CGO and SNS Optimization Algorithm for the Structures with Discontinuous and Continuous Variables

This study aims to find discontinuous and continuous approaches to reducing the size of planar truss structures with a specified shape and topology. The member's section area has assumed to be a decision variable, and the objective function is to minimize their weight. The member stresses and node displacements are the constraints that must maintain within the allowed limits for each condition. Chaos game optimization (CGO) and social network search (SNS) algorithms were used to optimize four well-known planar truss structures. In discontinuous-size cases, the results of the social network search (SNS) algorithm are the most cost-effective. However, the results of the chaos game optimization (CGO) algorithm are the most cost-effective in continuous-size cases.


Introduction
Developing a material structure that bears loads optimally is defned as structural optimization [1]. In engineering design, for example, the primary goal is to comply with basic standards while simultaneously achieving the best economic outcomes, i.e., selecting design parameters that meet all design criteria at the lowest possible cost. Optimization is a method for resolving problems like these. Commonly, the goal of structural optimization issues is to minimize objective function (typically the structural weight). Tis minimal design's value is limited to various problems based on factors such as displacements and stresses, as well as the structural member's minimum dimensions or cross-sectional areas. Te optimization problem may be classifed into two types according to whether the variables are continuous or discontinuous [2,3]: (i) Discontinuous optimization is an optimization problem considering discontinuous variables in which an object, such as an integer, must be found from a countable set.
(ii) A continuous optimization problem is one in which an optimal value must obtain from a continuous function. It might be limited or multimodal.
In recent decades, the ubiquity of trusses as structural systems has made their optimization an important engineering endeavor. Design ideas due sustainable, reliable, and cost-efective for the structure have been the impetus for the many developments of optimization procedures. Terefore, a trustworthy optimization strategy is always needed to tackle a structural optimization issue. Metaheuristic approaches are generic optimization tools that do not need a continuous objective function(s) or gradient information to direct the search process in this respect. Because of this positive quality, researchers in several sectors of science and engineering employ these approaches extensively [4][5][6][7][8][9]. Metaheuristics are also often applied to handle structural optimization problems, and studies have shown that these algorithms are efective in these situations [10][11][12][13][14][15]. In discontinuous design variables, most known optimization approaches are inefective because they interpret design variables as continuous. To deal with the discontinuous character of design variables, a few mathematical programming-based methodologies have been developed [2,3]. On the other hand, in real-world structures, the number of members, the degrees of freedom for every node, and the stifness matrix can compound the problem optimization. Terefore, in truss optimization, metaheuristic algorithms are more desirable because they can solve problems in fewer structural analyses. Te genetic algorithm (GA) and other novel heuristic algorithm-based optimization methodologies were proposed to achieve optimized designs for discontinuous structural systems because few mathematical programming technique-based approaches considered design variables as discontinuous. Many researchers have looked at GA-based optimization approaches, including Hajela and Lee [16], Rajeev and Krishnamoorthy [17], Camp et al. [18], Wu and Chow [19], Erbatur et al. [20], Pezeshk et al. [21]. In addition, various other studies are conducted in the feld of structure size optimization. Graph-based parameterization concept was employed by Giger and Ermanni [22]. Democratic PSO (DPSO) was used by Kaveh and Zolghadr [23]. A harmony search (HS) and a frefy algorithm (FA) were put to the test by Miguel and Fadel Miguel [24]. Also, Tejani et al. assessed the symbiotic organisms search (SOS) algorithm for structural optimization [25][26][27][28][29]. Te results show that the proposed adaptive SOS algorithm [25] is more efcient and reliable for the optimization of the structures under dynamic excitation. In the continuation of their research, a multiobjective adaptive symbiotic organisms search (MOASOS) is applied to truss optimization [27]. In this study, the weight of the truss and nodal displacement are objective functions. Te results of the MOASOS algorithm demonstrated that this algorithm can provide a competitive and better result than the previous studies. Also, structural optimization using the multiobjective plasma generation optimization (MOPGO) algorithm is studied by Kumar et al. [30].
A hybridized CSS and a sizable bang-huge crunch (CSS-BBBC) were introduced by Kaveh and Zolghadr [31]. Te adaptive ground fnite elements technique was introduced by Noilublao and Bureerat [32]. Truss constructions with discrete variables were optimized using the mine blast algorithm (MBA) by Sadollah et al. [33]. Te particle swarm optimizer (PSO), the heuristic particle swarm optimizer (HPSO), and the particle swarm optimizer with a passive congregation (PSOPC) were used by Li et al. [34,35] for the optimum design of pin-connected structures. Ho-Huu et al. [36] developed an adaptive elitist diferential evolution (AEDE) to optimize a truss with discrete design variables. Te centers and force formulation (CAFF) was used by Farshi and Alinia-Ziazi [37] for the sizing optimization of a truss. An artifcial bee colony (ABC) method was utilized by Hadidi et al. [38] to optimize the planar and space trusses. Eskandar et al. [39] used the water cycle algorithm (WCA) for the structural optimization of planar trusses. For size optimization of truss structures, Sangtarash et al. [40] presented a method known as the hybrid artifcial physics optimization and big bang-big crunch algorithm (HPBA). Kooshkbaghi and Kaveh [41] proposed the artifcial coronary circulation system algorithm (ACCSA) for sizing optimization of truss structures with continuous variables. An evolutionary algorithm based on the hybrid GA and PSO (HGAPSO) was developed by Kaveh and Malakoutirad [42] to solve force method-based simultaneous analysis and design problems for frame structures. An optimality criterion method (OCM) was used by Khan et al. [43] for large-scale structures. Yang et al. [44] proposed the computational efciency of accelerated particle swarm optimization combined with diferent chaotic maps for global optimization (CPSO3) with continuous and discrete variables. Azad and Hasancebi [45] presented an elitist self-adaptive step-size search (ESASS) for optimal sizing of truss structures based. Te enhanced colliding bodies optimization (ECBO) was introduced by Kaveh and Ilchi Ghazaan [46] for sizing optimization of truss structures with continuous and discrete variables. Talatahari and Azizi [47] have introduced a novel meta-heuristic approach termed the Chaos Game Optimization (CGO) Algorithm. In addition, Bayzidi et al. [48] recently introduced a novel meta-heuristic technique called Social Network Search (SNS). Te CGO algorithm's fundamental premise is based on certain chaos theory concepts. Te chaos game concept and fractal self-similarity difculties are considered. Te SNS algorithm replicates social network users' eforts to achieve more signifcant popularity by modeling choice emotions in expressing their thoughts. Te mixed continuous/discrete engineering optimization problems have been efectively solved using CGO and SNS, although they are fundamental methods. On the other hand, Singh et al. [49] present an improved version of the followthe-leader (iFTL) algorithm for the optimization of the truss problems.
Tis paper aims to determine the optimal discontinuous and continuous sizes for planar trusses of a certain design and topology so they may be constructed. It is necessary to conduct some case studies, both discontinuous and continuous in truss size, to assess the efcacy of each technique. For this aim, the very recent CGO and SNS algorithms are used in this paper. Te CGO and SNS algorithms will each be broken down and examined in further detail in the following sections. Finally, following the introduction of the trusses and the creation of the truss design optimization model, the outcomes of the two approaches are exhibited and compared with the previous study's conclusions. Te following is how the remainder of the paper is organized: in Section 2, the optimization problem is defned. In Section 3, the concepts of CGO and SNS are briefy described. Te continuous and discontinuous design variables used in CGO and SNS to optimize truss structure size are discussed in Section 4. Te CGO and SNS algorithms are used to optimize four wellknown truss designs in this section, with the results contrasted in discontinuous and continuous design variables. Finally, Section 5 provides a brief conclusion of the present study.

Description of the Truss Optimization Problems
Te objective of truss size optimization is to lower the structure's weight while adhering to stress and defection restrictions. As design variables, cross-sectional areas are chosen from a list of the allowable sections. Te problem can be formulated as follows: 2 Computational Intelligence and Neuroscience Subjected to: where the members of cross-sectional areas, denoted by the vector X, are taken into account (design variables), ρ i is the material density, W( X { }), and L i and X i are the weight of the truss, the length of members, and the cross-sectional area, respectively. z is the number of members, and n is the number of nodes. Te bottom displacement limit is denoted by δ min , the upper displacement limit is represented by δ max , and the displacement of node j is represented by δ j . Furthermore, the bottom limit of stress is represented by σ min , the upper limit stress is represented by σ max , and the stress of member p is represented by σ p . X p is the cross-sectional area of the member p that is considered from the allowable section. Te components in the allowable section list are ordered in ascending order.
For unconstrained optimization problems, CGO and SNS are presented. A penalty function method is used to deal with restrictions. In this method, the value of the objective function is determined by the sum of the constraint violation in this technique, and the constrained optimization problem is transformed into an unconstrained one using the following formula: where P is the unconstrained objective function, and ξ 1 and ξ 2 are the penalty function coefcients. In this study, ξ 1 and ξ 2 are both set to 1. K is the number of constraints, and Φ j is the number of constraints connected to j th constraint: Te lower and upper boundaries of the j th constraint are c j min and c j max , respectively, and the value of the j th constraint is c j .

Concepts of Chaos Game Optimization (CGO) and Social Network Search Algorithm (SNS)
3.1. CGO. Talatahari and Azizi [47] proposed the CGO algorithm. Te confguration of fractals utilizing chaos game approaches, as well as fractal self-similarity challenges, is examined in this program, which is based on chaos theory notions. Te CGO method is mathematically modeled using the fundamental concepts of fractals and chaos games. Tis mathematical model is predicated on the assumption that to complete the overall form of one, it is necessary to build several Sierpinski triangles by making use of seeds that are allowed to exist inside the search space. In this particular scenario, the production of fresh seeds inside a Sierpinski triangle is also used. Tree seeds are used to create a temporary triangle for each qualifying seed in the search area (X i ) [47]: (i) Te current position of the global best (GB) (ii) Te mean groups (MG i ), current position (iii) Te itch solution candidate's (X i ) position as the chosen seed Te following is the procedure for the frst seed: where X i is the i th solution candidate, GB is the global best so far discovered, and MG i is the average of some picked eligible seeds. Each of the β i and c i represents a random number of 0 or 1 for modeling the option of rolling a dice. At the same time, α i is the randomly generated factorial for modeling the mobility constraints of the seeds. Te second seed can move in a manner that is analogous to the movement of the frst seed in the direction of a point that is located on the connecting lines between the X i and the MG i . Despite this, the movement of the second seed is restricted by some factorials that were generated at random. Te following is the presentation of the second seed: where α i represents a random number of 0 or 1 for modeling the option of rolling a dice, and each of the β i and c i represents a random integer of 0 or 1 for modeling the mobility constraints of the seeds. Te remaining criteria are the same as for the frst seed. MG i came in third place. A random number-generating function that only produces two integers, 0 and 1, is used to demonstrate this. It is worth mentioning that the seed may migrate toward the connecting lines, X i and GB. Several random factorials, such as the following, may alternatively be used to accomplish this goal: A diferent procedure is used to create the fourth seed to execute the mutation phase in the position updates of the eligible seeds in the search space. Te seed's location is updated depending on specifc random changes in the Computational Intelligence and Neuroscience 3 selected decision factors. Te fourth seed can represent as follows: where k is a uniformly distributed random number in the range [0, 1], and R is a random integer between 1 and d. Four possible formulations for α i , which regulate the mobility limits of the seeds, are ofered to control and adapt the exploration and exploitation pace of the CGO algorithm: Rand is a uniformly distributed random number between 0 and 1, while δ and ε are random integers between 0 and 1. Te CGO algorithm's step-by-step approach is listed below, and the algorithm's pseudocode is shown in Figure 1.

SNS.
Te SNS method was proposed by Bayzidi et al. [48]. Tis algorithm is based on social networks, which are online platforms that allow people to connect electronically with one another. Users may learn about their favorite people's beliefs and opinions by following them on social media. Consequently, connecting with other network users may impact their opinions. Users are always looking for strategies to increase their network popularity, ensuring that connecting with and infuencing other users goes smoothly. Almost every metaheuristic algorithm employs a collection of procedures to generate new solutions. Te SNS algorithm generates a novel solution by employing one of four moods that resemble real-world social behavior. Te followings are the explanations and mathematical models for these operators (moods) [50].

Mood 1: Imitation.
Users may follow each other on social media, and when one publishes a new post, their followers will be notifed about the shared subject. Networks have become vital instruments for sharing information and ideas due to this property (propagation of views). Users of social media networks can keep track of their loved ones and celebrities. Tey will be aware of other people's responses if they follow the news. Tey will attempt to start a debate about the new event by imitating the viewpoint of another person if the new occurrence involves complex thoughts. Tis sentiment may be mathematically stated as follows [50]:

Mood 2: Conversation.
Users of social networks may electronically connect and discuss various topics. Individuals learn from one another and expand their understanding of events during the Talk, which takes the shape of private conversation. Users in conversation have a new perspective on events by hearing other people's perspectives. Lastly, because of the variations in ideas, they may construct a diferent picture of the problem, according to the following equation [50]:  Computational Intelligence and Neuroscience

Mood 3: Disputation.
Users discuss and defend their opinions on current events to others when in a disputation mood. Tis work is done on social media sites, such as in the comments and groups. Users may see various points of view from others in the comment area, which may be impacted by the stated reasons. Furthermore, users may form friendly relationships with one another, so they form a virtual group to share their viewpoints on a specifc topic. A random group of persons is identifed as a commentator or a member of a group to model this attitude, and the new afected perspective in the argument is as follows (12) [50]:

Mood 4: Innovation.
Users' ideas and experiences are sometimes refected in what they post. To put it another way, when a person considers a situation, he or she may be able to see it in a new light and get a better understanding of the problem's nature or gain a whole new perspective on it. A topic might have a variety of characteristics, each of which impacts the problem's comprehension. Consequently, modifying one of their ideas will afect the whole notion of the issue, resulting in a unique viewpoint. Tis notion is used to generate a new viewpoint via the innovation mood: where D is the total number of variables in the problem, and ub d and lb d are the lowest and highest values for the d th variable, whereas rand 1 and rand 2 are two random integers between 0 and 1. d is the d th variable selected between 1 and D. Te term n d new refers to a new viewpoint on the problem's d th dimension. x d j is an existing notion about the d th variable provided by another user (j th user who chose randomly and i ≠ j), which the i th user desires to change due to a new thought n d new . Finally, as x d inew , a new viewpoint of the dt h dimension will be formed. Te dimension x(x d inew ) will be formed as an interpolation dimension (x d inew ) is a dialogue between the new notion (n d new ) and the current (x d j ). Changing one dimension, x d inew creates a broad shift in the underlying notion and may be considered a new communication point of view. Te following model may be used to represent this process: Te SNS algorithm's step-by-step method is presented in Figure 2, along with the algorithm's pseudocode.

Numerical Examples
Te CGO and SNS are used in this part to solve discontinuous and continuous optimization benchmark problems using two well-known truss designs. Four wellknown planar truss structures were tested to ensure that the fndings were accurate and that the suggested algorithms were successful. To do the structural analysis, get the members' forces and node displacements, and optimize by using the CGO and SNS techniques, a program was built in the MATLAB software [51] programming environment. All runs are performed on a 64 bit computer with an Intel i7 (2.6 GHz) processor and 12 GB of RAM. Te population sizes of CGO and SNS were assumed to be 150 for the truss examples. (10 Members). In this study, the best confguration of a 10-bar planar truss is investigated, as shown in Figure 3. Table 1 contains the information that may be obtained on the design variables (nd � 10). Te number of degrees of freedom (DOF) in this case is eight. Tis truss is classifed into three sizes: loaded, discontinuous, and continuous in the following manner,

6-Node Planar Truss
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Te best designs presented so far are listed in Table 2, along with the study's results. As shown in Table 2, the SNS method yields signifcantly less weight than the CGO, and other algorithms are utilized by other researchers. Figures 4 and 5 depict the path of convergence of the CGO and the SNS algorithms for the six-node planar truss-case1. During the algorithm's execution, these diagrams show the structural weight loss process using the number of function evaluation (NFE).
In the discontinuous size situation, the outputs of the SNS algorithm are more cost-efective than the CGO algorithm, as shown in Figure 5. In addition, the statistical results of the SNS algorithm regarding average and standard deviation are signifcantly better than the CGO for the discontinuous size case. However, regarding analysis time (CPU) and convergence rate, the CGO is better than the SNS algorithm for the discontinuous size case.

Case 2.
Loads of p 1 and p 2 are 100 ("kips") and zero ("kips"), respectively, and the structure's members are chosen from the series' (X) interval as follows: Te study's conclusions are shown along with a summary of the best designs that have been made so far in Table 3. Te quantity of weight acquired from the CGO method is signifcantly smaller than that obtained from the SNS algorithm and other algorithms, as shown in Table 3. Figures 6 and 7 depict the path of convergence of the CGO and the SNS algorithms for the six-node planar truss-case 2. During the execution of the algorithm, these diagrams show the structural weight loss process using NFE.
Te outputs of the CGO algorithm are more costefective than the SNS method in the continuous-size situation, as shown in Figure 7. Furthermore, for the continuous-size situation, the statistical results of the CGO algorithm were superior to the SNS method regarding average and standard deviation, as well as analysis time (CPU) and convergence rate.

Case 3.
Loads of p 1 and p 2 are 100 ("kips") and 50 ("kips"), respectively, and the structure's members are chosen from the series' (X) interval as follows: Table 4 summarizes the fnest ideas provided so far and the study fndings. Te CGO technique yields considerably less weight than the SNS algorithm and other algorithms, as seen in Table 4. Figures 8 and 9 depict the convergence paths of the CGO and SNS algorithms for the six-node planar truss-case 3. Te structural weight reduction process in this form is tracked using the number of function evaluations, which demonstrates that the SNS and CGO are superior to other approaches in this regard.
Te outputs of the CGO algorithm are more cost-effective than the SNS method in the continuous-size situation, as shown in Figure 9. Furthermore, for the continuoussize situation, the statistical results of the CGO algorithm were superior to the SNS method regarding average and standard deviation, analysis time (CPU), and convergence rate. Table 3: Comparison of optimal results obtained for the six-node planar truss-case 2.

Design variables (in 2 )
Farshi and Alinia-Ziazi [37] Li et al. [35] Hadidi et al. [38] Eskandar et al. [39] Sangtarash et al. [40] Kooshkbaghi and Kaveh [41] Kaveh and Malakoutirad [42] Present study  (15 Members). Te second example of size optimization is a truss system with 15-bar, which is shown in Figure 10. Te design factors (nd � 15) are listed in A maximum number of 1000 iterations were set for comparison with other algorithms in the same circumstance. Table 6 compares the optimum design of CGO and SNS to PSO [34], PSOPC [34], HPSO [34], WCA [39], and MBA [33]. Table 6 shows that SNS achieved an optimum design value comparable to or better than other methods, whereas, in terms of the number of analyses, SNS and CGO are better than other methods. Figure 11 also shows the convergence route of the CGO and the SNS algorithms for an eight-node planar truss.
As shown in Figure 12, in the discontinuous-size situation, the outputs of the SNS algorithm are more cost-effective than the CGO method. Te statistical results of the SNS algorithm regarding average and standard deviation are signifcantly better than the CGO for the discontinuous size  Computational Intelligence and Neuroscience   case. However, the CGO outperforms the SNS algorithm regarding analysis time (CPU) and convergence rate. (17 Members). Te optimal design of a 17-bar planar truss is explored, as illustrated in Figure 13. Te information about the design variables (nd � 17) is presented in Table 7.

9-Node Planar Truss
Te cross-sectional areas of components are regarded as 17 sizing design factors, with the following minimum permissible values: Table 8 contains an overview of the fnest designs that have been provided so far, as well as the fndings of this study. Te amount of weight acquired from the CGO method is signifcantly smaller than the amount obtained from the SNS algorithm and other algorithms, as shown in Table 8. Figures 14 and 15 demonstrate the convergence paths of the CGO and SNS algorithms for a nine-node planar truss. Te NFE is used to monitor the structural weight reduction process in this form, which shows that the SNS and CGO in this term are better than other methods.
As shown in Figure 15, the outputs of the CGO method are more cost-efective than the SNS algorithm in the continuous-size situation. Also, the CGO algorithm got better statistical results for the continuous-size situation in terms of average, standard deviation, analysis time (CPU), worst weight, and convergence rate than the SNS technique. (45 Members). Te optimal design of a 45-bar planar truss, as illustrated in Figure 16, is evaluated in the last case. Te design factors (nd � 45) are listed in Table 9.

20-Node Planar Truss
For design reasons, the structure's components are divided into 23 groups according to Table 10, taking into account the symmetry of the structure. Tere is a lower limit for each size variable:  In the direction of X-axis and Y-axis |Δ j |⩽10 mmj � 3, 4, 5, 6, 7, 8Δ a � 10 mm Stress constraints: Table 10 contains a summary of the fnest ideas provided so far, as well as the fndings of this study. As shown in Table 10, the amount of weight gained by the CGO method is much less than that obtained by the SNS algorithm and other methods. Figure 17 and Figure 18 demonstrate the route of convergence of the CGO method and the SNS algorithm for the twenty-node planar truss. Te NFE is used to examine the structural weight reduction process in this form, which shows that the SNS and CGO in this term are better than other methods.
As shown in Figure 18, the outputs of the CGO algorithm are more cost-efective than the SNS method in the    −100(−445.374kN) Table 8: Comparison of optimal results obtained for the 9-node planar truss problem.

Summary Results.
To present a better comparison of the CGO and SNS algorithms, Table 11 shows the standard deviation, worst, best, mean, and CPU time of the outcomes. As previously stated, both algorithms are run 1500 times using a seed number and a population size of 150.

Conclusion
Te chaos game optimization approach and the social network search algorithm were used to analyze the continuous and discontinuous size optimization of planar trusses in this research. Te fndings of a computer analysis of four distinct kinds of planar trusses (6-node, 8-node, 9-node, and 20-node) that were evaluated under various degrees of discontinuous and continuous size and loading are presented in this study.
(1) Te solved cases demonstrate that the chaos game optimization algorithm and the social network search algorithm can solve the continuous and discontinuous size optimization problems and that these algorithms can fnd the best structural confguration faster than other methods. Other advantages of using the chaos game optimization algorithm with the social network search technique to solve structural optimization problems include a faster convergence rate, a better solution, and low computational efort. (2) In discontinuous sizes, the designs generated by the social network search algorithm are much more costefective than other designs, whereas, in continuoussize instances, the designs generated by the chaos