2. Multi-Aircraft Flight Planning Under Uncertainty Zehra Akyurt

3. 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.

4. Example 7 0 3 21 4 2 3 4 6 8 11

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

6. 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

7. Assumptions Aircraft are assumed to have equal velocity TFRs are assumed independent. 0 3 21 4 2 3 4 6 8 11 7 Stochastic program formulation not totally correct.

8. Space –Time Network 0 4 6 7 8 9 10 11 12 13 14 0 1 222 333333 x2x2 x1x1 x3x3 y1y1 y2y2 y3y3 z1z1 z3z3 z5z5 z2z2 z4z4 z6z6

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

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

11. 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 Obstacles in Formulation

12. Sample Problem Set p 1 =1,p 2 =0 c 1 =2,c 2 =3, all other arcs have capacity=5 3 airlines with 2,3 and 4 aircraft respectively = 9 aircraft 0 3 21 (4,5) (2,2) (3,5) (4,5) (6,5) (8,5) (11,5) (7,5) F=9

13. Results

14. Recall : Objectives were Min Total Travel Time Min Total Deviations

15. Questions?