# Social Network Search for Solving Engineering Optimization Problems

Authors: Hadi Bayzidi; Siamak Talatahari; Meysam Saraee; Charles-Philippe Lamarche
Journal: Computational Intelligence and Neuroscience
 (2021)

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## Abstract

In this paper, a new metaheuristic optimization algorithm, called social network search (SNS), is employed for solving mixed continuous/discrete engineering optimization problems. The SNS algorithm mimics the social network user’s efforts to gain more popularity by modeling the decision moods in expressing their opinions. Four decision moods, including imitation, conversation, disputation, and innovation, are real-world behaviors of users in social networks. These moods are used as optimization operators that model how users are affected and motivated to share their new views. The SNS algorithm was verified with 14 benchmark engineering optimization problems and one real application in the field of remote sensing. The performance of the proposed method is compared with various algorithms to show its effectiveness over other well-known optimizers in terms of computational cost and accuracy. In most cases, the optimal solutions achieved by the SNS are better than the best solution obtained by the existing methods.


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## Body

## 1. Introduction

Optimization is a part of the nature of human works, in which almost all of the human decisions go through an optimal process [1]. Optimization is embedded in the essence of the many branches of science, for example, a system with minimal energy in physics, the maximum profit in business, survival of the best organism in biology, and designing an engineering system that satisfies a set of constraints [2, 3]. Almost all of the engineering problems contain several nonlinear and complex constraints depending on the design criteria and safety rules.Over the last decades, various types of methods have been developed to solve constrained engineering problems. Two well-known groups of these methods are mathematical and metaheuristic methods. The idea of mathematical methods can be attributed to the development of the calculus of variations [4]. These methods employ the gradient of the objective function and constraints of the problem to find the optimal solution. The results of these methods are exact. However, these approaches search in a space near the starting point, which makes them sensitive to the initial starting point. In other words, just a correct starting point can lead to the global optima. In dealing with complex optimization problems, these methods are not suitable and frequently reach local solutions, and in some real applications, the gradient of the objective function and constraints is impossible to be calculated [5]. These drawbacks encourage researchers towards metaheuristic methods. Metaheuristic methods try to combine basic heuristic methods with randomization and rule-based theories, which are usually taken from natural phenomena such as evolution, swarm intelligence, and governing laws in different physics theories. Metaheuristic algorithms are approximate, but their results have high accuracy and are very close to the global optimum solution [6]. These methods are problem-independent, and the starting point does not determine the quality of the final solutions. Besides, these methods employ different operators to perform a global search in the space of the problem at an appropriate speed. These features have made them popular in recent decades. Also, these types of algorithms are among the most popular techniques that are employed for solving optimization problems in different fields, such as computer and electrical engineering [7], water, geotechnical and transport engineering [8], structure and infrastructures engineering [9], robotic [10], project and construction management [11], feature selection and data mining [12, 13], industrial and manufacturing [14], and medicine and biology [15].Glover [16] introduced the term metaheuristic firstly. The word metaheuristic is a combination of two old Greek words: meta and heuristic. The word heuristic has its origin in the old Greek work heuriskein, which means the art of discovering new strategies (rules) to solve problems. The suffix meta also is a Greek word that means “upper-level methodology” [17]. Almost every metaheuristic algorithm follows the general process shown in Figure 1. Algorithm steps cause fundamental differences in the performance of algorithms when faced with different problems. In the other words, algorithm steps represent the unique operators of each algorithm in which new solutions are generated. The operators of each algorithm refer to the optimal process of a particular phenomenon that those algorithms have imitated.Figure 1
The general form of optimization algorithms.According to the type of basic phenomena of each method, metaheuristic algorithms can be classified into four main categories: (1) evolutionary, (2) swarm intelligence, (3) physics-based, and (4) human-based algorithms. Evolutionary algorithms are motivated by natural evolution. Swarm intelligence algorithms model the natural behavior of animals in teamwork such as foraging and hunting. Physical phenomena and laws of science inspire physics-based algorithms, and finally, human-based algorithms mimic various optimal behaviors of humans in different conditions. Some of the most popular and novel algorithms are presented in Table1.Table 1
List of some popular and new metaheuristic algorithms.
AlgorithmReferencesEvolution strategy (ES)[18]Genetic algorithms (GA)[19]Ant colony optimization (ACO)[20]Particle swarm optimization (PSO)[21]Differential evolution (DE)[22]Cuckoo search (CS)[23]Bat algorithm (BA)[24]Charged system search (CSS)[25]Firefly algorithm (FA)[26]Eagle strategy (ES)[27]Krill herd algorithm (KH)[28]Flower pollination algorithm (FPA)[29]Grey wolf optimizer (GWO)[30]Optimization based on phylogram analysis (OPA)[31]Whale optimization algorithm (WOA)[32]Developed swarm optimizer (DSO)[33]Stochastic paint optimizer (SPO)[34]Chaos game optimization (CGO)[35, 36]Atomic orbital search (AOS)[37, 38]Material generation algorithm (MGA)[39]Crystal structure algorithm (CryStAl)[40]Social network search (SNS)[41]Each of these algorithms can behave differently when dealing with different problems, so that one particular algorithm may not solve some problems. Therefore, it is necessary to create a new high-performance optimization algorithm that is able to solve more types of problems. Novel metaheuristic methods are developed to find the optimal solution for complex and large-scale problems in less time than previous ones, with higher accuracy. These aims are satisfied by developing more robust algorithms that have a better ability to search the space of problems to find a better solution. In addition, this property arises from the right balance between exploration and exploitation of the proposed algorithm. Exploitation means searching around the current best solutions, while exploration tries to explore the search space more efficiently, often by randomization [42].In addition to inventing novel algorithms based on natural phenomena, developing new algorithms using hybridizing the operators of the current methods or modifying them is a hot topic in the field of metaheuristic algorithms. Firefly algorithm with chaos [43], hybrid particle swarm optimizer, ant colony strategy and harmony search scheme (HPSACO) [44], island-based cuckoo search with highly disruptive polynomial mutation (iCSPM) [45], quantum-behaved developed swarm optimizer (QDSO) [46], hybrid self-assembly with particle swarm optimization (SAPSO) [47], upgraded whale optimization algorithm (UWOA) [48], fuzzy controllers with slime mould algorithm (SMAF) [49], and hybrid invasive weed optimization-shuffled frog-leaping (SFLA-IWO) [50, 51] are some the newly developed hybrid or modified optimization algorithms.Social network search (SNS) algorithm is a robust metaheuristic algorithm that was innovated as a novel method for solving optimization problems, and its results showed that it is capable of outperforming various methods in dealing with different optimization problems [42]. The SNS algorithm simulates human behavior as users of a social network. Social network users can influence the opinions of other users on the network by sharing their views, opinions, and thoughts. Each of the users can also share their thoughts on the network and affect other people’s opinions. In other words, the SNS simulates particular moods where the views and opinions of users are influenced under their communications. This paper investigates the performance of the SNS algorithm using 14 constrained engineering optimization problems and a real application in the field of satellite image segmentation. The obtained results are compared with other optimizers in terms of best function value and number of function evaluations, and in most cases, the solutions of the SNS are better than the other methods.The rest of this paper is organized as follows. Section2 describes the SNS algorithm and constraint-handling technique. The performance of the SNS algorithm in solving optimization problems is evaluated against other methods in Section 3. Finally, conclusions are given in Section 4.


## 2. Materials and Methods

This section presents the general framework of the SNS algorithm and the utilized constraint-handling technique for solving engineering optimization problems.


### 2.1. Social Network Search (SNS)

Human beings are social species, which always try to communicate with each other. Social networks are virtual tools that were created for this goal with the advent of technology. The proposed SNS algorithm simulates the interactive behavior among users in social networks to achieve more popularity. Social networks are platforms where users can interact virtually with other users. Interacting with other users of the network may affect their opinions. The process of interacting with and influencing other users of the network goes through an optimal process so that users are always trying to increase their level of popularity on the network.The main property of social networks is that users can follow other persons, as shown in Figure2. If a user shares a new post, that person’s followers may be informed about the shared topic. This feature (fast propagation of views) has turned networks into a powerful tool for promoting information and ideas, which is due to having high connectivity of users in the social networks, as demonstrated in Figure 2.Figure 2
A general model for the social network.In social networks, users’ viewpoint can be affected by other views in different moods containing imitation, conversation, disputation, and innovation. One of these moods that look like real-world social behavior creates the new solution in the SNS algorithm. Description and mathematical modeling of these moods are as follows [42].


#### 2.1.1. Mood 1: Imitation

Imitation means that the views of other users are attractive, and usually, users try to imitate each other in expressing their opinions as follows:(1)Xinew=Xj+rand−1,1×R,R=rand0,1×r,r=Xj−Xi,where Xj represents the vector of the jth user’s view, which is selected randomly (i≠j), Xi is the view vector of the ith user, and rand−1,1 and rand0,1 are two random vectors in intervals [−1, 1] and [0, 1], respectively. In this mood, the new solution will be generated according to imitation space (Figure 3(a)), and this space is created using the radii of shock and popularity. The shock radius (R) reflects the amount of influence of the jth user, and its magnitude is considered as a multiple of r. The value of r shows the popularity radius of the jth user, which is calculated based on the difference in the opinions of the ith and jth users. Also, the final effect of the shock radius is reflected by multiplying its value to a random vector in the interval of [−1, 1], in which if the components of the random vector are positive, the shared view will be agreed with the jth opinion and vice versa. The process of the imitation mood is illustrated in Figure 3(a). As can be seen, using equation (1), the space of imitation will be formed, and then a point as a new view will be shared on the network.Figure 3
The process of different moods in the SNS algorithm. (a) Imitation. (b) Conversation. (c) Disputation. (d) Innovation.
(a)(b)(c)(d)


#### 2.1.2. Mood 2: Conversation

In social networks, users can communicate with each other and benefit from their conversation about different issues according to(2)Xinew=Xk+R,R=rand0,1×D,D=signfi−fj×Xj−Xi,where rand0,1 is a random vector in the interval [0, 1], Xj and Xk are the vectors of two randomly selected positions somehow i≠j≠k, and fi and fj are the objective functions of Xi and Xj, respectively. This mood models a state in which users learn from each other and increase their information about events. In conversation, users find a sight about a specific issue through other views, and finally, due to the differences in opinions, they can draw a new vision of the issue under discussion. Xk demonstrates the vector of the issue, which is randomly chosen to speak about it; also, R is the effect of chat, which is based on the differences of opinion and represents the change in their beliefs about the issue. D is the difference between the views of users. In addition, sign. is the sign function and signfi−fj determines the moving direction of Xk by comparing fi and fj. The process of this decision mood is shown in Figure 3(b). As can be seen, the user’s view about the issue changes as a result of conversations with the jth user. The changed opinion is considered as a new view to share with others. Changing the user’s view about the events is considered as the relocation of the events.


#### 2.1.3. Mood 3: Disputation

The disputation mood imagines a state where users explain their views about events to others and defend their opinion. In this situation, users see different views from other persons and may be influenced by the expressed reasons. The new affected view in disputation is as follows:(3)Xinew=Xi+rand0,1×M−AF×Xi,M=∑tNrXtNr,AF=1+roundrand,where rand0,1 is a random vector in the interval [0, 1] and Nr is a random integer between 1 and Nuser (Nuser is the population size or network size). Nr determines the number of users who participate in disputation, and participants are selected randomly. AF is the admission factor, which indicates the insistence from users on their opinion in discussions with other persons and is a random integer that can be either 1 or 2. round. is a function that rounds its input to the nearest integer number. The process of disputation mood is shown in Figure 3(c).


#### 2.1.4. Mood 4: Innovation

Sometimes a topic that users share on the networks comes from their new experiences and thoughts. In this mood, the new solution is developed by changing a randomly selected variable ofXi as follows:(4)Xinew=x1,x2,x3,…,xinewd,…,xD,xinewd=t×xjd+1−t×nnewd,nnewd=lbd+rand1×ubd−lbd,t=rand2,where d is the dth variable that is selected randomly in the interval [1, D], D is the number of problem variables, and rand1 and rand2 are two random numbers in the interval [0, 1]. Also, ubd and lbd are maximum and minimum values for the dth variable. nnewd represents the new idea about the selected dimension. xjd is the current idea about the dth variable presented by another user (jth user which selected randomly and i≠j) that ith user wants to change it because of the new idea (nnewd). Innovation models a state in which a person thinks about a specific issue, perhaps looks at that issue in a novel way, and is able to understand the nature of that problem more accurately or can find a completely different view about it. A particular subject may have distinct features, and each of them affects the understanding of the problem. As a result, by changing the idea about one of them (xid), the general concept of the subject will change, and a novel view will be achieved. xinewd is a new insight into the issue under consideration from the dth viewpoint and is replaced with the current view (xid). The outline of the construction of the new view is shown in Figure 3(d).In the SNS algorithm, only one of the predefined four models, so-called decision moods, will be selected and executed randomly for each user in each iteration of the algorithm. In other words, all of the moods described here are real-world behaviors of users in social networks, and it seems that the correct assumption is that only one of these moods occurs at a specific time (iteration) for each users. As a result, the chance of occurrence of these moods is considered to be small by using a random procedure with a uniform distribution.An important point is that the SNS algorithm has no specific parameter to be fine-tuned, and this feature is one of its superiority. In the third mood of the SNS algorithm, AF is defined as a random integer, and it can be considered a deterministic parameter whose value is generated randomly. To utilize the SNS algorithm, it just needs to determine the number of users (population size) and the maximum number of evaluations or iterations. The flowchart of the SNS algorithm is illustrated in Figure4. Besides, the MATLAB code of the SNS algorithm for solving engineering optimization problems is available in [52].Figure 4
The flowchart of the SNS algorithm.


### 2.2. Constraint-Handling Technique

Most engineering optimization problems aim to find optimal solutions under special conditions, which are usually based on resource limitations, design principles, and safety requirements. These special conditions are called constraints, and the main aim of constraint optimization is to find a feasible solution. A constrained optimization problems can be formulated as(5)minimize:fX,X=x1,x2,…,xd,subject to:giX≤0,i=1,2,…,ng,hjX=0,j=1,2,…,nh,lbk≤xk≤ubk,k=1,2,…,d,where the function fX is an objective function, X is a vector of solution variables (and they can be continuous, discrete, or mixed), giX and hjX are inequality and equality constraints, respectively, in which ng is the number of inequality constraints and nh is the number of equality constraints, d represents the number of variables, and lbk and ubk are the minimum and the maximum permissible values for the kth variable, respectively. The point worth mentioning is that a feasible solution satisfies all constraints. In contrast, infeasible solutions do not satisfy at least one constraint [53]. Also, in dealing with equality constraints, hjX=0 is replaced with an inequality hjX−δ≤0, where δ is a positive tolerance value. Another approach for handling the equality constraints is to replace hjX=0 with two inequality constraints of the type hjX≤0 and hjX≥0. This strategy facilitates convergence to the optimum design [54]. Therefore, all of the constraints can be transformed into inequality constraints.Metaheuristic algorithms cannot solve constraint optimization problems, directly. Therefore, it needs to equip them with an additional tool for handling constraints. A group of methods was developed for this proposal which is called constraint-handling techniques (CHTs). The CHTs enable the optimization methods to handle the objective function and constraints, simultaneously. CHTs are grouped into five categories: (1) penalty functions, (2) special representations and operators, (3) repair algorithms, (4) separation of objectives and constraints, and (5) hybrid methods [53]. The first method, penalty functions, is a simple and standard procedure for handling constraints. In the penalty function approach, a penalty term is added to the objective function, and then a constrained optimization problem is transformed into an unconstrained one. A penalty function can be formulated as follows:(6)FX=fX+PX,where FX is the fitness function, which expresses the unconstrained state of the constrained problem, fX is the objective function, and PX is the penalty term that denotes the violation of constraints and is calculated as follows:(7)PX=∑i=1ngαi×max0,giX+∑j=1nhβj×max0,hjX−δ,where max0,giX and max0,hjX−δ represent the value of the violations of the solutions according to the ith inequality and jth equality constraints, respectively. Also, αi and βj are penalty factors for these constraints, respectively. The magnitude of penalty factors affects the quality of answers, and the suitable penalty factors are problem-dependent.To solving constrained optimization problems, metaheuristic methods and CHT should be liked for recognizing feasible search space. Then, the optimizer should try to find the optimum or a near-optimum solution in the feasible region. Therefore, in each iteration of an algorithm, the fitness of the population is evaluated according to objective and constraint(s), and based on the calculated fitness function, the next generation of the population will be generated. In other words, the algorithm will identify the problem’s search space using the fitness of the current population.


## 2.1. Social Network Search (SNS)

Human beings are social species, which always try to communicate with each other. Social networks are virtual tools that were created for this goal with the advent of technology. The proposed SNS algorithm simulates the interactive behavior among users in social networks to achieve more popularity. Social networks are platforms where users can interact virtually with other users. Interacting with other users of the network may affect their opinions. The process of interacting with and influencing other users of the network goes through an optimal process so that users are always trying to increase their level of popularity on the network.The main property of social networks is that users can follow other persons, as shown in Figure2. If a user shares a new post, that person’s followers may be informed about the shared topic. This feature (fast propagation of views) has turned networks into a powerful tool for promoting information and ideas, which is due to having high connectivity of users in the social networks, as demonstrated in Figure 2.Figure 2
A general model for the social network.In social networks, users’ viewpoint can be affected by other views in different moods containing imitation, conversation, disputation, and innovation. One of these moods that look like real-world social behavior creates the new solution in the SNS algorithm. Description and mathematical modeling of these moods are as follows [42].


### 2.1.1. Mood 1: Imitation

Imitation means that the views of other users are attractive, and usually, users try to imitate each other in expressing their opinions as follows:(1)Xinew=Xj+rand−1,1×R,R=rand0,1×r,r=Xj−Xi,where Xj represents the vector of the jth user’s view, which is selected randomly (i≠j), Xi is the view vector of the ith user, and rand−1,1 and rand0,1 are two random vectors in intervals [−1, 1] and [0, 1], respectively. In this mood, the new solution will be generated according to imitation space (Figure 3(a)), and this space is created using the radii of shock and popularity. The shock radius (R) reflects the amount of influence of the jth user, and its magnitude is considered as a multiple of r. The value of r shows the popularity radius of the jth user, which is calculated based on the difference in the opinions of the ith and jth users. Also, the final effect of the shock radius is reflected by multiplying its value to a random vector in the interval of [−1, 1], in which if the components of the random vector are positive, the shared view will be agreed with the jth opinion and vice versa. The process of the imitation mood is illustrated in Figure 3(a). As can be seen, using equation (1), the space of imitation will be formed, and then a point as a new view will be shared on the network.Figure 3
The process of different moods in the SNS algorithm. (a) Imitation. (b) Conversation. (c) Disputation. (d) Innovation.
(a)(b)(c)(d)


### 2.1.2. Mood 2: Conversation

In social networks, users can communicate with each other and benefit from their conversation about different issues according to(2)Xinew=Xk+R,R=rand0,1×D,D=signfi−fj×Xj−Xi,where rand0,1 is a random vector in the interval [0, 1], Xj and Xk are the vectors of two randomly selected positions somehow i≠j≠k, and fi and fj are the objective functions of Xi and Xj, respectively. This mood models a state in which users learn from each other and increase their information about events. In conversation, users find a sight about a specific issue through other views, and finally, due to the differences in opinions, they can draw a new vision of the issue under discussion. Xk demonstrates the vector of the issue, which is randomly chosen to speak about it; also, R is the effect of chat, which is based on the differences of opinion and represents the change in their beliefs about the issue. D is the difference between the views of users. In addition, sign. is the sign function and signfi−fj determines the moving direction of Xk by comparing fi and fj. The process of this decision mood is shown in Figure 3(b). As can be seen, the user’s view about the issue changes as a result of conversations with the jth user. The changed opinion is considered as a new view to share with others. Changing the user’s view about the events is considered as the relocation of the events.


### 2.1.3. Mood 3: Disputation

The disputation mood imagines a state where users explain their views about events to others and defend their opinion. In this situation, users see different views from other persons and may be influenced by the expressed reasons. The new affected view in disputation is as follows:(3)Xinew=Xi+rand0,1×M−AF×Xi,M=∑tNrXtNr,AF=1+roundrand,where rand0,1 is a random vector in the interval [0, 1] and Nr is a random integer between 1 and Nuser (Nuser is the population size or network size). Nr determines the number of users who participate in disputation, and participants are selected randomly. AF is the admission factor, which indicates the insistence from users on their opinion in discussions with other persons and is a random integer that can be either 1 or 2. round. is a function that rounds its input to the nearest integer number. The process of disputation mood is shown in Figure 3(c).


### 2.1.4. Mood 4: Innovation

Sometimes a topic that users share on the networks comes from their new experiences and thoughts. In this mood, the new solution is developed by changing a randomly selected variable ofXi as follows:(4)Xinew=x1,x2,x3,…,xinewd,…,xD,xinewd=t×xjd+1−t×nnewd,nnewd=lbd+rand1×ubd−lbd,t=rand2,where d is the dth variable that is selected randomly in the interval [1, D], D is the number of problem variables, and rand1 and rand2 are two random numbers in the interval [0, 1]. Also, ubd and lbd are maximum and minimum values for the dth variable. nnewd represents the new idea about the selected dimension. xjd is the current idea about the dth variable presented by another user (jth user which selected randomly and i≠j) that ith user wants to change it because of the new idea (nnewd). Innovation models a state in which a person thinks about a specific issue, perhaps looks at that issue in a novel way, and is able to understand the nature of that problem more accurately or can find a completely different view about it. A particular subject may have distinct features, and each of them affects the understanding of the problem. As a result, by changing the idea about one of them (xid), the general concept of the subject will change, and a novel view will be achieved. xinewd is a new insight into the issue under consideration from the dth viewpoint and is replaced with the current view (xid). The outline of the construction of the new view is shown in Figure 3(d).In the SNS algorithm, only one of the predefined four models, so-called decision moods, will be selected and executed randomly for each user in each iteration of the algorithm. In other words, all of the moods described here are real-world behaviors of users in social networks, and it seems that the correct assumption is that only one of these moods occurs at a specific time (iteration) for each users. As a result, the chance of occurrence of these moods is considered to be small by using a random procedure with a uniform distribution.An important point is that the SNS algorithm has no specific parameter to be fine-tuned, and this feature is one of its superiority. In the third mood of the SNS algorithm, AF is defined as a random integer, and it can be considered a deterministic parameter whose value is generated randomly. To utilize the SNS algorithm, it just needs to determine the number of users (population size) and the maximum number of evaluations or iterations. The flowchart of the SNS algorithm is illustrated in Figure4. Besides, the MATLAB code of the SNS algorithm for solving engineering optimization problems is available in [52].Figure 4
The flowchart of the SNS algorithm.


## 2.1.1. Mood 1: Imitation

Imitation means that the views of other users are attractive, and usually, users try to imitate each other in expressing their opinions as follows:(1)Xinew=Xj+rand−1,1×R,R=rand0,1×r,r=Xj−Xi,where Xj represents the vector of the jth user’s view, which is selected randomly (i≠j), Xi is the view vector of the ith user, and rand−1,1 and rand0,1 are two random vectors in intervals [−1, 1] and [0, 1], respectively. In this mood, the new solution will be generated according to imitation space (Figure 3(a)), and this space is created using the radii of shock and popularity. The shock radius (R) reflects the amount of influence of the jth user, and its magnitude is considered as a multiple of r. The value of r shows the popularity radius of the jth user, which is calculated based on the difference in the opinions of the ith and jth users. Also, the final effect of the shock radius is reflected by multiplying its value to a random vector in the interval of [−1, 1], in which if the components of the random vector are positive, the shared view will be agreed with the jth opinion and vice versa. The process of the imitation mood is illustrated in Figure 3(a). As can be seen, using equation (1), the space of imitation will be formed, and then a point as a new view will be shared on the network.Figure 3
The process of different moods in the SNS algorithm. (a) Imitation. (b) Conversation. (c) Disputation. (d) Innovation.
(a)(b)(c)(d)


## 2.1.2. Mood 2: Conversation

In social networks, users can communicate with each other and benefit from their conversation about different issues according to(2)Xinew=Xk+R,R=rand0,1×D,D=signfi−fj×Xj−Xi,where rand0,1 is a random vector in the interval [0, 1], Xj and Xk are the vectors of two randomly selected positions somehow i≠j≠k, and fi and fj are the objective functions of Xi and Xj, respectively. This mood models a state in which users learn from each other and increase their information about events. In conversation, users find a sight about a specific issue through other views, and finally, due to the differences in opinions, they can draw a new vision of the issue under discussion. Xk demonstrates the vector of the issue, which is randomly chosen to speak about it; also, R is the effect of chat, which is based on the differences of opinion and represents the change in their beliefs about the issue. D is the difference between the views of users. In addition, sign. is the sign function and signfi−fj determines the moving direction of Xk by comparing fi and fj. The process of this decision mood is shown in Figure 3(b). As can be seen, the user’s view about the issue changes as a result of conversations with the jth user. The changed opinion is considered as a new view to share with others. Changing the user’s view about the events is considered as the relocation of the events.


## 2.1.3. Mood 3: Disputation

The disputation mood imagines a state where users explain their views about events to others and defend their opinion. In this situation, users see different views from other persons and may be influenced by the expressed reasons. The new affected view in disputation is as follows:(3)Xinew=Xi+rand0,1×M−AF×Xi,M=∑tNrXtNr,AF=1+roundrand,where rand0,1 is a random vector in the interval [0, 1] and Nr is a random integer between 1 and Nuser (Nuser is the population size or network size). Nr determines the number of users who participate in disputation, and participants are selected randomly. AF is the admission factor, which indicates the insistence from users on their opinion in discussions with other persons and is a random integer that can be either 1 or 2. round. is a function that rounds its input to the nearest integer number. The process of disputation mood is shown in Figure 3(c).


## 2.1.4. Mood 4: Innovation

Sometimes a topic that users share on the networks comes from their new experiences and thoughts. In this mood, the new solution is developed by changing a randomly selected variable ofXi as follows:(4)Xinew=x1,x2,x3,…,xinewd,…,xD,xinewd=t×xjd+1−t×nnewd,nnewd=lbd+rand1×ubd−lbd,t=rand2,where d is the dth variable that is selected randomly in the interval [1, D], D is the number of problem variables, and rand1 and rand2 are two random numbers in the interval [0, 1]. Also, ubd and lbd are maximum and minimum values for the dth variable. nnewd represents the new idea about the selected dimension. xjd is the current idea about the dth variable presented by another user (jth user which selected randomly and i≠j) that ith user wants to change it because of the new idea (nnewd). Innovation models a state in which a person thinks about a specific issue, perhaps looks at that issue in a novel way, and is able to understand the nature of that problem more accurately or can find a completely different view about it. A particular subject may have distinct features, and each of them affects the understanding of the problem. As a result, by changing the idea about one of them (xid), the general concept of the subject will change, and a novel view will be achieved. xinewd is a new insight into the issue under consideration from the dth viewpoint and is replaced with the current view (xid). The outline of the construction of the new view is shown in Figure 3(d).In the SNS algorithm, only one of the predefined four models, so-called decision moods, will be selected and executed randomly for each user in each iteration of the algorithm. In other words, all of the moods described here are real-world behaviors of users in social networks, and it seems that the correct assumption is that only one of these moods occurs at a specific time (iteration) for each users. As a result, the chance of occurrence of these moods is considered to be small by using a random procedure with a uniform distribution.An important point is that the SNS algorithm has no specific parameter to be fine-tuned, and this feature is one of its superiority. In the third mood of the SNS algorithm, AF is defined as a random integer, and it can be considered a deterministic parameter whose value is generated randomly. To utilize the SNS algorithm, it just needs to determine the number of users (population size) and the maximum number of evaluations or iterations. The flowchart of the SNS algorithm is illustrated in Figure4. Besides, the MATLAB code of the SNS algorithm for solving engineering optimization problems is available in [52].Figure 4
The flowchart of the SNS algorithm.


## 2.2. Constraint-Handling Technique

Most engineering optimization problems aim to find optimal solutions under special conditions, which are usually based on resource limitations, design principles, and safety requirements. These special conditions are called constraints, and the main aim of constraint optimization is to find a feasible solution. A constrained optimization problems can be formulated as(5)minimize:fX,X=x1,x2,…,xd,subject to:giX≤0,i=1,2,…,ng,hjX=0,j=1,2,…,nh,lbk≤xk≤ubk,k=1,2,…,d,where the function fX is an objective function, X is a vector of solution variables (and they can be continuous, discrete, or mixed), giX and hjX are inequality and equality constraints, respectively, in which ng is the number of inequality constraints and nh is the number of equality constraints, d represents the number of variables, and lbk and ubk are the minimum and the maximum permissible values for the kth variable, respectively. The point worth mentioning is that a feasible solution satisfies all constraints. In contrast, infeasible solutions do not satisfy at least one constraint [53]. Also, in dealing with equality constraints, hjX=0 is replaced with an inequality hjX−δ≤0, where δ is a positive tolerance value. Another approach for handling the equality constraints is to replace hjX=0 with two inequality constraints of the type hjX≤0 and hjX≥0. This strategy facilitates convergence to the optimum design [54]. Therefore, all of the constraints can be transformed into inequality constraints.Metaheuristic algorithms cannot solve constraint optimization problems, directly. Therefore, it needs to equip them with an additional tool for handling constraints. A group of methods was developed for this proposal which is called constraint-handling techniques (CHTs). The CHTs enable the optimization methods to handle the objective function and constraints, simultaneously. CHTs are grouped into five categories: (1) penalty functions, (2) special representations and operators, (3) repair algorithms, (4) separation of objectives and constraints, and (5) hybrid methods [53]. The first method, penalty functions, is a simple and standard procedure for handling constraints. In the penalty function approach, a penalty term is added to the objective function, and then a constrained optimization problem is transformed into an unconstrained one. A penalty function can be formulated as follows:(6)FX=fX+PX,where FX is the fitness function, which expresses the unconstrained state of the constrained problem, fX is the objective function, and PX is the penalty term that denotes the violation of constraints and is calculated as follows:(7)PX=∑i=1ngαi×max0,giX+∑j=1nhβj×max0,hjX−δ,where max0,giX and max0,hjX−δ represent the value of the violations of the solutions according to the ith inequality and jth equality constraints, respectively. Also, αi and βj are penalty factors for these constraints, respectively. The magnitude of penalty factors affects the quality of answers, and the suitable penalty factors are problem-dependent.To solving constrained optimization problems, metaheuristic methods and CHT should be liked for recognizing feasible search space. Then, the optimizer should try to find the optimum or a near-optimum solution in the feasible region. Therefore, in each iteration of an algorithm, the fitness of the population is evaluated according to objective and constraint(s), and based on the calculated fitness function, the next generation of the population will be generated. In other words, the algorithm will identify the problem’s search space using the fitness of the current population.


## 3. Results and Discussion

This section evaluates the performance of the SNS algorithm using 15 benchmark problems in various fields of engineering. One of these problems deals with the segmentation of satellite images in the field of remote sensing as a real application of metaheuristic algorithms in engineering. Each of these examples was run 30 times independently using the SNS algorithm, and the results are compared with different counterpart algorithms from the literature. In selecting the counterpart algorithm, an attempt has been made to use the results of newly developed methods.


### 3.1. Cantilever Beam

This problem is a structural engineering design example that is related to the weight optimization of a cantilever beam with a square cross section [55]. The beam is rigidly supported at one end, and a vertical force acts on the free node of the cantilever, as shown in Figure 5. The beam consists of five hollow square blocks with constant thickness, whose heights (or widths) are the decision variables, and the thickness is held fixed (here 2/3). This problem can be expressed analytically as follows:(8)minimize:fX=0.0624x1+x2+x3+x4+x5,subject to:gX=61x13+37x23+19x33+7x43+1x53−1≤0,variable range:0.01≤xi≤100,i=1,…,5.Figure 5
Schematic representation of cantilever beam.The best solutions for solving this problem obtained by the SNS and various methods are listed in Table2. It can be seen that the solution obtained by the SNS is better than that of the other methods. In addition, the SNS terminated after 12,000 evaluations. The statistical results of the SNS and other methods are listed in Table 3, and based on them, it can be seen that the SNS algorithm has obtained a more accurate answer in a smaller number of function evaluations (NFEs).Table 2
Best results of the cantilever beam design example.
AlgorithmVariablesConstraintx1x2x3x4x5g1XfXCS [18]6.00895.30494.50233.50772.1504−6.448E − 051.33999MFO [56]5.984875.316724.497333.513612.161624.182E − 091.33998ALO [57]6.018125.311424.488363.497512.15832−2.995E − 061.33995SOS [58]6.018785.303444.495873.498962.155641.393E − 041.33996SSA [59]6.015135.309304.495003.501422.152784.0578E − 091.33995SNS (present study)6.015455.310664.488003.505282.15428−3.2718E − 071.33995MFO: moth-flame optimization algorithm. ALO: ant lion optimizer. SSA: salp swarm algorithm.Table 3
Comparative results of SNS with other methods for the cantilever beam design problem.
AlgorithmWorstMeanBestSDNFEsSOS [58]NA1.339971.339961.1E − 515,000CGO [37]1.3406021.3400521.3399701.2245E − 04100,000AOS [39]1.4917111.3519541.3399570.02499743100,000MGA [40]1.34020111.34005261.33997566.99E − 05100,000SNS (present study)1.33995761.33995761.33995761.1102E − 1512,000


### 3.2. Optimal Design of I‐Shaped Beam

The other typical engineering optimization problem is the I-beam design problem, which aims to minimize the vertical deflection of the beam shown in Figure6. It simultaneously satisfies the cross-sectional area and stress constraints under given loads. The width of flange b=x1, the height of section h=x2, the thickness of the web tw=x3, and the thickness of the flange tf=x4 are variables of this problem. The maximum vertical deflection of the beam is fx=PL3/48EI when the length of the beam (L) and modulus of elasticity (E) are 5200cm and 523.104kN/cm2, respectively. The objective function and constraints of this problem are formulated as follows:(9)minimize:fX=5000x3x2−2x43/12+x1x43/6+2bx4x2−x4/22subject to:g1X=2x1x3+x3x2−2x4≤300,g2X=18x2×104x3x2−2x43+2x1x34x42+3x2x2−2x4+15x1×103x2−2x4x32+2x3x13≤56,variable range:10≤x1≤50,10≤x2≤80,0.9≤x3≤5,0.9≤x4≤5.Figure 6
A 3D view of beam design problem.Many optimizers have solved this nonlinearly constrained problem, and Table4 presents the best results of these methods. In addition, the statistical results for comparing the performance of the SNS methods are provided in Table 5. For this case study, the SNS needs 3600 function evaluations to reach these results, and it can be seen that the SNS performs superior compared to other methods.Table 4
Best results for the optimal design of I-shaped beam problem.
AlgorithmVariablesConstraintsx1x2x3x4g1Xg2XfXIARSM [60]79.9948.420.92.40.0869999−1.5245400.0131CS [18]80500.92.3216−0.012005−1.5700200.01307GWO [61]80500.92.3217−0.009059−1.5700710.0131EMGO-FCR [61]80500.92.3200−0.176000−1.5671790.0131SOS [58]80500.92.3217−0.000222−1.5702240.01307AEFA-C [61]79.967149.990.92.3164−0.560371−1.5595180.0131SNS (present study)80500.92.32170−1.5702280.0130741IARSM: improved adaptive response surface method. EMGO-FCR: ensemble meta model-based global optimization using fuzzy clustering. AEFA-C: artificial electric field algorithm.Table 5
Comparison of the results of SNS with the other methods for the I-shaped beam problem.
AlgorithmWorstMeanBestSDNFEsCS [18]0.013536460.01321650.01307470.00013455000SOS [58]NA0.01308840.01307414.0E − 55000AOS [39]0.01381400.01317880.01307411.555E − 04100,000SNS (present study)0.01307640.01307430.01307414.313E − 073600


### 3.3. Three-Bar Truss Design Problem

This case considers a 3-bar planar truss structure shown in Figure7. The volume of a statically loaded 3-bar truss is to be minimized subject to stress (σ) constraints on each of the truss members. The objective is to evaluate the optimal cross-sectional areas, A1=x1 and A2=x2. The mathematical formulation is given below:(10)minimize:fX=22x1+x2×l,subject to:g1X=2x1+x22x12+2x1x2P−σ≤0,g2X=x22x12+2x1x2P−σ≤0,g3X=12x2+x1P−σ≤0,l=100cm,P=2kN/cm3,σ=2kN/cm3,variable range:0≤x1,x2≤1.Figure 7
A schematic model of three-bar truss.The best results of different methods are presented in Table6. Also, Table 7 provides the statistical results of these algorithms. It can be seen that the best objective value of the SNS is equal or better than that of other methods. The required number of function evaluations (NFEs) for the SNS algorithm is 4800, which is much lower than that of other algorithms.Table 6
Best results of the three-bar truss design problem.
AlgorithmVariablesConstraintsx1x2g1Xg2Xg3XfXGA [62]0.7889150.4075699.6430E − 07−1.464873605−0.53512542263.8958857PSO [62]0.7886690.4082654.8650E − 07−1.464082376−0.535917137263.8958434ICA [62]0.7886250.4083898.4180E − 07−1.463941244−0.536057913263.8958452CS [18]0.788670.40902−2.9000E − 04−0.26853−0.73176263.9716WCA [5]0.7886510.4083160.0000E + 00−1.464024−0.535975263.895843GWO [62]0.7886480.4083253.3400E − 08−1.464014397−0.535985569263.8960063ALO [57]0.788662810.4082831−5.3170E − 12−1.464062005−0.53593799263.8958434MFO [56]0.788244770.40946697.7090E − 12−1.462717072−0.537282927263.8959796SSA [59]0.788665410.40827573.0000E − 10−1.464070360−0.53592963263.8958434WSA [62]0.7886830.408227−7.2400E − 07−1.464126180−0.53587454263.8958434SNS (present study)0.788684730.40822113.2978E − 08−1.4641325−0.5358675263.8958434WCA: water cycle algorithm. WSA: water strider algorithm.Table 7
Comparative results of SNS with other methods for the three-bar truss design problem.
AlgorithmWorstMeanBestSDNFEsGA [62]264.82080546263.96803663263.895885731.66862E − 0150,000PSO [62]264.58490296263.95741428263.895843411.36897E − 0150,000ICA [62]263.91413326263.89932689263.895845194.11693E − 0350,000CS [18]NA264.0669263.971569.00000E − 0515,000WCA [5]263.896201263.895903263.8958438.71000E − 055250GWO [62]263.90421778263.89795501263.896006311.61422E − 0350,000WSA [62]263.89743217263.89606687263.89584343.11960E − 0450,000CGO [37]263.8960068263.8958511263.8958432.51E − 05100,000AOS [39]263.895845263.895843263.8958438.26E − 09100,000SNS (present study)263.8958561263.8958462263.89584343.31056E − 064800


### 3.4. Tubular Column Design

This problem is an example of designing a uniform column of the tubular section to carry a compressive load at minimum cost. This problem has two design variables, the mean diameter of the columnd=x1 and the thickness of tube t=x2, which are shown in Figure 8. This column is made of a material with a yield stress of σy=500kgf/cm2 and a modulus of elasticity of E=0.85×1066kgf/cm2. The optimization model of this problem is given as follows:(11)minimize:fX=9.8x1x2+2x1,subject to:g1X=Pπx1x2σy−1≤0,g2X=8PL2π3Ex1x2x12+x22−1≤0,g3X=2.0x1−1≤0,g4X=x114−1≤0,g5X=0.2x2−1≤0,g6X=x28−1≤0,variable range:2≤x1≤14,0.2≤x2≤0.8.Figure 8
The 3D model of tubular column.According to the constraintsg1 and g2, the included stress in the column should be less than the buckling and yield stresses, respectively. Also, other constraints (g3, g4, g5, and g6) clamp the variables of the problem to the ranges of the variables. This problem was previously solved using various methods, and the best results of these methods and SNS are presented in Table 8. The SNS uses 1250 evaluations to solve this problem. In addition, the statistical results of some methods are reported in Table 9. According to these results, the SNS has found better results than other algorithms.Table 8
Best results of the tubular column example.
AlgorithmVariablesConstraintsx1x2g1Xg2Xg3Xg4Xg5Xg6XfXCS [18]5.451390.29196−0.0241−0.1095−0.633−0.610−0.315−0.63526.53217ISA [63]5.451156230.29196547−2.5E − 10−1.8E − 10−0.6331−0.6106−0.3149−0.63526.5313SNS5.451156230.29196547−2.6E − 10−1.8E − 10−0.6331−0.6106−0.3149−0.63526.4994969Table 9
Comparative results of SNS with other methods for the tubular column example.
AlgorithmWorstMeanBestSDNFEsISA [63]26.53226.53126.5311.70E − 043000CS [18]26.5397226.5350426.532171.93E − 0315,000FA [64]NA28.7426.522.083000AOS [39]26.6083136126.5316139926.531378281.0300E − 03100,000SNS (present study)26.4863709526.4863624926.486361472.2160E − 061250


### 3.5. Speed Reducer Design

In mechanical systems, one of the essential parts of the gearbox is the speed reducer, and it can be employed for several applications [65]. In this optimization problem (see Figure 9), the weight of the speed reducer is to be minimized with subject to 11 constraints. This problem has seven variables, face width b=x1, module of teeth m=x2, the number of teeth in the pinion z=x3, length of the first shaft between bearings l1=x4, length of the second shaft between bearings l2=x5, the diameter of first shafts d1=x6, and the diameter of second shafts d2=x7. The mathematical formulation of this problem is formulated as follows:(12)minimize:fX=0.7854x1x223.3333x32+14.9334x3−43.0934−1.508x1x62+x72+7.4777x63+x73+0.7854x4x62+x5x72,subject to:g1X=27x1x22x3−1≤0,g2X=397.5x1x22x32−1≤0,g3X=1.93x43x2x64x3−1≤0,g4X=1.93x53x2x74x3−1≤0,g5X=745x4/x2x32+16.9×106110x63−1≤0,g6X=745x5/x2x32+157.5×10685x73−1≤0,g7X=x2x340−1≤0,g8X=5x2x1−1≤0,g9X=x112x2−1≤0,g10X=1.5x6+1.9x4−1≤0g11X=1.1x7+1.9x5−1≤0,variable range:2.6≤x1≤3.6,0.7≤x2≤0.8,x3∈17,18,19,…,28,7.3≤x4,x5≤8.3,2.9≤x6≤3.9,5≤x7≤5.5.Figure 9
A schematic representation of speed reducer.This engineering problem has 11 constraints, seven nonlinear constraints and four linear inequality constraints, which are considered based on (1) bending stress of the gear teeth, (2) surface stress, (3) transverse deflections of the shafts, and (4) stresses in the shafts. The comparison of the best optimal solution with various optimization methods is given in Table10. The SNS method requires 3750 evaluations to find its solution. The statistical results of SNS and ten optimization methods are compared in Table 11. Among the compared optimization algorithms, the SNS has the lowest number of function evaluations while its results are better than those of the other methods.Table 10
Best results of the speed reducer design example.
CS [18]WCA [5]BA [66]ABC [67]APSO [68]SHO [69]SNSx13.501503.500003.500003.500003.501313.501593.50000x20.700000.700000.700000.700000.700000.700000.70000x317.0000017.0000017.0000017.0000018.0000017.0000017.00000x47.605007.300007.300017.300008.127817.300007.30000x57.818107.715327.715327.800008.042127.800007.71532x63.352003.350213.350213.350223.352453.351273.35021x75.287505.286655.286655.286685.287085.288745.28665g1X−0.07430−0.07392−0.07400−0.07392−0.12569−0.07434−0.07392g2X−0.19830−0.19800−0.19800−0.19800−0.28490−0.19836−0.19800g3X−0.43490−0.49917−0.49900−0.49917−0.34888−0.49980−0.49917g4X−0.90080−0.90464−0.90500−0.90156−0.89804−0.90162−0.90464g5X−0.00110−0.666850.000000.00000−0.66559−0.667170.00000g6X−0.000400.000000.00000−0.00063−0.00026−0.001170.00000g7X−0.70250−0.70250−0.70300−0.70250−0.68500−0.70250−0.70250g8X−0.000400.000000.000000.00000−0.00038−0.000450.00000g9X−0.58320−0.58333−0.58300−0.58333−0.58318−0.58314−0.58333g10X−0.08900−0.05133−0.05100−0.05133−0.14754−0.05111−0.05133g11X−0.013000.000000.00000−0.01070−0.04058−0.010560.00000fX3000.981002994.471072994.467102997.058413187.630492998.550702994.47107APSO: accelerated particle swarm optimization. SHO: spotted hyena optimizer.Table 11
Comparative results of SNS with other methods for the speed reducer design example.
AlgorithmWorstMeanBestSDNFEsCS [18]3009.003007.19973000.9814.96E + 00250,000ABC [67]NA2997.0584122997.0584120.00E + 0030,000WCA [5]2994.5055782994.4743922994.4710667.40E − 0315,150APSO [68]4443.017639003822.640624003187.630486003.66E + 0230,000SHO [69]3003.8892999.642998.55071.93E + 00NASSA3015.6626123005.5743772996.0217204.63E + 00NAWOA3233.5981243042.9150232996.6043404.08E + 01NACSS3106.2164513005.6589122996.4924784.86E + 00NACGO [37]2995.5049332994.4653972994.4436490.110282100,000FACSS [70]3006.4197462999.4137982996.37523764.82E + 00NASNS (present study)2994.47109922994.47106962994.47106627.00E − 063750FACSS: fuzzy adaptive charged system search.


### 3.6. Piston Lever

The main objective of this problem is to locate the piston components,H=x1, B=x2, D=x3, and X=x4, by minimizing the oil volume when the lever of the piston is lifted up from 0° to 45°, as shown in Figure 10. The formulation of this problem is given as follows:(13)minimize:fX=14πx32L2−L1,subject to:g1X=QLcosθ−R×F≤0,g2X=QL−x4−Mmax≤0,g3X=1.2L2−L1−L1≤0,g4X=x32−x2≤0,whereR=−x4x4sinθ+x1+x1x2−x4cosθx4−x22+x12,F=πPx324,L1=x4−x22+x12,L2=x4sinθ+x12+x2−x4cosθ2,θ=45°,Q=10,000lbs,L=240in,Mmax=1.8×106lbs in,P=1500psi,variable range:0.05≤x1,x2,x4≤500,0.05≤x3≤120.Figure 10
A model of piston lever problem.These inequality constraints consider the force equilibrium, the maximum bending moment of the lever, minimum piston stroke, and geometrical conditions. The best solutions obtained by SNS and some of the other algorithms are presented in Table12. In addition, the performance of the PSO [71], DE [71], GA [71], hybrid particle swarm optimization (HPSO) [71], HPSO with Q-learning [71], CS [18], ISA [63], CGO [37], MGA [40], AOS [39], and SNS is summarized in Table 13. The SNS algorithm obtains its results after 5000 evaluations, and its results are far better than those of other methods.Table 12
Best results of the piston lever example.
PSO [71]DE [71]GA [71]HPSO [71]CS [18]SNSx1133.3129.4250.0135.50.0500.050x22.442.433.962.482.0432.042x3117.14119.8060.03116.62120.000120.000x44.754.755.914.754.0854.083g1XNANANANA−1744.9120.000g2XNANANANA−600,000−600,000g3XNANANANA−117.185−117.187g4XNANANANA−4.50E − 04−9.74E − 12fX1221591611628.4278.412698349Table 13
Comparative results of SNS with other methods for the piston lever example.
AlgorithmWorstMeanBestSDNFEsPSO [71]29416612251.750,000DE [71]19918715914.250,000GA [71]21618516118.250,000HPSO [71]19718716213.450,000HPSO with Q-learning [71]16815112913.450,000CS [18]168.592040.23198.427159.055250,000ISA [63]610.6226.58.4111.212,500CGO [37]167.472808745.04865998.41281381367.24763100,000AOS [39]167.664986233.74127598.41914274293.46674724100,000MGA [40]167.473213432.46889258.41340665229.96370439100,000SNS (present study)167.472774724.31897438.41269834947.717926465000


### 3.7. Corrugated Bulkhead Design

This problem aims to minimize the weight of a corrugated bulkhead in a chemical tanker [72], in which the design variables are the width (x1), depth (x2), length (x3), and plate thickness (x4). The mathematical model of this optimization problem is given as follows:(14)minimize:fX=5.885x4x1+x3x1+x32−x22,subject to:g1X=−x4x20.4x1+x36+8.94x1+x32−x22≤0,g2X=−x4x220.2x1+x312+2.28.94x1+x32−x224/3≤0,g3X=−x4+0.0156x1+0.15≤0,g4X=−x4+0.0156x3+0.15≤0,g5X=−x4+1.05≤0,g6X=−x3+x2≤0,variable range:0≤x1,x2,x3≤100,0≤x4≤5.Tables14 and 15 compare the best and statistical results of SNS and other optimizers, respectively. According to these results, the SNS significantly improves the solution quality of other algorithms. In addition, the SNS method solves this problem after 3125 evaluations that is the lowest value among other methods.Table 14
Best results of the corrugated bulkhead design.
CS [18]VIGMM3 [61]AEFA-C [61]SNSx137.117949857.6923157.6927757.69230732x233.035021034.1476234.1329634.14762029x337.193947657.6923157.5529457.69230729x40.73062551.050001.050071.05g1X−23.3537699−0.25839−240.89634−240.6946226g2X−15.9738532−2.220E − 16−11.59051−1.46827E − 05g3X−0.00158548−9.769E − 15−6.864E − 05−5.80038E − 09g4X−0.00039992−5.551E − 16−2.250E − 03−6.20877E − 09g5X0.31937450.00000−7.581E − 05−6.50591E − 13g6X−4.15892660.68949−23.41997−23.544687fX5.8943316.84296−6.845846.842958019VIGMM3: vibration-based ideal gas molecular movement. AEFA-C: artificial electric field algorithm.Table 15
Comparative results of SNS with other methods for the corrugated bulkhead design.
AlgorithmWorstMeanBestSDNFEsFA [64]NA10.237.211.9512,000LF-FA [64]NA8.836.951.2612,000LS-LF-FA [64]NA7.446.860.6712,000AD-IFA [64]NA7.216.840.5812,000AOS [39]7.0669361867.0608083776.842958016.4911E − 04100,000SNS (present study)6.8430743996.8429798026.8429605152.0942E − 053125LF-FA: Levy flight firefly algorithm. LS-LF-FA: logarithmic spiral path flight firefly. AD-IFA: spiral-Levy flight firefly algorithm.


### 3.8. Design of Pressure Vessel

A cylindrical vessel is capped at both ends by hemispherical heads, as shown in Figure11. The objective is to minimize the total cost, including the cost of material, forming, and welding. This problem has four variables including the thickness of the shell Ts=x1, the thickness of the head Th=x2, the inner radius R=x3, and the length of the cylindrical section of the vessel, not including the head L=x4. In addition, x1 and x2 are integer multiples of 0.0625 in, while the other variables are continuous. The optimization problem can be expressed as follows:(15)minimize:fX=0.6224x1x3x4+1.7781x2x32+3.1661x12x4+19.84x12x3,subject to:g1X=−x1+0.0193x3≤0,g2X=−x2+0.00954x3≤0,g3X=−πx32x4−43πx33+1,296,000≤0,g4X=x4−240≤0,variable range:x1,x2∈1×0.0625,2×0.0625,3×0.0625,…,1600×0.0625,10≤x3,x4≤200.Figure 11
Schematic view of pressure vessel design.This problem has been used to evaluate the performance of many algorithms. Tables16 and 17 compare the best and statistical results of SNS and other algorithms, respectively. The SNS needs 6000 NFEs for solving this problem that is much lower than that of other algorithms.Table 16
Best results of the pressure vessel design.
G-QPSO [73]HPSO [74]CPSO [75]CDE [76]SAP [77]ABC [67]CS [18]EO [78]x10.812500.812500.812500.812500.812500.812500.812500.81250x20.437500.437500.437500.437500.437500.437500.437500.43750x342.0984042.0984042.0913042.0984040.3239042.0984542.0984542.09845x4176.63720176.63660176.74650176.63760200.00000176.63660176.63660176.63660g1X0.000000.000000.000000.00000−0.034320.000000.000000.00000g2X−0.03580−0.03580−0.00036−0.03580−0.05285−0.03588−0.03588−0.03588g3X−0.217903.12260−118.76870−3.70512−27.10585−0.00023−0.00005−0.00005g4X−63.36280−63.36340−63.25350−63.36230−40.00000−63.36340−63.36340−63.36340fX6059.72086059.71436061.07776059.73406288.74456059.71436059.71436059.71430MFO [56]GWO [31]WOA [33]APSO [68]IAPSO [68]NDE [79]MCEO [80]SNSx10.81250.8125000.8125000.81250.81250.81250.81250.81250x20.43750.4345000.4375000.43750.43750.43750.43750.43750x342.09844542.08918142.098269942.098442.098442.098445542.098445542.09845x4176.636596176.758731176.638998176.6374176.6366176.636595176.636596176.63660g1X−1.15E − 08−0.00017880−3.39E − 06−8.80E − 07−4.09E − 13−1.4E − 15−1.13E −  100.00000g2X−0.03588083−0.03296921−0.03588250−0.03588126−3.58E − 2−0.00035880−0.037564−0.03588g3X0.04023295−40.6168247−1.25270175−1.33153860−1.39E − 07−0.00000001−4.73E −  040.00000g4X−63.363404−63.241269−63.361002−63.3626−63.3634−0.63363404−63.3634−63.36340fX6059.71436051.56396059.74106059.724186059.714336059.714336059.71436059.71434G-QPSO: Gaussian quantum-behaved particle swarm optimization. CPSO: co-evolutionary particle swarm optimization. CDE: co-evolutionary differential evolution. SAP: self-adaptive penalty approach. EO: equilibrium optimizer. MCEO: multilevel cross entropy optimizer.Table 17
Comparative results of SNS with other methods for the pressure vessel design.
AlgorithmWorstMeanBestSDNFEsSAP [77]6308.156293.8436288.7457.41E + 003000HPSO [74]6288.6776099.93236059.71438.62E + 0181,000CDE [76]6059.7346085.236371.0464.30E + 01204,800CPSO [75]6363.80416147.13326061.07778.65E + 01200,000PSO [73]14076.3248756.68036693.72121.49E + 038000QPSO [73]8017.28166839.93266059.72094.79E + 028000G-QPSO [73]7544.49256440.37866059.72084.48E + 028000ABC [67]NA6245.3081446059.7147362.05E + 0230,000CS [18]6495.3476447.7366059.7143355.03E + 0215,000WOA [33]NA6068.056059.7416.57E + 016300APSO [68]7544.492726470.715686059.72423.27E + 02200,000EO [78]7544.49256668.1146059.71435.66E + 0215,000CGO [37]6330.9586856250.9573546247.6728191.07E + 01100,000SNS (present study)6410.0868866097.1002946059.7143359.28E + 016000


### 3.9. Design of Tension/Compression Spring

The tension/compression spring design problem is described in [81] for which the objective is to minimize the weight of a tension/compression spring, as shown in Figure 12. This problem is subject to constraints on minimum deflection, shear stress, surge frequency, limits on the outside diameter, and design variables. The design variables are the mean coil diameter D=x1, the wire diameter d=x2, and the number of active coils N=x3. The problem can be stated as(16)minimize:fX=x3+2x2x12,subject to:g1X=1−x23x371785x14≤0,g2X=4x22−x1x212566x2x13−x14+15108x12−1≤0,g3X=1−140.45x1x22x3≤0,g4X=x1+x21.5−≤0,variable range:0.05≤x1≤2,0.25≤x2≤1.3,2≤x3≤15.Figure 12
Tension/compression spring.Table18 compares the SNS with many optimization algorithms in terms of best optimization results, and Table 19 presents the statistical results of these algorithms. The SNS algorithm solves this problem in 9000 evaluations, and among the compared methods, just WOA [33] and MCEO [80] used a fewer number of evaluations, while their results are not as good as SNS.Table 18
Best results of tension/compression spring design.
AlgorithmVariablesConstraintsx1x2x3g1Xg2Xg3Xg4XfXCPSO [75]0.0517280.35764411.244543−8.25E − 04−2.52E − 05−4.051306−0.7270850.012674WCA [5]0.0516800.35650011.300400−1.60E − 13−7.90E − 14−4.053300−0.7278000.012665ABC [67]0.0517490.35817911.2037630.00E + 000.00E + 00−4.056663−0.7267130.012665SFOA [65]0.0518000.35900011.279000−3.24E − 06−3.58E − 07−4.060000−0.7260000.012700APSO [68]0.0525880.37834310.138862−1.55E − 04−8.33E − 04−4.089171−1.0690690.012700IAPSO [68]0.0516850.35662911.294175−1.97E − 10−4.64E − 10−4.053610−1.0916860.012665MFO [56]0.0519940.36410910.868421−4.10E − 063.04E − 06−4.068140−0.7226000.012667GWO [31]0.0516900.35673711.288850−7.91E − 05−7.51E − 06−4.053380−0.7277200.012666WOA [33]0.0512000.34520012.004000−5.60E − 04−3.00E − 05−4.027400−0.7357000.012676SHO [69]0.0511440.34375112.095500−3.30E − 041.16E − 05−4.025790−0.7367400.012674NDE [79]0.0516890.35671811.2889680.00E + 000.00E + 00−4.053785−0.7277280.012665SSA [59]0.0512070.34521512.004032−5.60E − 04−3.70E − 05−4.027410−0.7357200.012676SNS0.0515870.35426811.434058−1.37E − 08−3.18E − 10−4.048919−0.7294300.012665Table 19
Comparative results of tension/compression spring design.
AlgorithmWorstMeanBestSDNFEsCPSO [75]0.0129240.012730.01267475.20E − 05200,000HPSO [74]0.01271910.01270720.01266521.58E − 0581,000CDE [76]0.012790.0127030.01267022.70E − 05204,800PSO [73]0.0718020.0195550.0128571.17E − 0220,000QPSO [73]0.0181270.0138540.0126691.34E − 0320,000G-QPSO [73]0.0158690.0129960.0126666.28E − 0420,000WCA [5]0.0129520.0127460.0126658.06E − 0511,750ABC [67]NA0.0127090.0126651.28E − 0230,000APSO [68]0.0149370.0132970.01276.85E − 04120,000IAPSO [68]0.017828640.013676530.012665231.57E − 0320,000WOA [33]NA0.01270.01267633.00E − 044410MCEO [80]0.013509010.01271960.012660513.79E − 052000EO [78]0.0139970.0130170.0126663.91E − 0415,000SNS (present study)0.0127658730.0126847170.0126652462.38549E − 059000


### 3.10. Design of Welded Beam

This benchmark problem was introduced by Coello [77] and has been tackled by many researchers. As illustrated in Figure 13, the beam is under a vertical force. The goal of this problem is to find the minimum manufacturing cost of the welded beam. The problem is subject to seven constraints of stress, deflection, welding, and geometry. The variables are weld thickness h=x1, height l=x2, length t=x3, and bar thickness b=x4, as shown in Figure 13. The objective function can be mathematically be stated as(17)minimize:fX=1.10471x12x2+0.04811x3x414.0+x2,subject to:g1X=τX−τmax≤0,g2X=σX−σmax≤0,g3X=δX−δmax≤0,g4X=x1−x4≤0,g5X=P−PcX≤0,g6X=0.125−x1≤0,g7X=1.10471x12+0.04811x3x414.0+x2−5.0≤0,τX=τ′2+2τ′τ″x22R+τ″2,τ′=P2x1x2,τ″=MRJ,M=PL+x22,R=x224+x1+x322,J=22x1x2x224+x1+x322,σX⟶=6PLx4x32,δX⟶=6PL3Ex32x4,PcX⟶=4.013Ex32x46/36L21−x32LE4G,P=6000lb,L=14in,δmax=0.25in,E=30×106psi,G=12×106psi,τmax=13,600psi,σmax=30,000psi.variable range:0.1≤x1,x4≤2,0.1≤x2,x3≤10.Figure 13
Schematic of the welded beam structure with indication of design variables.Tables20 and 21 compare the best and statistical results of various optimizer in dealing with welded beam design problem. The SNS algorithm needs 9000 evaluations, which is the lowest NFE among other algorithms, while its results are better. In addition, the SNS algorithm has the lowest standard deviation that shows its robustness in solving this problem.Table 20
Best results of welded beam design.
BBO [62]PSO [62]ICA [62]WCA [5]ABC [67]EO [78]TEO [82]SSA [59]x10.18548600.2192920.2057990.2057280.2057300.20570000.205681000.2057000x24.31290003.4304163.4696343.4705223.4704893.47050003.472305003.4714000x38.43990308.4335599.0349509.0366209.0366249.03660009.035133009.0366000x40.23590200.2362040.2058060.2057290.2057300.20570000.205796000.2057000g1X−114.190503−0.1936420.0206880.0108320.0000000.0094704−0.60927688−0.8256803g2X−6.71716160.017873−0.0177630.119259−0.0000020.00149580.22468860−0.7106346g3X−0.0504000−0.016900−0.000007−0.0000010.000000−0.0000001−0.000115000.0000000g4X−3.2078633−3.276396−3.390414−3.390662−3.432984−3.3906592−3.39027259−3.3908203g5X−0.0604860−0.094292−0.080799−0.080728−0.080730−0.0807295−0.08068100−0.0807000g6X−0.2350000−0.235000−0.236000−0.235540−0.235540−0.2355403−0.23553783−0.2355381g7X−2639.74280−2668.48570−5.9519080.0576840.0000000.0008607−5.15644919−2.6033472fX1.91805501.8527201.7251351.7248561.7248521.72490001.725284001.7249100WSA [62]MCEO [80]CBO [62]GWO [62]SHO [69]WOA [33]CCSA [83]SNSx10.205730000.205729640.20572200.20567700.20556300.20539600.20570000.2057296x23.470489003.470488663.47041003.47089403.47484603.48429303.47020003.4704887x39.036624009.036623919.03727609.03855809.03579909.03742609.03620009.0366239x40.205730000.205729640.20573500.20573900.20581100.20627600.20570000.2057296g1X−0.025399590.00000363−0.0246881−0.0519206−1.4028657−21.5450190−0.00206100.0000000g2X−0.05312238−0.00002880−5.1106833−14.2018357−6.3837568−84.7713395−0.65695600.0000000g3X0.000000000.00000000−0.0000130−0.0000620−0.0002480−0.0008800−0.00000450.0000000g4X−3.39065616−3.39000000−3.3905161−3.3902412−3.3898697−3.3852837−3.3906250−3.3906591g5X−0.08073000−0.08072964−0.0807220−0.0806770−0.0805630−0.0803960−0.0807296−0.0807296g6X−0.23554035−0.23554032−0.2360000−0.2355503−0.2355421−0.2355825−0.2355406−0.2355403g7X−0.03155555−0.00001860−0.7536625−1.6632990−6.7607624−48.2829292−0.39068010.0000000fX1.724852001.724852301.72466301.72523201.72566101.73049901.72490001.7248523CCSA: conscious neighborhood-based crow search algorithm.Table 21
Comparative results of welded beam design.
AlgorithmWorstMeanBestSDNFEsPSO [62]3.8418452.6137851.852724.71E − 0150,000CPSO [75]1.7821431.7488311.7280241.29E − 02200,000HPSO [74]1.8142951.7490401.7248524.00E − 0281,000CDE [76]1.8241051.7681581.7334612.22E − 02204,800ICA [62]2.2377551.794331.7251351.10E − 0150,000BBO [62]3.6069332.6304121.9180554.11E − 0150,000FA [54]2.3455791.8786561.7312062.68E − 0150,000CSS [26]1.7594791.7396541.7248668.06E − 03NAWCA [5]1.7446971.7264271.7248564.29E − 0346,500ABC [67]NA1.7419131.7248523.10E − 0230,000MFO [62]1.7248521.7321091.9502413.40E − 0250,000SCA [62]1.7868631.8493641.9251623.47E − 0250,000APSO [68]1.9939991.8778511.7361937.61E − 0250,000SHO [69]1.7260641.7258281.7256612.87E − 04NASSA [62]1.7258861.8234262.2466381.28E − 0150,000GWO [62]1.7252321.726311.7284877.71E − 0450,000WOA [33]NA1.73201.7304990.02269900TEO [82]1.9311611.7680401.7252845.82E − 02NAEO [78]1.7367251.7264821.7248533.26E − 0315,000WSA [62]1.7250681.7249081.7248524.15E − 0550,000FACSS [70]1.7365651.7302231.7248536.363E − 7100,000T-CSS [84]1.7356561.7302121.7248602.00E − 06100,000SNS (present study)1.7250511.7248801.7248525.18E − 059000T-CSS: tribe-charged system search.


### 3.11. Design of Gear Train

The gear train design problem is an unconstrained discrete design problem in mechanical engineering and was introduced by Sandgren [85]. The purpose of this benchmark task is to minimize the gear ratio defined as the ratio of the angular velocity of the output shaft to the angular velocity of the input shaft. The number of teeth of gears nA=x1, nB=x2, nC=x3, and nD=x4 are considered as the design variables, and Figure 14 illustrates the 3D model of this problem. The mathematical formulation is provided as follows:(18)minimize:fX=16.931−x3x2x1x42,variable range:x1,x2,x3,x4∈12,13,14,…,60.Figure 14
The 3D model of gear train.The best results of 19 algorithms include the SNS are presented in Table22. It can be seen that all the algorithms find the optimum solution, except the PSO [62] and BBO [62]. In addition, the statistical results of 14 algorithms are compared in Table 23. The proposed method outperformed most of the other algorithms in terms of the mean, worst, SD, and NFEs.Table 22
Best results of the gear train design.
AlgorithmVariablesfXx1x2x3x4GA [62]491916432.70E − 12PSO [62]341320532.31E − 11ICA [62]431619492.70E − 12CS [18]431619492.70E − 12ABC [67]491619432.70E − 12MSFWA [86]491916432.70E − 12MBA [87]431619492.70E − 12BBO [62]532615512.31E − 11NNA [62]491619432.70E − 12GWO [62]491916432.70E − 12ISA [63]431916492.70E − 12APSO [68]431619492.70E − 12IAPSO [68]431619492.70E − 12MVO [88]431619492.70E − 12MFO [56]431916492.70E − 12ALO [57]491916432.70E − 12PSOSCALF [89]491916432.70E − 12WSA [62]431619492.70E − 12SNS (present study)431916492.70085714E − 12MBA: mine blast algorithm.Table 23
Comparative results of SNS with other methods for the gear train design.
AlgorithmWorstMeanBestSDNFEsGA [62]1.5247E − 081.6212E − 092.7009E − 123.2174E − 0950,000PSO [62]1.0222E − 067.9383E − 082.3078E − 111.8147E − 0750,000ICA [62]2.3576E − 098.0417E − 102.7009E − 127.7862E − 1050,000UPSO [90]N.A3.80562E − 082.700857E − 121.09E − 07100,000MFO [62]2.7265E − 087.5337E − 092.3078E − 119.3539E − 0950,000SCA [62]2.3576E − 098.8113E − 102.7009E − 126.4529E − 1050,000SSA [62]2.7265E − 081.9822E − 092.7009E − 124.5748E − 0950,000BBO [62]4.2018E − 074.5418E − 082.3078E − 117.2953E − 0850,000CBO [62]1.1173E − 082.1032E − 092.3078E − 112.4025E − 0950,000NNA [62]2.3576E − 096.1128E − 102.7009E − 126.1167E − 1050,000GWO [62]9.9216E − 103.3777E − 102.7009E − 124.0956E − 1050,000WOA [62]6.5123E − 099.6633E − 102.7009E − 121.1296E − 0950,000WSA [62]1.3616E − 091.6800E − 102.7009E − 123.8265E − 1050,000SNS (present study)1.36165E − 091.68012E − 102.700857E − 123.74894E − 1025,000


### 3.12. A Reinforced Concrete Beam Design

Amir and Hasegawa [91] presented a simplified optimization problem of designing a reinforced concrete beam, as shown in Figure 15. The beam is assumed to be simply supported with a span of 30 ft and subjected to a live load of 2000 lbf and a dead load of 1000 lbf, including the weight of the beam. The concrete compressive strength (σc) is 5 ksi, and the yield stress of the reinforcing steel (σy) is 50 ksi. The cost of concrete is $0.02/in2/linear ft, and the cost of steel is $1.0/in2/linear ft. To minimize the total cost of the structure, the area of the reinforcement As=x1, the width of the beam b=x2, and the depth of the beam h=x3 have to be determined. The structure should be proportioned to have a required strength based upon the ACI building code 318-77 as follows:(19)Mu=0.9Asσy0.8h1.0−0.59Asσy0.8bhσc≥1.4Md+1.7Ml,where Mu, Md, and Ml are the flexural strength, dead load, and live load moments of the beam, respectively. In this case, Md=1350in kip and Ml=2700in kip. The depth to width ratio of the beam is restricted to be less than or equal to 4. The optimization problem can be expressed as(20)minimize:fX=2.9x1+0.6x2x3,subject to:g1X=x2x3−4≤0,g2X=180+7.375x12x3−x1x2≤0,variable range:x1∈6,6.16,6.32,6.6,7,7.11,7.2,7.8,7.9,8,8.4,x2∈28,29,30,…,40,5≤x3≤10.Figure 15
A schematic view of reinforced concrete beam.It is clear that the variablesx1 and x2 are discrete, while x3 is continuous. The SNS method requires1000 evaluations to reach the optimum solution. Table 24 presents the results of optimum designs obtained by the SNS and other methods for this problem. In addition, the statistical results of FA [54], CS [18], AOS [39], and SNS are compared in Table 25. Obviously, the performance of the SNS method is better than other algorithms.Table 24
Best results of a reinforced concrete beam design.
GHN-EP [71]GA [92]FA [54]CS [18]ISA [63]AOS [39]SNSx16.327.206.326.326.326.326.32x234323434343434x38.6371808.04518.50008.50008.50008.58.5g1X−0.7745−2.8779−0.2241−0.2241−0.2241−0.22409498−0.22409411g2X−0.0635−0.0224000−1.00E − 075.27667E − 12fX362.00648366.1459359.2080359.2080359.2080359.2080359.2080GHN-EP: generalized Hopfield network-based extended penalty approach.Table 25
Comparative results of SNS with other methods for a reinforced concrete beam design.
AlgorithmWorstMeanBestSDNFEsFA [54]669.150460.706359.208080.7387025,000CS [18]NANA359.2080NA5000AOS [39]362.2535359.3306872359.20800.59614901100,000SNS (present study)362.634359.3222001359.20800.614985811000


### 3.13. Car Side Impact Design

On the foundation of the European Enhanced Vehicle-Safety Committee (EEVC) procedures, a car is exposed to a side impact, and the aim of this benchmark problem is minimizing the weight of the door using nine influence parameters including thicknesses of B-pillar inner=x1, B-pillar reinforcement =x2, floor side inner =x3, cross members =x4, door beam =x5, door beltline reinforcement =x6, roof rail =x7, materials of B-pillar inner =x8, floor side inner =x9, barrier height =x10, and hitting position =x11. Youn et al. [93] simplified this optimization problem’s analytical formulation and sped up computations using the global response surface methodology to approximate the structural weight and response to the impact. Figure 16 shows a model for the car side impact problem. According to the simplified models, the optimization problem is formulated as follows:(21)minimize:fX=1.98+4.90x1+6.67x2+6.98x3+4.01x4+1.78x5+2.73x7,subject to:g1X=1.16−0.3717x2x4−0.00931x2x10−0.484x3x9+0.01343x6x10−1≤0,g2X=46.36−9.9x2−12.9x1x2+0.1107x3x10−32≤0,g3X=33.86+2.95x3+0.1792x3−5.057x1x2−11.0x2x8−0.0215x5x10−9.98x7x8+22.0x8x9−32≤0,g4X=28.98+3.818x3−4.2x1x2+0.0207x5x10+6.63x6x9−7.7x7x8+0.32x9x10−32≤0,g5X=0.261−0.0159x1x2−0.188x1x8−0.019x2x7+0.0144x3x5+0.0008757x5x10+0.08045x6x9+0.00139x8x11+0.00001575x10x11−0.32≤0,g6X=0.214+0.00817x5−0.131x1x8−0.0704x1x9+0.03099x2x6−0.018x2x7+0.0208x3x8+0.121x3x9−0.00364x5x6+0.0007715x5x10−0.0005354x6x10+0.00121x8x11+0.00184x9x10−0.02x22−0.32≤0,g7X=0.74−0.61x2−0.163x3x8+0.001232x3x10−0.166x7x9+0.227x22−0.32≤0,g8X=4.72−0.5x4−0.19x2x3−0.0122x4x10+0.009325x6x10+0.000191x112−4≤0,g9X=10.58−0.674x1x2−1.95x2x8+0.02054x3x10−0.0198x4x10+0.028x6x10−9.9≤0,g10X=16.45−0.489x3x7−0.843x5x6+0.0432x9x10−0.0556x9x11−0.000786x112−15.7≤0,variable range:0.5≤x1,x2,x3,x4,x5,x6,x7≤1.5,x8,x9∈0.192,0.345,,−30≤x10,x11≤+30.Figure 16
A model of car side impact problem.The design of car side impact is also used as a benchmark problem to evaluate the performance of various methods. The best results of the SNS and these algorithms are presented in Table26. It should be noted that the results of other algorithms evaluated by Gandomi et al. [18, 54, 66] have different variables ranges, but in this paper, the variable ranges of [93] are utilized. Table 27 summarizes the statistical results obtained by the different optimization algorithms for the car side impact design problem. In this case, the SNS achieves its results with 20,000 NFEs. The CS method has better performance than all of the methods, according to the results presented in [18]. In addition, the best result of the SNS is better than those of other algorithms.Table 26
Best results of the car side impact design example.
PSO [54]DE [54]GA [54]CS [18]BA [66]SNSx10.500000.500000.500050.500000.500000.5x21.116701.116701.280171.116431.116701.115933208x30.500000.500000.500010.500000.500000.5x41.302081.302081.033021.302081.302081.302918991x50.500000.500000.500010.500000.500000.5x61.500001.500000.500001.500001.500001.5x70.500000.500000.500000.500000.500000.5x80.345000.345000.349940.345000.345000.345x90.192000.192000.192000.192000.192000.192x10−19.54935−19.5493510.3119−19.54935−19.54935−19.6388662x11−0.00431−0.004310.00167−0.00431−0.004311.49192E − 06g1X−0.617505−0.617505−0.43167−0.617423−0.61750525−0.61849499g2X−0.092707−0.092707−0.09839−0.092703−0.09270723−0.09273231g3X−0.100627−0.100627−0.13685−0.100625−0.10062705−0.10061401g4X−0.034209−0.034209−0.02699−0.034182−0.03420961−0.03418562g5X−4.278328−4.278328−3.77008−4.277761−4.27832783−4.28314394g6X−7.2838105−7.2838105−3.36103−7.282103−7.28381045−7.29404065g7X−0.0026365−0.0026365−0.000253.647E − 05−0.002636520g8X−2.425E − 05−2.425E − 05−8.6156E − 061.395E − 06−2.43E − 050g9X−0.9654269−0.9654269−0.58567−0.965154−0.96542696−0.96669762g10X−0.1666041−0.16660410.502506−0.166604−0.16660413−0.16739262fX22.8447422.8429822.8565322.8429422.8447422.84297965Table 27
Comparative results of SNS with other methods for the car side impact design example.
AlgorithmWorstMeanBestSDNFEsPSO [54]23.2135422.8942922.844740.1501720,000DE [54]24.1260623.2282822.842980.3445120,000GA [54]26.24057823.5158522.856530.6655520,000BA [66]23.2135422.8927322.844740.1738320,000FA [54]24.0662322.8937622.842980.1666720,000CS [18]23.2599822.8585822.842940.0761220,000WCA [94]23.3709337622.9751644222.843036481.9772E − 01NAMBA [94]23.4889421722.9364210422.843596401.5258E − 01NAER-WCA [94]24.4553128023.0699253422.843264613.502E − 01NAWOA [94]27.3608136824.8144861723.042162209.657E − 01NAFACSS [70]23.0536256222.9121235422.849074014.726E − 02NACSS [84]24.86356323.52326523.0073360.562345100,000T-CSS [84]23.80090422.90365322.8478480.078565100,000SNS (present study)23.1845493922.8814573622.842969540.10180121120,000ER-WCA: evaporation rate-based water cycle algorithm.


### 3.14. Cantilever Stepped Beam

This problem is a good benchmark to verify the capability of the optimization methods for solving continuous, discrete, and mixed variable structural design problems. This problem aims to minimize the volume of the beam. The width of segments (x1,x2,x3,x4,x5) and height of them (x6,x7,x8,x9,x10) are chosen to be the design variables. These ten variables are illustrated in Figure 17. Except for bending stress constraints, a specified aspect ratio is imposed such that the ratio of height to width in the segments of the beam is limited to be less than 20. The problem is formulated as follows:(22)minimize:fX=Dx1x6l1+x2x7l2+x3x8l3+x4x9l4+x5x10l5,subject to:g1X=6Pl5x5x102−σd≤0,g2X=6Pl5+l4x4x92−σd≤0,g3X=6Pl5+l4+l3x3x82−σd≤0,g4X=6Pl5+l4+l3+l2x2x72−σd≤0,g5X=6Pl5+l4+l3+l2+l1x1x62−σd≤0,g6X=Pl33E1I5+1I4+1I3+1I2+1I1−Δmax≤0,g7X=x10x5−20≤0,g8X=x9x4−20≤0,g9X=x8x3−20≤0,g10X=x7x2−20≤0,g11X=x6x1−20≤0,P=50000N,σd=14,000N/cm2,E=2×107N/cm2,Δmax=2.7cm,D=1.0,variable range:x1∈1,2,3,4,5,x2,x3∈2.4,2.6,2.8,3.1,1≤x4,x5≤5,x6∈30,31,32,…,65,x7,x8∈45,50,55,60,65,30≤x9,x10≤65.Figure 17
A 3D model of stepped cantilever beam.The first five constraints are related to the bending stresses in each beam segment that must be lower than the allowable limit (σd). Also, the deflection of the cantilever beam tip must be smaller than the limit deflection (Δmax). The aspect ratio between the height and width of the cross sections must be less than 20 and is applied by the last five constraints. Six of the variables (x1,x2,x3,x6,x7,x8) are discrete, and the rest of them (x4,x5,x9,x10) are continuous.The best and statistical results of the FA [54], thermal exchange optimization (TEO) [82], PSO [82], and SNS are presented in Tables 28 and 29, respectively. To solve this problem, the SNS needs 20,000 evaluations, which is lower than the NFEs of other compared methods, and at the same time, it outperforms all of them in terms of best, mean, worst, and SD.Table 28
Best results of the cantilever stepped beam.
FA [54]TEO [82]PSO [82]SNS (present study)x13353x23.13.13.13.1x32.62.62.62.6x42.2052.216290465311692.2045562.204555692x51.7501.769100853407631.7497571.749757012x660604760x755555555x850505050x944.09144.01442096962544.0911144.09111383x1034.99534.844577796611634.99514034.99514024g1X−1.83171281−33.157060−1.49E − 06−1.81899E − 12g2X−2.74873225−25.554426−2.38E − 06−1.81899E − 12g3X−153.846153−153.846154−153.8461538−153.8461538g4X−1203.41242−1203.412423−1203.412423−1203.412423g5X−111.111111−111.111111−419.1942055−111.1111111g6X−2.44816450−2.447883−2.44430588−2.448137336g7X−0.00285714−0.3037920−3.19744E − 14g8X−0.00408163−0.140500−4.54E − 09−3.55271E − 15g9X−0.76923076−0.769231−0.769230769−0.769230769g10X−2.25806451−2.258065−2.258064516−2.258064516g11X00−10.60fX63893.5263994.01891969393.43079663893.4307958715Table 29
Comparative results of SNS with other methods for the cantilever stepped beam.
AlgorithmWorstMeanBestSDNFEsFA [54]64262.9942064144.7531263893.52578175.9187950,000PSO [82]NANA69393.430796NA300,000TEO [82]NANA63994.018919NA300,000SNS (present study)63905.6952399763893.8874976063893.430795882.21E + 0020,000


### 3.15. Real Application of SNS in Remote Sensing (Segmentation of Satellite Images)

Image segmentation is an important topic in the field of remote sensing due to the increasing volume of collected images from satellites, airplanes, and other platforms [95]. Image segmentation aims to partition an image into several homogenous sections such that the combination of no two adjacent sections is homogenous. Segmentation is a difficult task due to poor resolution, unfavorable environmental conditions, ambiguous regions, and the presence of pixels with a weak local correlation in satellite images [96, 97]. Metaheuristic methods are proper tools that can deal with the difficulty of discovering the homogeneity measure in the images [98]. In this section, the SNS algorithm is employed for segmenting color satellite images, and then its results have been compared with different optimization algorithms.Thresholding techniques are the most common methods that are used as the objective function in image segmentation. In this study, Kapur’s entropy method is employed for this purpose, and its mathematical formulation is as follows:(23)H0=−∑i=0t1−1piω0log2piω0;H1=−∑i=t1t2−1piω1log2piω1;Hj=−∑i=tjtj+1−1piωjlog2piωj;Hm=−∑i=tmN−1piωmlog2piωm,where(24)ω0=∑i=0t1−1pi;ω1=∑i=t1t2−1pi;ωj=∑i=tjtj+1−1pi;ωm=∑i=tmN−1pi,where H0,H1,…,Hm are the entropy values of m + 1 various sections or classes, pi is the probability of the pixel intensity, and N is the total number of distinct intensity levels.The utilized image and its histogram patterns are shown in Figure18. This satellite image is taken from Pléiades Satellite Imagery to carry out the experimental study for segmentation. It can be seen that the histogram of the satellite image has a multimodal pattern, and it is very difficult to segment such an image that possesses immense information content.Figure 18
The utilized satellite image and its histograms (https://www.satpalda.com/gallery).The experiment was carried out 10 times to choose the best of each algorithm. Figure19 gives the segmented images for four different levels of thresholding (n) and compares the results of the SNS algorithm with PSO, Darwinian PSO (DPSO), ABS, CS, and CS algorithm with McCulloch’s method (CSMcCulloch). Also, Table 30 presents the comparison of threshold values between the SNS and other methods.Figure 19
Segmentation results of test images using metaheuristic algorithms for four different threshold levels.Table 30
Comparison of threshold values between algorithms.
nMethodsPSO [96]DPSO [96]ABC [96]540 82 123 164 20743 86 128 171 21140 82 121 162 20562 100 138 176 21765 104 142 180 21763 100 139 177 21754 93 131 171 21051 95 99 147 16656 95 134 173 214730 60 90 119 151 184 22033 66 97 129 160 191 22133 65 95 126 157 188 22052 80 105 133 161 189 21954 82 109 137 166 195 22554 81 109 137 166 195 22544 74 104 132 161 190 21943 73 103 132 161 190 21946 76 104 132 161 190 217922 44 68 94 120 148 175 203 23122 45 70 96 121 146 171 197 22422 48 73 99 125 152 179 204 23147 68 89 110 132 155 180 203 22944 63 82 103 125 147 171 197 22629 51 71 94 121 147 171 197 22821 48 75 101 128 155 181 206 23021 47 72 95 119 145 172 199 22521 48 75 101 126 151 177 201 2261117 38 59 78 98 119 140 162 183 207 23116 35 54 74 94 116 139 163 186 209 23116 32 50 70 91 113 136 159 182 206 23128 46 66 84 101 118 135 153 175 198 22444 63 82 101 119 138 157 176 194 214 23428 46 66 86 106 126 144 164 184 204 22518 36 56 77 99 121 141 161 181 202 22618 39 59 80 101 122 143 164 186 208 23018 37 56 76 96 118 140 162 185 208 230nCS [96]CSMcCulloch [96]SNS (present study)548 89 132 181 21045 88 130 172 20845 88 131 174 21549 91 133 179 21547 89 132 178 21243 57 71 97 17351 93 140 182 21948 88 133 173 20956 95 134 174 214742 81 110 145 182 221 24539 76 107 139 173 204 23235 66 96 126 157 189 22042 83 115 143 179 208 23940 77 108 142 174 203 23340 50 51 66 116 182 20643 77 109 143 181 217 24241 76 106 140 175 205 23446 75 103 132 161 191 223934 61 79 110 133 159 189 210 23325 51 75 103 130 156 182 209 23524 47 72 97 122 148 174 200 22525 56 77 109 137 161 191 210 24020 53 70 108 135 159 189 211 23812 12 14 48 62 115 157 186 19130 62 79 110 132 164 198 214 24529 61 75 109 130 163 194 213 24018 47 74 99 123 148 173 198 2261128 47 65 87 105 132 145 167 191 220 23923 45 64 84 105 129 145 164 186 211 23521 41 63 83 103 123 143 165 187 209 23225 48 69 91 110 132 147 165 191 214 24024 46 67 89 107 131 146 162 188 213 23811 12 40 61 83 100 110 180 217 219 25128 50 66 85 108 135 148 167 189 211 23522 48 65 85 106 130 147 163 185 212 23516 34 54 74 95 117 139 161 183 206 230Various criteria can be used for comparing the performance of metaheuristic algorithms in satellite image segmentation. Peak signal to noise ratio (PSNR) and feature similarity index (FSIM) are two quantitative performance metrics, which are utilized in this study [99]. PSNR measures the accuracy of the reconstructed image and is formulated as follows:(25)PSNR=10log1025521/mn∑j=1m∑k=1nXj,k−Xj,k'2,where mn is the size of image and X and X′ are the main and the processed images, respectively. In addition, FSIM is a criterion that calculates the similarity of the thresholded and original images as follows:(26)FISM=∑x∈XSLxPCmx∑x∈XPCmx,where SLx shows the similarity of images and PCmx is the phase congruency map. Tables 31 and 32 present the values of PSNR and FSIM metrics, respectively. According to these results, the SNS algorithm achieved better PSNR and FSIM values for all thresholding levels.Table 31
Comparison of PSNR between algorithms.
nPSO [96]DPSO [96]ABC [96]CS [96]CSMcCulloch [96]SNS520.388420.426720.470720.478120.481626.0094721.144921.188621.232721.15521.31626.1077922.247322.304822.334222.541822.749626.70821123.13923.231623.339923.544923.652727.6536Table 32
Comparison of FSIM between algorithms.
nPSO [96]DPSO [96]ABC [96]CS [96]CSMcCulloch [96]SNS50.94250.94270.94330.94020.94450.9665570.90970.91250.91410.90710.91740.9762690.9470.94790.94860.94410.95490.98178110.91750.91790.92140.91560.92760.98441This comprehensive study demonstrates that the developed SNS has competence among the other metaheuristic algorithms. Based on the results in solving classical engineering problems, it can be concluded that the SNS algorithm can perform superior to other algorithms in dealing with semi-real constrained problems. In addition, the image segmentation problem results show the SNS algorithm’s ability to solve real-world problems.


## 3.1. Cantilever Beam

This problem is a structural engineering design example that is related to the weight optimization of a cantilever beam with a square cross section [55]. The beam is rigidly supported at one end, and a vertical force acts on the free node of the cantilever, as shown in Figure 5. The beam consists of five hollow square blocks with constant thickness, whose heights (or widths) are the decision variables, and the thickness is held fixed (here 2/3). This problem can be expressed analytically as follows:(8)minimize:fX=0.0624x1+x2+x3+x4+x5,subject to:gX=61x13+37x23+19x33+7x43+1x53−1≤0,variable range:0.01≤xi≤100,i=1,…,5.Figure 5
Schematic representation of cantilever beam.The best solutions for solving this problem obtained by the SNS and various methods are listed in Table2. It can be seen that the solution obtained by the SNS is better than that of the other methods. In addition, the SNS terminated after 12,000 evaluations. The statistical results of the SNS and other methods are listed in Table 3, and based on them, it can be seen that the SNS algorithm has obtained a more accurate answer in a smaller number of function evaluations (NFEs).Table 2
Best results of the cantilever beam design example.
AlgorithmVariablesConstraintx1x2x3x4x5g1XfXCS [18]6.00895.30494.50233.50772.1504−6.448E − 051.33999MFO [56]5.984875.316724.497333.513612.161624.182E − 091.33998ALO [57]6.018125.311424.488363.497512.15832−2.995E − 061.33995SOS [58]6.018785.303444.495873.498962.155641.393E − 041.33996SSA [59]6.015135.309304.495003.501422.152784.0578E − 091.33995SNS (present study)6.015455.310664.488003.505282.15428−3.2718E − 071.33995MFO: moth-flame optimization algorithm. ALO: ant lion optimizer. SSA: salp swarm algorithm.Table 3
Comparative results of SNS with other methods for the cantilever beam design problem.
AlgorithmWorstMeanBestSDNFEsSOS [58]NA1.339971.339961.1E − 515,000CGO [37]1.3406021.3400521.3399701.2245E − 04100,000AOS [39]1.4917111.3519541.3399570.02499743100,000MGA [40]1.34020111.34005261.33997566.99E − 05100,000SNS (present study)1.33995761.33995761.33995761.1102E − 1512,000


## 3.2. Optimal Design of I‐Shaped Beam

The other typical engineering optimization problem is the I-beam design problem, which aims to minimize the vertical deflection of the beam shown in Figure6. It simultaneously satisfies the cross-sectional area and stress constraints under given loads. The width of flange b=x1, the height of section h=x2, the thickness of the web tw=x3, and the thickness of the flange tf=x4 are variables of this problem. The maximum vertical deflection of the beam is fx=PL3/48EI when the length of the beam (L) and modulus of elasticity (E) are 5200cm and 523.104kN/cm2, respectively. The objective function and constraints of this problem are formulated as follows:(9)minimize:fX=5000x3x2−2x43/12+x1x43/6+2bx4x2−x4/22subject to:g1X=2x1x3+x3x2−2x4≤300,g2X=18x2×104x3x2−2x43+2x1x34x42+3x2x2−2x4+15x1×103x2−2x4x32+2x3x13≤56,variable range:10≤x1≤50,10≤x2≤80,0.9≤x3≤5,0.9≤x4≤5.Figure 6
A 3D view of beam design problem.Many optimizers have solved this nonlinearly constrained problem, and Table4 presents the best results of these methods. In addition, the statistical results for comparing the performance of the SNS methods are provided in Table 5. For this case study, the SNS needs 3600 function evaluations to reach these results, and it can be seen that the SNS performs superior compared to other methods.Table 4
Best results for the optimal design of I-shaped beam problem.
AlgorithmVariablesConstraintsx1x2x3x4g1Xg2XfXIARSM [60]79.9948.420.92.40.0869999−1.5245400.0131CS [18]80500.92.3216−0.012005−1.5700200.01307GWO [61]80500.92.3217−0.009059−1.5700710.0131EMGO-FCR [61]80500.92.3200−0.176000−1.5671790.0131SOS [58]80500.92.3217−0.000222−1.5702240.01307AEFA-C [61]79.967149.990.92.3164−0.560371−1.5595180.0131SNS (present study)80500.92.32170−1.5702280.0130741IARSM: improved adaptive response surface method. EMGO-FCR: ensemble meta model-based global optimization using fuzzy clustering. AEFA-C: artificial electric field algorithm.Table 5
Comparison of the results of SNS with the other methods for the I-shaped beam problem.
AlgorithmWorstMeanBestSDNFEsCS [18]0.013536460.01321650.01307470.00013455000SOS [58]NA0.01308840.01307414.0E − 55000AOS [39]0.01381400.01317880.01307411.555E − 04100,000SNS (present study)0.01307640.01307430.01307414.313E − 073600


## 3.3. Three-Bar Truss Design Problem

This case considers a 3-bar planar truss structure shown in Figure7. The volume of a statically loaded 3-bar truss is to be minimized subject to stress (σ) constraints on each of the truss members. The objective is to evaluate the optimal cross-sectional areas, A1=x1 and A2=x2. The mathematical formulation is given below:(10)minimize:fX=22x1+x2×l,subject to:g1X=2x1+x22x12+2x1x2P−σ≤0,g2X=x22x12+2x1x2P−σ≤0,g3X=12x2+x1P−σ≤0,l=100cm,P=2kN/cm3,σ=2kN/cm3,variable range:0≤x1,x2≤1.Figure 7
A schematic model of three-bar truss.The best results of different methods are presented in Table6. Also, Table 7 provides the statistical results of these algorithms. It can be seen that the best objective value of the SNS is equal or better than that of other methods. The required number of function evaluations (NFEs) for the SNS algorithm is 4800, which is much lower than that of other algorithms.Table 6
Best results of the three-bar truss design problem.
AlgorithmVariablesConstraintsx1x2g1Xg2Xg3XfXGA [62]0.7889150.4075699.6430E − 07−1.464873605−0.53512542263.8958857PSO [62]0.7886690.4082654.8650E − 07−1.464082376−0.535917137263.8958434ICA [62]0.7886250.4083898.4180E − 07−1.463941244−0.536057913263.8958452CS [18]0.788670.40902−2.9000E − 04−0.26853−0.73176263.9716WCA [5]0.7886510.4083160.0000E + 00−1.464024−0.535975263.895843GWO [62]0.7886480.4083253.3400E − 08−1.464014397−0.535985569263.8960063ALO [57]0.788662810.4082831−5.3170E − 12−1.464062005−0.53593799263.8958434MFO [56]0.788244770.40946697.7090E − 12−1.462717072−0.537282927263.8959796SSA [59]0.788665410.40827573.0000E − 10−1.464070360−0.53592963263.8958434WSA [62]0.7886830.408227−7.2400E − 07−1.464126180−0.53587454263.8958434SNS (present study)0.788684730.40822113.2978E − 08−1.4641325−0.5358675263.8958434WCA: water cycle algorithm. WSA: water strider algorithm.Table 7
Comparative results of SNS with other methods for the three-bar truss design problem.
AlgorithmWorstMeanBestSDNFEsGA [62]264.82080546263.96803663263.895885731.66862E − 0150,000PSO [62]264.58490296263.95741428263.895843411.36897E − 0150,000ICA [62]263.91413326263.89932689263.895845194.11693E − 0350,000CS [18]NA264.0669263.971569.00000E − 0515,000WCA [5]263.896201263.895903263.8958438.71000E − 055250GWO [62]263.90421778263.89795501263.896006311.61422E − 0350,000WSA [62]263.89743217263.89606687263.89584343.11960E − 0450,000CGO [37]263.8960068263.8958511263.8958432.51E − 05100,000AOS [39]263.895845263.895843263.8958438.26E − 09100,000SNS (present study)263.8958561263.8958462263.89584343.31056E − 064800


## 3.4. Tubular Column Design

This problem is an example of designing a uniform column of the tubular section to carry a compressive load at minimum cost. This problem has two design variables, the mean diameter of the columnd=x1 and the thickness of tube t=x2, which are shown in Figure 8. This column is made of a material with a yield stress of σy=500kgf/cm2 and a modulus of elasticity of E=0.85×1066kgf/cm2. The optimization model of this problem is given as follows:(11)minimize:fX=9.8x1x2+2x1,subject to:g1X=Pπx1x2σy−1≤0,g2X=8PL2π3Ex1x2x12+x22−1≤0,g3X=2.0x1−1≤0,g4X=x114−1≤0,g5X=0.2x2−1≤0,g6X=x28−1≤0,variable range:2≤x1≤14,0.2≤x2≤0.8.Figure 8
The 3D model of tubular column.According to the constraintsg1 and g2, the included stress in the column should be less than the buckling and yield stresses, respectively. Also, other constraints (g3, g4, g5, and g6) clamp the variables of the problem to the ranges of the variables. This problem was previously solved using various methods, and the best results of these methods and SNS are presented in Table 8. The SNS uses 1250 evaluations to solve this problem. In addition, the statistical results of some methods are reported in Table 9. According to these results, the SNS has found better results than other algorithms.Table 8
Best results of the tubular column example.
AlgorithmVariablesConstraintsx1x2g1Xg2Xg3Xg4Xg5Xg6XfXCS [18]5.451390.29196−0.0241−0.1095−0.633−0.610−0.315−0.63526.53217ISA [63]5.451156230.29196547−2.5E − 10−1.8E − 10−0.6331−0.6106−0.3149−0.63526.5313SNS5.451156230.29196547−2.6E − 10−1.8E − 10−0.6331−0.6106−0.3149−0.63526.4994969Table 9
Comparative results of SNS with other methods for the tubular column example.
AlgorithmWorstMeanBestSDNFEsISA [63]26.53226.53126.5311.70E − 043000CS [18]26.5397226.5350426.532171.93E − 0315,000FA [64]NA28.7426.522.083000AOS [39]26.6083136126.5316139926.531378281.0300E − 03100,000SNS (present study)26.4863709526.4863624926.486361472.2160E − 061250


## 3.5. Speed Reducer Design

In mechanical systems, one of the essential parts of the gearbox is the speed reducer, and it can be employed for several applications [65]. In this optimization problem (see Figure 9), the weight of the speed reducer is to be minimized with subject to 11 constraints. This problem has seven variables, face width b=x1, module of teeth m=x2, the number of teeth in the pinion z=x3, length of the first shaft between bearings l1=x4, length of the second shaft between bearings l2=x5, the diameter of first shafts d1=x6, and the diameter of second shafts d2=x7. The mathematical formulation of this problem is formulated as follows:(12)minimize:fX=0.7854x1x223.3333x32+14.9334x3−43.0934−1.508x1x62+x72+7.4777x63+x73+0.7854x4x62+x5x72,subject to:g1X=27x1x22x3−1≤0,g2X=397.5x1x22x32−1≤0,g3X=1.93x43x2x64x3−1≤0,g4X=1.93x53x2x74x3−1≤0,g5X=745x4/x2x32+16.9×106110x63−1≤0,g6X=745x5/x2x32+157.5×10685x73−1≤0,g7X=x2x340−1≤0,g8X=5x2x1−1≤0,g9X=x112x2−1≤0,g10X=1.5x6+1.9x4−1≤0g11X=1.1x7+1.9x5−1≤0,variable range:2.6≤x1≤3.6,0.7≤x2≤0.8,x3∈17,18,19,…,28,7.3≤x4,x5≤8.3,2.9≤x6≤3.9,5≤x7≤5.5.Figure 9
A schematic representation of speed reducer.This engineering problem has 11 constraints, seven nonlinear constraints and four linear inequality constraints, which are considered based on (1) bending stress of the gear teeth, (2) surface stress, (3) transverse deflections of the shafts, and (4) stresses in the shafts. The comparison of the best optimal solution with various optimization methods is given in Table10. The SNS method requires 3750 evaluations to find its solution. The statistical results of SNS and ten optimization methods are compared in Table 11. Among the compared optimization algorithms, the SNS has the lowest number of function evaluations while its results are better than those of the other methods.Table 10
Best results of the speed reducer design example.
CS [18]WCA [5]BA [66]ABC [67]APSO [68]SHO [69]SNSx13.501503.500003.500003.500003.501313.501593.50000x20.700000.700000.700000.700000.700000.700000.70000x317.0000017.0000017.0000017.0000018.0000017.0000017.00000x47.605007.300007.300017.300008.127817.300007.30000x57.818107.715327.715327.800008.042127.800007.71532x63.352003.350213.350213.350223.352453.351273.35021x75.287505.286655.286655.286685.287085.288745.28665g1X−0.07430−0.07392−0.07400−0.07392−0.12569−0.07434−0.07392g2X−0.19830−0.19800−0.19800−0.19800−0.28490−0.19836−0.19800g3X−0.43490−0.49917−0.49900−0.49917−0.34888−0.49980−0.49917g4X−0.90080−0.90464−0.90500−0.90156−0.89804−0.90162−0.90464g5X−0.00110−0.666850.000000.00000−0.66559−0.667170.00000g6X−0.000400.000000.00000−0.00063−0.00026−0.001170.00000g7X−0.70250−0.70250−0.70300−0.70250−0.68500−0.70250−0.70250g8X−0.000400.000000.000000.00000−0.00038−0.000450.00000g9X−0.58320−0.58333−0.58300−0.58333−0.58318−0.58314−0.58333g10X−0.08900−0.05133−0.05100−0.05133−0.14754−0.05111−0.05133g11X−0.013000.000000.00000−0.01070−0.04058−0.010560.00000fX3000.981002994.471072994.467102997.058413187.630492998.550702994.47107APSO: accelerated particle swarm optimization. SHO: spotted hyena optimizer.Table 11
Comparative results of SNS with other methods for the speed reducer design example.
AlgorithmWorstMeanBestSDNFEsCS [18]3009.003007.19973000.9814.96E + 00250,000ABC [67]NA2997.0584122997.0584120.00E + 0030,000WCA [5]2994.5055782994.4743922994.4710667.40E − 0315,150APSO [68]4443.017639003822.640624003187.630486003.66E + 0230,000SHO [69]3003.8892999.642998.55071.93E + 00NASSA3015.6626123005.5743772996.0217204.63E + 00NAWOA3233.5981243042.9150232996.6043404.08E + 01NACSS3106.2164513005.6589122996.4924784.86E + 00NACGO [37]2995.5049332994.4653972994.4436490.110282100,000FACSS [70]3006.4197462999.4137982996.37523764.82E + 00NASNS (present study)2994.47109922994.47106962994.47106627.00E − 063750FACSS: fuzzy adaptive charged system search.


## 3.6. Piston Lever

The main objective of this problem is to locate the piston components,H=x1, B=x2, D=x3, and X=x4, by minimizing the oil volume when the lever of the piston is lifted up from 0° to 45°, as shown in Figure 10. The formulation of this problem is given as follows:(13)minimize:fX=14πx32L2−L1,subject to:g1X=QLcosθ−R×F≤0,g2X=QL−x4−Mmax≤0,g3X=1.2L2−L1−L1≤0,g4X=x32−x2≤0,whereR=−x4x4sinθ+x1+x1x2−x4cosθx4−x22+x12,F=πPx324,L1=x4−x22+x12,L2=x4sinθ+x12+x2−x4cosθ2,θ=45°,Q=10,000lbs,L=240in,Mmax=1.8×106lbs in,P=1500psi,variable range:0.05≤x1,x2,x4≤500,0.05≤x3≤120.Figure 10
A model of piston lever problem.These inequality constraints consider the force equilibrium, the maximum bending moment of the lever, minimum piston stroke, and geometrical conditions. The best solutions obtained by SNS and some of the other algorithms are presented in Table12. In addition, the performance of the PSO [71], DE [71], GA [71], hybrid particle swarm optimization (HPSO) [71], HPSO with Q-learning [71], CS [18], ISA [63], CGO [37], MGA [40], AOS [39], and SNS is summarized in Table 13. The SNS algorithm obtains its results after 5000 evaluations, and its results are far better than those of other methods.Table 12
Best results of the piston lever example.
PSO [71]DE [71]GA [71]HPSO [71]CS [18]SNSx1133.3129.4250.0135.50.0500.050x22.442.433.962.482.0432.042x3117.14119.8060.03116.62120.000120.000x44.754.755.914.754.0854.083g1XNANANANA−1744.9120.000g2XNANANANA−600,000−600,000g3XNANANANA−117.185−117.187g4XNANANANA−4.50E − 04−9.74E − 12fX1221591611628.4278.412698349Table 13
Comparative results of SNS with other methods for the piston lever example.
AlgorithmWorstMeanBestSDNFEsPSO [71]29416612251.750,000DE [71]19918715914.250,000GA [71]21618516118.250,000HPSO [71]19718716213.450,000HPSO with Q-learning [71]16815112913.450,000CS [18]168.592040.23198.427159.055250,000ISA [63]610.6226.58.4111.212,500CGO [37]167.472808745.04865998.41281381367.24763100,000AOS [39]167.664986233.74127598.41914274293.46674724100,000MGA [40]167.473213432.46889258.41340665229.96370439100,000SNS (present study)167.472774724.31897438.41269834947.717926465000


## 3.7. Corrugated Bulkhead Design

This problem aims to minimize the weight of a corrugated bulkhead in a chemical tanker [72], in which the design variables are the width (x1), depth (x2), length (x3), and plate thickness (x4). The mathematical model of this optimization problem is given as follows:(14)minimize:fX=5.885x4x1+x3x1+x32−x22,subject to:g1X=−x4x20.4x1+x36+8.94x1+x32−x22≤0,g2X=−x4x220.2x1+x312+2.28.94x1+x32−x224/3≤0,g3X=−x4+0.0156x1+0.15≤0,g4X=−x4+0.0156x3+0.15≤0,g5X=−x4+1.05≤0,g6X=−x3+x2≤0,variable range:0≤x1,x2,x3≤100,0≤x4≤5.Tables14 and 15 compare the best and statistical results of SNS and other optimizers, respectively. According to these results, the SNS significantly improves the solution quality of other algorithms. In addition, the SNS method solves this problem after 3125 evaluations that is the lowest value among other methods.Table 14
Best results of the corrugated bulkhead design.
CS [18]VIGMM3 [61]AEFA-C [61]SNSx137.117949857.6923157.6927757.69230732x233.035021034.1476234.1329634.14762029x337.193947657.6923157.5529457.69230729x40.73062551.050001.050071.05g1X−23.3537699−0.25839−240.89634−240.6946226g2X−15.9738532−2.220E − 16−11.59051−1.46827E − 05g3X−0.00158548−9.769E − 15−6.864E − 05−5.80038E − 09g4X−0.00039992−5.551E − 16−2.250E − 03−6.20877E − 09g5X0.31937450.00000−7.581E − 05−6.50591E − 13g6X−4.15892660.68949−23.41997−23.544687fX5.8943316.84296−6.845846.842958019VIGMM3: vibration-based ideal gas molecular movement. AEFA-C: artificial electric field algorithm.Table 15
Comparative results of SNS with other methods for the corrugated bulkhead design.
AlgorithmWorstMeanBestSDNFEsFA [64]NA10.237.211.9512,000LF-FA [64]NA8.836.951.2612,000LS-LF-FA [64]NA7.446.860.6712,000AD-IFA [64]NA7.216.840.5812,000AOS [39]7.0669361867.0608083776.842958016.4911E − 04100,000SNS (present study)6.8430743996.8429798026.8429605152.0942E − 053125LF-FA: Levy flight firefly algorithm. LS-LF-FA: logarithmic spiral path flight firefly. AD-IFA: spiral-Levy flight firefly algorithm.


## 3.8. Design of Pressure Vessel

A cylindrical vessel is capped at both ends by hemispherical heads, as shown in Figure11. The objective is to minimize the total cost, including the cost of material, forming, and welding. This problem has four variables including the thickness of the shell Ts=x1, the thickness of the head Th=x2, the inner radius R=x3, and the length of the cylindrical section of the vessel, not including the head L=x4. In addition, x1 and x2 are integer multiples of 0.0625 in, while the other variables are continuous. The optimization problem can be expressed as follows:(15)minimize:fX=0.6224x1x3x4+1.7781x2x32+3.1661x12x4+19.84x12x3,subject to:g1X=−x1+0.0193x3≤0,g2X=−x2+0.00954x3≤0,g3X=−πx32x4−43πx33+1,296,000≤0,g4X=x4−240≤0,variable range:x1,x2∈1×0.0625,2×0.0625,3×0.0625,…,1600×0.0625,10≤x3,x4≤200.Figure 11
Schematic view of pressure vessel design.This problem has been used to evaluate the performance of many algorithms. Tables16 and 17 compare the best and statistical results of SNS and other algorithms, respectively. The SNS needs 6000 NFEs for solving this problem that is much lower than that of other algorithms.Table 16
Best results of the pressure vessel design.
G-QPSO [73]HPSO [74]CPSO [75]CDE [76]SAP [77]ABC [67]CS [18]EO [78]x10.812500.812500.812500.812500.812500.812500.812500.81250x20.437500.437500.437500.437500.437500.437500.437500.43750x342.0984042.0984042.0913042.0984040.3239042.0984542.0984542.09845x4176.63720176.63660176.74650176.63760200.00000176.63660176.63660176.63660g1X0.000000.000000.000000.00000−0.034320.000000.000000.00000g2X−0.03580−0.03580−0.00036−0.03580−0.05285−0.03588−0.03588−0.03588g3X−0.217903.12260−118.76870−3.70512−27.10585−0.00023−0.00005−0.00005g4X−63.36280−63.36340−63.25350−63.36230−40.00000−63.36340−63.36340−63.36340fX6059.72086059.71436061.07776059.73406288.74456059.71436059.71436059.71430MFO [56]GWO [31]WOA [33]APSO [68]IAPSO [68]NDE [79]MCEO [80]SNSx10.81250.8125000.8125000.81250.81250.81250.81250.81250x20.43750.4345000.4375000.43750.43750.43750.43750.43750x342.09844542.08918142.098269942.098442.098442.098445542.098445542.09845x4176.636596176.758731176.638998176.6374176.6366176.636595176.636596176.63660g1X−1.15E − 08−0.00017880−3.39E − 06−8.80E − 07−4.09E − 13−1.4E − 15−1.13E −  100.00000g2X−0.03588083−0.03296921−0.03588250−0.03588126−3.58E − 2−0.00035880−0.037564−0.03588g3X0.04023295−40.6168247−1.25270175−1.33153860−1.39E − 07−0.00000001−4.73E −  040.00000g4X−63.363404−63.241269−63.361002−63.3626−63.3634−0.63363404−63.3634−63.36340fX6059.71436051.56396059.74106059.724186059.714336059.714336059.71436059.71434G-QPSO: Gaussian quantum-behaved particle swarm optimization. CPSO: co-evolutionary particle swarm optimization. CDE: co-evolutionary differential evolution. SAP: self-adaptive penalty approach. EO: equilibrium optimizer. MCEO: multilevel cross entropy optimizer.Table 17
Comparative results of SNS with other methods for the pressure vessel design.
AlgorithmWorstMeanBestSDNFEsSAP [77]6308.156293.8436288.7457.41E + 003000HPSO [74]6288.6776099.93236059.71438.62E + 0181,000CDE [76]6059.7346085.236371.0464.30E + 01204,800CPSO [75]6363.80416147.13326061.07778.65E + 01200,000PSO [73]14076.3248756.68036693.72121.49E + 038000QPSO [73]8017.28166839.93266059.72094.79E + 028000G-QPSO [73]7544.49256440.37866059.72084.48E + 028000ABC [67]NA6245.3081446059.7147362.05E + 0230,000CS [18]6495.3476447.7366059.7143355.03E + 0215,000WOA [33]NA6068.056059.7416.57E + 016300APSO [68]7544.492726470.715686059.72423.27E + 02200,000EO [78]7544.49256668.1146059.71435.66E + 0215,000CGO [37]6330.9586856250.9573546247.6728191.07E + 01100,000SNS (present study)6410.0868866097.1002946059.7143359.28E + 016000


## 3.9. Design of Tension/Compression Spring

The tension/compression spring design problem is described in [81] for which the objective is to minimize the weight of a tension/compression spring, as shown in Figure 12. This problem is subject to constraints on minimum deflection, shear stress, surge frequency, limits on the outside diameter, and design variables. The design variables are the mean coil diameter D=x1, the wire diameter d=x2, and the number of active coils N=x3. The problem can be stated as(16)minimize:fX=x3+2x2x12,subject to:g1X=1−x23x371785x14≤0,g2X=4x22−x1x212566x2x13−x14+15108x12−1≤0,g3X=1−140.45x1x22x3≤0,g4X=x1+x21.5−≤0,variable range:0.05≤x1≤2,0.25≤x2≤1.3,2≤x3≤15.Figure 12
Tension/compression spring.Table18 compares the SNS with many optimization algorithms in terms of best optimization results, and Table 19 presents the statistical results of these algorithms. The SNS algorithm solves this problem in 9000 evaluations, and among the compared methods, just WOA [33] and MCEO [80] used a fewer number of evaluations, while their results are not as good as SNS.Table 18
Best results of tension/compression spring design.
AlgorithmVariablesConstraintsx1x2x3g1Xg2Xg3Xg4XfXCPSO [75]0.0517280.35764411.244543−8.25E − 04−2.52E − 05−4.051306−0.7270850.012674WCA [5]0.0516800.35650011.300400−1.60E − 13−7.90E − 14−4.053300−0.7278000.012665ABC [67]0.0517490.35817911.2037630.00E + 000.00E + 00−4.056663−0.7267130.012665SFOA [65]0.0518000.35900011.279000−3.24E − 06−3.58E − 07−4.060000−0.7260000.012700APSO [68]0.0525880.37834310.138862−1.55E − 04−8.33E − 04−4.089171−1.0690690.012700IAPSO [68]0.0516850.35662911.294175−1.97E − 10−4.64E − 10−4.053610−1.0916860.012665MFO [56]0.0519940.36410910.868421−4.10E − 063.04E − 06−4.068140−0.7226000.012667GWO [31]0.0516900.35673711.288850−7.91E − 05−7.51E − 06−4.053380−0.7277200.012666WOA [33]0.0512000.34520012.004000−5.60E − 04−3.00E − 05−4.027400−0.7357000.012676SHO [69]0.0511440.34375112.095500−3.30E − 041.16E − 05−4.025790−0.7367400.012674NDE [79]0.0516890.35671811.2889680.00E + 000.00E + 00−4.053785−0.7277280.012665SSA [59]0.0512070.34521512.004032−5.60E − 04−3.70E − 05−4.027410−0.7357200.012676SNS0.0515870.35426811.434058−1.37E − 08−3.18E − 10−4.048919−0.7294300.012665Table 19
Comparative results of tension/compression spring design.
AlgorithmWorstMeanBestSDNFEsCPSO [75]0.0129240.012730.01267475.20E − 05200,000HPSO [74]0.01271910.01270720.01266521.58E − 0581,000CDE [76]0.012790.0127030.01267022.70E − 05204,800PSO [73]0.0718020.0195550.0128571.17E − 0220,000QPSO [73]0.0181270.0138540.0126691.34E − 0320,000G-QPSO [73]0.0158690.0129960.0126666.28E − 0420,000WCA [5]0.0129520.0127460.0126658.06E − 0511,750ABC [67]NA0.0127090.0126651.28E − 0230,000APSO [68]0.0149370.0132970.01276.85E − 04120,000IAPSO [68]0.017828640.013676530.012665231.57E − 0320,000WOA [33]NA0.01270.01267633.00E − 044410MCEO [80]0.013509010.01271960.012660513.79E − 052000EO [78]0.0139970.0130170.0126663.91E − 0415,000SNS (present study)0.0127658730.0126847170.0126652462.38549E − 059000


## 3.10. Design of Welded Beam

This benchmark problem was introduced by Coello [77] and has been tackled by many researchers. As illustrated in Figure 13, the beam is under a vertical force. The goal of this problem is to find the minimum manufacturing cost of the welded beam. The problem is subject to seven constraints of stress, deflection, welding, and geometry. The variables are weld thickness h=x1, height l=x2, length t=x3, and bar thickness b=x4, as shown in Figure 13. The objective function can be mathematically be stated as(17)minimize:fX=1.10471x12x2+0.04811x3x414.0+x2,subject to:g1X=τX−τmax≤0,g2X=σX−σmax≤0,g3X=δX−δmax≤0,g4X=x1−x4≤0,g5X=P−PcX≤0,g6X=0.125−x1≤0,g7X=1.10471x12+0.04811x3x414.0+x2−5.0≤0,τX=τ′2+2τ′τ″x22R+τ″2,τ′=P2x1x2,τ″=MRJ,M=PL+x22,R=x224+x1+x322,J=22x1x2x224+x1+x322,σX⟶=6PLx4x32,δX⟶=6PL3Ex32x4,PcX⟶=4.013Ex32x46/36L21−x32LE4G,P=6000lb,L=14in,δmax=0.25in,E=30×106psi,G=12×106psi,τmax=13,600psi,σmax=30,000psi.variable range:0.1≤x1,x4≤2,0.1≤x2,x3≤10.Figure 13
Schematic of the welded beam structure with indication of design variables.Tables20 and 21 compare the best and statistical results of various optimizer in dealing with welded beam design problem. The SNS algorithm needs 9000 evaluations, which is the lowest NFE among other algorithms, while its results are better. In addition, the SNS algorithm has the lowest standard deviation that shows its robustness in solving this problem.Table 20
Best results of welded beam design.
BBO [62]PSO [62]ICA [62]WCA [5]ABC [67]EO [78]TEO [82]SSA [59]x10.18548600.2192920.2057990.2057280.2057300.20570000.205681000.2057000x24.31290003.4304163.4696343.4705223.4704893.47050003.472305003.4714000x38.43990308.4335599.0349509.0366209.0366249.03660009.035133009.0366000x40.23590200.2362040.2058060.2057290.2057300.20570000.205796000.2057000g1X−114.190503−0.1936420.0206880.0108320.0000000.0094704−0.60927688−0.8256803g2X−6.71716160.017873−0.0177630.119259−0.0000020.00149580.22468860−0.7106346g3X−0.0504000−0.016900−0.000007−0.0000010.000000−0.0000001−0.000115000.0000000g4X−3.2078633−3.276396−3.390414−3.390662−3.432984−3.3906592−3.39027259−3.3908203g5X−0.0604860−0.094292−0.080799−0.080728−0.080730−0.0807295−0.08068100−0.0807000g6X−0.2350000−0.235000−0.236000−0.235540−0.235540−0.2355403−0.23553783−0.2355381g7X−2639.74280−2668.48570−5.9519080.0576840.0000000.0008607−5.15644919−2.6033472fX1.91805501.8527201.7251351.7248561.7248521.72490001.725284001.7249100WSA [62]MCEO [80]CBO [62]GWO [62]SHO [69]WOA [33]CCSA [83]SNSx10.205730000.205729640.20572200.20567700.20556300.20539600.20570000.2057296x23.470489003.470488663.47041003.47089403.47484603.48429303.47020003.4704887x39.036624009.036623919.03727609.03855809.03579909.03742609.03620009.0366239x40.205730000.205729640.20573500.20573900.20581100.20627600.20570000.2057296g1X−0.025399590.00000363−0.0246881−0.0519206−1.4028657−21.5450190−0.00206100.0000000g2X−0.05312238−0.00002880−5.1106833−14.2018357−6.3837568−84.7713395−0.65695600.0000000g3X0.000000000.00000000−0.0000130−0.0000620−0.0002480−0.0008800−0.00000450.0000000g4X−3.39065616−3.39000000−3.3905161−3.3902412−3.3898697−3.3852837−3.3906250−3.3906591g5X−0.08073000−0.08072964−0.0807220−0.0806770−0.0805630−0.0803960−0.0807296−0.0807296g6X−0.23554035−0.23554032−0.2360000−0.2355503−0.2355421−0.2355825−0.2355406−0.2355403g7X−0.03155555−0.00001860−0.7536625−1.6632990−6.7607624−48.2829292−0.39068010.0000000fX1.724852001.724852301.72466301.72523201.72566101.73049901.72490001.7248523CCSA: conscious neighborhood-based crow search algorithm.Table 21
Comparative results of welded beam design.
AlgorithmWorstMeanBestSDNFEsPSO [62]3.8418452.6137851.852724.71E − 0150,000CPSO [75]1.7821431.7488311.7280241.29E − 02200,000HPSO [74]1.8142951.7490401.7248524.00E − 0281,000CDE [76]1.8241051.7681581.7334612.22E − 02204,800ICA [62]2.2377551.794331.7251351.10E − 0150,000BBO [62]3.6069332.6304121.9180554.11E − 0150,000FA [54]2.3455791.8786561.7312062.68E − 0150,000CSS [26]1.7594791.7396541.7248668.06E − 03NAWCA [5]1.7446971.7264271.7248564.29E − 0346,500ABC [67]NA1.7419131.7248523.10E − 0230,000MFO [62]1.7248521.7321091.9502413.40E − 0250,000SCA [62]1.7868631.8493641.9251623.47E − 0250,000APSO [68]1.9939991.8778511.7361937.61E − 0250,000SHO [69]1.7260641.7258281.7256612.87E − 04NASSA [62]1.7258861.8234262.2466381.28E − 0150,000GWO [62]1.7252321.726311.7284877.71E − 0450,000WOA [33]NA1.73201.7304990.02269900TEO [82]1.9311611.7680401.7252845.82E − 02NAEO [78]1.7367251.7264821.7248533.26E − 0315,000WSA [62]1.7250681.7249081.7248524.15E − 0550,000FACSS [70]1.7365651.7302231.7248536.363E − 7100,000T-CSS [84]1.7356561.7302121.7248602.00E − 06100,000SNS (present study)1.7250511.7248801.7248525.18E − 059000T-CSS: tribe-charged system search.


## 3.11. Design of Gear Train

The gear train design problem is an unconstrained discrete design problem in mechanical engineering and was introduced by Sandgren [85]. The purpose of this benchmark task is to minimize the gear ratio defined as the ratio of the angular velocity of the output shaft to the angular velocity of the input shaft. The number of teeth of gears nA=x1, nB=x2, nC=x3, and nD=x4 are considered as the design variables, and Figure 14 illustrates the 3D model of this problem. The mathematical formulation is provided as follows:(18)minimize:fX=16.931−x3x2x1x42,variable range:x1,x2,x3,x4∈12,13,14,…,60.Figure 14
The 3D model of gear train.The best results of 19 algorithms include the SNS are presented in Table22. It can be seen that all the algorithms find the optimum solution, except the PSO [62] and BBO [62]. In addition, the statistical results of 14 algorithms are compared in Table 23. The proposed method outperformed most of the other algorithms in terms of the mean, worst, SD, and NFEs.Table 22
Best results of the gear train design.
AlgorithmVariablesfXx1x2x3x4GA [62]491916432.70E − 12PSO [62]341320532.31E − 11ICA [62]431619492.70E − 12CS [18]431619492.70E − 12ABC [67]491619432.70E − 12MSFWA [86]491916432.70E − 12MBA [87]431619492.70E − 12BBO [62]532615512.31E − 11NNA [62]491619432.70E − 12GWO [62]491916432.70E − 12ISA [63]431916492.70E − 12APSO [68]431619492.70E − 12IAPSO [68]431619492.70E − 12MVO [88]431619492.70E − 12MFO [56]431916492.70E − 12ALO [57]491916432.70E − 12PSOSCALF [89]491916432.70E − 12WSA [62]431619492.70E − 12SNS (present study)431916492.70085714E − 12MBA: mine blast algorithm.Table 23
Comparative results of SNS with other methods for the gear train design.
AlgorithmWorstMeanBestSDNFEsGA [62]1.5247E − 081.6212E − 092.7009E − 123.2174E − 0950,000PSO [62]1.0222E − 067.9383E − 082.3078E − 111.8147E − 0750,000ICA [62]2.3576E − 098.0417E − 102.7009E − 127.7862E − 1050,000UPSO [90]N.A3.80562E − 082.700857E − 121.09E − 07100,000MFO [62]2.7265E − 087.5337E − 092.3078E − 119.3539E − 0950,000SCA [62]2.3576E − 098.8113E − 102.7009E − 126.4529E − 1050,000SSA [62]2.7265E − 081.9822E − 092.7009E − 124.5748E − 0950,000BBO [62]4.2018E − 074.5418E − 082.3078E − 117.2953E − 0850,000CBO [62]1.1173E − 082.1032E − 092.3078E − 112.4025E − 0950,000NNA [62]2.3576E − 096.1128E − 102.7009E − 126.1167E − 1050,000GWO [62]9.9216E − 103.3777E − 102.7009E − 124.0956E − 1050,000WOA [62]6.5123E − 099.6633E − 102.7009E − 121.1296E − 0950,000WSA [62]1.3616E − 091.6800E − 102.7009E − 123.8265E − 1050,000SNS (present study)1.36165E − 091.68012E − 102.700857E − 123.74894E − 1025,000


## 3.12. A Reinforced Concrete Beam Design

Amir and Hasegawa [91] presented a simplified optimization problem of designing a reinforced concrete beam, as shown in Figure 15. The beam is assumed to be simply supported with a span of 30 ft and subjected to a live load of 2000 lbf and a dead load of 1000 lbf, including the weight of the beam. The concrete compressive strength (σc) is 5 ksi, and the yield stress of the reinforcing steel (σy) is 50 ksi. The cost of concrete is $0.02/in2/linear ft, and the cost of steel is $1.0/in2/linear ft. To minimize the total cost of the structure, the area of the reinforcement As=x1, the width of the beam b=x2, and the depth of the beam h=x3 have to be determined. The structure should be proportioned to have a required strength based upon the ACI building code 318-77 as follows:(19)Mu=0.9Asσy0.8h1.0−0.59Asσy0.8bhσc≥1.4Md+1.7Ml,where Mu, Md, and Ml are the flexural strength, dead load, and live load moments of the beam, respectively. In this case, Md=1350in kip and Ml=2700in kip. The depth to width ratio of the beam is restricted to be less than or equal to 4. The optimization problem can be expressed as(20)minimize:fX=2.9x1+0.6x2x3,subject to:g1X=x2x3−4≤0,g2X=180+7.375x12x3−x1x2≤0,variable range:x1∈6,6.16,6.32,6.6,7,7.11,7.2,7.8,7.9,8,8.4,x2∈28,29,30,…,40,5≤x3≤10.Figure 15
A schematic view of reinforced concrete beam.It is clear that the variablesx1 and x2 are discrete, while x3 is continuous. The SNS method requires1000 evaluations to reach the optimum solution. Table 24 presents the results of optimum designs obtained by the SNS and other methods for this problem. In addition, the statistical results of FA [54], CS [18], AOS [39], and SNS are compared in Table 25. Obviously, the performance of the SNS method is better than other algorithms.Table 24
Best results of a reinforced concrete beam design.
GHN-EP [71]GA [92]FA [54]CS [18]ISA [63]AOS [39]SNSx16.327.206.326.326.326.326.32x234323434343434x38.6371808.04518.50008.50008.50008.58.5g1X−0.7745−2.8779−0.2241−0.2241−0.2241−0.22409498−0.22409411g2X−0.0635−0.0224000−1.00E − 075.27667E − 12fX362.00648366.1459359.2080359.2080359.2080359.2080359.2080GHN-EP: generalized Hopfield network-based extended penalty approach.Table 25
Comparative results of SNS with other methods for a reinforced concrete beam design.
AlgorithmWorstMeanBestSDNFEsFA [54]669.150460.706359.208080.7387025,000CS [18]NANA359.2080NA5000AOS [39]362.2535359.3306872359.20800.59614901100,000SNS (present study)362.634359.3222001359.20800.614985811000


## 3.13. Car Side Impact Design

On the foundation of the European Enhanced Vehicle-Safety Committee (EEVC) procedures, a car is exposed to a side impact, and the aim of this benchmark problem is minimizing the weight of the door using nine influence parameters including thicknesses of B-pillar inner=x1, B-pillar reinforcement =x2, floor side inner =x3, cross members =x4, door beam =x5, door beltline reinforcement =x6, roof rail =x7, materials of B-pillar inner =x8, floor side inner =x9, barrier height =x10, and hitting position =x11. Youn et al. [93] simplified this optimization problem’s analytical formulation and sped up computations using the global response surface methodology to approximate the structural weight and response to the impact. Figure 16 shows a model for the car side impact problem. According to the simplified models, the optimization problem is formulated as follows:(21)minimize:fX=1.98+4.90x1+6.67x2+6.98x3+4.01x4+1.78x5+2.73x7,subject to:g1X=1.16−0.3717x2x4−0.00931x2x10−0.484x3x9+0.01343x6x10−1≤0,g2X=46.36−9.9x2−12.9x1x2+0.1107x3x10−32≤0,g3X=33.86+2.95x3+0.1792x3−5.057x1x2−11.0x2x8−0.0215x5x10−9.98x7x8+22.0x8x9−32≤0,g4X=28.98+3.818x3−4.2x1x2+0.0207x5x10+6.63x6x9−7.7x7x8+0.32x9x10−32≤0,g5X=0.261−0.0159x1x2−0.188x1x8−0.019x2x7+0.0144x3x5+0.0008757x5x10+0.08045x6x9+0.00139x8x11+0.00001575x10x11−0.32≤0,g6X=0.214+0.00817x5−0.131x1x8−0.0704x1x9+0.03099x2x6−0.018x2x7+0.0208x3x8+0.121x3x9−0.00364x5x6+0.0007715x5x10−0.0005354x6x10+0.00121x8x11+0.00184x9x10−0.02x22−0.32≤0,g7X=0.74−0.61x2−0.163x3x8+0.001232x3x10−0.166x7x9+0.227x22−0.32≤0,g8X=4.72−0.5x4−0.19x2x3−0.0122x4x10+0.009325x6x10+0.000191x112−4≤0,g9X=10.58−0.674x1x2−1.95x2x8+0.02054x3x10−0.0198x4x10+0.028x6x10−9.9≤0,g10X=16.45−0.489x3x7−0.843x5x6+0.0432x9x10−0.0556x9x11−0.000786x112−15.7≤0,variable range:0.5≤x1,x2,x3,x4,x5,x6,x7≤1.5,x8,x9∈0.192,0.345,,−30≤x10,x11≤+30.Figure 16
A model of car side impact problem.The design of car side impact is also used as a benchmark problem to evaluate the performance of various methods. The best results of the SNS and these algorithms are presented in Table26. It should be noted that the results of other algorithms evaluated by Gandomi et al. [18, 54, 66] have different variables ranges, but in this paper, the variable ranges of [93] are utilized. Table 27 summarizes the statistical results obtained by the different optimization algorithms for the car side impact design problem. In this case, the SNS achieves its results with 20,000 NFEs. The CS method has better performance than all of the methods, according to the results presented in [18]. In addition, the best result of the SNS is better than those of other algorithms.Table 26
Best results of the car side impact design example.
PSO [54]DE [54]GA [54]CS [18]BA [66]SNSx10.500000.500000.500050.500000.500000.5x21.116701.116701.280171.116431.116701.115933208x30.500000.500000.500010.500000.500000.5x41.302081.302081.033021.302081.302081.302918991x50.500000.500000.500010.500000.500000.5x61.500001.500000.500001.500001.500001.5x70.500000.500000.500000.500000.500000.5x80.345000.345000.349940.345000.345000.345x90.192000.192000.192000.192000.192000.192x10−19.54935−19.5493510.3119−19.54935−19.54935−19.6388662x11−0.00431−0.004310.00167−0.00431−0.004311.49192E − 06g1X−0.617505−0.617505−0.43167−0.617423−0.61750525−0.61849499g2X−0.092707−0.092707−0.09839−0.092703−0.09270723−0.09273231g3X−0.100627−0.100627−0.13685−0.100625−0.10062705−0.10061401g4X−0.034209−0.034209−0.02699−0.034182−0.03420961−0.03418562g5X−4.278328−4.278328−3.77008−4.277761−4.27832783−4.28314394g6X−7.2838105−7.2838105−3.36103−7.282103−7.28381045−7.29404065g7X−0.0026365−0.0026365−0.000253.647E − 05−0.002636520g8X−2.425E − 05−2.425E − 05−8.6156E − 061.395E − 06−2.43E − 050g9X−0.9654269−0.9654269−0.58567−0.965154−0.96542696−0.96669762g10X−0.1666041−0.16660410.502506−0.166604−0.16660413−0.16739262fX22.8447422.8429822.8565322.8429422.8447422.84297965Table 27
Comparative results of SNS with other methods for the car side impact design example.
AlgorithmWorstMeanBestSDNFEsPSO [54]23.2135422.8942922.844740.1501720,000DE [54]24.1260623.2282822.842980.3445120,000GA [54]26.24057823.5158522.856530.6655520,000BA [66]23.2135422.8927322.844740.1738320,000FA [54]24.0662322.8937622.842980.1666720,000CS [18]23.2599822.8585822.842940.0761220,000WCA [94]23.3709337622.9751644222.843036481.9772E − 01NAMBA [94]23.4889421722.9364210422.843596401.5258E − 01NAER-WCA [94]24.4553128023.0699253422.843264613.502E − 01NAWOA [94]27.3608136824.8144861723.042162209.657E − 01NAFACSS [70]23.0536256222.9121235422.849074014.726E − 02NACSS [84]24.86356323.52326523.0073360.562345100,000T-CSS [84]23.80090422.90365322.8478480.078565100,000SNS (present study)23.1845493922.8814573622.842969540.10180121120,000ER-WCA: evaporation rate-based water cycle algorithm.


## 3.14. Cantilever Stepped Beam

This problem is a good benchmark to verify the capability of the optimization methods for solving continuous, discrete, and mixed variable structural design problems. This problem aims to minimize the volume of the beam. The width of segments (x1,x2,x3,x4,x5) and height of them (x6,x7,x8,x9,x10) are chosen to be the design variables. These ten variables are illustrated in Figure 17. Except for bending stress constraints, a specified aspect ratio is imposed such that the ratio of height to width in the segments of the beam is limited to be less than 20. The problem is formulated as follows:(22)minimize:fX=Dx1x6l1+x2x7l2+x3x8l3+x4x9l4+x5x10l5,subject to:g1X=6Pl5x5x102−σd≤0,g2X=6Pl5+l4x4x92−σd≤0,g3X=6Pl5+l4+l3x3x82−σd≤0,g4X=6Pl5+l4+l3+l2x2x72−σd≤0,g5X=6Pl5+l4+l3+l2+l1x1x62−σd≤0,g6X=Pl33E1I5+1I4+1I3+1I2+1I1−Δmax≤0,g7X=x10x5−20≤0,g8X=x9x4−20≤0,g9X=x8x3−20≤0,g10X=x7x2−20≤0,g11X=x6x1−20≤0,P=50000N,σd=14,000N/cm2,E=2×107N/cm2,Δmax=2.7cm,D=1.0,variable range:x1∈1,2,3,4,5,x2,x3∈2.4,2.6,2.8,3.1,1≤x4,x5≤5,x6∈30,31,32,…,65,x7,x8∈45,50,55,60,65,30≤x9,x10≤65.Figure 17
A 3D model of stepped cantilever beam.The first five constraints are related to the bending stresses in each beam segment that must be lower than the allowable limit (σd). Also, the deflection of the cantilever beam tip must be smaller than the limit deflection (Δmax). The aspect ratio between the height and width of the cross sections must be less than 20 and is applied by the last five constraints. Six of the variables (x1,x2,x3,x6,x7,x8) are discrete, and the rest of them (x4,x5,x9,x10) are continuous.The best and statistical results of the FA [54], thermal exchange optimization (TEO) [82], PSO [82], and SNS are presented in Tables 28 and 29, respectively. To solve this problem, the SNS needs 20,000 evaluations, which is lower than the NFEs of other compared methods, and at the same time, it outperforms all of them in terms of best, mean, worst, and SD.Table 28
Best results of the cantilever stepped beam.
FA [54]TEO [82]PSO [82]SNS (present study)x13353x23.13.13.13.1x32.62.62.62.6x42.2052.216290465311692.2045562.204555692x51.7501.769100853407631.7497571.749757012x660604760x755555555x850505050x944.09144.01442096962544.0911144.09111383x1034.99534.844577796611634.99514034.99514024g1X−1.83171281−33.157060−1.49E − 06−1.81899E − 12g2X−2.74873225−25.554426−2.38E − 06−1.81899E − 12g3X−153.846153−153.846154−153.8461538−153.8461538g4X−1203.41242−1203.412423−1203.412423−1203.412423g5X−111.111111−111.111111−419.1942055−111.1111111g6X−2.44816450−2.447883−2.44430588−2.448137336g7X−0.00285714−0.3037920−3.19744E − 14g8X−0.00408163−0.140500−4.54E − 09−3.55271E − 15g9X−0.76923076−0.769231−0.769230769−0.769230769g10X−2.25806451−2.258065−2.258064516−2.258064516g11X00−10.60fX63893.5263994.01891969393.43079663893.4307958715Table 29
Comparative results of SNS with other methods for the cantilever stepped beam.
AlgorithmWorstMeanBestSDNFEsFA [54]64262.9942064144.7531263893.52578175.9187950,000PSO [82]NANA69393.430796NA300,000TEO [82]NANA63994.018919NA300,000SNS (present study)63905.6952399763893.8874976063893.430795882.21E + 0020,000


## 3.15. Real Application of SNS in Remote Sensing (Segmentation of Satellite Images)

Image segmentation is an important topic in the field of remote sensing due to the increasing volume of collected images from satellites, airplanes, and other platforms [95]. Image segmentation aims to partition an image into several homogenous sections such that the combination of no two adjacent sections is homogenous. Segmentation is a difficult task due to poor resolution, unfavorable environmental conditions, ambiguous regions, and the presence of pixels with a weak local correlation in satellite images [96, 97]. Metaheuristic methods are proper tools that can deal with the difficulty of discovering the homogeneity measure in the images [98]. In this section, the SNS algorithm is employed for segmenting color satellite images, and then its results have been compared with different optimization algorithms.Thresholding techniques are the most common methods that are used as the objective function in image segmentation. In this study, Kapur’s entropy method is employed for this purpose, and its mathematical formulation is as follows:(23)H0=−∑i=0t1−1piω0log2piω0;H1=−∑i=t1t2−1piω1log2piω1;Hj=−∑i=tjtj+1−1piωjlog2piωj;Hm=−∑i=tmN−1piωmlog2piωm,where(24)ω0=∑i=0t1−1pi;ω1=∑i=t1t2−1pi;ωj=∑i=tjtj+1−1pi;ωm=∑i=tmN−1pi,where H0,H1,…,Hm are the entropy values of m + 1 various sections or classes, pi is the probability of the pixel intensity, and N is the total number of distinct intensity levels.The utilized image and its histogram patterns are shown in Figure18. This satellite image is taken from Pléiades Satellite Imagery to carry out the experimental study for segmentation. It can be seen that the histogram of the satellite image has a multimodal pattern, and it is very difficult to segment such an image that possesses immense information content.Figure 18
The utilized satellite image and its histograms (https://www.satpalda.com/gallery).The experiment was carried out 10 times to choose the best of each algorithm. Figure19 gives the segmented images for four different levels of thresholding (n) and compares the results of the SNS algorithm with PSO, Darwinian PSO (DPSO), ABS, CS, and CS algorithm with McCulloch’s method (CSMcCulloch). Also, Table 30 presents the comparison of threshold values between the SNS and other methods.Figure 19
Segmentation results of test images using metaheuristic algorithms for four different threshold levels.Table 30
Comparison of threshold values between algorithms.
nMethodsPSO [96]DPSO [96]ABC [96]540 82 123 164 20743 86 128 171 21140 82 121 162 20562 100 138 176 21765 104 142 180 21763 100 139 177 21754 93 131 171 21051 95 99 147 16656 95 134 173 214730 60 90 119 151 184 22033 66 97 129 160 191 22133 65 95 126 157 188 22052 80 105 133 161 189 21954 82 109 137 166 195 22554 81 109 137 166 195 22544 74 104 132 161 190 21943 73 103 132 161 190 21946 76 104 132 161 190 217922 44 68 94 120 148 175 203 23122 45 70 96 121 146 171 197 22422 48 73 99 125 152 179 204 23147 68 89 110 132 155 180 203 22944 63 82 103 125 147 171 197 22629 51 71 94 121 147 171 197 22821 48 75 101 128 155 181 206 23021 47 72 95 119 145 172 199 22521 48 75 101 126 151 177 201 2261117 38 59 78 98 119 140 162 183 207 23116 35 54 74 94 116 139 163 186 209 23116 32 50 70 91 113 136 159 182 206 23128 46 66 84 101 118 135 153 175 198 22444 63 82 101 119 138 157 176 194 214 23428 46 66 86 106 126 144 164 184 204 22518 36 56 77 99 121 141 161 181 202 22618 39 59 80 101 122 143 164 186 208 23018 37 56 76 96 118 140 162 185 208 230nCS [96]CSMcCulloch [96]SNS (present study)548 89 132 181 21045 88 130 172 20845 88 131 174 21549 91 133 179 21547 89 132 178 21243 57 71 97 17351 93 140 182 21948 88 133 173 20956 95 134 174 214742 81 110 145 182 221 24539 76 107 139 173 204 23235 66 96 126 157 189 22042 83 115 143 179 208 23940 77 108 142 174 203 23340 50 51 66 116 182 20643 77 109 143 181 217 24241 76 106 140 175 205 23446 75 103 132 161 191 223934 61 79 110 133 159 189 210 23325 51 75 103 130 156 182 209 23524 47 72 97 122 148 174 200 22525 56 77 109 137 161 191 210 24020 53 70 108 135 159 189 211 23812 12 14 48 62 115 157 186 19130 62 79 110 132 164 198 214 24529 61 75 109 130 163 194 213 24018 47 74 99 123 148 173 198 2261128 47 65 87 105 132 145 167 191 220 23923 45 64 84 105 129 145 164 186 211 23521 41 63 83 103 123 143 165 187 209 23225 48 69 91 110 132 147 165 191 214 24024 46 67 89 107 131 146 162 188 213 23811 12 40 61 83 100 110 180 217 219 25128 50 66 85 108 135 148 167 189 211 23522 48 65 85 106 130 147 163 185 212 23516 34 54 74 95 117 139 161 183 206 230Various criteria can be used for comparing the performance of metaheuristic algorithms in satellite image segmentation. Peak signal to noise ratio (PSNR) and feature similarity index (FSIM) are two quantitative performance metrics, which are utilized in this study [99]. PSNR measures the accuracy of the reconstructed image and is formulated as follows:(25)PSNR=10log1025521/mn∑j=1m∑k=1nXj,k−Xj,k'2,where mn is the size of image and X and X′ are the main and the processed images, respectively. In addition, FSIM is a criterion that calculates the similarity of the thresholded and original images as follows:(26)FISM=∑x∈XSLxPCmx∑x∈XPCmx,where SLx shows the similarity of images and PCmx is the phase congruency map. Tables 31 and 32 present the values of PSNR and FSIM metrics, respectively. According to these results, the SNS algorithm achieved better PSNR and FSIM values for all thresholding levels.Table 31
Comparison of PSNR between algorithms.
nPSO [96]DPSO [96]ABC [96]CS [96]CSMcCulloch [96]SNS520.388420.426720.470720.478120.481626.0094721.144921.188621.232721.15521.31626.1077922.247322.304822.334222.541822.749626.70821123.13923.231623.339923.544923.652727.6536Table 32
Comparison of FSIM between algorithms.
nPSO [96]DPSO [96]ABC [96]CS [96]CSMcCulloch [96]SNS50.94250.94270.94330.94020.94450.9665570.90970.91250.91410.90710.91740.9762690.9470.94790.94860.94410.95490.98178110.91750.91790.92140.91560.92760.98441This comprehensive study demonstrates that the developed SNS has competence among the other metaheuristic algorithms. Based on the results in solving classical engineering problems, it can be concluded that the SNS algorithm can perform superior to other algorithms in dealing with semi-real constrained problems. In addition, the image segmentation problem results show the SNS algorithm’s ability to solve real-world problems.


## 4. Conclusion

The social network search (SNS) is a newly developed metaheuristic algorithm that mimics the behavior of social network users in expressing their opinions. In the present study, the SNS algorithm was employed for solving 14 semi-real constrained optimization problems and one real-world application in the field of remote sensing at a relatively low computational cost. From the comparative study, the SNS has shown its potential to handle various constrained optimization problems, and its performance is much better than other state-of-the-art algorithms in terms of the selected performance metrics. This is partly because there are no parameters to be fine-tuned in the SNS. In addition, it is worth mentioning that a simple penalty function method is used for constraint handling, while other compared methods maybe used advanced methods for this task.This algorithm uses four moods of users in the social networks, namely, imitation, conversation, disputation, and innovation. Users are influenced to express their new views using these four moods simulated from real-world behaviors of users in social networks that randomly accrue for each of them. As further studies, different modifications can be employed to improve the performance of the SNS. Some of these editions are listed below:(i)
In the course of iterations, each user is affected during a randomly selected mood. Developing this random selection to an adaptive selection may affect the performance.(ii)
In the imitation mood, the new view is created inside the imitation space. A new model for this space can have a high impact.(iii)
The shock radius (R) and popularity radius (r) are two important key parameters for improving the imitation mood output.(iv)
In the imitation, conversation, and innovation moods, a random user (Xj) is selected. The selection of this user affects extremely the performance of the SNS. Another selection mechanism can be useful.(v)
The subject (Xk) in conversation mood has an effective impact on the quality of the newly generated solutions.(vi)
In conversation mood, the direction and size of movements are affected bysignfi−fj. The change of this factor in an adaptive way that affects the size of movements is desirable.(vii)
In the disputation mood, a random number of users are considered. Different strategies can be integrated with this mood. For example, different neighborhood topologies can be used. In addition, dynamic regrouping schema can be useful to improve the performance of disputation mood.(viii)
A new mood can be designed to improve the ability of the SNS by modeling another specific situation in social networks.Hybridization of the proposed algorithm with other popular algorithms is a common way to benefit from the idea of different metaheuristics to develop a more robust optimization algorithm. In addition, the ability of this algorithm should be examined in dealing with other complex real-world optimization problems in different branches of science.



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