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Multi-Aircraft Flight Planning Under Uncertainty

Multi-Aircraft Flight Planning Under Uncertainty. Zehra Akyurt. Problem Description. Multiple aircraft belonging to different airlines Possibility of facing Temporary Flight Restrictions (TFR)s en route TFR reduces capacity of airspace which it covers

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Multi-Aircraft Flight Planning Under Uncertainty

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  1. Multi-Aircraft Flight Planning Under Uncertainty Zehra Akyurt

  2. Problem Description • Multiple aircraft belonging to different airlines • Possibility of facing Temporary Flight Restrictions (TFR)s en route • TFR reduces capacity of airspace which it covers • Need to find optimal routes for aircraft given stochastic travel conditions.

  3. Example 7 6 4 2 3 4 0 1 2 3 8 11

  4. Multi-Objectives • Minimize Cost: Minimize total expected travel time of all aircraft. • Maximize Equity: Minimize the expected differences of total time traveled, between airlines.

  5. 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

  6. Assumptions • Aircraft are assumed to have equal velocity • TFRs are assumed independent. 7 6 4 2 3 4 0 1 2 3 8 11 Stochastic program formulation not totally correct.

  7. 0 4 6 7 8 9 10 11 12 13 14 0 x1 x2 1 y1 z1 y2 z3 2 2 2 z5 z2 z4 z6 3 3 3 3 3 3 x3 y3 Space –Time Network

  8. Program Formulation Objective-1 Objective-2 Conservation of Flow Constraints Arc Capacity Constraints Non- Anticipativity Constraints

  9. Obstacles in Formulation • Used Xpress-MP to test model • Second objective contains absolute value Added additional constraints to overcome this obstacle (see Chvatal) Example:

  10. Obstacles in Formulation • Program is now linear! • Had to add integer constraints • Program is no longer linear, nor convex • Used two general methods to solve the two-objective integer program: • Weighting method • Constraint method

  11. Sample Problem Set • 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 (7,5) (6,5) (4,5) (2,2) (3,5) F=9 (4,5) 0 1 2 3 (8,5) (11,5)

  12. Weighting Method Constraint Method Weight of Time Time Deviation Time Constraint Time Deviation 1 92 7.75 100 99 0 0.9 92 2.25 99 99 0 0.8 92 2.25 98 94 1 0.7 92 2.25 97 95 1 0.6 92 2.25 96 95 1 0.5 92 2.25 95 95 1 0.4 92 2.25 94 94 1 0.3 94 1 93 93 1.75 0.2 94 1 92 92 2.25 0.1 99 0 0 99 0 Results

  13. Results Recall: Objectives were • Min Total Travel Time • Min Total Deviations

  14. Questions?

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