A Spring Search Algorithm Applied to Engineering Optimization Problems

Abstract

At present, optimization algorithms are used extensively. One particular type of such algorithms includes random-based heuristic population optimization algorithms, which may be created by modeling scientific phenomena, like, for example, physical processes. The present article proposes a novel optimization algorithm based on Hooke’s law, called the spring search algorithm (SSA), which aims to solve single-objective constrained optimization problems. In the SSA, search agents are weights joined through springs, which, as Hooke’s law states, possess a force that corresponds to its length. The mathematics behind the algorithm are presented in the text. In order to test its functionality, it is executed on 38 established benchmark test functions and weighed against eight other optimization algorithms: a genetic algorithm (GA), a gravitational search algorithm (GSA), a grasshopper optimization algorithm (GOA), particle swarm optimization (PSO), teaching–learning-based optimization (TLBO), a grey wolf optimizer (GWO), a spotted hyena optimizer (SHO), as well as an emperor penguin optimizer (EPO). To test the SSA’s usability, it is employed on five engineering optimization problems. The SSA delivered better fitting results than the other algorithms in unimodal objective function, multimodal objective functions, CEC 2015, in addition to the optimization problems in engineering.

1. Introduction

As the demand for quick and accurate solutions to ever increasingly complex problems expands, classical methods are being substituted for more robust approaches. One proposal is the use of heuristic random-based algorithms in place of searching the defined problem space exhaustively [1,2,3,4,5]. Heuristic algorithms are applicable to a variety of scientific fields, such as: logistics [6], bioinformatics [7], data mining [8], chemical physics [9], energy [10], security [11], electrical engineering [12,13,14,15,16], energy carriers [17,18] as well as other fields that aim to discover the optimal solution.

Each population-based algorithm may represent the conveying of data along with the interlinkage between elements in a different way. For example, genetic algorithms simulate evolution [19] while annealing algorithms, thermodynamics [20]; immunity algorithms, the human immune system [21]; colony optimization strategies, ants’ search for food [22]; and particle swarm optimization approaches, the behavior of birds while searching for food [23].

There are many laws of nature that may serve as inspiration, such as Newton’s universal law of gravitation, Hooke’s spring law, the laws of motion, the laws of energy and mass conservation, as well as the laws that dictate the electromagnetic force. A novel optimization algorithm was proposed based on Hooke’s spring law, with its corresponding precursory results detailed in [24]. Such algorithm is detailed and analyzed in the current paper with its improved equations. The SSAs capabilities were evaluated through 23 benchmark test functions, as well as a group of problems mentioned in the Constrained Single Objective Real-Parameter Optimization Technical Report, ‘CEC’2015’. The SSA was further corroborated by being weighed against eight established algorithms found in the literature, as well as being used to solve a selection of engineering problems.

Section 2 provides a greater insight to other established optimization approaches. Section 3 elucidates Hooke’s law while Section 4 outlines the SSA. Section 5 assesses the algorithm’s search ability while Section 6, its proficiency. Section 7 includes the outcome of the evaluation through the standard benchmark test functions. Section 8 includes the implementation of the algorithm on select engineering design problems. Finally, Section 9 encompasses conclusions.

2. A Brief History of Intelligent Algorithms

An algorithm is considered intelligent when it finds a suitable answer or solution to a problem whose main characteristic is optimization in the shortest possible time and using the least amount of available information [25]. Using a more complete definition, the heuristic method is a strategy that sacrifices part of the information to reach a solution in the shortest possible time and with good precision [26]. Usually, heuristic algorithms are very frequently based on natural processes, that is, typically biological processes or laws that explain physical phenomena. This approach has been widely considered in the last ten years, and numerous algorithms have been suggested. These algorithms have been classified into different categories, such as swarm-based algorithms, evolution-based algorithms, and physics-based algorithms.

2.1. Swarm-Based Algorithms

These techniques were developed from the analysis of several processes that exist naturally, such as the growth or symbiosis of plants, the feeding behavior of insects, and the behavior and social organization of animals [27]. The particle swarm optimization (PSO) algorithm is an indeterminate (random) search method that was developed around 1995 to support functional optimization [28]. This algorithm was developed by analyzing and taking a reference to the movement that birds develop as a group (team) when looking for food. The algorithm is based on the premise that a group of birds looks for food at random and that there is only one portion of food in the area (search space) in question, but none of the birds know where the food is. One of the most successful strategies could be: for the birds to follow the bird that is closest to the food, and in sequence to be the bird most likely to find the food. This strategy is, in fact, the source of the algorithm. In the algorithm, each solution, called a particle, is equivalent to a bird in the bird movement algorithm. Each particle (bird) has an arbitrary value calculated by a success function. Each particle (bird) also has a speed that controls the particle (bird). By continuing to search for optimal particles, the agent continues to move in the solution space. The firefly algorithm (FA) is an algorithm inspired by a natural system; in this case, based on swarms for projects where limited optimal solutions are sought [29]. The algorithm is inspired by the analysis of the radiation behavior of these insects. The firefly lives in groups and changes from low light to higher light intensity. The firefly algorithm generates a rhythmic light and passes through each different light pattern or behavior among these insects. The firefly algorithm uses these lights for two main purposes: finding mates to mate with and looking for food. These lights can also serve as a protection mechanism or strategy. The whale optimization algorithm (WOA) [30] is another nature-inspired optimization algorithm; as the name implies, this algorithm mimics the social behavior that humpback whales have. The most surprising thing about humpback whales is their hunting strategy. This food search strategy is called a bubble net feeding method. Humpback whales like to hunt schools of krill or small fish near the surface of the ocean. It has been analyzed that this foraging is done by creating distinctive bubbles along a circle or route in the form of the number “9”. Some of the other swarm-based algorithms are: artificial bee colony (ABC) [31], bat-inspired algorithm (BA) [32], spotted hyena optimizer (SHO) [33], cuckoo search (CS) [34], monkey search (MS) [35], group optimization (GO) [36], artificial fish-swarm algorithm (AFSA) [37], hunting search (HS) [38], moth-flame optimization algorithm (MFO) [39], dolphin partner optimization (DPO) [40], orientation search algorithm (OSA) [41], binary orientation search algorithm (BOSA) [42], dice game optimizer (DGO) [43], shell game optimization (SGO) [44], hide objects game optimization (HOGO) [45], donkey theorem optimization (DTO) [46], following optimization algorithm (FOA) [47], rat swarm optimizer (RSO) [48], darts game optimizer (DGO) [49], football game based optimization (FGBO) [50], grey wolf optimizer (GWO) [51], grasshopper optimization algorithm (GOA) [52], coupled spring forced bat algorithm (SFBA) [53], adaptive granularity learning distributed particle swarm optimization (AGLDPSO) [54], multi leader optimizer (MLO) [55], doctor and patient optimization (DPO) [56], and emperor penguin optimizer (EPO) [57].

2.2. Evolution-Based Algorithms

An algorithm is considered evolutionary when the algorithm combines aspects of natural selection and continuity of coordination. These algorithms are based on structures that simulate the rules of selection, recombination, change, and survival, similar to genetics, hence the adjective algorithm. These structures are based on genetic sets. In this method, the environment determines each person’s coordination or performance in a population and uses the most consistent individuals to reproduce [58]. Evolutionary algorithms are random search procedures that use genetic mechanisms and natural selection [59]. Genetic algorithms (GA) were developed as a method that seeks optimization, starting from fundamental basic operations in genetic biology [60]. The first record of using these concepts to create an optimization method occurred in 1967 [61]. GA is a particular type of evolution algorithm that exploits basic biological concepts such as inheritance and mutation [62] and has had satisfactory results in different scientific domains.

On the other hand, differential evolution (DE) [63] is an algorithm that also seeks intelligent optimization based on populations introduced in 1995 [64]. The initial version of this algorithm was used to solve problems with continuous variables, and interpretations of this algorithm have been implemented over time to solve optimization problems with discrete variables [65]. Other algorithms based on the theory of evolution have been developed, examples of which are: evolutionary programming (EP) [66], biogeography-based optimizer (BBO) [67], enhanced quantum-inspired differential evolution algorithm (IQDE) [68], genetic programming (GP) [69] and evolution strategy (ES) [70].

2.3. Physics-Based Algorithms

Physics-based algorithms, as the name implies, are inspired by the laws of physics. Simulated annealing (SA) is one of the best known and most popular optimization algorithms. SA was developed in 1983 [71], inspired by metals’ annealing, for example, in the artisan process to make swords or knives in the past. The process consists of first heating the metal to a very high temperature and then cooling it by gradually reducing the temperature so that the metal hardens and becomes harder. In this process, when the temperature of the metal increases, the speed of atomic movement increases dramatically and, in the next step, the gradual reduction in temperature causes the formation of specific patterns based on the location of its atoms [20]. Drastic temperature change is one of the adjustment parameters of this algorithm. The gravitational search algorithm (GSA) [72] is inspired by the universal law of gravitation developed by Isaac Newton. In this algorithm, objects such as planets in a galaxy are defined as search agents. The optimal region, similar to a black hole, absorbs the planets. Information about the fitness of any object is stored as gravity and mass inertia. The exchange of information and the effects of objects on each other is governed by the attractive force of gravitation [73]. Several algorithms have been developed based on laws and/or theories of physics, such as: charged system search (CSS) [74], galaxy-based search algorithm (GBSA) [75], curved space optimization (CSO) [76], ray optimization (RO) algorithm [77], artificial chemical reaction optimization algorithm (ACROA) [78], small world optimization algorithm (SWOA) [79], central force optimization (CFO) [80], black hole (BH) [81] and big-bang big-crunch (BBBC) [82].

3. Spring Force Law

If a force that moves an object in a closed path (forward and backward) is not affected by the object’s trajectory, that force is conservative. Another method for diagnosing conservative forces is that the work done by the force in different paths is equal to the difference between the initial and final points. The spring force is a type of conservative force [83].

Consider a spring that imposes a force on a particle with mass ‘m’. The particle moves horizontally in the x direction. When a particle is at the origin (x = 0), the spring is balanced. An external force ($F_{ext}$) influences the object in the anti-clock wise direction of the spring. The external force is always equal to the spring force. Thus, the particle is always in balance.

Consider that the particle is moved a distance x from its initial location x = 0. When the external factor imposes a force $F_{ext}$ on the particle, the spring offers a resistance force F_s on the particle. This force can be described by the spring force or Hooke’s law as follows:

$$F_s=-kx$$

(1)

where k is the spring force constant and $x$ denotes the spring displacement (strain or compression) from the balance point. Most real springs properly follow Hooke’s law up to a limit [84].

The system behavior is investigated in an isolated system as shown in Figure 1. It is assumed that only the spring force is imposed on the object.

In Figure 1, forces imposed on object j can be grouped into two, $F_{tota{l}_R}$, which variables as described in Equation (2), is the sum of forces imposed from the right and $F_{tota{l}_L}$, which, as shown in Equation (3), is the sum of forces imposed from the left. It is necessary to mention that the springs that are attached to the object from either right or left, are also attached to robust points at their other ends.

$$F_{tota{l}_R}^j= {\displaystyle \sum }_{i=1}^{n_R}{K}_{i,j}{x}_{i,j}$$

(2)

$$F_{tota{l}_L}^j= {\displaystyle \sum }_{l=1}^{n_L}{K}_{l,j}{x}_{l,j}$$

(3)

where, $n_R$ and $n_L$ are the number of left and right spring forces, $x_{i,j}$ and $x_{l,j}$ show the distance between the object j and the fixed left and right points, $K_{i,j}$ and $K_{l,j}$ are the spring stiffness coefficients between the object j and the fixed points.

The object is initially balanced with no force exerted on it. Then, by applying the spring forces, the object is pulled from the right and the left. Considering the magnitude of these forces, the spring either shifts to left or right until the system reaches a new equilibrium position. It is apt to mention that if the right and left forces are equal, the object remains at its original position.

Considering the stiffness coefficient of the springs that are connected to the object, two new parameters may be defined as below:

$$K_{equa{l}_R}^j= {\displaystyle \sum }_{i=1}^{n_R}{K}_{i,j}$$

(4)

$$K_{equa{l}_L}^j= {\displaystyle \sum }_{l=1}^{n_L}{K}_{l,j}$$

(5)

$K_{equa{l}_R}^j$ and $K_{equa{l}_L}^j$ are the right and the left constants of the spring, respectively. Considering Equation (1) the displacement values at each side may be defined as follows:

$$d{X}_R^j=\frac{F_{tota{l}_R}^j}{K_{equa{l}_R}^j}$$

(6)

$$d{X}_L^j=\frac{F_{tota{l}_L}^j}{K_{equa{l}_L}^j}$$

(7)

here, $d{X}_R^j$ and $d{X}_L^j$ are the displacement values of object j to the right and left, respectively. Therefore, the total displacement may be defined as follows:

$$X^j=d{X}_R^j+d{X}_L^j$$

(8)

$d{X}^j$ is the final object j displacement value that may be a positive or negative value.

$$X^j=X_0^j+d{X}^j$$

(9)

Equation (9), $X^j$ relates to the location of the new balance point of the system and object j. Besides, $X_0^j$ is the initial balance of object j.

By simulating Hooke’s law within the discrete time domain, a new optimization algorithm called the spring optimization was designed, which is explained further in the following section.

4. Spring Search Algorithm (SSA)

In this article, the spring search algorithm is run in an artificial system with discrete time. The system space is the defined domain of problem. It is possible to apply the spring force law as a tool to convey information. The designed optimization may be applied to solve any optimization problem, as long as the answer can be defined as a position within space, and its similarity to other problem answers can be expressed as spring stiffness comparisons. The stiffness of the spring is established relative to the objective function.

SSAs consist of two general steps: making a discrete time artificial system within the problem space by defining the initial positioning of objects, determining the governing laws and arranging parameters; letting the algorithm run until it reaches a stop.

4.1. Setting the System, Determining Laws and Arranging Parameters

First, the system space is determined with its multi-dimensional coordinate system in which the problem is defined. Any point in space may be the solution of the optimization problem. Search factors are sets of objects that are attached to each other by some springs. Each object has its own position as well as stiffness coefficients pertaining to the springs attached to it. The object’s position is a point in space where the solution of the problem may be found. The values of the springs are computed regarding the stiffness of both objects attached.

After setting the system, its governing laws are determined. Only the spring force laws and movement laws are observed in the system. The general patterns of these rules are similar to natural laws and are defined below.

In physics, mechanical and elastic science, Hooke’s law is an approximation that shows that any change in an object’s length is directly proportional to its load. Most materials follow this law with reasonable accuracy, except when the force is lower than its elasticity. Any deviation from Hooke’s law increases with the deformation quantity, such that with numerous deformations, when the material trespasses its linear elastic domain, Hooke’s law loses its application. The present article assumes that Hooke’s law is valid for all of the time observed.

The present location of any object equals the sum coefficient of its previous locations and its displacements according to the laws of motion. Any object displacement may be determined with the aid of the spring force law.

Consider a system as a set of m objects where the position of each object is a point in the search space and a solution to the problem.

$$X_i=\left(x_i^1,\dots ,x_i^d,\dots ,x_i^n\right)$$

(10)

The position of an object i of dimension d is designated $x_i^d$ in Equation (10). The initial positions of the objects are defined within the search space randomly. These objects tend to return to an equilibrium position by means of the forces exerted by the spring.

Equation (11) is employed in order to compute the spring stiffness coefficient.

$$K_{i,j}=K_{max}\times \left|F_n^i-F_n^j\right|\times \max \left(F_n^i,F_n^j\right)$$

(11)

In Equation (11), $K_{i,j}$ is the spring stiffness coefficient among objects i and j, $K_{max}$ is the maximum value of the spring stiffness coefficient, and is determined according to the type of problem in question, $F_n^i$ and $F_n^j$ are the normalized objective functions of objects i and j respectively. Equations (12) and (13) are used in order to normalize the objective function.

$${F^{\prime }}_n^i=\frac{f_{obj}^i}{\min \left(f_{obj}\right)}$$

(12)

$$F_n^i=\min \left({F^{\prime }}_n^i\right)\times \frac{1}{{F^{\prime }}_n^i}$$

(13)

In the above equations, F_obj is the objective function and $f_{obj}^i$ is the objective function value for the object i.

In the m variable problem, it is possible to assume that the problem has m dimensions and that there is a coordinate for each dimension. Therefore, it is possible to draw a system on its related coordinate based on each variable. Each coordinate’s strong points at the left or right side of the object are determined by comparing the objective function quantities. Stronger points related to each object mean that they are positioned at more optimal positions. Therefore, each coordinate has two general summative forces: the sum of the right, Equation (14) and the sum of the left, Equation (15). Both are applied to the object j.

$$F_{tota{l}_R}^{j_{,d}}={\displaystyle \sum }_{i=1}^{n_R^d}{K}_{i,j}{x}_{i,j}^d$$

(14)

$$F_{tota{l}_L}^{j_{,d}}={\displaystyle \sum }_{l=1}^{n_L^d}{K}_{l,j}{x}_{l,j}^d$$

(15)

In the above equations, $F_{tota{l}_R}^{j_{,d}}$ stands for the sum of the right forces and $F_{tota{l}_L}^{j_{,d}}$ the sum of the left forces imposed on object j within the dimension d. $n_R^d$ and $n_L^d$ are the d dimension right and left strong points; $x_{i,j}^d$ and $x_{l,j}^d$ represent the distance of the object ‘j’ from the right and left strong points; $K_{i,j}$ and $K_{l,j}$ are the spring stiffness coefficient attached to the object j and the strong points.

Now, considering Hooke’s law in the dimension d:

$$d{X}_R^{j,d}=\frac{F_{tota{l}_R}^{j,d}}{K_{equa{l}_R}^j}$$

(16)

$$d{X}_L^{j,d}=\frac{F_{tota{l}_L}^{j,d}}{K_{equa{l}_L}^j}$$

(17)

here, $d{X}_R^{j,d}$ and $d{X}_L^{j,d}$ are the displacement of object j to the right and the left of dimension d, respectively. The total displacement may be calculated as follows:

$$d{X}^{j,d}=d{X}_R^{j,d}+d{X}_L^{j,d}$$

(18)

where $d{X}^{j,d}$ is the final displacement of object j in the dimension d. The direction may be in the positive or negative x direction.

$$X_{new}^{j,d}=X_0^{j,d}+r_1 \times d{X}^{j,d},$$

(19)

Equation (19) relates $X_{new}^{j,d}$ to both the new position and balance point of the system, along with the d dimension of object j. Additionally, $X_0^{j,d}$ is the initial balance point of object j along with the d dimension. Here, $r_1$ is a random number with a uniform distribution within the span of $\left[0–1\right]$, which is used to preserve the random mode.

In the last step, the objects and springs after reaching balance have a small displacement due to slipping, which is simulated in Equation (20).

$$X^{j,d}=X_{new}^{j,d}+\frac{2\times \left(T-t\right)}{T} \times \left(-0.2+r_2\times 0.4\right)\times X_{new}^{j,d}$$

(20)

here, $X^{j,d}$ is the updated position of d dimension of object j, $T$ is the maximum number of iteration, $t$ is the counter of iteration, and $r_2$ is a random number with a uniform distribution within the span of $\left[0–1\right]$, which is used to preserve the random mode.

4.2. Time Passing and Parameters Updating

While initially forming a system, each object is randomly placed within a point in space where it may be the problem’s answer. At any given time, objects are assessed and their displacement is calculated using Equations (11) through (18). Thereafter, the object is placed in the new computed position. The parameter of interest is the spring stiffness coefficient that is updated at each stage based on Equation (11). The stop condition is established after the algorithm has been run for a given time. The steps of the SSA are as follows and a flowchart encompassing them is shown in Figure 2:

1-

Start

2-

Determine the system environment and the problem information

3-

Create the initial population of objects

4-

Evaluate and normalize fitness function or objective function

5-

Update parameter K

6-

Formulate the spring force and the laws of motion for each object

7-

Compute object displacement quantities

8-

Update object locations

9-

Repeat steps 4 through 8 until the stop condition is satisfied

10-

Print best solution

11-

End

5. Properties of the Proposed SSA

The above algorithm is a proposed optimization method that applies the spring force law. The strategy of the algorithm proposed is to use the spring fitness coefficient. In this algorithm, a set of objects search the space randomly using the spring force as a tool to transfer information between objects. Under the influence of other objects, each may arrive at a rough understanding of its surrounding space. The algorithm must be navigated such that the objects’ location improves as time passes.

Thus, springs with more fitness coefficients can be attached to those with better fitness functions. The springs attract other objects to themselves, allowing a suitable force to be exerted on each object. Objects tend to move towards better conditions as time goes by. Accordingly, objects placed in better locations must take slower and shorter steps. As an object arrives in a better condition, its stiffness coefficient increases. The stiffer objects search their surrounding environment with more precision. Indeed, this behavior, known here as the adjustment, is similar to arranging the learning rate in a neural network. The spring’s stiffness coefficient becomes smaller with the passing of time as a result of the spring’s force. Another reason why the spring stiffness coefficient decreases with time is that the objects tend to concentrate around better locations and need to search space with smaller and more precise steps.

Each object can influence the radius of its neighborhood according to its fitness value. As indicated in Figure 3, each object can move as influenced by the spring forces imposed on it.

6. Exploration and Exploitation in SSA

The optimization method must address two issues: exploration and exploitation. In the exploration aspect, the algorithm must have enough power to search the problem search space well and not be limited to only a few specific locations. The algorithm tackles exploitation by focusing on exploring optimal locations. Before running a population algorithm, it is necessary to search the designated space comprehensively. Hence, the algorithm must focus on searching for the solution’s general area during initial iterations, though as the time passes, it must locate itself more efficiently through aid from the population findings [85].

SSAs can search space by considering the suitable number of objects. The way the optimization algorithm improves its detection power is through the spring force effect by varying the spring’s stiffness coefficient and consequently controlling the spring forces among the objects. During initial iterations, the problem needs thorough searches, and as time passes, the population arrives at better results and the spring stiffness coefficient value is controlled. At the initial time, a proper value is chosen, and as time passes, that value decreases though the spring stiffness coefficient (Equation (11)) until it arrives at its minimum value.

7. Experimental Results and Discussion

This section presents the results from the evaluation of the SSA’s performance on twenty-three standard benchmark test functions. A detailed description of these benchmark functions is presented below. Furthermore, the results are compared to eight existing optimization algorithms.

7.1. Benchmark Test Functions

The performance of the SSA was assessed by using 23 benchmark test functions [86]. The experimentation was done on MATLAB version R2014a (8.3.0.532) in a Microsoft Windows 7 environment using a 64 bit Core i-7 processor with 2.40 GHz and 16 GB main memory. The average and standard deviation of the best optimal solutions are displayed in

Table 1. Results for SSA and other algorithms in Unimodal test functions.

		GA	PSO	GSA	TLBO	GOA	GWO	SHO	EPO	SSA
F1	Ave	1.95 × 10−12	4.98 × 10−09	1.16 × 10−16	3.55 × 10−02	2.81 × 10−01	7.86 × 10−10	4.61 × 10−23	5.71 × 10−28	6.74 × 10−35
	std	2.01 × 10−11	1.40 × 10−08	6.10 × 10−17	1.06 × 10−01	1.11 × 10−01	8.11 × 10−09	7.37 × 10−23	8.31 × 10−29	9.17 × 10−36
F2	Ave	6.53 × 10−18	7.29 × 10−04	1.70 × 10−01	3.23 × 10−05	3.96 × 10−01	5.99 × 10−20	1.20 × 10−34	6.20 × 10−40	7.78 × 10−45
	std	5.10 × 10−17	1.84 × 10−03	9.29 × 10−01	8.57 × 10−05	1.41 × 10−01	1.11 × 10−17	1.30 × 10−34	3.32 × 10−40	3.48 × 10−45
F3	Ave	7.70 × 10−10	1.40 × 1001	4.16 × 1002	4.91 × 1003	4.31 × 1001	9.19 × 10−05	1.00 × 10−14	2.05 × 10−19	2.63 × 10−25
	std	7.36 × 10−09	7.13 × 1000	1.56 × 1002	3.89 × 1003	8.97 × 1000	6.16 × 10−04	4.10 × 10−14	9.17 × 10−20	9.83 × 10−27
F4	Ave	9.17 × 1001	6.00 × 10−01	1.12 × 1000	1.87 × 1001	8.80 × 10−01	8.73 × 10−01	2.02 × 10−14	4.32 × 10−18	4.65 × 10−26
	std	5.67 × 1001	1.72 × 10−01	9.89 × 10−01	8.21 × 1000	2.50 × 10−01	1.19 × 10−01	2.43 × 10−14	3.98 × 10−19	4.68 × 10−29
F5	Ave	5.57 × 1002	4.93 × 1001	3.85 × 1001	7.37 × 1002	1.18 × 1002	8.91 × 1002	2.79 × 1001	5.07 × 1000	5.41 × 10−01
	std	4.16 × 1001	3.89 × 1001	3.47 × 1001	1.98 × 1003	1.43 × 1002	2.97 × 1002	1.84 × 1000	4.90 × 10−01	5.05 × 10−02
F6	Ave	3.15 × 10−01	9.23 × 10−09	1.08 × 10−16	4.88 × 1000	3.15 × 10−01	8.18 × 10−17	6.58 × 10−01	7.01 × 10−19	8.03 × 10−24
	std	9.98 × 10−02	1.78 × 10−08	4.00 × 10−17	9.75 × 10−01	9.98 × 10−02	1.70 × 10−18	3.38 × 10−01	4.39 × 10−20	5.22 × 10−26
F7	Ave	6.79 × 10−04	6.92 × 10−02	7.68 × 10−01	3.88 × 10−02	2.02 × 10−02	5.37 × 10−01	7.80 × 10−04	2.71 × 10−05	3.33 × 10−08
	std	3.29 × 10−03	2.87 × 10−02	2.77 × 1000	5.79 × 10−02	7.43 × 10−03	1.89 × 10−01	3.85 × 10−04	9.26 × 10−06	1.18 × 10−06

,

Table 2. Results for SSA and other algorithms in Multimodal test functions.

		GA	PSO	GSA	TLBO	GOA	GWO	SHO	EPO	SSA
F8	Ave	−5.11 × 1002	−5.01 × 1002	−2.75 × 1002	−3.81 × 1002	−6.92 × 1002	−4.69 × 1001	−6.14 × 1002	−8.76 × 1002	−1.2 × 1004
	std	4.37 × 1001	4.28 × 1001	5.72 × 1001	2.83 × 1001	9.19 × 1001	3.94 × 1001	9.32 × 1001	5.92 × 1001	9.14 × 10−12
F9	Ave	1.23 × 10−01	1.20 × 10−01	3.35 × 1001	2.23 × 1001	1.01 × 1002	4.85 × 10−02	4.34 × 10−01	6.90 × 10−01	8.76 × 10−04
	std	4.11 × 1001	4.01 × 1001	1.19 × 1001	3.25 × 1001	1.89 × 1001	3.91 × 1001	1.66 × 1000	4.81 × 10−01	4.85 × 10−02
F10	Ave	5.31 × 10−11	5.20 × 10−11	8.25 × 10−09	1.55 × 1001	1.15 × 1000	2.83 × 10−08	1.63 × 10−14	8.03 × 10−16	8.04 × 10−20
	std	1.11 × 10−10	1.08 × 10−10	1.90 × 10−09	8.11 × 1000	7.87 × 10−01	4.34 × 10−07	3.14 × 10−15	2.74 × 10−14	3.34 × 10−18
F11	Ave	3.31 × 10−06	3.24 × 10−06	8.19 × 1000	3.01 × 10−01	5.74 × 10−01	2.49 × 10−05	2.29 × 10−03	4.20 × 10−05	4.23 × 10−10
	std	4.23 × 10−05	4.11 × 10−05	3.70 × 1000	2.89 × 10−01	1.12 × 10−01	1.34 × 10−04	5.24 × 10−03	4.73 × 10−04	5.11 × 10−07
F12	Ave	9.16 × 10−08	8.93 × 10−08	2.65 × 10−01	5.21 × 1001	1.27 × 1000	1.34 × 10−05	3.93 × 10−02	5.09 × 10−03	6.33 × 10−05
	std	4.88 × 10−07	4.77 × 10−07	3.14 × 10−01	2.47 × 1002	1.02 × 1000	6.23 × 10−04	2.42 × 10−02	3.75 × 10−03	4.71 × 10−04
F13	Ave	6.39 × 10−02	6.26 × 10−02	5.73 × 10−32	2.81 × 1002	6.60 × 10−02	9.94 × 10−08	4.75 × 10−01	1.25 × 10−08	0.00 × 1000
	std	4.49 × 10−02	4.39 × 10−02	8.95 × 10−32	8.63 × 1002	4.33 × 10−02	2.61 × 10−07	2.38 × 10−01	2.61 × 10−07	0.00 × 1000

and

Table 3. Results for SSA and other algorithms in Multimodal test functions with low dimension.

		GA	PSO	GSA	TLBO	GOA	GWO	SHO	EPO	SSA
F14	Ave	4.39 × 1000	2.77 × 1000	3.61 × 1000	6.79 × 1000	9.98 × 1001	1.26 × 1000	3.71 × 1000	1.08 × 1000	9.98 × 10−01
	std	4.41 × 10−02	2.32 × 1000	2.96 × 1000	1.12 × 1000	9.14 × 10−1	6.86 × 10−01	3.86 × 1000	4.11 × 10−02	7.64 × 10−12
F15	Ave	7.36 × 10−02	9.09 × 10−03	6.84 × 10−02	5.15 × 10−02	7.15 × 10−02	1.01 × 10−02	3.66 × 10−02	8.21 × 10−03	3.3 × 10−04
	std	2.39 × 10−03	2.38 × 10−03	7.37 × 10−02	3.45 × 10−03	1.26 × 10−01	3.75 × 10−03	7.60 × 10−02	4.09 × 10−03	1.25 × 10−05
F16	Ave	−1.02 × 1000	−1.02 × 1000	−1.02 × 1000	−1.01 × 1000	−1.02 × 1000	−1.02 × 1000	−1.02 × 1000	−1.02 × 1000	−1.03 × 1000
	std	4.19 × 10−07	0.00 × 1000	0.00 × 1000	3.64 × 10−08	4.74 × 10−08	3.23 × 10−05	7.02 × 10−09	9.80 × 10−07	5.12 × 10−10
F17	Ave	3.98 × 10−01	3.98 × 10−01	3.98 × 10−01	3.98 × 10−01	3.98 × 10−01	3.98 × 10−01	3.98 × 10−01	3.98 × 10−01	3.98 × 10−01
	std	3.71 × 10−17	9.03 × 10−16	1.13 × 10−16	9.45 × 10−15	1.15 × 10−07	7.61 × 10−04	7.00 × 10−07	5.39 × 10−05	4.56 × 10−21
F18	Ave	3.00 × 1000	3.00 × 1000	3.00 × 1000	3.00 × 1000	3.00 × 1000	3.00 × 1000	3.00 × 1000	3.00 × 1000	3.00 × 1000
	std	6.33 × 10−07	6.59 × 10−05	3.24 × 10−02	1.94 × 10−10	1.48 × 1001	2.25 × 10−05	7.16 × 10−06	1.15 × 10−08	1.15 × 10−18
F19	Ave	−3.81 × 1000	−3.80 × 1000	−3.86 × 1000	−3.73 × 1000	−3.77 × 1000	−3.75 × 1000	−3.84 × 1000	−3.86 × 1000	−3.86 × 1000
	std	4.37 × 10−10	3.37 × 10−15	4.15 × 10−01	9.69 × 10−04	3.53 × 10−07	2.55 × 10−03	1.57 × 10−03	6.50 × 10−07	5.61 × 10−10
F20	Ave	−2.39 × 1000	−3.32 × 1000	−1.47 × 1000	−2.17 × 1000	−3.23 × 1000	−2.84 × 1000	−3.27 × 1000	−2.81 × 1000	−3.31 × 1000
	std	4.37 × 10−01	2.66 × 10−01	5.32 × 10−01	1.64 × 10−01	5.37 × 10−02	3.71 × 10−01	7.27 × 10−02	7.11 × 10−01	4.29 × 10−05
F21	Ave	−5.19 × 1000	−7.54 × 1000	−4570	−7.33 × 1000	−7.38 × 1000	−2.28 × 1000	−9.65 × 1000	−8.07 × 1000	−10.15 × 1000
	std	2.34 × 1000	2.77 × 1000	1.30 × 1000	1.29 × 1000	2.91 × 1000	1.80 × 1000	1.54 × 1000	2.29 × 1000	1.25 × 10−02
F22	Ave	−2.97 × 1000	−8.55 × 1000	−6.58 × 1000	−1.00 × 1000	−8.50 × 1000	−3.99 × 1000	−1.04 × 1000	−10.01 × 1000	−10.40 × 1000
	std	1.37 × 10−02	3.08 × 1000	2.64 × 1000	2.89 × 10−04	3.02 × 1000	1.99 × 1000	2.73 × 10−04	3.97 × 10−02	3.65 × 10−07
F23	Ave	−3.10 × 1000	−9.19 × 1000	−9.37 × 1000	−2.46 × 1000	−8.41 × 1000	−4.49 × 1000	−1.05 × 1001	−3.41 × 1000	−10.53 × 1000
	std	2.37 × 1000	2.52 × 1000	2.75 × 1000	1.19 × 1000	3.13 × 1000	1.96 × 1000	1.81 × 10−04	1.11 × 10−02	5.26 × 10−06

. For each benchmark test function, the SSA utilizes 20 independent runs, in which each run employs 1000 iterations.

7.2. Algorithms Used for Comparison

In order to prove the potency of the SSA, it is also compared to eight optimization algorithms on unimodal, multimodal, fixed-dimension multimodal and composite optimization. They were assessed by solving a set of minimization problems introduced in the Constrained Single Objective Real-Parameter Optimization Technical Report, ‘CEC’2015’ [86]. To validate the performance of the SSA, the eight optimization algorithms included: GA [87], PSO [29], GSA [73], TLBO [88], GOA [53], GWO [52], SHO [34], and EPO [58].

The parameter values of optimization algorithms are given in the next table.

Parameter definition

1: GA  2: Population size N = 80  3: Crossover 0.9  4: Mutation 0.05  5: PSO  6: Swarm size S = 50  7: Inertia weight decreases linearly from 0.9 to 0.4  8: C1(individual-best acceleration factor) increases linearly from 0.5 to 2.5  9: C2(global-best acceleration factor) decreases linearly from 2.5 to 0.5 10: GSA 11: Objects number N = 50 12: Acceleration coefficient (a = 20) 13: Initial gravitational constant (G0 = 100) 14: TLBO 15: Swarm size S = 50 16: GOA 17: Search Agents N = 100 18: $C_{max}=1$ 19: $C_{min}=4\times {10}^{-5}$ 20: l = 1.5 and f = 0.5 21: GWO 22: Wolves number = 50 23: a variable decreases linearly from 2 to 0 24: SHO 25: Search Agents N = 80 26: M Constant $\left[0.5, 1\right]$ 27: Control Parameter (h) $\left[5, 0\right]$ 28: EPO 29: Search Agents N = 80 30: Temperature Profile $\left[1, 1000\right]$ 31: A Constant $\left[-1.5, 1.5\right]$ 32: Function S() $\left[0, 1.5\right]$ 33: Parameter M = 2 34: Parameter f $\left[2, 3\right]$ 35: Parameter l $\left[1.5, 2\right]$

7.2.1. Evaluation of Unimodal Test Function with High Dimensions

Functions F₁ to F₇ are unimodal. The mean results of 20 independent runs of the algorithm are displayed in Table 1. These results show that the SSA has a better performance in all F₁ to F₇ functions than other algorithms.

7.2.2. Evaluation of Multimodal Test Functions with High Dimensions

In multimodal functions F₈ to F₁₃, by increasing the function dimensions, the number of local responses increased exponentially. Therefore, arriving at the minimum response of these functions is hard to achieve. In these types of functions, arriving at a response close to the ideal response denotes the algorithm’s high capability in avoiding wrong local responses. The results obtained from assessing F₈ to F₁₃ after 20 runs of the SSA as well as the other algorithms compared are presented in Table 2. The SSA demonstrated the best performance of all the functions analyzed.

7.2.3. Evaluation of Multimodal Test Functions with Low Dimensions

Functions F₁₄ to F₂₃ have both low dimensions in addition to low local responses. Results obtained from 20 runs of each algorithm, are shown in Table 3. These results indicate that the SSA has a suitable performance in all functions F₁₄ to F₂₃.

7.2.4. Evaluation of CEC 2015 Test Functions

This section is devoted to real approaches and techniques for solving single objective optimization problems. All of these test functions are minimization problems.

Table 4. Results for SSA and other algorithms in CEC 2015.

		GA	PSO	GSA	TLBO	GOA	GWO	SHO	EPO	SSA
Cec-1	Ave	8.89 × 1006	3.20 × 1007	7.65 × 1006	1.47 × 1006	4.37 × 1005	2.02 × 1006	2.28 × 1006	6.06 × 1005	1.50 × 1005
	std	6.95 × 1006	8.37 × 1006	3.07 × 1006	2.63 × 1006	4.73 × 1005	2.08 × 1006	2.18 × 1006	5.02 × 1005	1.21 × 1005
Cec-2	Ave	2.97 × 1005	6.58 × 1003	7.33 × 1008	1.97 × 1004	9.41 × 1003	5.65 × 1006	3.13 × 1005	1.43 × 1004	4.40 × 1003
	std	2.85 × 1003	1.09 × 1003	2.33 × 1008	1.46 × 1004	1.08 × 1004	6.03 × 1006	4.19 × 1005	1.03 × 1004	1.34 × 1008
Cec-3	Ave	3.20 × 1002	3.20 × 1002	3.20 × 1002	3.20 × 1002	3.20 × 1002	3.20 × 1002	3.20 × 1002	3.20 × 1002	3.20 × 1002
	std	2.78 × 10−02	1.11 × 10−05	7.53 × 10−02	3.19 × 10−02	9.14 × 10−02	8.61 × 10−02	7.08 × 10−02	3.76 × 10−02	1.16 × 10−06
Cec-4	Ave	6.99 × 1002	4.39 × 1002	4.42 × 1002	4.18 × 1002	4.26 × 1002	4.09 × 1002	4.16 × 1002	4.11 × 1002	4.04 × 1002
	std	6.43 × 1000	7.25 × 1000	7.72 × 1000	1.03 × 1001	1.17 × 1001	3.96 × 1000	1.03 × 1001	1.71 × 1001	5.61 × 1000
Cec-5	Ave	1.26 × 1003	1.75 × 1003	1.76 × 1003	1.09 × 1003	1.33 × 1003	8.65 × 1002	9.20 × 1002	9.13 × 1002	9.81 × 1002
	std	1.86 × 1002	2.79 × 1002	2.30 × 1002	2.81 × 1002	3.45 × 1002	2.16 × 1002	1.78 × 1002	1.85 × 1002	1.06 × 1002
Cec-6	Ave	2.91 × 1005	3.91 × 1006	2.30 × 1004	3.82 × 1003	7.35 × 1003	1.86 × 1003	2.26 × 1004	1.29 × 1004	1.05 × 1003
	std	1.67 × 1005	2.70 × 1006	2.41 × 1004	2.44 × 1003	3.82 × 1003	1.93 × 1003	2.45 × 1004	1.15 × 1004	1.05 × 1003
Cec-7	Ave	7.08 × 1002	7.08 × 1002	7.06 × 1002	7.02 × 1002	7.02 × 1002	7.02 × 1002	7.02 × 1002	7.02 × 1002	7.02 × 1002
	std	2.97 × 1000	1.32 × 1000	9.07 × 10−01	9.40 × 10−01	1.10 × 1000	7.75 × 10−01	7.07 × 10−01	6.76 × 10−01	5.50 × 10−01
Cec-8	Ave	5.79 × 1004	6.07 × 1005	6.73 × 1003	2.58 × 1003	9.93 × 1003	3.43 × 1003	3.49 × 1003	1.86 × 1003	1.47 × 1003
	std	2.76 × 1004	4.81 × 1005	3.36 × 1003	1.61 × 1003	8.74 × 1003	2.77 × 1003	2.04 × 1003	1.98 × 1003	1.34 × 1003
Cec-9	Ave	1.00 × 1003	1.00 × 1003	1.00 × 1003	1.00 × 1003	1.00 × 1003	1.00 × 1003	1.00 × 1003	1.00 × 1003	1.00 × 1003
	std	3.97 × 1000	5.33 × 1000	9.79 × 10−01	5.29 × 10−02	2.20 × 10−01	7.23 × 10−02	1.28 × 10−01	1.43 × 10−01	1.51 × 10−03
Cec-10	Ave	4.13 × 1004	3.42 × 1005	9.91 × 1003	2.62 × 1003	8.39 × 1003	3.27 × 1003	4.00 × 1003	2.00 × 1003	1.23 × 1003
	std	2.39 × 1004	1.74 × 1005	8.83 × 1003	1.78 × 1003	1.12 × 1004	1.84 × 1003	2.82 × 1003	2.73 × 1003	1.51 × 1003
Cec-11	Ave	1.36 × 1003	1.41 × 1003	1.35 × 1003	1.39 × 1003	1.37 × 1003	1.35 × 1003	1.40 × 1003	1.38 × 1003	1.35 × 1003
	std	5.39 × 1001	7.73 × 1001	1.11 × 1002	5.42 × 1001	8.97 × 1001	1.12 × 1002	5.81 × 1001	2.42 × 1001	1.01 × 1001
Cec-12	Ave	1.31 × 1003	1.31 × 1003	1.31 × 1003	1.30 × 1003	1.30 × 1003	1.30 × 1003	1.30 × 1003	1.30 × 1003	1.30 × 1003
	std	1.65 × 1000	2.05 × 1000	1.54 × 1000	8.07 × 10−01	9.14 × 10−01	6.94 × 10−01	6.69 × 10−01	7.89 × 10−01	1.50 × 10−01
Cec-13	Ave	1.35 × 1003	1.35 × 1003	1.30 × 1003	1.30 × 1003	1.30 × 1003	1.30 × 1003	1.30 × 1003	1.30 × 1003	1.30 × 1003
	std	3.97 × 1001	4.70 × 1001	3.78 × 10−03	2.43 × 10−04	1.04 × 10−03	5.44 × 10−03	1.92 × 10−04	2.76 × 10−04	6.43 × 10−05
Cec-14	Ave	8.96 × 1003	9.30 × 1003	7.51 × 1003	7.34 × 1003	7.60 × 1003	7.10 × 1003	7.29 × 1003	4.25 × 1003	3.22 × 1003
	std	6.32 × 1003	4.04 × 1002	1.52 × 1003	2.47 × 1003	1.29 × 1003	3.12 × 1003	2.45 × 1003	1.73 × 1003	2.12 × 1002
Cec-15	Ave	1.63 × 1003	1.64 × 1003	1.62 × 1003	1.60 × 1003	1.61 × 1003	1.60 × 1003	1.61 × 1003	1.60 × 1003	1.60 × 1003
	std	3.67 × 1001	1.12 × 1001	3.64 × 1000	1.80 × 10−02	1.13 × 1001	2.66 × 1000	4.94 × 1000	3.76 × 1000	5.69 × 10−01

shows the performance of the SSA as well as the other algorithms on the CEC 2015 test. Table 4 exhibits how the SSA is the most efficient optimizer of all of the benchmark test functions studied.

8. SSA for Engineering Design Problems

In this section, the SSA is applied to five constrained engineering design problems.

8.1. Pressure Vessel Design

The mathematical model of this problem was adapted from a paper by Kannan and Kramer [89].

Table 5. Comparison results for pressure vessel design problem.

Algorithms		Optimum Variables			Optimum Cost
	Ts	Th	R	L
SSA	0.778099	0.383241	40.315121	200.00000	5880.0700
EPO	0.778210	0.384889	40.315040	200.00000	5885.5773
SHO	0.779035	0.384660	40.327793	199.65029	5889.3689
GOA	0.778961	0.384683	40.320913	200.00000	5891.3879
GWO	0.845719	0.418564	43.816270	156.38164	6011.5148
TLBO	0.817577	0.417932	41.74939	183.57270	6137.3724
GSA	1.085800	0.949614	49.345231	169.48741	11,550.2976
PSO	0.752362	0.399540	40.452514	198.00268	5890.3279
GA	1.099523	0.906579	44.456397	179.65887	6550.0230

and

Table 6. Statistical results for pressure vessel design problem.

Algorithms	Best	Mean	Worst	Std. Dev.	Median
	SSA	5880.0700	5891.3099	024.341	5883.5153
EPO	5885.5773	5887.4441	5892.3207	002.893	5886.2282
SHO	5889.3689	5891.5247	5894.6238	013.910	5890.6497
GOA	5891.3879	6531.5032	7394.5879	534.119	6416.1138
GWO	6011.5148	6477.3050	7250.9170	327.007	6397.4805
TLBO	6137.3724	6326.7606	6512.3541	126.609	6318.3179
GSA	11,550.2976	23,342.2909	33,226.2526	5790.625	24,010.0415
PSO	5890.3279	6264.0053	7005.7500	496.128	6112.6899
GA	6550.0230	6643.9870	8005.4397	657.523	7586.0085

show the performance of the SSA along with the other algorithms. The SSA provides an optimal solution at (0.778099, 0.383241, 40.315121, 200.00000), with a corresponding fitness value of 5880.0700.

8.2. Speed Reducer Design Problem

This problem is modeled mathematically in [90,91]. The results of the optimization problem are presented in

Table 7. Comparison results for speed reducer design problem.

Algorithms				Optimum Variables				Optimum Cost
	b	m	p	l1	l2	d1	d2
SSA	3.50123	0.7	17	7.3	7.8	3.33421	5.26536	2994.2472
EPO	3.50159	0.7	17	7.3	7.8	3.35127	5.28874	2998.5507
SHO	3.506690	0.7	17	7.380933	7.815726	3.357847	5.286768	3001.288
GOA	3.500019	0.7	17	8.3	7.8	3.352412	5.286715	3005.763
GWO	3.508502	0.7	17	7.392843	7.816034	3.358073	5.286777	3002.928
TLBO	3.508755	0.7	17	7.3	7.8	3.461020	5.289213	3030.563
GSA	3.600000	0.7	17	8.3	7.8	3.369658	5.289224	3051.120
PSO	3.510253	0.7	17	8.35	7.8	3.362201	5.287723	3067.561
GA	3.520124	0.7	17	8.37	7.8	3.366970	5.288719	3029.002

and

Table 8. Statistical results for speed reducer design problem.

Algorithms	Best	Mean	Worst	Std. Dev.	Median
SSA	2994.2472	2997.482	2999.092	1.78091	2996.318
EPO	2998.5507	2999.640	3003.889	1.93193	2999.187
SHO	3001.288	3005.845	3008.752	5.83794	3004.519
GOA	3005.763	3105.252	3211.174	79.6381	3105.252
GWO	3002.928	3028.841	3060.958	13.0186	3027.031
TLBO	3030.563	3065.917	3104.779	18.0742	3065.609
GSA	3051.120	3170.334	3363.873	92.5726	3156.752
PSO	3067.561	3186.523	3313.199	17.1186	3198.187
GA	3029.002	3295.329	3619.465	57.0235	3288.657

. The optimal solution was provided by the SSA at (3.50123, 0.7, 17, 7.3, 7.8, 3.33421, 5.26536) with a corresponding fitness value equal to 2994.2472.

8.3. Welded Beam Design

The mathematical model of a welded beam design was adapted from [31]. The results to this optimization problem are presented in

Table 9. Comparison results for welded beam design problem.

Algorithms	Optimum Variables				Optimum Cost
	h	l	t	b
SSA	0.205411	3.472341	9.035215	0.201153	1.723589
EPO	0.205563	3.474846	9.035799	0.205811	1.725661
SHO	0.205678	3.475403	9.036964	0.206229	1.726995
GOA	0.197411	3.315061	10.00000	0.201395	1.820395
GWO	0.205611	3.472103	9.040931	0.205709	1.725472
TLBO	0.204695	3.536291	9.004290	0.210025	1.759173
GSA	0.147098	5.490744	10.00000	0.217725	2.172858
PSO	0.164171	4.032541	10.00000	0.223647	1.873971
GA	0.206487	3.635872	10.00000	0.203249	1.836250

and

Table 10. Statistical results for welded beam design problem.

Algorithms	Best	Mean	Worst	Std. Dev.	Median
SSA	1.723589	1.725124	1.727211	0.004325	1.724399
EPO	1.725661	1.725828	1.726064	0.000287	1.725787
SHO	1.726995	1.727128	1.727564	0.001157	1.727087
GOA	1.820395	2.230310	3.048231	0.324525	2.244663
GWO	1.725472	1.729680	1.741651	0.004866	1.727420
TLBO	1.759173	1.817657	1.873408	0.027543	1.820128
GSA	2.172858	2.544239	3.003657	0.255859	2.495114
PSO	1.873971	2.119240	2.320125	0.034820	2.097048
GA	1.836250	1.363527	2.035247	0.139485	1.9357485

. The SSA provides an optimal solution at (0.205411, 3.472341, 9.035215, 0.201153) with a corresponding fitness value equal: 1.723589.

8.4. Tension/Compression Spring Design Problem

The mathematical model of this problem was adapted from [31]. The results to this optimization problem are displayed in

Table 11. Comparison results for tension/compression spring design problem.

Algorithms	Optimum Variables			Optimum Cost
	d	D	p
SSA	0.051087	0.342908	12.0898	0.012656987
EPO	0.051144	0.343751	12.0955	0.012674000
SHO	0.050178	0.341541	12.07349	0.012678321
GOA	0.05000	0.310414	15.0000	0.013192580
GWO	0.05000	0.315956	14.22623	0.012816930
TLBO	0.050780	0.334779	12.72269	0.012709667
GSA	0.05000	0.317312	14.22867	0.012873881
PSO	0.05010	0.310111	14.0000	0.013036251
GA	0.05025	0.316351	15.23960	0.012776352

and

Table 12. Statistical results for tension/compression spring design problem.

Algorithms	Best	Mean	Worst	Std. Dev.	Median
SSA	0.012656987	0.012678903	0.012667902	0.001021	0.012676002
EPO	0.012674000	0.012684106	0.012715185	0.000027	0.012687293
SHO	0.012678321	0.012697116	0.012720757	0.000041	0.012699686
GOA	0.013192580	0.014817181	0.017862507	0.002272	0.013192580
GWO	0.012816930	0.014464372	0.017839737	0.001622	0.014021237
TLBO	0.012709667	0.012839637	0.012998448	0.000078	0.012844664
GSA	0.012873881	0.013438871	0.014211731	0.000287	0.013367888
PSO	0.013036251	0.014036254	0.016251423	0.002073	0.013002365
GA	0.012776352	0.013069872	0.015214230	0.000375	0.012952142

. The SSA provides the optimal solution at (0.051087, 0.342908, 12.0898), with a corresponding fitness value of 0.012656987.

8.5. Rolling Element Bearing Design Problem

The mathematical model of this problem is adapted from [92]. The results of this optimization problem are included in

Table 13. Comparison results for rolling element bearing design problem.

Algorithms		Optimum Variables									Opt. Cost
	Dm	Db	Z	fi	fo	KDmin	KDmax	ε	e	ζ
SSA	125	21.41890	10.94113	0.515	0.515	0.4	0.7	0.3	0.02	0.6	85,067.983
EPO	125	21.40732	10.93268	0.515	0.515	0.4	0.7	0.3	0.02	0.6	85,054.532
SHO	125.6199	21.35129	10.98781	0.515	0.515	0.5	0.68807	0.300151	0.03254	0.62701	84,807.111
GOA	125	20.75388	11.17342	0.515	0.515000	0.5	0.61503	0.300000	0.05161	0.60000	81,691.202
GWO	125.6002	21.32250	10.97338	0.515	0.515000	0.5	0.68782	0.301348	0.03617	0.61061	84,491.266
TLBO	125	21.14834	10.96928	0.515	0.515	0.5	0.7	0.3	0.02778	0.62912	83,431.117
GSA	125	20.85417	11.14989	0.515	0.517746	0.5	0.61827	0.304068	0.02000	0.624638	82,276.941
PSO	125	20.77562	11.01247	0.515	0.515000	0.5	0.61397	0.300000	0.05004	0.610001	82,773.982
GA	125	20.87123	11.16697	0.515	0.516000	0.5	0.61951	0.301128	0.05024	0.614531	81,569.527

and

Table 14. Statistical results for Rolling element bearing design problem.

Algorithms	Best	Mean	Worst	Std. Dev.	Median
SSA	85,067.983	85,042.352	86,551.599	1877.09	85,056.095
EPO	85,054.532	85,024.858	85,853.876	0186.68	85,040.241
SHO	84,807.111	84,791.613	84,517.923	0137.186	84,960.147
GOA	81,691.202	50,435.017	32,761.546	13,962.150	42,287.581
GWO	84,491.266	84,353.685	84,100.834	0392.431	84,398.601
TLBO	83,431.117	81,005.232	77,992.482	1710.777	81,035.109
GSA	82,276.941	78,002.107	71,043.110	3119.904	78,398.853
PSO	82,773.982	81,198.753	80,687.239	1679.367	8439.728
GA	81,569.527	80,397.998	79,412.779	1756.902	8347.009

and prove that the SSA provides an optimal solution at (125, 21.41890, 10.94113, 0.515, 0.515, 0.4, 0.7, 0.3, 0.02, 0.6) with a corresponding fitness value equal to 85,067.983.

9. Conclusions

There are many optimization problems in the different scientific domains that must be solved using the algorithms with each case’s necessary characteristics. In this project, a new optimization algorithm called spring search algorithm (SSA) was developed; Hooke’s law describes the starting point or basic concept. The search agents of the proposed method are weights that are connected by several springs. The system starts from a transitory situation or state and stabilizes at the equilibrium point, according to the law of the spring.

To evaluate the algorithm and purchase it, almost 40 standard objective functions, including unimodal and multimodal functions, in addition to CEC2015, were used to assess the performance of the proposed algorithm in solving optimization problems of a different nature. To review and analyze the algorithm’s results, these were compared with eight widely known optimization algorithms: GA, PSO, GSA, TLBO, GWO, GOA, SHO, and EPO.

The results in the functions show the superior exploration and exploitation capabilities of SSA compared to other optimization algorithms, for both unimodal and multimodal functions. The same occurs with the simulations using the SSA algorithm and the eight algorithms selected for comparison in the case of CEC2015, which shows the SSA’s high aptitude to solve this type of problem to optimize the function. In addition to the work carried out on the almost 40 functions, the SSA algorithm was evaluated in five engineering design optimization problems to evaluate the performance in solving optimization problems in real situations, showing that it is much more competitive than other algorithms.

For future work, it is suggested to develop a binary version of the SSA algorithm and apply this algorithm to multi-objective problems. Likewise, extending the concept of Hooke’s law to more complex models with more adjustment parameters, which, even though it will reduce simplicity, could generate additional advantages.