Performance Comparison of Recent Population-Based Metaheuristic Optimisation Algorithms in Mechanical Design Problems of Machinery Components

Abstract

The optimisation of complex engineering design problems is highly challenging due to the consideration of various design variables. To obtain acceptable near-optimal solutions within reasonable computation time, metaheuristics can be employed for such problems. However, a plethora of novel metaheuristic algorithms are developed and constantly improved and hence it is important to evaluate the applicability of the novel optimisation strategies and compare their performance using real-world engineering design problems. Therefore, in this paper, eight recent population-based metaheuristic optimisation algorithms—African Vultures Optimisation Algorithm (AVOA), Crystal Structure Algorithm (CryStAl), Human-Behaviour Based Optimisation (HBBO), Gradient-Based Optimiser (GBO), Gorilla Troops Optimiser (GTO), Runge–Kutta optimiser (RUN), Social Network Search (SNS) and Sparrow Search Algorithm (SSA)—are applied to five different mechanical component design problems and their performance on such problems are compared. The results show that the SNS algorithm is consistent, robust and provides better quality solutions at a relatively fast computation time for the considered design problems. GTO and GBO also show comparable performance across the considered problems and AVOA is the most efficient in terms of computation time.

1. Introduction

An engineering design problem comprises of designing products that meet the functional requirements by considering a multitude of values for decision variables [1]. With the increase in the complexity of the design and its application, the number of decision variables increases along with the candidate solutions [2]. Therefore, it is highly challenging to use exhaustive search methods to solve the optimisation problems [3]. To overcome this issue, it is possible to use the class of optimisation algorithms that fall under the category of metaheuristics such that the design space is explored with a guiding heuristic technique [4].

Metaheuristic algorithms can be classified into the following branches: evolutionary, physics-based, plant-based, swarm intelligence, and human-based algorithms [5]. The evolutionary algorithms mimic the principle of evolution and include genetic algorithm [6], differential evolution [7,8], genetic programming [9], biogeography-based optimisation [10], evolution strategies [11], etc. The physics-based algorithms are derived from real-world physical processes and comprise of simulated annealing [12], Henry’s gas solubility optimisation [13], atom search optimisation [14], black hole algorithm [15], colliding bodies optimisation [16] and many others. The plant-based metaheuristics mimic the behaviour of the plants and include flower pollination algorithm [17], and invasive weed optimisation [18]. The swarm optimisation algorithms are inspired from collective bodies or entities in nature and include particle swarm optimisation [19], ant colony optimisation [20], tunicate swarm algorithm [21], manta ray foraging optimization algorithm [22], sailfish optimizer [23], barnacles mating optimizer algorithm [24], firefly algorithm [25], artificial bee colony algorithm [26], dolphin echolocation [27], salp swarm optimisation algorithm [28] and cuckoo search [29]. The human-based algorithms include but are not limited to queuing search algorithm [30] and kidney-inspired algorithm [31].

Population-based metaheuristics start with a population of candidate solutions that are either selected at random or introduced using a custom script. The algorithms then perform global exploration and local exploitation and are able to provide acceptable good solutions for complex problems within reasonable computation time. They are easy to implement and have the capability to bypass the local optima [32]. In recent years, a number of novel algorithms have been proposed in the literature, such as African Vultures Optimisation Algorithm (AVOA) [33], Crystal Structure Algorithm (CryStAl) [34], Human-Behaviour Based Optimisation (HBBO) [35], Gradient-Based Optimiser (GBO) [36], Gorilla Troops Optimiser (GTO) [37], Runge–Kutta optimiser (RUN) [38], Social Network Search (SNS) [39], and Sparrow Search Algorithm (SSA) [40]. However, a comparison of these new-generation metaheuristic optimisation algorithms on their applicability and performance on various engineering design problems has not been evaluated. Therefore, in this paper, the above-mentioned eight new-generation population-based metaheuristic optimisation algorithms are compared and evaluated on five different classical mechanical machinery component design problems such as the tension/compression spring design, crane hook design, reduction gear design, pressure vessel design and hydrostatic thrust bearing design. To the best of the authors’ knowledge, this is the first benchmark research study that quantitatively and qualitatively evaluates the selected eight new-generation metaheuristic optimisation algorithms by applying them in various mechanical machinery component design problems. This article contributes to knowledge by providing a comprehensive comparison and discussion of the applicability of the selected algorithms.

The rest of the paper is organised as follows. Section 2 briefly explains the meta-heuristics optimisation algorithms used in this study. Section 3 describes the selected mechanical component design problems, and the results of the computer experiments. Section 4 discusses the findings of the paper, and provides an outlook for the future work. Finally, Section 5 concludes the paper.

2. Optimisation Algorithms

In this section, the selected optimisation algorithms are briefly explained. All technical details associated with these algorithms can be found in the foundation references.

2.1. African Vultures Optimisation Algorithm (AVOA)

The AVOA, proposed by [33], simulates the foraging and navigation behaviours of the African vultures. The algorithm comprises of an initial population of vultures and the first and second best solutions are considered. The algorithm comprises of four phases: (i) determining the best vulture in a group, (ii) rate of starvation of vultures, (iii) exploration and (iv) exploitation. The rate of starvation of vultures helps determine whether they are in exploration phase or exploitation phase. Two different strategies for exploration are used and the strategy is chosen with a parameter ‘$P_1$’. For the exploitation phase, two different strategies of rotating flight and siege-fight strategies are employed depending on the parameter ‘$P_2$’. The AVOA is found to perform faster than other algorithms due to lower computational complexity. From our understanding, we believe that this algorithm has not been applied elsewhere with the exception of the foundation paper.

2.2. Crystal Structure Algorithm (CryStAl)

The CryStAl proposed by [34] was inspired by the principles of the formation of crystals in nature. It is a parameter-free novel metaheuristic algorithm in which the balance between the exploration and exploitation phases is not controlled by parameters. During initialisation of the algorithm, the solution candidates are considered to be crystals and those which occupy the corners are called the main crystals. The new positions are determined by four kinds of position updates: (i) simple cubicle, (ii) cubicle with mean crystals, (iii) cubical with best crystals, and (iv) cubicle with best and mean crystals. The exclusion of the parameter to tune the exploration and exploitation phases can help overcome the problems associated with local optima entrapment and convergence. The authors believe that this algorithm has not been applied in any research problem apart from the foundation paper.

2.3. Human Behavior-Based Optimisation Algorithm (HBBO)

The HBBO algorithm was developed based on the behaviour of human beings and has five main steps: (i) initialisation, (ii) education, (iii) consultation, (iv) field changing probability, and (v) finalisation [35]. In the initialisation step, the individual human beings who form the population are spread across different fields. Each field has an expert individual who has the best function value and the members of the field educate and improve themselves to move around the best individual. The individuals of the various fields who are not the expert individuals meet with a random advisor from the society to change the solution variables. On finding a better solution, the existing solution is replaced with the better one. The individuals are capable of changing the field based on a probability value. The finalisation step involves calculating the fitness function and checking the stopping criteria. The algorithm is tested against benchmark problems to understand its reliability, result accuracy and convergence speed. The HBBO has recently been used in variety of research domain including manufacturing cell design [41] and cryptography [42], etc.

2.4. Gradient-Based Optimiser (GBO)

The GBO, initially developed by [36], is a population-based meta-heuristic algorithm. It uses two operators, the Gradient Search Rule (GSR) and Local Escaping Operator (LEO) to explore the search space. The gradient search rule, which is based on Newton’s method, promotes the exploration capability of the GBO. It comprises of a direction of movement term to enable movement towards promising solutions. The gradient-based optimisation benefits from two search methods that enable the exploration and exploitation, respectively. The local escaping operator allows the algorithm to escape from local optimum using six different values of the solution candidates. It comprises of adaptive parameters that allows the smooth transition from exploration to exploitation. The GBO has been used in a variety of applications, such as economic load dispatch problem [43], parameter extraction in photovoltaic models [44] and feature selection [45], etc.

2.5. Gorilla Troops Optimiser (GTO)

The gorilla troops optimiser is a novel meta-heuristic that was inspired by the social behaviour of gorillas and was proposed by [37]. The exploration phase of the algorithm comprises of three different mechanisms: (i) migration to unknown place, (ii) movement to other gorillas which balances the exploration and exploitation, and (iii) migration to a known location. The silverback gorilla represents the best solution and the candidate solutions move towards this best solution. Follow the silverback gorilla and competition for female gorillas are two mechanisms employed for the exploitation phase. The search history, convergence behavior, the average fitness of the population, and the trajectory of the first gorilla are considered for evaluation of the algorithm. From our understanding, we believe that this algorithm has not been applied elsewhere with the exception of the foundation paper.

2.6. Runge–Kutta Optimisation (RUN)

The Runge–Kutta optimizer, proposed by [38], was derived from the ideas of the Runge–Kutta method which is used in the field of mathematics to solve ordinary differential equations. It is a population-based optimisation algorithm which employs the Enhanced Solution Quality (ESQ) to avoid getting trapped in local optima and improve the solution quality. The exploration and exploitation phases are improved with the help of the scale factor. The convergence rate of the algorithm is improved by providing the choice of choosing the new solution based on its quality and thereby increasing the efficiency of global search while ensuring good quality local search. The balance between the exploration and the exploitation phases is provided with the help of two random variables. The performance of the RUN algorithm was qualitatively assessed by three metrics: (i) search history, (ii) trajectory graph, and (iii) convergence curve and the algorithm was compared with benchmark problems to assess its ability to explore and exploit the design space efficiently. From a review of literature, it is understood that this algorithm is yet to be applied on other design problems apart from those mentioned in the original work.

2.7. Social Network Search (SNS)

The SNS is a novel meta-heuristic algorithm that was inspired by the behaviour of users across the social network platforms [39]. Four different moods are considered in the algorithm: (i) imitation, (ii) conversation, (iii) disputation, and (v) innovation. The search agents are the social media users and one of the four moods will be chosen randomly for each user. Depending on the considered mood, the opinions of the user will change. The decision to share a view will depend on whether it is better than the current one. The algorithm comprises of three levels: initialisation, increasing popularity and termination. It is also important to note that the SNS algorithm uses non-parametric statistical methods. The algorithm is novel and it is believed that it has not been applied to any problems apart from the work done in the foundation paper.

2.8. Sparrow Search Algorithm (SSA)

The group wisdom, foraging and anti-predation behaviour of sparrows are considered for the novel swarm intelligence algorithm known as sparrow search algorithm [40]. The population is divided into producers and scroungers. The producers have two modes of search depending on whether a predator is nearby or not. The scroungers monitor the producers and compete with them for food. If a sparrow is at the edge of the group, it constantly tries to get a better position to avoid the predators. On the other hand, if a sparrow is at the centre of the group, it moves closer towards its neighbours. SSA is seen to have good performance in diverse search spaces. The SSA has been recently employed in various applications, including but not limited to: optimal model parameters identification of the Proton Exchange Membrane Fuel Cell (PEMFC) [46], optimal brain tumor diagnosis [47], pulmonary nodule detection [48], carbon price forecasting [49] and 3D route planning for UAVs [50].

3. Computer Experiments

In this section, the performance of the above-mentioned metaheuristic optimisation algorithms is analysed by performing the five design problems, namely: tension/compression spring design, crane hook design, reduction gear design, pressure vessel design and hydrostatic thrust bearing design. The mathematical formulation of each design study can be found in the Appendix A. All computer operations were performed in MATLAB environment, where the computer was configured with Intel(R) Core (TM) i7-11850H CPU @2.60 GHz and 32 GB RAM.

3.1. Parameter Settings

In the computer experiments, an exterior penalty function is deployed to deal with design constraints. The parameter settings of each optimisation algorithms were selected based on the values provided by the foundation papers that were referenced in the Section 2; they are provided in

Table 1. Specific parameter settings of used algorithms.

Algorithm	Parameter Settings
AVOA	$P_1$ = 0.6, $P_2$ = 0.4, $P_3$ = 0.6, $L_1$ = 0.8, $L_2$ = 0.2,
	w = 2.5,
CryStAl	Global parameters (FEs and NP)
HBBO	$N_{field}$ = 30, $k_1$ = 0, $k_2$ = 2.5, $\sigma$ = 0.2
GBO	${\beta }_{min}$ = 0.2, ${\beta }_{max}$ = 1.2, $pr$ = 0.5
GTO	p = 0.03, $\beta$ = 3, w = 0.8
RUN	Global parameters (FEs and NP)
SNS	Global parameters (FEs and NP)
SSA	$PD$ = 20%, $SD$ = 10%, $ST$ = 0.8

Notation: AVOA: $P_1$, $P_2$ and $P_3$ are control parameters determining the probability of selecting mechanism in the exploration phase, exploitation phase of the first part and the exploitation phase of the second phase, respectively. $L_1$ and $L_2$ are probability parameters for selecting the first and second-best vultures, and w is a parameter determining the disruption of the exploration and exploitation phases. HBBO: $k_1$ and $k_1$ are weight factors, $\sigma$ is the consultation factor, and $N_{field}$ is the number of initial fields GBO: $pr$ is the probability ratio and ${\beta }_{min}$ and ${\beta }_{min}$ are the GBO’s control parameters to balance algorithm’s exploration and exploitation processes. GTO: p is a parameter to select the mechanism of migration to an unknown location, $\beta$ and w are input parameters. SSA: $PD$ is the number of the producers, $SD$ is the number of sparrows who perceive the danger, and $ST$ represents the safety threshold.

. The maximum iteration number and the population size for each case study are selected based on the complexity of the problem and are given in

Table 2. The number of function evaluations $FEs$, maximum iteration number $t_{max}$ and the population size ($NP$) for each design optimisation problem.

Problem	NP	${\mathit{t}}_{\mathit{m}\mathit{a}\mathit{x}}$	FEs
Tension/compression spring design	35	500	17,500
Crane hook design	35	850	29,750
Reduction gear design	50	1250	62,500
Pressure vessel design	50	1000	50,000
Hydrostatic thrust bearing design	100	2000	200,000

along with the number of function evaluations (FEs). In order to analyse the robustness of the selected optimisation algorithms, each case is repeated 100 times and the best, mean, worst, standard deviation, convergence history, and the mean computation times are recorded. The best results obtained in each case are presented in bold in Tables 4, 6, 8, 10 and 12.

3.2. Tension/Compression Spring Design

The tension/compression spring design problem depicted in Figure 1 is a continuous constrained problem. The objective of this design problem is to minimise the weight of the coil spring subjected to the constraints of shear stress, surge frequency, and minimum deflection. The problem contains three design variables namely: the number of active coils ($x_1$), the diameter of the winding ($x_2$), and diameter of the wire ($x_3$).

The best solutions for each algorithm are presented in

Table 3. Best optimum solution of each algorithm for the tension/compression spring design optimisation problem.

Var.	Algorithm
	AVOA	CryStAl	HBBO	GBO	GTO	RUN	SNS	SSA
$f_{min}$	0.012669	0.012718	0.012665	0.012665	0.012665	0.012665	0.012665	0.012666
$x_1$	0.052157	0.050431	0.051688	0.051617	0.051673	0.051789	0.051699	0.051454
$x_2$	0.368082	0.327152	0.356695	0.354992	0.356338	0.359122	0.356956	0.351091
$x_3$	10.652547	13.285471	11.290315	11.390874	11.311287	11.149446	11.275058	11.626627

and the results of the simulation are provided in

Table 4. Statistical findings of the bench-marked algorithms for the tension/compression spring design problem.

Algorithm	Best	Mean	Worst	Std	CPU Time (s)
AVOA	0.012669196719	0.013560633588	0.015508926159	0.000828257932	0.00298611
CryStAl	0.012718059192	0.012857705948	0.013166450230	0.000099876120	0.0427
HBBO	0.012665232805	0.012983204879	0.015342442232	0.000546532591	0.03068571
GBO	0.012665327053	0.012725601218	0.013046308894	0.000082488889	0.00758664
GTO	0.012665237347	0.012736771587	0.013274672049	0.000114193785	0.00583106
RUN	0.012665468642	0.013161559938	0.017773399457	0.001091488094	0.01252020
SNS	0.012665308324	0.012700310221	0.012908393770	0.000053324273	0.00449754
SSA	0.01266624413	0.013359346602	0.017773159421	0.001336305394	0.00506082

for the tension/compression spring design problem. From Figure 2, it can be seen that for the best replication run, all the algorithms are able to converge to the global optimum approximately around 70 iterations. From Table 4, SNS performs well for across all the criteria, however, AVOA performs better in terms of the computation time. SNS is the most robust with a value of 5.3324 × ${10}^{-5}$, followed by GBO, CryStAl, GTO, HBBO, AVOA, RUN and SSA. The performance of the GBO is similar to that of the SNS; this could be due to the use of a Local Escaping Operator (LEO) which significantly changes the position of a solution. Although the HBBO has the best result across the replication runs, the standard deviation indicates that it is not able to consistently arrive at the global optimum for this particular design problem.

3.3. Crane Hook Design

The crane hook is a rectangular wire ring that is clipped and bent to give the final hook shape. The aim of the crane hook design optimisation is to minimise the volume of the hook subjected to the constraints of yield stress and geometric feasibility. There are three design parameters, namely: the outer radius of the hook ($x_1$), the inner radius of the hook ($x_2$), and width of the hook ($x_3$). Schematic view of the crane hook is given in Figure 3.

Table 5. Best optimum solution of each algorithm for the crane hook design optimisation problem.

Var.	Algorithm
	AVOA	CryStAl	HBBO	GBO	GTO	RUN	SNS	SSA
$f_{min}$	37.501763	37.505239	37.502003	37.501763	37.501764	37.501770	37.501765	37.501763
$x_1$	3.389569	3.393468	3.380692	3.389197	3.389747	3.390378	3.389217	3.389350
$x_2$	1.500000	1.500000	1.500000	1.500000	1.500000	1.500000	1.500000	1.500000
$x_3$	1.292017	1.288449	1.300485	1.292370	1.291849	1.291252	1.292352	1.292226

and

Table 6. Statistical findings of the bench-marked algorithms for the crane hook design problem.

Algorithm	Best	Mean	Worst	Std	CPU Time (s)
AVOA	37.501768304530	37.509521006356	37.567143974448	0.012437419152	0.00475095
CryStAl	37.502652718785	37.528776144364	37.614401549196	0.021833637274	0.02640220
HBBO	37.501773097093	37.573060310021	37.976041870924	0.122369512361	0.03530030
GBO	37.501763283040	37.501857914596	37.504126932995	0.000349234645	0.01275198
GTO	37.501763262652	37.503112132524	37.509524542183	0.001892413110	0.00894737
RUN	37.501775668620	37.508273179197	37.546955429412	0.010504431267	0.02143182
SNS	37.501763577814	37.501955818081	37.502840983287	0.000297921175	0.00738974
SSA	37.501763937287	37.502795252003	37.509275937416	0.001709946677	0.00797318

present the best solution and the statistical simulation results obtained by the algorithms for the crane hook design problem. Figure 4 indicates that for the best replication run, all the algorithms are able to converge to the global optimum approximately around 500 iterations. From Table 6, the SNS is the most robust and is followed by GBO, SSA, GTO, RUN, AVOA, CryStAl and HBBO. GBO has the best mean across the 100 replication runs. The mean value for the SNS algorithm is very close to the GBO with only a difference of 2.9792 × ${10}^{-4}$. It can also be seen that AVOA has the best performance in terms of the computation time followed by SNS and SSA.

3.4. Reduction Gear Design

The reduction gear design problem is one of the widely used engineering design optimisation benchmark problems and is initially conceptualised by [51]. Figure 5 depicts the schematic view of the reduction gear problem. The objective of this design problem is to find an acceptable near-optimal set of variables, i.e., that can provide a minimum total weight under eleven geometric constraints. The problem has seven design variables, namely: ($x_1$) the face width, ($x_2$) the module of the teeth, ($x_3$) the number of teeth on the pinion, ($x_4$) the length of the first shaft between bearings, ($x_5$) the length of the second shaft between bearings, ($x_6$) the diameter of the first shaft and ($x_7$) the diameter of the second shaft. The constraints are (i) the limits of the bending stress on the gear teeth, (ii) surface stress, (iii) transverse deflections of both shafts and (iv) stress in both shafts. Please note that the third variable in this design problem is an integer while the remaining are continuous variables.

For the reduction gear design, the algorithms slightly different ‘best’ solutions and they are presented in

Table 7. Best optimum solution of each algorithm for the reduction gear design optimisation problem.

Var.	Algorithm
	AVOA	CryStAl	HBBO	GBO	GTO	RUN	SNS	SSA
$f_{min}$	2994.355035	2999.199820	2994.355026	2994.355026	2994.355026	2994.360057	2994.355026	2994.355026
$x_1$	3.500000	3.502645	3.500000	3.500000	3.500000	3.500004	3.500000	3.500000
$x_2$	0.700000	0.700000	0.700000	0.700000	0.700000	0.700000	0.700000	0.700000
$x_3$	17.000000	17.000000	17.000000	17.000000	17.000000	17.000000	17.000000	17.000000
$x_4$	7.300000	7.300000	7.300000	7.300000	7.300000	7.300001	7.300000	7.300000
$x_5$	7.715320	7.776127	7.715320	7.715320	7.715320	7.715415	7.715320	7.715320
$x_6$	3.350215	3.355539	3.350215	3.350215	3.350215	3.350216	3.350215	3.350215
$x_7$	5.286654	5.288404	5.286654	5.286654	5.286654	5.286656	5.286654	5.286654

;

Table 8. Statistical findings of the bench-marked algorithms for the reduction gear design problem.

Algorithm	Best	Mean	Worst	Std	CPU Time (s)
AVOA	2994.355035177460	2996.948906023790	3010.205238324880	3.050912549282	0.00199605
CryStAl	2999.199820220130	3025.816152450150	3167.006692311780	23.589811855532	0.00682740
HBBO	2994.355026112020	2995.873515410200	3033.632877394140	7.534482000137	0.00670371
GBO	2994.355026112020	2994.355026112010	2994.355026112020	0.000000000004	0.00293696
GTO	2994.355026112020	2996.909938595160	3016.654294988220	5.793856810156	0.00263637
RUN	2994.360057223500	2996.941885889620	3005.175063824030	2.571268627965	0.00529230
SNS	2994.355026112020	2994.355026112010	2994.355026112020	0.000000000004	0.00225359
SSA	2994.355026112020	2994.355026112010	2994.355026112020	0.000000000004	0.00206693

presents the best solution and simulation results obtained by the eight algorithms. From Figure 6, for the best replication run, all algorithms are capable of converging to the global optimum around 600 iterations. From Table 8, GBO, GTO, SNS and SSA have good performance across all evaluation criteria. GBO is the most robust with a value of 3.635 × ${10}^{-12}$ followed by SNS, SSA, RUN, AVOA, GTO, HBBO and CryStAl. AVOA has the best performance in terms of the computation time and it is closely followed by SSA and SNS. GTO does find the global optimum with a relatively fast computation time, however, the standard deviation indicates that it is not as robust as a few other algorithms with the considered parameter values for the reduction gear design problem.

3.5. Cylindrical Pressure Vessel Design

The pressure vessel design problem is initially conceptualised by [52] which is one of the most used design optimisation benchmarks. The objective of this problem is to minimise the total manufacturing cost of the cylindrical pressure vessel which is a combination of material, forming and welding costs. The pressure vessel has covered at the split ends while the head part has a hemispherical shape. Figure 7 shows the schematic representation of the problem. This problem has four variables, namely: thickness of the shell ($x_1$), thickness of the head ($x_2$), inner radius ($x_3$) and the length of the vessel ($x_4$). The problem has four inequalities.

Table 9. Best optimum solution of each algorithm for the cylindrical pressure vessel design optimisation problem.

Var.	Algorithm
	AVOA	CryStAl	HBBO	GBO	GTO	RUN	SNS	SSA
$f_{min}$	5885.373899	5974.379050	5885.331251	5885.331290	5885.331251	5885.358639	5885.331259	5885.334152
$x_1$	0.778193	0.782667	0.778168	0.778168	0.778168	0.778170	0.778168	0.778170
$x_2$	0.384661	0.393180	0.384649	0.384649	0.384649	0.384655	0.384649	0.384650
$x_3$	40.320911	40.525039	40.319619	40.319619	40.319619	40.319678	40.319619	40.319705
$x_4$	199.982005	199.890718	200.000000	199.999998	200.000000	199.999554	200.000000	199.998801

and

Table 10. Statistical findings of the bench-marked algorithms for the cylindrical pressure vessel design problem.

Algorithm	Best	Mean	Worst	Std	CPU Time (s)
AVOA	5885.3738987482	6435.3490970377	7318.9955182938	495.3062860591	0.00205490
CryStAl	5974.3790500167	6470.6393698594	7102.3066251881	256.7609412380	0.01141518
HBBO	5885.3312508567	6184.9051478586	7318.9989210708	347.8356798794	0.01408385
GBO	5885.3312900824	5912.3347543230	6309.3266189585	68.5231352046	0.00535753
GTO	5885.3312508567	6209.8831784614	7318.9989210708	433.7313628757	0.00415526
RUN	5885.3586393254	6889.2020918829	7319.1357489398	628.5456318607	0.00902184
SNS	5885.3312588460	5967.6346717171	7318.2220337882	250.1319816538	0.00297717
SSA	5885.3341524728	6411.1266934476	7318.9989210708	479.2420166123	0.00303397

present the best solution for each algorithm and the statistical simulation results obtained by the algorithms for the pressure vessel design problem, respectively. From Figure 8, for the best replication run, it can be seen that all the algorithms are able to converge to the global optimum approximately around 200 iterations. In particular, as seen from Table 10, the GBO algorithm is the most robust with a standard deviation value of 68.523 for the pressure vessel design problem and has good scores across all the evaluation criteria. It is followed by the SNS, CryStAl, HBBO, GTO, SSA, AVOA and RUN. The GBO algorithm has a local escaping operator that prevents entrapment at the local optima; this could the reason for the robustness of the algorithm. In terms of the time to converge, SSA, HBBO and AVOA converge earlier, approximately around 80 iterations. In terms of the mean computation time across the replications, similar to the previous design problems (see Table 6 and Table 8), AVOA has the best performance. It can be seen that SNS also has a similar performance in terms of the computation time in addition to having a low standard deviation. In SNS, clear a distinction between the exploration and exploitation phases is not provided; the balance between the two phases is not controlled using parameters. Instead, the candidate solutions of the population are randomly subjected to one of the four strategies. Therefore, the exploration and exploitation phases happen at random depending on the chosen strategy which avoids the entrapment at local optima; this could be the reason why the SNS has a slightly higher computation time despite its robust performance.

3.6. Hydrostatic Thrust Bearing Design

In this design optimisation problem defined by [53], four design variables are considered as follows: bearing step radius ($x_1$), recess radius ($x_2$), viscosity ($x_3$) and flow rate ($x_4$). The objective of the optimisation is to minimise the power loss during the operation of the hydrostatic plain bearing. Figure 9 shows the schematic representation of the problem. The seven nonlinear constraints of weight capacity $g_1\left(\overrightarrow{x}\right)$, inlet oil pressure $g_2\left(\overrightarrow{x}\right)$, oil temperature rise $g_3\left(\overrightarrow{x}\right)$, oil film thickness $g_4\left(\overrightarrow{x}\right)$, step radius $g_5\left(\overrightarrow{x}\right)$, exit loss significance $g_6\left(\overrightarrow{x}\right)$ and contact pressure $g_7\left(\overrightarrow{x}\right)$ are also considered (please see Appendix A).

Table 11. Best optimum solution of each algorithm for the Hydrostatic thrust bearing design optimisation problem.

Var.	Algorithm
	AVOA	CryStAl	HBBO	GBO	GTO	RUN	SNS	SSA
$f_{min}$	19,544.66481	20,551.94918	23,183.54393	19,505.31331	19,505.76493	19,505.85672	19,515.50557	20,459.41775
$x_1$	5.959052191	6.038211653	6.398078596	5.955780499	5.955888118	5.955827384	5.956719465	6.042305703
$x_2$	5.3926286	5.452684743	5.874142492	5.38901305	5.389131987	5.38905445	5.389912069	5.4780081
$x_3$	5.38 × ${10}^{-6}$	5.57 × ${10}^{-6}$	6.73 × ${10}^{-6}$	5.36 × ${10}^{-6}$	5.36 × ${10}^{-6}$	5.36 × ${10}^{-6}$	5.36 × ${10}^{-6}$	5.74 × ${10}^{-6}$
$x_4$	2.286644923	2.539761897	4.066687785	2.269656072	2.269723697	2.269807988	2.272161645	2.647407856

and

Table 12. Statistical findings of the bench-marked algorithms for the Hydrostatic thrust bearing design problem.

Algorithm	Best	Mean	Worst	Std	CPU Time (s)
AVOA	19,544.6648069058	23,956.5558577224	42,889.0448729199	3127.6848031821	0.021506546
CryStAl	20,551.9491788596	22,834.6295959337	24,761.4009721265	1016.4245763020	0.103373525
HBBO	23,183.5439311853	31,259.1001178412	48,556.9957076442	4573.4127562409	0.094215091
GBO	19,505.3133077224	20,886.4642416408	26,025.4186699601	1481.9044873111	0.044018051
GTO	19,505.7649269808	22,182.0691330824	28,734.8082159540	1890.1924714863	0.038249415
RUN	19,505.8567225382	20,680.5525292687	33,034.8218788356	3044.6733188831	0.085310708
SNS	19,515.5055737337	19,536.7881264808	19,600.1259774218	16.5978846836	0.027272087
SSA	20,459.4177472381	31,040.7176967004	78,295.6327390864	8912.0581928620	0.025494880

present the best solution for each algorithm and the statistical simulation results obtained by the algorithms for the hydrostatic thrust bearing design problem, respectively. From Figure 10, for the best replication run, it can be seen that all the algorithms are able to converge to the global optimum approximately around 600 iterations. In particular, as seen from Table 12, the SNS algorithm is the most robust with a standard deviation value of 16.597 for the hydrostatic thrust bearing design problem and has good scores across all the evaluation criteria. It is followed by the CryStAl, GBO, GTO, RUN, AVOA, HBBO, and SSA. In terms of the time to converge, AVOA converges earlier, approximately around 500 iterations. It can be seen that SNS also has a similar performance in terms of the computation time in addition to having a low standard deviation. Although SNS is the most robust, the best value of the objective function is relatively higher than the values found by the GBO, GTO, and RUN algorithms.

4. Discussion

4.1. Observations from the Benchmark Study

This research study applies eight different metaheuristic algorithms on five mechanical component design problems to provide an insight into the efficiency and robustness of the algorithms for design optimisation. Following a detailed analysis of the various algorithms, it was understood that the performance of the algorithms vary across the different applications.

Across the selected mechanical design problems, it can be seen that SNS and GBO have very good results across the evaluation criteria. SNS performs very well in the tension/compression spring design, crane hook design, reduction gear design and hydrostatic thrust bearing design problems. For the pressure vessel design problem, however, GBO performs better than SNS. In the SNS algorithm, the candidate solutions represent the users and their moods are selected at random. The choice of mood greatly impacts the exploration and exploitation behaviour of the algorithm. Although it means that the algorithm might not have a strong sense of direction towards the global optimum, the computation time is still fast. The prevention of local optima entrapment facilitated by the structure of the algorithm might ensure that the algorithm is consistently able to find the global optimum across the 100 replications.

The GBO algorithm includes an escaping mechanism that introduces randomness to prevent entrapment in local optima. This suggests that the algorithm is able to provide a reasonably good performance and is comparable to SNS across all the design problems. SSA shows good performance overall and has good computation time and comparable robustness; in particular, it performs very well for the reduction gear design problem. However, in the hydrostatic thrust bearing design problem, SSA performed significantly poorly. This might be owing to the incompatibility of the selected algorithm parameters for this design case, which is the most complicated of the five case studies. It can also be seen that the HBBO, GTO, RUN and SSA algorithms also perform well and are able to find the near-optimal solutions in a majority of the problems. GTO also has good computation time across the design problems and this can be attributed to the movement to a known location and other gorillas during the exploration phase and the directional focus and best position update provided by the following of the silverback gorilla and competition for females, respectively, during the exploitation phase. The CryStal algorithm, on the other hand, performed consistently across all specified mechanical component design problems. However, it was observed that it performed badly in terms of convergence to the global optimum when compared to other algorithms. Because it is intended to be a simple and easy-to-use algorithm with no internal or external parameters to tune, a modified version of CryStal with improved exploration/exploration balance could be a feasible optimisation technique, particularly in mechanical component design optimisation problems.

The AVOA has consistently good performance across all five design optimisation problems in terms of computation time. As explained previously, the reason behind the fast computational speed of AVOA can be attributed to the two different exploration strategies which are chosen based on the satiation of the vulture, the best vulture’s position and the boundaries of the problem. Additionally, the exploitation phase employs four different strategies for local searches. AVOA is able to reach the near-optimal solutions fairly quickly; the performance across the replications, however, suggests that the AVOA is less robust than SNS and GBO. Therefore, for design problems which are time-sensitive and the variables need not be selected to a very high level degree of precision, AVOA is a good choice due to its fast computation time.

4.2. A Comparison with Traditional Optimisation Techniques

A further benchmark of the algorithms with two widely used meta-heuristics, GA and PSO was done and provided in Figure 11, such that the readers can appreciate and compare the performance of the novel algorithms with the classical widely used optimisation algorithms. The parameters of the GA are selected as follows: distribution index for crossover $=20$, distribution index for mutation $=20$, crossover probability $=0.5$, mutation probability $=0.2$. On the other hand, PSO parameters are selected as follows: inertia weight $=1$, inertia weight damping ratio $=0.99$, personal learning coefficient $=1.5$ and global learning coefficient $=2$. The population size and maximum number of iterations are same as highlighted in Table 2.

From Figure 11, there are four spider plots; the best scores for each algorithm acquired over 100 replications show if the algorithm is capable of discovering solutions that are near to the global optimum. A lower mean value of the observations across replications indicates the consistency of the algorithms under consideration. The radar chart displaying standard deviation can be used to evaluate the algorithm’s repeatability. The obtained results for GA and PSO for each design problem are given in

Table 13. Statistical findings of the GA and PSO for the selected design problems.

GA
Problem	Best	Mean	Worst	Std
Tension/compression spring	0.0127	0.0147	0.0194	0.0018
Crane hook	37.5020	37.5097	37.5534	0.0092
Reduction gear	2997.7740	3006.4162	3017.4886	4.7139
Pressure vessel	6188.2862	8392.3701	12,661.0607	1003.2803
Hydrostatic thrust bearing	20,569.5064	26,767.7566	51,065.2918	5032.2942
PSO
Tension/compression spring	0.0127	0.0133	0.0164	0.0009
Crane hook	37.5018	37.5109	37.5918	0.0172
Reduction gear	2994.3550	2994.3550	2994.3550	0.0000
Pressure vessel	5909.6257	6277.7288	7182.9114	307.6918
Hydrostatic thrust bearing	19,819.4262	26,077.9339	36,586.4885	4305.7754

. Figure 11 shows that, with the exception of the crane hook design problem, which is the simplest case, most recent optimisation algorithms outperform GA. PSO, on the other hand, performs well across all of the specified design optimisation problems. However, the majority of the selected recent optimisation methods outperformed PSO in terms of both accuracy and repeatability. Comparing with recent metaheuristics, as seen in Figure 11’s mean and standard deviation plots, PSO produced slightly poor results, excluding the reduction gear design optimisation problem. In terms of finding the global optimum, nearly all recent metaheuristic algorithms outperformed PSO and GA in various design challenges, with the exceptions of HBBO and CryStal. Nearly all of the selected recent metaheuristics outperform GA and PSO in terms of overall performance over 100 repetitions. This explicitly showcases that the superiority of new-generation population-based meta-heuristics over the vanilla versions of traditional GA and PSO optimisation algorithms.

4.3. Limitations and Future Work

The presented work considers the eight recent population-based meta-heuristic algorithms with the optimisation parameters selected from the foundation parameters. Performing a detailed sensitivity analysis to identify the best parameters for each algorithm was not done. A more detailed analysis will be done as part of the future work. Additionally, an extensive comparison of the performance of the algorithms on other design problems and engineering applications will also be done. From the presented comparison, it can be concluded that each algorithm has its strengths and weaknesses, and there is no distinct meta-heuristic that outperforms all others. Therefore, there is the possibility of combining the best of different algorithms to provide hybrid-approaches that will help overcome any issues with the these meta-heuristic approaches. Although the various tables and figures presented in this paper may be useful to practitioners when selecting an algorithm, it may be more beneficial to select a suitable one using a formal multi-criteria decision-making (MCDM) approach that allows quantitative comparison of the various optimisation algorithms based on their applicability on a specific type of engineering design problem. As a consequence, future study will evaluate various algorithms using MCDM techniques in order to provide a fair analysis that might be used to improve algorithm selection.

5. Conclusions

The complex mechanical design problems involve multiple objectives with a combination of variables along with the consideration of a multitude of constraints and boundary conditions. Therefore, the problem of identifying optimum design for a specific application involves rigorous exploration of the design space which might not be feasible within a practical time-frame. For this purpose, metaheuristics can be applied to such problems to search the design space with a guiding heuristic that ensures a near-optimal acceptable solution within a reasonable time-frame. In this research study, eight novel metaheuristic algorithms, namely AVOA, RUN, HBBO, CryStAl, GBO, GTO, SNS, and SSA, are applied on five different mechanical component design problems and their ability to choose acceptable solutions across multiple runs of optimisation, the robustness of the algorithms and computation time are calculated. From the performance comparison, it was understood that the SNS, GBO and GTO algorithms perform well across all mechanical component design problems, and AVOA algorithm has good performance in terms of computation time. In machinery component design optimisation problems, the SNS method is shown to be the best suited choice in terms of solution quality, robustness, and convergence speed with given parameter settings.