Multi-Objective Crystal Structure Algorithm (MOCryStAl): Introduction and Performance Evaluation

In many optimization problems, the main goal is to improve a single performance index in which a minimum or maximum value of this index fully reflects the quality of the response obtained from a system. However, in some cases, it is impossible to rely solely on a single index, so a multi-objective optimization problem with multiple performance indicators is considered where the values of all of them should be optimized simultaneously. The mentioned process requires a multi-objective optimization algorithm that can deal with the complexity of problems with simultaneous indexes. This paper presents the multi-objective version of a recently proposed metaheuristic algorithm called Crystal Structure Algorithm (CryStAl) which was inspired by the principles underlying the formation crystal structures. For the performance evaluation of this algorithm which is called MOCryStAl, the benchmark problems of the Completions on Evolutionary Computation (CEC) on multi-objective optimization, called CEC-09, are utilized. Some real-world engineering design problems are used to evaluate the efficiency of the proposed approach. The results demonstrate that the proposed methods can provide excellent results in dealing with the considered multi-objective problems.


I. INTRODUCTION
Optimization is the art of finding the best answer among a set of possible solutions under some predefined conditions. It is used for decision-making in various areas such as engineering, management, economics, and finance [1]- [4]. The complexity and interdependency of advanced engineering systems require an analyst with, at least, a general understanding of the system to assist in the optimization of production, laboratory, store, or service system. Besides, the interaction of the subsystems should be considered in the study of a system to ensure its integrity and optimality. In addition, the technical specifications and limitations of system components as well as existing uncertainties should be determined and considered when defining the sought-after goals. These goals often require multidisciplinary optimization and modeling The associate editor coordinating the review of this manuscript and approving it for publication was Dongxiao Yu .
Multi-objective optimization is an area of multi-criteria decision-making in which multiple objective functions need to be optimized simultaneously. Multi-objective optimization VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/  is also known by other names such as multi-objective programming, vector optimization, multi-criteria optimization, multi-attribute optimization, and Pareto optimization. Multiobjective optimization methods are used in many branches of science and engineering. These methods are particularly used to achieve optimal decisions in problems in which striking a balance between two or more conflicting goals is in perspective. In most engineering applications, processes, and systems, designers make decisions based on different conflicting goals. For example, in a typical vehicle design process, in addition to aiming to achieve the highest attainable performance, engineers make effort to minimize fuel consumption and environmental pollution at the same time.
In such cases, because more than one objective functions must be considered, it is necessary to consider the application of multi-objective optimization methods. The most important feature of such methods is that they provide system engineers and designers with more than one candidate solution (i.e. a possible answer to the problem) each of which will show the balance between the different objective functions. In recent decades, researchers have proposed the multiobjective versions of some well-known metaheuristic algorithms where the searching techniques of corresponding single-objective algorithms have been modified to deal with multiple objective functions. The multi-objective version of the Genetic Algorithm (GA) [29] as Non-Dominated Sorting Genetic Algorithm (NSGA), proposed by Srinivas and Deb, alongside the improved version of this method called NSGA-II [30], are amongst the primary multi-objective optimization methods in the area of evolutionary computation.
Furthermore, researchers in the field of artificial intelligence have proposed a range of multi-objective developments such as the Multiple Objective Particle Swarm Optimization (MOPSO) by Coello and Lechuga [31], Multi-Objective Evolutionary Algorithm (MOEA) by Zhang and Li [32], Multi-Objective Ant Colony Optimization (MOACO) by Alaya et al. [33], and Multi-Objective Simulated Annealing (MOSA) by Smith et al. [34]. Besides, the multi-objective versions of some other recently proposed metaheuristic algorithms have also been proposed, such as the multi-objective seagull optimization algorithm [35], multiobjective forest optimization algorithm [36], multi-objective whale optimization algorithm with differential evolution [37], multi-objective crow search algorithm [38], and Multi-objective Slap Swarm Algorithm (MSSA) [39].
Based on the Crystal Structure Algorithm (CryStAl) developed recently by Talatahari et al. [40], here we propose the multi-objective version of CryStAl, abbreviated as MOCryStAl, as a multi-objective metaheuristic algorithm. In this algorithm, the geometric principles of crystal Algorithm 1 Pseudo-Code of CryStAl [19] procedure structures, including the concepts of lattice and basis in the configuration of crystals, are in perspective. For the performance evaluation of this algorithm, the benchmark problems of the Completions on Evolutionary Computation (CEC) on multi-objective optimization, called CEC-09 [41], are utilized. Some real-world engineering design problems are used to evaluate the efficiency of the proposed approach. The results demonstrate that MOCryStAl is capable of providing excellent results in dealing with the considered multiobjective problems.

A. CRYSTAL STRUCTURE ALGORITHM (CryStAl)
Crystalline solids and their rich structural symmetries have inspired the conception and design of many man-made structures, mechanisms, and artworks [42]- [54]. The crystal structure algorithm (CryStAl) is a recently proposed metaheuristic algorithm in which the geometric principles of crystal structures are in perspective. As described in [40], the Bravais model is one of the frequently referenced configurations of crystals in which a periodic crystal structure is determined by an infinite lattice shape in which any lattice point is defined by the positions of related lattice points with a vector r = n i a i , where n i is an integer, a i is the shortest vector along the principal crystallographic directions, and i is the number of crystal corners. At first, the initialization process of the algorithm is formulated in which a random generation of the The objective function for all crystals obtain The non-dominated solution answers update The archive according to the found nondominated answers If the archive is full use the grid mechanism to delete the cur rent archives add the new answer to the archive endif If any of the new added solutions to the archive is located outside the hypercubes update the grids endif t = t + 1 end while; Return Archive end procedure initial candidate solutions are determined as follows:  where n is the number of initial candidate solutions or crystals, d is the dimension of the problem; x j i (0) is the initial position of the crystals; x j i,min and x j i,max are the minimum and maximum bounds in the considered problem; and ξ is a randomly generated number in the interval of [0,1].
Based on the concept of 'basis' in crystallography, all the crystals at the corners are considered as the main crystals, Cr main , determined randomly by considering the initially-created crystals (candidate solutions). It should be noted that the random selection process for each step is determined by omitting the current Cr. The crystal with the best configuration is determined as Cr b while the mean values of randomly-selected crystals are denoted by F c .
The position updating process for the crystals, in which the basis and lattice principles are utilized, are presented in Fig.1, where Cr new represents the new position, Cr old represents the old position, and r, r 1 , r 2 , and r 3 denote randomly-generated coefficient.
A boundary control flag is determined for considering the violating variables during the optimization process. In contrast, a maximum number of iterations or objective function evaluation can be considered for termination purposes. The pseudo-code of CryStAl is presented in Algorithm 1.

B. MULTI-OBJECTIVE CRYSTAL (MOCryStAl)
The Multi-Objective Crystal Structure Algorithm, abbreviated as MOCryStAl, is developed in this paper with the goal of solving some problems more effectively. In order to perform multi-objective optimization by MOCryStAl, we integrate three components. The employed components are very similar to those of (MOPSO) by Coello and Lechuga [31]. This algorithm has three capabilities for performing multi-objective optimizations as follows: The employed external archive effectively saves the best nondominated solutions obtained so far. The grid mechanism and selection leader component maintain the diversity of the archive during optimization. The probability of removing a solution rises in direct proportion to the hypercube's total number of solutions (segments). If the archive is full, the most crowded segments are chosen from which a solution is randomly deleted to make space for the new solution. In an exceptional circumstance, a solution is added outside the hypercubes. In this situation, all components are extended to include the new solutions. As a result, the components of several alternative solutions can be changed.
In the hope of finding a solution that is close to the global optimum, the search leaders guide the other search agents to the more promising areas of the search space. The solutions in a multi-objective search space, however, cannot easily be compared due to the Pareto optimality theories discussed above. The procedure for selecting leaders addresses this problem. As previously stated, there is a database of the best non-dominated solutions found recently. The leader selection component chooses the least congested VOLUME 9, 2021  regions of the search space and offers the best particles as non-dominated solutions. Each hypercube is chosen using a roulette-wheel methodology expressed by the following equation where C is a constant number higher than one and N is the variety of acquired Pareto optimal answers in the ith section. As can be concluded from Eq. (3), less congested hypercubes have a higher probability of suggesting new leaders. The chance of selecting a hypercube to select leaders from improves as the number of obtained solutions in the hypercube decreases. Importantly, the MOCryStAl approach is based on the CryStAl method for convergence. If one chooses a solution from the archive, the CryStAl approach will most likely be able to increase its already good consistency.
However, finding the Pareto optimal solutions among a large variety of responses is generally challenging. By using archive maintenance with the leader function collection, the problems can be solved. The pseudo-code of MOCryStAl is presented in Algorithm 2.

II. RESULTS AND DISCUSSION
In this section, the efficiency of the suggested technique is evaluated using performance measures and case studies, including unconstrained and constrained bi-and tri-objective mathematics (CEC-09) and real-world engineering design problems. The ability of multi-objective optimizers to handle problems with non-convexity and non-linearity is tested using these problems and mathematical functions. MATLAB 2021a was used to code the algorithm. On the computer, the following features are used to carry out the current work: the CPU is 2.3 GHz (an Intel Core i9 computer platform), and the RAM is 16 GB 2400 MHz DDR4 with Macintosh (macOS BigSur).

A. PERFORMANCE METRICS
The following four measures are used to assess the results of the algorithms:        front achieved by different many-objective algorithms.

B. EXPERIMENTAL SETUP
This section compares MOPSO, MSSA, and MOMVO to MOCryStAl, and the best figure of a collection of Pareto optimal is shown. Table 1 lists the initial settings for each algorithm. Each experiment had a maximum of 1000 iterations and 100 populations. As illustrated in Tables 2-4 and appendices A and B, this section puts the proposed algorithm to the test in 25 different case studies, including seventeen unconstrained and constrained bi-and tri-objective mathematical problems, as well as eight real-world engineering design problems.

1) DISCUSSION OF THE CEC-09 TEST FUNCTION
In Table 5, the comparative and statistical results of different multi-objective approaches alongside the proposed algorithm, namely MOCryStAl, are presented. The mentioned comparison metrics such as the GD, S, MS, and IGD are utilized in dealing with the CEC-09 problems. It turned out that MOCryStAl can outrank the other approaches, with regard to the IGD index, in five of the problems while the other methods, such as MSSA, also produce very competitive results.
Concerning GD, which are calculated in Table 6, MOCryS-tAl can provide acceptable results in most cases, including the five complex CEC-09 problems. The average results of UF8 in MOCryStAl have total differences of 90%, 80%, and 28% from MOPSO, MSSA, and MOMVO, respectively, which demonstrates the capability of the proposed   multi-objective algorithm in dealing with these sorts of complex problems.
Based on the results in Tables 7 and 8    fixed-dimension. In Table 9, the comparative results of the GD performance metric are demonstrated. The capability of the proposed methods in outranking the other multi-objective algorithms in four of these problems is demonstrated.
Regarding the IGD metric in dealing with the multimodal benchmark functions with fixed-dimension, the MOPSO, MSSA, and MOMVO are capable of providing best results for only one or two of the considered test problems. At the same time, the proposed MOCryStAl is capable of outranking the other methods in four of these complex test problems which demonstrate its capability in dealing with these sorts of complex problems.
In other metrics such as the MS and S indices, MOCryStAl is able to provide even more acceptable results than the IGD index, while this algorithm is capable of outranking the other methods in most of the considered test problems.
The true and obtained Pareto fronts for the ZDT and DTLZ problems by means of the proposed MOCryStAl method are illustrated in Fig. 4 and Fig. 5, respectively. It can be seen that this algorithm can create better solutions with a closer distance to the Pareto front.

3) DISCUSSION OF THE ENGINEERING PROBLEMS
Based on the fact that the novel multi-objective algorithms should be evaluated by means of some difficult realworld optimization problems, the capability of the proposed MOCryStAl is assessed through some other optimization problems. I these cases, the GD (Table 13), IGD (Table 14),   (Table 15), and S (Table 16) metrics are also utilized for performance evaluation while the results demonstrate the superiority of the proposed algorithm in most of the cases. In Fig. 6 and Fig. 7, the true and obtained Pareto fronts for these problems are represented through the proposed MOCryStAl method. This algorithm can create better solutions with a closer distance to the Pareto front.

III. CONCLUSION AND FUTURE WORK
This paper presented the multi-objective version of the Crystal Structure Algorithm (CryStAl) as a recently proposed metaheuristic algorithm inspired by some geometric principles of crystal structures including the lattice and basis in the configuration of crystals. For the performance evaluation of this algorithm, the benchmark problems of the Completions on Evolutionary Computation (CEC) on multi-objective optimization called CEC-09 were utilized. Some real-world engineering design problems were used to evaluate the efficiency of the proposed MOCryStAl approach. This paper demonstrates that MOCryStAl is capable of outranking the other approaches considering the IGD index in five of the CEC-09 problems while the other methods such as MSSA also produce very competitive results. Concerning GD, the average results of UF8 in MOCryStAl had total differences of 90%, 80%, and 28% from the results of MOPSO, MSSA, and MOMVO, respectively, which demonstrate the capability of the proposed multi-objective algorithm in dealing with such challenging problems. By considering the true and obtained Pareto fronts for the CEC-09, ZDT, and DTLZ problems, it is concluded that the proposed MOCryStAl method can create better solutions with a closer distance from the Pareto front.

APPENDIX A: ONSTRAINED MULTI-OBJECTIVE TEST PROBLEMS (USED IN THIS PAPER) CONSTR
There are two constraints and two design variables in this problem, which have a convex Pareto front.
Srinivasan and Deb [55] suggested a continuous Pareto optimal front for the following problem: Binh and Korn [56] were the first to propose this problem as follows: Osyczka and Kundu [57] proposed five distinct regions for the OSY test issue. There are also six constraints and six design variables to consider as follows:

APPENDIX B: CONSTRAINED MULTI-OBJECTIVE ENGINEERING PROBLEMS (USED IN THIS PAPER) THE FOUR-BAR TRUSS DESIGN PROBLEM
The 4-bar truss design problem [58], in which the structural volume (f 1 ) and displacement (f 2 ) of a 4-bar truss should be minimized, is a well-known problem in the structural optimization field. There are four design variables (x 1 − x 4 ) connected to the cross-sectional area of members 1, 2, 3, and 4, as shown in the equations below: Minimize

DISK BRAKE DESIGN PROBLEM
Ray and Liew [59] proposed the disc brake design issue, which has multiple constraints. Stopping time (f 1 ) and brake mass (f 2 ) for a disc brake are the two objectives to be minimized. The inner radius of the disc (x 1 ), the outer radius of the disc (x 2 ), the engaging force (x 3 ), and the number of friction surfaces (x 4 ) as well as five constraints, are shown in the following equations:

SPEED REDUCER DESIGN PROBLEM
The weight (f 1 ) and stress (f 2 ) of a speed reducer should be minimized in the speed reducer design issue, which is wellknown in the field of mechanical engineering [58] and [60]. There are seven design variables: gear face width (x 1 ), teeth module (x 2 ), number of teeth of a pinion (x 3 integer variable), distance between bearings 1 (x 4 ), distance between bearings 2 (x 5 ), the diameter of shaft 1 (x 6 ), and the diameter of shaft 2 (x 7 ), as well as eleven constraints.
Minimize In addition, he has been actively involved in the area of engineering optimization, especially in evolutionary algorithms. Using comprehensive finite element analyses, he has also investigated the use of different shaped sections in real steel frames. Recently, he has been engaged in research in the area of engineering optimization, especially in solving large-scale and practical structural design problems.
MAHDI AZIZI received the Ph.D. degree in structural engineering from the University of Tabriz. He has published many research papers in the fields of structural optimization, metaheuristic algorithms, and structural vibration control, where his main purpose has been developing and hybridizing metaheuristic algorithms for different applications. He has recently proposed some novel metaheuristic algorithms for optimization purposes. He teaches some basic and advanced courses of structural engineering in different universities. He also received a Postdoctoral Fellowship in structural optimization from the University of Tabriz.
SIAMAK TALATAHARI received the Ph.D. degree in structural engineering from the University of Tabriz. He subsequently joined the University of Tabriz (one of the top 10 universities in Iran), where he is currently an Associate Professor. His main research interests include data science (DS), machine learning (ML), and artificial intelligence (AI) and their applications in engineering. On the basis of his extensive research on the introduction, improvement, hybridization, and applications of DS/AI/ML methods for solving engineering problems, he has published over 120 refereed international journal articles, three edited books in Elsevier, and eight chapters in international books, with more than 8000 citations to his publications. He is honored by many academic awards, such as recognized as a Top One Percent Scientist of the World in the field of engineering and computer sciences for several years and one of the 70 Most Influential Professors in the history of Tabriz University. He was also recognized as the Distinguished Scientist of Iranian Forefront of Sciences, the most Prominent Young Engineering Scientist, a Distinguished Researcher, a Top Young Researcher, the most Acclaimed Professor, and a Top Researcher and Teacher. In addition, he has been selected to receive the TWAS Young Affiliateship from the Central and South Asia Region and Elite Awards from the Iranian Elites Organization. He served as the lead or a guest editor for some special issues of different journals.
POOYA SAREH received the B.Sc. degree (Hons.) in aerospace engineering from Sharif University of Technology, Tehran, Iran, the M.Sc. degree in mechanical engineering from the University of Sheffield, U.K., and the Ph.D. degree in engineering (structural mechanics) from the University of Cambridge, U.K., in 2014.
He subsequently worked as a Postdoctoral Research Associate in robotics at Imperial College London, U.K. He was a Lecturer in engineering design with the Department of Aeronautics, Imperial College London, from 2016 to 2018, a Visiting Lecturer at the Royal College of Art, U.K., and a Lecturer in industrial design and creative arts at the Division of Industrial Design, University of Liverpool, U.K., from 2018 to 2020. He is currently an Assistant Professor (a Lecturer in U.K. system) and the Director of the Creative Design Engineering Laboratory (Cdel), and the Programme Director of Advanced Mechanical Engineering and Design Programmes at the Department of Mechanical, Materials, and Aerospace Engineering, University of Liverpool.