Transcript A TOLL OPTIMIZATION PROBLEM

Bilevel Programming Approaches to
Revenue Management and Price Setting
Problems
Gilles Savard, École Polytechnique de Montréal, GERAD, CRT
Collaborators: P. Marcotte and C. Audet, L. Brotcorne, M. Gendreau,
J. Gauvin, P. Hansen, A. Haurie, B. Jaumard, J. Judice, M. Labbé, D.
Lavigne, R. Loulou, F. Semet, L. Vicente, D.J. White, D. Zhu
Students: so many including J.-P. Côté, V. Rochon, A. Schoeb, É.
Rancourt, F. Cirinei, M. Fortin, S. Roch, J. Guérin, S. Dewez, K. Lévy
16 janvier, 2004
Bilevel approaches to
revenue management
Outline
The revenue management problem
The bilevel programming problem
A price setting paradigm
… applied to toll setting
… a TSP instance
… applied to airline
Conclusion
16 janvier, 2004
Bilevel approaches to
revenue management
The revenue management problem
…the optimal revenue management of
perishable assets through price
segmentation (Weatherford and Bodily 92)
Fixed (or almost) capacity
Market segmentation
Perishable products
Presales
High fixed cost
Low variable cost
16 janvier, 2004
Bilevel approaches to
revenue management
The revenue management problem
RM Business process
Forecasting
Schedule with capacity
Pricing
Booking limits
Seat sales
16 janvier, 2004
Bilevel approaches to
revenue management
The revenue management problem
Some issues in airline industry:
How to design the booking classes?
 Restriction, min stay, max stay, service, etc…
… at what price?
 Willingness to pay, competition, revenue, etc…
… how many tickets?
 Given the evolution of sales (perishables)
… at what time?
 Given the inventory and the date of flight
16 janvier, 2004
Bilevel approaches to
revenue management
The revenue management problem
Evolution of Pricing & RM
1960’s: AA starts to use OR models
for RM decisions
1970’s: AA develops SABRE, providing
automatic update of availability and prices
1980’s: First RM software available
1990’s: RM grows, even beyond airlines
(hotel, rail, car rental, cruise, telecom,…)
2000’s: networks
16 janvier, 2004
Bilevel approaches to
revenue management
The revenue management problem
Decision Support Tools focus on
booking limits BUT mostly ignore pricing
Complex problem:
Must take into account its own action and
the competition, as well as passenger behaviour
Highly meshed network (hub-and-spoke)
OD-based vs. Leg-based approach
Data intensive
16 janvier, 2004
Bilevel approaches to
revenue management
The revenue management problem
 «Pricing has been ignored»
P. Belobaba (MIT)
 « Interest in RRM … is rising dramatically … RRM should
be one of the top IT priorities for most retailers »
AMR Research
 «Pricing Decision Support Systems will spur the next
round of airline productivity gains»
L. Michaels (SH&E)
16 janvier, 2004
Bilevel approaches to
revenue management
The revenue management problem
Until recently, capacity allocation and pricing
were performed separately: capacity allocation
is based on average historical prices; pricing is
done without considering capacity.
However, there is a strong duality relationship
between these two aspects.
A bilevel model combines both aspects while
taking into account the topological structure of
the network.
16 janvier, 2004
Bilevel approaches to
revenue management
The revenue management problem
Maximize expected revenue
by determining over time
the
the
the
the
products
prices
inventory
capacity
pricing
seat inventory
overbooking
taking into account
the market response
16 janvier, 2004
forecasting…
Bilevel approaches to
revenue management
Outline
The revenue management problem
The bilevel programming problem
A price setting paradigm
… applied to toll setting
… a TSP instance
… applied to airline
Conclusion
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming problem
Leader
min f1 ( x, y )

s.t . g1 ( x, y )  0
Follower
 y  argmin f 2 ( x, z )

s.t.
g 2 ( x, z )  0

16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming problem
… or MPEC problems
min f ( x, y )
s.t. g1 ( x, y )  0
IV
y  Y ( x )  y : g 2 ( x, y )  0
F ( x, y ), y  y '  0, y ' Y ( x )
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming problem
A linear instance…
y
F2
F1
x’
16 janvier, 2004
x’’
x
Bilevel approaches to
revenue management
Bilevel programming problem
Typically non convex, disconnected and strongly
NP-hard (HJS92) (even for local optimality (VSJ94))
Optimal solution  pareto solution (HSW89, MS91)
Steepest descent: BLP linear/quadratic (SG93)
Many instances:
Linear/linear (HJS92, JF90, BM90)
Linear/quadratic (BM92)
Convex/quadratic (JJS96)
Bilinear/bilinear (BD02, LMS98, BLMS01)
Bilinear/convex
Convex/convex

16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming problem
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming problem
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming model
Resolution approaches
Combinatorial approaches (global solution)
Lower level structure: combinatorial structure
Descent approaches (on the bilevel model)
Sensitivity analysis (local approach) (Outrata+Zowe)
Descent approaches (on an approximated
one-level model)
Model still non convex (e.g. penalization of the
second level KKT conditions) (Scholtes+Stöhr)
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming model
1. Combinatorial approaches: convex/quadratic
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming model
KKT
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming model
The one level formulation:
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming model
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming model
B&B: the subproblem and the relaxation
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming model
An efficient B&B algorithm can be developed by
Exploiting the monotonicity principle
Using two subproblems (primal and dual) to drive the
selection of the constraints
Efficient separation schemes
Using degradation estimation by penalties
Using cuts
Size (exact solution): 60x60 to 300X150
Heuristics: 600x600 (tabou, pareto)
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming model
2. Descent approach within a trust region approach (BC)
A good trust region model to bilevel program is
a bilevel program that
is easy to solve (combinatorial lower-level structure)
is a good approximation of the original bilevel
program
Such a non convex submodel (with exact
algorithm) can track part of the non convexity of
the original problem
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming model
Potential models:
Resolution
Approximation
++++
----
quad/lin
+++
---
conv/lin
++
--
lin/quad
+++
++
++
+++
+
++++
lin/lin
quad/quad
con/quad
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming model
Notations
actual
( x , y( x ))
real
( x , y )
current ( x , y )
( x , y )
predicted
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming model
Classic steps:
 Solve approximation  x k
f ( x k )  f ( x k )
actual
Let  k 

predicted f ( x k )  f ( x k )
 k   :
k   :
  k   :
x k 1  x k ,  k 1  1  k
2
x k 1  x k ,  k 1  2 k
x k 1  x k ,  k 1   k
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming model
With a linesearch step (to guaranty a strong
stationary point)
 if  k   min
then x k 1  arg min f ( xk )



where  2 j  k : j  1,,  log k 
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming model
 The sets X and Y are compact
 f and its gradientare Lipschitzcontinuous
on X  Y with resp.constantl and l '
 F and its Jacobian are Lipschitzcontinuous
on X  Y with resp.constantL and L'
 F is uniformly stronglymonotoneon Y , with modulus
b-stationary convergence
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming model
16 janvier, 2004
Bilevel approaches to
revenue management
Outline
The revenue management problem
The bilevel programming problem
A price setting paradigm
… applied to toll setting
… a TSP instance
… applied to airline
Conclusion
16 janvier, 2004
Bilevel approaches to
revenue management
A generic price setting model
T: tax or price vector
x: level of taxed activities
y: level of untaxed activities
16 janvier, 2004
Bilevel approaches to
revenue management
A generic price setting model
If the revenue is proportional to the activities we obtain
the so-called bilinear/bilinear problem:
16 janvier, 2004
Bilevel approaches to
revenue management
A generic price setting model
16 janvier, 2004
Bilevel approaches to
revenue management
A generic price setting model
16 janvier, 2004
Bilevel approaches to
revenue management
A generic price setting model
1. The one level formulation: combinatorial approach
16 janvier, 2004
Bilevel approaches to
revenue management
A generic price setting model
16 janvier, 2004
Bilevel approaches to
revenue management
A generic price setting model
2. One level formulation: continuous approach
16 janvier, 2004
Bilevel approaches to
revenue management
A generic price setting model
The combinatorial equivalent problem…
The continuous equivalent problem…
16 janvier, 2004
Bilevel approaches to
revenue management
Outline
The revenue management problem
The bilevel programming problem
A price setting paradigm
… applied to toll setting
… a TSP instance
… applied to airline
Conclusion
16 janvier, 2004
Bilevel approaches to
revenue management
… on a transportation network
Pricing over a network
16 janvier, 2004
Bilevel approaches to
revenue management
… on a transportation network
5
1
1
1
2
3
1
4
5
10
Free arcs
Toll arcs
T
Leader
max Tx
Follower
min (c+T)x + dy
Ax+By=b
x,y >=0
Toll vector
x
Toll arcs flow
y
Free arcs flow
16 janvier, 2004
Bilevel approaches to
revenue management
… on a transportation network
A feasible solution...
5
1
1+4
2
1+1
3
1+8
4
5
10
PROFIT = 4
16 janvier, 2004
Bilevel approaches to
revenue management
… on a transportation network
…the optimal solution.
5
1
1 +4
2
1 -1
3
1 +4
4
5
10
PROFIT = 7
16 janvier, 2004
Bilevel approaches to
revenue management
… on a transportation network
The algorithms:
Branch-and-cut approach on various MIPpaths and/or arcs reformulations (LMS98,
LB, SD, DMS01)
Primal-dual approaches (BLMS99, BLMS00, AF)
Gauss-Seidel approaches (BLMS03)
16 janvier, 2004
Bilevel approaches to
revenue management
… on a transportation network
Replacing the lower level problem by its optimality
conditions, the only nonlinear constraints are:
We can linearize this term (exploiting the
shortest paths):
16 janvier, 2004
Bilevel approaches to
revenue management
… on a transportation network
1. A MIP formulation
16 janvier, 2004
Bilevel approaches to
revenue management
… on a transportation network
2. Primal-dual approach (LB)
16 janvier, 2004
Bilevel approaches to
revenue management
… on a transportation network
Step 1: Solve for T and λ (Frank-Wolfe)
Step 2: Solve for x,y
Step 3: Inverse optimisation
Step 4: Update the M1 and M2
16 janvier, 2004
Bilevel approaches to
revenue management
Outline
The revenue management problem
The bilevel programming problem
A price setting paradigm
… a toll setting problem
… a TSP instance
… applied to airline
Conclusion
16 janvier, 2004
Bilevel approaches to
revenue management
TSP: a toll setting problem?
TSP: given a graph G=(V,E) and the length vector
c, find a tour that minimizes the total length.
min  cij xij
i
s.t.
j
x
ij
 1 j
ij
 1 i
i
x
j
xij  0,1
ij
subt our eliminat ion constraints
16 janvier, 2004
Bilevel approaches to
revenue management
TSP: a toll setting problem?
Find a toll setting problem such that
the profit for the leader is maximized
the shortest path for the user is a tour
the length of the tour is minimized
16 janvier, 2004
Bilevel approaches to
revenue management
TSP: a toll setting problem?
5
3
2
2
1
4
Optimal tour: length 8
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Bilevel approaches to
revenue management
TSP: a toll setting problem?
 1  cij / lmax
-1 + 5/10
-1 + 3/10
-1 + 2/10
-1 + 2/10
-1 + 2/10
-1 + 1/10
-1 + 1/10
-1 + 4/10
4
-1 + 4/10
16 janvier, 2004
t  2  cij / lmax
*
ij
Bilevel approaches to
revenue management
TSP: a toll setting problem?
16 janvier, 2004
Bilevel approaches to
revenue management
TSP: a toll setting problem?
16 janvier, 2004
Bilevel approaches to
revenue management
TSP: a toll setting problem?
Miller-Tucker-Zemlin lifted (DL91)
16 janvier, 2004
Bilevel approaches to
revenue management
TSP: a toll setting problem?
16 janvier, 2004
Bilevel approaches to
revenue management
TSP: a toll setting problem?
16 janvier, 2004
Bilevel approaches to
revenue management
3. TSP: a toll setting problem?
16 janvier, 2004
Bilevel approaches to
revenue management
TSP: a toll setting problem?
Sherali-Driscoll OR02
16 janvier, 2004
Bilevel approaches to
revenue management
Outline
The revenue management problem
The bilevel programming problem
A price setting paradigm
… applied to toll setting
… applied to telecommunication
… applied to airline
Conclusion
16 janvier, 2004
Bilevel approaches to
revenue management
Key Features of the model
Fares are decision variables, not static input
Fare Optimization is OD-based, not leg-based
All key agents taken into account:
AC and its resource management (fleet, schedule)
Competition fares
Detailed passenger behaviour (fare, flight duration,
departure time, quality of service, customer inertia, etc.)
Interaction among agents
AC maximizes revenue over entire network
Passengers minimize Pax Perceived Cost (PPC)
16 janvier, 2004
Bilevel approaches to
revenue management
Key Features of the model
Pricing at fare basis code level
Demand implied by rational customer reaction
to fares (AC and competition)
Demand
vs behavioural approach
16 janvier, 2004
Bilevel approaches to
revenue management
Key Features of the model
Full accounting of interconnectedness (overlapping
routes and markets, available capacity, ‘hub-andspoke’)
FEBRUARY 2ND-8TH 2002
“The danger for BA is that hacking away at its
network, and pulling out of loss-making routes,
could dry up traffic that uses those routes to
gain access to profitable transatlantic flights.”
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel Model Structure
MAX
Revenue = Fare
X
#Pax
Subject To
Market Share Objectives
Upper Level
(AC‘s RM strategy)
Revenue Objectives
Bounds on Fares
MIN
Pax Perceived Costs
Subject To
Lower Level
(Pax reaction)
Aircraft Capacities
Booking Limits
Demand
16 janvier, 2004
Bilevel approaches to
revenue management
Assignment Model
 Based on a multicriteria formulation
 Customer segmentation according to behavioural criteria
 Criteria






Fare
Flight duration (direct vs connecting flight)
Quality of service
Customer inertia
Fare restrictions
Departure time, frequency, etc.
 Perceived cost for passenger  :
16 janvier, 2004
Bilevel approaches to
revenue management
Parameter Distribution
Continuous Case
Discrete Case
 (, )
VOT
VOQ
Demand
100
80
60
40
20
0
 : VOT
 : VOQ
16 janvier, 2004
Grp 1
Grp 2
Grp 3
Grp 4
Bilevel approaches to
revenue management
Perceived cost
 (, )
 f , (1, 1 )  Tf ,  2Df  2.2Q
 : VOT
:
VOQ
 f , (1, 1 )  Tf ,  2Df  2.2Q
 f , (2 , 2 )  Tf ,  6Df  7Q
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel Model
Revenue = Fare x Number of Passengers
Market Shares
Upper Level
Bounds
Perceived Cost
Aircraft Capacities
Lower Level
Booking Limits
Demand
16 janvier, 2004
Bilevel approaches to
revenue management
Illustrative Example
Network
Structure
• 2 markets
• YUL-SFO
• YUL-ATL
YUL
• 2 pax segments per market
• Business (QoS sensitive)
• Leisure (price sensitive)
YYZ
• 2 Pax Perceived Cost (PPC) criteria
• Fare
• Quality of service (QoS)
SFO
ATL
• 1 fare per airline per market
16 janvier, 2004
Bilevel approaches to
revenue management
Illustrative Example (Data)
Supply side
Flights
Leg
Aircraft
(YUL, YYZ)
B767-300
200
(YYZ, SFO)
A320-100
130
(YUL, SFO)
A330
-
(YYZ, ATL)
A319
110
(YUL, ATL)
MD-81
Fare Structure
Capacity
-
Flight
Leg
Fare
QoS
AC1
(YUL, YYZ),
(YYZ, SFO)
$F1
50
UA
(YUL, SFO)
$1000
90
AC2
(YUL, YYZ),
(YYZ, ATL)
$F2
60
DL
(YUL, ATL)
$850
80
Demand Side
Pax
Segment
Market
Demand
QoS $
factor
S1
[YUL, SFO]
100
5
S2
[YUL, SFO]
450
1
S3
[YUL, ATL]
60
8
S4
[YUL, ATL]
385
1
16 janvier, 2004
Bilevel approaches to
revenue management
Illustrative example (Objective)
Action: Maximize AC’s Network Revenues
Find fares F1 and F2 that yield maximum revenue
maximize Revenue = (F1 x Pax1) + (F2 x Pax2)
where Pax1 and Pax2 denote Pax numbers attracted to
flights AC1 and AC2, respectively
16 janvier, 2004
Bilevel approaches to
revenue management
Illustrative example (Reaction)
Reaction: Minimize PPC on each market
Pax Perceived Cost
Segment
AC flight
Competition flight
S1 (SFO)
$F1 + (5 x 50) = $F1 + $250
$1000 + (5 x 90) = $1450
S2 (SFO)
$F1 + (1 x 50) = $F1 + $50
$1000 + (1 x 90) = $1090
S3 (ATL)
$F2 + (8 x 60) = $F2 + $480
$850 + (8 x 80) = $1490
S4 (ATL)
$F2 + (1 x 60) = $F2 + $60
$850 + (1 x 80) =
16 janvier, 2004
$930
Bilevel approaches to
revenue management
Illustrative Example (Revenue)
Local analysis of YUL-SFO market
Case 1: F1  $1040
Segments S1 and S2 fly AC1
Revenue: $1040 x 130 = $135 200
Case 2: F1  $1040 and F1  $1200
Only segment S1 flies AC1
Revenue: $1200 x 100 = $120 000
Case 3: F1  $1200
Segments S1 and S2 fly the competition
Revenue: $0
16 janvier, 2004
Bilevel approaches to
revenue management
Illustrative Example (Strategies)
Strategy
Fares (pax)
Revenue
Gain
Match competition’s fares
F1 = $1000
F2 = $850
(130)
(70)
$189 500
Base
Local analysis (SFO first)
F1 = $1040
F2 = $870
(130)
(70)
$196 100
+3.5%
Local analysis (ATL first)
F1 = $1200
F2 = $870
(90)
(110)
$203 700
+7.5%
Network solution (optimal)
F1 = $1200
F2 = $870
(100)
(100)
$207 000
+9.2%
Network solution after
competition matches
leader solution
F1 = $1400
F2 = $890
(100)
(100)
$229 000
Virtuous
Spiral
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Bilevel approaches to
revenue management
Revenue Function
Continuous Case
Discrete Case
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Bilevel approaches to
revenue management
Real-life Instance
Thousands of O-D pairs (markets)
More than 20 fare basis codes per
market
Hundreds of legs per day
Hub-and-spoke structure
Highly meshed network
Extended planning horizon
Capacity
16 janvier, 2004
Bilevel approaches to
revenue management
Model Resolution
 Discrete approach
 Combinatorial heuristics
 Branch-and-cut exact algorithms
 Continuous approach
 Sub-gradient based ascent method
 Hybrid approach
 Phase 1: coarse discrete approximation
 Phase 2: further optimization (fine tuning)
16 janvier, 2004
Bilevel approaches to
revenue management
Parameter Calibration
Procedure based on Hierarchical Inverse Optimization
Estimation from historical data
Same order of complexity (NP-Hard)
Calibration performed off-line on a regular basis
16 janvier, 2004
Bilevel approaches to
revenue management
Issues
Continuous vs. discrete
Design of decomposition techniques to deal
with the curse of dimensionality
Extremal solutions vs discretization
The dynamic of the process
Interaction with databases
Live scenarios
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Bilevel approaches to
revenue management
Conclusion
Bilevel programming is a rich class of
problems
Interests in both theoretical and practical
issues
Keeping the structure and the meaning of
the model of each agent
The natural way of modeling the yield
management problem
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Bilevel approaches to
revenue management
Additional Material
16 janvier, 2004
Bilevel approaches to
revenue management
Bilevel programming model
16 janvier, 2004
Bilevel approaches to
revenue management
Continuous Example
Uncapacitated, leg-based
( TVS )
SH
TKO
VAN
MTL
( TMV )
NY
Air Canada
United Airlines
China Airlines
Japan Airlines
16 janvier, 2004
Bilevel approaches to
revenue management
Continuous Example
(continued)
Flights
Label
Airlines
Path
Fare
Travel time
F1
UA + JAL
MTL-NY-TKO-SH
$1320
36 hrs
F2
AC + CA
MTL-VAN-SH
$TMV + $720
26 hrs
F3
AC + AC
MTL-VAN-SH
$TMV + $TVS
18 hrs
Objective
Find fares TMV and TVS that yield maximum revenue for Air Canada
max R ( TMV , TVS ) = ( TMV ) x ( X2 ) + ( TVS + TMV ) x ( X3 )
where X2 and X3 denote the number of passengers on flights F2 and F3
16 janvier, 2004
Bilevel approaches to
revenue management
Continuous Example
(lower level)
Customer Reaction
Flight
F1
Fare + (
Perceived travel cost
($1320)
x
Delay)
F1
F2
+ (36 x )
F2
($TMV + $720) + (26 x )
F3
($TMV + $TVS) + (18 x )
F3
1 = (1/10) [ TMV – 600 ]
F3
F2
F1
2 = (1/18) [ TMV + TVS – 1320 ]
3 = (1/8) [ TVS – 720 ]
1 2
16 janvier, 2004
3
max
Bilevel approaches to
revenue management

Continuous Example
Flow assignment
(flow assignment)
1

0
3
X2 = 1000 x 
1
max
X3 = 1000 x 
3
X1 = 1000 x
h() d
h() d
h() d
where h(·) denotes the density function associated to the VOT
parameter distribution
Assumption
 Parameter  is uniformly distributed over the interval [0, 90]
16 janvier, 2004
Bilevel approaches to
revenue management
Continuous Example
(solution)
Solution analysis
Region A : 0 < 1 < 3 < max
All flights carry flow
Revenue function: (5/18) [– 4 (TMV)2 + 6000 TMV – 5 (TVS)2 + 7200 TVS]
Optimal solution: TMV = $683 TVS = $787
Optimal revenue: $1 334 000
Region B : 0 < 2 < max and 1  3
Only flights F1 and F3 carry flow (flight F2 dominated)
Revenue function: (50/81) [– (TMV + TVS )2 + 2940 (TMV + TVS )]
Optimal solution: Any combination such that TMV + TVS = $1470
Optimal revenue: $1 334 000
16 janvier, 2004
Bilevel approaches to
revenue management
Continuous Example
(solution)
Contours of revenue function
TMV
 Revenue function is
piecewise quadratic
B
 It is not globally concave
A
 It may be nondifferentiable at the
boundary of the polyhedral regions
 Solution: TMV + TVS = $1470
 Optimal revenue: $1 334 000
TVS
16 janvier, 2004
Bilevel approaches to
revenue management