Multi-objective optimization: A method for selecting the optimal solution from Pareto non-inferior solutions

Based on the concept of performance-price ratio, we propose a quantitative method to solve multi-objective optimization problems. A new hypothesis is established in this paper: market rules that seek a higher-performing product with a lower price are used to compare and select Pareto non-inferior solutions. After carefully observing the distribution of the Pareto front, we find that the distribution is monotonically increasing or decreasing. This means that different variability exists in the Pareto front and that new inherent disciplines can be found. Based on this discovery, we use the performance-price ratio as a reference to construct the average variability that adjacent non-inferior solutions correspond to the objective function values. Then, the sensitivity ratio that is similar to the performance-price ratio is obtained, and a quantitative method is developed to evaluate Pareto non-inferior solutions. Two important achievements are derived: (1) based on the sensitivity ratio, a new subset of the Pareto non-inferior solution set is formed in accordance with the dominance relationship. The number of Pareto non-inferior solutions is reduced, and the bias degree corresponding to every Pareto non-inferior solution is obtained for different objectives. Thus, it is convenient for decision makers to select Pareto non-inferior solutions based on their preferences. (2) In the new subset of Pareto non-inferior solutions, the solution that corresponds to the minimal absolute value difference of the sensitivity ratio for different optimization objectives is defined as an unbiased and good solution. Accordingly, we obtain the optimal solution that is acceptable for every objective. Finally, the method is illustrated with a numerical example.