An Adaptive Tabu Search Algorithm for the Multi-Objective Node

Placement Problem In Heterogeneous Networks

Ons Abdelkhalek

, Saoussen Krichen

and Adel Guitouni

LARODEC Laboratory, Institut Sup

erieur de Gestion, University of Tunis, 2000 Le Bardo, Tunisia

Faculty of Law, Economics and Management, University of Jendouba, Jendouba, Tunisia

Gustavson School of Business, University of Victoria, Victoria B.C, Canada

Keywords:

Heterogeneous Network Management, Tabu Search Algorithm, Multi-Objective Optimization Problem,

Genetic Algorithm.

Abstract:

The Multi–objective Node Placement (MONP) problem focuses on extending an existing communication in-

frastructure with new wireless heterogeneous network components while achieving cost effectiveness and ease

of management. This extention aims to broaden the coverage and handle demand ﬂuctuations. In this paper,

the MONP problem is modeled as a multi–objective optimization problem with three objectives: maximizing

the communication coverage, minimizing active nodes and communication devices costs and maximizing of

the total capacity bandwidth in the network. As the MONP problem is N P –Hard, we present a meta–heuristic

based on the Tabu Search approach speciﬁcally designed for multi–objective problems in wireless networks.

An empirical validation of the model is deﬁned based on a selection of a real and large set of instances and sup-

ported by a performance comparison between the suggested algorithm and a multi–objective genetic algorithm

(MOGA). All tests are performed on a real simulation environment for the maritime surveillance application.

We show empirically that the proposed approach is more relevant to solve the MONP problem regarding each

objective in term of cardinality-based performance index.

1 INTRODUCTION

A heterogeneous network involves multiple resources

(e.g., relays, antennas, etc.) and contains different

types of sub-networks with different communication

devices (CDs) (e.g., cellular, radio, wiﬁ or Ad Hoc

networks) with varying functions and power level.

This integration takes advantage of various network-

ing techniques, such as the coverage of the cellular

systems, bandwidth of the wired networks and the

ﬂexibility of the mobile ad hoc networks. Several

studies addressed the integration of various architec-

tures in heterogeneous platforms, as LAN and wire-

less LAN (WLAN) (Bahri et al., 2005) (Niyato et

al., 2009), Wi-Fi and Wi-Max (Ting et al., 2009),

and the integration of Ad hoc and cellular networks

in (Hongyi et al., 2011). Nevertheless, published re-

search on optimization algorithms for heterogeneous

network extension seems much more limited. Most

studies reported in the literature focused only on the

extension of one existing homogeneous network in-

frastructure.

In our proposed model, we try to report with ﬁ-

delity all the aspects of the heterogeneity in merging

multiple network technologies by its degree of relia-

bility and how closely it captures the features of the

signal quality constraints. In this paper, we address

the multi-objective node placement (MONP) (Ab-

delkhalek et al., 2011, 2013) problem. The MONP

problem considers the following settings: a set of can-

didate sites representing the potential placement of

nodes, the trafﬁc distribution estimated by using em-

pirical prediction models and the signal quality prop-

agation model. Other aspects are also taken into ac-

count, such as an existing heterogeneous network in-

frastructure, hardware cost, signal quality and service

coverage.

The purpose is to ﬁnd the convenient way to build

and connect the network. In fact the optimal assign-

ment includes the efﬁcient number, position, CDs’ na-

ture and connections between active nodes in a special

area of coverage while taking into consideration mul-

tiple environmental constraints. The problem deals

with two aspects of the network management. First

we aim to maximize the coverage area within a het-

erogeneous network. To achieve this goal we attempt

249

Abdelkhalek O., Krichen S. and Guitouni A..

An Adaptive Tabu Search Algorithm for the Multi-Objective Node Placement Problem In Heterogeneous Networks.

DOI: 10.5220/0004834102490256

In Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems (ICORES-2014), pages 249-256

ISBN: 978-989-758-017-8

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

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

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

,...,n

}

Set D Set of communication devices {d

,...,d

}

M Set of predeﬁned candidate sites of interest

,...,l

}

R Set of test points (or receivers) {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.

= |{r

}|, if ∀ f , ∃i ∈ N, ∃k ∈ M and ∃d ∈ D

where S

f ,i,k

≥ θ

φ, otherwise.

D×D

Input matrix,T

= 1 if CD d and d

can

communicate

The maximum number of transmitters

assigned to a node with CD d

Cost of a node n

f ,i,k

Signal strength between a node n

with CD

d in CS l

and TP r

Power of the communication device d.

, G

Antenna gains of TP r

and node n

λ The carrier wavelength

f ,k

Euclidean distance from r

to CS l

(σ

,θ

) data rate demand (in Erlang) and signal

threshold of TP r

for the CD d

(α

,β

) Coordinates of the TP r

(α

,β

) Coordinates of the CS l

Initial existing networks infrastructure with

CD d

Maximum capacity for Z

(bandwidth)

Decision x

= 1 if a node n

with a CD d is assigned

Variables to CS l

i j

= 1 if a node n

is assigned to an other

node n

with a CD d

i f

= 1 if TP r

is assigned to a node

with CD d

2.2 Outline of the Problem Formulation

The MONP problem (Abdelkhalek, 2011) is formu-

lated as follows:

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Max Z

(X) =

∑

d=1

∑

i=1

∑

k=1

(1)

Min Z

(X) =

∑

d=1

∑

i=1

+ c

)

∑

k=1

(2)

Max Z

(X) = Min

{d,i6= j}

i j

(3)

s.t.

i j

= x

∀i 6= j, ∀k 6= k

with t

= 1 (4)

and d

k,k

≤ Max(w

)

∑

f =1

∑

k=1

i f

≤ s

∀i,d (5)

∑

i=1

∑

j=1

∑

d=1

i j

≤ T

∀i 6= j,∀Z

(6)

∑

i∈N/{ j}

i j

≤NL

∀ j ∈ {1,...,N} (7)

∑

d=1

≥1 ∀i ∈ N (8)

∑

k=1

= 1 ∀i ∈ {1,... , N},∃d ∈ D (9)

∑

i=1

≤ 1 ∀k ∈ {1, ...,M}, ∃d ∈ D (10)

∑

i=1

∑

k=1

i f

≤ 1 ∀d ∈ D (11)

∑

i=1

i j

≥ 1 ∀ j,t and j 6= i (12)

∑

i=1

≥ 1 ∀d (13)

i f

i j

∈ {0,1} ∀i,k, f , j,d (14)

The MONP problem addresses three main objectives

for the problem:

• Maximizing the communication coverage by max-

imizing the number of covered TPs in the area of

interest (see equation (1)). Thus, it ensures that

the received signal power from a given location

has met the received sensitivity of the TP.

• Minimizing active nodes and communication de-

vices costs through equation (2), by reducing the

number of deployed nodes and CDs.

• Maximizing of the total capacity bandwidth in the

network by maximizing the amount of trafﬁc held

by the network in equation (3). This leads to max-

imize the total minimum capacity bandwidth de-

ployed in all the network in order to ensure the

QoS.

These objectives are subject to two main sets of con-

straints:

• Communication Node Constraints: It represents

all constraints related to the trafﬁc carried in the

network, receivers sensitivity threshold of the mo-

bile and the mobility of users in the area of in-

terest. Equation (4) ensures that two nodes n

and n

can be connected only if their CDs d and

can communicate and their Euclidean distance

k,k

from CS l

to CS l

is less or equal to the

maximum power range between w

and w

re-

lated to CDs d and d

. In equation (5), the sum of

the total data rate demand of the covered TPs by

a node n

with CD d cannot exceed the capacity

of the CD. We assume in equation (6) that the

total trafﬁc capacity of all nodes connected to an

existing network Z

, should not exceed the maxi-

mum capacity of this network T

. Links capacity

constraints are presented in equation (7) where we

assume that all nodes connected to node n

should

not exceed the maximum capacity of links NL

al-

lowed for n

and related to his CD.

• Connection Node Constraints: It ensures the con-

nection between different entities in the network

based on their CD’s range, CS’ distance, link con-

straints,..,etc. Equations (8), (9) and (10) ensures

simultaneously that we can assign more than one

CD to a node n

, each nodes n

is assigned to one

CS l

and, ﬁnally, that each CS l

is assigned to

at most one node n

. We assume in equation (11),

(12) and (13) that each TP r

can be assigned to

at most one node n

with CD d, that each node

should be connected at least to another node

and ,ﬁnally, that at least one node n

should be

connected to the existing networks Z

in order to

ensure that all sub-networks are inter-connected.

X represents the efﬁcient solution with a combi-

nation of our decision variables.

Due to the N P -hardness of the MONP problem,

no exact solution can be ﬁnd to generate the Pareto

optimal front. We propose to apply an adapted MOTS

algorithm in order generate a promising sample of po-

tentially efﬁcient solutions.

3 THE TABU SEARCH

ENCODING ALGORITHM FOR

THE MONP PROBLEM

We propose an adaptation of the basic TS (Hansen,

2000) algorithm especially designed to the MONP

problem in order to generate the non-dominated solu-

tion set. TS starts from an initial feasible solution and

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moves repeatedly from a solution to a neighbor one.

Let s ∈ S be a current solution and N(s) the neighbor-

hood of s.

A solution s is encoded as a discrete vector of size

N. Each vector’s position represents a combination

between the index k = 1,...,M of the CSs and the in-

dex d = 1,...,D of the CD to which a node is assigned.

Figure 1 illustrates an example of a solution encoding

with 19 deployed nodes and 2 different CDs. Through

this Figure, we can see that every node has been as-

signed to one CS and has at least one CD. Every CS

has a unique index that indicates its position in the

map. Every node is assigned to a different CS.

3.1 Initial Solution

The initial solution is built based in a greedy based al-

gorithm. Nodes’ placement are randomly picked from

the set of CSs {1, 2, ...,M}, then we assign a CS’s in-

dex. CD’s types are piked from the set of available

CDs D based on the demand distribution. After as-

signing a CS, we search for the maximum number of

neighbors TP for each communication type and as-

sign a CD accordingly. The number of other nodes

to which they are connected is then constructed based

on constraints (4). If the initial solution is not fea-

sible, the set of constraints described above are used

through an adjustment process in order to drive the

search toward feasible solutions. In this process, a

modiﬁcation of the node assignment is made in order

to fulﬁll each constraint sequentially.

3.2 Neighborhood Exploration and

Evaluation

Each move consists in modifying one variable of the

vector X in an iteration it. The following moves are

considered for our model:

• remove a node n

from a selected CS l

• move a node n

which is already installed at a

given location l

to a vacant location l

• assign a node n

the nearest CS l

• assign a new CD d to a node n

for a certain CS l

• assign a node n

to an existing initial network Z

We only consider moves that preserve the connection

and communication constraints. Each time a move is

applied, the neighbor is evaluated based on the three

objective functions detailed above. A randomly non-

dominated solution from the neighborhood replace

the current solution even if it is not improving in order

to escape from the local optima. The whole process

is stopped if a given number of iterations it

max

have

been accomplished or when no improvement is per-

formed on the set of efﬁcient solutions obtained after

a certain number of iterations.

3.3 Tabu List

In order to escape the local optima and prevent re-

visiting previously explored solutions, a tabu list L is

used. This list keeps track of the last |L| modiﬁca-

tions during the search process. Over multiple runs,

TS is able to ﬁnd non–dominated solutions that repre-

sent good placement of a set of nodes from the list of

CS. At each step of the algorithm, a list that contains

non–dominated solutions P

is generated when the

MONP optimization problem is solved. The tabu list

L may prohibit attractive neighbors that have not yet

been generated. Hence, it is necessary to use a pertur-

bation criterion to accept forbidden neighbors. Our

perturbation criterion consists in choosing randomly

a tabu neighbor from the P

list.

The MOTS is compared to the MOGA, previ-

ously applied to the MONP problem for the maritime

surveillance application and detailed in (Abdelkhalek

et al., 2011). The main steps of the algorithm are sum-

marized as follows:

Begin

L := {}, P

0, it := 0,

Generate randomly a feasible starting solution S,

Repeat

Determining the best move s to its neighbor s

∈ N(s)

Determine the number of iterations for which the node is tabu

If Fitness(X

) Dominates Fitness(X) Then

← P

∪ {s

}

update Tabu List L

End If

it++

Until Stopping criterion

End

4 EXPERIMENTATIONS AND

NUMERICAL RESULTS

Results are run using a testbed simulator for real data

instances called Inform Lab (IL) (Abdelkhalek et al.,

2013). Each instance is solved with 30 independent

runs. A maximum number of iterations it

max

is set

to 500 iterations for each run. The algorithm stops

when no improvement is performed on the objective

functions after 100 iterations or when the maximum

number of iteration is reached. The number of iter-

ations that a node can be tabu is set between [5,10].

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Figure 1: An example of the TS encoding scheme for a heterogeneous network extension.

These parameters were selected after multiple exper-

imentations. The existing heterogeneous network in-

frastructure includes 8 nodes and 3 CDs in Vancouver

Island area.

The free space signal propagation model proposed

in (Ting et al., 2009) is used in the experimental de-

sign. Moreover, different region sizes, locations and

densities of TPs are considered. For each size of the

area, three instances of the problem were generated

with different region size. We deploy simultaneously

10, 20 and 30 nodes. For each instance nine different

TPs distribution are applied as presented in Tables 3,

2 and 4. Furthermore, ten different CD’s settings are

used in order to ensure a heterogeneity in the network

connections (see Table 1). The frequency and ranges

are used to compute the signal strength and the band-

width to optimize Z

. The number of CSs is always

greater than the number of nodes to place. Three dif-

ferent settings are used: for 10 active nodes we plan

17 CSs, 100 CSs for 20 active nodes and 183 CSs for

30 active nodes. A total of 54 different problem in-

stances were generated for the tests.

For each instance, we report the following mea-

sures: The average CPU time, number of non dom-

inated solutions (|P

|) and values of the three ob-

jective functions for both MOTS and MOGA where

is the number of covered TPs, Z

the total cost of

deployed CD and nodes, and ﬁnally Z

represents the

total minimum capacity bandwidth deployed in all the

Table 1: Communication Devices’ Settings.

Link # Bandwidth Frequency Range

(kbit/s) (Hz) (km)

1 400 1E09 300

2 4000 8.23E09 3600

3 1000 6E09 2000

4 600 1.75E09 500

5 800 2E09 600

6 3000 3E09 500

7 500 1.5E09 2000

8 4000 2E09 3000

9 1000 8E09 800

10 2000 7E09 1000

Figure 2: CPU in term of the number of TPs.

network.

Because of the N P -hardness of the MONP prob-

lem, potentially Pareto-optimal solutions are gener-

ated. In order to compare the performance of our two

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Table 2: Computational performance of MONP for 10 nodes and 17 CSs.

Pbs. TPs CDs MOGA MOTS

| CPU(s) Z

1 100 5 4 3 100 271907 400 4 5 100 271900 400

2 10 4 4 100 271230 1000 3 7 100 271222 1000

3 127 5 4 4 127 270271 400 5 7 127 270260 500

4 10 4 5 127 282275 2000 5 10 127 282265 2000

5 156 5 3 4 156 269173 400 4 7.5 156 269125 500

6 10 5 5 156 284970 500 6 11 156 285190 500

7 178 5 4 3 178 271026 800 5 7 178 271136 1000

8 10 7 6 178 278942 600 6 9 178 278930 600

9 225 5 4 5 223 275473 400 3 12 225 275402 500

10 10 5 7 225 280714 600 6 23 225 280504 800

11 278 5 3 5 272 275594 400 4 13 276 275590 400

12 10 2 9 277 283442 500 4 29 278 283412 600

13 400 5 5 7 400 270507 400 4 19 392 270517 500

14 10 5 12 400 281589 800 6 30 400 281459 1000

15 506 5 6 8 490 270778 500 4 20 498 270768 600

16 10 5 20 500 282749 4000 5 55 506 282650 3000

17 625 5 7 15 580 271530 400 9 35 622 271538 400

18 10 7 17 591 283484 2000 9 48 625 283564 1000

Table 3: Computational performance of MONP for 20 nodes and 100 CS.

Pbs. TPs CDs MOGA MOTS

| CPU(s) Z

1 100 5 3 6 100 471677 400 3 10 100 471665 400

2 10 3 8 100 484913 3000 2 15 100 484895 1000

3 127 5 3 7 127 468713 400 4 15 127 468688 400

4 10 5 9 127 482211 600 5 20 127 482013 800

5 156 5 4 9 156 470028 400 5 22 156 470128 400

6 10 4 12 156 487340 1000 5 30 156 487250 1000

7 178 5 4 9 178 477123 400 3 19 178 477083 400

8 10 5 13 178 492783 800 4 33 178 492780 500

9 225 5 3 11 218 468145 400 4 30 222 468045 400

10 10 2 17 220 482626 400 3 37 225 482603 1000

11 278 5 4 14 270 470694 400 3 38 277 470630 400

12 10 4 17 275 476757 400 3 40 278 476650 500

13 400 5 6 17 398 467905 600 7 42 400 467899 500

14 10 4 24 400 490869 500 6 78 400 490779 600

15 506 5 5 21 480 473621 400 5 69 500 473660 400

16 10 5 42 495 479550 500 7 188 503 479525 400

17 625 5 7 30 605 463591 800 8 72 608 463513 800

18 10 10 38 608 488099 500 10 96 612 486005 400

approaches, we use a simple cardinality-based index

namely Coverage of two sets index (C) (Zitzler and

Thiele, 1999). It’s used to compare the relative domi-

nance (i.e. coverage) between the two non dominated

solution sets, deﬁned as:

C(S

) = |{s

∈ S

;∃s

∈ S

 s

}|/|S

where S

and S

represent the set of non-

dominated solutions generated simultaneously by the

MOTS and MOGA. Table 5 reports the performance

of the MOTS and MOGA according to the C index.

In fact, two main results are pointed out: the domi-

nance regarding each objective function Z

(coverage

/ Objective) and the relative dominance that reﬂects

the performance of each method regarding to all ob-

jectives simultaneously. To this end, we count the set

of non-dominated solutions generated by both algo-

rithms. Then we compute, for each instance, the av-

erage dominance between the two sets (C(S

) and

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Table 4: Computational performance of MONP for 30 nodes and 183 CS.

Pbs. TPs CDs MOGA MOTS

| CPU(s) Z

1 100 5 3 9 100 807153 400 3 19 100 807140 400

2 10 2 13 100 811918 2000 3 25 100 811820 1000

3 127 5 3 10 127 806046 400 3 28 127 806015 500

4 10 4 14.5 127 842980 1000 4 40 127 842953 800

5 156 5 3 11 156 788932 400 4 39 156 788920 400

6 10 3 17 156 806433 3000 4 47 156 806360 2000

7 178 5 3 13.5 178 798788 400 4 36 178 798741 400

8 10 3 18 178 821985 2000 4 48 178 821935 2000

9 225 5 3 13 218 784453 400 3 30 223 784257 500

10 10 4 24 222 834403 600 5 65 225 834323 600

11 278 5 4 25 275 795268 400 5 62 276 795250 400

12 10 6 31 278 833327 800 5 86 278 833225 500

13 400 5 6 26 385 801875 400 5 96 395 801690 400

14 10 6 45 400 831033 1000 5 137 400 831009 1000

15 506 5 6 30 491 778785 800 7 75 500 778750 600

16 10 5 47 500 840161 500 6 230 502 840053 800

17 625 5 12 44 612 803887 600 11 200 620 803967 600

18 10 10 64 615 822732 400 11 258 625 822546 400

C(S

)) for all objectives. The results obtained by

the two approaches are compared. Thus, we can state

the following remarks:

• Starting from 225 TPs, we can notice that in

96% MOTS generates better solutions (i.e. 24 in-

stances out of 25). Moreover, in more than 50%

of the problem instances (i.e. 29 instances out of

54), both MOTS and MOGA reach the total net-

work coverage in terms of TPs covered.

• The cost is proportional to the size of the problem

instance. We can notice that the more TPs we de-

ploy in the area, the more expensive the cost of

our placement. This is due to the heterogeneity of

the network and the CDs’ cost. As can be gath-

ered from Table 5, the proposed MOTS is able to

ﬁnd solutions that are mostly better than MOGA.

In fact, Z

for MOTS are on average 33% better

than those proposed by MOGA.

• Regarding the maximization of the minimum

bandwidth, in 46.29% (i.e. 25 instances out of 54)

we get the perfect equality with the two methods.

However on the 29 remaining instances, MOGA

got better solutions on 18 instances (i.e. 62%).

• As we can see in Figure 3, MOGA requires less

CPU time. This gap becomes more important as

the problem size increase since MOTS still has

to go through several iterations due to its track-

ing process using tabu lists. This behavior is due

to the numerous iteration that MOTS has to go

through.

• For large instances, MOTS has better results then

the MOGA algorithm in terms of network cover-

age. Also, we can notice that 100% of TPs cov-

erage were achieved in 53% of the problem in-

stances.

• If we rate the total dominance of the two algo-

rithms for the set of non dominated solutions P

(Table 5), we clearly see that MOTS dominates

MOGA in 37.96% comparing to 34.25% for the

MOGA among the whole set of 54 problem in-

stances.

• We can clearly see from Tables 3, 2 and 4, that the

number of non dominated solutions P

is related

to the number of active nodes deployed in the net-

work. The more nodes we activate, the bigger is

the number of non dominated solution.

Table 5: Comparison of the coverage based index obtained

using MOTS and MOGA.

Solution Coverage / Objective Average

approaches Z

Coverage

MOTS 96% 66% 38% 37.96%

MOGA 4% 33% 62% 34.25%

Based in the numerical results reported in tables 3,

2, 4 and 5, we can note that the MOGA failed to meet

100% of TPs coverage for large scaled problems. This

is due to the repairing process used in MOGA to han-

dle a large set of constraints in our problem formu-

lation. However, we can clearly see that MOTS out-

performs MOGA to minimize the network’s cost and

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255

maximize the total network coverage. This can be ex-

plained by the capacity of MOTS to explore a large set

of solutions where MOGA, due to its crossover and

mutation process, can reduce considerably the feasi-

ble region.

5 CONCLUSIONS

Throughout this paper, we studied the MONP prob-

lem. This problem consists of deciding to the loca-

tion, the number and the interconnection of nodes in

order to extend an existing network. It also aims to

decide of the optimal type and number of deployed

CDs to ensure the heterogeneity of the infrastructure.

As the problem is multi–objective and heterogeneous,

no benchmarks exist. We thus generate three groups

of problem instances to test our approach based on

the number of active nodes deployed in the network

and the number of associated candidate sites. We

presented two resolution approaches that iteratively

solve the MONP problem. Our computational exper-

iments show that the tabu–based heuristic produced

solutions that were in 37.96% better than those pro-

duced with the genetic algorithm. These are consid-

ered as promising results if we take into considera-

tion the difﬁculty and complexity of the problem that

we have studied. However, the MOGA still get bet-

ter results on the execution time. Other research lines

should be carried out in future work to assess the per-

formance of our method.

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