Opposition-based learning multi-verse optimizer with disruption operator for optimization problems

Multi-verse optimizer (MVO) algorithm is one of the recent metaheuristic algorithms used to solve various problems in different fields. However, MVO suffers from a lack of diversity which may trapping of local minima, and premature convergence. This paper introduces two steps of improving the basic MVO algorithm. The first step is using opposition-based learning (OBL) in MVO, called OMVO. The OBL aids to speed up the searching and improving the learning technique for selecting a better generation of candidate solutions of basic MVO. The second stage, called OMVOD, combines the disturbance operator (DO) and OMVO to improve the consistency of the chosen solution by providing a chance to solve the given problem with a high fitness value and increase diversity. To test the performance of the proposed models, fifteen CEC 2015 benchmark functions problems, thirty CEC 2017 benchmark functions problems and seven CEC 2011 real-world problems were used in both phases of the enhancement. The second step, known as OMVOD, incorporates the disruption operator (DO) and OMVO to improve the accuracy of the chosen solution by giving a chance to solve the given problem with a high fitness value while also increasing variety. Fifteen CEC 2015 benchmark functions problems, thirty CEC 2017 benchmark functions problems and seven CEC 2011 real-world problems were used in both phases of the upgrade to assess the accuracy of the proposed models.


Introduction
Population-based algorithms and local search-based algorithms are examples of metaheuristic algorithms Altabeeb et al. 2021). Particle swarm optimization (PSO) Kennedy and Eberhart (1995) and ant colony optimization (ACO) Dorigo et al. (2006) are examples of population-based algorithms that work on a group of solutions at a time (Hassan et al. 2021). At each iteration, the best solutions were often used to add one or more new solutions. Hill climbing is one of the most popular local search-based B Mohammad Shehab moh.shehab12@gmail.com 1 Faculty of Computer Sciences and Informatics, Amman Arab University, Amman 11953, Jordan 2 algorithms . While local search-based algorithms focus on one solution and look to improve it using neighborhood solutions (Shehab et al. 2017a), hill climbing is one of the most common local search-based algorithms (Koziel and Yang 2011). The only disadvantage is that they choose to concentrate on exploitation over discovery, which raises the chances of being trapped in local optima (Abualigah et al. 2017). These approaches are suitable for finding promising areas in the search area. However, they are not so good at leveraging the search space field that is searched (Shehab et al. 2017b). Population-based approaches are classified as evolutionary computing and swarm intelligence (Abdelmadjid et al. 2013;Şahin and Abualigah 2021). Both approaches are focused on the natural biological evolution of natural creatures' social interaction behavior. Particle swarm optimization is an example of a swarm-based algorithm (PSO) Kennedy and Eberhart (1995).
This work highlights a recent stochastic population-based algorithm, called multi-verse optimizer (MVO) Mirjalili et al. (2016). MVO is bio-inspired from the multi-verse theory in physics, where it includes unique concepts models such as white, black, and wormholes. These concepts increase the ability of the exploration and exploitation search mechanisms, as well as the diversity. Table 1 presents a brief description of the comparison between MVO and popular metaheuristic algorithms, though MOV suffers from slow searching, local minima and premature convergence Mirjalili et al. 2017).
Therefore, there are several methods applied to solve these drawbacks including the enhanced MVO proposed in ADHIM et al. (2019); the authors improved the basic MVO by introducing a new version, called EMVO, to achieve high accuracy and efficiency of the requirement prioritization. EMVO is based on exchanging the information between the current solutions. EMVO starts by calculating the warm hole probability. Then, it calculates the rate of the traveling space. However, EMVO suffers from minimum simplicity, so it needs the maximum number of functions. The results illustrated that the EMVO achieved better accuracy of the solutions comparing with different existing techniques. Jangir et al. (2017) proposed a new technique, that is, hybridization particle swarm optimization (PSO) and multiverse optimizer (MVO), called HPSO-MVO, to determine the optimal reactive power dispatch (ORPD). The aim of HPSO-MVO is taking advantage of PSO to enhance the exploitation search, as well as the MVO to improve the exploration search in the uncertain environment (Shehab et al. 2016). The speed and locations of the particles are updated based on the position of the universe in each repetition. The experiment results show that the HPSO-MVO is outperformed the original MVO and PSO algorithm. Geng et al. (2019) proposed an improved multi-objective multi-verse optimizer (IMOMVO) to find the optimal solution of the reentrant hybrid flow shop scheduling problem (RHFSP) with maximum tardiness, objectives of makespan and idle energy consumption. IMOMVO includes Latin hypercube sampling (LHS) for the initialization of the population. Then, it updated the position of candidate solutions based on Levy flight. Moreover, it utilized logical selfmapping to chaotic local search. The authors proved the performance of IMOMVO using a set of benchmark functions and comparing it with various techniques such as MOMVO, MOPSO, NSGA-II and MOALO. Hu et al. (2016) employed the Levy flights with MVO, namely, LFMVO to find the optimal solution of the numerical and engineering optimization problems. The aim of utilizing the levy flights is to enhance the exploration search in the large-scale search space. The experiment results showed that the convergence speed and the quality of the solutions of LFMVO are outperformed various algorithms in the literature.
However, the drawback of these proposed methods is that they take into account just one factor to be enhanced (i.e., diversity, local minima and premature convergence). Also, most of the approaches that utilized an enhanced MVO algorithm contain more parameters that must be tuned. Therefore, the weaknesses of MVO still didn't solve, perfectly. Thus, an alternative enhanced method for the MVO algorithm should be introduced to cover all limitations, completely.
This paper highlights the two main weaknesses recognized in the performance trajectory of the basic version of the MVO: diversity of the population, trapping of local minima and premature convergence. Because of these weaknesses, MVO requires further refinements, to be enhanced the procedure search process of the basic MVO and combinations with other techniques. The following points are summarized the main contributions of this work.
-A new version including MVO and opposition-based learning is developed to improve the premature convergence problem by optimizing the initial population. -Applied the disruption operator to enhance the diversity of the population and the exploration ability. -Evaluating the performance of the proposed algorithm by comparing it with other published methods in the literature using CEC 2015 benchmark function problems, CEC 2017 benchmark function problems and CEC 2011 real-world problems.
The following is how the sections of this paper are structured. Section 2 displays some similar works. The opposition-based learning (OBL) technique, the interruption operator (DO) and the multi-verse optimizer (MVO) algorithm are all introduced in Section 3. After that, in Sect. 4, the proposed methods are listed. The experimental findings and discussions are presented in Sect. 5. Section 6 concludes with recommendations for the future.

Related works
To improve the efficiency of the simple Krill Herd (KH) algorithm, Wang et al. (2016) used opposition-based learning (OBL), position clamping (PC) and Cauchy mutation (CM). OBL speeds up the method's convergence, while PC and heavy-tailed CM aid KH in escaping local optima. The results show that among various OKH variants, the KH with OBL, CM and PC operators has the best score.  used dynamic topology and biogeography-based optimization to solve computational optimization problems based on learning to improve elephant herding optimization (EHO). When compared to other algorithms in the literature, the proposed algorithm proved to be more effective. It also outperformed the competition in the simple traveling salesman dilemma (TSP).
Suffer from premature convergence Guo et al. (2013) Rapidly converged Zhang and Sun (2002) Strength Balance between exploration and exploitation Hussien et al. (2020) Balance between intensification and diversification Abualigah et al. (2020a) Deal with the complex fitness landscape Hussien et al.
Evaluation is relatively expensive Zingg et al. (2008) Suffers from partial optimism Bai (2010) Get stuck on local optima Milad (2013) Needs huge memory resources Kulturel-Konak et al. (2003) The Cuckoo search (CS) algorithm has been improved in version . The developers in  used I-PKL-CS or self-adaptive intelligence learning. By combining the power of human knowledge learning and community knowledge learning with a threshold statistics learning approach, the proposed algorithm achieves a strong balance between discovery and manipulation. I-PKL-CS is a competitive new form of the algorithm, according to the findings.
The authors used the hierarchical step size cuckoo search algorithm and the Q-Learning step size and genetic operator in . (DMQL-CS). The DMQL-CS algorithm uses a step size management technique to analyze the individual multi-step evolution effect and learn the individual optimum step size by estimating the Q function value.
Using the opposition-based learning (OBL) strategy and OBSSA, Li et al. Hussien (2021) improved the Salp Swarm Algorithm (SSA) shortcomings, which can fail in avoiding local optima and have a sluggish convergence curve. The proposed algorithm was divided into two stages: the first stage used OBL to improve the initialization stage. The second stage used OBL in the population updating phase in each iteration. The findings show that OBSSA can compete with other algorithms in the literature in terms of efficiency.
Using the opposition-based learning (OBL) strategy, dubbed OBSSA, Hussien (2021) improved the shortcomings of the Salp Swarm Algorithm (SSA), which may fail to resist local optima and have a sluggish convergence curve. The proposed algorithm was divided into two stages: the first stage used OBL to improve the initialization stage. The second stage used OBL in the population updating phase in each iteration. The findings show that OBSSA can compete with other algorithms in the literature in terms of efficiency. Feng et al. (2018) implemented Monarch Butterfly Optimization (MBO) with Gaussian perturbation using opposition-based learning (OBL) (OMBO). The OBL technique is used on half of the population in the late stages of evolution. In contrast, Gaussian perturbation is used on the part of the population with low fitness in each evolution. OBL ensures OMBO's faster convergence, and Gaussian perturbation prevents OMBO from being trapped at a local optimum. The OMBO generated high-quality solutions, according to the findings of the experiments.
To solve the premature convergence and fragile balance between manipulation and discovery of the CS algorithm, named TOB-DCS,  used Taguchi oppositionbased scan. The Taguchi search strategy generated random generalized learning based on opposing relationships to improve the algorithm's exploration ability. The dynamic evaluation strategy reduced the number of function evaluations while also speeding up convergence. As a result, the TOB-DCS could strike a good balance between discovery and extraction, as shown by the results.

Preliminaries
This section presents a brief introduction for the oppositionbased Learning (OBL) strategy, disruption operator (DO) and the multi-verse optimizer (MVO) algorithm.

Opposition-based learning
Opposition-based learning (OBL) mechanism represented is finding the opposite solution for each current solution in the population. After that, it is using their fitness function to select the best solution (ElAziz et al. 2017).

Opposite number
The basics of the OBL method determined by assuming x ∈ [a, b] (a and b refer to the upper boundary and the lower boundary of the problem, respectively) and a, b ∈ R. Thus, the opposite point of x indicated byx op can be acquired as Tizhoosh (2005): The following subsection shows determining the opposite numbers with a higher dimension.

Opposite vector in d-dimensional
In case, opposite points d-dimensional space need to be computed as The calculation process is shown in following equation (Rahnamayan et al. 2008): where a j and b j refer to the lower and the upper boundaries of the element x j ∈ x, respectively. Finally, in the optimization method, the current solution x is chosen if f(x) is better than f (x op ) ; otherwise, x is chosen. So, the population of the solutions is updated based on the best values of x andx op .

Disruption operator
The disturbance operator (D op ) was inspired by the astrophysics effect, which it summarizes in "When the total mass (m) of a swarm of gravitationally bound particles is too close to a large object (M), the swarm will be broken apart." Similarly, when gravitational forces pull together a rigid body, it approaches a much larger entity harwit2006astrophysical. The full use of the D op is defined as the following equation (Liu et al. 2014): where Dis i, j indicates the Euclidean distance between two components, i t h and j t h. The Ψ ( a, b ) refers to a random number that is uniformly distributed located in [a, b]. Consequently, when the two components are too close together, then D op < 1, so the components will converge toward the origin.

Multi-verse optimizer (MVO)
MVO is one of the efficient metaheuristic algorithms proposed by Mirjalili et al. (2016) in 2016. MVO is inspired by the big bang theory that led to the birth of a universe. The algorithm and their mathematical models have been established based on three main concepts: white hole, black hole and wormhole. As many proposed metaheuristic algorithms, MVO focused on the exploration process using the white hole and black hole concepts, as well as the exploitation using the wormhole concept. The following steps describe the main operations of the MVO algorithm.

Population Initialization
The population is a subset of solutions, in MVO each candidate solution is called a Universe where represent as a vector of actual elements. For instance, where i refers to the number solutions in the population and d refers to the problem's dimension ). The following matrix shows a group of solutions in a certain population P.
where n refers to the number of solutions in the population P i .

Exploration
As mentioned previously, the white and black hole controls the exploration process in MVO. The mechanism of exchanging between the universes starts by calculating the fitness value which is called inflation rate for each universe. After that, the white hole (i.e., solution x k ) is selected using the fitness proportionate selection scheme. Equation 5 describes the exchange technique between the chosen universe x k and the black hole (i.e., solution x i ).
where j refers to the number of the elements in the universe, r1 is a random number locates between [0,1] and N 1(U i indicates the normalize fitness value of the i th universe. Equation 6 presents the normalization method for fitness value. where A i and A oi refer to the i th element of the vector after standardization and before standardization, respectively. n indicates the number of elements in the universe. It is worth to mention that the MVO algorithm doesn't utilize the normalization in the fitness proportionate selection scheme. The normalization is used the fitness value based on the decision in Eq. 5. Consequently, since the value of r1 locates between [0,1], the normalization of the fitness value should be located in the same range (i.e., [0,1]).

Exploitation
The wormhole in the MVO algorithm is responsible for the exploitation process in the search space. Equa-tion 7 represents the mechanism of the wormhole's utilization.
where X j indicates the j th solution of the best universe; lb j and ub j refer to the lower bound and upper bound of the j th element, respectively. r2, r3 and r4 are random numbers that fall in the ranges of [0, 1]. The transmission processes from discovery to exploitation are regulated by the traveling distance rate (TDR) and the wormhole presence likelihood (WEP). The WEP and TDR change their values over the direction of the iterations, as seen in Eqs. 8 and 9 .
where l refers to the current iteration, L refers to the maximum number of iterations, min is the minimum possible values of WEP where the common value is 0 and max is the maximum possible values of WEP where the common value is 2. p indicates the exploitation intensifier ( Fig. 1). It can be noticed that in Eq. 8, the WEP's value increases in each iteration. Thus, that enhanced the exploitation process, while in Eq. 9, the value of TDR reducing which affects the local search process in each iteration. The pseudo-code of the MVO algorithm is shown in Algorithm 1.

The proposed method
This section presents two new methods for enhancing basic MVO.

Opposition-based learning and multi-verse optimizer
This section presents the first improvement which combining the basic MVO with the OBL to enhance the ability of MVO in the exploration search and achieve the optimal value, rapidly. The proposed version is called OMVO. The following points show the two main stages of the OMVO: 1. Initial stage: The OMVO starts by initializing the population using the MVO strategy (i.e., . The OBL is utilized to find the opposite solution for each actual element (solution) in the vector. After that, it calculates the fitness function each solution Then, it determines the best N solutions from the union of the two populations (i.e., the actual and the opposite solutions).

Updating stage:
This stage presents the updating and evaluating the solutions based on Eq.6. Then, it determines the best solution and its fitness value. In contrast, the OBL mechanism keeps a group of the updated solutions from the MVO and calculating their opposite fitness values. After that, the best solution will be compared with the OBL solutions and the OMVO will choose the best to update the population in the next iteration. These processes will repeat until achieving the maximum number of iterations (i.e., stop criterion).

Improving OMVO using disruption operator
This section shows enhancing both the diversity and the convergence rate of the OMVO using the D op which is called OMVOD. The proposed algorithms are applied to improve its exploration and exploitation capabilities ). Thus, the disruption operator in OMVOD is depended on the distances between the elements at the given time in the search space. Therefore, the D op will be applied if the ratio of distances between the element i and the nearest element, and the ratio of distances between the element i and the start element is lower than the threshold C, see the following equation. where R i,nbd refers to the Euclidean distance between i th element and it's nearest element, R i,best is refers to the Euclidean distance between i th element and star element, C is defined as the following equation.
where the θ is a parameter that is either a constant or a function of generations; the GenIndex and MaxGen indicate the generation index and the maximum number of generations, respectively. The position of every element that satisfies Eq.10 will update according to the following equation: where U (-0.5, 0.5) returns a uniformly distributed pseudorandom number in the interval [-0.5, 0.5].
is the position of element i that should be disrupted, and ρ refers to a small number.

Experiments and results
In this section, comprehensive experiments are conducted using three sets of optimization problems to test the performance of the proposed algorithms (Abualigah et al. 2021a, b). These problems are divided into three main parts; Part 1: Experiments on CEC 2015, which contains fifteen benchmark functions (Shehab et al. 2017b). Part 2: Experiments on CEC 2017, which contains thirteen benchmark functions (Shehab et al. 2018). And Part 3: Experiments on CEC 2011 real-world problem, which includes seven real-world optimization problems .
The values of MVO's parameters are the same in Mirjalili et al. (2016). The maximum number of iterations is 1000, and the number of solutions N is set to thirteen with a dimension space value equal to the dimension of the given problem. Thus, all of the proposed algorithms were run thirteen times over each optimization problem for statistical analysis purposes. Furthermore, both experimental groups run on MATLAB 2016 running on Windows 7 on a PC with a Core 2 Duo processor and 16 GB of RAM. The below are the organizations and analyses of proposed approaches for these research problems:

Part 1: Experiments on CEC 2015
In this section, the proposed algorithms are tested. The findings of other published approaches in the literature are compared using CEC 2015 benchmark function problems. Table 2 contains the specifics of the CEC 2015 benchmark feature issues.
These benchmark functions are divided into four categories. The first is the Unimodal functions (from F1 to F2 in Table 2) used to evaluate the proposed algorithms' optimization accuracy and convergence speed ). In the given search field, these functions have a single final solution. Meanwhile, the Simple Multimodal Functions (from F3 to F9 in Table 2) test the efficiency of the proposed algorithms in evading the exploitation optimum and providing the global solution (Mohammad . The third form is Hybrid Functions (from F10 to F12 in Table  2), which evaluates the proposed algorithms' optimization efficiency and convergence acceleration. Finally, the Composition Functions (from F13 to F15 in Table 2) are the fourth kind. They are used to evaluate the optimization efficiency and convergence acceleration of the proposed algorithms.
Four versions of the proposed MVO are investigated using CEC 2015 benchmark function problems as shown in Tables 3-6 and Figure 2 (from A to O). These experiments are conducted to find the best vision from the given algorithms, which are Basic Multi-Verse Optimizer called BMVO, multiverse optimizer with disruption operator called MVOD, Opposition based Multi-Verse Optimizer called OMVO and Opposition-based learning Multi-Verse Optimizer with disruption operator called OMVOD. When the dimension space is set to 10, the efficiency of the proposed algorithms (BMVO, MVOD, OMVO and OMVOD) is shown in Table 3. The findings in Table 3 show that the proposed algorithm (OMVOD) outperformed other proposed methods (MVOD, OMVO) and the simple MVO in the vast majority of instances (BMVO). The suggested algorithm (OMVOD) generated the best results (F1 and F2). For four of the seven functions, the proposed algorithm (OMVOD) generated the best results in the second group (F4, F5, F8 and F9). In the third group, the proposed algorithm (OMVOD) outperformed the competition in three scenarios (all given functions). Finally, OMVOD achieved the best performance in the fourth group (F13 to F15). These research problems were used to assess discovery and extraction potential. We deduced from their findings that MVO's modification accomplished its primary goal of improving its efficiency.
The efficiency of the proposed algorithms (BMVO, MVOD, OMVO, and OMVOD) is shown in Table 4 when the dimension space is expanded to 30. The findings in Table 4 show that the proposed algorithm (OMVOD) outperformed other proposed methods (MVOD, OMVO) and the simple MVO in the vast majority of instances (BMVO). The suggested algorithm (OMVOD) generated the best results (F1 and F2). For seven out of seven functions, the proposed algorithm (OMVOD) generated the best results in the second group (all functions). In the third group, the proposed algorithm (OMVOD) outperformed the competition in two of three scenarios (F11 and F12).
OMVOD achieved the best performance in the fourth group (F13 to F15). These optimization problems were used to assess discovery and exploitation using an expanded dimension space (30). We deduced from their findings that the MVO modification, especially the proposed algorithm (OMVOD), achieved its primary goal of improving efficiency. The convergence behavior of the proposed algorithms on CEC 2015 test functions is seen in Figure 2(A-O). Furthermore, the findings seen in the sub-figures demonstrated  that the planned OMVOD conference to the best solution prevents premature convergence. The statistic rank test, Friedman rank test, for the proposed algorithms on CEC 2015 test functions with 10 and 30 dimension space (Dim) is shown in Table 5. The findings indicate that, in general, the suggested algorithms generated comparable results. Furthermore, when the dimension is set to 10, the proposed OMVOD received the highest mean rank value (17), followed by OMVO (35), MVOD (41) and BMVO (51). In addition, when the dimension is set to 30, the proposed OMVOD has the best mean rank value (16), followed by OMVO (35), MVOD (41) and BMVO (47). Furthermore, when the dimension value is set at 10, the proposed OMVOD has the highest final ranking value (ranked first), followed by OMVO (ranked second), MVOD (ranked third) and BMVO (ranked fourth) (ranked as the fourth). Furthermore, when the dimension value is set to 30, the proposed OMVOD has the highest final ranking value (ranked first), followed by OMVO (ranked second), MVOD (ranked third) and BMVO (ranked fourth) (ranked as the fourth), as shown in Table 5. When we merged the oppositionbased learning approach into the MVO with a disruption operator, we found that the best proposed variant is opposition-based learning multi-verse optimizer with a disruption operator called OMVOD. The second-best suggested approach, opposition-based learning with MVO (OMVO), showed that the opposition-based learning technique was better than the disruption operator. It gave the MVO more flexibility while maintaining its diversity of solutions.
When we merged the opposition-based learning approach into the MVO with a disruption operator, we found that the best proposed variant is opposition-based learning multiverse optimizer with a disruption operator called OMVOD. It outperformed other published approaches that used the same benchmark functions in the literature.

Part 2: Experiments on CEC 2017
In this section, the proposed algorithms are tested, and the findings of other published approaches in the literature are compared using CEC 2017 benchmark function problems. Table 7 shows the specifics of the CEC 2017 benchmark feature issues.
These benchmark functions are divided into four categories. The first is the Unimodal functions (from F1 to F3 in Table 7), which are used to evaluate the proposed algorithms' optimization accuracy and convergence speed. Furthermore, in the given search field, these functions provide a single final solution. Meanwhile, the Simple Multimodal Functions (from F4 to F10 in Table 7) are used to test the efficiency of the proposed algorithms in evading the exploitation optimum and   2015) Aydilek (2018) Aydilek (2018)

2,2565E+03
R a n k  providing the global solution. The third form is Hybrid Functions (from F11 to F20 in Table 7), which are used to evaluate the proposed algorithms' optimization efficiency and convergence acceleration. Finally, the Composition Functions (from F13 to F15 in Table 7) are the fourth kind, and they are used to evaluate the optimization efficiency and convergence acceleration of the proposed algorithms. When the dimension space is set to 10, Table 8 displays the efficiency of the proposed algorithms (i.e., BMVO, MVOD, OMVO, and OMVOD) on the CEC 2017 benchmark function problems. The findings in Table 8 confirmed that the proposed algorithm (OMVOD) outperformed other proposed methods (MVOD, OMVO) and the simple MVO in the majority of instances (BMVO). The suggested algorithm (OMVOD) generated the best results (F1 and F3).
In all cases (seven out of seven functions), the suggested algorithm (OMVOD) generated the best results in the second group (F4 to F10). The suggested algorithm (OMVOD) outperformed the competition in three out of three cases (all functions from F11 to F20). Finally, overall situations, OMVOD received the best performance in the fourth group (F21 to F30). These research problems were used to assess discovery and extraction potential. The performance of the proposed algorithms (i.e., BMVO, MVOD, OMVO and OMVOD) on CEC 2017 benchmark function problems is shown in Table 9 when the dimension space is set to 30.
The results shown in Table 9 are a continuation of the earlier results. The findings showed that the proposed algorithm (OMVOD) outperformed other proposed methods (MVOD, OMVO) and the simple MVO in every case (BMVO). We inferred from the results that adding the opposite dependent learning method to the MVO with the disruption operator achieved its key goals to improve its efficiency. On CEC 2017 benchmark function problems, Table 9 shows the efficiency of the proposed algorithms (i.e., BMVO, MVOD, OMVO and OMVOD) when the dimension space is set to 30. The results shown in Table 9 suggested that the proposed algorithm (OMVOD) outperformed other proposed methods (MVOD, OMVO) and the simple MVO in all situations (BMVO). In any case, in the first range, the suggested algorithm (OMVOD) produced the best results (F1-F3).
In all cases (seven out of seven functions), the suggested algorithm (OMVOD) generated the best results in the second group (F4-F10). The suggested algorithm (OMVOD) outperformed the competition in three out of three cases (all functions from F11 to F20). Finally, overall situations, OMVOD received the best performance in the fourth group (F21 to F30). These research problems were used to assess discovery and extraction potential. The performance of the proposed algorithms (i.e., BMVO, MVOD, OMVO and OMVOD) on CEC 2017 benchmark function problems is seen in Table 9 when the dimension space is set to 30.
According to the Friedman ranking test, the findings in Table 10 are devoted to the results seen earlier. Where the dimensions are equal to 10 and 30, the results showed that the proposed algorithm (OMVOD) outperformed other proposed methods (MVOD, OMVO) and the simple MVO (BMVO) in both situations. Therefore, we inferred from the results that adding the opposite dependent learning method to the MVO with the disruption operator achieved its key goals to improve its efficiency. Figure 3 depicts the distribution effects of the suggested processes. The distribution results for multiple functions from all groups are presented. The distributions of the obtained results by the proposed approach (OMVOD) are much better than the other comparative approaches, as seen in all sub-figures. Furthermore, all OMVOD's results have a high density near the optimum solutions, demonstrating that OMVOD's efficiency with these modifications is superior.
In every scenario, the proposed OMVOD produced better results than other related approaches (it got fourteen best cases out of thirteen). In general, the results suggest that the proposed algorithm is better for obtaining the best global  optimum solution. The statistic rank test, Friedman rank test, for the proposed algorithms on CEC 2017 test functions using 30 dimension space value is also seen in Table 11. The findings show that the suggested algorithms outperformed the competition. In addition, the suggested OMVOD received the highest mean rank value (2.13), followed by DVBA (3.10), GSA (3.46), GSO (4.23), PSO (4.63), HPSOFF (4.86), FA (6.46), FFPSO (8.00) and MVO (8.03). The proposed OMVOD received the highest overall ranking rating (ranking first), followed by DVBA (ranking second), GSA (ranking third), GSO (ranking fourth), PSO (ranking fifth), HPSOFF (ranking sixth), FA (ranking seventh), FFPSO (ranking eighth) and MVO (ranking ninth) (ranked the ninth). When we merged the opposition-based learning approach into the MVO with a disruption operator, we found that the best-proposed variant is opposition-based learning multiverse optimizer with a disruption operator called OMVOD. When contrasted to other published approaches in the literature that used the same benchmark functions, OMVOD produced better performance.

Part 3: Experiments on CEC 2011 real-world problems
Additionally, for the standard benchmark functions presented in the sections earlier, seven more real-world problems  -RWP01: optimal control of a non-linear stirred tank reactor -RWP02: spread spectrum radar polly phase code design -RWP03: large-scale transmission pricing problem -RWP04: dynamic economic dispatch (DED) problem -RWP05: hydrothermal scheduling problem -RWP06: hydrothermal scheduling problem -RWP07: hydrothermal scheduling problem Seven real-world problems were used to evaluate the proposed method's success and equate it to other related approaches from Wang (2018). To test the efficiency of many algorithms, these problems can be handled as continuous or discrete constrained problems. Five complementary approaches are used in this article to compare the findings of the proposed process. The population size is set to 30 for the experiments in this section. The maximum number of iterations is set to 1000. In Table 12, the results of 50 runs on seven RWPs are presented.
The proposed OMVOD algorithm outperformed the competition on six out of seven RWPs, as seen in Table 12 (except  RWP06). The BBO algorithm, on the other hand, just got the best solution on one out of seven RWPs. Furthermore, the suggested OMVOD algorithm obtained the mean best results on six out of seven RWPs in terms of mean best results (except RWP06). The BBO algorithm, on the other hand, had the best results on one out of seven RWPs. The consequence of the outcomes of these two scenarios is continuity. On five out of seven RWPs, the proposed OMVOD algorithm got the worst scores (except RWO02 and RWP06). The BBO algo-rithm, on the other hand, got the worst best results on one of seven RWPs (RWO02), while the ABC got the worst best results on one of seven RWPs (RWP06). Finally, on five out of seven RWPs, the proposed OMVOD algorithm obtained the best STD results (except RWO01 and RWP06). The BBO algorithm, on the other hand, had the better STD results on two out of seven RWPs. In general, as compared to other related approaches, the proposed OMVOD algorithm generated the best overall results by 22 times. We concluded that the proposed method (OMVOD) could solve the optimization problems from different disciplines efficiently, reflecting the effect of the opposite-based learning method and disruption operator in developing the optimization process to reach the optimal solution effectively.

Conclusions and future directions
Using the multi-verse optimizer (MVO) algorithm, this paper introduced new alternative approaches. Two key stages are included in the proposed methods: To improve its exploitation quest, the basic MVO is combined with opposition-based learning (OBL) in the first phase, resulting in OMVO. The disruption operator (DO) is examined in the second phase to increase the discovery search of the OMVO while preserving the variety of the solutions, which is referred to as OMVOD.
Twenty-five CEC 2015 benchmark functions problems, thirty CEC 2017 benchmark functions problems and seven CEC 2011 real-world problems are included in the experiments. The proposed algorithms' findings are comparable to those of many other algorithms that have been published in the literature. The best, average, and standard deviation of the fitness values is used to determine the efficacy of each algorithm. Compared to other related algorithms, the findings revealed that the OMVOD version is almost the best  Aydilek (2018) Topal and Altun (2016) Aydilek ( (2016) Aydilek (2018)    optimizer in all test problems. In summary, solving realworld problems showed that the proposed OMVOD has a promising potential to be very useful in solving structural design problems with unfamiliar search spaces. Furthermore, the suggested OMVOD encourages discovery and extraction phases to be balanced while maintaining solution diversity. It does, however, suffer from a late convergence flaw.
In future works, we will consider the performance of other algorithms for new hybrid versions. We will also utilize them in the different optimization problems and multi-objective problems to achieve better results. Furthermore, we will also apply the other representative computational intelligence algorithms such as Monarch Butterfly Optimization (MBO), Earthworm Optimization Algorithm (EWA), Elephant Herding Optimization (EHO), Moth Search (MS) Algorithm, Arithmetic Optimization Algorithm (AOA) and Aquila Optimizer (AO).

Author Contributions
The authors contributed to each part of this paper equally. The authors read and approved the final manuscript.

Funding
The authors have not disclosed any funding.
Data Availability Enquiries about data availability should be directed to the authors.