role of stochastic optimization in revenue management. topaloglu.pdf · other application settings...

Role of Stochastic Optimization in

Revenue Management

Huseyin TopalogluSchool of Operations Research and Information Engineering

Cornell University

Revenue Management

Revenue management involves making most use of limitedinventories of perishable resources under random demand

Airlines form a typical application settingResources correspond to capacities on flight legsCustomers arrive randomly over time

Other application settings include hotels, car rentals, advertising, fashion retail

Primary tradeoff is whether to give resources to a currently available customer that is willing to pay a low price or keep the resources with the hope of a future customer that may be willing to pay a high price

Revenue Management

$$$ $$$$$$$$

Capacity Allocation Dynamic Pricing

Assortment Offering

Decision

Mechanism

Revenue Management

Static, Single Resource Dynamic, Single Resource

Dynamic, Network of Resources

Modeling

Detail

Revenue Management

CapacityAllocation

DynamicPricing

AssortmentOffering

StaticSingle Leg

DynamicSingle Leg

DynamicNetwork

Network Revenue Management

Requests arrive over time

Make an acceptance or rejection decision for each request

Accepted requests consume capacities on one or more flight legs

Deterministic Linear Programming Approximation

: Set of flight legs in the airline network

: Set of itineraries

: Probability of getting a request for itinerary j at time t

: Revenue from itinerary j

: 1 if itinerary j uses flight leg i0 otherwise

: Total available capacity on flight leg i

L

J

pjt

fj

aij

flightsdepart

bookingsstart

ci


Assume that the numbers of itinerary requests take on their expected values

ZLP = max∑

j∈J

fj wj

st∑

j∈J

aij wj ≤ ci i ∈ L (µ∗i )

0 ≤ wj ≤∑

t∈T

pjt j ∈ J


are called bid prices

They characterize the value of a seat

Accept a request for itinerary j if

...subject to capacity availability

As time progresses, it is customary to refresh the bid prices by resolving the linear program

Optimal objective value of the linear program provides an upper bound on the performance of any nonancitipatorypolicy

{µ∗i : i ∈ L}

fj ≥∑

i∈L

aij µ∗i

Deterministic

Linear

Program

Decomposition by

Displacement

Adjustments

Decomposition by

Fare

Allocations

Dynamic Programming Formulation

: Set of flight legs in the airline network

: Set of itineraries

: Probability of getting a request for itinerary j at time t

: Revenue from itinerary j

: 1 if itinerary j uses flight leg i0 otherwise

: Remaining capacity on flight leg i at time t

: 1 if a request for itinerary j is accepted at time t0 otherwise

L

J

pjt

fj

aij

xit

ujt

Dynamic Programming Formulation

To obtain approximations to value functions, decomposethe dynamic programming formulation by the flight legs

Vt(xt) = max∑

j∈J

pjt

{fj ujt + Vt+1(xt − ujt

∑i∈L aij ei)

}

st aij ujt ≤ xit i ∈ L, j ∈ J

ujt ∈ {0, 1} j ∈ J

Decomposition by Fare Allocations

fj

λi′j

λi′′j

λi′′′j

: Fare allocation of itinerary j over flight leg iλij

∑

i∈L

λij = fj


fj

Revenue allocations allow us to solve single-leg problems

Single-leg problems provide an upper bound on exact value functions

∑

i∈L

vit(xit | λ) ≥ Vt(xt)

as long as ∑

i∈L

λij = fj j ∈ J


vit(xit | λ) = max∑

j∈J

pjt

{λij ujt + v

it+1(xit − aij ujt |λ)

}

st aij ujt ≤ xit j ∈ J

ujt ∈ {0, 1} j ∈ J

We obtain upper bounds on the optimal expected revenue

To obtain the tightest possible upper bound, we solve

∑

i∈L

vit(xit | λ) ≥ Vt(xt)∑

i∈L

vi1(ci |λ) ≥ V1(c)

Tightest possible upper bound is better than the one from deterministic linear programming approximation

convex in λ

ZLP ≥∑

i∈L

vi1(ci |λ∗) ≥ V1(c)


minλ

∑

i∈L

vi1(ci | λ)

st∑

i∈L

λij = fj j ∈ J

We can obtain a good revenue allocation from deterministic linear programming approximation

max∑

j∈J

fj wj

Computing Good Fare Allocations

st∑

j∈J

aij wj ≤ ci i ∈ L

0 ≤ wj ≤∑

t∈T

pjt j ∈ J


max∑

j∈J

fj wψj


st∑

j∈J

aij wij ≤ ci i ∈ L

0 ≤ wij ≤∑

t∈T

pjt i ∈ L, j ∈ J

wψj − wij = 0 j ∈ J , i ∈ L


max∑

j∈J

fj wψj

∑

i∈L

β∗ij = fj j ∈ J


st∑

j∈J

aij wij ≤ ci i ∈ L

0 ≤ wij ≤∑

t∈T

pjt i ∈ L, j ∈ J

wψj − wij = 0 j ∈ J , i ∈ L (β∗ij)

If we have the exact value functions, then we can make the decisions at time t by solving

Making Decisions

It is optimal to accept a request for itinerary j at time t when there is enough capacity and

fj + Vt+1(xt −∑

i∈L aij ei) ≥ Vt+1(xt)

Vt(xt) = max∑

j∈J

pjt


∑i∈L aij ei)

}


ujt ∈ {0, 1} j ∈ J

If we have the exact value functions, then we can make the decisions at time t by solving

Making Decisions

It is optimal to accept a request for itinerary j at time t when there is enough capacity and

fj ≥ Vt+1(xt) − Vt+1(xt −∑

i∈L aij ei)

Vt(xt) = max∑

j∈J

pjt


∑i∈L aij ei)

}


ujt ∈ {0, 1} j ∈ J

We accept a request for itinerary j at time t when there is enough capacity and

Making Decisions

fj ≥∑

i∈L

vit+1(xit |λ)−∑

i∈L

vit+1(xit − aij |λ)

fj ≥∑

i∈L

aij

[vit+1(xit | λ)− v

it+1(xit − 1 |λ)

]

bid price offlight leg i

An airline network with one hub serving multiple spokes

Numerical Performance

There is a high-fare and a low-fare itinerary connecting each origin-destination pair

Compare deterministic linear programming approximation(DLP), decomposition with fare allocations from the deterministic linear program (DFA-DLP) and decomposition with best fare allocations (DFA-OPT)

Comparing Upper Bounds

0

5

10

15

20

1 2 3 4 5 6 7 8 9 10 11 12

lowload

mediumload

highload

lowload

mediumload

highload

4 spokes 8 spokes

DFA-DLP

DFA-OPT

% gap with DLP

Comparing Expected Revenues

0

5

10

15

20

1 2 3 4 5 6 7 8 9 10 11 12

lowload

mediumload

highload

lowload

mediumload

highload

4 spokes 8 spokes

DFA-DLP

DFA-OPT

% gap with DLP

Deterministic

Linear

Program

Decomposition by

Fare

Allocations

Decomposition by

Displacement

Adjustments

Decomposition by Displacement Adjustments

Start from the deterministic linear program to decompose the dynamic programming formulation

ZLP = max∑

j∈J

fj wj

st∑

j∈J

aij wj ≤ ci i ∈ L (µ∗i )

0 ≤ wj ≤∑

t∈T

pjt j ∈ J


Start from the deterministic linear program to decompose the dynamic programming formulation

st∑

j∈J

aij wj ≤ ci

0 ≤ wj ≤∑

t∈T

pjt j ∈ J

ZLP = max∑

j∈J

[fj −

∑

k∈L\{i}

akj µ∗k

]wj +

∑

k∈L\{i}

µ∗k ck

We can solve the dynamic programming formulation of the single-leg problem over flight leg i

Single-leg problem provides an upper bound on exact value functions


vit(xit) +∑

k∈L\{i}

µ∗k xkt ≥ Vt(xt)

vit(xit ) = max∑

j∈J

pjt

{[fj −

∑

k∈L\{i}

akj µ∗k

]ujt + v

it+1(xit − aij ujt)

}

st aij ujt ≤ xit j ∈ J

ujt ∈ {0, 1} j ∈ J

Upper bounds are tighter than the one from the deterministic linear programming approximation

Apply the same idea for each flight leg i and approximate the value function by


∑

i∈L

vit(xit)

Vt(x t)

ZLP ≥ vi1(ci) +

∑

k∈L\{i}

µ∗k ck ≥ V1(c)

An airline network with one hub serving multiple spokes

Numerical Performance

There is a high-fare and a low-fare itinerary connecting each origin-destination pair

Compare deterministic linear programming approximation (DLP), decomposition with fare allocations from the deterministic linear program (DFA-DLP), decomposition with best fare allocations (DFA-OPT) and decomposition by displacement adjustments (DDA)

0

5

10

15

20

1 2 3 4 5 6 7 8 9 10 11 12

Comparing Upper Bounds

lowload

mediumload

highload

lowload

mediumload

highload

4 spokes 8 spokes

DFA-DLP

DFA-OPT

DDA

% gap with DLP

0

5

10

15

20

1 2 3 4 5 6 7 8 9 10 11 12

Comparing Expected Revenues

lowload

mediumload

highload

lowload

mediumload

highload

4 spokes 8 spokes

DFA-DLP

DFA-OPT

DDA

% gap with DLP

We can obtain better policies by using value function approximations that are non-separable by flight legs

It is important to incorporate additional constraints on various performance measures, such as occupancy

Maximizing expected revenue is reasonable when dealing with large airlines, but incorporating risk becomes important for smaller businesses

Number of dimensions of the state vector becomes even larger when we explicitly deal with overbooking and reservation fees

Deterministic problem becomes large when we deal with dynamic pricing and assortment planning

Directions for Improvement

Partial List of References

Talluri, K. T. and van Ryzin, G. J. (2004), The Theory and Practice of Revenue Management, KluverAcademic Publishers. (Book)

Phillips, R. L. (2005), Pricing and Revenue Optimization, Stanford University Press, Stanford, CA. (Book)

Simpson, R. W. (1989), Using network flow techniques to find shadow prices for market and seat inventory control, Technical report, MIT Flight Transportation Laboratory Memorandum M89-1, Cambridge, MA. (Network, capacity allocation)

Williamson, E. L. (1992), Airline Network Seat Control, PhD thesis, Massachusetts Institute of Technology, Cambridge, MA. (Network, capacity allocation)

Brumelle, S. L. and McGill, J. I. (1993), "Airline seat allocation with multiple nested fare classes", Operations Research 41, 127-137. (Single-leg, capacity allocation)

Gallego, G. and van Ryzin, G. (1994), “Optimal dynamic pricing of inventories with stochastic demand over finite horizons", Management Science 40(8), 999-1020. (Single-leg, dynamic pricing)

Gallego, G. and van Ryzin, G. (1997), “A multiproduct dynamic pricing problem and its applications to yield management”, Operations Research 45(1), 24-41. (Network, dynamic pricing)

Talluri, K. and van Ryzin, G. (1998), “An analysis of bid-price controls for network revenue management”, Management Science 44(11), 1577-1593. (Network, capacity allocation)

Talluri, K. and van Ryzin, G. (1999), “A randomized linear programming method for computing network bid prices”, Transportation Science 33(2), 207-216. (Network, capacity allocation)

Bertsimas, D. and Popescu, I. (2003), “Revenue management in a dynamic network environment”, Transportation Science 37, 257-277. (Network, capacity allocation, overbooking)

Talluri, K. and van Ryzin, G. (2004), "Revenue management under a general discrete choice model of consumer behavior", Management Science 50(1), 15-33. (Single-leg, assortment planning)

Partial List of References

Gallego, G., Iyengar, G., Phillips, R. and Dubey, A. (2004), Managing flexible products on a network, Computational Optimization Research Center Technical Report TR-2004-01, Columbia University. (Network, assortment planning)

Karaesmen, I. and van Ryzin, G. (2004), "Overbooking with substitutable inventory classes", Operations Research 52(1), 83-104. (Network, overbooking)

van Ryzin, G. and Vulcano, G. (2008), "Computing virtual nesting controls for network revenue management under customer choice behavior", Manufacturing & Service Operations Management 10(3), 448-467. (Network, assortment planning)

Zhang, D. and Cooper, W. L. (2005), "Revenue management for parallel flights with customer choice behavior", Operations Research 53(3), 415-431. (Network, assortment planning)

Adelman, D. (2007), "Dynamic bid-prices in revenue management", Operations Research 55(4), 647-661. (Network, capacity allocation)

Liu, Q. and van Ryzin, G. (2008), "On the choice-based linear programming model for network revenue management", Manufacturing & Service Operations Management 10(2), 288-310. (Network, assortment planning)

Kunnumkal, S. and Topaloglu, H. (2008), "A refined deterministic linear program for the network revenue management problem with customer choice behavior", Naval Research Logistics Quarterly 55(6), 563-580. (Network, assortment planning)

Topaloglu, H. (2009), "Using Lagrangian relaxation to compute capacity-dependent bid-prices in network revenue management", Operations Research 57(3), 637-649. (Network, capacity allocation)

Zhang, D. and Adelman, D. (2009), "An approximate dynamic programming approach to network revenue management with customer choice", Transportation Science 42(3), 381-394. (Network, assortment planning)

role of stochastic optimization in revenue management. topaloglu.pdf · other application settings...

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