# A Whale Optimization Algorithm with Convergence and Exploitability Enhancement and Its Application

Authors: Xiaodan Liang; Zijian Zhang
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

The whale optimization algorithm (WOA) is inspired by humpback whale social behavior and WOA is a popular swarm intelligence algorithm. Yet, the WOA does have certain flaws, such as a restricted global search capability and an inconsistent convergence speed, and when dealing with complex optimization problems, WOA is easy to fall into local optimum. To address the WOA’s shortcomings, a modified whale optimization algorithm with convergence and exploitability enhancement called MWOA-CEE is proposed. Three operators, opposition-based learning, exponentially decreasing function, and elite-guided Cauchy mutation, are integrated into the WOA to enhance the exploration performance and avoid local optimum. First, the opposition-based learning used the opposition-based concept to generate the opposite position of the candidate solution, which can enhance the convergence ability and quality of the solution. Then, the exponentially decreasing function balances the exploitation and exploration phases and boosts the exploitation capability. In the end, the elite-guided Cauchy mutation generated a wide disturbance near the current candidate solution, which improves solutions diversity and increases the algorithm’s global search capability. MWOA-CEE is validated on 19 optimization benchmark functions and 14 CEC2014 benchmark functions and used to identify IIR systems. It is compared to seven of the commonly used algorithms. The results prove that MWOA-CEE outperforms the other algorithms in terms of final solution quality and convergence rate.


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## Body

## 1. Introduction

Over the years, many nature-inspired optimization techniques based on metaheuristics have grabbed great attention from researchers to solve various problems in both mathematics and engineering. This kind of algorithm is mostly enlightened by the biological or physical phenomena of an animal. There are three sorts of approaches: evolution-based, physics-based, and swarm-based. The evolution-based approach is inspired by the law of natural evolution, for example, Genetic Algorithm (GA) [1], Genetic Programming [2], Evolution Strategy (ES) [3], and Biogeography-Based Optimization (BBO) [4].Physics-based approaches are inspired by the method of mimicry physics. The most famous algorithms in this field are Simulated Annealing (SA) [5], Curved Space Optimization [6], Gravitational Local Search Algorithm (GSA) [7], Central Force Optimization [8], Ray Optimization (RO) algorithm [9], and Brain Storm Optimization (BSO) [10].The swarm-based approach mirrors group species behavioral patterns. Particle Swarm Optimization (PSO) [11] is the often-used algorithm. PSO works by replicating the movement of organisms in a flock of birds to locate the ideal position. PSO employs a series of particles (candidate solutions) that are around the objective space to get a solution (optimal location). Simultaneously, other possible solutions track the optimal solution in their pathways. Ant Colony Optimization [12] is another noteworthy swarm-based approach that models the foraging behavior of many ant species. The bat algorithm (BA) [13] is also a prominent swarm-based algorithm. A bat algorithm combined with cuckoo search and Sugeno fuzzy inertia weight (UBCSIW) [14] was introduced in 2020. The cuckoo search boosts the exploration performance of the bat algorithm. The Sugeno fuzzy inertia weight boosts the diversity of bat population and results in a stable result. A newly introduced swarm-based algorithm called naked mole-rat (NMR) algorithm [15] was published in 2019. This strategy is derived from the naked mole-rat in nature. The experiment results indicated the performance of NMR in exploitation and exploration of the solutions.The whale optimization algorithm (WOA) [16] has been recently introduced. This algorithm is inspired by the natural hunting behaviors of humpback whales. To find prey, WOA employs three operators: encircling prey, researching prey, and attacking prey. There is a lot of modification to WOA. Generally, these studies could be divided into two categories. One is to apply the WOA to solve some practical issues. The other is to improve the performance of WOA with some strategies. In 2020, a hybridized WOA with GWO dubbed WOAGWO [17] was introduced, with the GWO hunting mechanism inserted in the WOA exploitation phase. GWO’s hunting mechanism is designed to keep whales moving from unpromising places and to boost exploration performance. In [18], Lévy flight trajectory-based WOA (LWOA) is presented. The Lévy flight trajectory is useful for boosting population variety against premature convergence and strengthening the performance of escaping the local optimum. L. Ma et al. in [19] suggested an adaptive localized strategy for large-scale issue solutions. This method is capable of balancing the convergence and variety of solutions in the goal space in an adaptive manner. The balanced variant of WOA (BWOA) introduced in [20] integrated Lévy flight and chaotic local search into WOA to enhance the inclusive exploratory and neighborhood-informed capacities of the conventional technique. In a paper proposed in 2019 by Rohit Salgotra et al. [21], three modified algorithms were introduced. The three concepts are used to enhance the exploration and hold the variety of solutions. The proposed algorithm was tested on the CEC2005 benchmark functions and obtained a promising result. Trivedi et al. [22] integrated a randomization strategy into an adaptive whale optimization algorithm (AWOA). The results reveal that AWOA is superior to normal WOA as regards the effectiveness. In a quadratic interpolation-based WOA [23], to boost exploitation ability and search accuracy, a modified exploration approach and quadratic interpolation are applied. In 2019, RDWOA [24] was suggested as a strengthened form of WOA. To speed up convergence, a random spare or random replacement technique is utilized. Furthermore, a mechanism of doubly adaptive weight is adopted to boost exploration ability in the early stage and exploitative ability in the later stage. Trivedi et al. [25] proposed a hybrid PSO-WOA. The PSO operator is used in exploitation phase and the basic WOA operator is adopted to explore an uncertain environment. The experimental results reveal the effectiveness of PSO-WOA. Based on different searching pathways and perceptual disturbance, Sun et al. [26] presented a modified WOA (PDWOA). Several spiral curves have been tested, and the results reveal that the Archimedean spiral curve with equal pitch is more effective than other spiral curves. A hybrid SA-WOA algorithm was presented by Mafarja and Mirjalili [27]. The Simulated Annealing (SA) method is included in the algorithm to improve the exploitation of WOA by finding the most promising regions found by basic WOA.WOA could also be applied to many real-world cases. A novel variation of WOA with a pattern search (PS) algorithm was proposed by Bentouati et al. [28]. The results show that the suggested method for addressing the optimal power flow problem is effective and performs well. The WOA [29] is used in 3D scanning processes for providing two critical parameters: scanning distance and incidence angle values. The effectiveness of WOA is verified by the statistical results. In a paper proposed by Aljarah et al. [30] in 2018, WOA was applied to train Neural Networks with a single hidden layer. The results reveal that the suggested WOA-based training algorithm outperforms the other compared algorithms as regards convergence speed. To improve the flow shop construction scheme, Zhang et al. [31] presented an improved whale optimization algorithm (EWOA). To increase the robustness and efficiency, the augmented WOA with Lévy flight was adopted, and this suggested model was also used to improve the building phases and zones in order to achieve a more efficient flow shop construction design. An adaptive reference vector reinforcement learning (RVRL) approach [32] is proposed for industrial copper burdening optimization. The proposed approach is not only competitive for benchmark problems but also effective in solving real-world instances. To estimate the parameters of solar cells, Oliva et al. [33] employed an updated chaotic whale optimization technique. The solution of complicated nonlinear and multimodal goal functions is part of the design problem. The proposed algorithm solves the design problem with the chaotic map and automatically adapts parameters.For many years, the IIR model has been an active field of research, and it has been utilized for applications in signal processing. Due to the entanglement of multimodal error surfaces, IIR model identification is a difficult and complex optimization task. To reduce the multimodal error function, a realistic, efficient, and resilient global optimization approach is required. Several attempts have been made in recent years to investigate alternate architectures and methods for adaptive IIR filters. Mohammadi et al. [34] have suggested an efficient optimization approach for digital IIR system identification based on inclined planes system optimization in 2017. Sarangi et al. [35] produced Crazy cat swarm optimization (Crazy-CSO) algorithm to optimize the unknown plant. Majhi and Panda [36] have also proposed utilizing PSO and Bacterial Foraging Optimization (BFO) to efficiently identify complicated nonlinear dynamic plants. The novel algorithm LMS-GA [37] is used to solve the digital filters. This approach combines the standard LMS algorithm and Genetic Algorithm. The statistical result on the IIR and FIR system identification verifies the performance of the proposed LMS-GA.Although WOA has a high rate of convergence, the effectiveness of WOA in jumping out of locally optimal solutions is nonideal. In order to improve global search capability of WOA, a modified WOA named MWOA-CEE is proposed in this research. There are three operators introduced in MWOA-CEE. An exponentially decreasing function design is used to keep a balance between exploitation and exploration. The search performance is improved using an opposition-based learning technique and elite-guided Cauchy mutation. Experiments on 19 basic optimization benchmark functions, 14 CEC2014 benchmark functions, and three applications on IIR system identification indicate the effectiveness of MWOA-CEE.The rest of this article is arranged as follows: The fundamental WOA algorithm is described in Section2. Section 3 describes the suggested algorithm as well as the operators. Section 4 contains the parameter settings and experimental outcomes. The conclusions are given in Section 5.


## 2. Whale Optimization Algorithm

The fundamental algorithm mimics whale behavioral tendencies. The three basic phases of the WOA algorithm are encircling prey, bubble-net assaulting, and searching for prey. The mathematical models of three strategies are described in the following subsections.


### 2.1. Encircling Prey

Encircling prey is the beginning for all procedures. Whales detect the position of prey and encircle it. This process is simulated by WOA. The global optimal solution is treated as the prey, while the other candidate solutions modify their places in reference to the global optimal solution.The location of the candidate solution,S⇀i+1, is calculated by the two following equations:(1)D⇀=C⇀⋅Sbest⇀i−S⇀i,(2)S⇀i+1=Sbest⇀i−A⇀⋅D⇀,where Sbest⇀i is the global optimal solution, S⇀i denotes the position of candidate solution in the current generation, i refers to the number of current iterations, and A⇀ and C⇀ are coefficient vectors, which are calculated as follows:(3)A⇀=2a⇀⋅r⇀−a⇀,(4)C⇀=2r⇀,where a⇀ decreases linearly from 2 to 0 over iterations and r⇀ is randomly generated vector range in [0, 1]. The global optimal solution gained in the current generation can be used to adjust the position of the candidate solution. The position of candidate solution could be updated via altering the values of A⇀ and C⇀.


### 2.2. Bubble-Net Attacking Method

Two methods are used in this step and they are designed as follows.


#### 2.2.1. Shrinking Encircling Mechanism

Equations (2) and (3) define the mathematical model of the shrinking encircling process. The value of A⇀ depends on the change of a⇀. In other words, by assigning a random number to A⇀ in [−1, 1], the new position of the candidate solution will be found between the current solution and global optimal solution.


#### 2.2.2. Spiral Updating Position

This technique calculates the new position of the candidate solution. The distance between the present candidate solution and global optimal solution is first calculated by equation (5). Then the new position is generated by equation (6).(5)D′⇀=Sbest⇀i−S⇀i,(6)S⇀i+1=D′⇀⋅ebl⋅cos2πl+Sbest⇀i,where b are constants that define the geometry of the logarithmic spiral and l is randomly generated range in [−1, 1], respectively.There is a 50% probability of successful exploitation (attacking the prey) by using either a shrinking mechanism or a spiral model, and this is manipulated by a random numberpro ∈ [0, 1]. The mathematical formula is(7)S⇀i+1=Sbest⇀i−A⇀⋅D⇀,D′⇀⋅ebl⋅cos2πl+Sbest⇀i,pro<0.5,pro≥0.5.


### 2.3. Search for Prey

Whales search at random based on their position. By using equation (3), these processes can be utilized in WOA. A⇀ is designed as a random value less than 1 or higher than −1; this makes a chance for WOA to execute a random search. Thus, the purpose of this method is to enhance WOA’s exploratory capabilities. When A⇀ is greater than 1, it enables WOA to execute a wide search. The following formula is the model:(8)D⇀=C⇀⋅S⇀ri−S⇀iS⇀i+1=S⇀ri−A⇀⋅D⇀,where S⇀r denotes a random position of whale, A⇀ is less than 1, and the global optimal solution of the current iteration is selected for updating candidate solutions.


## 2.1. Encircling Prey

Encircling prey is the beginning for all procedures. Whales detect the position of prey and encircle it. This process is simulated by WOA. The global optimal solution is treated as the prey, while the other candidate solutions modify their places in reference to the global optimal solution.The location of the candidate solution,S⇀i+1, is calculated by the two following equations:(1)D⇀=C⇀⋅Sbest⇀i−S⇀i,(2)S⇀i+1=Sbest⇀i−A⇀⋅D⇀,where Sbest⇀i is the global optimal solution, S⇀i denotes the position of candidate solution in the current generation, i refers to the number of current iterations, and A⇀ and C⇀ are coefficient vectors, which are calculated as follows:(3)A⇀=2a⇀⋅r⇀−a⇀,(4)C⇀=2r⇀,where a⇀ decreases linearly from 2 to 0 over iterations and r⇀ is randomly generated vector range in [0, 1]. The global optimal solution gained in the current generation can be used to adjust the position of the candidate solution. The position of candidate solution could be updated via altering the values of A⇀ and C⇀.


## 2.2. Bubble-Net Attacking Method

Two methods are used in this step and they are designed as follows.


### 2.2.1. Shrinking Encircling Mechanism

Equations (2) and (3) define the mathematical model of the shrinking encircling process. The value of A⇀ depends on the change of a⇀. In other words, by assigning a random number to A⇀ in [−1, 1], the new position of the candidate solution will be found between the current solution and global optimal solution.


### 2.2.2. Spiral Updating Position

This technique calculates the new position of the candidate solution. The distance between the present candidate solution and global optimal solution is first calculated by equation (5). Then the new position is generated by equation (6).(5)D′⇀=Sbest⇀i−S⇀i,(6)S⇀i+1=D′⇀⋅ebl⋅cos2πl+Sbest⇀i,where b are constants that define the geometry of the logarithmic spiral and l is randomly generated range in [−1, 1], respectively.There is a 50% probability of successful exploitation (attacking the prey) by using either a shrinking mechanism or a spiral model, and this is manipulated by a random numberpro ∈ [0, 1]. The mathematical formula is(7)S⇀i+1=Sbest⇀i−A⇀⋅D⇀,D′⇀⋅ebl⋅cos2πl+Sbest⇀i,pro<0.5,pro≥0.5.


## 2.2.1. Shrinking Encircling Mechanism

Equations (2) and (3) define the mathematical model of the shrinking encircling process. The value of A⇀ depends on the change of a⇀. In other words, by assigning a random number to A⇀ in [−1, 1], the new position of the candidate solution will be found between the current solution and global optimal solution.


## 2.2.2. Spiral Updating Position

This technique calculates the new position of the candidate solution. The distance between the present candidate solution and global optimal solution is first calculated by equation (5). Then the new position is generated by equation (6).(5)D′⇀=Sbest⇀i−S⇀i,(6)S⇀i+1=D′⇀⋅ebl⋅cos2πl+Sbest⇀i,where b are constants that define the geometry of the logarithmic spiral and l is randomly generated range in [−1, 1], respectively.There is a 50% probability of successful exploitation (attacking the prey) by using either a shrinking mechanism or a spiral model, and this is manipulated by a random numberpro ∈ [0, 1]. The mathematical formula is(7)S⇀i+1=Sbest⇀i−A⇀⋅D⇀,D′⇀⋅ebl⋅cos2πl+Sbest⇀i,pro<0.5,pro≥0.5.


## 2.3. Search for Prey

Whales search at random based on their position. By using equation (3), these processes can be utilized in WOA. A⇀ is designed as a random value less than 1 or higher than −1; this makes a chance for WOA to execute a random search. Thus, the purpose of this method is to enhance WOA’s exploratory capabilities. When A⇀ is greater than 1, it enables WOA to execute a wide search. The following formula is the model:(8)D⇀=C⇀⋅S⇀ri−S⇀iS⇀i+1=S⇀ri−A⇀⋅D⇀,where S⇀r denotes a random position of whale, A⇀ is less than 1, and the global optimal solution of the current iteration is selected for updating candidate solutions.


## 3. The Proposed Approach

### 3.1. Framework

To address the poor performance on jumping out of the local optimum in basic WOA and enhance the exploration capacity of WOA, this section introduces MWOA-CEE, a modified whale optimization algorithm with convergence and exploitability enhancements. Algorithm 1 describes the pseudocode of MWOA-CEE.Algorithm 1: MWOA-CEE.
(i)
Input: Population sizeN(ii)
The max value of iteration(iii)
Output: The best solutionS∗(iv)
Initializing population Si (i ← 1, 2, …, N),k←10(1)
fori∈1: Ndo(2)
if (rand <0.6)(3)
Generate new solution with OBL(4)
end if(5)
end for(6)
Calculate the fitness of all solutions(7)
Seek the best solutionS∗ with the best fitness(8)
while the stopping criterion is not fulfilled do(9)
Sort the solution by fitness and divide into ES and NES(10)
Updatea and lambda(11)
a←amin+amax−amin×e−kt/Max_iter(12)
lambda←2×e−2t/Max_iter(13)
sus←e−2t/Max_iter(14)
for1i∈1: Ndo(15)
computeA, C, l and generate pro(16)
jrand ← randint (1, m)(17)
for2j∈1 to mdo(18)
ifpro ≤ lambdathen(19)
if rand (0, 1) ≤ 0.9andj = = jrandthen(20)
CM=1/5+1/πarctanC⋅EPbesti−Si(21)
ifsus ≥ sprothen(22)
D←C⋅Srj−Sij(23)
S⇀ij←EPbestj−A⋅D−CM⋅EPbestj(24)
Else(25)
Sij←EPR1j−CM∗EPR2j−NEPR1j−CM∗EPR3j−NEPR2j(26)
end if(27)
Else(28)
Select a random solutionXrand(29)
D←C⋅Srj−Sij(30)
Sij←Srj−A⋅D(31)
end if(32)
elseifpro > lambdathen(33)
if rand (0, 1) ≤ 0.5then(34)
D←C⋅Sbestj−Sij(35)
Sij←Sbestj−A⋅D(36)
Else(37)
D′←Sbestj−Sij(38)
Sij←D′⋅ebl⋅cos2πl+Sbestj(39)
end if(40)
end for2(41)
end for1(42)
Adjust boundary solutions(43)
fori∈1: Ndo(44)
if (rand <0.6)(45)
Generate new solutionOi with OBL(46)
ifOi is suitable than Sithen(47)
Si ← Oi(48)
end if(49)
end if(50)
end for(51)
Find global optimalS∗ with the best fitness(52)
Updatek(53)
end whileAt first, opposition-based learning (OBL) will be utilized in the initialization step. The fitness values of all solutions are calculated subsequently. Then, the solutions that obtained the superior fitness value are defined as elite solutions (ES). The other solutions that obtained inferior fitness value are defined as nonelite solutions (NES). The solution that achieved the best fitness value is defined as the global optimum. Second, the information of global optimum is applied to the Cauchy mutation. The solution is updated with Cauchy mutation or the original WOA method according to a probabilitylambda. pro is a random number ranging between [−1, 1]. If pro is lower than lambda, the Cauchy mutation and elite strategy are used to update the solutions. If pro is greater than lambda, the encircling prey and spiral updating position are used to update the new solutions. If a random number ranging between [0, 1] is less than 0.5, the encircling prey is used to generate solutions. Otherwise, the new solutions are generated by the spiral updating position. From the beginning, more exploration is needed, so lambda is relatively great; as the iteration progresses, lambda decreases exponentially. Finally, the OBL is used again. The opposite positions of all solutions are generated. The solutions with better fitness values are chosen and the global optimum is updated for the next iteration.The OBL is utilized in the initialization step and works for improving the solutions quality. The exponentially decreasing function can balance the exploitation and exploration capabilities in the mid-term of the algorithm, and the exponentially decreasing function boosts the convergence rate. In the solution updating step, the elite-guided Cauchy mutation generates additional disturbance for solutions and leads the solutions to the promising areas. In one loop, the three strategies do not conflict and cooperate with each other to produce better solutions.In the proposed MWOA-CEE, three methods are integrated. The opposition-based learning method (lines 1–5 of Algorithm 1) is used to initialize the population and randomly update solutions in each iteration. Elite guidance mechanism (lines 9 and 25) generates additional perturbation for solutions in order to explore promising spots. The exponentially decreasing function (line 10) is applied to balance the ratio of the exploitation phase and exploration phase. The Cauchy mutation (lines 18–20) possesses more power in jumping out of the local optimum and converging to the global optimum. These three methods are introduced in detail in the following subsections.


### 3.2. Opposition-Based Learning Method

The opposite position (s′) for any candidate solution (s) in the population is calculated using the opposition-based concept [38]. Subsequently, the fitness value of these two solutions is calculated. The one that obtained the better fitness value will be chosen. This comparison helps WOA to find accurate solutions. This strategy has been demonstrated to be effective. [39, 40].In the initialization and solution updating processes of each iteration, opposition-based learning (OBL) is employed. The first step of the concept is to calculate the opposite position (s′) of the source position (s) by the following equation:(9)s′=sub+slb−s,where sub and slb are the upper and lower bounds of the variable.The second step is to calculate the fitness value of the opposite position and source position.The final step is to compare the new opposite position and the source position and choose the superior one.In initialization step, OBL can generate new solutions on opposition side and enhance the diversity in the search space. In solution updating step, OBL can help WOA improve the search performance and jump out of the local optimal solutions.


### 3.3. Exponentially Decreasing Function

In MWOA-CEE,a⇀ decreases exponentially from 2 to 0 and is formulated as follows [41]:(10)a=amin+amax−amin×e−kt/Max_iter,where amin and amax denote the minimum and maximum of a⇀, current iteration is indicated by t, and Max_iter is the maximum iteration. A large value of a⇀ generates a large step size in exploitation phase, which can improve the searching speed. A small step size with a small value of a⇀ can be helpful for exploration phase. k is set to 10 at the beginning. Then k also decreases exponentially from 2 to 0 by the following equation:(11)k=kmin+kmax−kmin×e−kt/Max_iter.The above decreasing function adopted by MWOA-CEE would boost the exploitation of the basic WOA.


### 3.4. Elite-Guided Cauchy Mutation

The elite-guided Cauchy mutation consists of two parts. The elite guidance mechanism divides candidate solutions (S) into two distinguishing solutions. Some of candidate solutions that obtained superior fitness value are defined as elite populations (EP). Other solutions that obtained inferior fitness value are defined as nonelite populations (NEP). The number of EP isτ∗S. The parameter of τ is randomly chosen from [0.3, 0.5], as recommended in [42]. This may properly balance diversity and convergence speed and considerably improve optimization performance. EP∪NEP equals the candidate solutions.The Cauchy mutation generates a wide disturbance near the current candidate solution. As a result, escaping the local optimum is easier for Cauchy mutation. The Cauchy mutation is utilized to improve solutions diversity. The Cauchy mutation increases the algorithm’s global search capability. Cauchy mutation has been confirmed as an effective method [43].In the presented approach, Cauchy mutation is used in the solution updating step. In the basic WOA, equations (2) and (6) are used to update candidate solutions. In the present method, the new solution updating function with Cauchy mutation is defined as follows:(12)CM=15+1πarctanC⋅EPbest⇀i−S⇀i,(13)S⇀i+1=EPbest⇀i−A⇀⋅D⇀−CM⋅EPbest⇀i,where EPbest⇀i contributed to the best solution in EP which is global optimal solution from the tth iteration. CM is calculated by the distance between the global optimal solution and the current solution S⇀i. The Cauchy mutation adds the disturbance information to the tth iteration best solution EPbest⇀i, which makes it easier to escape the local optimum.In another solution updating strategy, the solutions will be randomly chosen from the EP and NEP. The solution from EP is defined asEPR. The solution from NEP is defined as NEPR. The formula is as follows [44]:(14)S⟶i+1=EPR1⇀i−CM∗EPR2⇀i−NEPR1⇀i−CM∗EPR3⇀i−NEPR2⇀i,where CM is a number calculated by equation (13). EPR1⇀i, EPR2⇀i, and EPR3⇀i are randomly chosen from EP, NEPR1⇀i and NEPR2⇀i are randomly chosen from NES. EPR2⇀i−NEPR1⇀i and EPR3⇀i−NEPR2⇀i provided promising directions for the candidate solution. The formula of updating new solutions is(15)S⇀i+1=EPbest⇀i−A⇀⋅D⇀−C⋅EPbest⇀i,EPR1⇀i−C⋅EPR2⇀i−NEPR1⇀i−C⋅EPR3⇀i−NEPR2⇀i,sus≥spro,otherwise,where sus decreases exponentially from 1 to 0. By altering the value of spro, a different updating strategy could be employed. The lower spro speeds up the convergence of the solutions. The greater spro enhanced the exploration of the solutions.


### 3.5. Computational Complexity of MWOA-CEE

The complexity of the algorithm represents the efficiency of the execution of the algorithm. The solution update step given in Algorithm 1 (lines 9–52) dominates the computational complexity of MWOA-CEE in one iteration. In Algorithm 1,N solutions Si, i ∈ {1, 2, …, N}, need to be calculated for generated new solution Si+1, and each dimension (m) is considered. Thus, total O(m ⋅ N + N) are required. Since m > 1, the overall complexity of one generation of MWOA-CEE is approximately O(m ⋅ N).


## 3.1. Framework

To address the poor performance on jumping out of the local optimum in basic WOA and enhance the exploration capacity of WOA, this section introduces MWOA-CEE, a modified whale optimization algorithm with convergence and exploitability enhancements. Algorithm 1 describes the pseudocode of MWOA-CEE.Algorithm 1: MWOA-CEE.
(i)
Input: Population sizeN(ii)
The max value of iteration(iii)
Output: The best solutionS∗(iv)
Initializing population Si (i ← 1, 2, …, N),k←10(1)
fori∈1: Ndo(2)
if (rand <0.6)(3)
Generate new solution with OBL(4)
end if(5)
end for(6)
Calculate the fitness of all solutions(7)
Seek the best solutionS∗ with the best fitness(8)
while the stopping criterion is not fulfilled do(9)
Sort the solution by fitness and divide into ES and NES(10)
Updatea and lambda(11)
a←amin+amax−amin×e−kt/Max_iter(12)
lambda←2×e−2t/Max_iter(13)
sus←e−2t/Max_iter(14)
for1i∈1: Ndo(15)
computeA, C, l and generate pro(16)
jrand ← randint (1, m)(17)
for2j∈1 to mdo(18)
ifpro ≤ lambdathen(19)
if rand (0, 1) ≤ 0.9andj = = jrandthen(20)
CM=1/5+1/πarctanC⋅EPbesti−Si(21)
ifsus ≥ sprothen(22)
D←C⋅Srj−Sij(23)
S⇀ij←EPbestj−A⋅D−CM⋅EPbestj(24)
Else(25)
Sij←EPR1j−CM∗EPR2j−NEPR1j−CM∗EPR3j−NEPR2j(26)
end if(27)
Else(28)
Select a random solutionXrand(29)
D←C⋅Srj−Sij(30)
Sij←Srj−A⋅D(31)
end if(32)
elseifpro > lambdathen(33)
if rand (0, 1) ≤ 0.5then(34)
D←C⋅Sbestj−Sij(35)
Sij←Sbestj−A⋅D(36)
Else(37)
D′←Sbestj−Sij(38)
Sij←D′⋅ebl⋅cos2πl+Sbestj(39)
end if(40)
end for2(41)
end for1(42)
Adjust boundary solutions(43)
fori∈1: Ndo(44)
if (rand <0.6)(45)
Generate new solutionOi with OBL(46)
ifOi is suitable than Sithen(47)
Si ← Oi(48)
end if(49)
end if(50)
end for(51)
Find global optimalS∗ with the best fitness(52)
Updatek(53)
end whileAt first, opposition-based learning (OBL) will be utilized in the initialization step. The fitness values of all solutions are calculated subsequently. Then, the solutions that obtained the superior fitness value are defined as elite solutions (ES). The other solutions that obtained inferior fitness value are defined as nonelite solutions (NES). The solution that achieved the best fitness value is defined as the global optimum. Second, the information of global optimum is applied to the Cauchy mutation. The solution is updated with Cauchy mutation or the original WOA method according to a probabilitylambda. pro is a random number ranging between [−1, 1]. If pro is lower than lambda, the Cauchy mutation and elite strategy are used to update the solutions. If pro is greater than lambda, the encircling prey and spiral updating position are used to update the new solutions. If a random number ranging between [0, 1] is less than 0.5, the encircling prey is used to generate solutions. Otherwise, the new solutions are generated by the spiral updating position. From the beginning, more exploration is needed, so lambda is relatively great; as the iteration progresses, lambda decreases exponentially. Finally, the OBL is used again. The opposite positions of all solutions are generated. The solutions with better fitness values are chosen and the global optimum is updated for the next iteration.The OBL is utilized in the initialization step and works for improving the solutions quality. The exponentially decreasing function can balance the exploitation and exploration capabilities in the mid-term of the algorithm, and the exponentially decreasing function boosts the convergence rate. In the solution updating step, the elite-guided Cauchy mutation generates additional disturbance for solutions and leads the solutions to the promising areas. In one loop, the three strategies do not conflict and cooperate with each other to produce better solutions.In the proposed MWOA-CEE, three methods are integrated. The opposition-based learning method (lines 1–5 of Algorithm 1) is used to initialize the population and randomly update solutions in each iteration. Elite guidance mechanism (lines 9 and 25) generates additional perturbation for solutions in order to explore promising spots. The exponentially decreasing function (line 10) is applied to balance the ratio of the exploitation phase and exploration phase. The Cauchy mutation (lines 18–20) possesses more power in jumping out of the local optimum and converging to the global optimum. These three methods are introduced in detail in the following subsections.


## 3.2. Opposition-Based Learning Method

The opposite position (s′) for any candidate solution (s) in the population is calculated using the opposition-based concept [38]. Subsequently, the fitness value of these two solutions is calculated. The one that obtained the better fitness value will be chosen. This comparison helps WOA to find accurate solutions. This strategy has been demonstrated to be effective. [39, 40].In the initialization and solution updating processes of each iteration, opposition-based learning (OBL) is employed. The first step of the concept is to calculate the opposite position (s′) of the source position (s) by the following equation:(9)s′=sub+slb−s,where sub and slb are the upper and lower bounds of the variable.The second step is to calculate the fitness value of the opposite position and source position.The final step is to compare the new opposite position and the source position and choose the superior one.In initialization step, OBL can generate new solutions on opposition side and enhance the diversity in the search space. In solution updating step, OBL can help WOA improve the search performance and jump out of the local optimal solutions.


## 3.3. Exponentially Decreasing Function

In MWOA-CEE,a⇀ decreases exponentially from 2 to 0 and is formulated as follows [41]:(10)a=amin+amax−amin×e−kt/Max_iter,where amin and amax denote the minimum and maximum of a⇀, current iteration is indicated by t, and Max_iter is the maximum iteration. A large value of a⇀ generates a large step size in exploitation phase, which can improve the searching speed. A small step size with a small value of a⇀ can be helpful for exploration phase. k is set to 10 at the beginning. Then k also decreases exponentially from 2 to 0 by the following equation:(11)k=kmin+kmax−kmin×e−kt/Max_iter.The above decreasing function adopted by MWOA-CEE would boost the exploitation of the basic WOA.


## 3.4. Elite-Guided Cauchy Mutation

The elite-guided Cauchy mutation consists of two parts. The elite guidance mechanism divides candidate solutions (S) into two distinguishing solutions. Some of candidate solutions that obtained superior fitness value are defined as elite populations (EP). Other solutions that obtained inferior fitness value are defined as nonelite populations (NEP). The number of EP isτ∗S. The parameter of τ is randomly chosen from [0.3, 0.5], as recommended in [42]. This may properly balance diversity and convergence speed and considerably improve optimization performance. EP∪NEP equals the candidate solutions.The Cauchy mutation generates a wide disturbance near the current candidate solution. As a result, escaping the local optimum is easier for Cauchy mutation. The Cauchy mutation is utilized to improve solutions diversity. The Cauchy mutation increases the algorithm’s global search capability. Cauchy mutation has been confirmed as an effective method [43].In the presented approach, Cauchy mutation is used in the solution updating step. In the basic WOA, equations (2) and (6) are used to update candidate solutions. In the present method, the new solution updating function with Cauchy mutation is defined as follows:(12)CM=15+1πarctanC⋅EPbest⇀i−S⇀i,(13)S⇀i+1=EPbest⇀i−A⇀⋅D⇀−CM⋅EPbest⇀i,where EPbest⇀i contributed to the best solution in EP which is global optimal solution from the tth iteration. CM is calculated by the distance between the global optimal solution and the current solution S⇀i. The Cauchy mutation adds the disturbance information to the tth iteration best solution EPbest⇀i, which makes it easier to escape the local optimum.In another solution updating strategy, the solutions will be randomly chosen from the EP and NEP. The solution from EP is defined asEPR. The solution from NEP is defined as NEPR. The formula is as follows [44]:(14)S⟶i+1=EPR1⇀i−CM∗EPR2⇀i−NEPR1⇀i−CM∗EPR3⇀i−NEPR2⇀i,where CM is a number calculated by equation (13). EPR1⇀i, EPR2⇀i, and EPR3⇀i are randomly chosen from EP, NEPR1⇀i and NEPR2⇀i are randomly chosen from NES. EPR2⇀i−NEPR1⇀i and EPR3⇀i−NEPR2⇀i provided promising directions for the candidate solution. The formula of updating new solutions is(15)S⇀i+1=EPbest⇀i−A⇀⋅D⇀−C⋅EPbest⇀i,EPR1⇀i−C⋅EPR2⇀i−NEPR1⇀i−C⋅EPR3⇀i−NEPR2⇀i,sus≥spro,otherwise,where sus decreases exponentially from 1 to 0. By altering the value of spro, a different updating strategy could be employed. The lower spro speeds up the convergence of the solutions. The greater spro enhanced the exploration of the solutions.


## 3.5. Computational Complexity of MWOA-CEE

The complexity of the algorithm represents the efficiency of the execution of the algorithm. The solution update step given in Algorithm 1 (lines 9–52) dominates the computational complexity of MWOA-CEE in one iteration. In Algorithm 1,N solutions Si, i ∈ {1, 2, …, N}, need to be calculated for generated new solution Si+1, and each dimension (m) is considered. Thus, total O(m ⋅ N + N) are required. Since m > 1, the overall complexity of one generation of MWOA-CEE is approximately O(m ⋅ N).


## 4. Experiment Results and Analysis

### 4.1. Benchmark Suite

The proposed method employs 19 basic benchmark functions [45] F1–F19, 14 CEC2014 benchmark functions [46] F20–F33, and 3 IIR system identification cases F34–F36. The unimodal functions F1–F7 have just one global optimum solution. The multimodal functions F8–F12 have a lot of local optimum solutions. These functions evaluate whether the algorithm is capable of resolving the premature convergence problem. F13–F19 are the fixed-dimension multimodal functions, which evaluate the convergence property of an algorithm. The CEC2014 benchmark functions include 6 hybrid functions F20–F25 and 8 composition functions F26–F33.


### 4.2. Parameter Setting

To test the effectiveness of the proposed MWOA-CEE, it is compared with basic classical WOA, OWOA [47], OEWOA [21], IWOA [48], PSOGSA [49], and SOA [50] on 19 basic benchmark functions and WOA, OWOA, OEWOA, IWOA, PSOGSA, JADE [51], and SOA on 14 CEC2014 benchmark functions. In basic benchmark functions, the parameter of spro is set to 0.1. As the function needs just a few elite candidate solutions, the more elite candidate solutions may deteriorate the performance of MWOA-CEE. Meanwhile, for the more complicated test function like CEC2014 benchmark functions, the more elite candidate solutions mean more chance of jumping out of the local optimum. So, spro is set to 0.95 for the CEC2014 benchmark functions. The comparison between different spros is presented in Tables 1 and 2. The search region dimension D of CEC2014 benchmark functions and the population size are all set to 30. The other parameters of the proposed algorithm and comparison algorithms are shown in Table 3. All the experiments are performed on MATLAB R2020a on a computer with AMD Ryzen 7 4800H with Radeon Graphics 2.90 GHz and 16 GB RAM.Table 1
Different sets ofspro on 19 basic benchmark functions.
Functionspro = 0.1spro = 0.3spro = 0.5spro = 0.7spro = 0.9F11.71E − 1142.50E − 731.26E − 431.32E − 271.87E − 14F21.47E − 611.28E − 402.99E − 258.10E − 171.37E − 10F36.34E − 559.19E − 341.01E − 146.22E − 0518.99514289F43.24E − 485.01E − 296.70E − 142.57E − 050.022538805F526.3676372425.4548431625.5454186926.2617038326.41747162F60.0146471240.0112444940.0111169690.0325561990.543991875F70.0001100110.0001614410.0005240010.0008843890.003077991F800000F93.26E − 153.26E − 156.81E − 156.21E − 132.14E − 08F100001.85E − 160.045160548F110.0013621740.0008580840.0007843010.0042932160.015263065F120.0605394190.0134531920.0403232830.0202171920.154827139F131.6593709441.6593709451.6593709440.9980038381.659370944F140.0003107490.0003086330.0003076480.000308620.000307909F15−1.021085567−1.031628451−1.031628449−1.031628445−1.031628446F160.3979502380.3979052460.3978880030.3978882730.397889819F1733.0000001173.0000001393.0000000013.000000027F18−3.862775649−3.862778732−3.862780729−3.862780944−3.862779198F19−3.280912551−3.241929685−3.202901276−3.282192145−3.321794249Best95632Table 2
The results of different sets ofspro on CEC2014 benchmark functions.
Functionspro = 0.15spro = 0.35spro = 0.55spro = 0.75spro = 0.95F204.34E + 068.10E + 063.52E + 065.71E + 067.18E + 06F211.73E + 072.39E + 073.59E + 061.75E + 061.65E + 06F222.04E + 032.03E + 032.01E + 031.93E + 031.94E + 03F234.52E + 044.26E + 042.62E + 043.34E + 045.42E + 04F243.59E + 062.98E + 061.03E + 061.22E + 061.80E + 06F252.92E + 032.92E + 032.86E + 032.85E + 033.17E + 03F262.63E + 032.65E + 032.59E + 032.59E + 032.60E + 03F272.60E + 032.60E + 032.60E + 032.60E + 032.60E + 03F282.70E + 032.70E + 032.70E + 032.70E + 032.70E + 03F292.77E + 032.73E + 032.73E + 032.70E + 032.70E + 03F303.55E + 033.38E + 033.42E + 033.43E + 033.45E + 03F314.41E + 034.64E + 034.50E + 034.42E + 034.31E + 03F329.93E + 068.41E + 061.15E + 071.21E + 073.61E + 06F334.72E + 051.42E + 058.02E + 041.26E + 057.88E + 04Best13546Table 3
The parameter setting of algorithms.
No.AlgorithmParameter setting1WOAb = 1; a⇀∈2,0 (decreases linearly)2OWOAb = 1; a⇀∈2,0 (decreases linearly)3OEWOAb = 1; a⇀∈2,0 (decreases linearly)4IWOAb = 1; CR = 0.9; a⇀∈2,0 (decreases nonlinearly)5PSOGSAG0=1; C1 = 0.5; C2 = 1.5, α = 23, ω = rand [0, 1]6JADEp = 0.2, c = 0.1; μcr = 0.5, μf = 0.57SOAA ∈ [2, 0], fc = 28MWOA-CEEb=1; k = 10; a⇀∈2,0 (decreases exponentially)


### 4.3. Experiment Results

Table4 describes the experiment results of MWOA-CEE and its competitors. The best fitness, mean fitness, and standard deviation are used to measure which algorithm is the best. Figure 1 describes their convergence curves.Table 4
Results of MWOA-CEE and comparison algorithms on 19 basic benchmark functions.
FunctionWOAOWOAOEWOAIWOAPSOGSASOAMWOA-CEEF1Best2.83E − 844.60E − 868.77E − 1036.78E − 310.3306838081.95E − 143.57E − 120Mean4.28E − 747.95E − 755.13E − 881.94E − 281025.5959016.61E − 122.78E − 112Std2.29E − 732.66E − 741.59E − 874.53E − 283042.7991698.95E − 121.18E − 111F2Best5.18E − 581.19E − 573.10E − 663.23E − 200.0027746923.33E − 091.66E − 64Mean8.80E − 511.38E − 521.86E − 597.23E − 1920.651326562.44E − 085.06E − 61Std3.38E − 503.33E − 528.36E − 591.13E − 1835.683617192.98E − 081.73E − 60F3Best15577.9892510474.4302913500.625760.0020257344439.8681592.98E − 072.98E − 63Mean45147.6318342743.1852536998.31926.88802038315376.300420.0002124955.66E − 50Std16573.4640813398.6757914259.7371316.169708596198.3621850.0006704062.17E − 49F4Best2.101505020.9358583832.3510929016.06E − 0530.416878050.0001005924.69E − 50Mean51.6811970255.8011499250.493812250.00114905563.114451510.0179450581.25E − 47Std26.1563242227.3720143529.093643130.00155285621.010544290.0524517883.62E − 47F5Best27.2686784326.8395570627.3223248525.015252584.7818564327.1344356925.23117729Mean28.0057579727.7788546328.194887626.195050545336124.23728.2423224126.15813865Std0.4777004680.4732261080.3755871281.00391483529207381.060.513018830.56687665F6Best0.1070626570.0573942880.0955265020.0002860911.7569423012.3915486250.006984223Mean0.33914930.2364315150.4720182010.6182625692357.9123333.2094363420.011989604Std0.2177415040.1204899280.1875016940.4202676554280.6280770.4087260290.002751251F7Best8.31E − 056.48E − 051.31E − 050.0011143530.0841043180.0007464634.12E − 05Mean0.0040536990.0012285410.0006035560.0044897360.1944930130.0037007920.00030277Std0.0041261550.0015643230.0008316770.0026762040.0651733570.0026479340.000248748F8Best000090.540909783.41E − 130Mean7.58E − 1501.89E − 1535.20661486149.28098154.1649859950Std2.88E − 1401.04E − 1425.6368343332.94381196.748114680F9Best8.88E − 168.88E − 168.88E − 161.51E − 1414.309746919.956300538.88E − 16Mean4.56E − 154.44E − 154.80E − 154.06E − 1417.3680659819.960968033.49E − 15Std2.38E − 152.64E − 152.53E − 151.14E − 141.3860853320.0016705641.60E − 15F10Best00000.1596966834.30E − 140Mean00.0090347210.0150802740.01054064622.457485090.0146505740Std00.0494852050.0825980620.01788495838.431334630.0364655460F11Best0.0045466070.0042640680.0058909960.0116140487.990113750.1427923050.000456486Mean0.0281647480.0120484510.0201199110.04702090518.969076860.3366950320.001157217Std0.0313310560.0063772730.0109238010.0316992637.4300226490.1410750210.001365987F12Best0.0676049010.1305744490.1851836320.42143902920.98878011.5323191160.005454466Mean0.5693368540.3172071980.4610704790.867234398145.42676362.0786580960.018223123Std0.2553124820.1631018830.1828098840.246047227284.87378560.2054131020.013388545F13Best0.9980038380.9980038380.9980038380.9980038380.9980038380.9980038380.998003838Mean2.7961194362.0542509141.9522702334.6499422923.8656905312.5367495221.394824102Std3.2851957441.4198092881.8436911244.4541172064.3204902782.9215326710.807208015F14Best0.0003095360.0003092060.000310650.0003074860.0004607820.0003213220.000307512Mean0.0006753430.0008320430.0007075870.0057004580.0048989240.0011584150.000341407Std0.0002894260.0005928840.0004675180.0089952460.0078945750.0002325070.000166965F15Best−1.031628453−1.031628453−1.031628453−1.031628453−1.031628453−1.031628409−1.031628453Mean−1.031628453−1.031628453−1.031628453−1.031628453−1.031628453−1.031625794−1.031628353Std1.32E − 095.41E − 106.14E − 102.18E − 105.68E − 162.67E − 061.41E − 07F16Best0.3978873580.3978873580.3978873580.3978873580.3978873580.3978933190.397887369Mean0.3978946350.3978907850.3979013380.3978875680.3978873580.3980683350.397913948Std1.31E − 054.84E − 064.52E − 053.99E − 0700.0001781927.73E − 05F17Best3.0000000153.0000000033.000000007333.0000000043Mean3.0000421673.0000499643.0001437828.4000001615.73.0000910043.000002175Std0.000109970.0001381370.0004678720.5503589514.788509050.0001459974.88E − 06F18Best−3.862781338−3.862781855−3.862781142−3.862782146−3.862782148−3.860730401−3.862782Mean−3.856384903−3.860930234−3.85569394−3.861945455−3.862782148−3.855357337−3.862768163Std0.0067851090.0042340010.0317041360.0023913352.34E − 150.0015687861.37E − 05F19Best−3.321538276−3.321772806−3.321653036−3.321994988−3.321995172−3.191987194−3.321955125Mean−3.25327714−3.253025606−3.237354937−3.281605624−3.268292808−3.04292584−3.293175831Std0.0838083240.0883023170.0957852980.0581109030.0635432980.046977140.052130247+/ = /−16/1/216/1/217/0/217/0/216/0/319/0/0−/−/−Figure 1
Convergence curves of MWOA-CEE in comparison to other algorithms on 19 basic benchmark functions.
(a)(b)The MWOA-CEE obtained the overall best result inF1–F4, F6–F14, and F17. For F5, the best fitness value of MWOA-CEE is slightly worse than that of IWOA, while the mean fitness value of MWOA-CEE is better than that of IWOA. For F10, the MWOA-CEE and WOA get similar results. In benchmark suites F15 and F16, the MWOA-CEE performs slightly worse than OEWOA. For F18, the MWOA-CEE performs only slightly worse than PSOGSA. For F19, the mean fitness value of MWOA-CEE is better compared to other competitors and ranks first. The standard deviation value of SOA is slightly better than that of MWOA-CEE. However, as shown in Figure 1, the MWOA-CEE method has a substantially more rapid convergence rate compared to the other algorithms. For most of the benchmark functions, the convergence curves of MWOA-CEE are the most stable. So, that means the Cauchy mutation is useful for escaping the local optimum. The elite guidance mechanism leads the current solutions to the promising region. The exponentially decreasing function also speeds up the convergence rate and makes the performance of the algorithm stable.As shown in Figure1, for the unimodal functions that have just one global optimum solution like F1–F7 and multimodal functions with many local optimal solutions like F8–F12, the convergence speed of MWOA-CEE is the fastest and the most stable. However, IWOA fails to escape the local optimal solutions. The results obtained by IWOA are also nonideal. It is shown that not only is the DE operator insufficient for jumping out of local optimal, but also the stability of the DE operator is impaired for the multimodal functions. For the fixed-dimension multimodal functions like F13–F19, IWOA fails to solve F14 and F17. But MWOA-CEE in solving such problems is efficient. Therefore, the Cauchy mutation does improve the characteristic of jumping out of the local optimal regions and the OBL augments the search performance of the algorithm. In addition, the exponential decrease of a⇀ avoids premature convergence, and, for some functions with only one global optimum, the exponential decrease also enhances the convergence speed.The results analyzed by the Wilcoxon rank-sum test are shown in Table4. In the Wilcoxon rank-sum test, a significance level is set to 0.05. In Table 4, “+,” “−,” and “=” indicate that the result of the proposed algorithm MWOA-CEE is considerably better than, significantly worse than, and statistically similar to the compared algorithms, respectively. For most test functions, the performance of MWOA-CEE is significantly better than or statistically similar to those of the other variants of WOA.In Figure1, the convergence performance of MWOA-CEE for most of the functions is better compared to other comparison algorithms. For the first 13 functions, not only does MWOA-CEE converge to the global optimum, but also the convergence rate is the fastest and the most stable. For some functions like F15–F19, MWOA-CEE is statistically similar to other algorithms.The boxplots are presented in Figure2. For most of the functions, the stability of MWOA-CEE is better than or statistically similar to the stabilities of the other variants of WOA. Only for F15 and F16 is the stability of MWOA-CEE worse than the stability of IWOA. The convergence curves and the boxplots indicate that the improvement has been proven efficient in solving such problems. The OBL is useful for achieving better solution quality and helping the algorithm escape the local optimum. The exponentially decreasing function boosts the convergence rate and enhances the exploration ability. The Cauchy mutation and elite guidance mechanism do help algorithm escape the local optimum.Figure 2
Boxplots of MWOA-CEE in comparison to other algorithms on 19 basic benchmark functions.
(a)(b)Table5 presents the p values of the Wilcoxon rank-sum test on 19 basic benchmark functions of MWOA-CEE and other comparison algorithms. The p values are lower than 0.05 for most of the functions, which means the proposed MWOA-CEE is superior to other comparison algorithms. Therefore, this statistical value verifies that the enhancement of the MWOA-CEE is prominent.Table 5
p values of Wilcoxon rank-sum test obtained on 19 basic benchmark functions.
FunctionWOAOWOAOEWOAIWOAPSOGSASOAF11.73E − 061.73E − 061.73E − 061.73E − 061.73E − 061.73E − 06F21.73E − 061.73E − 066.14E − 011.73E − 061.73E − 061.73E − 06F31.73E − 061.73E − 061.73E − 061.73E − 061.73E − 061.73E − 06F41.73E − 061.73E − 061.73E − 061.73E − 061.73E − 061.73E − 06F51.73E − 061.73E − 061.92E − 067.50E − 011.73E − 061.92E − 06F61.73E − 061.73E − 061.73E − 063.18E − 061.73E − 061.73E − 06F71.73E − 061.04E − 031.47E − 011.73E − 061.73E − 061.73E − 06F85.00E − 011.00E + 001.00E + 002.56E − 061.73E − 061.73E − 06F96.84E − 021.37E − 012.76E − 021.55E − 061.73E − 061.73E − 06F101.00E + 005.00E − 011.00E + 006.10E − 051.73E − 061.73E − 06F111.73E − 061.73E − 061.73E − 061.73E − 061.73E − 061.73E − 06F121.73E − 061.73E − 061.73E − 061.73E − 061.73E − 061.73E − 06F139.84E − 031.96E − 022.70E − 029.84E − 036.42E − 031.29E − 03F142.35E − 061.49E − 051.97E − 053.93E − 019.32E − 061.73E − 06F152.88E − 061.92E − 062.35E − 061.73E − 061.73E − 061.73E − 06F167.66E − 013.87E − 029.37E − 021.92E − 061.73E − 063.41E − 05F171.89E − 041.74E − 046.89E − 052.58E − 033.11E − 052.88E − 06F182.88E − 061.24E − 054.73E − 061.48E − 021.73E − 061.73E − 06F194.99E − 033.68E − 027.27E − 034.41E − 019.26E − 011.73E − 06Table6 presents the results of the MWOA-CEE and other comparison algorithms on CEC2014 benchmark functions. For F21, F23, F24, F26, F27, F28, and F29, the result of the mean fitness obtained by MWOA-CEE is significantly better compared to other comparison algorithms. The standard deviation value of MWOA-CEE is slightly worse than that of PSOGSA on function F22. For F31, the proposed MWOA-CEE is worse than PSOGSA and ranks second. When compared to basic WOA and its typical variants, the suggested MWOA-CEE surpasses others and meets better results for the CEC2014 functions. Furthermore, the p values of the Wilcoxon rank-sum test on CEC2014 benchmark functions are presented in Table 7. For most of the functions, the p values of MWOA-CEE are less than 0.05, which indicated the effectiveness of the improvement.Table 6
Results of MWOA-CEE and comparison algorithms on 14 CEC2014 benchmark functions.
FunctionWOAOWOAOEWOAIWOAJADEPSOGSASOAMWOA-CEEF20Mean24174477.3324924917.3831360431.623199351.0515073451.564220034.8935655905.8145591895.356Std14207710.6222144873.9223091823.011851859.4912982323.473529882.4344140641.7773765520.914F21Mean10468095.736803675.87825122073.475681820.40847242914.3511991280743864328.493813478Std14703468.4511548387.5929319333.1215032669.1109128551.5340333599.729758288.083889471.19F22Mean2023.7944722033.759172056.1888991959.9531151963.0266341993.1133662015.4849461958.06618Std50.856779570.6198188256.087586737.0493581821.567620984.783502836.7389328433.08233462F23Mean242462.3831139688.4851168749.375538465.1809941602.9384256726.3580741833.7301231844.9218Std270187.7187138438.4222149794.528224991.2555927467.3196337884.4173117646.3382515406.6095F24Mean13078553.7814721084.5622536411.161507545.96640598.0574814380.351523780.031356397.03Std12497008.9912552859.6517577681.472116767.9365561887.26510667959.391760616.3521538004.54F25Mean3372.8884463348.3534413209.124132834.3673393093.2791693181.5479882926.4492842817.24135Std324.4097994320.1919597278.5844211187.1737108136.543769320.0357952187.5291062196.9935527F26Mean2720.2569052721.0058052730.8183482639.0875192630.6618062695.4478772681.0356412626.01187Std58.2640404728.8739789338.6436567411.262999828.4609995175.0691358225.4826191351.99787637F27Mean2610.8688082604.9918052605.4865522606.3038142636.1283732681.3809952600.0784942600.05961Std6.4863962357.9547291675.41043239113.308216619.25046283431.10500340.033130850.078456985F28Mean2716.2428882716.9084792716.4628962717.0303252713.5484722724.2360482715.6603592700Std19.873464217.8362038320.242075154.728696313.85203333810.2076010713.602273995.29E − 07F29Mean2717.2470542703.9615992708.0837942761.735652701.69022782.814982701.5339632700.765Std37.6843249718.1441787625.0027971289.341842291.57875800670.56929920.6805068890.13821941F30Mean3822.360273862.3555943818.2570193582.2209783569.6469253816.2590473902.204673507.1465Std368.2525357358.6079282347.7093214279.5051762198.261812309.9478831113.519936309.3860242F31Mean5699.8932985678.4400575622.8899744610.9573623897.804024870.462594583.0014774537.966845Std694.0775185804.9001091545.7985072528.6770144108.409638477.0569603490.860482545.6677578F32Mean13293436.1120691600.7422253640.921886952.726940620.5237424564.4019468915.7174552588.933Std9194739.29821370823.1616674043.513970903.1512236064.256535124.6713882608.9975658861.4F33Mean636059.9173676948.6649641817.866464763.275238473.5619936073.5064222250.943104754.8516Std498027.7531650625.0113498969.504366800.5159317000.608735326.65592149301.483870325.12469Table 7
p values of Wilcoxon rank-sum test obtained on CEC2014 benchmark functions.
FunctionWOAOWOAOEWOAIWOAJADEPSOGSASOAF203.18E − 063.88E − 064.73E − 061.32E − 026.64E − 041.65E − 018.29E − 01F212.07E − 021.71E − 016.34E − 069.78E − 028.92E − 056.87E − 021.92E − 06F223.11E − 054.73E − 061.73E − 066.14E − 012.29E − 011.02E − 011.92E − 06F233.18E − 062.35E − 063.52E − 065.44E − 011.53E − 014.99E − 033.33E − 02F242.13E − 063.88E − 062.88E − 069.10E − 018.47E − 066.88E − 018.61E − 01F253.18E − 062.60E − 065.31E − 055.72E − 012.60E − 051.60E − 041.96E − 02F261.49E − 051.73E − 061.73E − 064.05E − 011.59E − 011.29E − 031.64E − 05F271.73E − 061.92E − 061.73E − 061.73E − 061.73E − 061.73E − 062.30E − 02F281.02E − 012.07E − 024.72E − 021.73E − 061.73E − 061.73E − 061.73E − 06F296.88E − 011.40E − 021.71E − 033.88E − 046.56E − 021.73E − 063.18E − 06F302.96E − 031.29E − 037.71E − 044.78E − 014.28E − 018.94E − 045.75E − 06F311.02E − 055.79E − 053.18E − 067.04E − 012.37E − 053.16E − 027.19E − 01F324.20E − 042.16E − 055.75E − 066.27E − 024.28E − 027.52E − 022.58E − 03F334.73E − 062.35E − 066.34E − 066.84E − 036.89E − 052.22E − 046.16E − 04


### 4.4. Application on IIR System Identification

#### 4.4.1. IIR System Identification

The purpose of system identification is to build a predictive model for an unidentified plant. The black box tactic is an efficient approach to address the model identification problem. The mathematical model is generated solely from the input and output signals. Figure3 depicts the IIR paradigm for the system identification procedure.Figure 3
System identification process.The following transfer function is used to depict an IIR system:(16)YΖXΖ=α0+α1Ζ−1+α2Ζ−2+⋯+αmΖ−m1+β1Ζ−1+β2Ζ−2+⋯+βnΖ−n.In the IIR model, the general form is formulated as(17)yη=∑i=1Nαi⋅yη−i+∑j=0Mβj⋅xη−j,where x (η) and y (η) are the input signal. The numerator and denominator orders are represented by M and N, respectively, where αi and βj are the coefficients of IIR filter which define the poles and zeroes of IIR system. The mean square error of input and output is defined by the cost function f (θ) which is defined as follows:(18)minimizefθ=1S∑n=1Seη2,where S denotes the number of samples employed in the experiments. θ = {a1, …, an, b1, …, bm} represents the unknown exact parameters of the IIR mathematical model, and e(η) = (d(η)−y(η)).Three commonly used IIR identification scenarios, named IIR1 [52], IIR2 [53], and IIR3 [54], were chosen for our experimental study.The plant for IIR1 is a sixth-order system, with fourth-order IIR filter to be constructed as follows:(19)HPΖ−1=1−0.4Ζ−1−0.65Ζ−4+0.26Ζ−61+0.77Ζ−2+0.8498Ζ−4−0.6486Ζ−6,HMΖ−1=α0+α1Ζ−1+α2Ζ−2+α3Ζ−3+α4Ζ−41+β1Ζ−1+β2Ζ−2+β3Ζ−3+β4Ζ−4.The second example, IIR2, is a fourth-order plant that has been modeled with third order.(20)HPΖ−1=1−0.9Ζ−1+0.81Ζ−2−0.729Ζ−31−0.04Ζ−1−0.2775Ζ−2+0.2101Ζ−3−0.14Ζ−4,HMΖ−1=α0+α1Ζ−1+α2Ζ−21+β1Ζ−1+β2Ζ−2+β3Ζ−3.The unknown plant in the third scenario IIR3 is a sixth-order plant, whereas the IIR filter is of fourth order.(21)HPΖ−1=0.1084+0.5419Ζ−1+1.0837Ζ−2−1.0837Ζ−3+0.5419Ζ−4+0.1084Ζ−51−0.9853Ζ−1−0.9738Ζ−2−0.3864Ζ−3−0.1112Ζ−4−0.0113Ζ−5,HMΖ−1=α0+α1Ζ−1+α2Ζ−2+α3Ζ−3+α4Ζ−41+β1Ζ−1+β2Ζ−2+β3Ζ−3+β4Ζ−4.


#### 4.4.2. Simulation Results

In simulations, 30 search agents are chosen for all the five algorithms, the max iterations are set to 500, and the runtimes are set to 30. Tables8–10 show the outcomes of solving three IIR system identification cases. The best fitness, mean fitness, and standard deviation of test results are all presented. The convergence curves and boxplots are shown in Figures 4–6.Table 8
Simulation results of WOA, OWOA, OEWOA, IWOA, PSOGSA, SOA, and MWOA-CEE on IIR1.
WOAOWOAOEWOAIWOAPSOGSASOAMWOA-CEEIIR1Best0.00131150.0025790.005626580.000765840.0012110320.000204540.001148047Mean0.027100520.021170690.02281180.013182410.0120032790.018068980.011403406Std0.019380150.011255990.010943280.013324790.0096227530.019050870.006004819Table 9
Simulation results of WOA, OWOA, OEWOA, IWOA, PSOGSA, SOA, and MWOA-CEE on IIR2.
WOAOWOAOEWOAIWOAPSOGSASOAMWOA-CEEIIR2Best0.0036753130.0032315580.0042826430.0039615910.0032394470.004034370.003795381Mean0.0117339840.0115650130.0135535250.0104574280.0075904240.009690450.004944563Std0.012572910.0093219350.0143235070.0046572490.0039294310.007755090.001468054Table 10
Simulation results of WOA, OWOA, OEWOA, IWOA, PSOGSA, SOA, and MWOA-CEE on IIR3.
WOAOWOAOEWOAIWOAPSOGSASOAMWOA-CEEIIR3Best0.0009659990.0031125110.0042512890.0003550890.0002388780.002371320.000229781Mean0.1893503450.1162357430.1603759380.0943829590.0606508890.08695440.038437271Std0.1611242090.095838530.1063430620.1024844440.1115446370.108397660.061115939Figure 4
Results of MWOA-CEE and comparison algorithms on IIR1. (a) Convergence graph of the algorithms. (b) Boxplots of the algorithms.
(a)(b)Figure 5
Results of MWOA-CEE and comparison algorithms on IIR2. (a) Convergence graph of the algorithms. (b) Boxplots of the algorithms.
(a)(b)Figure 6
Results of MWOA-CEE and comparison algorithms on IIR3. (a) Convergence graph of the algorithms. (b) Boxplots of the algorithms.
(a)(b)As shown in Table8, the IWOA obtained the best fitness value. The MWOA-CEE performs somewhat worse than the IWOA and ranks second. However, MWOA-CEE achieved the best mean fitness and standard deviation. The mean fitness of MWOA-CEE is significantly better than those of other algorithms. The standard deviation of MWOA-CEE is slightly better than that of OEWOA. Figure 4 indicates that the MWOA-CEE has the fastest convergence speed when compared to other approaches.Tables9 and 10 show the statistics of IIR2 and IIR3, respectively. Table 9 shows that the suggested MWOA-CEE outperforms all other methods, while OWOA surpasses MWOA-CEE on the best fitness value. The mean fitness and standard deviation of MWOA-CEE are better than those of other algorithms. Table 10 shows that MWOA-CEE has excellent search capabilities and discovered the optimal solution, while other algorithms became trapped in local optimum. For IIR3, the IWOA gets the worse result. As illustrated in Figure 6, the convergence curves of MWOA-CEE are more stable than those of other methods.For IIR2 and IIR3, not only does MWOA-CEE outperform other algorithms in terms of average convergence speed and precision, but also the solution structure of MWOA-CEE is better compared to other algorithms. Although MWOA-CEE did not obtain the best optimal solution in the IIR1 and IIR2 cases, MWOA-CEE is still quite competitive among the other algorithms.From Table11, in all three cases, the p value is mostly less than 5%, which shows that MWOA-CEE is significantly superior to other comparison algorithms. According to the acquired data, MWOA-CEE has a greater capacity to recognize and identify IIR unknown systems compared to the other comparison algorithms. This illustrates that MWOA-CEE achieved great optimization competence and effectiveness.Table 11
p values of WOA, OWOA, OEWOA, IWOA, PSOGSA, and SOA on IIR system identification cases.
WOAOWOAOEWOAIWOAPSOGSASOAIIR12.41E − 047.16E − 041.24E − 058.45E − 019.10E − 011.59E − 01IIR22.37E − 051.74E − 041.80E − 051.80E − 053.61E − 038.92E − 05IIR32.22E − 047.16E − 044.86E − 054.28E − 029.26E − 011.66E − 02


## 4.1. Benchmark Suite

The proposed method employs 19 basic benchmark functions [45] F1–F19, 14 CEC2014 benchmark functions [46] F20–F33, and 3 IIR system identification cases F34–F36. The unimodal functions F1–F7 have just one global optimum solution. The multimodal functions F8–F12 have a lot of local optimum solutions. These functions evaluate whether the algorithm is capable of resolving the premature convergence problem. F13–F19 are the fixed-dimension multimodal functions, which evaluate the convergence property of an algorithm. The CEC2014 benchmark functions include 6 hybrid functions F20–F25 and 8 composition functions F26–F33.


## 4.2. Parameter Setting

To test the effectiveness of the proposed MWOA-CEE, it is compared with basic classical WOA, OWOA [47], OEWOA [21], IWOA [48], PSOGSA [49], and SOA [50] on 19 basic benchmark functions and WOA, OWOA, OEWOA, IWOA, PSOGSA, JADE [51], and SOA on 14 CEC2014 benchmark functions. In basic benchmark functions, the parameter of spro is set to 0.1. As the function needs just a few elite candidate solutions, the more elite candidate solutions may deteriorate the performance of MWOA-CEE. Meanwhile, for the more complicated test function like CEC2014 benchmark functions, the more elite candidate solutions mean more chance of jumping out of the local optimum. So, spro is set to 0.95 for the CEC2014 benchmark functions. The comparison between different spros is presented in Tables 1 and 2. The search region dimension D of CEC2014 benchmark functions and the population size are all set to 30. The other parameters of the proposed algorithm and comparison algorithms are shown in Table 3. All the experiments are performed on MATLAB R2020a on a computer with AMD Ryzen 7 4800H with Radeon Graphics 2.90 GHz and 16 GB RAM.Table 1
Different sets ofspro on 19 basic benchmark functions.
Functionspro = 0.1spro = 0.3spro = 0.5spro = 0.7spro = 0.9F11.71E − 1142.50E − 731.26E − 431.32E − 271.87E − 14F21.47E − 611.28E − 402.99E − 258.10E − 171.37E − 10F36.34E − 559.19E − 341.01E − 146.22E − 0518.99514289F43.24E − 485.01E − 296.70E − 142.57E − 050.022538805F526.3676372425.4548431625.5454186926.2617038326.41747162F60.0146471240.0112444940.0111169690.0325561990.543991875F70.0001100110.0001614410.0005240010.0008843890.003077991F800000F93.26E − 153.26E − 156.81E − 156.21E − 132.14E − 08F100001.85E − 160.045160548F110.0013621740.0008580840.0007843010.0042932160.015263065F120.0605394190.0134531920.0403232830.0202171920.154827139F131.6593709441.6593709451.6593709440.9980038381.659370944F140.0003107490.0003086330.0003076480.000308620.000307909F15−1.021085567−1.031628451−1.031628449−1.031628445−1.031628446F160.3979502380.3979052460.3978880030.3978882730.397889819F1733.0000001173.0000001393.0000000013.000000027F18−3.862775649−3.862778732−3.862780729−3.862780944−3.862779198F19−3.280912551−3.241929685−3.202901276−3.282192145−3.321794249Best95632Table 2
The results of different sets ofspro on CEC2014 benchmark functions.
Functionspro = 0.15spro = 0.35spro = 0.55spro = 0.75spro = 0.95F204.34E + 068.10E + 063.52E + 065.71E + 067.18E + 06F211.73E + 072.39E + 073.59E + 061.75E + 061.65E + 06F222.04E + 032.03E + 032.01E + 031.93E + 031.94E + 03F234.52E + 044.26E + 042.62E + 043.34E + 045.42E + 04F243.59E + 062.98E + 061.03E + 061.22E + 061.80E + 06F252.92E + 032.92E + 032.86E + 032.85E + 033.17E + 03F262.63E + 032.65E + 032.59E + 032.59E + 032.60E + 03F272.60E + 032.60E + 032.60E + 032.60E + 032.60E + 03F282.70E + 032.70E + 032.70E + 032.70E + 032.70E + 03F292.77E + 032.73E + 032.73E + 032.70E + 032.70E + 03F303.55E + 033.38E + 033.42E + 033.43E + 033.45E + 03F314.41E + 034.64E + 034.50E + 034.42E + 034.31E + 03F329.93E + 068.41E + 061.15E + 071.21E + 073.61E + 06F334.72E + 051.42E + 058.02E + 041.26E + 057.88E + 04Best13546Table 3
The parameter setting of algorithms.
No.AlgorithmParameter setting1WOAb = 1; a⇀∈2,0 (decreases linearly)2OWOAb = 1; a⇀∈2,0 (decreases linearly)3OEWOAb = 1; a⇀∈2,0 (decreases linearly)4IWOAb = 1; CR = 0.9; a⇀∈2,0 (decreases nonlinearly)5PSOGSAG0=1; C1 = 0.5; C2 = 1.5, α = 23, ω = rand [0, 1]6JADEp = 0.2, c = 0.1; μcr = 0.5, μf = 0.57SOAA ∈ [2, 0], fc = 28MWOA-CEEb=1; k = 10; a⇀∈2,0 (decreases exponentially)


## 4.3. Experiment Results

Table4 describes the experiment results of MWOA-CEE and its competitors. The best fitness, mean fitness, and standard deviation are used to measure which algorithm is the best. Figure 1 describes their convergence curves.Table 4
Results of MWOA-CEE and comparison algorithms on 19 basic benchmark functions.
FunctionWOAOWOAOEWOAIWOAPSOGSASOAMWOA-CEEF1Best2.83E − 844.60E − 868.77E − 1036.78E − 310.3306838081.95E − 143.57E − 120Mean4.28E − 747.95E − 755.13E − 881.94E − 281025.5959016.61E − 122.78E − 112Std2.29E − 732.66E − 741.59E − 874.53E − 283042.7991698.95E − 121.18E − 111F2Best5.18E − 581.19E − 573.10E − 663.23E − 200.0027746923.33E − 091.66E − 64Mean8.80E − 511.38E − 521.86E − 597.23E − 1920.651326562.44E − 085.06E − 61Std3.38E − 503.33E − 528.36E − 591.13E − 1835.683617192.98E − 081.73E − 60F3Best15577.9892510474.4302913500.625760.0020257344439.8681592.98E − 072.98E − 63Mean45147.6318342743.1852536998.31926.88802038315376.300420.0002124955.66E − 50Std16573.4640813398.6757914259.7371316.169708596198.3621850.0006704062.17E − 49F4Best2.101505020.9358583832.3510929016.06E − 0530.416878050.0001005924.69E − 50Mean51.6811970255.8011499250.493812250.00114905563.114451510.0179450581.25E − 47Std26.1563242227.3720143529.093643130.00155285621.010544290.0524517883.62E − 47F5Best27.2686784326.8395570627.3223248525.015252584.7818564327.1344356925.23117729Mean28.0057579727.7788546328.194887626.195050545336124.23728.2423224126.15813865Std0.4777004680.4732261080.3755871281.00391483529207381.060.513018830.56687665F6Best0.1070626570.0573942880.0955265020.0002860911.7569423012.3915486250.006984223Mean0.33914930.2364315150.4720182010.6182625692357.9123333.2094363420.011989604Std0.2177415040.1204899280.1875016940.4202676554280.6280770.4087260290.002751251F7Best8.31E − 056.48E − 051.31E − 050.0011143530.0841043180.0007464634.12E − 05Mean0.0040536990.0012285410.0006035560.0044897360.1944930130.0037007920.00030277Std0.0041261550.0015643230.0008316770.0026762040.0651733570.0026479340.000248748F8Best000090.540909783.41E − 130Mean7.58E − 1501.89E − 1535.20661486149.28098154.1649859950Std2.88E − 1401.04E − 1425.6368343332.94381196.748114680F9Best8.88E − 168.88E − 168.88E − 161.51E − 1414.309746919.956300538.88E − 16Mean4.56E − 154.44E − 154.80E − 154.06E − 1417.3680659819.960968033.49E − 15Std2.38E − 152.64E − 152.53E − 151.14E − 141.3860853320.0016705641.60E − 15F10Best00000.1596966834.30E − 140Mean00.0090347210.0150802740.01054064622.457485090.0146505740Std00.0494852050.0825980620.01788495838.431334630.0364655460F11Best0.0045466070.0042640680.0058909960.0116140487.990113750.1427923050.000456486Mean0.0281647480.0120484510.0201199110.04702090518.969076860.3366950320.001157217Std0.0313310560.0063772730.0109238010.0316992637.4300226490.1410750210.001365987F12Best0.0676049010.1305744490.1851836320.42143902920.98878011.5323191160.005454466Mean0.5693368540.3172071980.4610704790.867234398145.42676362.0786580960.018223123Std0.2553124820.1631018830.1828098840.246047227284.87378560.2054131020.013388545F13Best0.9980038380.9980038380.9980038380.9980038380.9980038380.9980038380.998003838Mean2.7961194362.0542509141.9522702334.6499422923.8656905312.5367495221.394824102Std3.2851957441.4198092881.8436911244.4541172064.3204902782.9215326710.807208015F14Best0.0003095360.0003092060.000310650.0003074860.0004607820.0003213220.000307512Mean0.0006753430.0008320430.0007075870.0057004580.0048989240.0011584150.000341407Std0.0002894260.0005928840.0004675180.0089952460.0078945750.0002325070.000166965F15Best−1.031628453−1.031628453−1.031628453−1.031628453−1.031628453−1.031628409−1.031628453Mean−1.031628453−1.031628453−1.031628453−1.031628453−1.031628453−1.031625794−1.031628353Std1.32E − 095.41E − 106.14E − 102.18E − 105.68E − 162.67E − 061.41E − 07F16Best0.3978873580.3978873580.3978873580.3978873580.3978873580.3978933190.397887369Mean0.3978946350.3978907850.3979013380.3978875680.3978873580.3980683350.397913948Std1.31E − 054.84E − 064.52E − 053.99E − 0700.0001781927.73E − 05F17Best3.0000000153.0000000033.000000007333.0000000043Mean3.0000421673.0000499643.0001437828.4000001615.73.0000910043.000002175Std0.000109970.0001381370.0004678720.5503589514.788509050.0001459974.88E − 06F18Best−3.862781338−3.862781855−3.862781142−3.862782146−3.862782148−3.860730401−3.862782Mean−3.856384903−3.860930234−3.85569394−3.861945455−3.862782148−3.855357337−3.862768163Std0.0067851090.0042340010.0317041360.0023913352.34E − 150.0015687861.37E − 05F19Best−3.321538276−3.321772806−3.321653036−3.321994988−3.321995172−3.191987194−3.321955125Mean−3.25327714−3.253025606−3.237354937−3.281605624−3.268292808−3.04292584−3.293175831Std0.0838083240.0883023170.0957852980.0581109030.0635432980.046977140.052130247+/ = /−16/1/216/1/217/0/217/0/216/0/319/0/0−/−/−Figure 1
Convergence curves of MWOA-CEE in comparison to other algorithms on 19 basic benchmark functions.
(a)(b)The MWOA-CEE obtained the overall best result inF1–F4, F6–F14, and F17. For F5, the best fitness value of MWOA-CEE is slightly worse than that of IWOA, while the mean fitness value of MWOA-CEE is better than that of IWOA. For F10, the MWOA-CEE and WOA get similar results. In benchmark suites F15 and F16, the MWOA-CEE performs slightly worse than OEWOA. For F18, the MWOA-CEE performs only slightly worse than PSOGSA. For F19, the mean fitness value of MWOA-CEE is better compared to other competitors and ranks first. The standard deviation value of SOA is slightly better than that of MWOA-CEE. However, as shown in Figure 1, the MWOA-CEE method has a substantially more rapid convergence rate compared to the other algorithms. For most of the benchmark functions, the convergence curves of MWOA-CEE are the most stable. So, that means the Cauchy mutation is useful for escaping the local optimum. The elite guidance mechanism leads the current solutions to the promising region. The exponentially decreasing function also speeds up the convergence rate and makes the performance of the algorithm stable.As shown in Figure1, for the unimodal functions that have just one global optimum solution like F1–F7 and multimodal functions with many local optimal solutions like F8–F12, the convergence speed of MWOA-CEE is the fastest and the most stable. However, IWOA fails to escape the local optimal solutions. The results obtained by IWOA are also nonideal. It is shown that not only is the DE operator insufficient for jumping out of local optimal, but also the stability of the DE operator is impaired for the multimodal functions. For the fixed-dimension multimodal functions like F13–F19, IWOA fails to solve F14 and F17. But MWOA-CEE in solving such problems is efficient. Therefore, the Cauchy mutation does improve the characteristic of jumping out of the local optimal regions and the OBL augments the search performance of the algorithm. In addition, the exponential decrease of a⇀ avoids premature convergence, and, for some functions with only one global optimum, the exponential decrease also enhances the convergence speed.The results analyzed by the Wilcoxon rank-sum test are shown in Table4. In the Wilcoxon rank-sum test, a significance level is set to 0.05. In Table 4, “+,” “−,” and “=” indicate that the result of the proposed algorithm MWOA-CEE is considerably better than, significantly worse than, and statistically similar to the compared algorithms, respectively. For most test functions, the performance of MWOA-CEE is significantly better than or statistically similar to those of the other variants of WOA.In Figure1, the convergence performance of MWOA-CEE for most of the functions is better compared to other comparison algorithms. For the first 13 functions, not only does MWOA-CEE converge to the global optimum, but also the convergence rate is the fastest and the most stable. For some functions like F15–F19, MWOA-CEE is statistically similar to other algorithms.The boxplots are presented in Figure2. For most of the functions, the stability of MWOA-CEE is better than or statistically similar to the stabilities of the other variants of WOA. Only for F15 and F16 is the stability of MWOA-CEE worse than the stability of IWOA. The convergence curves and the boxplots indicate that the improvement has been proven efficient in solving such problems. The OBL is useful for achieving better solution quality and helping the algorithm escape the local optimum. The exponentially decreasing function boosts the convergence rate and enhances the exploration ability. The Cauchy mutation and elite guidance mechanism do help algorithm escape the local optimum.Figure 2
Boxplots of MWOA-CEE in comparison to other algorithms on 19 basic benchmark functions.
(a)(b)Table5 presents the p values of the Wilcoxon rank-sum test on 19 basic benchmark functions of MWOA-CEE and other comparison algorithms. The p values are lower than 0.05 for most of the functions, which means the proposed MWOA-CEE is superior to other comparison algorithms. Therefore, this statistical value verifies that the enhancement of the MWOA-CEE is prominent.Table 5
p values of Wilcoxon rank-sum test obtained on 19 basic benchmark functions.
FunctionWOAOWOAOEWOAIWOAPSOGSASOAF11.73E − 061.73E − 061.73E − 061.73E − 061.73E − 061.73E − 06F21.73E − 061.73E − 066.14E − 011.73E − 061.73E − 061.73E − 06F31.73E − 061.73E − 061.73E − 061.73E − 061.73E − 061.73E − 06F41.73E − 061.73E − 061.73E − 061.73E − 061.73E − 061.73E − 06F51.73E − 061.73E − 061.92E − 067.50E − 011.73E − 061.92E − 06F61.73E − 061.73E − 061.73E − 063.18E − 061.73E − 061.73E − 06F71.73E − 061.04E − 031.47E − 011.73E − 061.73E − 061.73E − 06F85.00E − 011.00E + 001.00E + 002.56E − 061.73E − 061.73E − 06F96.84E − 021.37E − 012.76E − 021.55E − 061.73E − 061.73E − 06F101.00E + 005.00E − 011.00E + 006.10E − 051.73E − 061.73E − 06F111.73E − 061.73E − 061.73E − 061.73E − 061.73E − 061.73E − 06F121.73E − 061.73E − 061.73E − 061.73E − 061.73E − 061.73E − 06F139.84E − 031.96E − 022.70E − 029.84E − 036.42E − 031.29E − 03F142.35E − 061.49E − 051.97E − 053.93E − 019.32E − 061.73E − 06F152.88E − 061.92E − 062.35E − 061.73E − 061.73E − 061.73E − 06F167.66E − 013.87E − 029.37E − 021.92E − 061.73E − 063.41E − 05F171.89E − 041.74E − 046.89E − 052.58E − 033.11E − 052.88E − 06F182.88E − 061.24E − 054.73E − 061.48E − 021.73E − 061.73E − 06F194.99E − 033.68E − 027.27E − 034.41E − 019.26E − 011.73E − 06Table6 presents the results of the MWOA-CEE and other comparison algorithms on CEC2014 benchmark functions. For F21, F23, F24, F26, F27, F28, and F29, the result of the mean fitness obtained by MWOA-CEE is significantly better compared to other comparison algorithms. The standard deviation value of MWOA-CEE is slightly worse than that of PSOGSA on function F22. For F31, the proposed MWOA-CEE is worse than PSOGSA and ranks second. When compared to basic WOA and its typical variants, the suggested MWOA-CEE surpasses others and meets better results for the CEC2014 functions. Furthermore, the p values of the Wilcoxon rank-sum test on CEC2014 benchmark functions are presented in Table 7. For most of the functions, the p values of MWOA-CEE are less than 0.05, which indicated the effectiveness of the improvement.Table 6
Results of MWOA-CEE and comparison algorithms on 14 CEC2014 benchmark functions.
FunctionWOAOWOAOEWOAIWOAJADEPSOGSASOAMWOA-CEEF20Mean24174477.3324924917.3831360431.623199351.0515073451.564220034.8935655905.8145591895.356Std14207710.6222144873.9223091823.011851859.4912982323.473529882.4344140641.7773765520.914F21Mean10468095.736803675.87825122073.475681820.40847242914.3511991280743864328.493813478Std14703468.4511548387.5929319333.1215032669.1109128551.5340333599.729758288.083889471.19F22Mean2023.7944722033.759172056.1888991959.9531151963.0266341993.1133662015.4849461958.06618Std50.856779570.6198188256.087586737.0493581821.567620984.783502836.7389328433.08233462F23Mean242462.3831139688.4851168749.375538465.1809941602.9384256726.3580741833.7301231844.9218Std270187.7187138438.4222149794.528224991.2555927467.3196337884.4173117646.3382515406.6095F24Mean13078553.7814721084.5622536411.161507545.96640598.0574814380.351523780.031356397.03Std12497008.9912552859.6517577681.472116767.9365561887.26510667959.391760616.3521538004.54F25Mean3372.8884463348.3534413209.124132834.3673393093.2791693181.5479882926.4492842817.24135Std324.4097994320.1919597278.5844211187.1737108136.543769320.0357952187.5291062196.9935527F26Mean2720.2569052721.0058052730.8183482639.0875192630.6618062695.4478772681.0356412626.01187Std58.2640404728.8739789338.6436567411.262999828.4609995175.0691358225.4826191351.99787637F27Mean2610.8688082604.9918052605.4865522606.3038142636.1283732681.3809952600.0784942600.05961Std6.4863962357.9547291675.41043239113.308216619.25046283431.10500340.033130850.078456985F28Mean2716.2428882716.9084792716.4628962717.0303252713.5484722724.2360482715.6603592700Std19.873464217.8362038320.242075154.728696313.85203333810.2076010713.602273995.29E − 07F29Mean2717.2470542703.9615992708.0837942761.735652701.69022782.814982701.5339632700.765Std37.6843249718.1441787625.0027971289.341842291.57875800670.56929920.6805068890.13821941F30Mean3822.360273862.3555943818.2570193582.2209783569.6469253816.2590473902.204673507.1465Std368.2525357358.6079282347.7093214279.5051762198.261812309.9478831113.519936309.3860242F31Mean5699.8932985678.4400575622.8899744610.9573623897.804024870.462594583.0014774537.966845Std694.0775185804.9001091545.7985072528.6770144108.409638477.0569603490.860482545.6677578F32Mean13293436.1120691600.7422253640.921886952.726940620.5237424564.4019468915.7174552588.933Std9194739.29821370823.1616674043.513970903.1512236064.256535124.6713882608.9975658861.4F33Mean636059.9173676948.6649641817.866464763.275238473.5619936073.5064222250.943104754.8516Std498027.7531650625.0113498969.504366800.5159317000.608735326.65592149301.483870325.12469Table 7
p values of Wilcoxon rank-sum test obtained on CEC2014 benchmark functions.
FunctionWOAOWOAOEWOAIWOAJADEPSOGSASOAF203.18E − 063.88E − 064.73E − 061.32E − 026.64E − 041.65E − 018.29E − 01F212.07E − 021.71E − 016.34E − 069.78E − 028.92E − 056.87E − 021.92E − 06F223.11E − 054.73E − 061.73E − 066.14E − 012.29E − 011.02E − 011.92E − 06F233.18E − 062.35E − 063.52E − 065.44E − 011.53E − 014.99E − 033.33E − 02F242.13E − 063.88E − 062.88E − 069.10E − 018.47E − 066.88E − 018.61E − 01F253.18E − 062.60E − 065.31E − 055.72E − 012.60E − 051.60E − 041.96E − 02F261.49E − 051.73E − 061.73E − 064.05E − 011.59E − 011.29E − 031.64E − 05F271.73E − 061.92E − 061.73E − 061.73E − 061.73E − 061.73E − 062.30E − 02F281.02E − 012.07E − 024.72E − 021.73E − 061.73E − 061.73E − 061.73E − 06F296.88E − 011.40E − 021.71E − 033.88E − 046.56E − 021.73E − 063.18E − 06F302.96E − 031.29E − 037.71E − 044.78E − 014.28E − 018.94E − 045.75E − 06F311.02E − 055.79E − 053.18E − 067.04E − 012.37E − 053.16E − 027.19E − 01F324.20E − 042.16E − 055.75E − 066.27E − 024.28E − 027.52E − 022.58E − 03F334.73E − 062.35E − 066.34E − 066.84E − 036.89E − 052.22E − 046.16E − 04


## 4.4. Application on IIR System Identification

### 4.4.1. IIR System Identification

The purpose of system identification is to build a predictive model for an unidentified plant. The black box tactic is an efficient approach to address the model identification problem. The mathematical model is generated solely from the input and output signals. Figure3 depicts the IIR paradigm for the system identification procedure.Figure 3
System identification process.The following transfer function is used to depict an IIR system:(16)YΖXΖ=α0+α1Ζ−1+α2Ζ−2+⋯+αmΖ−m1+β1Ζ−1+β2Ζ−2+⋯+βnΖ−n.In the IIR model, the general form is formulated as(17)yη=∑i=1Nαi⋅yη−i+∑j=0Mβj⋅xη−j,where x (η) and y (η) are the input signal. The numerator and denominator orders are represented by M and N, respectively, where αi and βj are the coefficients of IIR filter which define the poles and zeroes of IIR system. The mean square error of input and output is defined by the cost function f (θ) which is defined as follows:(18)minimizefθ=1S∑n=1Seη2,where S denotes the number of samples employed in the experiments. θ = {a1, …, an, b1, …, bm} represents the unknown exact parameters of the IIR mathematical model, and e(η) = (d(η)−y(η)).Three commonly used IIR identification scenarios, named IIR1 [52], IIR2 [53], and IIR3 [54], were chosen for our experimental study.The plant for IIR1 is a sixth-order system, with fourth-order IIR filter to be constructed as follows:(19)HPΖ−1=1−0.4Ζ−1−0.65Ζ−4+0.26Ζ−61+0.77Ζ−2+0.8498Ζ−4−0.6486Ζ−6,HMΖ−1=α0+α1Ζ−1+α2Ζ−2+α3Ζ−3+α4Ζ−41+β1Ζ−1+β2Ζ−2+β3Ζ−3+β4Ζ−4.The second example, IIR2, is a fourth-order plant that has been modeled with third order.(20)HPΖ−1=1−0.9Ζ−1+0.81Ζ−2−0.729Ζ−31−0.04Ζ−1−0.2775Ζ−2+0.2101Ζ−3−0.14Ζ−4,HMΖ−1=α0+α1Ζ−1+α2Ζ−21+β1Ζ−1+β2Ζ−2+β3Ζ−3.The unknown plant in the third scenario IIR3 is a sixth-order plant, whereas the IIR filter is of fourth order.(21)HPΖ−1=0.1084+0.5419Ζ−1+1.0837Ζ−2−1.0837Ζ−3+0.5419Ζ−4+0.1084Ζ−51−0.9853Ζ−1−0.9738Ζ−2−0.3864Ζ−3−0.1112Ζ−4−0.0113Ζ−5,HMΖ−1=α0+α1Ζ−1+α2Ζ−2+α3Ζ−3+α4Ζ−41+β1Ζ−1+β2Ζ−2+β3Ζ−3+β4Ζ−4.


### 4.4.2. Simulation Results

In simulations, 30 search agents are chosen for all the five algorithms, the max iterations are set to 500, and the runtimes are set to 30. Tables8–10 show the outcomes of solving three IIR system identification cases. The best fitness, mean fitness, and standard deviation of test results are all presented. The convergence curves and boxplots are shown in Figures 4–6.Table 8
Simulation results of WOA, OWOA, OEWOA, IWOA, PSOGSA, SOA, and MWOA-CEE on IIR1.
WOAOWOAOEWOAIWOAPSOGSASOAMWOA-CEEIIR1Best0.00131150.0025790.005626580.000765840.0012110320.000204540.001148047Mean0.027100520.021170690.02281180.013182410.0120032790.018068980.011403406Std0.019380150.011255990.010943280.013324790.0096227530.019050870.006004819Table 9
Simulation results of WOA, OWOA, OEWOA, IWOA, PSOGSA, SOA, and MWOA-CEE on IIR2.
WOAOWOAOEWOAIWOAPSOGSASOAMWOA-CEEIIR2Best0.0036753130.0032315580.0042826430.0039615910.0032394470.004034370.003795381Mean0.0117339840.0115650130.0135535250.0104574280.0075904240.009690450.004944563Std0.012572910.0093219350.0143235070.0046572490.0039294310.007755090.001468054Table 10
Simulation results of WOA, OWOA, OEWOA, IWOA, PSOGSA, SOA, and MWOA-CEE on IIR3.
WOAOWOAOEWOAIWOAPSOGSASOAMWOA-CEEIIR3Best0.0009659990.0031125110.0042512890.0003550890.0002388780.002371320.000229781Mean0.1893503450.1162357430.1603759380.0943829590.0606508890.08695440.038437271Std0.1611242090.095838530.1063430620.1024844440.1115446370.108397660.061115939Figure 4
Results of MWOA-CEE and comparison algorithms on IIR1. (a) Convergence graph of the algorithms. (b) Boxplots of the algorithms.
(a)(b)Figure 5
Results of MWOA-CEE and comparison algorithms on IIR2. (a) Convergence graph of the algorithms. (b) Boxplots of the algorithms.
(a)(b)Figure 6
Results of MWOA-CEE and comparison algorithms on IIR3. (a) Convergence graph of the algorithms. (b) Boxplots of the algorithms.
(a)(b)As shown in Table8, the IWOA obtained the best fitness value. The MWOA-CEE performs somewhat worse than the IWOA and ranks second. However, MWOA-CEE achieved the best mean fitness and standard deviation. The mean fitness of MWOA-CEE is significantly better than those of other algorithms. The standard deviation of MWOA-CEE is slightly better than that of OEWOA. Figure 4 indicates that the MWOA-CEE has the fastest convergence speed when compared to other approaches.Tables9 and 10 show the statistics of IIR2 and IIR3, respectively. Table 9 shows that the suggested MWOA-CEE outperforms all other methods, while OWOA surpasses MWOA-CEE on the best fitness value. The mean fitness and standard deviation of MWOA-CEE are better than those of other algorithms. Table 10 shows that MWOA-CEE has excellent search capabilities and discovered the optimal solution, while other algorithms became trapped in local optimum. For IIR3, the IWOA gets the worse result. As illustrated in Figure 6, the convergence curves of MWOA-CEE are more stable than those of other methods.For IIR2 and IIR3, not only does MWOA-CEE outperform other algorithms in terms of average convergence speed and precision, but also the solution structure of MWOA-CEE is better compared to other algorithms. Although MWOA-CEE did not obtain the best optimal solution in the IIR1 and IIR2 cases, MWOA-CEE is still quite competitive among the other algorithms.From Table11, in all three cases, the p value is mostly less than 5%, which shows that MWOA-CEE is significantly superior to other comparison algorithms. According to the acquired data, MWOA-CEE has a greater capacity to recognize and identify IIR unknown systems compared to the other comparison algorithms. This illustrates that MWOA-CEE achieved great optimization competence and effectiveness.Table 11
p values of WOA, OWOA, OEWOA, IWOA, PSOGSA, and SOA on IIR system identification cases.
WOAOWOAOEWOAIWOAPSOGSASOAIIR12.41E − 047.16E − 041.24E − 058.45E − 019.10E − 011.59E − 01IIR22.37E − 051.74E − 041.80E − 051.80E − 053.61E − 038.92E − 05IIR32.22E − 047.16E − 044.86E − 054.28E − 029.26E − 011.66E − 02


## 4.4.1. IIR System Identification

The purpose of system identification is to build a predictive model for an unidentified plant. The black box tactic is an efficient approach to address the model identification problem. The mathematical model is generated solely from the input and output signals. Figure3 depicts the IIR paradigm for the system identification procedure.Figure 3
System identification process.The following transfer function is used to depict an IIR system:(16)YΖXΖ=α0+α1Ζ−1+α2Ζ−2+⋯+αmΖ−m1+β1Ζ−1+β2Ζ−2+⋯+βnΖ−n.In the IIR model, the general form is formulated as(17)yη=∑i=1Nαi⋅yη−i+∑j=0Mβj⋅xη−j,where x (η) and y (η) are the input signal. The numerator and denominator orders are represented by M and N, respectively, where αi and βj are the coefficients of IIR filter which define the poles and zeroes of IIR system. The mean square error of input and output is defined by the cost function f (θ) which is defined as follows:(18)minimizefθ=1S∑n=1Seη2,where S denotes the number of samples employed in the experiments. θ = {a1, …, an, b1, …, bm} represents the unknown exact parameters of the IIR mathematical model, and e(η) = (d(η)−y(η)).Three commonly used IIR identification scenarios, named IIR1 [52], IIR2 [53], and IIR3 [54], were chosen for our experimental study.The plant for IIR1 is a sixth-order system, with fourth-order IIR filter to be constructed as follows:(19)HPΖ−1=1−0.4Ζ−1−0.65Ζ−4+0.26Ζ−61+0.77Ζ−2+0.8498Ζ−4−0.6486Ζ−6,HMΖ−1=α0+α1Ζ−1+α2Ζ−2+α3Ζ−3+α4Ζ−41+β1Ζ−1+β2Ζ−2+β3Ζ−3+β4Ζ−4.The second example, IIR2, is a fourth-order plant that has been modeled with third order.(20)HPΖ−1=1−0.9Ζ−1+0.81Ζ−2−0.729Ζ−31−0.04Ζ−1−0.2775Ζ−2+0.2101Ζ−3−0.14Ζ−4,HMΖ−1=α0+α1Ζ−1+α2Ζ−21+β1Ζ−1+β2Ζ−2+β3Ζ−3.The unknown plant in the third scenario IIR3 is a sixth-order plant, whereas the IIR filter is of fourth order.(21)HPΖ−1=0.1084+0.5419Ζ−1+1.0837Ζ−2−1.0837Ζ−3+0.5419Ζ−4+0.1084Ζ−51−0.9853Ζ−1−0.9738Ζ−2−0.3864Ζ−3−0.1112Ζ−4−0.0113Ζ−5,HMΖ−1=α0+α1Ζ−1+α2Ζ−2+α3Ζ−3+α4Ζ−41+β1Ζ−1+β2Ζ−2+β3Ζ−3+β4Ζ−4.


## 4.4.2. Simulation Results

In simulations, 30 search agents are chosen for all the five algorithms, the max iterations are set to 500, and the runtimes are set to 30. Tables8–10 show the outcomes of solving three IIR system identification cases. The best fitness, mean fitness, and standard deviation of test results are all presented. The convergence curves and boxplots are shown in Figures 4–6.Table 8
Simulation results of WOA, OWOA, OEWOA, IWOA, PSOGSA, SOA, and MWOA-CEE on IIR1.
WOAOWOAOEWOAIWOAPSOGSASOAMWOA-CEEIIR1Best0.00131150.0025790.005626580.000765840.0012110320.000204540.001148047Mean0.027100520.021170690.02281180.013182410.0120032790.018068980.011403406Std0.019380150.011255990.010943280.013324790.0096227530.019050870.006004819Table 9
Simulation results of WOA, OWOA, OEWOA, IWOA, PSOGSA, SOA, and MWOA-CEE on IIR2.
WOAOWOAOEWOAIWOAPSOGSASOAMWOA-CEEIIR2Best0.0036753130.0032315580.0042826430.0039615910.0032394470.004034370.003795381Mean0.0117339840.0115650130.0135535250.0104574280.0075904240.009690450.004944563Std0.012572910.0093219350.0143235070.0046572490.0039294310.007755090.001468054Table 10
Simulation results of WOA, OWOA, OEWOA, IWOA, PSOGSA, SOA, and MWOA-CEE on IIR3.
WOAOWOAOEWOAIWOAPSOGSASOAMWOA-CEEIIR3Best0.0009659990.0031125110.0042512890.0003550890.0002388780.002371320.000229781Mean0.1893503450.1162357430.1603759380.0943829590.0606508890.08695440.038437271Std0.1611242090.095838530.1063430620.1024844440.1115446370.108397660.061115939Figure 4
Results of MWOA-CEE and comparison algorithms on IIR1. (a) Convergence graph of the algorithms. (b) Boxplots of the algorithms.
(a)(b)Figure 5
Results of MWOA-CEE and comparison algorithms on IIR2. (a) Convergence graph of the algorithms. (b) Boxplots of the algorithms.
(a)(b)Figure 6
Results of MWOA-CEE and comparison algorithms on IIR3. (a) Convergence graph of the algorithms. (b) Boxplots of the algorithms.
(a)(b)As shown in Table8, the IWOA obtained the best fitness value. The MWOA-CEE performs somewhat worse than the IWOA and ranks second. However, MWOA-CEE achieved the best mean fitness and standard deviation. The mean fitness of MWOA-CEE is significantly better than those of other algorithms. The standard deviation of MWOA-CEE is slightly better than that of OEWOA. Figure 4 indicates that the MWOA-CEE has the fastest convergence speed when compared to other approaches.Tables9 and 10 show the statistics of IIR2 and IIR3, respectively. Table 9 shows that the suggested MWOA-CEE outperforms all other methods, while OWOA surpasses MWOA-CEE on the best fitness value. The mean fitness and standard deviation of MWOA-CEE are better than those of other algorithms. Table 10 shows that MWOA-CEE has excellent search capabilities and discovered the optimal solution, while other algorithms became trapped in local optimum. For IIR3, the IWOA gets the worse result. As illustrated in Figure 6, the convergence curves of MWOA-CEE are more stable than those of other methods.For IIR2 and IIR3, not only does MWOA-CEE outperform other algorithms in terms of average convergence speed and precision, but also the solution structure of MWOA-CEE is better compared to other algorithms. Although MWOA-CEE did not obtain the best optimal solution in the IIR1 and IIR2 cases, MWOA-CEE is still quite competitive among the other algorithms.From Table11, in all three cases, the p value is mostly less than 5%, which shows that MWOA-CEE is significantly superior to other comparison algorithms. According to the acquired data, MWOA-CEE has a greater capacity to recognize and identify IIR unknown systems compared to the other comparison algorithms. This illustrates that MWOA-CEE achieved great optimization competence and effectiveness.Table 11
p values of WOA, OWOA, OEWOA, IWOA, PSOGSA, and SOA on IIR system identification cases.
WOAOWOAOEWOAIWOAPSOGSASOAIIR12.41E − 047.16E − 041.24E − 058.45E − 019.10E − 011.59E − 01IIR22.37E − 051.74E − 041.80E − 051.80E − 053.61E − 038.92E − 05IIR32.22E − 047.16E − 044.86E − 054.28E − 029.26E − 011.66E − 02


## 5. Conclusions

This paper suggests three strategies for improving the basic WOA’s shortcoming in exploration and convergence. The exploitation and exploration phases are balanced by an exponentially decreasing function. To enhance the capacity of the search phase and the ability to jump out of the local optimal solution, OBL and elite-guided Cauchy mutation operators are incorporated into the algorithm. These strategies avoid being stuck in the local optimum, speed up the convergence rate, and rapidly enhance the performance of original WOA. Seven state-of-the-art algorithms are used for comparison with the MWOA-CEE. As the experimental results show, the proposed algorithm surpasses other comparison algorithms in 19 basic benchmark functions. To evaluate the algorithm’s capacity to solve more complex functions, the proposed MWOA-CEE is tested on 14 CEC2014 benchmark functions. In most of the cases, the MWOA-CEE is significantly superior to other comparison algorithms. In order to further reveal the universality of practical engineering problems, three IIR system identification problems are employed. The experimental results demonstrate that MWOA-CEE is robust and efficient for solving this kind of engineering problem. The above benchmark functions and IIR model identification problems illustrate that the MWOA-CEE approach not only outperforms the original WOA algorithm but also outperforms other algorithms. As a result, the suggested MWOA-CEE has high efficiency and great performance.Despite the fact that MWOA-CEE has achieved remarkable achievements, there are certain drawbacks. For example, the MWOA-CEE is not ideal in terms of convergence accuracy and stability in some circumstances (F20 and F32 in CEC2014 benchmark functions). As a result, there is still potential for development in terms of convergence precision and stability. On the one hand, new strategies for improving the performance of MWOA-CEE and dealing with more complicated, high-dimensional situations and variable population should be investigated in future studies. On the other hand, the approach may be used to tackle optimization issues in a variety of domains.



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