Role of Stochastic Optimization in Revenue Management. Huseyin Topaloglu School of Operations Research and Information Engineering Cornell University

Transcription

1 Role of Stochastic Optimization in Revenue Management Huseyin Topaloglu School of Operations Research and Information Engineering Cornell University

2 Revenue Management Revenue management involves making most use of limited inventories of perishable resources under random demand Airlines form a typical application setting Resources correspond to capacities on flight legs Customers arrive randomly over time Other application settings include hotels, car rentals, advertising, fashion retail Primary tradeoff is whether to give resources to a currently available customer that is willing to pay a low price or keep the resources with the hope of a future customer that may be willing to pay a high price

3 Revenue Management Capacity Allocation Dynamic Pricing $ $$ $ $$$ $$ $ $ Assortment Offering Decision Mechanism

4 Revenue Management Static, Single Resource Dynamic, Single Resource Dynamic, Network of Resources Modeling Detail

5 Revenue Management Static Single Leg Dynamic Single Leg Dynamic Network Capacity Allocation Dynamic Pricing Assortment Offering

6 Network Revenue Management Requests arrive over time Make an acceptance or rejection decision for each request Accepted requests consume capacities on one or more flight legs

7 Deterministic Linear Programming Approximation L J p jt f j a ij c i : Set of flight legs in the airline network : Set of itineraries : Probability of getting a request for itinerary j at time t : Revenue from itinerary j : 1 if itinerary j uses flight leg i 0 otherwise : Total available capacity on flight leg i bookings start flights depart

8 Deterministic Linear Programming Approximation Assume that the numbers of itinerary requests take on their expected values Z LP =max st f j w j a ij w j c i i L (µ i) 0 w j t T p jt

9 Deterministic Linear Programming Approximation {µ i :i L} are called bid prices They characterize the value of a seat Accept a request for itinerary j if f j i La ij µ i...subject to capacity availability As time progresses, it is customary to refresh the bid prices by resolving the linear program Optimal objective value of the linear program provides an upper bound on the performance of any nonancitipatory policy

10 Allocations Decomposition by Fare Decomposition by Displacement Deterministic Linear Program Adjustments

11 Dynamic Programming Formulation L J p jt f j a ij x it u jt : Set of flight legs in the airline network : Set of itineraries : Probability of getting a request for itinerary j at time t : Revenue from itinerary j : 1 if itinerary j uses flight leg i 0 otherwise : Remaining capacity on flight leg i at time t : 1 if a request for itinerary j is accepted at time t 0 otherwise

12 Dynamic Programming Formulation V t (x t )=max { } p jt f j u jt +V t+1 (x t u jt i L a ije i ) st a ij u jt x it u jt {0,1} i L, To obtain approximations to value functions, decompose the dynamic programming formulation by the flight legs

13 Decomposition by Fare Allocations f j

14 Decomposition by Fare Allocations λ i j λ i j f j λ i j λ ij : Fare allocation of itinerary j over flight leg i λ ij =f j i L

15 Decomposition by Fare Allocations Revenue allocations allow us to solve single-leg problems vt i (x } it λ)=max p jt {λ ij u jt +vt+1 i (x it a ij u jt λ) Single-leg problems provide an upper bound on exact value functions vt i (x it λ) V t (x t ) as long as st a ij u jt x it u jt {0,1} i L i L λ ij =f j

16 Decomposition by Fare Allocations We obtain upper bounds on the optimal expected revenue vt i (x it λ) V t (x t ) v1 i (c i λ) V 1 (c) i L i L To obtain the tightest possible upper bound, we solve min v1 i (c i λ) convex in λ λ i L st λ ij =f j i L Tightest possible upper bound is better than the one from deterministic linear programming approximation Z LP i Lv i 1 (c i λ ) V 1 (c)

17 Computing Good Fare Allocations We can obtain a good revenue allocation from deterministic linear programming approximation max f j w j st a ij w j c i i L 0 w j t T p jt

18 Computing Good Fare Allocations We can obtain a good revenue allocation from deterministic linear programming approximation max f j w ψj st a ij w ij c i i L 0 w ij t T w ψj w ij =0 p jt i L,, i L

19 Computing Good Fare Allocations We can obtain a good revenue allocation from deterministic linear programming approximation max f j w ψj st a ij w ij c i i L 0 w ij t T p jt i L, w ψj w ij =0, i L (β ij ) i L β ij =f j

20 Making Decisions If we have the exact value functions, then we can make the decisions at time t by solving { } V t (x t )=max p jt f j u jt +V t+1 (x t u jt i L a ije i ) st a ij u jt x it u jt {0,1} i L, It is optimal to accept a request for itinerary j at time t when there is enough capacity and f j +V t+1 (x t i L a ije i ) V t+1 (x t )

21 Making Decisions If we have the exact value functions, then we can make the decisions at time t by solving { } V t (x t )=max p jt f j u jt +V t+1 (x t u jt i L a ije i ) st a ij u jt x it u jt {0,1} i L, It is optimal to accept a request for itinerary j at time t when there is enough capacity and f j V t+1 (x t ) V t+1 (x t i L a ije i )

22 Making Decisions We accept a request for itinerary j at time t when there is enough capacity and f j i L v i t+1(x it λ) i Lv i t+1(x it a ij λ) f j i L ] a ij [vt+1 i (x it λ) vt+1 i (x it 1 λ) bid price of flight leg i

23 Numerical Performance An airline network with one hub serving multiple spokes There is a high-fare and a low-fare itinerary connecting each origin-destination pair Compare deterministic linear programming approximation (DLP), decomposition with fare allocations from the deterministic linear program (DFA-DLP) and decomposition with best fare allocations (DFA-OPT)

24 Comparing Upper Bounds 20 DFA-DLP DFA-OPT 15 low medium high low medium high % gap with DLP spokes 8 spokes

25 Comparing Expected Revenues 20 DFA-DLP DFA-OPT 15 low medium high low medium high % gap with DLP spokes 8 spokes

26 Allocations Decomposition by Fare Decomposition by Displacement Deterministic Linear Program Adjustments

27 Decomposition by Displacement Adjustments Start from the deterministic linear program to decompose the dynamic programming formulation Z LP =max st f j w j a ij w j c i i L (µ i) 0 w j p jt t T

28 Decomposition by Displacement Adjustments Start from the deterministic linear program to decompose the dynamic programming formulation Z LP =max st [f j a kj µ k ]w j + µ k c k a ij w j c i k L\{i} k L\{i} 0 w j p jt t T

29 Decomposition by Displacement Adjustments We can solve the dynamic programming formulation of the single-leg problem over flight leg i v i t (x it)=max {[ p jt f j ] } a kj µ k u jt +vt+1 i (x it a ij u jt ) st a ij u jt x it u jt {0,1} k L\{i} Single-leg problem provides an upper bound on exact value functions vt i (x it)+ µ k x kt V t (x t ) k L\{i}

30 Decomposition by Displacement Adjustments Upper bounds are tighter than the one from the deterministic linear programming approximation Z LP v i 1(c i )+ k L\{i} µ kc k V 1 (c) Apply the same idea for each flight leg i and approximate the value function V t (x t) by vt i (x it) i L

31 Numerical Performance An airline network with one hub serving multiple spokes There is a high-fare and a low-fare itinerary connecting each origin-destination pair Compare deterministic linear programming approximation (DLP), decomposition with fare allocations from the deterministic linear program (DFA-DLP), decomposition with best fare allocations (DFA-OPT) and decomposition by displacement adjustments (DDA)

32 Comparing Upper Bounds 20 DFA-DLP DFA-OPT DDA 15 low medium high low medium high % gap with DLP spokes 8 spokes

33 Comparing Expected Revenues 20 DFA-DLP DFA-OPT DDA 15 low medium high low medium high % gap with DLP spokes 8 spokes

34 Directions for Improvement We can obtain better policies by using value function approximations that are non-separable by flight legs It is important to incorporate additional constraints on various performance measures, such as occupancy Maximizing expected revenue is reasonable when dealing with large airlines, but incorporating risk becomes important for smaller businesses Number of dimensions of the state vector becomes even larger when we explicitly deal with overbooking and reservation fees Deterministic problem becomes large when we deal with dynamic pricing and assortment planning

35 Partial List of References Talluri, K. T. and van Ryzin, G. J. (2004), The Theory and Practice of Revenue Management, Kluver Academic Publishers. (Book) Phillips, R. L. (2005), Pricing and Revenue Optimization, Stanford University Press, Stanford, CA. (Book) Simpson, R. W. (1989), Using network flow techniques to find shadow prices for market and seat inventory control, Technical report, MIT Flight Transportation Laboratory Memorandum M89-1, Cambridge, MA. (Network, capacity allocation) Williamson, E. L. (1992), Airline Network Seat Control, PhD thesis, Massachusetts Institute of Technology, Cambridge, MA. (Network, capacity allocation) Brumelle, S. L. and McGill, J. I. (1993), "Airline seat allocation with multiple nested fare classes", Operations Research 41, (Single-leg, capacity allocation) Gallego, G. and van Ryzin, G. (1994), Optimal dynamic pricing of inventories with stochastic demand over finite horizons", Management Science 40(8), (Single-leg, dynamic pricing) Gallego, G. and van Ryzin, G. (1997), A multiproduct dynamic pricing problem and its applications to yield management, Operations Research 45(1), (Network, dynamic pricing) Talluri, K. and van Ryzin, G. (1998), An analysis of bid-price controls for network revenue management, Management Science 44(11), (Network, capacity allocation) Talluri, K. and van Ryzin, G. (1999), A randomized linear programming method for computing network bid prices, Transportation Science 33(2), (Network, capacity allocation) Bertsimas, D. and Popescu, I. (2003), Revenue management in a dynamic network environment, Transportation Science 37, (Network, capacity allocation, overbooking) Talluri, K. and van Ryzin, G. (2004), "Revenue management under a general discrete choice model of consumer behavior", Management Science 50(1), (Single-leg, assortment planning)

36 Partial List of References Gallego, G., Iyengar, G., Phillips, R. and Dubey, A. (2004), Managing flexible products on a network, Computational Optimization Research Center Technical Report TR , Columbia University. (Network, assortment planning) Karaesmen, I. and van Ryzin, G. (2004), "Overbooking with substitutable inventory classes", Operations Research 52(1), (Network, overbooking) van Ryzin, G. and Vulcano, G. (2008), "Computing virtual nesting controls for network revenue management under customer choice behavior", Manufacturing & Service Operations Management 10(3), (Network, assortment planning) Zhang, D. and Cooper, W. L. (2005), "Revenue management for parallel flights with customer choice behavior", Operations Research 53(3), (Network, assortment planning) Adelman, D. (2007), "Dynamic bid-prices in revenue management", Operations Research 55(4), (Network, capacity allocation) Liu, Q. and van Ryzin, G. (2008), "On the choice-based linear programming model for network revenue management", Manufacturing & Service Operations Management 10(2), (Network, assortment planning) Kunnumkal, S. and Topaloglu, H. (2008), "A refined deterministic linear program for the network revenue management problem with customer choice behavior", Naval Research Logistics Quarterly 55(6), (Network, assortment planning) Topaloglu, H. (2009), "Using Lagrangian relaxation to compute capacity-dependent bid-prices in network revenue management", Operations Research 57(3), (Network, capacity allocation) Zhang, D. and Adelman, D. (2009), "An approximate dynamic programming approach to network revenue management with customer choice", Transportation Science 42(3), (Network, assortment planning)