Airline Network Revenue Management by Multistage Stochastic Programming

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Airline Network Revenue Management by Multistage Stochastic Programming K. Emich, H. Heitsch, A. Möller, W. Römisch Humboldt-University Berlin, Department of Mathematics Page 1 of 18 GOR-Arbeitsgruppe Revenue Management and Dynamic Pricing, Grünwald bei München, 13.2.2009

Introduction Airline revenue management deals with strategies for controlling the booking process within a network of flights under stochastic demand with the objective of maximizing the expected revenue. Often the booking process is controlled by seat protection levels or (so-called) bid prices. Aims: Development of a scenario-based stochastic programming model for airline network revenue management; Approximate representation of the multivariate booking demand processes by scenario trees; Numerical computations for the node-based stochastic integer program (using CPLEX); Lagrangian decomposition of the node-based stochastic integer program; algorithm design and numerical experience. Page 2 of 18

Notation Input data π n : probability of node n; stochastic (as scenario tree): : passenger demand; d n i,j,k γi,j,k n : cancelation rates; deterministic: f b i,j,k,t(n) : fares; f c i,j,k,t(n) : refunds; C l,m : capacity; Variables b n i,j,k : bookings; c n i,j,k : cancelations; : cumulative bookings; Bi,j,k n Ci,j,k n Pi,j,k n : cumulative cancelations; : protection level; : slack variables; : auxiliary integer variables; z P,n i,j,k, zd,n i,j,k z n i,j,k Indices t = 0,..., T : data collection points; i = 1,..., I: origin-destination-itin.; j = 1,..., J: fare classes; k = 1,..., K: points of sale; l = 1,..., L: legs; I l : index set of itineraries; m = 1,..., M(l): compartments; J m (l): index set of fare classes; n = 0,..., N: nodes; t(n): time of node n; n : preceding node of node n; Time horizon and data collection points (dcp): Day of Departure Booking Horizon Booking Interval 0 1... T DCPs Page 3 of 18

Approximation of the demand by scenario trees The passenger demand and cancelation rate process {ξ t } T t=0 = {(d i,j,k,t, γ i,j,k,t )} T t=0 is approximated by a finite number of scenarios forming a scenario tree. n = 0 t = 0 ξ n n N+ (n) n N T t = 1 t(n) T Scenario tree with T = 4, N = 22 and 11 leaves n = 0 root node, n unique predecessor of node n, path(n) = {1,..., n, n}, t(n) := path(n), N + (n) set of successors to n, N T := {n N : N + (n) = } set of leaves, path(n), n N T, scenario with (given) probability π n, π n := ν N + (n) πν probability of node n, ξ n realization of ξ t(n). Page 4 of 18

Generation of multivariate scenario trees General strategy: (i) Development of a stochastic model for the data process ξ (parametric [e.g. time series model], nonparametric [e.g. resampling from statistical data]) and generation of simulation scenarios; (ii) Construction of a scenario tree out of the simulation scenarios by recursive scenario reduction and bundling over time such that the optimal expected revenue stays within a prescribed tolerance. Page 5 of 18 Implementation: GAMS-SCENRED

t = 1 t = 2 t = 3 t = 4 t = 5 t = 1 t = 2 t = 3 t = 4 t = 5 t = 1 t = 2 t = 3 t = 4 t = 5 Page 6 of 18 t = 1 t = 2 t = 3 t = 4 t = 5 t = 1 t = 2 t = 3 t = 4 t = 5 t = 1 t = 2 t = 3 t = 4 t = 5 Illustration of the forward tree construction for an example including T=5 time periods starting with a scenario fan containing N=58 scenarios <Start Animation>

Airline network revenue management model (node representation) Objective max (P n i,j,k ) { N J K π n I n=0 i=1 j=1 k=1 [ ] } fi,j,k,t(n) b bn i,j,k f i,j,k,t(n) c cn i,j,k Constraints Cumulative bookings Bi,j,k 0 := B i,j,k 0 ; C0 i,j,k := C i,j,k 0 ; Bn i,j,k := Bn i,j,k + bn i,j,k Cumulative cancelations C n i,j,k = γn i,j,k Bn i,j,k + 0.5 Passenger demands and protection levels b n i,j,k dn i,j,k ; bn i,j,k P n i,j,k Bn i,j,k + Cn i,j,k Leg capacity limits K i I l j J m (l) k=1 P n i,j,k C l,m (n N T 1 ) Cancelations c n i,j,k = Cn i,j,k Cn i,j,k (disjunctive constraints) Integrality and nonnegativity constraints B n i,j,k, Cn i,j,k, P n i,j,k Z; bn i,j,k 0; cn i,j,k 0 Page 7 of 18

Airline network revenue management model (final) Objective max (P n i,j,k ) { N J K π n I n=0 i=1 j=1 k=1 [ ] } fi,j,k,t(n) b bn i,j,k f i,j,k,t(n) c cn i,j,k Constraints Cumulative bookings Bi,j,k 0 := B i,j,k 0 ; C0 i,j,k := C i,j,k 0 ; Bn i,j,k := Bn i,j,k + bn i,j,k Cumulative cancelations C n i,j,k = γn i,j,k Bn i,j,k + 0.5 Passenger demands b n i,j,k + zb,n i,j,k = dn i,j,k Cancelations c n i,j,k = Cn i,j,k Cn i,j,k Protection levels Bi,j,k n Cn i,j,k + zp,n i,j,k = P n i,j,k Number of bookings (disjunctive constraints) (κ > 0, adequately large) 0 z b,n i,j,k (1 zn i,j,k )dn i,j,k 0 z P,n i,j,k zn i,j,k κ zn i,j,k {0, 1} Leg capacity limits K i I l j J m (l) k=1 P n i,j,k C l,m (n N T 1 ) Integrality and nonnegativity constraints B n i,j,k, Cn i,j,k, P n i,j,k Z; bn i,j,k 0; cn i,j,k 0 Page 8 of 18

Comments: large scale structured integer linear program solvable by a standard solver (e.g. CPLEX) in reasonable time for smaller networks when neglecting integer constraints Dimensions: (S number of scenarios) 4IJKN continuous variables, IJK(N + 1 S) + 2IJKN integer (neglected) variables, IJKN binary variables 7IJK(N 1) + L n N T 1 l=1 M(l) constraints Protection levels (P n i,j,k ) n N have the same tree structure as the input data The (deterministic) protection levels of the first stage may be taken as a basis for the computer reservation system At the next dcp a new scenario tree has to be generated and the problem is resolved etc. Page 9 of 18

Airline revenue management: Numerical example Hub-and-Spokes Network #Legs 6 #ODIs 12 #Compartments 2 #Fare Classes 6 #POS 1 #DCPs 14 A H Hub C B Page 10 of 18 Tree and Size #Scenarios 92 #Nodes 1017 #cont. Variables 506.016 #bin. Variables 73.224 #Constraints 513.660

Data and numerical results dcp 0 1 2 3 4 5 6 7 8 9 10 11 12 13 d 182 126 84 56 35 21 14 10 7 5 3 2 1 0 Data collection points and days to departure The booking demand process is modeled by a non-homogeneous Poisson process. The cumulative booking requests B i,j,k (t) are independent and given by B i,j,k (t) = G i,j,k t 0 Γ(a i,j,k + b i,j,k ) ( τ ) ai,j,k 1 ( 1 τ ) bi,j,k 1 dτ, T (Γ(a i,j,k ) + Γ(b i,j,k )) T T where the integrands correspond to Beta distributions and the total number of booking requests G i,j,k is assumed to have a Gamma distribution. Page 11 of 18 800 Fares ODI:12 14 Optimal Protection Level of the First Stage ODI: 12 600 12 10 400 8 6 200 4 2 0 B1 B2 E1 E2 E3 E4 0 B1 B2 E1 E2 E3 E4

Numerical Results (continued) CPLEX-Results Version 8.1 MIP Gap 0.001 Solution Status Optimal #MIP Nodes passed 0 (root) 70 1 Cancellation Rate Cumulative Demand ODI: 12 Fare Class: 3 60 50 50 40 Computing time Total: 3 : 29.8 min (Intel Celeron, 2.0 GHz, Linux) Protection Level ODI: 12 Fare Class: 3 40 30 20 10 30 20 10 Page 12 of 18 2 4 6 8 10 12 2 4 6 8 10 12 70 1 Cancellation Rate Cumulative Demand ODI: 12 Fare Class: 6 60 50 50 40 Protection Level ODI: 12 Fare Class: 6 40 30 20 10 2 4 6 8 10 12 30 20 10 2 4 6 8 10 12

Lagrangian decomposition in airline revenue management Idea: Dualization of leg capacity limits Lagrangian function Λ: N I J Λ(λ, P ) := π n = = n=0 + i=1 n N T 1 π n I i=1 J j=1 I i=1 j=1 L l=1 K k=1 ( K N k=1 n N T 1 π n J j=1 n=0 M(l) m=1 ( ) f b,n i,j,k bn i,j,k f c,n i,j,k cn i,j,k λ n l,m i Il j J m (l) k=1 π n ( f b,n i,j,k bn i,j,k f c,n i,j,k cn i,j,k M(l) l L i m=1 δ j,l,m λ n l,mp n K Λ i,j,k (λ, P i,j,k ) + k=1 where L i = {l : i I l } and δ j,l,m = i,j,k + n N T 1 π n { 1 j Jm (l) 0 otherwise K C l,m P n ) n N T 1 π n L l=1 M(l) m=1 i,j,k L l=1 λ n l,mc l,m M(l) m=1 λ n l,mc l,m Page 13 of 18

Dual function D: D(λ) = sup Λ(λ, P ) P I J K = sup Λ i,j,k (λ, P i,j,k ) + P i,j,k i=1 j=1 k=1 n N T 1 π n L l=1 M(l) m=1 The function D is convex nondifferentiable and decomposable. Dual problem: inf D(λ) λ The relative duality gap is small (theory by Bertsekas 82). Subgradients: [ D(λ)] n l,m = π n C l,m i Il K j J m (l) k=1 Pi,j,k n λ n l,mc l,m Page 14 of 18 The Lagrange multipliers λ n l,m, n N t, may be interpreted as bid prices at t for leg l and compartment m. However, they are presently only available for n N T 1.

Dual solution algorithm Algorithm: Solve the dual problem Compute function values and subgradients Search for a primal feasible solution Solve decomposed subproblems Solution of the dual problem by a bundle subgradient method (e.g. proximal bundle method by Kiwiel or Helmberg) Solution of the subproblems by dynamic programming on scenario trees. Primal-proximal heuristic to determine a good primal feasible solution (e.g. by Daniilidis and Lemaréchal). Numerical results (computing times): Solving the Dual 4:52 min Heuristic 5:16 min Page 15 of 18

A realistic mid-size airline network example 25 20 15 10 5 0 0 10 20 30 40 50 ODI-Leg-Matrix RM problem dimensions #ODIs 54 #ODI-Fareclass-POS 489 #Legs 27 #Leg-Compartments 54 #DCPs 23 Tree and Size #Scenarios 98 #Nodes 1.441 #Variables 3.473.367 #Constraints 2.774.445 #Coupling Constr. 5.238 Page 16 of 18

Conclusions and future work We presented an approach to airline network revenue management using a scenario tree-based dynamic stochastic optimization model. The approach starts from a finite number of demand scenarios and their probabilities, requires no assumptions on the demand distributions except their decision-independence. Stochastic programming approaches lead to solutions that are more robust with respect to perturbations of input data. However, the models have higher complexity. Future work: Implementation refinements of the decomposition scheme Numerical test runs for mid-size networks Page 17 of 18 (URL: www.math.hu-berlin.de/~romisch, Email: romisch@math.hu-berlin.de)

References A. Daniilidis, C. Lemaréchal: On a primal-proximal heuristic in discrete optimization, Mathematical Programming 104 (2005), 105 128. D. Dentcheva, W. Römisch: Duality gaps in nonconvex stochastic optimization, Mathematical Programming 101 (2004), 515 535. K. Emich: Optimale Ertragssteuerung einer Fluggesellschaft unter unsicherer Nachfrage mittels Lagrange-Relaxation, Diplomarbeit, Humboldt- Universität Berlin, 2007. H. Heitsch, W. Römisch: Scenario tree modeling for multistage stochastic programs, Mathematical Programming 118 (2009), 371 406. Page 18 of 18 A. Möller, W. Römisch, K. Weber: Airline network revenue management by multistage stochastic programming, Computational Management Science 5 (2008), 355 377. K.T. Talluri, G.J. van Ryzin: The Theory and Practice of Revenue Management, Kluwer, Boston 2004.

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