A novel lifetime scheme for enhancing the convergence performance of salp swarm algorithm

Abstract

The performance of any meta-heuristic algorithm depends highly on the setting of dependent parameters of the algorithm. Different parameter settings for an algorithm may lead to different outcomes. An optimal parameter setting should support the algorithm to achieve a convincing level of performance or optimality in solving a range of optimization problems. This paper presents a novel enhancement method for the salp swarm algorithm (SSA), referred to as enhanced SSA (ESSA). In this ESSA, the following enhancements are proposed: First, a new position updating process was proposed. Second, a new dominant parameter different from that used in SSA was presented in ESSA. Third, a novel lifetime convergence method for tuning the dominant parameter of ESSA using ESSA itself was presented to enhance the convergence performance of ESSA. These enhancements to SSA were proposed in ESSA to augment its exploration and exploitation capabilities to achieve optimal global solutions, in which the dominant parameter of ESSA is updated iteratively through the evolutionary process of ESSA so that the positions of the search agents of ESSA are updated accordingly. These improvements on SSA through ESSA support it to avoid premature convergence and efficiently find the global optimum solution for many real-world optimization problems. The efficiency of ESSA was verified by testing it on several basic benchmark test functions. A comparative performance analysis between ESSA and other meta-heuristic algorithms was performed. Statistical test methods have evidenced the significance of the results obtained by ESSA. The efficacy of ESSA in solving real-world problems and applications is also demonstrated with five well-known engineering design problems and two real industrial problems. The comparative results show that ESSA imparts better performance and convergence than SSA and other meta-heuristic algorithms.

1 Introduction

Meta-heuristic algorithms have become quite popular in solving engineering design problems (Gandomi et al. 2013; Wang et al. 2014), modeling of chemical processes (Sheta et al. 2019), diagnosis of medical images (Braik et al. 2019) and many other problems (Mirjalili et al. 2017; Wang 2018). In recent years, many meta-heuristic algorithms have been proposed and have received excellent acceptance by researchers in the artificial intelligence community due to their prominence in generating low-cost and robust solutions for complex optimization and non-deterministic polynomial problems (Mirjalili et al. 2017). The literature review of meta-heuristic algorithms reveals their efficiency and reliability in solving real-world problems in various fields of study. Ant colony optimization (ACO) algorithm (Maniezzo 1992) inspired by the foraging behavior of ants is very effective in solving structural optimization problems (Luh and Lin 2009), genomics (Greene et al. 2008) and also in traffic area control problems (Sattari et al. 2014). Particle swarm optimization (PSO) is a well-known optimization algorithm that mimics the social behavior of fish schooling or bird flocking (Eberhart and Kennedy 1995). The performance of PSO in solving several complex problems such as image processing (Omran et al. 2006), electric power systems (AlRashidi and El-Hawary 2008) and offshore heavy oil reservoir (Wang and Qiu 2013) is distinctly confirmed in the literature. Salp swarm algorithm (SSA) (Mirjalili et al. 2017) is a recently evolved meta-heuristic algorithm, motivated by the dynamic swarming behavior of salps while searching for food in the oceans (Mirjalili et al. 2017). SSA has been used successfully to solve a large number of real-world problems in many branches of science such as extracting the parameters of the electrical equivalent circuit of a photovoltaic system cell (Abbassi et al. 2019), color image segmentation (Xing and Jia 2019), optimization of software-defined networks (Ateya et al. 2019) and training the weights and biases of a feed-forward neural network (FFNN) (Bairathi and Gopalani 2019). Artificial bee colony (ABC) algorithm (Karaboga and Basturk 2007), cuckoo search (CS) algorithm (Yang and Deb 2009), firefly algorithm (FA) (Yang 2009), bat algorithm (BA) (Yang 2010a), fruit fly optimization algorithm (FOA) (Pan 2012), gray wolf optimization (GWO) (Mirjalili et al. 2014), whale optimization algorithm (WOA) (Mirjalili and Lewis 2016) and moth search (MS) algorithm (Wang 2018) are some of the well-known swarm intelligence-based algorithms. Many of these algorithms have been used successfully to address many problems in several areas such as medical diagnoses (Wang et al. 2017), process control (dos Santos Coelho and Mariani 2012), image processing (El Aziz et al. 2017), text clustering (Rashaideh et al. 2018), feature selection and classification (Hegazy et al. 2018; Ibrahim et al. 2019) and many other applications (Mavrovouniotis et al. 2017; Ali et al. 2019).

With the development of several optimization algorithms, it is difficult to identify which algorithm is most appropriate to solve a given problem. This is due to that most of the meta-heuristic algorithms work on a generalized concept and do not have domain knowledge specific to each problem (KS and Murugan 2017). Many parameters have to be tuned for any meta-heuristic algorithm, which can provide more flexibility and robustness for an optimization algorithm, but this process requires careful initialization (Kumar et al. 2015). However, all meta-heuristic algorithms have their dependent parameters, where the performance of an algorithm is largely dependent on the values of its parameters. These algorithms’ dependent parameters may have a large influence on the effectiveness of algorithms’ search agents in exploring and exploiting the search space. The process of finding the best parameter setting for an algorithm is expected to enhance the applicability of an algorithm to solve a range of optimization problems in various areas of science. Parameter setting problem for meta-heuristic algorithms reveals the importance of presenting a robust mechanism that can automatically choose the best parameter setting for an algorithm to achieve its optimality in solving a given problem (Yang et al. 2013). However, parameter setting control for meta-heuristic algorithms is still an open problem.

Several parameter setting techniques have recently been evolved for several meta-heuristic algorithms as a part of enhancing optimization algorithms. They tend to show significantly improved performance compared to their meta-heuristic algorithms’ counterparts in solving benchmark functions or particular optimization problems. Many researchers first identified the parameters’ values of meta-heuristics in solving an optimization problem based on trial and error paradigm (Fallahi et al. 2014; Jain et al. 2019). For instance, Dobslaw (2010) used an automated Design of Experiment (DoE) framework to identify sensible initial parameter settings for problem instances for various meta-heuristic algorithms such as PSO and ACO. Also, a parameter setting method for ACO based on DoE was presented to solve 7 traveling salesman problem instances (Fallahi et al. 2014), where different solutions were obtained for the same problem using different parameter settings. Geem and Sim (2010) proposed a parameter-setting-free (PSF) technique to find the best parameter setting for the harmony search (HS) algorithm. Khadwilard et al. (2012) investigated the performance of FA for solving a job shop scheduling problem with various parameter settings. A method for self-tuning algorithms so that an algorithm to be tuned can be used to automatically tune its parameters was presented by Yang et al. (2013). They used FA to tune itself, where the authors claimed that this method of self-tuning algorithm worked well. Using an upper-level meta-heuristic to decide the most appropriate set of parameters for a low-level meta-heuristic was presented in Crawford et al. (2013). Crawford et al. (2013) applied a genetic algorithm (GA) as an upper-level meta-heuristic to optimize the parameter values of ACO and scatter search meta-heuristics for solving a particular problem. The idea is to transfer the parameter setting effort of one algorithm to another algorithm.

1.1 SSA challenges

The parameter setting for the dominant parameter of SSA is the foremost problem of SSA, where inappropriate values for the factors of this parameter may drive SSA to stagnate in local optima. Another problem of SSA is that it does not explicitly keep track of its best positions in former generations, which bounds its exploitation ability and leads to early convergence in some basic benchmark functions. Even though global and local search capabilities of SSA are somewhat convincing of randomization behavior and dynamic swarming behavior of salps, these search capabilities are sometimes limited which causes the solutions to get fall into local optima in some optimization problems.

To overcome these flaws, many researchers have developed hybridized versions of SSA with other algorithms to improve performance compared to their original counterparts. For example, Sayed et al. (2018) employed a hybrid approach between Chaos theory and SSA (CSSA) to solve a set of basic benchmark functions. The obtained results showed that CSSA can improve the convergence rate for SSA. Hegazy et al. (2018) proposed an improved salp swarm algorithm (ISSA) by controlling the parameters of SSA using the DoE framework to enhance the exploration process of SSA for solving feature selection problems. Ibrahim et al. (2019) proposed a hybrid approach between SSA and PSO (SSAPSO) to solve feature selection problems on datasets obtained from the UCI machine learning repository. Ali et al. (2019) proposed a hybrid approach between SSA and weighted $L_{1}$-norm optimization to design 2nd to 4th wideband infinite impulse response (IIR) digital differentiators (DDs). This hybridization approach was presented to enhance the exploration of SSA at the initial stages and exploitation at later stages by iteration-level hybridization process to get the global optimum solution. However, these techniques proposed for enhancing the performance of SSA are either very preliminary or computationally complex and require a relatively large computational cost.

From the literature, SSA is a promising optimization algorithm with room for possible improvement and employment in various disciplinary areas. For example, there is no proper discussion about the performance of SSA regarding its tuning parameters. Also, most applications of SSA on real-world problems have not explored the proper parameter settings of SSA. Finally, how do we control the exploration process of SSA up to the extent that SSA does not become so slow to converge? Therefore, a more reliable and computationally efficient approach to enhance SSA is required, but with the simplicity and speed of SSA. This is the motivation behind this study, in which a novel enhancement to SSA is proposed as given below.

1.2 Motivations of the proposed work

In this paper, to overcome the above shortcomings of SSA, we propose a novel lifetime convergence scheme for SSA, referred to as enhanced SSA (ESSA), to reinforce the exploration and exploitation capabilities of SSA. The proposed ESSA works in two stages: In the first stage, a new position updating mechanism is proposed to control the movement of salps in the search space. In this process, the salps that store the coordinates of the best positions that have achieved so far in previous generations are incorporated into the updating position process. In the next stage, a new dominant parameter different from that used in SSA is proposed in ESSA. Then, an innovative methodology to automatically self-tuning this dominant parameter is proposed. This is the main goal of this work, where a novel lifetime convergence process is incorporated with the proposed ESSA in a self-tuning scheme to ensure the ability of exploration at the initial iterations and the capacity of exploitation at the later iterations of the evolutionary process of ESSA. These improvements made to SSA assist the salps to explore different promising areas in the deep ocean and exploit each search area to find food sources. Further, they shall provide great potential for the salps to avoid stagnation in local optima and help them to move toward the global optimum solution with increased precision.

1.3 Contributions

The contribution of this work are as follows:

• Proposing a new position updating process for SSA, referred to as ESSA, that could improve the swarming behavior of the salps.

• Integrating a new lifetime convergence scheme with ESSA, which can be updated at each iteration loop of ESSA to reach optimal convergence. This is an innovation process where ESSA is used to tune itself. It is anticipated that this process will ultimately increase the exploration and exploitation abilities of ESSA to obtain the global optimum (i.e., food), rapidly and potentially efficiently and

• The performance of ESSA is compared with other state-of-the-art meta-heuristic algorithms on basic benchmark functions and several computational engineering design problems commonly used by meta-heuristic optimization algorithms in the literature.

In Sect. 2, the original SSA is described in detail. Section 3 then presents the proposed ESSA and provides a description of the lifetime convergence process proposed. The evaluation, convergence and statistical test results are presented in Sect. 4. Section 5 presents the results of ESSA on engineering design problems compared to other algorithms. Section 6 then presents the results of ESSA on modeling industrial problems. Section 7 presents an analysis and discussion of the obtained results with concluding comments and future scope in Sect. 8.

2 Basics of SSA

Salp swarm algorithm is a new meta-heuristic algorithm inspired by the unique and superior behavior of salps while searching for food in deep ocean (Mirjalili et al. 2017). Foraging behavior of salps is characterized by splitting the salp chain into a leader and followers that move toward food sources into deep ocean. That is, the first position in the chain is for the leader, while the remaining positions in the chain are for the followers. The leader guides the followers while searching for food sources, denoted by F. For n salps in a d-dimensional search space, the position of all salps is defined by a matrix $x_n^d$. Consider population of salps of size n, the position of the ith salp in SSA can be given as shown in Eq. (1).

$$\begin{aligned} x_i= \left( x_i^1, x_i^2,\ldots , x_i^k, \ldots , x_i^{d-1}, x_i^d\right) \end{aligned}$$

(1)

where i represents the ith salp in the population of SSA, d stands for the dimension of the ith salp and $x_i^k$ corresponds to the position of the ith salp in the kth dimension of the search space. Equation (2) can be used to assign the initial position of each salp in the salp chain.

$$\begin{aligned} x_i= ub^{k} + r \times (ub^{k} - lb^{k}) \end{aligned}$$

(2)

where $lb^{k}$ and $ub^{k}$ are the lower and upper bounds of the ith salp in the kth dimension, respectively, and r is a randomly generated number in the range from 0 to 1.

The position of the leader salp at the first position in the kth dimension, $x_1^k$, in the salp chain is updated according to Eq. (3) (Mirjalili et al. 2017).

$$\begin{aligned} x_1^k = {\left\{ \begin{array}{ll} F^k + c_1[(ub^k-lb^k)c_2 + lb^k] \qquad c_3\ge 0.5\\ F^k - c_1[(ub^k-lb^k)c_2 + lb^k] \qquad c_3 < 0.5 \end{array}\right. } \end{aligned}$$

(3)

where $F^k$ identifies the food source position in the kth dimension, $ub^k$ denotes the upper bound of the kth dimension, $lb^k$ denotes the lower bound of the kth dimension, and $c_2$ and $c_3$ are random numbers generated uniformly in the interval from 0 to 1.

Equation (3) states that the leader in SSA only updates its location in regard to the source of food F. The $c_2$ and $c_3$ parameters decree if the following position of the salp approach $\infty$ or $-\infty$ with the next movement. The $c_1$ parameter is defined as shown in Eq. (4) (Mirjalili et al. 2017).

$$\begin{aligned} c_1 = 2e^{-(\frac{4l}{L})^2} \end{aligned}$$

(4)

where l and L identify the current iteration and the maximum number of iterations, respectively. The $c_1$ parameter is the leading and dominant parameter in SSA, as it creates a balance between exploration and exploitation of the search space. The position of the followers is updated according to Eq. (5).

$$\begin{aligned} x_i^k = \frac{1}{2}(x_1^k + x_i^{k-1}) \end{aligned}$$

(5)

where $i\ge 2$, $x_i^k$ and $x_i^{k-1}$ stand for the positions of the ith follower salp in the kth and $k-1$th dimensions, respectively, and $x_1^k$ represents the position of the leader salp in the kth dimension.

The fitness function is evaluated at each iteration within the evolutionary process loop of SSA. The position updating processes of the leader and followers are continued until the stopping condition is reached. The basic steps of SSA can be described by the iterative procedural steps illustrated in Algorithm 1.

As shown in Algorithm 1, if any of the salps moves out of the search area, it will be returned to the boundaries based upon the simulated steps of SSA. All the steps of SSA in Algorithm 1 except the initialization step are executed repeatedly at each loop until the termination condition is met. It is observed that the food source, F, is updated throughout the evolutionary process of SSA since it is very likely that the salp chain will find a better solution by exploring and exploiting the surrounding environment. The leader and followers can move toward the global optimum throughout iterations.

3 Proposed ESSA

This section details the proposed position updating process for SSA. Then, it explains the proposed self-tuning method applied to improve parameter setting and convergence process for the proposed algorithm.

3.1 Proposed mathematical model for ESSA

In SSA and ESSA, the position of the food source needs to be updated iteratively. The position of the leader and the accompanying salps can be updated at dimension k as given below:

$$\begin{aligned} x_{i,l+1}^k = F^k + \bigtriangleup x \end{aligned}$$

(6)

where $\bigtriangleup x$ represents the new update and $x_{i,l+1}^k$ stands for the position of the ith salp at iteration $l+1$ at dimension k.

The term $\bigtriangleup x$ needs to be chosen carefully to achieve two essential aspects: exploration and exploitation of the search space. Further, this term should support the convergence path of the search agents (i.e., salps) toward the food source. Tuning $\bigtriangleup x$ is a significant matter that has a foremost impact on the overall performance of ESSA. This term plays a vital role in exploring the search space by helping the individuals of ESSA to move efficiently to the food. Therefore, it is useful to support updating this term with a lifetime convergence mechanism to achieve a proper convergence process and keep the exploration rate high throughout iterations.

In this work, $\bigtriangleup x$ was presented as a function of a lifetime parameter, $c_1$, as shown in Eq. (7).

$$\begin{aligned} x_{l+1}= & {} f(F, \bigtriangleup x )\nonumber \\ \bigtriangleup x= & {} f(x_l, c_1, ub, lb, c_2) \end{aligned}$$

(7)

where F represents the position of the food source, $x_l$ represents the current solution to a given problem at iteration l, $c_2$ is a randomly generated number in the range from 0 to 1 and f is a nonlinear mapping function from a given solution, $x_l$, to a new solution $x_{l+1}$.

In the position updating process of ESSA in Eq. (7), ESSA tends to generate a new solution $x_{l+1}$ at iteration $l+1$ for a given problem that is better than the current solution $x_{l}$ at iteration l. In ESSA, the positions of the salp leader and its accompanying salps are updated with a mathematical formula different from that used in SSA. The update mechanism of salps in ESSA is given in Eq. (8).

$$\begin{aligned} x_{i,l+1}^k={\left\{ \begin{array}{ll} F^k + c_1 x_{i,l}^k \left[ \left( ub^k-lb^k\right) c_2+lb^k\right] &{}\quad c_3 \ge 0.5\\ F^k - c_1 x_{i,l}^k \left[ \left( ub^k-lb^k\right) c_2+lb^k\right] &{}\quad c_3 < 0.5 \end{array}\right. } \end{aligned}$$

(8)

where $i \le n/2$ indicates the leader of the salp chain and the accompanying salps of the leader, $x_{i,l+1}^k$ and $x_{i,l}^k$ are the next and current positions of the salp leader and its accompanying salps at iterations $l+1$ and l, respectively, the parameter $c_1$ was used to dominate the movement of salps of ESSA and is reformulated in ESSA as shown below:

$$\begin{aligned} c_1= & {} \lambda _1 e^{K} \nonumber \\ K= & {} -\lambda _2 \sqrt{\lambda _3\times l} \end{aligned}$$

(9)

where the coefficients $\lambda _1$, $\lambda _2$ and $\lambda _3$ are used to update $c_1$ at each iteration loop of ESSA.

In ESSA, the position of the followers, representing the $i > n/2$ salps, are automatically updated using Eq. (5). The positions of the population of salps are updated iteratively with the help of the proposed lifetime convergence process. This process is used to control the parameter $c_1$ through the evolutionary process of ESSA throughout iterations. This parameter consists of three coefficients, $\lambda _1$, $\lambda _2$ and $\lambda _3$, that are needed to update iteratively to automatically tune $c_1$ using ESSA itself.

These three coefficients $\lambda _1$, $\lambda _2$ and $\lambda _3$ are proposed to provide a proper setting for $c_1$, where the proposed lifetime convergence process can empower the salp chain to explore more search space and exploit each promising area while searching for food sources. This is to reach an effective convergence process for ESSA, which can further improve the performance of ESSA in solving optimization problems. This proposed parameter setting method is described below.

3.2 Parameter setting-based ESSA

The proposed ESSA was used to tune the coefficients of the parameter $c_1$ using itself. This process is referred to as a self-tuning method. Mathematically speaking, we can describe the mathematical model of ESSA as:

$$\begin{aligned} x_{i,l+1}^k = f\left( F, x_{i,l}^k, \mathbf {p}, ub, lb, c_2 \right) \end{aligned}$$

(10)

where the parameter vector $\mathbf {p}=(\lambda _1, \lambda _2$, $\lambda _3)$. As the parameter vector $\mathbf {p}$ is the target of the proposed self-tuning parameter method, the representation formula shown in Eq. (10) can be reduced to:

$$\begin{aligned} x_{i,l+1}^k = f\left( x_{i,l}^k, \mathbf {p} \right) \end{aligned}$$

(11)

It is known that there is a global optimum, $f_\mathrm{min}(x)$, for each optimization problem to be optimized. This optimality depends on both ESSA itself and the optimization problem, $\phi$, to be addressed. That is, the optimality to be realized is to maximize the performance of:

$$\begin{aligned} \zeta = f(\phi , \mathbf {p}) \end{aligned}$$

(12)

This optimality is then denoted as:

$$\begin{aligned} \zeta _{*} = f_{*}(\phi , \mathbf {p}_{*}) \end{aligned}$$

(13)

where the parameter $\mathbf {p}_{*}$ is the optimal parameter setting for $c_1$.

For an optimization problem, the goal is to find the global optima $f_{*}$ for a function f(x) in a d-dimensional search space. That is, to minimize the function:

$$\begin{aligned} f(x) = (x_1, x_2, \ldots , x_d) \end{aligned}$$

(14)

To optimize a problem, there is a need to find $f_\mathrm{min}$ which may be close to the actual global optimum $f_{*}$. For a predetermined tolerance $\rho$, this may require $L_{\rho }$ iterations to realize $\left| f_\mathrm{min} - f_{*} \right| \le \rho$. Obviously, $L_{\rho }$ will largely depend on both the problem objective and the coefficients $\lambda _1, \lambda _2$ and $\lambda _3$ of vector $\mathbf {p}$ that can be used to tune $c_1$.

The aim of this self-tuning method is to find $\mathbf {p}_{*}$ in order to achieve the best setting for $c_1$ with the minimum number of iterations $L_{\rho }$. This idea could be addressed as a multi-objective optimization problem with two objectives: The first objective is for the given problem, $\phi$, and the second objective is for the number of iterations, $L_{\rho }$. This parameter tuning method can be written as follows.

$$\begin{aligned} \text {Minimize}&\quad f(\mathbf {x}) \nonumber \\ \text {Minimize}&\quad L_{\rho } = \text {ESSA}(f(x), \mathbf {p}) \end{aligned}$$

(15)

where $L_{\rho }$ is the number of iterations carried out to achieve a pre-identified tolerance $\rho$ so that the solution $f_\mathrm{min}$ is close enough to the true global solution $f_{*}$, respecting that $\left| f_\mathrm{min}-f_{*} \right| \le \rho$

This means that for tolerance $\rho$, there will be a pool of best parameter settings with a minimum $L_{\rho }$. But since $\rho$ is usually given, the normal way to solve this problem is to use the so-called $\lambda _1, \lambda _2$ and $\lambda _3$-constraints or $\rho$-constraint method. So, for a given $\rho \ge 0$, we can change one of the targets (i.e., $f(\mathbf {x})$) into a constraint, so that Eq. (16) becomes a single-objective optimization problem with a constraint as shown in Eq. (16).

$$\begin{aligned} \text {Minimize}&L_{\rho } = ESSA(f(x), \mathbf {p}) \end{aligned}$$

(16)

Subject to:

$$\begin{aligned} f(\mathbf {x}) \le \rho \end{aligned}$$

(17)

Ideally suited, the vector $\mathbf {p}_{*}$ must be adequately robust, where any slight variation in $\mathbf {p}_{*}$ should not significantly affect the accuracy of ESSA in solving various types of problems. This implies that $\mathbf {p}_{*}$ should fall into a flat range, not at a sharp peak in the parameter landscape.

For ESSA to converge properly, we employed exponential formulas for $\lambda _1, \lambda _2$ and $\lambda _3$ similar to that used for $c_1$ in Eq. (4), as shown in Eqs. (18), (19) and (20), respectively.

$$\begin{aligned} \lambda _1= & {} \theta _1 e^{-(\frac{l}{L})^2}, \quad \theta _1 \in (1, 10) \end{aligned}$$

(18)

$$\begin{aligned} \lambda _2= & {} \theta _2 e^{-(\frac{l}{L})^2} \quad \theta _2 \in (0.25, 2) \end{aligned}$$

(19)

$$\begin{aligned} \lambda _3= & {} \theta _3 e^{-(\frac{l}{L})^2} \quad \theta _3 \in (1, 3) \end{aligned}$$

(20)

where the parameters to be tuned in this work become $\theta _1, \theta _2$ and $\theta _3$, where the parameters $\lambda _1$, $\lambda _2$ and $\lambda _3$ can be obtained from Eqs. (18), (19) and (20), respectively, and then $c_1$ using Eq. (9).

3.3 Time complexity of ESSA

The computational complexity of ESSA can be given as:

$$\begin{aligned} {\mathcal {O}}(v(L(sd+ cs+ms^{2}))) \end{aligned}$$

(21)

where v represents the number of runs, L is the number of iterations, s displays the number of solutions, d shows the number of variables of the optimization problem (i.e., dimension), c denotes the cost of the objective function and m stands for the number of objectives.

3.4 Space complexity of ESSA

The space complexity of ESSA in terms of the memory space depends on the parameters of both the search agents and the dimension of the given problem. This identifies the amount of space that ESSA needs during the initialization process. Hence, the space complexity of ESSA can be defined as:

$$\begin{aligned} {\mathcal {O}} (sd) \end{aligned}$$

(22)

ESSA also requires additional space for other parameters such as $\lambda _1, \lambda _2,\lambda _3$, $\mathbf {p}$ and $c_1$, but this extra space is not critical, and therefore, the space complexity for ESSA remains within the range given in Eq. (22).

The parameters in Eq. (21) have an effect on the speed of ESSA while searching for the global optimal solution. The search time to find the global optimum increases as the values for these parameters increase. The number of iterations and search agents required to satisfactorily find the global solution depends on the complexity of the given problem. Therefore, the values of these parameters should be chosen carefully to precisely solve the problem.

3.5 Analysis of ESSA

The parameters $\lambda _1$, $\lambda _2$ and $\lambda _3$ in Eq. (9) were tuned through Eqs. (18), (19) and (20), respectively. These values were iteratively updated by a self-tuning method-based ESSA itself to reach the optimality, so that $c_1$ can promote the exploration and exploitation of ESSA, where the global minimum $f_{*}$ can be obtained. The main procedures of ESSA can be summarized by the iterative steps described in Algorithm 2.

In Algorithm 2, ESSA first initializes the population of salps concerning the upper and lower bounds of the problem variables. Then this algorithm computes the objective value for each salp and updates the parameters $\lambda _1, \lambda _2$ and $\lambda _3$ using Eqs. (18), (19) and (20), respectively. The objective value and these parameters are updated inside the evolutionary process of ESSA. These parameters need to decrease gradually in terms of $\theta _1, \theta _2$ and $\theta _3$, respectively. These parameters are necessary to update the vector $\mathbf {p}$. The next step is to use $\mathbf {p}$ to update the parameter $c_1$ using Eq. (9) and update the position of the leader and followers using Eqs. (8) and (5), respectively. This process is repeated at each iteration loop of ESSA, allowing ESSA to update this parameter by ESSA itself. Essentially, this self-tuning parameter method-based ESSA achieves two goals at once: finding the best setting for the parameter $c_1$ and finding the optimal objective value. The food source and position of the leader and followers will be updated during the optimization process because the salp chain is very probably to pursue a better solution by exploring and exploiting the surrounding area. The imitations of salps’ behavior in Algorithm 2 show that the salp chain movement modeled in the proposed ESSA can explore and exploit the area around both stationary and moving food sources. So, the salp chain has the potency to move toward the global optimum throughout iterations. Based on the above illustration of the strengths of the proposed ESSA, it is expected to be effective in solving optimization problems.

Fig. 1

Parameter space of the unimodal benchmark functions

Fig. 2

Parameter space of the multimodal benchmark functions

Fig. 3

Parameter space of the fixed-dimension multimodal benchmark functions

4 Experimental results and discussion

Many evaluation experiments were conducted in this section to clarify the theoretical assertions discussed the proposed ESSA in the previous section.

• 1. First, twenty-three standard benchmark functions are used to evaluate the performance of the proposed algorithm.

• 2. Second, qualitative results are obtained and presented to assess the general efficacy of the proposed ESSA in optimizing a collective set of benchmark test functions.

• 3. Third, the efficiency of the proposed ESSA is compared with the original SSA and other compelling meta-heuristic algorithms.

• 4. Fourth, convergence curves are provided to substantiate the accuracy and reliability levels of ESSA compared to SSA and other algorithms.

4.1 Benchmark test functions

It is always beneficial to use a set of standard testbeds with different characteristics to conveniently and confidently test the performance of any new algorithm on different benchmark functions and compare it with other algorithms. The diversity of test functions allows observing and testing the ability of any new algorithm from different perspectives.

A set of twenty-three basic benchmark functions are used to demonstrate the efficiency of the proposed algorithm compared to the standard SSA and other existing algorithms in the literature. These functions can be grouped into three main categories: unimodal (Dhiman and Kumar 2017), multimodal (Mirjalili 2015) and fixed-dimension multimodal (Dhiman and Kumar 2017). These functions are mathematically described in “Appendix A.” A detailed description of the characteristics of the unimodal functions, ($F_1$–$F_7$), multimodal functions ($F_8$–$F_{13}$) and fixed-dimension multimodal functions ($F_{14}$–$F_{23}$) is presented in Table 14, where Dim indicates the dimension of the function space, range stands for the search space limits of the function and $f_\mathrm{min}$ indicates the best-reported value.

The unimodal test functions have only one global optimum with no local optima. These functions are highly suitable to explore the convergence behavior and exploitative efficiency of the proposed algorithm. Multimodal and fixed-dimension multimodal test functions face the existence of several local optimum solutions and more than one global optimum. These functions are very appropriate to verify local optimum avoidance and study the explorative behavior of the proposed algorithm. All of these benchmark functions represent minimization problems.

Figures 1, 2 and 3 show the landscapes of unimodal, multimodal and fixed-dimension multimodal benchmark functions in two-dimensional versions, respectively.

To verify the accuracy of the proposed ESSA, its results on the functions described in “Appendix A” are compared to the results of the standard SSA (Mirjalili et al. 2017) and the results of six well-known algorithms: moth-flame optimization (MFO) (Mirjalili 2015), multi-verse Optimizer (MVO) (Mirjalili et al. 2016), sine–cosine algorithm (SCA) (Mirjalili 2016), gravitational search algorithm (GSA) (Rashedi et al. 2009), GA (Bonabeau et al. 1999) and HS (Geem et al. 2001). The parameter settings of these algorithms are given in the following subsection.

4.2 Experimental setup

The parameter settings of the proposed ESSA and other algorithms such as SSA, MFO, MVO, SCA, GSA, GA and HS are tabulated in Table 1. The parameter settings given in Table 1 are set according to the literature.

Table 1 Parameter setup of the compared algorithms

The number of search agents and the maximum number of iterations of ESSA and each algorithm is set to 1000 and 30, respectively. The stopping criterion for each algorithm is set according to the maximum number of iterations except for ESSA that is also considered a tolerance threshold.

In the proposed ESSA, we used the same initialization procedure as the other comparative algorithms. In this sense, the parameter settings for ESSA concerning the number of search agents and a maximum number of iterations are similar to each algorithm and are similar to what was reported in the literature. For other control parameters of each algorithm, the values in the latest version of each algorithm are employed to ensure the best performance. For instance, each algorithm has its control parameters and each algorithm is generally distinguished based on its solution update strategy, but these parameter settings are important to be the same to provide a fair comparison between ESSA and other algorithms.

4.3 Performance comparison

The mean and standard deviation (STD) measures were used to compare the performance of ESSA to those of the standard SSA and other optimization algorithms on different benchmark functions. Each of these algorithms is run 30 independent times on each test function, to compute meaningful statistics. Each run conducts 1000 times of iterations. The mean and STD results are computed at the last iteration of each algorithm on each test function to get the optimal solution and provide a fair comparison among all comparative algorithms.

4.3.1 Evaluation of unimodal functions

The functions $F_1$–$F_{7}$ are unimodal and allow algorithms to assess their exploitation capability. The mean and STD results obtained by ESSA, the standard SSA and other meta-heuristic algorithms on unimodal Functions $F_1$–$F_7$ are given in Table 2. The best results are shown in bold throughout the paper.

Table 2 Computed results for the unimodal function-based ESSA and other algorithms

Table 3 Computed results for the multimodal functions based on the proposed algorithm and other algorithms

Table 2 shows that the proposed ESSA is capable of delivering encouraging results, and outperforms the standard SSA and other algorithms in all of the unimodal functions. The better average values on these functions reveal that ESSA performs better than other algorithms on average and that the STD results reveal that this superiority is stable. Due to the existence of a single optimum in the unimodal functions, these functions can benchmark exploitation and convergence speed of algorithms. Therefore, the results in Table 2 show that ESSA avails from high exploitation and convergence speed of the unimodal functions.

4.3.2 Evaluation of multimodal functions

Multimodal functions may involve many local optima that can grow exponentially. These functions have the ability to assess the exploration capability of algorithms. Table 3 shows the mean and STD results obtained by ESSA and other algorithms over 30 independent runs for the functions ${F}_8$–${F}_{13}$.

As per the results in Table 3, it may be observed that ESSA again reports significantly better results than those obtained by SSA and other algorithms for most of the test functions including ${F}_9$–${F}_{13}$, and it is very competitive in the other test functions. These findings confirm that ESSA has a high value in terms of exploration power. The better results can be observed in both mean and STD measures, which point out how robust and well ESSA is when addressing these functions.

4.3.3 Evaluation of fixed-dimension functions

The mean and STD results of the fixed-dimension multimodal benchmark functions, $F_{14}$ to $F_{23}$, obtained by ESSA and other algorithms over 30 independent runs, are presented in Table 4.

Table 4 Computed results for the fixed-dimension multimodal functions based upon ESSA and other algorithms

The results in Table 4 show that the proposed ESSA has arrived at average results for the functions $F_{14}$–$F_{19}$ which are considerably better than those obtained by the standard SSA. There is no significant difference between the results of the proposed ESSA and those obtained by the standard SSA on functions $F_{20}$, $F_{21}$, $F_{22}$ and $F_{23}$. Moreover, the difference is very small on functions $F_{15}$ and $F_{16}$. Moreover, the mean results reported by ESSA outperform those reported by other algorithms. This confirms that ESSA has successfully identified the best global solutions and can efficiently explore the search space. This high exploration gives ESSA the power to avoid many local optima in a fixed-dimension multimodal search space.

In short, the average results and small STD values in Tables 2, 3 and 4 affirm that ESSA is robust, is stable and has large capabilities in exploring and exploiting the search space, even if the search space is complex.

4.4 Convergence analysis

Convergence curves are the most widespread qualitative results for optimization algorithms in the literature. In convergence curves, the best solutions obtained so far are generally stored at each iteration loop of the meta-heuristic algorithms. Thereafter, each convergence curve is drawn as a line to be eligible to describe how well an algorithm approximates the global optimum solution over a predetermined number of iterations. The convergence analysis aims to provide a deeper understanding of the exploitation and exploration processes of the proposed ESSA. Moreover, the convergence analysis results can judge the performance of ESSA compared to other algorithms. The convergence curves of ESSA, SSA, MFO, MVO, SCA, GSA, GA and HS for some benchmark test functions are presented in Fig. 4.

Fig. 4

Comparison of convergence curves of ESSA and other meta-heuristic algorithms obtained for some benchmark functions

The convergence curves in Fig. 4 divulge that ESSA delivers encouraging convergence responses in all of the evaluated test functions. In these curves, ESSA exhibits three featured converging behaviors while optimizing these benchmark functions. In the first few iteration steps, ESSA experiences sudden changes and converges very slowly to reach the most promising areas of the search space due to the adaptive self-parameter setting mechanism of the parameter $c_1$. This means that ESSA has a modest feebleness in exploring the search space in the first few iterations. This behavior is evident in functions $F_9$ and $F_{11}$. In the second converging behavior and beyond the next 150 iterations, ESSA gradually converges toward the global optimal solutions. This means that the exploration process of ESSA is fast and capable of locating the global solution using a modest number of iterations. This behavior is clearly evident in functions $F_1$–$F_{4}$. The third converging behavior can be inferred in the final iteration steps, where there is a clear convergence to find the global optimal solution or near-global optimal solution (i.e., food source) as obviously observed in functions $F_{10}$, $F_{13}$ and $F_{15}$. This efficiency of ESSA in the convergence curves is attributed to the proposed self-tuning parameter method, which allows the individuals of ESSA to explore the search space and move around the food source to locate the global optimum.

By comparing the convergence curves obtained by ESSA to those obtained by other algorithms, ESSA provides a rapid convergence than other algorithms in most of the test functions such as $F_1$–$F_4$ and $F_9$–$F_{11}$, and it is very competitive to others in the other test functions. On the other hand, GSA presented promising convergence curves better than ESSA in some functions such as $F_5$, $F_7$, $F_{12}$, $F_{13}$ and $F_{15}$.

Figure 5 shows the plot of $c_1$ and the variations of $\lambda _1$, $\lambda _2$ and $\lambda _3$ over the course of iterations of ESSA for some unimodal, multimodal and fixed-dimension multimodal functions. The values of these parameters used to plot $c_1$ were taken as 4, 0.44 and 1.1, respectively, where these values represent the optimal values for these tuned parameters in most of the optimization problems. Figure 5 shows that the variations of the parameters $\lambda _1, \lambda _2$ and $\lambda _3$ are very small at the beginning, and then, the values of these parameters fluctuate relatively large at the middle before approaching to small variations at the end. This narrow variation range of these parameters implies that the search agents of ESSA exploit each promising area to find the global optimum solution. These parameters need to reduce gradually in terms of $\theta _1$, $\theta _2$ and $\theta _3$, respectively. However, the best settings for these parameters mostly depend on the nature of the problem. The following comments can be drawn based upon these results:

• The optimal setting for $\lambda _1, \lambda _2$ and $\lambda _3$ in $c_1$ often depends on the nature of the problem, and there is no unique best setting for these parameters for all problems. However, Fig. 5 shows that proper values for these parameters can be chosen as $4 \pm 0.1, 0.44 \pm 0.1$ and $1.1 \pm 0.1$, respectively. The choice of these values should work appropriately for most problems, but not all problems.

• The small variations of the parameters $\lambda _1, \lambda _2$ and $\lambda _3$ mean that the actual setting of these parameters is not critical to a given problem, and therefore, there is no need to use a very large number of iterations and search agents to reach very optimal settings.

• Some parameters are more sensitive than others. In the present case, $\lambda _3$ needs more fine-tuning than $\lambda _1$ and $\lambda _2$, due to its smaller variations.

Fig. 5

Plot curves for the parameter $c_1$ ($\lambda _1 = 4.0$, $\lambda _2 =0.44$ and $\lambda _3 =1.1$) and the variations of these parameters over the course of iterations

4.5 Statistical test analysis

Statistical tests are necessary to ensure that the results in Tables 2, 3, 4 are not generated by chance. These tests are essential to demonstrate the reliability and accuracy of ESSA in obtaining the average and STD results in all of the 30 independent runs. To compare the average results of the algorithms presented in Tables 2, 3 and 4, and emphasize the significance of the results, statistical tests including Friedman’s and Holm’s tests (Pereira et al. 2015) are conducted to identify whether the results obtained by ESSA deviate from the results of other algorithms in a statistically significant manner.

Friedman’s test aims to determine whether there is a considerable difference between the findings of the various algorithms. This test is based on a null hypothesis that there is no remarkable difference in the performance of the comparative algorithms. The best performing algorithm gets the lowest rank, while the worst performing algorithm obtains the highest rank. In case the p value divulged by Friedman’s statistical test is equal or less than the significance level, here is 0.05, the null hypothesis is rejected, indicating that there are significant differences between the accuracy of the comparative algorithms. Friedman’s test is typically followed by a post hoc test such as Holm’s method to consider a pair-wise comparison of the algorithms. In general, the lowest ranked algorithm is used as a control algorithm for the post hoc test method. A summary of the statistical test results obtained using Friedman’s test on the mean results given in Tables 2, 3 and 4 is presented in Table 5.

Table 5 A summary of the statistical results obtained using Friedman’s test on the results shown in Tables 2, 3 and 4

According to the statistical results in Table 5, ESSA is the best performing algorithm among all other algorithms. Looking at the significance level of $\alpha = 0.05$, ESSA is ranked first in the test functions, followed in turn by SSA, GA, MVO, HS, MFO, GSA and SCA in the last rank.

The p value calculated by Friedman’s test based on the mean results in Tables 2, 3 and 4 is 4.06632E−7. The null hypothesis of equivalent performance is rejected to affirm if there are statistically considerable differences between the performance of the comparative algorithms. Holm’s method is then conducted as a posttest procedure to determine whether there are statistically significant differences between ESSA (i.e., the control algorithm) and other algorithms. The statistical results obtained using Holm’s method are given in Table 6.

Table 6 Results of Holm’s method based on the statistical mean results of the unimodal, multimodal and fixed-dimension multimodal benchmark functions

The results of Friedman’s and Holm’s test methods in Tables 5 and 6 illustrate the satisfactory performance and reliability of ESSA. These results demonstrate that ESSA will be proficient in solving complex real-world problems.

5 ESSA for engineering design problems

The efficiency of the proposed ESSA in solving real-world problems is demonstrated by experimenting with well-known examples of constrained engineering design problems. In the following subsections, ESSA was presented to solve five engineering design problems: the welded beam design problem, the tension/compression spring design problem, the three-bar truss design problem, the I-beam design problem and the cantilever beam design problem. These design problems have a range of constraints, so there is a need to equip ESSA with a constraint handling method. In this paper, to conduct a fair comparison between ESSA and algorithms reported in the literature, a death penalty function was used in a way analogous to that described in Yang (2010b). This constraint handling method has a low computational cost and does not use the information of infeasible solutions. The parameter setting of ESSA in solving the design problems below is the same as those presented in Table 1.

5.1 Welded beam design problem

The main goal of this classical design problem is to design a welded beam for the structure shown in Fig. 6 to achieve the minimum fabrication cost (Wang et al. 2014).

Fig. 6

A schematic structure of the welded beam

Table 7 A comparison of the results obtained by ESSA and other algorithms for the welded beam problem

As shown in Fig. 6, the welded beam composes of a beam A and a weld required to attach it to the member B. This design problem is subject to four constraints: shear stress ($\tau$), bending stress in the beam ($\theta$), buckling load on the bar ($P_c$) and end deflection of the beam ($\delta$). To solve this problem, there is a need to find the possible combination of the four structural parameters of the welded beam design: thickness of the weld (h), length of the clamped bar (l), height of the bar (t) and thickness of the bar (b). These structural design parameters can be represented by a vector as: $\vec {x} = [x_1, x_2, x_3, x_4]$, where $x_1, x_2, x_3$ and $x_4$ represent h, l, t and b, respectively. The mathematical formula of the cost function to be minimized for this design problem can be defined as follows:

$$\begin{aligned} f(\vec {x}) = 1.10471x^2_1x_2 +0.04811x_3x_4(14.0+x_2) \end{aligned}$$

(23)

Subject to the following constraints,

$$\begin{aligned} g_1(\vec {x})= & {} \tau (\vec {x})-\tau _\mathrm{max}\le 0\\ g_2(\vec {x})= & {} \sigma (\vec {x})- \sigma _\mathrm{max} \le 0\\ g_3(\vec {x})= & {} x_1 -x_4 \le 0\\ g_4(\vec {x})= & {} 1.10471x^2_1 +0.04811x_3x_4(14.0+x_2) -5.0 \le 0\\ g_5(\vec {x})= & {} 0.125- x_1 \le 0\\ g_6(\vec {x})= & {} \delta (\vec {x})- \delta _\mathrm{max} \le 0\\ g_7(\vec {x})= & {} P-P_c(\vec {x}) \le 0\\ \end{aligned}$$

where the other parameters of the beam design are determined as follows:

$$\begin{aligned} \tau (\vec {x})= & {} \sqrt{((\tau ')^2 + (\tau '')^2)+\frac{2\tau '\tau '' x_2}{2R}}, \tau '=\frac{p}{\sqrt{2}x_1x_2}\\ \tau ''= & {} \frac{MR}{J}, M=P\left( L+\frac{x_2}{2}\right) , R=\sqrt{(\frac{x_1+x_3}{2})^2+\frac{x_2^2}{4}}\\ J= & {} 2\left\{ \sqrt{2}x_1x_2\left[ \frac{x_2^2}{12}+(\frac{x_1+x_3}{2})^2\right] \right\} , \sigma (\vec {x})=\frac{6PL}{x_4x_3^2}\\ \delta (\vec {x})= & {} \frac{4PL^3}{Ex_4x_3^3}, P_c(\vec {x})=\frac{4.013\sqrt{EGx^2_3x_4^6/36}}{L^2}\left( 1-\frac{x^3}{2L}\sqrt{\frac{E}{4G}}\right) \end{aligned}$$

where $P =6000$ lb, $L =14$ in., $\delta _\mathrm{max} = 0.25$ in., $E =30\times 10^6$ psi, $G = 12\times 10^6$ psi, $\delta _\mathrm{max} = 13{,}600$ psi, $\sigma _\mathrm{max} = 30{,}000$ psi.

In this design problem, the ranges of the variables h, l, t and b were taken as $0.1\le x_1\le 2$, $0.1\le x_2\le 10$, $0.1\le x_3\le 10$ and $0.1\le x_4\le 2$, respectively. The welded beam design problem was solved by many algorithms such as SSA (Mirjalili et al. 2017), MVO (Mirjalili et al. 2016), SCA (Mirjalili 2016), MFO (Mirjalili 2015), CS algorithm (Gandomi et al. 2013), ACO (Kaveh and Talatahari 2010), GSA (Rashedi et al. 2009), improved PSO (IPSO) (Cagnina et al. 2008), ABC (Karaboga and Basturk 2008), evolutionary strategy (ES) (Mezura-Montes and Coello 2008), coevolutionary PSO (He and Wang 2007), differential evolution (DE) algorithm (Mezura-Montes et al. 2007) and simulated annealing (SA) (Hedar and Fukushima 2006). The comparative results for the best solution obtained by ESSA and other algorithms for this design problem are shown in Table 7. The results in Table 7 indicate that ESSA produced an optimal design for the welded beam problem by finding the optimal cost of approximately 1.72485, which is similar to the cost obtained by IPSO (Cagnina et al. 2008) and ABC (Karaboga and Basturk 2008).

5.2 Tension–compression spring design problem

The second engineering problem is a tension/compression spring with three parameters as shown in Fig. 7.

Fig. 7

A schematic of the tension/compression spring problem

Table 8 A comparison of the results obtained by ESSA and other algorithms for the tension/compression spring design problem

The goal of this design problem is to reduce the weight of a tension/compression spring design. This problem is subject to a number of constraints, including shear stress, surge frequency and minimum deflection. The parameters in this design problem can be defined as follows: wire diameter (d), mean coil diameter (D) and number of active coils (N), which are, respectively, represented by vector parameters as $\vec {x} = [x_1, x_2, x_3]$. The function $f(\vec {x}) = (x_3 +2)x_2x^2_1$ to be optimized is subject to the following constraints:

$$\begin{aligned} g_1(\vec {x})= & {} 1-\frac{x^3_2x_3}{71785x^4_1}\le 0\\ g_2(\vec {x})= & {} \frac{4x^2_2-x_1x_2}{12566(x_2x^3_1-x^4_1)}+\frac{1}{5108x^2_1}-1 \le 0\\ g_3(\vec {x})= & {} 1-\frac{140.45x_1}{x^2_2x_3} \le 0\\ g_4(\vec {x})= & {} \frac{x_1+x_2}{1.5} -1\le 0 \end{aligned}$$

The ranges of the parameters, d, D and N, were taken as follows: $0.05\le x_1\le 2.0$, $0.25\le x_2\le 1.3$ and $2.0\le x_3\le 15.0$, respectively. This problem was widely addressed in the literature by many algorithms such as spotted hyena optimizer (SHO) (Dhiman and Kumar 2017), SSA (Mirjalili et al. 2017), GWO (Mirjalili et al. 2014), MVO (Mirjalili et al. 2016), SCA (Mirjalili 2016), MFO (Mirjalili 2015), GSA (Rashedi et al. 2009), ES (Mezura-Montes and Coello 2008), coevolutionary PSO (He and Wang 2007), DE (Huang et al. 2007) and improved HS (IHS) (Mahdavi et al. 2007). A comparison of the best solution obtained by ESSA and other algorithms for this design problem is shown in Table 8. The results in Table 8 reveal the merits of ESSA in solving this problem by producing an optimal design. The proposed ESSA obtained a design for this problem with a cost of about 0.0126654, which is slightly better than the design obtained by similar algorithms such as GWO (Mirjalili et al. 2014).

5.3 Three-bar truss design problem

The three-bar truss engineering design problem (Gandomi et al. 2013) consists of two design variables, where the structural parameters in this problem are shown in Fig. 8.

Fig. 8

Three-bar truss design problem

Table 9 A comparison of the results obtained by ESSA and other algorithms for the three-bar truss design problem

The aim of this problem is to design a truss with three bars to minimize the volume of the truss structure for an objective function setup as:

$$\begin{aligned} f(\vec {x}) = \left( 2\sqrt{2}x_1+x_2\right) L \end{aligned}$$

(24)

This problem is subject to the following constraints:

$$\begin{aligned} g_1(\vec {x})= & {} \left( \frac{2\sqrt{2}x_1+x_2}{\sqrt{2}x_1^2+2x_1x_2} \right) P\le 2\\ g_2(\vec {x})= & {} \left( \frac{1}{x_1+\sqrt{2}x_2} \right) P\le 2\\ g_3(\vec {x})= & {} \left( \frac{2}{\sqrt{2}x_1^2+2x_1x_2}\right) P\le 2 \end{aligned}$$

Fig. 9

I beam design problem

Table 10 A comparison of the results obtained by ESSA and other algorithms for the I-beam design problem

where $0\le x_1\le 1.0$ and $0\le x_2\le 1.0$. The other constants are $L = 100$ cm and $P = 2$ kN/cm$^2$. The three-bar truss design problem was solved by many researchers using different algorithms such as SSA (Mirjalili et al. 2017), CS algorithm (Gandomi et al. 2013), mine blast algorithm (MBA) (Sadollah et al. 2013), PSO with DE (PSO-DE) (Liu et al. 2010) and DE with dynamic stochastic selection (DEDS) (Zhang et al. 2008). Table 9 shows a comparison of the best solutions obtained by ESSA and other algorithms previously reported in the literature for the three-bar truss design problem. According to Table 9, ESSA can find an optimal design for the three-bar truss problem with a cost of about 263.8958433.

5.4 I-beam design problem

The capability of ESSA was also used to solve the real I-beam engineering design problem that deals with four structural parameters (Gandomi et al. 2013). The main objective of this problem is to minimize the vertical deflection of the I-beam design shown in Fig. 9.

This design problem simultaneously meets the cross-sectional area and stress limitations under given loads. The four parameters of this problem are listed as: b, h, $t_w$ and $t_f$, where the length of the beam (L) and modulus of elasticity (E) are 5.200 cm and 523.104 kN/cm$^2$, respectively. The objective function defined to minimize the vertical deflection $f(x) = PL^3/48EI$ was considered as given in the following formula:

$$\begin{aligned} \text {Minimize:} f(\vec {x}) = \frac{5000}{\frac{x_3(x_2-2x_4)^3}{12} +\frac{x_1x_4^3}{6}+2x_1x_4\frac{(x_2-x_4)^2}{4}} \end{aligned}$$

Fig. 10

A cantilever beam design problem

Table 11 A comparison of the results obtained by ESSA and other algorithms for the cantilever beam design problem

The problem is subject to cross-sectional area less which would be than 300 cm$^2$, as defined below:

$$\begin{aligned} g_1(\vec {x}) = 2x_1x_3+x_3(x_2-2x_4)\le 300 \end{aligned}$$

where the design spaces of the variables are defined as: $10\le x_1\le 50, 10\le x_2\le 80, 0.9\le x_3\le 5.0$ and $0.9\le x_4\le 5.0$. There are several optimization algorithms in the literature previously presented to solve this design problem. These algorithms respected the aforementioned constraint and used the same range of design variables. The results of ESSA for the I-beam design problem design are compared with the standard SSA (Mirjalili et al. 2017) and many other algorithms such as symbiotic organism search (SOS) (Cheng and Prayogo 2014), CS algorithm (Gandomi et al. 2013), adaptive response surface method (ARSM) (Wang 2003) and inherited ARSM (IARSM) (Wang 2003). A comparison of the best solutions obtained by ESSA and other algorithms for the I-beam design problem is presented in Table 9. Table 10 shows that ESSA is capable of finding an optimal design for the I-beam design problem with a minimum cost of 0.0066259581655.

5.5 Cantilever beam design problem

The fifth engineering design problem is a cantilever beam that consists of five elements, each with a hollow cross section with a constant thickness (Gandomi et al. 2013). The beam is strictly supported as shown in Fig. 10, and there is an external vertical force acting at the free end of the cantilever.

The goal of this problem is to minimize the weight of a cantilever beam while setting an upper limit on the vertical displacement of the free end. The design variables are the heights (or widths) of the cross section of each element. The lower limits of these variables are too small and the upper limits are too large so they do not become active in the problem. To solve this optimization problem, there is a need to find a possible combination of the five parameters of this design. The cost function of this design problem can be formulated as follows:

$$\begin{aligned} f(\vec {x}) = 0.0624(x_1+x_2+x_3+x_4+x_5) \end{aligned}$$

where this problem is subject to the following constraint,

$$\begin{aligned} g_1(\vec {x}) = \frac{61}{x_1^3}+\frac{37}{x_2^3}+\frac{19}{x_3^3}+\frac{7}{x_4^3}+\frac{1}{x_5^3} \end{aligned}$$

The ranges of the variables in this design problem were considered as follows: $1\le x_i\le 10$, where $i\in {1, 2, 3, 4, 5}$. Several algorithms in the literature were used to solve the cantilever beam design problem such as SSA (Mirjalili et al. 2017), SOS (Cheng and Prayogo 2014), generalized convex approximation ($\hbox {GCA}_I$, $\hbox {GCA}_II$) (Gandomi et al. 2013) and CS (Gandomi et al. 2013). A comparison of the solutions obtained by ESSA and other algorithms for the cantilever beam design problem is presented in Table 11. The results in Table 11 affirm that ESSA produced an optimal design for the cantilever beam problem with an optimal cost of 1.33995634, which is the lowest cost compared to the others.

6 Modeling of complex nonlinear systems

This section shows the applicability of ESSA to model two real industrial problems. These systems are a metal cutting system and an industrial winding machine system.

6.1 Metal cutting system

The system modeling in this problem aims to design a cutting temperature experiment for the metal cutting system presented in Fig. 11 (Zhou et al. 1998).

Fig. 11

A schematic diagram of the metal cutting system tool

The metal cutting tool used here is P05 horny alloy steel and the workpiece metal is 38CrNi3Mo. A commonly cutting temperature model used for P05 is presented in Eq. (25) (Zhou et al. 1998; Faris and Sheta 2016).

$$\begin{aligned} T_r= ka^xf^yv^z \end{aligned}$$

(25)

where $T_r$ identifies the output of the temperature model, k is a parameter where its value depends on the machined material, a represents the depth of cut section in millimeters (mm), f represents the cutting feed rate in mm/rev, v identifies the speed of cutting in meter/minute (m/min) and x, y and z are three parameters whose values depend on the material of the cutting tool.

Fig. 12

Convergence curves over thirty experiments for parameter estimation of the cutting tool P05

In this study, ESSA was used to develop a model for a cutting temperature system using publicly available data (Yi-jian et al. 2005). More specifically, ESSA was used to estimate k, x, y, z parameters of the P05 model shown in Eq. (25). The main goal of the developed model is to minimize the error between the actual metal cutting temperature system (T) and the counterpart estimated model ($T_r$). The fitness function is defined in Eq. (26).

$$\begin{aligned} \text {fitness}= \sum _{i=1}^{n}\left[ T(i)-T_r(i)\right] ^2 \end{aligned}$$

(26)

where n represents the number of data samples. The variance-accounted-for (VAF) measure defined in Eq. (27) was used to assess the performance of the developed models.

$$\begin{aligned} \text {VAF} = \Big [1-\frac{\mathrm{var}(T-T_r)}{var(T)}\Big ] \times 100\% \end{aligned}$$

(27)

where var(y) is the variance of the actual data of the metal cutting temperature system (T) and var$(T-T_r)$ is the variance of the difference between the actual data, T, and the estimated data ($T_r$). The convergence curve obtained by ESSA compared to that obtained by SSA in the modeling of P05 is shown in Fig. 12. Figure 12 shows that ESSA presented faster convergence better than the convergence provided by the standard SSA. The model responses of P05 along with the true P05 obtained by ESSA and SSA are presented in Fig. 13. Table 12 compares the VAF values and the estimated values of k, x, y, z, obtained by ESSA with those obtained by other algorithms such as SSA, PSO and GA. The findings in Table 12 confirm that ESSA has remarkably reported a better VAF rate than those reported by the standard SSA, PSO and GA.

Fig. 13

Evaluation responses of the models developed by ESSA and SSA for the cutting tool P05

Table 12 A comparison of the results obtained by ESSA and other algorithms for P05 of a metal cutting temperature system

6.2 An industrial winding process

The industrial winding process under consideration is usually encountered in real web conveyance systems (Bastogne et al. 1998; Rodan et al. 2017). The structural diagram of this process is shown in Fig. 14.

As shown in Fig. 14, the winding process targeted in this study consists of three reels, referred to as reel 1, reel 2 and reel 3. These reels are controlled by three DC motors known as $M_1$, $M_2$ and $M_3$, respectively. Reels 1 and 3 are coupled to DC motors driven by set-point currents $I_{1}$ at motor 1 and $I_{3}$ at motor 2. Moreover, tension meters are placed to measure strip tensions in the web between reel 1 and reel 2, referred to as $T_{1}$, and between reel 2 and reel 3, referred to as $T_{3}$. In this process, there is a dynamo tachometer to measure the angular speeds of reels 1, 2 and 3, referred to as $S_{1}$, $S_{2}$ and $S_{3}$), respectively. The angular velocity of motor $M_2$, or denoted as $\Omega _2$, is measured using a dynamo tachometer. The main input variables of this process are: $S_{1}$, $S_{2}$, $S_{3}$, $I_1$ and $I_3$, and the outputs are: $T_{1}$ and $T_{3}$.

The training dataset for each model developed for $T_{1}$ and $T_3$ consisted of 1250 data samples for each input variable. The dataset in the testing process consisted of 1250 samples for each input variable. These datasets are described in Nozari et al. (2012). The goal of this modeling problem is to capture the main characteristics of the correct output responses for this process given its input and output parameters.To solve this problem using linear estimates, there is a need to create linear models for $T_{1}$ and $T_{3}$. These models are defined in Equations  and , respectively.

$$\begin{aligned} T_1(t)= & {} T_1(t-1)+\alpha _1 S_1(t)+\alpha _2 S_2(t)\nonumber \\&+\alpha _3 S_3(t) + \alpha _4 I_1(t)+\alpha _5 I_3(t) \\ T_3(t)= & {} T_3(t-1)+\beta _1 S_1(t)+\beta _2 S_2(t)\nonumber \\&+ \beta _3 S_3(t) + \beta _4 I_1(t)+\beta _5 I_3(t) \end{aligned}$$

Fig. 14

A schematic diagram of the winding process

where $T_{1}(t-1)$ and $T_{3}(t-1)$, respectively, are the preceding values for $T_{1}$ and $T_3$ at time $t-1$, and $\alpha _1$–$\alpha _5$ and $\beta _1$–$\beta _5$ are the weights of the linear models developed for $T_{1}(t)$ and $T_{3}(t)$, respectively.

The mean absolute percentage error (MAPE) criterion, defined in Eq. (28), was used as a fitness function for this modeling problem.

$$\begin{aligned} MAPE = \frac{1}{n}{\sum _{i=1}^{n} \left| \frac{y_i- {\hat{y}}_i}{y_i}\right| }\times 100\% \end{aligned}$$

(28)

where n represents the number of experimental data values, y identifies the actual values of a real experiment and ${\hat{y}}$ represents the estimated values generated by the models. In the development of $T_{1}$ and $T_3$ models, the maximum number of iterations and the number of search agents for ESSA and SSA were set to 1000 and 30, respectively. The convergence curves obtained by ESSA compared to the corresponding curves obtained by SSA in the modeling of $T_{1}$ and $T_3$ are shown in Fig. 15.

Fig. 15

Convergence curves over thirty experiments for linear $T_1$ and $T_3$ model-based ESSA and SSA

The convergence curves in Fig. 15 confirm the reliability of the proposed ESSA in reaching the minimum errors over those obtained by SSA. The performance of ESSA in tracking the output data of $T_{1}$ and $T_3$ is shown in Fig. 16 and 17, respectively, along with the actual responses of $T_{1}$ and $T_3$, for both training and test cases.

Fig. 16

Training and testing responses of the linear ${T}_{1}$ model developed by ESSA

Fig. 17

Training and testing responses of the linear ${T}_{3}$ model developed by ESSA

The simulation results in Figs. 16 and 17 demonstrate the validity and appropriateness of ESSA in reliably modeling the web tension, $T_{1}$ and $T_3$, of the winding process. Table 13 presents a comparative study of the performance of the proposed ESSA and the standard SSA. According to results in Table 13, ESSA achieved VAF rates for $T_{1}$ and $T_3$ models significantly better than those obtained by the standard SSA.

Table 13 A comparison of the results obtained by ESSA and SSA in developing linear models for $T_{1}$ and $T_{3}$ over 30 runs

7 Analysis of the results

The preceding sections used a set of benchmark functions, a set of five engineering problems and two real industrial problems with various characteristics to assertively measure and confirm the performance of ESSA. In this work, ESSA is different from SSA in that it uses the current positions of the search agents in the position updating mechanism. Also, the dominant parameter $c_1$ of ESSA does not depend on the maximum number of iterations as that of SSA. Moreover, in ESSA, we have successfully developed a novel lifetime convergence framework by self-tuning the parameter $c_1$ using ESSA itself. This is to find the best setting for $c_1$ so that ESSA can perform the best for a broad range of problems of varying complexity.

Inspecting the results of the unimodal, multimodal and fixed-dimension multimodal functions in Tables 2, 3 and 4, respectively, we may conclude that ESSA performs better than other algorithms in the majority of test functions. The higher mean values indicate that ESSA carries out better than others on average, and the standard deviation values confirm that this supremacy is stable. These results denote how excellently and robust ESSA is when solving these functions. Due to the existence of a single optimum in the unimodal functions, they can only measure exploitation and convergence of algorithms. Therefore, the results in Table 3 showed that ESSA takes advantage of the high exploitation and convergence speed of these functions.

As aforementioned, multimodal functions confront the presence of many local optimums in addition to more than one global optimum, so these functions are highly efficient to benchmark the ability of the algorithms to eschew local optimum and test their exploration capability. Therefore, the results in Tables 3 and 4 showed that ESSA benefits from large exploration.

This arises from the fact that the search agents of ESSA tend to communicate with each other, so they are not easily attracted to a local solution. The connection between the search agents of ESSA also allows it to explore the search space and gradually moves the salp chain toward the global optimal. The superior convergence of ESSA is attributed to the position updating process of salps around the best solutions obtained so far.

The entire salp chain converges toward the global optimum solution proportionate to the iteration number due to the proposed lifetime tuning mechanism for the parameter $c_1$ of ESSA based on ESSA itself. On the other hand, the results of the p values in Table 6 generated from Friedman’s and Holm’s tests show that the superiority of ESSA is statistically significant. The p values of Holm’s test also evidence how important the proposed algorithm is on the functions $F_1$–$F_{23}$. The outcomes of Friedman’s test also prove that the results of ESSA are statistically significant because the majority of the p values are less than the significance level (i.e., 0.05).

Despite the success of ESSA in solving basic benchmark functions of varying complexity, we conducted a more extensive test to assess its ability to solve engineering design problems and real industrial problems. This is to explore its reliability and verify the robustness of the proposed lifetime convergence scheme that is performed using the proposed self-tuning method. The results of ESSA on engineering design problems in Tables 7, 8, 9, 10 and 11, and industrial problems in Tables 12 and 13 show that ESSA produced stable performance and outperformed other algorithms. These engineering and industrial problems present very challenging test problems for any meta-heuristics search algorithm. This makes them very analogous to the actual search space that ESSA might encounter when solving real-world challenging and practical problems. Therefore, researchers always view meta-heuristic search algorithms as generally effective means to solve challenging problems of growing size. We know that each algorithm has its advantages such as flexibility, simplicity and limitations such as complexity and parameter setting. As analyzed and discussed above, ESSA has many key features; a natural question is: What are the limitations of the proposed ESSA? Based upon the position updating process, the lifetime convergence scheme and the obtained results of ESSA, we can sum up the limitations of ESSA as given in the following subsection.

7.1 Limitations of the proposed ESSA

It is worth mentioning here that the exploitation of ESSA attends to be better than its exploration. This can be deduced from the results produced from the unimodal functions compared to those of the multimodal functions. This is related to the position updating proposed and the proposed self-tuning method of this algorithm, where sudden changes in the solutions are small at the beginning of the execution of this algorithm. Despite this fact, the results in Tables 2, 3 and 4 confirm that this is not a serious concern, as the explorative behavior of ESSA is also very plausible. Mere exploration does not ensure locating the global optimum solution, where an appropriate balance is needed between exploration and exploitation. The computation time of ESSA is not a critical issue as it can vary according to the nature and complexity of the problem. Several meta-heuristic algorithms use fixed pre-tuned parameters. Contrarily, ESSA uses one main control parameter, which is tuned using ESSA itself as iterations proceed. This offers a way for ESSA to automatically turn from exploration to exploitation as the optimal solution approaches. This known limitation of meta-heuristic algorithms was addressed and provides an advantage to ESSA over other algorithms.

8 Conclusion and future works

This paper has proposed an improved variant of the salp swarm algorithm (SSA) or referred to as enhanced SSA (ESSA). ESSA is generally distinguished based on its strategy to update the salps position and the proposed self-tuning convergence method. The major elements that contribute to the exploitation and convergence abilities of ESSA are the proposed self-tuning method for the main parameter of ESSA and the position updating mechanism that works on the best positions acquired so far in the search space. These featured elements in ESSA allow its salps to effectively explore the search space and identify the global optimum. To verify the efficacy of ESSA, many experiments were conducted on basic benchmark test functions. The outstanding performance of ESSA on these functions evinces the exploitation and exploration capabilities of ESSA in addition to its convergence behavior. The results of ESSA were compared with SSA and other well-known algorithms in terms of the average and standard deviation values. ESSA has proved to be an effective candidate that outperforms many other algorithms. The convergence behavior of ESSA was analyzed and found to be converging rapidly compared to others. The ability of ESSA in solving challenging problems is manifested by solving five well-known engineering design problems and two real industrial processes. The proposed self-tuning method-based ESSA could be effective for parameter control, so a more general framework for both parameter setting and control could be expanded for a broad range of real-world applications. Future work could be directed to substantiate the robustness of ESSA to address multi-objective, continuous and discrete optimization problems.

Appendix A: Objective test functions

A detailed description of the unimodal benchmark test functions ($F_1$–$F_7$), multimodal benchmark test functions ($F_8$–$F_{13}$) and fixed-dimension multimodal benchmark test functions ($F_{14}$–$F_{23}$) is given in Table 14.

Table 14 Characteristics of benchmark functions