multi-aircraft flight planning under uncertainty zehra akyurt
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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
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xdxdp
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mnnji
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n tAj
nti
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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?