An adaptive boundary-based selection many-objective evolutionary algorithm with density estimation

Abstract

Many-objective evolutionary algorithms often struggle to strike a balance between convergence and diversity when solving many-objective optimization problems. Consequently, an adaptive boundary-based selection many-objective evolutionary algorithm with density estimation, MaOEA/ABS-DE, is proposed. Specifically, the algorithm initially utilizes prior information regarding the adaptive estimation of the shape of the Pareto front to perform coordinate transformation on the population, thus guiding the population towards the approximate boundary of the current Pareto front. Subsequently, a novel approach for estimating the neighboring density of individuals is introduced by utilizing the distribution information of the transformed population. Furthermore, adaptive boundary-based selection and shifted-based density estimation selection strategies are designed for the environmental selection process. The environmental selection process is completed through a two-step selection. In the first step, a candidate pool is formed by selecting solutions with lower neighboring density, which aims to maintain a broad search for the population in the objective space. In the second step, the SDE is introduced to compare the convergence of individuals in the candidate pool, and those with the optimal SDE values are selected. Finally, extensive comparative experiments are conducted on two benchmark test suites, MaF (Many-objective Functions) and WFG (Walking Fish Group), as well as on two practical cases. The MaOEA/ABS-DE is compared with five representative many-objective evolutionary algorithms, including 1by1EA, NSGA-III, MaOEA-CSS, RVEA, and SPEA2 + SDE. Experimental results demonstrate that the proposed algorithm outperforms or at least matches the performance of the five algorithms on 87.04%, 92.59%, 87.04%, 88.89%, and 87.04% of the test instances, respectively.

1 Introduction

In the intricate and complex scenarios of the real world, the attainment of optimal decision-making is centered on the profound comprehension of practical problems and their accurate abstraction into an optimization model. Subsequently, systematic exploration of solutions to this model is conducted through the utilization of advanced optimization algorithms, leading to the formulation of scientific and efficient decision-making schemes. Owing to its universality and effectiveness, this approach to optimization problems is widely adopted across diverse industries, serving as a crucial driving force for promoting development in various fields. Optimization models and their solution algorithms play a pivotal role in facilitating the efficient operation of production systems. For lot streaming in flowshop scheduling problem, Beren [1] has developed an optimization model aimed at minimizing the makespan, subject to constraints on machine capability and limited waiting time. Additionally, he has also proposed an optimization model that focuses on minimizing the average flow time for the production system. Given the NP-hard nature of such problems, approaches based on genetic algorithms (GA) have been designed to address them [2]. Furthermore, for the U-shaped assembly line balancing and parts feeding problem [3], an optimization model with dual objectives has been constructed: minimizing operational costs and maximizing workload balance. Subsequently, an exact solution method has been proposed for this model. These optimization problems often focus on the consideration of a single or dual objective. However, in practical applications, to make informed and comprehensive decisions, it is necessary to delve deeper and comprehensively consider the complex interactions and impacts of multiple dimensions and factors. Consequently, such problems necessitate the construction of a many-objective optimization model, which aims to simultaneously optimize more than three objectives. For instance, in the field of engineering, the design process necessitates the consideration of factors such as cost, performance, and reliability, among others [4]. In the realm of manufacturing and production, factors like production efficiency, cost, and delivery time must be taken into account [5, 6]. Furthermore, in the field of environmental protection, attention must be paid to pollution reduction, resource conservation, and the enhancement of energy efficiency [7]. Regarding such optimization problems, obtaining a unique optimal solution is not feasible. Instead, a set of satisfactory solutions that take into account all objectives is achieved, commonly referred to as Pareto-optimal solutions. The challenge lies in obtaining a set of uniformly distributed Pareto solution sets that converge as closely as possible to the true Pareto front [8]. Compared to other optimization algorithms, multi-objective or many-objective evolutionary algorithms (MOEAs or MaOEAs) initiate the search from a diverse population of initial points, rather than a single initial point. This approach enhances the likelihood of discovering a set of globally optimal solutions during the optimization process. Furthermore, MOEAs or MaOEAs rely on the evaluation information of objective functions to steer the search process, eliminating the need for gradient information related to the problem. Consequently, MOEAs or MaOEAs demonstrate robust performance and efficient search capabilities in solving such problems, particularly for combinatorial optimization problems that exhibit NP-hardness characteristics [9, 10]. Over the past two decades, a vast variety of MOEAs have been proposed to solve MOPs and have shown good performance. Typical algorithms include the NSGA-II [11], SPEA2 [12]. However, the performance of these algorithms deteriorates drastically when dealing with MaOPs. This can be mainly attributed to the following reasons: Firstly, as the number of objectives increases, the proportion of non-dominated solutions in a population grows exponentially, thereby posing challenges to the selection criteria based on the Pareto dominance principle in promoting the convergence of solutions. Secondly, maintaining population diversity by utilizing information in high-dimensional objective spaces becomes increasingly difficult. To address these challenges, researchers have successfully proposed several outstanding MaOEAs, which are broadly categorized into the following categories.

The first category is dominance-based MaOEAs. The most straightforward approach is to modify Pareto dominance relations, which improve the selection pressure by proposing new dominance theories. The core idea of these new dominance relationships is to relax the strict Pareto dominance by expanding the region of dominance. There are three main approaches to constructing the dominance region. The first approach involves dividing the objective space into grids, and then determining the dominance region based on the grid coordinates of each solution in the objective space, such as, grid dominance [13, 14]. The second approach divides the space into sub-regions by introducing a set of uniformly distributed weight vectors, and then determines the extent of the dominance region based on the aggregation function values within each sub-region. Examples of this approach include SDEA dominance (Scalarization-based dominance evolutionary algorithm) [15], RPS dominance (Reference points strengthened dominance relation) [16], and DR-Dominance (decomposition and rotation) [17]. These dominance relationships primarily design diverse aggregation functions to assess the convergence of solutions and then compare the aggregation function values of individuals within the same sub-region to determine the dominance level. The third approach adapts the determination of the dominance region based on the angle between objective vectors across the population, such as GPO (Generalized Pareto Optimality) [18], angle dominance [19], CSDR (controlled strengthen dominance relationship) [20], SDR (strengthen dominance relationship) [21], and CDR (convergence-based dominance relationship) [22].

The second category is indicator-based MaOEAs. Some performance indicators have been proposed, including the inverted generational distance (IGD) indicator [23], the hypervolume (HV) indicator [24], the R2 indicator [25], and the I_ε+ indicator [26]. These indicators are easy to integrate into MaOEAs. MaOEA/IBP [27] uses the I_ε+ indicator in the environment selection process, and then enhances the search effectiveness of the algorithm by selecting the two individuals with the most comparable search orientations via the fitness function. MaOEA-ITS [28] utilizes a modified version of the IGD indicator for environment selection. It constructs the necessary reference points by incorporating the cutting plane intercept. ANMPSO [29] combines the R2 indicator with a new global best solution selection mechanism to select individuals with better diversity and convergence. HAGA [30] introduces a novel hypervolume-driven selection mechanism. This mechanism divides the objective space into grids and computes the hypervolume within each grid based on the solutions present. This significantly reduces the computational cost associated with HV evaluation. The difficulty with algorithms based on IGD and R2 indicators lies in the collection of an appropriate reference set for calculating the indicator values. The computational complexity of the HV indicator increases exponentially with the increase in the number of objectives. The commonality among algorithms utilizing IGD, HV, and R2 indicators is the reliance on the contributions of individuals to the overall population's indicator values as a criterion for environmental selection.

The third category is decomposition-based MaOEAs. These algorithms employ two primary decomposition approaches. The first approach involves the introduction of a group of weight vectors to decompose MaOPs into a series of scalar subproblems. MOEA/D [31] is the most representative algorithm within this category. To further enhance the performance of MOEA/D in solving MaOEAs, numerous variants of MOEA/D have combined objective decomposition with other selection strategies. For example, θ-DEA [32] classifies the current population based on a set of reference points, and then uses an aggregation function for the solutions of the same type to rank the solutions so that the solutions in the population move closer to the reference points; In MaOEA/D-OPI [33], two scalarization functions are provided. Among them, the vertical projection value of the solution on the weight vector is used as a convergence measure, while the vector angle between the solution and the weight vector serves as a diversity measure; In MOEA/D-2PBI [34], two penalty parameter values in MOEA/D with the PBI function is proposed to enhance both convergence and diversity. Although this algorithm has been found to enhance its running speed, it fails to consider conflicts among weight vectors and does not effectively address the irregular PF problem. The second decomposition approach involves the predefinition of a set of reference vectors that partition the objective space into a series of subspaces, aimed at simultaneously exploring objective space regions and consequently preserving the diversity of the population, such as SPEA-CSS [35], NSGA-III [36] and RVEA [37]. However, it is difficult for predefined weight vectors to match the irregular Pareto front. Therefore, some methods for adaptively adjusting weight vectors have been proposed to improve the robustness of the algorithms. The methods for reference vector adjustment include: converting population individuals into reference vectors based on the angle between them [38,39,40], generating reference vectors through machine learning to predict the distribution of the Pareto front [41, 42], as well as combining fixed and non-fixed reference vectors to generate new ones [43, 44]. Considering that reference vectors can maintain the population diversity to a certain extent, the key to such algorithms is to find directions that can promote population convergence.

The fourth type is MOEAs with special convergence and diversity management strategies. The knee points on the PF facilitate the convergence of the population. Therefore, KnRVEA [45] combines a reference vector adaptive strategy with a knee points distribution adjustment strategy. In high-dimensional objective spaces, the angle between objective vectors or cosine similarity distance is often used to maintain the population diversity. This is primarily achieved by selecting a pair of solutions with the minimum objective vector angle or the maximum cosine similarity distance during each iteration, and then eliminating the one with poorer convergence or diversity, such as MaOEA-CSS [46], AnD [47], ACS-MOEA [48], PeEA [49]. Furthermore, NAEA [50] has designed a density estimation method based on the angle between objective vectors. Subsequently, niche techniques are employed to eliminate individuals with higher crowding degrees, thereby maintaining population diversity. In PSEA [51], to the diversity of the population, a novel strategy with the second auxiliary angle approach is devised for computing the neighborhood density. However, the angle-based selection strategy exhibits a preference for solutions located in the central region of the Pareto Front (PF). To enhance both the convergence and diversity of the population, some MaOEAs with multistage search strategies have been proposed [52,53,54]. These primarily involve dividing the entire evolutionary process into multiple stages, each stage favoring either convergence or diversity. The crucial aspect of these search strategies lies in the division of stages. Finally, to address irregular Pareto Front problems, algorithms with adaptive clustering [55,56,57] have emerged as promising approaches. Additionally, reference point adaptive adjustment methods [58,59,60] have also shown potential in addressing such challenges.

Table 1 provides a summary of the MaOEAs for solving MaOPs. Although most many-objective evolutionary algorithms improve the ability to solve MaOPs, they still have limitations in striking a balance between population convergence and diversity when dealing with irregular PFs. These limitations can be summarized as follows: 1) Dominance-based algorithms primarily rely on convergence metrics to enhance selection pressure, lacking effective mechanisms for maintaining diversity. 2) Algorithms based on decomposition and metrics such as IGD and R2 heavily depend on the distribution of reference vectors or points. Since the Pareto front of the problem is unknown, and when the population is far from the Pareto front, there will be a significant number of dominated individuals. Consequently, the reference vectors or points adjusted based on the population distribution may not accurately reflect the distribution of the Pareto front, potentially misleading the search direction of the population. 3) In high-dimensional objective spaces, algorithms based on the HV indicator require expensive evaluation costs, while I_ε+ is not strictly Pareto compliant. 4) Algorithms with special diversity and convergence management mechanisms often employ Euclidean distance or angular distance to design density estimation methods. However, in high-dimensional objective spaces, Euclidean distance is prone to generating solutions that exhibit resistance to domination. Density estimation methods based on angles tend to favor solutions located in the central region of the Pareto front. Therefore, these individual density measurement methods are not suitable for problems with a large number of objectives and are inadequate in handling different shapes of PFs. If prior information on estimating the shape of the Pareto front for the optimization problem can be leveraged to design density evaluation methods for individual neighborhoods, the capability of MOEAs in addressing MaOPs with various PF characteristics will be enhanced. Motivated by the above considerations, we attempt to design a more efficient algorithm to address the difficulties posed by these problems. The proposed algorithm addresses the difficulty in achieving uniform population distribution in irregular PFs while maintaining a better balance between convergence and diversity.

Table 1 A summary of MaOEAs for solving MaOPs

The contributions of this paper from theoretical perspective are mainly as follows.

• 1) The proposed algorithm designs a coordinated mechanism that incorporates adaptive boundary-based and shifted-based density estimation selection strategies in the environmental selection process. The former is employed to retain a well-distributed solution set as a candidate pool. To prevent an overemphasis on diversity without compromising convergence, the latter introduces the density estimation of individuals as a criterion for selecting solutions from the candidate pool, further safeguarding the population convergence. Through the sequential and collaborative employment of these two strategies, individuals that possess both excellent convergence and diversity are retained to continuously participate in the evolution process.

• 2) In the first selection strategy, a novel individual distribution measurement method is introduced. This distribution measurement employs prior information from adaptive estimation of the PF to perform coordinate transformations for the population. Subsequently, the Euclidean distance between the transformed individuals is utilized to design the neighborhood density estimation of individuals. Consequently, the robustness of the algorithm in solving MaOPs with various PF characteristics is enhanced. In the second selection strategy, individuals with good convergence are further selected from the candidate pool, indicating that the proposed algorithm places greater emphasis on local information in the objective space, thus further strengthening the convergence of the population.

The contributions of this paper from practical perspective is mainly as follows.

• 1) The performance of the proposed algorithm has been extensively evaluated on MaF and WFG benchmark test suites, as well as two practical application problems. Its effectiveness and practicality have been verified through simulations and case studies. The results and findings of this study can provide practical optimization tools and methods for similar real-world optimization problems in fields such as engineering design, production scheduling, and resource allocation.

The rest of the paper is organized as follows: Sect. 2 introduces the basic concepts in MOEAs. Then, Sect. 3 describes the proposed algorithm in detail. Following that, to demonstrate the superiority of the suggested algorithm, experimental designs and findings are provided in Sects. 4 and 5, respectively. Two engineering applications are given in Sect. 6. Finally, conclusions and future works are presented in Sect. 7.

2 Basic concepts

(1) Many-objective optimization problem (MaOP): many-objective optimization problem refers to an optimization problem containing more than three objective functions. In a MaOP, some sub-objectives may necessitate maximization, whereas others necessitate minimization. As a convention, it is customary to convert the requirements of all sub-objectives into minimization. Therefore, A MaOPs can be defined in the following mathematical terms:

$$\begin{gathered} min \, {\mathbf{F}}\left( {\mathbf{x}} \right) = \left( {f_{1} \left( {\mathbf{x}} \right),...,f_{M} \left( {\mathbf{x}} \right)} \right)^{T} \hfill \\ {\mathbf{x}} = \left( {x_{1} ,...x_{D} } \right)^{T} \in \Omega \hfill \\ \end{gathered}$$

(1)

where D is the number of decision variables; M is the number of objective functions;${\mathbf{x}} = (x_{1} ,...x_{D} )^{T}$ is a decision vector composed of D decision variables; $\Omega$ is D- dimensional decision space; F(x) represents an objective vector composed of M objective functions; $f_{m} \left( {\mathbf{x}} \right)$ represents the i-th objective function value.

(2) Pareto dominance relationship: For any two solutions, ${\mathbf{x}} \in \Omega$ and ${\mathbf{y}} \in \Omega$, if the condition in Eq. (2) is satisfied, solution ${\mathbf{x}}$ Pareto-dominates solution ${\mathbf{y}}$, denoted as ${\mathbf{x}} \prec {\mathbf{y}}$.

$$\begin{gathered} \forall m \in \left\{ {1,2,...,M} \right\},f_{m} \left( {\mathbf{x}} \right) \le f\left( {\mathbf{y}} \right) \hfill \\ \exists m \in \left\{ {1,2,...,M} \right\},f_{m} \left( {\mathbf{x}} \right) < f\left( {\mathbf{y}} \right) \hfill \\ \end{gathered}$$

(2)

From the Eq. (2), it can be inferred that for any two solutions, the probability of one Pareto-dominating the other is 0.5^M−1. Therefore, as the number of objectives increases, the probability of one solution dominating another decreases exponentially.

(3) Pareto optimal solution: For a solution ${\mathbf{x}}^{ * } \in \Omega$, if ${}^{\neg }\exists \, {\mathbf{x}} \in \Omega$, such that ${\mathbf{x}} \prec {\mathbf{x}}^{ * }$, then ${\mathbf{x}}^{ * }$ is referred to as a Pareto optimal solution or non-dominated solution.

(4) Pareto optimal set (PS): The collection of all Pareto optimal solution is referred to as the Pareto optimal set, denoted as $\text{PS}= \{ {\mathbf{x}}^{ * } |{}^{\neg }\exists \forall {\mathbf{x}} \in \Omega ,{\mathbf{x}} \prec {\mathbf{x}}^{ * } \}$.

(5) Pareto optimal front (PF): The collection obtained by mapping all Pareto optimal solutions from the Pareto optimal set to the objective space is termed the Pareto optimal front, denoted as $\text {PF}= \{ F({\mathbf{x}})|{\mathbf{x}} \in \text{PS}\}$.

(6) Ideal point: The point defined by the minimum values across M objectives is termed the ideal point, its mathematical definition is provided below.

$${\varvec{z}}^{ * } = (z_{1}^{*} ,z_{2}^{*} ,...,z_{M}^{*} )$$

(3)

$$z_{m}^{ * } = \min \{ f_{m} ({\mathbf{x}}_{1} ),f_{m} ({\mathbf{x}}_{2} ),...,f_{m} ({\mathbf{x}}_{N} )\} \, m = 1,2,...,M$$

(4)

where $N$ is the size of the population P. $f_{m} ({\mathbf{x}}_{i} )$ is the m-th objective value of the solution ${\mathbf{x}}_{i}$; $z_{m}^{ * }$ is the minimum values on the m-th objective.

(7) Nadir point: The point determined by the maximum values across M objectives among non-dominated solutions is referred to as the nadir point. It is defined by the following mathematical terms.

$${\varvec{z}}^{nar} = (z_{1}^{nar} ,z_{2}^{nar} ,...,z_{M}^{nar} )$$

(5)

$$z_{m}^{nar} = \max \{ f_{m} ({\mathbf{x}}_{1} ),f_{m} ({\mathbf{x}}_{2} ),...,f_{m} ({\mathbf{x}}_{N} )\} \, m = 1,2,...,M$$

(6)

where $N$ is the size of the population P. $f_{m} ({\mathbf{x}}_{i} )$ is the m-th objective value of the solution ${\mathbf{x}}_{i}$; $z_{m}^{nar}$ is the maximum values of the m-th objective among non-dominated solutions.

(8) extreme point: For each coordinate axis, the individual closest to the axis direction is termed an extreme point. For an optimization problem with M objectives, it has M extreme points. For the i-th axis direction, its extreme point is calculated by the following formula:

$$E_{i} = \{ {\mathbf{x|x}} = arg \, min{\text{ dist(}}{\mathbf{x,e}}_{i} {)}\} \, i{ = 1,2,} \ldots {,}M$$

(7)

where ${\mathbf{e}}_{i}$ is the unit vector in the i-th axis direction; x is a solution from the population; $dist {(}{\mathbf{x,e}}_{i} {)}$ is the Euclidean distance between x and the unit vector e_i in the objective space.

(9) Normalization: Due to the different ranges among objectives, it is necessary to normalize each population member. For any solution x, the normalization for its i-th objective is as follows:

$$f_{i} ({\mathbf{x}}) = \frac{{f_{i} ({\mathbf{x}}) - z_{m}^{*} }}{{z_{m}^{nar} - z_{m}^{*} }} \, i = 1,2,...,M$$

(8)

3 Proposed MaOEA/ABS-DE

In this section, we present the general framework of the proposed algorithm MaOEA/ABS-DE, followed by a thorough explanation of its key components. These include the mating selection process, the adaptive boundary-based fitness and density evaluation mechanisms, as well as the environmental process.

3.1 Framework of MaOEA/ABS-DE

Algorithm 1 describes the general framework of the proposed MaOEA/ABS-DE, which comprises the following steps: Firstly, an initial parent population P₀, consisting of a number N of individuals is randomly generated in the decision space; Secondly, the offspring population Qt is generated through the application of evolutionary operators, encompassing selection, crossover, and mutation. Thirdly, these offspring are merged with the parent individuals to form a combined population R_t, with a total population size of 2N. Fourthly, the distribution index and convergence index are determined using the fitness function and density estimation function, respectively. Finally, the environmental selection mechanism selects the N individuals exhibiting superior convergence and diversity from the combined population R_t for inclusion in the subsequent generation. This iterative process continues until a termination condition is satisfied. Lastly, to present a more intuitive demonstration of the process of the proposed algorithm in solving many-objective optimization problems, as well as the key components involved in MaOEA/ABS-DE, Fig. 1 depicts the flowchart outlining the specific implementation process of the proposed algorithm. The key components of the MaOEA/ABS-DE are described in detail below.

Algorithm 1 General Framework of MaOEA/ABS-DE

Fig. 1

The flowchart of the specific implementation process for the proposed algorithm

3.2 Mating selection strategy and reproduction

After the establishment of the parent population, the mating selection process is executed to identify promising individuals from P_t. To ensure the production of high-quality offspring, it is imperative to select parental individuals that exhibit both a significant degree of diversity and convergence. The aim of mating selection is to identify high-quality parents to form a mating pool for reproduction. In MaOEA/ABS-DE, mating selection process utilizes a binary tournament selection strategy to identify optimal parents. This binary tournament selection method effectively prevents stagnation in local optima and excels at maintaining the genetic diversity of offspring. Consequently, it has been widely employed in MOEAs.

The mating selection process in MaOEA/ABS-DE is given in Algorithm 2. Firstly, the P_t is normalized based on ideal and nadir points. Subsequently, both the neighborhood density of individuals and convergence evaluation are computed. For an individual x from P_t, its neighborhood density is determined as follows:

$${\Phi }({\mathbf{x}}) = \mathop {\min }\limits_{{{{\varvec{\Omega}}} \in P_{t} ,{{\varvec{\Omega}}} \ne {\mathbf{x}}}} \frac{1}{{d_{ABD} \left( {{\mathbf{x}},{{\varvec{\Omega}}}} \right) + 1}}$$

(9)

where Ω denotes an individual from the population P_t. d_ABD(x, Ω) represents the ABD between x and Ω, as detailed in Sect. 3.3. The neighborhood density of an individual is inversely proportional to the distance between that individual and its nearest neighbor. A smaller aggregate distance indicates a larger neighborhood density for a given individual x. To accurately measure the neighborhood relationships among individuals in high-dimensional objective spaces and enhance the robustness of MOEAs in solving various characteristic MaOPs, the ABD is introduced in this study. The convergence indicator is measured by shift-based density estimation, as elaborated in Sect. 3.4. Subsequently, two individuals, x and y, are randomly sampled from the population P_t. The individual exhibiting the lower values of both neighborhood density and convergence indicator is chosen for mating. In the event of a tie on both indices, one individual is randomly selected. After mating selection, the parental individuals are utilized to generate Q_t through simulated binary crossover (SBX) [61] and polynomial mutation (PM) [62]. These offspring individuals are then combined with the parent population P_t to form a new population R_t.

Algorithm 2 Mating Selection

3.3 Calculation of adaptive boundary-based distribution

A key problem in maintaining good population diversity lies in estimating the neighborhood density of individuals. Most existing MaOEAs tackle this issue by calculating Euclidean distances, cosine values of two objective vectors. However, many of these methods are inadequate for problems involving a large number of objectives and are unable to handle PFs of diverse shapes. To overcome these limitations, this paper designs an adaptive boundary distribution (ABD) for measuring the neighborhood relationship among individuals. Specifically, the distribution measure relies on Lp-norm distance and the PF geometry shape. The PF geometry shape is mathematically defined as:

$$\left( {\mathop \sum \limits_{i = 1}^{M} \left( {f_{i} ({\mathbf{x}})} \right)^{p} } \right)^{1/p} = 1$$

(10)

where M is the number of objective and p is the curvature of PF geometry shape.

In this paper, the curvature of PF is estimated by measuring the Lp-norm distances from non-dominated individuals to the origin [63]. The pseudocode for curvature estimation is detailed in Algorithm 3. The specific steps are as follows:

Step 1: A uniformly distributed set of p-values, denoted as P = {p¹, p², _…, pⁿ}, is constituted. For $\forall p^{j} \in P$, the Lp-norm distances of all non-dominated individuals, denoted as $D({\mathbf{x}}|p^{j} ) = \{ d({\mathbf{x}}_{1} |p^{j} ),d({\mathbf{x}}_{2} |p^{j} ),...,d({\mathbf{x}}_{N} |p^{j} )\}$, are calculated.

$$d({\mathbf{x}}_{i} |p^{j} ) = \left( {\sum\limits_{m = 1}^{M} {(f_{m} } ({\mathbf{x}}_{i} ))^{{p^{j} }} } \right)^{{{1 \mathord{\left/ {\vphantom {1 {p^{j} }}} \right. \kern-0pt} {p^{j} }}}} ,p^{j} > 0$$

(11)

where M is the number of objectives; ${\mathbf{x}}_{i}$ is the i-th non-dominated individual from the population R_t; $f_{m} ({\mathbf{x}}_{i} )$ is the m-th objective value of individual ${\mathbf{x}}_{i}$.

Step 2: For $\forall p^{j} \in P$, the standard deviation of the $D({\mathbf{x}}|p^{j} )$, is calculated.

$$V({\mathbf{x}}|p^{j} ) = std(D({\mathbf{x}}|p^{j} )) \, j = 1,2,...,n$$

(12)

The smaller the value of $V({\mathbf{x}}|p^{j} )$, the closer $p^{j}$ is to the curvature of approximate PF formed by the non-dominated individuals.

Step 3: Determine the p-value that corresponds to the smallest standard deviation among the Lp-norm distances, and designate it as the ideal curvature of the approximate PF constructed by the non-dominated individuals.

For parameter p, a set of selectable values needs to be preset the range of p for the PF shape estimation is set to {0.1, 0.2, 0.3, …, 10.0}. In principle, the range of p should be (0, + ∞). However, since the p-value cannot be zero and the range of contour variation is small when p > 10, setting the p-value range to [0.1, 10] is a reasonable choice. The interval is set to 0.1, as the range of interval 0.1, as it ensures the accuracy of the p value estimation due to the small range [63].

After approximating the curvature of the PF shape, the ABD between a pair of individuals in R_t is computed as a distribution indicator. The corresponding pseudo-code is provided in Algorithm 4. Initially, the transformed coordinate is calculated for each individual x in R_t.

$$\tilde{f}_{i} ({\mathbf{x}}) = \frac{{f_{i}^{\prime } ({\mathbf{x}})}}{{\left( {\mathop \sum \nolimits_{u = 1}^{M} f_{u}^{\prime } ({\mathbf{x}})^{p} } \right)^{1/p} }},i = 1, \ldots ,M$$

(13)

Subsequently, the Euclidean distance between the two transformed coordinates is calculated as the adaptive boundary distribution.

$$d_{{{\text{ABD}}}} ({\mathbf{x}},{\mathbf{y}}) = \sqrt {\mathop \sum \limits_{i = 1}^{M} \left( {\tilde{f}_{i} ({\mathbf{x}}) - \tilde{f}_{i} ({\mathbf{y}})} \right)^{2} } ,{\mathbf{x}},{\mathbf{y}} \in P_{t}$$

(14)

For example, as depicted in Fig. 2, six individuals, namely A, B, C, D, E, and F, are considered. Once curvature of the PF shape is identified, these six individuals are consecutively transformed to A', B', C', D', E', and F' which lie on the boundary of PF. Subsequently, the Euclidean distance between each pair of transformed individuals is calculated as a metric that indicates the distribution of individuals.

Fig. 2

Illustration of the boundary-based distribution

Algorithm 3 Main procedure of the PF curvature estimation

Algorithm 4 Distribution calculation

3.4 Calculation of shift-based density estimation

The behind idea of Shift-based density estimation (SDE) is based on a straightforward principle: it attempts to move individuals with good convergence closer to those with poor convergence by comparing their convergence across each objective [64]. This is done in the hopes that individuals with poor convergence will be assigned a higher density value, which will result in their preferential elimination during the evolutionary process. Consequently, SDE serves as a tool for quantifying the convergence of individuals. When measuring the SDE of individual p ∈ P_t, the initial step involves comparing the convergence of individual p with that of another individual q ∈ P_t across each objective. Subsequently, the individual q will be moved to a new position:

$$f_{i}{\prime} ({\mathbf{q}}) = \left\{ {\begin{array}{*{20}r} \hfill {f_{i} ({\mathbf{p}})} & \hfill {{\text{ if }}f_{i} ({\mathbf{q}}) < f_{i} ({\mathbf{p}})} \\ \hfill {f_{i} ({\mathbf{q}})} & \hfill {\text{ otherwise }} \\ \end{array} } \right.$$

(15)

where f_i(p) and f_i(q) denote the j-th objective function value for individuals p and q respectively; i ∈ (1, 2, ···, M), where M is the number of objectives; $f_{i}{\prime} ({\mathbf{q}})$ represents the i-th objective value of individual q after being transformed to its new position. After the above movement, the SDE of individual p can be determined as follows:

$$D({\mathbf{p}}) = \frac{1}{{\mathop \sum \nolimits_{i = 1}^{k} d\left( {{\mathbf{p}},{{\varvec{\Omega}}}_{i} } \right) + 2}}$$

(16)

where D(p) denotes the SDE of individual p; d (p, Ω_i) denotes the Euclidean distance between individual p and its i-th neighboring individual Ω_i; and k is set to $\sqrt{\text{N}}$. The higher the SDE value for an individual, the stronger convergence properties it exhibits. Figure 3 illustrates the SDE to further understand it. Four non-dominated individuals, A (0,10), B (7,4), C (10,0), and D (6,9), are used to estimate the density of D. First, individuals A (0,10), B (7,4) and C (10,0) are moved to A'(6,10), B'(7,9) and C'(10,9) respectively, as f₁(A) = 0 < f₁(D) = 6, f₂(B) = 4 < f₂(D) = 9 and f₂(C) = 0 < f₂(D) = 9. Subsequently, individuals A' and B' are identified as the two closest neighbors. Consequently, the SDE of D can be calculated.

Fig. 3

Illustration of shift-based density estimation in two-dimensional objective spaces

3.5 Environmental selection process with a two-step selection

Similar to other MOEAs, MaOEA/ABS-DE selects N elite individuals from the combined population R_t during the environmental selection process to serve as the parent population for the next generation. However, the environmental selection strategy of the MaOEA/ABS-DE differs from other algorithms. Its core lies in the two-step collaborative mechanism formed by the adaptive boundary-based selection and the shifted-based density estimation selection strategies, where adaptive boundary-based selection as the former step selection criterion and the SDE as the latter step selection criterion. The former strategy is employed to select a well-distributed solution set to form a candidate pool, while the latter incorporates SDE to identify the most convergent individuals from the candidate pool. Through the sequential collaboration of these two strategies, N individuals are selected one by one for the next generation population P_t+1.

The pseudocode of Algorithm 5 shows the environmental selection process of the proposed algorithm. The following steps provide further clarification of the pseudocode:

• The first step selection process based on adaptive boundary select strategy

  • Step 1: P_t+1 is initialized to contain the individual that exhibits the best convergence in R_t.

  • Step 2: All non-dominated individuals in R_t are identified and assembled into a set Q_N.

  • Step 3: The minimum d_ABD for each non-dominated individual x in R_t is calculated. A larger value of d_min for individual x suggests that it lies in a sparse region.

  • Step 4: The extreme solutions can promote the population towards approximating or covering the boundary of the PF. Consequently, their d_min values are set to + ∞ to facilitate their direct selection into P_t+1.

  • Step 5: All non-dominated individuals are sorted in descending order based on their d_min values. At this point, a temporary set R is introduced.

  • Step 6: If the number of non-dominated individuals is less than N −|P_t|, then all non-dominated individuals in R_t are moved to the set R. Otherwise, the two individuals exhibiting the highest d_min values in Q_N, denoted as u and v, are moved to the set R.

• The second step selection process based on SDE selection strategy

• • Step 7: The SDE is utilized to measure the convergence of individuals.

  • Step 8: The individual r exhibiting the best convergence in R is identified.

  • Step 9: The individual r is moved to P_t+1, while being removed from R_t.

The above steps are repeated until the population size P_t+1 reaches N.

Next, the necessity and advantages of the a two-step collaborative mechanism in the proposed algorithm, which incorporates the adaptive boundary selection strategy and the shifted-based density estimation selection strategy, will be discussed, Specifically, we will highlight the roles of these strategies in striking a balance between population convergence and diversity.

When the number of non-dominated solutions is scant, it signifies poor population convergence. Consequently, in the selection strategy of the first step, priority should be given to individuals that facilitate population convergence. Nevertheless, compared to other individuals in the population, non-dominated individuals exhibit better convergence properties. Therefore, incorporating all non-dominated solutions into the candidate pool can promote the convergence of the population towards the PF. When the number of non-dominated solutions is abundant, it suggests that the population is gradually approaching the PF. Consequently, individuals that promote population diversity should be preferentially selected. Based on the content introduced in Sect. 3.4, adaptive boundary-based distribution considers the shape information of the PF, thereby providing a more precise measurement for the neighborhood density of individuals. When compared to other individuals, finding a pair of individuals with the maximum d_min value indicates that the algorithm has identified individuals in the most sparse region. If individuals can repeatedly be selected from sparse regions, the diversity of the final population in search directions can be maintained effectively. However, if the individual with the maximum d_min value is directly selected, the algorithm will continuously emphasize diversity, lacking maintenance of convergence. Therefore, in the selection strategy of the second step, the convergence of all individuals in the candidate pool is compared to determine which one to be retained for the next generation. Based on Eqs. (15) and (16), it can be observed that the SDE of an individual compares the convergence information of other individuals across each objective. Subsequently, the convergence of an individual is measured through density estimation after its position is transformed. Therefore, the SDE of an individual comprehensively takes into account the convergence of all population members across all objectives. If individuals in the sparsest regions with the best SDE values can be preserved during each iteration, it can simultaneously maintain population diversity while further promoting population convergence.

During the environmental selection process, the MaOEA/ABS-DE algorithm does not rely on any weight vectors or reference points. Consequently, it exhibits the following advantages in solving MaOPs: it avoids the limited generality that arises from the difficulty of aligning the distribution of weight vectors with the shape of the Pareto front, as encountered in decomposition-based methods; it also overcomes the limitations that stem from the challenges in accurately computing indicator values due to the difficulty in adopting reference points for IGD and R2 metrics. Furthermore, in comparison to MOEAs that incorporate diversity management mechanisms based on objective vector angles and Euclidean distances, the designed distribution metric considers prior information on the shape of the PF front, making it more suitable for solving MaOPs with various characteristics.

Algorithm 5 Environmental selection (_Rt,d_ABD,Con)

4 Experimental design

4.1 Benchmark test questions

To comprehensively validate the versatility of the proposed algorithm for solving MaOPs, simulation experiments are conducted on two benchmark test suites, WFG [65] and MaF [66]. These two benchmark test suites are widely used for testing the performance of MOEAs, and they can be scaled to any number of objectives. Different test problems exhibit distinct characteristics, such as linear, mixed (convex and concave), multimodal, disconnected, biased, scaled, or degenerate. Table 2 presents the characteristics of all test problems from WFG and MaF benchmark test suites.

Table 2 The characteristics of and decision variables for each test instance

4.2 Performance metrics

In order to quantitatively assess the performance of algorithms in solving MaOPs, two widely used performance indicators in MOEAs [50, 67], namely the inversed generational distance (IGD) [68] and the diversity metric (DM) [69], the IGD metric is designed to evaluate the convergence and diversity of the solutions obtained by an algorithms, while the DM metric is designed to evaluate the diversity of the solutions obtained by an algorithms. A smaller IGD value indicates better convergence and diversity of the obtained solution set, while a larger DM value signifies superior diversity of the obtained solution set.

When an algorithm is chosen to solve a MaOP, its IGD can be obtained by calculating the distance between the Pareto-optimal solutions obtained by the algorithm and a set of reference points uniformly sampled from the true Pareto front. The formula for computing the IGD metric is as follows:

$$IGD(P) = \frac{1}{{\left| {P^{ * } } \right|}}\mathop \sum \limits_{{z^{ * } \in P^{ * } }} d\left( {z^{ * } ,P} \right)$$

(17)

$$d\left( {z^{ * } ,P} \right) = \mathop {\min }\limits_{z* \in P*,v \in P} d(z*,v)$$

(18)

where P is an approximate pareto optimal solution set obtained by an algorithm in the objective space; P* is a set of uniformly distributed solution set from the true Pareto front. |P^∗| is the total number of solutions in P*; d(z^∗, P) denotes the minimum Euclidean distance between the solution z^∗ ∈ P^∗ from the true Pareto front and solutions from P.

The DM metric is designed to evaluate the uniformity and spread of solutions obtained by algorithms, which takes values within the range [0, 1]. The larger the value of the DM metric, the better the distribution of the solutions. According to the original literature, the parameters of grid division for calculating the metric are all set to three. For a detailed introduction to this DM metric, refer to [69].

4.3 Comparison algorithms

To assess the performance of the proposed MaOEA/ABS-DE, five representative MaOEAs are selected for comparison: 1by1EA [70], NSGA-III [36], MaOEA-CSS [46], RVEA [37], and SPEA2 + SDE [64]. These five algorithms cover the main categories in MOEAs. Specifically, NSGA-III and RVEA are decomposition-based algorithms, while SPEA2 + SDE is an indicator-based algorithm. Both 1by1EA and MaOEA-CSS feature novel selection strategies. A brief introduction to each algorithm follows.

For 1by1EA, solutions are selected one by one in the environment selection process. Initially, convergence is the primary consideration, and only one solution with the best convergence indicator value is chosen. Subsequently, once a solution is selected, its neighboring solutions will be penalized. Through iterations of these two processes, a solution set with satisfactory convergence and distribution can be obtained.

NSGA-III is an improved version based on NSGA-II. It involves generating a set of uniformly distributed reference vectors using a two-layer method, which partitions the objective space into uniformly distributed subregions. Subsequently, a niche selection operator is designed to select solutions from each subregion.

MaOEA-CSS introduces a novel coordinated selection strategy in both the mating selection and environment selection processes. Additionally, to tackle the curse of dimensionality in MaOPs, distance-based convergence indicators and angle-based diversity indicators are proposed in these two selection steps, respectively.

RVEA divides the objective space into a series of subregions using a set of reference vectors. However, unlike the fixed reference vectors used in NSGA-III, the reference vectors in RVEA require dynamic adjustment. According to the scale of the objective value and mixed reference vectors, an adaptive method for dynamically adjusting the reference vector distribution is designed in RVEA. In addition, an angular penalty distance (APD) is proposed to select solutions from each subregion to strike a compromise between convergence and diversity.

SPEA2 + SDE introduces a shift-based density estimation (SDE) strategy based on SPEA2. Unlike traditional density estimation, which only considers the distribution of individuals in a population, SDE enhances selection pressure by simultaneously considering diversity and convergence of individuals. Additionally, SED adheres to Pareto dominance relations.

4.4 Parameter settings

1) Genetic operations: Based on the corresponding references of all comparison algorithms, we employ the simulated binary crossover (SBX) [71] and polynomial mutation (PM) [72] to generate offspring individuals. The parameters for SBX and PM are detailed in Table 3, where D represents the number of decision variables.

Table 3 Parameters for the crossover and mutation operators

2) Population size: To conduct fair comparative experiments with decomposition-based algorithms or reference points-based algorithms, such as NSGA-III and RVEA, the population size is determined based on these algorithms. According to [73], the parameters for population size and the generation of reference vectors are summarized in Table 4.

Table 4 Setting for the population size

3) Termination criterion: The termination criterion for each algorithm on each test problem is a predefined maximum number of iterations. For test problems with 5, 10, and 15 objectives, all algorithms are terminated when the maximum number of iterations reaches 1000, 1500, and 1800, respectively. Each algorithm runs 20 times independently on each test problem.

4) Parameters of the comparison algorithms: All parameters involved in the compared algorithms are set according to the settings in their respective original reference literature.

5) Statistical testing method: To compare the performance of algorithms on a test problem, the Wilcoxon rank-sum test [74] is utilized based on the IGD and DM results (mean and variance) obtained from 20 runs of all algorithms on that test problem. The test determines statistically whether one algorithm performs better than others. To make the statistical results more intuitive, the symbols " + ", "≈", and "-" respectively indicate that the result obtained by a comparison algorithm is significantly better than the proposed algorithm, statistically similar to the proposed algorithm, or significantly worse than the proposed algorithm.

5 Experimental results and discussions

In this section, the performance of the proposed algorithm is investigated. All experiments are conducted in four parts. Firstly, the effectiveness of the key components of the proposed algorithm, namely the adaptive boundary-based selection strategy and the shifted-based density estimation selection strategy, is verified. Secondly, to test the competitiveness of the proposed algorithm in balancing convergence and diversity, a comparative study of MaOEA/ABS-DE with five representative MaOEAs on WFG and MaF benchmark test problem sets is presented. Thirdly, the running time of all competing algorithms in solving MaOPs is compared. Finally, to validate the practicality of the proposed algorithm, all the competing algorithms are applied to optimize two specific engineering application examples.

5.1 Effectiveness of the proposed two strategies

To validate the effectiveness of the designed two-step selection mechanism, two variants were designed based on the proposed algorithm for comparative studies. Specifically, Variant 1, denoted as MaOEA/v1-DE, differs from the proposed algorithm in that it removes adaptive boundary-based fitness calculations in the first step of selection and instead employs Euclidean distance calculations between individuals to assess their distribution. Variant 2, designated as MaOEA/ABS-v2, differs from the proposed algorithm in that it utilizes the Euclidean distance from individuals in the objective space to the origin instead of SDE in the second step of selection. Table 5 presents the experimental results of the two variants and the proposed algorithm on several test problems. It is evident that the performance of the proposed algorithm significantly outperforms its two variants. The reasons for these experimental results are explained below.

Table 5 IGD results obtained for MaOEA/ABS-DE and its variants

For variant 1, the distribution metric based on Euclidean distance fails to accurately measure the distribution of individuals in high-dimensional objective spaces. This makes it challenging to maintain the search direction of the population in the objective space, often resulting in the population converging to only local regions of the PF, rather than the entire front. In contrast, the proposed algorithm employs an adaptive boundary selection strategy that considers prior information on the shape of the PF. This enables a more precise selection of individuals with good distribution, forming the candidate pool. As for variant 2, although it retains the adaptive boundary selection strategy, the second step of selection emphasizes convergence by using the Euclidean distance from individuals to the origin. This makes it difficult to accurately select individuals that promote convergence. By contrast, the proposed algorithm utilizes the SDE, which comprehensively considers the convergence of all population members across all objectives. Consequently, variant 2 exhibits inferior performance in terms of population convergence compared to the proposed algorithm.

Based on the comprehensive analysis, both the adaptive boundary-based selection strategy and the SDE-based selection strategy are indispensable in the proposed algorithm. This is because the two-step collaborative mechanism formed by these strategies effectively balances the population convergence and diversity for solving MaOPs. The verification of the effectiveness of strategies establishes a theoretical foundation for the extensive comparative simulation experiments conducted in the subsequent section.

5.2 Comparative analysis of algorithms on WFG test problems

The IGD and DM results of the six algorithms on the WFG test suite are summarized in Tables 6 and 7, respectively. By comparing the performance of each pair of algorithms based on the statistical data presented in these tables, Figs. 4 and 6 depict the performance scores of the six algorithms in terms of IGD and DM, respectively. Notably, a lower score signifies better algorithm performance. The next specific analysis is as follows.

Table 6 IGD results obtained by the algorithm on the WFG test problem

Table 7 DM results obtained by the algorithm on the WFG test problem

Fig. 4

The average IGD performance score obtained by all algorithms on WFG

Among the 27 test instances, MaOEA/ABS-DE achieved the best results for the IGD indicator in 24 of them. Experimental results clearly show that the overall performance of the proposed algorithm is the best, especially on WFG2, WFG3, WFG4, WFG5, and WFG7, where the IGD indicator of MaOEA/ABS-DE achieves optimal results for all test instances. Additionally, MaOEA/ABS-DE outperforms 1by1EA, MaOEA-CSS, and SPEA2 + SDE on all test instances of WFG. 1by1EA excessively focuses on the population convergence. When selecting an individual to retain, MaOEA-CSS either focuses solely on convergence or takes into account the distribution of individuals. In contrast, the proposed algorithm simultaneously considers both convergence and diversity in a two-step environmental selection process. Although both SPEA2 + SDE and MaOEA/ABS-DE employ SDE, SPEA2 + SDE directly ranks all individuals based on their SDE values, making it challenging to maintain diversity within the population. On the other hand, MaOEA/ABS-DE utilizes an adaptive boundary selection strategy prior to SDE, effectively preserving diversity in search directions. Consequently, MaOEA/ABS-DE outperforms 1by1EA, MaOEA-CSS, and SPEA2 + SDE on WFG. From the results presented in Fig. 4, it can be observed that MaOEA/ABS-DE attains the lowest average performance score. This is due to the fact that six of the nine test problems in WFG exhibit regular PF shapes, particularly concave shapes (WFG4-WFG9). These PF shapes demonstrate regular distributions without any disconnected, degenerated, or mixed features. Consequently, the shapes of their PF can be approximately estimated beforehand. By utilizing this prior information of the PF, the proposed algorithm can effectively steer the population towards the PF, ensuring a more efficient and accurate search process. The test problems WFG1-3 exhibit irregular PF shapes, posing significant challenges in estimation of their curvature. However, MaOEA/ABS-DE stands out as a superior algorithm in comparison to others. This superiority is attributed to the collaborative interaction between the two selection strategies designed, enabling MaOEA/ABS-DE to achieve optimal results. Furthermore, it is noteworthy that NSGA-III and RVEA also demonstrate satisfactory performance. This is due to the uniformly distributed reference vectors in NSGA-III and RVEA, which are capable of matching the PF of these regular test problems, thereby facilitating good performance. Additionally, RVEA incorporates an APD metric that relies on reference vectors, allowing the population to converge as closely as possible towards the direction of the reference vectors. Consequently, RVEA outperforms NSGA-III on the WFG problems, making it a superior choice in these instances.

To visually compare the quality of solution sets obtained by various algorithms in high-dimensional objective spaces, Fig. 5 presents parallel coordinate plots of the solution sets achieved by six algorithms on the 15-objective WFG6. The solution set obtained by 1by1EA is distributed solely along the boundary regions of the PF, due to its preference for solutions located in the boundary areas of individual coordinate axes. Conversely, the solution set achieved by MaOEA-CSS is concentrated in the central region of the PF, as MaOEA-CSS selects solutions based on the objective vector angles, which tend to favor solutions in the central region of the PF. NSGA-III achieves the lowest number of non-dominated solutions, which are sparsely distributed over the PF. This is because the Pareto dominance relationship employed in NSGA-III fails to exert sufficient selection pressure on the population. The solution set achieved by SPEA2 + SDE does not adequately cover the first four objectives. Compared to the other four algorithms, RVEA achieves the second-best solution set, trailing only behind MaOEA/ABS-DE. This is mainly because the PF of WFG6 is concave, and the adaptively generated reference vectors in RVEA slightly compromise the diversity of the solution set on such regular test problems. As can be observed from Fig. 5, the solution set achieved by MaOEA/ABS-DE exhibits the best convergence and diversity, as the proposed algorithm utilizes coordinate transformation to shift solutions to the approximate PF. By leveraging position information from the transformed individuals, adaptive boundary-based selection and SDE-based selection strategies are designed to maintain both population diversity and convergence.

Fig. 5

Final population distribution obtained by each algorithm on 15-objective WFG6

Given the detailed analysis of the experimental results from an IGD perspective, and the fact that the DM indicator solely measures the degree of uniform distribution among population members in the objective space without assessing the convergence of the population, the results in Table 7 will be briefly analyzed. It can be observed from Table 7 that MaOEA/ABS-DE obtains the optimal results for the DM indicator. Figure 6 provides a more intuitive comparison of the ability of each algorithm to maintain uniform population distribution. It can be seen that MaOEA/ABS-DE effectively maintains the distribution of solutions while ensuring convergence. Taken together, the above analysis and experimental results demonstrate that the proposed algorithm not only achieves the best convergence for solving WFG problems but also exhibits a superior distribution of solutions.

Fig. 6

The average DM performance score obtained by all algorithms on WFG

5.3 Comparative analysis of algorithms on MaF test problems

The MaF test problems are characterized by irregular Pareto fronts. Compared to WFG, MaF poses greater challenges to MaOEAs in balancing the convergence and diversity of the population. Consequently, to further evaluate the performance of the proposed algorithm, Tables 8 and 9 present the IGD and DM results obtained by six algorithms on MaF1-10. A detailed analysis of the experimental results follows.

Table 8 IGD results obtained by the algorithm on the MaF test problem

Table 9 DM results obtained by the algorithm on the MaF test problem

The PF of MaF1 exhibits a linear shape but is inverted in the objective space. MaOEA-CSS and 1by1EA achieved the best results. The performance of MaOEA/ABS-DE surpassed both RVEA and NSGA-III. This is because the inverted PF renders most of the reference vectors in RVEA and NSGA-III ineffective. MaF2 and MaF4 exhibit concave PF shapes, posing a challenge to algorithms to converge simultaneously on all objectives. The proposed MaOEA/ABS-DE demonstrates the best performance. For such problems with simple PF shapes, the adaptive boundary selection strategy in the proposed algorithm can accurately guide solutions towards the PF, while the SDE emphasizing convergence effectively maintains the convergence of the population. MaF5 and MaF10 are primarily used to test the ability of algorithms to maintain population diversity. MaOEA/ABS-DE achieves the best performance on these two test problems. MaF7 exhibits a disconnected PF, and the proposed MaOEA/ABS-DE demonstrates the best performance. As the number of objectives increases, the competitiveness of NSGA-III and RVEA decreases. This is because the number of PF sections on MaF increases exponentially with the increase in the number of objectives, rendering most reference vectors in RVEA and NSGA-III ineffective. For MaF6, 1by1EA, MaOEA-CSS, and RVEA outperformed MaOEA/ABS-DE overall. This is due to the fact that the PF of MaF6 degenerates into a low-dimensional concave surface in high-dimensional objective spaces. These three algorithms employ the objective vector angle to design the environmental selection strategy, which can simultaneously retain solutions in both the central and boundary regions of the concave surface. For MaF8, the proposed algorithm achieves the best performance, with RVEA performing the worst, followed by NSGA-III. This is due to the inverted PF, rendering most reference vectors in RVEA and NSGA-III ineffective. For MaF9, SPEA2 + SDE achieves the best results. This is because, regardless of the number of objectives, the optimal region of the PF in MaF9 is a two-dimensional plane. The Pareto dominance employed by SPEA2 + SDE promotes the convergence of solutions.

To visually compare the quality of solution sets obtained by all algorithms on problems characterized by mixed PF, Fig. 7 exhibits the distribution of non-dominated solutions obtained by all algorithms on MaF10 with 10 objectives. It can be observed from Fig. 7 that the solution sets obtained by 1by1EA, MaOEA-CSS, and RVEA are mostly distributed in the boundary regions of the PF, thus indicating poor spread. SPEA2 + SDE achieves comparable convergence with the proposed algorithm but exhibits inferior spread. According to the true PF of MaF10, the solution set obtained by the proposed algorithm exhibits the best quality, followed by NSGA-III. Although the PF of MaF10 is mixed and not disconnected, with most regions being concave, the uniformly distributed reference vectors in NSGA-III facilitate the spread of the population. The environmental selection mechanism of the proposed algorithm first considers the spread of the population and then simultaneously accounts for the convergence and spread of individuals in the second-stage selection. Consequently, it focuses on the exploitation of various local regions.

Fig. 7

Final population distributions obtained by each algorithm on the 10-objective MaF10

From the experimental results presented in Tables 8 and 9, MaOEA/ABS-DE outperforms other algorithms on 15 test instances out of the 27 MaF instances in terms of IGD. Similarly, in terms of DM, MaOEA/ABS-DE demonstrates superior performance in 12 test instances compared to other algorithms. Figures 8 and 9 provide a statistical overview of the IGD and DM performance scores of the six comparative algorithms. The results depicted in these figures also confirm that the proposed algorithm exhibits the best overall performance. Taken together, the experimental results and analysis indicate that the proposed algorithm can effectively balance the convergence and diversity of the solution set when solving MaF problems with complex PFs.

Fig. 8

The average IGD performance score obtained by all algorithms on MaF

Fig. 9

The average DM performance score obtained by all algorithms on MaF

5.4 Comparison of the running time of all algorithms

Due to limited space, Fig. 10 only presents the average runtime of six algorithms on WFG1-2 and MaF1-2. It can be observed that MaOEA-CSS and SPEA2 + SDE consume significantly more time than the other four algorithms. MaOEA/ABS-DE exhibits a moderate level of time consumption. NSGA-III and RVEA, the two algorithms based on reference points, exhibit the most ideal time consumption. The time consumption of 1by1EA is also satisfactory, only slightly exceeding that of NSGA-III and RVEA. The reasons for these results are easily explainable.

Fig. 10

The running time of all algorithms on MaF1-2 and WFG1-2

During the environmental selection process, both MaOEA-CSS and SPEA2 + SDE repeatedly select a pair of the most similar solutions from the combined population P_t, and subsequently delete one based on the distribution indicator of individuals. This selection-deletion operation is then repeatedly executed, resulting in significant time consumption for both algorithms. Additionally, in the mating selection process, SPEA2 + SDE selects parent solutions solely based on the fitness measure of solutions, whereas MaOEA-CSS selects parents based on both the ASF and the minimum objective vector angle, incorporating a probability parameter. Consequently, MaOEA-CSS consumes more time compared to SPEA2 + SDE. Although 1by1EA also involves repeated selection operations, it primarily selects multiple solutions simultaneously based on convergence metrics. As a result, 1by1EA exhibits higher selection efficiency than MaOEA-CSS and SPEA2 + SDE. NSGA-III and RVEA utilize reference vectors to form niches during the environmental selection process, and then select individuals from each niche simultaneously, achieving the highest efficiency in the selection process. MaOEA/ABS-DE involves pre-selecting non-dominated solutions from the combined population based on Pareto dominance relations during the environmental selection process, followed by a two-step selection for the remaining solutions. Therefore, among these six algorithms, MaOEA/ABS-DE exhibits moderate time consumption.

From the perspective of runtime, although the proposed algorithm MaOEA/ABS-DE may not be the most efficient, it achieves the best performance in terms of accuracy for solving MaOPs. Compared with the three relatively efficient algorithms, NSGA-III, RVEA, and 1by1EA, the runtime of MaOEA/ABS-DE is not excessively high, yet its accuracy in solving MaOPs significantly surpasses those of these three algorithms. Furthermore, both MaOEA-CSS and SPEA2 + SDE not only consume significantly more time than the proposed algorithm, but also their accuracy in solving MaOPs is far lower than that of MaOEA/ABS-DE. Therefore, overall, when solving MaOPs, the proposed MaOEA/ABS-DE is an approach worthy of promotion.

6 Real-word case studies

In this section, two practical problems are employed to further test the practicality of the proposed algorithm. The first optimization problem originates from automotive design optimization in the industrial field and belongs to the category of continuous optimization [75]. The second optimization problem is the many-objective traveling salesman problem [76], which falls into the realm of combinatorial optimization and is widely encountered in production scheduling. Both of these optimization problems are many-objective optimization problems. Based on the characteristics of the mathematical models for these two problems, the chromosome encoding method for the first optimization problem adopts real-valued encoding, while the chromosome encoding method for the second optimization problem employs symbolic encoding. Firstly, these two optimization models are programmed into a computer-recognizable programming language. Subsequently, the algorithm proposed in this paper and the five comparison algorithms are also programmed into a computer-recognizable programming language, with the source code of the five comparison algorithms being open-source. Finally, the designed algorithm and the five comparison algorithms are used on a computer to solve these two problems respectively. The average IGD and DM results obtained from 20 runs of all algorithms on these two practical problems are used as the comparison criteria. Since the calculation of IGD requires a reference set, and the true PF of practical problems is unknown, the non-dominated solutions obtained by the six algorithms are merged, and the non-dominated solutions from the merged set are then used as the reference set.

6.1 Crash safety design of vehicles

The optimization problem aims to simultaneously optimize three objectives by optimizing five design parameters of the vehicle front structure. Specifically, the first objective focuses on minimizing the mass of the vehicle, the second on maximizing acceleration during a full frontal crash, and the third on minimizing toe board intrusion during an offset-frontal collision. For a detailed introduction to this problem, please refer to [75]. For this case, the maximum number of iterations is set to 1000, and the population size is set to 105. Table 10 presents the average IGD and DM obtained by running all algorithms on this problem 20 times. Observations indicate that the proposed algorithm, MaOEA/ABS-DE, attains the optimal outcomes in terms of both IGD and DM. For a further visual comparison of the quality of the solution sets obtained by the six algorithms, Fig. 11 displays the non-dominated solutions obtained by each algorithm. It is noteworthy that the solution set obtained by MaOEA/ABS-DE dominates most of the non-dominated solutions obtained by other algorithms. Furthermore, the PF formed by the non-dominated solutions obtained by MaOEA/ABS-DE exhibits a continuous and uniform distribution in the objective space. Therefore, from Fig. 11 and Table 10, it can be concluded that the proposed algorithm achieves the best performance in terms of both convergence and distribution.

Table 10 IGD and DM values obtained by six algorithms in vehicle safety design optimization

Fig. 11

Pareto optimal solutions obtained by all algorithms on vehicle safety design optimization

6.2 Many-objective traveling salesman problem

The many-objective traveling salesman problem, belonging to the category of combinatorial optimization problems, is widely encountered in the field of production scheduling. This multi-objective traveling salesman problem involves 30 decision variables and requires the simultaneous optimization of five objectives. For a detailed introduction to this problem, please refer to [76]. For this case, the maximum number of iterations is set to 1000, and the population size is set to 210.

Table 11 exhibits the IGD and DM values obtained from 20 runs of all algorithms on this problem. In terms of IGD, the proposed algorithm, MaOEA/ABS-DE, achieves the optimal results. In terms of DM, it rankes second only to SPEA2 + SDE. NSGA-III and RVEA exhibit poor performance, which is attributed to the irregular nature of the PF in the real-world optimization problems, making it challenging for uniformly distributed reference vectors to adapt to such scenarios. Figure 12 demonstrates the distribution of objective values across all algorithms, with 1 representing 1by1EA, 2 representing NSGA-III, 3 representing MaOEA-CSS, 4 representing RVEA, 5 representing SPEA2 + SDE, and 6 representing MaOEA/ABS-DE. It can be observed from Fig. 12 that MaOEA/ABS-DE achieved the optimal objective values for Objectives 1, 2, 4, and 5.

Table 11 IGD and DM values obtained by six algorithms in MOTSP with 5 objectives

Fig. 12

The box plot of objective values obtained by six algorithms on MOTSP with 5 objectives

7 Conclusion

To strike a better balance between population diversity and convergence, an adaptive boundary selection-based many-objective evolutionary algorithm with density estimation, MaOEA/ABS-DE, is proposed. This algorithm proposes a two-step environmental selection process incorporating two strategies. By estimating the curvature of the PF corresponding to the current population, population is transformed to the boundary of the current PF. Subsequently, the adaptive boundary-based selection serves as the first step for selecting individuals into a candidate pool. To further enhance the convergence of the population, the shift-based density estimation selection strategy acts as the second step for selecting individuals from the candidate pool.

The effectiveness of the proposed two-step selection process is validated on WFG and MaF test problems. Simulation results that compare our algorithm with five representative MOEAs on WFG and MaF test problems demonstrate its robustness in handling various MaOPs with regular and irregular PFs. Finally, the proposed algorithm exhibits significant practicality in two real-world cases involving both continuous and discrete variables. Looking ahead, there is promise for further application of proposed algorithm in practical industrial scenarios, thereby providing strong support for management decisions, assisting managers in achieving more meticulous resource allocation, and significantly enhancing production efficiency.