Material Generation Algorithm: A Novel Metaheuristic Algorithm for Optimization of Engineering Problems

: A new algorithm, Material Generation Algorithm (MGA), was developed and applied for the optimum design of engineering problems. Some advanced and basic aspects of material chemistry, speciﬁcally the conﬁguration of chemical compounds and chemical reactions in producing new materials, are determined as inspirational concepts of the MGA. For numerical investigations purposes, 10 constrained optimization problems in different dimensions of 10, 30, 50, and 100, which have been benchmarked by the Competitions on Evolutionary Computation (CEC), are selected as test examples while 15 of the well-known engineering design problems are also determined to evaluate the overall performance of the proposed method. The best results of different classical and new metaheuristic optimization algorithms in dealing with the selected problems were taken from the recent literature for comparison with MGA. Additionally, the statistical values of the MGA algorithm, consisting of the mean, worst, and standard deviation, were calculated and compared to the results of other metaheuristic algorithms. Overall, this work demonstrates that the proposed MGA is able provide very competitive, and even outstanding, results and mostly outperforms other metaheuristics.


Introduction
Optimization techniques have been proposed for the optimum design of different problems of everyday life in order to increase the efficiency of systems and human resources. Most of the design problems in nature are complex, with multiple design variables and constraints that classical optimization algorithms, such as gradient-based algorithms, cannot handle. As a solution, numerous artificial intelligence experts have introduced new algorithms with better performance in different fields. Regarding the recent developments in technology, new optimization methods offering higher efficiency, greater accuracy, and increased speed rate are required to deal with difficult optimization problems.
Based on the mentioned concerns about the capabilities of optimization algorithms, a "metaheuristic" approach has been proposed by optimization experts [1] for solving different optimization problems. 'Metaheuristic' refers to specific solution techniques, where higher-level strategies are implemented into the main searching process of the optimization algorithms to provide a powerful searching method with specific capabilities, including the avoidance of entrapment in local optimal solutions. The history of developing different metaheuristic approaches as solutions in different optimization fields can be classified into five different time periods. A brief summary of these historical time periods is presented in Table 1.
by multiple methods in order to demonstrate its capability as an independent algorithm among the other methods while this kind of proposing a testing the algorithms is not the only aim of this area. The proposed novel algorithm can be of a great help in the situations that the other alternatives cannot reach to a reasonable response in dealing with a considered problem so there should be other alternatives in order to have a good chance to provide a well-designed plan for the industry and even human-related actions in the everyday life. A brief outline of this work is as follows: Section 2 discusses the inspirational concept and mathematical model of the MGA optimization algorithm. In Section 3, the problem statements, including the selected mathematical and engineering optimization problems utilized to test the proposed MGA as a novel metaheuristic algorithm, are presented. In Sections 4 and 5, the numerical results of the MGA algorithm and other alternative metaheuristic methods in dealing with the considered mathematical and engineering optimization problems are presented. In Section 6, the key findings of this research work are concluded, future research directions are suggested.

Material Generation Algorithm
In this section, the inspiration of MGA as a novel metaheuristic algorithm and the mathematical model of this algorithm are presented.

Inspiration
A material is a mixture of multiple substances comprised of the stuffs of the universe with volume and mass. The material generation process concerns the capability of different substances to merge with each other in order to generate new materials with higher functionality and improved energy levels. Elements are the basic building blocks of the materials, which cannot be broken into parts or even changed into other elements. Materials are engineered on an atomic, nano-, micro-, or macro-scale in order to control the specific properties and improve the performance of a material. Uniquely-generated materials are classified based on their general properties and specific characteristics and according to physical and chemical changes that influence a material's behavior.
Material chemistry is one of the most important disciplines in the material research field. Material engineers study the configuration of materials in order to improve the specific characteristics of materials, developing new ones that are more sustainable and also superior to the previous ones. Chemical changes in materials are achieved by reacting and combining various chemicals. In general, the chemical properties are altered by the transferring or sharing of electrons between atoms of different materials, specifically, chemical bonds formed between materials result in such modifications. In this work, three main concepts of material chemistry (compounds, reactions, and stability) were considered to formulate a metaheuristic optimization algorithm.

Chemical Compound
Most chemical elements in the universe are created through combinations with other elements. With that being, a few chemical elements exist freely in nature. Compounds are formed by combining multiple chemicals via chemical bonds, or the transferring or sharing of electrons, which result in one of the following: -Ionic compounds are created when electrons are transferred from the atoms of one element to those of another. -Covalent compounds form when electrons are shared between atoms of different elements.
In addition, ionic compounds contain multiple ions that are held together by the electrostatic force called ionic bonding. Although these compounds are neutral in nature, they consist of some negatively-and positively-charged ions, called anions and cations, respectively. The evaporation, precipitation, or freezing of the constituent ions are the main factors in the process of producing ionic compounds. When an atom or a small group of atoms starts to lose or gain electrons, an ionic compound forms according to the ionic bonding and charged particles. As an example, the formation of sodium chloride, also known as table salt, is depicted in Figure 1. In the process of electron transformation, a sodium (neutral) becomes a sodium cation (Na + ) when it loses one electron. In addition, Cl becomes a chloride anion (Cl − ) when it gains an electron. Thus, table salt is a solid aggregation of Na + and Cl − ions, which attract each other due to opposite charges.
In addition, ionic compounds contain multiple ions that are held together by the electrostatic force called ionic bonding. Although these compounds are neutral in nature, they consist of some negatively-and positively-charged ions, called anions and cations, respectively. The evaporation, precipitation, or freezing of the constituent ions are the main factors in the process of producing ionic compounds. When an atom or a small group of atoms starts to lose or gain electrons, an ionic compound forms according to the ionic bonding and charged particles. As an example, the formation of sodium chloride, also known as table salt, is depicted in Figure 1. In the process of electron transformation, a sodium (neutral) becomes a sodium cation (Na+) when it loses one electron. In addition, Cl becomes a chloride anion (Cl − ) when it gains an electron. Thus, table salt is a solid aggregation of Na+ and Cl-ions, which attract each other due to opposite charges. Covalent compounds form when an atom of a chemical element shares an electron with another element's atom, which usually occurs between nonmetal elements and results in an electrically neutral atom. Figure 2 displays the formation of a covalent compound that leads to the hydrogen atom. As an example, assuming that two hydrogen atoms begin approaching each other, the nucleus of one atom strongly attracts the electron of the other one. A covalent bond is achieved when a specific distance between the nuclei is reached, and the electrons are equally shared. The net repulsion between nuclei is ignored due to the greater net attraction.  Covalent compounds form when an atom of a chemical element shares an electron with another element's atom, which usually occurs between nonmetal elements and results in an electrically neutral atom. Figure 2 displays the formation of a covalent compound that leads to the hydrogen atom. As an example, assuming that two hydrogen atoms begin approaching each other, the nucleus of one atom strongly attracts the electron of the other one. A covalent bond is achieved when a specific distance between the nuclei is reached, and the electrons are equally shared. The net repulsion between nuclei is ignored due to the greater net attraction.
In addition, ionic compounds contain multiple ions that are held together by the electrostatic force called ionic bonding. Although these compounds are neutral in nature, they consist of some negatively-and positively-charged ions, called anions and cations, respectively. The evaporation, precipitation, or freezing of the constituent ions are the main factors in the process of producing ionic compounds. When an atom or a small group of atoms starts to lose or gain electrons, an ionic compound forms according to the ionic bonding and charged particles. As an example, the formation of sodium chloride, also known as table salt, is depicted in Figure 1. In the process of electron transformation, a sodium (neutral) becomes a sodium cation (Na+) when it loses one electron. In addition, Cl becomes a chloride anion (Cl − ) when it gains an electron. Thus, table salt is a solid aggregation of Na+ and Cl-ions, which attract each other due to opposite charges. Covalent compounds form when an atom of a chemical element shares an electron with another element's atom, which usually occurs between nonmetal elements and results in an electrically neutral atom. Figure 2 displays the formation of a covalent compound that leads to the hydrogen atom. As an example, assuming that two hydrogen atoms begin approaching each other, the nucleus of one atom strongly attracts the electron of the other one. A covalent bond is achieved when a specific distance between the nuclei is reached, and the electrons are equally shared. The net repulsion between nuclei is ignored due to the greater net attraction.

Chemical Reaction
Chemical reactions are the process of transforming one material into another while the chemical equations are used to represent chemical reactions, where the resulting products will have different properties than the starting materials (reactants/reagents), and intermediate materials (in some particular cases).
An example of a chemical reaction is depicted in Figure 3, in which the magnesium wire (Mg) and oxygen gas (O 2 ) yield powdery magnesium oxide (MgO). As presented in the left bulb, a fine magnesium filament is surrounded by oxygen before the reaction occurs. As the reaction proceeds, the white colored powdery magnesium oxide coats the bulb's inner surface, which is demonstrated in the right bulb. In this reaction, heat and light are also produced as intermediate materials but are not concerned in this description. The chemical equation of the presented chemical reaction is as follows: where s and g stand for solid and gas, respectively.

Chemical Reaction
Chemical reactions are the process of transforming one material into another while the chemical equations are used to represent chemical reactions, where the resulting products will have different properties than the starting materials (reactants/reagents), and intermediate materials (in some particular cases).
An example of a chemical reaction is depicted in Figure 3, in which the magnesium wire (Mg) and oxygen gas (O2) yield powdery magnesium oxide (MgO). As presented in the left bulb, a fine magnesium filament is surrounded by oxygen before the reaction occurs. As the reaction proceeds, the white colored powdery magnesium oxide coats the bulb's inner surface, which is demonstrated in the right bulb. In this reaction, heat and light are also produced as intermediate materials but are not concerned in this description. The chemical equation of the presented chemical reaction is as follows: where s and g stand for solid and gas, respectively.

Chemical Stability
Stability is one of the more important properties of materials in real-world applications. When generating new materials with different characteristics, it is important to consider the stability of the chemical compounds and reactions in different situations. In terms of chemical stability, chemicals have the tendency to resist changes, such as decomposition, due to internal factors and external influences such as heat, air, light, and pressure. Chemical stability is the resistance of a material to change in the presence of other chemicals. A stable chemical product refers to one that has not been specifically reactive in the environment and retains its properties over a specific period of time. Comparatively, unstable chemical materials easily decompose, corrode, polymerize, explode, or burn under certain conditions. When producing new chemical materials, the processes of transferring or sharing electrons within the initial materials will occur in such a way that the end product will be stable and applicable during a specific period of time.

Mathematical Model
In order to conduct an optimum design procedure, an optimization algorithm is developed in this section based on the mentioned principles of material chemistry. The basic concepts of the chemical compounds, reactions, and stability are utilized in order to develop and formulate a well-defined mathematical model for the new algorithm. Considering that many natural evolution algorithms establish a predefined population of solu-

Chemical Stability
Stability is one of the more important properties of materials in real-world applications. When generating new materials with different characteristics, it is important to consider the stability of the chemical compounds and reactions in different situations. In terms of chemical stability, chemicals have the tendency to resist changes, such as decomposition, due to internal factors and external influences such as heat, air, light, and pressure. Chemical stability is the resistance of a material to change in the presence of other chemicals. A stable chemical product refers to one that has not been specifically reactive in the environment and retains its properties over a specific period of time. Comparatively, unstable chemical materials easily decompose, corrode, polymerize, explode, or burn under certain conditions.
When producing new chemical materials, the processes of transferring or sharing electrons within the initial materials will occur in such a way that the end product will be stable and applicable during a specific period of time.

Mathematical Model
In order to conduct an optimum design procedure, an optimization algorithm is developed in this section based on the mentioned principles of material chemistry. The basic concepts of the chemical compounds, reactions, and stability are utilized in order to develop and formulate a well-defined mathematical model for the new algorithm. Considering that many natural evolution algorithms establish a predefined population of solution candidates that are evolved through random alterations and selection, MGA determines a number of materials (Mat) comprised of multiple periodic table elements (PTEs). In this algorithm, a number of materials is considered as the solution candidates (Mat n ), which are comprised of some elements represented as decision variables (PTE j i ). The mathematical presentation of these two aspects is as follows: where d is the number of elements (decision variables) in each material (solution candidates); and n is the number of materials considered to be the solution candidates.
In the first stage of the optimization process, PTE j i is determined randomly while the decision variables bounds are defined based on the considered problem. The initial positions of PTEs are determined randomly in the search space as follows: where PTE

Modeling Chemical Compound
To mathematically model the chemical compounds, all PTEs are assumed to be in the ground state, which can be externally excited by the magnetic fields, absorption of energy from photons or light and interactions with different colliding bodies or particles regarding ions or other individual electrons. Due to the different stabilities of elements, they have a tendency to lose, gain, or even share electrons with other PTEs, resulting in ionic or covalent compounds. To model the ionic and covalent compounds, d random PTEs are selected using the initial Mat (Equation (1)). For the selected PTEs, the processes of losing, gaining, or sharing electrons are modeled through the probability theory. To fulfill this aim, a continuous probability distribution is utilized for each PTE to configure a chemical compound, which is considered as a new PTE, as follows: where r 1 and r 2 are uniformly distributed random integers in the intervals of [1, n] and [1, d], respectively; PTE r 2 r 1 is a randomly selected PTE from the Mat; e − is the probabilistic component for modeling the process of losing, gaining or sharing electrons represented with normal Gaussian distribution in the mathematical model; and PTE k new is the new material.
The newly-created PTEs are utilized for producing a new material (Mat new 1 ), which is then added to the initial material list (Mat) as a new solution candidate: Then, the overall solution candidates are combined and presented as follows: Processes 2021, 9, 859 A schematic presentation of the described process for the configuration of new materials based on the concept of chemical compounds (ionic and covalent) is depicted in Figure 4.
A schematic presentation of the described process for the configuration of new materials based on the concept of chemical compounds (ionic and covalent) is depicted in Figure 4. The probabilistic approach for determining − is modeled through normal Gaussian distribution, which is important in statistics and often used in the natural and social sciences to represent real-valued random variables with unknown distributions. The probability of selecting a new element ( ) regarding the randomly selected initial element ( 1 2 ) is presented as follows: where is the mean, median or expectation of the distribution correspond to the selected random ( 1 2 ); is the standard deviation, which is set to unity in this paper; 2 is the variance; and is the natural base or Naperian base of the natural logarithm.

Modeling Chemical Reaction
Chemical reactions are sort of production process in which different chemical changes are determined in order to produce different products with modified properties even different from the initial reactants. In order to mathematically model the process of producing new materials by the chemical reaction concept, an integer random number ( ) is determined regarding the number of materials of the initial are considered for participating in a chemical reaction. Then, integer random numbers ( ) are generated to determine the positions of the selected materials in the initial so, the new solutions are linear combinations of the other solutions. For each material, a participation factor ( ) is also calculated since different materials would participate in the reactions with different amounts. A schematic presentation of the described process is depicted in Figure 5, and the mathematical presentation is as follows: The probabilistic approach for determining e − is modeled through normal Gaussian distribution, which is important in statistics and often used in the natural and social sciences to represent real-valued random variables with unknown distributions. The probability of selecting a new element (PTE k new ) regarding the randomly selected initial element (PTE r 2 r 1 ) is presented as follows: where µ is the mean, median or expectation of the distribution correspond to the selected random PTE (PTE r 2 r 1 ); σ is the standard deviation, which is set to unity in this paper; σ 2 is the variance; and e is the natural base or Naperian base of the natural logarithm.

Modeling Chemical Reaction
Chemical reactions are sort of production process in which different chemical changes are determined in order to produce different products with modified properties even different from the initial reactants. In order to mathematically model the process of producing new materials by the chemical reaction concept, an integer random number (l) is determined regarding the number of materials of the initial Mat are considered for participating in a chemical reaction. Then, l integer random numbers (mj) are generated to determine the positions of the selected materials in the initial Mat so, the new solutions are linear combinations of the other solutions. For each material, a participation factor (p) is also calculated since different materials would participate in the reactions with different amounts. A schematic presentation of the described process is depicted in Figure 5, and the mathematical presentation is as follows: where Mat m is the mth randomly selected material from the initial Mat; p m is the normal Gaussian distribution for the mth material participation factor; and Mat new 2 is the new material produced by the chemical reaction concept.
Processes 2021, 9, x FOR PEER REVIEW 8 of 34 where is the ℎ randomly selected material from the initial ; is the normal Gaussian distribution for the ℎ material participation factor; and 2 is the new material produced by the chemical reaction concept.

Modeling Chemical Stability
As previously described, the principle of material stability concerns the tendency of natural systems to seek local and general equilibria at all structural levels. Material stability is mathematically represented by determining the quality of the solutions as . Materials with the highest stability levels alongside the ones with lowest stability levels are equivalent to the best and worst fitness values of all solution candidates in the optimization runs.
Considering the chemical compound and chemical reaction configuration approaches, the overall solution candidates are combined as follows: Moreover, the stability levels of the initial material and newly0produced materials should be considered in order to decide whether or not the new materials should be included in the overall material list ( ) corresponding to the solution candidates. The quality of new solution candidates is then compared to the initial ones, whereby the new materials should be substituted by initial materials with worst fitness values corresponding to worst stability levels.
For boundary violation control, a flag is determined in order to control the violating solution candidates while a maximum number of iteration or objective function evaluation can be considered as stopping criteria. The flowchart of the MGA algorithm is presented in Figure 6.

Modeling Chemical Stability
As previously described, the principle of material stability concerns the tendency of natural systems to seek local and general equilibria at all structural levels. Material stability is mathematically represented by determining the quality of the solutions as Mat. Materials with the highest stability levels alongside the ones with lowest stability levels are equivalent to the best and worst fitness values of all solution candidates in the optimization runs.
Considering the chemical compound and chemical reaction configuration approaches, the overall solution candidates are combined as follows: Moreover, the stability levels of the initial material and newly0produced materials should be considered in order to decide whether or not the new materials should be included in the overall material list (Mat) corresponding to the solution candidates. The quality of new solution candidates is then compared to the initial ones, whereby the new materials should be substituted by initial materials with worst fitness values corresponding to worst stability levels.
For boundary violation control, a flag is determined in order to control the violating solution candidates while a maximum number of iteration or objective function evaluation can be considered as stopping criteria. The flowchart of the MGA algorithm is presented in Figure 6.

Problem Statement
In this section, a brief description of the considered design examples is presented. Regarding the fact that these examples are categorized as constrained optimization problems, the general formulations of these kinds of optimization problems are presented as follows: where f (x) is considered as the objective function of the optimization problem that can be considered to be maximized or minimized; g i (x) and h j (x) are the ith and jth inequality and equality constraint, respectively; x is the position vector related to the optimization variables; and n and m are the total number of inequality and equality constraints, respectively. In most cases, the equality constraints can be transformed into inequality constraints by considering the following: where ε is a predefined small positive number, which is typically near to zero. In this work, ε was set to 0.0001.

Mathematically-Constrained Problems
The mathematical problems of the CEC 2017 benchmark suite are presented in Table 2, while the specific details and mathematical formulations were presented in detail by Wu et al. [39]. In order to evaluate the results of the proposed MGA, the statistical results of different state-of-the-art metaheuristic algorithms regarding the considered constrained problems were derived of the recent literature [38][39][40].

Engineering Design Problems
The second type of constrained problems included 15 well-known engineering problems, which have been solved by different optimization algorithms. A brief description of these design examples is presented in Table 3, and the specific details of each example are provided in the following subsections. These examples have also been benchmarked by Kumar et al. [43] regarding the CEC 2020 engineering design scheme.

Numerical Results of Mathematical Problems
The numerical results based on the CEC 2017 benchmark problems by means of the MGA and other alternatives in dealing with the described constrained problems with different dimensions of 10, 30, 50, and 100 are presented in this section. For comparison, a total of 25 optimization runs was performed, including a maximum number of function evaluations (20,000 × D), where D is the problem dimension. These results are presented in Tables 4-7  Regarding the fact that the considered problems of the CEC 2017 benchmark suite are all the latest problems in the evolutionary computation field with higher levels of complexity and difficulties while there are few approaches that can provide acceptable results in dealing with these problems. In this regard, the reported results by MGA are marginal because there are not any better results for the considered problems in the literature so the MGA calculated the latest reported results which demonstrate the capability of this algorithm in competing with other methods.
In order to have a better perspective on the performance of different metaheuristic algorithms in dealing with the CEC 2017 benchmark problems, the box plots which are derived of the analysis of the variance (ANOVA), which were conducted for the normalized values of the reported bests, means, standard deviations (Std), and worsts for different dimensions of 10, 30, 50, and 100 in Figures 7-10. It can be concluded that the MGA has competitive performance in dealing with these problems.               rithm in competing with other methods. In order to have a better perspective on the performance of different metaheuristic algorithms in dealing with the CEC 2017 benchmark problems, the box plots which are derived of the analysis of the variance (ANOVA), which were conducted for the normalized values of the reported bests, means, standard deviations (Std), and worsts for different dimensions of 10, 30, 50, and 100 in Figures 7-10. It can be concluded that the MGA has competitive performance in dealing with these problems.  rithm in competing with other methods.
In order to have a better perspective on the performance of different metaheuristic algorithms in dealing with the CEC 2017 benchmark problems, the box plots which are derived of the analysis of the variance (ANOVA), which were conducted for the normalized values of the reported bests, means, standard deviations (Std), and worsts for different dimensions of 10, 30, 50, and 100 in Figures 7-10. It can be concluded that the MGA has competitive performance in dealing with these problems.      Based on the provided results for the MGA and other state-of-the-art approaches in the evolutionary computation field, the AGA is capable of competing with these excellent algorithms while in some cases even MGA outperforms the others. this performance in dealing with CEC 2017 lead to the fact that MGA's mathematical model is well-established model in which the global and local search are conducted with no need to any parameters to be tuned. In other words, this algorithm does not need any internal parameters to be defined prior to the optimization process which makes this algorithm a best choice in Based on the provided results for the MGA and other state-of-the-art approaches in the evolutionary computation field, the AGA is capable of competing with these excellent algorithms while in some cases even MGA outperforms the others. this performance in dealing with CEC 2017 lead to the fact that MGA's mathematical model is well-established model in which the global and local search are conducted with no need to any parameters to be tuned. In other words, this algorithm does not need any internal parameters to be defined prior to the optimization process which makes this algorithm a best choice in dealing with complex problems in which there are not any information about the complexity level of the problem. Additionally, the MGA generates only two new solution candidates in each iteration which makes the algorithm to require less computational efforts for optimization purposes. Hence, these aspects can be of great importance when the MGA is compared to the other metaheuristic algorithms in the evolutionary computation field. In other words, MGA is a parameter free optimization approach with less computational cost, which makes this algorithm different form the other approaches, while the inspirational concept of this algorithm is also unique.

Numerical Results of Engineering Problems
The numerical results of MGA considering the previously-described engineering design problems are presented in this section. In this regard, the results of other metaheuristics in dealing with these design examples were taken from the literature in order to make fair judgments.
The comparative results of the speed reducer design engineering problem, including the obtained design (decision) variables related to the best optimum configuration determined by different methods, are presented in Table A1. In addition, the statistical results, such as the best, mean, and worst fitness values alongside the standard deviation, are presented in Table 8. The results of different metaheuristics show that the best results of MGA are better than the best results of the other approaches in dealing with this design example. The MGA is also capable of providing better statistical results, including mean and standard deviation. The Friedman statistical test results are also presented in Table A2 for comparative purposes. Considering the spring design problem, the best and statistical results of different metaheuristics, including the obtained design variables related to the best optimum design, are presented in Tables 9 and A3, respectively. It should be mentioned that MGA is capable of obtaining very competitive results for this constrained engineering design problem. It also should be mentioned that MGA yields better statistical results in terms of the mean, worst fitness values alongside the standard deviation than the results of other metaheuristics. The Friedman statistical test results are also presented in Table A4 for comparative purposes.
Tables 10 and A5 present the final and statistical results obtained by the different methods for the pressure vessel engineering design problem, respectively. From these tables, the best result of the MGA method is better than the results of the other approaches. By comparing the statistical results, it is obvious that MGA has better performance in statistical analysis, especially the mean, and worst fitness values alongside the standard deviation. The Friedman statistical test results are also presented in Table A6 for comparative purposes.  The results of the welded beam design problem in Tables 11 and A7 show that MGA is capable of converging to better results than the other approaches. Although the maximum difference between the best results of MGA and the other approaches is only about 4%, MGA is capable of providing better statistical results, including the mean, worst fitness values alongside standard deviation. The Friedman statistical test results are also presented in Table A8 for comparative purposes. In Table A9, the final design of different methods and MGA for the three-bar truss design problem, including the obtained design variables, are presented. Table 12 displays the statistical results. Considering the results reported by previous researchers, it is clear that MGA yields very competitive results for this engineering design problem. MGA determined the best optimum value that has been reported thus far, according to the literature, for the considered design example. It also should be noted that the statistical results, including the mean and standard deviation, for the MGA are much better than the results of other approaches. The Friedman statistical test results are also presented in Table A10 for comparative purposes.
The results of the multiple disk clutch brake design problem solved by MGA and other approaches are summarized in Table A11 [47,50,65,68]. The statistical results are presented in Table 13. Accordingly, MGA is capable of calculating very impressive results compared to the other metaheuristics. The maximum and minimum differences between the results of MGA and other metaheuristics are about 49% and 24%, which demonstrates the capability of this algorithm in dealing with multiple disk clutch brake design problem. In addition, the statistical results, including the mean and worst fitness values, demonstrate that MGA can yield extremely better results than the other approaches. The Friedman statistical test results are also presented in Table A12 for comparative purposes.  The final results of different metaheuristics in dealing with the planetary gear train design problem, one of the most important and well-established constrained optimization problems, are presented in Tables 14 and A13. By comparing the best results of MGA with other approaches, it can be concluded that MGA can yield outstanding results. Although MGA is also capable of providing better statistical results for the mean and worst fitness values alongside standard deviation results cannot be compared since they have yet to be reported in the literature. The Friedman statistical test results are also presented in Table A14 for comparative purposes. For the step-cone pulley engineering design problem, the final results of different metaheuristics are presented in Table A15, and the statistical results are provided in Table 15. By comparing the best results, it can be concluded that MGA can yield very impressive results for this constrained engineering problem. The maximum difference between the mean results of MGA and other approaches is about 31%. The Friedman statistical test results are also presented in Table A16 for comparative purposes. In Table 16, the comparative results of different metaheuristics in dealing with the hydrostatic thrust bearing design problem, including the obtained design and its related best optimum configuration, are presented. Table 17 displays the statistical results. It can be concluded that MGA is capable of converging to better results than the other approaches. The maximum difference between the best results of MGA is about 29%, where MGA yielded better statistical results for the mean, worst fitness values alongside the standard deviation than the other approaches. The Friedman statistical test results are also presented in Table 18 for comparative purposes. The optimum results of different metaheuristics in dealing with the ten-bar truss design problem are presented in Tables 19 and 20. By comparing the best results, it can be concluded that MGA is capable of outperforming other metaheuristics approaches. Until now, the best value obtained for this example was 529.25, which has been overcome by MGA with 529.12. This indicates the capability of MGA to provide remarkable results for some complex constrained design problems. The results of different methods for the rolling element bearing design problem are presented in Tables 21 and 22. It is clear that the best result of the MGA in this case is better than those of other approaches in the literature. Regarding the fact that this problem is a maximization optimization problem, MGA is also capable of providing remarkable statistical results.   Table 23 display the comparative and statistical optimization results of multiple optimization algorithms and MGA in dealing with the gear train design problem. It is obvious that MGA outranks the other optimization algorithms, Specifically, MGA obtained a perfect best of zero, which has not been obtained by other metaheuristics, confirming the capability of MGA to yield the lowest possible value in this case. The Friedman statistical test results are also presented in Table A18 for comparative purposes. Considering the steel I-shaped beam as one of the most well-formulated design problems, the final and statistical optimization results of multiple metaheuristics are presented in Tables 24 and 25, respectively. By comparing these optimum results, MGA outranked all other well-known algorithms that have been reported recently.  The final results of different metaheuristics for the piston lever design problem, a frequently occurring optimization problem, are presented in Table A19. The statistical results, including the best, mean, and worst fitness values alongside standard deviation, are presented in Table 26 for comparative purposes. Based on the results, MGA is capable of providing better statistical (mean, worst, and standard deviation of the results) and greatly outranked the other algorithms in terms of the best results. The Friedman statistical test results are also presented in Table A20 for comparative purposes. Considering the cantilever beam engineering design problem, the optimization results of the different optimization algorithms are all presented in Tables 27 and 28. By comparing the best results of these methods, it can be concluded that MGA is capable of achieving better results. According to the literature, recently-developed algorithms can yield 1.34, at best, for this example. Herein, we found that MGA is capable of providing even better result (1.33997) by conducting a better searching procedure. The statistical results of other optimization algorithms are not reported in the literature; thus, the remarkable results of MGA are beneficial for future works.  By comparing the p-values of the Friedman statistical test which are presented in the table of results by Chi-sq., it is concluded that for the piston lever design example, the lowest p-values is determined which demonstrates the fact that for this example, there are noticeable difference between the results of different approaches. However, the p-values of other examples are also near a mean of 9 which represents the stability of the conducted optimization runs and the statistical tests ( Figure 11).

Conclusions
In this paper, the Material Generation Algorithm (MGA) is presented as a new metaheuristic for different applications and various optimization problems. In this regard, 25 constrained design problems were considered to evaluate MGA, including 10 mathematically-constrained problems presented by the Competitions on Evolutionary Computation (CEC 2017) and 15 well-known engineering design problems. For comparative purposes, the best results of different metaheuristic algorithms, such as state-of-the-art me-

Conclusions
In this paper, the Material Generation Algorithm (MGA) is presented as a new metaheuristic for different applications and various optimization problems. In this regard, 25 constrained design problems were considered to evaluate MGA, including 10 mathematically-constrained problems presented by the Competitions on Evolutionary Computation (CEC 2017) and 15 well-known engineering design problems. For comparative purposes, the best results of different metaheuristic algorithms, such as state-of-the-art metaheuristics from CEC 2017, were selected for comparative purposes. Considering the results of MGA in dealing with the mathematical problems, it should be noted that this algorithm is capable of providing very competitive results in different dimensions. In addition, MGA yielded very impressive results in all of constrained engineering design problems compared to the previously reported algorithms. Specifically, the highest difference of about 24% between the best results of MGA and the best results reported thus far in the literature was found for the multiple disk clutch brake engineering design problem. For the three-bar truss design problem, MGA can provide very competitive results and, importantly, nearly the best results reported thus far. For the tension or compression spring, pressure vessel and rolling element bearing problems, the best results were higher for MGA than the best reported results.
While the proposed MGA has been proven to be a powerful method, different applications of this method are suggested for future research. It should be mentioned that the capability of this optimization approach can be controlled in dealing with some complex real-world and even computationally-expensive optimization problems. In addition, some other challenges, such as improving the general formulation of this method and hybridizing with other approaches, should be investigated properly.     4 Ray and Liew [58] 4.00 5 Gandomi et al. [45] 4.00 Chi-sq. 9.4000 Prob > Chi-sq.