A novel enhanced flow regime algorithm using opposition-based learning

Abstract

Metaheuristics are widely used in science and industry because it as a high-level heuristic technique can provide robust or advanced solutions compared to classical search algorithms. Flow Regime Algorithm is a novel physics-based optimization approach recently proposed, and it is one of the candidate algorithms for solving complex optimization problems because of its few parameter configurations, simple coding, and good performance. However, the population that initialized randomly may have poor diversity issues, resulting in insufficient global search, and premature convergence to local optimum. To solve this problem, in this paper, a novel enhanced Flow Regime Algorithm based on opposition learning scheme is proposed. The proposed algorithm introduces the opposition-based learning strategy into the generation of some populations to enhance the global search performance while maintaining a fast convergence rate. In order to verify the performance of the proposed algorithm, 23 benchmark numerical optimization functions were studied experimentally in detail and compared with six well-known algorithms. Experimental results show that the proposed algorithm outperforms all other metaheuristic algorithms in all unimodal functions with higher accuracy, and can obtain competitive results on more multimodal cases. A statistical comparison shows that the proposed algorithm has superiority. Finally, that the proposed algorithm can achieve higher quality alignment compared to most other metaheuristic-based systems and OAEI ontology alignment systems.

Keywords

Meta-heuristic algorithms flow regime algorithm opposition-based learning benchmark functions

1 Introduction

Metaheuristic algorithms are also regarded as advanced heuristics, and the ideas of algorithms are often inspired by the laws of physics in nature, the schemes of biological evolution, and the gregarious behavior of animals [1, 2]. These algorithms are based on natural properties and implement the logical exploration and exploitation through mathematical modeling. Exploration is a globally divergent search of the solution space to maximize new possible solutions, and exploitation is the process of searching the space locally to find a better solution within the neighborhood of the current best individual [3]. For example, particle swarm optimization algorithm (PSO) is inspired by bird flock behavior to achieve balanced global search and local search by inertia weights w [4, 5]. Grasshopper optimization algorithm (GOA) is inspired by grasshopper colony foraging behavior, and the exploration and exploitation processes is balanced by adaptively adjusting comfort zone coefficients [3]. Moth-flame optimization algorithm (MFO) is inspired by moth navigation flight to maintain exploration and exploitation processes by modeling a spiral equation [6].

Since this advanced global technique is a stochastic optimization method that relies on generated random characteristics of this random population make the algorithm with high performance and a strong optimality driver. This is also the root cause of meta-heuristic algorithms being able to solve problems that classical search techniques cannot solve. As a result, it is widely used in many engineering, science and mathematics optimization problems, such as the welded beam design [7], drone obstacle avoidance problem [8], network structure optimization [9], feature selection [10], image segmentation [11], multi-objective interval number planning [12], etc.

Flow regime algorithm (FRA) is a recently proposed novel physics-based algorithm, the idea of which is mainly inspired by classical fluid mechanics and flow regimes [13]. Flow regimes are usually divided into laminar and turbulent flows by a parameter called the Reynolds number. Inspired by these two flow regimes, the laminar flow is simulated as the local search of the algorithm, and turbulent flow is simulated as the global search of the algorithm. A Reynolds-like variable is formulated and called search type factor (STF) which balance the exploration and exploitation process. In detail, STF dynamically adjusts the probability of global and local searches as the number of iterations changes. However, random populations may have poor diversity problems, which cause the algorithm to enter local search prematurely, that is, the phenomenon of premature convergence. To solve this problem, in this work, a scheme of opposition-based learning (OBL) [14] is used to enhance the exploration capacity of the FRA. The proposed algorithm is called OLFRA. At the beginning of each generation of fluid particle renewal, the diversity of the population is increased by the opposite estimation of a certain probability of the population, so that more candidate solutions are searched, and the convergence rate can be accelerated.

The main contributions for this work are as follows:

An improved flow regime algorithm using opposition-based learning is proposed.

The performance of the proposed algorithm has been studied on both unimodal and multimodal functions.

The proposed algorithm has been studied for solving the ontology alignment problem.

The rest of the paper is organized as follows: Section 2 provides an overview of relevant work. Section 3 introduces FRA and basic theories for OBL. The proposed novel FRA based on opposite learning is described in Section 4. Next, in Section 5, OLFRA is performed a detailed experimental study, and the experimental results are analyzed. A real application is studied in Section 6. In Section 7, the work of this paper is summarized and future plans are given.

2 Related work

The opposition-based learning scheme is the opposite solution for the current solution, and it is calculated through the idea of estimation and inverse estimation, so that those particles who are far away from the optimal solution are redirected. This strategy shortens the search time, and improves the probability of obtaining the optimal solution. In recent years, OBL has been applied to a variety of meta-heuristic algorithms, and a large number of experiments have proved to be one of the most effective strategies for improving global search performance. Ewees et al. [15] used the strategy of opposition-based learning to update half of the grasshopper population to solve the issues of the basic grasshopper optimization algorithm (GOA) of premature convergence and slow movements. Joshi et al. [16] proposed a gravitational search algorithm embedded with chaotic sequence based OBL to enhance its search ability. Si et al. [17] balances the exploration and exploitation capacity of the meta-heuristic algorithm by using OBL strategy, thereby improving its performance. Sarkhel et al. [18, 19] improved the performance of the OBL theory by taking into account each component of the candidate and integrated it into the harmonic search algorithm to obtain superior performance compared to the original OBL. Si et al. [20] designed five variants of the Salp Swarm Algorithm (SSA) based on the OBL scheme, and the experimental results show that embedding in the SSA using different OBL schemes can enhance the exploration capability of the algorithm and obtain better performance. Similarly, Wang et al. [21] used orthogonal opposition-based learning to enhance the comprehensive performance of the salp swarm algorithm. Dhargupta et al. [22] proposed combining the OBL with Grey Wolf Optimization (GWO) to enhance the exploration capability of the algorithm while maintaining a fast convergence rate. Omran et al. [23] used the OBL scheme to improve the performance of Particle Swarm Optimization (PSO). Yu et al. [29] used chaotic mapping and opposition-based learning to improve global search-ability for sparrow search algorithm. Zhao et al. [30] introduced the opposition-based learning mechanism into continuous ant colony optimization (ACO) to solve the problem of falling into local optima and low accuracy. Sharma et al. [31] used the opposition-based learning scheme for initializing the population of firefly algorithm (FA) and achieved better experiment results. The scheme improves the diversity of the population for the PSO by combining the randomly initialized population x and the opposite population $\tilde{x}$ as the initial population. Bacanin et al. [32] proposed an improved Bat Algorithm (BA) to solve the energy-efficient clustering problems by using opposition-based initialization strategy. Balande et al. [33] utilized the opposite-based learning approach to improve the convergence speed and the population quality of the teaching-Learning-Based algorithm. Wang et al. [34] improved equilibrium optimizer algorithm through incorporating teaching-learning-based algorithm and opposition-based learning. Tair et al. [35] proposed an improved whale optimisation algorithm by combining chaotic and opposition-based learning scheme.The improved algorithm achieved excellent performance in machine learning Feature selection.

3 Basic notations and theory

3.1 Flow Regime Algorithm (FRA)

FRA is a recently proposed physics-based meta-heuristic [13]. The idea of this algorithm is inspired by classical fluid mechanics and flow regimes. Specifically, the algorithm simulates two states of the flow regime: laminar and turbulent. Turbulence is simulated as a global search for FRA, and laminar flow is simulated as a local search. The algorithm balances exploration and exploitation by formulating a Reynolds number-like Selection Type Factor (STF). This Selection Type Factor (STF) is defined as follows: $\begin{matrix} {STF}^{i} = 3.2 \times 10^{6} \times \frac{Maxit}{n} \times \frac{∥ {gbest}_{n} - currParticle ∥}{∥ ({particle}_{r 1} - {particle}_{r 2}) ∥} \end{matrix}$ (1)

This 3.2 × 10⁶ is used here to establish similarity between the critical Reynolds number and the critical STF. The Maxit represents the maximum number of iterations. This gbest_n represents the global best solution found so far, n represents the current number of iterations, currParticle represents the current fluid particle, and particle_r1 and particle_r2 represent two randomly selected particles. To select the global and local search processes of the algorithm, the following conditions are defined: $\begin{matrix} x_{n + 1}^{i} = x_{n}^{i} + γ Levy \times \\ ({gbest}_{n} - x_{n}^{i}) \times \frac{0.37}{\sqrt[5]{STF}}, STF ⩾ 3.2 \times 10^{5} \end{matrix}$ (2) $\begin{matrix} x_{n + 1}^{i} = x_{n}^{i} + γ Rand \times \\ ({gbest}_{n} - x_{n}^{i}) \times \frac{4.96}{\sqrt{STF}}, STF < 3.2 \times 10^{5} \end{matrix}$ (3)

When the STF is greater than or equal to 3.2 × 10⁵, this indicates that the current particle is far from the starting point of the boundary layer. Further, the particle is in the turbulent region, that is, the algorithm enters the global search phase. Here Levy represents the random number generated by the Levy distribution. The γ is a scaling factor. When the STF is less than 3.2 × 10⁵, this indicates that the current particle is near the boundary layer starting point. This also indicates that the particle is in the laminar flow region, that is, the algorithm enters the local search phase. Here Rand is a random number generated by the Gaussian distribution.

3.2 Opposition Based Learning (OBL)

The random populations may have poor diversity problems, resulting in the defect of premature convergence of meta-heuristic algorithms. According to the review of Section 1, it is shown that the scheme for opposition-based learning can improve the diversity of populations, thereby enhancing the exploration capacity of the algorithm, and improving the convergence accuracy and convergence rate [16]. The opposition-based scheme is a better approximation of a candidate solution, which is achieved by considering the estimation and inverse estimation of one candidate solution in a solution space [14]. Specifically, the opposite vector $\tilde{x} = ({\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n})$ of an n-dimensional candidate solution vector x = (x₁, x₂ ⋯ x_n), x_i ∈ [a_i, b_i] is defined as follows: $\begin{matrix} {\tilde{x}}_{i} = a_{i} + b_{i} - x_{i}, i = 1, 2, \dots, n \end{matrix}$ (4)

Here a_i and b_i represent the upper and lower bounds of the ith dimension in the search space. For example, x = (0.026, 0.068) is a two-dimensional point in the interval [– 1,1], and this opposite point is $\tilde{x} = (- 0.026, - 0.068)$ .

4 Proposed new algorithms

Since FRA is a novel physics-based meta-heuristic algorithm, there is currently a lack of research related to it, especially to improve performance with opposition-based learning scheme. This phenomenon motivated our interest in this work research. After the functional experimental study of the basic FRA algorithm, it is found that poor solutions are obtained in more cases, for example, for the unimodal function Rosenbrock, the optimal value found by the algorithm is only 4.85E+00. The reason for this result is that the new particle learns from the best solution gbest, but the best solution may only reach the local optimal situation due to the random population diversity problem. In order to increase the diversity of the initial population, and enhance the exploration capacity, in this work, the OBL scheme is used in the initialization phase of a population, where a complete population is composed of opposite populations and random populations. In addition, in the global search phase, we modify the last part of equation (2) to use $4.37 / \sqrt[5]{STF}$ to increase the amount of change in the new particle. The intention is to find more promising solutions. The flowchart of this proposed algorithm is shown in Fig. 1.

Fig. 1

Flowchart of OLFRA

First, the algorithm uses the OBL scheme to directly initialize population $X_{obl}^{i}$ according to a certain proportion of population, and the remaining population X_r randomly generated according to the Gaussian distribution. Finally, a complete population is represented as $X = {X_{obl}^{i}, X_{r}}$ . Next, the fitness value of each particle is calculated, and the current best solution gbest is found. In the main loop, the algorithm calculates the selection type factor STF value by randomly selecting two particles. Finally, the best solution gbest is updated to the global optimal solution. This proposed algorithm OLFRA is summarized in Algorithm 1.

Algorithm 1: Summary of OLFRA
1: Initialize maximum number of iterations(maxIter), population size (N), OBL ratio (OR), parameter γ.
2: Initialize the population through OBL according to the OR. $X_{obl}^{i} (i = 1, 2, \dots, n), n = N \times OR$
3: Initialize the remaining population through a Gaussian distribution of random numbers. X_r (i = 1, 2, ⋯ , z) , z = N - n
4: Find the best particle gbest from $X_{obl}^{i}, X_{r}$
5: while (t < maxIter)
6: for all particles
7: Select two random particle_r1and particle_r2
8: Calculate the STF by using Equation (1)
9: IfSTF ⩾ 3.2 × 10⁵
10: Perform global search using Equation (2)
11: Else
12: Perform local search using Equation (3)
13: End if
14: End for
15: Evaluate particles and updating gbest
16: t = t+1;
17: End while
18: Return gbest
19: End

5 Experiments

In this section, 23 standard benchmark functions [24] are used to evaluate the convergence accuracy and convergence rate for the proposed algorithm OLFRA. These functions are tabulated in Table 1, Table 2, and Table 3 according to their categories: unimodal functions, multimodal functions, and multimodal functions with fixed dimensions. Specifically, functions with only one extreme point are called unimodal functions such as F1-F7. Functions with multiple extreme points are called multimodal functions such as F8-F23, where F14-F23 is a multimodal function with a fixed number of dimensions. These functional features enable evaluation of the convergence accuracy, convergence rate, and jump-out optimal performance of a meta-heuristic.

Table 1
Unimodal functions

FID Function Type Domain Dim Optimal

F1 $f (x) = \sum_{i = 1}^{n} x_{i}^{2}$ Unimodal [–100,100] 10 0

F2 $f (x) = \sum_{i = 1}^{n} | x_{i} | + \prod_{i = 1}^{n} | x_{i} |$ Unimodal [–10,10] 10 0

F3 $f (x) = \sum_{i = 1}^{n} {(\sum_{j = 1}^{i} | x_{i} |)}^{2}$ Unimodal [–100,100] 10 0

F4 f (x) = max{ |x_i|, 1 ⩽ i ⩽ n } Unimodal [–100,100] 10 0

F5 $f (x) = \sum_{i = 1}^{n - 1} [100 {(x_{i + 1} - x_{i}^{2})}^{2} + {(x_{i} - 1)}^{2}]$ Unimodal [–30,30] 10 0

F6 $f (x) = \sum_{i = 1}^{n} {([x_{i} + 0.5])}^{2}$ Unimodal [–100,100] 10 0

F7 $f (x) = \sum_{i = 1}^{n} {ix}_{i}^{4} + random [0, 1]$ Unimodal [–1.28,1.28] 10 0

FID	Function	Type	Domain	Dim
F1	$f (x) = \sum_{i = 1}^{n} x_{i}^{2}$	Unimodal	[–100,100]	10
F2	$f (x) = \sum_{i = 1}^{n} \| x_{i} \| + \prod_{i = 1}^{n} \| x_{i} \|$	Unimodal	[–10,10]	10
F3	$f (x) = \sum_{i = 1}^{n} {(\sum_{j = 1}^{i} \| x_{i} \|)}^{2}$	Unimodal	[–100,100]	10
F4	f (x) = max{ \|x_i\|, 1 ⩽ i ⩽ n }	Unimodal	[–100,100]	10
F5	$f (x) = \sum_{i = 1}^{n - 1} [100 {(x_{i + 1} - x_{i}^{2})}^{2} + {(x_{i} - 1)}^{2}]$	Unimodal	[–30,30]	10
F6	$f (x) = \sum_{i = 1}^{n} {([x_{i} + 0.5])}^{2}$	Unimodal	[–100,100]	10
F7	$f (x) = \sum_{i = 1}^{n} {ix}_{i}^{4} + random [0, 1]$	Unimodal	[–1.28,1.28]	10

Table 2

Multimodal functions

FID	Function	Type	Domain	Dim	Optimal
F8	$f (x) = \sum_{i = 1}^{n} - x_{i} sin (\sqrt{\| x_{i} \|})$	Multimodal	[–500,500]	10	– 418.9829×Dim
F9	$f (x) = \sum_{i = 1}^{n} [x_{i}^{2} - 10 cos (2 π x_{i}) + 10]$	Multimodal	[–5.12,5.12]	10	0
F10	$f (x) = - 20 exp (- 0.2 \sqrt{\frac{1}{n} \sum_{i = 1}^{n} x_{i}^{2}}) - exp (\frac{1}{n} \sum_{i = 1}^{n} cos (2 π x_{i})) + 20 + e$	Multimodal	[–32,32]	10	0
F11	$f (x) = \frac{1}{4000} \sum_{i = 1}^{n} x_{i}^{2} - \prod_{i = 1}^{n} cos (\frac{x_{i}}{\sqrt{i}}) + 1$	Multimodal	[–600,600]	10	0
F12	$f (x) = \frac{π}{n} {10 \sin^{2} (π y_{1}) + \sum_{i = 1}^{n - 1} {(y_{i} - 1)}^{2} [1 + 10 \sin^{2} (π y_{i + 1})] + {(y_{n} - 1)}^{2}} + \sum_{i = 1}^{n} u (x_{i}, 10, 100, 4)$ $u (x_{i}, a, k, m) = {\begin{matrix} k {(x_{i} - a)}^{m}, x_{i} > a \\ 0, - a ⩽ x_{i} ⩽ a \\ k {(- x_{i} - a)}^{m}, x_{i} < - a \end{matrix}$	Multimodal	[–50,50]	10	0
F13	$f (x) = 0.1 {\sin^{2} (3 π x_{1}) + \sum_{i = 1}^{n} {(x_{i} - 1)}^{2} [1 + \sin^{2} (3 π x_{i} + 1)] + {(x_{n} - 1)}^{2} [1 + \sin^{2} (2 π x_{n})]} + \sum_{i = 1}^{n} u (x_{i}, 5, 100, 4)$	Multimodal	[–50,50]	10	0

Table 3

Multimodal functions with fix dimension

FID	Function	Type	Domain	Dim	Optimal
F14	$f (x) = {(\frac{1}{500} + \sum_{j = 1}^{25} \frac{1}{j + \sum_{i = 1}^{2} {(x_{i} - a_{ij})}^{6}})}^{- 1}$	Multimodal	[–65.536,65.536]	2	1
F15	$f (x) = \sum_{i = 1}^{11} {[a_{i} - \frac{x_{1} (b_{i}^{2} + b_{i} x_{2})}{b_{i}^{2} + b_{i} x_{3} + x_{4}}]}^{2}$	Multimodal	[–5,5]	4	0.0003
F16	$f (x) = 4 x_{1}^{2} - 2.1 x_{1}^{4} + \frac{1}{3} x_{1}^{6} + x_{1} x_{2} - 4 x_{2}^{2} + 4 x_{2}^{4}$	Multimodal	[–5,5]	2	–1.0316
F17	$f (x) = {(x_{2} - \frac{5.1}{4 π^{2}} x_{1}^{2} + \frac{5}{π} x_{1} - 6)}^{2} + 10 (1 - \frac{1}{8 π}) cos x_{1} + 10$	Multimodal	[–5,5]	2	0.398
F18	$f (x) = [1 + {(x_{1} + x_{2} + 1)}^{2} (19 - 14 x_{1} + 3 x_{1}^{2} - 14 x_{2} + 6 x_{1} x_{2} + 3 x_{2}^{2})]$ $\times [30 + (2 x_{1} - 3 x_{2}) \times (18 - 32 x_{1} + 12 x_{1}^{2} + 48 x_{2} - 36 x_{1} x_{2} + 27 x_{2}^{2})]$	Multimodal	[–2,2]	2	3
F19	$f (x) = - \sum_{i = 1}^{4} c_{i} exp (- \sum_{j = 1}^{3} a_{ij} {(x_{j} - p_{ij})}^{2})$	Multimodal	[1,3]	3	–3.86
F20	$f (x) = - \sum_{i = 1}^{4} c_{i} exp (- \sum_{j = 1}^{6} a_{ij} {(x_{j} - p_{ij})}^{2})$	Multimodal	[0,1]	6	–3.32
F21	$f (x) = - \sum_{i = 1}^{5} {[(X - a_{i}) {(X - a_{i})}^{T} + c_{i}]}^{- 1}$	Multimodal	[0,10]	4	–10.1532
F22	$f (x) = - \sum_{i = 1}^{7} {[(X - a_{i}) {(X - a_{i})}^{T} + c_{i}]}^{- 1}$	Multimodal	[0,10]	4	–10.4028
F23	$f (x) = - \sum_{i = 1}^{10} {[(X - a_{i}) {(X - a_{i})}^{T} + c_{i}]}^{- 1}$	Multimodal	[0,10]	4	–10.5363

The results of each function are compared with basic FRA and the other state-of-the-art meta-heuristic algorithms, such as Biogeography-Based Optimization (BBO) [26], Moth-Flame Optimization (MFO) [6], Grasshopper Optimization Algorithm (GOA) [3], Bat Algorithm (BA) [27] and Firefly Algorithm (FA) [28]. For fairness of comparison, the common parameters involved in each algorithm: population size (N) is taken as 100, and the maximum number of iterations is 500. The detailed parameters of each algorithm are in Table 4. Each algorithm is run independently 20 times, and the mean (Mean) and standard deviation (Std) of the fitness function results are calculated as evaluation metrics. For a function, an algorithm performs best when it found the smallest mean value, and has better stability when it achieved the smaller the standard deviation if the mean is the same [22]. In the results comparison, the results for GOA, BA and FA were taken from our previous work [25]. These experiments were performed on Intel(R) Core (TM) i7-10700 CPU @ 2.90 GHz,2.90 GHz and 8 GB RAM. Each algorithm is implemented and executed in MATLAB 2017 under the Windows 10 Professional.

Table 4

The parameter setting for all algorithm

Algorithm	Configuration of parameter
OLFRA	OR = 0.6, γ= 0.5
FRA	γ= 0.5
BBO	KeepRate = 0.2, KeepN = KeepRateN,NewN* = N- KeepN,α= 0.9,Mutation = 0.1,ImmigrationR = 1- linspace(1,0,N)
MFO	b = 1
GOA	cMax = 1, cMin = 0.00004
BA	Loudness A = 0.5, Pulse rate = 0.5, MinFrequency = 0, MaxFrequency = 2
FA	alpha = 1.0, gamma = 0.01, Random reduction factor = 0.97, Attraction constant = 1.0

5.1 Significance test

Friedman’s test is a nonparametric statistical process whose goal is to detect whether there is a significant difference between two or more algorithms [36]. In this subsection, Friedman’s test is used to analyze the statistical significance of OLFRA and 6 other algorithms. The data samples for each algorithm are from the average value in Tables 5, 6 and 7. The null hypothesis indicates that there are no differences in the algorithms. If the null hypothesis is rejected, it indicates that there is a significant difference between the performance of all algorithms. The significance level is 0.05. The result of this experiment is χ= 37.151(The test statistics is approximated by using Chi-squared distribution) and p-value = 0.000002. From the results of this experiment, it can be seen that since the p-value is less than 0.05, the null hypothesis is rejected. This also shows that there is an extremely significant statistical difference, that is, it is statistically significant. Further, this discrepancy suggests the existence of at least one important set of outcomes.

Table 5
Results of OLFRA and other well know metaheuristics in unimodal functions

FID Index OLFRA FRA BBO MFO GOA BA FA

F1 Mean 6.58E-18 5.52E+00 1.9973E-06 1.73105E-14 2.90652E-06 1.09439E+03 8.49185E+03

Std 1.06E-17 1.50E+01 1.95165E-06 1.85496E-14 2.35340E-06 9.16417E+02 1.58896E+03

F2 Mean 9.24E-10 6.95E-10 1.81487E-04 2.34565E-09 6.98209E-03 5.82710E+00 6.14283E-06

Std 1.03E-09 1.31E-09 6.51647E-05 1.81267E-09 2.71745E-02 6.09603E+00 6.78355E-07

F3 Mean 3.16E-16 2.21E+02 1.13328E-02 5.83927E-04 4.70317E-02 2.48119E+03 7.98919E+03

Std 8.32E-16 5.74E+02 1.13410E-02 1.37528E-03 4.47591E-02 1.55937E+03 1.76732E+03

F4 Mean 1.19E-09 2.02E-01 1.00390E-02 2.08307E-04 2.93396E-03 1.94196E+01 5.09357E+01

Std 1.77E-09 2.15E-01 2.70557E-03 2.35905E-04 3.71191E-03 6.75339E+00 4.20524E+00

F5 Mean 6.04E-17 4.85E+00 4.80768E+00 4.52178E+03 4.74352E+01 7.45728E+03 7.37692E+06

Std 1.60E-16 6.99E+00 2.44854E+00 2.01216E+04 9.50617E+01 1.28938E+04 5.15138E+06

F6 Mean 1.73E-17 6.07E+00 2.45855E-06 1.06061E-14 2.45167E-06 1.17495E+03 8.43910E+03

Std 2.73E-17 1.18E+01 1.71425E-06 1.05568E-14 1.50313E-06 8.09874E+02 2.16920E+03

F7 Mean 1.37E-04 2.10E-03 5.42294E-04 3.19621E-03 2.86813E-03 2.43276E-01 1.50123E-02

Std 1.36E-04 2.94E-03 2.73448E-04 1.54543E-03 1.58811E-03 1.24734E-01 9.49153E-03

FID	Index	OLFRA	FRA	BBO	MFO	GOA	BA	FA
F1	Mean	6.58E-18	5.52E+00	1.9973E-06	1.73105E-14	2.90652E-06	1.09439E+03	8.49185E+03
	Std	1.06E-17	1.50E+01	1.95165E-06	1.85496E-14	2.35340E-06	9.16417E+02	1.58896E+03
F2	Mean	9.24E-10	6.95E-10	1.81487E-04	2.34565E-09	6.98209E-03	5.82710E+00	6.14283E-06
	Std	1.03E-09	1.31E-09	6.51647E-05	1.81267E-09	2.71745E-02	6.09603E+00	6.78355E-07
F3	Mean	3.16E-16	2.21E+02	1.13328E-02	5.83927E-04	4.70317E-02	2.48119E+03	7.98919E+03
	Std	8.32E-16	5.74E+02	1.13410E-02	1.37528E-03	4.47591E-02	1.55937E+03	1.76732E+03
F4	Mean	1.19E-09	2.02E-01	1.00390E-02	2.08307E-04	2.93396E-03	1.94196E+01	5.09357E+01
	Std	1.77E-09	2.15E-01	2.70557E-03	2.35905E-04	3.71191E-03	6.75339E+00	4.20524E+00
F5	Mean	6.04E-17	4.85E+00	4.80768E+00	4.52178E+03	4.74352E+01	7.45728E+03	7.37692E+06
	Std	1.60E-16	6.99E+00	2.44854E+00	2.01216E+04	9.50617E+01	1.28938E+04	5.15138E+06
F6	Mean	1.73E-17	6.07E+00	2.45855E-06	1.06061E-14	2.45167E-06	1.17495E+03	8.43910E+03
	Std	2.73E-17	1.18E+01	1.71425E-06	1.05568E-14	1.50313E-06	8.09874E+02	2.16920E+03
F7	Mean	1.37E-04	2.10E-03	5.42294E-04	3.19621E-03	2.86813E-03	2.43276E-01	1.50123E-02
	Std	1.36E-04	2.94E-03	2.73448E-04	1.54543E-03	1.58811E-03	1.24734E-01	9.49153E-03

Table 6

Results of OLFRA and other well know metaheuristics in multimodal functions

FID	Index	OLFRA	FRA	BBO	MFO	GOA	BA	FA
F8	Mean	– 4.19E+03	– 4.18E+03	– 3.79799E+03	– 3.28507E+03	– 2.85775E+03	– 4.55398E+30	– 3.45422E+03
	Std	1.00E-12	1.30E+01	2.76531E+02	2.57874E+02	3.53461E+02	1.89866E+31	1.70164E+02
F9	Mean	0.00E+00	6.05E-01	3.63161E+00	1.48320E+01	1.92524E+01	2.64660E+01	1.29345E+00
	Std	0.00E+00	7.18E-01	1.86211E+00	1.02458E+01	1.00914E+01	1.02457E+01	1.21214E+00
F10	Mean	8.71E-10	8.15E-01	5.35269E-04	4.82158E-08	4.05824E-01	1.18306E+01	1.78076E+01
	Std	9.95E-10	1.03E+00	2.04735E-04	4.33680E-08	7.33561E-01	1.27765E+00	9.81946E-01
F11	Mean	3.89E-17	7.17E-01	4.14728E-02	1.55964E-01	3.29984E-01	1.62792E+01	7.42628E+01
	Std	1.10E-16	4.55E-01	2.33803E-02	9.91267E-02	1.79887E-01	8.07141E+00	1.78471E+01
F12	Mean	2.28E-18	2.83E-01	3.0938E-08	1.55504E-02	4.72271E-01	8.38161E+02	1.18424E+07
	Std	7.72E-18	4.62E-01	5.6336E-08	6.95433E-02	1.12417E+00	3.69254E+03	7.04618E+06
F13	Mean	3.31E-18	1.56E-01	6.14664E-08	2.19747E-03	4.94840E-03	7.43999E+04	4.31527E+07
	Std	6.86E-18	2.24E-01	3.63799E-08	4.50912E-03	7.42449E-03	2.85921E+05	2.47970E+07

Table 7

Results of OLFRA and other well know metaheuristics in multimodal functions with fix dimension

FID	Index	OLFRA	FRA	BBO	MFO	GOA	BA	FA
F14	Mean	9.98E-01	1.19E+00	2.87409E+00	9.98004E-01	9.98004E-01	6.78400E+00	2.39482E+00
	Std	1.97E-16	8.42E-01	2.18852E+00	0.00E+00	4.35236E-16	6.01704E+00	1.60260E+00
F15	Mean	1.40E-03	7.90E-03	6.89864E-04	8.52240E-04	4.73216E-03	1.92242E-03	8.63180E-03
	Std	5.19E-04	1.37E-02	1.86703E-04	2.30204E-04	8.02033E-03	1.71643E-03	9.91865E-03
F16	Mean	– 1.03E+00	– 6.45E-01	– 1.03163E+00	– 1.03163E+00	– 1.03163E+00	– 9.90820E-01	– 1.03163E+00
	Std	1.30E-03	3.38E-01	2.92232E-15	2.27813E-16	1.24432E-12	1.82500E-01	6.78791E-15
F17	Mean	5.77E-01	6.83E-01	3.97887E-01	3.97887E-01	3.97887E-01	3.97887E-01	3.97887E-01
	Std	2.25E-01	1.96E-01	3.05533E-14	0.00E+00	1.81586E-11	2.28975E-10	1.61992E-15
F18	Mean	1.46E+01	2.04E+01	4.35000E+00	3.00000E+00	3.00000E+00	3.00000E+00	3.00000E+00
	Std	1.43E+01	1.27E+01	6.03738E+00	8.70472E-16	1.52956E-11	1.76574E-08	4.73883E-14
F19	Mean	– 3.01E-01	– 2.89E-01	– 3.00479E-01	– 3.86278E+00	– 3.00479E-01	– 3.8241E+00	– 2.87184E-01
	Std	1.14E-16	3.94E-02	1.13906E-16	2.27813E-15	1.13906E-16	1.72852E-01	1.37001E-02
F20	Mean	– 2.68E+00	– 2.24E+00	– 3.27444E+00	– 3.21766E+00	– 3.21842E+00	– 3.2506E+00	– 3.30416E+00
	Std	3.07E-01	5.28E-01	5.97586E-02	4.72682E-02	7.97478E-02	5.97589E-02	4.35562E-02
F21	Mean	– 1.02E+01	– 9.76E+00	– 6.14478E+00	– 7.63429E+00	– 7.75713E+00	– 4.0036E+00	– 1.01532E+01
	Std	3.13E-15	7.46E-01	3.47702E+00	3.25944E+00	3.10781E+00	1.87852E+00	3.09717E-12
F22	Mean	– 1.04E+01	– 1.02E+01	– 7.37194E+00	– 9.68716E+00	– 9.49368E+00	– 5.2215E+00	– 1.04029E+01
	Std	2.14E-05	4.06E-01	3.46677E+00	2.20862E+00	2.26439E+00	3.20207E+00	2.56798E-12
F23	Mean	– 1.05E+01	– 1.01E+01	– 9.05145E+00	– 9.27966E+00	– 8.96254E+00	– 4.9515E+00	– 1.05364E+01
	Std	1.20E-05	7.37E-01	2.69718E+00	2.61629E+00	2.85784E+00	3.37269E+00	2.14138E-12

To further analyze whether the proposed OLFRA algorithm provides significant results, this Wilcoxon test is performed to make pairwise comparisons with OLFRA. This null hypothesis is that the performance of the two algorithms is equivalent to a significance level of 0.05. If the comparison between the two algorithms results in a p-value less than 0.05, then there is a significant difference between them. As seen in Table 8, except for MFO, the proposed algorithm provides statistically significant results in all cases.

Table 8

Wilcoxon Test Result

	FRA	BBO	MFO	GOA	BA	FA
χ	275.000	206.000	192.000	223.000	227.000	232.000
p-value	0.000031	0.038620	0.100486	0.009730	0.006791	0.004250

5.2 Results and analysis

For convergence accuracy, the function calculation results in Table 1, Table 2 and Table 3 correspond to Table 5, Table 6 and Table 7, respectively, where the optimal solution is marked in bold. It is clear from Table 5 that the proposed algorithm outperforms all other meta-heuristic algorithms in convergence accuracy. Compared with the basic FRA, OLFRA greatly improves the accuracy, which also shows that embedding the OBL initialized population increases the population diversity and enhances the exploration ability of the algorithm. For multimodal functions, it is as can be seen from Table 6, that the proposed algorithm obtains better results than other meta-heuristics except F8. This also proves that the diversity problem of random initial population can be solved using the OBL model, thereby providing more promising solutions for the algorithm. For multimodal functions with fixed dimensions, the results of Table 7 show that the proposed algorithm obtains better solutions on F14, F15, F16, F21, F22, and F23. This also shows that OLFRA can improve the probability of the algorithm jumping out more local optimal in the complex functions of multi-extreme points, and thus obtain better solutions.

Regarding the convergence performance of each algorithm, the convergence rate of each algorithm is shown in Fig. 2, Fig. 3, and Fig. 4 according to the category of function: unimodal function, multimodal function, and multimodal function with fixed dimensionality. The abscissa of each plot represents the number of iterations, and the ordinate represents the best fitness value so far. It can be clearly seen from Fig. 2 that the proposed algorithm improves the rate of convergence of the basic algorithm to a better solution. This performance improvement is mainly due to the fact that OBL changes the direction of the random population, because if the random population is close to the optimal solution, then the algorithm converges faster. From the convergence curve of Fig. 3, it can be seen that the proposed algorithm has a faster convergence rate except for the function F8 compared to other meta-heuristic algorithms. For F8, however, only BA gets the best performance. For multimodal functions with fixed dimensions, it can be seen from the results of Fig. 4 that OLFRA maintains a rate of convergence towards a better solution on functions F14, F16, F17, F21, F22, and F23.

Fig. 2

Convergence curves for all algorithms in unimodal functions 1–7.

Fig. 3

Convergence curves for all algorithms in unimodal functions 8–13.

Fig. 4

Convergence curves for all algorithms in unimodal functions with fix dimension 14–23.

Through the above analysis of the results, it can be concluded that initializing the population based on OBL can improve the exploration ability of the algorithm, so as to avoid premature convergence and discover the local optimal solution. This is mainly because meta-heuristics use random populations as a starting point. Further, if the random point is in the opposite direction of the best, then the algorithm is prone to stagnation. If the random point is in the direction of the optimal solution, then the algorithm will converge quickly.

6 Real application

The ontology alignment problem is a challenging such as an effective combination of multiple simple matchers. Ontology alignment is the process of finding a set of entity correspondences in a given two or more domain ontologies. This process is explained in Fig. 5. In recent years, the use of swarm intelligence algorithms to solve the ontology alignment problem has been proven to be an effective method [38–40]. To solve the problem of effective combination of multiple matchers. In this work, the ontology alignment problem is transformed into an optimization problem. High-quality alignment is achieved by optimizing an optimal set of combined weights. First, this optimization problem is defined as follows:

Fig. 5

The ontology alignment process [37].

$Given \vec{x} = [x_{1}, x_{2} \dots x_{n}]$ $Maximizef (\vec{x}) = α \frac{| x |}{Sum (O_{1}, O_{2})} + (1 - α) \frac{\sum_{i = 1}^{| x |} d_{i}}{| x |}$ $Variables range 0 ⩽ x ⩽ 1$

Here $\vec{x}$ is used to represent the combined weight. $f (\vec{x})$ represents the objective function. |x| represents the corresponding quantity found for the current solution. Sum (O₁, O₂) is used to represent the total number of candidates corresponding from two ontologies. α is a constant factor.

The o and o′ is two ontologies. A is an input alignment.A′ is an alignment result.

In order to fairly evaluate the performance of the algorithm, the common parameters are uniformly set to N (population size) = 20, MaxIter (Termination) = 10, Dim (number of variables) = 4, ub(upperBound) = 1 and lb (lowerBound) = 0. Other intrinsic parameter configurations for each algorithm are shown in Table 9.

Table 9

The parameter setting for all algorithm on ontology alignment

Algorithm	Configuration of parameter
OLFRA	OR = 0.5, γ= 0.3
FRA	γ= 0.3
BBO	KeepRate = 0.2, KeepN = KeepRateN,NewN* = N- KeepN,α= 0.9,Mutation = 0.1,ImmigrationR = 1- linspace(1,0,N)
MFO	b = 1
GOA	cMax = 1, cMin = 0.00004
BA	Loudness A = 0.5, Pulse rate = 0.5, MinFrequency = 0, MaxFrequency = 2
FA	alpha = 1.0, gamma = 0.01, Random reduction factor = 0.97, Attraction constant = 1.0

To verify the performance of the proposed algorithm in the application, Anatomy track is used to perform matching tasks and each algorithm runs independently 5 times. This average is listed in Table 10. The results are compared to FRA, BBO, MFO, GOA, BA and FA. In terms of F-measure, BBO got the best results. The proposed OLFRA, FRA and GOA obtained suboptimal results. The main reason for this is that the proposed algorithm only changes the initialized population. The renewal phase of the population has not been modified. The BA and FA get the worst results. For Optimal F-measure, the proposed OLFRA, BBO and MFO can all achieve high-quality alignment results of 0.86.

Table 10

Comparison results of OLFRA with other ontology alignment system

Evaluate\Algorithm	OLFRA	FRA	BBO	MFO	GOA	BA	FA
Precision	0.950	0.918	0.939	0.822	0.950	0.323	0.271
Recall	0.757	0.776	0.780	0.774	0.756	0.594	0.489
F-measure	0.842	0.841	0.852	0.791	0.842	0.396	0.348
Optimal F	0.860	0.850	0.860	0.860	0.850	0.593	0.357

In order to better compare the performance of the proposed algorithm to solve the ontology alignment problem. SANOM, Lily, Wiktionary, ALIN, FCAMap-KG and DOME systems from OAEI (Ontology Alignment Evaluation Initiative) were selected. The results of this comparison are shown in Table 11. From this result, it can be seen that the proposed OLFRA outperforms other systems except SANOM.

Table 11

Comparison results of OLFRA with other ontology alignment system

Evaluate\Algorithm	OLFRA	SANOM	Lily	Wiktionary	ALIN	FCAMap-KG	DOME
Precision	0.950	0.888	0.873	0.968	0.974	0.996	0.996
Recall	0.757	0.844	0.796	0.730	0.698	0.631	0.615
F-measure	0.842	0.865	0.833	0.832	0.813	0.772	0.760

7 Conclusions and future work

In this paper, a novel Flow Regime Algorithm using opposition-based learning scheme is proposed. This OBL scheme is embedded in the initialization phase of the population. The algorithm directly initializes the opposite vector as a partial fluid particle by using the OBL strategy, and the remaining population is initialized by a random number of Gaussian distributions. In order to verify the performance of the proposed algorithm, 23 standard benchmark functions are used to conduct detailed experimental studies, and the experimental results show that embedding OBL into the initial population can increase the diversity of the population, thereby enhancing the convergence accuracy and convergence rate of the algorithm. Compared to the other state-of-the-art proposed meta-heuristic algorithms, the proposed OLFRA achieved better performance in most cases. Statistical analysis also shows that the proposed algorithm provides statistically significant results in some cases. The application in ontology alignment problems can demonstrate that the proposed algorithm can achieve higher quality alignment compared to most other meta heuristic-based systems and OAEI ontology alignment systems. In future work, we will further improve the performance of FRA, especially in more practical problems.

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