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Global multiobjective optimization using evolutionary methods: An experimental analysis 117Figure 1 graphically describes the Pareto-dominance concept for a minimization problemwith two objectives (k1 and k2). Figure 1(a) shows the location of several solutions. The filledcircles represent non-dominated solutions, while the non-filled ones symbolize dominatedsolutions. Figure 1(b) shows the relative distribution of the solutions in reference to s. Thereexist solutions that are worse (in both objectives) than s, better (in both objectives) than s, andindifferent (better in one objective and worse in the other).In this paper, we address all of these issues for four MOEAS, SPEA2, NSGA-II, PESA anda hybrid version, msPESA, mainly based on PESA and NSGA-II. From the simulation resultson a number of difficult test problems, we find that msPESA outperforms three other contemporaryMOEAs in terms of finding a diverse set of solutions and in converging near the globalPareto-optimal set.In the remainder of the paper, we briefly mention these MOEAs in Section II. Thereafter,Section III presents simulation results and compares msPESA with three other elitist MOEAs(SPEA2, NSGA-II and PESA). Finally, we outline the conclusions of this paper.2. Multiobjective Optimization Algorithms ImplementedWe mentioned earlier that, along with convergence to the Pareto-optimal set, it is also desiredthat a MOEA maintains a good spread of solutions in the obtained set of solutions. In thefollowing, we describe this issues in each algorithm.SPEA2 (The Strength Pareto Evolutionary Algorithm). SPEA2 [6] combines, in the samefitness value, dominance information about the individual (with rank and count of dominance)as well as the density information about its niche, computed by the nearest neighbortechnique. Therefore SPEA2 also implements fitness-sharing. As the file size is fixed, whenthe number of non-dominated individuals exceeds the file size, a clearing process is followed,considering the smallest distance to the neighbors. SPEA2 uses a fine grain assignment fitnesstechnique incorporating the density information that is useful for the selection process.In addition, the size of the file is fixed and the file fills up with dominated individuals whenthere are not enough non-dominated ones. The clustering technique is replaced (in comparisonwith SPEA) by a method which has similar characteristics but does not eliminate extremesolutions (see [6]). Finally, with SPEA2 only the members of the file participate in the matingprocess.NSGA-II (Non-dominated Sorting Genetic Algorithm-II). NSGA-II [3] proposes a newpartial order between two solutions defined by the crowded comparison operator. The crowdedoperator acts in the following way: given two solutions, their rankings are first checked. If therankings are different, the solution with the lower ranking is assumed to be the best one. Ifthey are equal, the density information is checked. In this case, the solution with the less populatedniche is assumed to be the best. Although both mechanisms seek to maintain diversity,NSGA-II does not use the crowding mechanism of niches. NSGA-II uses tournament selectionbased on the defined operator. In the algorithm operation (see [3]), when fronts are added, thesize of the allowed population can be exceeded. In this case, the least diversified solutions ofthe last front are eliminated. This is a clearing procedure. Even so, this clearing is subject to thedominance relation as established in the definition of the comparison operator. In conclusion,while SPEA2 presented a hybrid operation between fitness-sharing and crowding, NSGA-IIuses a combination of fitness-sharing and clearing as its niching mechanism.PESA (The Pareto Envelope-based Selection Algorithm). PESA [4] is a mixed algorithmbetween PAES [9] and SPEA [5]. It uses a small internal population and a (usually) largerexternal population. The external population is actually the archive which stores the currentapproximation to the Pareto front, and the internal population are new candidate solutions