Strategic Management in Dynamic Environments MGMT 690Beginning D.docx

Strategic Management in Dynamic Environments MGMT 690
Beginning Development of Global Strategies
3-4 Pages
Mike, one of the marketing strategists on your team, stops at
your office door wanting to talk. “We use fabrics that are made
domestically; however, there are issues with using these same
fabrics globally. There are laws and regulations that prevent us
from shipping these fabrics to other countries. This is a huge
concern. One of our primary selling points is the consistency of
quality of our product.”
You confirm Mike’s concern, “That’s an excellent point,” you
say. “Now you’ve just given yourself and our team more work
for the presentation. I’m sure that will come up. One of the
board members used to run a textile plant in China.”
Mike nods his head in agreement. “I imagine textiles will not be
the only resource concern,” he says.
Consider the following in your response:
· Why should resources be a concern in a global strategy?
· What resources may be a concern in the country you selected?
· How will this impact the decision to move to the country that
you selected?
· How will this impact your competitive strategy in your global
market?
MUST USE ACADEMIC SOURCES SUCH AS GOOGLE
SCHOLAR, GOVERNMENT, SCHOLARLY REVIEWED ETC.
European Journal of Operational Research 241 (2015) 502–512
Contents lists available at ScienceDirect

European Journal of Operational Research
journal homepage: www.elsevier.com/locate/ejor
Innovative Applications of O.R.
Solving air traffic conflict problems via local continuous
optimization
Clément Peyronne a,∗ , Andrew R. Conn b, Marcel Mongeau
c,d, Daniel Delahaye c,d
a Capgemini, 15 av. du Dr Maurice Grynfogel, 31000 Toulouse,
France
b IBM, T.J. Watson Research Center, P.O. Box 218, Yorktown
Heights, NY 10598, USA
c ENAC, MAIAA, F-31055 Toulouse, France
d Université de Toulouse, IMT, F-31400 Toulouse, France
a r t i c l e i n f o
Article history:
Received 29 August 2012
Accepted 31 August 2014
Available online 28 September 2014
Keywords:
Air traffic conflict problem
B-splines
Continuous optimization

Genetic algorithms
Semi-infinite programming
a b s t r a c t
This paper first introduces an original trajectory model using B-
splines and a new semi-infinite program-
ming formulation of the separation constraint involved in air
traffic conflict problems. A new continuous
optimization formulation of the tactical conflict-resolution
problem is then proposed. It involves very few
optimization variables in that one needs only one optimization
variable to determine each aircraft trajec-
tory. Encouraging numerical experiments show that this
approach is viable on realistic test problems. Not
only does one not need to rely on the traditional, discretized,
combinatorial optimization approaches to this
problem, but, moreover, local continuous optimization methods,
which require relatively fewer iterations
and thereby fewer costly function evaluations, are shown to
improve the performance of the overall global
optimization of this non-convex problem.
© 2014 Elsevier B.V. All rights reserved.
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1. Introduction
This introduction describes the importance of the conflict-
resolution problem, followed by some details on related work.
1.1. Context
Air traffic management (ATM) aims at ensuring smooth running
of
the transportation system under safety and schedule alignment
con-
straints while keeping flights on schedule. In order to reach this
goal,
air traffic is planned at different time frames. Strategical
planning is
done several months before take-off and consists of assigning
flight
plans for a whole day of traffic with an emphasis on an even
distribu-
tion of aircraft density in space and time. Pre-tactical planning
then
updates the strategical planning trajectories using information
such
as weather or airspace congestion. It takes place two hours
before the
aircraft reaches the considered airspace. Tactical planning, the

subject
of this study, is performed within a 20-minute time horizon and
con-
sists mainly of conflict detection and resolution. This tactical
planning
work has always been done by air traffic controllers who are in
charge
of the most critical aspect of ATM, namely, ensuring sufficient
sep-
aration between airplanes. Air traffic controllers are responsible
for
∗ Corresponding author. Tel.: +33 627025703.
E-mail addresses: [email protected] (C. Peyronne),
[email protected] (A. R. Conn), [email protected] (M.
Mongeau),
[email protected] (D. Delahaye).
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http://dx.doi.org/10.1016/j.ejor.2014.08.045

0377-2217/© 2014 Elsevier B.V. All rights reserved.
nsuring the respect of regulatory separation rules that are
currently
nautical miles horizontally and 1000 feet vertically.
A conflict situation happens when an aircraft enters the standard
afety zone of another aircraft (a situation where a regulatory
sepa-
ation rule is not respected). Our work focuses on tactical
planning,
hich is currently handled by air traffic controllers on each
airspace
ector. To solve a conflict, controllers can use a set of
maneuvers: off-
et, turning point, speed change, and flight-level change. For
instance,
ig. 1 shows maneuvers implying direction changes.
As a consequence of increasing traffic, controllers in charge of
an
irspace sector must handle more and more flights (see Fig. 2).
The
urrent approach is to decrease the size of the control sectors in
order
o compensate for the growth of traffic. However, the traffic is
reach-

ng the point where a decrease of the size of sectors is no longer
effi-
ient. In reality, ATM has already used every available resource
in an
ttempt to increase airspace capacity. However, from now to
2030, air
raffic is expected to increase by a factor of two or three
(SESAR Joint
ndertaking, 2009). Consequently, ATM will have to deal with
this
verload while ensuring at least similar standards of safety
(SESAR
oint Undertaking, 2009). As is illustrated in Fig. 2, the
difference of
raffic capacity between 1970 and 2010 is rather dramatic.
In this context, the long-term vision aims at lowering the
workload
f the air traffic controllers by reducing the conflict-resolution
task.
n example is the 4D-trajectory concept, which consists of
defining
recisely a trajectory in space and time. One option is to create
an

utomatic conflict-resolution tool to provide advisory solutions
to
he controller. Some previous work has been done in the
direction
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C. Peyronne et al. / European Journal of Operational Research
241 (2015) 502–512 503
Initial trajectories Turning point Offset
Fig. 1. Two maneuvers implying direction changes.
Fig. 2. Evolution of traffic, number of sector, controllers and
the number of flights per
controller.
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f automatic-conflict resolution (see Bosc, Durand and Maugis
1997)
ut to our knowledge, none has been used operationally yet.
.2. Previous related work
One approach for air traffic automatic conflict resolution uses
nav-
gation functions (Dimarogonas & Kyriakopoulos, 2005; Roussos
&
yriakopoulos, 2009). What follows is a presentation of the
repul-
ive force technique that inspired the navigation function
methods
the repulsive force technique is much more intuitive and easier
to

Obstacle
Potential line
Aircraft
Repulsive
force
Sliding force
Sliding
Fig. 3. Repulsive force between an aircraft and an
omprehend). A repulsive force technique considers the airspace
as
potential field, and aircraft as particles navigating in it.
Negative
harges represent obstacles for the aircraft (other aircraft,
congested
reas). The destination is associated with a positive charge. As a
result,
ach aircraft is attracted by its destination while being repulsed
by
bstacles, as is illustrated by Fig. 3. Each trajectory is
determined in
ts own functional space so that an aircraft cannot be attracted by
the
estination of another aircraft. This enables the automatic

generation
f conflict-free trajectories.
Navigation function methods were shown to ensure collision
voidance while connecting the departure and destination points.
The
ajor drawback of navigation functions is that the obtained
solution
oes not necessarily respect ATM constraints such as the
particular
ounded speed (an aircraft cannot fly below or above a certain
range
f speed), or trajectory smoothness. Furthermore, they can lead
to
ajor delays and overcosts as they tolerate large deviations from
the
irect route, i.e. the straight line between the departure and
arrival
oints.
Optimization methods have also been used in air traffic
automatic
onflict resolution. In Pallottino, Feron and Bicchi (2002), two
ap-
roaches based on a local optimization method are presented.

One
pproach considers using only speed changes, and the other
relies
olely on direction changes. However, the speed-change
approach
annot solve every conflict situation (simply consider a face-to-
face
ituation for example), and their direction-change approach is re-
tricted to straight-line maneuvers.
Further developments are presented in Alonso-Ayuso, Escudero,
nd Martin-Campo (2011) combining speed and altitude changes
and
sing a mixed-integer linear optimization approach. This method
pro-
ides very interesting results both from the point of view of
computa-
ional time and the quality of resolution on conflict situation
involving
p to 50 aircraft (which are not all involved in the same conflict
how-
ver). In Alonso-Ayuso et al. (2011) a method is presented that
relies
n exact optimization, which is a significant advantage.

However, the
se of altitude changes is a drawback as it induces costly
maneuvers
hat are avoided by air traffic controllers due to their high costs
in fuel
hat are unacceptable for airline companies.
Remaining optimization methods for this problem rely on
heuris-
ics. The authors of Durand and Alliot (1995) and Médioni,
Durand and
lliot (1994) obtain relatively good results on real traffic using
Genetic
lgorithms (GA). However, their approach is restricted to offset
and
urning-point maneuvers, i.e. piecewise-linear trajectories.
Similarly,
sing ant colony optimization and modelling trajectories as a
path
n a graph, Durand and Alliot (2009) and Olive (2006) are
confined
Aircraft 1
Aircraft 2
force

Sliding
force
Potential line
Potential line
Relative speed 2/1
Relative speed 1/2
obstacle (left), between two aircraft (right).
504 C. Peyronne et al. / European Journal of Operational
Research 241 (2015) 502–512
α4start
α1start
α2start
α3start
α4end
α3end
α1end
α2end
Aircraft 3

Aircraft 2 Aircraft 4
Aircraft 1
Fig. 4. A typical input of the problem: one conflict involving
trajectory segments
(in bold).
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to produce piecewise-linear trajectories. Finally, another recent
so-
lution for tactical conflict resolution, presented in Dougui,
Delahaye,

Puechmorel and Mongeau (2012), called LPA (Light
Propagation Al-
gorithm), is inspired by an analogy with light propagation. LPA
uses
a Branch-and-Bound technique to build trajectories and obtains
good
results on a full day of traffic over France. However, the
trajectories
are built sequentially, which can cause significant deviations for
some
aircraft. For more detail on air traffic conflict detection and
resolution,
see the reviews by Delahaye and Puechmorel (2013) and Kuchar
and
Yang (2000).
1.3. Limitation of scope
There are three means to solve an air traffic conflict: speed
changes, direction changes and altitude changes (of course,
reduc-
ing the traffic also naturally decrease the number of conflict).
This
paper focuses on tactical en-route conflict resolution, which in
our

case means we consider a 20-minute time horizon. Thus, this
study
considers only direction changes, as speed changes are not as
effi-
cient on such a short time frame, and as altitude changes are
only
used as a last resort by air traffic control (because of their cost
and
the passenger discomfort they engender). That is why, in this
paper,
each aircraft is also assumed to preserve its imposed vertical
profile.
Consequently, as the study concentrates on en-route traffic,
aircraft
are considered to fly at a stable altitude, which means solving
the
conflicts solely in two dimensions.
This paper presents a tractable and practical method to solve
tac-
tical conflicts on the considered time horizon, involving only a
small
part of the trajectory (corresponding to the considered 20-
minute
window). One of the contributions of this paper is to model

trajec-
tories with B-splines. The overall trajectory is to be managed by
an
algorithm that aggregates the separate 20-minute windows. To
treat
a large air traffic instance (for example the French en-route air
traffic),
the conflict solver presented in the paper is to be applied at
regular
time intervals using a moving time-window. Furthermore, the
solver
can be applied on different geographical zones at the same time.
See
Dougui et al. (2012) for more details on this moving time-
window
process. For these reasons, computational efficiency is an
important
aspect of conflict resolution. The concept of a moving time-
window
(decreasing the number of considered aircraft) and the need for
com-
putational efficiency are the main reasons why local
optimization has
been tested in this study.

1.4. Problem description and overview
As already mentioned, the context of this study is the tactical
plan-
ning phase. The aim here is to obtain, from a conflicted
situation, an
optimal conflict-free solution by deviating the trajectories
smoothly
and as little as possible.
This paper introduces an original trajectory model using B-
splines
and a new semi-infinite programming formulation of the
separation
constraint involved in air traffic conflict problems. Another
related
contribution consists of a new continuous-optimization
formulation
of the tactical conflict-resolution problem that involves very
few op-
timization variables: only one real optimization variable per
trajec-
tory. Finally, encouraging numerical experiments are obtained
us-
ing genetic algorithms, a finite-difference interior-point

method, and
derivative-free optimization.
The input data of the problem consist of :
1) N: the number of aircraft involved in the conflict
2) αistart and α
i
end
: the initial and final positions of the aircraft i (i =
1, 2, . . . , N) respectively, as illustrated in Fig. 4.
3) vi: the speed of aircraft i, which is assumed constant.
Note that each piece of trajectory considered is first to be
modelled
by a B-spline, defined by a single continuous parameter. The
output
hould consist of smooth conflict-free trajectories, with
trajectory i
oing from αistart to α
i
end
at constant speed vi.
. B-spline trajectory model and decision variables
The use of smooth trajectories is not possible in today’s
perational context as air traffic control is currently restricted to

tra-
ectories involving linear segments connected with small
constant-
urvature turns. However, the challenging research projects
SESAR
nd NextGen consider smooth trajectories as an option for the
future
light-Management Systems. The ability of future FMS (Flight
Man-
gement System) to fly such trajectories opens new opportunities
for
ptimization with respect to environmental criterion. In this
context,
he use of cubic B-splines (smooth piecewise-cubic polynomials)
to
esign aircraft trajectories allows one to describe an aircraft
trajectory
eviation between the points αstart and αend with a single
continuous
arameter. These parameters (one per aircraft) will be the
optimiza-
ion variables in this study. Furthermore, it should be clear that
it
s more desirable in practice to use smooth trajectories. This
section

escribes the main ingredients of B-spline theory in a more
general
ontext.
.1. Elements of B-spline theory
B-splines are parameterized curves determined by a set of
points
alled control points. One considers here a set of control points
Xk, Yk) ∈ R2 . One can define γ (s), the B-spline curve
determined
y the control points (Xk, Yk) as follows:
(s) = (γx(s), γy(s)), s ∈ [0, Ncp − 1],
here s is the natural parameter of the B-spline, and Ncp is the
number
f control points. The B-spline curve is obtained as a linear
combina-
ion of the B-spline function basis (Bk)k=−1,...,Ncp . Each
element, Bk, of
he function basis is a cubic-polynomial function. The basis
function
0 is the interpolation natural cubic spline of the following
points:
−2, 0),(−1, 16 ), (0, 23 ), (1, 16 ), (2, 0) centered on 0, and the
remaining
lements of the basis are obtained by simple translations of B0.

241 (2015) 502–512 505
Fig. 5. The four B-spline basis functions relevant for the
interval [1, 2].
γ
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γ
Start Point
End PointMoveable Point
Fig. 6. A B-spline trajectory determined by three control points.
i
B
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B
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The B-spline basis functions used to calculate the B-spline
curve
(s) for s ∈ [1, 2] are the ones with a non-null value on this
interval
1, 2] (in bold in Fig. 5). Thus, the unique 2D B-spline curve
that is
etermined by the Ncp control points (Xk, Yk) along with three
so-
alled phantom points is given by the following linear
combination of

he Bk’s : γx(s) =
∑Ncp+1
k=−1 XkBk(s) and for γy(s) =
∑Ncp+1
k=−1 YkBk(s).
The B-spline fitting curve is a piecewise cubic polynomial
function
f s, where the knots, sk, are the points where the pieces join.
Denote
k
x ,
dγ kx
ds
,
d2γ kx
ds2
and
d3γ kx
ds3
respectively the B-spline desired values: the
-spline desired first, second and third derivative values for s =
sk.
he following requirements yield standard B-spline properties for
nit knot intervals:
γ kx =

Xk+1 + 4Xk + Xk−1
6
,
dγ kx
ds
= Xk+1 − Xk−1
2
,
d2γ kx
ds2
= Xk+1 − 2Xk + Xk−1,
d3γ kx
ds3
= Xk+2 − 3Xk+1 + 3Xk − Xk−1. (1)
ince the Taylor’s expansion is exact for polynomials (providing
nough terms are taken) one then obtains the B-spline value for
all
∈ [sk, sk+1]:
x(s) = γ kx + (s − sk)
dγ kx
ds
+ (s − sk)
2
2

d2γ kx
ds2
+ (s − sk)
3
6
d3γ kx
ds3
. (2)
igher-degree derivatives are zeros, as γx(s) is a cubic B-spline.
As
e use three control points (αstart and αend and one movable,
α(u)),
here are three knots, inducing two sub-intervals. The trajectory
γ (s)
s represented on the interval [0, 2] and on the two sub-intervals
0, 1] and [1, 2]. For more details, the B-spline theory is
described, for
xample, in Duncan (2005).
In the following, the indices i and j will be used to determine
the
onsidered aircraft, and k will be used to determine the
considered
ontrol point. Thus, γ i stands for the trajectory of the ith
aircraft, and

k for the value of γ at the knot s for the considered trajectory.
k
B-spline fitting is a very efficient tool for trajectory modelling
n terms of both fitting quality and computational time.
Moreover,
-splines feature interesting properties such as C2-continuity,
which
s crucial for modelling flyable smooth aircraft trajectories. In
addi-
ion, a very attractive property is the fact that, by construction,
the
-splines minimize the quantities:
∫
(γ ′′x (s))2ds and
∫
(γ ′′y (s))2ds. This
s important in an operational context as it induces low energy
con-
umption and passenger comfort.
B-splines have already been used in trajectory optimization and
ne can find an example in Milam, Mushambi and Murray
(2000).
.2. Optimization variables defining the trajectories
In this study, one models a trajectory with a cubic B-spline de-

ned by three control points: the start and end points (given input
ata), plus a middle control point (Fig. 6). This middle control
point
s called the movable control point, as it will be used to deviate
the
rajectory. The exact position of this control point will be
monitored
y a single real-valued parameter. The vector of decision
variables
f the optimization problem we are about to define will be made
of
hese real-valued parameters.
One wants a compromise between allowing the trajectory to
eviate freely (in any direction) from the direct route in order to
void conflicts, and staying as close as possible to the direct
route.
or that purpose, one defines a fixed maximal bandwidth
(interval
Research 241 (2015) 502–512
Dmax

Dinit
pimid
d
αistart
αi(ui)
αiend
Fig. 7. Direct route, maximal deviation bandwidth and modified
trajectory.
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p
r

c
w
t
a
s
w
n
p
⎧ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎩
γ
t
w
(
l
v
[−Dmax, Dmax]) that depends on the direct route length (Dinit).
The
trajectory will be imposed to stay within the maximal bandwidth
(see Fig. 7).
The middle control point coordinates corresponding to the

trajectory of aircraft i (i = 1, 2, . . . , N) is denoted αi(ui). Its
exact posi-
tion will be determined using the parameter ui which will
represent
a bandwidth proportion. It is defined as: ui = ± dDmax
(percent), where
Dmax = λDinit, d is the distance between the control point
αi(ui) and
the direct route for aircraft i, and λ is a user-defined proportion.
In
the numerical tests, λ is set to 0.3, so as to expect a maximal
distance
increase of about 10 percent from the original trajectory (this
was
observed empirically).
The optimization variable of the problem is therefore the vector
u whose ith component is that percentage ui corresponding to
tra-
jectory i = 1, 2, . . . , N. The search space is therefore the
hypercube
[−100, 100]N .
In order to calculate the deviation of each aircraft (which will
be
required for computing the objective function of the
optimization
formulation of the next section), one must first build the
trajectories

driven by the above-defined control points. From a given
deviation
percentage vector, u, one calculates the control point locations,
and
the B-splines driven by them. In order to illustrate this, consider
the
trajectory of a single aircraft, i. First, let us define the direction
vector
δi := αi
end
− αistart of the ith direct route. The middle of the original
trajectory direct route is called the site and is denoted by pi
mid
. The
middle control point αi is placed on the line orthogonal to the
direct
route and intersecting the direct route at pi
mid
(see Fig. 7). In other
words,
pimid = αistart +
1
2
δi. (3)

The vector of norm Dimax (maximal deviation bandwidth of the
ith
aircraft) that is orthogonal to the direct route, and, by
convention,
oriented toward the left of the aircraft trajectory, is denoted by
qi ∈
R
2. More precisely, it is obtained by solving the system:{
(δi)T qi = 0
||qi|| = Dimax.
(4)
The first line of this system ensures the orthogonality between q
and δ,
while the second line defines the length of q. Solving this
system leads
to two possible vectors q. To choose between those two
possibilities,
we define a convention stating that qi should be oriented toward
the
left of the original trajectory. Consequently, if ui > 0, the ith
aircraft
will turn left. Hence, one can write the following relation
between the
control point αi(u)and the ith optimization variable ui
corresponding

to aircraft i:
αi(ui) = pimid + uiqi. (5)
Thus, given ui, one knows the three required control points (α
i
start,
αi(ui)and α
i
end
), and can compute the B-spline defining the ith aircraft
trajectory.
. Optimization problem formulation
The problem consists in designing, if possible, an optimal
conflict-
ree situation. This section proposes an optimization formulation
of
his problem whose constraints will require that there is no
conflict.
he objective function of the problem can be defined in several
ways:
ne can consider minimizing the total deviation distance or total
fuel
onsumption for instance. This paper concentrates on finding
conflict-
ree trajectories that minimize the average total deviation

distance
with respect to the direct routes). The formulation presented can
asily be adapted to other objective functions.
.1. Formulation
A new idea introduced in this paper is to consider the
continuous
eparation distance as a constraint, making the approach more
direct
han previous methods that rely on a discretization of the
trajectories,
ot only for numerical purposes but in the actual formulation of
the
roblem to be solved. However, this constraint must be respected
at
ll times, leading to an infinite number of constraints. As
described
bove, one uses u, the vector containing all the middle control
point
ocations, ui, i = 1, 2, . . . , N as the optimization variable.
Let f (u) be the objective function representing the average trav-
lled distance for every aircraft (details are given in Section 3.2),
which
s written as follows:

(u) = 1
N
N∑
i=1
f i(ui), (6)
here f i(ui) is related to the travelled distance of the aircraft i
ollowing its modified trajectory. This function is further
explained
n Section 3.2.
In order to guarantee a conflict-free situation, one must ensure
hat all aircraft are, at least, τ := 5 nautical miles away from
each
ther (in practice, τ = 5.001 is taken to give a margin tolerance).
he separation-norm constraint can be expressed explicitly by
im-
osing, for any pair of aircraft i and j whose trajectories are
denoted
espectively γ i(ui; s) and γ
j(uj; s):
ij(u; t) := ‖γ i(ui; s(t))− γ j(uj; s(t))‖22 ≥ τ 2, ∀ t ∈ [0, tijmin],
(7)
here t
ij
min

:= min(ti
end
, t
j
end
), with ti
end
is the arrival time of aircraft i
o its end point αend
i
. The natural spline parameter, s, and the time, t,
re related by the following bijective relation between s and t:
(t) = θ−1(ui; t) and t = θ(ui; s), (8)
here θ(ui; s) =
∫ s
0
√( dγx(ui,ξ )
dξ
)2 + ( dγx(ui,ξ )
dξ
)2
dξ relates time and the
atural spline parameter, s, through the arc-length closed-form

ex-
ression.
Therefore, the problem can be formulated as:
min
u
f (u) = 1
N
N∑
i=1
f i(ui)
s.t. cij(u; t) ≥ τ 2 ∀ t ∈ [0, tij
min
]; i = 1, . . . , N − 1;
j = i + 1, . . . , N.
(9)
In order to compute cij(u; t), one needs to evaluate γ i(ui; s(t))
and
j(uj; s(t)) at time t. This is achieved in practice by sampling the
rajectory in time, noting that one has for each s, the values (xi,
yi, di, t),
here di is the distance travelled by aircraft i whose co-ordinates
are
xi, yi) at time t. These values are sampled with respect to time,
using

inear interpolations (di and t are exact; and xi and yi are
approximated
ia sampling).
241 (2015) 502–512 507
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> 5Nm
Fig. 8. Different trajectory configurations leading at most to:
zero conflict (left), one
conflict (center), and two conflicts (right).
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.2. The objective function and its derivatives
In this sub-section, the objective function is defined and closed-
orm expressions of the objective-function derivatives with
respect
o the optimization variables are obtained. Recall that the
objective-

unction is expressed in (6), and that the optimization variables
re meant to describe the middle control-point locations (αi(ui) =
i
mid
+ uiqi).
Since f i(u) depends only on the ith u component, the ith partial
erivatives of the objective-function is:
∂ f
∂ui
= 1
N
∂ f i
∂ui
.
s described in the previous section, each trajectory can be rep-
esented by a parametric curve in R2: γ i(ui; s) =:
(
γ ix(ui;s)
γ iy(ui;s)
)
, where
∈ [0, 2] (= [0, Ncp − 1]), and it is defined by a B-spline driven

by
hree control points corresponding to s ∈ {0, 1, 2}.
To define the objective function, let us define:
ˆi(ui) :=
∫ 2
0
√√√√(dγ ix(ui; s)
ds
)2
+
(
dγ iy(ui; s)
ds
)2
ds.
nd,
i(ui) :=
∫ 2
0
(
dγ ix(ui; s)
ds

)2
+
(
dγ iy(ui; s)
ds
)2
ds. (10)
he former gives the total distance travelled by the aircraft i
while the
atter represents an energy. Minimizing the energy Ti also
minimizes
ˆi, therefore, one can rather minimize Ti in order to avoid the
square
oot (see, for example do Carmo 1992, pp. 190–200). We choose
the
otal energy rather than the travelled distance as a criterion in
order
o ease the derivatives calculation.
In order to have a normalized objective function f (u), one
defines
he functions f i(ui) ∈ [0, 1] as follows:
i(ui) = T
i(ui)− Ti(0)
Ti(1)− Ti(0) . (11)

tedious calculation, detailed in Appendix A, yields the closed-
form
erivatives, for i = 1, . . . , N:
∂
∂ui
f i = 1
Ti(1)− Ti(0)
(
32
15
ui
((
qix
)2 + (qiy)2)
)
.
.3. The constraint functions
Inspired by Conn and Gould (1987) and Visweswariah, Haring
and
onn (2000), one can reformulate the constraint function cij as:
ij(u) =
∫ tij
min

0
max{τ 2 − cij(u; t); 0}dt = 0.
oing so, one transforms the semi-infinite constraints (each con-
traints is defined over a time interval [0, t
ij
min
]) into a single equal-
ty constraint (see also Stein 2012 for a more theoretical survey
of
emi-infinite optimization). This constraint ensures that the
separa-
ion norm is respected between aircraft i and j. Indeed, if, for
some pair
f aircraft (i, j), there is a conflict, then τ − cij(u; t) > 0 during
some
ime interval, leading to a non-zero integral (constraint
violation).
ne can rewrite Cij(u) as:
ij(u) =
∫
ij
(
τ 2 − cij(u; t))dt, (12)

ith
ij := ⋃
κ
[t
ij
inκ
, t
ij
outκ
], where the union is over each time interval
t
ij
inκ
, t
ij
outκ
] during which the aircraft i and j are in conflict (violat-
ng the separation constraint), and where κ is the number of such
ime intervals. Since there is only one movable control point for
each
rajectory (which means an aircraft cannot do more than one
turn),
here can only be at most κ = 2 violating time intervals for each
pair of
ircraft (see Fig. 8). Consequently, without loss of generality,
one can

ssume there are at most three possible configurations
corresponding
o κ = 0, 1, 2 depending on the situation. A situation with κ
conflicts
orresponds to κ violating intervals.
Standard optimization methods commonly require providing the
bjective-function and constraint derivatives. No satisfying
results
ave been obtained for the computation of the constraint
derivatives
lthough we are working on such an improvement (and have suc-
essfully provided derivatives for the objective function). Thus,
in this
tudy, when constraint derivatives will be required by an
optimiza-
ion method, we shall be content with finite-difference gradients.
. Optimization methods
This section details the different optimization methods applied
to
he problem. First, a genetic optimization method (GA) is used
to han-
le the combinatorial aspect of the problem, as this optimization
ap-

roach is the most used for conflict resolution problems. One can
next
ake advantage of the fact that the B-spline trajectory model
allows
ne to apply a standard local continuous optimization method
such as
nterior-point methods. Finally, because of the relatively large
num-
er of function evaluations required for the use of finite-
difference
pproximations of the gradients, it is natural to try one of the
modern
erivative-free optimization method.
.1. Genetic algorithm and problem-specific genetic operators
The first optimization method considered to solve the problem
is
enetic algorithms (GA; Goldberg, 1989). We emphasize that this
is
lso a contribution of this paper since we use the formulation (9).
Our implementation of GA selects the best individuals of the
pop-
lation at each iteration using a deterministic (λ,μ)-tournament
se-
ection which randomly selects λ individuals and keeps the μ

best
lements, where λ > μ are user-defined parameters. This step is
re-
eated until a new intermediate population is completed. Genetic
perators (crossover, mutation or nothing) are then applied with
pecified user-defined probabilities (noted respectively pc, pm
and,
− pc − pm). Ultimately, one obtains the next generation of
chromo-
omes. This generational process is repeated until some user-
defined
ermination condition has been reached. In this study, a maximal
umber of generations is imposed.
The chromosome encoding used here represents the trajectories
f N aircraft using a vector of N real numbers (the u ’s, see Fig.
9).
i
Research 241 (2015) 502–512
Aircraft 1 u1
Aircraft 2 u2
... ...

Aircraft 19 u19
Aircraft 20 u20
Fig. 9. Chromosome encoding for a 20-aircraft instance.
w
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It respects the locality principle: two individuals close to each
other
in the search space represent close solutions. The crossover
operator
is used to mix the features of two good individuals (good from
the
point of view of candidates for improving upon the current
solution),
called parents, from the previous generation. It consists of
picking
the most conflicted aircraft of each parent (more precisely, we
seek

for the highest value of
∑
j �=i Cij(u)among all aircraft i = 1, . . . , N), say
aircraft i for one parent and aircraft j for the other, and of
modifying
its B-spline trajectory over [s0, s2] using a barycentric
transformation
of the two parents’ corresponding movable control point (Fig.
10).
The crossover results in two children that are expected to yield
local
improvement.
The mutation operator, used to diversify the genes in the popu-
lation in order to explore widely the search space, consists, in
this
application context, of choosing randomly one movable control
point
and to assign to it a new value chosen randomly (using a
uniform
distribution) in the interval (percent) [−100; 100] (see Fig. 11).
Finally, the fitness, which quantifies the ability of an individual
to
solve the problem is defined as follows:
f (u) = 1

N
N∑
i=1
f i(ui)+ ω
N∑
i=1
N∑
j=1
Cij(u), (13)
CROSSOVER
Parent 2Parent 2
Child 1 Child 2
Fig. 10. Parent chromosomes and children obtained via a
barycentric transformation (a
MUTATION
Fig. 11. Initial and mutated chromosome a
here ω is a penalization parameter, weighting the importance of
easibility with respect to optimality, set empirically by the user.
This penalty function is also invoked in order to enable the use
f the derivative-free optimization method BOBYQA described
in
ection 4.2.1.

.2. Local optimization method
This section details briefly the local optimization methods
applied
nd the reasons why they were chosen.
.2.1. Derivative-free optimization
The lack of closed-form expressions of the constraint deriva-
ives naturally leads to derivative-free optimization methods
(Conn,
cheinberg & Vicente, 2009). The method chosen here is
Powell’s
OBYQA (Bound Optimization BY Quadratic Approximation;
Powell,
009), one of the most effective derivative-free optimization
methods
vailable. It is based on a trust-region model described in
Chapter 10
f Conn et al. (2009). Note that this class of methods does not
han-
le constraints directly (except for simple bounds).
Consequently, the
enalized objective function defined in (13) is used.
For our numerical experiments, we use all the default values

roposed in Powell (2009) for the various parameters defining
the
ethod.
.2.2. Local differentiable optimization
In fact, a closed-form expressions of the derivatives of the ob-
ective function can be obtained here (this calculation is detailed
in
ppendix A). One can therefore consider applying a standard
local
ifferentiable optimization method. We choose constrained
interior-
oint methods because they are state-of-the-art methods for non-
inear programming. The numerical experiments are conducted
us-
ng the Matlab routine fmincon (Byrd, Gilbert & Nocedal, 2000).
The
CROSSOVER
veraging the chosen bandwidth percentage) and their
corresponding trajectories.
MUTATION
nd their corresponding trajectories.

241 (2015) 502–512 509
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a
1
2
3
4
5
6
t
t
a
Fig. 12. Roundabout test problem configuration for N = 6
aircraft.

(
p
5
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s
p

1
a
a
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1
2
3
4
5
radient of the objective function is supplied to this routine.
How-
ver, the derivatives of the constraint functions (12) (which are
not
rivial to obtain) are approximated automatically by fmincon
using
nite differences. Current work aims at obtaining a closed form
of the
radient of the constraint functions.
Here again, we use in our tests the default values provided by
atlab for the various parameters involved in the algorithm.
. Numerical results

This section presents comparative numerical results obtained
with
he different optimization methods proposed above. First, an
aca-
emic test problem, called the roundabout, is presented. Then, a
more
ealistic problem (so-called operational-like test problem) is
proposed.
lthough rather artificial, both test problems feature some opera-
ional aspects of real-life problems and thereby allows one to
test the
iability of our methodology.
The results we obtain with GA are used as a reference. Indeed,
e expect GA, considering its ergodicity property, to be able to
reach
he neighborhood of any desired point of the search space within
the
large) number of iterations allowed. Consequently, with a
significant
umber of function evaluations, we expect GA to find a solution
whose
alue is relatively close to the optimal value. Moreover, GA is
known

o be fit to handle the air traffic conflict problem (Médioni et al.,
994), which makes it a good reference to compare with the
results
f our local optimization approach. To summarize, we shall
compare
he results we obtain with local optimization method with
(fmincon)
nd without derivatives (BOBYQA), to the ones we obtain with
our
A implementation.
The proposed genetic algorithm is implemented in Java. As
entioned above, the differentiable local optimization method
used
s the routine fmincon from the Matlab Optimization toolbox.
The
erivative-free optimization method BOBYQA is coded in fortran
77
nd is called from Matlab via a mex interface. As a consequence
of
his difference in programming language, computational time
cannot
e considered as an objective comparison criterion. We rather
rely

ere on comparing the number of function evaluations, which
rep-
esents most of the computational time spent by the optimization
ethods. This corresponds to common practice in black-box opti-
ization where the objective and/or constraint functions are
costly
o evaluate. However, in order to give an idea of the order of
computa-
ion time involved, let us simply present the calculation time for
one
valuation of both the objective and constraint functions for a
same
oint, and on a same traffic situation. In both cases (Java and
Matlab),
e use a 2.53 GHz processor Intel Core 2 Duo on a Ubuntu 12.04
LTS
perating system. Using Java, one evaluation requires 7
milliseconds
hile with Matlab, it needs 26 milliseconds.
Here are the parameter values used to implement the different
bove-mentioned algorithms:
) Population size: 100; number of generations: 100 (hence GA

will
evaluate 10,000 times the objective and constraint functions)
) Mutation probability: pm = 0.3
) Crossover probability: pc = 0.6
) Constraint penalization parameter: ω = 0.01
) Stopping criterion of the local optimization methods: ||uk+1 −
uk|| < 10−6
) Approximate global optimization value tolerance for GA f
− f <
�f := 10−4, for feasible solutions, where f
is the best solution
value found by GA. This f
value will be used as a comparative
quality criterion in the numerical results.
The local optimization methods fmincon and BOBYQA start
with
he best feasible point from 100 randomly-generated points from
he search space ([−1, 1]N). These 100 extra function
evaluations
re taken into account in the function-call counts for these
methods
in a suitable parallel environment, choosing many different
starting
oints could be relatively inexpensive).

.1. Roundabout test problem
First, a simple but difficult academic test with conflict
situations,
hat is widely used for air traffic conflict problems, is
considered. Each
nstance of the roundabout problem involves N aircraft
uniformly
istributed on a circle of radius 100 nautical miles. Each of the N
ircraft flies to the diametrically opposed point at a common
speed
for N even, each point on the circle has an outgoing and an
incoming
rajectory, see Fig. 12).
This problem is convenient to test conflict-resolution methods
be-
ause, due to its symmetry, there is an expected solution. Indeed,
air
raffic controller common practice would then make all the
aircraft
nvolved in the conflict turn to the same side. In the sequel, this
obvi-
us solution is referred to as a global solution of the problem.
This test

roblem is widely used because automatic air traffic conflict
resolu-
ion algorithms are not designed to take advantage of this
particular
ymmetry.
The roundabout test problems considered here involve succes-
ively N = 2, 4, 6, 8, and 16 aircraft. Recall that the chosen start
oint of the local optimization methods is the best point among a
00 randomly-generated points of [−1, 1]N . GA initial
population will
lso be randomly generated in [−1, 1]N .
The computational results obtained with the three optimization
pproaches (GA, fmincon, and BOBYQA) on this academic
problem
eveals that:
) GA and BOBYQA always (for N = 2, 4, 6, 8, and 16) finds a
global
minimum (all aircraft turn to the same side).
) For N = 2, fmincon always finds a global optimum.
) For N = 4, fmincon finds a local minimum for which
objective-
function value vary in [f
, 3 × f
] depending on the chosen starting
point.

) For N = 6 fmincon finds a local minimum for which objective-
function value vary in [f
, 8 × f
].
) For N = 8 and 16 aircraft, a local optimum is not always found
with
fmincon, depending on the chosen starting point.
Research 241 (2015) 502–512
Fig. 13. Operational-like test problem configuration for N = 6
aircraft.
i
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l
m
solution.
No further detail is given as this is an academic problem that
does not
correspond to real-world problems.
5.2. Operational-like test problem
In order to create test problems that are more realistic, an
operational-like test problem can be created by introducing
some
perturbations in the roundabout test problem. This study is
limited
to conflicted situations involving N = 6 aircraft, as according to
air
traffic controller experience, conflicts involving more than four
air-
craft are not typical in en-route traffic (only one four-aircraft
conflict
happens over France each year). For each aircraft, one randomly
dis-
places (or not) the position of the start and end points on the
circle
by 20◦ to obtain a conflict situation involving various crossing
angles,

losing thereby the special symmetry of the roundabout test
problem
(see Fig. 13). A randomly-generated speed vi ∈ [550; 650]
knots (nau-
tical miles per hour) is assigned to each aircraft i, i = 1, . . . , N,
during
the generation of the test-cases (for the academic roundabout
test
problem, the speed of each aircraft was vi = 600 knots).
Following
this methodology, 100 different instances were generated. Each
op-
timization method is applied to these 100 instances in order to
make
an elementary statistically-valid comparative study.
Table 1
Comparative numerical results over the 100 instances.
Feasibility success rate (percent)
Average objective-function value
Median objective-function value
Objective-function value difference with respect to f
Number of function evaluations
To compare the results of the different methods on these 100
nstances, GA results are taken as a reference, as, with 10,000

func-
ion evaluations, it is expected to find a (nearly) globally
optimal value
for each of these instances. Several criteria are taken into
account.
irst, the percentage of success of the local optimization method
is
eported (whether it converges to a locally-optimal solution or
hether it fails). The objective-value statistics (mean, median,
worst,
est) are taken over the 100 instance runs. For each instance, the
dif-
erence in the objective-function values is calculated with
respect to
when the local method converges to a feasible solution. The
best
respectively worst) objective-function value differences with
respect
o f
presented in the table correspond to instances for which fmin-
on and BOBYQA performed the best (respectively worst).
Finally, the
tatistics on the number of function evaluations required to
converge

re displayed. The results are summarized in Table 1.
The percentage of success for the local optimization methods
s high. This shows that local optimization methods are a viable
lternative for automatic conflict resolution. The objective-
function
ifference reveals that a local method can even find better
feasible
bjective function values than genetic algorithms. More
precisely,
mincon obtains better results (in terms of the objective-function
alue) than GA in 48 instances (over the 95 solutions fmincon
finds),
hich is very promising. BOBYQA obtains better results (again,
in
erms of the objective-function value) than GA for 23 instances,
hich is still significant. A discussion on the corresponding num-
er of function evaluations is given below. Of course, in the
remain-
ng cases, the local methods converged to high (i.e. bad)
objective-
unction values (local minima) when compared with GA, which
is

hy there is a discrepancy between the median and the average of
the
bjective-function value difference (a gap of 0.30 can be
considered as
ignificantly bad).
As mentioned previously, GA performs 10,000 function
valuations here, although it finds a feasible (conflict-free)
solution
efore. After how many function evaluations does GA find such a
easible solution? The first conflict-free solution is discovered
by
A quickly (110 evaluations on average, roughly equivalent to
the
ure random-search starting-point strategy of fmincon and
BOBYQA).
owever, such feasible solutions are of poor quality. In any case,
he global optimum found by GA requires much more time to be
ound with an �f precision (6 times the average number of
function
valuations needed by fmincon, and 15 times for BOBYQA). The
lo-
al optimization methods are therefore more efficient than GA
for

nding a feasible solution in terms of computational time for
compa-
able quality. We remark however that GA seems to be more sta-
le than fmincon, as it finds feasible solutions for 98 out of the
00 test cases, and the two cases where GA does not find a fea-
ible solution, the GA solution is close to being feasible (the
con-
traint violation is of order 10−2, when, for fmincon infeasible
so-
ution, the violation is significantly larger). However, BOBYQA
is the
ost stable methods among these three as it always find a feasible
GA fmincon BOBYQA
98 95 100
0.044 0.049 0.091
0.044 0.070 0.11
Mean − 0.029 0.067
Median − 0.006 0.047
Worst − 0.34 0.29
Best − −0.054 −0.069
Mean 10000 1135 400
Median 10000 820 393
Worst 10000 6825 1033

Best 10000 178 194
241 (2015) 502–512 511
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C(
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T
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Finally, note that the number of functions reported for fmincon
is
trongly impacted by the fact that it relies on finite-difference
con-
traint gradients (in N dimensions, each finite-difference
gradient
pproximation requires N + 1 function evaluations, and gradients
are
eeded at least once at every iteration). Indeed, the average num-
er of iterations is only 54. Once the constraint gradients will be

omputed (current research work), one can expect a much lower
umber of function evaluations for local differentiable
optimization
ethods.
. Conclusion
Standard approaches to air traffic conflict resolution problems
ely on discretization of the search space yielding a
combinatorial
ptimization formulation of the problem and, thereby,
computation-
ntensive optimization methods must be used. This paper shows
that
he use of an original B-spline trajectory model involving only
one
ontinuous variable per aircraft, and of a semi-infinite
programming
ormulation of the separation constraints, permits one to obtain
good
esults via local optimization methods. A major advantage of a
local
ptimization methods is that it requires much fewer (costly)
function

valuations. Moreover, in a safety operational context, competent
uthorities may prefer decision-aid tools based on a deterministic
ptimization methods with an underlying convergence theory and
a
ore efficient use of function evaluations.
Note that the optimization formulation introduced here involves
nly differentiable functions. Also, one expects that, ultimately,
it
ill not be necessary to rely either on derivative-free optimiza-
ion methods, or on finite-difference variants of classical
differen-
iable local optimization methods. Current work concentrates on
omputing closed-form constraint derivatives in order to
decrease
ignificantly the computational time required by local
optimization
ethods. Finally, the elementary random-search start-point selec-
ion step combined here with the local optimization method
could
e replaced with a more adapted hybrid two-phase (global/local)
ethod.

cknowledgments
This work has been supported by French National Research
Agency
ANR) through COSINUS program (project ID4CS no. ANR-09-
COSI-
05) and through JCJC program (project ATOMIC no. ANR 12-
JS02-
09-01). The authors are indebted to Professor Stéphane
Puechmorel
or his help on the calculation and computation of the
derivatives,
nd to Laurent Lapasset for interesting discussions on the
operational
spects of the problem. The authors are grateful to two
anonymous
eviewers for many detailed and helpful comments.
ppendix A. Objective-function derivatives
Recall that the expression for the objective function is given by
6), (10) and (11), and that given a control point list, αistart, α
i(ui)
nd αi
end
, the x components of the B-spline approximation associated

ith movable control point α(ui) for aircraft i is given by γ
i
x(ui; s).
or notational simplicity, in the sequel, the aircraft index i is
dropped
n the notation, except on the optimization variable ui, with
respect
o which we differentiate. Moreover, the factor 1
Ti(1)−Ti(0) , being a
ormalization constant, it is not included in the following
calculation.
ach term in the integral (in the expression of Ti(ui)) can be
considered
eparately. The calculation is only detailed for the term
involving γ ix .
sing the Taylor’s expansion expressed in Eq. (2), one can obtain
i
x(ui; s) for each s in the interval [sk, sk+1]. This Taylor’s
expansion is
xact as the B-spline curve γ ix(ui; s) is a third-degree
polynomial.
The trajectory model involves three control points and two
phan-
om control points. Thus, the B-spline is to be expressed on two
ntervals, [s0, s1] and [s1, s2].
Let us begin with the case for which s ∈ [s0, s1] := [0, 1].

Using (2)
ith k = 0, one obtains:
i
x(ui; s) = γ 0x + (s − s0)
dγ 0x
ds
+ (s − s0)
2
2
d2γ 0x
ds2
+ (s − s0)
3
6
d3γ 0x
ds3
.
o calculate our B-spline for s ∈ [0, 2], five control points are
used: the
hantom point X−1 := 2X0 − X1 = 2X0 − (pimidx + uiq
i
x), the starting
oint X0 := αstartx , the movable control point X1 := αix(ui) =
pimidx +
iq

i
x, the ending point X2 := αendx and the second phantom point
3 := 2αendx − (pimidx + uiq
i
x). To compute on each interval [sk, sk+1],
he B-spline method uses four control points Xk−1, Xk, Xk+1,
Xk+2. In
ur case, when s ∈ [s0, s1] = [0, 1] (i.e. for k = 0), the B-spline
com-
utation uses X−1, X0, X1, and X2, and when s ∈ [s1, s2] = [1,
2] (i.e.
or k = 1), the computation uses X0, X1, X2, and X3. The
dependen-
ies in ui are determined accordingly. In the following, for
notational
implicity, we will use X0 for αstartx , and X2 for αendx . Thus,
for all
∈ [s0, s1] = [0, 1], one has:
i
x(ui; s) = X0 +
(
pimidx + uiqix − X0
)
s
+ (X2 − 2pimidx − 2uiqix + X0) s
3
6

.
hen, one can obviously deduce, by differentiating the expression
bove, that, for all s ∈ [s0, s1]:
dγ ix(ui; s)
ds
= (pimidx + uiqix − X0) + (X2 − 2pimidx − 2uiqix + X0) s
2
2
.
onsequently, for all s ∈ [s0, s1]:
dγ ix(ui; s)
ds
)2
= (pimidx + uiqix − X0)2
+ s
4
4
(
X2 − 2pimidx − 2uiqix + X0
)2
+ s2(pimidx + uiqix − X0)
× (X2 − 2pimidx − 2uiqix + X0).

t is necessary to calculate the integral
∫ smax
0
( dγ ix(ui;s)
ds
)2 + ( dγ iy(ui;s)
ds
)2
ds
n order to obtain Ti. As mentioned before, only the terms in γ ix
are
etailed, and, in this part, the calculation focuses on the interval
where
∈ [0, 1]:
1
0
(
dγ ix(ui; s)
ds
)2
ds =
∫ 1
0
[(

)2
+ s
4
4
(
)2
+ s2(pimidx + uiqix − X0)
× (X2 − 2pimidx − 2uiqix + X0)
]
ds
= (pimidx + uiqix − X0)2
+ 1
5
(
X2
2
− pimidx − uiqix +
X0
2
)2

+ 1
3
(
)
hen, differentiating the above expression with respect to ui,
ne obtains the closed-form expression of the integral for
∈ [s0, s1] := [0, 1] :
∂
∂ui
∫ 1
0
(
dγ ix(ui; s)
ds
)2
ds = qix
[
16
15
(

pimidx + uiqix
) − 6
5
X0 + 2
15
X2
]
.
Research 241 (2015) 502–512
F
R
A
A
B
C
C
D
D

d
D
D
D
D
G
K
M
M
O
P
P
R
S
S
V
The calculation now concentrates on the interval where s ∈
[s1, s2] (k = 1). Using Taylor’s expansion (2) on this interval,
one obtains:
γ ix(u; s) = γ 1x (s1)+ (s − s1)

dγ 1x (s1)
ds
+ (s − s1)
2
2
d2γ 1x (s1)
ds2
+ (s − s1)
3
6
d3γ 1x (s1)
ds3
.
Now, this value has to be calculated for s ∈ [s1, s2] = [1, 2],
where the
optimized control point is Xk = pmidx + uiqx . Consequently,
due to the
change of k from 0 to 1 for this interval, one has: Xk−1 := X0 =
αstartx ;
Xk+1 := X2 = αendx ; Xk+2 := 2X2 − pmidx − uiqx. Thus, we
obtain for all
s ∈ [1, 2]:
γ ix(ui; s) =
1

6
(
X2 + 4pimidx + 4uiqix + X0
) + (s − 1)
2
(X2 − X0)
+ (s − 1)
2
2
(
)
+ (s − 1)
3
6
(
2pimidx + 2uiqix − X2 − X0
)
.
Differentiating with respect to s, one obtains for all s ∈ [1, 2]:
dγ ix(ui; s)
ds
= 1

2
(X2 − X0)+ (s − 1)
(
)
+ (s − 1)
2
2
(
)
, (A.1)
and therefore,(
dγ ix(ui; s)
ds
)2
= 1
4
(X2 − X0)2 + (s − 1)2
(
)2

+ (s − 1)
4
4
(
)2
+ (s − 1)(X2 − X0)
(
)
+ (s − 1)
2
2
(X2 − X0)
(
)
+ (s − 1)3(2pimidx + 2uiqix − X2 − X0)
Integrating this value over [1, 2] yields:∫ 2
1
(dγ ix(ui; s)
ds
)2

ds = 1
4
(X2 − X0)2 + 1
3
(
)2
+ 1
20
(
)2
+ 1
2
(X2 − X0)
(
)
+ 1
6
(X2 − X0)
(

)
+ 1
4
(
)
One then differentiates with respect to ui:
∂
∂ui
∫ 2
1
(
dγ ix(ui; s)
ds
)2
ds = qix
[
16
15
(pimidx + uiqix)+
2

15
X0 − 6
5
X2
]
.
As mentioned earlier, similar formulas can be obtained for the
terms in γ iy . Thus, the exact formula for the objective-function
deriva-
tive is written as follows:
∂
∂ui
f i(ui) = qix
[
16
15
(
pimidx + uiqix
) − 6
5
X0 + 2
15
X2

]
+ qix
[
16
15
(
pimidx + uiqix
) + 2
15
X0 − 6
5
X2
]
+ qiy
[
16
15
(
pimidy + uiqiy
) − 6
5
Y0 + 2
15

Y2
]
+ qiy
[
16
15
(
pimidy + uiqiy
) + 2
15
Y0 − 6
5
Y2
]
= 32
15
qix
(
pimidx + uiqix
) + 32
15
qiy(p

i
midy
+ uiqiy)
− 16
15
qix(X2 + X0)−
16
15
qiy(Y2 + Y0)
inally, since by construction, we have pi
midx
= X2+X02 , and pimidy =
Y2+Y0
2 , the ith component of the gradient vector ∇ if (u) is:
∂
∂ui
f (u) = 1
N(Ti(1)− Ti(0))
(
32
15
ui
((
qix

)2 + (qiy)2)
)
.
eferences
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CATS: A complete air traffic sim-
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lonso-Ayuso, A., Escudero, L. F., & Martin-Campo, F. J.
(2011). Collision avoidance in
the air traffic management: A mixed integer linear optimization
approach. IEEE
Transactions on Intelligent Transportation Systems, 12, 47–57.
yrd, R. H., Gilbert, J. C., & Nocedal, J. (2000). A trust region
method based on inte-
rior point techniques for nonlinear programming. Mathematical
Programming, 89,
149–185.
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for semi-infinite pro-
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onn, A. R., Scheinberg, K., & Vicente, L. N. (2009).
Introduction to derivative-free opti-
mization. MPS-SIAM series on optimization. Society for

Industrial and Applied Math-
ematics Philadelphia, PA: USA. 173–205.
elahaye, D., & Puechmorel, S. (2013). Modeling and
optimization of air traffic. Wiley-
ISTE: USA.
imarogonas, D. V., & Kyriakopoulos, K. J. (2005). A feedback
stabilization and collision
avoidance scheme for multiple independent nonholonomic non-
point agents. In:
Proceedings of the IEEE international symposium on robotics
and automation (pp. 820–
825). IEEE: USA.
o Carmo, M. P. (1992). Riemannian geometry. Birkhäuser:
Switzerland.
ougui, N., Delahaye, D., Puechmorel, S., & Mongeau, M.
(2013). A light-propagation
model for aircraft trajectory planning. Journal of Global
Optimization, 56(3),
873–895.
uncan, M. (2005). Applied geometry for computer graphics and
computer-aided design
(2nd ed.) Springer: UK.
urand, N., & Alliot, J. M. (2009). Ant colony optimization for
air traffic conflict res-
olution. In: Proceedings of the 8th USA/Europe air traffic
management research and
development seminar. Napa, USA.

urand, N., Alliot, J. M., & Noailles, J. (1996). Automatic
aircraft conflict resolution using
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96). ACM: USA.
oldberg, D. E. (1989). Genetic algorithms in search
optimization and machine learning.
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uchar, J. K., & Yang, L. C. (2000). A review of conflict
detection and resolution modeling
methods. IEEE Transactions on Intelligent Transportation
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édioni, F., Durand, N., & Alliot, J. M. (1995). Air traffic
conflict resolution by ge-
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(pp. 370–383). Springer: USA.
ilam, M. B., Mushambi, K., & Murray, R. M. (2000). A new
computational approach to
real-time trajectory generation for constrained mechanical
systems. In: Proceedings
of the 39th IEEE conference on decision and control. IEEE:
USA.
live, X. (2006). Résolution de conflits par algorithmes
stochastiques parallèles (Con-
flict resolution via parallel stochastic algorithms) (Ph.D.
thesis). Toulouse, France:

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allottino, L., Feron, E., & Bicchi, A. (2002). Conflict resolution
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management systems solved with mixed integer programming.
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owell, M. J. D. (2009). The Bound Optimization BY Quadratic
Approximation (BOBYQA)
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oussos, G., & Kyriakopoulos, K. J. (2009). Towards constant
velocity navigation and
collision avoidance for autonomous nonholonomic aircraft-like
vehicles. In: Pro-
ceedings of the Joint 48th IEEE conference on decision and
control and 28th Chinese
control conference. IEEE: USA.
ESAR Joint Undertaking (2009). SESAR Joint Undertaking
Brochure: Today’s partner
for tomorrow’s aviation. Technical Report. Brussels, Belgium.
tein, O. (2012). How to solve a semi-infinite optimization
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isweswariah, C., Haring, R., & Conn, A. R. (2000). Noise
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http://refhub.elsevier.com/S0377-2217(14)00758-9/bib001
http://refhub.elsevier.com/S0377-2217(14)00758-
9/bib023Solving air traffic conflict problems via local
continuous optimization1 Introduction1.1 Context1.2 Previous
related work1.3 Limitation of scope1.4 Problem description and
overview2 B-spline trajectory model and decision variables2.1
Elements of B-spline theory2.2 Optimization variables defining
the trajectories3 Optimization problem formulation3.1
Formulation3.2 The objective function and its derivatives3.3
The constraint functions4 Optimization methods4.1 Genetic
algorithm and problem-specific genetic operators4.2 Local
optimization method4.2.1 Derivative-free optimization4.2.2
Local differentiable optimization5 Numerical results5.1
Roundabout test problem5.2 Operational-like test problem6

ConclusionAcknowledgmentsAppendix A Objective-function
derivativesReferences
Assignment: Security Working Group (WG)
Throughout this course, you will be working on several aspects
of security management that will result in a Comprehensive
Security Management Plan for an organization of your
choosing. This plan will allow you to assess the security
concerns of your organization and propose needs and changes.
The Comprehensive Security Management Plan will introduce
methodologies that can be applied to enterprise security design.
Each week, you will complete a part of the Comprehensive
Security Management Plan, and the final draft will be due at the
end of the course. This is the course Key Assignment that you
will make contributions to each week.
Project Selection
The first step will be to select a real or hypothetical
organization as the target for your Comprehensive Security
Management Plan document. This organization will be used as
the basis for each of the assignments throughout the course and
should conform to the following guidelines:
· Nontrivial: The selected organization should be large enough
to allow reasonable exercise of the security management
analysis and planning processes.
· Domain knowledge: You should be familiar enough with the
organization to allow focus on the project tasks without
requiring significant time for domain education.
· Accessibility: You should have good access to the people and
other information that is related to the organization because this
will be an important part of the process.
· The selected organization may already have security
management in place, but it may still be used as the basis for
the projects in this course.
· The selected organization must have a need for some kind of
security management as part of its operations.
· Feel free to identify a hypothetical organization that meets the

requirements.
· You may make any necessary assumptions to fulfill the
requirements of organization selection.
Select an existing organization or identify a hypothetical
organization that fits these requirements, and submit your
proposal to your instructor before proceeding further with the
assignments in the course. Approval should be sought within the
first several days of the course through an e-mail proposal to
your instructor.
Assignment
The first task in this process will be to select an organization or
identify a hypothetical organization to use as the basis of the
projects. Next, you will create the shell document for the final
project deliverable that will be worked on during each unit.
While you proceed through each project phase, content will be
added for each section of the document to gradually complete
the final project. Appropriate research should be conducted to
support the development of the document, and assumptions may
be made.
For the first phase of the Comprehensive Security Management
Plan document, you will create an enterprise organizational
chart in the first document section. A proposed security working
group (WG) organization and its ties to the enterprise will be
added. Finally, include a 1-page discussion of the flow of
information, decision-making communication, and
responsibilities of the chief security officer (CSO). Create the
skeleton for the Comprehensive Security Management Plan as
follows:
· Use Word
· Title Page
· Course number and name
· Project name
· Your name
· Date
· Table of Contents (TOC)
· Use an auto-generated TOC.

· This must be on a separate page.
· This must be a maximum of 3 levels deep.
· Be sure to update the fields of the TOC before submitting your
project.
· Section Headings (Create each heading on a new page with
"TBD" as content, except for the sections that are listed under
"New Content.")
· Project Outline (Week 1)
· Security Requirements (Week 1)
· Security Business Requirements (Week 2)
· Security Policy (Week 3)
· System Design Principles (Week 4)
· The Training Module (Week 5)
· References
The following are the project deliverables for Week 1:
· New Content
· Project Outline and Security Requirements
· Include a brief description of the real or hypothetical
organization in which the Comprehensive Security Management
Plan will be implemented.
· Include the company's size, location(s), and other pertinent
information.
· Initial Security Projects
· Corporate organizational chart
· WG structure and ties added to corporate organizational chart
· Memo discussing communication flows with WG
· Name the document "yourname_CS654_IP1.doc."

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