An integrated schedule planning and revenue management model

Transcription

1 An integrated schedule planning and revenue management model Bilge Atasoy Michel Bierlaire Matteo Salani LATSIS - 1 st European Symposium on Quantitative Methods in Transportation Systems September 07, / 19

2 Motivation Demand responsive transportation systems Better representation of demand Appropriate demand models Flexibility in supply New concept: Clip-Air (Sponsored by EPFL - Middle East.) Integration of supply-demand interactions in transportation models 2/ 19

3 Objectives Comparative analysis between standard fleet and Clip-Air Atasoy, Salani, Bierlaire, and Leonardi, 2012 Development of appropriate demand models Atasoy and Bierlaire, 2012 Development of integrated schedule design and fleet assignment model and revenue management (supply-demand interactions) Atasoy, Salani, and Bierlaire, 2012 Solution techniques for the resulting decision problems 3/ 19

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 a 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 No-revenue represented by the subset I s I s for segment s. 4/ 19

5 Itinerary choice model Demand for class h for each itinerary i in market segment s: exp(v i (p i,z i,β)) d i = D s 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: exp(v j (p j,z j,β)) b i,j = exp(v k (p k,z k,β)) k I s \{i} 5/ 19

6 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. Further details in Atasoy and Bierlaire (2012). 6/ 19

7 Estimation results β fare β time non-stop one-stop non-stop one-stop β morning economy business Price elasticity of demand: E P i price i = P i price i price i P i An example for a non-stop itinerary price elasticity for economy is 2.03 and for business for a one-stop itinerary price elasticity for economy is 2.14 and for business 7/ 19

8 Integrated schedule planning and revenue management Fleet assignment Schedule design Mandatory flights Optional flights Schedule planning Revenue management Pricing-demand Spill-recapture Capacity allocation Business seats Economy seats 8/ 19

9 Integrated model - Schedule planning Max (d i t i,j + t j,i b j,i )p i C k,f x k,f : revenue - cost (1) h H s S h i (Is \I s ) j Is j (Is \I s 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 A a x k,f R k : fleet availability k K (5) f CT y k,a,mine a = y k,a,maxe + a : cyclic schedule k K,a A (6) π h k,f = Q k x 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) Itinerary-based fleet assignment Spill and recapture 9/ 19

10 Integrated model - Revenue management δ i,f d i s S h i (Is \I s ) j Is i j δ i,f t i,j + j Is j (Is \I s ) i j δ i,f t j,i b j,i π h k,f : capacity h H,f F (10) k K t i,j d i : total spill h H,s S h,i (I s \ I s ) (11) exp(v d i = D i (p i,z i,β)) s exp(v j (p j,z j,β)) : logit demand h H,s Sh,i I s (12) j Is b i,j = exp(v j (p j,z j,β)) exp(v k (p k,z k,β)) : recapture ratio h H,s Sh,i (I s \ I s ),j I s (13) k Is \{i} d i d i : realized demand h H,s S h,i I s (14) LB i p i UB i : bounds on price h H,s S h,i I s (15) t i,j 0 h H,s S h,i (I s \ I s ),j I s (16) b i,j 0 h H,s S h,i (I s \ I s ),j I s (17) π h k,f 0 h H,k K,f F (18) 10/ 19

11 Integrated model The resulting model is a mixed integer nonlinear problem Nonlinearity is due to the explicit supply-demand interactions The model is implemented in AMPL and BONMIN solver is used BONMIN does not guarantee optimality We consider a sequential approach as a reference model to evaluate the integrated model: Fleet assignment is optimized with estimated demand/price Revenue is optimized with the resulting capacity 11/ 19

12 Added value of the integration - Sequential vs integrated Sequential approach (SA) Integrated model - % Change Profit Pax. Flights Seats Profit Pax. Flights Seats 1 15, , % 33.50% , , % % , % 14.18% ,311 1, ,186 1, % -3.80% ,054 1, , % , ,656 1, ,983 2, ,920 2, % -0.97% ,902 1, , % 5.83% 10 1, , , % 16.69% , ,100 1, % -2.72% ,428 1, % 4.94% ,347 1, % 1.40% ,251 1, Data instances are derived from ROADEF 2009 dataset.

13 Heuristic method Available solvers are able to converge on instances with up to 4 airports and about 35 flights. We devised a heuristic procedure based on two simplified versions of the model: FAM LS : price-inelastic schedule planning model MILP Explores new fleet assignment solutions based on a local search Price sampling Variable neighborhood search (VNS) REV LS : Revenue management with fixed capacity NLP Optimizes the revenue for the explored fleet assignment solution 13/ 19

14 Heuristic method Require: x 0, y 0, d 0, p 0, t 0, b 0, π 0, time max, n min, n max, notimpr, tabulistsize g := 0, time := 0, n fixed := n min, notimpr := 0, z := INF, tabulist := /0 repeat p g := Price sampling(t g 1, p g 1, d g 1 ) {d g,b g } := Logit model(p g ) L := Fixing(x g 1, t g 1, n fixed ) {x g,y g,π g,t g } := solve z FAM LS (p g,d g,b g,l) if ( x g / tabulist) then tabulist := tabulist x g {p g,d g,b g,π g,t g } := solve z REV LS (x g,y g ) if (z REV LS z ) then Update z Intensification: n fixed := n fixed + 1 when n fixed < n max notimpr := 0 else if (notimpr == 3) then Diversification: n fixed := n fixed 1 when n fixed > n min notimpr := notimpr 1 end if end if g := g + 1 until time time max 14/ 19

15 Local search Neighborhood solutions are visited based on the spill rather than a fully random search Price sampling: A random price is drawn for each itinerary If the spilled passengers are higher than the average decrease the price Otherwise increase the price Fixing FAM solutions - VNS: The itineraries are sorted according to their spilled number of passengers Low spill value associated flights have a higher probability to be fixed to their current aircraft If the solution is improved more assignments are fixed and vice versa. 15/ 19

16 Performance of the heuristic The omitted instances are the ones where the sequential approach has the same solution as the integrated model. SA Integrated model Best solution by BONMIN Heuristic - Avg. over 5 runs Flights Profit Profit Time (sec) %dev. from %imp. Time (sec) %time max 43,200 BONMIN over SA max 3,600 red ,372 37, % 5.55% % ,990 46,037 2, % 4.66% % ,901 70,904 2, % 1.11% % ,186 87,212 42, % 3.59% % , ,791 12, % 0.30% % ,920 94,203 1, % 0.30% % , ,544 7, % 0.43% % , ,575 37, % 0.83% % ,347 96,486 17, % 3.36% % ,100 38, % 2.98% % ,369 53, % 1.45% % , ,467 31, % 0.71% % , ,434 4, % 0.91% 1, % , ,789 4, % 0.17% % , ,364 22, % 0.30% 1, % , ,598 42, % 0.88% % , ,731 31, % 0.29% 1, % 16/ 19

17 Improvement due to the local search SA Random Neighborhood % Improvement neighborhood based on spill Profit Profit Time(sec) Profit Time(sec) Quality of Reduction the solution in time 2 35,372 37, , % 4 43,990 44, , % ,901 No imp. over SA 70, % ,186 85,335 1,649 87, % , , , % 11 93,920 No imp. over SA 94, % ,902 No imp. over SA 858, % ,428 No imp. over SA 138, % ,347 96, , % ,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, % - 17/ 19

18 Conclusions and future work Integrated schedule planning and revenue management More efficient schedule planning with the information on supply-demand interactions Heuristic Obtaining upper bound in order to more appropriately quantify the performance of the heuristic Further solution methods for the resulting mixed integer nonlinear problem Decomposition methods FAM and REV models 18/ 19

19 Thank you for your attention! 19/ 19

20 Clip-Air 20/ 19