A Statistical Modeling Approach to Airline Revenue. Management

Transcription

1 A Statistical Modeling Approach to Airline Revenue Management Sheela Siddappa 1, Dirk Günther 2, Jay M. Rosenberger 1, Victoria C. P. Chen 1, 1 Department of Industrial and Manufacturing Systems Engineering The University of Texas at Arlington Campus Box Arlington, TX USA 2 Sabre Research Group Bebelstrasse Witten GERMANY June 30,

2 Abstract Revenue management (RM) aims to maximize a company s revenue by allocating the right seat to the right customer. In this paper, we present an approach based on a Markov decision process (MDP) formulation. Our approach involves an off-line phase that derives a policy for accepting/rejecting customer booking requests, and an on-line phase that conducts the actual decisions as the booking requests arrive. To enable a computationally-tractable solution method, the off-line phase consists of three components: (1) identification of realistic ranges of remaining seat capacity at different points in time, (2) solutions to deterministic and stochastic linear programming problems that provide upper and lower bounds, respectively, on the MDP value function, and (3) estimation of the upper and lower bound value functions using statistical modeling. This value function approximation is then used to determine the RM accept/reject policy. Prior versions of this statistical modeling approach have employed remaining seat capacity ranges from zero to the capacity of the aircraft. In reality, actual remaining capacities are near capacity when the booking process begins and near zero when the flights depart. Thus, our modified version uses realistic ranges to enable a more accurate statistical model, leading to a better RM policy. 2

3 Before deregulation in 1979, airlines were managed by the Civil Aeronautics Board (Bailey et al. 1985), which dictated the routes to be flown and the fares to be charged to the customers. The costs were passed on to the passengers with guaranteed profit levels to the airlines. Carriers simply accepted the passengers on a first-come-first-serve basis. There were limited booking classes, and there was little control over revenues. After deregulation, there was tremendous growth in the number of certified airlines and high pressure on pricing. Airline carriers began to explore ways to compete effectively, and different approaches in revenue management (RM) evolved. Since then, airlines have expanded their efforts in RM to increase their revenue. Revenue management, also known as yield management, is defined as, Selling the right seat at the right time to the right passenger for the right price (Ben 1995). RM is applied in various transportation sectors, such as auto rentals, ferries, rail, tour operators, cargo, and cruises. Other areas, like hotel/resorts, extended stay hotel, health care, manufacturing apparel, and companies that produce perishable goods etc., also use RM (Bodily and Pfeifer 1973). Airlines have developed a complex and diverse fare structure. They offer a variety of fares to meet different classes of customers. Airlines use restrictions to establish different classes of services. For example, customers who buy tickets well in advance (21 days) will get the tickets for a lower price than customers who buy the ticket on the day of departure. Seats on the same flight are sold at different fares to different customers. There is competition among the airline carriers to expand and explore RM to improve their revenue. American Airlines, for example, reported an increase in revenue of 5% due to improved RM methods in 1992, which translated to $1.4 billion over a 3-year period (Smith et al. 1992). 1 Revenue Management Process Overview A leg is a flight that travels non-stop from an origin to a destination. In an airline reservation system, customers request a particular itinerary, which consists of one or more legs. The booking process typically starts three months prior to the date of departure. A customer makes a booking request by bidding a price for a desired itinerary. Once the request is placed, an airline representative uses a computer reservation system to decide if the request is to be accepted or rejected. The customer s requested price is compared with a threshold calculated by the airline that represents the fair market value. If the customer s bid is higher than the airline s threshold value, then the request is accepted; otherwise, the request is rejected. Usually airlines update their prices at certain specific dates during the booking process called 3

4 reading dates. These dates get closer to each other as the day of departure gets closer (see Figure: 1). Booking requests rejected for any demand class due to unavailability or filled seats are called spilled demand. Figure 1 about here. 1.1 Problem Definition Airline RM deals with managing the inventory of the demand classes offered on each itinerary, so as to maximize revenue. In this paper, we concentrate on the seat inventory control problem, also called the yield management problem. Given the flight capacities and schedule, can we accept the request placed by a customer at time τ during the booking process? Our approach seeks to achieve a better RM policy by using statistical methods to approximate the value functions of a Markov decision process (MDP). The value function approximation is conducted off-line and makes use of deterministic and stochastic linear programming problems from the RM literature. In the next section, we provide the relevant background on airline RM. The statistical modeling approaches are described in Sections 3 and 4. Our computational results on a real airline hub are presented in Section 5, and concluding remarks are given in Section 6. 2 RM Methodology The research in this paper is based on the statistical modeling approach of Chen et al. (2003) and Günther (1998). They formulated the RM model as an MDP, similar to that of Lautenbacher and Stidham (1999). Traditionally MDP has been solved using Stochastic Dynamic Programming (SDP), which provides a superior RM policy, but SDP is computationally intensive. Hence, Chen et al. (2003) developed a new statistical modeling approach, motivated by the Orthogonal Array (OA) and Multivariate Adaptive Regression Splines (MARS) SDP method of Chen et al. (1999), to estimate upper and lower bounds of the MDP value functions. The bounds are estimated at reading dates, these dates divide the booking period into smaller intervals called reading periods. 4

5 2.1 RM Literature and Background Littlewood (1972) was the first to address the RM problem of computing booking limits for a single leg with two demand classes. His rule: Sell the discount seats as long as the revenue from the low fare passengers is greater than or equal to the product of marginal revenue from full fare and probability that full fare demand does not exceed the remaining capacity. Belobaba (1987) extended this rule to multiple demand classes. He introduced the term EMSR (Expected Marginal Seat Revenue). The EMSR method produces optimal booking limits for the two-demand class problem, and it is easy to implement RM as an MDP In the airline booking process, the decision to accept or reject a current booking request (BR) depends on the remaining seat capacity, the time the request was placed, the itinerary and demand class requested, and other characteristics of the current request, but it does not depend on decisions made about previous booking requests. Hence, RM can be classified as an MDP (Lautenbacher and Stidham 1999). The MDP formulation for the RM problem divides the three-month booking period into t MDP time intervals, with at most one booking request per interval. These intervals are indexed in decreasing order, i = t MDP,..., 1, 0, where i = 1 denotes the first interval immediately preceding departure, and i = 0 is at departure. Each reading period can have multiple booking requests while each MDP interval can have at most one booking request. The state vector x i holds the remaining leg capacities at the beginning of time interval i. Let p f i (g) denote the probability that a request for g seats, for demand class f occurs in time interval i; p i(0) denotes the probability of no booking requests in time interval i; and G f is the maximum size of a group request for demand class f. Suppose the booking process is in state x at the beginning of time interval i. If a booking request for g seats that arrives during time interval i is accepted, then a new state x is reached at the beginning of time interval i 1, where x subtracts g seats from the legs involved in the requested itinerary and is greater than or equal to zero. Let F i (x), for x 0, denote the optimal value function, the maximum expected revenue collected over time intervals i through departure when the system is at state x at the beginning of time interval i. Then F 0 (x) = 0 for all x. Thus, the MDP 5

6 value functions can be written as: G m f F i (x) = p f i (g) f=1 g=1 max{gr f + F i 1 (x ), F i 1 (x)}, if (x 0), F i 1 (x), otherwise. The fair market value (FMV) of a group of requested seats is defined as the difference in the value function of rejecting the request versus accepting the request: FMV = F i 1 (x) F i 1 (x ), where x Dynamic Programming Approach Ladany and Bedi (1977) and Hersh and Ladany (1978) developed dynamic programming formulations to allocate seats for two flight legs. They discuss overbooking and cancellations for flights with one intermediate stop. Ladany and Bedi (1977) simplified the approach by removing all conditioning on current bookings. Rothstein (1971) formulated the RM problem as a nonhomogeneous Markovian sequential decision process considering overbooking. Lee and Hersh (1993) developed a discrete-time dynamic model to find an optimal booking policy. Their analysis showed that for problems with more than two booking classes and no multiple seat bookings, the optimal booking policy can be reduced to two sets of critical values: (1) booking capacity and (2) decision periods. Lautenbacher and Stidham (1999) solved the single-leg problem without overbooking using a discrete-time MDP. They link the dynamic customers of different demand classes book at the same time and static demand for different demand classes arrives separately in a predetermined order through a dynamic program common to both. Subramanian et al. (1999) took overbooking, cancellations and no shows into consideration while solving for the seat allocation problem using the MDP for a single flight leg with multiple demand classes. They showed that: (1) booking limits need not be monotonic in the time remaining until departure, (2) it would be optimal to accept a low-demand class and reject a high demand class passenger because of differing cancellation refunds, and (3) the optimal policy depends upon both the total capacity and the remaining capacity of the flight. Zhang and Cooper (2005) formulated a simultaneous seat-inventory control problem of a set of parallel flights between a common origin and destination with dynamic customer choice among the flights as an extension of the classic multiperiod, single-flight block demand revenue management model. They proposed a simulation-based techniques for solving the stochastic optimization problem. 6

7 2.1.3 Bid Price Approach Bid pricing is practiced by most of the airlines. In the bid price approach, a threshold or bid price is assigned to each flight leg. If a customer s booking request is greater than or equal to the sum of the bid prices along the desired itinerary, then the request is accepted; otherwise it is rejected; see Figure 2. Consider the following example: Suppose a customer bids $1220 for the itinerary he wishes to travel. Table 1 gives the threshold values set by the airlines for each of the legs on the itinerary. Since the total price bid by the customer ($1220) is greater than the sum of the bid prices for the legs $1109, the request is accepted. Figure 2 about here. Table 1 about here. Bid pricing is easy to implement and requires storage of only a single bid price for each flight leg. It gives a nested itinerary and demand class specific control policy, and it is easy to manage the inventory. Despite all these advantages it has the disadvantage that it is difficult to estimate/determine good bid prices. Frequent revisions are required with re-optimization and re-forecasting. Some of the methods used to estimate bid prices are discussed in Sections and Deterministic Bid Price Approach Define the following notation: T = the total number of reading dates, indexed by t, where t = T represents the first reading date. r = the vector of fares associated with each demand class. u = a seat allocation decision vector for all demand classes. x t = a vector of remaining seat capacities at reading date t. d t = a vector of remaining demand at reading date t. A = a 0-1 itinerary-leg matrix, with one if that itinerary includes the leg. 7

8 Given the number and position of reading dates, the flight schedule and capacities, a deterministic (DET) linear programming problem is solved at reading date t to obtain the seat allocations aggregated over reading dates t through departure. It is modeled as below. (DET) max ru (1) s.t. Au x t (2) 0 u E[d t ]. (3) The dual of this problem will provide bid prices for each flight leg. Gallego and Van Ryzin (1994) used a network model to compute bid prices. Every time a request is accepted, remaining seat capacity is updated, and at reading dates t < T updated capacity values are used to solve (DET) to generate new bid prices. Results show that revenue increases by increasing the number of the reading dates. The higher the number of reading dates, the higher the accuracy of the bid prices, and hence, more revenue can be captured. However, more reading dates requires more computation Stochastic Bid Price Approach In addition to the notation above, define the following: u ft = a seat allocation decision vector for demand class f at reading date t, where u t is the corresponding vector for all demand classes. d ft = a random variable of the demand for demand class f on reading date t. A stochastic model, also called the probabilistic nonlinear programming model (PNLP) is considered when demand is a random variable. This model is also referred to as the stochastic (STOCH) network model and is as given below. t m (STOCH) max r f E[min(d fτ, u fτ )] (4) τ=1 f=1 ( t ) s.t. A u τ x t (5) τ=1 u ft 0. (6) Again, the dual will provide bid prices for each flight leg. Talluri and Van Ryzin (1999) analyzed a randomized version of deterministic linear programming to compute network bid prices. Their method is more difficult to implement than the (DET) method. It consists of simulating 8

9 the itinerary demand with a sequence of realizations, and solving (DET) to allocate capacities to itineraries for each realization. The dual prices from the sequence are averaged to form a bid price approximation. Hersh and Ladany (1978) presented a two-stage stochastic programming model to overcome the shortcomings of the (DET) and (STOCH) models. The first stage allocates capacity to all the demand classes, and the second stage models capacity utilization. Their simulation results show that this provides better revenue improvements than a linear programming approach. They also prove that their approach is prone to less error than those resulting from the linear programming method Hybrid Approach Curry (1990) combined both the EMSR and mathematical programming approaches. The EMSR approach accounts for computerized reservation system nesting, but only controls seat inventory, by controlling leg bookings. Mathematical programming handles realistically large problems and accounts for multiple origin-destination (OD) itineraries and side constraints. Curry developed equations to solve the RM problem, when demand classes are nested on an OD itinerary, and inventory is not shared among the ODs. Cooper and Homem-de Mello (2006) studied policies that combine both mathematical programming and MDP methods. They employed a simple allocation policy when far from the time of departure and developed a detailed decision rule close to departure. They used sampling-based stochastic optimization methods to solve the formulation. The solution was capable of using deterministic optimization techniques. They employed an MDP solution for a portion of the booking process rather than approximations of MDP value functions. Their results showed that the hybrid policies perform well for two-leg problems, but their approach cannot be used for larger networks. Bertsimas and Boer (2005) developed an algorithm that addressed different issues, like demand uncertainty, nesting and the dynamic nature of the booking process. They combined a stochastic gradient algorithm and approximate dynamic programming ideas to improve the initial booking limits. Talluri and Van Ryzin have worked on developing discrete choice models for RM since Their aim is to capture the buy-up and buy-down behavior of the customers (Talluri and Van Ryzin to appear). 9

10 3 The Statistical Modeling Approach to RM Motivated by the successful application of orthogonal array experimental designs and multivariate adaptive regression splines in stochastic dynamic programming (Chen et al. 1999), Chen et al. (2003) proposed an MDP based OA-MARS approach to RM. In this approach, the RM problem is solved in two parts, off-line and on-line. The off-line or the statistical modeling module derives the RM accept/reject policy while the on-line or booking module simulates the actual decisions. Their model assumes: 1. The booking process starts ninety days before the day of departure. 2. Flight capacities and the flight schedule are known. 3. There is no overbooking or cancellation. 3.1 Statistical Modeling Module The steps involved in this module are: 1. The reading dates are chosen and remaining seat capacities are initially set equal to the flight capacities for the flight legs. 2. An OA experimental design is constructed to provide discretized coverage of the remaining seat capacity state space. The state space ranges from zero to the plane capacities of the flights in the network. 3. For each of the discretization points, the (DET) model and the (STOCH) model are solved. The (DET) model is proved to provide an upper bound on the MDP value function, denoted by Ft U (x) (Talluri and Van Ryzin 1998) and the (STOCH) model is proved to provide a lower bound, denoted by Ft L (x) (Günther 1998). This loop is repeated at all reading dates. 4. For each reading date, a MARS approximation is fit separately to estimate the (DET) and (STOCH) revenues over the entire state space. Thus a total 2T different MARS approximations are generated. Figure 3 illustrates the procedure followed in the statistical modeling module. The essential statistical models ˆF L t and ˆF U t are now made available for the on-line booking module. Figure 3 about here. 10

11 3.2 Booking Module A fair market value for a booking request of group size g, for demand class f at time τ is estimated using Pessimistic = ˆF L τ (x) ˆF U τ (x ) (7) Optimistic = ˆF U τ (x) ˆF L τ (x ) (8) Fair Market Value = Pessimistic + Optimistic 2 (9) In Figure 4, the RM policy is defined as, accept the booking request only if the requested fare is greater than the fair market value. Figure 4 about here. In the next section, we identify appropriate ranges for remaining seat capacities to reduce the modeling domain and enable more accurate MARS approximations. The statistical modeling and booking modules described in Sections 3.1 and 3.2 are modified as follows: The off-line phase consists of a revised statistical modeling module that conducts a preprocessing simulation to identify realistic ranges of the remaining seat capacity state variables and then builds statistical models of the (DET) and (STOCH) revenue functions to estimate bounds on the value function of the MDP. The on-line phase uses the statistical models from the off-line phase in the RM policy to make the booking decisions similar to the statistical modeling approach. 4 Revised Statistical Modeling Module In the revised version of the statistical modeling module, realistic ranges of the remaining capacity state variable are generated, instead of the same ranges (from zero to capacity) throughout the booking period. Intuitively, these ranges should be close to the capacity at the beginning of the booking period and move closer to zero towards departure. 4.1 Generation of Realistic State Space In the statistical modeling module of Section 3.1, the state space remains the same for all the reading dates. Hence, the design points are spread out over a wider region than required. We know that, in practice, one is unlikely to find 11

12 an empty flight on the day of the departure or a full flight 90 days prior to the day of departure. In order to be more realistic, we estimate the possible/realistic ranges for each reading date. These are called trust regions. Demand scenarios are generated based on real data. Remaining flight capacity is initialized to the actual flight capacity. The (DET) model, as described in Section 2.1.4, is employed at the reading dates to generate bid prices. The RM policy which states, accept booking request only if the fare is greater than the bid price is used to make decisions on accepting/rejecting the request. Upon accepting the request, remaining seat capacity is updated to remaining capacity minus the booking request s group size g. At each reading date, an optimization model is solved to obtain updated bid prices, and the process repeats until the flight departs. Demand scenarios are simulated many times and at the end of each reading date, remaining seat capacities are recorded. Figure 5 shows the generation of the trust regions. Remaining seat capacities obtained at each reading date over the entire simulation are used to determine the maximum and minimum capacities at those reading dates. Figure 5 about here. To estimate the number of simulation runs needed to obtain good realistic ranges, a simulation for an initial sample size of s = 30 was run. The resulting data was used to estimate the standard deviation of remaining capacity, σ. Desired sample size was then estimated using a confidence interval approach, (2zα/2 ) σ 2 s =, (10) E where E is 5% of the expected value of the sample plus or minus the confidence coefficient times the standard error. A total of 85 simulation runs were conducted. 4.2 Approximation of the Value Functions The remaining seat capacity state spaces are set according to the empirically-derived realistic ranges. Similar to step 2 of the statistical modeling module in Section 3.1, an OA experimental design is employed to identify discretization points in the realistic state spaces for each reading date. Otherwise, steps 3 and 4 in Section 3.1 remain essentially the same (see Figure 3). As in Sections and 2.1.5, the (DET) model is used to provide an upper bound on the MDP value function, and the (STOCH) model is used to provide a lower bound. Solving the deterministic model is a straightforward LP, 12

13 but there are different approaches for solving the stochastic model. The next section describes the approach used in this paper. 4.3 Solving the Stochastic Network Optimization Model In the (STOCH) model, consider E[min(d ft, u ft )] of the objective function. Olinick and Rosenberger (2003) showed that this function is concave. Expanding it using a Taylor series about a constant u 0 0 we have, E[min(d ft, u ft )] = E[min(d ft, u 0 )] + (u ft u 0 ) E[min(d ft, u 0 )] + o( u ft u 0 2 ). (11) From the definition of expected value we know that, for any discrete random variable L, E[L] = lp(l). Let, b(d ft ) = min(d ft, u 0 ). For simplicity, let d ft = d for the purposes derivation. Hence, u 0 E[min(d, u 0 )] = dp(d) + d=0 d=u 0+1 u 0 p(d), (12) where p(d) is the probability of demand d. Demand is assumed to follow a compound Poisson process with arrival rate λ. Let H be the cumulative distribution function for the Poisson distribution and h be the probability mass function for the Poisson distribution. Hence, u 0 E[min(d, u 0 )] = de λ λ d /d! + u 0 d=0 u 0 d=u 0+1 = λ e λ λ d 1 /(d 1)! + u 0 d=1 u 0 1 = λ e λ λ d /d! + u 0 [1 d=0 e λ λ d /d! (13) u 0 d=0 d=u 0+1 e λ λ d /d! (14) e λ λ d /d!] (15) = λh(u o 1) + u 0 [1 H(u o )]. (16) Using finite differences, E[min(d, u 0 )] is estimated by E[min(d, u 0 )] = [(u 0 + 1) (u 0 + 1)H(u 0 + 1) + λh(u 0 )] [u 0 u 0 H(u 0 ) + λh(u 0 1)] (17) = λh(u 0 ) u 0 h(u 0 + 1) H(u 0 + 1) + 1. (18) Define w ft, as the decision vector for the expected number of passengers for demand class f at reading date t, such that w ft E[min(d ft, u ft )]. 13

14 Substituting all the above into the original (STOCH) model, the final (STOCH) model obtained is as below. t m max r f w fτ (19) τ=1 f=1 ( t ) s.t A u τ x t (20) τ=1 w fτ u fτ [λh(u 0 ) u 0 h(u 0 + 1) H(u 0 + 1) + 1] λh(u 0 1) u 0 [λh(u 0 ) u 0 h(u 0 + 1) H(u 0 + 1) + H(u 0 )] f = 1, 2,...m, τ = 0, 1,...t, u 0 R + (21) u fτ w fτ 0 f = 1, 2,...m, τ = 0, 1,...t. (22) The set (21) has an infinite number of constraints, so it is computational intractable to solve this linear programming problem exactly as stated. However, we can approximate this linear programming problem by replacing the infinite set R + by a finite set in which u 0 = 1, 2,..., u, where u in a practical upper bound on the values u fτ, for all f = 1,..., m and τ = 0,..., t. In our computational experiments in Section 5, we set u to be Computational Results Our methodology was tested on the 31-leg airline hub application used by Chen et al. (2003). Their application used fifteen reading dates, included 123 itineraries, and had a maximum demand of 50. Data on flight capacities and demand distribution parameters were provided by a domestic airline carrier. An OA experimental design, was used to generate 31 2 design/discretization points. Demand scenarios were generated based on the data given. Using all this information the revised statistical modeling module was executed, and the resulting RM policy was employed in a simulated booking process. The results obtained from 2000 simulation runs are given in Table 2. Table 2 about here. The column labeled Load is the nominal load factor, defined to be the quotient of total requested capacity over available capacity. Airlines use nominal load factors of up to 150%. The column CV contains the coefficient of variations considered for each load factor. The average revenues generated in the simulation are reported for four RM 14

15 methods: the (DET) bid price approach, the (STOCH) bid price approach, the original statistical modeling approach of Section 3, and the revised statistical modeling approach of Section 4. The standard error is given below each average revenue, followed by the percentage increase in average revenue compared with the (DET) bid price approach. 6 Concluding Remarks It can be seen in Table 2 that the original statistical modeling approach is better than both the (DET) and (STOCH) bid price approaches, and the revised statistical modeling approach shows improvement over the original. We can also observe that the (STOCH) bid price approach performs better than the (DET) bid price approach in certain instances. These results not only demonstrate the benefit of a statistical modeling MDP based approach, but, further, the benefit of using realistic trust regions for the state space. In on-going research, we are relaxing the assumption of no overbooking or cancellation. It is possible to estimate the number of seats to be overbooked, called overbooking pad, for each flight leg before the booking process starts. In the statistical modeling approach, the state space would be modified to be the sum of the actual flight capacity and the overbooking pad. Using a MARS approximation of revenue, we are studying the use of a combined Newton s and steepest ascent method to estimate the optimal overbooking pad. 15

16 References Bailey, E. E., Graham, R. D. and D, K. P.: 1985, Deregulating the Airlines, Cambridge Mass: MIT Press 25. Belobaba, P.: 1987, Survey paper: Airline yield management an overview of seat inventory control, Transportation Science 21, Ben, V.: 1995, Origin and Destination Yield Management, Handbook of Airline Economics. Bertsimas, D. and Boer, S. d.: 2005, Simulation-based booking limits for airline revenue management, Operations Research 53, Bodily, S. E. and Pfeifer, P. E.: 1973, Overbooking decision rules, OMEGA 20, Chen, V. C. P., Günther, D. and Johnson, E. L.: 2003, Solving for an optimal airline yield management policy via statistical learning, Journal of the Royal Statistical Society, Series C 20, Chen, V. C. P., Ruppert, D. and Shoemaker, C. A.: 1999, Applying experimental design and regression splines to high dimensional continues state stochastic dynamic programming., Operations Research 47, Cooper, W. L. and Homem-de Mello, T.: 2006, A class of hybrid methods for revenue management, working paper. Curry, E. R.: 1990, Optimal airline seat allocation with fare classes nested by origin and destinations, Transportation Science 24, Gallego, G. and Van Ryzin, G.: 1994, A multi-product dynamic pricing problem and its applications to network yield management, Operations Research. Günther, D.: 1998, Airline Yield Management, PhD thesis, Georgia Institute of Technology. Hersh, M. and Ladany, S.: 1978, Optimal seat allocation for flights with one independent stop, Computers and Operations Research pp Ladany, S. P. and Bedi, D. N.: 1977, Dynamic rules for flights with an intermediate stop, The International Journal of Management Science 5,

17 Lautenbacher, J. and Stidham, S. J.: 1999, The underlying markov decision process in the single-leg airline yieldmanagement problem, Transportation Science 33. Lee and Hersh, M.: 1993, A model for dynamic airline seat inventory control with multiple seat bookings, Transportation Science 27, Littlewood, K.: 1972, Forecasting and control passenger booking, AGIFORS Symposium Proceedings, Nagoya, Japan, pp Olinick, E. V. and Rosenberger, J. M.: 2003, Optimizing revenue in cdma networks under demand uncertainty, Technical Report 03-EMIS-03. Rothstein, M.: 1971, An airline overbooking model, Transportation Science 5, Smith, B., Leimkuhler, J. and Darrow, R.: 1992, Yield management at american airlines, Interfaces 22, Subramanian, J., Stidham, S. and Lautenbacher, C. J.: 1999, Airline yield management with overbooking, cancellations and no-shows, Transportation Science 33. Talluri, K. and Van Ryzin, G.: 1998, An analysis of bid- price control for network revenue management, Management Science 44, Talluri, K. and Van Ryzin, G.: 1999, A randomized linear programming method for computing network bid prices, Transportation Science 33, Talluri, K. and Van Ryzin, G.: to appear, Revenue management under a general discrete choice model of consumer behavior, Transportation Science. Zhang, D. and Cooper, W. L.: 2005, Revenue management for parallel flights with customer-choice behavior, Operations Research 53,

18 Table Caption Table: 1 Example calculation of a bid price. Table: 2 Average revenues from 2000 simulations of the 31-leg hub using four methods: DET = Deterministic Bid Price, STOCH = Stochastic Bid Price, STAT = Statistical Modeling Approach, REV STAT = Revised Statistical Modeling Approach. Results are shown for load factors of 75%, 120%, and 150%, and various coefficient of variation (CV) values. Standard errors are given in parentheses, and percent increase in average revenue from DET is shown.

19 Figure Caption Figure: 1 Figure: 2 Figure: 3 Figure: 4 Figure: 5 Representation of reading dates. Flow chart representing a general bid price approach. Flow chart representing the statistical modeling module. Flow chart representing the booking module. Flow chart representing generation of reduced realistic state spaces.

20 Table 1: Example calculation of a bid price. Flight Leg Airline Bid Price ($) Total 1109

21 Table 2: Average revenues from 2000 simulations of the 31-leg hub using four methods: DET = Deterministic Bid Price, STOCH = Stochastic Bid Price, STAT = Statistical Modeling Approach, REV STAT = Revised Statistical Modeling Approach. Results are shown for load factors of 75%, 120%, and 150%, and various coefficient of variation (CV) values. Standard errors are given in parentheses, and percent increase in average revenue from DET is shown. Load (%) CV DET STOCH STAT REV STAT (586) (533.6) (503.5) ( 486.5) -0.12% 0.01% 0.07% (680.4) ( 762.6) (623) ( 645.3) 0.22% 0.82% 1.35% (769.2) (858.7) (845.4) (764.4) 0.03% 0.52% 0.89% (586) (508) (455.2) (467.5) -0.75% 2.7% 2.86% (603.2) ( 548.4) ( 457.7) ( 428.4) -0.51% 3.76% 4.90% (702.6) (774.0) (736.8) (632.7) 1.45% 3.09% 5.82% (591.1) ( 500.6) (524.6) (425.65) -0.29% 0.46% 1.57% (609.4) (770.1) (643.9) (700.6) -6.90% 3.04% 3.65% (679.0) (779.3) (756.9) (737.8) % 3.84% 4.35%

22 Figure 1: Representation of reading dates.

23 Figure 2: Flow chart representing a general bid price approach.

24 Figure 3: Flow chart representing the statistical modeling module.

25 Figure 4: Flow chart representing the booking module.

26 Figure 5: Flow chart representing generation of reduced realistic state spaces.