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to maximize the number of users receiving signal.
However, this will be constrained by the minimiza-
tion of the deployed entities’ cost. Second, we lead
to improve the quality of signal and maximize infor-
mation flows in the network as we minimize cost of
deployed devices in the network.
It’s obvious that the problem is highly combinato-
rial with an enormous number of possible combina-
tions and conflicting objectives. The problem is mod-
eled as a multi-objective optimization problem sub-
ject to system constraints. If we reduce the proposed
problem to the Antennas Placement Problem (APP)
or Transmitter Placement Problem (TPP) (Lee et al.,
2000) (Ting et al., 2009), we will clearly conclude
that it is an N P–Hard problem. Therefore, heuristic
approaches can be considered to solve the problem.
We developed a multi-objective tabu search (MOTS)
approach (Hansen, 2000) given its ability to tackle the
high complexity of similar problems and to generate
a promising approximation of the efficient set.
As the MONP is newly modeled multi-objective
and heterogeneous, no benchmarks exist. To test
our approach, we generated 54 different real prob-
lem instances with varying region sizes, locations,
density of test points (TPs) and number of active
nodes. We compare the MOTS algorithm to the
Multi-objective Genetic Algorithm (MOGA) (Ab-
delkhalek et al., 2011). The empirical application is
validated in a maritime surveillance application with
a simulation environment called Inform Lab (IL) (Ab-
delkhalek et al., 2013) using real data instances. An-
tennas are represented by nodes in maritime platforms
(i.e. helicopters, ships, boats,..) and CDs represent all
the equipment capable to ensure the communication
between different technologies (i.e. radio, cellular,
WLAN,..).
The remaining of this paper is organized as fol-
lows. In Section 2, we provide a brief description of
the problem modeling. Section 3 presents the adapted
MOTS algorithm to solve the MONP problem. The
performance of the proposed algorithm is presented in
Section 4 and compared with the MOGA on a bench
of realistic problem set.
2 THE MULTI–OBJECTIVE
NODE PLACEMENT PROBLEM
IN A HETEROGENEOUS
NETWORK
The MONP problem (Abdelkhalek et al., 2011, 2013)
consists to find the appropriate placements for a set of
nodes in an existing heterogeneous network Z
d
using
a set of pre-defined candidate sites (CSs) as potential
locations. For each selected CS, find the appropriate
node and CDs, as well as the suitable ad hoc connec-
tion strategies between the new deployed node and the
existing infrastructure. All these choices must satisfy
a set of conflicting objectives and constraints.
2.1 Notation
The following table explains the notation related to
the mathematical formulation.
Indexing N Set of nodes {n
1
,...,n
N
}
Set D Set of communication devices {d
1
,...,d
D
}
M Set of predefined candidate sites of interest
{l
1
,...,l
M
}
R Set of test points (or receivers) {r
1
,...,r
R
}
Parameters τ = (p,s, Each CD has a set of characteristics related
c,t,w, b) to the infrastructure and to the sub node,
where: p represents the power, s denotes the
capacity between nodes and TPs, c is the
cost between CDs (includes the technologies’
cost deployed to connect the two devices), t
denotes the CD’s type (see Table 1), w
denotes the transmission range related to
a CD, and b denotes the bandwidth between
two different nodes when connecting
the infrastructure.
a
kd
= |{r
f
}|, if ∀ f , ∃i ∈ N, ∃k ∈ M and ∃d ∈ D
where S
d
f ,i,k
≥ θ
d
f
φ, otherwise.
T
D×D
Input matrix,T
dd
0
= 1 if CD d and d
0
can
communicate
NL
d
The maximum number of transmitters
assigned to a node with CD d
C
i
Cost of a node n
i
S
d
f ,i,k
Signal strength between a node n
i
with CD
d in CS l
k
and TP r
f
p
d
Power of the communication device d.
G
f
, G
i
Antenna gains of TP r
f
and node n
i
λ The carrier wavelength
d
f ,k
Euclidean distance from r
f
to CS l
k
(σ
d
f
,θ
d
f
) data rate demand (in Erlang) and signal
threshold of TP r
f
for the CD d
(α
f
,β
f
) Coordinates of the TP r
f
(α
k
,β
k
) Coordinates of the CS l
k
Z
d
Initial existing networks infrastructure with
CD d
T
Z
d
Maximum capacity for Z
d
(bandwidth)
Decision x
d
ik
= 1 if a node n
i
with a CD d is assigned
Variables to CS l
k
y
d
i j
= 1 if a node n
i
is assigned to an other
node n
j
with a CD d
w
d
i f
= 1 if TP r
f
is assigned to a node
n
i
with CD d
2.2 Outline of the Problem Formulation
The MONP problem (Abdelkhalek, 2011) is formu-
lated as follows:
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