Keywords: multiobjective optimization, evolution strategy, rank-niche fitness function.

Many evolution-based algorithms have been developed to solve optimization problems in recent years. Most of these algorithms were based on genetic algorithms(GA). Owing to the nature of simultaneous searches from many points in the design space, the evolution-based solvers are deemed to be better than mathematical programming methods to locate the global optimum solution. Since the evolution algorithms search the design space from many different points, they have another advantage to find Pareto-optimal solutions for multiobjective optimization problems in a single run. The efficiency of these evolution-based methods in finding Pareto-optimal solutions is in general better than mathematical programming methods. Fonseca and Fleming [1] gave an overview of evolutionary algorithms in multiobjective optimization and classified those approaches into three categories: the plain aggregating approach, the population-based non-Pareto approach and the Pareto based approach. In general the last category generates the best results.

The evolution strategy(ES) which was developed by Rechenberg [2] in 1973 is another important evolutionary algorithm other than the GA. Three evolutionary steps are included in ES. The first one is recombination which blends genetic materials from two randomly chosen individuals from the parent generation. The second step is mutation which causes self change of genetic materials based on the random number of a normal distribution. The last step is selection which chooses the best individuals resulted from previous two evolution steps to survive. The key step to generate many converged and evenly distributed Pareto-optimal solutions is in the selection step. This paper proposes the following fitness function to be used as the selection basis.

where , and are the rank, the niche and the constraint violation of the th individual, respectively. , and are the maximum rank, the maximum niche and the maximum constraint violation in the generation, respectively. The fitness function so defined contains three important information of each individual. The first term includes the normalized superiority of the individual. The second term shows the normalized neighbourhood crowding status of the individual. The third term contains the normalized information of feasibility. After the first two operations, offspring are generated. Those offspring are subjected to Pareto domination checks with an externally saved non-dominated elites. The external elite set is continually updated through this checking process generation after generation. The parents of next generation are generated using selection scheme. i.e. the best individuals with lowest fitness values are chosen to be the parents of next generation. The non-dominated individuals in the elite set at the last generation are the final Pareto-optimal solutions of the problem. In order to have evenly distributed Pareto-optimal solutions the number of individuals in the elite set is limited to a finite number. If the number in the elite set exceeds the designated number, an anti-crowding procedure using clustering technique [3] is applied to reduce the individuals in the elite set to the given number.

Preliminary results show this approach indeed generates many evenly distributed Pareto-optimal solutions on the Pareto front for several test problems. References

1

C.M. Fonseca, P.J. Fleming, "An overview of evolutionary algorithms in multiobjective optimisation", Evolutionary Computation, 3(1), 1-16, 1995. doi:10.1162/evco.1995.3.1.1

2

I. Rechenberg, "Evolutionsstrategie: Optimierung technisher systeme nach prinzipien der biologischen evolution", Frommann-Holzboog Berlag, Stuttgart, 1973.

3

J.N. Morse, "Reducing the size of the nondominated set: Pruning by clustering", Computer & Operation Research, 7, 55-65, 1980. doi:10.1016/0305-0548(80)90014-3

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