| Multi-objective optimization problems(MOPs)widely exist in industrial production process,such as the environmental and economic dispatch problem for the power system,the rolling system load distribution optimization problem,the car robust design problem and the water resource management optimization problem.Traditional multi-objective optimization algorithms,which are based on the Pareto dominance,have achieved a better performance to address MOPs.However,these algorithms are difficult for solving many-objective optimization problems(MaOPs)which are commonly involved three or more than three objectives.Evolutionary multi-objective optimization algorithms can solve MaOPs with the appearance of the decomposition based scalarizing functions and the different performance indicators.R2 indicator is a performance indicator,which can be formulated via given a set of uniformly distributed weight vectors and the proper scalarizing function.R2 indicator is suitable for solving MaOPs in terms of some characteristics,such as weak Pareto dominance,and easy to calculate.Finally,a set of trade-off solutions will be provided for the decision makers to improve the quality of the product and enhance the efficiency of the enterprise production.In this dissertation,we pay our attention on R2 indicator based multi-objective evolutionary algorithms,which are suitable for solving MaOPs.The main contents and results are given as follows:(1)Balancing convergence and diversity is difficult for R2 indicator based generational evolutionary algorithm to address MaOPs.We propose an R2 indicator and reference vector guided many-objective evolutionary algorithm via adopting the reference vector guided secondary selection strategy.On the one hand,the R2 indicator based selection method is introduced to select the first rank candidate solutions,which can be seen as the convergence candidate solutions.On the other hand,the reference vector guided selection method is introduced to prune the remaining candidate solutions to enhance the algorithm diversity.The two-stage selection strategy can well balance the convergence and diversity.To deal with the normalization for R2 indicator,the adaptive reference vector updated method is embedded into our two-stage selection strategy.The R2 indicator and reference vector guided evolutionary algorithm is suitable for MaOPs.The experimental results have validated that the proposed algorithm can well solve MaOPs.(2)Balancing convergence and diversity is difficult for R2 indicator based steady-state evolutionary algorithm,we propose an R2 indicator and decomposition based steady-state evolutionary algorithm via introducing the objective space decomposition strategy to address MaOPs.Firstly,we judge whether the ideal point is updated or not while introducing a new candidate solution.Then,we comprehensively consider how to balance the convergence and diversity for the R2 indicator based selection strategy.The selection strategy of the whole algorithm which is divided into five cases is fully studied.The experimental results have validated that the R2 indicator and decomposition based steady-state evolutionary algorithm can well solve MaOPs through comparing with the related algorithms and the corresponding variants.(3)Based on the multi-objective particle swarm optimizer to deal with multi-frontal MOPs,we propose an R2 indicator and decomposition based multi-objective particle swarm optimizer.Firstly,we introduce R2 indicator as an implicitly elitist maintenance strategy to prune the candidate solutions.Then,the R2 indicator and the PBI(Penalty-based boundary intersection,PBI)decomposition method are adopted to select the global best leader and update the personal best leader,respectively.Consequently,the Gaussian learning strategy and the elitist learning strategy are embedded into our proposed algorithm to help the algorithm jump out of the local PF.The experimental results have validated that R2-MOPSO is suitable for the multi-frontal MOPs.R2-MOPSO is the foundation for solving MaOPs.(4)It is difficult for PSO to balance the convergence and diversity for solving MaOPs.Meanwhile,how to balance the exploitation and exploration for solving MaOPs is difficult to solve.We propose an R2 indicator and Kuixing guided many-objective particle swarm optimizer to address MaOPs because the traditional multi-objective particle swarm optimizer cannot well solve MaOPs.Firstly,a new bi-level archive maintenance strategy is introduced to balance the convergence and diversity.Secondly,a new leader selection strategy based on Kuixing is the bridge between the objective space and the decision variable space.The modified velocity updated equation based on Kuixing is introduced to balance the exploration and exploitation.Finally,the Gaussian learning strategy and the elitist learning strategy are embedded into our proposed algorithm to help the algorithm jump out of the local PF.The experimental results have validated that the R2 indicator and Kuixing guided many-objective particle swarm optimizer can well solve MaOPs. |
