An adaptive adjacent maximum distance crossover operator for multi-objective algorithms

Most genetic operators use random mating selection strategy and fixed rate crossover operator to solve various optimization problems. In order to improve the convergence and diversity of the algorithm, an adaptive adjacent maximum distance crossover operator is proposed in this paper. A new mating selection strategy (distance-based mating selection strategy) and an adaptive mechanism (adaptive crossover strategy based on population convergence) are adopted. Distance-based mating selection strategy purposefully selects parents to produce better offspring. Adaptive crossover strategy based on population convergence increases the convergence speed of the algorithm by controlling the crossover probability. The proposed crossover strategy is evaluated on the simulated binary crossover operators of non-dominated sorting genetic algorithm II and multi-objective evolutionary algorithm based on decomposition. The performance of the algorithm is verified on a series of standard test problems. Finally, the optimization results of the improved algorithm using adaptive adjacent maximum distance crossover operator and the standard algorithm are compared and analyzed. The experimental results show that the algorithm using adaptive adjacent maximum distance crossover operator has better optimization results.


Introduction
In recent years, multi-objective optimization algorithm has received extensive attention (Gong and Jiao 2009;Liu et al. 2018). With the development of multi-objective optimization and several applications like design of digital filters, image processing, and other relevant fields, there are many problems to be solved. In genetic algorithm, individuals evolve to a higher fitness direction through selection, crossover and mutation to find the optimal solution of the optimization problem. In NSGA-II (Deb et al. 2002a,b) algorithm and MOEA/D (Zhang and Li 2007) algorithm, individuals evolve to a higher dominance level through selection, crossover and mutation until all solutions are non-dominated solutions. It can be seen that these steps are the most important operations in genetic operators (Kora and Yadlapalli 2017). In previous studies, the parents of crossover operators in some multi-objective optimization algorithms were randomly selected. If a certain strategy is introduced into mating selection and individuals are purposefully selected for crossover operation, better offspring will be produced. Therefore, it is necessary to introduce some mating selection strategies to select proper parents and produce better offspring. Most of the existing crossover operators use fixed crossover probability, but the algorithm needs different crossover probability in different iteration stages. The inappropriate crossover probability will greatly affect the convergence speed of the algorithm. Therefore, it is necessary to adopt appropriate adaptive strategy to allocate appropriate crossover probability in different stages of algorithm iteration.
The improvement method proposed in this paper is an evolutionary technology, which has been widely used in various fields. In recent years, ET has been widely used in the field of filters (Agrawal et al. 2021). Kumar et al. (2018) proposed a new efficient method for implementing the Hilbert transform using an all-pass filter, based on fractional derivatives (FDs) and swarm optimization. In this method, a variant of the swarm-intelligence-based multimodal search space technique, known as the constraint-factor particle swarm optimization, is exploited for finding the suitable values for the FD and x 0 . In addition, Agrawal et al. (2017) determined the best value of FD constraint and passband reference point by minimizing the sum of passband error (E p ) and stopband error (E s ) of IIR filter by using particle swarm optimization (PSO), constraint factor inertial PSO (CFI-PSO) and quantum PSO, artificial bee colony algorithm and cuckoo search technology and other different evolutionary techniques. In the field of filters, Agrawal et al. (2018) also proposed a new design method of digital bandpass and band stop infinite impulse response (IIR) filters with approximate linear phase response. In this study, the modern heuristic technology called cuckoo search (CS) optimization is used to determine the optimal value of FDs and reference frequency simultaneously to minimize the fitness function. Agrawal et al. (2019) also improved the quantum particle swarm optimization (QPSO) technology by taking advantage of the novelty of the scout bee search and replacement mechanism in the artificial bee colony (ABC) algorithm.
Evolutionary algorithm maintains population diversity through crossover operator and mutation operator, and its convergence speed is also affected by these two main operators. Through the crossover operation, the genetic material between solutions is structurally and randomly exchanged, and it is possible that a ''good'' solution can produce a ''better'' solution. As the only index of crossover operator, the value of crossover probability is very important. When the crossover probability is large, the search ability of the algorithm can be enhanced, but the convergence of the algorithm will be reduced; when the crossover probability is small, the global search ability of the algorithm will be greatly reduced. One of the demerits of NSGA-II and MOEA/D algorithms is the use of fixed rate crossover operators. Throughout the whole iterative evolution process, we can know that different crossover rates are required in different stages of algorithm iteration. At the beginning of the algorithm iteration, if the crossover probability is small, it is difficult for the population to produce excellent individuals; at the later stage of the algorithm iteration, most of the solutions have large fitness values or are non-dominated solutions. At this time, if a large crossover probability is adopted, the excellent solutions are easy to be destroyed and the algorithm is easy to fall into local optimization. Adaptive crossover strategy (Srinivas and Deb 1994) can solve the above problems. In this paper, an adaptive crossover strategy is proposed, which can select the appropriate crossover rate at different stages of algorithm iteration.
In some crossover operators, parents participating in crossover are randomly selected. In genetics, parents with greater genetic differences are more likely to produce excellent offspring, while ''close relatives'' with similar genes are more likely to produce worse offspring. Therefore, in order to transfer the excellent genes of the parents to the offspring and make the offspring perform better, this paper introduces an adjacent maximum distance crossover strategy which can select a pair of individuals with the largest genetic difference between adjacent individuals as parents.
In the basic NSGA-II and MOEA/D algorithms, they solve all kinds of problems with fixed crossover operator rate and random mating selection. Although NSGA-II and MOEA/D are excellent multi-objective evolutionary algorithms, they may produce unsatisfactory results on some problems due to the defects of fixed rate and random mating selection. The adaptive crossover mechanism can dynamically adjust the crossover probability according to the degree of population convergence, and the offspring generated by adjacent maximum distance crossover is better. This paper combines these two mechanisms and proposes an adaptive adjacent maximum distance crossover operator. In order to solve the defects of random mating selection and fixed rate crossover operator, further improve the convergence and diversity of evolutionary algorithm and enhance the search ability of population, an adaptive adjacent maximum distance crossover operator is introduced into NSGA-II algorithm and MOEA/D algorithm. On this basis, two variant algorithms are proposed: modified non-dominated sorting genetic algorithm II with adaptive adjacent maximum distance crossover (AAM-NSGA-II) and modified multi-objective evolutionary algorithm based on decomposition with adaptive adjacent maximum distance crossover (AAM-MOEA/D). Finally, the performance of the improved algorithm on a series of test problems is tested to verify the performance of the adaptive adjacent maximum distance crossover operator on the two algorithms.
The rest of this paper is organized as follows: Sect. 2 reviews the development of multi-objective optimization algorithms. Section 3 introduces the research background. In this section, the principles of NSGA-II and MOEA/D as well as the principles of the simulated binary crossover operator used are introduced in detail. Section 4 introduces the multi-objective optimization problem model, the basic principle of fixed-rate crossover operator, and the specific principle of the proposed new operator in detail. Section 5 is the experimental design. This section describes the test problems and Pareto frontier characteristics used in the experiment in detail, and introduces the performance index and specific experimental parameter settings. Section 6 is the experimental results and analysis. Finally, the conclusion and outlook for the future are Sect. 7.

Related work
In the past decade, multi-objective optimization algorithms have developed rapidly, and many excellent algorithms have been proposed. Some classical multi-objective optimization algorithms include: multi-objective particle swarm optimization (MOPSO) (Coello et al. 2004), strength Pareto evolutionary algorithm (SPEA) (Zitzler and Thiele 1999), multi-objective optimization algorithm based on decomposition (MOEA/D) (Zhang and Li 2007), nondominated sorting genetic algorithm (NSGA) (Srinivas and Patnaik 1994), etc. On this basis, researchers in related fields put forward many improvements. Chen et al. (2018) introduced two crossover operations into particle swarm optimization algorithm and proposed PSOCO (Particle Swarm Optimization algorithm with crossover operation). The algorithm constructs high-quality samples to guide the evolution of particles by crossover operating the local optimal position of each particle. Raquel and Naval (2005) integrated crowding distance into PSO (Particle Swarm Optimization), maintained the diversity of non-dominated solutions in external archives through crowding distance mechanism and mutation operator, so that PSO algorithm can solve multi-objective optimization problems. Junfeng Dong et al. (2019) combined with the idea of individual neighborhood, applied the idea of neighborhood in density clustering algorithm DBSCAN (density-based spatial clustering of applications with noise) to the sorting mechanism, and solved the defect of uneven distribution of Pareto front of crowding distance sorting mechanism adopted by NSGA-II algorithm. Jensen (2003) applied the binary search method to non-dominated sorting, and reduced the time complexity of NSGA2 from O (GMN 2 ) to O (GNlog MÀ1 N). For the case of large population size and few targets, the algorithm can save a lot of processing time. Li et al. (2014) combined Pareto domination with decomposition-based method. The proposed MOEA/DD (multiobjective evolutionary algorithm based on decomposition and dominance) algorithm uses their advantages to balance the convergence and diversity of the evolutionary process. Castro et al. (2017) combined the advantages of CMA-ES (covariance matrix adaptation evolution strategy) and MOEA/D, proposed a new multi-objective CMA-ES: MOEA/D-CMA (MOEA/D with covariance matrix adaption evolution strategy), then introduced the Pareto advantage update mechanism of MOEA/DD, and further proposed MOEA/DD-CMA (MOEA/DD with covariance matrix adaption evolution strategy).
So far, in order to improve the performance of EAS, many effective improvements have been proposed for crossover operator and mutation operator. Deep et al. (2007) proposed Laplace crossover operator (LX), which is a real coded crossover operator, avoiding the problem of high computational cost of binary coding. By combining this operator with MPTM (Makinen, Periaux and Toivanen mutation) and NUM (non-uniform mutation) mutation operator, they proposed two genetic algorithms: LX-MPTM and LX-NUM. Koohestani (2020) applies the improved partial mapping crossover (IPMX) to genetic algorithm. IPMX can efficiently generate offspring, which significantly improves the efficiency of replacement-based genetic algorithm. Kiraz et al. (2020) proposed a new allparent crossover operator: collective crossover operator. All individuals in the population participate in the crossover at the same time and produce an offspring. The impact of each individual on the offspring is determined by its An adaptive adjacent maximum distance crossover operator for multi-objective algorithms 7421 fitness value. Iqbal et al. (2020) combined PMX (Partiallymatched crossover) with CX (Cycle Crossover) and proposed a complete mapping crossover operator (CMX), which avoids the random cutting problem of existing mapping operators. Inspired by the sinusoidal motion of waves, Varun Kumar and Panneerselvam (2017) and others proposed a sinusoidal motion crossover operator (SMC) and applied it to the genetic algorithm to solve the vehicle routing problem (VRP). This method can produce two generations at the same time, and the chromatin produced is more random. Hassanat et al. (2019) studied the influence of crossover and mutation ratio on the performance of the algorithm. The method proposed in this study makes the crossover and mutation ratio change linearly in the search process and has been verified in the traveling salesman problem (TSP): when the population size is large, the method works very well. By studying the migration behavior of monarch butterflies in nature, Wang et al. (2018) proposed monarch butterfly optimization algorithm (MBO) to solve various global optimization tasks. However, MBO has not obtained a better solution in some test problems. Therefore, Wang et al. introduced greedy strategy and adaptive crossover operator into MBO algorithm and proposed a variant algorithm: GCMBO (MBO with greedy strategy and selfadaptive crossover operator). Greedy strategy can make better individuals inherit to the next generation, and adaptive crossover operator can significantly improve the diversity of the population in the later stage of search. Deng et al. (2017) proposed an adaptive differential evolution rotation crossover operator, which can generate test vectors in accordance with Levy distribution under the control of adaptive crossover parameters and rotation control vectors. The operator is applied to differential evolution variants and JADE-RCO (a variant of Differential Evolution with rotating crossover operator) algorithm to verify the superiority of the operator. Different crossover operators are suitable for solving different problems. Zhao et al. (2019) proposed a multi population adaptive crossover strategy, which divides the original population into multiple populations, assigns different crossover operators to each sub population, and designs corresponding management strategies to adaptively adjust the scale of each sub population.
It can be seen that in the field of multi-objective optimization, scholars have constantly proposed some excellent and classic algorithms, and many latecomers have made improvements to these algorithms. Although there are some excellent crossover schemes and adaptive strategies, the inherent defects of crossover operators have not been solved. Therefore, this study combines the proposed crossover strategy with adaptive strategy to enhance the performance of the algorithm.

NSGA-II
The NSGA-II (Deb et al. 2002a, b) algorithm, proposed by Deb et al., is an improvement of the standard NSGA. NSGA algorithm has the following defects: high computational complexity of non-dominated sorting, lack of elitism, and need to specify shared parameters. In order to solve these defects, three innovations are proposed in NSGA-II: fast non-dominated sorting, fast congestion distance estimation and congestion comparison operator.
The iterative process of NSGA-II is as follows: assume that the current algebra is t and the population size is N. The parent population P t of this generation was crossed and mutated to generate population P c and P m and merge population R t =P t [P c [P m : The population R t is non-dominated sorted, the individuals are assigned to different nondominated fronts (F 1 , F 2 , etc.), and the crowding distance is sorted for each front. A new species group S t =[ l i¼1 F i such that the number of individuals in S t is equal to or greater than N. If S t j j=N, then the next-generation parent viduals in F l are retained, the remaining individuals are deleted, and the next-generation parent population P tþ1 =[ l i¼1 F i is obtained, let t = t ? 1 and repeat the above process until the number of iterations reaches the maximum. The NSGA-II algorithm flow is given in Algorithm 1.

MOEA/D
MOEA/D (Zhang and Li 2007) was proposed by Qingfu Zhang et al. in 2007. The decomposition-based method can decompose a multi-objective optimization problem into several scalar optimization subproblems (a series of single objective optimization problems or multiple multi-objective subproblems) and optimize them at the same time. Each subproblem is assigned several neighbors, and each subproblem is optimized only by using the information of adjacent subproblems. This method makes the computational complexity of MOEA/D in each generation lower than MOGLS and NSGA-II algorithms.
The core idea of MOEA/D is to optimize the local through the cooperation of adjacent subproblems, so as to advance the overall population to the Pareto optimal front. The optimal solution of the subproblem is the Pareto optimal solution of the multi-objective problem. The iterative process is shown in Algorithm 2.
An adaptive adjacent maximum distance crossover operator for multi-objective algorithms 7423

Simulated binary crossover (SBX)
Simulated binary crossover (Pan et al. 2021) is a single point binary crossover operator, which is widely used in real coded multi-objective evolutionary algorithm. The steps of generating generation t ? 1 individuals c 1 and c 1 from generation t individuals x 1 and x 2 are as follows: Step 1: generate a random number u i [(0,1); Step 2: take the ratio of the difference between the two individuals of generation t ? 1 and the difference between the two individuals of generation t as the uniform distribution factor: Step 3: using the following probability density function Pðb i Þ calculate b qi , so that under the corresponding probability density curve, from 0 to b qi the area of the interval is equal to u i .
parameter g c is called the crossover distribution index and is any nonnegative real number.
Step 4: calculate the two offspring of generation t ? 1: From formula (4): It can be seen from formula (3) and formula (5) that the distance between offspring individuals is proportional to the distance between parent individuals, and this distance ratio is a decreasing function of g c . The crossover distribution index g c has the following characteristics on the generation of offspring individuals: if the value of g c is large, the probability of offspring individuals close to their parents is large; If the value of g c is small, it is more likely that the offspring will be far away from the parent. 4 Adaptive adjacent maximum distance crossover operator

Mathematical model of multi-objective optimization problem
The general multi-objective optimization problem is set as the minimization problem, and its mathematical model is as follows: where m is the number of targets, X is the decision space, and n is the number of decision variables.

Fixed rate crossover operator
Most multi-objective optimization algorithms use fixed rate crossover operators (Yi et al. 2018). Taking single point crossing as an example, the process of generating offspring is shown in Algorithm 3.

Principle of adaptive adjacent maximum distance crossover operator
The adaptive mechanism proposed in this paper is based on population convergence. In the early stage of the algorithm iteration, the individuals in the population are randomly distributed in the decision space. At this time, using a large An adaptive adjacent maximum distance crossover operator for multi-objective algorithms 7425 crossover probability can increase the probability of producing excellent offspring and accelerate the convergence speed in the early stage of the algorithm; With the continuous iteration of the algorithm, there are more and more solutions in the non-dominated frontier F 1 , and the overall convergence degree of the population is higher and higher. At this time, in order to prevent excellent individuals (nondominated solutions and individuals with high dominance) from being destroyed by crossover operation and prevent the algorithm from falling into local optimization, a smaller crossover probability should be adopted. And the adaptive strategy holds that when the algorithm iterates to a certain extent and the number of non-dominated solutions of the population accounts for a large proportion of the population, the algorithm can be considered to have entered the late stage of iteration. Therefore, the principle of adaptive strategy is to reduce the crossover rate of the algorithm to a certain value when the algorithm meets the following two conditions at the same time: 1. The current number of iterations t of the algorithm reaches half of the total number of iterations t max . 2. The number of individuals in the non-dominated frontier NF 1 accounts for more than 50% of the whole population.
The core idea of adjacent maximum distance crossover is that the greater the parental difference, the better the offspring. In this paper, the Euclidean distance between individuals is used to measure individual differences. Maximum distance crossover selects the pair of individuals with the largest distance in the population as the crossover parents, and judges whether to cross according to the crossover probability. Then select the individual with the largest distance among the other individuals and repeat the above steps until all individuals in the population are traversed. However, when the parental individuals with the largest distance are crossed, the probable offspring produced by the parental individuals will appear between the two parental individuals, which is easy to cause the population to cluster together, resulting in a decline in the population distribution. The adjacent maximum distance crossing can avoid the above drawbacks. Adjacent maximum distance crossover first randomly selects a crossover parent, and then selects the nearest individual among the adjacent individuals of this crossover parent as the second crossover parent. This can improve the convergence and maintain the diversity of the population.
Adjacent individuals are defined as follows: for all individuals in the population, the k-th objective function value is sorted in ascending or descending order. Find the position of the k-th objective function value of the first crossover parent after sorting, and its adjacent individual is called its adjacent individual. If the first crossover parent is in the middle of the sorting list, there are two adjacent individuals; If it is at the beginning or end of the sorted list, there is only one adjacent individual. If the objective function has M objectives, each individual can have at most 2 M adjacent individuals.
The strategy proposed in this paper is a general strategy, which can be combined with the crossover operators such as single point crossover, uniform crossover, and simulated binary crossover.
The flow of adaptive adjacent maximum distance crossover operator is shown in Algorithm 4. Firstly, the current iteration times t and the proportion of non-dominated individuals pF 1 are obtained, and whether the algorithm has entered the later stage of iteration is judged to determine whether to change the crossover probability. The number of individuals participating in crossover n C is obtained according to the crossover probability. Sort on each objective function value. Select the first crossover parent. Find all adjacent individuals of parent 1, calculate the Euclidean distance between parent 1 and all adjacent individuals, and select the individual with the largest distance from parent 1 as parent 2. After that, the two cross parents were cross operated. Repeat the above steps until the number of individuals participating in the crossover is n C .

Test problem
The performance of adaptive adjacent maximum distance crossover operator is verified on ZDT (Zitzler et al. 2000), DTLZ (Deb et al. 2002a, b) and other test problems (Tian et al. 2019;Cheng et al. 2015). Table 1 summarizes the characteristics of Pareto optimal front for different types of test problems.

Performance index
In order to evaluate the performance of the algorithm, inverted generational distance (IGD) (Coello and Cortés 2005) and hypervolume (HV) (Zitzler and Thiele 1999) are selected as performance indicators.

Inverted Generational Distance (IGD)
IGD is an index to measure the distance between real Pareto front individuals and Pareto frontier individuals generated by the algorithm. The lower the IGD value, the better the algorithm performance. It is defined as follows: where PF Ã is the real Pareto frontier and PF Ã j j is the number of individuals in the real Pareto front, d i represents the minimum Euclidean distance from each point in the real Pareto front in the target space to the known front.

Hypervolume (HV)
The HV index represents the volume of the hypercube surrounded by the Pareto solution obtained by the algorithm and the reference point in the target space. The higher the HV value, the better the algorithm performance. It is defined as follows: where d represents the Lebesgue measure, which is used to measure volume. PF j j is the number of individuals in the Pareto solution set generated by the algorithm. v i represents ? / -/ = 0/6/0 Fig. 1 Convergence characteristics of IGD mean of NSGA-II and AAM-NSGA-II on ZDT series test problems An adaptive adjacent maximum distance crossover operator for multi-objective algorithms 7429 the supervolume formed by the reference point and the i-th solution.

Parameter setting
The parameters of each ZDT test problem are shown in Table 2. The ZDT test problem has only two objective functions (Zitzler and Thiele 1999). In this experiment, the adaptive adjacent maximum distance crossover operator is applied to NSGA-II algorithm to verify the performance of the improved operator in the algorithm based on non-dominated sorting; And it is applied to MOEA/D algorithm to verify the performance of the operator in the decomposition-based algorithm. NSGA-II and MOEA/D algorithms adopt simulated binary crossover, and the crossover probability of each algorithm is set as follows: the crossover rate of NSGA-II and MOEA/D was p c ¼ 0:5. The initial crossover probability of AAM-NSGA-II and AAM-MOEA/D is p c ¼ 0:5, and the reduced crossover probability is p c ¼ 0:1. In the crossover operator, the value range of crossover probability is [0,1]. In the early stage of the algorithm iteration, the crossover probability is set to 0.5 to obtain the balance between the convergence and search ability of the algorithm. In the later stage of the algorithm iteration, with the higher and higher degree of population convergence, the value of crossover probability should be smaller and smaller.

Experimental results and analysis
The experiment is carried out on the evolutionary multiobjective optimization platform PlatEMO-v2.8 (Tian et al. 2017) based on MATLAB. The Wilcoxon signed-rank test (Derrac et al. 2011) is used to compare the performance differences of the algorithms. Among them, '' ? '', ''-'', and '' = '' , respectively, represent significantly superior, significantly inferior, and indistinguishable from the algorithm proposed in this paper. The index result of each algorithm is the mean and standard deviation obtained by running 20 times independently. When the two algorithms obtain the same mean, the algorithm with smaller standard deviation has better effect. The best results obtained on each test case are highlighted.
6.1 Application of adaptive adjacent maximum distance crossover operator on NSGA-II Table 3 shows the average value and standard deviation of IGD index of 20 independent runs of NSGA-II algorithm on ZDT benchmark problem before and after improvement. Obviously, the performance of the improved algorithm is better than that of the standard NSGA-II, because AAM-NSGA-II obtained six best IGD values in six groups of tests. AAM-NSGA-II algorithm achieves better results on ZDT3 test problem with discontinuous Pareto front and ZDT6 test problem with uneven front distribution. And the performance of standard NSGA-II algorithm in ZDT4 test problem is not good. Through comparison, it can be seen that the improved operator greatly improves the performance of NSGA-II on ZDT4 test problem. Figure 1 shows the convergence characteristics of IGD mean obtained by NSGA-II and AAM-NSGA-II on ZDT series test problems. In ZDT1, ZDT2 and ZDT3 test, AAM-NSGA-II proposed in this paper obtains a smaller IGD index value at the early stage of algorithm evaluation and keeps the IGD index value smaller than that of NSGA-II in subsequent evaluation. At the initial stage of algorithm evaluation, AAM-NSGA-II performed worse than NSGA-II on ZDT4, ZDT5 and ZDT6 test problems. However, after the algorithm is evaluated to a certain extent, for ZDT5 and ZDT6, the performance of AAM-NSGA-II is 3.8652e -1 -9.21e -4 3.8732e -1 7.05e -4 ? / -/ = 0/5/1 Fig. 2 Convergence characteristics of HV mean of NSGA-II and AAM-NSGA-II on ZDT series test problems An adaptive adjacent maximum distance crossover operator for multi-objective algorithms 7431 very similar to that of NSGA-II, and the performance of AAM-NSGA-II is slightly better than that of NSGA-II algorithm, and the final performance of the former in ZDT4 test problem is much better than that of NSGA-II algorithm.
In order to further prove that AAM-NSGA-II algorithm can obtain solutions with better diversity and convergence, Table 4 gives the average value and standard deviation of HV index of NSGA-II and AAM-NSGA-II algorithm in 20 independent runs on ZDT benchmark problem. Among them, the HV index value obtained by NSGA-II algorithm on ZDT3 test problem is better than that obtained by AAM-NSGA-II algorithm, but according to the result of Wilcoxon signed rank test as '' = '', there is no difference in performance between the two. In other test problems, the performance of AAM-NSGA-II algorithm is better than the former. Figure 2 shows the convergence characteristics of HV mean of two algorithms on ZDT series test problems. The comparison results on ZDT1, ZDT2 and ZDT5 show that the HV index value of AAM-NSGA-II is always better than that of NSGA-II, and the improvement effect on ZDT5 test is very good. The comparison results of the two algorithms in zdt4 and zdt6 show that the HV index value obtained by MaF2 11 2.7264e -3 -7.96e -5 2.6557e -3 9.25e -5 MaF3 11 7.1849e ? 1 = 6.05e ? 1 7.6480e ? 1 8.49e ? 1  AAM-NSGA-II in the middle of algorithm evaluation is slightly worse than that of NSGA-II, and the HV index value at the end of evaluation is better than the latter. However, the performance of AAM-NSGA-II on ZDT3 test problem is not as good as NSGA-II. Based on the above comparative analysis of NSGA-II and AAM-NSGA-II algorithms on IGD and HV indexes, it can be concluded that the adaptive adjacent maximum distance crossover operator has more advantages than the crossover operator with fixed crossover rate and random mating selection. The improved AAM-NSGA-II algorithm can obtain solutions with good convergence and diversity on most ZDT problems.
In addition, this paper also carried out a comparison experiment between the improved NSGA-II algorithm and the basic NSGA-II algorithm on other test problems. Tables 5 and 6 show the comparison results of IGD and HV indexes of the two algorithms. The objective number of all test problems in the table is 2, and dimension D is given in the table. Set the number of evaluations to 20,000. From the index comparison results, it can be seen that AAM-NSGA-II algorithm can get better results than NSGA-II algorithm in most test problems. From the comparison results of IGD indicators, except that there is no difference between the results of the two algorithms on DTLZ3, 4 and MaF3 test problems, the other results show that the performance of the improved algorithm is better. From the HV index comparison results, the performance of the two algorithms on DTLZ4, MaF2 and MaF3 test problems is similar, and the comparison results of other test problems show that the performance of the improved algorithm is better.
6.2 Application of adaptive adjacent maximum distance crossover operator on MOEA/D Table 7 shows the IGD mean and standard deviation of the two algorithms. On the whole, AAM-MOEA/D shows advantages in ZDT1-ZDT5 test problems, and it can be seen that the introduction of adaptive near maximum distance crossover operator greatly improves the performance of the algorithm in ZDT2 and ZDT4 test problems. According to Wilcoxon signed rank test, their performance in ZDT5 test is very close. Only in ZDT6 test, the performance of MOEA/D is better than that of AAM-MOEA/ D algorithm. Figure 3 shows the convergence characteristics of the IGD mean value evaluated 20,000 times by MOEA/D and AAM-MOEA/D algorithms on ZDT1-ZDT6 test problems. By observing the images, it can be seen that the performance of AAM-MOEA/D on ZDT1-ZDT5 is better than that of MOEA/D algorithm, and the performance on ZDT2 is always much better than that of MOEA/D algorithm. However, according to the convergence curve of ZDT6 test problem, although the performance of the algorithm proposed in this paper is better than MOEA/D in the early stage of algorithm evaluation, it is not as good as MOEA/D algorithm in the later stage of evaluation. By observing the convergence curve of IGD index of the algorithm on ZDT5 test problem, it can be seen that the standard MOEA/D algorithm cannot converge on this kind of problem. However, it can be seen that the introduction of adaptive adjacent maximum distance crossover operator reduces the IGD index value of the algorithm on the ZDT5 test problem and improves the performance of the algorithm. Table 8 shows the mean and standard deviation of 20 independent runs of the HV index obtained by MOEA/D and AAM-MOEA/D on ZDT1-6 test problems. From the analysis of Table 8, it can be seen that the results obtained by the AAM-MOEA/D algorithm in ZDT1, ZDT2, ZDT4 and ZDT5 are better than the algorithm before the improvement. AAM-MOEA/D has similar results to MOEA/D on the ZDT3 test problem with discontinuous Pareto front. However, the optimal HV target value is not obtained on the ZDT6 test problem with uneven Pareto front, which shows that the performance of the AAM-MOEA/D algorithm is not excellent on such problems. Figure 4 shows the HV index convergence curves of MOEA/D and AAM-MOEA/D algorithms evaluated Tables 9 and 10 show the performance indexes of MOEA/D algorithm and AAM-MOEA/D algorithm on some test problems. The comparison results of IGD indicators show that there is little difference between the two algorithms in IMMOEA_F1 test. In addition, AAM-MOEA/D algorithm has better performance on test problems except DTLZ2 and MaF2. HV index results show that AAM-MOEA/D algorithm performs worse than MOEA/D on DTLZ2 and MaF2, and has similar performance on DTLZ3 and IMMOEA_F1. In addition, the performance of AAM-MOEA/D algorithm is better on other test problems.

Conclusion
Aiming at the defects of fixed crossover rate and random mating selection in most algorithms, in order to improve the convergence and diversity of the algorithm, this paper makes the following improvements to the crossover operator: an adaptive mechanism is adopted in the crossover operator, and an appropriate crossover probability is adopted in different stages of algorithm iteration. The adjacent maximum distance crossover strategy is introduced to select the individual with the largest difference among the adjacent individuals as the crossover parent, so as to enhance the ability of genetic operators to generate excellent solutions. Finally, the improved crossover operator is introduced into NSGA-II algorithm and MOEA/D algorithm. The index data and image comparison results on the ZDT test problem show that compared with NSGA-II algorithm and MOEA/D algorithm using simulated binary crossover, AAM-NSGA-II algorithm and AAM-MOEA/D algorithm introducing adaptive adjacent maximum distance crossover operator overcome the defects of random mating selection and fixed rate crossover operator. The improved algorithm can obtain better offspring, obtain more 6.8954e -1 -1.62e -2 6.9353e -1 4.55e -3 ZDT2 2.6103e -1 -6.95e -2 3.6602e -1 5.60e -2 ZDT3 5.9732e -1 = 4.29e -2 5.9292e -1 2.76e -2 ZDT4 5.7792e -1 -7.32e -2 6.2171e -1 4.92e -2 ZDT5 7.3786e -1 -6.93e -3 7.5266e -1 1.10e -2 ZDT6 3.7129e -1 ? 5.13e -3 3.2929e -1 1.26e -2 Fig. 4 Convergence characteristics of HV mean of MOEA/D and AAM-MOEA/D on ZDT series test problems appropriate crossover probability in different stages of algorithm iteration, and effectively speed up the convergence speed of the algorithm.
For future work, we should try to improve another important operation of genetic operator: mutation operator. Or the adaptive adjacent maximum distance crossover strategy proposed in this paper is applied to mutation operator and crossover operator at the same time to further study the performance of this strategy. And introduce the improved strategy into more types of algorithms to study the generality of the strategy. Finally, it is necessary to evaluate the performance of AAM-NSGA-II and AAM-MOEA/D in practical application. 2020JC-44. Data availability Enquiries about data availability should be directed to the authors.

Declarations
Conflict of interest The authors declare that there are no conflicts of interest regarding the publication of this paper. ? / -/ = 2/8/1 ? / -/ = 2/7/2 An adaptive adjacent maximum distance crossover operator for multi-objective algorithms 7437