Transcript Equilibrium problems with equilibrium constraints: Houyuan Jiang Danny Ralph

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
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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)
xR ,F : R  R .
Find x such that
n
n
n
Min ( x, F ( x))  0
x  0, F ( x)  0, x F ( x)  0
T
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)
x  Rn , y  Rm ,
f : R nm  R ,
g : R nm  R p , h : R nm  R q
F : R nm  R m
x -- upper level variable
Min f ( x, y )
s.t. g ( x, y )  0
y -- lower level variable
h ( x, y )  0
y  0, F ( x, y )  0, y T F ( x, y )  0
Leader
Controls
Responses
Followers’
equilibrium system
MPEC is a modelling tool for
the Stackelberg leaderfollower game where
followers play a game with a
given input from the leader.
Bi-level programs
(BP)
Min f ( x, y )
s.t.
g ( x, y )  0
h ( x, y )  0
x -- upper level variable
y -- lower level variable
Min u ( x, y )
s.t.
v ( x, y )  0
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.
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.
Leaders’
equilibrium system
Controls
Responses
Research questions:
 Existence of solutions
 Uniqueness
 Sensitivity analysis
Followers’
equilibrium system
 Computational methods
Existing MPEC/BP models in RM
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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
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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
Vk ( x)  E  max {rk yk  Vk 1 ( x  yk )
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
0 yk {d k , x}
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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
Max ( u1 ,..., u K )
 K

E  rk xk 
 k 1

K
s.t.
u
k 1
k
C
u k  0, k
where
xK  min( d K , u K ),
K
xk  min( d k ,  ul 
l k
K
x )
l  k 1
l
Is the new formulation equivalent to
the DP formulation?
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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
K
xk  min( d k ,  ul 
l k

K
x )
l  k 1
l
We are looking for optimal protection levels so that the expected total
revenue is maximized.
It is a stochastic MPEC
Max ( u1 ,..., u K )
 K

E  rk xk 
 k 1

K
s.t.
u
k 1
k
C
u k  0, k
xK  min( d K , u K ),
K
xk  min( d k ,  ul 
l k
K
 x ), k  1,..., K  1
l  k 1
l
An equivalent BP formulation
Max ( u1 ,..., u K )

 K
E  rk xk 

 k 1
K
s.t.
u
k 1
k
C
u k  0, k
K
max ( x1 ,..., x K )
c
k 1
K
s.t.
x
l k
l
k
xk

K
u
l k
xk  d k , k
xk  0, k
Where 0 < c1 < c2 < … < cK
l
, k
Classical inventory control models in
networks
r
Max
j
xj
Deterministic
Linear Program
j
Ax  C
s.t.
0  x j  d j , j
Max

 rj E min( u j , d j )
j
s.t.
Au  C
u j  0, j

Probabilistic
Nonlinear Program
Virtual nesting control over networks
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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
 J

E  r j x j 
 j 1

Max (..., u1 s ,..., u K s ,...)
K
s.t.
u
k 1
ks
 C s , s (leg )
u ks  0, k ( virtual class ), s
J
max ( x1 ,..., x J ,..., y ks ,...)
s.t.
y ks 
K
y
l k
ls
c
j 1
j
xj
a
x , k , s
sj
j
j: class k on leg s

K
u
l k
ls
,k , s
0  x j  d j , j
A stochastic programming
formulation of Higle and Sen (2003)
E h(u , d ) 
Max (..., u1 s ,..., u K s ,...)
K
s.t.
u
k 1
ks
 C s , s (leg )
u ks  0, k (class) , s
h(u , d )  max ( x1 ,..., x J )
s.t.
a
K
r
k 1
x
sj
j
j: class k on leg s
j
xj
 u ks , k , s
0  x j  d j , j
The inventory control problem under
competition
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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?
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Max ( u )
E rL z L  rH z H
s.t.
z L  min( u  , d L  max( 0, d L  u  ))

z H  min( C  xL , d H
 max( 0, d H  (C  z L ))
An EPEC formulation
 Accepted bookings from
xk
its own demand
 Accepted bookings from
yk
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
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Smoothing and other MPEC methods are applied
to approximate MPEC (and EPEC) by MP (and
NCP): Local optimal solutions vs global optimal
solutions.
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Monte Carlo sampling (sample-path optimization)
methods for handling stochastic demand: largescale problems vs accuracy of approximations.