# Dynamic Stochastic Model for a Single Airport Ground Holding Problem Avijit Mukherjee

```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 &micro; 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 ; &micro; = (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; &micro;1 = (τ 1 − 1)
1
χ2
= τ ;&micro; = 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 &times; ⎨⎢ ∑ ∑ (t − Arr f )&times; ( X f ,t − X f ,t −1 )⎥ + λ &times; ∑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 ≤ &micro; 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
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} &times; ⎨⎢
q ∈ {1..Q} ⎪⎢ f ∈ {1..F } t = Arr f
⎩⎣
q
⎛
⎜ t − RBS
f
⎜
⎝
⎫
2
⎤
q
q
⎞
T
⎪
⎥+λ&times;
q
⎟ &times; (X
−
X
)
∑W t ⎬
f ,t
f ,t −1 ⎥
⎟
t =1 ⎪
⎠
⎥⎦
⎭
31
Multi-Criteria Optimization
⎫
⎧⎡
⎤
q
q
T +1
⎪
⎪⎢
⎥
(
)
) +
t − Arrf &times; ( X
−X
∑
⎪
⎪⎢ ∑
f ,t
f , t −1 ⎥
⎪
⎪⎢ f ∈ Φ t = Arr
⎥
f
⎦
⎪
⎪⎣
⎡
⎤ ⎪
2
⎪
q
q
q
1
T
+
⎢
⎥ ⎪
⎪
⎛
⎞
)⎥ + ⎬
Min
−X
∑ P{q} &times; ⎨weight &times; ⎢ ∑
∑ ⎜ t − RBS ⎟ &times; ( X
f
f ,t
f , t −1
q ∈ {1..Q} ⎪
⎢ f ∈ Φ t = Arr f ⎝
⎥ ⎪
⎠
⎣
⎦ ⎪
⎪
T
⎪
⎪
q
λ
&times;
∑
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
```