A Parametric Analysis of Air Traffic Flow Control Options Under Weather Uncertainty

A Parametric Analysis of Air Traffic Flow Control Options Under Weather Uncertainty

November 17, 2002

INFORMS Annual Meeting

Lisa Grignon, IE, University of Washington

Aslaug Haraldsdottir, ATM Group, The Boeing Company

Joyce Yen, IE, University of Washington

Zelda Zabinsky, IE, University of Washington

November 17, 2002 INFORMS Conference 2

Objective

Conduct a parametric analysis on flow control options to gain a better understanding of

their effects on a dynamic air traffic system

We would like to compare the trade-off between ground delay and air delay given

uncertainties in the weather prediction


Agenda

• Air Traffic Background

• Simple Policy-Based Approach

• Stochastic Optimization Formulation

• Next Steps


Flow Control Decisions

• A collaborative decision is made between Air Traffic Control (ATC), the Airline Operational Control (AOC), and affected centers

• Flow control options result in either some form of ground delay or air delay

• Two major flow control options– Ground holding (delay on the ground)

– Miles-in-Trail (delay in the air)


Question

How do we make decisions regarding ground delay and air delay to minimize

either the total delay or the cost of delay?


Probabilistic Problem• Consider a set of planes arriving at a single airport

N

N-1

2

1

AirportQueue

(RunwaySystem)

Flights

Queuing Phenomenon

Due to weather uncertainty, there is a probabilistic reduction of capacity, airport arrival rate (AAR)


Simple Policy-Based Approach to Determine Ground and Air Delay

• Reschedule the flights based on the probabilistic arrival rate and calculate the ground delay

• Using the arrival rate profiles, assess the actual landing times and calculate the expected air delay, using FIFO policy

• Use these calculations to estimate the expected total delay


Probabilistic Arrival Rate Example

Probability Time 1 Time 2 Time 3 Time 4

Demand 30 40 20 0

AAR Profile 1 0.5 10 30 40 90

AAR Profile 2 0.1 30 10 40 90

AAR Profile 3 0.4 10 10 40 90

Original AAR 40 40 40 90

Most Likely AAR 10 30 40 90

Expected AAR 12 20 40 90


Expected Delay Under Three Policy Decisions

0

10

20

30

40

50

60

70

80

90

Original AAR Most Likely AAR Expected AAR

Del

ay E[Air Delay]

Ground Delay


Cost of Delay

0.00

50.00

100.00

150.00

200.00

250.00

300.00

350.00

400.00

Air $ = Ground $ Air $ = 2xGround $ Air $ = 5xGround $

Co

st Original AAR

Most Likely AAR

Expected AAR


Summary of Policy-Based Approach

A more conservative decision (lower AAR) results in a higher expected delay, but may result in lower expected costs

Although we can choose the best of the three policies based on cost information, not all possible policies have been considered


Stochastic Optimization Formulation

• Pose as a stochastic optimization problem

• Consider flights arriving at a single airport

• Aggregate flights in groups based upon the original scheduled arrival

• Octavio Richetta and Amedeo Odoni (1993,1994)– Min E[Cost of ground delay] + E[Cost of air delay]

– Dynamic formulation


Modification of Objective Function

• In addition to cost of ground delay and air delay, the value of the system should include the utility of the flights based on their total delay

• This new objective would be a utilitarian point of view; good for both ATC and AOC


Utility of Total Delay

• Total delay is important

• Utility of a flight is based on total delay

• A two hour delay results in 50% utility

Delay Time of Flight

Utility ofFlight

100 %

50 %

2 hours

Utility of a Flight as a function of Total Delay


Modified Stochastic Optimization Problem

Decision variables determine the AAR for each time period and distribute the arrival rate to flights according to their originally scheduled arrival time

First stage decisions (Xij) reschedule the arrival time of flights from i to j

Recourse decisions (Ωijk) assign actual arrival time k (which may differ from the original arrival time i or rescheduled arrival time j)


Mathematical Formulation (TEX slide)

kjiSX

TkX

TjiX

TkMS

MS

TiNX

ijCjkCikU

ijkkij

k

i

k

ijijkij

T

jkijkij

k

ik

k

ijijkk

T

ii

T

ii

i

T

ijij

T

i

T

ij

T

jkgaijk

,,0,,

1,,10)(

,,1,

1,,1

,,1

subject to

)]()()([max

1

1

1

11

1

1

1 1


Example Problem

Probability Time 1 Time 2 Time 3 Time 4 *Demand 30 40 20 0

AAR Profile 1 0.5 10 30 40 90AAR Profile 2 0.1 30 10 40 90AAR Profile 3 0.4 10 10 40 90

Original AAR 30 40 20 0Most Likely AAR 10 30 40 10Expected AAR 12 20 40 18

Stoch. Opt. AAR 30 10 40 10

* Time 4 is the slack time where we have the ability to recover the schedule


Comparison of Policies

0100200300400500600700800

Syste

m V

alu

e

Original AAR

Most Likely AAR

Expected AAR

Stoch. Opt. AAR


Parametric Analysis• Utility of a flight

– Constant– Slightly decreasing– Moderate decrease,

such that a 2 hour delay results in 50% utility

Delay Time of Flight

Utility ofFlight

100 %

50 %

2 hours

Utility of a Flight as a function of Total Delay

Constant

Slight

Moderate

• Delay costs– Air delay cost = Ground

delay cost– Air delay cost =

2*Ground delay cost– Air delay cost =

5*Ground delay cost


Analysis of Utility• Linear

– Constant, slightly decreasing, moderately decrease

• Monic Quadratic– Slightly decreasing, moderately decrease

• Nonmonic Quadratic– Slightly decreasing, moderately decrease

2 4 6 8 10

20

40

60

80

100


Parametric ExperimentsAir Delay $ =

Ground Delay $Air Delay $ =

2xGround Delay $Air Delay $ =

5xGround Delay $

Linear Constant LC, 1x LC, 2x LC, 5x

Linear Slightly Decreasing LS, 1x LS, 2x LS, 5x

Linear Moderately Decreasing

LM, 1x LM, 2x LM, 5x

Monic Quadratic Slightly Decreasing

mQS, 1x mQS, 2x mQS, 5x

Monic Quadratic Moderately Decreasing

mQM, 1x mQM, 2x mQM, 5x

Nonmonic Quadratic Slightly Decreasing

nmQS, 1x nmQS, 2x nmQS, 5x

Nonmonic Quadratic Moderately Decreasing

nmQM, 1x nmQM, 2x nmQM, 5x


Comparison of Delay for Functions with Varying Cost of Air Delay

0

20

40

60

80

100

120

1x 2x 5x 1x 2x 5x 1x 2x 5x

Linear Constant Linear ModeratelyDecreasing

Monic QuadraticModerately Decreasing

Del

ay E[Air Delay]

Ground Delay



Linear Constant


Parametric ExperimentsAir Delay $ =

Ground Delay $Air Delay $ =

2xGround Delay $Air Delay $ =

5xGround Delay $

Linear Constant LC, 1x LC, 2x LC, 5x

Linear Slightly Decreasing LS, 1x LS, 2x LS, 5x


LM, 1x LM, 2x LM, 5x

Monic Quadratic Slightly Decreasing

mQS, 1x mQS, 2x mQS, 5x


mQM, 1x mQM, 2x mQM, 5x

Nonmonic Quadratic Slightly Decreasing

nmQS, 1x nmQS, 2x nmQS, 5x

Nonmonic Quadratic Moderately Decreasing

nmQM, 1x nmQM, 2x nmQM, 5x


Comparison of Delay for 2x Cost with Linear Utility Functions

0

10

20

30

40

50

60

70

80

90

Constant Slightly Decreasing Moderately Decreasing

Del

ay E[Air Delay]

Ground Delay

Constant Slightly Decreasing Moderately Decreasing


Comparison of Delay for 2x Cost with Moderately Decreasing Functions

0

10

20

30

40

50

60

70

80

90

Linear Monic Quadratic Nonmonic Quadratic

Del

ay E[Air Delay]

Ground Delay

2 4 6 8 10

20

40

60

80

100


Comparison of Delay for 2x Cost with Slightly Decreasing Utility Functions

0

10

20

30

40

50

60

70

80

90

Linear Monic Quadratic Nonmonic Quadratic

Del

ay E[Air Delay]

Ground Delay

2 4 6 8 10

20

40

60

80

100


Summary of Parametric Analysis

• When cost of air delay is greater than cost of ground delay, choose a lower AAR

• Steeper utility function offsets the high cost of air delay

• Similar functions yield similar results

• Policy decisions depend on objective function properties


Future Work

• Multiple decision points with updated information (dynamic formulation)

• Solve on larger, representative problem

• Explore error in forecasting

A Parametric Analysis of Air Traffic Flow Control Options Under Weather Uncertainty

Technology

Transcript of A Parametric Analysis of Air Traffic Flow Control Options Under Weather Uncertainty