A novel grasshopper optimization algorithm based on swarm state difference and its application

Abstract

The grasshopper optimization algorithm (GOA) has received extensive attention from scholars in various real applications in recent years because it has a high local optima avoidance mechanism compared to other meta-heuristic algorithms. However, the small step moves of grasshopper lead to slow convergence. When solving larger-scale optimization problems, this shortcoming needs to be solved. In this paper, an enhanced grasshopper optimization algorithm based on solitarious and gregarious states difference is proposed. The algorithm consists of three stages: the first stage simulates the behavior of solitarious population learning from gregarious population; the second stage merges the learned population into the gregarious population and updates each grasshopper; and the third stage introduces a local operator to the best position of the current generation. Experiments on the benchmark function show that the proposed algorithm is better than the four representative GOAs and other metaheuristic algorithms in more cases. Experiments on the ontology matching problem show that the proposed algorithm outperforms all metaheuristic-based method and beats more the state-of-the-art systems.

Keywords

Meta-heuristic algorithms grasshopper optimization algorithm solitarious and gregarious states chemotaxis operator ontology matching

1 Introduction

Since the metaheuristic algorithms have higher performance and greater optimization driving force [1], these techniques have been widely used in a variety of applications, such as image segmentation [2], feature selection [3], very large scale integration (VLSI) circuits [4], hypergraph path [5], and data mining [6]. Meta-heuristics can solve problems that cannot be solved by conventional methods, and it aims to provide sufficiently good solutions for optimization problems. Therefore, meta-heuristics are also called high-level heuristics technique or global search technique or nature-inspired optimization algorithms [1]. Over the past decade, many nature-inspired optimization algorithms have been proposed such as particle swarm optimization, whale optimization algorithm, bacterial foraging optimization algorithm, firefly optimization algorithm, bat optimization algorithm and artificial bee colony optimization algorithm.

Grasshopper optimization algorithm (GOA) is a new swarm intelligence-based algorithm. Further, it is a bio-inspired optimization technology by simulating the behaviors of grasshopper swarm foraging [7]. The GOA algorithm is also a population-based stochastic global search technology. Compared with other population-based intelligence algorithms, this algorithm can well balance the exploration and exploitation process through adaptive adjustment of the comfort zone coefficient. In particular, the movement of a grasshopper needs its own position, the current best target position, and the all grasshoppers position. As the result, the algorithm is able to fully explore and exploit the search space. Specifically, GOA has the advantage of high exploration and local optima avoidance due to the high repulsive force between grasshoppers. The attraction forces between the grasshoppers and the adaptive comfort zone prompt GOA to maintain a strong exploitation. However, the small step moves of grasshopper lead to slow convergence speed. When solving larger-scale optimization problems, this shortcoming needs to be resolved. Considering larger research space, we try to improve the convergence rate of GOA through the biological behavior of grasshopper swarm.

In this work, two states of gregarious and solitarious are transplanted into the original GOA [8, 9]. The density information of the population is used to divide the original population into two subpopulations to simulate the gregarious and solitarious behavior. The mechanism accelerates the speed of convergence to the optimal solution.

The remainder of this discussion is organized as follows: Section 2 outlines existing works. The proposed algorithm SGBGOA is described in Section 3. Section 4 implemented a series of detailed experiments. Section 5 summarizes this work and gives a plan for the future.

2 Related works

2.1 Basic GOA

The grasshopper optimization algorithm [7, 22] is one of the most recent metaheuristic algorithms, which simulates the movement behavior of grasshoppers by a mathematically model. The mathematical model is defined as follows:

$X_{i} = S_{i} + G_{i} + A_{i}$ (1) while X_i is the position for the ith grasshopper, S_i indicates the social interaction, G_i represents the gravity force of the ith grasshopper, and A_i is the wind direction. The social interaction function S is defined as follows:

$s (r) = u e^{\frac{- r}{l}} - e^{- r}$ (2) where u is the strength of attraction; l indicates the attractive length scale. The r is the distance between the i-th and the j-th grasshopper and is calculated as follows:

$r_{ij} = | x_{j} - x_{i} |$ (3)

To solve optimization problem, a mathematical model is defined as follows:

$X_{i} = c_{1} (\sum_{\begin{matrix} j = 1 \\ j \neq i \end{matrix}}^{N} c_{2} \frac{δ - θ}{2} s (d_{ij}) \hat{d_{ij}}) + \hat{T_{d}}$ (4)

X_i is the position vector of the i-th grasshopper; c₁ is similar to the inertia weight of particle swarm optimization, which balances the exploration and exploitation; δ and θ are the upper and lower bounds of the search range; s is the attraction function; d_ij is the distance between the i-th grasshopper and the j-th grasshopper; $\hat{T_{d}}$ is the currently searched optimal target value; The unit vector $\hat{d_{ij}}$ from the i-th grasshopper to j-th grasshopper is defined as follow:

$\hat{d_{ij}} = \frac{x_{j} - x_{i}}{d_{ij}}$ (5)

x_j is the position vector of the j-th grasshopper; x_i is the position vector of the i-th grasshopper; c₂ is an internal parameter, which is the decreasing coefficient to shrink the comfort zone, repulsion zone and attraction zone; The c₁ and c₂ is defined as follows:

$c_{1} = c_{2} = c_{\max} - t \frac{c_{\max} - c_{\min}}{L}$ (6) where c_max and c_min indicate the maximum value and minimum value of the parameter c, respectively; t is the current iteration number; L is the maximum number of iterations;

2.2 Some variants of GOA

The grasshopper optimization algorithm has attracted widespread attention from scholars due to its powerful search capabilities. In recent years, many researchers have proposed different improved variants from the tuning of parameters, the topology structure of the population and the hybridizing evolutionary strategy. This three categories are discussed below.

For the variants of GOA based on parameter tuning, since this comfort zone coefficient c is a key parameter to maintain a proper balance between exploration and exploitation, the turning of this parameter has important research value. Algamal et al. [10] proposed an improved grasshopper optimization algorithm (PGOA). In PGOA, a parameter turning equation is proposed to improve the exploitation process of the algorithm, which can be quickly reduced in a small number of iterations. Arora et al. [11] employ chaos map to provide the parameters c₁ and c ₂ (CGOA). In this study, different chaotic maps are embedded into the GOA algorithm to tune the parameters adaptively. Experimental result shows that the Circle map is the most effective method for providing the parameters c₁ and c₂ at the same time.

For the topological structure of the population, it affects the diversity of the population and the exploration of the GOA. Ewees et al. [12] proposed a grasshopper optimization algorithm based on opposition-based learning (OBLGOA). The opposite learning strategy [23] is used in 50% of the population, and the learned individuals are judged according to the fitness value, and the good individuals are propagated to the next iteration. Bala et al. [13] proposed a grasshopper optimization algorithm with simple attraction and repulsion. The algorithm divides the population into two sub-populations, the best 25% of individuals are selected to the top group, and the remaining 75% of individuals are combined to the bottom group. Subsequently, two individuals are selected randomly from top and bottom group according to a certain probability. Next, the absolute value of the difference between the two individuals is defined as diff. If each gene in diff less than 3, the algorithm performs a repulsion. If each gene in diff greater than 7, the algorithm performs an attraction.

For the variants of GOA based on hybridizing evolutionary strategy. Luo et al. [32] proposed an improved hybrid grasshopper optimization algorithm. The Gaussian mutation and Levy-flight strategy are used to improve the search performance of GOA. In addition, the OBL strategy is also used for all agents in the current population. El-shorbagy et al. [14] proposed a grasshopper optimization algorithm based on hybrid genetic algorithm. The improved algorithm mainly combines exploitation of the GA and exploration of the GOA.

3 Proposed new algorithm

The grasshopper optimization algorithm has been widely used in artificial intelligence and engineering applications.

In this paper, the two state differences between the solitarious and gregarious behaviors of grasshopper swarm are transplanted into GOA to enhance the performance while improving the convergence rate, called SGBGOA. The motivation and the principle of the SGBGOA are explained in the following two sections respectively.

3.1 Motivation for GOA improvement

Since the original grasshopper optimization algorithm uses a small step and slow movement manner, the convergence speed of the GOA is slow when calculating large optimization problems. In order to solve this problem, this work studies the biological behavior of grasshopper swarms.

Topaz et al. [8] claimed that grasshoppers exhibit two phases of mutual conversion: gregarious and solitarious. The individuals with solitarious state are repelled by other grasshoppers, and gregarious grasshoppers are attracted to the conspecifics and formed large aggregations such as marching hopper bands. These two behavioral states strongly depend on the local population density. In sparse food environment, the gregarious grasshoppers will transition to solitarious state. In a dense food environment, the solitarious grasshoppers will transition to gregarious state. Furthermore, some studies have shown that grasshoppers of raising alone have acquired most of the behavioral characteristics of the gregarious state within 4–8 hours of crowding, including the tendency to aggregate [26, 27]. The characteristic encourages the tendency of other grasshoppers to transform into gregarious state. Inspired by these studies, a novel enhanced grasshopper optimization algorithm based on the gregarious and solitarious states difference is proposed. The population of the original grasshopper optimization algorithm is divided into two subpopulations: sparse population and dense population. The sparse population represents the solitarious state. The dense population represents the gregarious state. In the solitarious state, since the solitarious individual can obtain most of the characteristics of the population from the gregarious state, the algorithm determines whether the solitarious individual is attracted or repelled by calculating the repulsive force between the solitarious individual and gregarious individual. It is worth noting that individuals with small repulsive forces, i.e., conspecifics, are easily attracted to the gregarious state. Individuals with high repulsive force, i.e., non-conspecifics, move into the opposite direction, and them are saved in sparse population. In the gregarious state, the learned sparse population are merged with the dense population to form a large aggregation. The details of the algorithm are introduced in the following subsections.

3.2 SGBGOA algorithm

Based on the above analysis. Three learning stages are designed: the sparse population learn from dense population, each grasshopper learns from all population, and a local operator that learn from the best target. For the first stage, we propose two learning equations, which are used to learn from the conspecifics and move into the opposite direction for the non-conspecifics. The opposite movement strategy for non-conspecifics is also indicated mutual repulsion. For the second stage, the sparse population learned by the first stage is combined with the dense population, and then each individual employs the update equation to learn from all other individuals. The algorithm updates the dense population in each iteration. The mechanism realize the transition process from the solitarious to gregarious state. Finally, a chemotactic operator is applied to the optimal target position of the current generation to enhance the local search ability of the algorithm. The specific steps are described as follows:

Stage 1: Sparse population learn from dense population

First, the grasshopper population is initialized randomly and the fitness value of each grasshopper is calculated; the grasshopper population is divided into two subpopulations: sparse population and dense population. Specifically, taking the minimization problem as an example, the grasshopper position is sorted in ascending order according to the fitness of the grasshopper. The first 50% of the individuals are selected into the dense population, and the last 50% are selected to the sparse population. Second, sparse population learn from dense population. In order to determine the repulsive force between individual of sparse population and dense population, a repulsive force equation is used [8]. It is defined as follows:

$R_{f} un (d) = β \times e^{\frac{- d}{ρ}}$ (7) where β is interaction amplitudes; d is the distance between the i-th grasshopper and the j-th grasshopper; ρ is interaction length scales.

In order to realize the process of sparse population learning from dense population, two equations are proposed to model this process. Specifically, the Equation (8) represent learning from the conspecifics, and the Equation (9) represent learning from the opposite direction of the non-conspecifics. The two equations are defined as follows:

$\begin{matrix} x_{new} = & x + cauchyRand 1 \times (x_{r} - x) \\ + cauchyRand 2 \times (T - x) \end{matrix}$ (8)

$x_{new} = x - rand \times R_{fun (d_{ij})} \times (x_{r} - x)$ (9)

Next, two individuals are selected randomly from the dense population. two repulsive forces are calculated between individual of the sparse population and these two individuals by Equation (7) respectively. The two repulsive forces are denoted as Rep1 and Rep2, respectively. If Rep1 is less than Rep2, the individual of the sparse population learns from the dense individual through Equation (8). Otherwise, it learns from the opposite direction of the dense individual with large repulsive force through Equation (9). After that, if the learned individual is better than the current individual, the position of the individual of the sparse population is updated. This detailed learning process is described in Algorithm 1

Algorithm 1: The sparse population learn from the dense population
Input: sparse population is represented as spop; dense population is
represented as dpop;
Output: sparse population after learning
2: Fori = 1: size(spop)
3: sx = selectSpop();
4: Randomly select two individuals dx1 and dx2 from dpop
5: Calculate the repulsive force repf1 between sx and dx1;
6: Calculate the repulsive force repf2 between sx and dx2;
7: Ifrepf1 < repf2
8: sx learns from dx1 according to Equation (8);
9: Else
10: sx learns in the opposite direction of dx2 according to Equation (9);
11: End
12: newFit = f(x(i));
13: If newFit < f(sx)
14: spop(i) = x(i); // update ith individual in sparse population
15: End
16: End

Stage 2: Each grasshopper learns from all population

The sparse population learned by the first stage are combined with the dense population to form aggregates. The position of each grasshopper in the aggregates is updated by Eq. (4). That is, the position of each grasshopper is updated based on its own current position, the target position, and the position of all other grasshoppers. After that, the fitness values of all grasshoppers are recalculated and sorted in ascending order. The top 50% of individuals are selected to update the dense population.

Stage 3: A local chemotaxis operator

In order to enhance the local search capability of the SGBGOA, a chemotaxis operator [25] is used to further exploitation feasible solutions near the current best target location. In detail, the chemotaxis operator comes from the bacterial foraging optimization algorithm, which has strong local search capabilities. Chemotaxis operators include tumbling and swimming operators [24]. Swimming represents the pattern of individuals moving in a certain direction. The individual changes the direction of movement by tumbling. If the fitness value is not improved, it moves several steps in the new direction until the maximum number of moving or a stable fitness value is reached. The chemotactic operation of the ith individual is shown as follows:

$X_{i} (t + 1) = X_{i} (t) + C (i) \frac{φ (i)}{\sqrt{φ^{T} (i) φ (i)}}$ (10) where X_i (t + 1) indicates the position of the ith individual in t + 1 generation; C (i) is the step size of the movement; φ indicates a unit vector with random direction. In SGBGOA, the algorithm generates a random number uniformly distributed from 0 to 1. If the random number is greater than p, the chemotaxis operator is executed to find feasible solution near the best target location. This p is defined asfollows:

$p = c_{\max} - t \frac{c_{\max} - c_{\min}}{maxIter}$ (11) Algorithm 2 shows the chemotactic operator.

Algorithm 2: Chemotactic Operator
1: Initialize the number of chemotaxis steps Nc, the step size of
swimming Ns, C(i) is the step size of the movement;
2: If rand > pthen
3: Tlast = TargetFitness;
4: X(t) = TargetPosition(t);
5: Perform the tumbling operation through Equation (10)
6: Tnew = f(x) //compute new target fitness by using new solution x;
7: m = 0 // The counter of swimming
8: while (m < Ns)
9: m = m + 1;
10: If Tnew < Tlast
11: m = Ns;
12: Else
13: Perform the tumbling operation through Equation (10);
14: Tnew = f(x);
15: End
16: End while
17: TargetPosition(t) = x(t);
18: TargetFitness = Tnew;
19: End

Algorithm 3 shows the SGBGOA algorithm

Algorithm 3: Summary of SGBGOA
1: Initialize maximum number of iterations(maxIter),
cmax, cmin, the population X_i (i = 1, 2, ⋯ , n);
2: Calculate the fitness of each individual
3: T = find the best target fitness
4: Divide the population into sparse and dense population
5: while (t < maxIter)
6: Update c using Equation (6)
7: Perform algorithm 2 to obtain a new subpopulation
8: Combine the new subpopulation and dense population
9: for each search individual
10: Normalize the distances between individuals in [1, 4]
11: Update the position of the current individual by using
the Equation (4)
12: End for
13: Update dense population by sorting.
14: Perform algorithm 3 near the current best target position.
15: Update target position and T if there is a better solution.
16: t = t + 1;
17: End while
18: End

4 Experiments

In this section, three experimental tasks are implemented to verify the performance of the proposed SGBGOA algorithm. Before the experiment, the twenty-three benchmark functions are described. The parameters of each algorithm are set. Regarding the first task, the proposed algorithm SGBGOA is compared with three representative GOA variants and the other metaheuristic techniques in test functions. The mean (Mean) and standard (Std) deviation are calculated separately for each algorithm. For the second task, the convergence speed of the proposed algorithm is compared with other metaheuristic algorithm on each test function. All algorithms are executed independently on each test function 20 times, and the average is calculated. For the third task, a real-world application, i.e., ontology alignment is tested. Ontology alignment is implemented as an optimization problem by using the proposed SGBGOA, named SGBGOA-OM. Section 4.3.3 studies the maximum number of iterations required for the algorithm to find a high-quality alignment to prove whether the proposed algorithm can speed up the completion of the task. For alignment quality, the results are compared with the original GOA, three representative GOA variants and the other metaheuristic techniques such as CPSO [15]. In order to further verify the effect of the SGBGOA-OM method, several state-of-the-art ontology alignment methods are selected.

4.1 Experiment 1: Numerical experiments

Twenty-three benchmark functions [7] are used in this experiment, which include seven unimodal functions and sixteen multimodal functions. Furthermore, the unimodal function has only one extreme point in their domain. In contrast, the multimodal function has multiple extreme points and multiple local optima. Therefore, the feature is most suitable for evaluating the performance of the proposed algorithm, especially the ability to jump out of the local optimum. The detailed information of these functions is described in Table 1.

Table 1
The benchmark functions

FID Benchmark 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

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

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, 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

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

4.1.1 Parameters settings

The parameters involved in the algorithms include the own parameters and public parameters. In order to make each algorithm have the best performance, the own parameters of each algorithm adopt original paper. The common parameters of each algorithm, i.e., the maximum number of iterations (maxIter) and the size of population (N), are set to the same value. In benchmark function experiment, maxIter = 500 and N = 100. The detailed parameter settings of each algorithm are shown in Table 2.

Table 2
The parameter setting for all algorithm

Algorithm Configuration of parameter

SGBGOA cMax = 1, cMin = 0.00004, Ns = 6, α= 0.4, Division ratio = 50% of population

OBLGOA cMax = 1, cMin = 0.00004, OBL ratio = 50% of population

PGOA w_max = 1

CGOA wMax = 20, wMin = 1e-10

GOA cMax = 1, cMin = 0.00004

BA Loudness A = 0.5, Pulse rate = 0.5, Min Frequency = 0, Max Frequency = 2

FA alpha = 1.0, gamma = 0.01, Random reduction factor = 0.97, Attraction constant = 1.0

Algorithm	Configuration of parameter
SGBGOA	cMax = 1, cMin = 0.00004, Ns = 6, α= 0.4, Division ratio = 50% of population
OBLGOA	cMax = 1, cMin = 0.00004, OBL ratio = 50% of population
PGOA	w_max = 1
CGOA	wMax = 20, wMin = 1e-10
GOA	cMax = 1, cMin = 0.00004
BA	Loudness A = 0.5, Pulse rate = 0.5, Min Frequency = 0, Max Frequency = 2
FA	alpha = 1.0, gamma = 0.01, Random reduction factor = 0.97, Attraction constant = 1.0

4.1.2 Results and analysis

In this experiment, each test function is executed independently by using SGBGOA, OBLGOA, PGOA, CGOA, GOA, Bat Algorithm (BA), and Firefly Algorithm (FA), and the mean and standard deviation are calculated. The results are shown in Table 3. Compared with other GOA algorithms in terms of mean, it can be clearly seen from the results that the proposed SGBGOA algorithm outperforms other GOA algorithms in all functions except F1, F4, F7, F8, F10, F15, and F20. It is worth noting that all algorithms have not achieved the optimal value on the function F8. For functions F14, F17, F16, and F18, the all GOA algorithms found the same optimal value. Concerning unimodal functions (F1-F7), the SGBGOA algorithm obtains better solution on F2, F3, F5 and F6. It performs poorly on F1, F4 and F7 functions compared to the second best OBLGOA. Concerning multimodal functions (F8-F23), the proposed SGBGOA algorithm finds better solutions for all functions except F8, F10 and F20. On the contrary, although the OBLGOA algorithm obtains the same solution on F14, F17, and F18, it obtains a poor solution in solving most multimodal functions. The possible reason is that OBLGOA only uses the OBL learning strategy to change the diversity of the population. Therefore, the exploitation ability of the algorithm has not been improved. As a result, when solving functions with multiple extreme points, the exploration and exploitation process of this algorithm may not be well balanced.

Table 3
Comparison results of SGBGOA, OBLGOA, PGOA, CGOA, GOA, BA and FA

FID SGBGOA OBLGOA PGOA CGOA GOA BA FA

F1 Mean 2.27998E-06 4.22801E-07 1.04865E+04 1.56262E+03 2.90652E-06 1.09439E+03 8.49185E+03

Std 1.91203E-06 2.60657E-07 1.49575E+03 1.18394E+03 2.35340E-06 9.16417E+02 1.58896E+03

F2 Mean 6.66869E-04 9.33705E-03 2.71820E+01 2.15224E+01 6.98209E-03 5.82710E+00 6.14283E-06

Std 4.22887E-04 2.64949E-02 8.59679E+00 8.73922E+00 2.71745E-02 6.09603E+00 6.78355E-07

F3 Mean 2.15126E-05 1.04955E-04 1.01056E+04 2.13212E+03 4.70317E-02 2.48119E+03 7.98919E+03

Std 1.78827E-05 2.39969E-04 2.93868E+03 1.85008E+03 4.47591E-02 1.55937E+03 1.76732E+03

F4 Mean 1.17866E-03 3.46676E-04 5.15103E+01 2.52165E+01 2.93396E-03 1.94196E+01 5.09357E+01

Std 6.68965E-04 1.34844E-04 5.83727E+00 1.44414E+01 3.71191E-03 6.75339E+00 4.20524E+00

F5 Mean 5.34257E+00 7.99207E+00 1.45764E+07 5.22756E+05 4.74352E+01 7.45728E+03 7.37692E+06

Std 2.36408E+00 5.53218E-01 7.24520E+06 1.48650E+06 9.50617E+01 1.28938E+04 5.15138E+06

F6 Mean 1.46108E-06 2.77997E-06 9.98141E+03 1.90052E+03 2.45167E-06 1.17495E+03 8.43910E+03

Std 7.35315E-07 1.47059E-06 2.14701E+03 1.89431E+03 1.50313E-06 8.09874E+02 2.16920E+03

F7 Mean 1.18584E-03 1.79750E-04 2.86757E+00 1.02619E-02 2.86813E-03 2.43276E-01 1.50123E-02

Std 6.36529E-04 8.37165E-05 1.50318E+00 5.51931E-03 1.58811E-03 1.24734E-01 9.49153E-03

F8 Mean –3.15564E+03 –2.93515E+03 –2.17969E+03 –2.60407E+03 –2.85775E+03 –4.55398E+30 –3.45422E+03

Std 2.72093E+02 2.51652E+02 2.06061E+02 4.42765E+02 3.53461E+02 1.89866E+31 1.70164E+02

F9 Mean 7.81260E+00 9.11010E+00 6.89775E+01 1.06022E+01 1.92524E+01 2.64660E+01 1.29345E+00

Std 3.83606E+00 5.91614E+00 1.72766E+01 6.43574E+00 1.00914E+01 1.02457E+01 1.21214E+00

F10 Mean 1.98273E-01 2.79041E-04 1.82222E+01 1.12729E+01 4.05824E-01 1.18306E+01 1.78076E+01

Std 4.91647E-01 8.85242E-05 7.23458E-01 3.48719E+00 7.33561E-01 1.27765E+00 9.81946E-01

F11 Mean 1.04488E-01 1.91780E-01 8.68240E+01 7.64495E+00 3.29984E-01 1.62792E+01 7.42628E+01

Std 6.01825E-02 1.13388E-01 2.26206E+01 6.83514E+00 1.79887E-01 8.07141E+00 1.78471E+01

F12 Mean 4.14393E-07 1.64789E-02 1.44750E+07 1.28317E+06 4.72271E-01 8.38161E+02 1.18424E+07

Std 4.17866E-07 7.13913E-02 9.85898E+06 2.55631E+06 1.12417E+00 3.69254E+03 7.04618E+06

F13 Mean 2.85875E-06 6.15827E-03 5.17887E+07 4.39950E+06 4.94840E-03 7.43999E+04 4.31527E+07

Std 1.10639E-05 1.29778E-02 2.74941E+07 8.23063E+06 7.42449E-03 2.85921E+05 2.47970E+07

F14 Mean 9.98004E-01 9.98004E-01 1.05943E+01 3.80241E+00 9.98004E-01 6.78400E+00 2.39482E+00

Std 3.41719E-16 3.63788E-16 6.12005E+00 3.37614E+00 4.35236E-16 6.01704E+00 1.60260E+00

F15 Mean 9.72121E-04 3.49819E-03 3.50471E-02 1.38534E-02 4.73216E-03 1.92242E-03 8.63180E-03

Std 1.73348E-03 7.27087E-03 2.97185E-02 1.38647E-02 8.02033E-03 1.71643E-03 9.91865E-03

F16 Mean –1.03163E+00 –1.03155E+00 –8.12521E-01 –9.36787E-01 –1.03163E+00 –9.90820E-01 –1.03163E+00

Std 4.34678E-10 2.34291E-04 3.55435E-01 1.88228E-01 1.24432E-12 1.82500E-01 6.78791E-15

F17 Mean 3.97887E-01 3.97887E-01 4.94286E-01 5.43834E-01 3.97887E-01 3.97887E-01 3.97887E-01

Std 1.21605E-13 5.16553E-12 2.57903E-01 1.74633E-01 1.81586E-11 2.28975E-10 1.61992E-15

F18 Mean 3.00000E+00 3.00000E+00 7.34382E+00 3.17984E+00 3.00000E+00 3.00000E+00 3.00000E+00

Std 5.69667E-12 1.12261E-11 8.87360E+00 4.41092E-01 1.52956E-11 1.76574E-08 4.73883E-14

F19 Mean –3.85838E+00 –3.00479E-01 –5.46357E-02 –3.00479E-01 –3.00479E-01 –3.82413E+00 –2.87184E-01

Std 3.72742E-03 1.13906E-16 4.79634E-02 1.13906E-16 1.13906E-16 1.72852E-01 1.37001E-02

F20 Mean –3.26255E+00 –3.26819E+00 –2.25251E+00 –3.20880E+00 –3.21842E+00 –3.25066E+00 –3.30416E+00

Std 6.09909E-02 8.94432E-02 4.21025E-01 1.33741E-01 7.97478E-02 5.97589E-02 4.35562E-02

F21 Mean –9.77968E+00 –8.89009E+00 –1.53434E+00 –4.88163E+00 –7.75713E+00 –4.00361E+00 –1.01532E+01

Std 1.67042E+00 2.24460E+00 7.75145E-01 3.15516E+00 3.10781E+00 1.87852E+00 3.09717E-12

F22 Mean –1.04029E+01 –9.08235E+00 –1.71915E+00 –5.12478E+00 –9.49368E+00 –5.22153E+00 –1.04029E+01

Std 1.29432E-09 2.34676E+00 6.06982E-01 3.23872E+00 2.26439E+00 3.20207E+00 2.56798E-12

F23 Mean –1.02684E+01 –8.65071E+00 –1.96980E+00 –4.31923E+00 –8.96254E+00 –4.95155E+00 –1.05364E+01

Std 1.19870E+00 2.63657E+00 1.12057E+00 2.57724E+00 2.85784E+00 3.37269E+00 2.14138E-12

FID		SGBGOA	OBLGOA	PGOA	CGOA	GOA	BA	FA
F1	Mean	2.27998E-06	4.22801E-07	1.04865E+04	1.56262E+03	2.90652E-06	1.09439E+03	8.49185E+03
	Std	1.91203E-06	2.60657E-07	1.49575E+03	1.18394E+03	2.35340E-06	9.16417E+02	1.58896E+03
F2	Mean	6.66869E-04	9.33705E-03	2.71820E+01	2.15224E+01	6.98209E-03	5.82710E+00	6.14283E-06
	Std	4.22887E-04	2.64949E-02	8.59679E+00	8.73922E+00	2.71745E-02	6.09603E+00	6.78355E-07
F3	Mean	2.15126E-05	1.04955E-04	1.01056E+04	2.13212E+03	4.70317E-02	2.48119E+03	7.98919E+03
	Std	1.78827E-05	2.39969E-04	2.93868E+03	1.85008E+03	4.47591E-02	1.55937E+03	1.76732E+03
F4	Mean	1.17866E-03	3.46676E-04	5.15103E+01	2.52165E+01	2.93396E-03	1.94196E+01	5.09357E+01
	Std	6.68965E-04	1.34844E-04	5.83727E+00	1.44414E+01	3.71191E-03	6.75339E+00	4.20524E+00
F5	Mean	5.34257E+00	7.99207E+00	1.45764E+07	5.22756E+05	4.74352E+01	7.45728E+03	7.37692E+06
	Std	2.36408E+00	5.53218E-01	7.24520E+06	1.48650E+06	9.50617E+01	1.28938E+04	5.15138E+06
F6	Mean	1.46108E-06	2.77997E-06	9.98141E+03	1.90052E+03	2.45167E-06	1.17495E+03	8.43910E+03
	Std	7.35315E-07	1.47059E-06	2.14701E+03	1.89431E+03	1.50313E-06	8.09874E+02	2.16920E+03
F7	Mean	1.18584E-03	1.79750E-04	2.86757E+00	1.02619E-02	2.86813E-03	2.43276E-01	1.50123E-02
	Std	6.36529E-04	8.37165E-05	1.50318E+00	5.51931E-03	1.58811E-03	1.24734E-01	9.49153E-03
F8	Mean	–3.15564E+03	–2.93515E+03	–2.17969E+03	–2.60407E+03	–2.85775E+03	–4.55398E+30	–3.45422E+03
	Std	2.72093E+02	2.51652E+02	2.06061E+02	4.42765E+02	3.53461E+02	1.89866E+31	1.70164E+02
F9	Mean	7.81260E+00	9.11010E+00	6.89775E+01	1.06022E+01	1.92524E+01	2.64660E+01	1.29345E+00
	Std	3.83606E+00	5.91614E+00	1.72766E+01	6.43574E+00	1.00914E+01	1.02457E+01	1.21214E+00
F10	Mean	1.98273E-01	2.79041E-04	1.82222E+01	1.12729E+01	4.05824E-01	1.18306E+01	1.78076E+01
	Std	4.91647E-01	8.85242E-05	7.23458E-01	3.48719E+00	7.33561E-01	1.27765E+00	9.81946E-01
F11	Mean	1.04488E-01	1.91780E-01	8.68240E+01	7.64495E+00	3.29984E-01	1.62792E+01	7.42628E+01
	Std	6.01825E-02	1.13388E-01	2.26206E+01	6.83514E+00	1.79887E-01	8.07141E+00	1.78471E+01
F12	Mean	4.14393E-07	1.64789E-02	1.44750E+07	1.28317E+06	4.72271E-01	8.38161E+02	1.18424E+07
	Std	4.17866E-07	7.13913E-02	9.85898E+06	2.55631E+06	1.12417E+00	3.69254E+03	7.04618E+06
F13	Mean	2.85875E-06	6.15827E-03	5.17887E+07	4.39950E+06	4.94840E-03	7.43999E+04	4.31527E+07
	Std	1.10639E-05	1.29778E-02	2.74941E+07	8.23063E+06	7.42449E-03	2.85921E+05	2.47970E+07
F14	Mean	9.98004E-01	9.98004E-01	1.05943E+01	3.80241E+00	9.98004E-01	6.78400E+00	2.39482E+00
	Std	3.41719E-16	3.63788E-16	6.12005E+00	3.37614E+00	4.35236E-16	6.01704E+00	1.60260E+00
F15	Mean	9.72121E-04	3.49819E-03	3.50471E-02	1.38534E-02	4.73216E-03	1.92242E-03	8.63180E-03
	Std	1.73348E-03	7.27087E-03	2.97185E-02	1.38647E-02	8.02033E-03	1.71643E-03	9.91865E-03
F16	Mean	–1.03163E+00	–1.03155E+00	–8.12521E-01	–9.36787E-01	–1.03163E+00	–9.90820E-01	–1.03163E+00
	Std	4.34678E-10	2.34291E-04	3.55435E-01	1.88228E-01	1.24432E-12	1.82500E-01	6.78791E-15
F17	Mean	3.97887E-01	3.97887E-01	4.94286E-01	5.43834E-01	3.97887E-01	3.97887E-01	3.97887E-01
	Std	1.21605E-13	5.16553E-12	2.57903E-01	1.74633E-01	1.81586E-11	2.28975E-10	1.61992E-15
F18	Mean	3.00000E+00	3.00000E+00	7.34382E+00	3.17984E+00	3.00000E+00	3.00000E+00	3.00000E+00
	Std	5.69667E-12	1.12261E-11	8.87360E+00	4.41092E-01	1.52956E-11	1.76574E-08	4.73883E-14
F19	Mean	–3.85838E+00	–3.00479E-01	–5.46357E-02	–3.00479E-01	–3.00479E-01	–3.82413E+00	–2.87184E-01
	Std	3.72742E-03	1.13906E-16	4.79634E-02	1.13906E-16	1.13906E-16	1.72852E-01	1.37001E-02
F20	Mean	–3.26255E+00	–3.26819E+00	–2.25251E+00	–3.20880E+00	–3.21842E+00	–3.25066E+00	–3.30416E+00
	Std	6.09909E-02	8.94432E-02	4.21025E-01	1.33741E-01	7.97478E-02	5.97589E-02	4.35562E-02
F21	Mean	–9.77968E+00	–8.89009E+00	–1.53434E+00	–4.88163E+00	–7.75713E+00	–4.00361E+00	–1.01532E+01
	Std	1.67042E+00	2.24460E+00	7.75145E-01	3.15516E+00	3.10781E+00	1.87852E+00	3.09717E-12
F22	Mean	–1.04029E+01	–9.08235E+00	–1.71915E+00	–5.12478E+00	–9.49368E+00	–5.22153E+00	–1.04029E+01
	Std	1.29432E-09	2.34676E+00	6.06982E-01	3.23872E+00	2.26439E+00	3.20207E+00	2.56798E-12
F23	Mean	–1.02684E+01	–8.65071E+00	–1.96980E+00	–4.31923E+00	–8.96254E+00	–4.95155E+00	–1.05364E+01
	Std	1.19870E+00	2.63657E+00	1.12057E+00	2.57724E+00	2.85784E+00	3.37269E+00	2.14138E-12

Compared with other metaheuristic techniques BA and FA in terms of mean, the proposed SGBGOA achieves the best performance for most of the functions except F2, F9, F20, F21, and F23. For functions F16, F17, F18, F22, these algorithms found the same optimal value. For unimodal functions (F1-F7), The SGBGOA only fails to obtain a good solution on the function F2. Concerning multimodal functions (F8-F23), the proposed SGBGOA does not reach the optimal value on the function F8, F9, F20, F21, and F23.

Based on the above analysis, although the accuracy for a few functions is slightly lower than OBLGOA, the SGBGOA algorithm can obtain better accuracy compared to OBLGOA, PGOA, CGOA and GOA algorithm. The proposed algorithm outperforms other meta-heuristic algorithms in most functions. The main reason is that the proposed algorithm is able to quickly discover the feasible area by adopting the mechanism of the sparse population learning from dense population and to find the approximate optimal solution by using the local chemotaxis operator. Moreover, this proposed algorithm has a good balance for exploration and exploitation.

4.2 Experiment 2: Convergence performance of SGBGOA

In this experiment, the convergence process of the SGBGOA algorithm is performed on the 23 functions and compared with based on GOA and other metaheuristic techniques. Figure 1 shows the convergence curves of each algorithm. The abscissa indicates the number of iterations, and the ordinate indicates the fitness. This figure clearly shows that the proposed SGBGOA algorithm speeds up the convergence to the optimal solution in all functions except functions F2,F7,F8, and F9. Improvement of the convergence speed is mainly due to the continuous learning of sparse population from dense population, which reduces the probability of learning from poor individuals. Furthermore, this strategy speeds up the movement to the global optimal solution. This is more conducive to the exploitation process of the local search strategy, which is to avoid premature convergence and fall into the local optimum. On the basis of the above analysis, it can be concluded that the improvement strategy of the SGBGOA algorithm is successful. This strategy can improve the accuracy and convergence speed of the original algorithm.

Fig. 1

Convergence curves for all algorithms.

4.3 Experiment 3: Application examples

In this subsection, the proposed SGBGOA algorithm is tested in different challenges of large ontology alignment problem. Specifically, a large real-world ontology Anatomy track and two Large Biomedical tracks are used to evaluate the performance of the SGBGOA-OM method. First of all, this ontology alignment problem and ontology alignment are formally defined as an optimization problem. Then the experimental results of the three matching tasks are analyzed by using three evaluation indicators: Precision, Recall and F-measure.

4.3.1 Mathematical model of ontology alignment

Ontology alignment is an important process in data integration [16]. The evolutionary algorithms have proven to be an effective method for ontology alignment [28, 29]. In this experiment, the ontology alignment process is transformed into an optimization problem. Furthermore, a suboptimal alignment is found through the SGBGOA algorithm. For details, please refer to our previous work [31]. The ontology alignment method is named SGBGOA-OM.

The purpose of ontology alignment is to find a correspondences set between the given ontology concepts. This target can be defined as a function f: $\begin{matrix} M = f (O_{1}, O_{2}, p, u) \end{matrix}$ where M is a set of correspondences. O₁ and O₂ are two ontologies, p is a set of parameters, u indicates a set of resources and similarity measure algorithms [33].

The mathematical model of ontology alignment as an optimization problem is defined by the following equations:

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

where $\vec{x}$ represents a set of weight vectors combined with basic matchers. It also corresponds to a candidate alignment, which is found in the solution space formed by the combined basic matchers. |x| represents the correspondence size in the candidate alignment. Sum (O₁, O₂) represents the sum of all candidate mappings. α is a weighting factor.

4.3.2 Parameter configuration

The configuration of the parameter is shown in Table 4.

Table 4
The parameter setting for all algorithm

Algorithm Configuration of parameters

SGBGOA cMax = 1, cMin = 0.01, Ns = 6, α= 0.4, N = 20, Division ratio = 50% of population

OBLGOA cMax = 1, cMin = 0.01, N = 20, OBL ratio = 50% of population

PGOA cMax = 1, cMin = 0.01, N = 20, w_max = 1

CGOA cMax = 1, cMin = 0.01, N = 20, wMax = 20, wMin = 1e-10

GOA cMax = 1, cMin = 0.01, N = 20

BA Loudness A = 0.5, Pulse rate = 0.5, Min Frequency = 0, MaxFrequency = 2

FA alpha = 1.0, gamma = 0.01, Random reduction factor = 0.97, Attraction constant = 1.0

Algorithm	Configuration of parameters
SGBGOA	cMax = 1, cMin = 0.01, Ns = 6, α= 0.4, N = 20, Division ratio = 50% of population
OBLGOA	cMax = 1, cMin = 0.01, N = 20, OBL ratio = 50% of population
PGOA	cMax = 1, cMin = 0.01, N = 20, w_max = 1
CGOA	cMax = 1, cMin = 0.01, N = 20, wMax = 20, wMin = 1e-10
GOA	cMax = 1, cMin = 0.01, N = 20
BA	Loudness A = 0.5, Pulse rate = 0.5, Min Frequency = 0, MaxFrequency = 2
FA	alpha = 1.0, gamma = 0.01, Random reduction factor = 0.97, Attraction constant = 1.0

4.3.3 Study the efficiency of the proposed method

The maximum number of iterations (MaxIter) required for the SGBGOA-OM to find a high-quality alignment is studied to prove whether the proposed algorithm can speed up the completion of the task. Table 5 shows the average results of 10 independent runs at different iteration times for each algorithm on Anatomy track. It can be clearly seen that the proposed SGBGOA-OM method can find a high-quality alignment as F-measure = 0.855 in the case of only MaxIter = 10. Furthermore, the proposed method can quickly converge to the optimal solution, thereby improving the execution efficiency of ontology alignment.

Table 5
Comparison results with other better algorithms under different iteration. The times are in milliseconds

MaxIter SGBGOA-OM OBLGOA-OM GOA-OM

F-measure Time F-measure Time F-measure Time

10 0.855 36948 0.804 33204 0.834 25803

30 0.785 89688 0.841 79914 0.842 57165

50 0.855 249644 0.840 130263 0.842 90946

MaxIter	SGBGOA-OM	OBLGOA-OM	GOA-OM
10	0.855	36948	0.804	33204	0.834	25803
30	0.785	89688	0.841	79914	0.842	57165
50	0.855	249644	0.840	130263	0.842	90946

4.3.4 Anatomy track

Table 6 shows the comparison results with based on GOA algorithms and the other metaheuristic techniques. Each algorithm is independently executed 10 times, and the average value is calculated. In terms of F-measure, the SGBGOA-OM algorithm obtains the best results. Further, the SGBGOA-OM method can find the suboptimal alignment, i.e., the value of the optimal F-measure is 0.867. Since the SGBGOA-OM method does not use background knowledge in the execution process, five ontology alignment methods from OAEI such as POMAP++ [30], SANOM [30], Lily [17], Wiktionary [18], ALIN [19], FCAMap-KG [20] and DOME [21], are selected to compare with the SGBGOA-OM method. As shown in Table 7, the SGBGOA-OM method outperforms all other methods except POMAP++ and SANOM in terms of F-measure. This result shows that the proposed algorithm can find a high-quality alignment with a fast convergence rate.

Table 6
Comparison results of SGBGOA-OM with other algorithms

Algorithm SGBGOA-OM OBLGOA PGOA CGOA GOA CPSO BA FA

Precision 0.9363 0.9475 0.9673 0.9668 0.9513 0.9687 0.30 0.275

Recall 0.7868 0.7538 0.7183 0.6900 0.7558 0.7386 0.546 0.497

F-measure 0.8550 0.8397 0.8243 0.8012 0.8423 0.8379 0.376 0.354

Optimal Fmeasure 0.8670 0.8440 0.8360 0.8500 0.8460 0.849 0.593 0.363

Algorithm	SGBGOA-OM	OBLGOA	PGOA	CGOA	GOA	CPSO	BA	FA
Precision	0.9363	0.9475	0.9673	0.9668	0.9513	0.9687	0.30	0.275
Recall	0.7868	0.7538	0.7183	0.6900	0.7558	0.7386	0.546	0.497
F-measure	0.8550	0.8397	0.8243	0.8012	0.8423	0.8379	0.376	0.354
Optimal Fmeasure	0.8670	0.8440	0.8360	0.8500	0.8460	0.849	0.593	0.363

Table 7

Comparison results of SGBGOA-OM with other systems participated OAEI-2019

Methods	POMAP++	SANOM	SGBGOA-OM	Lily	Wiktionary	ALIN	FCAMap-KG	DOME
Precision	0.919	0.888	0.9363	0.873	0.968	0.974	0.996	0.996
Recall	0.877	0.844	0.7868	0.796	0.730	0.698	0.631	0.615
F-measure	0.897	0.865	0.8550	0.833	0.832	0.813	0.772	0.760

Based on the above analysis, it can be concluded that the proposed algorithm is able to achieve better alignment than the other algorithms. It can improve the efficiency of the original GOA in practical applications, especially in large-scale complexproblems.

4.3.5 Large biomedical track

The FMA-NCI-small task is to match the FMA fragment with 3696 classes and the NCI fragment with 6488 classes.

The FMA-SNOMED-small task is to match the FMA fragment with 10157 classes and the SNOMED fragment with 13412 classes. All results are the average of 5 independent executions of each algorithm. The proposed SGBGOA-OM method is compared with GOA-based algorithms, the other meta-heuristic algorithms and the systems from OAEI¹ competition. The results of the FMA-NCI-small task are given in Table 8, in which it can clearly observe that the proposed SGBGOA-OM method outperforms all metaheuristic-based methods, DOME, and AGM in terms of F-measure. Compared with FCAMapKG and LogMapLt, it achieves competitive results. Table 9 shows the comparison results on the FMA-SNOMED-small task. The proposed SGBGOA-OM method obtains the highest quality alignment and is significantly better than other meta-heuristic-based methods and the competition systems fromOAEI 1 .

Table 8
The results of the alignment on largebio task FMA-NCI small

Algorithm SGBGOA-OM OBLGOA PGOA CGOA GOA BA FA FCAMapKG LogMapLt DOME AGM

Precision 0.969 0.587 0.00 0.588 0.588 0.484 0.485 0.967 0.967 0.984 0.495

Recall 0.78 0.465 0.00 0.465 0.459 0.572 0.574 0.817 0.819 0.766 0.481

F-measure 0.864 0.519 0.00 0.519 0.515 0.524 0.526 0.886 0.887 0.861 0.488

Optimal Fmeasure 0.867 0.867 0.00 0.865 0.863 0.527 0.529 N/A N/A N/A N/A

Algorithm	SGBGOA-OM	OBLGOA	CGOA	GOA	BA	FA	FCAMapKG	LogMapLt	DOME	AGM
Precision	0.969	0.587	0.588	0.588	0.484	0.485	0.967	0.967	0.984	0.495
Recall	0.78	0.465	0.465	0.459	0.572	0.574	0.817	0.819	0.766	0.481
F-measure	0.864	0.519	0.519	0.515	0.524	0.526	0.886	0.887	0.861	0.488
Optimal Fmeasure	0.867	0.867	0.865	0.863	0.527	0.529	N/A	N/A	N/A	N/A

Table 9

The results of the alignment on largebio task FMA-SNOMED small

Algorithm	SGBGOA-OM	OBLGOA	PGOA	CGOA	GOA	BA	FA	FCAMapKG	LogMapLt	DOME	AGM
Precision	0.865	0.934	0.972	0.971	0.952	0.516	0.522	0.973	0.968	0.988	0.463
Recall	0.662	0.51	0.280	0.266	0.414	0.577	0.585	0.222	0.208	0.198	0.365
F-measure	0.748	0.639	0.423	0.406	0.553	0.545	0.551	0.362	0.342	0.330	0.408
Optimal Fmeasure	0.776	0.748	0.617	0.606	0.739	0.548	0.559	N/A	N/A	N/A	N/A

5 Conclusions and future work

In this paper, a novel grasshopper optimization algorithm based on solitarious and gregarious states difference is proposed to enhance the performance of the original GOA algorithm. In the proposed algorithm, a novel strategy of sparse population learning from dense population is proposed to improve the global search ability of the GOA. A chemotaxis operator is used near the best target location to improve the local search capability of the GOA. The SGBGOA algorithm is executed on 23 benchmark functions. The experimental results show that the proposed algorithm can obtain a better solution than several representative GOAs and the other metaheuristic techniques. Finally, the proposed algorithm is implemented in different challenges of ontology alignment problem. Experimental results demonstrate that the SGBGOA can improve the efficiency and quality in large ontology alignment tasks compared to other algorithms. In the future, we intend to study the performance of SGBGOA in the other application.

Footnotes

Acknowledgments

This work is supported by the Ministry of Education-China Mobile Joint Fund Project under Grant No. MCM2020J01.

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