Gilles Savard,

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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

2
Outline

• The revenue management problem
• The bilevel programming problem
• A price setting paradigm
• applied to toll setting
• a TSP instance
• applied to airline
• Conclusion

3
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

4
The revenue management problem

• RM Business process
• Forecasting
• Schedule with capacity
• Pricing
• Booking limits
• Seat sales

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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

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The revenue management problem

• Evolution of Pricing RM
• 1960s AA starts to use OR models
  for RM decisions
• 1970s AA develops SABRE, providing
  automatic update of availability and prices
• 1980s First RM software available
• 1990s RM grows, even beyond airlines
  (hotel, rail, car rental, cruise, telecom,)
• 2000s networks

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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

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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 (SHE)

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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.

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The revenue management problem

• Maximize expected revenue
• by determining over time
• the products
• the prices
• the inventory
• the capacity
• taking into account
• the market response

pricing
seat inventory
overbooking
forecasting
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Outline

• The revenue management problem
• The bilevel programming problem
• A price setting paradigm
• applied to toll setting
• a TSP instance
• applied to airline
• Conclusion

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Bilevel programming problem
Leader
Follower
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Bilevel programming problem
or MPEC problems
IV
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Bilevel programming problem
A linear instance
F2
x
x
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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

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Bilevel programming problem
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Bilevel programming problem
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Bilevel programming model

• Resolution approaches
• Combinatorial approaches (global solution)
• Lower level structure combinatorial structure
• Descent approaches (on the bilevel model)
• Sensitivity analysis (local approach)
  (OutrataZowe)
• Descent approaches (on an approximated one-level
  model)
• Model still non convex (e.g. penalization of the
  second level KKT conditions) (ScholtesStöhr)

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Bilevel programming model
1. Combinatorial approaches convex/quadratic
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Bilevel programming model
KKT
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Bilevel programming model
The one level formulation
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Bilevel programming model
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Bilevel programming model
BB the subproblem
and the relaxation
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Bilevel programming model

• An efficient BB 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)

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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

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Bilevel programming model

• Potential models

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Bilevel programming model

• Notations

real
actual
current
predicted
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Bilevel programming model

• Classic steps

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Bilevel programming model

• With a linesearch step (to guaranty a strong
  stationary point)

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Bilevel programming model
b-stationary convergence
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Bilevel programming model
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Outline

• The revenue management problem
• The bilevel programming problem
• A price setting paradigm
• applied to toll setting
• a TSP instance
• applied to airline
• Conclusion

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A generic price setting model
T tax or price vector x level of taxed
activities y level of untaxed activities
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A generic price setting model
If the revenue is proportional to the activities
we obtain the so-called bilinear/bilinear problem
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A generic price setting model
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A generic price setting model
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A generic price setting model
1. The one level formulation combinatorial
approach
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A generic price setting model
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A generic price setting model
2. One level formulation continuous approach
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A generic price setting model
The combinatorial equivalent problem
The continuous equivalent problem
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Outline

• The revenue management problem
• The bilevel programming problem
• A price setting paradigm
• applied to toll setting
• a TSP instance
• applied to airline
• Conclusion

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on a transportation network
Pricing over a network
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on a transportation network
5
1
1
1
1
2
3
4
5
10
Leader max Tx Follower min (cT)x
dy AxByb
x,y gt0
T Toll vector x Toll arcs flow y
Free arcs flow
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A feasible solution...
on a transportation network
5
1 4
1 1
1 8
1
2
3
4
5
10
PROFIT 4
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on a transportation network
the optimal solution.
5
1
1
1
4
- 1
4
1
2
3
4
5
10
PROFIT 7
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on a transportation network
The algorithms

• 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)

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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)
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on a transportation network
1. A MIP formulation
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on a transportation network
2. Primal-dual approach (LB)
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on a transportation network
Step 1 Solve for T and ? (Frank-Wolfe) Step 2
Solve for x,y Step 3 Inverse optimisation Step
4 Update the M1 and M2
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Outline

• The revenue management problem
• The bilevel programming problem
• A price setting paradigm
• a toll setting problem
• a TSP instance
• applied to airline
• Conclusion

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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.

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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

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TSP a toll setting problem?
5
3
2
2
1
4
Optimal tour length 8
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TSP a toll setting problem?
-1 5/10
-1 3/10
-1 2/10
-1 2/10
-1 1/10
-1 2/10
-1 1/10
-1 4/10
-1 4/10
4
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TSP a toll setting problem?
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TSP a toll setting problem?
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TSP a toll setting problem?
Miller-Tucker-Zemlin lifted (DL91)
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TSP a toll setting problem?
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TSP a toll setting problem?
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3. TSP a toll setting problem?
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TSP a toll setting problem?
Sherali-Driscoll OR02
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Outline

• The revenue management problem
• The bilevel programming problem
• A price setting paradigm
• applied to toll setting
• applied to telecommunication
• applied to airline
• Conclusion

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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)

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Key Features of the model

• Pricing at fare basis code level
• Demand implied by rational customer reaction to
  fares (AC and competition)

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Key Features of the model

• Full accounting of interconnectedness
  (overlapping routes and markets, available
  capacity, hub-and-spoke)

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Bilevel Model Structure
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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 ?

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Parameter Distribution

• Continuous Case

• Discrete Case

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Perceived cost
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Bilevel Model
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Illustrative Example

• Network Structure

• 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

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Illustrative Example (Data)
Supply side Flights
Fare Structure
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
Leg Aircraft Capacity
(YUL, YYZ) B767-300 200
(YYZ, SFO) A320-100 130
(YUL, SFO) A330 -
(YYZ, ATL) A319 110
(YUL, ATL) MD-81 -
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
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Illustrative example (Objective)

• Action Maximize ACs 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

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Illustrative example (Reaction)

• Reaction Minimize PPC on each market

Pax Perceived Cost 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
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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

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Illustrative Example (Strategies)

Strategy Fares (pax) Revenue Gain
Match competitions 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
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Revenue Function

• Continuous Case

• Discrete Case

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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

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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)

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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

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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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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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Additional Material

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Bilevel programming model
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Continuous Example

• Uncapacitated, leg-based

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Continuous Example (continued)

• Flights

• Objective

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Continuous Example (lower level)

• Customer Reaction

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Continuous Example (flow assignment)

• Flow assignment

• Assumption
• Parameter ? is uniformly distributed over the
  interval 0, 90

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Continuous Example (solution)

• Solution analysis
• Region A 0 lt ?1 lt ?3 lt ?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 lt ?2 lt ?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

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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