Multi-Aircraft Flight Planning Under

Uncertainty

Zehra Akyurt

Problem Description

Multiple aircraft belonging to different airlinesPossibility of facing Temporary Flight Restrictions (TFR)s en routeTFR reduces capacity of airspace which it coversNeed to find optimal routes for aircraft given stochastic travel conditions.

Example

7

0 3214 2 3

4 6

8

11

Multi-Objectives

Minimize Cost: Minimize total expected travel time

of all aircraft.

Maximize Equity: Minimize the expected differences of

total time traveled, between airlines.

Stochastic Program

Will use a multi-stage scenario based stochastic program formulation.

What will a scenario be? A joint realization of all the TFRs.

What will a stage be? Any point at which new decisions

must be made

Assumptions• Aircraft are assumed to have equal

velocity • TFRs are assumed independent.

0 3214 2 34 6

811

7

Stochastic program formulation not totally correct.

Space –Time Network

0 4 6 7 8 9 10 11 12 13 14

0

1

2 2 2

333 333

x2x1

x3

y1

y2

y3

z1

z3

z5

z2z4z6

Program Formulation

. node upto tionsrepresenta identicalh wit

, scenarios and , 0

,, ,

, , 0

,

,

s.t

Min

Min

scenariounder airline from j)(i, arc n arcon flow ofamount

,,

,,

,,

,,

,,

,,

)(,

,

)(,

,

),(

,

),(

,

k

),(

,

k

th,

i

kklxx

klncx

klxx

kFx

kFx

xdxdp

xdp

klx

mnnji

lknji

lk

nji

lknji

lk

n

nhj

lk

n

nji

lk

l

n tAj

nti

lk

l

n sAj

njs

lk

l n Aji

nij

lknij

n Aji

nij

lknij

k

n l Aji

nij

lknij

k

nij

lk

ba

ba

Objective-1

Objective-2

Conservation of Flow Constraints

Arc Capacity Constraints

Non-AnticipativityConstraints

Obstacles in Formulation

Used Xpress-MP to test modelSecond objective contains absolute value

Added additional constraints to overcome this obstacle (see Chvatal)

Example:

dycaxb Min

dcxe

dcye

baxe

baxe

ee

2

2

1

1

21

s.t.

Min

Program is now linear! Had to add integer constraintsProgram is no longer linear, nor convexUsed two general methods to solve the two-objective integer program:

Weighting methodConstraint method

Obstacles in Formulation

Sample ProblemSet• p1=1,p2=0• c1=2,c2=3, all other arcs have capacity=5• 3 airlines with 2,3 and 4 aircraft respectively = 9 aircraft

0 321(4,5) (2,2) (3,5)

(4,5) (6,5)

(8,5)

(11,5)

(7,5)

F=9

0990

0990.1

2.2592921940.2

1.7593931940.3

194942.25920.4

195952.25920.5

195962.25920.6

195972.25920.7

194982.25920.8

099992.25920.9

0991007.75921

DeviationTimeTime Constraint DeviationTimeWeight of Time

 Constraint Method  Weighting Method

Results

ResultsRecall: Objectives were

Min Total Travel Time

Weighting Method

0

2

4

6

8

10

90 92 94 96 98 100

Travel Time

Dev

iatio

ns

Min Total Deviations

Constraint Method

0

0.5

1

1.5

2

2.5

90 92 94 96 98 100

Travel Time

Dev

iatio

ns

Questions?