Optimization Models for Off-loading Flow Constrained Areas
Ming Zhong, Michael O. Ball, University of Maryland, College Park
Introduction
Multi-Objective Integer Programming
Model
The Federal Aviation Administration (FAA) and the
aviation community within the U.S. have recently raised
new operational concepts for identifying, implementing and
managing Flow Constrained Areas (FCA) based on
Collaborative Routing (CR). The main purpose of the study
is to determine how the FAA can meet its objectives while
giving NAS users as much flexibility as possible when
choosing routes.
Given: P total number of involved flights.
ai number of involved flights for airline company i.
∑a
i
=P
ri = target allocated number of flights for airline company i.
Objectives: (1) Efficiency: Max∑ x
f
¾3-dimensional volume
of airspace associated
with a time interval.
¾FCAs are defined in
response to
1) severe weather
2) capacity/demand
imbalance due to
excess demand.
∑
f
| y i − ri |
i
Main Constraints:
Reduced capacity:
∑
f ∈U
FCA
Flight conservation:
x f ≤ ck
Target allocation:
Reduce by 4
Reduce by 3
Reduce by 2
0
Normal
Normal
2
4
6
8
10 12 14 16 18 20 22 24 26 28 30 32
∀j, ∀k
Ongoing Research: Stochastic Model
j ,k
∑
x f + yi = ai
Goal:
¾Give flexibilities to airline companies
¾Improve system efficiency and take into account
the uncertainty and dynamics of demand on FCAs.
∀i
ri = a i ∑ y i / P
∀i
ii
Alternative Network Flow Model
1. Single sector case
The underlying matrix is totally unimodular (TU) and problem can
be efficiently solved as a pure network flow model.
(C1,0)
28
24
20
16
12
8
4
0
Number of offloaded flights
f ∈ Ai
# of flights
Reduced capacity
during FCA for a
sector
Min
or Min ∑ (1 − x f )
Effcient frontiers
Sum of the deviations
Decision variables:
xf = 1, if this flight stays in this area, 0 otherwise.
yi = number of offloaded flights for airline company i.
(2) Equity:
Weather block
¾FCA consists of 31 sectors, involving 45 airline
companies/classes associated with 283 flights.
¾Data has maximum simultaneous number of flights
of 7, and the reduced capacity changes from 3 to 5.
i
f
What is a Flow
Constrained Area
(FCA)
Computational Results
(capacity, cost)
(C1,0)
Two-stage decision procedure:
¾Flexibility: give tokens to airline companies for
some flights in the first stage.
¾Utilization: assign the rest time resource to
maximize system utilization.
Event time instance
(1,-1)
(1,-1)
Reduced sector capacity
Involved flight
time
Problem
Given:
¾ Starting and ending times of FCA
¾ A set of involved flights F
¾ Entrance and exit times for each flight at each sector
¾ The reduced sector capacity in this FCA area.
Goal:
Partition the involved flights into two groups:
1) maintain original routes,
2) reroute outside this area
Time axis
Mathematical model:
Decision variables:
2. Multiple sector case
The underlying matrix is not totally unimodular (TU) but problem
can be solved as a network flow model with side constraints.
(C2,0)
Xf =1 if the flight is pre-allowed to use FCA, 0 otherwise
Y ф f = 1 if the flight is adjusted to pass the FCA under
scenario ф while Xf =0; 0 no adjustment.
Objectives: (1) Predictability:Max ∑ x
(C2,0)
f
f
(C1,0)
(1,-1)
(C1,0)
Represents sector change
(1,-1)
Each layer represents
one sector
Side constraint: x = w
Time axis
φ φ
(2) Utilization: Max ∑ w f x f + ∑ ∑ p y
f
f
φ
Main Constraints:
Reduced capacity
∀ k,∀ j ,∀ φ
∑ x f + aφ f yφ f ≤ ck
f ∈U
j ,k
f