In practical engineering, complex multi-objective problems often lead to losses in the convergence and diversity of optimization results. To address this issue, an auxiliary point evolutionary algorithm based on KnEA (A Knee Point Driven Evolutionary Algorithm) is proposed. This algorithm utilizes inflection points with obvious trends in the evolution process as the centers of neighborhood spaces. By analyzing the relationship between inflection points in the neighborhood space and those of the previous generation, the potential of the neighborhood space is quantitatively evaluated. Additionally, a strategy of adding auxiliary points based on the evolution potential of the neighborhood space is introduced to enhance the spatial search ability during evolution. In the population environment selection, inflection points and auxiliary points are preferentially chosen to evolve the next generation, thereby balancing the convergence and diversity of the population. The proposed algorithm is compared with popular algorithms on multi-objective test problems, and the experimental results demonstrate its significant advantages in multi-objective optimization. Finally, the algorithm is applied to the optimization design of a two-stage helical cylindrical gear reducer.
1. Introduction
Multi-objective optimization problems have become one of the crucial issues in various fields such as engineering, biology, and ecology. In recent years, multi-objective problems have shown an increasingly complex trend in their forms, thus imposing stricter requirements on the optimization performance of multi-objective optimization algorithms in practical problems. During the multi-objective optimization process, excessive diversity maintenance may result in insufficient convergence of solutions. Conversely, excessive convergence maintenance can lead to the loss of solution diversity.
To maintain diversity and convergence, various strategies have been adopted in the population evolution process. One category is the multi-subpopulation strategy. Among them, the grid partitioning strategy has significant advantages in dividing the population. For example, in [3], a grid partitioning memetic clustering analysis was used to enhance population diversity. However, it is difficult to determine appropriate parameter settings using the grid strategy. Another category is the adaptive selection of individuals with obvious trends on the Pareto front to adjust the diversity and convergence of the population. In [4], a method of gradually selecting individuals with the best convergence to enter the next generation in the environment selection was proposed to improve the population convergence and diversity. In [5], inflection points with obvious trends were used as the preferred individuals in the environment selection, and the neighborhood range of the inflection points was adaptively partitioned to increase the convergence rate and maintain diversity.
Despite the ability to improve the selection pressure of the population through individuals with obvious advantages, the problem of insufficient convergence still persists when facing complex problems. Therefore, to better maintain population diversity and improve convergence in solving complex multi-objective problems, an auxiliary point evolutionary algorithm based on KnEA, named AKnEA (KnEA-Based Assistant Point Evolution Algorithm), is proposed. AKnEA divides the target space into multiple subspaces by establishing a spatial environment centered around each inflection point. It judges the evolution potential of the target space by analyzing the evolutionary distance between consecutive generations of inflection points. To enhance the competitive ability of the potential space during evolution, additional neighbors of the inflection points are selected to enter the next generation, thereby continuously approaching the true Pareto front. To verify the performance of the AKnEA algorithm, it is compared with three popular evolutionary algorithms on the WFG series of test functions. Finally, to prove the applicability of the algorithm to practical engineering problems, it is applied to the optimization design of a two-stage helical cylindrical gear reducer.
2. KnEA Principle
In KnEA, inflection points are identified by the distance from non-dominated solutions to the extreme point hyperplane, and an adaptive strategy is employed to adjust the neighborhood size of each inflection point. The identification of inflection points and their neighborhoods is illustrated in Figure 1.
2.1 Identification of Inflection Points
Figure 1 shows the method of identifying inflection points in a bi-objective minimization problem. The line connecting points A and I is defined as the extreme line L. If the distance from a solution to the extreme line is the maximum within its neighborhood, then that point is identified as an inflection point. In the figure, points A, C, G, H, and I are identified as inflection points. For a bi-objective minimization problem, the extreme line L is defined as follows:
where the specific values of , , and are determined by the extreme points.
The distance formula from a solution to the extreme line is as follows:
2.2 Neighborhood Adjustment
In the environment selection, inflection points are preferentially selected to enter the next generation to enhance convergence. After each iteration, the neighborhood size of the inflection points is adaptively adjusted. The neighborhood size is adjusted based on the ratio of the number of identified inflection points to the total number of non-dominated solutions. In the early stage of evolutionary optimization, the neighborhood size decreases rapidly, leading to a significant increase in the number of identified inflection points. When the iteration reaches a certain level, the neighborhood size remains constant.
3. Auxiliary Point Evolutionary Algorithm Based on KnEA
3.1 Neighborhood Space Evolution Potential
To provide more effective guidance for the population evolution direction, a neighborhood potential strategy is proposed. The potential of the neighborhood space is analyzed based on the neighborhood space where the inflection point is located. The representation of the neighborhood space potential is shown in Figure 2.
In Figure 2, the shaded area represents the neighborhood space centered around inflection point C. The size of the neighborhood space is represented by the cosine similarity between the individual near the neighborhood boundary and inflection point C. The formulas for the upper and lower bounds of the neighborhood space size are as follows:
where and represent the upper and lower bounds of the neighborhood space, respectively.
To analyze the potential value of the current inflection point’s neighborhood space, it is explored whether there are inflection points from the previous generation in this neighborhood space. There are two cases: one is that no previous generation inflection point is found in the neighborhood space, and the other is that a previous generation inflection point is detected. When there is only one previous generation inflection point in the neighborhood space, the Euclidean distance between and the current inflection point is calculated to quantitatively analyze the potential of this neighborhood space.
3.2 Auxiliary Point Strategy
In complex multi-objective problems, the population may experience excessive diversity and insufficient convergence during evolution. Therefore, enhancing the search intensity in the potential space of the target space can further improve the convergence of the evolutionary algorithm.
After evaluating the potential of the inflection point’s neighborhood space, auxiliary points are adaptively added to the potential space based on the potential evaluation result to improve the search ability. The relationship between the average Euclidean distance between the individuals in the neighborhood and the inflection point and the distance is used as the basis for determining whether to add auxiliary points to the space. The formula for calculating the average distance is as follows:
where is the index of the inflection point, and is the Euclidean distance from the -th inflection point to the inflection point in the previous generation within the neighborhood. When , a random individual within the neighborhood is selected as an auxiliary point to enter the next generation.
3.3 Algorithm Flow
The flowchart of the proposed AKnEA algorithm is shown in Figure 3.
- Initialization: Set the total number of iterations and initialize the population.
- Non-dominated Sorting: Sort all non-dominated solutions in the population.
- Inflection Point Identification and Neighborhood Adjustment: Identify inflection points and adjust their neighborhood sizes.
- Neighborhood Space Potential Evaluation: Evaluate the potential of the neighborhood space of each inflection point.
- Auxiliary Point Selection: Based on the potential evaluation result, select auxiliary points if necessary.
- Environmental Selection: Select inflection points and auxiliary points to enter the next generation.
- Termination Condition Judgment: If the termination condition is met (e.g., reaching the maximum number of iterations), output the population; otherwise, return to step 2.
4. AKnEA Simulation Test
4.1 Test Functions and Parameter Settings
The WFG series of test functions are selected to verify the performance of the AKnEA algorithm. The proposed algorithm is tested and compared with MOEAD, MMOPSO, and KnEA on the WFG series of test functions with 3 and 4 objectives.
The parameter settings of the compared algorithms are adopted from the original literature. In the algorithm’s solution process for the test functions, the number of evaluations is set to 10,000, and the number of individuals in the population is set to 100. All tests are independently run 30 times, and the average and standard deviation of the performance index values are recorded. Additionally, the Wilcoxon rank-sum test with a significance level of 0.05 is used to statistically analyze the experimental results, where the symbols “+”, “-“, and “≈” indicate that the algorithm’s index result is significantly better, significantly worse, and statistically similar to that of the AKnEA algorithm, respectively.
4.2 Algorithm Performance Evaluation Index
The performance of the algorithm is evaluated by the Inverted Generational Distance (IGD). For each test problem, approximately 5,000 points are uniformly sampled on the true Pareto front to calculate the IGD value. A smaller IGD value indicates better algorithm performance.
4.3 Test Results Analysis
The IGD standard average, variance, and rank-sum test results of the MOEAD, MMOPSO, KnEA, and AKnEA algorithms on the 3-objective WFG4 – 7 test functions are shown in Table 1. It can be seen from the shaded part that the IGD values of the proposed algorithm are smaller than those of the other compared algorithms. From the rank-sum test results, the performance of AKnEA is significantly better than that of the other algorithms. Compared with KnEA, on the 3-objective WFG7 test function, the standard deviation of the proposed algorithm is significantly smaller than that of KnEA. Therefore, the proposed algorithm also has an advantage in the stability of the optimization effect.
| Problem | MOEAD | MMOPSO | KnEA | AKnEA |
|---|---|---|---|---|
| WFG4 | 2.8955e – 1 (1.23e – 2)- | 3.0679e – 1 (9.53e – 3)- | 2.5756e – 1 (8.13e – 3)- | 2.4173e – 1 (6.72e – 3) |
| WFG5 | 2.7446e – 1 (9.33e – 3)- | 2.9072e – 1 (8.92e – 3)-(7.48e – 3)- | 2.5088e – 1 (5.88e – 3) | |
| WFG6 | 3.2832e – 1 (2.23e – 2)- | 3.5647e – 1 (4.35e – 2)-(1.77e – 2)- | 3.0730e – 1 | 2.8979e – 1 (1.51e – 2) |
| WFG7 | 4.1946e – 1 | 2.9189e – 1 (5.26e – 2)-(1.03e – 2)-(1.12e – 2)- | 2.5228e – 1 | 2.3008e – 1 (5.11e – 3) |
| +/-/= | 0/4/0 | 0/4/0 | 0/4/0 |
To further test the performance of the algorithm on high-dimensional complex problems, the proposed algorithm is tested on the 4-objective WFG4 – 7. The test results of the four algorithms are shown in Table 2. The experimental results again verify the superiority of AKnEA. More specifically, the optimization effects of MOEAD and MMOPSO on the 4-objective problems are significantly worse than that of the proposed algorithm. In addition, compared with KnEA, the standard deviation values of the optimization results of the proposed algorithm are smaller than those of KnEA.
| Problem | MOEAD | MMOPSO | KnEA | AKnEA |
|---|---|---|---|---|
| WFG4 | 8.1244e – 1 (1.05e – 1)- | 7.7874e – 1 (2.24e – 2)- | 7.2068e – 1 (1.65e – 2)- | 6.9642e – 1 (1.24e – 2) |
| WFG5 | 9.2846e – 1 (8.66e – 2)- | 7.8969e – 1 (2.11e – 2)- | 7.0798e – 1 (2.19e – 2)- | 6.8301e – 1 (1.23e – 2) |
| WFG6 | 1.0240e + 0 (1.44e – 1)- | 9.1594e – 1 (4.51e – 2)- | 7.5895e – 1 (2.81e – 2)- | 7.2923e – 1 (2.73e – 2) |
| WFG7 | 1.0483e + 0 | 7.9151e – 17.1677e – 1 (1.56e – 1)-(2.53e – 2)-(2.03e – 2)- | 6.8315e – 1 (1.40e – 2) | |
| +/-/= | 0/4/0 | 0/4/0 | 0/4/0 |
To better illustrate the performance of AKnEA, the Pareto fronts of the AKnEA algorithm on the 3-objective and 4-objective WFG6 test problems are shown in Figure 4. It can be seen from the figure that the proposed algorithm has good performance in maintaining both population diversity and convergence.
(a) WFG6: 3 objectives
(b) WFG6: 4 objectives
Figure 4: Pareto Fronts of AKnEA on WFG6 with 3 and 4 Objectives
5. Engineering Example
A two-stage helical cylindrical gear reducer is composed of transmission components, shaft parts and their accessories, and the gearbox and its accessories. The various components interact with each other, and even contradictions may arise in the parameter design. Therefore, in the design process of the components, the relationships between different requirements need to be comprehensively considered.
5.1 Reducer Multi-Objective Optimization Model
5.1.1 Design Variables
where and are the normal module of the two gears; and are the number of teeth of the high-speed and low-speed stage pinions of the reducer, respectively; is the transmission ratio of the high-speed stage of the reducer; and are the helix angles of the reference circle; and and are the tooth width coefficients.
5.1.2 Objective Functions
To reduce the production cost of the reducer, reducing the volume and weight of the reducer is an important optimization objective. Therefore, minimizing the center distance of the reducer and the thickness of the gears are the two optimization objectives. The objective functions are as follows。
