In recent years, performance indicators were introduced as a selection mechanism in multi-objective evolutionary algorithms (MOEAs). A very attractive option is the R2 indicator due to its low computational cost and weak-Pareto compatibility. This indicator requires a set of utility functions, which map each objective to a single value. However, not all the utility functions available in the literature scale properly for more than four objectives and the diversity of the approximation sets is sensitive to the choice of the reference points during normalization. In this paper, we present an improved version of a MOEA based on the R2 indicator, which takes into account these two key aspects, using the achievement scalarizing function and statistical information about the population's proximity to the true Pareto optimal front. Moreover, we present a comparative study with respect to some other emerging approaches, such as NSGA-III (based on Pareto dominance), ∆p-DDE (based on the ∆p indicator) and some other MOEAs based on the R2 indicator, using the DTLZ and WFG test problems. Experimental results indicate that our approach outperforms the original algorithm as well as the other MOEAs in the majority of the test instances, making it a suitable alternative for solving many-objective optimization problems.

The incorporation of performance indicators as the selection mechanism of a multi-objective evolutionary algorithm (MOEA) is a topic that has attracted increasing interest in the last few years. This has been mainly motivated by the fact that Pareto-based selection schemes do not perform properly when solving problems with four or more objectives. The indicator that has been most commonly used for being incorporated in the selection mechanism of a MOEA has been the hypervolume. Here, however, we explore the use of the R2 indicator, which presents some advantages with respect to the hypervolume, the main one being its low computational cost. In this paper, we propose a new MOEA called Many-Objective Metaheuristic Based on the R2 Indicator (MOMBI), which ranks individuals using a utility function. The proposed approach is compared with respect to MOEA/D (based on scalarization) and SMS-EMOA (based on hypervolume) using several benchmark problems. Our preliminary experimental results indicate that MOMBI obtains results of similar quality to those produced by SMS-EMOA, but at a much lower computational cost. Additionally, MOMBI outperforms MOEA/D in most of the test instances adopted, particularly when dealing with high-dimensional problems having complicated Pareto fronts. Thus, we believe that our proposed approach is a viable alternative for solving many-objective optimization problems. 1 The Hypervolume (also known as the S metric or the Lebesgue Measure) of a set of solutions measures the size of the portion of objective space that is dominated by those solutions collectively [6].

Evolutionary multiobjective optimization has been a research area since the mid-1980s, and has experienced a very significant activity in the last 20 years. However, and in spite of the maturity of this field, there are still several important challenges lying ahead. This paper provides a short description of some of them, with a particular focus on open research areas, rather than on specific research topics or problems. The main aim of this paper is to motivate researchers and students to develop research in these areas, as this will contribute to maintaining this discipline active during the next few years.

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