BULETINUL INSTITUTULUI POLITEHNIC DIN IAI

Publicat de

Universitatea Tehnică „Gheorghe Asachi” din Ia%i

Tomul LVI (LX), Fasc. 1, 2010

SecŃia

AUTOMATICĂ %i CALCULATOARE

COMPARATIVE STUDY OF MULTIOBJECTIVE

GENETIC ALGORITHMS

BY

MIHAELA SIMONA CÎRCIU and FLORIN LEON

Abstract. The objective of this paper is to study three algorithms for solving

multiobjective optimization problems. The first algorithm is a weight<based genetic

algorithm, which consist in the application of a scaling function to the weighted sum of

objectives. This is the simplest extension of a simple genetic algorithm. The determination

of weights that the user must choose and the impossibility to discover all the solutions for

problems with non<convex Pareto<optimal front are the main disadvantages of this

algorithm. Vector evaluated genetic algorithm (VEGA) was the first genetic algorithm

proposed for multiobjective optimization. This is a non<Pareto approach based on the

selection of some relevant groups of individuals, each group being assigned an objective.

The method tends to “gather” results around the extremes of the solution space producing

a sub<optimal Pareto<optimal front convergence. The third algorithm NSGA<II is based on

Pareto dominance to exploit the search space towards the optimal front. It is an efficient

algorithm used in many studies. The algorithms are implemented in an application that

allows solving any multiobjective problem with one of the three algorithms presented

above and therefore can be used as a framework for a comparative study regarding the

performance of different optimization methods.

Key words: multiobjective optimization, genetic algorithms, VEGA, NSGA<II.

2000 Mathematics Subject Classification: 80M50, 58E17.

1. Introduction

“Evolutionary algorithms” is a generic term used to denote any

stochastic search algorithm that uses mechanisms inspired by the biological

evolution and genetic operators such as mutation, crossover, selection and the

fittest survival.

The solutions proposed for optimization problem are playing the role of

individuals in a population. The solutions with high fitness will have higher and

Mihaela Simona Cîrciu and Florin Leon

36

probability to survive and will reproduce superior solutions to the evolutionary

process.

There are three main types of Evolutionary Algorithms, among others:

Genetic Algorithms, Evolutionary Strategies, and Evolutionary Programming.

All three were developed independently, but the general framework is the same.

They evolve a population of candidate solutions to the given problem to

find optimal solutions, using operators inspired by natural genetic variation

(creating new individuals through the crossover and mutation), natural selection and

fittest survival (competition for reproduction and resources among individuals).

However, each type has several variants due to different origins and

their targets toward specific domains. Genetic algorithms use crossover and

mutation and are suitable for optimization problems. Evolutionary strategies use

real parameters. At first evolutionary strategies did not use crossover operator, it

have been introduced later. Therefore, their framework is similar to that of

genetic algorithms with real parameters.

Multiobjective problems are usually more difficult, but they find many

applications in real<life situations.

We organize our paper as follows. In section 2, we present an overview

of the genetic algorithms. In section 3 we describe the three algorithms used for

comparison: the weighted genetic algorithm, VEGA and NSGA<II. Finally, in

section 4, we illustrate some case studies with three especially devised

multiobjective optimization problems, used to compare the performance of the

algorithms under investigation.

2. Genetic Algorithms

Many real world problems involve simultaneously optimization of

multiple objectives, often contradictory. In a multiobjective optimization

problem, it is not possible to find the best solution with respect to all objectives.

A solution may be optimal regarding one objective, but inferior regarding

another objective. The goal is to find a set of optimal solutions, known as

Pareto<optimal set. These solutions are optimal in sense that no other solutions

in the search space are better considering all the objectives. Since none of the

solutions in the set is better than any other solution, with respect to all

objectives, any one of them is an acceptable solution.

A multiobjective genetic algorithm can be a modified version of a

simple genetic algorithm, designed to solve multiobjective optimization

problems. The ability to use evolutionary algorithms to solve multiobjective

optimization problems was suggested by Rosenberg in his dissertation [1]. He

suggested using a genetic algorithm for finding the chemistry of a population of

single<celled organisms with multiple properties or objectives. Since a genetic

algorithm works with a population of candidate solutions, the dominance

criteria can be used to adjust the search toward the Pareto<optimal front and

Bul. Inst. Polit. Ia%i, t. LVI (LX), f. 1, 2010

37

multiple Pareto<optimal solutions can be obtained in a single run. This is the

main reason that makes genetic algorithms suitable for multi objective

optimization [2].

Genetic algorithms operate with a collection of chromosomes, called the

population. The population is randomly initialized. As the search evolves, the

population includes better and better solutions and then it converges, meaning

that it is dominated by a single solution [3]. However, a simple can only provide

one solution for a given combination of weights, unlike a dedicated MOGA,

which can provide a set of solutions simultaneously.

2.1. Genetic Algorithm Structure

It is considered that the length and number of chromosome used are

constant and the population P(t+1) at time t+1 is obtained by taking all descendants

of the population P(t) and then delete chromosomes of previous generation [4].

Genetic algorithm structure is the following:

Step 1: t ← 0;

Step 2: Initialize the population P(t) with random values;

Step 3: Evaluate chromosomes from population P(t). For this purpose is

used a performance function problem dependent;

Step 4: While the termination condition is not satisfied run the

following steps:

Step 4.1: Select chromosomes from P(t) which will promote to the new

generation. Let P

1

be the set of selected chromosome (the intermediate population).

Step 4.2: Apply genetic operators to chromosomes from the recombination

pool of P(t) (crossover and mutation). Let P

2

be the population obtained, i.e. the

offspring of population P(t). Delete from P

1

offspring parents. Chromosome

remained in P

1

are included in P

2

. Build a new generation: P(t+1) ← P

2

; delete

all chromosome from P(t);t ← t+1; Evaluate P(t). A commonly used

termination condition is reaching a specified number of generations, although

there are other termination conditions that test the convergence of the GA by

evaluating the fitness of the population. The result of the algorithm is given by

the better individual from the last generation. In reality, nothing guarantees that the

best individual will pass its genes into the next generation. Therefore, it may be

beneficial to retain the best individual in every generation, a process called elitism.

2.2. Multiobjective Genetic Algorithms

In contrast to single optimization, in multiobjective optimization both

fitness and selection must support several objectives. Therefore, multiobjective

algorithms differ from simple genetic algorithm in the way the fitness

assignment and selection works. Several different variants of multiobjective

Mihaela Simona Cîrciu and Florin Leon

38

algorithms have been introduced with different fitness assignment and selection

strategies. Based on fitness assignment and selection strategies, multiobjective

algorithms can be classified as aggregation based approach, population based

approach and Pareto based approach [2].

The first multiobjective genetic algorithm, vector evaluated algorithm

(VEGA) was proposed by Schaffer [8]. Afterwards, several multiobjective

evolutionary algorithms were developed, such as: Multiobjective Genetic

Algorithm (MOGA), Niched Pareto Genetic Algorithm (NPGA), Weight<based

Genetic Algorithm (WBGA), Random Weighted Genetic Algorithm (RWGA),

Non<dominated Sorting Genetic Algorithm (NSGA), Strength Pareto

Evolutionary Algorithm (SPEA), Pareto<Archived Evolution Strategy (PAES),

Fast Non<dominated Sorting Genetic Algorithm (NSGA<II). Although there are

many variations of multiobjective genetic algorithms these cited are well known

and credible algorithms that have been used in many applications and their

performances were tested in several comparative studies. Generally,

multiobjective genetic algorithms differ based on their fitness assignment

procedure, elitism or diversification [3].

In this study, three algorithms are chosen for discussion: the weighted

genetic algorithm, the vector evaluated genetic algorithm (VEGA), and the fast

non<dominated sorting genetic algorithm (NSGA<II). In the following we

describe each of the three algorithms.

3. Classes of Multiobjective Genetic Algorithms

3.1. Weighted Genetic Algorithm

Since the simple genetic algorithm is based on a scalar function to guide

the search, the most intuitive approach for using a genetic algorithm to solve a

multiobjective optimization problem is to combine all objectives into a single

objective problem using one of the traditional aggregating functions methods.

Then simple genetic algorithm is used to solve the problem. An example of this

approach is a linear sum of weights, which consists of adding all objective

functions together using different weights for each one. Thus, the multiobjective

optimization problem is transformed into a scalar optimization problem of the

form:

∑

=

k

i

i

w

1

where,

i

w are the weights representing the relative importance of

the objectives. Usually, 1

1

=

∑

=

k

i

i

w [2].

Even if the idea of the algorithm is simple, it introduces a difficult

question. What values of weights must one use? Of course, there is no unique

answer to this question. The answer depends on the importance of each

objective in the context of the problem. The weight of an objective is usually

Bul. Inst. Polit. Ia%i, t. LVI (LX), f. 1, 2010

39

chosen in proportion to its importance.

This method doesn’t require any changes to the mechanism of a simple

genetic algorithm. Therefore, it is simple, efficient and easy to implement. It can

be used to solve simple multiobjective optimization problems with few

objective functions and convex search space. Method suffers from the following

difficulties [2]: a Pareto optimal solution is specific to the preference parameters

used in converting a multiobjective optimization problem into a single

optimization problem. In order to find different Pareto optimal solutions, the

parameters must be changed and a new problem to be solved. Thus, in order to

find n different Pareto<optimal solutions, n different single objective

optimization problem need to be formed and solved. Another disadvantage is

that the method is sensitive to the choice of weights and requires the user to

have knowledge about the problem for generate correct parameters.

The main drawback is that for some problems which exhibit a non<

convex Pareto<optimal front, this approach is unable to find all the solution, no

matter how the weights are chosen.

However, this algorithm is easier to implement than specific

multiobjective optimization algorithms, and if the importance of each objective

is known, it may be a simple method to find acceptable solutions.

3.2. Vector Evaluated Genetic Algorithm (VEGA)

VEGA is the first genetic algorithm used to approximate the Pareto

optimal set by a set of non<dominated solutions; it was implemented by

Schaffer [8]. The name is appropriate for multiobjective optimization, because

the algorithm evaluates an objective vector.

VEGA is a straightforward extension of a simple genetic algorithm for

multiobjective optimization. Since a number of objectives (M) have to be

handled, Schaffer thought of dividing the genetic algorithm population into M

equal subpopulations randomly, in each iteration. Each subpopulation is

assigned a fitness based on a different objective function. In this way, each of

the M objective functions is used to evaluate some members in the population.

This algorithm emphasizes solutions which are good for individual

objective functions. Schaffer uses the proportional selection operator.

In order

to find intermediate solutions, Schaffer allowed crossover between any two

solutions in the entire population. In this way, a crossover between two good

solutions, each corresponding to a different objective may find offspring which

are good compromised solutions between the two objectives. The mutation is

applied to each individual as usual.

Criticisms of VEGA [7] include the following arguments. VEGA is very

simple and easy to implement, since only the selection mechanism has to be

modified. One of its main advantages is that, despite its simplicity, can generate

several solutions in one run of the algorithm. However, this shuffling and

Mihaela Simona Cîrciu and Florin Leon

40

merging of all the subpopulations that VEGA performs corresponds to

averaging the fitness components associated with each of the objectives. Since

Schaffer uses proportional fitness assignment, these fitness components are in

turn proportional to the objectives themselves. Therefore, the resulting expected

fitness corresponds to a linear combination of the objectives where the weights

depend on the distribution of the population at each generation. This means that

VEGA has the same problems than the algorithm previously discussed.

3.3. Fast Non6dominated Sorting Genetic Algorithm II (NSGA6II)

The Non<dominated Sorting Genetic Algorithm was proposed by Srinivas

and Deb [9]. In this study it is discussed an improved version of NSGA, called

NSGA<II, which is a fast multiobjective algorithm with elitism proposed by Deb

et al. [6] to remove the weaknesses of NSGA. The main features of NSGA<II,

such as low computational complexity, parameter less diversity preservation,

elitism and real valued representation are described in the following [2].

Low Computational Complexity: The NSGA<II requires at most O(mN

2

)

computational complexity, which is lower compared to O(mN

2

) of NSGA. The

procedure for finding non<dominated front used in NSGA<II is similar to the

non<dominated sorting procedure suggested by Goldberg [5] except that a better

bookkeeping strategy is used to make it more efficient. In this bookkeeping

strategy, every solution from the population is compared with a partially filled

population for domination instead of with every other solution in the population

as in the NSGA. Initially, the first solution from the population is kept in a set

P’. Then, each solution i is compared with all solution in P’ one by one. If the

solution i dominates any solution j of P’ then the solution j is removed from

P’. Otherwise, if the solution i is dominated by any solution j in P’, then the

solution is ignored. If solution i is not dominated by any solution in P’ then it

is saved in P’. Therefore, the set P’ grows with non<dominated solutions.

When all solutions of the population is checked, the solutions in P’ constitute

the non<dominated set. To find the other fronts, the non<dominated solutions

in P’ will be reduced from P and the procedure is repeated until all solutions in

P are ranked.

Diversity preservation: To maintain diversity among solutions, the

NSGA<II replaces the fitness sharing approach in the NSGA with a crowded

comparison approach, which does not require any user<defined parameter. In the

crowded comparison approach, every solution in the population has two

attributes: a non domination rank (i

rank

) and a crowding distance (i

distance

). The

crowding distance of a solution i serves as an estimate of the size of the largest

cuboid enclosing the point i without including any other point in the population.

Fig. 1 [2] illustrates the crowding distance calculation for solution i in

its front, which is the average side<length for the cuboid enclosing solution i

(shown with a dash box).

Bul. Inst. Polit. Ia%i, t. LVI (LX), f. 1, 2010

41

The crowded tournament selection operator which is used to guide the

search towards a spread<out Pareto<optimal front is defined as follows [2]: A

solution i wins a tournament with another solution j if solution i has a better

rank (i

rank

< j

rank

) or i and j has the same rank but solution i has a better

crowding distance than solution j (i

rank

= j

rank

and i

distance

> j

distance

). If i and j has

the same rank and the same crowding distance then one of them is randomly

chosen as a winner.

c

≺

is the crowded comparison operator and is formally

defined as i

c

≺

j if (

rank

rank

ji

<

) or

rankrank

ji

=

( and i

distance

> j

distance

).

Fig. 1 − Crowding distance calculation.

Elitism in NSGA<II is ensured by comparing the current population

with previously found best non<dominated solutions (kept in P’) and by

combining the parent and the child populations to form a combined population

with size 2N. Then the population combined is sorted according to non<

domination. The best solutions are solutions belonging to the nest non<

dominated front F

1.

If the size of F

1

is smaller than N, then all solutions in F

1

are

selected for the new population. The remaining solutions of the new populations

are selected from subsequence non<dominated fronts in order of their ranking

F

2

,

F

3

,

and so on. This procedure is continued until N solutions are selected for

the new population. To choose exactly N members, the solutions in the last front

are sorted using crowded comparison operator in descending order.

Fig. 2 [2] illustrates this procedure. The new population is now used for

selection, crossover and mutation to create a new population of size N [2], [6].

Fig. 2 − Elitism in NSGA<II.

Mihaela Simona Cîrciu and Florin Leon

42

Real valued representation: NSGA<II supports real<valued representation,

in which a chromosome is represented as a vector of variables in real<valued

numbers. Using a real<valued representation, NSGA<II can achieve arbitrary

precision in the obtained solutions (the precision would depend on the machine)

and it can handle problems having a continuous search space. In real<coded

genetic algorithms the main challenge is how to create a new pair of offspring

vectors from a pair of real valued parent vectors and how to mutate them [2].

NSGA<II uses arithmetic crossover which linear combines two parent

chromosomes to produce two offspring as shown in the following equations,

where a is a real number between 0 and 1:

(1) offspring

1

= a S parent

1

+ (1 – a) 5 parent

2

(2) offspring

2

= (1 – a) S parent

1

+ a S parent

2

The mutation operator replaces the value of all genes with uniformly

distributed random value in the range specified by the user.

The diversity among non dominated solutions is introduced by using the

crowding comparison procedure which is used in the tournament selection

during the population reduction phase. Since the solutions compete with their

crowding distance, no extra niching parameter is required. Although the

crowding distance is calculated in the objective function space, it can also be

implemented in the parameter space [6].

4. Case Studies

There are several features that may cause difficulties for

multiobjective genetic algorithms in converging to the Pareto optimal font and

maintaining diversity within the population. Multimodality, deception and

isolated optima are known problems in single objective optimization. Certain

characteristics of the Pareto<optimal front may prevent a multiobjective

algorithm from finding diverse Pareto optimal solutions: non<convexity,

discreteness and non< uniformity. The proposed test functions used in this

study reflect these problems.

Each test considers two objectives, each objective demanding the

minimization of a different objective function.

4.1. Convex Pareto6Optimal Front

The first set of test functions has a convex Pareto<optimal front, when

the goal is to simultaneously minimize the functions f and h:

1

xf =

2

2

1

x

xg +=

gfh

⋅

=

Bul. Inst. Polit. Ia%i, t. LVI (LX), f. 1, 2010

43

Each algorithm used the following parameters: population size: N=200,

number of generations: G=100, mutation rate: p

m

= 0.03, and crossover rate:

p

c

=0.9.

The non<dominated solutions obtained by each algorithm are displayed

in Fig. 3. For the weight<based GA, the weights were chosen in the interval

[0,1], and several runs were performed for each combination.

Fig. 3 − Non<dominated solutions obtained for the problem with

a convex Pareto<optimal front.

For a convex front, all the algorithms obtain a good approximation of

the Pareto<optimal front.

4.2. Concave Pareto6Optimal Front

The second set of test functions has a non<convex Pareto<optimal front:

1

xf =

30,

1

1

1

=⋅+=

∑

=

nx

n

g

n

i

i

g

g

f

h ⋅

−=

2

2

1

Mihaela Simona Cîrciu and Florin Leon

44

The algorithms use the same parameters as before. The non<dominated

solutions obtained by each algorithm are visualized in Fig. 4.

Fig. 4 − Non<dominated solutions obtained for the problem

with a concave Pareto<optimal front.

NSGA<II and VEGA obtain a good approximation of Pareto<optimal

front. However, the weighted genetic algorithm is unable to find the

intermediate solutions.

4.3. Discontinuous Pareto6Optimal Front

The third set of test functions has a discontinuous Pareto<optimal front:

1

xf =

2

2

1

x

xg +=

( )f

g

f

g

f

h ⋅⋅−−=

π

6sin1

2

2

The introduction of the sine function causes a discontinuity in the

Pareto<optimal front, although in the parameter space there is no discontinuity.

Each algorithm uses the same parameters as before. The non<dominated

solutions obtained by each algorithm are displayed in Fig. 5.

Bul. Inst. Polit. Ia%i, t. LVI (LX), f. 1, 2010

45

Fig. 5 − Non<dominated solutions obtained for the problem

with a discontinuous Pareto<optimal front.

For this problem, only NSGA<II approximates the correct Pareto<

optimal front. The weighted genetic algorithm can find some solution clusters in

the neighborhood of the true Pareto fronts. In this case, VEGA doesn’t provide

good solutions.

4. Conclusions

The weighted genetic algorithm works well on the convex front, but has

difficulties on a non<uniform or concave front. Another difficulty is to choose

the weights, which requires the user to have knowledge about the problem. The

algorithm obtains a single final solution, thus to find different Pareto optimal

solutions, the algorithms has to be run with several sets of weights.

Vector evaluated genetic algorithm exceeds the performance of the

weighted genetic algorithm. The problem for this algorithm is that final

solutions tend to converge to the optimal values of each objective, and “lose”

middle solutions. That’s because of the fitness assignment. It is observed that

among the final solutions are dominated solutions, thus was implemented a non<

dominated sorting function that removes these false solutions.

NSGA<II is the most efficient algorithm, exceeds the performance of

both algorithms and obtains a good approximate of Pareto<optimal front

regardless of its shape. Real<world engineering problems involve simultaneously

optimizing multiobjective where trade<offs are important. All the problems will

require some customization of the genetic algorithms approach to handle the

Mihaela Simona Cîrciu and Florin Leon

46

objectives. It is imagined that the Pareto solutions identified by genetic

algorithms would be reduced to a representative small set for the designer or

engineer to further investigate. Therefore, for most implementations it is not

vital to find every Pareto optimal solution, but rather, efficiently and reliably

identify Pareto optimal solutions across the range of interest for each objective

function.

A c k n o w l e d g e m e n t s. This work was supported in part by CNCSIS

grant 316/2008, contract no. 671/2009, Behavioural Patterns Library for Intelligent

Agents Used in Engineering and Management.

Received: November 22, 2009 “Gheorghe Asachi” Technical University of Ia=i,

Department of Computer Science and

Information Technology

e>mail: fleon@cs.tuiasi.ro

R E F E R E N C E S

1. Rosenberg R.S., Simulation of Genetic Populations with Biochemical Properties.

Ph. D. Diss., University of Michigan, 1967.

2. Tran K.D., An Improved Multi>Objective Evolutionary Algorithm with Adaptable

Parameters. International Journal of Intelligent Systems Technologies and

Applications archive, Vol. 7, 4, 347−369, 2009.

3. Konak A., Coit D.W., Smith A.E., Multi>Objective Optimization Using Genetic

Algorithms: A Tutorial. Reliability Engineering & System Safety, Vol. 91, 9,

September 2006, 992−1007, 2006.

4. Leon F., InteligenŃă artificială > principii, tehnici, aplicaŃii. Tehnopress, Ia%i, 2006.

5. Goldberg D.E., Deb K., A Comparative Analysis of Selection Schemes Used in

Genetic Algorithms. Foundations of Genetic Algorithms (1991).

6. Deb K., Pratap A., Agarwal S., Meyarivan T., A Fast and Elitist Multiobjective

Genetic Algorithm: NSGA>II. Proceedings of the Parallel Problem Solving

from Nature VI Conference, 2002.

7. Coello C.A., Lamont G.B., Van Veldhuizen D.A., Evolutionary Algorithms for

Solving Multi>Objective Problems. Second Edition, Springer, 2007.

8. Schaffer D.J., Multiple Objective Optimization with Vector Evaluated Genetic

Algorithms. Proceedings of the international conference on genetic algorithm

and their applications, 1985.

9. Srinivas N., Deb K., Multiple Objective Optimization Using Nondominated Sorting

in Genetic Algorithms. Evolutionary Computation, 2, 3, 221−248, 1994.

10. Deb Kalyanmoy, Multi>Objective Optimization Using Evolutionary Algorithms.

Wiley, 2001.

11. Zitzlet E., Deb K., Thiele L., Comparison of Multiobjective Evolutionary

Algorithms: Empirical Results. Technical report 70, Computer Engineering and

Networks Laboratory (TIK), Swiss Federal Institute of Technology (ETH)

Zurich, 2000.

Bul. Inst. Polit. Ia%i, t. LVI (LX), f. 1, 2010

47

STUDIU COMPARATIV ASUPRA UNOR ALGORTIMI

GENETICI MULTIOBIECTIV

(Rezumat)

Lucrarea prezintă studiul a trei algoritmi pentru rezolvarea problemelor

multiobiectiv. Primul algoritm ales pentru studiu este un algoritm genetic vectorial, care

este cea mai simplă extensie a unui algoritm genetic simplu. Problema multiobiectiv

este transformată într<o problemă cu un singur obiectiv prin combinarea tuturor

obiectivelor folosind o sumă de ponderi. Problema este apoi rezolvată cu un algoritm

genetic simplu. De%i simplu, acest algoritm are câteva dezavantaje. Primul este alegerea

ponderilor care sunt introduse de utilizator, ceea ce presupune o bună cunoa%tere a

problemei de rezolvat. Ponderile sunt alese, de obicei în funcŃie de importanŃa

obiectivului. Un alt dezavantaj este faptul că în final algoritmul obŃine o singură soluŃie,

de aceea pentru a obŃine mai multe soluŃii optime e necesară rularea algoritmului cu mai

multe seturi de ponderi. Algoritmul genetic vectorial cu ponderi este sensibil la forma

frontului Pareto %i nu reu%e%te să determine soluŃii pe un front concav.

Al doilea algoritm ales pentru studiu este algoritmul genetic evaluat vectorial,

primul algoritm genetic propus pentru optimizarea multiobiectiv. Acesta este o abordare

non<Pareto bazată pe selecŃia unor grupuri relevante de indivizi, fiecărui grup fiindu<i

aplicat un obiectiv. Dacă numărul de obiective este M, algoritmul împarte populaŃia în

M subpopulaŃii %i atribuie fiecăreia o funcŃie obiectiv. SubpopulaŃiile sunt apoi unite. Se

aplică operatorii genetici: operatorul de selecŃie proporŃională, operatorul de încruci%are,

care este aplicat între oricare doi indivizi. Aceasta permite ca din două soluŃii bune în

raport cu obiective diferite să se obŃină soluŃii copii ce sunt compromisuri bune.

Algoritmul este mai eficient decât algoritmul vectorial, dar este %i el sensibil la forma

frontului Pareto.

Algoritmul cu sortare nedominată este unul dintre primii algoritmi evolutivi. În

lucrarea de faŃă este prezentată o variantă îmbunătăŃită a acestuia, NSGA<II. Acesta este

un algoritm bazat pe dominanŃa Pareto pentru ghidarea căutării către frontul optim. Este

cel mai eficient algoritm dintre cei trei studiaŃi, aproximând corect orice front Pareto,

indiferent de forma acestuia.

Cei trei algoritmi sunt implementaŃi într<o aplicaŃie ce permite rezolvarea

oricărei probleme multiobiectiv %i care poate fi utilizată ca o platformă pentru studierea

comparativă a performanŃelor diferitelor metode de optimizare.

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