A Stochastic Programming Approach to Incorporating Weather Uncertainty into Air Traffic Flow Managem

1
A Stochastic Programming Approach to
Incorporating Weather Uncertainty into Air
Traffic Flow Management
Zelda B. Zabinsky University of
Washington November 4, 2003
2
Overview

• Incorporating Weather Uncertainty in Airport
  Arrival Rate (ARR) Decisions
• General Assignment Problem
• Schedule Recovery Problem under Weather
  Uncertainty

3
Background

• Collaborative research between UW and Boeing on
  air traffic management (ATM) under temporary
  capacity constraints
• Contribute to National Flow Model (NFM)
• Dynamic simulation environment representing NAS
• Evaluation of ATM operational concepts
• Two Aspects of ATM with weather uncertainty
• Airport Arrival Rate (ARR) Decisions
• Schedule Recovery Problem

4
Incorporating Weather Uncertainty in Airport
Arrival Rate Decisions

• Objectives
• Determine optimal airport arrival rates
  (capacity)
• Investigate the trade-off between ground delay
  and air delay given uncertainties in the weather
  prediction
• Examine, How do inaccuracies in weather
  forecasts affect flow decisions?

5
Flow Control Decisions

• A collaborative decision is made between Air
  Traffic Control (ATC), the Airline Operational
  Control (AOC), and affected centers
• Flow control options result in either some form
  of ground delay or air delay
• Two major flow control options
• Ground holding (delay on the ground)
• Miles-in-Trail (delay in the air)

6
Decision Representation

• Single airport with multiple arrivals
• How to make delay decisions to minimize total
  delay or cost of delay?

7
Stochastic Optimization Formulation Assumptions

• Due to weather uncertainty, there is a
  probabilistic reduction of capacity, airport
  arrival rate (AAR)
• Model Assumptions
• Single airport
• Flights aggregated by scheduled arrival
• Previous work
• Octavio Richetta and Amedeo Odoni (1993,1994)
• Min ECost of ground delay ECost of air
  delay

8
Stochastic Optimization Formulation Utility
Function

• New objective function included utility of flight
  as function of total delay

9
Stochastic Optimization-Objective Function

• Two sets of decision variables
• First stage decisions (Xij) reschedule the
  arrival time of flights from i to j, by ground
  delay
• Recourse decisions ( ) assign actual arrival
  time k under scenario q indicating ground, air
  and total delay
• Probability of scenario q, (pq ) weather
  uncertainty

Ground Delay
Air Delay
Original Arrival i
Actual Arrival k
Rescheduled Arrival j
10
Stochastic Optimization-Constraints
11
A Simple Test Case
Time 4 is the slack time where we have the
ability to recover the schedule
12
Comparison of Policies
13
A More Realistic Test Case

• Sixteen time period model - 15 min intervals

Based On Official Airline Guide Boston Logan
Airport Arrival Data Demand for Monday 8AM to
12PM
14
Scenario Setup

• Forecast gives capacity for each time period
• Five capacity cases (each with three possible
  forecasts) created to represent various weather
  conditions
• Fair Weather
• Late Storm
• Intense Storm
• Mid-time Storm
• Unpredictable Weather
• Four probability cases represent different
  distributions of capacity forecasts
• Twenty scenarios
• Examined Three Utility Cases
• Ground Delay Air Delay (1X)
• 2( Ground Delay) Air Delay (2X)
• 5( Ground Delay) Air Delay (5X)

15
Makeup of Total Delay
One Unit of Delay 15 min
16
Summary of Insights

• Decisions sensitive to value of total delay and
  relative costs of air delay and ground delay
• If only minimize cost of air and ground (and
  ignore total delay), assign more ground delay and
  not value opportunity to take advantage of
  clearing weather
• When air delay cost gt ground delay cost,
  schedules more ground delay
• Unpredictable Late Storm scheduling longer
  delays
• As relative cost of air delay increases see more
  flights rescheduled in later time periods

17
Generalized Assignment Problem (GAP) with
Forecasted Resource Capacities

• Objectives
• Identify stochastic programming formulations of a
  specific resource-constrained generalization of
  the assignment problem with capacity uncertainty
• Establish exact and approximate solution
  strategies to solve resulting problems
• Evaluate solution performance on a set of random
  test problems

18
Resource-Constrained Assignment
Tasks
Agents
Resources
Assignment Costs
Resource Usages
1
1
1
2
2
2
Resource Capacities
i
j
r
I
J
R
19
Deterministic CCGAP Formulation

• Find the minimum-cost set of assignments subject
  to resource capacities and one-to-many matching

• Possible Sources of Uncertainty
• Assignment costs
• Presence or absence of individual tasks
• Presence or absence of agents
• Amount of resources needed to process tasks
• Resource capacities

20
Stochastic CCGAP Formulation

• Incorporate capacity uncertainty in the objective
  using an expected second-stage value function

• EQ(X) includes information on a set of recourse
  actions that guarantees feasibility under the
  actual set of resource capacities

21
Three Formulations

• Simple Recourse on Amount of Infeasibilities
• Excess capacity usage is allowed
• Each unit of excess usage is penalized
• Simple Recourse on Number of Infeasibilities
• Excess capacity usage is allowed
• Each resource with excess usage is penalized
• Simple Recourse on Cancellations
• Excess capacity usage is not allowed
• Existing task-agent assignments are allowed to be
  cancelled
• Tasks without any agents are penalized

22
Solution Approach

• Branch-and-bound methodology
• Search tree with I levels
• Each level corresponds to a task
• Branching corresponds to fixing each tasks
  assignment to an agent
• Lower bounds
• Obtained using Lagrangian relaxation on the
  capacity constraints
• Resulting formulation has a trivial solution
• Subgradient search at every node for tighter
  bounds
• Upper bounds
• Heuristic based on Lagrangian relaxation solution

23
Schedule Recovery Problem under Weather
Uncertainty
Problem Need for strategies to address weather
uncertainty
24
Simple Example

• 16 flight legs in 4 itineraries (I)
• A-H-C-H-A, B-H-C-H-B,A-H-D-H-A, B-H-D-H-B
• First legs depart at the same time
• 37 recovery options for each leg (J)
• 3 alternative routes ?12 alternative departure
  times for each route
• Cancellation
• 1,536 resources (R)
• 32 system elements
• 17 airspace sectors
• 5 airports(gates, arrivals and departures)
• 48 time slices

25
Simple Example

• Single storm from 0100 hrs to 0300 hrs
• Uncertain location
• Five Alternative Forecasts (F5)

26
Performance Evaluation

• Actual value of revised schedules
• Measure arrival delays under actual weather
• Test under five possible actual weather scenarios
• Assuming each forecast is actual weather
• Average actual value across possible realizations

27
Preliminary Results
28
Original Schedules
4
11
Capacity 5 aircraft
1
8
15
A
C
Capacity 3 aircraft
5
12
2
9
16
Capacity 1 aircraft
H
6
13
3
10
17
D
B
7
14
29
Average Capacities
11
4
Capacity 5 aircraft
1
8
15
A
C
Capacity 3 aircraft
5
12
2
9
16
Capacity 1 aircraft
H
6
13
3
10
17
D
B
7
14
30
Stochastic Programming (SRA)
11
4
Capacity 5 aircraft
1
8
15
A
C
Capacity 3 aircraft
5
12
2
9
16
Capacity 1 aircraft
H
6
13
3
10
17
D
B
7
14
31
Conclusions

• Stochastic programming techniques produce
  solutions with high expected performance for many
  ATM problems
• Stochastic programming can be used in an extended
  Generalized Assignment Problem to provide robust
  solutions
• Stochastic programming approximate solution takes
  less computational time and provides better
  values than a deterministic solution with average
  resource capacities