and d) reciprocal track.
distribution and, at the same time, improve demand and capacity
balancing, which is of interest to the ANSP. A reasonable assumption
in all this is that the ATC controllers are proficient and effective in
conflict resolution during the tactical phase.
Therefore, because there is more than one objective function to be
optimized simultaneously, we are dealing with a multiobjective
optimization problem [40,41].
2.
Pareto-Optimal Solutions:
ϵ-Constraint Method
If, in a nontrivial multiobjective optimization problem, there is no
single solution that simultaneously optimizes each objective, then the
objective functions are said to be conflicting. In such cases, a possibly
infinite number of Pareto-optimal solutions are possible. A solution is
efficient, or Pareto optimal, if none of the objective functions can be
improved in value without degrading some of the other objective
values [42]. Without additional information on preferences, all
Pareto-optimal solutions are considered to be equally good.
According to Van Veldhuizen and Lamont [43], the final solution of
a multiobjective problem is the result of two processes: optimization
and decision. Depending on the decision stage in which the decision
maker expresses their preference, multiobjective methods may be
classified as
“a priori,” “interactive,” or “a posteriori.”
In this study, we used an a posteriori method because such methods
inform the decision maker about the context of any alternatives before
making the final decision. This type of method requires a lot of
computation; however, the advantage is that the decision maker need
only be involved in the decision phase.
The specific method used was the
ε-constraint [44] method, which
provides a representative subset of Pareto-optimal solutions, or
Pareto frontier, for each individual conflict. The multiobjective
problem may be expressed as follows:
min
xϵΣ
f
1
x; f
2
x; : : : ; f
p
x
(7)
where x is vector of decision variables; f
1
x; f
2
x; : : : ; f
p
x are
the p objective functions; and Σ is the solution space, defined as the
set of all possible points of an optimization problem that satisfy the
problem
’s constraints.
In the
ε-constraint method, one of the objective functions is optimized
using the other objective functions as constraints. These are incorporated
into the restrictions part of the model, as shown in the following:
min f
1
x
such that
f
2
x ≥ ε
i
(8)
To find the set of Pareto-optimal solutions, the problem is solved
several times by selecting different values of the constraint vector
ε
i
with
i ϵ1; 2; : : : ; n, where n is the number of variables.
In this study, the
“likelihood of ATC resolution” function is used as a
constraint. The selection of the
ε
i
values means that a specific region of
the objective space is out of bounds. Therefore, the optimal solution
(the one that minimizes additional fuel consumption) must come from
the remaining workable solutions (Fig. 11). For a specific constraint,
we must find the optimal solution for all values of the constraints vector
ε
i
. This will give us the Pareto frontier for that particular conflict.
Once the Pareto frontiers have been calculated for all conflicts, it is
necessary to analyze the solutions in every frontier in order to find the
global optimal solution for all conflicts.
3.
Decision-Based Design
There are two main steps in decision-based design:
1) Generate the option space: in other words, the Pareto frontiers
for each conflict.
2) Select the best option [45].
Choosing from among the option spaces is not a trivial matter, but
rather a function of tradeoffs and compromises. It involves
populating a number of optimal solutions along the Pareto frontiers of
each conflict, and then selecting one based on the values of the
attributes for the given solutions.
In the Pareto frontiers of each conflict, as well as the solutions that
satisfy the requirement for minimum fuel and maximum likelihood of
ATC resolution, there are also intermediate solutions. These solutions
imply different levels of tradeoff, in which one objective is more fully
satisfied to the detriment of another. The tradeoff is given by the slope
of the curve, tan
α, connecting two solutions:
tanα f
2
x
j
− f
2
x
i
∕f
1
x
j
− f
1
x
i
(9)
where f
2
x
i
and f
2
x
j
are the likelihood of ATC resolution of
solutions i and j; and f
1
x
j
and f
1
x
i
are the additional fuel
consumed by solutions i and j.
A small value of
α implies that an increase in the likelihood of ATC
resolution is achieved in exchange for a slight increase in fuel
consumption. Conversely, a high value of
α implies a dramatic
increase in fuel consumption is required to increase the likelihood of
ATC resolution. We used Pareto frontiers slopes (tan
α) to populate
the frontiers of each conflict with a number of optimal solutions. This
was done using an iterative process as follows:
1) Choose a specific value of tan
α.
2) Use it to populate the Pareto frontier of each conflict.
3) Select
the solution
giving minimum additional
fuel
consumption. If the slope is
≥ tan α, then this is the optimal solution.
4) Otherwise, (if the slope is less than the selected tan
α) choose the
next value from the Pareto frontier.
5) Repeat the process until the optimal solutions have been found.
Solutions with small values of tan
α are closer to satisfying the
criterion for minimum additional fuel consumption. Conversely, high
values of tan
α are closer to satisfying the criterion for the maximum
likelihood of ATC resolution (Fig. 12).
Table 2
Breakdown of temporal deviation of conflicts as a function of relationship
between aircraft
Same track
Crossing tracks
(45
–90 deg)
Crossing tracks
(90
–135 deg)
Reciprocal track
Minimum
% Conflicts
Minimum
% Conflicts
Minimum
% Conflicts
Minimum
% Conflicts
−3.8
1.7
−3.9
1.3
−2.3
3.1
−4.5
1.9
−2.3
3.0
−2.7
2.3
−1.4
5.3
−3.0
2.1
−1.4
4.8
−1.8
3.3
−0.8
7.3
−1.9
3.2
−0.7
7.7
−1.1
5.6
−0.3
10.1
−1.2
4.6
−0.2
10.1
−0.6
6.4
0.0
11.4
−0.7
7.4
0.1
10.8
−0.3
8.2
0.3
8.2
−0.2
10.7
0.4
9.8
0.0
11.2
0.6
7.2
0.1
11.1
0.9
7.8
0.3
8.9
0.9
6.8
0.4
9.3
1.4
5.8
0.6
6.3
1.4
5.1
0.9
9.8
2.1
4.7
1.1
6.0
2.1
3.1
1.6
6.6
3.3
3.5
1.8
4.8
3.0
2.7
2.3
3.5
5.2
1.9
3.0
2.6
4.2
1.4
3.7
2.6
622
CALVO-FERNÁNDEZ ET AL.
Downloaded by UNIV. OF ARIZONA on March 14, 2017 | http://arc.aiaa.org | DOI: 10.2514/1.G000691
To detect the values of tan
α used in this study, the k-means
algorithm is applied to the histogram with Pareto frontiers slopes for
all conflicts.
Summarizing, in this section, we applied the criteria of minimum
fuel consumption and maximum likelihood to ATC resolution. We
then analyzed the tradeoff solutions using different slope values. The
results are given in the following section.
III.
Results
A.
Development of a Conflict-Resolution Model Using a Data-Driven
Methodology
This data-driven study was carried out using operational data (flight
plans and radar tracks) gathered over a period of 72 days from the
continental Spain FIR. More than 4200 flights per day were involved.
The database of historical conflicts contained the conflicts
detected in each scenario. These scenarios were generated at time
intervals of 5 min, giving 288 scenarios per day. To test the reliability
of the data from the
“5 min” intervals, we also calculated the conflicts
for
“1 min” intervals over one day and compared the two sets of data
obtained. The results showed that only 1.10% of the conflicts
detected using 1 min intervals were not detected when 5 min intervals
were used. Therefore, to reduce the computation required, 5 min
intervals were used throughout the study.
1.
Conflict Detection Based on Historical Data
The 4-D trajectory was calculated for each scenario using
sampling points taken every 5 s. We used the
“pairwise
conflict detection algorithm.
” Table 1 gives the breakdown of
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