Dynamic Stochastic Model for
a Single Airport Ground
Holding Problem
Avijit Mukherjee
Mark Hansen
University of California at Berkeley
1
Literature
‰ Static Integer Programming
‰Many papers on deterministic problem
‰Stochastic problem studied in Richetta
and Odoni (1993); Hoffman (1997); Ball et
al.(2003)
‰Equity issues addressed by Vossen et al.
(2002)
2
Literature
‰ Dynamic Models: Richetta and Odoni
(1994)
‰Inability to revise previously assigned ground
delays
‰Objective function is to minimize expected delay
cost
‰No longer Linear Programming Problem if nonlinear measure of delay introduced
‰Can handle specific type of scenario tree
3
Research Contributions
‰ Dynamic Stochastic Model for SAGHP
‰Ability to revise ground delays of some flights
(non-departed)
‰Can handle any generalized scenario tree
‰ Alternative Objective Functions
‰Expected Squared Deviation from RBS Allocation
‰ Multi-Criteria Optimization
4
Capacity Scenario Tree
Scenario 1 (p=0.3)
P(Scenario 2)=1 Scenario 2 (p=0.2)
P(Scenario 2)=0.4
P(Scenario 2)=0.2
Scenario 3 (p=0.4)
Scenario 4 (p=0.1)
1
2
3…….
τ1
τ2
τ3……..
T
5
Decision Making Process
ξ1
ξ2
ξ3
ξ4
1
τ1
df
τ2
τ3
af
T
6
Decision Variables
X
⎧⎪1 if flight f is planned to arrive by time period
= ⎨ t under scenario q;
f ,t
⎪⎩0 otherwise
q
f ,t
X
2
f ,t
1
X
1
f ,t
=X
2
f ,t
0
X
1
df
af
7
Model Formulation
‰ Objective Function
‰Min. Expected Total Cost of Delay (sum of
ground and airborne delays)
‰ Major Constraints
‰Number of arrivals during any time interval
(period) less than airport capacity
‰Coupling Constraints: decisions cannot be based
on a particular scenario until it is completely
realized
8
Model Parameters and Input Data
{1..T + 1} : set of time periods of uniform duration, T
being the planning horizon
Φ = {1..F } : Set of Flights
Dep f ∈{1..T } : scheduled departure time period of flight f
Arr f ∈ {1..T } : scheduled arrival time period of flight f
λ : Cost ratio between airborne and ground delay
9
Θ : set of capacity scenarios
Pq : Probabilit y of occurrence of scenario q ∈ Θ
q
M t : Airport arrival capacity at time period t
under capacity scenario q
q
M T +1 is set to a high value for all q ∈ Θ
10
B = total number of branches of the scenario tree; B ≥ Θ
Ν = number of scenarios represente d by ith branch; i ∈ {1..B}
i
The scenarios represented by branch i is given by set
i
i
i
i
1
k
Νi
k
Ω = {S ,.., S ,.., S }, S ∈ Θ
i
The time periods corresponding to start and end nodes of
a branch are given by
oi and µ i ; i ∈ {1..B}
11
B = 7; Θ = {χ1 , χ 2 , χ 3 , χ 4 }; Θ = 4
χ1
i=4
i = 2; Ν = 2; Ω = {χ , χ };
2
2
1 2
ο = τ ; µ = (τ − 1)
2 1 2
2
i =6
i =3
i = 1; Ν 1 = 4 ; Ω1 = { χ 1 , χ 2 , χ 3 , χ 4 };
2
3…….
i = 5 ; Ν 5 = 1;
χ3
Ω 5 = { χ 2 };
i = 7 χο 5
ο1 = 1; µ1 = (τ 1 − 1)
1
χ2
= τ ;µ = Τ
2 5
4
τ1
τ2
τ3……..
T+1
12
Decision Variables
X
⎧⎪1 if flight f is planned to arrive by the end of
= ⎨ time period t under scenario q;
f ,t
q ∈ Θ, f ∈ Φ,
⎪⎩0 otherwise
q
t ∈ { Arrf ..T + 1}
⎧1 if flight f is released for departure by the end of
⎪
q
q ∈ Θ, f ∈ Φ,
Y f ,t = ⎨ time period t under scenario q;
⎪0 otherwise
t ∈ {Dep f ..T + 1}
⎩
W tq = number of aircraft subject to airborne queuing delay
at time t for one or more time periods, under scenario q
13
Objective Function
T
T +1
q
q
⎧⎪⎡
⎫
⎤
q⎪
Min ∑ Pq × ⎨⎢ ∑ ∑ (t − Arr f )× ( X f ,t − X f ,t −1 )⎥ + λ × ∑W t ⎬
t =1
q∈{1..Q}
⎪⎩⎢⎣ f ∈{1.. F } t = Arr f
⎪⎭
⎥⎦
Constraints
Decision Variables Non Decreasing
q
q
X f , t − X f , t −1 ≥ 0 ; ∀f ∈ Φ, q ∈ Θ, t ∈ { Arr f ..T + 1}
Planned Departure Time of Flights
⎧ q
q
⎪
Y f , t = ⎨ X f , t + Arr f − Dep f ; if
⎪1 otherwise
⎩
t + Arr f − Dep f ≤ T
14
Arrival Capacity
⎛
⎞
W tq−1 − W tq + ∑ ⎜ X qf , t − X qf , t −1⎟ ≤
⎠
f ∈Φ ⎝
M tq ; t ∈ {1..T + 1}, q ∈ Θ
Feasibility Conditions
W 0q = W Tq +1 = 0
X qf ,T +1 = 1
∀f ∈ Φ, q ∈ Θ
Coupling Constraints for Ground Holding Decision Variables
Si
i
i
Sk
S1
Ni
Y f , t = .... = Y f , t = ........ = Y f , t ;
f ∈ Φ, t ∈ {1..T }; S i ∈ Ω i : Ν i ≥ 2 and ο i ≤ t ≤ µ i
k
15
Static vs Dynamic Formulation
‰ Static Stochastic Model (Ball et al 2003, RichettaOdoni 1993) is a special case of dynamic model.
‰ The decisions are taken once at the beginning of
day and not revised later.
is same for all q ∈ Θ; i.e., superscript q in the decision variables
can be dropped.
q
Therefore, X f ,t can be denoted as X f ,t ; ∀q ∈ Θ
X
q
f ,t
Similarly,Y f ,t = Y f ,t ; ∀q ∈ Θ
q
16
Example
‰ 4 Scenarios, 4
Decision Stages
‰ 13 Time Periods
‰ 13 Flights
‰ Cost Ratio λ = 5
3
ξ3 ξ 4
2
Capacity
ξ1 ξ 2
1
0
0
1
2
3
4
5
6
7
8
9
10
11
12
13
time period
Probability Mass Function : P{ξ1} = 0.5; P{ξ 2 } = 0.3; P{ξ 3} = 0.1; P{ξ 4 } = 0.1
17
Scenario Tree
ξ1
ξ2
ξ3
ξ4
1
2
3
4
5
6
7
8
9
10
11
12
13
18
Flight
No.
Dep.
Arr.
Decision Richetta-Odoni
Stage
Model
MukherjeeHansen Model
(Optimal
Solution 1)
ξ1
ξ 2 ξ3
ξ 4 ξ1
ξ 2 ξ3
MukherjeeHansen Model
(Optimal
Solution 2)
ξ4
ξ1
ξ 2 ξ3
ξ4
2
6
7
1
3
3
3
3
1
2
5
5
1
2
4
4
3
2
8
1
0
0
0
0
1
1
1
1
1
1
1
1
7
5
9
1
0
0
0
0
0
0
0
0
0
0
0
0
8
7
9
2
0
3
3
3
0
1
2
2
0
1
3
3
12
9
11
3
0
0
1
1
0
0
1
1
0
0
1
1
19
Expected Cost of Delay
12
Expected Delay Cost
10
8
Airborne Delay
Ground Delay
6
4
2
0
Richetta-Odoni
Mukherjee-Hansen
20
Cumulative Arrivals at DFW
July14_2003
cumulative number of arrivals
350
300
250
200
150
100
50
0
0:00 1:00 2:00 3:00 4:00 5:00 6:00 7:00 8:00 9:00 10:00 11:00 12:00
time of day
21
Case 1: Baseline
35
ξ1 ξ 2 ξ 3 ξ 4 ξ 5 ξ 6
15
0:00
8:00
8:30
9:00
9:30
10:00 10:30
12:00
Time of Day
Probability Mass Function
P{ξ1 } = 0.4; P{ξ 2 } = 0.2; P{ξ 3 } = 0.1; P{ξ 4 } = 0.1; P{ξ 5 } = 0.1; P{ξ 6 } = 0.1
Cost Ratio λ = 3
22
Scenario Tree for Baseline Case
ξ1
ξ2
ξ3
ξ4
ξ6
ξ5
23
Other Cases
Case 2: Change in Cost Ratio. λ = 25
Case 3: Change in PMF
P{ξ1} = 0.1; P{ξ 2 } = 0.1; P{ξ3} = 0.1; P{ξ 4 } = 0.1; P{ξ5 } = 0.2; P{ξ 6 } = 0.4
Case 4: Early Branching. Scenarios are realized 30
minutes earlier
24
Results: Baseline Case
Baseline Case
Airborne Delay
Ground Delay
Expected Cost of Delay (Aircraft-Hr)
40
35
‰ Mukherjee-Hansen
Model
‰ Ground delays more
severe
‰ Less airborne delays
‰ Total expected cost
least
30
25
20
15
10
5
0
Ball et al.
Ricehtta-Odoni
Mukherjee-Hansen
‰ Delay reduction
compared to Static
Model
‰ 10% in MukherjeeHansen Model
‰ 2% in Richetta-Odoni
25
Planned Arrival Rates in Baseline Case
Time Period
Mukherjee-Hansen Model
Richetta-Odoni Model
ξ2
ξ3
ξ4
ξ5
ξ6
35
34
34
34
34
34
5
14
14
14
14
14
14
18
18 18
12
12
12
12
12
12
12
12
12 12
20
21
13
13
13
13
33
33
33 33
12
12
20
20
20
20
ξ5
ξ6
ξ1
ξ2
ξ3
ξ4
9:00AM9:15AM
33
25
25
25
25 25
9:15 AM-9:30
AM
16
5
5
5
5
9:30AM-9:45AM
12
26
18
9:45AM10:00AM
20
25
10:00AM10:15AM
12
12
ξ1
26
Cases 2 and 3
Expected Cost of Delay (Aircraft-Hr)
50
Case2
Case3
45
‰ No airborne delays
40
35
‰ Static model plans for
worst scenario
30
25
20
15
10
5
0
Ball et al.
Ricehtta-Odoni
Mukherjee-Hansen
‰ Dynamic models
adaptive to changing
conditions
‰ Delays are higher in
case 3 due to high
probability of worse
conditions
27
Case 4: 30 Minutes Early Information
Expected Delay Cost (Aircraft-Hr)
40
Airborne Delay
Ground Delay
35
30
‰ Static model produces
same delays as in
baseline case
25
‰ Value of early information
20
15
10
5
0
Ball et al.
Ricehtta-Odoni
Mukherjee-Hansen
‰ Mukherjee-Hansen
Model
‰ Least delays
‰ Absorbs most of the delay
by ground holding
28
Perfect Information Case
ξ1
ξ2
ξ3
ξ4
ξ5
ξ6
0:00
12:15
29
Mukherjee-Hansen
Richetta-Odoni
Ball et al.
Perfect Information
350
300
Percentage
250
200
150
100
50
0
0
1
2
Case #
3
4
30
Alternative Objective Functions
Minimizing Expected Squared Deviation from RBS
Allocation
⎧⎡
T +1
⎪⎢
Min
∑
∑
∑ P{q} × ⎨⎢
q ∈ {1..Q} ⎪⎢ f ∈ {1..F } t = Arr f
⎩⎣
q
⎛
⎜ t − RBS
f
⎜
⎝
⎫
2
⎤
q
q
⎞
T
⎪
⎥+λ×
q
⎟ × (X
−
X
)
∑W t ⎬
f ,t
f ,t −1 ⎥
⎟
t =1 ⎪
⎠
⎥⎦
⎭
31
Multi-Criteria Optimization
⎫
⎧⎡
⎤
q
q
T +1
⎪
⎪⎢
⎥
(
)
) +
t − Arrf × ( X
−X
∑
⎪
⎪⎢ ∑
f ,t
f , t −1 ⎥
⎪
⎪⎢ f ∈ Φ t = Arr
⎥
f
⎦
⎪
⎪⎣
⎡
⎤ ⎪
2
⎪
q
q
q
1
T
+
⎢
⎥ ⎪
⎪
⎛
⎞
)⎥ + ⎬
Min
−X
∑ P{q} × ⎨weight × ⎢ ∑
∑ ⎜ t − RBS ⎟ × ( X
f
f ,t
f , t −1
q ∈ {1..Q} ⎪
⎢ f ∈ Φ t = Arr f ⎝
⎥ ⎪
⎠
⎣
⎦ ⎪
⎪
T
⎪
⎪
q
λ
×
∑
Wt
⎪
⎪
⎪
⎪ t =1
⎪
⎪
⎭
⎩
32
Work in Progress
‰ Reformulating the model as a minimum
cost network flow problem.
‰ Ability to handle time varying
unconditional probabilities of the
capacity scenarios
33
Acknowledgements
‰ Authors are thankful to Prof. Mike Ball
for his thoughtful suggestions on this
research.
34