三目标优化:一种计算Pareto非劣解相对于各优化目标偏向度及其进一步分析的方法

考虑到理性决策者通常以获得高性价比结果为最佳选择,本文基于求解三目标优化问题得到的Pareto非劣解进一步分析.以"性价比"概念为基础,建立了Pareto前沿各点排序的基本规则,定义了相邻点概念,并明确了相邻点选择的方法.根据Pareto前沿各点与其相邻点的分布特点,计算得到了Pareto前沿各点的变化率;设计了灵敏比概念,得到了各Pareto非劣解相对各优化目标的偏向程度.本文的创新性贡献有3点:①利用三目标Pareto前沿灵敏比形成的新支配关系,进一步得到了比Pareto非劣解集范围更小的子集;②首次量化出三目标优化问题的Pareto非劣解相对于各优化目标的偏向度;③给出了各Pareto非劣解偏向于各优化目标的不平衡度,得到了不平衡度最小的解.最后,通过具体算例演示了上述计算过程,并与多种常用方法的计算结果进行了对比分析,验证了文中所述方法的可行性和有效性.本文研究成果对于进一步认识Pareto非劣解所具有的重要特性,深化三目标优化问题的求解是一次重要的理论推进.

Tri-objective optimization problems: A method of calculating the bias degree and further analysis of each Pareto non-inferior solution corresponding to each objective

Considering the rational decision makers usually make the best choice to obtain high cost-performance results, further analysis is conducted based on the Pareto non-inferior solutions that are derived by solving a tri-objective optimization problem. Based on the concept of cost-performance ratio, the basic rule of ordering the Pareto front points is established, the concept of the adjacent points is defined, and the method for the selection of adjacent points is clarified. Based on the distribution feature of the Pareto front points and their adjacent points, the Pareto front change rate is calculated. The concept of sensitivity ratio is defined, and the bias degree of each Pareto non-inferior solution corresponding to each objective is calculated. The innovations are as follows:(i) a new dominance relationship that is formed by the Pareto front sensitivity ratio of the tri-objective optimization problem is used to obtain a subset that has a smaller range than the Pareto non-inferior solution set; (ii) the bias degree that corresponds to each Pareto non-inferior solution for each objective is quantified for the tri-objective optimization problems for the first time; (iii) the unbalance degree, which corresponds to each Pareto non-inferior solution for each objective, is also derived. Finally, the above calculation process is demonstrated by calculating numerical examples, and the results are compared with that obtained by other common methods. The results illustrate that the proposed method in this paper is feasible and valid. This research is a significant theoretical advancement for understanding the important features of Pareto non-inferior solutions and for solving tri-objective optimization problems.