This article focuses on the multi – objective optimization problem in engineering. Aiming at the complex multi – objective problems that lead to the loss of convergence and diversity of optimization results, an improved algorithm AKnEA (KnEA – Based Assistant Point Evolution Algorithm) is proposed. The article first introduces the background and significance of multi – objective optimization, then elaborates on the principle of the KnEA algorithm and the improvement ideas of AKnEA in detail. Subsequently, through simulation tests and engineering examples, the performance of AKnEA is verified. Finally, the research results are summarized, and prospects for future research directions are put forward.
1. Introduction
1.1 The Significance of Multi – Objective Optimization
Multi – objective optimization problems are widespread in various fields such as engineering, biology, and ecology [1]. In engineering design, for example, when designing a product, multiple performance indicators often need to be considered simultaneously, such as minimizing cost, maximizing performance, and improving reliability. These objectives usually conflict with each other. For instance, in the design of a car engine, increasing power may lead to higher fuel consumption and larger emissions. Therefore, finding a balance among these objectives is crucial for obtaining an optimal design solution.
1.2 Challenges in Multi – Objective Optimization
As multi – objective problems become more complex, traditional optimization algorithms face great challenges. One of the main problems is maintaining the balance between the convergence and diversity of solutions during the optimization process. Excessive emphasis on diversity may result in insufficient convergence, meaning that the solution cannot approach the optimal value effectively. On the contrary, over – focusing on convergence may lead to a loss of diversity, causing the algorithm to fall into local optima and unable to find a better global solution [2].
1.3 Research Objectives and Methods
The main objective of this research is to develop an efficient multi – objective optimization algorithm that can better handle complex problems and maintain the balance between convergence and diversity. To achieve this goal, this article proposes the AKnEA algorithm. The research methods include theoretical analysis of the algorithm, simulation tests using standard test functions, and application verification in practical engineering problems.
2. KnEA Algorithm Principle
2.1 Knee Point Identification in KnEA
In the KnEA algorithm, knee points play a crucial role. Knee points are identified by calculating the distance from non – dominated solutions to the hyperplane formed by extreme points. For a two – objective minimization problem, as shown in Figure 1, the line connecting the two extreme points A and I is regarded as the extreme value line L. If the distance from a solution to the extreme value line L is the maximum in its neighborhood, this point is identified as a knee point. In Figure 1, points A, C, G, H, and I are identified as knee points. [Insert Figure 1: Knee Points and Neighborhoods in KnEA]
Mathematically, for the extreme value line \(ax + by + c = 0\) (where the values of a, b, and c are determined by the extreme points), the distance formula from a solution \(C(x_C,y_C)\) to the extreme value line L is: \(d(C, L)=\begin{cases}\frac{\vert ax_{C}+by_{C}+c\vert}{\sqrt{a^{2}+b^{2}}}&\text{if }ax_{C}+by_{C}+c < 0\\-\frac{\vert ax_{C}+by_{C}+c\vert}{\sqrt{a^{2}+b^{2}}}&\text{otherwise}\end{cases}\)
2.2 Adaptive Adjustment of Neighborhood Size
After identifying the knee points, the KnEA algorithm adaptively adjusts the neighborhood size of each knee point. The neighborhood size is adjusted according to the ratio of the number of identified knee points to the total number of non – dominated solutions. In the early stage of evolutionary optimization, the neighborhood size decreases rapidly, which leads to a significant increase in the number of found knee points. When the iteration reaches a certain level, the neighborhood size remains unchanged. This adaptive adjustment mechanism helps the algorithm balance exploration and exploitation during the optimization process.
2.3 Role of Knee Points in Environment Selection
In the environment selection process of the KnEA algorithm, knee points are preferentially selected to enter the next generation. This is because knee points are often located in areas with potential for further improvement in the objective space. By giving priority to knee points, the algorithm can guide the population to move towards better solutions and improve the convergence speed of the algorithm.
3. AKnEA Algorithm Improvement
3.1 Neighborhood Space Evolution Potential Analysis
3.1.1 Concept of Neighborhood Potential
The AKnEA algorithm proposes a neighborhood potential strategy to better guide the population evolution direction. The potential of the neighborhood space is analyzed based on the neighborhood space where the knee point is located. As shown in Figure 2, the potential of the neighborhood space is related to whether there are knee points from the previous generation in the current neighborhood space. [Insert Figure 2: Neighborhood Potential in AKnEA]
3.1.2 Calculation of Neighborhood Potential
The size of the neighborhood space is represented by the cosine similarity between the individuals near the neighborhood boundary and the knee point. To analyze the potential value of the current knee – point neighborhood space, two cases are considered. If there is no previous – generation knee point in the neighborhood space, it indicates that this area may be a new exploration area. If there is a previous – generation knee point and there is only one such point, the Euclidean distance d between the previous – generation knee point \(C’\) and the current knee point C is calculated to quantitatively analyze the potential of the neighborhood space. A smaller distance d may indicate a more promising neighborhood space for further exploration.
3.2 Assistant Point Strategy
3.2.1 Motivation for the Assistant Point Strategy
In the face of complex multi – objective problems, the population may experience excessive diversity and insufficient convergence during the evolution process. To address this issue, the AKnEA algorithm proposes an assistant point strategy. By increasing the search intensity in potential spaces, the convergence of the evolutionary algorithm can be further improved.
3.2.2 Determination of Assistant Points
After evaluating the potential of the knee – point neighborhood space, assistant points are adaptively added to the potential space according to the evaluation results. The relationship between the average Euclidean distance Hm between individuals in the neighborhood and the knee point and the distance d is used as the basis for determining whether to add an assistant point. The formula for calculating the average distance Hm is: \(Hm = mean\left(\sum_{i = 1}^{n}h_{i}^{j}\right)\) When \(d>Hm\), an individual in the neighborhood is randomly selected as an assistant point to enter the next generation. This strategy can enhance the exploration ability of the algorithm in potential spaces and promote the population to approach the true Pareto front.
3.3 Algorithm Flow
The flow of the AKnEA algorithm is shown in Figure 3. First, the total number of iterations and other parameters are set, and the non – dominated solutions are selected to enter the next generation. Then, the knee points are initialized. In each iteration, the non – dominated sorting of the previous – generation knee points is performed, and assistant points are added according to the neighborhood potential and the assistant point strategy. Finally, the new population is obtained, and the algorithm continues to iterate until the termination condition is met. [Insert Figure 3: Flow Chart of AKnEA Algorithm]
4. AKnEA Simulation Test
4.1 Test Functions and Parameter Settings
4.1.1 Selection of Test Functions
The WFG series of test functions is selected to verify the performance of the AKnEA algorithm [3]. The WFG series includes a variety of complex multi – objective test problems, which can comprehensively evaluate the performance of the algorithm in different scenarios. The AKnEA algorithm is compared with three popular evolutionary algorithms, MOEAD [4], MMOPSO [5], and KnEA, on 3 – objective and 4 – objective WFG series test functions.
4.1.2 Parameter Settings
The parameters of the compared algorithms adopt the recommended values in the original literature. In the process of solving the test functions, the number of evaluations is set to 10000, and the number of individuals in the population is set to 100. All tests are independently run 30 times, and the average value and standard deviation of the performance index values are recorded. In addition, the Wilcoxon rank – sum test with a significance level of 0.05 is used to statistically analyze the experimental results. The symbols “+”, “ – ”, and “≈” represent that the performance index result of the compared algorithm 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) [6]. For each test problem, about 5000 points are uniformly sampled on the true Pareto front to calculate the IGD value. The smaller the IGD value, the better the performance of the algorithm. The IGD value can comprehensively measure the convergence and diversity of the solution set obtained by the algorithm, reflecting how close the solution set is to the true Pareto front and how evenly the solutions are distributed.
4.3 Test Results Analysis
4.3.1 Results on 3 – Objective WFG4 – 7 Test Functions
The IGD standard average values, variances, 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 of the table that the IGD values of the proposed AKnEA algorithm are all smaller than those of the other compared algorithms. From the results of the rank – sum test, 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, indicating that the proposed algorithm has an advantage in the stability of the optimization effect. [Insert Table 1: Test Results On WFG4 – 7 with 3 Objectives]
4.3.2 Results on 4 – Objective WFG4 – 7 Test Functions
To further test the performance of the algorithm on high – dimensional complex problems, the proposed algorithm is tested on the 4 – objective WFG4 – 7 test functions. The test results of the four algorithms are shown in Table 2. The experimental results once again verify the superiority of AKnEA. Specifically, the optimization effects of MOEAD and MMOPSO on 4 – objective problems are significantly worse than those of the proposed algorithm. In addition, compared with KnEA, the standard deviation values of the optimization results of the proposed algorithm are all smaller than those of KnEA, indicating that the proposed algorithm can obtain more stable optimization results in high – dimensional complex problems. [Insert Table 2: Test Results on WFG4 – 7 with 4 Objectives]
4.3.3 Visualization of AKnEA Performance
To more vividly display 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 a good performance in maintaining the diversity and convergence of the population. The solutions obtained by the AKnEA algorithm are evenly distributed on the Pareto front, and are closer to the true Pareto front, indicating that the algorithm can effectively find a set of high – quality solutions in multi – objective optimization problems. [Insert Figure 4: Parato Frontier on WFG6 with AKnEA Objective 3 and 4]
5. Engineering Example: Optimized design of two-stage helical cylindrical gear reducer
5.1 Reducer Multi – Objective Optimization Model
5.1.1 Design Variables
The design variables of the two – stage helical cylindrical gear reducer are defined as: \(X=\left[m_{s1},m_{n2},z_{1},z_{3},i_{1},\beta_{1},\beta_{2},\psi_{d1},\psi_{d2}\right]^{T}=\left[x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7},x_{8},x_{9}\right]^{T}\) where \(m_{n1}\), \(m_{n2}\) are the normal module of the two gears; \(z_{1}\), \(z_{3}\) are the number of teeth of the high – speed – stage small gear and the low – speed – stage small gear of the reducer, respectively; \(i_{1}\) is the transmission ratio of the high – speed stage of the reducer; \(\beta_{1}\), \(\beta_{2}\) are the helix angles of the pitch circles; \(\psi_{d1}\), \(\psi_{d2}\) are the face – width coefficients. The number of teeth of the high – speed – stage large gear is calculated by \(z_{2}=i_{1}z_{1}\), and the number of teeth of the low – speed – stage large gear is calculated by \(z_{4}=7.3776z_{3}/i_{1}\).
5.1.2 Objective Functions
In the design of a belt – driven conveyor, the transmission power of the reducer is 4.15kW, the maximum input speed and output speed are given, and the total transmission ratio \(i_{all}=7.3776\). The AKnEA algorithm is used for multi – objective optimization design of the reducer. The population size is set to 200, and the number of evaluations is set to 50000. To reduce the production cost of the reducer, minimizing the center distance and the gear thickness are set as the two optimization objectives. The objective functions are as follows: \(min f_{1}(X)=\frac{m_{s1}z_{3}(1 + i_{1})}{2\cos\beta_{1}}+\frac{m_{n2}z_{4}(1 + 7.3776/i_{1})}{2\cos\beta_{2}}\) \(min f_{2}(x)=\psi_{d1}d_{1}+\psi_{d2}d_{3}=\frac{\psi_{d1}m_{n1}z_{1}}{\cos\beta_{1}}+\frac{\psi_{d2}m_{s2}z_{3}}{\cos\beta_{2}}\) The relationship curve between the two objectives of the reducer is shown in Figure 5. It can be seen from the figure that the center distance and the tooth width are inversely proportional. In practical engineering, considering the conflict relationship between the objectives, the design scheme of the reducer can be selected on the relationship curve according to actual needs. [Insert Figure 5: Curve of Relationship Between Center Distance and Tooth Width]
