Aircraft wing rib component optimization using artificial neural network–assisted superb fairy-wren algorithm

1 Introduction

Optimization techniques have found a wide range of applications in various domains, especially engineering, electronics, and plant handling systems, for many years. One of the major techniques that researchers have used many times is the classical optimization technique. This method is effectively able to solve the various challenges. However, as time passes, the complexity and nature of the problem fluctuate. Hence, classical optimization methods are unable to attain globally optimized solutions with effective convergence rates, handle multimodal fitness functions, and consume high computational time. These several aspects are well-covered by nature-inspired algorithms known as metaheuristics optimizers. These algorithms are inspired by physics-based phenomena, swam intelligence, human behaviors, optic theory, and evolutionary aspects [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31]. Accordingly, multiobjective versions of these algorithms are established and referred to as foundational algorithms. For instance, particle swarm optimization algorithms and genetic algorithms are some of the well-established algorithms. Multiobjective genetic algorithms, multiobjective symbiosis organism optimizers, and multiobjective ant lion optimizers are some of the benchmark algorithms in the domain of multiobjective optimization [7], [8], [9], [10].

Various subcategories and multistrategies enhance the metaheuristics algorithms to improve their exploration and exploitation capabilities and realize global optimum solutions. Some of the most recently observed strategies include leader selection–based strategy, meta-perception, advanced intelligence methods, and artificial neural networks [11], [12], [13], [14], [15]. Moreover, dynamic oppositional–based techniques, levy flight approach, chaotic maps, and dynamic random walk are well established and conventional techniques used to improve the various phases of the algorithms. Furthermore, some of the most recently developed algorithms are the superb fairy-wren optimizer [16], divine religious optimizer [17], starfish optimizer [18], and catch fish optimization algorithm [19]. Apart from the developed algorithms, there are several new domains over which these algorithms are tested for optimization. For instance, feature selection problems, topology planning problems, corrosion characteristics in offshore plants, and the medical field. This being said modification of the algorithms provides potential opportunities in the optimization domain, especially in industrial optimization for various component designs. Hence, the article studies modified superb fairy-wren optimization algorithms for various discipline optimization challenges and compare the available literature results [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30].

2 Artificial neural network–assisted superb fairy-wren optimization algorithm

The superb fairy-wren optimizer (SFWO) was established from the inspiration of the different territorial establishment techniques of the fairy-wren birds, including giving birth to small fairy wren, feeding cycle, and their techniques to prevent small ones from the predators by generating different voices and alarms. Accordingly, the algorithm is categorized by the different life-cycle stages of the superb fairy wren, such as young growth, breeding, feeding, and preventing enemies. Moreover, the algorithm mathematical section is categorized based on the exploration that covers the young bird’s growth and feeding cycle. In contrast, the exploitation phase includes techniques to save the small birds from predators [16].

The initialization of the SFWO was realized by selecting the random populations of the young birds. Accordingly, the position of each fairy wren is updated by Equation (1) for better exploration of the algorithm [16].

During the birth stage and feeding stage, fairy wren initially asses the paternity test to prevent invasion. Moreover, the eggs are incubated by the number of fairy wrens around the year and are treated as a part of teaching to feed and look out for the young ones. However, once the young birds are closer to maturity, their activities increase, and they can change their position after a particular time. The same can be given by Equations (2) and (3) [16].

with X_b: current optimum location; C: constant value as 0.8.

Further, the optimizer considers the exploitation phase as a technique of fairy wren to protect against predators. These include various alarm sounds, continuous flapping of their wings to misguide predators, and alteration of the positions to attain a safer location. Moreover, they quickly escape from the predator’s target, causing a slight change in the position of the member of the fairy wren. The same can be given by Equation (6) [16].

with l: step size in case of levy flight operator, k: equivalence factor in case of adaptive flight, and X_b: controlling the directional movement of birds.

Accordingly, the adaptive flight factor can be given by Equation (7).

with w: frequency value of the alarm call to prevent from the predators.

Hence, the algorithm is executed by considering three prime stages of the fairy wren: young bird growth, feeding, and protection against predators. Accordingly, the computational complexity of the optimizer can be given by Equation (8) [16].

The studied optimizer is modified using an artificial neural network (ANN) approach as shown in Figure 1.

Figure 1:

Artificial neural network flowchart.

The ANN works based on the multiple inputs taken from the given source that leads to the processing of each input and provides output. Hence, each input is processed accordingly, and the output is analyzed. In the case of metaheuristics algorithms, ANN processes each input or source function from the search space and explores them better than a standalone optimizer. Accordingly, each possibility provides an optimized value and is compared on a global scale. This being said the modified algorithm can attain the global optimum solution with a superior convergence rate. Furthermore, the modified artificial neural network–based superb fairy-wren optimization algorithm is studied and tested over industrial components.

3 Industrial component optimization using the modified fairy-wren optimizer

In the present section, industrial challenges of various disciplines are optimized using a modified superb fairy-wren optimizer (MSFWO). Each challenge’s fitness functions, constraints, and design parameters are well-defined. Accordingly, a wide range of problems are selected for optimization. For instance, economic optimization of the thermal heat exchanger, structural optimization of the reinforced concrete beam, economic optimization of the welded beam, mechanical design optimization of piston lever assembly, and aircraft wing component optimization. The comparison in terms of the results was made with passing vehicle search (PVS), salp swarm algorithm (SS), gray wolf optimizer (GWO), symbiotic organism search (SOS), ant lion optimizer (ALO), artificial flora (AF), and grasshopper optimization algorithm (GOA).

3.1 Economic optimization of fin tube heat exchanger (FTHE)

With a broad heat duty capability, heat exchangers (HEs) are well-known heat recovery machines utilized in several sectors. Additionally, FTHE’s expanded surface area over the cylindrical tube increases the rate of heat transmission, as shown in Figure 2. FTHEs are typically composed of stainless steel and aluminum. Both longitudinal and transverse augmentations are possible for the fins. The FTHE is used explicitly in industry for process heating purposes. Figure 1 displays a crisp three-dimensional view of FTHE. The optimization of the FTHE considers the particular application, which is to lower the temperature of the water-processed air. Inlet water and hot air temperatures (40 °C and 104 °C) and the predicted air temperature (51 °C) at the output are the particular parametric circumstances. Air and water volumetric flows are kept at 58 kg/s and 39 kg/s, respectively [1], [2].

Figure 2:

Fin and tube heat exchanger, a) computer-aided layout and b) line diagram with design variables.

The main obstacle facing the heat recovery units and companies is the whole cost of the FTHE. However, when designing the heat exchanger, thermal-hydraulic considerations are equally crucial. The heat exchanger’s overall cost is optimized and chosen as the goal function. Therefore, the FTHE’s initial cost and any maintenance or handling costs are considered when calculating the final economic factors. Table 1 lists the tier ranges and values of the seven choice variables that impact the design and a number of intricate constraints that are taken into consideration for the FTHE’s optimization, as shown in Table 1 [2]:

Table 1:

Design parameters and constraints for the fin and tube heat exchanger.

	Variables	Maximum range	Minimum range	Limits
Parameters	De- external diameter of tube	6.9	13
	Numbers of rows of tubes	1	6
	Height of heat exchanger	4.5	80
	Width	3	5
	Pitch length wise	12.7	32
	Lateral pitch	20.4	30.8
	Placing of fins	1	8.7
Imposed conditions	Pressure drops in tubes	–	–	≤4.5
	Pressure drops in fins	–	–	≤0.03
	Heat exchanger weight	–	–	≤475
	Ration of width to diameter	–	–	>60

The economics optimization of the FTHE was examined in this article using a modified superb fairy-wren optimization (MSFWO) algorithm. Additionally, the results obtained were contrasted with a few benchmark algorithms published in the literature to guarantee the MSFWO’s performance level. The statistical findings from the MSFWO and eight other MHs that were compared are shown in Table 2. Table 2 shows that MSFWO achieves the best outcomes (at the lowest cost) with a noteworthy success rate. Additionally, it can be verified that the MSFWO’s target standard deviation is significantly lower than the findings of the FTHE problem.

Table 2:

Results obtained for the proposed algorithm and comparison.

Optimizers	Superior	Average	Inferior	Deviations
MSFWO	3,466.80	3,467.25	3,467.95	0.829
Starfish optimizer (SFO)	3,469.97	3,471.93	3,475.63	1.17
Geyser inspired optimizer (GIO)	3,472.97	3,473.45	3,474.81	1.09
Hippopotamus optimizer (HO)	3,475.91	3,469.15	3,467.42	0.596

As a result, the MSFWO’s results are trustworthy and competitive compared to the other algorithms. Accordingly, MSFWO attained the best value of 3,466.80 for the fitness function with a standard deviation of 0.829. The findings for both the MSFWO-imposed limits and the optimal values of the design variables are shown in Table 2. Given the encouraging MSFWO results, the new MH (metaheuristic) can be used in various engineering design application domains.

3.2 Structural optimization of reinforced concrete beam

This study’s seventh test case is considered a framework in its reduced form, as seen in Figure 3. The structure’s weight is supposed to be supported to minimize the overall structural cost. As seen in Table 3, nearly every method that was studied was able to produce the best values. According to Table 3, MSFWO performs better than the others regarding statistical findings, with mean and SD values of 359.2080 and 0.1263, respectively, and the least function evaluation of 10,000. The MSFWO discovered the best solutions, such as SFO, GIO, and HO algorithms.

Figure 3:

The reinforced concrete beam structure with design dimensions.

Table 3:

Results for the reinforced concrete beam problem.

Optimizers	Superior	Average	Inferior	Deviations	FEs
MSFWO	359.2080	359.502	360.102	0.1263	10,000
SFO	359.986	360.8932	361.6340	1.42032	10,000
GIO	360.5123	361.2087	362.2115	0.0007	10,000
HO	361.0123	362.5407	362.6340	1.5863	10,000

3.3 Volumetric optimization of piston lever design

Piston lever design is investigated in this case study, which aims to minimize oil volume as the piston lever rises from 0° to 45°, as seen in Figure 4. Many inequalities are bound, such as geometrical limitations, the stroke’s highest length, the bending moment’s peak value, and equilibrium forces acting over the reciprocating components. In contrast, MSFWO discovered the best solutions overall. Additionally, the statistical analysis in Table 4 shows that the optimum functional value attained correlates to the maximum consistency of MSFWO. However, the MSFWO obtains the lowest function evaluation while achieving the superior SD value of 0.6539.

Figure 4:

Computer-assisted design of piston lever assembly.

Table 4:

Optimized fitness function values and comparison.

Optimizers	Superior	Average	Inferior	Deviations	FEs
MSFWO	8.4220	8.58236	8.5936	0.6539	25,000
SFO	8.4563	8.6536	8.6985	1.2536	25,000
GIO	8.4869	8.9632	8.7253	1.9863	25,000
HO	8.4986	8.98632	8.7563	2.536	25,000

3.4 Economic optimization of pressure vessel

The final test scenario taken into consideration for the MSFWO’s performance check in this work is a pressure vessel’s design challenge that consists of mixed integers, as seen in Figure 5. This is one of the often-utilized test examples to minimize overall costs, including material, welding, and manufacturing costs.

Figure 5:

Pressure vessel design.

In contrast to other algorithms, the MSFWO, SFO, GIO, and HO algorithms are found to identify the optimum, which is shown in Table 5. Additionally, the algorithms SFO, GIO, and HO do not yield near-optimal results. In terms of mean, SD, and function evaluation values, MSFWO performs better statistically overall than the others. With the least function evaluation of 15,000, the suggested hybrid MSFWO method thus has a significantly higher success rate and resilience than other algorithms under consideration.

Table 5:

Fitness function values comparison.

Optimizers	Superior	Average	Inferior	Deviations	FEs
MSFWO	6,059.714	6,060.375	6,060.987	0.986	15,000
SFO	6,061.718	6,068.375	6,072.236	1.896	15,000
GIO	6,063.511	6,076.369	6,073.973	1.912	15,000
HO	6,064.714	6,104.502	6,075.492	2.012	15,000

3.5 Aircraft wing rib component (AWRC) optimization using modified superb fairy-wren optimizer

Many of the aircraft’s parts are optimized to lower the aircraft’s overall weight and counteract the pollutants it emits. This rib is an essential component of any aircraft wing structure. In this case, the weight of the AWRC has been optimized through shape optimization using the MSFWO approach. Additionally, the AWRC’s goal is to have the least amount of bulk while being able to withstand comparable stress circumstances. The provided optimization task can be finished by optimizing the topology for the optimal shape and then structurally reducing the volume to lower the mass. Figure 6 displays the first design.

Figure 6:

Initial design of rib.

In the first step of structural optimization, the weight of the AWRC was employed as a restriction. In the second stage, volume was reduced, optimizing the structures and frames of the system. Figure 7 displays the optimized figure with the optimum design variables (Table 6). In contrast, Figure 8 validates the final design of the AWRC. The entire formulation with decision parameters and constraints can be found in Equations (9) and (10).

Figure 7:

Limiting conditions of the rib.

Table 6:

Design variables.

Variables of the rib	Interval/ranges-including end values
Y1	5–14
Y2	6–16
Y3	4–12

Figure 8:

Optimal design of the aircraft rib.

Objective function to be optimized:

(9) Minimum .   F y = mass  y

(10) Stress cnstraints : σ max ≤ σ permissible

Whereas, in Equation (10), the constraints in terms of stresses are represented by σ_max. 60 MPa is the permissible stress value.

The results for the identical problem as determined by the five well-known MHs algorithms are listed in Table 7, together with the statistical results for the AWRC as determined by MSFWO. Consequently, the MSFWO algorithm achieved better statistical results than the other optimizers. Furthermore, the MSFWO attains the lowest weight of 968 g with the smallest standard deviation of 3. Consequently, the weight has been successfully decreased by the selected algorithm.

Table 7:

Statistical analysis results.

	Best mass (g)	Mean	Worst	Std	Stress (MPa)	NFE
Initial design	1,200	–	–	–	54	–
New design	1,120	–	–	–	52	–
Geyser inspired optimizer (GIO)	1,085	1,095	1,120	24	56	100
Hippopotamus optimizer (HO)	1,075	1,085	1,118	18	52	100
Starfish optimizer	1,085	1,090	1,116	14	54	100
Crayfish optimizer	1,005	1,075	1,112	12	59	100
MSFWO	968	970	976	3	58	100

4 Conclusions

This study proposed a Modified Artificial Neural Network–Assisted Superb Fairy-Wren Optimization Algorithm (MSFWOA) to enhance industrial component optimization. By integrating ANNs, the algorithm significantly improves exploration–exploitation balance, leading to more efficient global optimization. The algorithm was tested across various industrial case studies, demonstrating superior performance in minimizing costs, structural weight, and material usage compared to conventional metaheuristics. MSFWOA consistently achieved optimal or near-optimal solutions with enhanced convergence speed and lower standard deviation. The results validate its robustness, reliability, and potential applicability in engineering design and optimization. Future research could explore further modifications, hybridization with other algorithms, and expansion into additional industrial domains.