Enhanced hippopotamus optimization algorithm and artificial neural network for mechanical component design

1 Introduction

High-level, problem-independent optimization frameworks known as metaheuristics help underlying heuristics identify nearly optimum solutions to challenging optimization issues. Due to their versatility and capacity to tackle a wide range of optimization issues, such as high-dimensional, nonlinear, and non-convex problems that are frequently challenging or impossible to resolve using conventional deterministic techniques, these algorithms have become incredibly popular in recent years [1]. In order to investigate and take advantage of search spaces, metaheuristics imitate a variety of natural processes, including biological evolution, physical occurrences, and animal behavior. Metaheuristics are relevant to a wide range of real-world applications because they can handle both continuous and discrete issues and do not presuppose specific problem structures, in contrast to standard optimization methods that need gradient information [2], [3].

Modified optimization algorithms are an extension of traditional optimization methods or metaheuristics. These enhanced algorithms enhance performance by adding adaptive mechanisms, hybrid techniques, or specialized operators catering to the particulars of the task. Their goals are higher precision, quicker convergence, and improved solutions for space exploration and exploitation. This review of the literature looks at the most current developments in metaheuristic algorithms and emphasizes changes made recently to enhance optimization performance. In order to get around typical problems, including sluggish convergence rates, inadequate exploration, and premature convergence, academics have recently concentrated on altering conventional metaheuristics [4], [5], [6], [7], [8], [9], [10]. Some of the recently established modified and base versions of the metaheuristics algorithms are flood algorithms [11], multi-objective particle swarm optimization algorithms [12], Polar fox optimization algorithms [13], multi-objective cheetah optimizers [14], Hiking optimization algorithms [15] and Hippopotamus optimization algorithm [16]. The development of algorithms based on the inspiration of nature, evolutions, and physics-based are not limited to empowering the research domain of the optimization algorithm. However, the capabilities of each developed algorithm to attain a global optimum solution can be realized by applying it to wide interdisciplinary problems. However, convergence rates, local optima trap, and high computational times are some of the observed points in the newly developed algorithms. Accordingly, researchers developed the modified version of the expanded base optimizer to improve the performance and effectiveness in obtaining global optimum solution. This being said, the incorporation of oppositional-based learning techniques, artificial neural network techniques, chaotic maps, levy flight mechanism, and hybridization are some of the most observed techniques to improve the exploration and exploitation phase of the algorithm [17], [18], [19], [20], [21], [22], [23], [24], [25].

The present article introduces a modified artificial neural network-assisted hippopotamus optimization algorithm (MHOA) and applies to different engineering applications. The rest of the article is structured as follows: Section 2: understanding of HOA; Section 3: Applications of MHOA for engineering component optimization; and Section 4: conclusion.

2 Understanding of HOA

The optimizer was developed on the basis of the different survival techniques of the hippopotamuses that usually live alone in rivers or ponds. Accordingly, their defensive strategies towards different predators, position alteration, and their hiding strategies within the pond. The mathematical formulation of each phase, from searching to attaining a global solution, is explained in the sub-section [16].

The initial randomized solution or position of the hippopotamus can be given by Equation (1) [16].

with X_i: current position and LB and UB: lower-upper bounds and upper bound values are represented for the i-th candidate at j-th space or position vector.

Accordingly, in the exploration phase, a herd of the species consists of a male, dominant male or leader of the herd, female, and calves hippopotamuses. The entire group is directed by the actions of the dominant male hippopotamus, who accordingly arrange their position to protect themselves and establish the territory. Accordingly, the location of the male candidates amongst the herd in the reservoir can be written as per Equation (2) [16].

with, x^Mh _ij: The updated position of the hippopotamus with a dominant male is denoted by Dh. Random number ranges between 0 and one are denoted with y1, and I1 is an integer range from one to two.

The location of the female or immature hippopotamus (X^FBh _ij) within the group can be given by Equation (3) [16].

The baby hippopotamus is away from its mother if T is more than 0.6; otherwise, it is nearer to the mother. Moreover, h₁ is the randomly selected vector that operates according to the scenarios.

Another stage within the exploration is protecting against the predators of the individual herd and establishing the territory. However, immature or sick hippopotamuses may be trapped by the predators and separated from the herds. Accordingly, Equation (4), gives the location of the predator within the search domain [16].

with, r: random number ranging from 0 to 1. Moreover, the distance (D) of the predator to the particular hippopotamus can be given by Equation (5).

The third phase is the exploitation phase, in which the hippopotamus escapes from the predator and identifies the best suitable location for their herd as well as for the individual level. This pertains to pursuing the global optimum solution of the fitness function using the algorithm. Accordingly, the alteration in the position and identification of the safer location from the predator can be modeled by Equations (6) and (7). Moreover, in the formulated equations, t is denoted for the current iterations, and T is denoted for the maximum number of iterations [16].

with x^Hippo _ij: effective location of the hippopotamus after escaping from the predator, r: random vector that includes a sub-set of different random vectors ultimately leads to improvement of the quality of the exploitation phase.

The total computational complexity of the HOA can be given by Equation (8) [16].

3 Modified hippopotamus optimization algorithm (MHOA)

Artificial neural network strategies are enhanced in many sectors, including power systems, fuzzy logic circuits, thermal systems designs, and design optimizations. This technique relates to the neurons available in the human brain that decide the best or optimized action against the message received by the human brain out of the number of inputs. Moreover, the response time and optimized actions and input realize very little time. Hence, the same strategy is augmented with the available new HOA algorithm for further modifications and improvement in the available results and to attain globally optimized solutions.

The ANN strategies include the following components: neurons or nodes, activation function, layers, forward propagations, loss function, backpropagation, and gradient descent. Moreover, the potential variants in ANN are not limited to feedforward neural networks, recurrent neural networks, and convolution neural networks. The MHOA was compared with the base version and simultaneously with the other well-known, established results from the literature.

4 Applications of MHOA for engineering component optimization

The modified HOA was applied over the five different disciplined engineering, structural, automobile, manufacturing, and mechanical components for their different parameter optimizations. Moreover, the statistical results were compared with the published work in the literature vis-à-vis HOA results to confirm the performance of the modified version of the algorithm.

4.1 Structural optimization of 10-bar truss

The ten-bar truss structure consists of 10 trusses that comply to sustain the different loads acting from the various directions. The optimization’s primary objectives are to reduce the structure’s overall weight or to reduce the maximum nodal deflections. Accordingly, in the present case, as shown in Figure 1, a minimum mass of the structural elements was kept as an objective function. Accordingly, Table 1 shows the optimized values of the 10 trusses with fitness function values realized by MHOA with HOA, COA and Chimp. The optimized values of each truss member are more effective in the case of MHOA than the rest of the algorithms. Moreover, Table 2 illustrates the statistical results and comparisons made over the stated algorithms. MHOA shows dominance in terms of deviations and best results compared to other competitors, including HOA.

Figure 1:

Ten-Bar structural element.

Table 1:

Comparison of the best optimum solution for the 10-bar plane truss problem.

Variables	Algorithms
	MHOA	COA	Chimp
A1	33.500	33.500	33.500
A2	1.6200	1.6200	2.1300
A3	22.900	22.900	22.000
A4	14.200	14.200	18.800
A5	1.6200	1.6200	1.6200
A6	1.6200	1.6200	1.6200
A7	7.9700	7.9700	7.2200
A8	22.900	22.900	22.000
A9	22.000	22.000	22.000
A10	1.6200	1.6200	1.8000
Fitness function	5,490.7	5,490.7	5,567.4

Table 2:

Comparison of the best optimum solution for the of planetary gear-box design.

Variables	Algorithms
	MHOA	MRFO	Chimp	BMO	SAR	GBO
N1	36	38	50	28	36	37
N2	31	26	29	19	30	22
N3	29	23	26	16	28	20
N4	24	24	32	17	24	24
N5	25	21	24	14	21	27
N6	87	87	116	62	87	87
P	3	3	4	3	3	3
M1	1.750000	2	2	1.750000	1.750000	2.500000
M3	2	2	1.750000	1.750000	2	2.250000
fbest	0.52666	0.526895	0.526739	0.537058	0.527346	0.526280

4.2 Weight optimization of planetary gear-box design

Planetary gearboxes are widely used across various industries for efficient power transmission. To optimize their performance, minimizing the weight of the entire gear-train structure is essential. In formulating the problem, seven design variables (six continuous and one discrete) are defined, constrained by 11 critical linear and non-linear conditions. Figure 2 presents a 3D kinematic view of the gearbox. The optimized values of the decision variables and objective function are listed in Table 2. Among the optimizers, MHOA achieved the best objective function value.

Figure 2:

Planetary type gearbox design.

4.3 Robot gripper optimization using MHOA

In this case study of a robot gripper, seven decision parameters (a, b, c, d, e, f, l, δ) and six constraints are considered to minimize the difference between the maximum and minimum force exerted by the gripper. The variables are depicted in the gripper’s line diagram in Figure 3. Figure 4 shows a robot gripper. Table 3 displays the optimal function values and operating parameters, while Table 4 presents the analytical results. According to Table 3, MHOA achieved the best function value of 2.5437951.

Figure 3:

Robot gripper problem.

Figure 4:

Robot gripper.

Table 3:

Comparison of the best optimum solution for the hydro-static thrust bearing problem.

Variables	Algorithms
	MHOA	MRFO	Chimp	BMO	SAR	GBO
a	150.00000	148.0117	150.0000	148.9930	147.7556	149.997
b	149.88285	134.6299	149.8893	144.6811	141.7630	149.880
c	200.00000	198.9289	200.0000	175.8954	189.9823	199.999
e	0.0000000	13.20780	0.000000	0.000000	5.747826	3.585e-08
f	150.00000	139.2476	13.58781	148.9316	134.7398	149.999
l	100.94288	103.8090	104.1508	165.2107	109.5289	100.980
δ	2.2974148	2.388345	1.614771	2.631458	2.395397	2.30189
f best	2.5437951	2.764161	2.864835	4.118013	2.971362	2.54450

Table 4:

Parametric optimization results for the piston lever.

Parameters		Applied algorithms
	MHOA	MVO	SSA
H	0.05000000	0.0500000	0.0500000
B	2.04151359	2.0450926	2.0422810
X	120.000000	120.00000	119.91119
D	4.08302718	4.0835494	4.0845349
f best	8.41269883	8.4289960	8.4220378

4.4 Optimization of piston lever using MHOA

The fourth mechanical design optimization problem discussed in this study focuses on the piston lever design, with the goal of minimizing oil volume as the piston lever moves upward from 0° to 45°, while tracing the H, B, D, and x piston points (as shown in Figure 5). Several inequality constraints, such as force equilibrium, maximum piston stroke length, lever bending moment, and geometric restrictions, are considered.

Figure 5:

Three dimensional design of piston lever design.

4.5 Automobile component optimization using MHOA

The present section structurally optimized the vehicle diaphragm spring to reduce the overall weight of the component. It is the imperative component of any automotive system that absorbs energy loads and provides effective comfort drive. Accordingly, it is subject to uneven loads and complex stresses that lead to the failure of the diaphragm spring. The mathematical form of the fitness function and constraints can be given by Equations (9) and (10).

Minimization of mass of spring.

(9) F x = mass x

Imposed Constraint

(10) g x ≤ 0 ,

x i l ≤ x i ≤ x i u , i = 1

with x_i ^l and x_i ^u: decision parameters with lower and upper bound values, with constraints as g(x) of maximum stress acting over the diaphragm spring.

Accordingly, the initial design and limiting conditions of the diaphragm spring can be depicted in Figures 6 and 7, respectively.

Figure 6:

Initial design domain of the spring.

Figure 7:

Boundary conditions of the spring.

For the present study, five design parameters of the individual diaphragm of the spring are identified for the potential optimization using MHOA. Moreover, the bounds of each parameter are: 28<X₁<42, 2<X₂<6, 3<X₃<5, 6<X₄<12 and 36<X₅<48. Moreover, each design variable can be depicted in the Figure 8. Accordingly, Figure 9 shows the structurally optimized design of the studied system using MHOA. The statistical results obtained from the algorithm are compared with HOA and four other established algorithms and are tabulated in Table 5. This being said, MHOA dominated other algorithms in terms of best-optimized results with the least number of functional evaluations.

Figure 8:

x₁–x₅ design variables.

Figure 9:

Optimized diaphragm spring using modified hippopotamus optimizer.

Table 5:

Comparison of the best results for the vehicle diaphragm spring.

Optimization algorithm	Optimized-mass (gram)	Stress-constaints (MPa)
Initial design	312.6	268
Ship rescue optimizer	283	249
Cheetah optimizer	280	245
Starfish optimizer	275	240
HOA	273	240
MHOA	268	240

5 Conclusions

With the use of artificial neural networks (ANN), the modified hippopotamus optimization algorithm (MHOA) was able to optimize complicated engineering problems with higher convergence rates and more accurate solutions. The benefits of MHOA in a variety of fields, such as structural, mechanical, and automotive design optimizations, were demonstrated by a comparative study versus more established optimization approaches like HOA, starfish optimizer, ship rescue optimizer, and others. Better exploration–exploitation balance is promoted by the dynamic modification of algorithm parameters made possible by the integration of ANN, which is essential for preventing premature convergence. The statistical findings from several case studies attest to MHOA’s superiority in delivering accurate, reliable, and efficient solutions, opening up a potential path for upcoming optimization problems in a variety of domains.