Bilevel Bilevel approaches to approaches to revenue revenue management 16 janvier, 16 janvier, 2004 2004 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
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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)
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))
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