Equilibrium problems with equilibrium constraints: A new modelling paradigm for revenue management Houyuan Jiang Danny Ralph Stefan Scholtes The Judge Institute of Management University of Cambridge, UK

Outline  Reviews of various mathematical programming models  The inventory control model in a single-leg setting: From dynamic programming to MPEC.  The inventory control model in a network setting: From dynamic programming to MPEC.  The inventory control model under competition: From Nash equilibrium to EPEC.

Nonlinear complementarity problems (NCP) A standard modelling tool for problems in game theory including Nash equilibrium, general/Walrasian equilibrium, traffic/Wardrop equilibrium problems, etc.

Mathematical programs with equilibrium constraints (MPEC) Followers’ equilibrium system Leader Controls Responses x -- upper level variable y -- lower level variable MPEC is a modelling tool for the Stackelberg leader- follower game where followers play a game with a given input from the leader.

Bi-level programs (BP) Similar to MPEC, BP is a modelling tool for decision makings involving hierarchical structures where some constraints of the higher level problem are defined as a parametric optimization problem. Under some constraint qualifications of the lower level problem, BP is converted into an example of MPEC. x -- upper level variable y -- lower level variable

MPEC vs MP Is MPEC just a special case of MP? No. In fact standard constraint qualifications do not hold at any feasible point of MPCC, a special case of MPEC. Therefore, new theory and computational methods have to be studied. Much progress has been made on both theory and numerical algorithms for MPEC in the last decade.

Equilibrium problems with equilibrium constraints (EPEC) EPEC is an extension of MPEC to deal with multiple-leader and multiple-follower games. Followers’ equilibrium system Leaders’ equilibrium system Controls Responses Research questions:  Existence of solutions  Uniqueness  Sensitivity analysis  Computational methods

Existing MPEC/BP models in RM  J.P. Côté, P. Marcotte and G. Savard, A bilevel modelling approach to pricing and fare optimisation in the airline industry, Journal of Revenue and Pricing Management (2) (2003).  A.C. Lim, Transportation network design problems: An MPEC approach, PhD dissertation, Johns Hopkins University,  J.L. Higle and S. Sen, Stochastic programming model for network resource utilization in the presence of multi-class demand uncertainty, Technical Report, University of Arizona,  S. Kachani, G. Perakis, C. Simon, An MPEC approach to dynamic pricing and demand learning.

The static inventory control problem in a single-leg setting  Customers are divided into non-overlapping classes.  Demands of different classes are stochastic and independent.  Customers arrive in order from the lowest to the highest class.  No cancellations, no no-shows, no group bookings.  Nested booking control mechanism is used.  What are optimal protection levels?

A classical dynamic programming formulation  k: Index for customer classes,  r k : The ticket price for class k (r 1 > r 2 > … > r K )  D k : The random demand variable for class k  d k : A realization of D k  C: The total capacity of the flight  u k : The booking limit for class k  v k : The protection limit for class k and higher  V k (x): The optimal expected total revenue from class k and higher when the remaining capacity is x

A probabilistic nonsmooth nonlinear programming formulation

Is the new formulation equivalent to the DP formulation?  In the DP formulation, there are optimal protection levels or nested booking limits such that it is optimal to stop selling capacity to class k+1 in stage k+1 once the capacity remaining drops to the optimal protection level for k and higher.  This implies that for any demand scenario, in stage k, the number of allocation x k must be either the demand of class k in this scenario or the maximum number of seats available to this class in stage k, which is described by  We are looking for optimal protection levels so that the expected total revenue is maximized.

It is a stochastic MPEC

An equivalent BP formulation Where 0 < c 1 < c 2 < … < c K

Classical inventory control models in networks Deterministic Linear Program Probabilistic Nonlinear Program

Virtual nesting control over networks  In virtual nesting, products are clustered according to some criteria to form a number of virtual “classes” on each leg.  Each product is mapped into a virtual class on each leg.  Leg protection levels are applied to this virtual nesting control scheme.  Customers arrive from lower to higher in revenue order.  Considered in de Boer-Bertsimas (2001) and Talluri-van Ryzin (2003); solved using simulation based optimization.

A stochastic MPEC for the virtual nesting control

A stochastic programming formulation of Higle and Sen (2003)

The inventory control problem under competition  Considered in Li-Oum (1998) and Netssine-Shumsky (2003).  Two airlines  and  in a single-leg setting.  Two airlines have the same capacity.  There are two classes of customers: L and H.  Two airlines charge the customers the same prices.  Each airline has its original demand for each class of customers. If the demand cannot be satisfied, the customer will seek a booking from the rival airline.  What are optimal booking limits u  and u  for both airlines?

An EPEC formulation Accepted bookings from its own demand Accepted bookings from its competitors demand

Research questions We have only provided modelling frameworks, but have not fully explored the followings:  Existence  Uniqueness  Sensitivity analysis (results obtained)  Computational methods  Numerical experiments  Extensions  …

Remarks on computational methods  Smoothing and other MPEC methods are applied to approximate MPEC (and EPEC) by MP (and NCP): Local optimal solutions vs global optimal solutions.  Monte Carlo sampling (sample-path optimization) methods for handling stochastic demand: large- scale problems vs accuracy of approximations.