A Modified Equilibrium Optimizer Using Opposition-Based Learning and Teaching-Learning Strategy

Equilibrium Optimizer (EO) is a newly developed intelligent optimization algorithm inspired by control volume mass balance models. EO has been proven to have an excellent solution effect on some optimization problems, with the advantages of ease of implementation and strong adaptability. However, the original EO has some disadvantages when solving complex multimodal problems, including an immature balance between exploration and exploitation, the high probability of falling into local optima entrapment, and the slow rate of convergence. In order to address these shortcomings, a modified equilibrium optimizer (OTLEO) is proposed using teaching-learning-based optimization (TLBO) and opposition-based learning (OBL). These modifications aim to maintain the diversity of the solutions, expand the algorithm’s search range, improve exploration and exploitation capabilities, and avoid local optima. To verify and analyze the modified equilibrium optimizer algorithm’s performance, the OTLEO was tested for 32 classical benchmark functions. All algorithms are independently run 30 times in the same environment. Thereafter, the comparative evaluation against the OTLEO and other six representative algorithms is conducted. Four real-world engineering application problems, including multiple sequence alignment and so on, are used for additional validation. The experimental results, statistical tests, qualitative analysis, and stability analysis demonstrate that the proposed OTLEO outperforms the original EO and other algorithms.


I. INTRODUCTION
ious fields, including economics, mathematics, medicine, 26 and engineering. Using traditional mathematical optimization 27 methods such as Quasi-Newton, Conjugate Gradient, Fast 28 Steepest, and Sequential Quadratic Programming Methods 29 The associate editor coordinating the review of this manuscript and approving it for publication was Mauro Tucci .
to solve real-world non-convex problems is difficult [1], [2], 30 While the proposed metaheuristic algorithms do not use 31 derivative information or other mathematical traits such as 32 continuity, flexibility, and gradient-free nature, etc. [3]. The 33 employment of metaheuristic algorithms to solve optimiza- 34 tion issues has been praised and explored by many academics 35 for its advantages, including easy operation, flexible mecha- 36 nism, efficient solution, and the ability to avoid local optima. 37 Genetic interactions. To perform the optimization, Charged System 101 Search (CSS) [17] employs a combination of rules from 102 Coulomb's law of electrostatics and Newtonian mechan-103 ics. The fourth category is that of human-based algo-104 rithms, which are inspired by human interactions and human 105 behavior in societies. Teaching-Learning-Based Optimiza-106 tion (TLBO) [18], Competitive Algorithm (ICA) [19] and 107 Coronavirus Herd Immunity Optimizer (CHIO) [20] are sev-108 eral human-based algorithms. 109 Afshin Faramarzi et al. proposed the Equilibrium opti-110 mizer (EO) [9] in 2020 as a new intelligent optimiza-111 tion technique. The algorithm's characteristics include small 112 number of setup parameters, simple structure, improved 113 solving capabilities, and ease of implementation [21]. 114 Jingbo Wang et al. [22] proposed photovoltaic cell parame-115 ter estimation based on an improved equilibrium optimizer 116 algorithm, which can efficiently improve both optimiza-117 tion precision and reliability for estimating photovoltaic cell 118 parameters. Manoj Kumar Naik et al. [23] used an opposi-119 tion equilibrium optimizer based on context-sensitive entropy 120 dependency for multilevel thresholding of remote sens-121 ing images. Shaik et al. [24] enhanced the voltage distri-122 bution of the distribution system by reorganizing 106 the 123 DG layout with the equilibrium optimizer. Dinkar et al. [25] 124 solved Image Segmentation using Multilevel Threshold-125 ing by Opposition-based Laplacian Equilibrium Optimizer. 126 Joshi et al. [26] used the equilibrium optimizer to optimize 127 the allocation of the static varariable compensator to achieve 128 voltage control and reduce loss. 129 The EO algorithm has the disadvantages of being sensi-130 tive to control parameters, easily falling into local optimal 131 solutions, and low efficiency of algorithm execution when 132 solving complex multi-peaked optimization problems, and 133 thus cannot guarantee convergence to the optimal condition. 134 The search agents position update always depends on the 135 equilibrium state, which may restrict the exploration capa-136 bility of the EO [27]. In order to further improve the global 137 search capability of the EO algorithm, many scholars have 138 conducted in-depth studies and proposed different improve-139 ment methods. To avoid the algorithm trapped in a local 140 optimum solution, Gupta et al. [3] employed Gaussian muta-141 tion and the notion of population partitioning and recon-142 struction based an additional exploratory search mechanism, 143 expanding the algorithm's search strategy and improving the 144 search accuracy. Dinka et al. updated the candidate solu-145 tions by random wandering of the Laplace distribution and 146 mix different acceleration coefficients through a backward 147 learning mechanism to enhance the exploitation of the algo-148 rithm and speed up the convergence. Jia et al. [28] com-149 bined the EO and a heat exchange optimization algorithm. 150 By applying the EO to alter the DG layout, Shaik et al. [24] 151 improved the voltage distribution in the distribution sys-152 tem. To facilitate algorithm exploration and exploitation, 153 Chen et al. [29] presented an interruption-based multi-154 objective equilibrium optimization algorithm with muta-155 tion operators. Liu et al. [21] added an adaptive mutation 156 VOLUME 10,2022 proportional strategy, a mechanism of the whale optimizamizer (OTLEO)by integrating two strategies. In the particle   proposed OTLEO algorithm over other selected 214 algorithms.

215
The remainder of this paper is structured as follows: 216 Section II introduces the original EO, and in Section III 217 the proposed OTLEO is described in detail. The benchmark 218 function tests, experiments in which EO applies permutations 219 of two strategies and discussion of the results are presented 220 in Section IV. In Section V, the proposed OTLEO is applied 221 to four real-world engineering problems. Finally, conclusions 222 and future research directions are presented in Section VI.

224
The equilibrium optimizer is an optimization algorithm 225 inspired by the physical phenomenon of mass dynamic equi-226 librium in a control volume, in which the mass equilibrium 227 equation is used to describe the concentration of inactive 228 components in a control volume as a function of their various 229 sources and absorption mechanisms. The equation of mass 230 balance offers the fundamental mechanics for the preserva-231 tion of the entering, leaving and producing mass [31].
where C is the concentration inside the control volume (V ), 234 v dC dt is the rate change of mass in the control volume, Q is the 235 volumetric flow rate, C eq represents the concentration at an 236 equilibrium. G is the mass generation rate. A stable equilib-237 rium state is achieved when v dC dt is equal to 0. Representing 238 dC dt as f ( q v ) where f is a function and dC dt represents the inverse 239 of residence time, referred to λ here. Hence, Equation (1) can 240 be represented as follows: Integrate both sides in Equation (2) over time, gives: This results in: Note that F in the Equation (4) is calculated as follows: The initial start time and concentration are t 0 and c 0 , respec-249 tively. The above equations serve as the overall framework for 250 designing EO in this section. A particle in the EO corresponds 251 to a solution, and a particle's concentration corresponds to its 252 location in the PSO algorithm [9].

298
Each iteration updated the concentration of each particle by 299 selecting candidates at random from a pool of candidates 300 with the same probability. Each particle will underwent an 301 updating process until the optimization process was com- While time t is a function of iteration, and it decreased as the 309 number of iterations increases: Iter and Max_iter represent the current and maximum num-312 ber of iterations, respectively. The constant value a 2 was used 313 to control exploitation ability. We defined t 0 as follows: where a 1 represents a constant proportional to global explo-316 ration capability. The higher the a 1 value, the better the 317 exploration ability and, as a result, the lower the exploita-318 tion performance. Similarly, the higher the a 2 , the better 319 the exploitation and the lower the exploration [9]. In this 320 paper, a 1 and a 2 are equal to 2 and 1, respectively. These 321 constants are selected through empirical testing of a subset 322 of test functions.However, these parameters can be tuned 323 for other problems as needed [9]. The direction of explo-324 ration and exploitation is determined by the third component, 325 sign(r − 0.5). r is a random vector in the terval of [0, 1]. 326 Substitute Equation (10) into Equation (8): The generation rate (G) is one of the most important terms in 329 the proposed algorithm for providing an accurate solution by 330 improving the exploitation. and has the following definition: 331 where: where r 1 and r 2 are random numbers in [0, 1]. GCP is defined 336 as the Generation rate Control Parameter, and GP means 337 generation probability denoting the proportion of parti-338 cles that utilizes generation for state updating, respectively. 339 GP = 1 means that there will be no generation rate term par-340 ticipating in the optimization process. This state emphasizes 341 high exploration capability, and often leads to non-accurate 342 solutions. GP = 0 means that the generation rate term will 343 always be participating in the process, which increases the 344 stagnation probability in local optima. Based on empirical 345 testing, GP = 0.5 provides a good balance between explo-346 ration and exploitation phases [9]. Finally, the EO general 347 updating rule is as follows: The first term in the search equation Equation (15)

390
The teacher is considered the best solution obtained so far, 391 which can be selected from the equilibrium pool C eq , pool, 392 that is formed using the best candidates, C eq (1) , C eq(2) , C eq(3) ,

393
C eq(4) and C eq , ave. In this paper, we set C eq (1)  for i = 1 to number of particles (n) do 8: Calculate the fitness of i th particle;  28: 30: end for 31:
and C i is the opposite vector from the real 436 vector C i . By comparing the fitness function, the improved 437 position update formula is as follows: where f (C i ) and f ( C i ) are the fitness values of ith particle C i 440 and antiparticle C i .The flow chart is provided in Figure 1.

442
In this section, the overall performance of the proposed 443 OTLEO is verified from various aspects by using a collection  As an evolutionary algorithm, biogeography-based opti-487 mization (BBO) [35] is based on natural biogeography. 488 VOLUME 10, 2022

508
The OTLEO algorithm proposed in this paper is an 509 improved algorithm using teaching-learning-based optimiza- greatly improved, and the improved strategy proposed in this 542 paper can effectively improve the optimization accuracy of 543 the algorithm. On the unimodal function (F1-F9), OTLEO 544 all obtained the optimal solution. Although OTLEO does 545 not achieve the optimal solution compared with m-EO2 on 546 functions F5 and F7, it is better than m-EO2 on all other 547 18 functions. This shows that OTLEO still shows the best 548 stability and the strongest ability to find the optimal solution 549 on the whole, and maintains a remarkable balance between 550 exploration and exploitation. The OTLEO algorithm also 551 shows its superiority on the multimodal function. As shown in 552 Table 3, OTLEO achieves the optimal solution on 8 functions 553 (F10-F18) of the 10 multimodal functions. Compared with 554 the proposed OTLEO algorithm, the EO, m-EO1, m-EO2 555 and m-EO3 algorithms only obtain 5, 3, 3 and 3 optimal 556 solutions on multi-modal functions, respectively. The m-EO3 557 algorithm outperforms the other three algorithms on function 558 F12, but is significantly inferior to OTLEO. As shown in 559 Table 3, the performance of EO algorithm is slightly better 560 than m-EO1, m-EO2 and m-EO3 algorithms on multi-peak 561 functions, but there is still a big gap between EO algorithm 562 and the proposed OTLEO. As shown in the table, m-EO3 563 only obtains 4 optimal solutions, far less than the 18 optimal 564 solutions obtained by OTLEO. The reason is that in m-EO3 565 algorithm, the OBL optimization is carried out first. At this 566 time, the particles updated by the algorithm are not optimized, 567 and there is little variation between particles. Then the TLBO 568 is carried out. Even if the fitness of the whole population is 569 improved, the diversity of the population is not significantly 570 improved, as a result the algorithm does not improve the prob-571 lem of falling into local optimal solution. On the contrary, 572 the proposed OTLEO algorithm firstly used TLBO to select 573 the optimal particles and improved the fitness value of the 574 overall population. Then, the optimized whole population is 575 updated by OBL, and the population diversity is improved. 576 Therefore, OTLEO has a stronger ability to avoid falling into 577 local optimum than other algorithms. 578 Figure 2 shows the fitness curve of the five algorithms on 579 the benchmark function when the dimension of dim = 30. 580 We selected four functions as representatives, among which 581 F1, F2, and F6 are unimodal functions, and F10 is multi-582 modal. It can be seen from Figure 2 that OTLEO shows 583 the highest convergence speed on all functions, thus show-584 ing higher convergence. Compared with the other four 585 algorithms, OTLEO has higher convergence accuracy on 586 functions F1, F2, and F6, and has a stronger ability to con-587 verge to the optimal solution. This shows that the OTLEO 588 algorithm has strong performance in balancing exploitation 589 and exploration, and can better avoid falling into local opti-590 mum. Figure 2 illustrates that the TLBO strategy and the 591 OBL strategy introduced by OTLEO can ensure population 592 diversity and improve the robustness of the algorithm.

593
To sum up, the OTLEO proposed in this paper can effec-594 tively improve the optimization ability of the EO algorithm. 595 And the TLBO strategy takes priority and the OBL strat-596 egy follows, which can maximize the performance of the 597    the previous ones as the iteration progresses. The conclusion 627 drawn from the qualitative analysis is that OTLEO shows an 628 admirable performance in balancing the ability in exploration 629 and exploitation to improve the accuracy of global optimal 630 solution.

632
Unimodal benchmark functions have the characteristic of 633 having a unique minimum,which can be used to evaluate 634 the exploitation capability of algorithms. By contrast, mul-635 timodal benchmark functions exist multiple local optima in 636 search space, which are utilized to assess the performance 637 of preventing local optima entrapment. In order to obtain 638 the statistically comparison between the proposed OTLEO 639 and other algorithms, Table 5, 6 and 7 expose the experi-640 mental result of the seven algorithms on scalable benchmark 641 functions with dimensionalities of 30, 50 and100 respec-642 tively. The best value, the average value, and the standard 643 derivation are three metrics used to show the experimen-644 tal results.       According to the comparative analysis discussed in the pre-720 vious section, OTLEO shows the improvement among other 721 algorithms. To ensure the completeness, this section discusses 722 and analyses the statistical significance and stability. In order 723 to estimate the significant difference between OTLEO and 724 other algorithms, the Wilcoxon rank-sum test [36] and Fried-725 man's statistical test [37] are applied and the result are shown 726 in Table 8 and Table 10.

727
The Wilcoxon rank-sum test is used to verify the sig-728 nificant differences between the average values of samples, 729 with the significance level of α setting to 0.05. The value 730 of p less than 0.05 can be considered as rejecting the null 731 hypothesis. Table 8 shows the p-value which represents the 732 statistical probability obtained in Wilcoxon rank-sum testis. 733 In the table, the symbol ''+'', ''−'' and ''='' are used 734 to denote whether the null hypothesis should be accepted 735 or reject. ''+'' demonstrates that there exists significant 736 statistical difference, rejecting the null hypothesis. ''='' indi-737 cates the impossible comparison due to the similar sam-738 pled data. While ''−'' depicts that no statistical difference 739 exists and that the null hypothesis is accepted. As shown 740 in Table 8 Table 10 shows the 749 result of Friedman's statistical test. The proposed OTLEO 750 comes out on top for all dimensions of testing. Accordingly, 751 the OTLEO with better exploration and exploitation ability 752 shows a significant improvement compared with the other 753 6 algorithms.

754
In addition, the box-plot is used to check the proposed 755 OTLEO's stability in obtaining the optimal solution com-756 pared with algorithms. The box-plot for describing the degree 757 of data dispersion is shown in Figure 5. The tests are con-758 ducted for 30 dimensions. As seen from Figure 5, the OTLEO 759 has little change in average values and extreme values, and 760 no major outliers are generated. It is worth mentioning that 761 the OTLEO locates in the lowest position 17 times out of 20. 762 It reveals that the solution quality of OTLEO is superior than 763 other algorithms, with a significant stability.     Multiple sequence alignment (MSA) [38] is the simulta-793 neous comparison of two or more sequences that may be 794 VOLUME 10, 2022    and delete instructions to align sequences. Extracting infor-807 mation from sequence matching with HMMs models is a 808 way to use global information to improve the accuracy of 809 sequence matching, while being simpler to the data compared 810 to traditional sequence matching methods. In HMM mod-811 els, common training methods are based on statistical and 812 reestimation methods, such as the Baum-Welch algorithm; 813 the Baum-Welch algorithm makes the parameter estimation 814 of HMM models However, since Baum-Welch algorithm is 815 a local optimization algorithm based on the steepest gra-816 dient descent, it is very easy to get trapped in the local 817 X. Wang et al.: Modified EO Using OBL and Teaching-Learning Strategy fewer control parameters and fast convergence, when 831 they approach the global optimal solution, they tend to 832 slow down the search speed and decrease the search 833 accuracy, and are easily trapped in the local optimal 834 solution.

835
The OTLEO algorithm mentioned in this paper has strong 836 global exploitation capability, local exploration capability 837 and fast convergence speed. Therefore, we try to solve the 838 MSA problem by the OTLEO algorithm and verify the feasi-839 bility of the algorithm on the MSA problem through a large 840 number of experiments. 841 VOLUME 10, 2022   Based on the results of OTLEO training, the globally opti-883 mal Hidden Markov Model corresponding to the particles is 884 obtained. This model is then used to compare the sequences 885 using the Viterbi algorithm [45] to obtain the optimal pairing 886 results, and the results are evaluated using an objective func-887 tion based on the SOP (sum-of-pairs) [46] scoring system. 888 SOP scoring function as follows: where l i ,l j is the aligned sequence i and j. D is the distance 891 metric of two sequences. n is the number of sequences to be 892 aligned.

894
In this paper, four sequences from the first reference set of 895 the BALIBASE database [47] were selected as the target to 896 be aligned. We list the four sequences in Table 11. The BAL-897 IBASE database provides artificial high-quality 3D structural 898 overlay reference standard sequences and is widely used in 899 the field of multiple sequence alignment. The first column is the names of selected sequence sets, 901 N is the number of sequences in each set, the third column 902 lists the minimum and maximum lengths of the sequences in 903 each set and the fourth column lists the sequences' identity. 904 The algorithm is adapted by 30 independent experiments 905 with 200 iterations per experiment and the scoring function 906 SOP is used as the fitness of the algorithm. Table 11 records 907 the comparison of the two algorithms EO and OTLEO on four 908 sequences. Where 'avg' is the average score and 'best' is the 909 highest score during the iteration. Figure 7 records the SOP 910 score curves during the algorithm iterations.

911
Based on the data in Table 11 and the score curves in 912 Figure 6, it can be seen that OTLEO has better performance 913 on the MSA problem compared to the original EO. The SOP 914 scores obtained by the OTLEO algorithm are much higher 915 than those of the EO algorithm on all four sets of test sequence 916   is a continuous optimization problem that is challenged with 927 obstacles to achieve a balance between primary constrains 928 and related constrains. The key objective is to optimize fabri-929 cating cost. This section aims to solve the constrained WBD 930 problem by analyzing the results of the OTLEO algorithm 931 compared with other algorithms.

932
Four variables are considered in WDB including the 933 beam's length of (l), height (t), thickness(b) and the weld 934 thickness, respectively. Seven constraints consist of shear 935 stress constraints(τ ), beam end deflection constraints(δ), 936 bending stress constraints(θ), bar's buckling load(P c ) and 937 VOLUME 10, 2022 side constraints. The problem is mathematically detailed 938 below: where:  The pressure vessel design problem is an engineering opti-  The second line in Table 12 indicates the optimization 999 results of OTLEO and other algorithms on pressure ves-1000 sel design, including two indicators: best value and aver-1001 age value. It is worth noting that both WOA algorithm and 1002 OTLEO algorithm obtain the optimal solution. The experi-1003 mental results of this common benchmark structure optimiza-1004 tion problem show that, compared with other meta-heuristic 1005 algorithms, the performance of the proposed OTLEO algo-1006 rithm is much better than that of EO, PSO, BBO and 1007 TLBO under the same number of evaluations and runs, same 1008 as WOA. It also reveals that OTLEO shows good compatibil-1009 ity in mixed discrete-continuous problems.   obtains the optimal value in both the optimal index and the 1036 average index at the same time, illustrating the stability of its 1037 solution.

1039
In this paper, a new variant of EO called OTLEO using 1040 teching-learning-based optimization and position-based 1041 learning strategy is proposed. The proposed OTLEO algo-1042 rithm alleviate the shortcomings of the original EO, such 1043 as slow rate of convergence, low convergence accuracy, 1044 and tendency to fall into the local optimal. Through mutual 1045 learning among different particles, it helps to alleviate the 1046 problem of decreasing particle diversity as the number of 1047 iterations increases, maintain the diversity of the population, 1048 and improve the optimization accuracy.

1049
Validation and analysis of OTLEO's performance. The 1050 modified equilibrium optimizera algorithm is benchmarked 1051 with 32 classical functions. In terms of accuracy, conver-1052 gence speed, and stability, the experimental results show that 1053 OTLEO outperforms and is competitive with six state-of-the-1054 art algorithms.Based on objective fitness, standard deviation, 1055 and the Wilcoxon rank-sum test, OTLEO has better opti-1056 mization performance than other metaheuristic algorithms 1057 such as PSO, BBO, GWO, WOA, and TLBO.Moreover, 1058 four real-world engineering application problems are used 1059 for additional validations, and the experiments show that 1060 the improved algorithm has excellent performance in these 1061 problems.

1062
However, OTLEO also has the disadvantages of insuffi-1063 cient comparison accuracy and slow convergence speed in 1064 solving high-dimensional problems. In the future, binary, 1065 VOLUME 10, 2022 are considered to be applied to solve power dispatch prob-