Oppositional spotted hyena optimizer with mutation operator for global optimization and application in training wavelet neural network

Abstract

Success behind nature inspired evolutionary metaheuristic algorithms lies in its seemly combination of operator’s castoff for smooth balance between exploration and exploitation. The deficit in such combination leads to untimely convergence of an algorithm, simultaneously failed to attain global optimum by stocking in local optimum. This work represents atypical algorithm termed as OBL-MO-SHO to improve the performance of existing SHO. To deal with more intricate realistic problems and to enhance the explorative and exploitative strength of SHO, we have integrated the oppositional learning concept with mutation operator. The proposed algorithm OBL-MO-SHO (oppositional spotted hyena optimizer with mutation operator) reveals promising performance in terms of achieving global optimum and superior convergence rate which confirms its improved exploration and exploitation capability within searching region. To establish competency of proposed OBL-MO-SHO algorithm the same is appraised by means of standard functions set belongs to IEEE CEC 2017. The efficacy of said method has been proven by means of various performance metrics and the outcomes also compared with state-of-the-art algorithms. To scrutinize its uniqueness statistically, Friedman and Holms test has been performed as one non-parametric test. Additionally as an application to unravel real world intricate difficulties the said OBL-MO-SHO algorithm has been castoff to train wavelet neural network by considering datasets selected from UCI depository. The reported results unveils that the evolved OBL-MO-SHO might be one potential algorithm for enlightening different optimization difficulties effectively.

Keywords

Swarm intelligence spotted hyena optimizer opposition-based learning mutation operator optimization classification wavelet neural network (WNN)

1 Introduction

Optimization is course of generating all probable consequences confined to a predetermined search region by considering applied constraints and opt for the finest outcome. Currently optimization has been widely used across disciplines of engineering and sciences. To deal with intricate real world optimization glitches, nature influenced evolutionary metaheuristic methods are predominantly employed. The higher acceptance of such algorithms is due to its underlying advantages and disadvantages associated with deterministic algorithms which prefers a predefined input set of parameters which leads to unsuccessful attainment of global optimum. In addition to this metaheuristic methods make use of stochastic technique which in turn avoids local optima stagnation problem satisfactorily and succeeding towards global optimum [1]. In the year 1997 scientist Wolpert and Macready stated a theorem termed as “No free lunch theorem” which is the cause of evolvement of new metaheuristic algorithm. According to the theorem all the optimization difficulties can’t be resolved by a single metaheuristic algorithm due to their undergoing differences [2]. Metaheuristic algorithms are also termed as nature inspired evolutionary techniques because it draws their motivation from different natural phenomenon. Some popular approach follows Darwinians concept of evolution i.e. “legacy of good qualities” and competition for the “survival of the fittest”. Some popular approaches which falls to this category are, Genetic algorithm [3], Differential evolution [4] and evolutionary strategies [5]. The other categories of algorithms rely on locomotion, foraging and chasing pattern of swarm. Popular methods belong to this category are particle swarm optimization (PSO) [6], salp swarm algorithm (SSA) [7], grey wolf optimization (GWO) [8], improved salp swarm algorithm with space transformation search [9] and artificial bee colony algorithm (ABC) [10] etc. Apart from this some approaches are also rely on natural laws of universe e.g. gravitational search algorithm (GSA) [11], central force optimization algorithm (CRO) [12] and big bang big crunch algorithm [13] etc. The spotted hyena optimizer (SHO) is one of the recently developed nature inspired evolutionary algorithm proposed by Dhiman et al. for resolving constrained and unconstrained continuous optimization glitches. SHO imitates the social and hunting pattern of spotted hyena. Although there are a huge number of metaheuristic algorithms are available, but we have chosen SHO due to its superior explorative strength inside search region as compared to other state-of-the-art algorithm. The inspiration behind the work is to improve the explorative strength and to enhance the convergence rate of SHO. The ultimate aim of improved exploration is to disclose better solutions which are still unexplored within the search region. The metaheuristic approach is well accepted due to the underlying randomness associated for the selection of input factors [14]. Due to this advantage this is always a chance to improve the explorative strength for continuous optimization difficulties. Hence in the proposed OBL-MO-SHO algorithm we have integrated oppositional learning approach along with mutation operation to enhance the explorative and exploitative strength of classical SHO. The efficacy of the proposed method has been validated by using benchmark function sets belongs to IEEE CEC 2017 [15], along with the outcome is scrutinized over other recent algorithms. Furthermore, the proposed OBL-MO-SHO algorithm has been utilized as a trainer to train wavelet neural network over selected benchmark datasets taken from UCI depository to resolve genuine real world optimizations [16]. Though SHO is a very recently developed metaheuristic algorithm, but it is attracting researchers due to its capacity to resolve various complicated optimization tasks. The real world convoluted engineering complications can be tackled by hybridizing SHO with PSO [17] and also by using the multi objective version of SHO [18]. The same algorithm also castoff to solve various engineering problem and the practicability is demonstrated in high dimensional environment [19]. Feature selection task are also handled by considering binary version of SHO [20].The same algorithm SHO also used in electrical engineering to solve OPF problems [21]. It has been also castoff to solve monetary dispatch issue. To prove its effectiveness as a trainer Li et al. castoff SHO for training FFNN [22]. It has also verified as a successful trainer algorithm to train multilayer perceptron’s network concurrently proving its uniqueness by considering nonparametric tests [23]. The above applications in multidisciplinary fields within a very short span of time confirms its suitability for solving various complex optimization glitches.

The rest part of the paper is organized as follows: Subsection 2 characterises the details behind SHO algorithm, the oppositional learning concept, mutation operator and the proposed oppositional SHO with mutation operator algorithm. Subsection 3 characterises details about the benchmark functions used along with the environment inclusive of experimentation settings. Subsection 4 characterises the depth analysis of conquered results, comparison of conquered results with state-of-the-art algorithms, complexity and convergence analysis. Subsection 5 characterises the statistical significance of the proposed method by the consideration of nonparametric tests. Subsection 5 characterises the realistic application of proposed method as a trainer algorithm over chosen benchmark datasets by considering wavelet neural network. In conclusion the overall summary of the carried work and future directions has been presented.

2 Basic preliminaries

The current subsection represents social and chasing behaviour of hyenas and proposed oppositional SHO with mutation operator.

2.1 Outline of SHO

The spotted hyena or laughing hyena is a greatly successful social carnivorous animal who lives in groups and exhibits most complex social behaviour. Spotted hyenas are complicated, clever and capability to battle unendingly for foodstuff and territory. In the group the female members are the dominance over the male members. Newly born male member have to leave the group when becoming adult and joins in a new searched group as a lowest ranked member in the group to consume food. But the female members will assured a stable position in their own clan. Based on the social behaviour recently Dhiman et al. developed the Spotted Hyena Optimizer (SHO) as a nature inspired metaheuristic algorithm in 2017 [24]. The design of SHO also is based on the social relationship and haunting nature i.e. tracking, chasing, encircling and attacking of spotted hyenas which modelled mathematically to solve optimization problems. Due to the lack of knowledge about the search region priori, the location of prey is considered as the current best or objective as it is closed to optimum. The other search agents in the space update their locations after conforming the location of best candidate solution about the global optimum solution. The behaviour is represented by Equations (1) and (2).

$\vec{D_{ps}} = | \vec{B_{cv}} * \vec{P_{pv}} * I - \vec{PS} (CI) |$ (1)

$\vec{PS} (I + 1) = \vec{P_{pv}} (I) - \vec{E_{cv}} * \vec{D_{ps}}$ (2) Where, $\vec{D_{ps}}$ , denotes distance between target and spotted hyenaCI, denotes ongoing iteration $\vec{P_{pv}}$ , denotes location vector of target $\vec{PS}$ , denotes location vector of spotted hyena $\vec{B_{cv}}$ and $\vec{E_{cv}}$ , describes co-efficient vectors and denoted as: $\vec{B_{cv}} = 2 * \vec{ran d_{v 1}}$ $\vec{E_{cv}} = 2 * \vec{h_{ℓ}} * \vec{ran d_{v 2}} - \vec{h_{ℓ}}$ $\vec{h_{ℓ}} = 5 - (itern * (\frac{5}{maxitern}))$ Where,itern = 1,2,3, ... , maxitern $\vec{ran d_{v 1}}$ And $\vec{{rand}_{v 2}}$ denotes arbitrary vectors in the range [0,1] $\vec{h_{ℓ}}$ , linearly reduce from [5–0].

To outline the chasing behavior mathematically, the location of target is acknowledged to the finest agent which considered as optimum. The other search agent’s formulates a collection of reliable associates and holds back the finest consequence acquired until that point to alter their places. The Equations (3), (4), and (5) are accountable to alter place of spotted hyenas.

$\vec{D_{ps}} = | \vec{B_{cv}} * \vec{P_{bs}} - \vec{P_{rs}} |$ (3)

$\vec{P_{rs}} = \vec{P_{bs}} - \vec{E_{cv}} * \vec{D_{ps}}$ (4)

$\vec{C_{ro}} = \vec{P_{rs}} + \vec{P_{rs + 1}} + \dots + \vec{P_{rs + NC}}$ (5) Where $\vec{P_{bs}}$ designates the finest search agent and $\vec{P_{rs}}$ designates the place of other agents within search area. $\vec{C_{ro}}$ , defines collection of all optimal results.

$\begin{matrix} NC = {Count}_{num} \\ (\vec{P_{bs}}, \vec{P_{bs + 1}}, \vec{P_{bs + 2}} + \dots + (\vec{P_{bs}} + \vec{MR})) \end{matrix}$ Where NC indicates the populace of spotted hyenas, MR specifies arbitrary vector, limiting from [0, 1] and num indicates sum of candidate solutions.

The ultimate finest result can be estimated by Equation (6).

$\vec{PS} (CI + 1) = \frac{\vec{C_{ro}}}{NC}$ (6) Where $\vec{PS} (CI + 1)$ characterizes the most favorable consequence which is acquired from previously observed value accomplished from distinct iteration and others will alter their own places in accordance with observed finest result attained.

2.2 The OBL concept

In case of stochastic algorithms which relies on randomness, offers optimum result and quicker convergence if the optimum solution is not far away. But assuming a worst scenario where the global optimum is present just in the opposite direction of considered random initial guesses position, at this point the optimization process will consume more time or sometimes stuck in local optimum. So considering pre-knowledge, is impossible for a best random guess, the search process should continue among all directions or more accurately, in the opposite directions. The concept was first introduced by Tizhoosh in the year of 2005 [25]. Opposite numbers using OBL model can be stated as follows:

Let p∈[m, n], then opposite number can be expressed as,

$p^{\to} = m + n - p$ (7) Where p signifies a real number and m, n signifies the bounds of the search area.

2.3 Mutation operator

To enhance the explorative strength of evolutionary based stochastic methods mutation operator has been used [26]. Here we have used normal distributed uniform mutation operator rather random mutation operator. Because by applying random mutation operators, to the stochastic algorithms, it leads towards local optima stagnation many times due to random selection. So to avoid limitations associated with random mutation, normal distributed uniform mutation has been integrated with oppositional learning for achieving superior outcome. This approach leads to balanced motion considering lesser to larger mutants around the best search agent in ongoing as well as opposite directions. Mathematically, normal distributed uniform mutation can be represented as,

$B_{j}^{k} = s_{j} + m (0, σ^{2}) = s_{j} + σ * m (0, 1)$ (8) Where s_j signifies un-mutated spotted hyena population and σ signifies mutated factor. So position of entire jthspotted hyena in kthdimension can be represented as,

$s_{j}^{k} = s_{j} + [SRD * s_{j} * m (0, 1)]$ (9) Where considered SRD value is 0.2.

2.4 Proposed oppositional SHO with mutation operator (OBL-MO-SHO)

The current segment depicts the proposed oppositional learning mutated spotted hyena optimizer algorithm (OBL-MO-SHO). This OBL-MO-SHO incorporates the oppositional learning model along with normal distributed uniform mutation operator represented in Equation (7 & 9). In the first step, oppositional learning method has been applied by considering possible candidate solution to generate reversed candidate solutions for further iteration. At the outcome the approach improves the exploration ability by taking advantage of initial randomly achieved solutions as well as opposite solutions also, which improves the rate of convergence. In the second approach, the normal distributed uniform mutation operator is applied in continuation with oppositional learning model to establish counter balance between exploration and exploitation. To achieve the purpose smaller mutant factors has been utilized for current and reverse directions, which boost the controlling factors for better exploitation and speed up the convergence rate with growth of iterations. The considered mutant is 0.2. The pseudocode for proposed OBL-MO-SHO algorithm is represented in Table 1.

Table 1
Proposed Oppositional SHO with Mutation Operator

Pseudocode of OBL-MO-SHO algorithm

Mention entire parameters and declare h_ℓ, b_cv,E_cvand NC

Evaluate the fitness of each distinct agent

Concurrently evaluate by oppositional-mutated model as shown in Eqs. (7 and 9)

Allocate P_bs = finest search agent

Make a cluster of optimal values, denoted as C_ro

while (cnt < highest count of repetitions) do

for individual search agent in the population do

Modify placement of ongoing search agents in population byEquation (6)

end for

Update factors: h_ℓ, b_cv, E_cvand NC

Check for exceeding beyond boundary limits by any of search agents: if so fine-tune

Determine fitness of modified search agents individually

Alter P_bsvalue, if there persists superior result as compare to earlier finest result

Modify cluster C_roin accordance with P_bs

cnt = cnt + 1

end while

return P_bs

Pseudocode of OBL-MO-SHO algorithm
Mention entire parameters and declare h_ℓ, b_cv,E_cvand NC
Evaluate the fitness of each distinct agent
Concurrently evaluate by oppositional-mutated model as shown in Eqs. (7 and 9)
Allocate P_bs = finest search agent
Make a cluster of optimal values, denoted as C_ro
while (cnt < highest count of repetitions) do
for individual search agent in the population do
Modify placement of ongoing search agents in population byEquation (6)
end for
Update factors: h_ℓ, b_cv, E_cvand NC
Check for exceeding beyond boundary limits by any of search agents: if so fine-tune
Determine fitness of modified search agents individually
Alter P_bsvalue, if there persists superior result as compare to earlier finest result
Modify cluster C_roin accordance with P_bs
cnt = cnt + 1
end while
return P_bs

3 Standard functions and Numerical Experimentation settings

IEEE CEC 2017 benchmark functions set has been employed to endorse the proficiency of presented OBL-MO-SHO [15]. The set covers 28 diversified constrained optimization difficulties essentially unimodal, multimodal, composite and hybrid in nature. The results reported after executing the proposed method inclusive of SHO, by allowing 500 reiterations for individual trial and 25 such trials considered for problem dimension 10 and 30 over 20 search agents. The searching region boundary values has been considered as per CEC 2017 as: [–10, 10], [–20, 20], [–50, 50] and [–100, 100]. For the simulation purpose we have considered tool as (MATLAB 2016a) with system specification as Intel (R) Core i7 CPU @ 3.7GHz, Attached memory 4 GB inclusive of OS 64 bit.

4 Outcome analysis

The recorded results expressed are according to the substructure provisioned in IEEE CEC 2017. The obtained result from existing SHO and proposed OBL-MO-SHO algorithm has been represented by assuming 28 constrained functions set pertaining to CEC 2017 applied over dimensions 10 and 30, conferred in Tables 2 and 3. The diverse efficacy measures experienced by executing objective functions from CEC 2017 are accorded in respect of minimum, maximum, mean, median and STD (standard deviation) value. The CEC 2017 function set incorporates 4 varieties of functions such as unimodal, multimodal, hybrid and composite. Unimodal functions are well suited to probe the exploitation competence and multimodal functions are convenient for exploration proficiency. Unimodal functions are appeared as F₁ to F₃ and F₄ to F₁₀ signifies multimodal function in the CEC 2017 objective functions set. The proposed OBL-MO-SHO illustrates superior outcomes as counter to SHO algorithm, which confirms the concrete compliance of the said method. The rest of the functions i.e. F₁₁ to F₂₈falls to the category of hybrid and composite functions in nature. The evolvement of new metaheuristic algorithms day by day is due to the cause for attaining global optimum, by escaping successfully from local minima stuck problem. This important characteristics of metaheuristic algorithm has been assessed by both hybrid and composite functions. In our reported result from Tables 2 and 3, Proposed OBL-MO-SHO outperforms over SHO in all directions.

Table 2
Mean, Standard Deviation, Median, Minimum, Maximum error value of objective functions conquered by OBL-MO-SHO and SHO for IEEE CEC 2017 problems having dimension 10. Bold mark indicates superior result

Benchmark functions Algorithm Mean Median Std Dev Minimum Maximum

1 SHO 6.20E+03 5.33E+03 2284.466 4.91E+03 1.08E+04

OBL-MO-SHO 4.89E+03 4.63E+03 2365.041 2.69E+03 9.13E+03

2 SHO 6.12E+03 6.41E+03 2171.331 2.74E+03 8.43E+03

OBL-MO-SHO 5.34E+03 6.01E+03 1829.201 2.50E+03 6.99E+03

3 SHO 4.83E+03 5.37E+03 1407.224 2.93E+03 6.25E+03

OBL-MO-SHO 4.06E+03 4.39E+03 1542.287 2.06E+03 6.08E+03

4 SHO 2.38E+02 2.27E+02 26.26785 2.21E+02 2.77E+02

OBL-MO-SHO 2.36E+02 2.27E+02 24.66272 2.18E+02 2.72E+02

5 SHO 3.15E+05 2.65E+05 267437.2 5.37E+04 6.78E+05

OBL-MO-SHO 3.03E+05 2.45E+05 277365.4 4.30E+04 6.78E+05

6 SHO 5.07E+02 5.31E+02 130.7192 3.19E+02 6.53E+02

OBL-MO-SHO 4.96E+02 5.01E+02 123.6232 3.18E+02 6.31E+02

7 SHO –2.22E+02 –2.10E+02 52.09287 –2.88E+02 –1.79E+02

OBL-MO-SHO –1.41E+02 –1.34E+02 28.8603 –1.81E+02 –1.14E+02

8 SHO 10.692 10.95 2.562659 7.09 13.67

OBL-MO-SHO 10.4 10.91 2.51994 6.85 13.31

9 SHO 7.51 7.91 1.089709 5.95 8.27

OBL-MO-SHO 7.51 7.91 1.089709 5.95 8.27

10 SHO 28.0275 27.475 9.367769 18.49 38.67

OBL-MO-SHO 25.115 23.61 12.05025 14.89 38.35

11 SHO –2.51E+02 –2.25E+02 77.23535 –3.86E+02 –1.97E+02

OBL-MO-SHO –2.25E+02 –2.25E+02 4.4721 –2.27E+02 –2.25E+02

12 SHO 4.97E+03 5.38E+03 874.09 3.97E+03 5.57E+03

OBL-MO-SHO 4.92E+03 5.33E+03 910.8421 3.88E+03 5.56E+03

13 SHO 4.83E+08 4.73E+08 2.35E+08 1.56E+08 8.05E+08

OBL-MO-SHO 4.11E+08 4.33E+08 2.42E+08 1.52E+08 7.80E+08

14 SHO 20.635 20.7 0.249332 20.29 20.85

OBL-MO-SHO 20.295 20.515 0.494132 19.56 20.59

15 SHO 39.8075 40.27 6.562567 32.32 46.37

OBL-MO-SHO 39.685 40.21 6.591269 32.12 46.2

16 SHO 1.69E+02 1.70E+02 8.869423 1.58E+02 1.78E+02

OBL-MO-SHO 1.64E+02 1.63E+02 10.90871 1.52E+02 1.77E+02

17 SHO 2.208 2.19 0.230152 1.92 2.52

OBL-MO-SHO 2.188 2.13 0.232207 1.91 2.51

18 SHO 3.99E+03 4.29E+03 1447.927 2.22E+03 5.18E+03

OBL-MO-SHO 3.89E+03 4.20E+03 1517.418 2.03E+03 5.15E+03

19 SHO 30.654 31.74 3.053036 26.39 33.5

OBL-MO-SHO 29.94 31.66 3.434734 25.08 33.48

20 SHO 2.17 2.14 0.096609 2.09 2.31

OBL-MO-SHO 1.69 1.565 0.296311 1.5 2.13

21 SHO 5.21E+03 5.08E+03 828.77 4.34E+03 6.32E+03

OBL-MO-SHO 4.96E+03 4.84E+03 703.15 4.34E+03 5.82E+03

22 SHO 2.16E+08 2.36E+08 96465033 8.89E+07 3.05E+08

OBL-MO-SHO 2.11E+08 2.31E+08 94378507 8.49E+07 2.97E+08

23 SHO 20.71 20.71 0.148997 20.51 20.89

OBL-MO-SHO 20.524 20.55 0.216979 20.25 20.75

24 SHO 37.97667 39.64 4.463814 32.92 41.37

OBL-MO-SHO 37.32333 38.82 4.000779 32.79 40.36

25 SHO 1.65E+02 1.67E+02 31.51058 1.32E+02 1.96E+02

OBL-MO-SHO 1.60E+02 1.59E+02 36.28935 1.26E+02 1.95E+02

26 SHO 2.176667 2.32 0.361866 1.53 2.46

OBL-MO-SHO 2.146667 2.305 0.400832 1.41 2.45

27 SHO 4.09E+03 3.80E+03 1779.963 2.39E+03 6.37E+03

OBL-MO-SHO 3.98E+03 3.67E+03 1731.521 2.33E+03 6.24E+03

28 SHO 28.355 28.41 3.042526 24.75 31.85

OBL-MO-SHO 24.695 24.295 1.891851 23.15 27.04

Benchmark functions	Algorithm	Mean	Median	Std Dev	Minimum	Maximum
1	SHO	6.20E+03	5.33E+03	2284.466	4.91E+03	1.08E+04
	OBL-MO-SHO	4.89E+03	4.63E+03	2365.041	2.69E+03	9.13E+03
2	SHO	6.12E+03	6.41E+03	2171.331	2.74E+03	8.43E+03
	OBL-MO-SHO	5.34E+03	6.01E+03	1829.201	2.50E+03	6.99E+03
3	SHO	4.83E+03	5.37E+03	1407.224	2.93E+03	6.25E+03
	OBL-MO-SHO	4.06E+03	4.39E+03	1542.287	2.06E+03	6.08E+03
4	SHO	2.38E+02	2.27E+02	26.26785	2.21E+02	2.77E+02
	OBL-MO-SHO	2.36E+02	2.27E+02	24.66272	2.18E+02	2.72E+02
5	SHO	3.15E+05	2.65E+05	267437.2	5.37E+04	6.78E+05
	OBL-MO-SHO	3.03E+05	2.45E+05	277365.4	4.30E+04	6.78E+05
6	SHO	5.07E+02	5.31E+02	130.7192	3.19E+02	6.53E+02
	OBL-MO-SHO	4.96E+02	5.01E+02	123.6232	3.18E+02	6.31E+02
7	SHO	–2.22E+02	–2.10E+02	52.09287	–2.88E+02	–1.79E+02
	OBL-MO-SHO	–1.41E+02	–1.34E+02	28.8603	–1.81E+02	–1.14E+02
8	SHO	10.692	10.95	2.562659	7.09	13.67
	OBL-MO-SHO	10.4	10.91	2.51994	6.85	13.31
9	SHO	7.51	7.91	1.089709	5.95	8.27
	OBL-MO-SHO	7.51	7.91	1.089709	5.95	8.27
10	SHO	28.0275	27.475	9.367769	18.49	38.67
	OBL-MO-SHO	25.115	23.61	12.05025	14.89	38.35
11	SHO	–2.51E+02	–2.25E+02	77.23535	–3.86E+02	–1.97E+02
	OBL-MO-SHO	–2.25E+02	–2.25E+02	4.4721	–2.27E+02	–2.25E+02
12	SHO	4.97E+03	5.38E+03	874.09	3.97E+03	5.57E+03
	OBL-MO-SHO	4.92E+03	5.33E+03	910.8421	3.88E+03	5.56E+03
13	SHO	4.83E+08	4.73E+08	2.35E+08	1.56E+08	8.05E+08
	OBL-MO-SHO	4.11E+08	4.33E+08	2.42E+08	1.52E+08	7.80E+08
14	SHO	20.635	20.7	0.249332	20.29	20.85
	OBL-MO-SHO	20.295	20.515	0.494132	19.56	20.59
15	SHO	39.8075	40.27	6.562567	32.32	46.37
	OBL-MO-SHO	39.685	40.21	6.591269	32.12	46.2
16	SHO	1.69E+02	1.70E+02	8.869423	1.58E+02	1.78E+02
	OBL-MO-SHO	1.64E+02	1.63E+02	10.90871	1.52E+02	1.77E+02
17	SHO	2.208	2.19	0.230152	1.92	2.52
	OBL-MO-SHO	2.188	2.13	0.232207	1.91	2.51
18	SHO	3.99E+03	4.29E+03	1447.927	2.22E+03	5.18E+03
	OBL-MO-SHO	3.89E+03	4.20E+03	1517.418	2.03E+03	5.15E+03
19	SHO	30.654	31.74	3.053036	26.39	33.5
	OBL-MO-SHO	29.94	31.66	3.434734	25.08	33.48
20	SHO	2.17	2.14	0.096609	2.09	2.31
	OBL-MO-SHO	1.69	1.565	0.296311	1.5	2.13
21	SHO	5.21E+03	5.08E+03	828.77	4.34E+03	6.32E+03
	OBL-MO-SHO	4.96E+03	4.84E+03	703.15	4.34E+03	5.82E+03
22	SHO	2.16E+08	2.36E+08	96465033	8.89E+07	3.05E+08
	OBL-MO-SHO	2.11E+08	2.31E+08	94378507	8.49E+07	2.97E+08
23	SHO	20.71	20.71	0.148997	20.51	20.89
	OBL-MO-SHO	20.524	20.55	0.216979	20.25	20.75
24	SHO	37.97667	39.64	4.463814	32.92	41.37
	OBL-MO-SHO	37.32333	38.82	4.000779	32.79	40.36
25	SHO	1.65E+02	1.67E+02	31.51058	1.32E+02	1.96E+02
	OBL-MO-SHO	1.60E+02	1.59E+02	36.28935	1.26E+02	1.95E+02
26	SHO	2.176667	2.32	0.361866	1.53	2.46
	OBL-MO-SHO	2.146667	2.305	0.400832	1.41	2.45
27	SHO	4.09E+03	3.80E+03	1779.963	2.39E+03	6.37E+03
	OBL-MO-SHO	3.98E+03	3.67E+03	1731.521	2.33E+03	6.24E+03
28	SHO	28.355	28.41	3.042526	24.75	31.85
	OBL-MO-SHO	24.695	24.295	1.891851	23.15	27.04

Table 3

Mean, Standard Deviation, Median, Minimum, Maximum error value of objective functions conquered by OBL-MO-SHO and SHO for IEEE CEC 2017 problems having dimension 30. Bold mark indicates superior result

Benchmark functions	Algorithm	Mean	Median	Std Dev	Minimum	Maximum
1	SHO	3.82E+04	3.37E+04	19594.45	2.12E+04	6.43E+04
	OBL-MO-SHO	3.27E+04	3.04E+04	12436.64	2.14E+04	4.85E+04
2	SHO	5.49E+04	5.01E+04	23194.59	3.41E+04	8.53E+04
	OBL-MO-SHO	4.30E+04	3.81E+04	16992.62	2.98E+04	6.60E+04
3	SHO	1.66E+05	1.48E+05	71210.37	1.00E+05	2.67E+05
	OBL-MO-SHO	1.64E+05	1.48E+05	73208.45	9.22E+04	2.66E+05
4	SHO	9.70E+02	1.01E+03	122.0983	8.04E+02	1.12E+03
	OBL-MO-SHO	9.68E+02	1.00E+03	121.494	8.04E+02	1.12E+03
5	SHO	4.98E+06	4.82E+06	1548798	3.35E+06	6.68E+06
	OBL-MO-SHO	4.98E+06	4.82E+06	1548798	3.35E+06	6.68E+06
6	SHO	2.59E+03	2.54E+03	434.0891	2.11E+03	3.16E+03
	OBL-MO-SHO	2.50E+03	2.41E+03	445.4492	2.07E+03	3.12E+03
7	SHO	–2.58E+02	–2.57E+02	50.63596	–3.27E+02	–1.95E+02
	OBL-MO-SHO	–2.58E+02	–2.57E+02	50.32594	–3.27E+02	–1.96E+02
8	SHO	16.2525	16.725	2.523561	12.8	18.76
	OBL-MO-SHO	15.6875	16.19	2.214473	12.75	17.62
9	SHO	12.388	12.6	0.486487	11.79	12.88
	OBL-MO-SHO	12.388	12.6	0.486487	11.79	12.88
10	SHO	40.811	40.715	4.555499	35.51	46.4
	OBL-MO-SHO	40.7825	40.625	4.556595	35.5	46.38
11	SHO	–4.32E+02	–3.97E+02	125.3695	–6.18E+02	–2.90E+02
	OBL-MO-SHO	–4.32E+02	–3.97E+02	125.3695	–6.18E+02	–2.90E+02
12	SHO	1.95E+04	1.95E+04	846.0693	1.86E+04	2.06E+04
	OBL-MO-SHO	1.94E+04	1.93E+04	888.8194	1.84E+04	2.05E+04
13	SHO	2.20E+09	2.35E+09	4.33E+08	1.57E+09	2.53E+09
	OBL-MO-SHO	2.15E+09	2.27E+09	4.05E+08	1.56E+09	2.49E+09
14	SHO	21.224	21.22	0.02881	21.2	21.27
	OBL-MO-SHO	21.14	21.18	0.06442	21.04	21.19
15	SHO	46.3125	47.045	2.623831	42.56	48.6
	OBL-MO-SHO	46.18	46.905	2.66427	42.38	48.53
16	SHO	6.63E+02	6.63E+02	20.90255	6.38E+02	6.88E+02
	OBL-MO-SHO	6.61E+02	6.61E+02	21.10884	6.37E+02	6.87E+02
17	SHO	5.65	5.4	0.80666	4.98	7.05
	OBL-MO-SHO	5.59	5.33	0.8	4.97	6.99
18	SHO	1.77E+04	1.82E+04	1873.277	1.50E+04	1.93E+04
	OBL-MO-SHO	1.76E+04	1.81E+04	1844.813	1.49E+04	1.91E+04
19	SHO	1.13E+02	1.14E+02	2.160247	1.10E+02	1.15E+02
	OBL-MO-SHO	1.10E+02	1.10E+02	3.511885	1.06E+02	1.13E+02
20	SHO	10.09	10.345	0.575442	9.23	10.44
	OBL-MO-SHO	9.4575	9.38	0.269119	9.23	9.84
21	SHO	1.86E+04	1.89E+04	1429.161	1.66E+04	2.00E+04
	OBL-MO-SHO	1.82E+04	1.83E+04	1276.715	1.65E+04	1.96E+04
22	SHO	1.94E+09	1.92E+09	3.27E+08	1.57E+09	2.36E+09
	OBL-MO-SHO	1.90E+09	1.87E+09	3.15E+08	1.56E+09	2.31E+09
23	SHO	21.2	21.17	0.056569	21.15	21.29
	OBL-MO-SHO	21.15	21.16	0.057879	21.08	21.23
24	SHO	48.53	49.54	3.150627	43.05	50.87
	OBL-MO-SHO	47.178	48.27	3.042083	42.44	50.01
25	SHO	6.60E+02	6.82E+02	50.17968	5.85E+02	6.91E+02
	OBL-MO-SHO	6.56E+02	6.78E+02	52.58961	5.77E+02	6.89E+02
26	SHO	5.754	5.68	0.269963	5.48	6.18
	OBL-MO-SHO	5.696	5.63	–0.21836	5.47	6.01
27	SHO	1.79E+04	1.92E+04	2281.082	1.53E+04	1.93E+04
	OBL-MO-SHO	1.75E+04	1.91E+04	2858.321	1.42E+04	1.92E+04
28	SHO	1.17E+02	1.17E+02	3.741657	1.13E+02	1.22E+02
	OBL-MO-SHO	1.13E+02	1.14E+02	1.707825	1.11E+02	1.15E+02

4.1 Consideration with state-of-the-art metaheuristic algorithms

The metaheuristic methods differ among each other on various factors, starting from number of regulating parameters, framework of algorithm and flow of execution. So the purpose of each evolved algorithm is specific in its use to resolve optimization complications. But to analyse the explorative power, exploitative power, convergence rate and figure out the differences of considered algorithms, such methods have executed over unimodal, multimodal, composite and hybrid functions. All the algorithms chosen have passed through an open resemblance among each other by employing standard functions belonging to CEC 2017 over dimension 10. The absolute count of iterations over the functions is restricted to 10⁴∗specified dimensions. The examined algorithms such as SCA [27], MFO [28], GWO [8], ALO [29], WOA [30] inclusive of proposed OBL-MO-SHO and their observed outcomes has been conferred in Table 4. From the announced outcome proposed OBL-MO-SHO algorithm outperforms over others.

Table 4
Scrutinizing performance of state-of-art algorithms for IEEE CEC 2017 benchmark complications having dimension 10. Bold mark specifies superior outcome

Benchmark functions GWO WOA SCA MFO ALO OBL-MO-SHO

1 2.01E+04 1.22E+04 1.37E+04 1.37E+04 2.50E+04 4.89E+03

2 2.51E+04 1.11E+04 1.06E+04 1.72E+04 1.64E+04 5.34E+03

3 2.33E+04 1.03E+04 2.28E+04 2.23E+04 1.07E+04 4.06E+03

4 2.65E+02 1.72E+02 2.86E+02 2.11E+02 2.15E+02 2.36E+02

5 7.36E+05 1.86E+05 3.85E+05 4.80E+05 3.71E+05 3.03E+05

6 6.81E+02 8.61E+02 9.61E+02 1.20E+03 8.14E+02 4.96E+02

7 –7.50E+00 –2.14E+02 –1.43E+02 5.40E+01 –1.17E+02 –1.41E+02

8 5.51E+01 –2.04E+01 5.29E+01 2.29E+01 2.18E+01 10.4

9 8.38E+00 –0.60E+00 9.48E+00 7.35E+00 5.09E+00 7.51

10 2.01E+01 4.01E+01 4.04E+01 4.79E+01 4.18E+01 25.115

11 –1.05E+02 –2.08E+02 –1.49E+02 –1.98E+02 –3.75E+02 –2.25E+02

12 1.99E+04 1.11E+04 1.61E+04 1.24E+04 2.17E+04 4.92E+03

13 1.04E+10 1.61E+09 7.01E+09 1.68E+09 5.42E+09 4.11E+08

14 2.11E+01 2.13E+01 2.12E+01 2.11E+01 2.08E+01 20.295

15 6.20E+01 7.95E+01 7.58E+01 7.09E+01 8.45E+01 39.685

16 3.21E+02 3.24E+02 3.55E+02 3.25E+02 2.75E+02 1.64E+02

17 3.18E+00 4.58E+00 5.09E+00 4.74E+00 4.33E+00 2.188

18 9.55E+03 1.05E+04 1.15E+04 1.45E+04 2.24E+04 3.89E+03

19 3.68E+01 3.36E+01 3.37E+01 3.95E+01 3.27E+01 29.94

20 3.34E+00 3.01E+00 3.09E+00 2.50E+00 2.97E+00 1.69

21 1.09E+04 1.55E+04 1.55E+04 1.50E+04 5.67E+03 5.73E+03

22 2.08E+09 2.75E+09 2.39E+09 4.45E+09 3.88E+09 2.11E+08

23 2.19E+01 2.16E+01 2.20E+01 2.22E+01 2.13E+01 20.524

24 5.85E+01 6.90E+01 6.65E+01 5.83E+01 7.46E+01 37.32

25 2.56E+02 3.28E+02 3.09E+02 3.69E+02 3.57E+02 1.60E+02

26 4.84E+00 4.25E+00 4.09E+00 4.94E+00 5.99E+00 2.14

27 1.44E+04 1.77E+04 1.55E+04 1.10E+04 1.28E+04 3.98E+03

28 3.86E+01 3.45E+01 3.64E+01 3.59E+01 3.14E+01 24.69

Benchmark functions	GWO	WOA	SCA	MFO	ALO	OBL-MO-SHO
1	2.01E+04	1.22E+04	1.37E+04	1.37E+04	2.50E+04	4.89E+03
2	2.51E+04	1.11E+04	1.06E+04	1.72E+04	1.64E+04	5.34E+03
3	2.33E+04	1.03E+04	2.28E+04	2.23E+04	1.07E+04	4.06E+03
4	2.65E+02	1.72E+02	2.86E+02	2.11E+02	2.15E+02	2.36E+02
5	7.36E+05	1.86E+05	3.85E+05	4.80E+05	3.71E+05	3.03E+05
6	6.81E+02	8.61E+02	9.61E+02	1.20E+03	8.14E+02	4.96E+02
7	–7.50E+00	–2.14E+02	–1.43E+02	5.40E+01	–1.17E+02	–1.41E+02
8	5.51E+01	–2.04E+01	5.29E+01	2.29E+01	2.18E+01	10.4
9	8.38E+00	–0.60E+00	9.48E+00	7.35E+00	5.09E+00	7.51
10	2.01E+01	4.01E+01	4.04E+01	4.79E+01	4.18E+01	25.115
11	–1.05E+02	–2.08E+02	–1.49E+02	–1.98E+02	–3.75E+02	–2.25E+02
12	1.99E+04	1.11E+04	1.61E+04	1.24E+04	2.17E+04	4.92E+03
13	1.04E+10	1.61E+09	7.01E+09	1.68E+09	5.42E+09	4.11E+08
14	2.11E+01	2.13E+01	2.12E+01	2.11E+01	2.08E+01	20.295
15	6.20E+01	7.95E+01	7.58E+01	7.09E+01	8.45E+01	39.685
16	3.21E+02	3.24E+02	3.55E+02	3.25E+02	2.75E+02	1.64E+02
17	3.18E+00	4.58E+00	5.09E+00	4.74E+00	4.33E+00	2.188
18	9.55E+03	1.05E+04	1.15E+04	1.45E+04	2.24E+04	3.89E+03
19	3.68E+01	3.36E+01	3.37E+01	3.95E+01	3.27E+01	29.94
20	3.34E+00	3.01E+00	3.09E+00	2.50E+00	2.97E+00	1.69
21	1.09E+04	1.55E+04	1.55E+04	1.50E+04	5.67E+03	5.73E+03
22	2.08E+09	2.75E+09	2.39E+09	4.45E+09	3.88E+09	2.11E+08
23	2.19E+01	2.16E+01	2.20E+01	2.22E+01	2.13E+01	20.524
24	5.85E+01	6.90E+01	6.65E+01	5.83E+01	7.46E+01	37.32
25	2.56E+02	3.28E+02	3.09E+02	3.69E+02	3.57E+02	1.60E+02
26	4.84E+00	4.25E+00	4.09E+00	4.94E+00	5.99E+00	2.14
27	1.44E+04	1.77E+04	1.55E+04	1.10E+04	1.28E+04	3.98E+03
28	3.86E+01	3.45E+01	3.64E+01	3.59E+01	3.14E+01	24.69

4.2 Complexity and convergence analysis

The commencing of parameters related to SHO as well as OBL-MO-SHO taken time as 0(TIN∗DN), where TINconnotes complete iteration total for generating arbitrary masses and DNconnotes the inputted problem measure. Fitness estimation consumes extreme time as 0(maxitern∗TIN∗DN), where maxiternconnotes complete reiterations sum. Over-all optimal counting entails time as 0(maxitern∗NoH), where NoHconnotes hyena population. The process, fitness estimation and re-initialization of parameters lingers until desired consequences attained, which consumes time as 0(n). Hence the complete time disbursed by OBL-MO-SHO algorithm is 0(n∗TIN∗maxitern∗DN∗NoH).

The convergence chart of proposed OBL-MO-SHO and existing SHO algorithm has been presented as shown in Fig. 1 in view of CEC 2017 problems set over dimension 10. The flat axis of the chart illustrates complete reiterations and perpendicular axis illustrates customary value functions indicated. Scrutinising the presented chart, proposed OBL-MO-SHO algorithm proves it’s worth concerning to faster convergence rate towards conquering global peak value, at the same time not falling in local optima stagnation.

Fig. 1

Convergence curve acquired from benchmark functions from IEEE CEC 2017.

5 Statistical analysis for performance assessment

Statistical analysis has been carried out to certify that the underlying data interpreted accurately and the established relationships are significant and not just a possibility occurrences. Here we have proved the notable distinction exists among considered metaheuristic methods. To achieve the purpose we have considered two nonparametric test, first one is Friedman’s test and later one is Holms test. To carry out Friedman’s test, we consider one null hypothesis H₀stated by: there exists no substantial heterogeneity among inspected metaheuristic methods. To establish the hypothesis considered significance level ∝ is 0.05. For evaluation purpose one rank is allocated to all metaheuristic method used, ranging from 1 to m based on the result obtained from benchmark function sets. The average of rank can be computed by Equation 10.

$\begin{matrix} {AVR}_{i} = \\ number of datasets \end{matrix}$ (10)

The Friedman’s statistics can be computed by Equation 11.

$F_{s} = \frac{(m - 1) Y_{s}^{2}}{m (T - 1) - (Y_{s}^{2})}$ (11)

Where $Y_{s}^{2} = \frac{12 m}{T (T + 1)} [\sum_{i} {AVR}_{i}^{2} - \frac{T \to {(T + 1)}^{2}}{4}]$

Here m depicts the entire number of objective test functions set and T depicts castoff optimization methods count. The Friedman’s statistics F_sis disseminated specifying to F-distribution inclusive of (T–1) and (T–1)(m–1)degree of freedom. The computed degree of freedom considering 6 metaheuristic algorithms and 28 objective functions set is in the scope 5 to 135. The perceived critical value for F(5,135)over αtaken as 0.05 is 4.3650 [31]. The assessed F_svalue is 16.640 which is much greater than observed critical value. So the null hypothesis is rejected due to observed greater F_svalue than critical value, confirming presence of notable distinctions among castoff algorithms. Then to establish the uniqueness of proposed method we have to go for another test i.e. Holms test. The null hypothesis may be stated by: the algorithm compared behaves equally. For this we require z_r value which is reflected in Equation 12.

$z_{r} = \frac{R_{i} - R_{j}}{SE}$ (12)

Where $SE = \sqrt{\frac{T (T + 1)}{6 \to m}}$

By considering the above computed z_rvalue possibly of p_rvalue can be determined from normal distribution table [32]. The considered hypothesis may be allowed or discarded by computing the reported p_rvalues with $\frac{α}{(T - 1)}$ values. From Table 6 all observed p_r values are lesser as compared to $\frac{α}{T - 1}$

Table 5

Average rank of recent metaheuristic algorithms concentrated on average error obtained from objective functions pertaining to CEC 2017

Problem	GWO	WOA	SCA	MFO	ALO	OBL-MO-SHO
1	2.01E+04(5)	1.22E+04(2)	1.37E+04(3.5)	1.37E+04(3.5)	2.50E+04(6)	4.89E+03(1)
2	2.51E+04(6)	1.11E+04(3)	1.06E+04(2)	1.72E+04(5)	1.64E+04(4)	5.34E+03(1)
3	2.33E+04(6)	1.03E+04(2)	2.28E+04(5)	2.23E+04(4)	1.07E+04(3)	4.06E+03(1)
4	2.65E+02(5)	1.72E+02(1)	2.86E+02(6)	2.11E+02(2)	2.15E+02(3)	2.36E+02(4)
5	7.36E+05(6)	1.86E+05(1)	3.85E+05(4)	4.80E+05(5)	3.71E+05(3)	3.03E+05(2)
6	6.81E+02(2)	8.61E+02(4)	9.61E+02(5)	1.20E+03(6)	8.14E+02(3)	4.96E+02(1)
7	–7.50E+00(5)	–2.14E+02(1)	–1.43E+02(2)	5.40E+01(6)	–1.17E+02(4)	–1.41E+02(3)
8	5.51E+01(6)	–2.04E+01(1)	5.29E+01(5)	2.29E+01(4)	2.18E+01(3)	10.4(2)
9	8.38E+00(5)	–0.60E+00(1)	9.48E+00(6)	7.53E+00(4)	5.09E+00(2)	7.51(3)
10	2.01E+01(1)	4.01E+01(3)	4.04E+01(4)	4.79E+01(6)	4.18E+01(5)	25.115(2)
11	–1.05E+02(6)	–2.08E+02(3)	–1.49E+02(4)	–1.98E+02(3)	–3.75E+02(1)	–2.25E+02(2)
12	1.99E+04(5)	1.11E+04(2)	1.61E+04(4)	1.24E+04(3)	2.17E+04(6)	4.92E+03(1)
13	1.04E+10(6)	1.61E+09(1)	7.01E+09(5)	1.68E+09(2)	5.42E+09(4)	4.11E+08(1)
14	2.11E+01(3.5)	2.13E+01(6)	2.12E+01(5)	2.11E+01(3.5)	2.08E+01(2)	20.295(1)
15	6.20E+01(2)	7.95E+01(5)	7.58E+01(4)	7.09E+01(3)	8.45E+01(6)	39.685(1)
16	3.21E+02(3)	3.24E+02(4)	3.55E+02(6)	3.25E+02(5)	2.75E+02(2)	1.64E+02(1)
17	3.18E+00(2)	4.58E+00(4)	5.09E+00(6)	4.74E+00(5)	4.33E+00(3)	2.188(1)
18	9.55E+03(2)	1.05E+04(3)	1.15E+04(4)	1.45E+04(5)	2.24E+04(6)	3.89E+03(1)
19	3.68E+01(5)	3.36E+01(3)	3.37E+01(4)	3.95E+01(6)	3.27E+01(2)	29.94(1)
20	3.34E+00(6)	3.01E+00(4)	3.09E+00(5)	2.50E+00(2)	2.97E+00(3)	1.69(1)
21	1.09E+04(3)	1.56E+04(6)	1.55E+04(5)	1.50E+04(4)	5.67E+03(2)	4.96E+03(1)
22	2.08E+09(2)	2.75E+09(4)	2.39E+09(3)	4.45E+09(6)	3.88E+09(5)	2.11E+08(1)
23	2.19E+01(4)	2.16E+01(3)	2.20E+01(5)	2.22E+01(6)	2.13E+01(2)	20.524(1)
24	5.85E+01(3)	6.90E+01(5)	6.65E+01(4)	5.83E+01(2)	7.46E+01(6)	37.32(1)
25	2.56E+02(2)	3.28E+02(4)	3.09E+02(3)	3.69E+02(6)	3.57E+02(5)	1.60E+02(1)
26	4.84E+00(4)	4.25E+00(3)	4.09E+00(2)	4.94E+00(5)	5.99E+00(6)	2.14(1)
27	1.44E+04(4)	1.77E+04(6)	1.55E+04(5)	1.10E+04(2)	1.28E+04(3)	3.98E+03(1)
28	3.86E+01(6)	3.45E+01(3)	3.64E+01(5)	3.59E+01(4)	3.14E+01(2)	24.69(1)
Average Rank(Ar)	4.12	3.17	4.33	4.25	3.57	1.39

Table 6

Outcome conquered from Holm’s Test

Rank	Algorithm	z_r value	p_r value	$\frac{α}{(k - i)}$	Hypothesis
1	SCA	–10	0.00004	0.01	Rejected
2	MFO	–8	0.00004	0.0125	Rejected
3	GWO	–6	0.00004	0.016	Rejected
4	ALO	–4	0.00004	0.025	Rejected
5	WOA	–2	0.02018	0.05	Rejected

(T - 1) values, leads towards rejection of the hypothesis. From the above analysis, the proposed OBL-MO-SHO algorithm proves its supremacy among all considered metaheuristic methods.

6 Wavelet neural network (WNN)

Wavelet theory was developed by Grossman and Morlett in the year of 1980 [33]. There are 7 mother wavelet functions available. The challenge is to incorporate the benefits of wavelet transform concept and neural network to resolve realistic complications. So the term wavelet neural network (WNN) is the combination of two approaches i.e. neural network and wavelet transformations. This was suggested by Q. Zhang in the year of 1992 [34]. WNN employs mother wavelet functions as activation function rather than considering sigmoid. Basically WNN comprises of 3 layers: input, hidden and output layer [35]. Input layer contains the descriptive parameters initialized to the network. The middle layer i.e. the hidden layer consists of hidden units. The hidden units are similar to the neurons present in sigmoid based neural network and termed as wavelons. The inputted parameters to the hidden layer are converted to, expanded and interpreted form of mother wavelet functions. At the end the desired outcome is computed in the output layer. We have considered the Gaussian wavelet function for the computation purpose, which is presented in Equation 13.

$Ψ = \frac{h}{\sqrt{2 π}} exp (- \frac{h^{2}}{2})$ (13)

Here SHO algorithm and suggested OBL-MO-SHO algorithm has been cast-off to train wavelet neural network together with other metaheuristic algorithms such as DE, GA, PSO and GWO. For classification purpose 8-benchmark datasets has been picked from UCI depository [16]. The observed outcome has been analysed in respect of numerous performance metrics represented in Tables 9 and 10. The value of RMSE can be considered by determining the variance between desired outcome and estimated outcome observed from WNN symbolised in Equation (14). Simple architecture of WNN and narrative of diverse datasets are indicated in Fig. 2 and Tables 7 and 8 as well. For any meta-heuristic algorithm reduced RMSE value, reduced standard deviation and increased average value endorses its solid exploitable ability. From the reported result in Table 9, superior outcome approves the suitability of proposed OBL-MO-SHO algorithm concerning to exploitation. The enhanced specificity and sensitivity consequence connotes the proper classification skill of the recommended technique as classifier. The finest accuracy value conquered in respect to all other castoff algorithms presented in Table 10, endorses supremacy of proposed method for accomplishing global best at the same time not trapped in local optima stuck which favours the solid approval of OBL-MO-SHO algorithm.

Fig. 2

Basic architecture of WNN.

Table 7

Structure of WNN for diverse datasets

Classification of datasets	Number of Attributes	MLP structure
Balloon	4	4-4-1
Cancer	9	9-9-1
Diabetes	8	8-8-1
Glass	9	9-9-1
Iris	4	4-4-1
Seed	7	7-7-1
Soybean	35	35-35-1
Vehicle	18	18-18-1

Table 8

Depiction of cast-off standard datasets

Classification datasets	No. of attributes	No. of training samples	No. of test samples	No. of class
Balloon	4	20	20	2
Cancer	9	683	120	2
Diabetes	8	768	150	2
Glass	9	214	42	6
Iris	4	150	30	3
Seed	7	210	42	3
Soybean	35	47	12	4
Vehicle	18	846	160	4

Table 9

Performance assessment of OBL-MO-SHO w.r.t SHO, DE, GA, GWO and PSO

Dataset /Algorithm		DE	GA	GWO	PSO	SHO	OBL-MO-SHO
Balloon	MIN RMSE	0.4820	0.4827	0.4384	0.4740	0.4334	0.4331
	AVG	0.5123	0.5246	0.4985	0.5011	0.4987	0.5218
	STD	0.0213	0.0287	0.0411	0.0219	0.0287	0.0128
	SPECIFICITY	0.4	0.7	0.7	0.7	0.7	0.8
	SENSITIVITY	0.8	0.7	0.8	0.8	0.8	0.8
	PREVALENCE	45	40	40	40	40	40
Cancer	MIN RMSE	0.4067	0.4033	0.2988	0.3999	0.3943	0.3911
	AVG	0.4356	0.4512	0.3312	0.4242	0.4526	0.4412
	STD	0.0188	0.0312	0.0281	0.0192	0.0412	0.0117
	SPECIFICITY	0.95	0.57	0.96	0.7	0.93	0.93
	SENSITIVITY	0.96	0.8	0.98	0.7	1	1
	PREVALENCE	47.5	50	49.16	47.5	50	50
Diabetes	MIN RMSE	0.4752	0.4806	0.4670	0.4775	0.4611	0.4609
	AVG	0.5201	0.5588	0.5321	0.5116	0.5244	0.5214
	STD	0.0391	0.0452	0.0512	0.0327	0.0399	0.0314
	SPECIFICITY	0.68	0.62	0.35	0.5712	0.6428	0.6428
	SENSITIVITY	0.68	0.84	0.95	0.8713	0.86	0.86
	PREVALENCE	34.28	42.14	47.85	43.57	40	40
Glass	MIN RMSE	0.2367	0.2258	0.2418	0.2677	0.4061	0.4047
	AVG	0.3589	0.3789	0.3113	0.3654	0.4685	0.4427
	STD	0.0778	0.0819	0.0485	0.0511	0.0357	0.0189
Iris	MIN RMSE	0.4052	0.4143	0.4081	0.4343	0.4018	0.3997
	AVG	0.4415	0.4527	0.4658	0.4958	0.4645	0.4377
	STD	0.0213	0.214	0.0321	0.0421	0.0421	0.0212
Seed	MIN RMSE	0.4374	0.4380	0.4276	0.4849	0.4270	0.4252
	AVG	0.5562	0.5878	0.5587	0.5988	0.4985	0.5089
	STD	0.0621	0.0598	0.0486	0.0613	0.0571	0.479
Soybean	MIN RMSE	0.4798	0.4702	0.4671	0.4765	0.4628	0.4608
	AVG	0.5012	0.5214	0.4977	0.5010	0.4923	0.5583
	STD	0.0216	0.0311	0.0213	0.0211	0.158	0.0541
Vehicle	MIN RMSE	0.3804	0.3014	0.1525	0.4012	0.4731	0.4719
	AVG	0.4125	0.3339	0.1623	0.4312	0.5146	0.5498
	STD	0.0213	0.0156	0.0023	0.0098	0.0239	0.0689

Table 10

Accuracy conquered from standard data sets

Dataset /Algorithm		DE	GA	GWO	PSO	SHO	OBL-MO-SHO
Balloon	Error Rate (%)	30	45	30	35	20	15
	Accuracy (%)	70	65	70	65	80	85
Cancer	Error Rate (%)	29.16	13.33	10	6.66	3.33	3.33
	Accuracy (%)	70.84	86.67	90	93.34	96.67	96.67
Diabetes	Error Rate (%)	29.28	37.85	40	30	27.85	27.14
	Accuracy (%)	70.72	62.15	60	70	72.15	72.86
Iris	Error Rate (%)	10	43.33	30	16.66	8	7.33
	Accuracy (%)	90	56.67	70	83.34	92	92.67
Seed	Error Rate (%)	28.20	33.33	43.58	38.46	17.94	15.38
	Accuracy (%)	71.8	66.67	56.42	61.54	82.06	84.62
Soybean	Error Rate (%)	16.66	25	25	25	8.33	0
	Accuracy (%)	83.34	75	75	75	91.67	100
Vehicle	Error Rate (%)	21.87	19.37	16.25	23.75	13.12	12.5
	Accuracy (%)	78.13	80.63	83.75	76.25	86.88	87.5
Glass	Error Rate (%)	19.04	21.42	21.42	19.04	14.28	11.90
	Accuracy (%)	80.96	78.58	78.58	80.96	85.72	88.1

$RMS E_{val} = \sqrt{\frac{\sum_{p = 1}^{N} (Y_{p}^{q} - T_{p}^{q})}{N}}$ (14)

7 Conclusion

The proposed OBL-MO-SHO algorithm is a refined variant of SHO algorithm. The prime concern of the developed algorithm is to amplify the exploration strength as well as achieving better exploitation capability. The said method has been verified in terms of exploration, exploitation, avoidance from local optima stuck, convergence speed and application in realistic environments. The various performance measures reported by scrutinizing proposed OBL-MO-SHO, are superior in comparison with SHO and other considered algorithms. Finally observing and analysing the outcomes reported by OBL-MO-SHO and existing SHO, some significant conclusive points are highlighted as follows:

The integration of oppositional learning with mutation operator enhances the explorative strength of proposed algorithm which confirms exploration about more search region than SHO.

Successfully avoids local optima stagnation problem and faster convergence speed confirms its better exploration ability.

Statistically proved by Friedman and Holm’s test, confirming its uniqueness among other state-of-art algorithms.

Successfully applied to resolve realistic classification problems, as a trainer algorithm to train wavelet neural network.

Observed lesser RMSE and higher average value confirms its solid exploitative strength.

Superior accuracy attained confirms its improved ability not to trap in local optima.

In future, the proposed algorithm will be applied to resolve various realistic problems. In addition to this it might be applied to train all higher order neural networks to establish as a suitable trainer algorithm.

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