Opposition-based multi-objective whale optimization algorithm with multi-leader guiding

During recent decades, evolutionary algorithms have been widely studied in optimization problems. The multi-objective whale optimization algorithm based on multi-leader guiding is proposed in this paper, which attempts to offer a proper framework to apply whale optimization algorithm and other swarm intelligence algorithms to solving multi-objective optimization problems. The proposed algorithm adopts several improvements to enhance optimization performance. First, search agents are classified into leadership set and ordinary set by grid mechanism, and multiple leadership solutions guide the population to search the sparse spaces to achieve more homogeneous exploration in per iteration. Second, the differential evolution and whale optimization algorithm are employed to generate the offspring for the leadership and ordinary solutions, respectively. In addition, a novel opposition-based learning strategy is developed to improve the distribution of the initial population. The performance of the proposed algorithm is verified in contrast to 10 classic or state-of-the-arts algorithms on 20 bi-objective and tri-objective unconstrained problems, and experimental results demonstrate the competitive advantages in optimization quality and convergence speed. Moreover, it is tested on load distribution of hot rolling, and the result proves its good performance in real-world applications. Thus, all of the aforementioned experiments have indicated that the proposed algorithm is comparatively effective and efficient.


Introduction
During the past three decades, evolutionary algorithms (EAs) have aroused extensive attention in solving optimization problems as their great performance in adaptability and efficiency, especially in those problems that are difficult to deal with by conventional mathematical methods. Such algorithms construct evolutionary process by simulating natural phenomena and evolutionary mechanisms, which have various applications in science and engineering domains. Some classical algorithms include genetic algorithm (GA) (Holland 1992), particle swarm optimization (PSO) (Eberhart 1995), and differential evolution (DE) (Storn 1997). Some state-of-the-arts algorithms include artificial bee colony (ABC) (Karaboga 2005), Cuckoo search (CS) (Yang 2009), whale optimization algorithm (WOA) .
However, most optimization problems in science and engineering domains involve multiple objectives, which are more difficult to solve as the optimization objectives conflict with each other. Hence, there is no single solution to achieve optimum for all objectives simultaneously, because improving one of objectives may result in deterioration of others. In this situation, the solution for multiobjective optimization problems (MOPs) is a trade-off set, called Pareto solution set.
In 1967, Rosenberg proposed the idea of using evolutionary-based search to solve MOPs, but failed to realize. Since then, the research domain that solving MOPs with EAs (MOEAs) has gradually attracted wide attention.  Schaffer (1985) proposed vector evaluated genetic algorithms (VEGA), which combined GA with multi-objective optimization (MOO). In 1989, Goldberg proposed to combine Pareto theory in economics with EAs to solve MOPs, which is of great significance to the follow-up research. Fonseca (1993) proposed multi-objective genetic algorithm (MOGA) and the core idea is to sort the individuals in the population based on Pareto domination. Srinivas and Deb (1994) proposed non-dominated sorting genetic algorithm (NSGA), which classifies individuals into multiple layers based on the concept of non-dominated sorting. The improved version of the strength Pareto evolutionary algorithm (SPEA2) (Zitzler 2001) is proposed, which has excellent performance in evenness of solution set. Fast non-dominated sorting genetic algorithm (NSGA-II) is proposed by Deb et al. (2002), which is an improved version of NSGA. Fast non-dominated sorting is employed to grade the individuals and crowding distance comparison operator is employed to assess the distribution of individuals with the same ranking. NSGA-II has great performance in both convergence and distribution, which is one of the most famous MOEAs so far.  proposed improved NSGA-II, named NSGA-III, which introduces reference points to the NSGA-II framework.
Besides the above MOO based on Pareto domination, decomposition-based and indicator-based MOO are also popular frameworks. For the former, the problem is decomposed into multiple single-objective subproblems and those subproblems are parallel optimized. The selection operation is performed through aggregation functions rather than Pareto domination, such as multi-objective evolutionary algorithm based on decomposition (MOEA/ D) (Zhang 2007) and collaborative indicator-based operator selection for MOEA based on decomposition (MOEA/ D-IHO) (Maghawry 2020). For the latter, a certain indicator is adopted to calculate fitness, and the individuals are compared in terms of fitness, such as hypervolume-based many-objective optimization algorithm (HypE) (Bader 2011) and two-stage R2 indicator-based evolution algorithm (TS-R2EA) (Liu 2020).
With the development of MOEAs, increasing researchers have focused on solving the complicated MOP, such as many-objective problem (Yuan 2015), expensive multiobjective problem (Pan 2018), multi-model multi-objective problem (Yue 2017), dynamic multi-objective problem (Zhang et al. 2020a). Meanwhile, many researchers have paid attention to applying MOEA in real-world applications, such as trajectory optimization (Chai 2018(Chai , 2021, overlapping community detection (Tian et al. 2019a), maneuver planning (Chai 2020), etc. Moreover, there is some novel and interesting research. Zhang et al. (2020b) applied determinantal point processes to MOEAs and Chai (2017) combined MOEAs with game theory. Coello (2004) proposed multi-objective particle swarm optimization (MOPSO). The external archive is introduced to store and retrieve the Pareto solutions obtained. Two key operations are how to maintain the external archive and how to select the leader. For the first operation, comparing the dominance relations between new generation Pareto solutions and the archive members, and the non-dominated solutions are conserved. If the archive is full, the grid mechanism is triggered to divide the objective space into several sub regions and deleting the solution in the sub region with the most individuals. For another operation, the crucial thought is that the search agents are guided toward the relatively sparse region. Based on the criterion, the solution in the least crowded regions is prone to be selected as the leader. MOPSO provides a reference framework for other swarm intelligence algorithms to deal with MOPs. Since then, with the development of swarm intelligence algorithms, a variety of swarm intelligence algorithms have been extended to solve MOPs. Some of these well-known algorithms include multi-objective cuckoo search (MOCS) (Yang 2013), multi-objective bat algorithm (MOBA) (Yang 2011), multi-objective grey wolf optimizer (MOGWO) , multi-objective whale optimization algorithm (MOWOA) (Wang 2019), multi-objective multi-verse optimizer (MOMVO) (Mirjalili 2017).
However, the traditional way to apply swarm intelligence optimization algorithms to solve MOPs is selecting an individual from the external archive as the leader to guide optimization direction. In per iteration, more attention is paid to searching in certain subspace, which might cause a bad diversity. Thus, the multi-leader selection criterion is introduced to avoid the above problem in this paper.
In this paper, the main contributions are as follows: 1) A new Opposition-based learning (OBL) strategy is proposed to offer an appropriate solution when the point and the opposite do not dominate each other, which is more suitable for multi-objective optimization. 2) A novel selection framework of leadership set is proposed, and multi-leader solutions are selected by grid mechanism to guide optimization direction. 3) Hybrid evolutionary strategy is employed to generate offspring, which combines WOA with DE organically.
The remaining part is organized as follows. Section 2 addresses the related definitions of MOO and the standard OBL strategy. In Sect. 3, the proposed algorithm is presented in details. Section 4 presents numerical experiment and analysis of tests with a series of competitive algorithms on benchmarks. Section 5 presents the optimization instance of load distribution on hot continuous rolling. Section 6 draws the conclusions and introduces the future research directions.

Multi-objective optimization problems
Multi-objective optimization refers to the optimization of a problem with more than one objective function. The minimization problem is studied as an example in this paper. Without loss of generality, it can be formulated as follows: where the vector x ¼ ðx 1 ; x 2 ; . . .; x D Þ 2 S R D claims to the decision variables,S represents the decision space,D is the number of decision variables,½l i ; u i are the boundaries of the i th variable, the vector fðxÞ 2 X R M claims to the objective variables,X represents the objective space,M is the number of objective variables. A solution x S is called Pareto optimal solution iff: :9y 2 S; y 0 x.
Step 1 Step 3 Step 2 Step 4 Opposition-based multi-objective whale optimization algorithm with multi-leader guiding 15133 Definition 3 Pareto optimal set: The Pareto optimal set (PS) is defined as: PS ¼ fx 2 S :9y 2 S : j y 0 xg.

Opposition-based learning strategy
The initial position distribution of the population will affect the convergence speed and optimization efficiency. The OBL strategy was proposed by Tizhoosh (2006), which has been proved to be effective in the optimization field by an abundant literature in single-objective optimization problems (SOPs). The original solution and its opposite solution are completely symmetric in decision space, and there must be a solution closer to the optimal solution between both of them. The OBL strategy is employed to improve the distribution of the initial population when there is no prior knowledge, thus accelerating the convergence and improving optimization efficiency. The definition of the opposite position is given below: Opposite position: Given a point x ¼ ðx 1 ; x 2 ; . . .; x D Þ in decision space with x i 2 ½l i ; u i ; i ¼ 1; 2; . . .; D. The Opposite position of x is denoted by 3 The proposed algorithm

A novel opposition-based learning strategy
The original population is randomly generated in the decision space, then the corresponding opposite population is generated utilizing the OBL strategy. Compare the fitness between the original individual and the opposite in order. If x is better than x 0 , the original individual is selected. Otherwise, the opposite individual is selected. It is easy to select the better solution from the original individual and the opposite in the SOPs. However, in MOPs, it is difficult to select a better solution without adding extra computation when the original and the opposite do not dominate each other. In this case, a solution is commonly selected at random between the original and the opposite. Moreover, with the increase of objective variables, the probability of this phenomenon is increasing dramatically. In extreme cases, there are no dominant relationships between all the original solutions and the corresponding opposite solutions, and the OBL strategy is out of work and degenerated to the normal random initialization. An improved OBL strategy is addressed by Wang et al. (2016). The main idea is that dividing population into 2 parts, the former is generated randomly, while the latter is generated in terms of OBL. The improved method optimizes the distribution of the solutions but fails to eliminate the points with poor performance.
Borrowing the idea from the above, a novel OBL strategy is proposed for the initialization of MOO. The main procedure is given as follows: Step 1 The first half population P 1 is generated at random in the decision space.
Step 2 The rest half population P 2 is generated according to P 1 based on standard OBL strategy.
Step 3 Combing the P 1 and P 2 , then comparing the original individual and the opposite. If there is a dominant relationship between them, saving the non-dominated solution only. Otherwise, save both of them.
Step 4 Calculating the size of the rest population, then supplementary population is randomly generated to meet population size.
The simple explaining of the above procedure is shown in Fig. 1.
The novel OBL strategy is more suitable for MOPs. On the one hand, the initialization population with good convergence can be obtained, because the inferior solution is deleted between the original individual and the opposite. On the other hand, the distribution of population can be ensured, because the original solution and its opposite solution are completely symmetric in decision space.

Selection of leadership solutions and generation of offspring
The selection of the leadership solution is important for multi-objective optimization algorithm based on swarm intelligence, which affects diversity and convergence. In previous studies, only an optimal solution is usually selected from external archive as the optimization leader, which easily makes the solutions converge prematurely in a small area near the optimal solution in each iteration, thereby affecting the diversity of the solution set. Hence, in this paper, a multi-leader selection criterion is proposed. The main steps of selection criterion are given as follows.
First, calculating the minimum and maximum regarding the i th objective (denoted as f min i and f max i ) (only considering Pareto front solutions), and the upper and lower boundaries u i and l i are calculated as follows: where e is a small positive number, and e is 0.01. Each dimension is divided equally based on the upper and lower boundaries. The coordinates of an individual in i th objective are calculated as follows: where K represents the number of partition in each objective dimension, and K is 5 in this paper. : d e is the function of rounding up. Therefore, any solution in Pareto optimal front can be uniquely divided into a grid.
Finally, calculating the crowding distance of the Pareto front, and the individual with the largest crowding distance in each grid is selected to join the leadership set, as is depicted in Fig. 2.
For example, the P1, P2 and P5 are divided into the same grid in Fig. 2. Because of not a non-dominated solution, P5 is ruled out. Then, calculate the crowding distance of P1 and P2. P3 and P2 are adjacent to P1, while P1 and P4 are adjacent to P2. Thus, the crowding distance of P1 and P2 is X1 ? Y1 and X2 ? Y2, respectively. Obviously, the latter value is larger, so P2 is selected to add to the leadership set. The others may be deduced by analogy. A solution is selected from each grid with Pareto solutions and added to the leadership set. It is worthwhile to note that normalization should be executed before calculating crowding distance.
After the leadership solutions are selected by grid mechanism, the DE is employed to generate the offspring for the leadership solutions, while WOA is used to generate offspring for the ordinary solutions except for leadership solutions. For each ordinary agent, the individual is randomly selected from the leadership solutions as prey, and the ordinary agent is guided to conduct encirclement, exploration and hunting. If the number of Pareto solutions is more than 5, the above operation is executed, otherwise all the offspring are generated by DE. Opposition-based multi-objective whale optimization algorithm with multi-leader guiding 15135 DE algorithm includes 3 operators, namely mutation, crossover and selection. The mutation strategy of ''DE/ rand/1'' and binomial crossover are employed in detail.
The definition of ''DE/rand/1'' is as follows: where v i is the mutation vector.x r1 ,x r2 and x r3 are randomly selected parent vectors. r 1 ,r 2 and r 3 are distinct integers randomly selected from the range of 1; NP ½ , and NP is cardinality of population. F is a mutation parameter with the range of 0; 2 ½ , and usually F equals 0.5. The definition of binomial crossover is as follows: where u ij is the i th components of the offspring vector.randðÞ is a uniform random number with the range of 0; 1 ½ and generated independently for each j and each i. CR is a crossover probability with the range of 0; 1 ½ , and usually CR equals 0.5. k is an integer randomly generated with the range of 1; D ½ to ensure that at least one component derives from the mutation vector for offspring vector. D is the dimension of decision variables.

Whale optimization algorithm
In this part, the mathematical model of encircling prey, exploring prey and hunting prey is introduced.

Encircling prey
As the location of prey (the optimal position) is unknown, assuming that the current optimal agent is the target prey. After the target is defined, other search agents update the position according to the target prey. The mathematical model is described by the following equations: where t indicates the current iteration,X indicates the position vector,X Ã indicates the current optimal solution, A and C are coefficient vectors and they are calculated by the following equations: where r 1 and r 2 are random vectors with the range of ½0; 1,ã is linearly decreased from 2 to 0 as the iteration progressed. If A j j 1, search agents approach and encircle the prey, which is in the stage of local searching.

Exploring prey
The vector A decides the behavior of search agents in WOA. If A j j[ 1, search agents explore prey. The mathematical model is described as follows: where X rand represents the position vector of an agent selected randomly in current iteration. Unlike the phase of encircling prey, the position of the search agent is updated based on a random agent instead of the current optimal agent. With the increase of A j j, the agents explore prey in a broader space, which is in the stage of global searching.
3) Hunting prey. The humpback whales hunt along a spiral path, and the mathematical model is described as follows: where D 0 represents the distance from agent to the prey,b is a constant for defining the shape of the logarithmic spiral,l is a random number with the range of ½À1; 1.
To simulate those simultaneous behaviors, a probability p is used to choose between either encircling prey or hunting prey. Generally, p is 0.5. The mathematical model is described as follows: where r is a random number with the range of ½0; 1.

Elite selection
After the generation of offspring agents, combining the parent agents and offspring agents, then selecting specified number of individuals from the combining population based on the non-dominant sorting method and maximum crowding distance criterion. The elite selection strategy in this paper is similar to NSGA2, and the specific process is represented as follows: (1) The combined population is divided into several subsets with different ranking according to the nondominant sorting of individuals.
(2) Each subset is sequentially stored in the next generation population according to the ranking until the specified number is just exceeded. (3) The crowding distance of generation population in the last saved ranking is calculated, and the individuals with the minimum crowding distance are successively deleted until the population size reaches the requirements.
The main cycle procedure is shown in Fig. 3. Firstly, the leadership set is selected by the proposed selection criterion from the parent individuals. Next, the DE algorithm is employed to generate the offspring for the leadership solutions, while WOA is employed for the ordinary solutions. Then, combining the parent and offspring solutions. Finally, the parent set in the next iteration is selected according to non-dominant sorting and crowding distance sorting.
By combination of OBL, multi-leader guided strategy and hybrid evolutionary operator, the pseudo code of the proposed algorithm is provided as follows:

Motivation of improvement strategies
In the former part of this section, the proposed algorithm has been introduced in detail. In this part, we explain why these improvements are adopted. The advantages of the novel OBL strategy have been introduced in Sect. 3.1.
Compared to ordinary leader guiding mechanism, multileader guiding strategy based on gridding is more conducive to the searching process in sparse space, because search agents are divided into several teams and search for different space instead of paying all attention to the same place. Figure 4 depicts the schematic diagram of the two mechanisms mentioned above (It is worth noting that only the update position of the dominated solution is given for clarity). It is obvious that multi-leader guiding strategy is beneficial to maintaining the diversity of the population. In addition, the proposed algorithm does not need any additional external archive.
For evolutionary process, hybrid operators combined WOA with DE are adopted. WOA has an advantage in the exploitation process due to the existence of leaders, while DE is excellent in the exploration process. Thus, it helps to maintain the balance between exploration and exploitation by hybrid operators, which improves the capacity of convergence.
Algorithm: Multi-objective whale optimization algorithm based on multi-leader guiding (MOWOAMLG) Input: population size N, iterations Itermax. Output: Pareto set. 1: Begin 2: Generate initial population with the proposed OBL as showed in Section 3.1.

18:
End 19: End 20: Combine children solutions and parent solutions and select elite as showed in Section 3.4 .

21:
End 22: Return the elite in the last cycle.

23: End
Opposition-based multi-objective whale optimization algorithm with multi-leader guiding 15137

Computational complexity of MOWOAMLG
Computational complexity is defined as the total times of searches executed in per iteration, which is one of the important evaluation criteria for optimization algorithms. Population initialization. Generating half population and calculating the opposite population: oðNDÞ; Dominant relationship comparison: oðMNÞ; Population complement: oðNDÞ.
where N, D, M and K indicate the size of population, the dimensions of decision variables, the number of optimization objectives and the number of divisions of grid, respectively. There is no doubt that K is less than N. Hence, the overall computational complexity of MOWOAMLG is oðMN 2 Þ, which is identical to several famous algorithms, such as NSGA2, MOPSO.
All computational experiences are implemented on MATLAB R2018b under the window environment with an Intel core i5-9600 3.7 GHz processor and 16 GB RAM. The population size is set to 100. The algorithm stops after 10,000 times of function evaluations (FEs).

Performance metric
In order to assess the Pareto front obtained by different algorithms, 4 metrics are selected to quantify the performance.
a. Inverted Generation Distance (IGD) The metric describes the minimum distance from each individual in the true Pareto optimal solution set P Ã to the obtained Pareto solution set P. This measure is formulated as follows: where distðp; PÞ is the minimize Euclidean distance between a point p 2 P Ã and the solution set P. P Ã j j is the cardinality of P Ã . The metric can assess comprehensively both convergence and diversity. The lower the IGD value is, the better the obtained solution set is.

Fig. 5 continued
Opposition-based multi-objective whale optimization algorithm with multi-leader guiding 15139 The metric describes the approaching degree between the true Pareto optimal solution set P Ã and the obtained Pareto solution set P. This measure is formulated as follows: where distðp; P Ã Þ is the minimize Euclidean distance between a point p 2 P and the solution set P Ã . P j j is the cardinality of P. The smaller the GD value is, the closer the obtained solution set is to the true set, the better the convergence of the solution set is.
c. Spacing Measure (SP) The metric describes the evenness of the obtained solution set P. This measure is formulated as follows: j j is the cardinality of P. The lower the SP value is, the more even the distribution of individuals in the obtained solution set is.
d. Coverage Over the Pareto Front (CPF) The main idea of CPF is to reduce the dimensionality of the objective space through projection, then evaluate the diversity of the projected solution set. The metric evaluates the evenness and spread of a solution set simultaneously. The specific calculation rule can be found in the literature (Tian et al. 2019b). The upper the CPF value is, the more excellent the diversity of solution set is. In particular, the CPF value is 1, if the diversity of the solution set is extremely perfect.

Discussion of the results
The statistical results are provided to compare the performance with other comparison algorithms. Thirty random independent experiments are performed for each test algorithm under each test problem to weaken randomness. And the mean and standard deviation of the mentioned metrics are presented. Moreover, the Mann-Whitney test (also known as Wilcoxon rank sum test) is employed to compare the metric results obtained by the proposed algorithm and other comparison algorithms under a significance level of 0.05. '' ? '','' = '','' -'' represent the WOMOWMLG is better than, similar to and worse than the competition algorithm, respectively. And the result is statistically counted as '' w/t/l'', which signifies that compared to the corresponding competition algorithm, the proposed algorithm wins on w functions, ties on t functions and loses on l functions. In addition, the Kruskal-Wallis test is    The ranking evaluation is determined as follows: where, RSðAl metr i jp j Þ is the ranking score for the algorithm Al i in terms of the metr on the problem p j .KWðAl i Þ is the Kruskal-Wallis rank sum for the algorithm Al i .N is the times of repeated experiments.
Firstly, the results of the IGD are calculated. More attention should be paid to this metric because of the capacity to assess convergence and diversity simultaneously. Therefore, the boxplot of the IGD is presented in Fig. 5. It is convenient to analyze the distribution of results and identify outliers based on the plot. For reasons of space and clarity, only the corresponding boxplot for the top 5 algorithms are provided for each test problem.
The IGD result on bi-objective problems is shown in Table 1 and the best value for each test instance is highlighted in the form of bold. As per the statistics results on ZDT1, MOWOAMLG is better than all competitions except MOWOA, and the gap is tiny between MOWOAMLG and MOWOA. As for the ZDT2 and ZDT3, MOWOAMLG surpasses all other competitive algorithms. Noting that the best performance of IGD belongs to MOCS on UF1, UF2 and UF7, but MOWOAMLG outperforms the other algorithms and obtains the second rank. The mean statistics and Mann-Whitney test results show that MOWOAMLG exceeds other competitors on UF4, UF5 and UF6. And the corresponding boxplot in Fig. 5 shows MOWOAMLG is significantly lower and narrower than others on UF4. The best performance is achieved by SPEA2 on Kursawe, and MOWOAMLG obtained a similar result to GDE3, which outperforms other competitors. Although the optimal results on the Poloni and Schaffer1 are obtained by SPEA2, MOWOAMLG achieves competitive performance only next to SPEA2.
The IGD result on tri-objective problems is shown in Table 2. As per the results on DTLZ1, MOWOAMLG demonstrates the best effect. SPEA2 surpasses all other competitions on DTLZ2 and DTLZ5. Nevertheless, MOWOAMLG obtains competitive performance, which performs only worse than SPEA2, according to the Mann-Whitney test. As for the DTLZ4, MOWOAMLG is a little worse than GDE3, but the Mann-Whitney test shows that the performance of them is similar. MOWOAMLG outstrips all other algorithms on DTLZ6. It is noteworthy that NSGA2 is worse than MOWOAMLG according to the mean of IGD on DTLZ4, while the Mann-Whitney test shows that NSGA2 surpasses MOWOAMLG. The reason for the contradictory results can be revealed by the boxplot. There is an extreme outliers for NSGA2, which deteriorates the statistical results. Due to the same reason, MOWOAMLG is significantly better than the SPEA2 on the DTLZ4 and DTLZ6 problems based on the mean metric, while both of them are similar based on the Mann-Whitney test. The mean and variance statistics are strongly influenced by extreme outliers. Thus, it is unreasonable to compare algorithms just by calculating the mean and variance statistics, especially when the distribution of the results is seriously inconsistent with a normal distribution. Nonparametric statistics such as the Mann-Whitney test weaken the impact of extreme outliers because the test does not directly focus on the values, but rather compares the order of the data. As per the results on Viennet3, although Fig. 6 The ranking of IGD metric (bar represents mean, line represents standard deviation) Opposition-based multi-objective whale optimization algorithm with multi-leader guiding 15143  MOWOAMLG fails to receive the best results, its performance is only dominated by GDE3 algorithm. From the above analysis, it is easy to find that MOWOAMLG obtains competitive performance. To be specific, MOWOAMLG achieves the optimal results on 9 problems, obtains the second rank on 10 problems and the third rank on 1 problem. For the SPEA2 and MOCS, although they achieve the optimal results on several problems, respectively, they obtain unsatisfactory even awful results on certain problems. Figure 6 illustrates the statistical results of ranking score over all the test problems among 11 algorithms, and the bar graph and line graph describe the mean and standard deviation, respectively. Figure 6 clearly indicates that the top 5 algorithms are MOWOAMLG, GDE3, SPEA2, MOCS and NSGA2 in terms of IGD. According to the standard deviation of ranking score, MOWOAMLG and GDE3 have advantages in stability, while the SPEA2 and MOCS are weak.
The statistics of the GD are presented in Tables 3 and 4. This metric evaluates the capacity of the obtained solution set to approach the true Pareto front, namely the performance of convergence. From the mean of GD, MOWOAMLG and MOEAD obtain the optimal results on 4 of the 20 test problems, respectively, while NSGA2, SPEA2 and MOGWO achieve the best results on 3 test problems, respectively. In terms of the Mann-Whitney test, MOEAD has slightly better performance in convergence than MOWOAMLG. However, the stability of MOEAD is poor, mainly in the following two aspects: (1) Performance on the same test problem varies greatly. The standard deviation of MOEAD is larger, especially in Schaffer2, Poloni and DTLZ7. (2) Performance on different test problems varies greatly. MOEAD achieved good results on UF6, Viennet3 and others, while has the almost worst performance on ZDT series among all the competitors. For the Mann-Whitney test, as long as there are enough numbers of good results in MOEAD, no matter how bad the terrible results are, the overall performance of MOEAD is superior on this problem. But for the Kruskal-Wallis test, due to all competitors involved, the terrible results in MOEAD affect the ranking strongly. Thus, MOEAD is worse than MOWOAMLG and GDE3 in terms of ranking score, namely MOWOAMLG and GDE3 rank higher than MOEAD when considering all competitors. On the contrary, MOWOAMLG and GDE3 are more outstanding in stability. Figure 7 clearly indicates that the top 5 algorithms are MOWOAMLG, GDE3, MOEAD, SPEA2, and NSGA2 in terms of GD.
The statistics of the SP are presented in Table 5 and 6, which evaluates the evenness of the obtained solution set. From the mean of SP, the SPEA2 and MOWOA obtain the optimal results on 7 and 5 test problems, respectively, while MOWOAMLG achieves the best results on 4 of the   20 test problems. The results of the Mann-Whitney test reveal that MOWOA performs well on bi-objective problems but bad on tri-objective problems, while SPEA2 has good performance on bi-objective and tri-objective problems. Figure 8 clearly indicates that the top 5 algorithms are SPEA2, MOWOAMLG, MOWOA, GDE3 and MOBA. Although MOWOAMLG is slightly worse than SPEA2 in average performance, it is more stable. A solution set with good diversity means the set has advantages in both evenness and spread. Whereas, the SP metric only evaluates the evenness of a solution set. The statistics of the CPF metric are presented in Tables 7 and 8, which evaluate the evenness and spread of the obtained solution set simultaneously. From the mean of CPF, the SPEA2 and MOWOAMLG achieve optimal results on 8 and 6 test problems, respectively. Compared MOWOAMLG with SPEA2 by Mann-Whitney test, the former performs better on bi-objective problems, while the latter performs better on tri-objective problems. Figure 9 clearly indicates that the top 5 algorithms are MOWOAMLG, SPEA2, GDE3, MOBA and NSMFO in terms of CPF. It is worthwhile to note that the MOWOA achieves third rank in terms of SP, while obtains bad results on CPF, indicating MOWOA has advantages in evenness but fails to maintain the spread. In contrast, MOWOAMLG achieves second and first rank, respectively, in terms of SP and CPF, which are competitive in evenness and spread. Figure 10 depicts the obtained Pareto fronts on 10 selected problems. In order to save space, the top 5 rank algorithms (based on IGD metric) are selected to display. The front shape of ZDT2 is concave. It is clearly observed that SPEA2, MOCS and NSGA2 show poor convergence, and GDE3 fails to perform well in local evenness. As a contrast, MOWOAMLG provides an excellent Pareto front. The Pareto front of ZDT3 consists of 5 disconnected convex curves. The Pareto front obtained by MOWOAMLG is in accordance with the ideal front perfectly, which is distinctly better than MOCS. As for UF1, all the obtained Pareto fronts are poor in distributedness, but the fronts of MOWOAMLG and MOCS are relatively broader than others. The Pareto front of UF5 is composed of a few discrete points, which is difficult to solve. All the algorithms have bad performance in convergence and diversity, but MOWOAMLG is superior to others because MOWOAMLG succeeds in finding 2 points on the Pareto front, and trying to search other regions. The Pareto front of the UF7 is linear, which is easy to converge to the optimal front. All the algorithms are able to approach the Pareto front, but fail to cover a few parts. The true Pareto front of Schaffer2 has two discontinuous parts, and the problem is easy to solve. All comparison algorithms have satisfying results, but for SPEA2 most points are clustered in one part, which results in sparse distribution on the other part. The front shape of DTLZ1 is linear but with multiple local Pareto fronts, which makes it easy to converge to local optimum. The results show that only the solution set obtained by MOWOAMLG is able to converge perfectly to the global optimum, which indicates its obvious capacity in convergence. The front shape of DTLZ4 is concave, biased and non-uniform, and it tends to test the ability to maintain diversity for MOEAs. All the algorithms have a similar front except SPEA2. SPEA2 has poor stability on this problem, and the obtained front might cover the ideal front perfectly or converge to the front boundary, which also can Fig. 7 The ranking of GD metric Opposition-based multi-objective whale optimization algorithm with multi-leader guiding 15147 Table 5 The be discovered on the statistics of IGD and CPF. The front shape of DTLZ6 is a curve, and all the 5 algorithms have satisfactory performances on it, which converge to the ideal Pareto front perfectly. It is clearly observed that MOWOAMLG has obvious advantages in evenness. The front shape of DTLZ7 contains 4 disconnected surfaces, and the optimal solution set is also disconnected in decision space. It is used to test the capacity for maintaining the distribution. It is observed that 5 algorithms have similar performances, and MOWOAMLG is competitive in convergence, but has general performance in evenness.

Comparison of convergence speed
In addition to the performance of the solution set, convergence speed is also an important metric to evaluate an algorithm. In this section, the convergence curve of IGD is provided to compare the convergence speed. The convergence curves of the top 5 rank algorithms on 9 selected test problems are depicted in Fig. 11. The results show that compared to other competitors, MOWOAMLG has significant advantages in convergence speed on the ZDT1, ZDT3, DTLZ6 and DTLZ7, and has similar or slightly good results on other problems. The comparison result indicates that MOWOAMLG is able to obtain good Pareto front in cost of fewer optimization iterations.

Parameter study
In the proposed algorithm, the number of partition K is introduced to control the grid division, which has some influence on the performance of convergence and diversity.
Specifically, a small partition number may cause premature convergence due to the loss of leadership group diversity, while a large number may lead to carelessness search due to the imbalance between the leadership set and ordinary set. Hence, the sensitivity analysis of the parameter K is performed in this subsection. Furthermore, contrast tests are carried out at K 2 ½2; 10 on 8 selected test problems. Table 9 presents the IGD values of the proposed algorithm with different partition numbers. The results indicate that a variation of partition number results in difference in performance, but when K [ 3, the gap is in apparent, which shows that the proposed algorithm demonstrates robust performance to the setting of partition number. The corresponding partition numbers when obtaining the best IGD values on 8 problems are counted, and most of the best partition number lies in the interval between 4 and 6. Therefore, a partition number of 5 is suggested in this paper.  Opposition-based multi-objective whale optimization algorithm with multi-leader guiding 15149 Table 6 The result of SP metric on tri-objective problems

Time consumption statistics
In order to compare the time consumption of various algorithms, Table 10 lists the mean value of the time consumption on several selected problems, averaged over 20 runs. It can be found that MOEAD consumes the least time and MOGWO consumes the most time. Although the proposed algorithm is not superior in time consumption, considering the optimization performance, it is acceptable.

Application study
Load distribution is the core problem on hot continuous rolling, which directly affects the rolling stability and the shape quality of strip in the process of finishing rolling. Load distribution is a typical multi-objective problem, thus MOWOAMLG is applied to solve it in this section. Here, a certain product of Baosteel 1880 hot strip mill is studied as an example. The steel type is Q235B, the width of strip is 1244 mm, the thickness of the inlet strip is 42.57 mm, the thickness of the finished product is 3.38 mm and the target crown is 0.04 mm. The exit thickness of all passes h ¼ ðh 1 ; h 2 ; Á Á Á ; h nÀ1 Þ T are taken as the decision variable to optimize load distribution. The rolling energy consumption, load balance and strip shape control are selected as optimization objectives. The multi-objective model is described as follows: where h i is export thickness of i th pass;h 0 is thickness of inlet strip;n is the total number of finishing pass;N i indicates rolling power (kW) of i th pass;P i indicates rolling force (kN) of i th pass;c i indicates rolling force rate of i th pass;C i h reflects export crown (lm) of i th pass;D i indicates optimal adjustment;I i reflects motor current (A) of i th stand;P m and I m indicate the maximum rolling force and motor current, respectively.
The MOWOAMLG and 4 other competitors (including GDE3, SPEA2, MOCS, NSGA2) are employed to solve the load distribution problem. The maximum iteration and population size are set to 50 and 100, respectively. For ease of comparison, only rolling energy consumption (f1) and load balance (f2) are selected as optimization objectives in this section. The obtained Pareto optimal fronts is depicted in Fig. 12. Figure 12 clearly indicates that compared to other algorithms, MOWOAMLG and MOCS have prominent advantages in spread and evenness. Moreover, MOWOAMLG is slightly better than MOCS. In order to avoid randomness, 30 independent experiments are performed. The following 3 metrics are calculated to quantify the performance. a. Coverage metric (CM) Fig. 8 The ranking of SP metric Opposition-based multi-objective whale optimization algorithm with multi-leader guiding 15151 The metric denotes the dominance relationship of Pareto solution from two set, as follows: where Q j j is the cardinality of Q, if is true, the value of h i is 1, otherwise the value is 0. CMðP; QÞ ¼ 0 means no solution in Q is dominated by P, while CMðP; QÞ ¼ 1 means all solutions in Q are dominated by P.

b. Maximum Spread (MS)
This metric describes the spread of the obtained Pareto solutions, as follows: where M represents the number of objectives, f max i and f min i are the maximum and minimum in i th objective, respectively. The upper value denotes the better spread. Note that the value of objectives should be normalized to avoid the difference of orders of magnitude.

c. Inverted Generation Distance (IGD)
This metric has been described previously, but it is difficult to calculate the true Pareto front for the actual problem. Thus, the above five algorithms are employed to solve the problem with the maximum iteration of 100 and the population size of 500. Ten independent experiments are performed and the Pareto front solutions are selected from all solutions obtained from experiments based on non-dominant sorting. A population with the size of 150 is obtained based on circle crowded sorting (Cheng et al. 2017), which is regarded as the true Pareto front. Normalization is also carried out before this process.
The statistical results of the above metrics are provided in Table 11. The statistical results of CM show that MOWOAMLG is slightly better than MOCS and obviously superior to the other 3 algorithms. More than half of the solutions obtained by the 3 algorithms are dominated by the solutions obtained by MOWOAMLG, but only less than 6 percent of solutions from MOWOAMLG are dominated by the solutions obtained from 3 algorithms. Thus, MOWOAMLG is competitive in convergence capacity. The results of MS indicate that MOWOAMLG, MOCS and GDE3 have similar spread capacity, which is significantly superior to SPEA2 and NSGA2. The low value of the IGD indicates that the powerful performance of MOWOAMLG in solving this actual problem. Mann-Whitney test is performed for the obtained solution sets in terms of MS and IGD, and the results indicate that MOWOAMLG is better than other 4 comparison algorithms on these 2 metrics. Moreover, time consumption is also counted. The proposed  The tri-objective optimization problem of load distribution is solved by MOWOAMLG. The maximum iteration and the population size are set to 50 and 300, respectively. The obtained Pareto front set is demonstrated in Fig. 13a. Then, the solutions which dominate the empirical solution are selected in Fig. 13b. After normalization, the weighted fitness of three objectives is calculated with the weight of 5:3:2 and the solution with the minimal weighted fitness is selected as the optimization result. The comparison results of empirical solution and optimization solution are demonstrated in Table 12. In terms of reduction of each pass, the reductions of the optimization solution in F1 and F3 decrease, while other passes increase. In terms of the 3 optimization objectives, the rolling power of the optimization solution reduces, the distribution of rolling load is more balanced and the shape control is more suitable contrast with the empirical solution. In general, the optimization solution is better than the empirical solution, which is appropriate to be used in actual industrial production.

Conclusion
A novel multi-objective optimization algorithm named MOWOAMLG is proposed in this paper. In MOWOAMLG, a multi-leader guiding strategy has been suggested to improve the performance, which is easy to extend to other similar algorithms. The hybrid evolutionary operators containing DE and WOA are employed to achieve the balance between exploration and exploitation. In addition, an OBL strategy which is more suitable for multi-objective optimization is employed for population initialization. Experimental results on a variety of standard MOPs have demonstrated the promising performance of the proposed algorithm, in comparison with several classic or state-of-the-arts algorithms. Moreover, the proposed algorithm is successfully applied to the load distribution problem on hot continuous rolling, which further verifies the effectiveness in real-world applications.
In the future, we intend to extend the proposed algorithm to solve constrained multi-objective problems. How to effectively utilize the information of the solutions which fail to satisfy the constraint condition but perform well is the core of dealing with the constrained problems. Moreover, how to increase the selection pressure of the Pareto solutions in many-objective optimization is also the focus of the future work. Fig. 9 The ranking of CPF metric Opposition-based multi-objective whale optimization algorithm with multi-leader guiding 15155 Fig. 10 The comparison of the obtained Pareto optimal front Fig. 11 The comparison of convergence speed Opposition-based multi-objective whale optimization algorithm with multi-leader guiding 15157  Fig. 12 The comparison of the obtained Pareto optimal front on load distribution  Fig. 13 The obtained Pareto optimal front on load distribution