Abstract
Multi-verse optimizer (MVO) is a novel nature-inspired algorithm that has been applied to solve many practical optimization problems. Nevertheless, the original MVO has problems of low convergence speed and accuracy of final solutions. Besides, the failure to strike a balance between exploration and exploitation and the easiness of falling into local optimum in the early stages makes MVO hard to converge. In this paper, we propose a novel hybrid algorithm called Hybrid Queuing Search algorithm with MVO (HQS-MVO) by introducing Queuing Search Algorithm (QSA) and Metropolis rule to overcome these shortcomings. The introduction of QSA is to improve the accuracy of final solutions. At the same time, the Metropolis rule is employed to prevent the algorithm from falling into the local optimum, thus improving the convergence speed of the original MVO. Then, we compare the performance of HQS-MVO on 30 benchmark functions of CEC2014 and 10 benchmark functions of CEC2019 with the other four related algorithms and three latest algorithms. The results show that HQS-MVO has the most accurate solutions in most cases compared with other seven algorithms in most cases, and gains the lowest standard deviations. Moreover, we make convergence curve of the eight algorithms. Compared with other algorithms, HQS-MVO shows outstanding performances and converge faster in general. Finally, we apply the proposed algorithm in a real engineering optimization problem and compare its performance with other algorithms, the results show that HQS-MVO is still the best one in problem of designing of gear train.
Introduction
Multi-verse optimizer (MVO) is a novel bio-inspired meta-heuristic optimization algorithm proposed in recent years, which is applied extensively to global optimization problems. Compared with conventional mathematical methods, which are easily trapped into local optimum, meta-heuristic algorithms can strike a balance between exploration and exploitation. Besides, meta-heuristic algorithms are usually nature-inspired. For instance, Genetic Algorithm (GA) was proposed based on evolutionary theory [1]. Queuing Search Algorithm (QSA) was inspired by human behaviors in queuing [2]. Particle Swarm Optimization algorithm (PSO) was inspired by the process of predatory behavior of fish or birds [3]. Grey Wolf Optimization Algorithm (GWO) was inspired by the hunting behavior of grey wolves [4]. Simulated Annealing (SA) was proposed based on the phenomenon of solid annealing in physics [5]. Gravitational Search Algorithm(GSA) was an algorithm in which search agents work following Newton’s law of gravity [6]. Inspired by the intelligent behavior of the honey bee swarm, Artificial Bee Colony algorithm (ABC) was proposed as a swarm intelligence algorithm [7].
Utilizing the concepts of white hole, black hole and worm hole in the multi-verse theory, the MVO was proposed by Seyedali Mirjalili in 2016 [8]. The main idea of MVO is to exchange objects in each universe depending on its inflation rate and travel through the worm hole. The universe with the highest inflation rate tends to exchange objects with other universes through the white hole to help them get a higher inflation rate. The process of white/black hole exchange is exploration while travelling through worm holes in the process of exploitation. MVO has the advantages of few parameters, low complexity and simple coding. Using the theory of white/black hole exchange, MVO has a strong capacity for global search. Therefore, MVO is widely applied to many practical global optimization problems, such as data clustering [9], the optimal configuration of wind/ photovoltaic / storage grid-connected microgrid [10], multi-threshold color image segmentation [11], task scheduling [12] and so on. Han-Dong Jia et al. [13] introduced MVO in wireless sensor network to help maintain population diversity and avoid the problem and being trapped into local optimum. Sivakumar Pothiraj et al. [14] hybridized the original MVO to fit for Floor Planning of 3D IC Design. Nour El Yakine Kouba et al. [15] introduced MVO to help optimize the suggested fuzzy-PID controller parameters. Abdul Majid Hasan et al. [16] utilized MVO for low-frequency oscillations damping. Indrajit N. Trivedi et al. [17] also solved the problem of the modern power system through MVO. Hossam Hassan Ali et al. [18] proposed application of MVO to design model predictive control, hoping to fit for multi-interconnected systems. Heba Ahmed Hassan et al. [19] introduced MVO into distributed generation and distribution static compensator to strengthen the integration of them. Prakash Chandra Sahu et al. [20] applied MVO to design the optimal parameters of fuzzy tilt controller.
Although MVO has wide applications in many fields to search for the optimum, its exploitation capacity is not so strong, making it hard to gain accurate solutions [21]. Moreover, compared with some other meta-heuristic optimization algorithms proposed in recent years, the convergence speed of MVO is a little slower [22]. Therefore, plenty of work has been done to improve the convergence speed as well as the accuracy of MVO, most of which can be separated into three main aspects: the hybridization of MVO and some other algorithms (such as K-means clusters, PSO, and so on), the change of search strategy and the adjustment of control parameter formulations. Besides, a handful of works focus on the combination of these three aspects.
Many works have focused on hybrid algorithms based on MVO to handle various optimum problems. M.Talaat et al. [23] proposed a hybrid model of artificial neural networks and MVO to develop a forecasting system. Pradeep Jangir et al. [24] combined MVO with PSO to enhance the capacity of exploitation of MVO in uncertain environments. Thi-Kien Dao et al. [9] proposed a hybrid MVO and feedforward neural network model to avoid being trapped into local optima. WenHao Lai et al. [25] introduced density-based spatial clustering of applications with noise to enhance the exploration capacity of MVO. Wendong Tang et al. [10] combined MVO with Lévy flight and differential evolution-based multi-objective to find feasible solutions. Zuriani Mustaffa et al. [26] demonstrated a hybrid MVO-Least Squares Support Vector Machines for accurate prediction. Jian Zhou et al. [27] proposed optimization of the random forest by the application of MVO to better classification performance. Ben Cao et al. [21] proposed a K-means MVO to increase convergence speed in the early stages of iteration. M. Naveed Iqbal et al. [28] presented a combination of MVO with sequential quadratic programming to get better solutions. Mostafa Meshkat et al. [29] proposed a hybrid MVO with selection and crossover operator to improve the performance of MVO. Laith Abualigah and Muhammad Alkhrabsheh [30] combined MVO with GA to optimize task scheduling. Mohamed Abdel-Basset et al. [31] introduced overlapping detection phase and grid quorum-based positioning strategy at the end of every iteration in MVO, hoping to encode universes as sensors coordinates in a two-dimensional interest region. Ahmed Fathy et al. [32] proposed an algorithm in which MVO was introduced to identify the optimal parameters of proton exchange membrane fuel cell. Xiaoyu Wang et al. [33] combined MVO with Nash nonlinear grey Bernoulli model to select the optimal parameters. Yilmaz Omer et al. [34] utilized SA at the end of each iteration and proposed a mechanism named black hole selection to increase exploration and exploitation. Datta Samik et al. [35] introduced Fire Fly Algorithm (FFA) at the beginning of each iteration to strengthen the convergence speed of conventional MVO. Sharif Naser Makhadmeh et al. [36] solved power scheduling problem by combining MVO with GWO.
Aside from the hybrid model, some other works also focused on modifying the search strategy. Jian Lin et al. [12] introduced insertion improvement to help MVO find the best operation sequence. Sarah E. Shukri et al. [37] improved MVO by saving the best solution of each iteration and feeding them back to the algorithm as new solutions after a certain number of iterations. Ammar Kamal Abasi et al. [38] introduced the neighborhood selection strategy to find the best neighbor solution and help improve exploitation capacity. Ewees Ahmed A et al. [39] adopted chaotic theory to help initialize the population. Kaifeng Geng et al. [40] utilized Latin hypercube sampling technology, Lévy flight and logical self-mapping to further improve MVO. Jinkun Luo et al. [41] adopted a multi-level guidance mechanism to substitute the original search mechanism of MVO and a weighted mutation disturbance method to improve the robustness of the algorithm. Ling-Ling Li et al. [42] introduced chaotic sequences to initialize the population and sine-cosine algorithm to get a higher convergence speed.
Besides, there are also a few works focusing on the adjustment of control parameter formulations. Siyue Liang et al. [43] adjusted the Travel Distance Rate(TDR) of worm hole travelling. Julakha Jahan Jui et al. [44] introduced sine-cosine algorithm to improve the process of worm hole travelling together with TDR. Xiaoyu Wang et al. [45] proposed a self-adaptive algorithm for Worm hole Existence Possibility (WEP), and TDR was amended to avoid being trapped into local optimum. Urvashi Chauhan et al. [46] made some similar adjustments to WEP and TDR, namely improved MVO.
Apart from the poor exploitation ability, the exploration ability of the original MVO is also needed to improve. It has been proposed that universes in the early iteration period tend to be far from the accurate solution, which may cost the algorithm more time to converge. The current exploration strategy may lead the algorithm to fall into the local optimum in the early stages [37]. Therefore, different methods should be taken to overcome these two problems. To enhance exploitation ability, we need to apply a strategy or mechanism after carrying out the original procedure of exploitation to improve the accuracy of final solutions. Besides, a mechanism should be introduced in the early period of algorithm to escape local optimum. Hence, to overcome these shortcomings, in this paper we propose a novel hybrid MVO by introducing several algorithms and combining them with MVO. As mentioned above, the original algorithm tends to fall into local optimum in the early stages of iterations, which needs a new mechanism to help escape local optimum and improve the convergence speed. According to the previous research above, hybridizations of MVO and other algorithms have broad applications. Meanwhile, combination of different algorithms can utilize their strengths of them. Hence, in this paper, we choose to hybridize MVO with other algorithms or mechanisms to promote its performance.
To enhance the exploration capacity, we introduce the Metropolis rule to prevent the algorithm from falling into local optimum in the early stages of iteration. The Metropolis rule is mainly applied in SA to help escape local optimum by allowing hill-climbing moves [47]. Many researchers focused on the application of the Metropolis rule to enhance the convergence speed of different algorithms, such as PSO [48], GA [49], group search optimizer [50], and so on. By depriving the result of a particular possibility in the early stages, the Metropolis rule enables algorithms to escape local optimum [49]. Consequently, we apply the Metropolis rule in this paper to improve the exploration ability of MVO. Besides, to further improve the convergence speed and accuracy of solutions of the algorithm, we introduce QSA for balancing exploration and exploitation as well as improving the accuracy of final results. According to the existing research, QSA has the best optimal results and convergence speed in many solutions compared with other algorithms [22]. Xiaolei Zheng et al. [51] introduced QSA into artificial neural network and successfully gained results with high accuracy. Meanwhile, the third part of QSA can help the algorithm further improve the solutions, while the first and second parts of QSA can help balance exploration and exploitation [2]. Hence, we will introduce the mechanisms of QSA to help improve the accuracy of the final solutions of the original MVO.
The search strategy proposed in this paper is to improve the convergence speed and accuracy of final solutions of the original MVO. The proposed algorithm is called Hybrid Queuing Search and Multi-Verse Optimizer (HQS-MVO), and the main contributions of this paper are summarized as follows: The Metropolis rule is employed in the exploration stage to keep the algorithm from falling into the local optimum. Queuing search algorithm is introduced in both the exploration and exploitation to enhance the convergence ability and accuracy of the algorithm. Experiments of HQS-MVO and other competitive algorithms on 30 benchmark functions of CEC2014 and 10 benchmark functions of CEC2019 demonstrate the effectiveness of robustness in optimization problems.
The remainder of this paper is organized as follows. Section 2 briefly introduces the original MVO, QSA and Metropolis rule. Section 3 introduces our improvement of MVO, namely HQS-MVO, in detail. In Section 4 we present the experimental results of HQS-MVO and compare it with seven competitive algorithms and some published results, and then we apply the algorithm in an engineering optimization problem to test its performance. Lastly, we make a conclusion of this paper in Section 5.
Related algorithms
Multi-verse optimizer (MVO)
Multi-verse optimizer, which was proposed by Seyedali Mirjalili, Seyed Mohammad Mirjalili and Abdolreza Hatamlou in 2016 [8], is a novel meta-heuristic algorithm inspired by the theory of multi-verse. There are three main concepts in MVO: black hole, white hole and worm hole. The white/black hole can receive/send objects from/to other universes, while the worm hole serves as a tunnel, enabling objects to travel to everywhere in the universe. The following rules are applied to the algorithm: Every universe is able to receive objects through the white hole and send objects through the black hole. Every universe has its inflation rates, and the possibility of having black holes rise in proportion to inflation rates, while the possibility of having white holes is in inverse proportion to inflation rates. Worm holes can exist anywhere regardless of inflation rates.
Based on the rules above, every universe exchange objects randomly with each other, and objects travel through worm holes at a certain Travel Distance Rate (TDR). The process of white/ black hole exchange is meant for exploration while travelling through worm hole is a process of exploitation. The exploration happens at every iteration of the algorithm, while the exploitation is determined by Worm hole Existence Possibility (WEP). WEP rises with the process of iteration to strike a balance between the exploration and exploitation. The schematic diagram of the MVO is presented in Fig. 1.
The schematic diagram of MVO.
Assume that
In the equation, d is the dimension of the variables, and n is the number of universes. In this way, the process of exploration through black/white hole is indicated as follows:
Worm hole can exist anywhere in the universe to improve the inflation rate, which means that the exploitation is performed regardless of the current inflation rate. The process of worm hole travelling as well as the exploitation can be expressed by the following formulation:
QSA is a meta-heuristic optimization algorithm inspired by human behavior in queuing and was proposed by Jinhao Zhang et al. in 2018 [2]. There are three parts in the algorithm, namely business 1, business 2 and business 3. All of these parts have the following principles: Some businesses must be handled, while some others are optional. Each business has three queues, and its leader, the leader with stronger ability, tends to spend less time handling businesses. All the customers can change their queues randomly, and whether they want to change their queues is affected by other customers, themselves and leaders.
The three different businesses in the algorithm all have their mechanisms. Details are as follows:
Business1 is a process that must be handled. In this business, there are three queues Q11, Q12 and Q13, and customers that have the three best fitness values are selected as leaders, namely A11, A12, A13. The following formulation can calculate the number of each queue:
Business2 is an optional process, and the probability of handling it, p
i
, can be calculated by the following formulation:
Business3 is the last part of the algorithm and is an optional process determined by p
i
, which is the same as business2. Different from business1 and business2, business3 doesn’t have any queues, and a customer is only affected by other customers. The effect formulation is as follows:
The Metropolis rule is an acceptance rule that is widely used in the SA algorithm. It is a mechanism that determines whether to accept the new solutions even though the fitness value of the old solution is better than the new one. The probability of accepting new solutions can be calculated by Equation (14).
Motivation
Numerous works have focused on the hybridization of MVO and other intelligent optimization algorithms. Even though hybridizations enable MVO to deal with many special kinds of problems, some hybrid algorithms have not improved their performance in many other issues in terms of accuracy or convergence speed [24, 25]. Hence, a new way to enhance the original MVO is needed to deal with common optimization problems.
The original MVO has two main problems: the weak capacity of exploitation [22] and easiness to fall into local optimum, which makes MVO hard to converge [21]. For one thing, as mentioned above, in the early stage of iteration, the initial universes are often far away from the global optimum. In this way, the mechanism of white/black hole exchange may make the algorithm converge at a low speed and easier to fall into local optimum. For another thing, worm hole travelling tends to miss the global optimum, which can’t guarantee the accuracy of the final solution. In this paper, we need to enable the algorithm to escape local optimum in the early stages and strengthen its exploitation ability at the end of iterations to improve the accuracy of final results.
As mentioned above, hybridizations of different algorithms with MVO can utilize their strengths and overcome the shortcomings of original algorithms if the algorithm is introduced correctly [28]. Based on the consideration above, a novel hybrid enhanced multi-verse optimizer is proposed by combining MVO with QSA. QSA, described in detail in the previous section, is composed of three mechanisms, namely business 1,2,3 [2]. Business 1 is utilized to enrich the population of search agents, while business 2 is mainly applied to balance exploration and exploitation. We divide business1, 2 into two groups at the beginning and end of MVO to improve its capacity of exploration and exploitation. Besides, we introduce the Metropolis rule at the end of each iteration to prevent the algorithm from falling into the local optimum. The Metropolis rule is a mechanism mainly applied in SA. By accepting worse solutions at certain possibilities, the Metropolis rule enables the algorithm to escape local optimum in the early stages of iterations [47], which will strengthen the exploration ability and improve the convergence speed of original algorithms.
Hybrid multi-verse optimizer with queuing search algorithm (HQS-MVO)
The proposed algorithm can be divided into four sections. The first section uses business1 and business2 of the QSA to help perform a preliminary search, thus narrowing the search scope to improve convergence speed. The second section is the original MVO. The last part uses the Metropolis rule to determine whether to accept the new solutions. Detailed descriptions are as follows.
Firstly, we initialize the population and parameters of the algorithm. Then, by carrying out business1 and business2 before exchanging objects through the black hole, which will help ensure the population diversity of the search agent, the exploration ability of the whole algorithm can be improved. Next, after conducting black/white hole exchange, business three is handled for further exploitation, namely local search. It should be noticed that even if the probabilities of carrying out business2 are all determined by Pr i , which can be calculated by Equation (10). However, the probability of handling business 2 decreases through the iterations. The black/white hole exchange, together with the switch from business2, will provide a better balance between exploration and exploitation. Lastly, at the early stage of iterations, the Metropolis rule is applied to help universes escape from the local optimum. The probability of rejecting new solutions decreases through iterations, which is the same as business2. The schematic diagram of the HQS-MVO is presented in Fig. 2.
The schematic diagram of HQS-MVO.
The HQS-MVO is proposed by introducing QSA and Metropolis rule. The HQS-MVO has excellent performance, and the pseudo-code is shown in Algorithm 2:
Assuming that N is the number of universes, D is the dimension of each universe, T is the maximum of iterations. The complexity of HQS-MVO is presented as follows: The time of random initialization operation is O (1). The process of business1 and business2 are O (N). The process of sorting universes is done by employing Quicksort algorithm, and it requires O (NlogN) time. The roulette wheel selection requires O (N) time.
Generally, the complexity of HQS-MVO is approximately equal to O (T × (N + N + NlogN + N × D × N)) ≈ O (T × N2 × D).
Experimental results
Benchmark functions
We use 30 benchmark functions of the CEC2014 [52] special session on real-parameter optimization and 10 benchmark functions of the CEC2019 to test the performance of HQS-MVO. There are four groups in CEC2014 benchmark functions, search ranges of these functions are [-100,100]. Unimodal Functions (F1 –F3) Simple Multimodal Functions (F4 –F16) Hybrid Function 1 (F17 –F22) Composition Functions (F23 –F30)
Details of CEC2019 benchmark functions are shown in Table 1.
Details of CEC2019 benchmark functions
Parameter settings
In this section, to prove the excellent performance of HQS-MVO, we compare it with the original MVO, HMVO [28], LBMVO [38] and PSO-MVO [24]. To further test its performance, we compare the proposed algorithm with some latest published method, namely WOA [53], GWO [4] and SCA [54]. This paper takes the following parameter settings: the population size N is set to 50, the dimension of each universe D is set to 30, the maximum number of iterations T is set to 6000. More details of the parameter settings of these algorithms are shown in Table 2.
Parameter settings of the algorithms involved in the control experiment
Parameter settings of the algorithms involved in the control experiment
The next part of this section will compare the final results of eight algorithms. The third part will present convergence curves of these algorithms on 30 CEC2014 benchmark functions and 10 CEC2019 benchmark functions. After that, we will compare the experimental results of HQS-MVO on CEC2014 with some published results for better validations. In the fifth part, we will utilize Wilcoxon signed-rank test to carry out statistical analysis. Then, we will carry out the parameter sensitivity analysis in the last part to evaluate the impact of some parameters.
The experimental results of HQS-MVO and other algorithms have been presented in the latter part of this section. During the experiments on CEC2014 and CEC2019, we set the number of runs per function to 30 and save the mean values and standard deviations, which are represented by “Mean” and “Std Dev”.
It is obvious that the HQS-MVO is better than the other seven algorithms, and details are discussed as follows: Unimodal Functions (F1 - F3). In this group, the HQS-MVO outperforms the other seven algorithms on all three functions, namely F1, F2, and F3. More specifically, in terms of accuracy, HQS-MVO is significantly better than original MVO, LBMVO, HMVO, PSO-MVO, WOA, GWO and SCA. Besides, the results gained by HQS-MVO have the lowest standard deviations, which has demonstrated the robustness of this algorithm on these three functions. The introduction of QSA has helped improve the exploitation ability and thus made the final global optimum more accurate. Additionally, the application of the Metropolis rule, as well as mechanism of business 1,2 has successfully enhanced the global search ability of the algorithm. Consequently, the proposed algorithm shows better robustness and is able to gain the best results. Simple Multimodal Functions (F4 - F16). It can be seen from Table 2 that HQS-MVO performs better than other algorithms in all cases in this part, and the other seven algorithms only outperform HQS-MVO on F6, F7, F11, F15 and F16 in terms of standard deviations. Specially speaking, HQS-MVO has overwhelming performances on functions like F4, F6, F8, F9, F10 and F11 compared with other seven algorithms. Meanwhile, HQS-MVO obtains the lowest standard deviations in most cases, which demonstrates that HQS-MVO has the best robustness in this section. Hybrid Functions (F17 - F22). These functions are combined with several random basic functions, which makes it difficult for the algorithm to search for accurate solutions. In this part, HQS-MVO is superior to all the other seven algorithms and has the most accurate results, which has confirmed that HQS-MVO is still the best one among the eight algorithms. Meanwhile, HQS-MVO still has the lowest standard deviations on these functions except F19. HMVO gains the best standard deviations on F19. It is worth noticing that solutions gained by HQS-MVO is much better than the other seven algorithms, especially on F17, F18, F20, F21, F22. As mentioned above, the introduction of QSA has successfully strengthened the exploitation ability of the original MVO [2]. Composition Functions (F23 - F30). In this group, HQS-MVO is the winner in all cases and still has the best performance in this group. Meanwhile, HQS-MVO shows overwhelming performances on F26 to F30 and is still the most robust in this part. On these functions, HQS-MVO has obtained much more accurate solutions and lower standard deviations than the other seven algorithms. Besides, HMVO performs the second best on F29 in terms of accuracy and robustness and gains the second lowest standard deviations on F27, F28 and F29.
The experimental results of CEC2019 benchmark functions have been presented in Table 4.
Experimental results of HQS-MVO and other 7 algorithms on CEC2019
Experimental results of HQS-MVO and other 7 algorithms on CEC2019
It can be observed from Table 4 that HQS-MVO is superior to other seven algorithms in half of the cases. Specially speaking, HQS-MVO outperforms other seven algorithms and has the most accurate results on F5, F6, F7, F8 and F9. Furthermore, it shows the best robustness on F3, F6, F7, F9 and F10. Generally speaking, although the proposed algorithms is inferior to GWO and SCA in few cases, HQS-MVO shows the best performance on these functions.
Figure 3 shows the logarithm of mean function error values of HQS-MVO and other seven competitive algorithms over 30 independent runs on some selected benchmark functions of CEC2014. They are made into convergence curves. The horizontal ordinate shows the logarithm values, and the longitudinal coordinates represent the number of iterations. According to the picture, HQS-MVO has the best convergence speed among the eight algorithms in most of the functions. More especially speaking, HQS-MVO shows overwhelming performance on F1, F9, F10, F18, F19, F28 while is only inferior to some of the other seven algorithms on F5 and F12. Besides, even though PSO-MVO shows a higher convergence speed at the early stage of iterations on F5 and F12, HQS-MVO has gained a more accurate result at last. In the early stages of iterations of F3, F16 and F21, HQS-MVO is inferior to the other seven algorithms but surpasses by them in the latter stages. Lastly, HQS-MVO does not achieve overwhelming performances on F7, F14, F23, F25 and F27. As mentioned above, the Metropolis rule can help the algorithm escape local optimum in the early stages and mechanisms of businesss1,2 help enrich the population of search agents and balance exploration and exploitation, thus improving the convergence speed of the algorithm.
The Convergence figure of HQS-MVO and other 7 competitive algorithms on selected test functions of CEC2014.
Similarly, the logarithm of mean function error values of HQS-MVO and other 7 competitive algorithms over 30 independent runs on some selected benchmark functions of CEC2019 are shown in Fig. 4. The HQS-MVO still shows the best performances in general. The HQS-MVO shows overwhelming convergence speed compared with other algorithms on F7. On F6, F8 and F9, the proposed algorithm converges faster than other algorithms in the later stages. The HQS-MVO does not show obvious advantage compared with other algorithms on F2, F4 and F10.
The Convergence figure of HQS-MVO, LBMVO, HMVO, PSO-MVO and MVO on selected test functions of CEC2019.
In order to prove that HQS-MVO is better than the other seven competitors, we utilize Wilcoxon signed-rank test to verify the result of comparison between HQS-MVO and the other seven algorithms. The sample to be tested in this part is the optimization results of eight algorithms running 30 times. The result is shown in Tables 5 6, where R+ represents the sum of ranks of which HQS-MVO outperforms the other seven algorithms, while R- is the sum of ranks that HQS-MVO is inferior to the other seven algorithms. p-value denotes the statistical significance of the test, namely whether to reject the statistical hypothesis (α = 0.05). Lastly, sig . represents the result of the test. “+” denotes that HQS-MVO is obviously better than its competitors while “≈” represents the result that HQS-MVO is similar to other algorithms, “-” means HQS-MVO is inferior to its competitors.
Statistical result of Wilcoxon signed-rank test on CEC2014
Statistical result of Wilcoxon signed-rank test on CEC2014
Statistical result of Wilcoxon signed-rank test on CEC2019
Results of HQS-MVO and other published results
It can be easily concluded from Table 5 and Table 6 that HQS-MVO shows a significantly better result than all of the other seven competitors. Besides, most of the R+ values highly surpass R- values, which demonstrates that HQS-MVO is a significantly better choice for searching for the global optimum compared with HMVO, LBMVO, PSO-MVO, original MVO, SCA, WOA and GWO in most cases. Taking experimental results and convergence speed above into consideration, HQS-MVO has the best robustness compared with those seven algorithms, and performs very well in terms of exploration. What’s more, HQS-MVO has successfully struck a balance between exploration and exploitation.
To sum up, based on the experimental results and convergence figures above, the introduction of QSA and Metropolis in HQS-MVO has significantly improved the convergence speed and accuracy of solutions because the Metropolis rule enables the algorithm to escape from local optimum in the early stages. QSA can enrich the population of search agents and balance exploitation and exploration at the same time.
To further validate the search ability of HQS-MVO, in this section we compare the experimental results of HQS-MVO with some published results on CEC2014. The proposed algorithm is compared with BA, Hus, GSA, BBO and HTA. Results are published by Ghanshyam G. Tejani et al. [55] and the dimensions are all set to 30. The results are presented in Table 6.
It can be observed from Table 6 that HQS-MVO is still the most competitive algorithm in comparison with others. Specially speaking, HQS-MVO ranks first in 15 functions, namely F1, F4, F6, F11, F13 and F14, F17, F19 to F22, F26 to F28 and F30. Although HQS-MVO does not show better performance than other algorithms in F5, F7, F12, F15, F16, F24 and F25, there isn’t much difference between experiment results of HQS-MVO and the best one.
Parameter sensitivity analysis
To explore the impact of some parameters on HQS-MVO, we utilize the sensitivity analysis in this part. The parameters needed to be tested are exploitation accuracy (p) and number of populations. The range of p is [1, 6], and decreases linearly at an interval of 0.1. Number of populations is set to 20, 30,40, 50, 60, 70, 80, 90 and 100. Each value of p and population size is tested on F13 of CEC2014 and results are shown in Figs. 5 6.
The influence of exploitation accuracy.
The influence of populations.
It can be observed from Fig. 5 that mean values and standard deviations of final solutions do not vary with the decrease of p value, which demonstrates that the exploitation accuracy does not show obvious influence on the exploitation ability and robustness of the algorithm. In Fig. 6, we can conclude that the mean values of final solutions show a slight decrease with the increases of populations. Specially speaking, with the increase of populations, the mean values get better and standard deviations are stable, which means that the increase of populations will slightly enhance the search ability and robustness of the algorithm.
In this part, we apply the HQS-MVO in a real engineering optimization problem and compare its performance with other algorithms.
Parameter Estimation for Frequency-Modulated (FM) Sound Waves
The purpose of this problem is to determine the optimal values of parameters for FM synthesizer [56]. The discrete variables are six decision parameters. The problem is defined as follows.
Comparison of results
To test the performance of HQS-MVO on this problem, we conduct 30 independent runs and compare the results with HMVO, LBMVO, PSO-MVO, MVO, DE and PSO. The populations of all the algorithms are set to 30, the iteration is set to 6000. The mean values, standard deviations and statistical results are presented in Table 6. The convergence curves are shown in Fig. 7.
Convergence curves of HQS-MVO and other 6 algorithms.
It can be observed from Table 6 and Fig. 7 that HQS-MVO has the better exploration ability in comparison with other competitions. Despite the fact that the proposed algorithm does not show overwhelming performance over PSO-MVO, LBMVO and MVO, it has the best convergence speed in early stages, which demonstrates its outstanding search ability.
The original MVO has many drawbacks in terms of convergence speed, solution accuracy and balance between exploration and exploitation. The introduction of QSA can guarantee the diversity of populations, thus improving the convergence speed and accuracy of the final solutions. Moreover, the Metropolis rule helps prevent the algorithm from falling into the local optimum. The two mechanisms above have greatly enhanced the capacity of exploitation and exploration of MVO. The innovations of HQS-MVO are as follows:
HQS-MVO employs the Queuing search algorithm to improve the capacity of global search, namely exploration, thus increasing the convergence speed of the algorithm.
The introduction of the Metropolis rule at the early stage of iterations has effectively prevented the algorithm from falling into the local optimum.
Results of HQS-MVO and other 6 algorithms on FM Parameter Estimation problem
It has been demonstrated by performance testing experiments on CEC2014 and comparison of HQS-MVO with the other seven algorithms that HQS-MVO performs excellently in dealing with different kinds of problems and functions. In this article, we prevent MVO from falling into local optimum in the early stage. Meanwhile, the introduction of three mechanisms of QSA can enrich the population of search agents of MVO and strike a better balance between exploration and exploitation.
In conclusion, we introduced Queuing search algorithm and Metropolis rule to improve the search ability of the HQS-MVO algorithm. Despite the excellent performance of HQS-MVO, there remains much improvement space. For instance, WEP and TDR in original MVO deserve more consideration in adjusting control parameter formulations. In the future study, relevant work will be carried out focusing on controlling parameter formulations and the introduction of some more novel algorithms, such as Grey Wolf Optimizer, and so on. From the Tables 3 4 we can observe that although the HQS-MVO shows better robustness than other algorithms, its standard deviation is still a little high, which may affect its performance in some unknown problems. Therefore, more work will be done to enhance its robustness to suit more unknown problem in the future.
Experimental results of HQS-MVO and other 7 algorithms on CEC2014 at D = 30
Footnotes
Acknowledgments
This research was funded by Major Project of Philosophy and Social Science Research in Colleges and Universities in Jiangsu province (2019SJZDA039), National Natural Science Foundation of China (No.72274099, No.71974100), Humanities and Social Sciences Fund of the Ministry of Education, China (No.22YJC630144), and NUIST Students' Platform for Innovation and Entrepreneurship Training Program (XJDC202210300107).