A stochastic approximation method for the single-leg revenue management problem with discrete demand distributions

We consider the problem of optimally allocating the seats on a single flight leg to the demands from multiple fare classes that arrive sequentially. It is well-known that the optimal policy for this problem is characterized by a set of protection levels. In this paper, we develop a new stochastic approximation method to compute the optimal protection levels under the assumption that the demand distributions are not known and we only have access to the samples from the demand distributions. The novel aspect of our method is that it works with the nonsmooth version of the problem where the capacity can only be allocated in integer quantities. We show that the sequence of protection levels generated by our method converges to a set of optimal protection levels with probability one. We discuss applications to the case where the demand information is censored by the seat availability. Computational experiments indicate that our method is especially advantageous when the total expected demand exceeds the capacity by a significant margin and we do not have good a priori estimates of the optimal protection levels.

[1]  Shaler Stidham,et al.  The Underlying Markov Decision Process in the Single-Leg Airline Yield-Management Problem , 1999, Transp. Sci..

[2]  Renwick E. Curry,et al.  Optimal Airline Seat Allocation with Fare Classes Nested by Origins and Destinations , 1990, Transp. Sci..

[3]  Yuri Ermoliev,et al.  Numerical techniques for stochastic optimization , 1988 .

[4]  John N. Tsitsiklis,et al.  Neuro-Dynamic Programming , 1996, Encyclopedia of Machine Learning.

[5]  P. Glynn,et al.  Stochastic optimization by simulation: numerical experiments with the M / M /1 queue in steady-state , 1994 .

[6]  Richard D. Wollmer,et al.  An Airline Seat Management Model for a Single Leg Route When Lower Fare Classes Book First , 1992, Oper. Res..

[7]  Dimitris Bertsimas,et al.  Simulation-Based Booking Limits for Airline Revenue Management , 2005, Oper. Res..

[8]  Yuri Ermoliev,et al.  Stochastic quasigradient methods. Numerical techniques for stochastic optimization , 1988 .

[9]  Itir Z. Karaesmen,et al.  Overbooking with Substitutable Inventory Classes , 2004, Oper. Res..

[10]  P. Glynn,et al.  Stochastic Optimization by Simulation: Convergence Proofs for the GI/G/1 Queue in Steady-State , 1994 .

[11]  W. T. Huh,et al.  Adaptive Capacity Allocation with Censored Demand Data : Application of Concave Umbrella Functions , 2006 .

[12]  G. Ryzin,et al.  Revenue Management Without Forecasting or Optimization: An Adaptive Algorithm for Determining Airline Seat Protection Levels , 2000 .

[13]  Lawrence W. Robinson,et al.  Optimal and Approximate Control Policies for Airline Booking with Sequential Nonmonotonic Fare Classes , 1995, Oper. Res..

[14]  Retsef Levi,et al.  Adaptive Data-Driven Inventory Control Policies Based on Kaplan-Meier Estimator for Censored Demand , 2010 .

[15]  W. Lieberman.  The Theory and Practice of Revenue Management , 2005 .

[16]  M. Fu,et al.  Optimization of( s, S ) inventory systems with random lead times and a service level constraint , 1998 .

[17]  Marvin Hersh,et al.  A Model for Dynamic Airline Seat Inventory Control with Multiple Seat Bookings , 1993, Transp. Sci..

[18]  Huseyin Topaloglu.  A Stochastic Approximation Method to Compute Bid Prices in Network Revenue Management Problems , 2008, INFORMS J. Comput..

[19]  Jeffrey I. McGill,et al.  Airline Seat Allocation with Multiple Nested Fare Classes , 1993, Oper. Res..

[20]  Paul Glasserman,et al.  Sensitivity Analysis for Base-Stock Levels in Multiechelon Production-Inventory Systems , 1995 .

[21]  Yuri M. Ermoliev.  Stochastic Quasigradient Methods , 2009, Encyclopedia of Optimization.

[22]  Martin Zinkevich,et al.  Online Convex Programming and Generalized Infinitesimal Gradient Ascent , 2003, ICML.

[23]  Huseyin Topaloglu,et al.  Using Stochastic Approximation Methods to Compute Optimal Base-Stock Levels in Inventory Control Problems , 2008, Oper. Res..

[24]  Michael C. Fu,et al.  Sample Path Derivatives for (s, S) Inventory Systems , 1994, Oper. Res..

[25]  Garrett J. van Ryzin,et al.  Computing Virtual Nesting Controls for Network Revenue Management Under Customer Choice Behavior , 2008, Manuf. Serv. Oper. Manag..

[26]  Garrett J. van Ryzin,et al.  Simulation-Based Optimization of Virtual Nesting Controls for Network Revenue Management , 2008, Oper. Res..

[27]  Harold J. Kushner,et al.  wchastic. approximation methods for constrained and unconstrained systems , 1978 .

[28]  Garrett J. van Ryzin,et al.  Stocking Retail Assortments Under Dynamic Consumer Substitution , 2001, Oper. Res..