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International Journal of Computing & Information Sciences Vol. 2, No. 1, April 2004 38

Evolutionary Algorithms for Multi-Criterion

Optimization: A SurveyAshish Ghosh

Machine Intelligence Unit, Indian Statistical Institute, Kolkata-108, India.

Satchidananda Dehuri

Department of Computer Science & Engineering,Silicon Institute of Technology, BBSR-24, India.

Abstract:In this paper, we review some of the most popular evolutionary algorithms and a systematic comparison among them.

Then we show its importance in single and multi-objective optimization problems. Thereafter focuses some of the multi-objective

evolutionary algorithms, which are currently being used by many researchers, and merits & demerits of multi-objectiveevolutionary algorithms (MOEAs). Finally, the future trends in this area and some possible paths of further research are

addressed.

Keywords: Evolutionary algorithms, multi-criterion optimization, Pareto-optimal solution, dominated and non-dominatedsolution.

Received:March 01, 2004| Revised: February 03, 2005| Accepted:March 10, 2005

1. Introduction

Evolutionary techniques have been used for the purposeof single-objective optimization for more than threedecades [1]. But gradually people discovered that manyreal world problems are naturally posed as multi-objective. Now a days multi-objective optimization is nodoubt a very popular topic for both researchers andengineers. But still there are many open questions in thisarea. In fact there is not even a universally accepteddefinition of optimum as in single objectiveoptimization, which makes it difficult to even compareresults of one method to another, because normally thedecision about what the best answer is corresponds tothe so called (human) decision-maker.

Since multi-criterion optimization requiressimultaneous optimization of multiple often competing orconflicting criteria (of objectives), the solution to such

problems is usually computed by combining them into asingle criterion optimization problem. But the resultingsolution to the single objective optimization problem isusually subjective to the parameter settings chosen by theuser [2], [3]. Moreover, since usually a classicaloptimization method is used, only one solution(hopefully a Pareto-optimal solution) can be found in onesimulation run. So, in-order to find multiple Pareto-optimal solutions, evolutionary algorithms are the bestchoice, because it deals with a population of solutions. Itallows to finding an entire set of Pareto-optimal solutions

in a single run of the algorithm. In addition to this,evolutionary algorithms are less susceptible to the shapeor continuity of the Pareto front.

The rest of the paper is organized as follows. Section2 gives a brief description of evolutionary algorithm.Section 3 addresses the key concept of multi-objectiveevolutionary algorithm. The different approaches ofMOEA and their merits and demerits are the subject ofsection 4. A discussion of the future perspectives isgiven in section 5. Section 6 concludes the article. Someapplications of multi-objective evolutionary algorithmsare also indicated.

2. Evolutionary AlgorithmsEvolution is in essence a two-step process of randomvariation and selection [4]. It can be modeledmathematically as ]))[((]1[ txvstx =+ , where the

population at time t, denoted as ][tx , is operated on byrandom variation v, and selection to give rise to a new

population ]1[ +tx . The process of evolution can bemodeled algorithmically and simulated on a computer.The modeled algorithm is known as evolutionaryalgorithms (EAs). Evolutionary computation is the fieldthat studies the properties of these algorithms and similar

procedures for simulating evolution on a computer. Thesubject is still relatively new and represents an effort to

bring together researchers who have been working inthese and other closely related fields.

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39 International Journal of Computing & Information Sciences Vol. 2, No. 1, April 2004

In essence, the EAs are used to describe computer-based problem solving algorithms that use computationalmodels of evolutionary processes as key elements in theirdesign and implementation. A variety of evolutionaryalgorithms have been evolved during the last 30 years.

The major ones are i) Genetic algorithms, ii)Evolutionstrategies, iii) Evolutionary programming and iv)Genetic programming. Each of these constitutes adifferent approach, however they are inspired by thesame principle of natural evolution. They all share acommon conceptual base of simulating the evolution ofindividualstructures via processes of selection,mutation,and crossover. More precisely, EAs maintain a

populationof structures that evolve according to rules ofselection and other operators, which are referred to asgenetic operators, such as crossoverand mutation. Let ushave a discussion about all these evolutionary algorithms.

2.1. Genetic AlgorithmsA genetic algorithm developed by J.H. Holland(1975)[5][1] is a model of machine learning, whichderives its behavior from a metaphor of the processes ofevolutionin nature. GAs are executed iteratively on a setof coded chromosomes, called a population, with three

basic genetic operators: selection, crossover andmutation. Each member of the population is called achromosome (or individual) and is represented by astring. GAs use only the objective function informationand probabilistic transition rules for genetic operations.The primary operator of GAs is the crossover. The figure1shows the basic flow diagram of a GA:

2.2. Evolutionary Strategy

Evolutionary strategies (ES) were developed in Germanyby I.Rechenberg and H.P.Schwefel [6-8]. It imitatesmutation, selection and recombination by using normallydistributed mutation, a deterministic mechanism forselection (which chooses the best set of offspring

individuals for the next generation) and a broadrepertoire of recombination operators. The primaryoperator of the ES is mutation. There are two variants ofselection commonly used in ES. In the elitist variant, the parents individual for new generation are selectedfrom the set of both parents and offspring at oldgeneration. This is calledplus strategy(+).

If parent individuals do not take part in the selection,then individuals of the next generation are selectedfrom offspring, which is called comma strategy anddenoted as (,). The notation ( ,+ ) is used tosubsume both selection schemes. Shows the basic

principle of ES.

AlgorithmBegin

t = 0initialize Ptevaluate Pt

while (termination condition not satisfied) dobegin

t = t+1select Ptfrom Pt-1mutate Ptevaluate Pt

endend

2.3 Evolutionary ProgrammingFogel (1960) proposed a stochastic optimizationstrategysimilar to GAs called evolutionary programming (EP)[9].EP is also similar to evolutionary strategies (ESs). Like

both ES and GAs, EP is a useful method of optimizationwhen other techniques such as gradient descent or directanalytical discovery are not possible. Combinatorial and

real-valued

function optimization,

in which theoptimization surface or fitness landscape is "rugged",possessing many locally optimal solutions, are wellsuited for evolutionary programming. The basic EPmethod involves 3 steps (repeat until a certain number ofiterations is exceeded or an acceptable solution isobtained): Generate the initial population of solutions

randomly. Each solution is replicated into a new population.

Each of these offspring solutions are mutatedaccording to a distribution of mutation types,

ranging from minor to extreme with a continuumof mutation types. The severity of mutation isFigure 1. Basic structure of genetic algorithm

yes

no

Intialize o ulation

Fitness assi nment

selection

crossover

mutation

output

Is perform-ance good

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Evolutionary Algorithms for Multi-criterion Optimization: A Survey 40

judged on the basis of the functional changeimposed on the parents.

Each offspring solution is assessed by computingit's fitness. Typically, a stochastic tournament isheld to determine N solutions to be retained for the

Population of solutions, although this isoccasionally performed deterministically. There isno requirement that the Population size be heldconstant, however, not that only a single offspringis generated from each parent.

It should be pointed out that EP typically does not useany crossover as a genetic operator.

2.4 Genetic programmingGenetic programming (GP) was introduced by John Koza(1992)[10-12]. GP is the extension of the genetic model

of learning into the space of programs. That is, theobjects that constitute the populationare not fixed-lengthcharacter strings that encode possible solutions to the

problem at hand, they are programs that, when executed,are the candidate solutions to the problem. These

programs are expressed in genetic programming as parsetrees, rather than as lines of code. Thus, for example, thesimple program cba *+ would be represented as:

or to be precise, as suitable data structures linkedtogether to achieve this effect. Because this is a verysimple thing to do in the programming language LISP,many genetic programmers tend to use LISP. There arestraightforward methods to implement GP using a non-LISP programming environment also.

The programs in the Population are composed ofelements from the function set and the terminal set,which are typically fixed sets of symbols selected to beappropriate to the solution of problems in the domain of

interest.In GP the crossover operation is implemented by

taking randomly selected subtrees in the individual's(selected according to fitness) and exchanging them. Itshould be noted that GP usually does not use anymutation as in genetic operator.

3. Multi-objective evolutionary algorithmsMulti-objective optimization methods as the namesuggests, deal with finding optimal solutions to problemshaving multiple objectives [13]. So in this type of

problems the user is never satisfied by finding onesolution that is optimum with respect to a single criterion.

The principle of a multi-criterion optimizationprocedure is different from that of a single criterionoptimization. In a single criterion optimization the maingoal is to find the global optimal solutions. However, in amulti-criterion optimization problem, there is more than

one objective function, each of which may have adifferent individual optimal solution. If there is asufficient difference in the optimal solutionscorresponding to different objectives then we say that theobjective functions are conflicting to each other. Multi-criterion optimization with such conflicting objectivefunctions gives rise to a set of optimal solutions, insteadof one optimal solution. The reason for the optimality ofmany solutions is that no one can be considered to be

better than any other with respect to all other objectivefunctions. These optimal solutions have a special namecalled Pareto-optimal solutions following the name of an

economist Vilfredo Pareto. He stated in 1896 a conceptaccording to his name, known as Pareto optimality.

The concept is that the solution to a multi-objectiveoptimization problem is normally not a single value but

instead a set of values also called the Pareto set. Letus illustrate thePareto optimal solution with the time &space complexity of an algorithm shown in the followingfigure. In this problem we have to minimize both time aswell as space complexity.

The point p represents a solution, which has a minimal

time, but the space complexity is high. On the other hand,the point r represents a solution with high timecomplexity but minimum space complexity. Considering

both objectives, no solution is optimal. So in this case wecant say that solution p is better than r. In fact, thereexist many such solutions like q also belong to thePareto optimal set and one cant sort the solution

according to the performance metrics considering both

+/ \

a */ \

b c

q

p

Figure 2. Pareto optimal solutions

Time

Pareto-optimal front

t

sSpace

r

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objectives. All the solutions, on the curve, are known asPareto-optimal solutions.

From the figure-2 it is clear that there existsolutions, which does not belong to the Pareto optimalset, such as a solution like t. The reason why t does

not belong to Pareto optimal set is obvious. If wecompare t with solution q, 't is not better than qwith respect to any of the objectives. So we can say thatt is a dominatedsolution or inferiorsolution.

Now we are clear about the optimality concept ofmulti-criterion. Based on the above discussions, we firstdefine conditions for a solution to become dominatedwith respect to another solution and then presentconditions for a set of solutions to become a Pareto-optimal set.

Let us consider a problem having m objectives (say

miif ,.....,3,2,1, = and m >1). Any two solutions

)1(

u and

)2(u

(having t decision variables each) can haveone of two possibilities-one dominates the other or none

dominates the other. A solution)1(

u is said to dominate

the other solution)2(

u , if the following conditions aretrue:

1. The solution )1(u is no worse (say the operator denotes worse and } denotes better) than

)2(u in all

objectives, or .....,3,2,1),)2(

())1(

( miuifuif =

2. The solution )1(u is strictly better than )2(u in atleast one objective, or )

)2(()

)1(( uifuif { for at

least one, i {1,2,3,.. , m }.

If any of the above condition is violated, the solution)1(

u does not dominate the solution)2(

u . If)1(

u

dominates the solution)2(

u , then we can also say that)2(

u is dominated by)1(

u , or)1(

u is non dominated by

)2(u , or simply between the two solutions, )1(u is thenon-dominated solution.

The following definitions ensure whether a set ofsolutions belongs to a local or global Pareto-optimal set:

Local Pareto-optimal setIf for every member u in a set S, no solution v

satisfying vu , where is a small positive

number, which dominates any member in the set S, thenthe solutions belonging to the set S constitute a localPareto-optimal set. The definition says that if we

perturbing the solution u in a small amount then the

resultant solution v dominates any member of that setthen the set is called local Pareto optimal set

Global Pareto-optimal setIf there exits no solution in the search space whichdominates any member in the set S, then the solutions

belonging to the set S constitute a global Pareto-optimalset.

Difference between non-dominated set & a

Pareto-optimal setA non-dominated setis defined in the context of a sampleof the search space (need not be the entire search space).In a sample of search points, solutions that are notdominated (according to the previous definition) by anyother solution in the sample space constitute the non-

dominated set. A Pareto-optimal set is a non-dominatedset, when the sample is the entire search space. Thelocation of the Pareto optimal set in the search space issometimes loosely called as the Pareto optimal region.

From the above discussions, we observe that thereare primarily two goals that a multi-criterion optimizationalgorithm must try to achieve:

1.Guide the search towards the global Pareto-optimal region, and

2. Maintain population diversity in the Pareto-

optimal front.

The first task is a natural goal of any optimization

algorithm. The second task is unique to multi-criterionoptimization.Multi-criterion optimization is not a new field of

research and application in the context of classicaloptimization. The weighted sum approach, -perturbationmethod, Goal programming, Tchybeshev method, min-max method and others are all popular methods oftenused in practice [14]. The core of these algorithms, is aclassical optimizer, which can at best, find a singleoptimal solution in one simulation. In solving multi-criterion optimization problems, they have to be usedmany times, hopefully finding a different Pareto-optimal

solution each time. Moreover, these classical methodshave difficulties with problems having non-convexsearch spaces.

Evolutionary algorithms (EAs) are a natural choicefor solving multi-criterion optimization problems becauseof their population-based nature. A number of Pareto-optimal solutions can, in principle, be captured in an EA

population, thereby allowing a user to find multiplePareto-optimal solutions in one simulation run. Althoughthe possibility of using EAs to solve multi-objectiveoptimization problems was proposed in the seventies.David Schaffer first implemented Vector evaluatedgenetic algorithm (VEGA) in the year 1984. There waslukewarm interest for a decade, but the major popularity

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of the field began in 1993 following a suggestion byDavid Goldberg based on the use of the non-dominationconcept and a diversity- preserving mechanism. Thereare various multi-criterions EAs proposed so far, bydifferent authors.

4. Different Approaches of MOEAFollowing popular approaches have been used bydifferent researchers for multi-objective optimization,each one having its merits and demerits. Weighted-sum-Approach (Using randomly generated

weights and Elitism) Schaffers Vector Evaluated Genetic Algorithm

(VEGA) Fonseca and Flemings Multi-Objective Genetic

Algorithm (MOGA)

Horn, Nafploitis, and Goldbergs Niched ParetoGenetic Algorithm (NPGA) Zitzler and Thieles Strength Pareto Evolutionary

Algorithm (SPEA) Srinivas and Debs Non-dominated Sorting Genetic

Algorithm (NSGA) Vector-optimized evolution strategy (VOES) Weight-based genetic algorithm (WBGA)

Sharing function approach Vector evaluated approach

Predator-prey evolution strategy (PPES) Rudolphs Elitist multi-objective evolutionary

algorithm (REMOEA) Elitist non-dominated sorting genetic algorithm

(ENSGA) Distance-Based Pareto genetic algorithm (DBPGA) Thermodynamical genetic algorithm (TGA) Pareto-archived evolution strategy (PAES)4.1. Weighted-Sum ApproachThis approach was proposed by Ishibuchi and Murata in1996[15]. Here they used randomly generated weightsand elitism. Let us see how this method works with the

help of the following algorithm. Suppose there are m objective functions miif ,....,3,2,1, = . Our task is to

optimize (i.e. maximize) all these objective.

Algorithm1. Generate the initial population randomly.2. Compute the value of the m objectives for each

individual in the population. Then determine whichare the non-dominated solutions and store them in aseparate population that we will call external todistinguish it from the current population, which we

call current.

3. If n represents the number of non-dominatedsolutions in external and p is the size of current,

then we select np pairs of parents using the

following procedure:

The fitness function used for each individual is)(....)(22)(11)( xmfmwxfwxfwxf ++= Ge

nerate m random number in the interval [0, 1] i.e.

mrrr ,....,2,1 . The weight values are determined as

follow

( )mrrrir

iw +++=

...21

mi ,....,3,2,1= . This ensure that all 0iw ,

mi ...,3,2,1= and that 1,...2,1 =mwww .

Select a parent with probability( )

=currentx currentfxf

currentfxfxp

)()(

)()()(

min

min ,

where minf is the minimum fitness in the current

population.

4. Apply crossover to the selected np pairs ofparents. Apply mutation to the new solutionsgenerated by crossover.

5. Randomly select n solutions from external. Thenadd the selected solutions to the np solutionsgenerated in the previous step to construct a

population of sizep .

6. Stop if a pre-specified stopping criterion is satisfied(e.g., the pre-specified maximum number ofgenerations has been reached). Otherwise, return tostep 2.

Merits & DemeritsThe advantage of this method is, its computationalefficiency, and its suitability to generate a strongly non-

dominated solution that can be used as an initial solutionfor other techniques.

Its main demerit is the difficulty to determine theweights that can appropriately scale the objectives whenwe do not have enough information about the problem,

particularly if we consider that any optimal pointobtained will be a function of such weights. Still moreimportant is the fact that this approach does not generate

proper Pareto-optimal solutions in the presence of non-convex search spaces regardless of the weights used.However the incorporation of elitism is an important

aspect of this method.

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4.2. Schaffers Vector Evaluated Genetic

Algorithm (VEGA )David Schaffer (1984)[16-18] extended GrefenstettesGENESIS program [Grefenstette 1984] to include

multiple objective functions. Schaffers approach was touse an extension of the simple genetic algorithm (SGA)that he called the vector evaluated genetic algorithm(VEGA), and that differed from the SGA in selectiononly. This operator was modified so that at eachgeneration a number of sub-populations were generated

by performing proportional selection according to eachobjective in turn. Thus for a problem with k objectives,k sub-populations of size N/k would be generated(assuming a total population size of N). These sub-

populations would be shuffled together to obtain a newpopulation of size N, on which the GA would apply, the

crossover and mutation operators in the usual way.Figure 3 shows the structural representation of thisprocess. Schaffer realized that the solutions generated byhis system were non-dominated in a local sense, becausetheir non-dominance was limited to the current

population, and while a local dominated individual is alsoglobally dominated, the converse is not true [Schaffer1985]. An individual who is dominated in one generationmay become dominated by an individual who emerges ina later generation. Also he noted a problem that ingenetics is known as speciation(i.e. we could have theevolution of species within the population which excel

on different aspects of performance). This problemarises because this technique selects individuals whoexcel in one dimension of performance, without lookingat the other dimensions. The potential danger doing it isthat we could have individuals with what Schaffer callsmiddling performance in all dimensions, which could

be very useful for compromise solutions, but that will notsurvive under this selection scheme, since they are not inthe extreme for any dimension of performance (i.e. theydo not produce the best value for any objective function,

but only moderately good values for all of them).Speciation is undesirable because it is against our goal of

finding a compromise solution. Schaffer tried tominimize this speciation by developing two heuristics the non-dominated selection heuristic (a wealthredistribution scheme), and the mate selection heuristic(across breeding scheme).

In the non-dominated selection heuristic, dominatedindividuals are penalized by subtracting a small fixed

penalty from their expected number of copies duringselection. Then the total penalty for dominatedindividuals was divided among the non-dominatedindividuals and was added to their expected number ofcopies during selection.

But this algorithm failed when the population hasvery few non-dominated individuals, resulting in a large

fitness value for those few non-dominated points,eventually leading to a high selection pressure. The mateselection heuristic was implemented by the following

procedure:Initially select an individual randomly from the

population. Then the corresponding mate was selected asthe individual, which has maximum Euclidean distance

from it.

But it failed too to prevent the participation of poorerindividuals in the mate selection. This is because ofrandom selection of the first mate and the possibility of alarge Euclidean distance between a champion and a less

fitted solution. Schafer concluded that the random mateselection is far superior than this heuristic.

Start

Initialize thePopulation

Evaluate forobjective

function f1

Evaluate forObjective

Function f2

Select N/ksub-

populationBased on f1

Combined all the sub-population

Suffle entirepopulation

Is performancesatisfactor ?

Evaluate forObjective

Function fk

Select N/ksub-

populationBased on f2

Select N/ksub-

populationBased on fk

Crossover

Mutation

Stop

es

no

Figure 3 Schematic Representation of VEGA

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Merits & DemeritsThe main advantage of this algorithm is its simplicity i.e.this approach is easy enough to implement. Richardson etal.(1989)[19] noted that the shuffling and merging of all

the sub-populations corresponds to averaging the fitnesscomponents associated with each of the objectives. SinceSchaffer used proportional fitness assignment, thesefitness components were in turn proportional to theobjectives themselves. Therefore, the resulting expectedfitness corresponds to a linear combination of theobjectives where the weights depend on the distributionof the population at each generation as shown byRichardson et al.

The main consequence of this is that when we have aconcave trade-off surface certain points in concaveregions will not be found through this optimization

procedure in which we are using just a linearcombination of the objectives, and it has been proved thatthis is true regardless of the set of weights used.Therefore, the main weakness of this approach is itsinability to produce Pareto-Optimal solutions in presenceof non-convex search spaces.

4.3. Fonseca and Flemings MOGAFonseca and Fleming (1993)[20-22] implementedGoldbergs suggestion in a different way. Let us firstdiscuss what is Goldbergs suggestion about Paretoranking.

Goldberg proposed the Pareto ranking scheme in(1989), where a solution x at generation t has acorresponding objective vector xu, and n is the

population size, the solutions rank is defined by thefollowing algorithm.

Algorithmcurr_rank = 1m = nWhile n != 0 do

For i= 1. . . ,m

doIf xuis non-dominatedRank (x, t) = curr_rankenddoFor i=1,.....,mdoIf rank (x, t)= curr_rankRemove x from populationn= n-1;enddocurr_rank = curr_rank +1m = n

endwhile

This means, in Pareto ranking, the whole populationis checked and all non-dominated individuals areassigned rank 1. Then remove these individuals fromthe population, which has rank 1. Again detect all non-dominated individual from the rest of the population and

assign rank 2. In this way the procedure goes on till allsolutions have been assigned the required rank.But in MOGA, the whole population is checked and allnon-dominated individuals are assigned rank 1. Otherindividuals are ranked by checking the non-dominance ofthem with respect to the rest of the population in thefollowing way.

For example, an individual xiat generation t, whichis dominated by pi

(t)individuals in the current generation.Its current position in the individual's rank can be given

byRank (xi, t) = 1 + pi

(t)

After completing the ranking procedure, then it is time toassign the fitness to each individual. Fonseca andFleming proposed two method of assigning fitness.

Rank-based fitness assignment method Niche-formation methods.Rank-based fitness assignment is performed in thefollowing way [1993]

1. Sort population according to rank2. Assign fitness to individuals by interpolating from

the best (rank 1) to the worst (rank n N) in the

usual way, according to some function, usually linearbut not necessarily.

3. Average the fitness of individuals with the samerank, so that all of them will be sampled at the samerate. This procedure keeps the global populationfitness constant while maintaining appropriateselection pressure, as defined by the function used.

As Goldberg and Deb have pointed out, this type ofblocked fitness assignment is likely to produce a largeselection pressure that might produce prematureconvergence. To avoid this, Fonseca and Fleming use the

second method (i.e. Niche-formation method) todistribute the population over the Pareto optimal region,

but instead of performing sharing on the parametervalues, they have used sharing on the objective functionvalues.

Merits & DemeritsThe main advantage of MOGA is that it is efficient andrelatively easy to implement. The performance of thismethod is highly dependent on the sharing factor (share).However Fonseca and Fleming have developed a good

methodology to compute such a value for their approach.

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dominated solutions having more dominated solution inthe combined population. After generating a populationfor next generation, it is required to update the external

population. The updating procedure is as follows:For each individual in the current population, check

whether it is dominated by, both current population andexternal population. If it is not dominated by both thenadd this individual to external population (EXTERNAL).Then all solutions dominated by the added one areeliminated from external population. In this way theexternal population is updated in every generation. Thefollowing algorithm shows the idea:

Algorithm1. Initialize the population of size n.2. Determine all the non-dominated solutions from

current population and store them in a separatepopulation called EXTERNAL.3. Combine the external with current population and

assign the fitness to individuals according to howmany solutions it dominates.

4. After assigning the fitness to all the individuals,select n individuals for next generation.

5. Apply crossover and mutation to get the newpopulation of size n.

6. Update the EXTERNAL population.7. Test whether the performance criterion is satisfied or

not. If not then goto step 3.Otherwise stop.

Zitzler and Thieles[25] solved the 0/1-Knapsackproblem using this approach. They have reported betterresults than other methods.

Merits & DemeritsThe main merits of this method are that, it shows theutility of introducing elitism in evolutionary multi-criterion optimization. Since this method does not usesharing factor share& tdomas in the MOGA and NPGA;this facilitates good advantages.

There exists a multi-objective problem where the

resulting Pareto-optimal front is non-convex. In that caseGA's success to maintain diverse Pareto-optimalsolutions largely depends on fitness assignment

procedure. This method does not converge to true Pareto-optimal solutions, because this method uses the fitnessassignment procedure, which is very sensitive to concavesurface.

4.6. Srinivas and Debs Non-dominated

Sorting Genetic Algorithm (NSGA)Non-dominated GAs varys from simple GAs only in the

way the selection operator in used. The crossover and

mutation operators remain as usual [28-30]. Let usdiscuss the selection operator they proposed.

For selection, two steps are needed. First, thepopulation is ranked on the basis of an individuals non-domination level and then sharing is used to assign

fitness to each individual.

Ranking individuals based on non

domination levelConsider a population of size n, each having m (>1)objectives function values. The following algorithm can

be used to find the non-dominated set of solutions:

AlgorithmFor i = 1 to n do

For j = 1 to n do

If (j != i) thenCompare solutions x(i)andx(j)for domination using twoconditions for all m objectives.if for any j , x(i)is dominated by x(j) ,mark x(i) as dominated

endifendfor

endfor

Solutions, which are not marked dominated, are non-dominated solution. All these non-dominated solutions

are assumed to constitute the first non-dominated front inthe population. In order to find the solutions belonging tothe second level of non-domination, we temporarilydisregard the solutions of the first level of non-domination and follow the above procedure. Theresulting non-dominated solutions are the solutions of thesecond level of non-domination. This procedure iscontinued till all solutions are classified into a level ofnon-domination. It is important to realize that the numberof different non-domination levels could vary between 1to n.

Fitness assignmentIn this approach the fitness is assigned to each individualaccording to its non-domination level. An individual in ahigher level gets lower fitness. This is done in order tomaintain a selection pressure for choosing solutions fromthe lower levels of non-domination. Since solutions inlower levels of non-domination are better, a selectionmechanism that selects individuals with higher fitness

provides a search direction towards the Pareto-optimalregion.

Setting a search direction towards the Pareto-optimal region is one of the two tasks of multi-objectiveoptimization. Providing diversity among current non-

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dominated solutions is also important in order to get agood distribution of solutions in the Pareto-optimal front.The fitness assignment is performed in two stages.

1. Assigning same dummy fitness to all the solutions ofa particular non-domination level.

2. Apply the sharing strategy.Now we discuss the details of these two stages:

First, all solutions in the first non-dominated frontare assigned a fitness equal to the population size. This

becomes the maximum fitness that any solution can havein any population. Based on the sharing strategy, if asolution has many neighboring solutions in the samefront, its dummy fitness is reduced by a factor and ashared fitness is computed. The factor depends on thenumber and proximity of neighboring solutions. Once allsolutions in the first front are assigned their fitness

values, the smallest shared fitness value is determined.Thereafter, the individuals in the second non-

domination level are all assigned a dummy fitness equalto a number smaller than the smallest shared fitness ofthe previous front. This makes sure that no solution in thesecond front has a shared fitness better than that of anysolution in the first front. This maintains a pressure forthe solutions to lead towards the Pareto-optimal region.The sharing method is again used among the individualsof second front and shared fitness of each individual isfound. This procedure is continued till all individuals areassigned a shared fitness.

After the fitness assignment method, use a stochasticremainder roulette-wheel selection for selecting Nindividuals. Thereafter apply the crossover and mutation.Shared fitness is calculated as follows:

Given a set of 1n solutions in the lth non-dominated

front each having a dummy fitness value 1f , the sharing

procedure is performed in the following way for each

solution 1,...,3,2,1 ni= .

1. Compute a normalized Euclidean distance measurewith another solution j in the lthnon-dominated front, asfollows:

2

1)()(

)()(

=

=

P

ps

p

u

p

j

p

i

p

ijxx

xxd

where P is the number of variables in the problem. The

parameter's)(u

px and)(s

px are the upper and lower

bounds of variable px .

2. This distance dij is compared with a pre-specifiedparameter shareand the following sharing function valueis computed:

( )

=otherwise

ifdijdijdsh

shareijshare

,0

,2

1)(

3. Increment j. If 1nj

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) are used as an individual in a population. Thus anindividual i results in two objective vector evaluations:

(i) df , calculated by using the decision variable of the

dominant genotype and (ii) rf , calculated by using the

decision variables of the recessive genotype. Let us seethe procedure of fitness evaluation and selection scheme:

Let n be the number of objective functions. Thenthe selection process is completed in n steps. For eachstep a user defined probability is used to choose anobjective. This vector can either be kept fixed or variedwith generations. If the mthobjective function is chosen,the fitness of an individual i s computed as a weighted

average of its dominant objective value imdf )( and

recessive objective value imrf )( in a 2 : 1ratio.

( ) ( )( )im

r

i

md

i ffF 31)(32 +=

For each selection step, the population is sortedaccording to each objective function and the best

( )( )thnn 1 portion of the population is chosen asparents. This procedure is repeated n times, every timeusing the survived population from the previous sorting.Thus the relationship between and is as follows:

( )( ) = mmm 1 All new solutions are copied into an external set.Which stores all non-dominated solutions found since thebeginning of the simulation run. After new solutions are

inserted into the external set, a non-domination check ismade on the whole external set and only new non-dominated solutions are retained. If the size of theexternal set exceeds a certain limit, a niching mechanismis used to eliminate closely packed solutions. This allowsmaintenance of diversity in the solutions of the externalset.

Merits & DemeritsWe observe that this algorithm performs a dominationcheck to retain non-dominated solutions and a nichingmechanism to eliminate crowded solutions. Thesefeatures are essential in a good MOEA. Since Kursawehas not simulated it for multi-objective optimization, itremains a challenging task to investigate how thisalgorithm will fare in more complex problems. Becauseof its complications it is not much used by currentresearchers.

4.8 Weightbased Genetic Algorithm

(WBGA)Hajela and Lin (1993)[32[33] proposed a weightbasedgenetic algorithm for multi-objective optimization. As

the name suggests, each objective function if is

multiplied by a weight iw . A GA string represents all

decision variables ix as well as their associated weights

iw . The weighted objective function values are then

added together to calculate the fitness of a solution. Eachindividual in a GA population is assigned a different

weight vector. The key issue in WBGAs is to maintaindiversity in the weight vectors among the populationmembers. In WBGAs the diversity in the weight vectorsis maintained in two ways: A niching method is used only on the sub-string

representing the weight vectors. Carefully chosen subpopulations are evaluated for

different predefined weight vectors, an approachsimilar to that of VEGA.

Sharing function approach

In addition to n decision variables the user codes aweight vector of size M in a string, the weights must bechosen in a way so as to make the sum of all weights inevery string one.

= jii www

Otherwise if only a few Pareto-optimal solutions aredesired, a fixed set of weight vectors may be chosen

before hand and an integer variable wx can be used to

represent one of the weight vectors, kw .A solution x(i) is then assigned a fitness equal to the

weighted sum of the normalized objective function

values computed as follows: ( )minmaxmin))(()( )()( jjjjix

wj

ifffixfwxF =

In order to preserve diversity in the weight vectors, asharing strategy is proposed:

)()( j

w

i

wij xxd =

The sharing function )( ijdsh is computed with a niching

parameter share. Next calculate the shared fitness F .

Algorithm1. For each objective function j, set upper and lower

bounds as maxjf and minjf .

2. For each solution .,...,3,2,1 ni= calculate thedistance

)()( kw

i

wik xxd =

with all solutions nk ...,3,2,1= . Then calculate thesharing function values as follows:

=otherwise

ifdddsh

shareikshareik

ik,0

,)(

Thereafter, calculate the niche count of the solution i as

= )( ikci dshn .

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3 For each solution ni ,....,2,1= follow the entireprocedure below.

Assign fitness iF according to )(ixF calculate the

shared fitness asciii

nFF =

.

When all population members are assigned fitness F ,the proportionate selection is applied to create the matingpool. Thereafter crossover and mutation operators areapplied on the entire string, including the sub-string

representing wx .

Merits & DemeritsSince a WBGA uses a singleobjective GA, not muchchange is needed to convert a simple GA implementationinto a WBGA one. Moreover, the complexity of the

algorithm (with a single extra variable wx ) is smallerthan other multi-objective evolutionary algorithms.

The WBGA uses a proportionate selection procedureon the shared fitness values. Thus, in principle, theWBGA will work in a straightforward manner in findingsolutions for maximization problems. However, ifobjective functions are to be minimized, they are requiredto be converted into a maximization problem. Moreover,for mixed types of objective functions (some are to beminimized and some are to maximized), complicationsmay arise in trying to construct a fitness function.

Weighted sum approaches share a common difficulty

in finding Pareto-optimal solutions in problems havingnon-convex Pareto-optimal region. Thus, a WBGA mayalso face difficulty in solving such problems.

Vector evaluated approachThis approach is similar to VEGA. First a set of Kdifferent weight vectors wk (where k= 1,2,.. K) arechosen. Thereafter, each weight vector wk is used tocompute the weighted normalized fitness for all n

population members. Among them the best Kn

members are grouped together into a subpopulation.

Perform selection, crossover and mutation on pkto createa new population of size Kn , if k < K, increment k by

one and go to step-2. Otherwise, combine all

subpopulations to create the new population kpp = .

If |p| < n, add randomly created solutions to make thepopulation size equal to n.

Merits & DemeritsAs in any weight-based method, knowledge of the weightvectors is also essential in this approach. This method isbetter in complexity that the sharing function approach,

because no pair-wise distance metric is required here.There is also no need for keeping any additional variable

for the weight vectors. However, since GA operators areapplied to each subpopulation independently, an adequatesubpopulation size is required for finding the optimalsolution corresponding to each weight vector. Thisrequires a large population size.

By applying both methods in a number of problems,investigators concluded that the vector-evaluatedapproach is better that the sharing function approach.

4.9. Predator-prey evolution strategy

(PPES)Laumanns et al. (1998)[34] suggested a multi-objectiveevolution strategy (ES), which is different from any ofthe methods discussed so far. This algorithm does not usea domination check to assign fitness to a solution. Insteadthe concept of a predator-prey model is used. Preys

represent a set of solutions (x(i)

, i=1,2,3.. ,n) which areplaced on the vertices of a undirected connected graph.First each predator is randomly placed on any vertex ofthe graph. Each predator is associated with a particularobjective function. Thus at least M predators areinitialized in the graph. Secondly, staying in a vertex, apredator looks around for preys in its neighboringvertices. The Predator catches a prey having the worstvalue of its associated objective function. When a preyx(i) is caught it is erased from the vertex and a newsolution is obtained by mutating (and recombining) arandom prey in the neighborhood of x(i). The new

solution is then kept in the vertex. After this event isover, the predator takes a random walk to any of itsneighboring vertices. The above procedure continues inparallel (or sequential) with all predators until a pre-specified number of iterations have elapsed.

Merits & Demerits

The main advantage of this method is its simplicity.Random walks and replacing worst solutions ofindividual objectives with mutated solutions are allsimple operations, which are easy to implement. Another

advantage of this algorithm is that it does not emphasizenon-dominated solutions directly. Experimentally shownthat the algorithm is sensitive to the choice of mutationstrength parameter and the ratio of predator and prey. Noexplicit operator is used to maintain a spread of solutionsin the obtained non-dominated set.

4.10. Thermodyanamical genetic

algorithm (TDGM)Kita et al. (1996) [35] suggested a fitness function, whichallows a convergence near the Pareto-optimal front and a

diversity-preservation among obtained solutions. Thefitness function is motivated from the thermodynamic

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equilibrium condition, which corresponds to theminimum of the Gibbs energy F, defined as follows:

HTEF = ,

where E is the mean energy, H is the entropy, and T is

the temperature of the system. The relevance of theabove term in multi-objective optimization is as follows.The mean energy is considered as the fitness measure.The second term HT controls the diversity existent in thepopulation members. In this way minimization of F

means minimization of E and maximization of HT. In

the parlance of multi-objective optimization, this meansminimization of the objective function (convergencetowards the Pareto-optimal set) and the maximization ofdiversity in obtained solutions. Since these two goals areprecisely the focus of an MOEA, the analogy between

the principle of finding the minimum free energy state ina thermodynamic system and the working principle of anMOEA is applicable.

Algorithm

1. Select a population size N, maximum number ofgenerations maxt and an annealing schedule )(tT

(monotonically non-increasing function) for thetemperature variation.

2. Set the generation counter t=0 and initialize arandom population )(tP of size N.

3. Using crossover & mutation operator, create theoffspring population )(tQ of size N from )(tP .

Combine the parent and offspring populationstogether and call this

)()()( tQtPtR = .

4. Create an empty set )1( +tP and set the individualcounter i of )1( +tP to one.

5. For every solution )(tRj , construct{ }jtPjtP += )1(),( and calculate the freeenergy fitness of the jth member as follows:

= )()()()( jHtTjEjF k where

),(/)( jtPEjE k and the entropy is

defined as =kk

k PPjH 11 log)( . The termkP1

is the proportion of bit l on the locus k of thepopulation ),( jtP .

After all 2N population members are assigned afitness, find the solution )(tRj which will

minimize F given above . Include )1( + tRj .

6. If Ni , then 1+= ii and go to step 5.

7. If maxtt , then 1+= tt and go to step 3.

The term Ek can be used as the non-dominated rank ofthe kthsolution in the population ),( jtP .

Merits & DemeritsSince both parents and offspring populations arecombined and the best set of N solutions are selectedfrom the diversity preservation point of view, thealgorithm is an elitist algorithm.

The TDGA requires a predefined annealingschedule T(t)- It is interesting to note that thistemperature parameter acts like a normalizationparameter for the entropy term need to balance betweenthe mean energy minimization and entropymaximization. Since this requires a crucial balancing actfor finding a good convergence and spread of solutions,an arbitrary T(t) distribution may not work well in mostproblems.

4.11. Pareto-archived evolution strategy

(PAES)Knowles and Corne (2000) suggested a MOEA, whichuses an evolution strategy. In its simplest form, the PAESuses a (1+1)- ES. The main motivation for using an EScame from their experience in solving real worldtelecommunications network design problem.

At first, a random solution x0 (call this a parent)is chosen. Using a normally distributed probabilityfunction with zero mean and with fixed mutation strengththen mutates it. Let us say that the mutated offspring isC0. Now both of these solutions are compared and thewinner becomes the parent of the next generation. Themain crux of the PAES lies in the way that a winner ischosen in the midst of multiple-objectives.

At any generation t, in addition to the parent P tand the offspring Ct, the PAES, maintains an archive ofthe best solutions found so far. Initially, this archive isempty and as the generations proceed, good solutions are

added to the archive and updated. However a maximumsize is always maintained. First the parent Pt and theoffspring Ct are compared for domination. It generatesthree scenarios: If Ptdominates Ctthe offspring Ctis not accepted and

a new mutated solution is created for furtherprocessing.

If Ct dominates Pt, the offspring is better than theparent. Then solution Ctis accepted as a parent of thenext generation and as copy of it is kept in thearchive.

If both Pt and Ct are non-dominated to each other,then the offspring is compared with the currentarchive:

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The offspring is dominated by a number of thearchive. This means this is not a worth candidate,so the parent Pt again mutated to find newoffspring for further processing.

The offspring dominates a member of thearchive. Then the dominated members aredeleted and the offspring is accepted. Theoffspring becomes the parent of the nextgeneration.

The offspring is not dominated by any memberof the archive and it also does not dominate anymember of the archive. In this case, it belongs tothat non-dominated front in which the archivesolution belongs. In such a case the offspring isadded to the archive only if there is a slotavailable in the latter.

The acceptance of the offspring in the archive does notqualify it to become the parent of the next generation.This is because as for the offspring Ct, the current parentPt is also a member of the archive.

To decide who qualifies as a parent of the nextgeneration, the density of solution in their neighborhoodis checked. The one residing in the least crowded area inthe search space qualifies as the parent. If the archive isfull, the above density-based comparison is performedbetween the parent and the offspring to decide whoremains in the archive. The density calculation isdifferent from the other methods we have discussed so

far.Each objective is divided into 2d equal divisions,

where d is a user defined depth parameter. In this way,the entire search space is divided into (2d)Munique, equalsized M dimensional hypercubes. The archived solutionsare placed in these hypercubes according to theirlocations in the objective space. There after the numberof solutions in each hypercube is counted. If the offspringresides in a less crowded hypercube than the parent, theoffspring becomes the parent of the next generation.Otherwise the parent solution continues to be the parentof the next generation.

Algorithm (Single iteration)

Parent, tP ,offspring tC and an archive tA . The archive

is initialized with the initial random parent solution.

1. If tC is dominated by any member of tA , settt PP =+1 ( tA is not updated). Process is complete.

Otherwise, if tC dominates a set of members from

At:

{ }iCAiiCD ttt = ,|)( , perform the followingstep.

)( ttt CDAA =

{ }ttt CAA =

tt CP =+1

Process is complete. Otherwise go to step 2.

2. Count the number of archived solutions in eachhypercube. The parent tP belongs to a hypercube

having pn solutions. While the offspring belongs to

a hypercube having cn solutions. The highest count

hypercube contains the maximum number ofarchived solutions.

If NAt , include the offspring in the archive or

{ }ttt CAA = and ),(1 ttt PCwinnerP =+ and

return. Otherwise (if NAt = ), check if Ctbelongsto the highest count hypercube. If yes reject Ct, set

tt PP =+1 , and return. Otherwise replace a random

solution r from the highest count hypercube with Ct:

{ }rAA tt =

{ }ttt CAA =

),(1 ttt PCwinnerP =+

The process is complete. The winner

( )tt PC , chooses Ct, if nc< np.Otherwise it chooses Pt.

Merits & DemeritsThe PAES has a direct control on the diversity that canbe archived in the Pareto optimal solutions step-2 of thealgorithm emphasizes the less populated the hypercubesto survive, thereby ensuring diversity. Choosing anappropriate value of d can control the size of thesehypercubes. In addition to choosing an appropriatearchive size (n), the depth parameter d, which directlycontrols the hypercube size, is also an importantparameter.

4.12. Rudolphs Elitist Multi-objective

Evolutionary Algorithm (REMOEA)Rudolph (2001)[36][37] suggested, but did not simulate,a multi-objective evolutionary algorithm that useselitism. Nevertheless, the algorithm provides a usefulplan for introducing elitism into a multi-objective EA. Inits general format, parents are used to create offspringusing genetic operators. Now there are two populations:the parent population Ptand the offspring population Qt.The algorithm works in three steps as follows:

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Algorithm1. Create the parent population Ptto create an offspring

population Qt. Find non-dominated solutions of Qtand put them in Q*. Delete Q*from Qtor Qt = Qt\Q*. Set

0== QP .

2. For each q Q*, find all solutions p Pt that aredominated by q, put these solutions of Pt in a set

D(q). If 0)( }qD , carry out the following:

)(qDPP tt =

{ }qPP = Q* = Q*/ {q}

Otherwise, if D(q)= 0 and q is non-dominated withall existing members of P(t), then perform thefollowing:

{ }qQQ = , Q* = Q* /{q}3. Form a combined population PPP tt =+1 . If

1+tP , then fill Pt+1 with elements from the

following sets in the order mentioned below until

( )tQQQtP *,,)1( =+ .

Merits & DemeritsRudolphs has proven that the above algorithm with apositive variation kernel of its search operators allowsconvergence (of at least one solution in the population) tothe Pareto-optimal front in a finite number of trials infinite search space problem. However, what was not beenproved is a guaranteed maintenance of diversity amongthe Paretooptimal solutions. If a populationbasedevolutionary algorithm converges to a single Pareto-optimal solution, it cannot be justified for its use overother classical algorithms. What makes evolutionaryalgorithms attractive is their ability to find multiplePareto-optimal solutions in one single run.

4.13. Elitist Non-dominated Sorting

Genetic Algorithm (ENSGA)Like Rudolphs algorithm, in ENSGA, the offspringpopulation Qt is first created by using the parentpopulation Pt. However, instead of finding the non-dominated front of Qt, only, first the two populations arecombined together to form Rt of size 2N. Then a non-dominated sorting is used to classify the entire populationRt. Though it requires more effort than Qt alone, butallows a global non-dominated check among theoffspring and parent solutions. Once the non-dominatedsorting is over, the new population is filled by solutionsof different non-dominated fronts, are at a time. Thefilling starts with the best non-dominated front and

continues with solutions of the second non-dominated

front, followed by the third non-dominated front, and soon.

Since the overall population size of Rt is 2N, not allfronts may be accommodating in N slots available in thenew population. All fronts, which could not beaccommodated, are simply deleted. When the lastallowed front is being considered, there may exist moresolutions in the last front than the remaining slots in the

new population. Instead of arbitrarily discarding somemembers from the last front, it would be wise to use aniching strategy to choose the members of the last front,which reside in the least crowded region in that front.

A strategy like the above does not affect theproceedings of the algorithm much in the early stage ofevolution. This is because, early on, there exist manyfronts in the combined population. It is likely thatsolutions of many good non-dominated fronts are alreadyincluded in the new population, before they add up to n.It then hardly matters which solutions is included to fill

up the population. However, during the later stages of thesimulation, it is likely that most solution in thepopulation lie in the best non-dominated front. It is alsolikely that in the combined population Rt of size 2N, thenumber of solutions in the first non-dominated frontexceeds N. The above algorithm then ensures thatniching will choose a diverse set of solutions from thisset. When the entire population converges to the Pareto-optimal front the continuation of this algorithms willensure a better spread among the solutions. Followingshows the step-by-step format of algorithms.

Pt+1

F1 F2 F3 F4

QtPt

F1 F2 F3

After non-dominated sorting

Reject

Fi ure 5. Structural re resentation of ENSGA

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Algorithm1. Combine parent and offspring populations and create

Rt= Pt U Qt. Perform a non-dominated sorting to Rtand identify different fronts: Fi, i= 1,2,etc.

2. Set new population P t+1= 0. Set a counter i=1.Until |P t+1| + Fi< N, perform P t+1 = P t+1Fiandi = i+1.

3. Perform the crowding-sort (Fi, < c) procedure(described below) and include the most widelyspread (N- |Pt+1|) solutions using the crowdingdistance values in the sorted Fito P t+1.

4. Create offspring Qt+1from P t+1by using the crowdedtournament selection, crossover and mutationoperators.

Let us see the following crowding-sort (F,).3. For m= 1,2,.. M, assign a large distance to the

boundary solutions, or dI1m = dIl

m = and for all

other solutions j = 2 to l-1, assigndjj

m= djj

m+ ((fm(Ij+1

m) fm(Ij-1

m))/(fm

max- fm

min))

The index Ij denotes the solution index of the jth

member in the sorted list.

Crowded Tournament Selection OperatorThe crowded comparison operator (dj.

Hence, the one residing in a less crowded area (with alarger crowding distance di) wins. The crowding distancedi can be computed in various ways. In the above wehave mentioned a method.

Merits & Demerits

The solutions are competing based on their crowdingdistances, no niching parameter is required here (such as

share needed in the MOEA , NSGAs & NPGAs). In theabsence of the crowded comparison operator, thisalgorithm also exhibits a convergence proof to thePareto-optimal solution set similar to that in Rudolphsalgorithm, but the population size would grow with the

generation counter. The elitism mechanism does notallow an already found Pareto-optimal solution to bedeleted. However when the crowded comparison isused to restrict the population size, the algorithm loses itsconvergence property.

4.14. Distance Based Pareto Genetic

Algorithm (DBPGA)Osyezka and Kundu (1995) [38][39] suggested an elitistGA a distancebased Pareto genetic algorithm (DPGA),which attempts to emphasize the progress towards the

Pareto optimal front and the diversity along theobtained front by using one fitness measure. Thisalgorithm maintains two populations, One standard GApopulation Ptwhere genetic operation are performed andanother elite population Etcontaining all non-dominatedsolutions found thus far. The working principle of thisalgorithm is as follows:

Algorithm1. Create an initial random population P0of size N and

set the fitness of the first solution as F 1. Setgeneration counter t=0.

2. If t=0, insert the first element of P0in an elite set E0= {1}. For each population member j 2 for t=0 andj 1 for t > 0, perform the following steps to assign afitness value.Calculate the distance dj(k)with each elite member k (with fitness em

(k), m= 1,2,k) as follows:

dj(k)= (( em

(k)- fm(j))/ (em

(k)))2

Find the minimum distance and the index of the elite

member closest to solution j:djmin= min dj

(k),Kj* = { k : dj

(k)= djmin}

If any elite member dominates solution j the fitnessof j is

Fj = max [0, F(e(kj*)- dmin]

otherwise, the fitness of j isFj= F(e

(kj*)) + dminand j is included in Et by eliminating all elitemembers that are dominated by j.

3. Find the maximum fitness value of all elite members:|Et|

Fmax= Max Fik=1

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All elite solutions are assigned fitness Fmax.4. If t < tmaxor any other termination criteria is satisfied,

the process is complete. Otherwise go to step-5.5. Perform selection, crossover and mutation on Pt and

create a new population Pt+1. Set t = t+1 and go to

step-2.

Note that no separate genetic processing is performed onthe elite population Etexplicitly. The fitness Fmaxof elitemembers is used in the assignment of fitness of solutionsof Pt.

Merits & DemeritsFor some sequence of fitness evaluations both goals ofprogressing towards the Pareto-optimal front andmaintaining diversity among solutions are archived

without any explicit niching method.Since the elite size is not restricted, so it will grow toany size. This will increase the computational complexityof the algorithm with generations. Here the choice of theinitial Fi is important. The DPGA fitness assignmentscheme is sensitive to the ordering of individual in apopulation.

5.Future research directionsAlthough a lot of work has been done in this area, mostof it has concentrated on application of conventional orad-hoc techniques to certain difficult problems.

Therefore, there are several research issues that stillremain to be solved. Some of which are discussed here.

1. Applicability of MOEA in more difficult real worldproblems.

2. The stopping criteria of MOGA, because it is notobvious to understand when the population hasreached a point from which no further improvementcan be reached.

3. Choosing the best solution from Pareto optimal set.4. Hybridization of multi-objective EAs.5. Although a lot of work has been done in this area but

the theoretical portion is not so much exploited. So atheory of evolutionary multi-objective optimizationis much needed, examining different fitnessassignment methods in combination with differentselection schemes.

6. The influence of mating restrictions might beinvestigated, although restricted mating is not verywidespread in the field of multi-objective EA.

6.Conclusion and DiscussionIn this paper, we have described an overview of the EAssuch as EP, ES, GP & GA. Then the importance of EA inMulti-objective optimization problems. We attempted to

provide a comprehensive review of the most popularevolutionary-based approaches to multi-objectiveoptimization problems, also a brief discussion of theiradvantages, disadvantages. Finally we discuss the mostpromising area for future research.

Recently multi-objective Evolutionary algorithms areapplied in various fields such as Engineering designproblems, Computer networks, Goal Programming, Gasturbine Engine controller design, and resource schedulingetc [40-43].

Other hybridizations typically enjoy the generic andapplication specific merits of the Individual MOEAtechniques that they integrate [44-47].

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About the Author

Ashish Ghosh is an Associate Professor of the Machine

Intelligence Unit at the Indian Statistical Institute, Calcutta. Hereceived the B.E. degree in Electronics andTelecommunication from the Jadavpur University, Calcutta in1987, and the M.Tech. and Ph.D. degrees in Computer Sciencefrom the Indian Statistical Institute, Calcutta in 1989 and 1993,respectively. He received the prestigious and most covetedYoung Scientists award in Engineering Sciences from theIndian National Science Academy in 1995; and in ComputerScience from the Indian Science Congress Association in 1992.He has been selected as anAssociateof the Indian Academy ofSciences, Bangalore in 1997. He visited the Osaka PrefectureUniversity, Japan with a Post-doctoral fellowship duringOctober 1995 to March 1997; and Hannan University, Japan as

a Visiting scholar during September-October, 1997. DuringMay 1999 he was at the Institute of Automation, ChineseAcademy of Sciences, Beijing with CIMPA (France)fellowship. He was at the German National Research Centerfor Information Technology, Germany, with a German Govt.(DFG) Fellowship during January-April 2000. DuringOctober-December 2003 he was a Visiting Professor at theUniversity of California, Los Angeles. He also visited variousUniversities/Academic Institutes and delivered lectures in

different countries including South Korea, Poland, and TheNetherland. His research interests include EvolutionaryComputation, Neural Networks, Image Processing, Fuzzy Sets

and Systems, Pattern Recognition, and Data Mining. He hasalready published about 55 research papers in internationally

reputed journals and referred conferences, has edited 4 books,and is acting as guest editor of various journals.

Satchidananda Dehuri is currently working as a lecturer inSilicon Institute of Technology, India. He received his M.Sc.(Mathematics) from Sambalpur University in 1998, andM.Tech. (Computer Science) from Utkal University in 2001.He is working toward the Ph.D. degree in the Department ofComputer Science and Application of Utkal University. He has

published 10 papers in various International Conferences. Hisresearch interests include pattern recognition, evolutionaryalgorithm for multi-criterion optimization, and Data Mining &Data Warehousing

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