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Genetic Algorithms for Multiobjective Optimization:Formulation, Discussion and Generalization∗
Carlos M. Fonseca† and Peter J. Fleming‡
Dept. Automatic Control and Systems Eng.University of SheffieldSheffield S1 4DU, U.K.
Abstract
The paper describes a rank-based fitness as-signment method for Multiple Objective Ge-netic Algorithms (MOGAs). Conventionalniche formation methods are extended to thisclass of multimodal problems and theory forsetting the niche size is presented. The fit-ness assignment method is then modified toallow direct intervention of an external deci-sion maker (DM). Finally, the MOGA is gen-eralised further: the genetic algorithm is seenas the optimizing element of a multiobjectiveoptimization loop, which also comprises theDM. It is the interaction between the twothat leads to the determination of a satis-factory solution to the problem. Illustrativeresults of how the DM can interact with thegenetic algorithm are presented. They alsoshow the ability of the MOGA to uniformlysample regions of the trade-off surface.
1 INTRODUCTION
Whilst most real world problems require the simulta-neous optimization of multiple, often competing, cri-teria (or objectives), the solution to such problems isusually computed by combining them into a single cri-terion to be optimized, according to some utility func-tion. In many cases, however, the utility function isnot well known prior to the optimization process. Thewhole problem should then be treated as a multiobjec-tive problem with non-commensurable objectives. Inthis way, a number of solutions can be found whichprovide the decision maker (DM) with insight into thecharacteristics of the problem before a final solution ischosen.
∗in Genetic Algorithms: Proceedings of the Fifth Inter-national Conference (S. Forrest, ed.), San Mateo, CA: Mor-gan Kaufmann, July 1993.
Multiobjective optimization (MO) seeks to optimizethe components of a vector-valued cost function. Un-like single objective optimization, the solution to thisproblem is not a single point, but a family of pointsknown as the Pareto-optimal set. Each point in thissurface is optimal in the sense that no improvementcan be achieved in one cost vector component that doesnot lead to degradation in at least one of the remainingcomponents. Assuming, without loss of generality, aminimization problem, the following definitions apply:
Definition 1 (inferiority)A vector u = (u1, . . . , un) is said to be inferior to v =(v1, . . . , vn) iff v is partially less than u (v p< u), i.e.,
∀ i = 1, . . . , n , vi ≤ ui ∧ ∃ i = 1, . . . , n : vi < ui
Definition 2 (superiority)A vector u = (u1, . . . , un) is said to be superior tov = (v1, . . . , vn) iff v is inferior to u.
Definition 3 (non-inferiority)Vectors u = (u1, . . . , un) and v = (v1, . . . , vn) are saidto be non-inferior to one another if v is neither infe-rior nor superior to u.
Each element in the Pareto-optimal set constitutesa non-inferior solution to the MO problem. Non-inferior solutions have been obtained by solving ap-propriately formulated NP problems, on a one at atime basis. Methods used include the weighted sumapproach, the ε-constraint method and goal program-ming. Within the goal programming category, thegoal attainment method has shown to be particu-larly useful in Computer Aided Control System Design(CACSD) (Fleming, 1985; Farshadnia, 1991; Fleminget al., 1992). Multicriteria target vector optimizationhas recently been used in combination with genetic al-gorithms (Wienke et al., 1992).
By maintaining a population of solutions, genetic al-gorithms can search for many non-inferior solutions inparallel. This characteristic makes GAs very attrac-tive for solving MO problems.
2 VECTOR EVALUATED GENETIC
ALGORITHMS
Being aware of the potential GAs have in multiob-jective optimization, Schaffer (1985) proposed an ex-tension of the simple GA (SGA) to accommodatevector-valued fitness measures, which he called theVector Evaluated Genetic Algorithm (VEGA). The se-lection step was modified so that, at each generation,a number of sub-populations was generated by per-forming proportional selection according to each ob-jective function in turn. Thus, for a problem with qobjectives, q sub-populations of size N/q each wouldbe generated, assuming a population size of N . Thesewould then be shuffled together to obtain a new popu-lation of size N , in order for the algorithm to proceedwith the application of crossover and mutation in theusual way.
However, as noted by Richardson et al. (1989), shuf-fling all the individuals in the sub-populations togetherto obtain the new population is equivalent to linearlycombining the fitness vector components to obtain asingle-valued fitness function. The weighting coeffi-cients, however, depend on the current population.
This means that, in the general case, not only willtwo non-dominated individuals be sampled at differ-ent rates, but also, in the case of a concave trade-offsurface, the population will tend to split into differ-ent species, each of them particularly strong in one ofthe objectives. Schaffer anticipated this property ofVEGA and called it speciation. Speciation is unde-sirable in that it is opposed to the aim of finding acompromise solution.
To avoid combining objectives in any way requires adifferent approach to selection. The next section de-scribes how the concept of inferiority alone can be usedto perform selection.
3 A RANK-BASED FITNESS
ASSIGNMENT METHOD FOR
MOGAs
Consider an individual xi at generation t which is dom-
inated by p(t)i individuals in the current population. Its
current position in the individuals’ rank can be givenby
rank(xi, t) = 1 + p(t)i .
All non-dominated individuals are assigned rank 1, seeFigure 1. This is not unlike a class of selection meth-ods proposed by Fourman (1985) for constrained opti-mization, and correctly establishes that the individuallabelled 3 in the figure is worse than individual labelled2, as the latter lies in a region of the trade-off which isless well described by the remaining individuals. The
13511211f1f21 35
11 2
1
1
f1
f2
Figure 1: Multiobjective Ranking
method proposed by Goldberg (1989, p. 201) wouldtreat these two individuals indifferently.
Concerning fitness assignment, one should note thatnot all ranks will necessarily be represented in the pop-ulation at a particular generation. This is also shownin the example in Figure 1, where rank 4 is absent.The traditional assignment of fitness according to rankmay be extended as follows:
1. Sort population according to rank.
2. Assign fitnesses to individuals by interpolatingfrom the best (rank 1) to the worst (rank n∗ ≤ N)in the usual way, according to some function, usu-ally linear but not necessarily.
3. Average the fitnesses of individuals with the samerank, so that all of them will be sampled at thesame rate. Note that this procedure keeps theglobal population fitness constant while maintain-ing appropriate selective pressure, as defined bythe function used.
The fitness assignment method just described appearsas an extension of the standard assignment of fitnessaccording to rank, to which it maps back in the case ofa single objective, or that of non-competing objectives.
4 NICHE-FORMATION METHODS
FOR MOGAs
Conventional fitness sharing techniques (Goldberg andRichardson, 1987; Deb and Goldberg, 1989) have beenshown to be to effective in preventing genetic drift, inmultimodal function optimization. However, they in-troduce another GA parameter, the niche size σshare,which needs to be set carefully. The existing theoryfor setting the value of σshare assumes that the solu-tion set is composed by an a priori known finite num-ber of peaks and uniform niche placement. Upon con-vergence, local optima are occupied by a number ofindividuals proportional to their fitness values.
On the contrary, the global solution of an MO prob-lem is flat in terms of individual fitness, and there isno way of knowing the size of the solution set before-hand, in terms of a phenotypic metric. Also, localoptima are generally not interesting to the designer,who will be more concerned with obtaining a set ofglobally non-dominated solutions, possibly uniformlyspaced and illustrative of the global trade-off surface.The use of ranking already forces the search to concen-trate only on global optima. By implementing fitnesssharing in the objective value domain rather than thedecision variable domain, and only between pairwisenon-dominated individuals, one can expect to be ableto evolve a uniformly distributed representation of theglobal trade-off surface.
Niche counts can be consistently incorporated into theextended fitness assignment method described in theprevious section by using them to scale individual fit-nesses within each rank. The proportion of fitness allo-cated to the set of currently non-dominated individualsas a whole will then be independent of their sharingcoefficients.
4.1 CHOOSING THE PARAMETER σshare
The sharing parameter σshare establishes how far aparttwo individuals must be in order for them to decreaseeach other’s fitness. The exact value which would allowa number of points to sample a trade-off surface onlytangentially interfering with one another obviously de-pends on the area of such a surface.
As noted above in this section, the size of the set of so-lutions to a MO problem expressed in the decision vari-able domain is not known, since it depends on the ob-jective function mappings. However, when expressedin the objective value domain, and due to the defini-tion of non-dominance, an upper limit for the size ofthe solution set can be calculated from the minimumand maximum values each objective assumes withinthat set. Let S be the solution set in the decisionvariable domain, f(S) the solution set in the objectivedomain and y = (y1, . . . , yq) any objective vector inf(S). Also, let
m = (miny
y1, . . . , miny
yq) = (m1, . . . , mq)
M = (maxy
y1, . . . , maxy
yq) = (M1, . . . , Mq)
as illustrated in Figure 2. The definition of trade-offsurface implies that any line parallel to any of the axeswill have not more than one of its points in f(S), whicheliminates the possibility of it being rugged, i.e., eachobjective is a single-valued function of the remainingobjectives. Therefore, the true area of f(S) will be lessthan the sum of the areas of its projections accordingto each of the axes. Since the maximum area of eachprojection will be at most the area of the correspond-ing face of the hyperparallelogram defined by m and
f3(m1,m2,m3)(M1,M2,M3)f2f1
f3
(m1, m2, m3)
(M1, M2, M3)
f2
f1
Figure 2: An Example of a Trade-off Surface in 3-Dimensional Space
M, the hyperarea of f(S) will be less than
A =
q∑
i=1
q∏
j=1
j 6=i
(Mj − mj)
which is the sum of the areas of each different face ofa hyperparallelogram of edges (Mj −mj) (Figure 3).
In accordance with the objectives being non-commensurable, the use of the ∞-norm for measuringthe distance between individuals seems to be the mostnatural one, while also being the simplest to compute.In this case, the user is still required to specify an indi-vidual σshare for each of the objectives. However, themetric itself does not combine objective values in anyway.
Assuming that objectives are normalized so that allsharing parameters are the same, the maximum num-ber of points that can sample area A without in-terfering with each other can be computed as thenumber of hypercubes of volume σq
share that can beplaced over the hyperparallelogram defined by A (Fig-ure 4). This can be computed as the difference in vol-ume between two hyperparallelograms, one with edges(Mi−mi +σshare) and the other with edges (Mi−mi),divided by the volume of a hypercube of edge σshare,i.e.
N =
q∏
i=1
(Mi − mi + σshare) −q∏
i=1
(Mi − mi)
σqshare
Conversely, given a number of individuals (points), N ,it is now possible to estimate σshare by solving the
f3(M1,M2,M3)f2f1(m1,m2,m3)
f3
(m1, m2, m3)
(M1, M2, M3)
f2
f1
Figure 3: Upper Bound for the Area of a Trade-off Surface limited by the Parallelogram defined by(m1, m2, m3) and (M1, M2, M3)
(q − 1)-order polynomial equation
Nσq−1share −
q∏
i=1
(Mi − mi + σshare) −q∏
i=1
(Mi − mi)
σshare= 0
for σshare > 0.
4.2 CONSIDERATIONS ON MATINGRESTRICTION AND CHROMOSOMECODING
The use of mating restriction was suggested by Gold-berg in order to avoid excessive competition betweendistant members of the population. The ability to cal-culate σshare on the objective domain immediately sug-gests the implementation of mating restriction schemeson the same domain, by defining the corresponding pa-rameter, σmating.
Mating restriction assumes that neighbouring fit in-dividuals are genotypically similar, so that they canform stable niches. Extra attention must therefore bepaid to the coding of the chromosomes. Gray codes,as opposed to standard binary, are known to be usefulfor their property of adjacency. However, the codingof decision variables as the concatenation of indepen-dent binary strings cannot be expected to consistentlyexpress any relationship between them.
On the other hand, the Pareto set, when representedin the decision variable domain, will certainly exhibitsuch dependencies. In that case, even relatively smallregions of the Pareto-set may not be characterized bya single, high-order, schema and the ability of matingrestriction to reduce the formation of lethals will beconsiderably diminished. As the size of the solution
f3(m1,m2,m3)s-share(M1,M2,M3)f2f1
f3
(m1, m2, m3)
σshare
(M1, M2, M3)
f2
f1
Figure 4: Sampling Area A. Each Point is σshare apartfrom each of its Neighbours (∞-norm)
set increases, an increasing number of individuals isnecessary in order to assure niche sizes small enoughfor the individuals within each niche to be sufficientlysimilar to each other.
Given a reduced number of individuals, the Pareto setof a given vector function may simply be too large forthis to occur. Since, on the other hand, the designer isoften looking for a single compromise solution to theMO problem, reducing the size of the solution set bydeciding at a higher level which individuals express agood compromise would help to overcome the prob-lems raised above.
5 INCORPORATING
HIGHER-LEVEL DECISION
MAKING IN THE SELECTION
ALGORITHM
When presented with the trade-off surface for a givenfunction, the decision maker (DM) would have to de-cide which of all of the non-dominated points to chooseas the solution to the problem. First, the regions of thePareto set which express good compromises accordingto some problem-specific knowledge would be identi-fied. Then, having a clearer picture of what is achiev-able, the idea of compromise would be refined until thesolution was found. As a consequence, a very preciseknowledge of the areas that end up being discarded isof doubtful utility. Only the “interesting” regions ofthe Pareto set need to be well known.
Reducing the size of the solution set calls for higher-level decision making to be incorporated in the selec-tion algorithm. The idea is not to reduce the scope ofthe search, but simply to zoom in on the region of the
Pareto set of interest to the DM by providing externalinformation to the selection algorithm.
The fitness assignment method described earlier wasmodified in order to accept such information inthe form of goals to be attained, in a similar wayto that used by the conventional goal attainmentmethod (Gembicki, 1974), which will now be brieflyintroduced.
5.1 THE GOAL ATTAINMENT METHOD
The goal attainment method solves the multiobjectiveoptimization problem defined as
minx∈Ω
f(x)
where x is the design parameter vector, Ω the feasibleparameter space and f the vector objective function,by converting it into the following nonlinear program-ming problem:
minλ,x∈Ω
λ
such thatfi − wiλ ≤ gi
Here, gi are goals for the design objectives fi, andwi ≥ 0 are weights, all of them specified by the de-signer beforehand. The minimization of the scalar λleads to the finding of a non-dominated solution whichunder- or over-attains the specified goals to a degreerepresented by the quantities wiλ.
5.2 A MODIFIED MO RANKINGSCHEME TO INCLUDE GOALINFORMATION
The MO ranking procedure previously described wasextended to accommodate goal information by alteringthe way in which individuals are compared with oneanother. In fact, degradation in vector componentswhich meet their goals is now acceptable provided itresults in the improvement of other components whichdo not satisfy their goals and it does not go beyondthe goal boundaries. This makes it possible for one toprefer one individual to another even though they areboth non-dominated. The algorithm will then identifyand evolve the relevant region of the trade-off surface.
Still assuming a minimization problem, consider twoq-dimensional objective vectors, ya = (ya,1, . . . , ya,q)and yb = (yb,1, . . . , yb,q), and the goal vector g =(g1, . . . , gq). Also consider that ya is such that it meetsa number, q − k, of the specified goals. Without lossof generality, one can write
∃ k = 1, . . . , q − 1 : ∀ i = 1, . . . , k ,
∀ j = k + 1, . . . , q , (ya,i > gi) ∧ (ya,j ≤ gj) (A)
which assumes a convenient permutation of the objec-tives. Eventually, ya will meet none of the goals, i.e.,
∀ i = 1, . . . , q , (ya,i > gi) (B)
or even all of them, and one can write
∀ j = 1, . . . , q , (ya,j ≤ gj) (C)
In the first case (A), ya meets goals k + 1, . . . , q and,therefore, will be preferable to yb simply if it domi-nates yb with respect to its first k components. Forthe case where all of the first k components of ya areequal to those of yb, ya will still be preferable to yb
if it dominates yb with respect to the remaining com-ponents, or if the remaining components of yb do notmeet all their goals. Formally, ya will be preferable toyb, if and only if
(
ya,(1,...,k) p< yb,(1,...,k)
)
∨(
ya,(1,...,k) = yb,(1,...,k)
)
∧[(
ya,(k+1,...,q) p< yb,(k+1,...,q)
)
∨
∼(
yb,(k+1,...,q) ≤ g(k+1,...,q)
)]
In the second case (B), ya satisfies none of the goals.Then, ya is preferable to yb if and only if it dominatesyb, i.e.,
ya p< yb
Finally, in the third case (C) ya meets all of the goals,which means that it is a satisfactory, though not nec-essarily optimal, solution. In this case, ya is preferableto yb, if and only if it dominates yb or yb is not satis-factory, i.e.,
(ya p< yb)∨ ∼(yb ≤ g)
The use of the relation preferable to as just described,instead of the simpler relation partially less than, im-plies that the solution set be delimited by those non-dominated points which tangentially achieve one ormore goals. Setting all the goals to ±∞ will make thealgorithm try to evolve a discretized description of thewhole Pareto set.
Such a description, inaccurate though it may be, canguide the DM in refining its requirements. When goalscan be supplied interactively at each GA generation,the decision maker can reduce the size of the solutionset gradually while learning about the trade-off be-tween objectives. The variability of the goals acts asa changing environment to the GA, and does not im-pose any constraints on the search space. Note thatappropriate sharing coefficients can still be calculatedas before, since the size of the solution set changes ina way which is known to the DM.
This strategy of progressively articulating the DMpreferences, while the algorithm runs, to guide thesearch, is not new in operations research. The maindisadvantage of the method is that it demands a highereffort from the DM. On the other hand, it potentiallyreduces the number of function evaluations requiredwhen compared to a method for a posteriori articula-tion of preferences, as well as providing less alternative
@@
@@
@@
@@DM
a priori
knowledgeGA
objective function values
fitnesses
(acquired knowledge)
results
Figure 5: A General Multiobjective Genetic Optimizer
points at each iteration, which are certainly easier forthe DM to discriminate between than the whole Paretoset at once.
6 THE MOGA AS A METHOD FOR
PROGRESSIVE ARTICULATION
OF PREFERENCES
The MOGA can be generalized one step further. TheDM action can be described as the consecutive evalu-ation of some not necessarily well defined utility func-tion. The utility function expresses the way in whichthe DM combines objectives in order to prefer onepoint to another and, ultimately, is the function whichestablishes the basis for the GA population to evolve.Linearly combining objectives to obtain a scalar fit-ness, on the one hand, and simply ranking individualsaccording to non-dominance, on the other, both corre-spond to two different attitudes of the DM. In the firstcase, it is assumed that the DM knows exactly whatto optimize, for example, financial cost. In the secondcase, the DM is making no decision at all apart fromletting the optimizer use the broadest definition of MOoptimality. Providing goal information, or using shar-ing techniques, simply means a more elaborated atti-tude of the DM, that is, a less straightforward utilityfunction, which may even vary during the GA process,but still just another utility function.
A multiobjective genetic optimizer would, in general,consist of a standard genetic algorithm presenting theDM at each generation with a set of points to be as-sessed. The DM makes use of the concept of Paretooptimality and of any a priori information availableto express its preferences, and communicates them tothe GA, which in turn replies with the next generation.At the same time, the DM learns from the data it ispresented with and eventually refines its requirementsuntil a suitable solution has been found (Figure 5).
In the case of a human DM, such a set up may requirereasonable interaction times for it to become attrac-tive. The natural solution would consist of speeding
up the process by running the GA on a parallel ar-chitecture. The most appealing of all, however, wouldbe the use of an automated DM, such as an expertsystem.
7 INITIAL RESULTS
The MOGA is currently being applied to the stepresponse optimization of a Pegasus gas turbine en-gine. A full non-linear model of the engine (Han-cock, 1992), implemented in Simulink (MathWorks,1992b), is used to simulate the system, given a num-ber of initial conditions and the controller parametersettings. The GA is implemented in Matlab (Math-Works, 1992a; Fleming et al., 1993), which means thatall the code actually runs in the same computation en-vironment.
The logarithm of each controller parameter was Grayencoded as a 14-bit string, leading to 70-bit long chro-mosomes. A random initial population of size 80 andstandard two-point reduced surrogate crossover andbinary mutation were used. The initial goal valueswere set according to a number of performance require-ments for the engine. Four objectives were used:
tr The time taken to reach 70% of the final outputchange. Goal: tr ≤ 0.59s.
ts The time taken to settle within ±10% of the finaloutput change. Goal: ts ≤ 1.08s.
os Overshoot, measured relatively to the final outputchange. Goal: os ≤ 10%.
err A measure of the output error 4 seconds after thestep, relative to the final output change. Goal:err ≤ 10%.
During the GA run, the DM stores all non-dominatedpoints evaluated up to the current generation. Thisconstitutes acquired knowledge about the trade-offsavailable in the problem. From these, the relevantpoints are identified, the size of the trade-off surfaceestimated and σshare set. At any time in the optimiza-
tr ts ov err0
0.2
0.4
0.6
0.8
1
Nor
mal
ized
obj
ectiv
e va
lues
Objective functions
0.59s 1.08s 10% 10%
Figure 6: Trade-off Graph for the Pegasus Gas TurbineEngine after 40 Generations (Initial Goals)
tion process, the goal values can be changed, in orderto zoom in on the region of interest.
A typical trade-off graph, obtained after 40 genera-tions with the initial goals, is presented in Figure 6and represents the accumulated set of satisfactory non-dominated points. At this stage, the setting of a muchtighter goal for the output error (err ≤ 0.1%) revealsthe graph in Figure 7, which contains a subset of thepoints in Figure 6. Continuing to run the GA, moredefinition can be obtained in this area (Figure 8). Fig-ure 9 presents an alternative view of these solutions,illustrating the arising step responses.
8 CONCLUDING REMARKS
Genetic algorithms, searching from a population ofpoints, seem particularly suited to multiobjective opti-mization. Their ability to find global optima while be-ing able to cope with discontinuous and noisy functionshas motivated an increasing number of applications inengineering and related fields. The development of theMOGA is one expression of our wish to bring decisionmaking into engineering design, in general, and controlsystem design, in particular.
An important problem arising from the simple Pareto-based fitness assignment method is that of the globalsize of the solution set. Complex problems can beexpected to exhibit a large and complex trade-off sur-face which, to be sampled accurately, would ultimatelyoverload the DM with virtually useless information.Small regions of the trade-off surface, however, can stillbe sampled in a Pareto-based fashion, while the deci-sion maker learns and refines its requirements. Nicheformation methods are transferred to the objectivevalue domain in order to take advantage of the prop-erties of the Pareto set.
tr ts ov err0
0.2
0.4
0.6
0.8
1
Nor
mal
ized
obj
ectiv
e va
lues
Objective functions
0.59s 1.08s 10% 0.1%
Figure 7: Trade-off Graph for the Pegasus Gas TurbineEngine after 40 Generations (New Goals)
tr ts ov err0
0.2
0.4
0.6
0.8
1N
orm
aliz
ed o
bjec
tive
valu
es
Objective functions
0.59s 1.08s 10% 0.1%
Figure 8: Trade-off Graph for the Pegasus Gas TurbineEngine after 60 Generations (New Goals)
0 1 2 3 4 570
70.5
71
71.5
72
72.5
73
73.5
Time (s)
Low
-pre
ssur
e sp
ool s
peed
(%
)
Figure 9: Satisfactory Step Responses after 60 Gener-ations (New Goals)
Initial results, obtained from a real world engineeringproblem, show the ability of the MOGA to evolve uni-formly sampled versions of trade-off surface regions.They also illustrate how the goals can be changed dur-ing the GA run.
Chromosome coding, and the genetic operators them-selves, constitute areas for further study. Redundantcodings would eventually allow the selection of the ap-propriate representation while evolving the trade-offsurface, as suggested in (Chipperfield et al., 1992).The direct use of real variables to represent an indi-vidual together with correlated mutations (Back et al.,1991) and some clever recombination operator(s) mayalso be interesting. In fact, correlated mutationsshould be able to identify how decision variables re-late to each other within the Pareto set.
Acknowledgements
The first author gratefully acknowledges support byPrograma CIENCIA, Junta Nacional de InvestigacaoCientıfica e Tecnologica, Portugal.
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