Ọdịgbo Metaheuristic Optimization Algorithm for Computation of Real-Parameters and Engineering Design Optimization

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INTRODUCTION
Humans and animals face challenges within their time and space of habitation, and they attempt to solve the challenges by making decisions and selecting and combining variables influencing the conditions. The challenges range from simple to difficult-complex ones, but the satisfaction derived from attaining the goal motivates effort for solution pursuant [1]. Engineering has availed very good solutions for small scaled problems using exact methods, but such fails when the problem becomes special and high dimension, become very costly and time consuming [2]. Meanwhile, the study of nature showed complex problems solved by meta-ideas and heuristics. The aesthetics that describes the meta-heuristics provide solutions that are near-optimal yet scalable with problem dimensions [3] despite the difficult procedural uncertainties [4]; the huge difficulty is associated with the mapping of routines called intelligence from rules or heuristics that describe events of nature which falls in a multidisciplinary field [5,6]. Research in this direction has yielded several methodologies for solving engineering problems, yet more are anticipated [6]. This work aims to address some multidisciplinary domain concerns; a significant gap in balancing exploitation and exploration in populations of solution search impacts the state-of-art. Also, most recent works have scantily described the critical analogies of the metaphors that reflect the aesthetics of the target nature's source with the derived mathematical models, while the majority favors hybridization. Also, only a handful of the existing algorithms had human behavior metaphors, which this work proposes. Based on life science, a simple category of existing solutions could be into biological and non-biological (abiotic) hybrids, Bio-Abiotic hybrids, Bio-Bio hybrids, and Abiotic -Abiotic hybrids; however other literature may use alternative categorizations such as Swarm, Evolutionary, and Human intelligence. Genetic algorithms (GA) led the natural biological methods [7]. Particle Swarm Optimization (PSO) is inspired by flocks of birds and schools of fish [8] [9]. A few others due to space constraints are; Artificial Bee Colony from bee foraging [10]; Ant Colony [11]. In literature, numerous applications of the metaheuristics includes scheduling, loading, packaging, design, and control [12], image processing, amongst numerous others. The abiotic category is based on artificial physical experiences, such as Tabu Search, which made use of the creation of a tabu list [13]; Water Evaporation Optimization (WEO), mimicking the evaporation of water [14,15]; JAYA mimicking the gravitation towards success [15]; Atomic Orbital Search (AOS) [16], etc. Some modified/hybrids are; Grey Wolf and PSO [17] gave (GWO-PSO), MOGSABAT [18] from the multiobjective gravitational search algorithm, and the echolocation ability of the bat algorithm [19]. Many other metaheuristic methods can be found [20,21]. ỌMOA is a new strategy proposed by this work; the data is from the human population shown in Section II. The aesthetics are based on informal learning. The mathematical relations are developed in Section III and experiments, results and discussions are also presented in Section III, while Section IV is the conclusion.
The data of this work is gathered from the Ndịgbo people's mercantilism. This ideology is found in major Market setups across the World, where Ndịgbo are found in huge populations Fig. 1(a) shows Ahịa [n+9] described in Fig. 1(b) explicitly; the lines show the nonlinear relationships. The inner layers are shops and are associated with entities enlisted. The local Ahịa are networked across major cities in Nigeria (Ibadan, Lagos, Onitsha, etc.) which affiliates to extensions in Countries like Japan and Germany. The Ahịa primary agent (humans) are ụmụ-ahịa, and secondary are ndi-ọsọ-ahịa, Mazi, Bankers, customers (Regular and Non-Regular), suppliers, forwarding and clearing and etc. Meanwhile, the number of decision variables in sales, storage, borrowing etc., varies with constraints of environments like the cash flow, religion, and local/global politics etc.; taking shop 9 -GermanyLine Fig. 2; it comprises 1 -Mazi, 5 -Ụmụ-ahịa, 1 -Onye-ọsọ-ahịa, 12 -Regular Customers, 100 -Emergency Customers and trading on Benz-Spare Parts as shown. In obtaining the adjacency list from the data, assumptions made included (1) (1/0 means connected/not-connected) www.ijacsa.thesai.org respectively; (2) also shop data is deterministic data at capture time, the network of the single shop nine(9) is modeled, and the simulation -Bayes graph is as shown The complex network of the shop (54500 edges, 364 nodes), on a market day named Ọrie; Fig. 3 is made. It depicts the intense cognitive field (energy) of nonlinear relationship maps responsible for transduced processes with experience (edges) of ụmụ-ahịa (nodes) that Ọrie day. Daily customer satisfaction time monotonically decreases with increasing edges, even with constraints in cycles. Beyond shop 9, thousands of shops contribute to the ahịa data; the computer structural model is shown in Fig. 4. Fig. 4. Ahịa environment across many shops. S1, S2, m1, m2, m3, m4, and M2 represents: shops 1, ndiọsọ-ahịa, nwa-ahịa-1, nwa-ahịa-2, nwa-ahịa-3, nwa-ahịa-4, and many other distance ahịa (markets). The updates are processes of the ụmụ-ahịa transforming on every market day (Ọrie, Afọr, Nkwọ, Eke). The colours is evidence uncertainty.

D. Initialization of Population (Market-Size)
ỌMOA; with the decision variables, Ịgba-ọsọ-ahịa and Ddimensions, the solution vector in the ahịa can be represented as (1).
The fitness value of each Mazi will be computed as a vector of (2); For instance, a new shop with new ụmụ-ahịa (ages of 3 and 8 yrs.), some constraints of this age group include (a) Nostalgic energy in early months, inherent childishness (comprises of untargetted and undirected energies): chaotic sleeping patterns, food pattern, and desires for the first few months persist. But discipleship (hands-on, disciplinary actions, corrections, task handling, rewards) between 1 and 5 years changes their energies to focused excitement; next is integrity and trust test; Each nwa-ahịa has a position and cost affected by such constraints and uncertainty in capacity, inductiveness, and reluctance. The population is generated using (3).
The Ф is a random generator, M is the market population (pop), and D is the dimension. The pseudocode is shown below.   They cooperate, meet set thresholds, and satisfy the customer to get his Gold. Initializing a new Mazi, a new shop, and his contribution to the solution space will be given by (4).

 
Combining equation (1) and (2) Some of the major constraints (3) as mentioned in Section II. n; the number of generationsthe stopping criteria, r indicates x is random. Mazi cost alone in trade without ụmụahịa in the cycle of Ahịa days (6) resolves to a fitness vector: Pi is the cost of Mazi in the population of P, i cycles of Ahịa days, and the transmute of Mazi's energies via the training processes. At the same time, the ụmụ-ahịa adopt the emitted energies originating from multidiscipline like phycology, social tactics, resilience, experience, transactional techniques, relationship with customers, banks, etc., The differences compared to theirs cut across domains and, by analogy, involve transduction [28]. This process is given by a resemblance of balancing potentials and kinetics. 1 2 (1, ) * ( ( , :)) * * ( ) Where ( ,:) Pp in (4) is the cost of the new population at time p, Ef is the energy factor, while the r X random ụmụahịa cognitive state of five analogous Bayesian energies interacting actively in a shop.
Equations (5), (6) and (7) are all nonlinear cognitive vectors, and ratios of series [1/8, 1/40, and 1/80] of time divisions (could take any ratio as they are probabilities of random events, recall a state ranges from 0 to 1), i, r remain the same; visually, a huge network ensues as shown below.   Current time t ; the previous timestamps as the selected node (60). Search in a generation gives (11 Where updates at ri  taken during iteration. The compact dynamics (12); mimics a rhythmic nodding to music and stratagem -ịgba, which gives.
Where is a vector of emergent solutions. The threshold facilitates ụmụ-ahịa exploration; disciple-Rhythmknown as discipleship compliance, given by (13).
Where m is a random number [0,1], delta has the same dimension and size as the solution but is pseudorandom. This cooperation serves as the bond linking one source to another [23,29]. Mazi; sometimes sacrifices profit for an improvedcustomer base and to escape the local optima trap by analogy as Ịgba-ọsọ-ahịa updates; objectives of the fitness bound the strategy as given in Fig. 8. Constraints are bounded as in A, N collapse to upperbound (UB) in B, and N collapse to lower-bound (LB) as shown in Fig. 8 Where (10); f Ui is the fitness function from the best cost of the discipleship and adjustments made (error correction), while f i is the best solution fitness of the original objective function, which is optimal.

F. Graphical Flow of ỌMOA
The ụmụ-ahịa can be considered as moving particles [30][31][32]. Mazi realization comes after generations of successful cycles [33]; rather than unhealthy competition, all ụmụ-ahịa depends on each other; The main body's pseudocode (2) during iteration is as follows:  The flow chart of Fig. 9 shows the methodology for applying the ỌMOA algorithm. The subsequent sections discuss applications.

III. APPLICATION OF ỌMOA ON BENCHMARK FUNCTIONS
Most metaheuristic algorithms use Pattern Matrix, and the solutions are identified as those that improved through the number of generations up until the convergence time of the simulation. ỌMOA inherent energy synergy principles.

1) Default parameters are used
2) 30 independent runs were used for Unconstraint Benchmark, 50 for constraint functions. The parameters for the engineering designs are as stated in the referenced literature provided 3) The total number of cost function evaluations is 1000·n·M, where M = 10 is the number of iterations. The logarithmic Scale was considered for visualization due to its convenience and compact.

4)
For the Constraint problems as depicted by the competition, a solution value less than 10 −8 is treated as zero; several performance indicators for solution values are used: best, worst, mean, and standard deviation (Std). Test for convergence time also provided;

A. Experiments and Comparison of Results
ỌMOA is compared with five of the best similar algorithms as shown in Table II  The list in Table II is a competitive group; notably, 4 -5 won the CEC 2017 competition [39,40].

B. Experiment 1: Difficult Unconstraint Benchmark
Functions ỌMOA is validated on the existing established algorithms listed in Table II with about 30 difficult functions chosen with modality (unimodal to check and confirm exploitation strength, multimodal for diversity or exploratory capability of ỌMOA), Separability (possible separable and non-separable) and then multi-dimensionality (confirming search and exploratory strength of ỌMOA). The performance averages are visualized using boxplots. Further, the significance and statistical students test (t-test) was conducted for all algorithms, with a time complexity test. A subset of test benchmark functions with varying degrees of difficulty is used to substantiate that ỌMOA can exploit and explore the solution space and find the solutions for optimum. In Table III, unconstraint benchmark test functions are categorized in modality, Separability, and Dimensionality (N), also: M is the modality, 0 -Uni-modal; 1 -Multimodal, S is the Separability, 0 -Non-Separable; 1 -Separable.

C. Benchmark -Unimodal and Separable Functions
To tighten the competitiveness, we identified the algorithms with the highest performances with a t-value above 0.05. ỌMOA and A3 (all best solutions in BOLD font) lead with equal best performance as shown in Table IV. The MS was next, followed by A1, HHO, and WFS.
ỌMOA and A3 leading showed good exploration, and exploitation strength, particularly of ỌMOA obtained the best optimal objective solutions before the completion of generations. Fig. 10 also shows consistent distributions with fewer outliers.

D. Unimodal and Non-Separable
The functions in this category include 30-dimensional problems Zakharov to Dixon Price with great complexity. Table V shows ỌMOA and A3 tops, followed by A2, A1, HHO, MS, and WFS. Besides Beale, which did not yield a better result, ỌMOA got even Easom, a problem with inherent complex nature.    The boxplot of Fig. 12 shows that the mean solutions distribution of the data of ỌMOA and A3 are tight, with little outliers equalling minimal deviation.
The convergence comparison of Fig. 13 confirms the summary made at the beginning of the subsection.

E. Multimodal and Separable
Complex structures, multiple, unequal hilltops, and valleys-shaped functions are tested as shown in Table VI. Besides the booth function, ỌMOA had remarkable exploratory abilities for the dimensionalities above n = 2 (i.e., n = 5, 10, 30) of the last three functions while tracking deeper than values provided by the global optima in literature for Holder Table, Michalewicz (2, 5, and 10). Rastrigin (n = 30) was also explored optimally by ỌMOA and HHO.   In Fig. 15, ỌMOA had made extra-advance to explore for solutions far better than all the compared algorithms in these problems. Even some of the solutions were far better optimum that set global values as the Holder   The boxplot in Fig. 16 shows the clear visuals, confirms ỌMOA better. The exploratory ability of the ỌMOA is evident from mean solutions distribution and standard deviations.
The convergence is a reflection of the ability of the tree depth of the network of markets embedded in the model visualized in Fig. 17. 288 | P a g e www.ijacsa.thesai.org

G. Statistical Test and Significance
Table VIII presents the entire statistical hypothesis test carried out to confirm the difference in mean and significance validation in the distribution of solutions by the algorithms on the 30 unconstraint benchmark functions. Table IX is the summary of the test conducted to prove the hypothesis of the performances of the experiment; 1: means ỌMOA (in black ink) is more significant, -1: gives better significance to the contender (another algorithm), while 0: depicts no significant difference in performance (contender, equal, ỌMOA).  The highest equal performance point, 13 is between ỌMOA and A3, with ỌMOA leading with 13 optimal solutions, more than A3's other 4 better performances. Next is HHO with 8 equal points, ỌMOA with 7 better optimal solutions, and HHO making 5 places. MS and A1 shared very close contest with EHO behind. Table X also is the presentation of the mean runtime measure given below. The performance of ỌMOA in this section was very high on benchmark complex unconstraint problems compared to contending methods.

H. Results and Statistical Testing with CEC 2017
This section reports ỌMOA on real-parameter single objective optimization challenging problems featured in Computational Evolution Computation -CEC 2017 with a statistical comparison between ỌMOA and winners of the competition.

I. Result of ỌMOA with CEC 2017
The values are the differences between the global optima and the ones obtained with ỌMOA for 10D, 30D, and 100D during every 51 runs, as shown in the Table XI, and competition is presented in Table XII for 50D 1) 10D, 30D, 50D and 100D Performances  The uni-modal functions EC1, EC2, and EC3 results were least expected within the number of functional evaluations provided, perhaps due to parameter tuning differences from recommended.
 EC7 -EC10 multimodal functions all attained global optima in all dimensions. ỌMOA also met 10D and 30D optimal values, with a minor difference for 50D and not too good 100D. EC5 solutions are not good in all dimensions; while EC6 10D was globally optimal, the rest dimensions were not impressive and inadequate for some ranges of solutions.

J. Time Complexity Analysis
The competition provided appropriate information on the modalities to compute the time complexity [39]. The observation and experimentation shown in Table XII of this work is as follows:  Evaluate a code consisting of basic arithmetic operation for 1,000,000 iterations and recode the time (T 0 ).

K. Comparison of ỌMOA with the Winners of CEC2017 (EC1-EC29)
The subsection presents a performance comparison between the 50D problem size for ỌMOA and other state-ofart high-performing algorithms, especially those that won the CEC2017 competition for real-parameter single objective optimization challenges, as shown in  Table XII; ỌMOA had remarkably shown better performance on most of the complex problems considered, as the last row shows. Also, ỌMOA showed better performances compared with the winners of the competition (competing method, equity, ỌMOA w/t/l), e.g., (EBOwithCMAR won 11, equal in 1 and ỌMOA won 17).

L. Benchmark Design Real Engineering CEC 2020 Single
Objective Problems Eight (8) difficult engineering design-constrained problems that exhibit functional inequality and equality constraints are considered; compared with state-of-the-art algorithms from CEC 2020 real-world optimization issues presented in [42][43][44]. Among the results presented are the experiments' statistical best, mean, median, worst, and standard deviations. Generally, all models follow a structure as shown in Eq. (15). www.ijacsa.thesai.org Where f is the fitness, xs' are the design variables, g is the constraint with less than equality (often greater than for maximization problems), and n is the number of constraints. The conversion of the functional constraint from inequality to equality transforms the problem into equation (16).
Where f p (x) is the penalized objective function. Highperforming state-of-the-art algorithms are adopted and compared against the design of certain engineering problems of Fig. 18 (a: Welded beam), (b: Pressure Vessel, and c: Compression Spring). The constraint violations are considered, and the penalty function method is used, which often transforms a constrained problem into an unconstrained continuous counterpart for ease of implementation.

M. Statistical Comparison of Results for Tension / Compression Spring Design Problem
The design problem in Fig 18 (c) aims at reducing the weight of the tension/compression spring without compromising domain properties like the shear stress, frequency wave, and displacement functionalities [45]. The control variables are wire diameter (x1), mean coil diameter (x2), and the number of coils (x3); the mathematical formulation is detailed in [42]. Upon the experiment, ỌMOA yielded the most optimal weight compared to the other highperforming algorithms within a minimal number of function evaluations. The result of the compared simulation is shown in Table XIV.

N. Statistical Comparison of the Results for Welded Beam Problem
The welded beam problem Fig. 18 (b) [46] is to minimize the cost of construction. The impacting constraints include shear stress( ); bending stress in the beam ( ); buckling load of the bar (P c ); end deflection of the beam ( ) and side constraints. The decision variables are (1) the thickness of the weld (x1), the length of the attached part of bar (x2), the height of the bar (x3) and the thickness of the bar (x4). The model formulation is given in [42]. And compared simulated statistical results in Table XV with parametric results in  Table XVI.  In less than 60000 functional evaluations of 10 runs, ỌMOA yielded a mean cost that is the most optimal for the welded beam in comparison while obeying the constraints.

O. Results for Pressure Vessel Design Problem
The Pressure Vessel Design objective in Fig. 18 (b) [47] is to minimize the cost associated with materials, building, and welding of a cylindrical vessel with capped ends and a hemispherical-shaped head. The impacting variables include the shell thickness x(1), the head thickness x(2), the inner radius x(3), and the length of the cylindrical section excluding the head x(4); the model formulation is given in [42].
The yield of ỌMOA on the Pressure Vessel design problem produced the best optimal mean value and had a far smaller number of function evaluations of Table XVII. ỌMOA met all the inequality constraints; best mean fitness as shown, followed by MCEO, WCA, and NM-PSO, respectively. However, contrary to the large number assigned to the penalty using the other algorithms, ỌMOA found better results with negligible penalty value for problems of spring and welding beam, and even no penalty was applied to vessel design.

P. Robot Gripper Problem
The complexity involved in manipulating the grippers to minimize the difference between the minimum and maximum forces of the robotic action is ongoing research. Seven design variables, geometric properties, with about seven inequality constraints, are targeted. The Mathematical formulations are found here in [48].   The experimental result of ỌMOA on the gripper problem showcases a new optimum as against the optimum global set value [42]; also better than the competing algorithms in comparison [44].

Q. Rolling Element Bearing
Five design variables that affect the optimal design of a rolling bearing with the capacity to carry load efficiently amidst nine inequality constraints are considered in the design. The mathematical derivations are provided by [42], while we show the schematics in Fig. 20.  With an optimum global set at (25287.918415), the experimental result of Table XIX shows that ỌMOA had set a better global optimum as it also performed better than the competing algorithms [44].

R. Gas Transmission Compressor Design (GTCD)
Four variables with one inequality constraint are targeted when designing the gas transmission compressor. The work [42] provides the mathematical formulation while we show the schematics in Fig. 21 and the solutions provided by many optimization state-of-the-art to designs. The results of the comparison for the experiment on GTCD are shown in Table XX. The experimental results show ỌMOA had a set a new global optimum than that set by the competition as the global optimum is (2.9648954173E+06) [42], with the other algorithms as presented in [49].

S. Himmelblau's Function
This nonlinear function has been used to test many novel metaheuristic algorithms; it has five main design variables and six inequality constraints to be handled, as shown in Himmelblau [50]. In Table XXI, we show the results of the performances of the metaheuristic algorithms used in comparison.
The experimental result shows that ỌMOA obtained a better minimum compared to the competing algorithms and set a new global optimum compared to the global presented by [42], which is −3.066554E+04, with the other algorithms as presented in [49].

T. Multiple Disk Clutch Brake Design Problem
The design objective is to minimize the mass of the multiple disk clutch brake, five decision variables with nine nonlinear constraints. The mathematical formulation is given in [42]. 29.1039 29.1039 -3.19E+04 Fig. 22. The schematic geometric representation of the multiple disc clutch design. Fig. 22 is schematics geometric representation of the clutch disc problem. However, the experimental results are shown in Table XXII. ỌMOA performed better than ABC, which was reported as best at the time of competition report, and others in this design problem and further set a much better global optimum than benchmarked in [42]; 0.23524245790; with the other algorithms as presented in [49].

IV. CONCLUSION
In this work, Ọdịgbo Metaheuristic Optimization Algorithm -ỌMOA, a new nature-inspired population-based metaphor, was proposed and used in experiments and engineering designs with very great performance. The idea stemmed from the informal learning pattern and discipleship, which is ingrained in the socio-cultural behavior of the indigenous peoples -the Ndigbo of a West African tribe is presented. The learners cope through practice and observation. The experiment conducted considered 30 benchmark unconstraint problems, 29 CEC 2017 (50D) real-parameter single objective constraint optimization, and about 8 engineering design constrained problems from CEC 2020; the results showed that OMOA had balanced exploitation and exploratory capacities with very good convergence time too. Comparison to the performance of other well-established stateof-the-art algorithms depicts the exceptional performance of the automata. The significant test also confirms the relative efficiency of ỌMOA with t-values and p-values presented in Table VIII and summarized in Table IX. The convergence time test using the Rastrigin function also shows OMOA had better speed than the contender in Table X. The competing algorithms were the most award winners in past competitions from 2017 till date. In all complex engineering problems presented, ỌMOA had performed remarkably well and had, in some cases, set new minimum attainable best solutions; Of interest are the new values better than the set global optimums in some functions and engineering designs (Clutch Disc, Himmelblau, GTCD, Rolling Bearing, Robotic Gripper).
The future direction is to further validate with the most recent CECs and design optimization problems in other fields. Meanwhile, ỌMOA shows merit to be considered in the current state-of-the-art.