selecting objective functions for multiobjective...

International Journal of Science, Engineering and Technology Research (IJSETR)

Volume 7, Issue 7, July 2018, ISSN: 2278 -7798

445 All Rights Reserved © 2018 IJSETR

Abstract— In most research work of wireless sensor network,

several objectives concerned with node deployment are issued

such as, coverage, network connectivity, network lifetime,

network latency, energy consumption, energy efficiency. For

redundant nature of random node deployment, realistic

optimization strategy is necessary to satisfy simultaneously

more than one objectives. In addition, appropriate objective

functions are necessary for decision variables of certain

solutions due to conflicting nature of each objective. In this

work, the two objective functions are considered for maximum

coverage and maximum energy efficiency in multiobjective

optimization problem. For selecting appropriate objective

functions, objective functions for global and local optimization

are tested by genetic algorithm in multiobjective optimization

problem. In simulation study, Pareto optimal solutions of

genetic algorithm for two objectives are analysized for right

choice of objective functions. The results illustrated in this work

are feasible in accordance with Pareto Front in objective space

for unconstrained multiobjective optimization.

Index Terms— Objective Function, Multiobjective

Optimization, Node Deployment, Wireless Sensor Network.

1) INTRODUCTION

For random deployment of nodes in wireless sensor network

(WSN), redundant deployment and active nodes scheduling

are adopted to achieve maximum coverage. Diverse

performance metrics of WSN are necessary to optimize in

random node deployment. For example, energy efficiency is

major concern in node replacement and network coverage is

important for quality of service (QoS). Because of conflicting

in performance metrics with each other, optimizing the

performance of WSN is vital in their real applications.

Among several performance metrics for WSN, more than one

of these metrics are chosen as optimization objectives and the

rest are constraints in optimization. To optimize two or more

objective functions at the same time, several algorithms have

been developed for multiobjective optimization [1] - [3].

Population based evolutionary algorithms (EAs) are

appropriate to solve multiobjective optimization problem.

Some components of EA are representation, fitness

assignment and population. To diverse optimization

performance metrics, genetic algorithms (GAs), one of major

EAs, have been used in multiobjective optimization. In

Manuscript received June, 2018.

Dr. Khin Kyu Kyu Win is with the Department of Electronic

Engineering of Yangon Technological University, Gyogone, Insein PO,

11011, Yangon, Myanmar.

Phyu Phyu Thant is with the Department of Electronic Engineering of

Yangon Technological University, Gyogone, Insein PO, 11011, Yangon,

Myanmar (corresponding author to provide phone: 09251167687.

solving problem, GA can deal with several objective

functions whether they are stationary or transient, linear or

nonlinear, and continuous or discontinuous. In most GAs,

representation of each solution is a sequence of populations. Each solution is assigned a fitness value that is given by an

objective function or fitness function. For solving diverse

multiobjective optimization problems, multiobjective GA is

implemented to be utilized Pareto optimal solution. The set of

this solution provides with a set of flexible trade-offs for

multiple objectives. Multiobjective GA can address for

optimal node deployment to satisfy multiple objectives in

WSN, such as maximum coverage, maximum energy

efficiency, minimum delay, etc [4]. For optimization problem

with multiple objectives in WSN, optimization algorithms

and objective functions are necessary to consider.

Optimization algorithms are more diverse than the types of objective functions. But, appropriate objective function is

more impact than the specific choice of optimization

algorithm.

In this paper, influences of different objective functions are

analysized by using multiobjective GA optimization

algorithm for node deployment in WSN. The rest of paper is

organized as follows. Section II includes the two specific

objective metrics and problem statement. Optimization

strategies are discussed in Section III. Analysis of objective

functions with simulated results are shown in Section IV.

Section V concludes the analysis on multiobjective

optimization for selection of appropriate objective functions.

2) PROBLEM STATEMENT AND OBJECTIVE MAT RICS

Different applications of WSN require specific QoS

requirements. Objective Metrics used for characterizing QoS are coverage area, delay, number of active nodes, bit-error

rate and network lifetime. In some case, lifetime of the

network reduces when more energy is consumed by the

nodes. The compromise between multiple metrics is needed

without conflicting each other.

1) Coverage The coverage in WSN corresponds to the sensing range of sensor node. The coverage quality of entire 2D region is most commonly considered in research work. In this area coverage, each point in the region is observed by at least one sensor node. Sensing disk model is normally defined for area coverage problem. All points within a disk model centered at the node are considered in node coverage [5]. Assumption is that the monitoring area A is divided into

m×n points in 2D and m×n monitoring target. Given sensor

nodes N is deployed in area A. For optimal coverage of whole

network, local neighborhood coverage of sensor node i with

coordinate (xi, yi) is computed as follow;

Selecting Objective Functions for Multiobjective

Optimization in Wireless Sensor Network

Khin Kyu Kyu Win, Phyu Phyu Thant




i

k

u

i

u

P

RN

m nN

(1)

ui represents special neighborhood of node i and Pk represents

combined coverage of target k. Nu is the number of

neighborhood. The optimization goal is maximization of

neighborhood coverage of Ri of node i.

2) Energy Efficiency

Energy efficiency is closely related to network lifetime. To

increase network lifetime, it is necessary to utilize sensor

nodes in an efficient manner. Given communication

bandwidth W, energy efficiency of node i is computed as

follows;

2log (1 )i

i

i

W

p

(2)

γi denotes the signal-to-interference-plus-noise ratio (SINR)

at the destination receiver relative to node i and pi is

transmission power of node i. The optimization goal is to

adjust sensing range of node i with minimization of SINR for

the sake of energy conservation.

3) OPTIMIZATION STRATEGIES

In this paper, multiobjective optimization problem is

determined the problem of finding a vector of two decision

variables which optimizes a vector function. The elements of

vector function represent the objective functions as

computable functions of decision variables. Given n

variables and m (m>1) objectives, multiobjective

optimization problem can be formulated as

1 2

min ( ) min[ ( ), ( ),......., ( )]m

f x f x f x f x (3)

with inequality constraints ( ) 0, 1,2,........,i ieg x i m or

equality constraints ( ) 0, 1,2,.........,i eqh x j m . It is noted

that nx R with nR being decision space and ( ) mf x R

with mR representing objective space. For the minimization of

m objectives 1 2( ), ( ),............, ( )mf x f x f x , it is important to a

achieve feasible solution. When this solution is not dominated by any other solutions in feasible space Ω, it is Pareto optimal solutions of objective functions [6].

Definition (Pareto Optimality): A point *x

is Pareto

Optimal if for every x

and 1,2,......., I k either,

*( ( ) ( ))i I i if x f x

Or, there is at least one i I such that *( ) ( )i if x f x

3) Multiobjective Genetic Algorithm

In general, genetic algorithms are based on genetic and

evolutionary theory and can be used for solving diverse

optimization problems. By operating on

generation-by-generation basis, a number of Pareto Optimal

solutions can be found throughout the evolution generations.

The Pseudo code for multiobjective genetic algorithm is as

follow [7].

Step 1: Initialize Population

Step 2: Evaluate Objective Values

Step 3: Assign Rank Based on Pareto Dominance

Step 4: Compute Niche Count

Step 5: Assign Linearly Scaled Fitness

Step 6: Assign Shared Fitness

Step 7: For i=1 to G

Selection via Stochastic Universal Sampling

Single Point Crossover

Mutation

Evaluate Objective Values Assign Rank Based on Pareto Dominance

Compute Niche Count

Assign Linearly Scaled Fitness

Assign Shared Fitness

End Loop

4) SIMULATED RESULTS AND DISCUSSIONS

In this analysis of multiobjective optimization problem,

solver is gamultiobj from optimization toolbox of Matlab. To

identify right objective functions for defined objective

metrics, two functions are used as a standard such as De

Jong’s function 2 and Rastrigin’s function. First function, De

Jong’s function 2 is a classic optimization problem and global minimum is inside a long, narrow, parabolic shape flat valley.

This function is with slow convergence when trying to

minimize. It has a unique minimum at the point (1, 1) when

the function value is zero. Visualization of this function is

shown in Fig. 1.

Figure 1. Plot of De Jong’s Function 2

The second function, Rastrigin’s function is highly

multimodal with cosine modulation to produce many local

minima. The location of minima is regularly distributed with

a global minimum at (0,0). This function can be visualized in

Fig. 2.

Figure 2. Plot of Rastrigin Function




Suppose the function of objective 1 is taken as De Jong’s

function 2, it is necessary to determine the objective function

for second objective to compromise two objectives. In this

work, second objective function is randomly selected and determined its appropriateness by using Pareto optimal

solutions. If the pattern of points (Pareto Front – PF) given

from Pareto optimal solutions of these two objective

functions in objective space are according to the illustrated

curve in Fig. 3, tradeoffs between objective 1 and objective 2

can be clearly seen.

Figure 3. Pareto Front of Unconstrained Multiobjective Optimization

Based on ideal solution in Fig. 3, a square indicates the

joint minimum of two objective functions. The remaining

solutions marked as solid circles are non-dominated and

dominated by at least one solution of the PF. If both objective

functions obtain ideal solution as an illustrated in PF curve,

standard function for objective 1 and selected function for

objective 2 is appropriate for decision making of

multiobjective optimization problem. Fig. 4 illustrated the PF of De Jong’s function 2 and selected objective function. The

selected objective function is defined as follow.

4 2 4 2 2

1 1 1 2 2 1 210f x x x x x x x mR

When simulation is used with multiobjective GA solver,

results in Fig. 5 illustrates the PF of Rastrigin’s function for

objective 1 tested with selected function for objective 2.

Comparing these two PFs in Fig. 4 and Fig. 5, the right choice

of De Jong’s function 2 and selected objective function can

satisfy simultaneously multiobjective optimization problem

using specific optimization algorithm GA. When testing the

simulation, the solver is gamultiobj, desired variables is two, population size is default value 50, and population fraction is

also default value 0.35. After selecting appropriate objective

functions, the desired PF is tested at different population size

from 50 to 200. Fig. 6, Fig. 7, and Fig. 8 show the results with

different population sizes. It shows that the PF cannot

guarantee minimization of two objectives at large dense in

population size.

The performance tests in Fig. 9, Fig. 10 and Fig. 11

illustrated with different population fraction of objective 1

and objective 2. Range of population fraction can be tested

from 0 to 1. From these results, no additional parameters are

required for Pareto converging using Genetic algorithm such

as different sizes of population. Fig. 12 and Fig. 13

demonstrated characteristics of desired PF when using

multiobjective optimization with genetic algorithm. These

figures show the ranking of genes and a set of solutions span

the entire Pareto optimal solution.

Figure 4. PF of Rastrigin’s Function and Defined Function

Figure 5. PF of De Jong’s Function 2 and Defined Function

Figure 6. Desired PF with Population Size-100


Objective 1

Objective 2

dominated

Non-dominated





Figure 9. Desired PF with Population Fraction-0.5



Figure 12. Desired PF and Rank based on Pareto Dominance

Figure 13. Pareto Distance and Spreading of Desired PF

5) CONCLUSION

Based on reasons of maximum coverage and maximum

energy efficiency in node deployment of WSN, multiojective optimization problem was considered in this paper.

Maximizing coverage and minimizing overlap between each

sensor node results large number of relay nodes for

communication. Therefore, depletion of energy at these

sensor nodes will be sooner and network lifetime will be

shorter. Such conflicting of multiple objectives can be solved

by using specific optimization algorithms or right choice of

objective functions. Thus in this work, selecting appropriate objective functions are analysized by using genetic algorithm

for multiobjective optimization problem. Analysis on

optimization of two objectives in 2D network and illustrated

results are confirmed by using Pareto optimal solutions and

Pareto Front. Future research will be on implementing

optimal node deployment with selected objective functions

and specific algorithm in heterogeneous network.

REFERENCES

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sensor networks based on particle swarm optimization with coherent

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6) E. Zitzler. K. Deb, and L. Thiele, “Comparison of multiobjective

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students and professionals,” Cambridge University Press, New York,

USA, 2015.

Khin Kyu Kyu Win received her BE(Electronics) degree from Yangon

Technological University, Yangon, Myanmar. She did postgraduate

study for M.Eng (Research) at school of Electrical and Electronic

Engineering, Nanyang Technological University, Singapore. Her PhD

degree was received from Electronic Engineering Department, Yangon

Technological University. Her research work is in the area of wireless

communication engineering. She is currently a lecturer at Yangon

Technological University and also a member of Myanmar Engineering

Association.

Phyu Phyu Thant received her BE(Electronics) degree from

Technological University (Thanlyin), Yangon, Myanmar. She is

currently doing postgraduate research for her master degree at

Electronic Engineering Department, Yangon Technological University.

Her research work is concerned wireless sensor network. She is also an

instructor at Technological University (Thanlyin).

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