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)

0)(,0)(,0

such that Find

.:,

xFxxFx

x

RRFRx

T

nnn

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

0))(,(Min xFx

Mathematical programs with equilibrium constraints (MPEC)

0),(,0),(,0

0),(

0),(s.t.

),(Min

:

:,:

,:

,,

yxFyyxFy

yxh

yxg

yxf

RRF

RRhRRg

RRf

RyRx

T

mmn

qmnpmn

mn

mn

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)

0),( s.t.

),(Min

0),(

0),(s.t.

),(Min

yxv

yxu

yxh

yxg

yxf

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) 23-36 (2003).

A.C. Lim, Transportation network design problems: An MPEC approach, PhD dissertation, Johns Hopkins University, 2002.

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, 2003.

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

)({max)( 1},{0

kkkkxdy

k yxVyrExVkk

k: Index for customer classes, rk: The ticket price for class k (r1 > r2 > … > rK) Dk: The random demand variable for class k dk: A realization of Dk

C: The total capacity of the flight uk: The booking limit for class k vk: The protection limit for class k and higher Vk(x): The optimal expected total revenue from class k and

higher when the remaining capacity is x

A probabilistic nonsmooth nonlinear programming formulation

),min(

),,min( where

,0

s.t.

Max

1

1

1),...,( 1

K

kl

K

klllkk

KKK

k

K

kk

K

kkkuu

xudx

udx

ku

Cu

xrEK

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 xk 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.

),min( 1

K

kl

K

klllkk xudx

It is a stochastic MPEC

1,...,1),,min(

),,min(

,0

s.t.

Max

1

1

1),...,( 1

Kkxudx

udx

ku

Cu

xrE

K

kl

K

klllkk

KKK

k

K

kk

K

kkkuu K

An equivalent BP formulation

kx

kdx

kux

xc

ku

Cu

xrE

k

kk

K

kl

K

klll

K

kkkxx

k

K

kk

K

kkkuu

K

K

,0

,

, s.t.

max

,0

s.t.

Max

1),...,(

1

1),...,(

1

1

Where 0 < c1 < c2 < … < cK

Classical inventory control models in networks

jdx

Ax

xr

jj

jjj

,0

C s.t.

Max

ju

Au

duEr

j

jjjj

,0

C s.t.

),min(Max

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

jdx

skuy

skxay

xc

sku

sCu

xrE

jj

K

kl

K

kllsls

skjjsjks

J

jjjyxx

ks

K

ksks

J

jjjuu

ksJ

Kss

,0

,,

,, s.t.

max

),classvirtual(,0

)leg(, s.t.

Max

legon class :

1,...),...,,...,(

1

1,...),...,(...,

1

1

A stochastic programming formulation of Higle and Sen (2003)

jdx

skuxa

xrduh

sku

sCu

duhE

jj

ksskj

jsj

K

kjjxx

ks

K

ksks

uu

J

Kss

,0

,, s.t.

max ),(

,(class) ,0

)leg(, s.t.

),(Max

legon class :

1),...,(

1

,...),...,(...,

1

1

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?

))(,0max(,min(

)),0max(,min( s.t.

Max)(

LHHLH

LLL

HHLLu

zCddxCz

udduz

zrzrE

An EPEC formulation

Accepted bookings from its own demand

Accepted bookings from its competitors demand

k

k

y

x

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.