Artificial ecosystem optimization by means of fitness distance balance model for engineering design optimization

Optimization techniques have contributed to significant strides in complex real-world engineering problems. However, they must overcome several difficulties, such as the balance between the capacities for exploitation and exploration and avoiding local optimum. An enhanced Artificial Ecosystem Optimization (AEO) is proposed incorporating Fitness Distance Balance Model (FDB) for handling various engineering design optimization problems. In the proposed optimizer, the combined FDB design aids in selecting individuals who successfully contribute to population-level searches. Therefore, the FDB model is integrated with the AEO algorithm to increase the solution quality in nonlinear and multidimensional optimization situations. The FDBAEO is developed for handling six well-studied engineering optimization tasks considering the welded beam, the rolling element bearing, the pressure vessel, the speed reducer, the planetary gear train, and the hydrostatic thrust bearing design problems. The simulation outcomes were evaluated compared to the systemic AEO algorithm and other recent meta-heuristic approaches. The findings demonstrated that the FDBAEO reached the global optimal point more successfully. It has demonstrated promising abilities. Also, the proposed FDBAEO shows greater outperformance compared to several recent algorithms of Atomic Orbital Search, Arithmetic-Trigonometric, Beluga whale, Chef-Based, and Artificial Ecosystem Optimizers. Moreover, it declares great superiority compared to various reported optimizers.


Introduction
To address large-scale real-world challenges, a substantial emphasis has increasingly been dedicated to handling restricted optimization problems.The word "optimization process" refers to establish the optimal values for specific system features to accomplish design, operation, or planning functions at the maximum possible quality and minimum feasible cost [1].In this regard, several approaches for dealing with restrictions in meta-heuristic computational frameworks have been presented [2].They have sparked a significant amount of attention and were utilized to solve a variety of optimization problems.They share characteristics, such as the searching technique, which includes two parts of diversification and intensification [3,4].The four basic forms of meta-heuristic methods are physicsbased techniques, evolutionary algorithms, swarm intelligence techniques, and human-based algorithms.Physics-based techniques are motivated by physical laws such as equilibrium optimization [5][6][7], Archimedes optimization algorithm [8,9], Henry gas solubility method [10], and motivate physics-based strategies.Evolutionary computing algorithms [11][12][13] have been designed depending on the modeling of biological evolutionary features such as crossovers, mutations, and selections.Swarm intelligence approaches, like particle swarm optimization (PSO) [14], cuckoo search algorithm [15], moth swarm algorithm [16], sand cat swarm optimization [17], jellyfish search technique [18], social spider algorithm [19], heap-based optimization [20], manta rays foraging optimizer [21], and artificial bee colony [22], are a series of techniques influenced by swarming and animal group behavior.
In [23], genetic algorithm (GA) and PSO have been employed to tackle nine limited mechanical engineering design optimization issues.To complement the searching efficiency, selection, crossover, and n-point randomized mutation operators of the GA were included with PSO depending on a dynamically adaptable inertia factor in this work.In [24], a butterfly algorithm (BA) was established and refined by including an exploitation stage of development that gives solutions additional chances to better themselves for tackling three mechanical design issues searching for the optimum design of tension/compression spring, welded beam, and gear train.In [25], the JAYA optimization approach has been used to optimize numerous features of a constrained design.The JAYA optimizer blends evolutionary algorithms' survivability of the strongest concept with the global optimum solution attraction of swarm intelligent techniques.In [26], coarse and fine-grained approaches are incorporated with Chaotic JAYA optimization using multiple populations considering a diffusion grid in tackling different restricted optimization engineering issues.In [27], JAYA has been developed and deployed to solve design engineering optimization challenges using three separate extensive learning methodologies.Despite these effective uses of the JAYA metaheuristic technique, it is susceptible to becoming stranded in local optima when addressing complicated optimization problems due to its single learning method and limited population knowledge [28].S. Sarhan et al. presented an upgraded slime mold algorithm (SMA) in 2022 that employs chaotic behavior and an elitist Artificial ecosystem optimization by means of fitness distance… grouping to solve engineering issues [29].In [29], it has been implemented in numerous benchmark datasets, including optimum power flow (OPF) in electrical networks [30] and combined heat and electricity dispatch [31].
Furthermore, recent literature emphasizes the need of developing meta-heuristic optimization techniques for solving several optimization problems such as Atomic Orbital Search (AOS) Algorithm [32,33], Arithmetic-Trigonometric Optimization Algorithm (ATOA) [34], Beluga whale optimization [35], Chef-Based Optimization Algorithm (CBOA) [36] and Artificial Ecosystem Optimizer (AEO) [37].The AEO algorithm requires minimal adjustments and is straightforward to apply to engineering disciplines.In order to achieve the optimal fitness value, AEO comprises three energy transfer processes in an ecosystem that involve production, consumption, and decomposition.In the first mechanism, the production operation enables AEO to generate a new member at random, but in the consumption framework, searching area exploration may be enhanced and exploitation could be conducted.Due to its strong global search capabilities and robustness, the AEO technique has been utilized to handle a variety of practical engineering optimization difficulties, like as DG and capacitors allocations in electrical distribution systems [38], demand side management of hybrid energy systems [39], combined heat and power dispatch [40], parameters identification of impulse response (IIR) filter [41], parameters estimation of proton exchange membrane fuel cell [42] and optimal electrical representation of PV cells [43].
Also, several other optimizers were introduced and developed for solving mechanical engineering design optimizations.In [44], a self-adaptive strategy has been introduced into differential evolution (DE) technique, updating the proportions of the crossover and mutation rates.Also, in this study of [44], an elitism has been incorporated to the DE algorithm to provide ability in retaining, in each iteration, the best-found solution.In [45], a comparative assessment of several algorithms, including PSO, gray wolf optimizer (GWO), moth-flame optimization (MFO), salp swarm algorithm (SSA), ant lion optimizer (ALO), water cycle algorithm (WCA), mine blast algorithm (MBA), artificial bee colony (ABC) and whale optimization algorithm (WOA), have been executed on six mechanical design problems.To reduce the weight of a single-stage spur gear, a queue search optimization (QSO) technique has been used to improve the primary spur gear characteristics [46].In addition to that, equilibrium optimizer (EO) has been adopted and developed to solve the robot gripper mechanism, hydrostatic thrust bearing, planetary gearbox, and ten-bar planar truss structure [47].It also utilized successfully for optimal integration of renewable photovoltaic sources and batteries in power systems [48,49] and allocation of biomass energy resources in distribution feeders [50].In [51] and [52], the artificial neural network method has been combined, respectively, with ALO and genetic algorithm to optimize the cutting parameters; the latter qualifies the minimum quantity lubrication cooling for sustainable machining, and the ceramic CC650.In [53], GWO has been utilized for shape designing the synchronous motors to minimize the flux saturation and maximize the operating force.In [54], social network search technique has been performed for the reactive power dispatch problem for minimizing the voltage variations and the transmission power losses.
The search for optimal solutions of the different engineering design problems is always one of the important challenges in industry.Therefore, there are a seeking for assessing, developing, and improving the optimization techniques in order to enhance the quality of the design to achieve the design requirements.In this paper, a novel enhanced, FDBAEO, Artificial Ecosystem Optimization with Fitness Distance Balance for handling various engineering design optimization problems.In the novel proposed technique, the FDB model is integrated with the AEO algorithm to increase the solution quality in nonlinear and multidimensional optimization situations.FDBAEO has been applied effectively for the OPF constrained by the transient stability limit in power systems [55].In this regard, six well-studied engineering optimization tasks were addressed.First, the welded beam design issue is addressed to reduce the cost of the beam while keeping limits on bending stress, shear stress, buckling load, and end deflection in mind.Second, the rolling element bearing design problem is addressed to maximize the rolling element bearing's dynamic load-carrying capability.In the third scenario, the pressure vessel design problem is tackled to reduce the overall cost, which includes material, forming, and welding expenses.In the fourth scenario, the speed reducer design problem is treated to minimize its weight responsive to limitations on bending stress of the gear teeth, surface stress, transverse shaft deflections, and shaft stresses.In the fifth scenario, the planetary gear train design is addressed to reduce the maximum gear ratio errors.Finally, the hydrostatic thrust bearing design problem is solved to minimize the hydrostatic thrust bearing's power loss.The simulation outcomes were evaluated in comparison to the systemic AEO algorithm and other published meta-heuristic approaches.The FDBAEO has demonstrated promising problem-solving abilities.This condition suggests that the design adjustments made during the AEO's decomposition phase were better suited for mimicking the algorithm's functioning in the actual world.The FDBAEO has demonstrated promising problem-solving abilities.The following are the primary contributions presented in this study: • A Fitness Distance Balance model emerged with Artificial Ecosystem Optimization to formulate a novel FDBAEO with better performance.• The standard AEO and the proposed FDBAEO have been assessed on well-studied engineering optimization tasks including the pressure vessel, speed reducer, welded beam, and rolling element bearing design problems.• The proposed FDBAEO shows greater outperformance compared to several recent algorithms of Atomic Orbital Search (AOS), Arithmetic-Trigonometric Optimization Algorithm (ATOA), Beluga whale optimization, Chef-Based Optimization Algorithm (CBOA), and Artificial Ecosystem Optimizer (AEO).• Furthermore, the suggested FDBAEO is stated to be more resilient and stable than other compared approaches.
This paper is divided into five parts: Section II illustrates the mathematical formulation of the AEO optimization, while Section III describes the processes of the suggested FDB model that is evolved using AEO.Section VI describes and analyses the findings achieved by standard AEO and the suggested FDBAEO in contrast to the newly created optimizers, while Section V contains the article's final remarks.

3
Artificial ecosystem optimization by means of fitness distance…

Artificial ecosystem optimizer
Three energy transfer strategies in an ecosystem are the main aspects of AEO which are consumption, production, and decomposition.The first strategy is production, where the production operator permits AEO to generate a new individual in a random way.The new individual is generated based on the knowledge information taken from both the best individual (Y B ) and a randomized individual (Y R ) that is generated in the search space randomly.Based on that, it can overcome the previous regarding individual.Thus, the production operator is mathematically formulated as in Eq. (1) [37]: where P M is the size of a population; the symbol (it) is the current iteration; T max is the maximal iterations' number; (UB) is the upper bound; (LB) is the lower bound; z and z 1 are random vectors of range of [0, 1].Furthermore, the term (z(UB-LB) + LB) and the term ((1− it/T max )z 1 ) reveal the position of an individual in the search space and a linear weight coefficient, respectively.
Levy flying is added to this algorithm in the consumption mechanism so that they can efficiently search the search space [33].It stimulates the food quest of many different species, including cuckoos and lions, because it can be dealt with as a mathematical operator.Levy flight represents a random walk that can explore the search region in an efficient way; meanwhile, it can attain the global optimum because some stages' durations are much longer while running slowly.However, there appear to be two drawbacks to this action: the requirement to change a myriad of settings and the complexity.As a result, Eq. (3) addresses the consumption factor, which is parameter-free random, and is added to improve the characteristic of Levy flight.
The consumption factor assists consumers to implement three consumption tactics.Herbivore is the first tactic as modeled in Eq. ( 4), where the producer can be eaten by the consumer.
Carnivore is the second tactic as modeled in Eq. (5), where the consumer can be eaten randomly only by a consumer with a sophisticated level of energy. (1) Omnivore is the third tactic as modeled in Eq. (6), where the producer and consumer can be eaten by the consumer randomly with a sophisticated level of energy.
The third strategy is decomposition, where x i position in a population of the ith individual can be updated by the decomposer x n position as depicted in Eq. ( 7), using the decomposition factor D = 3u, u ≈ N(0, 1) and the weight coefficients ( 2 ⋅ r 3 − 1 and r 3 ⋅ r 1 2 − 1 ).This process demonstrates exploitation to some level since it allows the next position of each individual to go around the decomposer (the best individual).This strategy can be formulated mathematically as in Eq. ( 7).

Artificial ecosystem optimization with fitness distance balance model
The goal of designing the FDB picking technique is to identify the individuals who will contribute the largest to the searching processes inside a population consistently and effectively.As a result, it can be assured that the variety and intensifying processes are performed in a balanced way.The distance between the solution individuals and the best option (X best ) would be determined using the Euclidean distance measure.As a result, each member's distance (Dis i ) from the best option is determined as follows: The grade for the solution alternatives is determined in the second stage of the FDB technique.The candidates' normalized fitness values (NFit) and normalized distance values (NDis) are employed in the grade computation.For each individual (i), they can be evaluated as follows: Based on that, the use of normalized numerical values is intended to avoid these two characteristics from dominating each other in the goal computation.Therefore, the grade (GR) of each individual (i) can be assessed as follows: (6)

Artificial ecosystem optimization by means of fitness distance…
After determining the grade of all individuals, a roulette wheel selection mechanism [56] is performed to pick a solution with a high chance of having a high grade (X FDB ).Therefore, the decomposition stage that is defined in Eq. ( 7) is upgraded by merging the FDB technique as follows: Usually, the optimization problem targets a minimization objective searching for the optimal control variables subject to several inequality constraints as follows: where Fit(x) refers to the targeted minimization objective; x represents the vector of control variables; g i (x) indicates the inequality constraints and N Con symbolizes their number.
The proposed FDBAEO methodology is developed to handle the constraints with the aid of penalty functions.Thus, the mathematical model in Eq. ( 13) is upgraded to the following model incorporating a penalized terms to the violated constraints: where λ i refers to a penalty coefficient which is related to each inequality constraint (i) and its value is very high and considered of 10 20 ; and Penalty i (x) indicates the penalty term related to each inequality constraint (i) which can be modeled as follows: Based on that model, the penalty term will be diminished if the regarding constraints are maintained.
Figure 1 depicts the designed FDBAEO workflow.It begins by arbitrarily producing a population.The initial searching individual changes its location about Eq. ( 1) at every repetition, but the remaining individuals have the comparable possibility of choosing Herbivore like per Eq. ( 4), Carnivore like per Eq. ( 5), or Omnivore like per Eq. ( 6) to modify their positions.It could be acceptable whenever an individual obtain a superior fitness attribute.After that, the FDB model is activated to pick a solution.To do that, each member's distance is evaluated from the best option as in Eq. ( 8).Then, the candidates' normalized fitness values and distance values are evaluated using Eqs.(9 and 10) while the grade for the solution alternatives is determined in the second stage of the FDB technique as in Eq. (11).The location of every member would then be adjusted using Eq.(12).During the upgrading phase, individuals would be generated at random in the searching universe when it is distant from the higher or lower boundaries.All ( 12) Artificial ecosystem optimization by means of fitness distance… modifications are carried out continuously until the AEO method with a termination condition is fulfilled.Finally, the option of the best candidate is achieved.

Application for engineering optimization problems
In this section, the proposed FDBAEO technique is applied to six well-studied engineering optimization tasks.First, the welded beam design issue is addressed to reduce the cost of the beam while keeping limits on bending stress, shear stress, buckling load, and end deflection in mind.Second, the rolling element bearing design problem is addressed to maximize the rolling element bearing's dynamic load-carrying capability.In the third scenario, the pressure vessel design problem is tackled to reduce the overall cost, which includes material, forming, and welding expenses.In the fourth scenario, the speed reducer design problem is treated to minimize its weight responsive to limitations on bending stress of the gear teeth, surface stress, transverse shaft deflections, and shaft stresses.In the fifth scenario, the planetary gear train design is addressed to reduce the maximum gear ratio errors.Finally, the hydrostatic thrust bearing design problem is solved to minimize the hydrostatic thrust bearing's power loss.The proposed FDBAEO is implemented compared to several recent algorithms of Atomic Orbital Search (AOS), Arithmetic-Trigonometric Optimization Algorithm (ATOA), Beluga whale optimization, Chef-Based Optimization Algorithm (CBOA), and Artificial Ecosystem Optimizer (AEO).For comparative assessment, the same circumstances are considered for all algorithms with 30 individuals, 300 iterations, and 50 separate runs.The simulations are run using MATLAB R2017b on a computer with an Intel(R) Core (TM) i7-7200U CPU (2.5 GHz) and 8 GB of RAM.

Scenario 1: welded beam design problem
The welded beam design issue is addressed to reduce the cost of the beam while keeping limits on bending stress, shear stress, buckling load, and end deflection in mind [2]. Figure 2 depicts a welded beam on a substrate.As shown, at a distance L from the substrate, a load weight P is supported by the beam.Upper and lower welds of length l 1 and thickness h are used to secure the beam to the substrate.The beam has a rectangular cross section with dimensions b and t.Steel is the substance of the beam.In this study, the minimization of the beam's manufacturing cost is considered under the applied load weight P. The weight is set at 6,000 pounds, and the distance is set at 14 inches.There are four continuous design variables which are the weld thickness (H), weld length (l 1 ), beam height (t), and beam width (b).The following is a more thorough representation of the problem: The proposed FDBAEO, AEO, AOS, Beluga, and CBOA are applied for Welded beam design problem for 50 separate runs.The optimal settings and fabrication cost's objective are tabulated in Table 1.Also, the convergence curves of the compared algorithms are displayed in Fig. 3.As shown, the proposed FDBAEO can achieve the least value of the considered objective of 1.63482 where AEO, AOS, ATOA, Beluga and CBOA obtain objective values of 1.63528, 1.647761, 1.844511, 1.819376 and 1.659462, respectively.This demonstrates the high efficiency of the proposed FDBAEO in finding the least fabrication cost which is obtained at a weld thickness of 0.185216 inches, weld length of 3.632255 inches, beam height of 9.51898 inches and beam width of 0.185412 inches.From Fig. 3, the AOS and CBOA converge much faster toward the near-optimal solution from initial iterations.
To assess the robustness of the proposed FDBAEO, AEO, AOS, Beluga and CBOA over 50 separate runs, Fig. 4 displays the distribution of their obtained fitness values.As shown, the proposed FDBAEO can achieve the least value of the considered objective over all the runs.To show more its success, Table 2 describes the distribution of the obtained fitness values of the proposed FDBAEO, AEO, AOS, ATOA, Beluga and CBOA through robustness indices.In this table, over the 50 separate runs, the minimum and maximum fitness values are extracted while the  mean fitness and the standard deviation are evaluated for each applied algorithm.From Table 2, the proposed FDBAEO represents the best performance with the least robustness indices of 1.63482 as the minimum, 1.642185 as the mean, 1.664656 as the maximum and 0.006752 as the standard deviation.Nevertheless, the proposed FDBAEO provides 100% constraints validation where all the constraints over all 50 separate runs are satisfied.The AEO and Beluga provide similar validations like the proposed FDBAEO.On the other side, the AOS, ATOA and CBOA achieve only 70%, 86% and 84% constraints validations, respectively.In this regard, compared to the basic AEO, the proposed FDBAEO shows   1 3 Artificial ecosystem optimization by means of fitness distance… great superiority with improvement of 0.03%, 14.13%, 52.36% and 40.05% in the minimum fitness, mean fitness, maximum fitness and standard deviation, respectively.Also, compared to the Beluga, the proposed FDBAEO shows greater outperformance with improvement of 10.14%, 58.16%, 80.07% and 99.6% in the minimum fitness, mean fitness, maximum fitness and standard deviation, respectively.Additionally, Table 3 compares the applied methods with other published algorithms for designing welded beams and displays the best results.As can be shown, the suggested FDBAEO exhibits more aptitude by outperforming the others in terms of objective value.

Scenario 2: rolling element bearing design problem
Figure 5 describes this engineering scenario where the rolling element bearing design problem is addressed to maximize the rolling element bearing's dynamic load-carrying capability [59].There are ten design variables and ten restrictions.The following is the problem formulation: The proposed FDBAEO, AEO, AOS, Beluga and CBOA are applied for this scenario where the regarding optimal settings and objective are tabulated in Table 4. Also, the convergence curves of the compared algorithms are displayed in Fig. 6.As shown, the proposed FDBAEO can achieve the greatest value of the considered objective of 85,538.445where AEO, AOS, ATOA, Beluga and CBOA obtain objective values of 85,535.833,85,162.69751,85,411.681,77,061.675and 85,475.013,respectively.
To assess the robustness of the proposed FDBAEO, AEO, AOS, Beluga and CBOA over 50 separate runs, Fig. 7 displays the distribution of their obtained fitness ( 28) f c = 37.91 Artificial ecosystem optimization by means of fitness distance… Additionally, all the compare algorithms provide 100% constraints validation.Compared to the basic AEO, the proposed FDBAEO shows great superiority with improvement of 0.0031%, 3.446%, 62.51% and 99.11% in the maximum fitness, mean fitness, minimum fitness and standard deviation, respectively.Compared to the AOS, the proposed FDBAEO shows greater outperformance with improvement of 0.44%, 18.38%, 50.4% and 99.67% in the minimum fitness, mean fitness, maximum fitness and standard deviation, respectively.Compared to the ATOA, the proposed FDBAEO shows greater outperformance with improvement of 0.148%, 39.66%, 47.01% and 99.39% in the minimum fitness, mean fitness, maximum fitness and standard deviation, respectively.Compared to the Beluga, the proposed FDBAEO shows greater outperformance with improvement of 9.91%, 25.14%, 60.91% and 99.48% in the minimum fitness, mean fitness, maximum fitness and standard deviation, respectively.Compared to the CBOA, the proposed FDBAEO shows greater outperformance with improvement of 0.074%, 2.7%, 3.77% and 95.24% in the minimum fitness, mean fitness, maximum fitness and standard deviation, respectively.
Besides, the best-obtained objectives are displayed in Table 6 that shows a comparison between the applied techniques and other reported algorithms for rolling element bearing design.As shown, the proposed FDBAEO demonstrates higher ability by achieving the maximum objective value compared to the others.

Scenario 3: pressure vessel design problem
The pressure vessel design problem is tackled to reduce the overall cost, which includes material, forming, and welding expenses.The problem objective is to minimize the fabrication costs of the design shown in Fig. 8 [63].Four variables must be For scenario 3, the proposed FDBAEO, AEO, AOS, Beluga and CBOA are applied whereas the regarding optimal settings and fabrication cost's objective   Artificial ecosystem optimization by means of fitness distance… respectively.This demonstrates the high efficiency of the proposed FDBAEO in finding the least fabrication cost.For this scenario, the distribution of the obtained fitness of the proposed FDBAEO, AEO, AOS, ATOA, Beluga and CBOA is plotted in Fig. 10 while Table 8 records their accompanied robustness indices.As shown, all the compare algorithms provide 100% constraints validation.The proposed FDBAEO represents the best performance with the least robustness indices of 5889.89 as the minimum, 5959.458 as the mean, 6473.76 as the maximum and 97.03 as the standard deviation.
The proposed FDBAEO shows great superiority on the minimum costs with improvement of 0.29%, 1.41%, 14.48%, 13.51% and 17.3% compared to AEO, AOS, ATOA, Beluga and CBOA, respectively.Based on the mean costs, the proposed FDBAEO shows great superiority with improvement of 5.98%, 12.6834%, 78.04%, 47.89% and 71.07%compared to AEO, AOS, ATOA, Beluga and CBOA, respectively.Based on the maximum costs, the proposed FDBAEO shows great superiority with improvement of 10.03%, 13.91%, %, 93.97%, 67.47% and 88.63% compared to AEO, AOS, ATOA, Beluga and CBOA, respectively.Based on the standard deviation of the costs, the proposed FDBAEO shows great superiority with improvement of 75%, 79.44%, 99.63%, 97.13% and 99.27% compared to AEO, AOS, ATOA, Beluga and CBOA, respectively.Artificial ecosystem optimization by means of fitness distance… Furthermore, the best-obtained targets are shown in Table 9, which compares the applied methodologies to previous documented pressure vessel design algorithms.The least objective value achieved by the suggested FDBAEO in comparison to the others displays superior ability, as can be shown (Fig. 11).

Scenario 4: speed reducer design problem
In the fourth scenario, the speed reducer design problem is solved by reducing its weight in response to constraints on the surface stress, shaft stresses, transverse shaft deflections, and bending stress of the gear teeth.There are seven design variables which are the number of teeth in the pinion (z), module of teeth (m), face width (b), length of the first (l 1 ) and second (l 2 ) shaft between bearings and the diameter of the first (d 1 ) and second (d 2 ) shafts.The problem formulation is as follows: Minimize Subject to: 27 − 1 ≤ 0 and For this scenario, the proposed FDBAEO, AEO, AOS, Beluga and CBOA are applied whereas their regarding optimal settings and fabrication cost's objective are tabulated in Table 10 and their corresponding convergence curves are displayed in Fig. 12.As shown, the CBOA and the proposed FDBAEO achieve the least value of the considered objective of 2996.67where AEO, AOS, ATOA and Beluga obtain objective values of 2996.86,2997.23,3038.84 and 3038.086,respectively.This demonstrates the high efficiency of the proposed FDBAEO in finding the least fabrication cost.

Scenario 5: planetary gear train design problem
In the fifth case study, the FDBAEO and AEO algorithms are established and used to solve the planetary gearbox design problem [64] as shown in Fig. 14.As long as eleven requirements are met, the design's goal is to reduce the maximum gear ratio errors.The problem formulation is as follows: Minimize There are eleven restrictions in the planetary gear train issue.The values of the outer diameters of the ring gear, planet-2, and idler-5 are each subject to three constraints.The details include: Also, there must be no contact between adjacent gears.
The following restriction must be considered in order to guarantee a perfect gear arrangement while maintaining equal space between the planet gears: Also, some constraints should be maintained as follows: For this scenario, the proposed FDBAEO and AEO are applied whereas their regarding optimal settings are tabulated in Table 12.As shown, both techniques can achieve the same objective.

Scenario 6: hydrostatic thrust bearing design problem
In the sixth scenario, the hydrostatic thrust bearing design problem is solved.The goal aims to reduce a hydrostatic thrust bearing's power loss [66] (see Fig. 15).The oil viscosity (μ), recess radius (R0), bearing step radius (R), and flow rate (Q)  Artificial ecosystem optimization by means of fitness distance… where For this scenario, the proposed FDBAEO and AEO are applied whereas their regarding optimal settings are tabulated in Table 14.As shown, the proposed FDBAEO obtains a lower objective target of 1616.2where the AEO finds a close objective value of 1616.95.

Conclusions
This article suggests a novel enhanced Artificial Ecosystem Optimization (AEO) incorporating Fitness Distance Balance Model (FDB) for handling various engineering design optimization problems.In the novel proposed technique, FDBAEO, the FDB model is integrated with AEO algorithm.The FDB technique is to identify the individuals who will contribute the largest to the searching processes inside a population in a consistent and effective manner.Therefore, it increases the solution quality in nonlinear and multidimensional optimization situations.The proposed FDBAEO is implemented compared to several recent algorithms of Atomic Orbital Search, Arithmetic-Trigonometric Optimization Algorithm, Beluga whale optimization, Chef-Based Optimization Algorithm and Artificial Ecosystem Optimizer.The compared algorithms are applied for six engineering design tasks of the welded beam design, the rolling element bearing design, the pressure vessel design problem, the speed reducer design problem, the planetary gear train design and the hydrostatic thrust bearing design problem.The simulation outcomes were evaluated in comparison to the systemic AEO algorithm and other recent meta-heuristic approaches.The results showed that the recommended approach was more effective in achieving the global optimal spot.The FDBAEO has shown potential problem-solving skills.

Fig. 3 Fig. 4
Fig. 3 Best Convergence curves of the compared algorithms for welded beam design problem

Fig. 9 Fig. 10
Fig. 9 Best Convergence curves of the compared algorithms for Pressure vessel design problem

Table 1
Optimal results of the proposed FDBAEO, AEO, AOS, ATOA, Beluga and CBOA for welded beam design problem

Table 2
Statistical indices of the proposed FDBAEO, AEO, AOS, ATOA, Beluga and CBOA for welded beam design problem

Table 3
Comparison of the best results between the applied techniques and other reported algorithms for welded beam design

Table 4
Optimal results of the proposed FDBAEO, AEO, AOS, ATOA, Beluga and CBOA for Rolling element bearing design Distribution of the obtained fitness values for the compared algorithms for Rolling element bearing design values while Table 5 describes their accompanied robustness indices.As shown, the proposed FDBAEO represents the best performance with the best robustness indices of 85,538.444as the minimum, 85,489.774as the mean, 85,222.971as the maximum and 53.148 as the standard deviation.

Table 6
Comparison of the best results between the applied techniques and other reported algorithms for rolling element bearing design Fig. 8 Pressure vessel design problem are tabulated inTable 7 and their corresponding convergence curves are displayed in Fig. 9.As shown, the proposed FDBAEO can achieve the least value of the considered objective of 5889.89where AEO, AOS, ATOA, Beluga and CBOA obtain objective values of 5907.03,5973.58,6887.29,6809.89 and 7121.99,

Table 7
Optimal results of the proposed FDBAEO, AEO, AOS, ATOA, Beluga and CBOA for Pressure vessel design problem

Table 8
Statistical indices of the proposed FDBAEO, AEO, AOS, ATOA, Beluga and CBOA for Pressure vessel design problem

Table 9
Comparison of the best results of the applied techniques and the reported algorithms for Pressure vessel design

Table 10
records their accompanied robustness indices.As shown, all the compare algorithms provide 100% constraints validation.The proposed FDBAEO represents Optimal results of the FDBAEO, AEO, AOS, ATOA, Beluga and CBOA for speed reducer design problem Distribution of the obtained fitness values for the compared algorithms for Speed reducer design problem

Table 13
Statistical indices of the proposed FDBAEO, AEO and other reported algorithms for planetary gear train design problem

Table 14
Optimal results of the proposed FDBAEO and AEO for hydrostatic thrust bearing design problem

Table 15
Statistical indices of the FDBAEO, AEO and reported algorithms for hydrostatic thrust bearing design problem