multi-objective optimization · multi-objective optimization metaheuristics evolutionary algorithms...

Multi-objective Optimization

Carlos A. Coello Coello

Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Pareto Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Multi-objective Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Multi-objective Evolutionary Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Recent Algorithmic Trends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9Use of Other Metaheuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

Artificial Immune Systems (AIS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10Particle Swarm Optimization (PSO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Ant Colony Optimization (ACO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11Other Metaheuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Some Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Some Current Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Cross-References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

Abstract

This chapter provides a short overview of multi-objective optimization us-ing metaheuristics. The chapter includes a description of some of the mainmetaheuristics that have been used for multi-objective optimization. Althoughspecial emphasis is made on evolutionary algorithms, other metaheuristics,such as particle swarm optimization, artificial immune systems, and ant colonyoptimization, are also briefly discussed. Other topics such as applications and

C. A. Coello Coello (�)Departmento de Computación, CINVESTAV-IPN, México D.F., Méxicoe-mail: [email protected]

© Springer International Publishing AG 2018R. Martí et al. (eds.), Handbook of Heuristics,https://doi.org/10.1007/978-3-319-07153-4_17-1

1

mailto:[email protected]

https://doi.org/10.1007/978-3-319-07153-4_17-1

2 C. A. Coello Coello

recent algorithmic trends are also included. Finally, some of the main researchtrends that are worth exploring in this area are briefly discussed.

KeywordsMulti-objective optimization � metaheuristics � evolutionary algorithms �optimization

Introduction

Metaheuristics have been widely used for solving different types of optimizationproblems (see, e.g., [88, 109, 198]).

One particular class of optimization problems involves having two or more(often conflicting) objectives which we aim to optimize at the same time. In fact,such problems, which are called “multi-objective” are quite common in real-worldapplications, and their solution has triggered an important amount of work withinOperations Research [135].

During the last 40 years, a large number of mathematical programming tech-niques have been developed to solve certain specific classes of multi-objectiveoptimization problems. However, such techniques have a relatively limited ap-plicability (e.g., some of them require the first or even the second derivativeof the objective functions and the constraints; others can only deal with convexPareto fronts, etc.). Such limitations have motivated the development of alternativeoptimization methods, from which metaheuristics have become a very popularalternative [43].

From the many metaheuristics currently available, evolutionary algorithmshave been, without doubt, the most popular choice for dealing with any sort ofoptimization problem [21, 85], and multi-objective optimization is, by no means,an exception. Thus, in this chapter, we will focus our discussion mainly on the useof evolutionary algorithms for solving multi-objective optimization problems.

The use of evolutionary algorithms for solving multi-objective optimizationproblems was originally hinted in 1967 [173], but the first actual implementationof what is now called a “multi-objective evolutionary algorithm (MOEA)” wasnot produced until the mid-1980s [178, 179]. However, this area, which is nowcalled “evolutionary multi-objective optimization,” or EMO, has experienced avery important growth, mainly in the last 20 years [41, 42, 52, 201]. The authormaintains the EMOO repository, which, as of December 8, 2017, contains over11,000 bibliographic entries, as well as public-domain implementation of someof the most popular MOEAs. The EMOO repository is located at https://emoo.cs.cinvestav.mx/.

The remainder of this chapter is organized as follows: In section “BasicConcepts”, we provide some basic concepts related to multi-objective optimization,which are required to make this chapter self-contained. The use of evolutionaryalgorithms in multi-objective optimization is motivated in section “Multi-objectiveEvolutionary Algorithms”. A short discussion on other bio-inspired metaheuristics

https://emoo.cs.cinvestav.mx/

https://emoo.cs.cinvestav.mx/

Multi-objective Optimization 3

that have also been used for multi-objective optimization is provided in section “Useof Other Metaheuristics”. Some of the main research topics which are currentlyattracting a lot of attention in the EMO field are briefly discussed in section“Multi-objective Evolutionary Algorithms”. A set of sample applications of MOEAsis provided in section “Some Applications”. Some of the main topics of researchin the EMO field that currently attract a lot of attention are briefly discussedin section “Some Current Challenges”. Such topics include the use of othermetaheuristics. Finally, some conclusions are provided in section “Conclusions”.

Basic Concepts

In this chapter, we focus on the solution of multi-objective optimization problems(MOPs) of the form:

minimize�f1.Ex/; f2.Ex/; : : : ; fk.Ex/

�(1)

subject to the m inequality constraints

gi .Ex/ � 0 i D 1; 2; : : : ; m (2)

and the p equality constraints

hi .Ex/ D 0 i D 1; 2; : : : ; p (3)

where k is the number of objective functions fi W Rn ! R. We call Ex D

Œx1; x2; : : : ; xn�T the vector of decision variables. We wish to determine from among

the set F of all vectors which satisfy (2) and (3) the particular set of valuesx�1 ; x

�2 ; : : : ; x

�n which yield the optimum values of all the objective functions.

Pareto Optimality

It is rarely the case that there is a single point that simultaneously optimizes allthe objective functions. In fact, this situation only arises when there is no conflictamong the objectives, which would make unnecessary the development of specialsolution methods, since this single solution could be reached after the sequentialoptimization of all the objectives, considered separately. Therefore, we normallylook for “trade-offs,” rather than single solutions when dealing with multi-objectiveoptimization problems. The notion of “optimality” normally adopted in this case isthe one originally proposed by Francis Ysidro Edgeworth [63] and later generalizedby the French economist Vilfredo Pareto [152]. Although some authors call thisnotion Edgeworth-Pareto optimality, we will use the most commonly adopted term:Pareto optimality.


We say that a vector of decision variables Ex� 2 F (i.e., a feasible solution) isPareto optimal if there does not exist another Ex 2 F such that fi .Ex/ � fi .Ex

�/

for all i D 1; : : : ; k and fj .Ex/ < fj .Ex�/ for at least one j (assuming that all the

objectives are being minimized).In words, this definition says that Ex� is a Pareto optimal solution if there exists

no feasible vector of decision variables Ex 2 F which would decrease some criterionwithout causing a simultaneous increase in at least one other criterion. Assumingthe inherent conflict normally present among (at least some) objectives, the use ofthis concept normally produces several solutions. Such solutions constitute the so-called Pareto optimal set. The vectors Ex� corresponding to the solutions includedin the Pareto optimal set are called nondominated. The image of the Pareto optimalset under the objective functions (i.e., the objective function values correspondingto the decision variables contained in the Pareto optimal set) is called Pareto front.

Multi-objective Evolutionary Algorithms

The core ideas related to the development of search techniques that simulate themechanism of natural selection (Darwin’s survival of the fittest principle) can betraced back to the early 1930s [75]. However, the three main techniques based onthis notion were developed during the 1960s: genetic algorithms [101], evolutionstrategies [184], and evolutionary programming [74]. These approaches, which arenow generically denominated “evolutionary algorithms,” have been found to be veryeffective for solving single-objective optimization problems [76, 89, 185].

The basic operation of an evolutionary algorithm (EA) is described next. First,a set of potential solutions (called “population”) to the problem being solved israndomly generated. Each solution in the population (called “individual”) encodesall the decision variables of the problem (i.e., each individual contains all thedecision variables of the problem to be solved). The user needs to define a measureof performance for each of the solutions. Such a measure of performance is called“fitness function” and will allow us to know how good is a solution with respect tothe others. Such a fitness function is normally a variation of the objective functionof the problem that we wish to solve (e.g., the objective function that we aimto optimize). Then, a selection mechanism must be applied in order to decidewhich individuals will “mate.” This selection process is generally stochastic andis normally based on the fitness contribution of each individual (i.e., the fittestindividuals have a higher probability of being selected). Upon mating, a set of“offspring” (or children) are generated. Such offspring are “mutated” (this operatorproduces a small random change, with a low probability, on the contents of anindividual) and constitute the new population to be evaluated at the next iteration(each iteration is called a “generation”). This process is repeated until reaching astopping condition (normally, a maximum number of generations defined by theuser) [64].


The main motivation for using EAs for solving multi-objective optimizationproblems relies on their population-based nature, which allows them to generate(if properly manipulated) several elements of the Pareto optimal set, in a singlerun. In contrast, mathematical programming techniques normally generate a singleelement of the Pareto optimal set per run. Additionally, the so-called multi-objectiveevolutionary algorithms (MOEAs) are less susceptible to the shape and continuityof the Pareto front and require less specific domain information to operate [41].

MOEAs extend a traditional (single-objective) EA in two main aspects:

• The selection mechanism: In this case, the aim is to select nondominatedsolutions and to consider all the nondominated solutions in a population to beequally good (unless there is some specific preference from the user, all theelements of the Pareto optimal set are equally good).

• A diversity maintenance mechanism: Because of stochastic noise, EAs tend toconverge to a single solution if run for a sufficiently long time [89]. In order toavoid this, it is necessary to block the selection mechanism in a MOEA, favoringthe diversity of solutions, so that several elements of the Pareto optimal set canbe generated in a single run.

Regarding selection, early MOEAs relied on the use of aggregating functions(mainly linear) [93] and relatively simple population-based approaches [179]. How-ever, such approaches have evident drawbacks (i.e., the use of linear aggregatingfunctions does not allow the generation of non-convex portions of the Pareto frontregardless of the weights combination that is adopted [49]). Toward the mid-1990s, MOEAs started to adopt variations of the so-called Pareto ranking selectionmechanism. This approach was originally proposed by David E. Goldberg in hisseminal book on genetic algorithms [89], and it consists of sorting the populationof an EA based on Pareto optimality, such that all nondominated individualsare assigned the same rank (or importance). The aim is that all nondominatedindividuals get the same probability of being selected and that such probability ishigher than the one corresponding to individuals which are dominated. Althoughconceptually simple, this sort of selection mechanism allows for a wide variety ofpossible implementations [41, 52].

Regarding diversity maintenance, a wide variety of methods have been proposedin the specialized literature to maintain diversity in a MOEA. Such approachesinclude fitness sharing and niching [53, 91], clustering [203, 230], geographicallybased schemes [120], the use of entropy [46, 117], and parallel coordinates [100],among others. In all cases, the core idea behind diversity maintenance mechanismsis to penalize solutions that are too close from each other in some space (i.e.,decision variable or objective function space or even both). Most MOEAs penalizesolutions that are too close from each other in objective function space, because itis normally aimed to have solutions well-distributed along the Pareto front.

Additionally, some researchers have proposed the use of mating restrictionschemes (which imposes rules on the individuals that can be recombined) [188,230].Furthermore, the use of relaxed forms of Pareto dominance has also become rela-


tively popular in recent years, mainly as an archiving technique which encouragesdiversity, while allowing the archive to regulate convergence (the most popular ofsuch mechanisms is, with no doubt, �-dominance [124], which has been adopted insome approaches such as �-MOEA [57]).

A third component of modern MOEAs is elitism, which normally consists ofusing an external archive (also called “secondary population”) that may (or maynot) interact in different ways with the main (or “primary”) population of the MOEAduring selection. The main purpose of this archive is to store all the nondominatedsolutions generated throughout the search process, while removing those thatbecome dominated later in the search (called local nondominated solutions). Theapproximation of the Pareto optimal set produced by a MOEA is thus the finalcontents of this archive. It is important to emphasize that the use of elitism isnot only advisable (the lack of elitism could make us lose nondominated solutionsgenerated during the search), but it is also required because of theoretical reasons(elitism is required in order to guarantee convergence of a MOEA to the Paretooptimal set as proved in [174]).

It is worth noticing that, in practice, external archives are normally bounded to acertain maximum number of solutions. This was originally done in some MOEAsthat used the external archive during the selection stage (see, e.g., [230]). In sucha case, allowing the size of the archive to grow too much dilutes the selectionpressure, which has a negative effect on the performance of the MOEA. However,most modern MOEAs bound the size of the external archive, even if the archiveis not used during the selection process, mainly because of practical reasons (thismakes it easier to compare results with respect to other MOEAs).

An important remark is that the use of a plus (+) selection is another possibleelitist mechanism. Under this sort of selection scheme, the population of parentscompetes with the population of offspring (both populations are of the same size),and then we keep only the best half. This sort of selection scheme has been relativelypopular in single-objective optimization and has been also adopted in some modernMOEAs (see, e.g., [56]), but it is less popular than the use of external archives.

Multi-objective Evolutionary Algorithms

In spite of the very large number of publications related to MOEAs that can befound in the literature, there is only a handful of algorithms that are actually usedby a significant number of researchers and/or practitioners around the world.

1. Strength Pareto Evolutionary Algorithm 2 (SPEA2): This is an updatedversion of the Strength Pareto Evolutionary Algorithm (SPEA) proposed in thelate 1990s [230] whose main features are the following. It adopts an externalarchive (called the external nondominated set), which stores the nondominatedsolutions previously generated and participates in the selection process (togetherwith the main population). For each individual in this archive, a strength value iscomputed. This strength value is proportional to the number of solutions that a


certain individual dominates. In SPEA, the fitness of each member of the currentpopulation is computed according to the strengths of all external nondominatedsolutions that dominate it. As the size of the external nondominated set growstoo much, this significantly reduces the selection pressure and slows down thesearch. In order to avoid this, SPEA adopts a clustering technique that prunes thecontents of the external nondominated set so that its size remains below a certain(pre-defined) threshold. SPEA standardized the use of external archives as theelitist mechanism of a MOEA, although this sort of mechanism had been usedbefore by other researchers (see, e.g., [107]). SPEA2 has three main differenceswith respect to the original SPEA [231]: (1) it incorporates a fine-grained fitnessassignment strategy which takes into account, for each individual, both thenumber of individuals that dominate it and the number of individuals by which itis dominated, (2) it uses a nearest neighbor density estimation technique whichguides the search more efficiently (i.e., a more efficient clustering algorithm isadopted), and (3) it uses an enhanced archive truncation method that guaranteesthe preservation of boundary solutions (this fixes a bug from the original SPEA).

2. Pareto Archived Evolution Strategy (PAES): This is perhaps the most simpleMOEA than can be possibly designed. It was proposed by Knowles and Corne[119], and it consists of a (1+1) evolution strategy (i.e., a single parent thatgenerates a single offspring through the application of mutation to the parent)in combination with a historical archive that stores the nondominated solutionspreviously found. This archive is used as a reference set against which eachmutated individual is being compared. Such (external) archive adopts a crowdingprocedure that divides objective function space in a recursive manner. Then, eachsolution is placed in a certain grid location based on the values of its objectives(which are used as its “coordinates” or “geographical location”). A map ofsuch a grid is maintained, indicating the number of solutions that reside in eachgrid location. When a new nondominated solution is ready to be stored in thearchive, but there is no room for it (the size of the external archive is bounded),a check is made on the grid location to which the solution would belong. Ifthis grid location is less densely populated than the most densely populatedgrid location, then a solution (randomly chosen) from this heavily populatedgrid location is deleted to allow the storage of the newcomer. This aims toredistribute solutions, favoring the less densely populated regions of the Paretofront. Since the procedure is adaptive, no extra parameters are required (exceptfor the number of divisions of the objective space).

3. Nondominated Sorting Genetic Algorithm II (NSGA-II): This is a heavilyrevised version of the Nondominated Sorting Genetic Algorithm (NSGA),which was originally proposed in the mid-1990s [193] as a straightforwardimplementation of the Pareto ranking algorithm described by Goldberg in hisbook [89]. NSGA was, however, slow and produced poorer results than othernon-elitist MOEAs available in the mid-1990s, such as MOGA [77] and NPGA[103]. NSGA-II adopts a more efficient ranking procedure than its predecessor.


Additionally, it estimates the density of solutions surrounding a particularsolution in the population by computing the average distance of two points oneither side of this point along each of the objectives of the problem. This valueis the so-called crowding distance. During selection, NSGA-II uses a crowdedcomparison operator which takes into consideration both the nondomination rankof an individual in the population and its crowding distance (i.e., nondominatedsolutions are preferred over dominated solutions, but between two solutions withthe same nondomination rank, the one that resides in the less crowded regionis preferred). NSGA-II does not use an external archive as most of the modernMOEAs in current use. Instead, the elitist mechanism of NSGA-II consistsof combining the best parents with the best offspring obtained (i.e., this is a(� C �)-selection used in evolution strategies). Due to its clever mechanisms,NSGA-II is much more efficient (computationally speaking) than its predecessor,and its performance is so good that it has become very popular, triggeringa significant number of applications and becoming some sort of landmarkagainst which new MOEAs have been compared during more than 10 years, inorder to merit publication. More recently, the Nondominated Sorting GeneticAlgorithm III (NSGA-III) [54] was introduced. NSGA-III still adopts the sameNSGA-II framework (i.e., it still performs a classification of the population innondominated levels). However, its mechanism to maintain diversity is basedon the use of a number of well-spread reference points which are adaptivelyupdated. The NSGA-III does not require any additional parameters (other thanthose associated to the genetic algorithm that is used as its search engine), andit is designed to deal with many-objective optimization problems (i.e., problemshaving 4 or more objectives).

4. Pareto Envelope-based Selection Algorithm (PESA): This algorithm wasproposed by Corne et al. [44] and uses a small internal population and a largerexternal (or secondary) population. PESA adopts the same adaptive grid fromPAES to maintain diversity. However, its selection mechanism is based on thecrowding measure. This same crowding measure is used to decide what solutionsto introduce into the external population (i.e., the archive of nondominatedvectors found along the evolutionary process). Therefore, in PESA, the externalmemory plays a crucial role in the algorithm since it determines not only thediversity scheme but also the selection performed by the method. There is alsoa revised version of this algorithm, called PESA-II [45], which is identical toPESA, except for the fact that a region-based selection is used in this case.In region-based selection, the unit of selection is a hyperbox rather than anindividual. The procedure consists of selecting (using any of the traditionalselection techniques [90]) a hyperbox and then randomly selecting an individualwithin such hyperbox. The main motivation of this approach is to reduce thecomputational costs associated with traditional MOEAs (i.e., those based onPareto ranking).

5. Multi-objective Evolutionary Algorithm based on Decomposition (MOEA/D):This approach was proposed by Zhang and Li [221]. The main idea of this


algorithm is to decompose a multi-objective optimization problem into severalscalar optimization subproblems which are simultaneously optimized. Thedecomposition process requires the use of weights, but the authors provide amethod to generate them. During the optimization of each subproblem, onlyinformation from the neighboring subproblems is used, which allows thisalgorithm to be effective and efficient. MOEA/D is generally considered oneof the most powerful MOEAs currently available, as has been evidenced byseveral comparative studies.

Recent Algorithmic Trends

Many other MOEAs have been proposed in the specialized literature (see, e.g.,[39, 57, 200, 202, 219]), but they will not be discussed here due to obvious spacelimitations. A more interesting issue, however, is to try to predict which sort ofMOEA will become predominant in the next few years.

Efficiency is, for example, a concern nowadays, and several approaches havebeen developed in order to improve the efficiency of MOEAs (see, e.g., [113]). Also,the use of fitness approximation, fitness inheritance, surrogates, and other similartechniques has become more common in recent years, which is a clear indicationof the more frequent use of MOEAs for the solution of computationally expensiveproblems (see, e.g., [118, 157, 166, 169, 177, 217, 222]).

However, the most promising research line within algorithmic design seems tobe the use of a performance measure in the selection mechanism of a MOEA. Thisresearch trend formally started with the Indicator-Based Evolutionary Algorithm(IBEA) [229], although this idea had been already formulated and used (in differentways) by other authors (see, e.g., [106, 120]). Nevertheless, the most representativeMOEA within this family is perhaps the S metric selection Evolutionary Multi-Objective Algorithm (SMS-EMOA) [19, 67]. SMS-EMOA was originally proposedby Emmerich et al. [67] and is based on NSGA-II. SMS-EMOA creates an initialpopulation, and then, it generates only one solution by iteration using the operators(crossover and mutation) of the NSGA-II. After that, it applies Pareto ranking.When the last front has more than one solution, SMS-EMOA uses the hypervolumecontribution to decide which solution will be removed. The Hypervolume (alsoknown as the S metric or the Lebesgue Measure) of a set of solutions measuresthe size of the portion of objective space that is dominated by those solutionscollectively. This is the only unary performance indicator which is known to bePareto compliant [232]. Beume et al. [19] proposed not to use the contribution to thehypervolume indicator when in the Pareto ranking we obtain more than one front.In that case, they proposed to use the number of solutions which dominate to onesolution (the solution that is dominated by more solutions is removed). The authorsargued that the motivation for using the hypervolume indicator is to improve thedistribution in the nondominated front and then it is not necessary in fronts differentto the nondominated front.


The hypervolume indicator has attracted a lot of attention from researchers dueto its interesting theoretical properties. For example, it has been proven that themaximization of the hypervolume is equivalent to finding the Pareto optimal set[73]. Empirical studies have shown that (for a certain number of points previouslydetermined) the maximization of the hypervolume does indeed produce subsets ofthe Pareto front which are well-distributed [67, 120].

However, there are also practical reasons for being interested in indicator-basedselection. The main one is that MOEAs such as SMS-EMOA seem to continueworking as usual as we increase the number of objectives, as opposed to Pareto-based selection mechanisms which are known to degrade quickly in problemshaving more than three objectives (this research area is known as many-objectiveoptimization). Although the reasons for the poor scalability of Pareto-based MOEAsrequires further study (see, e.g., [182]), the need for scalable selection mechanismshas triggered an important amount of research around indicator-based MOEAs.The main drawback of adopting the hypervolume in the selection mechanismof a MOEA is its extremely high computational cost. One possible alternativefor dealing with this high computational cost is to estimate the hypervolumecontribution. However, MOEAs designed around this idea (e.g., HyPE [7]) seemto have a poor performance with respect to those that adopt exact hypervolumecontributions. An alternative is to rely on other performance indicators. In thisregard, several researchers have proposed the design of selection mechanisms basedon performance measures such as �p [168, 183] and R2 [24, 94, 99, 154]. The useof the maximin [10, 132] is another intriguing alternative, as this expression seemsto be equivalent to the use of the �-indicator [233].

Use of Other Metaheuristics

A wide variety of other bio-inspired metaheuristics have become popular in thelast few years for solving optimization problems [43]. Multi-objective extensions ofmany of these metaheuristics already exist [41], but few efforts have been made toactually exploit the main specific features of each of them. There are also few effortsin trying to understand the types of problems in which each of these metaheuristicscan be more suitable.

Next, we briefly review three popular bio-inspired metaheuristics that are goodcandidates for being used as multi-objective optimizers, but many other choices alsoexist (see, e.g., [41]).

Artificial Immune Systems (AIS)

Our natural immune system has provided a fascinating metaphor for developing anew bio-inspired metaheuristic. Indeed, from a computational point of view, ourimmune system can be considered as a highly parallel intelligent system that isable to learn and retrieve previously acquired knowledge (i.e., it has “memory”),


when solving highly complex recognition and classification tasks. This motivatedthe development of the so-called artificial immune systems (AISs) during the early1990s [50, 51].

AISs were identified in the early 1990s as a useful mechanism to maintaindiversity in the context of multimodal optimization [78, 191]. Smith et al. [192]showed that fitness sharing can emerge when their emulation of the immune systemis used. Furthermore, the approach that they proposed turns out to be more efficient(computationally speaking) than traditional fitness sharing [53], and it does notrequire additional information regarding the number of niches to be formed.

Over the years, a wide variety of multi-objective extensions of AISs have beenproposed (see, e.g., [27, 32, 38, 79, 130, 155]). However, most of the algorithmictrends in MOEAs have had a delayed arrival in multi-objective AISs. Also, the highpotential of multi-objective AISs for pattern recognition and classification tasks hasbeen scarcely exploited.

For more information on multi-objective AISs, the interested reader is referredto [28, 80].

Particle Swarm Optimization (PSO)

This metaheuristic is inspired on the movements of a flock of birds seeking food,and it was originally proposed in the mid-1990s [116]. In the particle swarmoptimization algorithm, the behavior of each particle (i.e., individual) is affectedby either the best local (within a certain neighborhood) or the best global (i.e.,with respect to the entire swarm or population) individual. PSO allows particlesto benefit from their past experiences (a mechanism that does not exist in traditionalevolutionary algorithms) and uses neighborhood structures that can regulate thebehavior of the algorithm.

The similarity of PSO with EAs has made possible the quick developmentof an important number of multi-objective variants of this metaheuristic (see,e.g., [40, 139, 142, 153, 165, 216]). Unlike AISs, most algorithmic trends adoptedwith MOEAs have quickly been incorporated into multi-objective particle swarmoptimizers (MOPSOs). Nevertheless, few PSO models have been used for multi-objective optimization, and the study of the specific features that could makeMOPSOs advantageous over other metaheuristics in some specific classes ofproblems is still a pending task.

For more information on MOPSOs, the interested reader is referred to [11, 37,41, 61, 167].

Ant Colony Optimization (ACO)

This metaheuristic was inspired on the behavior observed in colonies of realants seeking for food. Ants deposit a chemical substance on the ground, calledpheromone [60], which influences the behavior of the ants: they tend to take those


paths in which there is a larger amount of pheromone. Therefore, pheromone trailscan be seen as an indirect communication mechanism used by the ants (which can beseen as agents that interact to solve complex tasks). This interesting behavior of antsgave rise to a metaheuristic called ant system, which was originally applied to thetraveling salesperson problem. Nowadays, the several variations of this algorithmthat have been developed over the years are collectively denominated by ant colonyoptimization (ACO), and they have been applied to a wide variety of domains,including continuous optimization problems.

Several multi-objective versions of ACO are currently available (see, e.g., [2,4, 70, 111, 137, 140, 170]). However, the algorithmic trends developed in MOEAshave had a slow delay in being incorporated into multi-objective ACO algorithms.Additionally, the use of alternative ACO models for multi-objective optimizationhas been relatively scarce.

For more information on multi-objective ACO, the interested reader is referredto [4, 41, 82].

Other Metaheuristics

Many other metaheuristics have been extended to deal with multi-objective opti-mization problems, including simulated annealing [48,65,189,190,194], differentialevolution [134, 138, 197, 199, 204, 207, 208], tabu search [30, 86, 95, 149], scattersearch [12, 15, 16, 158, 160, 211], and artificial bee colony [33, 58, 127, 159],among many others. However, their discussion was omitted due to obvious spaceconstraints.

Some Applications

Multi-objective metaheuristics have been extensively applied to a wide variety ofdomains. Next, we will provide a short list of sample applications classified in threelarge groups: (1) engineering, (2) industrial, and (3) scientific. Specific areas withineach of these large groups are also identified.

By far, engineering applications are the most popular in the current literature onmulti-objective metaheuristics. This is not surprising if we consider that engineeringdisciplines normally have problems with better understood mathematical models. Arepresentative sample of engineering applications is the following:

• Electrical engineering [161, 176]• Hydraulic engineering [186, 196]• Structural engineering [218, 224]• Aeronautical engineering [6, 172]• Robotics [115, 220]• Control [123, 131]• Telecommunications [105, 141]


• Civil engineering [145, 214]• Transport engineering [8, 223]

Industrial applications are the second most popular in the literature on multi-objective metaheuristics. A representative sample of industrial applications of multi-objective metaheuristics is the following:

• Design and manufacture [9, 17, 228]• Scheduling [26, 29]• Management [34, 69]

Finally, there are several publications devoted to scientific applications. Forobvious reasons, computer science applications are the most popular in the literatureon multi-objective metaheuristics. A representative sample of scientific applicationsis the following:

• Chemistry [13, 71, 126, 227]• Physics [14, 171]• Medicine [97, 122]• Computer science [146, 175]

This sample of applications should give at least a rough idea of the increasinginterest of researchers for adopting multi-objective metaheuristics in practically allkinds of disciplines.

Some Current Challenges

The existence of challenging, but solvable problems, is a key issue to preservethe interest in a research discipline. Although multi-objective optimization usingmetaheuristics is a discipline in which a very important amount of research hasbeen conducted, mainly within the last 15 years, several interesting problems stillremain open. Additionally, the research conducted so far has also led to new andintriguing topics. The following is a small sample of open problems that currentlyattract a significant amount of research within this area:

• Scalability: Although multi-objective metaheuristics have been commonly usedfor a wide variety of applications, they have certain limitations. For example, asindicated before, selection mechanisms based on Pareto optimality are known todegrade quickly as we increase the number of objectives. The reason is that, aswe increase the number of objectives, unless there is a significant increase in thepopulation size, all the individuals will quickly become nondominated, whichwill cause stagnation (i.e., no individual will be better than the others, whichmakes the selection mechanism totally useless). In fact, there is experimentalevidence that indicates that, when dealing with ten objectives, random sampling


performs better than Pareto-based selection [121]. There are alternative rankingtechniques that can be used to deal with these problems having four or moreobjectives [83, 129], but it is also possible to use relaxed forms of Paretodominance [59, 72], dimensionality reduction techniques [23, 128], or indicator-based selection mechanisms [19, 229]. It is worth indicating that scalability indecision variable space (i.e., the capability of multi-objective metaheuristics fordealing with large-scale problems) has been scarcely studied in the specializedliterature (see, e.g., [5, 62]).

• Incorporation of user’s preferences: It is normally the case that the user doesnot need the entire Pareto front of a problem but only a certain portion of it. Forexample, solutions lying at the extreme parts of the Pareto front are normallyunnecessary since they represent the best value for one objective, but the worstfor the others. Thus, if the user has at least a rough idea of the sort of trade-offs that he/she aims to find, it is desirable to be able to explore in more detailonly the nondominated solutions within the neighborhood of such trade-offs.This is possible, if we use, for example, biased versions of Pareto ranking [47]or some multi-criteria decision- making technique, from the many developedin Operations Research [22, 35, 150]. In spite of the importance of preferenceincorporation in real-world applications, the use of these schemes in multi-objective metaheuristics is still relatively scarce [31, 68, 81, 110].

• Dealing with expensive problems: There is an important number of real-world multi-objective problems for which a single evaluation of an objectivefunction may take several minutes, hours, or even days [6]. Evidently, suchproblems require special techniques to be solvable using MOEAs [177]. Themain approaches that have been developed in this area can be roughly dividedinto three main groups:

1. Use of parallelism: This is the most obvious approach given the currentaccess to low-cost parallel architectures (e.g., GPUs [18, 187]). It is worthnoting, however, that in spite of the existence of interesting proposals (see,e.g., [3, 136]), the basic research in this area has remained scarce, since mostpublications involving parallel MOEAs focus on specific applications or onparallel extensions of specific MOEAs, but rarely involve basic research.

2. Surrogates: In this case, knowledge of past evaluations of a MOEA isused to build an empirical model that approximates the fitness functions tobe optimized. This approximation can then be used to predict promisingnew solutions at a smaller evaluation cost than that of the original problem[114, 118]. Current functional approximation models include polynomials(response surface methodologies [162]), neural networks (e.g., multilayerperceptrons (MLPs) [102, 108, 156]), radial-basis function (RBF) networks[147, 205, 213], support vector machines (SVMs) [1, 20], Gaussian processes[25, 206], and Kriging [66, 163] models. The use of statistical modelsdeveloped in the mathematical programming literature in combination with


MOEAs is also an interesting alternative (see, e.g., the multi-objective P-algorithm [151, 226]).

3. Fitness approximation: The idea of these approaches is to predict the fitnessvalue of an individual without performing its actual evaluation. From theseveral techniques available, fitness inheritance has been the most popular inmulti-objective optimization [164], but the research in this area has remainedscarce.

• Theoretical foundations: Although an important effort has been made in recentyears to provide a solid theoretical foundation to this field, a lot of work is stillrequired, and the work done in the Operations Research community can providesome useful hints (see, e.g., [151, 225]). The most relevant theoretical work inthis area includes topics such as convergence [174, 209], archiving [180, 181],algorithm complexity [143, 144], and run-time analysis [87, 125].

• Constraint-handling: The use of multi-objective optimization concepts to han-dle constraints in single-objective evolutionary algorithms has been a relativelypopular research area (see, e.g., [84,98,133,195,210,215]). In contrast, however,the design of constraint-handling mechanisms for MOEAs has not been thatpopular (see, e.g., [92, 96, 104, 112, 148, 212]). This is remarkable, given theimportance of constraints in real-world applications, but this may be due tothe fact that some modern MOEAs (e.g., NSGA-II) already incorporated aconstraint-handling mechanism. Additionally, the use of benchmarks containingconstrained test problems (e.g., [55]) has not been very popular in the specializedliterature. Most of the current research in this area has focused on extendingthe Pareto optimality relation in order to incorporate constraints (e.g., givingpreference to feasibility over dominance, such that an infeasible solution isdiscarded even if it is nondominated). Also, the use of penalty functions that“punish” a solution for not being feasible are easy to incorporate into a MOEA[36]. However, topics such as the design of constraint-handling mechanismsfor dealing with equality constraints, the design of scalable test functions thatincorporate constraints of different types (linear, nonlinear, equality, inequality),and the study of mechanisms that allow an efficient exploration of constrainedsearch spaces in MOPs remain practically unexplored.

Conclusions

In this chapter, we have provided some basic concepts related to multi-objectiveoptimization using metaheuristics. This overview has included basic conceptsrelated to multi-objective optimization in general, as well as some algorithmicdetails, with a particular emphasis on multi-objective evolutionary algorithms.

Some of the recent algorithmic trends have also been discussed, and some sampleapplications have been addressed. In general, breadth has been favored over depth,but a significant number of bibliographic references are provided for those interestedin gaining an in-depth knowledge of any of the topics discussed herein.


The information provided in this chapter aims to serve as a general overview ofthe field and to motivate the interest of the reader for pursuing research in this area.As could be seen, several research opportunities are still available for newcomers.

Cross-References

�Ant Systems�Evolutionary Algorithms�General Concepts of Metaheuristics�Genetic Algorithms� Implementation of Metaheuristics�Metaheuristics� Particle Swarm Methods�Theoretical Analysis of Stochastic Search Algorithms

Acknowledgements The author acknowledges support from CONACYT project no. 221551.

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