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OPTIMIZED SECTORIZATION OF AIRSPACE WITH CONSTRAINTS
Huy Trandac, Philippe Baptiste - Heudiasyc Laboratory, UMR CNRS 6599, University of
Technology of Compiègne, Centre de Recherches de Royallieu,
BP 20529, F-60205 Compiègne cedex, France.
Vu Duong - Eurocontrol Experimental Centre, Centre de Bois des Bordes,
BP15, F-91222 Bretigny sur Orge cedex, France.

Abstract—In this paper we consider the Optimized
Airspace Sectorization Problem (ASP) with
constraints in which a given airspace is to be
partitioned into a number of sectors. The objective of
ASP is to minimize the coordination workload
between adjacent sectors. We proposed a constraint
programming approach to optimize the sectorization
that shall satisfy all specific constraints e.g. the
controllers’ workload is balanced among the sectors,
the sectors are not fragmented, aircraft can not enter
twice the same sector; aircraft cannot stay less than a
given amount of time in each sector crossed, sectors
cannot be fragmented etc.
Introduction
Sectorization is a fundamental architectural
feature of the Air Traffic Control (ATC) system. The
airspace is divided into a number of sectors, each of
them is assigned to a team of controllers (Control
Positions). Controllers of a given sector have (1) to
monitor the flights, (2) to avoid conflicts between
aircraft and (3) to exchange information with
adjacent sectors where aircraft have planned to go.
These tasks induce a workload which is often
decomposed into three corresponding parts [1, 2, 3]:
• The monitoring workload (MW) comes
from the cyclic checking of aircraft
trajectories.
• The conflict workload (CW) results from
resolution and avoidance of conflicts between
aircraft.
• The coordination workload (OW) is
basically related to the exchanges that have to
be performed between controllers of adjacent
sectors and pilots of aircraft that are crossing
through.

But the air traffic changes over the day. This
often leads to workload imbalance between the
sectors. Furthermore, it is desirable that there are
more the sectors (then more control positions) in the
dense traffic periods of the day than the weak
periods. Hence a tool to “dynamically” re-sectorize
the airspace (more precisely a part of airspace – e.g.
the sectors of a Air Traffic Control Center) is
required to cope with the evolution of the traffic.
When the sectors are designed, not only the
balance constraint must be hold (in term of
workload), but also that several following specific
constraints have to be taken into account:
• Convexity constraint. The same aircraft can
not enter twice the same sector. It is not
sensible, but it happened in the past, e.g.
national boundaries in European airspace.
For instance, the following case in the Figure
1 is not admissible:


Sector A
Sector B
traffic line
Sector boundary

Figure 1: Convexity Constraint

• Minimum distance constraint. The distance
between a sector border and a network node
must be not less than a given distance (see
Figure 2). This constraint ensures that the
controller has enough time to solve conflicts
which may occur at this node.

traffic lines
Sector B
Sector A
Sector boundary
critical point (too closed to boundary)

Figure 2: Minimum Distance Constraint
• Minimum sector crossing time constraint.
The aircraft must stay in each crossed sector
at least a given amount of time Tmin (see

2
Figure 3). This constraint ensures the
controller has enough time to control the
aircraft.

Sector B
Sector A Tmin
traffic line

Figure 3: Minimum Sector Crossing Time
Constraint
• Connectivity constraint. The sector can not
be fragmented. For example, the solution in
Figure 4 is not feasible.

Sector B
Sector A
Sector A

Figure 4: Connectivity Constraint
It is easy to see that, when the airspace is
sectorized, more the routes are cut, more the
coordination workload is induced. Hence, the
objective of the optimization of Airspace
Sectorization Problem (ASP) is to minimize the sum
of cut routes.
A genetic algorithm [4] to solve ASP has been
proposed in [1]. Chromosomes are defined as sets of
sectors’ center points; the sector is then defined as the
Voronoï diagram [5] associated to the set of center
points (i.e., a sector is a set of points that are closer to
its central point than to any other center points).
Voronoï-like sectors are geometrically convex but in
practice, sectors convex in the sense of routes (the
same route does not cross the same sector twice).
Hence a Voronoï-like sectorization might be sub-
optimal. Furthermore, sectors built by the Voronoï
diagram may lead to unfair load distribution.
Delahaye et al. [3] have tried to improve this
approach. A sector is defined by a set of connected
vertices of the network and the chromosome contains
all information needed to define the sectors. But
again, sectors built by synthesis of connected vertices
do not ensure the convexity constraint.
More recently, airspace has been divided in
small volume units and a sector is obtained by joining
some of these elementary units [6]. Unfortunately,
the most specific constraints can not be taken into
account and for instance, the sectors can be
fragmented in the solution.
Our initial investigation on this problem has
been published in [7], but without the connectivity
constraint. In this paper, we introduce a constraint-
programming formulation to solve ASP. Our goal is
to take into account all geometrical constraints, as
defined in the next section, when building sectors.
Our approach includes a heuristic for variables and
values ordering based on the notion of gain of
Kernighan/Lin heuristic [8] for Graph Partitioning
Problem. With this model, we can compute optimal
solutions for small size instances of ASP. For the
large size instances, we use a two-phase approach:
firstly, we apply a restricted Kernighan/Lin (RKL)
heuristic to find a “good” solution; and in the second
phase, we enter a re-optimization loop, relying on the
constraint programming model, that improves this
solution.
Modeling
In this section, we firstly propose a discrete
model for ASP. The airspace is modeled by a
valuated graph and a sector in a solution of ASP is
defined as a set of vertices, without geometrical
boundaries. And secondly, we propose a way to
compute the sectors’ boundaries from a solution of
ASP.
Airspace Sectorization and Graph
Partitioning
We rely on the following model: Airspace is
made of routes that cross each other. In the following,
G = (V, E) denotes the graph representing the
airspace, where:
• V (the set of vertices) is the set of beacons
and crossing points u,
• and E (the set of edges) is such that (u, v)
belongs to E if and only if there is a direct
route from u to v.
The graph G is valuated both on its vertices and
edges as follows (see Figure 5):
• cv : conflict workload, induced by the
conflicts that occur at v, is assigned to v
• me : monitoring workload belongs to an edge
e=(u,v). It is divided in two equal parts
mu=mv=me/2 that are assigned to u and v
• oe : coordination workload assigned to the
edge e. This workload is set to 0 if the
vertices of the edge are in the same sector.