Introduction

Recently, multi-objective optimization problems (MOPs) [1] have gained widespread attention due to the need to deal with real engineering problems with conflicting objectives [49]. In particular, problems with more than three optimization objectives are referred to as many-objective optimization problems (MaOPs) [2], such as the four-objective aerodynamic optimization of a cruise ship [3] and the seven-objective optimization of an automotive control system [4]. Moreover, for many real-world MOPs, their function evaluations (FEs) incur expensive costs, including computational costs [5], simulation modeling costs [6, 7], or experimental costs [8]. As a result, these problems are referred to as data-driven or expensive MOPs (EMOPs) [9, 10]. In expensive multi-objective or many-objective optimization problems (EMOMaOPs), the number of non-dominated solutions grows exponentially with the increase in the number of objectives, leading to a significant increase in conflicts between objectives. Additionally, these problems often exhibit characteristics like nonconvexity, multimodality, and black box [9]. Despite the widespread use of multi-objective evolutionary algorithms (MOEAs) in various engineering fields, directly applying MOEAs that rely on evaluating large-scale functions does not meet the requirement of low cost in realistic scenarios. Consequently, effectively solving EMOMaOPs presents a significant challenge.

To alleviate these dilemmas, surrogate-aided evolutionary algorithms (SAEAs) [11] have been proposed. For these SAEAs, they consist of three core components: the surrogate models, the optimizers, and the infill sampling criterion. These methods, which are key factors affecting the performance of SAEAs, are briefly described below (see Sect. “Related work and motivation” for a detailed overview).

  1. 1.

    Surrogate models: Surrogate models, also known as meta-models, can be used in SAEAs for approximate assessment to enable pre-screening of solutions [50]. The Kriging model [16, 18], often known as the Gaussian process [51], artificial neural networks (ANN) [13, 52, 53], and support vector machines (SVM) [14] are examples of common surrogate models. In addition, some composites of surrogate models can be used in order to improve the prediction [22, 23, 54]. As the dimension of the objective increases, approximation errors inevitably accumulate, which may lead the algorithm to evaluate inappropriate solutions [24].

  2. 2.

    Optimizers: Optimizers in SAEAs provide selection pressure to push populations towards the Pareto-optimal front (PF) [18]. Theoretically, the classification of optimizers is not significantly different from that of MOEAs, which are generally classified as advantage-based optimizers [25,26,27,28], decomposition-based optimizers [31, 32, 34, 55], and performance-metrics-based optimizers [35,36,37,38]. In addition, to improve the performance of SAEAs, a combination of optimizers can be considered [39].

  3. 3.

    Infill sampling criteria: Designing appropriate infill sampling criteria is one approach to improve the performance of SAEAs [23]. For the currently existing infill sampling criteria, they are generally designed based on hybrid performance metrics or pure metrics [13, 16, 18, 51]. Ideal infill sampling criteria should address the balance between exploration and extraction [56]. To achieve these balances, the designed criteria generally choose solutions with improved convergence, diversity and model accuracy [18].

Many excellent algorithms have been developed for EMOMaOPs, such as CPS-MOEA [12], CSEA [13], MCEAD [14], PCSAEA [15], KRVEA [16], EDNARMOEA [17], and KTA2 [18]. However, the effectiveness and area of application of existing algorithms are unsatisfactory, and there are still many challenges when dealing with EMOMaOPs. The first challenges is how to obtain more reliable candidates, especially when the objective dimension is high or the accuracy of the surrogate models is poor. The second one is how to design a more efficient adaptive sampling criterion that can meet different demands. Therefore, this paper proposes a two-risk archive algorithm (TRAA), which contains a new candidate solution mining strategy and a new adaptive sampling criterion for better handling of EMOMaOPs.

The important contributions of this paper are summarized below:

  1. 1.

    A candidate solution mining strategy is proposed to obtain high-risk and low-risk archives. Specifically, the strategy is based on two classes of optimization models and performs model search using KnEA [27] and NSGA-III [31], respectively, and then obtains the two archives through archive selection based on Kriging models. The strategy ensures the convergence, diversity and reliability of the candidate solutions.

  2. 2.

    A more comprehensively considered adaptive sampling criterion is designed. The criterion includes sampling environment assessment, demand assessment, and sampling strategy execution. The optimal sampling strategy can be selected based on the current state. This strategy ultimately achieves a significant increase in the performance of the algorithm by improving the most important performance in each iteration, such as comprehensive performance, convergence, diversity, or model uncertainty.

The rest of this paper is organized as follows. In the next section, the work related to the three parts of the SAEAs is summarized, followed by a description of the motivation for this work. In the following section, the EMOMaOPs and the Kriging model are introduced. Next section gives a detailed description of the proposed algorithm. In the following section, experimental results and analysis are presented. Finally, conclusions are given in the last section.

Related work and motivation

Related work

This section summarizes the related work on surrogate models, optimizers, and infill sampling criteria in SAEAs.


(1) Surrogate models: surrogate models can be divided into the following two categories based on their crucial role in SAEAs:


Classification models based on SAEAs: This class of approaches uses the surrogate model as a classifier to distinguish between classes of candidate individuals, e.g., to distinguish between good and bad individuals. CSEA [13] uses a single feed-forward neural network (FNN) to predict the dominance relationship between candidate and reference solutions, and then the reliability of the prediction is estimated by a validation set. In contrast, in CPS-MOEA [12], a classification tree is used to categorize candidate individuals as good or bad in order to reduce the number of expensive function evaluations. In addition, MCEAD [14] utilizes SVM to construct multiple classifiers, and each local classifier is trained as a corresponding scalarization function. PCSAEA [15] is another interesting classification-based SAEA, which uses FNN for two-by-two comparison of candidate solutions instead of directly predicting the fitness value of the solution, which helps to balance positive and negative samples.


Regression models based on SAEAs: This class of methods generally utilizes surrogate models to approximate the objective function, the aggregation function, and the Pareto rank. In ParEGO [19], a single Kriging model is used to approximate the aggregation function. However, this type of algorithm is not suitable for solving EMOMaOPs because suitable weight vectors are harder to obtain. Kriging models have also been utilized to predict the rank after non-dominated ordering, represented by the algorithm OREA [20]. In addition, the most common and most used method is the approximation of expensive objective functions with surrogate models. KRVEA [16] classifies the existing data based on their intrinsic relationship with reference vectors and subsequently selects real sample points in each class to train the Kriging model. In EDNARMOEA, a dropout neural network is used instead of Gaussian processes to improve the performance of the algorithm. An interesting strategy is given in the literature [21], which employs a method to enhance the predictive effect of the surrogate models through historical data. In HSMEA [22], to improve the reliability of the prediction, the best surrogate model is selected by comparing the root mean square errors of the four types of surrogate models. Another successful case is KTA2 [18], in which the Kriging model is used to construct one class of sensitive models and two classes of insensitive models. And the insensitive models are selected by the relationship between the predicted points of the sensitive models and the centroid to reduce the influence of the influential points and improve the predicted solutions. CSA-MaOEA [23] used a combination of surrogate model construction, using a combination of a global surrogate model for the whole data and multiple local surrogate models built using multiple sets of data after clustering, with the aim of enhancing the surrogate models’ prediction reliability.

For EMOMaOPs, although the amount of their training data is not enormous, it is significant how to train and utilize surrogate models to mine reliable data. Based on the current research, it can be summarized that there are several types of strategies: using improved surrogate models, combining multi-class surrogate models, enhancing local prediction, reducing data influence points, and data enhancement. Different strategies have different applicability to the same algorithm, and what is suitable for the designed algorithm is the most important.


(2) Optimizers: Theoretically, optimizers in SAEAs can also be classified into three categories, as in the classification of MOEAs.


Advantage-based optimizers: For this, the selection pressure can be increased by modifying the Pareto dominance relation or adding convergence-related criteria, such as fuzzy dominance [25], preference rank order [26], and knee points in KnEA [27]. The RSEA-based [28] solution strategy used in both CSEA [13] and REMO [29] also belongs to this category. This optimizer projects the solution in the objective space into a two-dimensional radial space to preserve diversity and enhance convergence. There is also NSGA-II [1], which is currently widely used in two- and three-objective, and this algorithm has also obtained good results in PA-EMaOEA [24] and SGMOO [30].


Decomposition-based optimizers: For example, the familiar NSGA-III [31] employs reference information in the environment selection to choose the individuals. In addition, RVEA [32] is used as the optimizer in KRVEA [16], while MOEA/D-EGO [33] and OREA [20] use MOEA/D [34].


Performance-metrics-based optimizers: Such optimizers generally use one or more metrics to evaluate the quality of different candidate individuals for subsequent environmental selection. Examples include SMSMOEA [35], methods based on IGD and R2 [36], and methods based on convergence and diversity metrics [37]. In addition, KTA2 [18] ingeniously employs the Two Arch2 [38] algorithm to obtain a convergence archive and a diversity archive as candidate solution sets for adaptive sampling.

In addition to the single use of the above three types of optimizers, a mixture of multiple types of optimizers can also be considered to improve the individual selection effect. In ESB-CEO [39], MOEA/D [34] and RVEA [32] are selected as optimizers to consider their convergence and diversity, respectively. It is also important to note that since SAEAs introduce surrogate models, the uncertainty of the surrogate models should be taken into account when using these optimizers to obtain candidate solutions. However, many SAEAs ignore this point, which may lead to poor-quality candidate individuals.


(3) Infill sampling criteria: Infill sampling criteria is another key element of SAEAs, which also affects their performance. These criteria are used to identify promising new alternatives and evaluate them with real functions to obtain new individuals, which are ultimately used to improve surrogate models and populations. Expected improvement (EI) [40] and probability of improvement (PoI) [41] are two examples of classical criteria. To improve the performance of SAEAs, many new infill sampling criteria have also been added in recent years. HSMEA [22] applies a diversity-preserving strategy based on reference vectors and a convergence-preserving strategy based on Euclidean distances, but it does not take into account the surrogate model uncertainty requirements. KRVEA [16] uses a diversity maintenance strategy based on reference vectors, a convergence maintenance strategy based on angular penalty distances, and an error-based model uncertainty adjustment strategy for the infill sampling criterion. CSEA [13] uses reference-point-based convergent sampling, classification-based diversity sampling, and uncertainty improvement sampling for its infill sampling criterion. In the sampling criteria of DDEA-MESS [42], global, local, and trust region sampling are used. Moreover, in the literature [39], an adaptive sampling strategy based on entropy search is used to improve diversity and convergence. Literature [43] proposes a strategy to measure uncertainty that takes into account the distance to the samples in the decision space and the approximation variance in the objective space. KTA2 [18] proposed an adaptive sampling criterion based on pure metrics, including convergence sampling, diversity sampling, and uncertainty sampling. Among them, uncertainty sampling relies on the uncertainty estimates provided by Kriging. CSA-MaOEA [23] employs a infill sampling strategy where each round of samples includes convergence-improving individuals, diversity-improving individuals, and uncertainty-improving individuals. This is a measure that can be taken when it is difficult to distinguish between demand states. In the infill criterion of GP-iGNG [44], convergence ranking is measured using the Euclidean distance from the clustering-based solution to the ideal point in the target space, while diversity ranking is measured using the minimum angle between the unassessed solutions and all the assessed non-inferior solutions in the same class.

Motivation

Our motivation is based on the following considerations.

The first aspect is how to obtain reliable candidate solutions. This requires a concerted effort of surrogate models, optimizers, and some specific strategies. Currently, in terms of obtaining candidate solutions, most SAEAs perform model searches based on surrogate models and optimizers to obtain candidate solutions, which ignores the uncertainty of surrogate model estimation. This approach has weak individual selection pressure and may select undesirable individuals or lose desirable individuals with high uncertainty. There are also differences in the prediction effects of surrogate models for different problems. When the prediction is good, the effect of uncertainty can be ignored; when the prediction is poor, consideration can be given to weakening the effect of uncertainty. Therefore, when obtaining candidate solutions, consider obtaining archives of two uncertainty levels in this paper: a high-risk archive and a low-risk archive. Furthermore, there is a difference in the reliability of candidate solutions from different surrogate models, and the combination of multiple types of surrogate models can improve the reliability of candidate solutions. Besides, combinations of multi-class optimizers can be considered to be used in SAEAs to improve their performance.

The second aspect is how to design more effective adaptive infill sampling guidelines. For EMOMaOPs, most existing infill sampling criteria for SAEAs consider population convergence, diversity, and surrogate model uncertainty demands. They use hybrid performance metrics or pure metrics to measure the potential benefits of these three aspects, aiming to improve populations as much as possible within limited FEs. However, most SAEAs do not distinguish or fully utilize the individual uncertainty information in the adaptive sampling stage. And their infill sampling criteria are not design based on different uncertainty information, and not combine hybrid and pure metrics. Thus, leave much room for improvement in the performance of the algorithms.

Preliminaries

Expensive multi/many-objective optimization problems

EMOMaOPs can be expressed as:

$$ \begin{array}{*{20}c} {\begin{array}{*{20}r} \hfill {minimize} & \hfill { \, F\left( x \right) = \left( {f_{1} \left( x \right), \ldots ,f_{m} \left( x \right)} \right)^{T} } \\ \hfill { \, s.t.} & \hfill {x = \left( {x_{1} , \ldots ,x_{d} } \right)^{T} } \\ \end{array} } \\ \end{array} $$
(1)

where \(x\) is a design variable, \(d\) is the variable's dimension, and \(x\in {R}^{d}\). The objective vector \(F(x)\) is made up of \(m\) objective functions \(f(x)\) that need to be optimized. Furthermore, it is frequently not possible to find an optimal solution that minimizes the objectives simultaneously, but rather a group of several optimal solutions, because there are typically conflicts between these objective functions. In the objective space, the ideal solution is called the PF, and in the decision space, it is called the Pareto-optimal set (PS).

Kriging model

The surrogate model used in this paper is Kriging, and it can provide uncertain information about the prediction points, which is highly beneficial for both obtaining candidate solutions and adaptive infill sampling. The related strategies of Kriging will be further discussed in the Sect. “Proposed algorithm”. If Kriging is to be constructed, the trial sample \(x\) and the corresponding response y need to be specified. Here it is assumed that \(x=\left[{x}_{1},{x}_{2},\dots,{x}_{d}\right]\) and \(y=\left[{y}_{1},{y}_{2},\dots,{y}_{n}\right]\), then the relationship between the objective response value and the design variable value in the Kriging model is shown in the following equation:

$$ \begin{array}{*{20}c} {y_{j} \left( x \right) = \sum\limits_{s = 1}^{p} {\beta_{s} } f_{s} \left( x \right) + z\left( x \right)} \\ \end{array} $$
(2)

where \(\sum\nolimits_{s = 1}^{p} {\beta_{s} } f_{s} \left( x \right)\) is a regression model that reflects the change in the process mean through \(p\) regressions, and \(z\left( x \right)\) is a model of a stochastic process with mean \(0\) and standard deviation \(\sigma\). For any two samples \(S^{i}\) and \(S^{j}\) in the sample, the covariance is:

$$ \begin{array}{*{20}c} {Cov\left[ {z\left( {S^{i} } \right),z\left( {S^{j} } \right)} \right] = \sigma^{2} {\mathbf{R}}\left( {\left[ {R\left( {S^{i} ,S^{j} } \right)} \right]} \right)} \\ \end{array} $$
(3)

where \({\mathbf{R}}\) is the \(N_{{\text{m}}} \times N_{{\text{m}}}\) correlation matrix, \({\mathbf{R}} = \left[ {\begin{array}{*{20}c} {R\left( {S^{1} ,S^{1} } \right)} & \cdots & {R\left( {S^{1} ,S^{{N_{{\text{m}}} }} } \right)} \\ \vdots & \ddots & \vdots \\ {R\left( {S^{{N_{{\text{m}}} }} ,S^{1} } \right)} & \cdots & {R\left( {S^{{N_{{\text{m}}} }} ,S^{{N_{{\text{m}}} }} } \right)} \\ \end{array} } \right]\); \(R\left( {S^{i} ,S^{j} } \right)\) is the correlation function, which controls the smoothness of the Kriging model and the correlation between data points and between quantitative observations. And in the work of this paper, the correlation function model is Gaussian:

$$ \begin{array}{*{20}c} {R\left( {S^{i} ,S^{j} } \right) = \exp \left( { - \sum\limits_{k = 1}^{d} {\theta_{k} } \left( {S_{k}^{i} - S_{k}^{j} } \right)^{2} } \right)} \\ \end{array} $$
(4)

where \(\theta_{k}\) is the \(k\)th hyperparameter and \(S_{k}^{i}\) and \(S_{k}^{j}\) are the \(k\)th coordinate values of \(S^{i}\) and \(S^{j}\), respectively.

Proposed algorithm

In this section, the proposed TRAA is described in detail. Subsequent content covers the TRAA framework, the candidate solution mining strategy for two-risk archives, and the adaptive sampling criterion.

TRAA framework

The framework of the proposed TRAA is shown in Fig. 1, and its pseudo-code is shown in Algorithm 1. TRAA is similar to existing SAEAs in its overall structure, including population initialization, surrogate models training, candidate solution set determination, adaptive sampling, population update and iteration. However, in terms of detail, there are significant differences between the TRAA and currently available SAEAs. In TRAA, in order to mine reliable promising candidate individuals, we designed a way to mine high-risk and low-risk archives. In addition, we design an adaptive sampling criterion based on two-risk archives to meet different needs for performance improvement.

Fig. 1
figure 1

TRAA framework

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Algorithm 1
figure a

The pseudo-code of TRAA

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According to Algorithm 1 and Fig. 1, TRAA can be further decomposed into subsequent steps:

Step 1: Initialization. The initial individuals of TRAA are initialized using Latin hypercube sampling with expensive objective functions for population initialization. In addition, assign \(HAr\) and \(LAr\) to the empty set and calculate \(ScE\). (corresponding to line 1).

Step 2: Mining two-risk archives. Get \(HAr\), \(HArErr\) and \(LAr\) by candidate solution mining strategy. And \(HAr\) is the high-risk archive, \(LAr\) is the low-risk archive, and \(HArErr\) is the uncertainty information corresponding to \(HAr\). This session includes surrogate model training and two archive determinations based on surrogate models. Details can be found in Algorithm 2. (corresponding to the line 3).

Step 3: Adaptive sampling. Adaptive sampling based on the current information to get the offspring \(Off\). Details can be found in Sect. “Adaptive sampling criterion”. (Corresponds to line 4).

Step 4: Update the population. Combine \(Pop\) and \(Off\) to get a new population \(Pop\). (Corresponds to line 5).

Step 5: Iterate and output results. Repeat steps 2 through 5 until the computational budget is exhausted. Then Output all nondominated solutions in pop to get Sol. (corresponding to lines 2–7).

Candidate solution mining strategy for two-risk archives

This section presents a candidate solution mining strategy for two-risk archives. The schematic diagram of this strategy is shown in Fig. 2 and its pseudo-code is shown in Algorithm 2. In order to obtain more reliable candidate solutions, a series of measures are applied to the strategy, such as combining multiple classes of Kriging models, controlling uncertainty and objective value of candidate solutions, and hybrid use of optimizers. These measures are intended to increase the pressure for individual selection, reduce the waste of expensive FEs, and improve the performance of the algorithm.

Fig. 2
figure 2

Candidate solution mining strategy for two-risk archives

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Algorithm 2
figure b

Candidate solution mining strategy for two-risk archives

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According to Fig. 2 and Algorithm 2, the session can be further subdivided into the following steps:

Step 1: Training two types of Kriging models. In this step, we use the DACE toolbox to train two types of Kriging models, which are \(KMR1\) using first-order regression model and \(KMR0\) using zero-order regression model. The reasons for using these two types of models are as follows: (1) \(KMR1\) and \(KMR0\) are the two commonly used types of Kriging, with better nonlinear fitting ability and less training time; (2) Both types of models provide an estimate of the uncertainty at the prediction point, which is useful for candidate solution selection; (3) The use of two types of Kriging combinations can improve the reliability of candidate solutions and improve the performance of TRAA. (Corresponding to lines 1–2).

Step 2: Determine \(\omega \). If \(HAr\) is empty, then \(\omega \) is set to \({\omega }_{cs}\). And if not, \(\omega \) is set to \({\omega }_{max}\). (Corresponding to line 3).

Step 3: KMR1-based two-archive mining. This step is realized as follows. First, a model search based on a non-conservative optimization model is performed to obtain the candidate archive \(HAr1\) and its uncertainty information \(HArE1\). Then, the mode of the maximum value of each objective in \(HAr1\) is calculated to obtain the score \(Score1\). Next, if \(Score1\) is greater than or equal to \(ScE\), a model search is performed based on the conservative optimization model to obtain the candidate solution set \(LAr1\) and its uncertainty information L \(ArE1\). If less than, then \(LAr1\) is the same as \(HAr1\) and L \(ArE1\) is the same as \(HArE1\). (corresponding to lines 4–10).

The following points need to be noted in step 3:

(1) Two optimization models. Two optimization models are involved in this step, namely the non-conservative optimization model and the conservative optimization model. The former replaces its objective functions with \(KMR1\) on the basis of Eq. (1) and the latter replaces them on the basis of Eq. (5). In conservative optimization models, the objective values and uncertainty estimates are restricted. This is mainly based on the following considerations: individuals that are close to or in the objective space of existing individuals are considered more trustworthy, and individuals with smaller uncertainty estimates are considered more trustworthy. These constraints are set mainly to mine more reliable and higher-quality candidate solutions.

$$ \begin{array}{*{20}c} {\begin{array}{*{20}r} \hfill {minimize} & \hfill { \, F\left( x \right) = \left( {f_{1} \left( x \right), \ldots ,f_{m} \left( x \right)} \right)^{T} } \\ \hfill { \, s.t.} & \hfill { \, \left\{ {\begin{array}{*{20}c} {x = \left( {x_{1} , \ldots ,x_{d} } \right)^{T} } \\ {f_{i}^{l\min } \le f_{i} \left( x \right) \le f_{i}^{l\max } , i = 1,..,m} \\ {Unc\left( x \right) \le ScE} \\ \end{array} } \right.} \\ \end{array} } \\ \end{array} $$
(5)

where \({f}_{i}^{lmin}\) and \({f}_{i}^{lmax}\) are the conservative upper and lower bounds of \({f}_{i}(x)\), respectively, \({f}_{i}^{lmin}={f}_{i}^{min}-\varphi ({f}_{i}^{max}-{f}_{i}^{min})\), \({f}_{i}^{lmax}={f}_{i}^{max}+\varphi ({f}_{i}^{max}-{f}_{i}^{min})\), \({f}_{i}^{min}\) is the minimum value of the \(i\)-th objective of all non-dominated solutions in \(Pop\), and \({f}_{i}^{max}\) is the maximum value of the \(i\)-th objective of all dominated solutions in \(Pop\); \(Unc(x)\) is the mode length of the uncertainty information provided by Kriging at this point.

  1. (2)

    Initial population. The combination of \(HAr^{\prime}\) and \(Pop\) is used as the initial population when acquiring \(HAr1\). \(HAr^{\prime}\) is obtained by re-evaluating the old \(HAr\) by \(KMR1\). The combination of \(LAr^{\prime}\) and \(Pop\) is used as the initial population when acquiring \(LAr1\). \(LAr^{\prime}\) is obtained by re-evaluating the old \(LAr\) by \(KMR1\).

  2. (3)

    Optimizers. The optimizer for the non-conservative optimization model is KnEA and the optimizer for the conservative optimization model is NSGA-III. The optimizers are employed in this way for the following considerations. KnEA introduces ‘knee points’ that are useful for expensive multi-/many-objective optimization, especially when surrogate models can approximate the true objective function well. NSGA-III introduces widely distributed based reference points, which is good for population diversity improvement when the objective dimension is high. NSGA-III is a good choice for model search when surrogate models are not predicting well. Therefore, considering the above, TRAA adopts KnEA and NSGA-III as the optimizers for the two optimization models, respectively.

Step 4: KMR0-based two-archive mining. This step is similar in principle to step 3. The only difference is in the type of surrogate models. This step uses KMR0, while step 3 uses KMR1. (corresponding to lines 11–17).

Step 5: Archive selection. In this step, the selection of archives is performed based on the uncertainty performance of each archive. Take the determination of \(HAr\) as an example. First, the mode length of the uncertainty information of each individual in \(HArE1\) and \(HArE0\) is calculated respectively. Then calculate the mean value of all the mode lengths. Finally, based on the calculation results, a better archive is selected as \(HAr\) from \(HAr1\) and \(HAr0\), and the \(HArErr\) is updated. The same principle applies for the determination of \(LAr\).

Adaptive sampling criterion

Based on the two-risk archives obtained, we elaborated an adaptive sampling strategy as shown in Fig. 3. The proposed adaptive sampling strategy can be further subdivided into the following steps:

Fig. 3
figure 3

Adaptive sampling criterion

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Step 1: Environmental classification. Categorize the current uncertainty environment where \(HAr\) is located. First, the uncertainty score of \(HAr\), \(Unc\), is calculated in the same way as in Algorithm 2, line 5 or line 11. Then determine whether \(Unc\) is greater than \(1.6\times {10}^{2\alpha }\). When \(Unc\) is greater than the set value, it is a controversial environment. Otherwise, it is a reliable environment.

Step 2: Demand assessment. When the environment is a reliable environment, the current demand is a hybrid performance demand. When the environment is controversial, the current demand needs to be determined by \(index\). The \(index\) is determined through the \(DivQL\) function, which is defined in the following equation. If \(index\) is 1, the the current demand is diversity demand. If \(index\) is 0, the the current demand is convergence demand.

$$ \begin{array}{*{20}c} {index = \left\{ {\begin{array}{*{20}c} {1,\frac{{\sum\nolimits_{i}^{{n_{P} }} {P_{i} } }}{{n_{P} }} < \frac{{\sum\nolimits_{j}^{{n_{L} }} {L_{j} } }}{{n_{L} }}} \\ {0,\frac{{\sum\nolimits_{i}^{{n_{P} }} {P_{i} } }}{{n_{P} }} \ge \frac{{\sum\nolimits_{j}^{{n_{L} }} {L_{j} } }}{{n_{L} }}} \\ \end{array} } \right.} \\ \end{array} $$
(6)

where \({P}_{i}\) is the minimum Euclidean length between the \(i\)-th individual and the other individuals in the \(Pop\) objective space, \({n}_{P}\) is the number of individuals in the \(Pop\), \({L}_{j}\) is the minimum Euclidean length between the \(j\)-th individual and the other individuals in the \(LAr\) objective space, and \({n}_{L}\) is the number of individuals in the \(LAr\).

Step 3: Main sampling execution. According to the demand state, the corresponding sampling is executed. The correspondence between the three sampling methods involved and the demands is shown in Fig. 3. \(TAr\) is obtained by the corresponding sampling strategy, and this archive is the set of selected individuals.

Step 4: Individual-quality verification and uncertainty sampling. Calculate \(nl\), which is the number of non-similar individuals to \(Pop\) in \(TAr\). If \(nl\) is less than or equal to \(1\), perform the uncertainty sampling strategy based on Euclidean distance and \(PAr\) is obtained by this sampling strategy. If \(nl\) is greater than \(1\), \(PAr\) is the same as \(TAr\).

Step 5: Delete similar individuals. Delete individuals in \(PAr\) that are similar to individuals in \(Pop\).

Step 6: Evaluate with real function. Evaluate all the remaining individuals in \(PAr\) with expensive real functions to get the offspring \(Off\).

Based on the foregoing, TRAA incorporates four types of sampling strategies. Each strategy is presented below.

  • Hybrid performance sampling strategy based on IGD+ (as shown in Algorithm 3.1): for reliable environments, in order to improve convergence and diversity, we directly adopt the hybrid performance sampling strategy based on IGD+ . First, \(HAr\) is merged with \(Pop\) to obtain the new set, and then the new set is non-dominated to obtain \(PF\). And update \(setPop\) and A. Then the IGD+ score is calculated for all selected cases (corresponding to line 5 of the algorithm). Based on the results of the calculation, the individual with the best IGD+ is selected. And update A and setPop. Repeat steps 5–8 in Algorithm 3.1 until the number of individuals satisfies the set \({N}_{p}\). Finally, make \(TAr\) the same as A.

    Algorithm 3.1
    figure c

    Hybrid performance sampling strategy based on IGD+

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  • Angle-based diversity sampling strategy (as shown in Algorithm 3.2): for diversity improvement, we employ an angle-based diversity sampling strategy, which improves diversity and also improves convergence to some extent. This approach is more effective compared to the target distance-based approach. In this strategy, we simultaneously consider the characteristics of the current population \(Pop\) and the future population \(LAr\). This is done by calculating the minimum angle between the future population \(LAr\) and the current comparison population and selecting the individual with the largest angle. Then repeat the selection operation for individuals until the number of individuals reaches the required \({N}_{p}\). Finally, update \(TAr\).

    Algorithm 3.2
    figure d

    Angle-based diversity sampling strategy

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  • Convergence sampling strategy based on \({I}_{\epsilon +}\) (as shown in Algorithm 3.3): to improve convergence, a strategy based on \({I}_{\epsilon +}\) is utilized to guide individual selection. Here, it is necessary to construct the \({I}_{\epsilon +}\)-based matrix \(I\), which can be computed using Eq. 7. If \({x}_{i}\) is dominated by other solutions in the aggregate, its metric can be computed using Eq. 8. Based on the above, in this convergence strategy, we first obtain \(I\), \(C\), \(F\), and \(IndexChoice\) (corresponding to lines 1–6). Then find the individual \({x}^{*}\) with the smallest \(F\) value based on \(IndexChoice\), update \(F\) again, and remove \({x}^{*}\) from \(IndexChoice\). Keep repeating the above steps until the number of individuals meets the set \({N}_{p}\). Finally, update \(TAr\).

    $$ \begin{aligned}& I = \left[ {\begin{array}{*{20}c} {I_{\varepsilon + } \left( {x_{1} ,x_{1} } \right)} & \cdots & {I_{\varepsilon + } \left( {x_{1} ,x_{j} } \right)} & \cdots & {I_{\varepsilon + } \left( {x_{1} ,x_{N} } \right)} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {I_{\varepsilon + } \left( {x_{i} ,x_{1} } \right)} & \cdots & {I_{\varepsilon + } \left( {x_{i} ,x_{j} } \right)} & \cdots & {I_{\varepsilon + } \left( {x_{i} ,x_{N} } \right)} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ {I_{\varepsilon + } \left( {x_{N} ,x_{1} } \right)} & \cdots & {I_{\varepsilon + } \left( {x_{N} ,x_{j} } \right)} & \cdots & {I_{\varepsilon + } \left( {x_{N} ,x_{N} } \right)} \\ \end{array} } \right],\\ & i,j = 1,2,3, \ldots ,N \\ \end{aligned} $$
    (7)

    where \(I_{\varepsilon + } \left( {x_{i} ,x_{j} } \right) = max\left( {F\left( {x_{i} } \right) - F\left( {x_{j} } \right)} \right)\), and it determines the minimum required distance ϵ for \(x_{i}\) to dominate \(x_{j}\).

    $$ \begin{array}{*{20}c} {F\left( {x_{i} } \right) = \mathop \sum \limits_{{x_{j} \in P {\setminus} \left\{ {x_{i} } \right\}}} - e^{{ - \frac{{I_{\varepsilon + } \left( {x_{i} ,x_{j} } \right)}}{{0.05{\text{C}}}}}} } \\ \end{array} $$
    (8)

    where \(x_{j} \in P {\setminus} \left\{ {x_{i} } \right\}\) and it denotes the solutions other than \(x_{i}\) in overall P.

    Algorithm 3.3
    figure e

    Convergent sampling strategy based on \({{\varvec{I}}}_{{\varvec{\epsilon}}+}\)

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  • Uncertainty sampling strategy based on Euclidean length (as shown in Algorithm 3.4): for uncertainty improvement, we use the Euclidean length-based approach to quantitatively score the uncertainty of each individual and then select the \({N}_{p}\) individuals with the largest uncertainty scores in \(HAr\).

    Algorithm 3.4
    figure f

    Uncertainty sampling strategy based on Euclidean length

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Experimental research

In this section, two types of benchmark problems, DTLZ [45] and WFG [46], are used to validate the effectiveness of the proposed method [18]. We use the same objective dimensions for testing as in [18], so benchmark problems with 3, 4, 6, 8 and 10 objectives are used to test the proposed method. Table 1 lists the decision variable dimensions and the objective dimensions for each test problem.

Table 1 Decision variable dimension \(D\) and target dimension \(M\) for each test problem
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To evaluate the performance of the method, the mean value of IGD+ [48] after 300 evaluations is used in this paper. Compared to IGD, IGD+ provides a better assessment of the performance of the non-dominated solution set. The smaller the IGD+ value, the more effective the method is. Besides, the Wilcoxon rank sum (WRS) test with a significance level of 0.05 is used to assess the difference in performance between the TRAA and the comparison algorithms. The symbols "+" and "−" indicate that our proposed TRAA method performs significantly better and worse than the comparison algorithm, respectively, while "≈" indicates that the two comparison algorithms are not statistically difference. All methods are tested on PlatEMO [47]. The specific parameter settings of TRAA are shown as follows.

  1. 1.

    Initial population size, \({N}_{I}\), is set to 100.

  2. 2.

    The uncertainty control parameter, \(\tau \), is set to 3.

  3. 3.

    The relevant function parameters of Kriging models, \({\theta }_{1}\) and \({\theta }_{0}\), are set to \({\left[\begin{array}{ccc}5& \cdots .& 5\\ \vdots & \cdots .& \vdots \\ 5& \cdots .& 5\end{array}\right]}_{[M\times D]}\)

  4. 4.

    The environmental classification parameter, \(\alpha \), is set to \(0\).

  5. 5.

    The initial number of iterations in the process of obtaining two-risk archives, \({\omega }_{cs}\), is set to \(250\).

  6. 6.

    The maximum iterations in the process of obtaining two-risk archives, \({\omega }_{max}\), is set to \(30\).

  7. 7.

    Increased number of samples per round in TRAA, \({N}_{p}\), is set to \(5\).

Parameter sensitivity analysis

In TRAA, there is a parameter \(\tau \) that controls the strength of the uncertainty of the candidate solution set. In the case of some candidate individuals with particularly large uncertainty, the general algorithms are prone to multiple invalid choices due to large individual uncertainty, which will seriously affect the effectiveness of the algorithms. Especially for some hard-to-converge and harder-to-predict MOPs, multiple invalid choices are particularly fatal. Because different values of \(\tau \) may have different effects on the actual performance of TRAA, a parameter sensitivity analysis of \(\tau \) is considered here. The test problem is chosen to be DTLZ1 with a multi-modal landscape, where individual objective functions are harder to predict and harder to converge. For this problem, the objective numbers \(M\) are set to 3, 4, 6, 8, and 10.

To assess the impact of different \(\tau \), we tested a range of values \(\tau \in [1, 2, 3, 4, 5]\). Figure 4 provides the Average IGD+ results of DTLZ1 treated by TRAA with different \(\tau \). It can be observed that the value of \(\uptau \) should not be set too large, and \(\tau = 5\) does not give a more satisfactory result. Additionally, from \(\tau = 5\) to \(\tau = 3\), the value of \(\tau \) decreases and a significant improvement in performance occurs, especially for these cases where the number of targets is less than or equal to 8. At \(\tau = 3\), good performance results are achieved. From \(\tau = 3\) to \(\tau = 1\), deterioration in performance occurs as \(\tau \) decreases, and this phenomenon is more pronounced at \(M =\) 3, 4, and 10. Therefore, the value of \(\tau \) should not be set too small, as it can limit the generation of diverse individuals. Based on the summarized analysis, set the value of \(\tau \) in the algorithm to 3 and use this as a general setting for all test instances.

Fig. 4
figure 4

Average IGD+ results of DTLZ1 treated by TRAA with different \(\tau \)

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In addition, there is a parameter α in TRAA which affects the choice of sampling strategy. Therefore, it is necessary to perform a parametric sensitivity analysis of the α. Here, DTLZ1 and DTLZ7 are selected for analysis. DTLZ1 has more demand for convergence improvement and DTLZ7 has more demand for diversity improvement. To assess the impact of different \(\alpha \), a range of values \(\alpha \) ∈ [− 2, − 1, 0, 1, 2] are tested. Figure 5 provides the Average IGD+ results of DTLZ1 and DTLZ7 treated by TRAA with different \(\alpha \). It can be observed that, in general, the value of \(\alpha \) should not be too large or too small, which is evident on \(\alpha \) = 2, − 1 and − 2. At objective dimensions of 3 and 6, the change of \(\alpha \) has a significant change in performance. At a objective dimension of 10, the change of \(\alpha \) does not produce a significant change in performance. On the whole, \(\alpha \) = 0 or 1 is significantly better than the other cases. It is also considered that \(\alpha \) = 0 is slightly better than \(\alpha \) = 1 at \(M\) = 3 and there is no significant difference between the two at \(M\) = 6. Therefore, the value of \(\alpha \) in the algorithm is set to 0 and this is used as a general setting for all test instances.

Fig. 5
figure 5

Average IGD+ results of DTLZ1 and DTLZ7 treated by TRAA with different \(\alpha \)

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Kriging model impact analysis

To demonstrate the advantages of using two-class Kriging models, three simple experiments are conducted. Three classes of methods are involved here, which are described as follows.

  • TRAA(K0) uses the generated data based on \(KMR0\) for subsequent operations, namely \(HAr\)←\(HAr0\), \(LAr\)←\(LAr0\) and \(HArErr\)←\(HArE0\).

  • TRAA(K1) uses the generated data based on \(KMR1\) for subsequent operations, namely \(HAr\)←\(HAr1\), \(LAr\)←\(LAr1\) and \(HArErr\)←\(HArE1\).

  • TRAA uses the methodology proposed in this paper, which requires the use of \(KMR0\) and \(KMR1\). The description of this part can be found in Sect. “Candidate solution mining strategy for two-risk archives”.

The IGD+ statistics of the 3-target DTLZ treated by TRAA(K0), TRAA(K1) and TRAA, respectively, are given in Table 2. As shown in Table 2, TRAA achieves better results in all tests.

Table 2 Statistical results of IGD+ values obtained for TRAA(K0), TRAA(K1) and TRAA
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Optimizer impact analysis

To test the improvement of TRAA in optimizer performance, four approaches are used for the research. Each approach is described below:

  • TRAA(N + K) performs model searches using NSGA-III for non-conservative optimization models and KnEA for conservative optimization models.

  • TRAA(N) performs model searches for non-conservative optimization models and conservative optimization models using NSGA-III.

  • TRAA(K) performs model searches for non-conservative optimization models and conservative optimization models using KnEA.

  • TRAA performs model searches using KnEA for non-conservative optimization models and NSGA-III for conservative optimization models.

The IGD+ statistics of the DTLZ3, DTLZ6, and DTLZ7 treated by TRAA(N + K), TRAA(N), TRAA(K) and TRAA are given in Table 3. As shown in Table 3, TRAA performed well overall. This indicates that the use of a proper combination of optimizers can improve the performance of the algorithm. In addition, the overall performance of the two algorithms using KnEA for the non-conservative optimization model is better. This is probably because KnEA’s knee-point-based strategy is more sensitive to key boundary points, facilitating the improvement of surrogate models and populations.

Table 3 Statistical results of IGD+ values obtained for TRAA(N + K), TRAA(N), TRAA(K), and TRAA
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Impact analysis of adaptive sampling criteria

To further discuss the role of adaptive sampling strategies in TRAA, we peel off the four core sampling strategies in TRAA to obtain four variants. The TRAA is then compared with the TRAA based on the four individual variants. The four variants are described below:

  • TRAA(I) only adopts the hybrid performance sampling strategy based on IGD+ , which disregards the uncertainty of candidate solutions and pursues diversity and convergence based on IGD+ indicator bootstrap**.

  • TRAA(C) only employs the convergent sampling strategy based on \({I}_{\epsilon +}\), which pursues convergence improvement with low prediction risk.

  • TRAA(D) only applies the angle-based diversity sampling strategy, which pursues diversity improvement under low prediction risk. In addition, due to the angle-based sampling criterion, there is some convergence improvement while diversity improvement.

  • TRAA(U) only utilizes the uncertainty sampling strategy based on Euclidean length, which emphasizes surrogate model uncertainty.

Based on the above, DTLZ1, DTLZ2, DTLZ4, DTLZ5, DTLZ7, WFG1, WFG2, WFG3, WFG6, and WFG9 of the three targets are tested using these methods. The superiority of the proposed criterion can be seen from the experimental results in Table 4. As we expected, TRAA(I) is suitable for problems where the PF is easy to predict (e.g., WFG2, DTLZ5), and performs poorly on problems like DTLZ1 where the PF is harder to predict. TRAA(C) has good results in problems where convergence is emphasized, but has fewer areas of application and overall poorer results due to lack of uncertainty and diversity improvement. TRAA(D) achieves good results on some problems where diversity needs are dominant or even where convergence needs are dominant, thanks to an angle-based sampling strategy that improves both diversity and convergence. On the other hand, considering only uncertainty, TRAA(U) performs poorly and hardly achieves any satisfactory results. TRAA combines the strengths of these variants and identifies high-quality solutions, thus increasing the effectiveness and the area of application of the algorithm. As expected, TRAA shows good performance in all tests.

Table 4 IGD+ statistical results of TRAA and four other variants, and the best results is highlighted
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Comparison over other algorithms

To validate the performance of the proposed TRAA and to facilitate the discussion of its effects, TRAA is compared with the state-of-the-art SAEAs, including KTA2 [18], CPS-MOEA [12], EDNARMOEA [17], KRVEA [16], CSEA [13], MCEAD [14], and PCSAEA [15]. The performance of these algorithms is compared on 80 MOPs (DTLZ and WFG) with different number of targets (3, 4, 6, 8, 10).

The average IGD+ values obtained for the seven advanced algorithms and TRAA over 30 independent runs from DTLZ1 to DTLZ7 are given in Table 5, where the best results are highlighted. The performance of the TRAA algorithm is compared with the other algorithms using the WRS test. The results show that TRAA achieves the best results in 25 of all 35 test instances. In contrast, KTA2, CPS-MOEA, EDNARMOEA, KRVEA, CSEA, MCEAD, and PCSAEA obtained the best results in the numbers of 2, 0, 3, 2, 2, 1, and 0. Of the seven algorithms compared in these comparisons, the best compared to TRAA is KTA2, which outperforms TRAA in only five aspects. This shows that TRAA is a highly competitive algorithm for solving EMOMaOPs.

Table 5 Statistical results of IGD+ obtained by 7 comparison algorithms and TRAA on 35 DTLZ test instances over 30 independent runs
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For both DTLZ1 and DTLZ3, which are strong convergence demand problems, TRAA obtains good results. To better demonstrate the effectiveness of TRAA, Fig. 6 shows the average IGD+ change of the four algorithms and TRAA based on 300 function evaluations for DTLZ1. DTLZ1 is a problem where the target value is more difficult to predict accurately under conditions of low FEs, and therefore SAEAs generally get poor convergence on this problem. However, based on Fig. 6, we can find that TRAA has a more significant decrease in IGD+ compared to the other four algorithms, especially when IGD+ is at a high level. This indicates that TRAA has good convergence ability for similar problems.

Fig. 6
figure 6

Average IGD+ variation of the four algorithms and TRAA for the DTLZ1 problem based on 300 function evaluations

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In the cases of DTLZ2, which requires more diversity, TRAA obtaines the best results in all five problems tested, which indicates that it has a good ability to improve the diversity of populations. However, Table 5 also shows the phenomenon that it does not achieve as good results as we expected when dealing with DTLZ4 problems. It only achieves similar best results for the five types of DTLZ4 problems with target numbers of 8 and 10, and worse than the best results for the rest. In contrast, EDNARMOEA and CSEA performed better on the DTLZ4 problems. This may require strengthening TRAA's ability to challenge trustworthy predicted PFs. In addition, when dealing with DTLZ7, TRAA gets the best results when the number of targets is 3, 4, 6, and 8, and performs averagely when the number of targets is 10. Figure 7 shows all non-dominated solutions corresponding to the median IGD+ values of the seven comparison algorithms and TRAA. From the figure, it can be seen that TRAA has a better ability to adjust diversity and maintain subpopulations in different regions compared to the other 7 algorithms. This is actually attributed to TRAA's infill sampling criterion, which enables adaptive sampling for regular/regular and continuous/discontinuous PFs.

Fig. 7
figure 7

All non-dominated solutions corresponding to the median IGD+ values of the seven comparison algorithms and TRAA in 30 independent runs for the 3-objective DTLZ7

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The experimental results of the DTLZ suite highlight the advantages of TRAA over the other algorithms in solving EMOMaOPs, and TRAA outperforms the other algorithms, which demonstrates the reliability of TRAA in solving EMOMaOPs.

To further insight into the performance of TRAA, the WFG suite is chosen as an additional benchmark test. Among them, WFG1 has the most conversion functions in WFG problems, which makes it difficult to achieve good diversity for MOEAs. WFG2 is a disconnected problem with irregular PFs. WFG3 is a single-mode indivisible problem. WFG4 has a multimodal landscape. WFG5 is a divisible problem with deceptions. WFG6 is a concave, unimodal, and non-separable problem. WFG7 is a separable, unimodal problem. WFG8 is an inseparable problem. WFG9 is a biased, simultaneously deceptive, and multimodal inseparable problem. Therefore, it is more difficult to get good solutions for the WFG problems than for the DTLZ problems.

The statistical results comparing the algorithms with TRAA for the WFG problems are summarized in Table 6, where the best results are highlighted. It is observable that TRAA is the best-performing algorithm overall, with 34 best results in a cumulative total of 45 test problems. It is followed by KTA2 (7), PCSAEA (2), CSEA (1), MCEAD (1), EDNARMOEA (0), KRVEA (0), and CPS-MOEA (0). Based on the experimental results, we can observe that TRAA performs extremely well on WFG2, 6, and 8 and well on WFG3, 4, 5, and 7, thanks to the candidate solution mining strategy and adaptive sampling criterion of TRAA. However, TRAA performs averagely on WFG1 and 9 problems. In contrast, KTA2 has good results when dealing with extremely complex problems such as WFG9. We speculate that this may be related to the fact that KTA2 uses diversity archives and convergence archives to improve the population. Therefore, a more comprehensive demand assessment and sampling strategy can be designed in the future. For WFG1, CSEA achieves better or statistically similar results over TRAA for a target number equal to 3 or 4, while being inferior to TRAA for the rest of the aspects. For WFG3, PCSAEA achieves better results than TRAA for a target number equal to 10. Furthermore, it is interesting to note that both WFG5 and WFG9 possess deception and TRAA shows good performance on WFG5 and poor performance on WFG9. The reason for this phenomenon may be the lack of a more effective strategy for improving trusted predicted PFs.

Table 6 Statistical results of IGD+ obtained by 7 comparison algorithms and TRAA on 45 WFG test instances over 30 independent runs
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In terms of the experimental results of the WFG suite, TRAA still achieves satisfactory results in general, which demonstrates its advantage over other algorithms in solving EMOMaOPs.

The running time of the algorithm is also a factor to be considered. The time cost should not be too much and needs to be within an acceptable range. Figure 8 gives the average running time results of TRAA and seven other state-of-the-art running algorithms. According to the results, CPSMOEA and MCEAD require less running time, followed by K-RVEA, PCSAEA, while the other four types of algorithms require slightly more time. In general, the TRAA can achieve similar levels of runtime as some advanced SAEAs. Although its running time is not particularly small, it is well within an acceptable range compared to the cost of expensive evaluations.

Fig. 8
figure 8

Average running time of TRAA and 7 comparison algorithms based on 30 independent runs

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Overall, according to the experimental results of DTLZ and WFG suite, the TRAA algorithm performs better overall compared to the other seven algorithms that are currently state-of-the-art. In addition, TRAA has clear advantages in terms of effectiveness and areas of application, and it also handles some MOPs with complex PFs or large prediction uncertainties better. The good performance of TRAA is mainly attributed to its inherent advantages:

  1. 1.

    TRAA introduces a high-risk archive and a low-risk archive as candidate solution sets. In the obtaining process of these two archives, an archive selection strategy based on two types of Kriging models is used to reduce the effect of uncertainty, and a combination of two optimizers is used to improve the performance of TRAA. The obtained two archives can provide more reliable candidate solutions for the subsequent different demand states.

  2. 2.

    Unlike existing sampling criteria, TRAA uses a four-state adaptive sampling criterion based on two archives. The criterion can perform the most appropriate sampling method and choose the appropriate solutions on demand. The designed four sampling methods improve the overall performance, convergence, diversity, and uncertainty of the surrogate model, respectively. The criterion is more competitive than some existing sampling criteria.

Conclusions

This paper proposes a two-risk archive algorithm, which contains a strategy for mining high-risk and low-risk archives and a four-state adaptive sampling criterion. The proposed candidate solution mining strategy combines two kinds of Kriging models, two optimization models, and two optimizers to mine more satisfying candidate solutions for different uncertainty requirements. In addition, TRAA employs a more comprehensively considered four-state adaptive sampling criterion that achieves overall performance improvement, convergence improvement, diversity improvement, and surrogate model uncertainty improvement on demand. The results from the widely used DTLZ and WFG test suites demonstrate that TRAA outperforms the current mainstream 7 SAEAs. Moreover, TRAA is highly adaptable to different types of problems, not limited to excelling in only one type. These applicable problems include some intractable problems with complex PFs or target values that are harder to predict. This research highlights the potential of TRAA, which is based on the two-risk archives and the adaptive sampling strategy, to address EMOMaOPs.

However, TRAA performs moderately in dealing with some fraudulent problems, which may require a more effective strategy for improving trusted predicted PFs. Additionally, due to the inherent limitations of Kriging, although it has good generalization ability, its model training and target value prediction are more time-consuming compared to RBFNN and SVR, especially when using multiple Kriging models or dealing with high-dimensional EMOMaOPs. A promising approach to address this problem is to combine Kriging with other surrogate models. Furthermore, the uncertainty estimation of the combined model can be handled by utilizing the intrinsic information between the predicted solutions and the real data in the decision space and the target space.