic transformation: V i = βln(p i ) + c i ms i = υ s exp(βln(p i ) + c i ) ms i = υ s + βp i + c i ms i ln(ms i), υ s ln(υ s ), p i ln(p i). This is applicable to any utility specification. 17/ 23

19 But... We need both ms i and ms i... cannot simply include ms i = exp(ms i ) 18/ 23

20 But... We need both ms i and ms i... cannot simply include ms i = exp(ms i ) We can penalize the deviation M(ms i exp(ms i ))2 18/ 23

21 But... We need both ms i and ms i... cannot simply include ms i = exp(ms i ) We can penalize the deviation M(ms i exp(ms i ))2 The revenue in the objective function... can use similar tricks 18/ 23

22 5 Illustrative Example I - Aggregate V 1 = βp 1 V 2 = βp 2 ms 1 = υ + βp 1 demand 100 p 2 = 200 Revenue alternative 1 alternative 2 optimize p if β = p1 = price of alternative 1: a 1 max 100ms 1 p 1 max exp(ms 1 + ln(p1)) max ms 1 + ln(p1) Transformation: max ms 1 + ln(p1) M(ms 1 exp(ms 1 ))2 19/ 23

23 Illustrative Example II - Socio-economics V 1,n = β n p V 2,n = β n p 2 Group 1: N 1 = 600,β 1 = 2 Group 2: N 2 = 400,β 2 = 0.1 Revenue p 2 = 2 optimize p 1 p1 = Price of alternative max R 1 + R 2 max 600ms 1,1 p ms 1,2 p 1 Transformation: R n = ln(n n ) + ms 1,n + ln(p 1) max R n M(R n exp(r n)) 2 M(ms 1,n exp(ms 1,n)) 2 n N 20/ 23

24 Back to the airline case study 980 flights, 2,197 itineraries, all flights have a capacity of 195 seats Same optimal prices are found for the following set of penalties: Revenue Computational Reformulated model (in millions) time (sec.) M=(100, ,000) M=(10,000-10,000) M=(1,000-10,000) M=(100-10,000) M=(10-10,000) M=(1,000-1,000) / 23

25 Conclusions The integrated model has promising results... which motivates the effort in devising solution methodologies Logarithmic transformation provides a concave formulation of the revenue problem... is flexible for extensions with socio-economics/more endogenous variables... is expected to facilitate efficient solution methodologies 22/ 23

26 Introduction Demand model Heuristic Results Thank you for your attention! 23/ 23 Transformation Conclusions

27 Logit behavior 24/ 23

28 Itinerary choice model Market share and demand for itinerary i in market segment s: ms i = exp(v i(p i,z i,β)) d i = D s ms i exp(v j (p j,z j,β)) j I s - D s is the total expected demand for market segment s. Spill and recapture effects: Capacity shortage passengers may be recaptured by other itineraries (instead of their desired itineraries) Recapture ratio is given by: b i,j = k I s \{i} exp(v j (p j,z j,β)) exp(v k (p k,z k,β)) 25/ 23

29 Itinerary choice model Value of time (VOT): VOT i = V i/ time i V i / cost i = β time cost i β cost For the same OD pair... VOT for economy, non-stop: 8 e/hour VOT for economy, one-stop: 19.8, 11, 9.2 e/hour VOT for business, non-stop: 21.7 e/hour 26/ 23

30 Spill and recapture model Forecasted demand for an itinerary is 120 Airline considers assigning a capacity of 100 to the associated flight Estimated spilled passengers is 20 If these people are redirected to other itineraries in the market what percantage will accept? 27/ 23

31 Results BONMIN Sequential Local search heuristic Integrated model approach (SA) Average over 5 replications Profit Time % deviation Time %deviation %impr. Time Profit Profit (sec) from BONMIN (sec) from BONMIN over SA (sec) 1 15, , % 1 15, % 0.00% , , % 1 37, % 5.55% , , % 1 50, % 0.00% ,037 2,807 43, % 1 46, % 4.65% ,904 1,580 69, % 1 70, % 1.11% ,311 1,351 82, % 1 82, % 0.00% ,212 32,400 84, % 1 87, % 3.59% ,819 8, , % 1 779, % 0.00% , , % 2 135, % 0.00% , , % 1 107, % 0.00% ,820 31,705 85, % 2 85, % 0.33% ,544 5, , % 1 858, % 0.43% ,881 32, , % 1 112, % 2.71% ,808 10,643 82, % 1 85, % 4.09% , , % 1 49, % 0.00% , , % 1 38, % 2.98% , , % 1 27, % 0.00% , , % 1 45, % 1.65% , , % 1 26, % 0.00% 1 28/ 23

32 Improvement due to the local search Sequential Random Neighborhood approach (SA) neighborhood based on spill % Improvement Profit Profit Time(sec) Profit Time(sec) Quality of Reduction the solution in time 2 35,372 37, , % 4 43,990 44, , % 88.88% 5 69,901 No imp. over SA 70, % ,186 85,335 1,649 87, % 96.36% 8 904, , , % 11 93,920 No imp. over SA 94, % ,902 No imp. over SA 858, % ,428 No imp. over SA 138, % ,347 96, , % 90.56% 16 37,100 38, , % 18 52,369 53, , % ,464 No imp. over SA 147, % ,169 No imp. over SA 219,136 1, % ,114 No imp. over SA 163, % ,615 No imp. over SA 227,284 1, % ,561 No imp. over SA 210, % ,136 No imp. over SA 470,494 1, % - 29/ 23

33 A small example 2 airports CDG-MRS 4 flights - all are mandatory 2 aircraft types: seats We start with an initial FAM solution: F1 F2 F3 F4 AC1 X X X X AC2 30/ 23

34 A small example - GBD iterations Iteration 1 Iteration 2 Sub Master Sub Master LB UB LB UB = AC1 AC2 AC1 AC2 F1 X F1 X F2 X F2 X F3 X F3 X F4 X F4 X Iteration 3 Iteration 4 Sub Master Sub Master LB UB LB UB = AC1 AC2 AC1 AC2 F1 X F1 X F2 X F2 X F3 X F3 X F4 X F4 X 31/ 23