Abstract
The multi-objective evolutionary algorithm based on decomposition (MOEA/D) has achieved remarkable success in regular optimizing multi-objective problems. In practice, there are lots of irregular problems. The projection of their Pareto fronts (PF) may not cover the whole triangle on the first octant. There is an empty PF (EPF) or complex PF area. These problems significantly reduce the performance of MOEA/D. In this paper, an improved algorithm is proposed to deal with such complex problems. The basic idea is: (1) the PBI method is used to ensure the convergence and simultaneously keep diversity over the triangle; (2) a non-dominated and maximum distance based selection is used to replace the dominated objective individuals selected by the PBI method. It is referred to selection based on the PBI and Non-dominated Maximum distance (MOEAD-PNM). First, the mathematical foundation between the penalty factor and contour line is deduced. Based on the foundation, the penalty factor setting method is proposed and the PBI-based method is used to select an individual for each reference vector. The reference vectors directing to the EPF induce “bad” solutions. Then, these solutions are gradually replaced by better individuals selected according to the non-dominated and maximum distance-based selection strategy. The proposed two-step selection method can guarantee the wideness and uniformity of the “good” solution set. Finally, the performance of the MOEAD-PNM algorithm and five classic algorithms on more than ten test problems are compared. The experimental results show the competitiveness and effectiveness of the proposed algorithm on these challenging problems.
Highlights
Multi-objective optimization problems (MOPs) exist widely in the real world, such as route planning [1], distribution networks [2], engineering structures [3], nuclear reactor radiation shielding [4], and novel drug design [5]. It is different from a single objective optimization problem, which can find an optimal solution
The population size of MOEAD-PNM in Table 2 refers to the population size selected in the PBI selection, excluding individuals selected by the non-dominated maximum distance selection operations
STRATEGY ANALYSIS OF THE MOEAD-PNM To verify the effectiveness of the proposed strategies, we respectively describe the evolution process of three different categories of irregular problems
Summary
Multi-objective optimization problems (MOPs) exist widely in the real world, such as route planning [1], distribution networks [2], engineering structures [3], nuclear reactor radiation shielding [4], and novel drug design [5]. Algorithms for solving the MOPs are mainly divided into three categories: Pareto dominance, indicators, and decomposition These algorithms have corresponding deficiencies when dealing with irregular problems. A set of predefined reference vectors uniformly distributed throughout the objective space is used in decomposition-based MOEAs. The optimal solution of a certain sub-problem is the intersection of the reference vector of the sub-problem and the real PF [18]. In literature [26], a reference vector adaptive strategy (ADEA) is proposed to deal with different scale problems. An improved algorithm based on the PBI decomposition is proposed for irregular multiobjective optimization problems. A precise θ calculation method for each reference vector is proposed and used It can simultaneously ensure diversity and convergence in each search direction.
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