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A Niched Pareto Genetic Algorithm for Multiobjective Optimization

Je rey Horn, Nicholas Nafpliotis, and David E. Goldberg

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As accepted for publication in the Proceedings of the First IEEE Conference on Evolutionary Computation, IEEE World Congress on Computational Intelligence, Volume 1, 1994 (ICEC '94) Piscataway, NJ: IEEE Service Center, pp. 82-87

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(camera-ready preprint) ICEC '94 c 1994 IEEE (pp. 82-87)

A Niched Pareto Genetic Algorithm for Multiobjective Optimization

Je rey Horn, Nicholas Nafpliotis, and David E. Goldberg

Abstract| Many, if not most, optimization problems have multiple objectives. Historically, multiple objectives have been combined ad hoc to form a scalar objective function, usually through a linear combination (weighted sum) of the multiple attributes, or by turning objectives into constraints. The genetic algorithm (GA), however, is readily modi ed to deal with multiple objectives by incorporating the concept of Pareto domination in its selection operator, and applying a niching pressure to spread its population out along the Pareto optimal tradeo surface. We introduce the Niched Pareto GA as an algorithm for nding the Pareto optimal set. We demonstrate its ability to nd and maintain a diverse \Pareto optimal population" on two arti cial problems and an open problem in hydrosystems.

Genetic algorithms (GAs) have been applied almost exclusively to single-attribute1 problems. But a careful look at many real-world GA applications reveals that the objective functions are really multiattribute. Typically, the GA user nds some ad-hoc function of the multiple attributes to yield a scalar tness function. Often-seen tools for combining multiple attributes are constraints, with associated thresholds and penalty functions, and weights for linear combinations of attribute values. But penalties and weights have proven to be problematic. The nal GA solution is usually very sensitive to small changes in the penalty function coe cients and weighting factors 9].

The authors are with the Illinois Genetic Algorithms Laboratory, University of Illinois at Urbana-Champaign, 117 Transportation Building, 104 South Mathews Ave., Urbana, IL 61801. Internet: je horn@uiuc.edu, nick-n@uiuc.edu, goldberg@vmd.cso.uiuc.edu. Phone: 217/333-2346, Fax: 217/2445705. The rst author acknowledges support from NASA under contract number NGT-50873, while the remaining authors acknowledge support provided by the U.S. Army under Contract DASG60-90-C-0153. 1 We use the terms \attribute", \objective", and \criteria" interchangeably to describe a scalar value to be maximized or minimized. \Decision variable" refers to the parameters of the problem encoded in the genome of the genetic algorithm.

A few studies have tried a di erent approach to multicriteria optimization with GAs: using the GA to nd all possible tradeo s among the multiple, con icting objectives. Such solutions are nondominated, in that there are no other solutions superior in all attributes. In attribute space, the set of non-dominated solutions lie on a surface known as the Pareto optimal frontier2. The goal of a Pareto GA is to nd a representative sampling of solutions all along the Pareto front. We assume the reader is familiar with the simple GA 3]. Here we review previous approaches to multiobjective optimization with GAs. In his 1984 dissertation 10], and later in 11], Scha er proposed his Vector Evaluated GA (VEGA) for nding multiple solutions to multiobjective (vector valued) problems. He created VEGA to nd and maintain multiple classi cation rules in a set covering problem. VEGA tried to achieve this goal by selecting a fraction of the next generation using one of each of the attributes (e.g., cost, reliability). Although Scha er reported some success, VEGA seems capable of nding only extreme points on the Pareto front, where one attribute is maximal, since it never selects according to tradeo s among attributes. In his review of GA history, including Scha er's VEGA, Goldberg 3] suggested the use of nondomination ranking and selection to move a population toward the Pareto front in a multiobjective problem. He also suggested using some kind of niching to keep the GA from converging to a single point on the front. A niching mechanism, such as sharing 5], would allow the GA to maintain individuals all along the non-dominated frontier. Fonseca and Fleming 2], and, independently, Horn and Nafpliotis 7], implemented Goldberg's two suggestions, and successfully applied the resulting algorithms to di cult, open problems. Fonseca and

2 We assume familiarity with the concept of Pareto optimality, but note here that the Pareto front often goes by the names Pareto optimal set, non-dominated frontier, e cient points, and admissible points.

II. Previous Work

I. Introduction

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Fleming found many good tradeo s in a four attribute gas turbine design problem. Horn and Nafpliotis concentrated on a series of two attribute problems, which we describe later in this paper. The speci cs of the Niched Pareto GA are localized to implementation of selection for the genetic algorithm. One of the most widely implemented selection techniques for GAs is tournament selection. In tournament selection a set of individuals is randomly chosen from the current population and the best of this subset is placed in the next population. By adjusting the size of the tournament we can exert some control over the amount of selection pressure and hence convergence speed. Thus the smallest tournament size of two (binary tournament) exhibits slower convergence than any larger tournament size. Tournament selection assumes that we want a single answer to the problem. After a certain number of generations the population will converge to a uniform one. To avoid convergence and maintain multiple Pareto optimal solutions, we have altered tournament selection in two ways. First we added Pareto domination tournaments. Second, when we have a non-dominant tournament (i.e., a tie), sharing is implemented to determine the winner. The binary relation of domination leads naturally to a binary tournament in which two randomly selected individuals are compared. If one dominates the other, it wins. Initially, we used such a small local domination criterion, but we soon found that it produced insu cient domination pressure. There were too many dominated individuals in later generations. It seemed that a sample size of two was too small to estimate an individual's true \domination ranking"3. Because we wanted more domination pressure, and more control of that pressure, we implemented a sampling scheme as follows. Two candidates for selection are picked at random from the population. A comparison set of individuals is also picked randomly from the population. Each of the candidates are then compared against each individual in the comparison set. If one candidate is dominated by the comparison set, and the other is not, the latter is selected for reproduction. If neither or both are dominated by the comparison set, then we must use

3 Note that any partial order determines a unique ranking, in which maximal individuals are ranked rst, then removed. The remaining individuals are reordered, and the maximal individuals of this set are ranked second, and removed, etc. This is the domination ranking scheme suggested by Goldberg 3].

III. The Niched Pareto GA

sharing to choose a winner, as we explain later. The sample size tdom (size of comparison set) gives us control over selection pressure, or what we call domination pressure. The performance of the Niched Pareto GA is somewhat sensitive to the amount of domination versus sharing pressure applied 7]. A problem will arise if both candidates are on the current non-dominated front since neither will be dominated. Even o the front, a small tdom could mean that neither appears dominated. And of course both could be dominated. How is a winner then chosen in such a \tie"? If we choose the winner at random, genetic drift will cause the population to converge to a single region of the Pareto front. To prevent this we implement a form of sharing when there is no preference between two individuals. Fitness sharing was introduced by Goldberg and Richardson 5], analyzed in detail by Deb 1], and applied successfully to a number of di cult and real world problems. The goal of tness sharing is to distribute the population over a number of di erent peaks in the search space, with each peak receiving a fraction of the population in proportion to the height of that peak4 . To achieve this distribution, sharing calls for the degradation of an individual's objective tness fi by a niche count mi calculated for that individual. This degradation is obtained by simply dividing the objective tness by the niche count to nd the shared tness: fi =mi . The niche count mi is an estimate of how crowded is the neighborhood (niche) of individual i. It is calculated over all individuals in the P current population: mi = j 2Pop Sh d i; j]], where d i; j] is the distance between individuals i and j and Sh d ] is the sharing function. Sh d] is a decreasing function of d i; j], such that Sh 0] = 1 and Sh d share ] = 0. Typically, the triangular sharing function is used, where Sh d] = 1 ? d= share for d share and Sh d] = 0 for d > share . Here share is the niche radius, xed by the user at some estimate of the minimal separation desired or expected between the goal solutions. Individuals within share distance of each other degrade each other's tness, since they are in the same niche. Thus convergence occurs within a niche, but convergence of the full population is avoided. As one niche \ lls up", its niche count increases to the point that its shared tness is lower than that of other niches. Fitness sharing was originally combined with tness proportionate (a.k.a., roulette wheel) selection. When sharing is combined with the more popular

4 The authors sometimes refer to this form of niching as tness proportionate sharing.

B. Sharing on the non-dominated frontier

A. Pareto domination tournaments

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tournament selection, however, the niched GA exhibits chaotic behaviour 8]. The wild uctuations in niche subpopulations induced by the \naive" combination of sharing and tournament selection can be avoided. Oei, Goldberg, and Chang 8] suggest the use of tournament selection with continuously updated sharing, in which niche counts are calculated not by using the current population, but rather the partly lled next generation population. This method was used successfully by Goldberg, Deb, and Horn 4] on a \niching-di cult" problem. Also in 4], it was found empirically that sampling the population was su cient to estimate the niche count and so avoid the O(N 2) comparisons needed to calculate exactly the mi . We incorporate both techniques (continously updated sharing and niche count sampling) in the Niched Pareto GA. In any application of sharing, we can implement genotypic sharing, since we always have a genotype (the encoding). But Deb's work 1] indicated that in general, phenotypic sharing is superior to genotypic sharing. Intuitively, we want to perform sharing in a space we \care more about", that is, some phenotypic space. Since we are interested in maintaining diversity along the phenotypic Pareto optimal front, which exists only in attribute space, it makes sense to perform our sharing in attribute space5 . When the candidates are either both dominated or both non-dominated, it is likely that they are in the same equivalance class (in the partial order induced by the domination relation). Because we are interested in maintaining diversity along the front, and most of the individuals in these equivalence classes can be labeled \equally" t, we do not implement any form of tness degradation according to the niche count. Instead, the \best t" candidate is determined to be that candidate which has the least number of individuals in its niche and thus the smallest niche count. We call this type of sharing equivalence class sharing6 . Figure 1 illustrates how this form of sharing should work between two non-dominated individuals. Here we are maximizing along the x-axis and minimizing on the y-axis. In this case the two candidates for selection are not dominated by the comparison set. Thus the two candidates are in the Pareto optimal subset (the dashed region) of the union of the comparison set and the candidates. From a Pareto point of view, neither candidate is preferred. But if we

5 To impose a meaningful metric on attribute space, the attributes should be scaled to the same numerical range, (e.g., 0 to 1). This is possible if we have some idea of the extreme values theoretically attainable by each attribute. 6 This technique might also be called at tness sharing, since the same e ect is produced in normal (single-attribute) tness sharing when two individuals have the exact same tness.

Comparison Set Individuals Candidate Individuals

Equivalence Class Region

Candidate 2

Niches Determined by Sigma Share Candidate 1

Figure 1: Equivalence class sharing. want to maintain useful diversity (i.e., a representative sampling of the Pareto frontier), it is apparent that it would be best to choose the candidate that has the smaller niche count. In this case, that is candidate 2.

IV. Application to Three Problems

A. Problem 1: A simple test function

We begin testing our new algorithm by constructing a simple, arti cial problem with an easily calculated Pareto optimal front. We call problem 1 \unitation versus pairs" for the two attributes of unitation and complementary adjacent pairs. Unitation Unit s] is simply the number of ones in the xed length bit string s, thus Unit 01110010] = 4. Pairs Prs s] is the number of pairs of adjacent complementary bits, either 01 or 10. Thus Prs 01110010] = 4. The problem is to maximize both attributes. Among strings of a particular unitation, those strings with the greatest \mixing" of ones and zeroes dominate those who \clump" their ones (e.g., 01010 dominates 01100, though both have the same unitation). In Figure 2, we plot the feasible region of the twodimensional attribute space Unit versus Prs for a 12-bit problem. P indicates a point on the Pareto front, while \-" indicates a feasible point that is dominated by some member(s) of the Pareto set. In Figure 3, we plot the initial population (generation 0) of 100 randomly generated 12-bit individuals7. The numbers plotted in attribute space are the numbers of individuals with attribute values corresponding to the coordinates in attribute space. Figure 4 shows the population after 100 generations. We can see that the GA has succeeded in nding all but one member of the Pareto set, and

7 Tournament size t dom = 10, niche size share = 2:0, singlepoint crossover probability pc = 0:9, and mutation rate pm = 0:01.

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Unitation 12 | P 11 | - P 10 | - - - P 09 | - - - - - P 08 | - - - - - - - P 07 | - - - - - - - - - P 06 | - - - - - - - - - - P 05 | - - - - - - - - - 04 | - - - - - - - 03 | - - - - - 02 | - - - 01 | - 00 | |||||||||||||||||||||||{ 0 1 2 3 4 5 6 7 8 9 10 11 12 Pairs

35 30 25 20 f22 15 10 5 -6 -4 -2 0 2 4 6 f21

Figure 5: Scha er's function F2, P = fx j 0

x

2g

Figure 2: Problem 1's discrete, two dimensional attribute space, with feasible (-) and Pareto (P) points indicated. that it is easy to nd points on the front), but not

Unitation Generation 0 12 | 0 11 | 0 2 10 | 0 0 1 4 09 | 0 0 1 2 1 2 08 | 0 0 0 4 8 1 2 1 07 | 0 0 1 2 5 3 6 1 0 0 06 | 0 0 2 2 6 3 5 6 0 0 0 05 | 0 0 6 2 2 2 1 6 0 0 04 | 0 2 0 1 1 3 0 1 03 | 0 0 0 0 0 1 02 | 0 1 0 0 01 | 0 0 00 | 0 |||||||||||||||||||||||{ 0 1 2 3 4 5 6 7 8 9 10 11 12

necessarily easy for the Niched Pareto GA. Since there are actually many more solutions at middle points on the front, and only one or two at each end point of the front, it should be harder to maintain equal size subpopulations at the extreme points. Next we compare our algorithm to Scha er's VEGA by running it on one of the test functions from Schaffer's dissertation 10]. This is the simple function F2, with a single decision variable, the real-valued x, and two attributes, f21 andf22 to be minimized: f21(x) = x2 f22(x) = (x ? 2)2 The decision variable is mapped to a 14-bit string as a binary-coded integer. Thus 00000000000000 = xmin = ?6:00 and 11111111111111 = xmax = 6:00. We plot f21 and f22 over this range of x in Figure 5. It is clear that the Pareto front is where the tradeo exists between the two functions. That is, for 0 x 2:00, one of the functions is decreasing to its best while the other is increasing away from its best. Like Scha er, we use a small population size N = 30. Our niche size share = 0:1 and tournament size tdom = 4. As Figure 6 illustrates, the Niched Pareto GA is able to maintain a fairly even spread of solutions along the Pareto front. There are a few dominated individuals in the population (to the right of x = 2:00), as in the VEGA run above, but most individuals are on the front. Although our population has several gaps in its distribution on the front, it appears more evenly distributed than generation 3 of the VEGA run8 . Most importantly, the Niched Pareto GA exhibits stability in this population distribution for many more generations than were indicated for VEGA (200 > 3). F2 is an easy problem for the GA: the initial population contains many individuals on the front already. However, this front is much denser than that of problem 1 above, challenging the Niched Pareto GA to maintain N subpopulations of size 1 along the front.

8 Scha

B. Problem 2: Scha er's F2

Pairs

Figure 3: Distribution of the randomly generated initial

population.

appears to be maintaining substantial subpopulations at each such point. Moreover, there are few (none here) dominated individuals in the current population. Although not shown, we have plotted population distributions over many generations, and noticed that the GA does indeed maintain roughly equal size subpopulations at each Pareto point over many generations. Dominated solutions regularly appear, due to crossover and mutation, but are not maintained. We have observed similar behaviour over many runs of the GA on di erent initial population distributions (all random, but using di erent random seeds). We have also successfully tried larger problems (l > 12), with correspondingly larger population sizes N, such as 400 individuals on a 28 bit problem 7]. Finally, we note that this problem is GA-easy (in

Unitation Generation 100 12 | 26 11 | 0 11 10 | 0 0 0 14 09 | 0 0 0 0 0 12 08 | 0 0 0 0 0 0 0 16 07 | 0 0 0 0 0 0 0 0 0 21 06 | 0 0 0 0 0 0 0 0 0 0 0 05 | 0 0 0 0 0 0 0 0 0 0 04 | 0 0 0 0 0 0 0 0 03 | 0 0 0 0 0 0 02 | 0 0 0 0 01 | 0 0 00 | 0 |||||||||||||||||||||||{ 0 1 2 3 4 5 6 7 8 9 10 11 12

Pairs

Figure 4: Stable subpopulations on the Pareto front.

er (1984) only gave results for generations 0, 1, and 3.

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6

0.15 Generation 0

f22 5 4 3 2 1 -1 6 5 f22 4 3 2 1 -1 1 2 1 2

f21

0.14

gen 0 =

VEGA

Average Volume Detected

0.13 0.12 0.11 0.1 0.09 0.08 0.07 0.06

X

f21 Niched Pareto GA

X

0.05 0 50 100 150 200 250 300 350 Number of Plumes Detected 400 450 500

Figure 6: VEGA versus the Niched Pareto GA. Top: VEGA

on F2's Pareto frontier, generation 3. Bottom: The Niched Pareto GA's distribution, generation 200. Figure 7: Initial population distribution, problem 3.

To challenge the Niched Pareto GA's ability to search for diverse tradeo s, we chose a larger, real-world (i.e., unsolved) application for our third test problem: optimal well placement for groundwater contaminant monitoring9. The problem is to place a set of k out of a possible w wells in order to maximize the number of detected leak plumes from a land ll into the surrounding groundwater, and to minimize the volume of cleanup involved. These two objectives con ict. Simply optimizing for the minimum volume of cleanup will give us an answer with attributes f0; 0g, where we detect no plumes and therefore have no volume of contaminant to clean up. If we maximize the number of detected plumes our volume of cleanup increases dramatically. It is important to note that this problem is intractable. The search space is of size (w ). In our k speci c example we have w = 396 and k = 20. The ? whole search space is then 396 which is 2:269 20 1033. This makes it impossible to know the actual Pareto optimal front from enumeration. Monte-Carlo simulation was used to develop a set of possible leak plumes, the set of wells that detect each plume, and the volume leaked when each well detected the contaminant plume. Using these data, we constructed a vector-valued tness function to return the number of plumes and average volume detected by any given set of wells. In our rst few runs, N = 2000, share = 40, tdom = 40, pc = 0:8, and no mutation. In Figure 7 one can see that the random initial population is distributed throughout the search space. Figure 8 shows that after 230 generations the Niched Pareto

9 This open problem was developed by Wayland Eheart and his colleagues at the Civil Engineering Department at the University of Illinois at Urbana-Champaign. We are grateful to Dr. Eheart and his students S. Ranjithan, P. Stork, and S. Cieniawski for helping us implement it.

C. Problem 3: Open problem in hydrosystems

Generation 230 0.15 gen 230 = 0.14 0.13 Average Volume Detected 0.12 0.11 0.1 0.09 0.08 0.07 0.06 0.05 0 50 100 150 200 250 300 350 Number of Plumes Detected 400 450 500

Figure 8: Final distribution, problem 3. GA has found an apparent front that is indeed improved over the initial population. It is promising to see that even after a large number of generations we are maintaining diversity over most of an apparent front. There is de nite improvement as to the location of the front and the decrease in the number of dominated individuals in the population. We do not know yet whether this is the actual Pareto optimal front or a sub-optimal front. But our rst few runs indicate that the equivalence-class sharing and dominance tournaments are working together. We have shown that a tradeo curve better than a random sampling can be developed by the Niched Pareto GA on an open problem. These preliminary results on the application of the niched, Pareto technique are encouraging. Fonseca and Fleming 2] have also reported initial success with a similar algorithm. But we have found that the performance of the Niched Pareto GA is sensitive to the settings of several parameters. In particular, it is important to have a large enough populaV. Discussion

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tion to search e ectively and to sample the breadth Pareto approach to multiobjective problems. of the Pareto front. Both 7] and 2] discuss the setReferences ting of share and population size together to yield 1] Deb, K. (1989). Genetic algorithms in mule ective sampling. timodal function optimization. MS thesis, But the behaviour of the Niched Pareto GA seems TCGA Report No. 89002. University of Alto be most a ected by the degree of selection presabama. sure applied. Just as tournament size tsize is critical to selection pressure and premature convergence in 2] Fonseca, C. M., & Fleming, P. J. (1993). Genetic algorithms for multiobjective optimizaa regular GA with tournament selection, so tdom dition: formulation, discussion and generalizarectly e ects the convergence of the Niched Pareto tion. Proceedings of the Fifth International GA. Horn and Nafpliotis 7] illustrate the e ects of Conference on Genetic Algorithms. Morgantoo little and too much dominance pressure. Here, Kau man, 416-423. we summarize their empirically-derived, order-of3] Goldberg, D. E. (1989). Genetic Algorithms in magnitude guidelines: Search, Optimization, and Machine Learning. tdom 1% of N; results in too many dominated Reading, MA: Addison-Wesley. solutions (a very fuzzy front). 4] Goldberg, D. E., Deb, K., & Horn, J. (1992). tdom 10% of N; yields a tight and complete Massive multimodality, deception, and genetic distribution. algorithms. Parallel Problem Solving From Natdom 20% of N; causes the algorithm to preture, 2, North-Holland, 37-46. maturely converge to a small portion of the 5] Goldberg, D. E., & Richardson, J. J. (1987). front. Alternative tradeo s were never even Genetic algorithms with sharing for multifound. modal function optimization. Genetic Algorithms and Their Applications: Proceedings of We have not yet addressed the critical issue of the Second ICGA, Lawrence Erlbaum Assosearch, but we have some intuitions. Our intuition ciates, Hillsdale, in the case of Pareto optimization is that the diver- 6] Horn, J., (1993). NJ, 41{49. chain analysis Finite Markov sity along the currently non-dominated frontier acof genetic algorithms with niching. Proceedings tually helps the search for new and improved tradeof the Fifth International Conference on Geo s, thus extending the frontier. Individuals from netic very di erent parts of the front might be crossed 7] Horn,Algorithms. Morgan-Kau man, 110-117. J., & Nafpliotis, N. (1993). Multiobjecto produce o spring that dominate a portion of the tive optimization using the niched Pareto gefront lying between their parents. That is, infornetic algorithm. IlliGAL Report No. 93005. Illimation from very di erent types of tradeo s could nois Genetic Algorithms Laboratory. Univerbe combined to yield other kinds of good tradeo s. sity of Illinois at Urbana-Champaign. Indeed, we see some evidence for this in problem K., Goldberg, D. E., J., 3. Because equivalence class sharing cannot be ex- 8] Oei, C. Tournament selection, & Chang, S. the (1991). niching, and pected to maintain more than one copy of an indipreservation of diversity. IlliGAL Report No. vidual (i.e., niche counts are approximately 1 for all 91011. Illinois Genetic Algorithms Laboratory. niches at steady state), and because we used high University of Illinois at Urbana-Champaign. crossover rates (typically 0.7-0.9), the maintenance 9] Richardson, J. T., Palmer, M. R., Liepins, of the front over hundreds of generations was largely G., & Hilliard, M. (1989). Some guidelines due to the constant generation and regeneration of for genetic algorithms with penalty funcindividuals on the front from the crossover of two tions. Proceedings of the Third International di erent parents. Therefore, most crosses of parConference on Genetic Algorithms. Morganents on or near the front yielded o spring also on Kau or near the front. This behaviour is evidence that 10] Scha man, 191-197. er, J. D., (1984). Some experiments in Pareto diversity helps Pareto search. machine learning using vector evaluated genetic Finally, we point out that the domination touralgorithms, Unpublished doctoral dissertation, nament does not rely strictly on a domination reVanderbilt lation, but rather on an antisymmetric, transitive 11] Scha er, J.University. Multiple objective opD., (1985). relation. Similarly, equivalence class sharing is usetimization with vector evaulated genetic algoful not only on the Pareto optimal frontier, but in rithms. In J. Grefenstette, ed., Proceedings of any equivalence class in a partial order. Thus the an International Conference on Genetic AlgoNiched Pareto GA can be used to search any parrithms and their Applications, 93{100. tially ordered space, not just those induced by the

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