bilevel approaches to revenue management 16 janvier, 2004 gilles savard, École polytechnique de...
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Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
16 janvier, 16 janvier, 20042004
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
Bilevel Programming Approaches to Revenue Management and Price Setting Problems
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Outline
The revenue management problemThe bilevel programming problemA price setting paradigm
… applied to toll setting… a TSP instance… applied to airline
Conclusion
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
The revenue management problem
…the optimal revenue management of perishable assets through price segmentation (Weatherford and Bodily 92)
Fixed (or almost) capacityMarket segmentationPerishable productsPresalesHigh fixed costLow variable cost
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
The revenue management problem
RM Business process
ForecastingSchedule with capacity PricingBooking limitsSeat sales
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
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, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
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 available1990’s: RM grows, even beyond airlines
(hotel, rail, car rental, cruise, telecom,…)
2000’s: networks
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
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, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
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, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
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, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
The revenue management problem
Maximize expected revenueby determining over time
the productsthe pricesthe inventorythe capacity
taking into account
the market response
pricing
seat inventory
overbooking
forecasting…
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Outline
The revenue management problemThe bilevel programming problemA price setting paradigm
… applied to toll setting… a TSP instance… applied to airline
Conclusion
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming problem
0),( s.t.
),(min
1
1
yxg
yxf
0),( s.t.
),(argmin
2
2
zxg
zxfy
Leader
Follower
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming problem
)(',0'),,(
0),(:)(
0),(s.t.
),(min
2
1
xYyyyyxF
yxgyxYy
yxg
yxf
… or MPEC problems
IV
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming problem
F1 y
x
F2
x’ x’’
A linear instance…
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
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, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming problem
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming problem
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
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, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming model
1. Combinatorial approaches: convex/quadratic
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming model
KKT
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming model
The one level formulation:
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming model
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming model
B&B: the subproblem
and the relaxation
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming model
An efficient B&B algorithm can be developed byExploiting the monotonicity principleUsing two subproblems (primal and dual) to
drive the selection of the constraints Efficient separation schemesUsing degradation estimation by penaltiesUsing cuts
Size (exact solution): 60x60 to 300X150 Heuristics: 600x600 (tabou, pareto)
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming model
A good trust region model to bilevel program is a bilevel program thatis 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
2. Descent approach within a trust region approach (BC)
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming model
Potential models:
Resolution Approximation
lin/lin ++++ ----
quad/lin +++ ---
conv/lin ++ --
lin/quad +++ ++
quad/quad ++ +++
con/quad + ++++
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming model
Notations
),( yx
),( yx
),( yx))(,( xyxactual
predicted
real
current
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming model
Classic steps:
kkk
k
kkk
k
kkkk
k
k
k
k
k
k
k
k
k
xx
xx
xx
xfxf
xfxf
x
11
11
11
,:
2,:
21,:
)()(
)()(
predicted
actualLet
ion approximat Solve
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming model
With a linesearch step (to guaranty a strong stationary point)
kkj
kkk
j
xfx
log,,1:2 where
)(minargthen if 1min
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming model
modulus with, on monotonestrongly uniformly is
' and constant resp. with on
continuous Lipschitzare Jacobian its and
' and constant resp. with on
continuous Lipschitzaregradient its and
compact are and sets The
YF
LLYX
F
llYX
f
YX
b-stationary convergence
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming model
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Outline
The revenue management problemThe bilevel programming problemA price setting paradigm
… applied to toll setting… a TSP instance… applied to airline
Conclusion
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
A generic price setting model
T: tax or price vector
x: level of taxed activities
y: level of untaxed activities
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
A generic price setting model
If the revenue is proportional to the activities we obtain the so-called bilinear/bilinear problem:
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
A generic price setting model
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
A generic price setting model
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
A generic price setting model
1. The one level formulation: combinatorial approach
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
A generic price setting model
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
A generic price setting model
2. One level formulation: continuous approach
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
A generic price setting model
The combinatorial equivalent problem…
The continuous equivalent problem…
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Outline
The revenue management problemThe bilevel programming problemA price setting paradigm
… applied to toll setting… a TSP instance… applied to airline
Conclusion
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
… on a transportation network
Pricing over a network
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
… on a transportation network
1 2 3 4
105
5
1 11
Toll arcs
Free arcs
Leader max Tx
Follower min (c+T)x + dy Ax+By=b x,y >=0
T Toll vector
x Toll arcs flow
y Free arcs flow
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
A feasible solution...
2 3 4
105
5
1 + 41
1 + 1 1 + 8
PROFIT = 4
… on a transportation network
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
… on a transportation network
2 3 4
105
5
11
1 1
PROFIT = 7
…the optimal solution.
+ 4
- 1 + 4
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
… on a transportation network
Branch-and-cut approach on various MIP-paths and/or arcs reformulations (LMS98, LB, SD, DMS01)
Primal-dual approaches (BLMS99, BLMS00, AF)
Gauss-Seidel approaches (BLMS03)
The algorithms:
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
… 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, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
… on a transportation network
1. A MIP formulation
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
… on a transportation network
2. Primal-dual approach (LB)
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
… 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, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Outline
The revenue management problemThe bilevel programming problemA price setting paradigm
… a toll setting problem… a TSP instance… applied to airline
Conclusion
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
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.
sconstraintn eliminatiosubtour
1,0
1
1s.t.
min
ijx
ix
jx
xc
ij
jij
iij
i jijij
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
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, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
TSP: a toll setting problem?
2
1
3
4
2
5
Optimal tour: length 8
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
TSP: a toll setting problem?
-1 + 2/10-1 + 1/10
-1 + 3/10
-1 + 4/10
-1 + 2/10
-1 + 5/10
4max
* /2 lct ijij
-1 + 1/10
-1 + 4/10
-1 + 2/10
max/1 lcij
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
TSP: a toll setting problem?
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
TSP: a toll setting problem?
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
TSP: a toll setting problem?
Miller-Tucker-Zemlin lifted (DL91)
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
TSP: a toll setting problem?
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
TSP: a toll setting problem?
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
3. TSP: a toll setting problem?
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
TSP: a toll setting problem?
Sherali-Driscoll OR02
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Outline
The revenue management problemThe bilevel programming problemA price setting paradigm
… applied to toll setting… applied to telecommunication… applied to airline
Conclusion
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
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 faresDetailed passenger behaviour (fare, flight duration,
departure time, quality of service, customer inertia, etc.)
Interaction among agentsAC maximizes revenue over entire networkPassengers minimize Pax Perceived Cost (PPC)
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
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, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Key Features of the model
““The danger for BA is that hacking away at its The danger for BA is that hacking away at its networknetwork, and pulling out of loss-making routes, , and pulling out of loss-making routes, could dry up traffic that uses those routes to could dry up traffic that uses those routes to gain access to profitable transatlantic flights.”gain access to profitable transatlantic flights.”
FEBRUARY 2ND-8TH 2002FEBRUARY 2ND-8TH 2002
Full accounting of interconnectedness (overlapping routes and markets, available capacity, ‘hub-and-spoke’)
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel Model Structure
Lower LevelLower Level(Pax reaction)(Pax reaction)
MIN MIN Pax Perceived Costs Pax Perceived Costs
Aircraft CapacitiesAircraft Capacities
Booking LimitsBooking Limits
DemandDemand
Subject ToSubject To
Upper LevelUpper Level(AC‘s RM strategy)(AC‘s RM strategy)
Market ShareMarket Share ObjectivesObjectives
Bounds on FaresBounds on Fares
Subject ToSubject To
Revenue Revenue ObjectivesObjectives
MAX MAX Revenue Revenue = Fare = Fare X X #Pax#Pax
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
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, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Parameter Distribution
Continuous Case Discrete Case
0
20
40
60
80
100
Grp 1 Grp 2 Grp 3 Grp 4
VOT VOQ Demand
((, , ))
: : VOQVOQ
: VOT: VOT
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Perceived cost ((, , ))
: : VOQVOQ
: VOT: VOT
QDT fff 2.22),( ,11,
QDT fff 2.22),( ,11,
QDT fff 76),( ,22,
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel Model
Upper LevelUpper Level
Lower LevelLower Level
RevenueRevenue = Fare x Number of Passengers = Fare x Number of Passengers
Perceived CostPerceived Cost
Aircraft CapacitiesAircraft Capacities
Booking LimitsBooking Limits
Market SharesMarket Shares
BoundsBounds
DemandDemand
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Illustrative Example
Network Structure
YULYUL
YYZYYZ
ATLATL
SFOSFO
• 2 markets• YUL-SFO• YUL-ATL
• 2 pax segments per market
• Business (QoS sensitive)• Leisure (price sensitive)
• 2 Pax Perceived Cost (PPC) criteria
• Fare• Quality of service (QoS)
• 1 fare per airline per market
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Illustrative Example (Data)
Leg Aircraft Capacity
(YUL, YYZ) B767-300 200
(YYZ, SFO) A320-100 130
(YUL, SFO) A330 -
(YYZ, ATL) A319 110
(YUL, ATL) MD-81 -
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
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
Supply side Flights Fare Structure
Demand Side
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
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, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
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) = $930
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Illustrative Example (Revenue)
Local analysis of YUL-SFO marketCase 1: F1 $1040
Segments S1 and S2 fly AC1Revenue: $1040 x 130 = $135 200
Case 2: F1 $1040 and F1 $1200Only segment S1 flies AC1Revenue: $1200 x 100 = $120 000
Case 3: F1 $1200Segments S1 and S2 fly the competitionRevenue: $0
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Illustrative Example (Strategies)
Strategy Fares (pax) Revenue Gain
Match competition’s fares
F1 = $1000 (130)F2 = $850 (70)
$189 500 Base
Local analysis (SFO first)
F1 = $1040 (130)F2 = $870 (70)
$196 100 +3.5%
Local analysis (ATL first)
F1 = $1200 (90)F2 = $870 (110)
$203 700 +7.5%
Network solution (optimal)
F1 = $1200 (100)F2 = $870 (100)
$207 000 +9.2%
Network solution after competition matches leader solution
F1 = $1400 (100)F2 = $890 (100)
$229 000 Virtuous
Spiral
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Revenue Function
Continuous Case Discrete Case
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Real-life Instance
Thousands of O-D pairs (markets)More than 20 fare basis codes per
marketHundreds of legs per dayHub-and-spoke structureHighly meshed networkExtended planning horizonCapacity
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
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, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
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, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
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
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
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
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Additional Material
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Bilevel programming model
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Continuous Example
Uncapacitated, leg-based
MTLMTLVANVANTKOTKOSHSH
NYNY
( TMV )
( TVS )
Japan AirlinesChina Airlines
United AirlinesAir Canada
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Continuous Example (continued)
Flights
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
AC + AC
AC + CA
UA + JAL
Airlines
MTL-VAN-SH
MTL-VAN-SH
MTL-NY-TKO-SH
Path
18 hrs$TMV + $TVS F3
26 hrs$TMV + $720F2
36 hrs$1320F1
Travel timeFareLabel
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Continuous Example (lower level)
Customer Reaction
($TMV + $TVS) + (18 x )F3
($TMV + $720) + (26 x ) F2
($1320) + (36 x ) F1
Perceived travel costFlight
31
Fare + ( x Delay)F1
F3
F2
F1F2
F3
2 max(1/8) [ TVS – 720 ] 3 =
(1/18) [ TMV + TVS – 1320 ]2 =
(1/10) [ TMV – 600 ]1 =
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Continuous Example (flow assignment)
Flow assignment
Assumption Parameter is uniformly distributed over the interval
[0, 90]
where h(·) denotes the density function associated to the VOT parameter distribution
h() d 31
1000 x
=X2
h() d 10
1000 x
=X1
max3
h() d1000 x
=X3
16 janvier, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Continuous Example (solution)
Solution analysisRegion 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, 200416 janvier, 2004 Bilevel approaches Bilevel approaches to revenue to revenue managementmanagement
Continuous Example (solution)
Contours of revenue function
Revenue function is piecewise quadratic
It is not globally concave
It may be nondifferentiable at the boundary of the polyhedral regions
Solution: TMV + TVS = $1470
Optimal revenue: $1 334 000
TM
V
TVS
A
B
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