The document discusses time-based metering for air traffic management. It proposes using mixed integer linear programming (MILP) to minimize total delay for aircraft arriving at an airport. The approach models aircraft trajectories through an airspace network as a series of streams and reference points, with separation constraints between aircraft. The objective is to schedule aircraft arrival times at the final approach fix to minimize delay from scheduled times. Constraints include separation times, maximum allowed delay times, and prohibiting overtaking within streams. The approach is intended to benchmark real-time planning tools and improve efficiency of air traffic metering.

1. MILP FOR TIME-BASED METERING IN AIR
TRAFFIC MANAGEMENT
Narendra Sharma
MS Defense
December 2, 2009
Human Factors and Systems Engineering Lab (HFSEL)
School of Industrial Engineering
Advisors: Prof. Steven Landry
Prof. Leyla Ozsen

2. OUTLINE
 Introduction
 Literature Review
 Proposed Approach
 Results and Discussion
 Conclusion and Future Research
2/37

3. AIR TRANSPORTATION SYSTEM
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A Typical Arrival Route
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4. ASPECTS OF AIR TRANSPORTATION SYSTEM
 Safety
 Capacity
 Fuel Efficiency
 Allowable Maximum Delay Time (AMDT)
 No Overtake (passing not allowed)
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5. MINIMUM TIME SEPARATION
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Minimum separation time between aircraft at reference point (RP)
STA: Scheduled time of arrival
5/37

6. NO OVERTAKE
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No overtake in jet route No overtake in stream
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7. METERING
 Distance-based metering
 Flow based metering
 Inefficient merging
 Ripple effect
 Time-based metering
 Trajectory level
 Scheduling at reference points
 Efficient merging
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8. MOTIVATION
 Aircraft are scheduled near terminal area
 Infeasibility (overtakes)
 Low altitude holding
 Time-based metering: En route and Terminal
 No benchmarks for real time planning tool such as McTMA
 No overtaking in same stream
 High altitude delays
8/37

9. RESEARCH PROBLEM AND PROPOSED APPROACH
Research Problem
 Objective: Minimize total delay
 Constraints:
 Separation constraints
 AMDT
 No overtaking within a stream
 Time window
Proposed Approach
 Mixed integer 0-1 programming using GAMS/CPLEX Software
9/37

10. CONTRIBUTIONS
 Mixed-integer 0-1 programming: En route + Terminal
 Benchmark real-time planning tools such as McTMA
 Mixed-integer 0-1 programming for Constrained Position Shifting
 Classification of literature
10/37

11. SCHEDULING POLICIES
 Existing Policies
 First-come-first-serve (FCFS)
 Passing Allowed (PA)
 Constrained Position Shifting (CPS)
 Proposed policy
 Time-based Scheduling (TBS)
11/37

12. RELATED WORK
En
Static/ Speed Time No
Paper Holding Precedence AMDT MPS route
Dynamic up window overtake
airspace
Abela et al. ✓ ✓ ✓
Static ✓
(1993)
Ernst et al. ✓ ✓ ✓ ✓ ✓
Static
(1999)
Beasley et al. ✓ ✓ ✓ ✓ ✓
Static
(2000)
Beasley et al. ✓ ✓ ✓ ✓ ✓
Dynamic
(2004)
Brinton ✓ ✓ ✓ ✓ ✓
Dynamic
(1992)
Balakrishnan ✓ ✓ ✓ ✓ ✓ ✓
Static
et al. (2006)
Bianco et al. ✓ ✓ ✓ ✓ ✓
Dynamic
(2006)
12/37

13. RELATED WORK (CONTD.)
En
Static/ Speed Time No
Paper Holding Precedence AMDT MPS route
Dynamic up window overtake
airspace
Psaraftis ✓ ✓
Static
(1980)
Dear et al. ✓ ✓ ✓
Dynamic
(1991)
Venkatakrishnan ✓ ✓ ✓
Both
et al. (1993)
Trivizas ✓ ✓
Static
(1998)
Saraf et al. ✓ ✓ ✓
Static
(2006)
Proposed ✓ ✓ ✓ ✓ ✓
Static
Approach
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14. CLASSIFICATION OF POLICIES
PA CPS FCFS TBS
Speed up
Holding
Time
✓ ✓ ✓
window
Precedence ✓ ✓ ✓
AMDT ✓ ✓ ✓ ✓
MPS ✓
No overtakes ✓ ✓ ✓
En route
✓
Airspace
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15. HYPOTHETICAL AIRSPACE NETWORK
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D rcs1s1 = r0 s 3
r0 s 2 s2 ! S
rcs 2 s 2 = r0 s 3
Rs : Set of reference points in stream s ∈S Rs = r0 s , r1s ,..., rcs s ( )
cs : Cardinality of set Rs
MP: Meter Point, MF: Meter Fix, FAF: Final Approach Fix, RW: Runway
SD: Set of streams farthest from RW in any route, SS: Final stream
15/37

16. ET Ar : Estimated arrival time of aircraft a at reference point r
a
3.2 L: A Large Variables
Decision Number
DECISION VARIABLES
3.2 Decision Variables
 1 if aircraft a arrives at reference point r at time t,
r
Uat = (3.1)
 0 otherwise

  if aircraft a is trailing aircraft a in stream s,
∑
a1 a2 tU
αs U= =
r 1 1 if aircraft a arrives at reference point r at time t,
1 2
r at : represents scheduled time of arrival (STA) of aircraft a
at 
 0otherwise
(3.2)
(3.1
t 0 otherwise

 1 if aircraft a is trailing aircraft a in stream s,
a1 a2 1 2
3.3 Objective =
αs Function (3.2
 0 otherwise
 
3.3 α sa1a2 : represents
Objective Function 
precedence variable between aircraft a1 and a2 in stream s
min tUat − ET Ar 
r
a (3.3)
s∈{SS} a∈As r
t∈Ta ,r∈{F AF }
 
The objective (equation (3.3)) is to
minimize total airborne delay of all the aircraft
the
16/37
 tUat −this Ar  are not changed
r
at the final approach fix (since the schedules are fixed at ET point
min a (3.3
s∈{SS} a∈As r
t∈Ta ,r∈{F AF }

17. OBJECTIVE FUNCTION
 Minimize total delay at runway
⎡ ⎤
min ∑ ∑ ⎢ ∑ tU at − ETAa ⎥ r r
s∈{SS} a∈As ⎢ t ∈Tar ,r ∈{FAF}
⎣ ⎥
⎦
STA of aircraft a at r
Summation over ETA of aircraft a at r
all aircraft
STA: Scheduled time of arrival
ETA: Estimated time of arrival
As: Set of aircraft in stream s
17/37

18. CONSTRAINTS
1. ∑ U at = 1,
r
∀a ∈A, ∀r ∈P(a)
t ∈Tar
P(a) : Reference points in the path of aircraft a
2. 0≤ ∑ t 2U at2 −
r2
∑ t1U at1 − ETEa,r1 ,r2 ≤ AMDTr1r2 ,
r1
t 2 ∈Tar2 t1 ∈Tar1
{ ( )}
∀a ∈As , ∀ ( r1 , r2 ) ∈ Rs | r1 = ris , r2 = r(i +1)s , 0 ≤ i cs , ∀(r1 , r2 ) ∈P(a), ∀s ∈S
ETE: Estimated time en route
3. ∑ tU 2
r
a2 t 2 − ∑ tU 1
r
a1t1 ≥ Lα sa2 a1 − L, α sa2 a1 = 1 t 2 t1
t 2 ∈Tar2 t1 ∈Tar1
∀ ( a1 , a2 ) ∈As , ∀r ∈Rs , ∀r ∈P(a1 ) ∩ P(a2 ), ∀s ∈S
18/37

19. CONSTRAINTS (CONTD.)
4. ∑ tU 2
r
a2 t 2 − ∑ tU 1
r
a1t1 ( ) ( )
≥ sepa1a2 α sa1a2 + sepa2 a1 α sa2 a1 ,
t 2 ∈Tar2 t1 ∈Tar1
∀ ( a1 , a2 ) ∈As , ∀r ∈P(a1 ) ∩ P(a2 ), ∀s ∈S
MITr Miles in-trail
r ∈Rs {FAF} sepa1a2 =
va1 Speed
r ∈{FAF} sepa1a2 = Wa1a2 Wake vortex
separation
19/37

20. CONSTRAINTS (CONTD.)
5. α s21 2 = α s11 2 , s1
aa aa
!# s2
r0 s1
∀ ( a1 , a2 ) ∈As1 ∩ As2 ,
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{ }
∀ ( s1 , s2 ) ∈ S | rcs1 s1 = r0 s2 , s1 ∉{SS}
!$ rcs1s1 = r0s2
s3
6. α sa1a2 + α sa2 a1 = 1,
∀ ( a1 , a2 ) ∈As , ∀s ∈S
7. (α a1 a2
s )(
− α sa2 a1 ETAa1 − ETAa2 0,
r r
)
∀ ( a1 , a2 ) ∈As , ∀r ∈P(a1 ) ∩ P(a2 ), r = r0 s , ∀s ∈SD
20/37

21. ASSUMPTIONS OF PROPOSED APPROACH
 Static/Snapshot case
 Constant speed throughout an aircraft flight
 Single runway
21/37

22. PHL AIRSPACE NETWORK
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23. STREAM FORMATION
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Arrival routes are combined to form streams (s1, s2, s3)
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24. PHL STREAM REPRESENTATION
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PHL arrival routes converted to streams. Streams are depicted (s1-s15)
24/37

25. PARAMETERS
 Excess Delay
 Associated with the reference points lying on the scheduling horizon
 Delays that can not be absorbed inside the horizon
 Total Delay
 Total delay for all aircraft at the runway
 Infeasibility
 Computed resultant sequence issues overtake
25/37

26. TBS SCHEDULING HORIZON
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27. FCFS SCHEDULING HORIZON
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28. CPS AND PA SCHEDULING HORIZON
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Dashed line represents scheduling horizon for CPS and PA approach
28/37

29. TOTAL DELAY FOR ALL AIRCRAFT AT RUNWAY
20 30 35
aircraft aircraft aircraft
Asterisk (*) marked solutions are Infeasible
29/37

30. MEDIAN OF NON-ZERO EXCESS DELAY PER
AIRCRAFT
20 30 35
aircraft aircraft aircraft
30/37

31. INFEASIBILITY
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• Dotted lines represent ETA and solid lines represent STA
• Scale on the figure represents arrival times at reference points
31/37

32. FCFS STA SCHEMATIC REPRESENTATION
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Timeline
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33. PA STA SCHEMATIC REPRESENTATION
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Timeline
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34. CONCLUSION AND FUTURE RESEARCH
 Results Review
 Trajectory level scheduling
 Efficient merging
 High altitude delays
 Reduced excess delays
 Future Research
 Dynamic case
 Multiple runway
34/37

35. ACKNOWLEDGMENT
 Committee Members:
Prof. Steven Landry, School of Industrial Engineering, Purdue University
Prof. Leyla Ozsen, Department of Decision Sciences, San Francisco State University
Prof. Nelson Uhan, School of Industrial Engineering, Purdue University
35/37

36. Thank You!
36/37

37. Questions
?
37/37

38. FCFS STA SCHEMATIC REPRESENTATION
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Timeline
38

39. PA STA SCHEMATIC REPRESENTATION
012 122 132 142 152 112 622 632 642
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Timeline
39

40. CPS STA SCHEMATIC REPRESENTATION
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40

41. TBS STA SCHEMATIC REPRESENTATION
789 899 8:9 8;9 89 889 =99 =:9 =;9
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41