1

Sensitivity Studies of Network Optimization with Displacement Adjusted Virtual Nesting using PODS.

Thomas Fiig, Revenue Management Development, Scandinavian Airlines System, Denmark.Hedegaardsvej 88, DK-2300 Copenhagen.

AGIFORS Reservations and Yield Management  Conference, BKK 2001

2

Simulation Set-up in PODS

H1(41)

H2(42)

432

1

109

87

65

1517

1614

1211 22

21

2019

18

2827 26

252423

33 32313029

39

38

37

36

3534

40

13

Network D

2 Airlines

2 hubs

20 Cities (west & east).

482 markeds.

2892 paths.

Realistic fares.

Realistic disutilities.

Airline A: O&D methods

Airline B: FCYM

3

DAVN-METHODS

Emsrb

DAVN(LP)

jiiDCFare

AdjForecast

by leg

(iterate until convergence)

LP opt.

NETBID

DC

ii

DCDC

Fare

Adj

ProBPDAVN(ProBP)

Path demand and fares

Forecastby path & fares

NETBID

Path demand and fares

Forecastby path & fares

LP opt.

NETBID

DC

4

DAVN-METHODS

Emsrb

DAVN(LP)

jiiDCFare

AdjForecast

by leg

(iterate until convergence)

LP opt.

NETBID

DC

ii

DCDC

Fare

Adj

ProBPDAVN(ProBP)

Path demand and fares

Forecastby path & fares

DAVN(LP)

Emsrb

DAVN(LP)

jiiDCFare

AdjForecast

by leg

Path demand and fares

Forecastby path & fares

LP opt.

DC

5

DAVN-METHODS

Emsrb

DAVN(LP)

jiiDCFare

AdjForecast

by leg

(iterate until convergence)

LP opt.

NETBID

DC

ii

DCDC

Fare

Adj

ProBPDAVN(ProBP)

Path demand and fares

Forecastby path & fares

ProBP

(iterate until convergence)

ii

DCDC

Fare

Adj

ProBP

EmsrbForecast

by leg

Forecastby path & fares

DC

Path demand and fares

6

DAVN-METHODS

Emsrb

DAVN(LP)

jiiDCFare

AdjForecast

by leg

(iterate until convergence)

LP opt.

NETBID

DC

ii

DCDC

Fare

Adj

ProBPDAVN(ProBP)

Path demand and fares

Forecastby path & fares

DAVN(ProBP)

(iterate until convergence)

ii

DCDC

Fare

Adj

DAVN(ProBP)

EmsrbForecast

by leg

Forecastby path & fares

DC

Path demand and fares

7

Overview

Effect of reoptimization frequency

ProBP bidprice

IV

Properties of displacement costs (DC)

DAVN (LP, ProBP)

Sensitivity of bucket location, and number of buckets

DAVN (LP)I

DESCRIPTIONYM-METHODSERIES

II

III Effect of noise in displacement costs. Robustness and optimality.

DAVN (LP, ProBP)

8

Simulation Study I

– Airline A, DAVN (LP-Method) vs. Airline B EMSRb

– Sensitivity on the number of buckets 6-40, fixed and demand equalized.

– Demand factor 1.0

9

-0,4

-0,3

-0,2

-0,1

0

0,1

0,2

0 10 20 30 40 50

#buckets

Rev

enu

e (%

)DAVN(LP) Sensitivity on # buckets

System wide buckets. Fixed values through legs and timeframes. Equidistant buckets.

Demand equalized buckets. Leg specific. Recalculated at each timeframe.

DAVN LP (8 buckets), demand equalized, corresponds to the base 0%

10

Summary of Study I

• Sensitivity on the number of buckets. – Large sensitivity on the number of buckets.

Revenue difference between 8 and 30 buckets = 0.1467%.

– Fixed system wide buckets limits is not a good idea.

11

Simulation Study II

– Airline A, DAVN (LP, ProBP) vs. Airline B EMSRb– Buckets are demand equalized and number=8.– Properties of displacement costs.– Demand factors 1.0

12

Distribution of DC, LP-Method

0

5000

10000

15000

0 125 250

$

#Ob

s

TF1

TF4

TF7

TF10

TF13

TF16

Displacement Costs in LPDistribution of displacement cost by timeframe (TF)

Early Time Frame

Departure

13

Properties DC (LP)

Average DC as function of TF. Note that flights open at intermediate TF and then closes at departure.

PCT of Displacement costs that are zero as function of TF. Note that 40% - 90% of the flights are wide open.

Average DC, LP-Method

020406080

100

0 5 10 15 20

TF

$

PCT DC=0 LP-Method

00,20,40,60,8

1

0 5 10 15 20

TF

PC

T

Departure

14

Distribution of DC, PROBP-Method

0

5000

10000

15000

20000

0 125 250

$

#Ob

s

TF1

TF4

TF7

TF10

TF13

TF16

Displacement Costs in PROBP

Distribution of displacement cost by timeframe (TF)

Departure

15

Properties DC (PROBP)

Average DC as function of TF. Note that DC are much more stable, although DC tends to increase towards dep.

PCT of Displacement costs that are zero as function of TF. Note that approx. 10%-70% of the flights are wide open.

Average DC, PROBP-Method

020406080

100

0 5 10 15 20

TF

$

PCT DC=0 PROBP-Method

0

0,2

0,4

0,6

0,8

0 5 10 15 20

TF

PC

T

Departure

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Summary of Study II

•Displacement Cost from LP– Distribution ragged. Approx 40%-90% of the DC are

zero.– Average of DC decrease at intermediate timeframes

and increases towards departure.•Displacement Cost from ProBP

– Distribution smooth. Between 10%-70% of the DC are zero.

– Average of DC is almost constant.

17

Simulation Study III

– Airline A, DAVN (LP, ProBP) vs. Airline B EMSRb– Buckets are demand equalized and number=8.– Demand factors 1.0– Sensitivity of random noise to the

displacement costs k-factor between 0 and 0.5.

– Temporal dependence of DC.In the simulation study the noise are introduced as:

randkDCDC facoldnew11

18

-0,2-0,15

-0,1-0,05

00,05

0,10,15

0 0,2 0,4 0,6

Noise k-factor

Re

v. in

pc

t

DAVN(LP)

DAVN(ProBP)

Random Noise to DC

ProBP is robust to addition of random noise. Even k-factors as high as 0.5. LP is sensitive.

Let us look at the distributions for k=0.3

DAVN LP (8 buckets), demand equalized, corresponds to the base 0%

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DC in LP

Distribution of DC, LP-Method

0

5000

10000

15000

0 125 250

$#O

bs

TF1

TF4

TF7

TF10

TF13

TF16

The noise has a smoothing effect on the distribution of the DC. The revenue is redu-ced by 0.03 %

Distribution of DC, LP (k=0.3)

0

5000

10000

15000

0 125 250$

#Ob

s

TF1

TF4

TF7

TF10

TF13

TF16

20

DC in ProBP

Distribution of DC, PROBP-Method

0

5000

10000

15000

20000

0 125 250

$#O

bs

TF1

TF4

TF7

TF10

TF13

TF16Again the noise has a smoothing effect on the distribution. Distribution of DC, PROBP (k=0.3)

0

5000

10000

15000

20000

0 125 250$

#Ob

s

TF1

TF4

TF7

TF10

TF13

TF16

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LP AVG(DC) with noise

DAVN(LP) with k=0.0; k=0.3; and k=0.5. Revenue is decreasing with noise.

k=0

15

20

25

30

35

40

0 4 8 12 16

TF

AV

G(D

C)

The revenue is decreasing with increasing noise.

k=0.3

k=0.5

k=0

22

30354045505560

0 4 8 12 16

TF

AV

G(D

C)

ProBP AVG(DC) with noise

DAVN(ProBP) with k=0.0; k=0.3; and k=0.5. Revenue is max in k=0.3.

The slope goes from (+) to (-) as maximum revenue is attained.

k=0

k=0.3

k=0.5

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Summary of Study III

• Random noise to DC for DAVN (LP, ProBP)– Identified that revenue maximum coincides with

constant DC as function of timeframe.– ProBP actually gets stabilized by random noise.– LP is very sensitive to noise, since most of the

DC are zero.– The time dependence of the DC can be used as

a quality measure.– Active use of smoothing techniques to forecasts

are possibly a way to generate smoother DC distributions, which in turns stabilizes convergence of the DC.

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Simulation Study IV

– Airline A, Bidprice (ProBP) vs. Airline B EMSRb– Demand factors 1.0– Study the revenue dependence of the

frequency of network optimization.

25

-0,4

-0,2

0

0,2

0,4

0,6

0 2 4 6 8

reopt every #book/leg

Rev

enu

e %

ProBP

Revenue as function of reopt.Breakeven for ProBP is <2 bookings per leg. Points correspond to reopt = 10, 100, 200, 500 and 1000

DAVN LP (8 buckets), demand equalized, corresponds to the base 0%

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Benchmarking

EMSRb corresponds to the base 0%

0,45

1,51

0,57

1,04

1,391,391,401,36

0,00

0,50

1,00

1,50

Rev

in p

ct.

BIDPRICEBOOKING LIMITS