bilevel programming approaches to revenue management and price setting problems

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


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



RM Business process

ForecastingSchedule with capacity PricingBooking limitsSeat sales



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



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



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



«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)



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.



Maximize expected revenueby determining over time

the productsthe pricesthe inventorythe capacity

taking into account

the market response

pricing

seat inventory

overbooking

forecasting…


Outline



Conclusion


Bilevel programming problem

0),( s.t.

),(min

1

1

yxg

yxf

0),( s.t.

),(argmin

2

2

zxg

zxfy

Leader

Follower



)(',0'),,(

0),(:)(

0),(s.t.

),(min

2

1

xYyyyyxF

yxgyxYy

yxg

yxf

… or MPEC problems

IV



F1 y

x

F2

x’ x’’

A linear instance…



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


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)



1. Combinatorial approaches: convex/quadratic



KKT



The one level formulation:



B&B: the subproblem

and the relaxation



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)



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)



Potential models:

Resolution Approximation

lin/lin ++++ ----

quad/lin +++ ---

conv/lin ++ --

lin/quad +++ ++

quad/quad ++ +++

con/quad + ++++



Notations

),( yx

),( yx

),( yx))(,( xyxactual

predicted

real

current



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



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

kkj

kkk

j

xfx

log,,1:2 where

)(minargthen if 1min



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


Outline



Conclusion


A generic price setting model

T: tax or price vector

x: level of taxed activities

y: level of untaxed activities



If the revenue is proportional to the activities we obtain the so-called bilinear/bilinear problem:



1. The one level formulation: combinatorial approach



2. One level formulation: continuous approach



The combinatorial equivalent problem…

The continuous equivalent problem…


Outline



Conclusion


… on a transportation network

Pricing over a 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


A feasible solution...

2 3 4

105

5

1 + 41

1 + 1 1 + 8

PROFIT = 4




2 3 4

105

5

11

1 1

PROFIT = 7

…the optimal solution.

+ 4

- 1 + 4



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:



Replacing the lower level problem by its optimality conditions, the only nonlinear constraints are:

We can linearize this term (exploiting the shortest paths):



1. A MIP formulation



2. Primal-dual approach (LB)



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


Outline


… a toll setting problem… a TSP instance… applied to airline

Conclusion


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



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



2

1

3

4

2

5

Optimal tour: length 8



-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



Miller-Tucker-Zemlin lifted (DL91)


3. TSP: a toll setting problem?



Sherali-Driscoll OR02


Outline


… applied to toll setting… applied to telecommunication… applied to airline

Conclusion


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)



Pricing at fare basis code level

Demand implied by rational customer reaction to fares (AC and competition)

Demand vs behavioural

approach



““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’)


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


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 :


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


Perceived cost ((, , ))

: : VOQVOQ

: VOT: VOT

QDT fff 2.22),( ,11,

QDT fff 2.22),( ,11,

QDT fff 76),( ,22,


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


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


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


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


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


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


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


Revenue Function

Continuous Case Discrete Case


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


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)


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


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


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


Additional Material


Continuous Example

Uncapacitated, leg-based

MTLMTLVANVANTKOTKOSHSH

NYNY

( TMV )

( TVS )

Japan AirlinesChina Airlines

United AirlinesAir Canada


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


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 =


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


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



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


TM

V

TVS

A

B

bilevel programming approaches to revenue management and price setting problems

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