High equilibrium optimizer for global optimization

Abstract

With the advent of the information age, people have higher requirements for basic algorithms. Meta-heuristic algorithms have received wide attention as a high-level strategy to study and generate fully optimized solutions to data-driven optimization problems. Using the advantage of equilibrium optimizer (EO) with better balance mode, combined with the strategy of memetic algorithm, different proportion of temperature is introduced in different stages. That is, EO and thermal exchange optimization (TEO) are fused to obtain a new highly balanced optimizer (HEO). While keeping the guiding strategy and memory mode unchanged of EO, the accuracy of optimization is greatly improved. 14 well-known benchmark functions and 7 selective algorithms were used for HEO evaluation comparison experiments. On the basis of the fitness function curve, the optimal solution and other experimental data are tested statistically. The experimental results show that the improved algorithm has high accuracy and stability, but at the cost of running a little more time. Application testing of complex engineering problems is also one of the main purposes of algorithm design. In this paper, three typical engineering design problems (three truss, welded beam and rolling bearing design) are tested and the experimental results show that this algorithm has certain competitiveness and superiority in classical engineering design.

Keywords

Equilibrium optimizer thermal exchange optimization memetic algorithm benchmark functions engineering design problems

1 Introduction

Algorithm refers to an accurate and complete description of the solution plan and a series of clear instructions to solve the problem which represents the strategy mechanism of using systematic methods to describe the solution. “Dichotomy” and “Newton iterative method” belong to approximate iterative method. Iterative algorithm is a basic method to solve problems by computer. In recent years, the iterative algorithm has attracted much attention because of the meta-heuristic optimization algorithm. Meta-heuristic algorithm is a basic method, independent of the existence of the problem. With its wide applicability, particle swarm optimization (PSO) [1], flower pollination algorithm (FPA) [2], whale optimization algorithm (WOA) [3] and other algorithms have been proposed successively. Most meta-heuristic algorithms are inspired by biological habits and physical phenomena in reality. Because the mathematical modeling of these phenomena is relatively simple, the introduction of meta-heuristic algorithm gradually enters the climax.

The pure meta-heuristic algorithm usually requires the author to have a deep understanding of a certain phenomenon or a certain concept so as to obtain a perfect mathematical modeling. This makes creation not so easy. At the same time, it is found that the advantages of an algorithm may have some defects, so people think of improving the pure algorithm for a certain problem. For example, chaos initialization [4] improved initialization, Levy flight strategy [5] improved particle step size, and Opposition-based Learning (OBL) strategy [6] enhanced optimal solution accuracy. The hybrid algorithm can also be regarded as an improvement of the algorithm, which combines the two algorithms to make up for each other’s shortcomings, so as to achieve the effect that one plus one is greater than two. The application of hybrid algorithm is also very extensive: hybrid gravitational search with multiverse optimization algorithm (GSA-MVO) is used to solve the threshold segmentation problem of color images [7], hybrid whale optimization with simulated annealing algorithm (WOA-SA) can be effectively applied to the problem of feature extraction [8], hybrid flower pollination with clonal selection algorithm (FPA-CSA) is applied to the estimation of photovoltaic parameters [9]. This paper mixes two algorithms based on physical concepts, equilibrium optimizer (EO) [10] and thermal exchange optimization (TEO) [11], and obtains a hybrid algorithm High equilibrium optimizer (HEO).

EO is the most recent algorithm released in November 2019. The inspiration came from a simple hybrid dynamic mass balance optimizer. TEO algorithm was proposed in 2017. Due to its effectiveness and wide applicability, it has also obtained some subsequent promotion and application. Such as improving TEO through Levy flight and other strategies and applying it to multi-threshold segmentation of color images [12]. Mixed with SOA, and use three-dimensional pulse-coupled neural network to process oil image segmentation [13].

The balance of exploration and development in EO has been better prepared. If you want to improve, the most effective way is to improve the accuracy of the different stages. In this regard, the incorporation of temperature changes with different weights in the two stages of quality acceptance and output can be considered as a reasonable approach. Combined with the memetic algorithm strategy which has attracted much attention in recent years, this makes the combination of EO and TEO possible. Through 14 basic functions and 3 practical engineering problems, the effectiveness of the improved algorithm is tested, and finally a conclusion is drawn.

The rest of this paper is structured as follows: In Section 2 describes the EO algorithm briefly; In Section 3, a brief description of TEO and the improved HEO algorithm is made; In Section 4 the experimental results of the test function are introduced and discussed; And the research aspects of solving practical engineering problems with different benchmarks are discussed and analyzed in Section 5; Finally, this paper summarizes the work and describes the perspective of future development.

2 Related works

With the advent of industry 4.0, the industrial network physical system has been widely concerned. Meta-heuristic algorithm, as the underlying algorithm, is also playing a role in the transition from theoretical breakthrough to practical technology in this core interdisciplinary research field of industry and academia [14, 15]. The innovation of meta-heuristic algorithm is increasing, and the application prospect is becoming more and more extensive. Taking the traditional and representative PSO algorithm as an example, by using it to optimize the parameters of LSSVM and combine it with fuzzy information granulation to realize the function of predicting the change range and trend of freight volume. Which provides a new method for the study of logistics system performance [16]. There are also multi-objective heuristic optimization algorithm is used to solve the controller optimization problem in intelligent control [17]. As an engineering application of similar work in this paper, the hybrid heuristic algorithm GA-IPA (Genetic algorithm and Interior point algorithms) is applied to the calculation of axisymmetric viscoelastic heat transport performance [18].

As can be seen from the literature review, scholars and experts have never stopped exploring. The NFL says there is no single algorithm that effectively solves all optimization problems [19]. This prompted us to try to come up with a new algorithm.

3 Equilibrium optimizer

A newly proposed dynamic mass balance optimizer EO inspired by a simple well-mixed dynamic mass balance optimizer on a control body, where the mass balance equation is used to describe the concentration of non-active components in the control body as a function of its various source and sink mechanisms. The mass balance equation can be expressed as follows. $V \frac{dC}{dt} = {QC}_{eq} - QC + G$ (1) $λ = \frac{Q}{V}$ (2) where C, $V \frac{dC}{dt}$ , Q and C_eq are the concentration, the rate of change of mass, the volumetric flow rate and the concentration at an equilibrium state, respectively. Here, λ represents the inverse of the residence time, which is a random vector in the interval of [0,1]. And finally, the correlation between time t and concentration C will get. $\frac{dC}{λ C_{eq} - λ C + \frac{G}{V}} = dt$ (3) $\int_{C_{0}}^{C} \frac{dC}{λ C_{eq} - λ C + \frac{G}{V}} = \int_{t_{0}}^{t} dt$ (4) $C = C_{eq} + (C_{0} - C_{eq}) F + \frac{G}{λ V} (1 - F)$ (5)

where C₀ and t₀ shows the initial start time and concentration. F is the exponential term which can help EO balance the exploration and exploitation stages. $\vec{F} = a_{1} sign (\vec{r} - 0.5) [e^{- \vec{λ} t} - 1]$ (6) $t = {(1 - \frac{Iter}{Max_iter})}^{(a_{2} \frac{Iter}{Max_iter})}$ (7) ${\vec{t}}_{0} = \frac{1}{\vec{λ}} ln (- a_{1} sign (\vec{r} - 0.5) [1 - e^{- \vec{λ} t}]) + t$ (8)

where Iter and Max _ iter present the current and the maximum number of iterations, respectively. a₁, a₂ are the constant value which equals 2 and 1, respectively. r is a random vector in [0,1].

The Generation rate G is presented as follows: $\vec{G} = {\vec{G}}_{0} e^{- \vec{k} (t - t_{0})}$ (9) where $\vec{G_{0}}$ defined as: $\vec{G_{0}} = \vec{GCP} (\vec{C_{eq}} - \vec{λ} \vec{C})$ (10) $\vec{GCP} = {\begin{matrix} 0.5 r_{1} & r_{2} \geq GP \\ 0 & r_{2} < GP \end{matrix}$ (11) where r₁ and r₂ indicate random numbers in [0,1]. GCP is defined as a Generation rate Control Parameter that primarily represents the likelihood of contributing to the update process. GP called Generation Probability which is set to 0.5 in the original article. Finally, the updating formula of position can be calculated as follows: $\vec{C} = {\vec{C}}_{eq} + (\vec{C} - {\vec{C}}_{eq}) \cdot \vec{F} + \frac{\vec{G}}{\vec{λ} V} (1 - \vec{F})$ (12) where V is the unit. And pseudocode of traditional EO for global optimization is given in Algorithm 1.

Algorithm 1: Pseudocode of EO

1. Begin

2. Initialize the position of each particle x_i, i = 1 … n;

3. Initialize the free parameters a₁ = 2 ; a₂ = 1 ; GP = 0.5;

4. While Iter < Max_iter

5. Four leading particles were selected by comparing the fitness function values:

{\vec{C}}_{eq 1}

{\vec{C}}_{eq 2}

C_{eq 3}^{⇀}

C_{eq 4}^{⇀}

{\vec{C}}_{av e}

{\vec{C}}_{eq 1}

{\vec{C}}_{eq 2}

C_{eq 3}^{⇀}

C_{eq 4}^{⇀}

)/4

7. Construct the equilibrium pool

{\vec{C}}_{eq . pool}

{\vec{C}}_{eq 1}

{\vec{C}}_{eq 2}

C_{eq 3}^{⇀}

C_{eq 4}^{⇀}

{\vec{C}}_{e q (ave)}

}

9. Accomplish memory saving (if Iter > 1)

10. Assign

t = {(1 - \frac{I ter}{Max_iter})}^{(a_{2} \frac{Iter}{Max_iter})}

11. For i = 1:n

12. Generate random vector of

\vec{λ}

\vec{r}

13. Construct

\vec{F} = a_{1} sign (\vec{r} - 0.5) [e^{- \vec{λ} t} - 1]

14. Construct

\vec{GCP} = {\begin{matrix} 0.5 r_{1} (r_{2} \geq GP) \\ 0 (r_{2} < GP) \end{matrix}

15. Construct

\vec{G_{0}} = \vec{GCP} (\vec{C_{eq}} - \vec{λ} \vec{C})

16. Construct

\vec{G} = \vec{G_{0}} . \vec{F}

17. Update concentrations

\vec{C} = {\vec{C}}_{eq} + (\vec{C} - {\vec{C}}_{eq}) . \vec{F} + \frac{\vec{G}}{\vec{λ} V} (1 - \vec{F})

18. End(For)

19. Iter = Iter+1

20. End while

21. End

4 Thermal exchange optimization

A new optimization algorithm based on Newton’s cooling law was proposed in 2017. The basic idea is that the heat loss rate of an object is proportional to the temperature difference of its surroundings.

In the model, some objects are defined as cooling objects, while others represent the environment. The formula for temperature updates between different objects is defined as: $T_{i}^{env} = (1 - (c_{1} + c_{2} \times (1 - t)) \times random) \times T_{i}^{' env}$ (13) $t = \frac{l}{L}$ (14) where c₁, c₂ are the controlling variables, $T_{i}^{env}$ is given by ${T^{'}}_{i}^{env}$ , which is the previous temperature of the object. l is the current iteration times, L is the maximum number of iterations.

The update formula can be expressed as $T_{i}^{new} = T_{i}^{env} + (T_{i}^{old} - T_{i}^{env}) exp (- β t)$ (15) $β = \frac{Cos t (object)}{Cos t (worst \begin{matrix} object \end{matrix})}$ (16)

The value of β of each object is obtained by Equation 16, and the value of β varies to determine the transformation of the temperature.

Compare the parameters Pro and rand to determine if you need to change the components of each cooling object. If rand < Pro, regenerate a particle by the following formula. This mechanism is used to avoid the algorithm falling into local optimum. $T_{i, j} = T_{i, min} + rand \times (T_{j, max} - T_{j, min})$ (17)

Where, T_j,max and T_j,min are the lower and upper bounds of the jth variable.

Algorithm 2: Pseudocode of TEO

1. Begin

2. Initialize the temperature T_i (i = 1, 2, …, n)

3. Initialize cmax,cmin, and maximum number of iterations L

4. Calculate the fitness function

5. W hilel < L

6. Update t and β

7. For i = 1 : n

8. Update the position of the current search agent by the Eq.15

9. Evaluate the fitness of all agents

10. Update Cost if there is a better solution;

11. End(For)

12. l = l+1

13. End while

14. End

4.1 Hybrid Algorithm of EO and TEO

Effective algorithm fusion can make the two or more algorithms of fusion show better performance. The existing fusion methods can be divided into probability fusion, multi-population and memetic algorithm. References [20, 21] and [8] are used as examples, respectively. In this paper an embedded hybrid optimization algorithm is proposed, which is involved in memetic algorithm. The novel algorithm can achieve a high precision solution than the two mixed algorithms.

EO and TEO are belong to the optimization algorithm based on the physical principle. The main purpose of EO algorithm is to balance the mass of the object into the output with time. In order to make the balance optimizer balance system more perfect, it can be considered that the mass input and output are also mixed with the change in heat. Which can be called HEO. In this paper, the combination of EO algorithm and TEO improves the search precision and retain the diversity of population especially in subsequent iterations. EO has made adequate preparations for balancing exploration and exploitation, and TEO here is more like a perfect factor to improve the accuracy of EO in both exploration and exploitation.

Therefore, the pseudocode of HEO algorithm can be expressed as flow.

Algorithm 3: Pseudocode of HEO

1. Begin

2. Initialize the search particles X_i (i = 1, 2, …, n)

3. Initialize the free parameters and basic factors

4. W hileIter < Max_iter

5. Complete the preparation phase of EO algorithm

6. For i = 1:n

7. Generate random vector of

\vec{λ}

\vec{r}

8. Construct

\vec{F} = a_{1} sign (\vec{r} - 0.5) [e^{- \vec{λ} t} - 1]

9. Update the TEO factor T_obj by the Equation 15

10. Construct

\vec{GCP} = {\begin{matrix} 0.5 r_{1} (r_{2} \geq GP) \\ 0 (r_{2} < GP) \end{matrix}

11. Construct

\vec{G_{0}} = \vec{GCP} (\vec{C_{eq}} - \vec{λ} \vec{C})

12. Construct

\vec{G} = \vec{G_{0}} . \vec{F}

13. If r₂ ≥ GP

14.

\vec{C} = T_{obj} * 0.9 + ({\vec{C}}_{eq} + (\vec{C} - {\vec{C}}_{eq}) . \vec{F} + \frac{\vec{G}}{\vec{λ} V} (1 - \vec{F})) * 0.1

15. Elseif r₂ < GP

16.

\vec{C} = ({\vec{C}}_{eq} + (\vec{C} - {\vec{C}}_{eq}) . \vec{F} + \frac{\vec{G}}{\vec{λ} V} (1 - \vec{F})) * 0.9 + T_{obj} * 0.1

17. End(If)

18. End(For)

19. Iter = Iter+1

20. End While

21. End

It is well known that there is heat transfer in the process of mass exchange. You can think of heat as always going from high mass to low mass. In the early stage of mass conversion, heat transfer dominates, and in the late stage, mass conversion dominates. Throughout the conversion phase, generation probability GP represents the probability of participation in the update of the control concentration based on the generation rate. It plays a very important regulatory role. When GP = 1, no generation rate term is involved in the optimization process. This state emphasizes high exploration capability and often leads to inaccurate solutions. When GP = 0, the generation rate term always participates in the process, increasing the locally optimal stagnation probability. Therefore, different proportions of temperature conversion factors are suitable for the two ranges [0,0.5] and [0.5,1]. As a rule of thumb, GP = 0.5 provides a good balance between the exploration and development phases. Furthermore, the flowchart of HEO algorithm is shown in Fig. 1.

Fig. 1

The flowchart of HEO algorithm.

5 Experimental results and discussion

In this section, in order to study the numerical efficiency performance of the proposed HEO algorithm, a set of 14 unconstrained functions are taken from the classic benchmark functions CEC2005 (Suganthan et al., May 2005). Which can be divided into two main types:

Unimodal (UM) functions: These functions are used to assess the solution precision and the convergence rate of the proposed algorithm. Which can be used to validate the exploitation (hardening) capabilities of different optimizers. As shown in Table 4 is (F1-F7).

Multi-modal (MM) functions: These functions can be used to verify the potential of the algorithm to exploration (diversify) and avoid falling into local optimum which are from F8 to F14.

The functional composition, dimension, range limitation and optimal position of functions has been given in the table. It is noteworthy that Dim represents the number of variables designed for mathematical functions.

5.1 Experimental setup

All the experimental series were carried out on MATLAB R2017b (The Math Works Inc., Natick, MA, USA), and the computer was configured as Intel(R) Pentium (R) CPU G4560 @3.50 GHz (Intel, Santa Clara, CA, USA), using Microsoft Windows 7 system (Microsoft, Redmond, WA, USA).

5.2 Comparison of each algorithm’s performance

In order to verify the performance of the proposed algorithm, the proposed HEO is compared with 7 other state of the art optimization algorithms according to the test function results. As the basic algorithm, EO and TEO are necessary to give the comparison results. GSA, MVO and SCA are the representatives in the meta-heuristic algorithm based on physical principles. SHO and GWO lead the population renewal pattern similar to EO. The details of the comparison algorithm are as Table 2, where the parameter values and references are given. The population size N is set to 30 and the number of iterations Max _ iter is set to 500 for fair comparison. Meanwhile, all experiments are conducted 30 times.

Table 1
Benchmark functions

Function Dim Range f_min

$f_{1} (x) = \sum_{i = 1}^{n} x_{i}^{2}$ 30 [-100,100] 0

$f_{2} (x) = \sum_{i = 1}^{n} {| x}_{i} | + \prod_{i = 1}^{n} | x_{i} |$ 30 [-10,10] 0

$f_{3} (x) = \sum_{i = 1}^{i} {(\sum_{j - 1}^{i} x_{j})}^{2}$ 30 [-100,100] 0

f₄ (x) = max {|x_i|, 1 ≤ I ≤ n} 30 [-100,100] 0

$f_{5} (x) = \sum_{i = 1}^{n - 1} [100 (x_{i + 1} - x_{i}^{2})^{2} + (x_{i} - 1)^{2}]$ 30 [-30,30] 0

$f_{6} (x) = \sum_{i = 1}^{n} ([x_{i} + 0.5])^{2}$ 30 [-100,100] 0

$f_{7} (x) = \sum_{i = 1}^{n} {ix}_{4} + random [0, 1)$ 30 [-1.28,1.28] 0

$F_{8} (x) = \sum_{i = 1}^{n}$ 30 [-5.12,5.12] 0

$F_{9} (x) = - 20 \exp - 0.2 \sqrt{\frac{1}{n} \sum_{i = 1}^{n} x_{i}^{2}}) - exp (\frac{1}{n} \sum_{i = 1}^{n} cos (2 π x_{i})) + 20 + e$ 30 [-32,32] 0

$F_{10} (x) = \frac{1}{4000} \sum_{i = 1}^{n} x_{i}^{2} - \prod_{i = 1}^{n} cos (\frac{x_{i}}{\sqrt{i}}) + 1$ 30 [-600,600] 0

$\begin{matrix} F_{11} = \frac{π}{n} {10 sin (π y_{1}) + \sum_{i = 1}^{n - 1} (y_{i} - 1)^{2} [1 + 10 {sin}^{2} (π y_{i + 1}) + (y_{n} - 1)^{2}} \\ + \sum_{i = 1}^{n} u (x_{i}, 10, 100, 4) + \sum_{i = 1}^{n} u (x_{i}, 10, 100, 4) \\ y_{i} = 1 + \frac{x_{i} + 1}{4} \\ u (x_{i}, a, k, m = {\begin{matrix} k (x_{i} - a)^{m} \\ 0 \\ k (- x_{i} - a)^{m} \end{matrix} \begin{matrix} x_{i} > a \\ - a < x_{i} < a \\ x_{i} < a \end{matrix} \end{matrix}$ 30 [-50,50] 0

$F_{12} (x) = 0.1 {{sin}^{2} (3 π x_{1}) + \sum_{i = 1}^{n} (x_{i} - 1^{2} [1 + {sin}^{2} (3 π x_{i} + 1)] + (x_{n} - 1)^{2} [1 + {sin}^{2} (2 π x_{n})]}$ 30 [-50,50] 0

$+ \sum_{i = 1}^{n} u (x_{i}, 5, 100, 4)$

$F_{13} = (\frac{1}{500} + \sum_{j = 1}^{25} \frac{1}{j + \sum_{i = 1}^{2} (x_{i} - a_{ij})^{6}})^{- 1}$ 2 [-65,65] 1

$F_{14} = \sum_{i = 1}^{11} [a_{i} - \frac{x_{1} (b_{i}^{2} + b_{i} x_{2})}{b_{i}^{2} + b_{i} x_{3} + x_{4}}]^{2}$ 4 [-5,5] 0.0003

Function	Dim	Range	f_min
$f_{1} (x) = \sum_{i = 1}^{n} x_{i}^{2}$	30	[-100,100]	0
$f_{2} (x) = \sum_{i = 1}^{n} {\| x}_{i} \| + \prod_{i = 1}^{n} \| x_{i} \|$	30	[-10,10]	0
$f_{3} (x) = \sum_{i = 1}^{i} {(\sum_{j - 1}^{i} x_{j})}^{2}$	30	[-100,100]	0
f₄ (x) = max {\|x_i\|, 1 ≤ I ≤ n}	30	[-100,100]	0
$f_{5} (x) = \sum_{i = 1}^{n - 1} [100 (x_{i + 1} - x_{i}^{2})^{2} + (x_{i} - 1)^{2}]$	30	[-30,30]	0
$f_{6} (x) = \sum_{i = 1}^{n} ([x_{i} + 0.5])^{2}$	30	[-100,100]	0
$f_{7} (x) = \sum_{i = 1}^{n} {ix}_{4} + random [0, 1)$	30	[-1.28,1.28]	0
$F_{8} (x) = \sum_{i = 1}^{n}$	30	[-5.12,5.12]	0
$F_{9} (x) = - 20 \exp - 0.2 \sqrt{\frac{1}{n} \sum_{i = 1}^{n} x_{i}^{2}}) - exp (\frac{1}{n} \sum_{i = 1}^{n} cos (2 π x_{i})) + 20 + e$	30	[-32,32]	0
$F_{10} (x) = \frac{1}{4000} \sum_{i = 1}^{n} x_{i}^{2} - \prod_{i = 1}^{n} cos (\frac{x_{i}}{\sqrt{i}}) + 1$	30	[-600,600]	0
$\begin{matrix} F_{11} = \frac{π}{n} {10 sin (π y_{1}) + \sum_{i = 1}^{n - 1} (y_{i} - 1)^{2} [1 + 10 {sin}^{2} (π y_{i + 1}) + (y_{n} - 1)^{2}} \\ + \sum_{i = 1}^{n} u (x_{i}, 10, 100, 4) + \sum_{i = 1}^{n} u (x_{i}, 10, 100, 4) \\ y_{i} = 1 + \frac{x_{i} + 1}{4} \\ u (x_{i}, a, k, m = {\begin{matrix} k (x_{i} - a)^{m} \\ 0 \\ k (- x_{i} - a)^{m} \end{matrix} \begin{matrix} x_{i} > a \\ - a < x_{i} < a \\ x_{i} < a \end{matrix} \end{matrix}$	30	[-50,50]	0
$F_{12} (x) = 0.1 {{sin}^{2} (3 π x_{1}) + \sum_{i = 1}^{n} (x_{i} - 1^{2} [1 + {sin}^{2} (3 π x_{i} + 1)] + (x_{n} - 1)^{2} [1 + {sin}^{2} (2 π x_{n})]}$	30	[-50,50]	0
$+ \sum_{i = 1}^{n} u (x_{i}, 5, 100, 4)$
$F_{13} = (\frac{1}{500} + \sum_{j = 1}^{25} \frac{1}{j + \sum_{i = 1}^{2} (x_{i} - a_{ij})^{6}})^{- 1}$	2	[-65,65]	1
$F_{14} = \sum_{i = 1}^{11} [a_{i} - \frac{x_{1} (b_{i}^{2} + b_{i} x_{2})}{b_{i}^{2} + b_{i} x_{3} + x_{4}}]^{2}$	4	[-5,5]	0.0003

Table 2

Parameters of the algorithms

Reference	Algorithm	Parameters setting
—	HEO	a₁ = [0, 1]; a₂ = [0, 1];GP=0.5; $\vec{λ} = [0, 1]$ ;
		$c_{1}, c_{2} = [\begin{matrix} 0 & or & 1 \end{matrix}]$ ; Pro = [0, 1].
[10]	EO	a₁ = [0, 1]; a₂ = [0, 1]; GP=0.5; $\vec{λ} = [0, 1]$ .
[11]	TEO	$c_{1}, c_{2} = [\begin{matrix} 0 & or & 1 \end{matrix}]$ ; Pro = [0, 1].
[22]	GSA	Alpha = 20; G₀ = 100; Rnorm = 2; Rpower = 1.
[23]	MVO	r₁, r₂, r₃, r₄ = [0, 1]; H = 0.5; WEP=[0.2,1];
		TDR = [0,1].
[24]	SHO	$\vec{h} = [0, 5]$ ; $\vec{M} = [0.5, 1]$ ; $\vec{B} = [0, 2]$ ; $\vec{E} = [0, 2]$ .
[25]	GWO	a = [0, 2]; r₁, r₂ = [0, 1].
[26]	SCA	r₁ = [0, 2]; r₂ = [0, 2π]; r₃ = [0, 2]; r₄ = [0, 1].

Table 3

Comparative results on benchmark functions

Functions	Alogrithm
	HEO	EO	TEO	GSA	MVO	SHO	GWO	SCA
f ₀₁	5.60E-196	8.95E-29	1.61E-102	9.18E+04	8.72E-01	1.42E-03	1.38E-28	7.26E+03
f ₀₂	0.00E+00	1.30E-16	2.45E-52	6.65E+10	8.21E+01	1.25E-28	7.36E-17	1.27E-108
f ₀₃	6.44E-198	2.60E-35	4.31E-50	1.82E+06	2.39E+02	5.59E+04	1.76E-08	4.72E+04
f ₀₄	1.22E-96	4.54E-17	4.00E-50	9.64E+01	2.20E+00	1.73E+01	4.13E-07	8.10E+01
f ₀₅	5.93E-02	2.87E+01	3.02E+01	2.20E+08	3.37E+02	1.96E+04	2.79E+01	7.80E+07
f ₀₆	1.11E-03	3.70E-10	7.74E+00	1.15E+05	1.51E+00	3.93E+00	1.03E-04	1.41E+03
f ₀₇	1.18E-05	4.10E-04	6.05E-03	3.15E+01	3.85E-02	1.03E+00	1.35E-03	6.76E+01
f ₀₈	-1.10E+149	-2.58E+05	-2.43E+03	-1.88E+02	-7.90E+03	-1.09E+04	-7.43E+33	-2.91E+03
F ₉	0.00E+00	0.00E+00	2.77E-01	2.78E+02	1.26E+02	0.00E+00	1.00E+00	0.00E+00
f ₁₀	8.88E-16	8.88E-16	2.02E-01	2.07E+01	1.35E+00	6.40E-01	9.33E-14	1.22E+01
f ₁₁	0.00E+00	0.00E+00	9.07E-01	8.81E+02	8.07E-01	2.53E-03	0.00E+00	2.61E+01
f ₁₂	1.79E-05	2.63E-05	9.51E-01	5.97E+08	2.07E+00	3.61E+01	3.97E-02	4.28E+07
f ₁₃	2.97E+00	8.04E-12	3.01E+00	1.52E+09	2.12E-01	8.73E+04	5.61E-01	2.54E+07
f ₁₄	1.02E+00	6.90E+00	6.94E+00	4.98E+02	1.99E+00	7.19E+00	1.08E+01	2.98E+00

Table 4

Computation time

Functions	Alogrithm
	HEO	EO	TEO	GSA	MVO	SHO	GWO	SCA
f ₀₁	4.93E-01	2.77E-01	7.81E-02	9.47E-01	1.78E-01	1.83E-01	1.11E-01	8.34E-02
f ₀₂	5.46E-01	2.95E-01	9.64E-02	9.61E-01	1.77E-01	2.03E-01	1.31E-01	1.03E-01
f ₀₃	9.51E-01	7.10E-01	4.96E-01	1.36E+00	5.92E-01	6.20E-01	5.28E-01	5.05E-01
f ₀₄	5.25E-01	2.77E-01	7.68E-02	9.48E-01	2.20E-01	1.83E-01	1.12E-01	8.59E-02
f ₀₅	5.57E-01	3.14E-01	1.08E-01	9.73E-01	2.00E-01	2.14E-01	1.38E-01	1.14E-01
f ₀₆	5.26E-01	2.76E-01	7.98E-02	9.55E-01	1.80E-01	1.85E-01	1.11E-01	8.55E-02
f ₀₇	6.14E-01	3.61E-01	1.63E-01	1.01E+00	2.55E-01	2.67E-01	1.94E-01	1.68E-01
f ₀₈	5.64E-01	3.10E-01	1.14E-01	9.83E-01	2.05E-01	2.20E-01	1.47E-01	1.21E-01
F ₉	5.20E-01	2.86E-01	9.44E-02	9.59E-01	2.46E-01	1.96E-01	1.18E-01	9.90E-02
f ₁₀	7.70E-01	3.02E-01	1.15E-01	9.80E-01	2.62E-01	2.14E-01	1.32E-01	1.18E-01
f ₁₁	5.61E-01	3.24E-01	1.39E-01	1.00E+00	2.38E-01	2.44E-01	1.57E-01	1.47E-01
f ₁₂	8.30E-01	5.82E-01	3.90E-01	1.26E+00	4.78E-01	5.16E-01	4.13E-01	3.95E-01
f ₁₃	9.63E-01	5.80E-01	3.91E-01	1.25E+00	4.77E-01	5.10E-01	4.10E-01	3.93E-01
f ₁₄	1.23E+00	9.97E-01	8.58E-01	1.17E+00	8.91E-01	9.89E-01	8.26E-01	8.64E-01

To data visualization, facilitate comparison and analysis, eight of the algorithm convergence curve as shown in Table 6. Mathematical statistics were made for the 30 optimal results obtained by each algorithm, and corresponding box graphs were generated. Due to space limitation, functions 1, 4 and 7 from UM function and 10, 13 from MM function are selected. According to the convergence curve, HEO is superior to other methods in most cases, and can maintain population diversity to some extent in the late period of renewal. HEO has advantages in both exploration and development, and it is likely to find a better solution in the exploration stage. EO is roughly second in each set of experiments, and sometimes gets a better optimal solution than HEO. GSA is probably because it’s an earlier algorithm, and it’s not perfect enough so every set of experiments is the worst, but its contribution to optimization cannot be denied.

Table 5

The STD values of benchmark functions

Functions	Alogrithm
	HEO	EO	TEO	GSA	MVO	SHO	GWO	SCA
f ₀₁	0.00E+00	4.44E-21	4.23E-101	1.36E+04	3.14E-01	1.52E+01	1.71E-27	5.04E+03
f ₀₂	2.75E-94	3.59E-12	2.06E-50	1.98E+19	7.04E+01	2.66E-04	8.91E-17	0.00E+00
f ₀₃	3.09E-106	1.45E-17	1.97E-48	2.34E+06	1.41E+02	2.51E+04	5.58E-04	2.03E+04
f ₀₄	2.02E-95	1.35E-09	1.20E-50	4.56E+00	9.82E-01	1.66E+01	9.10E-07	1.07E+01
f ₀₅	7.28E+00	1.43E+01	3.18E+00	7.26E+07	7.63E+02	2.69E+05	7.14E-01	4.32E+07
f ₀₆	1.34E-03	4.54E-09	5.42E-01	1.61E+04	2.96E-01	2.33E+01	4.15E-01	5.12E+03
f ₀₇	7.32E-05	9.89E-05	3.47E-03	2.62E+01	1.29E-02	5.66E-01	1.12E-03	5.41E+01
f ₀₈	Inf	5.59E+06	3.39E+02	1.14E+03	7.05E+02	1.67E+03	3.01E+35	3.45E+02
f ₉	0.00E+00	1.03E+01	2.44E-01	2.08E+01	3.92E+01	3.12E+01	3.77E+00	2.54E-01
f ₁₀	0.00E+00	5.90E-12	1.39E-01	2.28E-01	5.32E-01	5.87E+00	1.62E-14	4.66E+00
f ₁₁	0.00E+00	0.00E+00	3.40E-01	1.49E+02	8.14E-02	2.85E-01	5.91E-03	4.86E+01
f ₁₂	5.06E-05	3.57E-10	2.78E-01	3.35E+08	1.90E+00	4.36E+04	1.42E-02	1.40E+08
f ₁₃	1.21E+00	7.53E-01	2.98E-01	5.16E+08	9.24E-02	1.90E+05	2.12E-01	2.23E+08
f ₁₄	3.87E+00	5.68E+00	2.82E+00	1.88E+02	8.51E-01	5.60E+00	4.25E+00	4.85E+00

Table 6

The convergence curve and Boxplot

Functions	Convergence curve	Boxplot
f ₀₁
f₀₄
f₀₇
f ₁₀
f ₁₃

The running time in Table 4 can also be seen by the convergence curve. The shortest running time data has been marked in bold. According to the data results, TEO has the fastest running time among many algorithms and shows the fastest convergence in the curve. It is clear that HEO and EO time rank relatively low, both in the same order of magnitude. This indicates that the mixing of EO and TEO has been largely improved with less time sacrificed, which also benefits from the fast convergence of TEO.

In order to further evaluate the stability of the algorithm, the boxplot is selected on the graph, which is given after the corresponding convergence curve, and the lower box represents the excellent stability. HEO is always at the bottom and rarely fluctuates. STD values of 30 fitness function values are selected numerically, as shown in Table 5. Generally, the smaller the value of STD is, the better the stability is. HEO contains the largest number of minimum values, accounting for approximately 50%. The combination and correspondence of the two can prove the stability of HEO. SCA, SHO, and GSA are not very stable to some extent.

In order to prove that there are significant differences between algorithms. Wilcoxon rank sum test was further tested by non-parametric statistics [27], assuming a significance level of 5%. A more detailed description of the Wilcoxon rank-sum test can be found in literature. The experimental results are given in Table 7. A p value greater than 0.05 indicates no significant difference between the two groups, while a p value less than 0.05 indicates significant difference. All of the 90 cases achieved good results, indicating that HEO was significantly different from the other 7 algorithms.

Table 7

P-values of the Wilcoxon rank-sum test over 30 runs (p > 0.05 has been in bold)

Functions	HEOvsEO	HEOvsTEO	HEOvsGSA	HEOvsMVO	HEOvsSHO	HEOvsGWO	HEOvsGWO
f ₀₁	1.22E-115	2.91E-114	8.46E-105	1.59E-118	2.56E-142	1.43E-113	7.74E-07
f ₀₂	1.41E-156	4.59E-148	1.79E-124	5.12E-159	2.80E-150	2.26E-157	4.30E-129
f ₀₃	5.79E-165	4.33E-184	5.75E-165	5.79E-165	3.84E-166	2.23E-165	4.37E-188
f ₀₄	2.22E-152	3.24E-157	5.27E-142	1.28E-157	4.16E-153	1.43E-145	3.02E-161
f ₀₅	3.31E-156	1.26E-145	2.27E-162	1.91E-162	7.92E-163	6.55E-160	1.44E-167
f ₀₆	5.72E-117	7.46E-114	9.95E-115	1.48E-130	9.95E-89	2.01E-10	1.22E-62
f ₀₇	3.01E-161	5.39E-142	6.48E-161	2.50E-170	2.73E-171	5.38E-160	4.41E-169
f ₀₈	1.22E-115	2.91E-114	8.46E-105	1.59E-118	2.56E-142	1.43E-113	7.74E-07
f ₉	1.41E-156	4.59E-148	1.79E-124	5.12E-159	2.80E-150	2.26E-157	4.30E-129
f ₁₀	5.79E-165	4.33E-184	5.75E-165	5.79E-165	3.84E-166	2.23E-165	4.37E-188
f ₁₁	2.22E-152	3.24E-157	5.27E-142	1.28E-157	4.16E-153	1.43E-145	3.02E-161
f ₁₂	3.31E-156	1.26E-145	2.27E-162	1.91E-162	7.92E-163	6.55E-160	1.44E-167
f ₁₃	5.72E-117	7.46E-114	9.95E-115	1.48E-130	9.95E-89	2.01E-10	1.22E-62
f ₁₄	3.01E-161	5.39E-142	6.48E-161	2.50E-170	2.73E-171	5.38E-160	4.41E-169

6 HEO for classical engineering problems

The effectiveness of a good algorithm lies in its application. HEO has shown certain advantages in different types of functions in the previous section. Next, three engineering problems were selected for further testing. They are: welded beam design problem, three-bar truss design problem, rolling element bearing problem. For the sake of simplicity, all problems are implemented in MATLAB through the barrier penalty function. The results obtained by HEO and the corresponding results of different algorithms are listed in the table at the end of each engineering problem. The HEO runs independently for each project 30 times, with a selected remora population of 30 and an iteration of 500. Finally, make a corresponding evaluation for different issues.

6.1 Welded beam design problem

As it named, this problem deals with designing a welded beam to minimize the fabrication cost [28]. The minimization process is subject to some constraints such as shear stress, bending stress in the beam, buckling load on the bar, end deflection of the beam, and side constraints. This optimum design has four parameters: thickness of weld (h), length of the clamped bar (l), height of the bar (t), and thickness of the bar (b) as shown in Fig. 2. The mathematical formulation is also illustrated as follows:

Fig. 2

Schematic of the welded beam (Above: Engineering drawing, Below: 3D).

Consider	$\vec{x} = [\begin{matrix} x_{1} & x_{2} & x_{3} & x_{4} \end{matrix}] = [\begin{matrix} h & l & t & b \end{matrix}]$
Minimize	$f (\vec{x}) = 1.10471 x_{1}^{2} x_{2} + 0.04811 x_{3} x_{4} (14.0 + x_{2})$
	$g_{1} (\vec{x}) = τ (\vec{x}) - τ_{max} \leq 0$
	$g_{2} (\vec{x}) = σ (\vec{x}) - σ_{max} \leq 0$
Subject to	$g_{3} (\vec{x}) = δ (\vec{x}) - δ_{\max} \leq 0$
	$g_{4} (\vec{x}) = x_{1} - x_{4} \leq 0$
	$g_{5} (\vec{x}) = P - P_{c} (\vec{x}) \leq 0$
	$g_{6} (\vec{x}) = 0.125 - x_{1} \leq 0$
	$g_{7} (\vec{x}) = 1.10471 x_{1}^{2} x_{2} + 0.04811 x_{3}$
	x₄ (14.0 + x₂) -5.0 ≤ 0
Variable range	0.1 ≤ x₁ ≤ 2
	0.1 ≤ x₂ ≤ 10
	0.1 ≤ x₃ ≤ 10
	0.1 ≤ x₄ ≤ 2
	$τ (\vec{x}) = \sqrt{(τ^{'})^{2} + 2 τ^{'} τ^{″} \frac{x_{2}}{2 R} + (τ^{″})^{2}}$
	$τ^{'} = \frac{P}{\sqrt{2} x_{1} x_{2}}, τ^{″} = \frac{MR}{J}, M = P (L + \frac{x_{2}}{2})$
	$R = \sqrt{\frac{x_{2}^{2}}{4} + (\frac{x_{1} + x_{3}}{2})^{2}}$
where	$J = 2 {\sqrt{2} x_{1} x_{2} [\frac{x_{2}^{2}}{4} + (\frac{x_{1} + x_{3}}{2})^{2}]}$
	$σ (\vec{x}) = \frac{6 PL}{x_{4} x_{3}^{2}}, δ (\vec{x}) = \frac{6 {PL}^{3}}{{Ex}_{3}^{2} x_{4}}$
	$P_{c} (\vec{x}) = \frac{4.013 E \sqrt{\frac{x_{3}^{2} x_{4}^{6}}{36}}}{L^{2}} (1 - \frac{x_{3}}{2 L} \sqrt{\frac{E}{4 G}})$
	$P = 6000 lb, \begin{matrix} \end{matrix} L = 14 in, \begin{matrix} \end{matrix} E = 30 \times 1^{6} psi, \begin{matrix} \end{matrix}$
	$τ_{max} = 13600 psi, \begin{matrix} \end{matrix} σ_{max} = 30000 psi$
Consider	$\vec{x} = [\begin{matrix} x_{1} & x_{2} \end{matrix}] = [\begin{matrix} A_{1} & A_{2} \end{matrix}]$
Minimize	$f (\vec{x}) = (2 \sqrt{2} x_{1} + x_{2}) \times 1$
	$g_{1} (\vec{x}) = \frac{\sqrt{2} x_{1} + x_{2}}{\sqrt{2} x_{1}^{2} + 2 x_{1} x_{2}} P - σ \leq 0$
Subject to	$g_{2} (\vec{x}) = \frac{x_{2}}{\sqrt{2} x_{1}^{2} + 2 x_{1} x_{2}} P - σ \leq 0$
	$g_{3} (\vec{x}) = \frac{1}{\sqrt{2} x_{2} + x_{1}} P - σ \leq 0$
Variable range	0 ≤ x₁, x₂ ≤ 1
where	$\begin{matrix} 1 = 100 cm & P = 2 KN / {cm}^{2} & σ = 2 KN / cm \end{matrix}$

This problem has also tested and the results are shown in the Table 8. It is shown that the proposed algorithm can find the lowest cost design. Thus, it is reasonable to think that the proposed algorithm is feasible in solving such problems.

Table 8

Comparison results of tensions/compression spring design

Algorithm	Optimum variables				Optimum
	h	l	t	b	weight
HEO	0.2030	3.3607	9.0211	0.2058	1.7 22 447
MFO	0.2057	3.4703	9.0364	0.2057	1.72452
CBO	0.2057	3.4704	9.0372	0.2057	1.724663
IHS	0.2057	3.4705	9.0366	0.2057	1.7248
GWO	0.2057	3.4784	9.0368	0.2058	1.72624
MVO	0.2055	3.4731	9.0445	0.2057	1.72645
C.&M.	0.2060	3.4713	9.0202	0.2065	1.72822
WOA	0.2054	3.4843	9.0374	0.2063	1.730499

6.2 Three-bar truss design problem

As one of the most researched work cases [28], it is often used by various algorithms for testing. This task is also a minimization problem, mainly to find the minimum of the total weight of the structure. Figure 3 illustrates the shape of the truss and the associated forces on the structure. The figure contains two parameters: Area of strip 1 = Area of strip 3, and Area of strip 2. This problem can be mathematically described as follows:

Fig. 3

Schematic of the Three-bar truss design (Left: 3D, Right: Engineering drawing).

Table 9 shows the detailed results of other algorithms and proposed HEO. Based on the results in Table 9, HEO is significantly better than other optimizers, and HEO is observed to be exceptionally competitive.

Table 9

Comparison results of the welded beam design

Algorithm	Optimum variables		Optimum cost
	A ₁	A ₂
HEO	0.7880	0.4078	263.6524
Tsa	0.7880	0.4080	263.6800
PSO-DE	0.7887	0.4082	263.8958
HHO	0.7887	0.4083	263.8958
ALO	0.7887	0.4083	263.8958
SSA	0.7887	0.4083	263.8958
DEDS	0.7887	0.4082	263.8958
MVO	0.7886	0.4085	263.8958

6.3 Rolling element bearing problem

Different from the previous problems, this engineering problem is the maximum value of the dynamic bearing capacity of the solution target [29]. A total of 10 variables are included, a schematic of which is shown in Fig. 4. The test case is expressed as follows:

Fig. 4

Schematic of the pressure vessel [3].

Consider	$\vec{x} = [\begin{matrix} x_{1} & x_{2} & x_{3} & x_{4} \end{matrix}] = [\begin{matrix} T_{s} & T_{h} & R & L \end{matrix}]$
Minimize	$\begin{matrix} f (\vec{x}) = 0.6224 x_{1} x_{3} x_{4} + 1.7781 x_{2} x_{3}^{2} \\ \begin{matrix} \end{matrix} + 3.1661 x_{1}^{2} x_{4} + 19.84 x_{1}^{2} x_{3} \end{matrix}$
	g₁ (x) = - x₁ + 0.0193x₃ ≤ 0
Subject to	$g_{2} (\vec{x}) = - x_{3} + 0.00954 x_{3} \leq 0$
	$g_{3} (\vec{x}) = - π x_{3}^{2} x_{4} - \frac{4}{3} π x_{3}^{3} + 1296000 \leq 0$
	$g_{4} (\vec{x}) = x_{4} - 240 \leq 0$
	0 ≤ x₁ ≤ 99
Variable range	0 ≤ x₂ ≤ 99
	10 ≤ x₃ ≤ 200
	10 ≤ x₄ ≤ 200

This test results has shown in the following Table 10. As it represents, the proposed algorithm can reach the lowest cost design. Therefore, it is hopefully that the proposed algorithm is suitable for such problems.

Table 10

Comparison results of the pressure vessel design

Algorithm		GA4	TLBO	PVS	HHO	SHO	HEO
Optimum variables	D _m	125.71910	125.71910	125.71906	125.00000	125.00000	125.41900
	D _b	21.423000	21.425590	21.425590	21.000000	21.407320	21.456963
	Z	11.000000	11.000000	11.000000	11.092073	10.932680	10.908222
	f _i	0.5150000	0.5150000	0.5150000	0.5150000	0.5150000	0.5149094
	f ₀	0.5150000	0.5150000	0.5150000	0.5150000	0.5150000	0.5152510
	K _dmin	0.4159000	0.4242660	0.4004300	0.4000000	0.4000000	0.4087870
	K _dmax	0.6510000	0.6339480	0.6801600	0.6000000	0.7000000	0.6426149
	ɛ	0.3000043	0.3000000	0.3000000	0.3000000	0.3000000	0.3008210
	e	0.0223000	0.0688580	0.0799900	0.0504740	0.0200000	0.0286078
	ξ	0.7510000	0.7994980	0.7000000	0.6000000	0.6000000	0.6012870
	Optimum weight	81843.30	81859.74	81859.741	83011.883	85054.532	8 5070. 342

7 Conclusions

An algorithm combined with EO and TEO is put forward which called HEO. The good balance in EO combined with the high accuracy and rapidity of TEO improves the optimization accuracy of HEO in both exploration and exploitation stages. It can also be seen from the convergence curve that the ability of population renewal is also improved in the late iteration. The stability and accuracy have been improved, and the robustness of the algorithm has been strengthened. According to the evaluation of the experiment, HEO is better than EO and TEO. Accuracy has improved significantly, even better than algorithms that have similar bootstrap strategies, such as GWO and SHO. HEO algorithm has strong stability and feasibility, and even has strong competitiveness in engineering problems. The success of fusion also proves the effectiveness of memetic algorithm strategy. Substance exchange and heat transfer are actually a continuous process, but this paper uses the idea of leaders in weight ratio, so an adaptive weight ratio can be designed in the future to achieve a more suitable balance. And the algorithm can also be applied to other fields, such as image segmentation feature selection and multi-objective problem.

Footnotes

Acknowledgment

This work was supported under Educational research project for young and middle-aged teachers of Fujian Provincial Department of Education, Guiding project of Sanming science and technology plan, Sanming University introduces high-level talents to start scientific research projects (20YG14).

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