“An Airspace Planning and Collaborative
Decision Making Model (APCDM) Under Safety,
Workload, and Equity Considerations”
Hanif D. Sherali
Raymond W. Staats
Antonio A. Trani
NEXTOR Research Seminar - Falls Church, VA
June 3, 2003
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1
Overview
• Motivation & Background
• APCDM
•
•
•
•
Conflict Resolution
Sector Workloads
Collaborative Decision Making (CDM) Considerations
Overall Model Formulation
• Model Analyses
• Probabilistic Aircraft Encounter Model (PAEM) Analysis
• Parameterization & Sensitivity Analysis
• Conflict Constraint Formulation Analysis
• Research Directions
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Motivation & Background
• Delays: Space Launch, Weather Systems
• Congested Airspace: Safety and ATC Workload
• Distribute sector workloads
• Minimize en-route aircraft conflicts
• Airline Competition
•
•
•
•
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Fair allocation of constrained resources
New entrants and small/medium community service
Disparity in distribution of costs
Consumer expectations
3
APCDM
• Flight Plan Selection
• For each flight, select one flight plan from among
alternatives
• Minimize Flight Costs (Objective Function)
• Subject to Considerations (Penalty Terms in Objective
Function):
• Sector Workload
• Safety (Conflict Resolution)
• Decision Equity
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4
APCDM
Generate New Flight Plans
Column Generation
Record
Iterative Results
Adjust Parameters
INPUTS:
•Sector Geometries
•Flight Plans
•SUAs
•Weather Closures
Sector
Occupancies
AOM
Conflict
Analysis
PAEM
Conflict &
Workload
Constraint
Generation
CDM
Equity
Representation
Optimize
Flight Plan
Selection
INNER LOOP
OUTER LOOP
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APCDM
• Sector Occupancies
• Aircraft Conflict Analysis
• Stochastic with respect to aircraft trajectory
• Conflict risk thresholds
• Conflict & Workload Constraint Generation
• Continuous time formulation
• Two new classes of valid inequalities
• Sector workloads--average and peak workloads
• CDM Equity Representation
• Cost Model
• Collaboration Efficiency & Equity
• Mixed-Integer Programming Model
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6
Sector Occupancy
AIRSPACE OCCUPANCY MODEL
• Mathematical NAS representation
• 20 centers each divided into sectors
• Flight plans processed to determine sector occupancy time
intervals
• Occupancy data used:
• To characterize sector occupancy workloads
• As pre-processing data for PAEM conflict analysis
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7
Sector Workload
• Workload Characterization
• Average Occupancy throughout some horizon (ws)
• Peak Occupancy (ns)
• Occupancy Constraints:
• Prohibit selection of combinations of flight plans that cause
sector capacity ( ns ) to be exceeded
• Penalty Functions:
• Average Workload:
• Constant penalty
S
∑γ
s =1
s
ws
• Peak/Average Differential:
S
ns
∑∑ µ
s =1 n =0
sn
ysn
• Piecewise linear representation of increasing quadratic function
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8
Conflict Analysis
PROBABILISTIC AIRCRAFT ENCOUNTER MODEL
•
Proximity Shell Around Each Focal Aircraft
δz
ry
ajecto
r
t
t
f
a
ircr
focal a
δy
δx
• Moves with aircraft as it traverses its flight trajectory
• Conflict occurs when another intruder aircraft pierces the
focal aircraft’s proximity shell
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Conflict Analysis
PROBABILISTIC AIRCRAFT ENCOUNTER MODEL
• Aircraft Position & Trajectory Not Known With Certainty
• Weather Effects
• Navigation System Inaccuracy
• Pilot Error
planned trajectory
actual trajectory
• Bounded Error Regions → Probabilistic Trajectory Corridor
• Randomized Displacement Errors
• Wind-induced Displacement Errors
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10
Conflict Analysis
PROBABILISTIC AIRCRAFT ENCOUNTER MODEL
• For each pair of discretized error trajectory realizations
(for focal and intruder aircraft) we can compute the conflict risk:
P(ξ )
0.2
conflicts
0.3
0.5
1.0
conflict
probability
0
1
0.8
pthresh
0.6
0.4
0.2
0.0
0
1
conflicts
time
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0
?1
?2
?3
?4
1
11
Conflict Resolution Constraints
• Probabilistic conflicts generated by PAEM are fit into the
constraint structure of APCDM
• Constraints prohibit the selection of particular
combinations of flight plans
• Flight pairs that have a “fatal” conflict
• Flight combinations that exceed sector ATC conflict
resolution capability during any specified time interval
• Penalty function:
∑
( P ,Q )∈A
ϕ PQ zPQ
• Constant ϕPQ is determined by conflict geometry
• Polyhedral analysis of conflict constraint structure
• Derived classes of valid inequalities to tighten representation
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12
Conflict Resolution Constraints
C3 REPRESENTATION
• Define T NC as:
( P , Q, R ), P < Q < R : for some ( s ,k ), a subgraph of Gsk that is 


induced by the nodes P, Q, and R contains precisely two edges, 
 but no such subgraph for any ( s ,k ) contains three edges



•
( x , z ) :
zPQ ≤ 1,
∑

( P ,Q )∈M sk

xP + xQ + xR ≤ 2,
C3 = 

z PQ ≥ xP + xQ − 1,

z ≥ 0, x binary

P
Q
R


∀( P ,Q, R ) ∈ TNC  .


∀( P, Q ) ∈ A


∀( s, k )
• C3 tightens representation
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Conflict Resolution Constraints
C4 REPRESENTATION
• Focus is on underlying star-graph
convex hull constraints
Q
R
P
•
( x , z ) :



C4 = 




∑

( P , Q )∈M sk

*
z
≤
x
,
∀
r
=
1,...,
r
,
∀
P
∈
I

∑ ( PQ) P
P
.
Q∈ Jr ( P )

zPQ ≥ xP + xQ − 1, ∀( P ,Q ) ∈ A


z ≥ 0, x binary

zPQ ≤ 1, ∀( s, k )
W
• C4 is a provably tighter representation than C3
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CDM Considerations
• Optimal Individual Decisions vs. Optimal Group Decision
• Each participating airline’s decisions represent conflicting
objectives
• Possibly no feasible satisfying solution for these conflicting
objectives
• Inefficient overall use of the NAS
• Collaboration Efficiency
• Ratio of costs incurred by an airline due to resolution between
the group’s conflicting objectives to costs obtainable using the
airline’s individually optimized strategy
• Collaboration Equity
• Aggregate measure of disparity of costs incurred via group
decision
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CDM Considerations
FLIGHT PLAN COST MODEL
• Fuel Cost: (Ffp)
• Base of Aircraft Data (BADA) Operations Performance Model
• Fuel cost as a function of:
• Aircraft type
• Mass
• Flight envelope
• Aerodynamics
• Engine thrust
• Reduced power flight
• Fuel consumption
• Ground movement
• Delay Cost: D fp = ( t dfp )( dfc ) ( lf ) (δ )
•
•
•
•
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Length of delay ( t dfp )
Connection delay cost factor ( d cf )
Passenger load estimate ( l f )
Delay cost factor per passenger-minute (δ )
16
CDM Considerations
FLIGHT PLAN COST MODEL
• Total Flight Plan Cost: cfp = Ffp + Dfp
• Flight Cancellations
• Each flight has a “cancellation” surrogate flight plan
• Cancellation cost is greater than highest cost surrogate
c f 0 = max {F fp } + ( t df 0 )( d cf ) ( l f ) (δ )
p∈Pf
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CDM Considerations
COLLABORATION EFFICIENCY
• Airline Collaboration Cost:
• Ratio of total airline cost after resolving conflicting objectives
between all airlines to airline’s individually optimized flight
costs.
dα ( x ) =
∑ ∑c
f ∈Aα p ∈P f 0
∑c
f ∈ Aα
fp
x fp
*
f
• We impose dα(x) ≤ Dmax, a maximum CDM-based cost ratio,
for all airlines α =1,…, α
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CDM Considerations
COLLABORATION EFFICIENCY
• Airline Collaboration Efficiency:
1 if
Eα ( x) = 
0 if
• Function constructed such that
Dmax
• This yields:
Eα ( x) =
∑c
f ∈Aα
*
f
−
∑ ∑c
f ∈Aα p ∈Pf 0
fp
dα ( x) = 1
dα ( x) = Dmax
x fp
( Dmax − 1) ∑ c*f
, ∀α = 1,..., α
f ∈ Aα
• ω-Mean Collaboration Efficiency:
where ωα =
Aα
, ∀α =1,...,α ,
F
α
∑ω
α =1
α
∑ω
α
α
Eα ( x),
= 1, ωα ≥ 0, ∀α .
• ω-Mean Collaboration Inefficiency ( 1 − ∑ ωα Eα ( x) )
is penalized in the objective function
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α
19
CDM Considerations
COLLABORATION EQUITY
• Airline Collaboration Equity:
 α

E ( x) = Eα ( x) −  ∑ ωα Eα ( x) 
 α =1

e
α
• ω-Mean Collaboration Inequity:
α
x ≡ ∑ ωα Eαe ( x )
e
α =1
• Formulation linearizes the absolute value terms
• Penalty function minimizes disparities in efficiencies
(i.e. seeks a more equitable solution)
F
•
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F
µ = µ = µ0 ∑ c = ( 0.1) ∑ c*f
e
D
f =1
*
f
f =1
20
Model APCDM
F
∑∑c
min
f =1 p∈Pf 0
∑
subj to:
p∈Pf 0
S
fp
ns
x fp + ∑∑ µsn ysn + µ
s =1 n =0
D
α
∑ ω [1 − E
α =1
α
α
S
( x)] + µ x + ∑ γ s ws +
e
e
s =1
Conflict Resolution
Constraints (C4)
CDM Constraints
Dmax
Eα ( x ) =
ns ≤ ns , ∀s = 1,..., S
∑y
sn
= 1, ∀s = 1,..., S
n= 0
ns − ws = ∑ ny sn , ∀s = 1,..., S
n= 0
∑
( f , p)∈ Csi
∑
Q ∈J sk ( P)
z( PQ) ≤ rs xP , ∀ P ∈ Nsk :
ns
1
ws =
H
∑
( f , p)∈Ωs
t x fp , ∀s = 1,..., S
s
fp
x fp ≤ ns , ∀i = 1,..., I s , s = 1,..., S
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( P ,Q )∈ A
ϕ PQ z PQ
x fp = 1 ∀f = 1,..., F
Workload Constraints
ns
∑
J sk ( P) ≥ rs +1, ∀( s , k )
xP + xQ ≤ 1, ∀ ( P, Q) ∈ FC
xP + xQ − z PQ ≤ 1, ∀ ( P ,Q ) ∈ A
∑
( P,Q )∈ Msk
zPQ ≤ rs , ∀( s, k )
∑c
f ∈ Aα
*
f
−
∑ ∑c
f ∈Aα p ∈P f 0
( Dmax − 1) ∑ c*f
fp
x fp
, ∀α = 1,...,α
f ∈Aα
Eα ( x ) ≥ 0, ∀α = 1,..., α
 α

Eαe ( x ) = Eα ( x) −  ∑ ωα Eα ( x )  , ∀α = 1,...,α
 α =1

e
e
e
− Emax ≤ Eα ( x ) ≤ Emax , ∀α = 1,..., α
α
∑ω ν
α =1
α
= x ≤v
e
α
e
ν α ≥ − Eαe ( x) and ν α ≥ Eαe ( x), ∀α = 1,..., α
21
PAEM Analysis
• Eight Probabilistic Trajectory Displacement Sets
• 5 Randomized
• 3 Wind-induced
• 15 to 45 Realizations
• Nonlinear increasing
relationship
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PAEM Analysis
• Two-Dimensional Displacement Regions
• Minimal conflicts generated with vertical displacements
• FAA-imposed separation much greater than maximum
vertical deviations (±400 ft)
• Reduces number of realizations and computational effort
• Identified Intervals Versus Threshold Probability
69 deterministic conflicts
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559 deterministic conflicts
23
PAEM Analysis
• Baseline Threshold Probabilities
• Conservative
• Identify a reasonable number of probabilistic conflicts
• Comparable to probabilities observed for conflicts identified
in previous deterministic analyses
pthresh =
{ p1, p2 , pfatal }
=
{ 13 , 16 , 118}
• Sensitivity Analysis: Vary p1, p2, and pfatal proportionally
• Perturbations to the structures of induced conflict subgraphs
• Results demonstrate model insensitivity to moderate
changes in threshold probabilities
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APCDM Parameterization
Dmax
• Recall: dα ( x) ≤ Dmax
1
• Collaboration
Efficiency Curve
Eα ( x)
0
Dmax
Dmax
Dmax
Dmax
Dmax
dα ( x )
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APCDM Parameterization
Dmax
• Increasing Objective
Value: Two Factors
• Slope of Efficiency Curve
• Surrogate Selections
• Parameter
Influences Decision
when D max ≤ 1.20
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APCDM Parameterization
e
Emax
• Four instances run with unconstrained airline collaboration
equities
max Eαe ( x )
α
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APCDM Parameterization
e
Emax
• CDM-3 Analysis
• More stringent equity requirements induce reduced
collaboration efficiencies
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APCDM Parameterization
µ0
• Function of the CDM penalty terms in objective
• Mathematical incentive for maximizing collaboration
efficiencies and decision equity (i.e., mimimize “spread”)
• Should not dominate solution
• Definition: “CDM Improvement”
 α

α
 
0.5   ∑ ωα Eα ( x ) 
− ∑ ωα Eα ( x)   + 0.5 {x e }µ =0 − { xe }µ = µ 

0
0
0 
 µ0 = µ 0
 α =1
µ0 =0 
  α =1
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APCDM Parameterization
µ0
• Four APCDM instances tested using seven µ0 values
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Conflict Constraint Formulation Analysis
• Compactness of Representation: C3 versus C4
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Conflict Constraint Formulation Analysis
• CPLEX default cut
generation disabled
• Analysis
• Recommend C4
Formulation
• Specialized C4 cuts
superior to CPLEX
general-purpose cuts
w.r.t. APCDM
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Research Contributions
The APCDM with the following characteristics:
• Probabilistic Conflict Analysis
• Two alternative representations for trajectory errors
•
•
•
•
Continuous time formulation for conflict risk intervals
Two new classes of valid inequalities
Flight plan cost model
CDM Representation
• Examines distribution of costs as well as maximum spread of
costs
• Practical Applications
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Research Directions
• Alternative Utility Theory based equity considerations
• Flight plan generation
• Dynamic Airspace Issues
• Weather Systems
• Space Launch SUAs
• Dynamic Resectorization
• Strategic and tactical scenario tests
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Q
U
E
S
T
I
O
N
S
?
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