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Multiobjective evolutionary algorithms (MOEAs) with higher population diversity have been extensively presented in literature studies and shown great potential in the approximate Pareto front (PF). Especially, in the recent development of MOEAs, the reference line method is increasingly favored due to its diversity enhancement nature and auxiliary selection mechanism based on the uniformly distributed reference line. However, the existing reference line method ignores the nadir point and consequently causes the Pareto incompatibility problem, which makes the algorithm convergence worse. To address this issue, a multiobjective evolutionary algorithm based on the adaptive cross-reference line method, called MOEA-CRL, is proposed under the framework of the indicator-based MOEAs. Based on the dominant penalty distance (DPD) indicator, the cross-reference line method can not only solve the Pareto incompatibility problem but also enhance the population diversity on the convex PF and improve the performances of MOEA-CRL for irregular PF. In addition, the MOEA-CRL adjusts the distribution of the cross-reference lines directly defined by the DPD indicator according to the contributing solutions. Therefore, the adaptation of cross-reference lines will not be affected by the population size and the uniform distribution of cross-reference lines can be maintained. The MOEA-CRL is examined and compared with other MOEAs on several benchmark problems. The experimental results show that the MOEA-CRL is superior to several advanced MOEAs, especially on the convex PF. The MOEA-CRL exhibits the flexibility in population size setting and the great versatility in various multiobjective optimization problems (MOPs) and many-objective optimization problems (MaOPs).

Multiobjective optimization problems (MOPs), which have more than one conflicting objective to be optimized, can be defined as^{m} is the objective vector, ^{m} is the objective space, _{1}, …, _{n})^{T} ∈ ^{n} is the candidate solution, and _{1}, _{2} ∈ Ω come to two solutions in the feasible search space. Next, if and only if _{i} (_{1}) ≤ _{i} (_{2}) for each _{i} (_{1}) ≠ _{i} (_{2}) for ∃ _{1} dominates _{2}. If there is no any ^{∗}), ^{∗} will be called a global Pareto-optimal solution. The number of Pareto-optimal solutions is generally more than one in a multiobjective optimization problem, and the set of the Pareto-optimal solutions is named as the Pareto-optimal set. The Pareto-optimal set reflects the geometry of the Pareto front (PF) [

In recent years, many multiobjective evolutionary algorithms (MOEAs) have been proposed for solving multiobjective optimization problems in various fields [

Although the existing MOEAs have been proved to be effective in many practical applications [

The imbalance of convergence and diversity makes it difficult to provide searches on different levels in the objective space. Most parts of the objective space are easier to be searched than the rest. If the nondominated solutions are not uniformly distributed, the candidate solutions may be remained far away from each other and the population may be in danger of losing diversity. In particular, it is difficult to generate feasible solutions in the untapped objective space by genetic factors since the remote parents cannot generate good offspring solutions efficiently in multiobjective optimization [

The first type of MOEAs is established on the fundamentals of Pareto dominance theory. The MOEAs selecting the population of next generation by Pareto dominance theory prefer nondominated individuals. The Pareto dominance theory itself does not promote the preservation of individuals with the diversity in the objective space, so the crowded strategy, niche theory, and other auxiliary strategies for diversity enhancement are proposed in order to gain the expansion of the objective space and enhance diversity. In NSGA-II, the diversity is improved by the crowded distance [

The second type of MOEAs is built on the concept of decomposition. In MOEA/D [

The third category is known as the indicator-based MOEAs. The indicator-based theory guides the selection process by integrating the convergence and diversity into a single indicator. The advantage comparison method proposed by Sun et al. [

In addition, many reference line methods have been widely used to enhance the diversity of MOEAs in recent years. For instance, MOEAs based on the nondominated sorting method (NSGA-III [

In general, the reference line method is constructed by ideal points and reference points. However, the existing reference line methods seldom take into account the influence of the nadir point although the objective space of each generation is often limited between the ideal point and the nadir point. Therefore, the influence of the nadir point cannot be ignored. In addition, when dealing with the problems with convex PFs, the reference line method can cause Pareto incompatibility problems, which makes the convergence of the algorithm worse. Considering the excitations and the defects of reference line, we propose the method based on adaptive cross-reference line. Compared to the existing reference line method, the main new contributions of this work can be summarized as follows:

The concept of the cross-reference line is proposed, and a new MOEA called MOEA-CRL is proposed. It inherits the advantages of the ideal point reference line in convergence, adding the nadir point reference line to enhance diversity. The ideal point reference line is combined with the nadir point reference line to divide the objective space into multiple subspaces, and the unique contributing solution preserved in each subspace to ensure the uniform distribution of the Pareto solution set. Compared with the existing reference line methods, the proposed cross-reference line method performs better in terms of diversity.

Based on the cross-reference line, a dominant penalty distance (DPD) is proposed to solve the Pareto incompatibility problem caused by the reference line method. Compared with the existing reference line indicators, the DPD indicator combines the properties of the ideal point reference line and the nadir point reference line, which not only solves the Pareto incompatibility problem but also improves the performance of the MOEA-CRL on the convex PF.

The cross-reference line adaptation method is proposed to improve the performance of MOEA-CRL for irregular PFs. The cross-reference line adaptation method not only achieves the uniform distribution by uniformly sampling points on the unit hyperplane but also adaptively adjusts the distribution of the cross-reference lines according to the contributing solutions. Compared with the existing reference line adaptive method, it adjusts the distribution of the cross-reference lines according to the contributing solutions directly defined by the DPD indicator, so the adaptation of the cross-reference line can be not affected by the population size, and the uniform distribution of the cross-reference lines can be maintained.

The rest of this paper is organized as follows. In Section

In the MOEAs based on decomposition, the existing reference line method is more advanced compared with the reference point method [

Illustration of the example containing 5 candidate solutions which are on a concave PF and a convex PF. (a) Concave PF. (b) Convex PF.

In order to improve the evaluation limitations of the reference point method, the reference point-based method has been developed into various reference line methods [^{∗} separately. The distance between the candidate solution

Reference line based on ideal points.

The aggregation function is used as a fitness value function that weighs the merits of an individual. The aggregation function is usually a function of the individual

The aggregate optimization equation of the PBI function

PBI also uses the obtained ideal point _{1} and _{2} of the solution ^{T} in the two-dimensional objective space. In the PBI method, a candidate solution with a small _{1} is first considered as a better candidate solution close to the Pareto front. In addition, the distance _{2} from the weight vector _{2} multiplied by _{1}. In summary, a candidate solution with a small _{1} and _{2} is considered as a better candidate solution. The balance between _{1} and _{2} in

PBI reference line method [

The Pareto incompatibility problem means that the individual's reference line evaluation results may face conflicts with the results of the nondominated relationship during the iteration process. The reference line method of the PBI aggregation function can effectively improve the diversity of candidate solutions near the coordinate axis in convex PF by increasing the value of the parameter _{2} is much larger than the influence of _{1}. This method not only maintains the diversity of candidate solutions for convex PF close to the coordinate axis, but also quickly obtains the candidate solution

Example of Pareto incompatibility.

It can be clearly seen from Figure _{2} from the reference line, so the candidate solution set

The setting of the reference point in the aggregation function plays a key role in the performance of MOEA/D. In fact, different types of reference points may have different effects on the exploration behavior of MOEA/D. Most MOEA/D improvements use ideal points as reference points. As mentioned in [

As Wang et al. [^{∗} and the nadir point ^{nad} in the Chebyshev function has an important influence on the distribution of the optimal solution on the PF. In particular, in the case where the ideal point ^{∗} is used as a reference point, the optimal solutions of the subproblems of convex PF and concave PF are shown in Figures ^{∗}, if the nadir point ^{nad} is used as a reference point, the distribution directions of the optimal solutions on these PFs are reversed, as shown in Figures ^{∗} and the nadir point ^{nad} is complementary, using them as reference points at the same time may improve their performance, making them approximate convex PF and concave PF. In addition, if the nadir point ^{nad} is not used as a reference point, you may face greater diversity risks when it is not easy to maintain diversity.

Optimal solution distribution on PF when ideal point ^{∗} and nadir point ^{nad} are used as reference points.

The cross-reference line is formed by matching the ideal point reference line and the nadir point reference line one by one. As shown in Figure

The cross-reference line method.

It is worth noting that if a reference line is defined as the line between the ideal point and the nadir point, the distance between the candidate solution on the reference line and the ideal point reference line and the nadir point reference line is zero. Therefore, it will have an absolute advantage and break the fairness of candidate evaluation. In order to solve this problem, the connection between the ideal point and the nadir point is defined as the penalty line of the cross-reference line, and a certain additional penalty value is added to the candidate solutions that fall on the connection line. Therefore, the candidate solution on the penalty line can only be considered as a contributing solution to other cross-reference lines near the penalty line.

The DPD evaluation indicator of the cross-reference line method is based on the ideal point reference line distance _{∗} and the nadir point reference line distance _{nad}. Among them, the equation of the ideal point reference line distance _{∗} and the nadir point reference line distance _{nad} is as follows:

Based on the cross-reference line method, the dominant penalty distance (DPD) indicator is defined as the maximum value of the ideal point reference line distance (_{∗}) and the nadir point reference line weighted distance (_{nad}). The weighting factor _{∗} is the distance from a candidate solution to the ideal point reference line, _{nad} is the distance from a candidate solution to the nadir point reference line.

Taking the weighting factor _{∗} of the nadir point reference line. The angle area between the boundary line of _{nad}.

Cross-reference line DPD indicator under different PF when

The basic idea of the DPD indicator proposed in this paper is to combine the nadir point and the ideal point and use the cross-reference line as the evaluation reference. This method can not only effectively improve the diversity of candidate solutions near the coordinate axis in convex PF but also ensure the convergence under Pareto’s dominance theory. As shown in Figure

Evaluation method of the DPD indicator based on cross-reference lines.

The cross-reference line method enables MOEAs to ensure good convergence and diversity when dealing with various types of PF problems. As shown in Figure _{p} to a cross-reference line and the cross-reference line also has the smallest DPD_{p} to the candidate solution

Convergence and diversity of the cross-reference line method under different PF when

As shown in Figure

Cross-reference line method under different PF when

The cross-reference line method is an improvement to the reference line method, inheriting the advantages of the ideal point reference line in terms of convergence and adding the nadir point reference line to enhance diversity. The cross-reference line is the combination of the ideal point reference line and the nadir point reference line to divide the objective space into multiple subspaces, and unique candidate solution with the best convergence is kept in each subspace to ensure uniform distribution of the Pareto solution set.

As shown in Figure _{∗} of the candidate solution (3, 6) to the ideal point reference line is smaller than the distance _{nad} to the nadir point reference line, so the DPD indicator of (3, 6) is dominated by _{nad}. The distance _{∗} from the candidate solution (3, 5) to the ideal point reference line is greater than the distance _{nad} from the nadir point reference line, so the DPD indicator of (3, 5) is dominated by _{∗}. It can be calculated that the DPD value of (3, 6) is greater than (3, 5), so (3, 5) in the candidate solution set

Example of the cross-reference method to avoid Pareto incompatibility.

According to DPD evaluation indicator equation (_{p} indicator.

In this section, we first describe the overall framework of the proposed MOEA-CRL in detail. Then, the case study on the implementation of the adaptive cross-reference line method is demonstrated in detail. Finally, the environment selection based on the DPD indicator is illustrated in detail, and the differences on environment selection between MOEA-CRL and other MOEAs are analyzed.

In this section, the general framework of the MOEA-CRL will be elaborated through employing the cross-reference line method and the DPD indicator on the existing fundamentals of MOEAs. The cross-reference lines are formed by the intersection of the ideal point reference lines and the nadir point reference lines. As shown in Algorithm

_{R} (number of reference point and archive size),

[^{∗}, ^{nad}] ⟵

^{∗}, ^{nad});

[^{∗}, ^{nad}] ⟵ ^{∗}, ^{nad});

^{∗}, ^{nad});

The initialization provides preparation for MOEA-CRL. Firstly, a random initialized population is generated according to the initial parameters, which include the number of objectives

The optimization is the core of the MOEA-CRL. The mating pool is established according to the tournament selection strategy. The DPD is used as an evaluation indicator to calculate the fitness value _{p} of each candidate solution _{p} can be expressed as_{max} represents the maximum value of DPD in the population, _{p} of each individual is calculated separately. In the second step, the individual selection is employed by the tournament selection strategy, which can randomly select two candidate solutions for comparison and then retain the individual with the larger _{p}. In general, the mating pool with the number of

^{∗} (ideal point), ^{nad} (nadir point),

the fitness of each candidate solution is calculated;

Two candidate solutions

_{p} > _{q}

The adaptive cross-reference line method is a key step in the optimization. As shown in Algorithm

^{∗} (ideal point), ^{nad} (nadir point)

^{∗′} (new ideal point), ^{nad′} (new nadir point)

Duplicate candidate solutions are deleted in

Dominated candidate solutions are deleted in

_{p∈A}_{i} (^{∗}

^{∗′} ← min_{p∈A}_{i} (

^{∗′} ← ^{∗};

_{p∈A}_{i} (^{nad}

^{nad′} ← max_{p∈A}_{i} (

^{nad′} ← ^{nad};

_{i} (_{i} (^{∗},

_{i} (_{i} (^{nad},

DPD (_{i}) ←max (_{i} (_{i}), ^{∗}_{i} (_{i}));

^{con} ← {_{p∈A} DPD (

^{con};

_{p1∈A\A′} min_{p2∈A′} arccos (

^{valid} ← {^{con}: DPD (_{r′∈R} DPD (

^{valid};

_{p∈A′\R′} min_{r∈R′} arccos (

^{∗} and ^{nad};

In the second step of Algorithm

)/^{∗}

)

)

)

)

)

)

In the fourth step of Algorithm _{p} for a cross-reference line. Through the calculation of the DPD evaluation indicator, all contributing solutions are copied from ^{con} to the new archive _{p1∈A\A;p2∈A′} arccos (

The fifth and sixth steps of Algorithm ^{valid} is obtained. The valid reference points must satisfy two conditions at the same time: (1) the solution _{p} for the cross-reference line and (2) the cross-reference line with the solution _{p}. The calculation of the DPD indicator will be affected by the distance between the candidate solution ^{valid} is copied into the reference point set _{r∈R′} arccos (

Figure _{p1∈A\A; p2∈A′} arccos (^{valid} as shown in Figure

Illustration of the process of reference point adaptation. (a) Based on the DPD indicator, the four candidate solutions all meet the conditions of the contributing solution. (b) The four contributing solutions and two noncontributing solutions are copied to the new archive ^{valid}. (d) The four valid reference points from ^{valid} and the projection points of the two candidate solutions in the new file

The adaptation of reference points not only ensure their own uniformity but also reflect the geometric property of the PF. Therefore, after updating the reference points, the cross-reference lines can also adaptively update to improve the performances of MOEA-CRL for irregular PF.

The environment selection based on the DPD indicator presents as shown in Algorithm

population), ^{∗} (ideal point), ^{nad} (nadir point)

_{i} (_{i} (_{q∈A}_{i} (

_{min}

_{k}| >

_{p} ∈ _{k} DPD (_{k}\{

_{k} ← _{k}\{

_{k};

Although the selection of most decomposition-based evolutionary algorithms is guided by a set of reference points, the reference lines in MOEA-CRL have different purposes. In the MOEA-CRL, the cross-reference lines are adopted to calculate the DPD indicator to evaluate candidate solutions, but each candidate solution is associated with unique reference point in the decomposition-based MOEAs. Therefore, the population size of MOEA-CRL can be unequal to the number of reference points and is not necessarily the same as the method proposed by Das and Dennis [

In addition, MOEA-CRL adjusts the distribution of the cross-reference lines according to the contributing solutions directly defined by the DPD indicator, so the adaptation of the cross-reference line can be not affected by population size, and the uniform distribution of cross-reference lines can be maintained. Regardless of the size of the population, MOEA-CRL is always able to obtain uniformly distributed candidate solutions, providing the flexibility for population size settings. This conclusion is further evidenced by the empirical results in Section

In this section, the sensitivity analysis of the DPD weight coefficients is firstly conducted. It not only proves the validity of the cross-reference line method but also offers the best weight coefficient

In the experiment, 19 test problems from three widely used test suites, including DTLZ1-DTLZ7 [

In the MOEA-CRL, the maximum of the ideal point reference line and the nadir point reference line is selected as the DPD indicator, so the choice of the DPD weight coefficient

In this section, in order to study the effect of the weight coefficient

Sensitivity tests for DPD weight coefficient

It can be seen from Figure

In order to compare fairly with existing advanced algorithms, this article uses general parameter settings, as follows:

Setting of the reference point. The reference point generation of MOEA/D, NSGA-III, and RVEA is on the basis of the two-layer method proposed by Das and Dennis [

Relevant parameter settings of the competition algorithm. In MOEA/D, the size of neighborhood

Genetic operation. The crossover operators in all experiments in this experiment are analog binary crossovers, and the mutation operators are polynomial mutations [

Performance indicators. The convergence and the diversity of the solution sets are indicated by the IGD and the hyper-volume (HV). In the HV calculation, all individuals of the population have been normalized, then the normalized HV value is calculated with a reference point (1.1, 1.1, …, 1.1). The MOEA with a larger HV value has better performance than the other. In addition, in order to reduce the computational complexity and improve the computational efficiency, the Monte Carlo estimation method is adopted for problems with the objective number is 5 and 10, and the number of sampling points required for the calculation is set to 1,000,000. In the DPD calculation, approximately 5,000 uniformly distributed points are sampled at the PF by the two-layer method proposed by Das and Dennis [

Settings of the number of reference points for each number of objectives, where

Number of objectives ( | Parameter ( | Number of reference points/population size ( |
---|---|---|

3 | 13, 0 | 105 |

5 | 6, 0 | 210 |

10 | 3, 2 | 275 |

Settings of the number of objectives, the number of decision variables, and the maximal number of generations for each test problem.

Test problem | Pareto front | ||
---|---|---|---|

Regular Pareto front | |||

DTLZ1 | 3, 5, 10 | Linear | |

DTLZ3 | 3, 5, 10 | Concave | |

DTLZ2, 4 | 3, 5, 10 | Concave | |

WFG4-9 | 3, 5, 10 | Concave | |

Irregular Pareto front | |||

DTLZ5-6 | 3, 5, 10 | Mostly degenerate | |

DTLZ7 | 3, 5, 10 | Disconnected | |

WFG1 | 3, 5, 10 | Sharp tails | |

WFG2 | 3, 5, 10 | Disconnected | |

WFG3 | 3, 5, 10 | Mostly degenerate | |

MaF3 | 3, 5, 10 | Convex | |

MaF11 | 3, 5, 10 | Convex disconnected | |

MaF15 | 3, 5, 10 | 20^{∗}M | Convex large scale |

Statistical results (mean values and standard deviations) of IGD value obtained by MOEA/D, NSGA-III, RVEA, KnEA, and MOEA-CRL on DTLZ1-DTLZ7, WFG1-WFG9, MaF3, MaF11, and MaF15 with 3 objectives.

Problem | MOEA/D | NSGA-III | RVEA | KnEA | MOEA-CRL | ||
---|---|---|---|---|---|---|---|

DTLZ1 | 3 | 7 | 2.8508 | 1.8979 | 1.8978 | 5.0030 | 1.8977 |

DTLZ2 | 12 | 6.9661 | 5.0301 | 5.0301 | 6.6663 | 4.6814 | |

DTLZ3 | 12 | 1.0106 | 5.0394 | 5.0355 | 1.0817 | 4.7075 | |

DTLZ4 | 12 | 2.3255 | 1.3215 | 5.0300 | 1.5318 | 2.2800 | |

WFG4 | 3 | 12 | 2.7142 | 2.0405 | 2.0800 | 2.4949 | 1.9326 |

WFG5 | 2.8760 | 2.1444 | 2.1580 | 2.6120 | 2.0682 | ||

WFG6 | 2.9642 | 2.1871 | 2.2604 | 2.8667 | 2.1759 | ||

WFG7 | 2.7206 | 2.0414 | 2.0612 | 2.4178 | 1.9289 | ||

WFG8 | 3.1847 | 2.6527 | 2.8098 | 3.3436 | 2.6066 | ||

WFG9 | 2.7920 | 2.0537 | 2.0722 | 2.2643 | 1.9470 | ||

＋/−/≈ | 0/10/0 | 1/8/1 | 1/8/1 | 1/9/0 | |||

DTLZ5 | 3 | 12 | 1.2417 | 1.1730 | 5.8469 | 1.0462 | 5.7501 |

DTLZ6 | 12 | 1.2419 | 1.7437 | 5.9099 | 4.7828 | 4.2868 | |

DTLZ7 | 22 | 2.4700 | 7.0580 | 1.0489 | 8.4509 | 2.0368 | |

WFG1 | 3 | 12 | 2.1051 | 1.3620 | 1.5393 | 1.8632 | 1.4766 |

WFG2 | 12 | 2.1674 | 1.5105 | 1.6491 | 1.8398 | 1.5058 | |

WFG3 | 12 | 4.1686 | 8.8550 | 2.1668 | 9.6049 | 8.9822 | |

Convex | |||||||

MaF3 | 3 | 12 | 1.5083 | 4.3564 | 3.8402 | 1.4269 | 3.4876 |

MaF11 | 12 | 2.1673 | 1.5147 | 1.6463 | 1.8721 | 1.5082 | |

MaF15 | 5 | 4.5845 | 3.6301 | 9.0510 | 5.1952 | 2.6890 | |

＋/−/≈ | 1/8/0 | 3/4/2 | 1/8/0 | 1/8/0 |

Statistical results (mean values and standard deviations) of HV value obtained by MOEA/D, NSGA-III, RVEA, KnEA, and MOEA-CRL on DTLZ1-DTLZ7, WFG1-WFG9, MaF3, MaF11, and MaF15 with 5 objectives and 10 objectives.

Problem | MOEA/D | NSGA-III | RVEA | KnEA | MOEA-CRL | |
---|---|---|---|---|---|---|

DTLZ1 | 5 | 9.0873 | 9.7979 | 9.7984 | 6.2616 | 9.7988 |

10 | 9.7273 | 9.8648 | 9.9967 | 0.0000 | 9.9973 | |

DTLZ2 | 5 | 7.1112 | 8.1269 | 8.1252 | 7.9064 | 8.1626 |

10 | 6.2665 | 9.4539 | 9.6963 | 9.5650 | 9.7102 | |

DTLZ3 | 5 | 4.3566 | 7.7543 | 7.7757 | 3.9573 | 7.3975 |

10 | 6.4046 | 3.3590 | 9.6443 | 0.0000 | 9.6234 | |

DTLZ4 | 5 | 2.7690 | 7.3262 | 7.7683 | 7.6308 | 7.2000 |

10 | 6.2169 | 9.6721 | 9.6984 | 9.5637 | 9.7165 | |

WFG4 | 5 | 6.2968 | 8.0469 | 8.0565 | 7.8730 | 8.0575 |

10 | 5.8107 | 9.4578 | 9.4326 | 9.5750 | 9.1518 | |

WFG5 | 5 | 5.9294 | 7.6126 | 7.6092 | 7.4495 | 7.3465 |

10 | 5.3478 | 8.9907 | 8.9790 | 8.9664 | 8.4566 | |

WFG6 | 5 | 5.5065 | 7.4242 | 7.4662 | 7.2041 | 7.2008 |

10 | 4.7917 | 8.6935 | 8.6291 | 8.6854 | 8.7014 | |

WFG7 | 5 | 6.2952 | 8.0771 | 8.0685 | 7.9537 | 7.9284 |

10 | 6.1632 | 9.4638 | 9.4552 | 9.5737 | 9.4268 | |

WFG8 | 5 | 3.2668 | 6.9440 | 6.9749 | 6.6088 | 6.9786 |

10 | 5.2900 | 8.3115 | 7.4198 | 8.1031 | 8.5967 | |

WFG9 | 5 | 4.1801 | 7.6559 | 7.6952 | 7.6796 | 7.4665 |

10 | 5.3897 | 8.6420 | 8.7342 | 9.0581 | 8.3182 | |

＋/−/≈ | 0/20/0 | 10/8/2 | 10/5/5 | 8/10/2 | ||

DTLZ5 | 5 | 1.0968 | 1.0483 | 9.1890 | 7.1580 | 1.0808 |

10 | 9.7922 | 2.8809 | 9.0906 | 3.5304 | 8.2365 | |

DTLZ6 | 5 | 9.4634 | 5.7651 | 9.9670 | 9.1029 | 9.2650 |

10 | 9.8124 | 3.0230 | 9.2019 | 0.0000 | 9.1491 | |

DTLZ7 | 5 | 1.7168 | 2.3157 | 2.1343 | 2.4843 | 2.2732 |

10 | 4.5806 | 1.7116 | 1.3371 | 9.1243 | 7.4082 | |

WFG1 | 5 | 9.4334 | 9.7370 | 9.8276 | 9.9246 | 9.9835 |

10 | 5.0234 | 9.4008 | 9.9015 | 9.9756 | 7.4040 | |

WFG2 | 5 | 9.6566 | 9.9555 | 9.9404 | 9.9320 | 9.9683 |

10 | 9.9648 | 9.9700 | 9.8471 | 9.9306 | 9.9191 | |

WFG3 | 5 | 9.1998 | 1.6765 | 1.5967 | 7.7484 | 1.6083 |

10 | 7.5376 | 2.4516 | 0.0000 | 0.0000 | 4.0606 | |

MaF3 | 5 | 9.9646 | 9.9870 | 9.9895 | 8.8686 | 9.9975 |

10 | 9.9993 | 2.9125 | 9.8343 | 0.0000 | 9.9997 | |

MaF11 | 5 | 9.7508 | 9.9583 | 9.9397 | 9.9299 | 9.9751 |

10 | 9.3320 | 9.9641 | 9.8256 | 9.9284 | 9.9818 | |

MaF15 | 5 | 2.9995 | 0.0000 | 2.4759 | 0.0000 | 1.0571 |

10 | 3.0314 | 0.0000 | 1.9696 | 0.0000 | 2.6803 | |

＋/−/≈ | 5/10/3 | 4/12/2 | 5/10/3 | 4/14/0 |

Table

Figure

The nondominated solution set with the median IGD value among 30 runs obtained by MOEA/

Table

Figure

The nondominated solution set with the median HV value among 30 runs obtained by MOEA/D, NSGA-III, RVEA, KnEA, and MOEA-CRL on DTLZ1, DTLZ2, and MaF3 with 10 objectives.

Through the sensitivity analysis of the weight coefficient of the 3-objective problem, it can be concluded that the use of the nadir point does not weaken the convergence but increases the convergence pressure. However, as the objective dimension increases, the use of the nadir point will indeed cause the deterioration of convergence and even the failure to converge.

In the experiment, the population size N is set as same as the number of reference points since the reference points are generally associated with each candidate solution in most decomposition-based MOEAs. The number of reference points depends on the method of Das and Dennis [

The population size setting of MOEA-CRL proposed in this paper is flexible, and the number of reference points has less influence on it. The number of candidate solutions can be less than the number of reference points and also can be larger.

MOEA-CRL with different population sizes was tested on DTLZ1, DTLZ2, and MaF3 with three objectives. Figure

The nondominated solution sets of MOEA-CRL with population sizes of 35, 70, 105, 140, and 175 with 3 objectives.

In this paper, an evolutionary algorithm based on the adaptive cross-reference line method, called MOEA-CRL, is proposed to inherit the advantages of the ideal point reference line for better convergence and add the nadir point reference line for higher diversity. Especially, on the convex PF, MOEA-CRL solves the Pareto incompatibility problem and significantly enhances the population diversity. Furthermore, this paper proposed the DPD indicator based on the cross-reference lines. The properties of the ideal point reference line and the nadir point reference line are combined to solve the Pareto incompatibility problem as well as improve the performance of the MOEA-CRL on the convex PF. Based on the DPD evaluation strategy of the cross-reference line method, MOEA-CRL retains unique solution with the best convergence in each attraction region as a nondominated solution, which ensures that the Pareto solution set is distributed evenly. Finally, this paper proposed a cross-reference line adaptation method in order to enhance the performance of MOEA-CRL in dealing with the irregular problems.

The experimental results show the superiority of MOEA-CRL on the convex PF. It also has the competitiveness due to the adaptability of cross-reference lines while solving those MOPs and MaOPs with other types of PFs. Remarkably, the cross-reference line method is only used to calculate the DPD indicator. Therefore, the population size is irrelated to the number of the cross-reference lines, and subsequently, the population size setting is flexible. The proposed MOEA-CRL proves that the adaptive cross-reference line method is prospective for significantly improving the diversity especially in the convex PF.

In fact, the experimental results also clearly illustrate that the performance of MOEA-CRL deteriorates significantly with the increase of dimensions. That means that the cross-reference line method still poses the challenges in dealing with some research issues such as high-dimensional deterioration and more complex convex PF problems.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work was supported by the National Natural Science Foundation of China (nos. 51875419 and 51605345).