An effective solution to numerical and multi-disciplinary design optimization problems using chaotic slime mold algorithm

Abstract

Slime mold algorithm (SMA) is a recently developed meta-heuristic algorithm that mimics the ability of a single-cell organism (slime mold) for finding the shortest paths between food centers to search or explore a better solution. It is noticed that entrapment in local minima is the most common problem of these meta-heuristic algorithms. Thus, to further enhance the exploitation phase of SMA, this paper introduces a novel chaotic algorithm in which sinusoidal chaotic function has been combined with the basic SMA. The resultant chaotic slime mold algorithm (CSMA) is applied to 23 extensively used standard test functions and 10 multidisciplinary design problems. To check the validity of the proposed algorithm, results of CSMA has been compared with other recently developed and well-known classical optimizers such as PSO, DE, SSA, MVO, GWO, DE, MFO, SCA, CS, TSA, PSO-DE, GA, HS, Ray and Sain, MBA, ACO, and MMA. Statistical results suggest that chaotic strategy facilitates SMA to provide better performance in terms of solution accuracy. The simulation result shows that the developed chaotic algorithm outperforms on almost all benchmark functions and multidisciplinary engineering design problems with superior convergence.

1 Introduction

Meta-heuristic algorithms being simple and easy to implement are effectively applied to continuous, discrete, constrained, or unconstrained problems which were found hard to solve using conventional methods such as conjugate gradient, quadratic programming, and quasi-network methods. These meta-heuristic algorithms are single solution based which provides only one solution during optimization or population-based which mimics mostly natural phenomena and evolves a set of solutions during each iteration. These meta-heuristic algorithms are mainly categorized into four main groups: evolutionary, physics-based, human-based, and swarm intelligence type algorithms. Evolutionary algorithms (EAs) such as genetic algorithm (GA), differential evolution (DE) [1], genetic programming (GP) [2], and evolution strategy (ES) [3] mimics behaviors such as selection, recombination and mutation. The second class utilizes some physical laws such as gravitational search algorithm (GSA) [4], big-bang big-crunch (BBBC) [5], multi-verse optimizer (MVO) [6] and sine cosine algorithm (SCA) [7]. The third category mimics certain human behaviors which includes some of the well-known algorithms such as Tabu search (TS) [8], teaching learning based optimization (TLBO) [9], socio evolution and learning optimization (SELO). The last category of P-meta-heuristics uses collective or social intelligence that artificially simulates the behaviors such moving in swarms, flocks and herds. This Swarm Intelligence includes particle swarm optimization (PSO) [10], ant colony optimization (ACO) [11], artificial bee colony (ABC) [12], machine learning (ML) [13], bat-inspired algorithm (BA) [14], grey wolf optimization (GWO) [15], moth flame optimization (MFO) [16], artificial neural network (ANN) [17] and Harris Hawk optimizer (HHO) [18]. Machine-learning approach have great potential in solving numerical complexities involving large variables [19]. These algorithms have their own strategies but although they have two common search feature steps: diversification and intensification. The random search process is the explorative (diversification) process where the operators search the different regions more intensely to obtain the global optima. The local search process is the intensification step which is performed after the exploration phase to enhance solution accuracy. A proper balance between exploration and exploitation is the basic need to avoid any local minima entrapment. A large variety of meta-heuristic algorithms has been invented to solve various numerical and engineering optimization problems but still, there is always a scope of improvement in the existing research by utilizing no-free-lunch (NFL) theory. This theorem persuades that no method is the globally best and there is always scope for new more efficient algorithms.

The researchers are continuously working on different variants to employ innumerable sorts of advanced methods on various problems. The concise surveys of the different stochastic meta-heuristics and heuristic methods are presented in Table 1. The rest of the paper is arranged as: Sect. 2 includes a comprehensive review of the latest SMA variants. Section 3 comprises the basics of Physarum polycephalum and the mathematical model. In Sect. 4, standard benchmark test functions are included. Section 5 includes test results of the proposed algorithm and comparative analysis with well-known algorithms. Section 6 presents analysis of 10 multi-disciplinary problems. In Sect. 7, paper is concluded. Finally, limitations and future scope are explored in Sect. 8.

Table 1 Review of some existing algorithms

2 Literature survey of some recent SMA and chaotic variants

In this section, a comprehensive study of specific allied work has been presented to explore information regarding various recent advancements related to SMA and Chaotic strategies. The foraging behavior of Physarum polycephalum to discover new sites for food has been efficiently mimicked by many researchers to develop many new meta-heuristic algorithms with different platforms. Adamatzky et al. had experimentally proved that plasmodium (slime mold) could navigate through dissimilar channels without exploring all possible solutions. But, some irregularities are noticed that restrict the maze-solvers to compete with conventional architectures [50]. Nakagaki et al. presented mathematical statistics showing intelligent behavior of plasmodium wherein a tabular network is formed to find multiple food sources via the shortest path. It is observed that the proposed method not only explores different shapes but also provides overtime depending states [51]. Andrew Adamatzky and Jeff Jones effectively implemented the foraging capability to search optimal routes in ten urban areas and perform reconfiguring of transport networks [52]. Beekman et al. carried out an intensive study about the decision-making process of slime molds for choosing the optimal path for getting nutrition by searching the shortest path [53]. Burgin et al. designed the modal of a structural machine by adopting the inherent ability of molds to sense cell information. Further, it is noticed that structural machines have the potential to solve complex computational problems by implementing machine algorithms [54]. Daniel et al. had applied the SMA technique to solve the transportation issue due to growing urbanization. In this concern, SMA provides an optimal solution to minimize the travel path by searching the shortest path [55]. Houbraken et al. have developed an extended fault-tolerant algorithm by utilizing the slime mold concept to improve the fault-tolerant network in the telecommunication sector [56]. Kropat et al. presented a deterministic approach to solving single path and multi-path optimization problems under uncertainty. This research also explores the robustness of the SMA algorithm to tackle emergencies and avoiding disasters [57]. Abdel-Basset et al. have developed a hybrid algorithm by incorporating Harris’s Hawk’s algorithm with whale optimization algorithm (WOA). In this work, the image segmentation problem (ISP) related to the X-ray of an infected person due to Covid-19 has been improved [58]. Zhao et al. presented a new levy fight distributed parameters to enhance the performance of basic SMA [59]. Patino-Ramirez et al. described the influence of chemical composition on the morphology and dynamics of slime mold [60]. Kouadri et al. have analyzed the optimal power flow problem of a hybrid renewable system. This work also effectively improves system stability by integrating VAR compensators with the thermal-wind system [61]. GAO et al. have presented a hybrid algorithm by combining grey wolf optimizer with SMA and three different types of optimization problems [62]. Nguyen et al. incorporated SMA algorithm for handling hydropower generation and compared simulation results with other algorithms [63]. Davut et al. presented a novel solution to PID-controlled DC motor and AVR system by utilizing the exploitation capability of the SMA algorithm [64]. Chen et al. introduced the SMA algorithm for solving stochastic optimization problems and had effectively applied the SMA algorithm to solve benchmark mark and engineering design problems [65]. Recently, researchers are getting more attracted toward hybrid and chaotic variants as these strategies have the inherent capacity to enhance the local search process. A large number of chaos optimization algorithms (COAs) and hybrid COA have proven that chaos could easily escape from local minima as compared to classical stochastic optimization algorithms. There are various chaotic variants such as adaptive chaotic sine cosine algorithm [66], chaotic whale optimization algorithm [67], chaotic dragonfly algorithm [68], modified whale optimization algorithm [69], chaotic Krill Herd algorithm [32], binary grasshopper optimization algorithm [70], chaotic grey wolf optimization [71]. Some of the recent SMA and chaotic variants are explored in Table 2 to get more familiar with various concepts related to the proposed work.

Table 2 Review of some recent SMA and chaotic variants

From the above literature survey, it is noticed that a large variety of meta-heuristic and hybrid variants have been invented by the researchers to fix different types of stochastic complexities. Some real-world problems such as network foraging, fault-tolerant, transportation, structural machines, engineering design, image segmentation, optimal power flow, and feature selection were analyzed by various researchers using a heuristic approach. The solution accuracy of any algorithm depends on its capability to have a proper balance between intensification and diversification. Studies revealed that slow convergence is the common faintness of most heuristic algorithms. This ultimately gives rise to reduced computational efficiency. Thus, to improve the solution efficiency, a trend of developing hybrid algorithms is escalating fast. In addition, diverse chaotic strategies have been effectively incorporated by many researchers to optimize their specific objective function. The ultimate aim of these techniques is to provide an optimal solution for a pre-defined objective function. Recently, a chaotic variant of SMA using the “Chebyshev function” was presented by Zhao et al. [74]. In this work, 100 Monte Carlo experiments were performed using SMA and Chebyshev mapping. Only the “Sargan function” of the uni-modal test function was simulated for 100 iterations. “Sine Wave function” was simulated for multi-modal test function for the same number of iterations. It was noticed that the solution of these benchmark functions is not exploited to an appreciable level. To check the validity of this method, only two standard benchmark functions had been tested. It was noticed that the results given by Chebyshev and sine wave function were not capable of giving efficient solutions. In most cases, simulation results are subjected to premature convergence. Although the methodology of the proposed CSMA is similar to the basic variant but differs in terms of the selection of chaos map. In the proposed research the local search capability of basic SMA has been enhanced using “sinusoidal chaotic function”. The CSMA method has been effectively employed to evaluate global optimization, standard benchmark, and engineering design problems. The comparative analysis demonstrated in the result section revealed that the suggested method gives outstanding performance in terms of fitness evaluation and solution accuracy.

3 Chaotic slime mold algorithm

3.1 Background of proposed research

It was found that the organism’s behavior could be easily adopted and statistically modeled to handle unconstrained and non-convex mathematics. Investigators have endeavored to mimic the working guidelines to progress computations and algorithms. Slime molds have acknowledged ample courtesy in contemporary years. The slime mold points out in this article normally refers to the Physarum polycephalum which belongs to the species of order Physarales, subclass Myxogastromycetidae, class Myxomycetes, division Myxostelida. Since it was first classified as a fungus, named as “slime mould” [65]. Typically, the plasmodium forms a network of protoplasmic tubes connecting the masses of protoplasm at the food sources, which is efficient in terms of network length and elasticity [50]. During the relocation cycle, the front end reaches out into a fan-molded, trailed by an interconnected venous organization that permits cytoplasm to stream inside, as shown in Fig. 1. Molds use their venous network for searching multiple food sources thus secreting enzymes to trap the food centers. It may even cultivate to extra than 900 cm² when there is adequate food in the environment [57].

Fig. 1

Searching structure of Physarum polycephalum (slime mold)

In the case of food scarcity, the slime mold even flows vibrantly, that helps to understand how slime mold search, moves, and connect food in the changing environment. When a secretion approaches the target, slime can judge the positive and negative feedback and find the ultimate route to grasp food in a better way. This suggests that slime mold can construct a concrete path subject to the level of food concentration. It prefers to select the region of high food concentration. Depending upon the food concentration and environmental risk, the mold weighs the speed and decides to leave the old location, and begins its new search during foraging [58]. Slime mold adopts empirical rules based on currently available insufficient data to decide to initialize new search and exit present location while foraging. Even if a food source is available in abundance, mold may divide its biomass to exploit other resources on the information of some rich high-quality food information. It may dynamically adjust their search patterns as per the quality of food stock [59].

3.2 Basic slime mold algorithm

Step 1 In this step, mathematics for the slime mold behavior is formed and following rule is assigned to find updated position of during search for food. The criteria for this depends upon r and p. This is the contraction mode of mold:

$$ \overrightarrow {X(t + 1)} = \left\{ {\begin{array}{*{20}c} {\overrightarrow {{X_{b} (t)}} + \overrightarrow {{v_{b} }} \cdot \left( {\overrightarrow {W} \cdot \overrightarrow {{X_{A} (t)}} - \overrightarrow {{X_{B} \left( t \right)}} } \right)} & {r < p} \\ {\overrightarrow {{v_{c} }} \cdot \overrightarrow {X(t)} } & {r \ge p} \\ \end{array} } \right., $$

(1)

where $\overrightarrow {{v_{b} }}$ is a parameter with a range of $\left[ { - a\,,a} \right]\,\,,\,\,\overrightarrow {{v_{c} }}$ is the parameter which approaches linearly toward zero. ‘t’ is the current iteration, $\overrightarrow {{X_{b} }}$ is the location of each particle in region where odor is maximum, $\vec{X}$ is the mold’s location, $\vec{X}_{A}$ and $\vec{X}_{B}$ are the randomly selected variables from the swarm, $\overrightarrow {W}$ is the measure of weighs of masses.

The maximum limit of p is as follows:

$$ p = \tan h\left| {S(i) - {\text{DF}}} \right|, $$

(2)

where $i \in 1,2, \ldots ,n,$ $S(i)$ = fitness of $\vec{X}$, DF = overall fitness from all steps.

The equation of $\vec{v}_{b}$ as follows:

$$ \overrightarrow {{v_{b} }} \, = \,\left[ { - a\,,a} \right] $$

(3)

$$ a = \arctan h\left( { - \left( {\frac{t}{\max \_t}} \right) + 1} \right). $$

(4)

The equation of $\overrightarrow {W}$ is listed as follows:

$$ \overrightarrow {W({\text{smell}}\,{\text{index}}(i))} = \left\{ {\begin{array}{*{20}c} {1 + r \cdot \log \left( {\frac{{{\text{bF}} - S(i)}}{{{\text{bF}} - {\text{wF}}}} + 1} \right),} & {{\text{condition}}} \\ {1 - r \cdot \log \left( {\frac{{{\text{bF}} - S(i)}}{{{\text{bF}} - {\text{wF}}}} + 1} \right),} & {{\text{others}}} \\ \end{array} } \right., $$

(5)

$$ {\text{Smell}}\,{\text{Index}} = {\text{sort}}(S), $$

(6)

where $S(i)$ rank first half of the population, r is the random value in the interval of $\left[ {0,1} \right],$ ${\text{bF}}$ is the optimal fitness obtained in the current iterative process, ${\text{wF}}$ is the worst fitness value obtained in the iterative process, and Sort (s) function sorts fitness values.

Step 2 The equation for upgrading the positions of agents (i.e. to wrap food) is given as follows:

$$ \overrightarrow {{X^{*} }} = \left\{ {\begin{array}{*{20}c} {{\text{rand}} \cdot \left( {{\text{UB}} - {\text{LB}}} \right) + {\text{LB}},} & {{\text{rand}} < z} \\ {\overrightarrow {{X_{b} (t)}} + \overrightarrow {{v_{b} }} \cdot (W \cdot \overrightarrow {{X_{A} (t)}} - \overrightarrow {{X_{B} (t)}} ,} & {r < p} \\ {\overrightarrow {{v_{c} }} \cdot \overrightarrow {X(t)} ,} & {r \ge p} \\ \end{array} } \right., $$

(7)

where LB and UB are the search limits, and rand and r denote the random value in [1].

Step 3 With the up gradation in the search process, the value of $\overrightarrow {{v_{b} }}$ vibrantly changes between $\left[ { - a,a} \right]$ and $\overrightarrow {{v_{c} }}$ varies between $\left[ { - 1,1} \right]$ and at last shrinks to zero. This is known to be as ‘grabbling of food’.

3.3 Types of chaotic functions

The concept of probability distribution is captured by lot of meta-heuristics algorithms to gain randomness. Chaotic maps could be beneficial if randomness due to ergodicity, idleness and molding properties are replaced. These criteria’s are full filled by the following equation:

$$ y_{k + 1} = f(y_{k} ). $$

(8)

In Eq. (8), $y_{k + 1} \;{\text{and}}\;f(y_{k} )$ are the (k + 1)th and Kth chaotic number, respectively. The action of chaotic function is dependent on initial value $y_{0}$. The particular type of chaotic function will generate a solution within the standardized equations as shown in Table 3.

Table 3 Chaotic map functions

3.4 Algorithm of proposed work

The basic SMA is combined with the sinusoidal chaotic approach to further enhance the performance. The pseudo-code for method proposed is as shown in Fig. 2.

Fig. 2

Pseudo-code of chaotic slime mold algorithm

There are several complex optimization problems wherein mathematical reformulations restrict to perform algorithms efficiently. The proposed Chaotic SMA variant is advantageous over standard algorithms along with standard SMA. The convergence rate of stochastic methods was noticed not much efficient due to premature convergence. Whenever some additional functionality is incorporated with the original system, the performance starts getting degraded. After a thorough review of various SMA and chaotic variants, it is observed that the sinusoidal function to be the most appropriate to improve exploitation. The proposed algorithm utilizes a sinusoidal chaotic function to intensify the search capacity of classical SMA and optimize the objective fitness of various problems. The chaotic approach enables SMA to regulate the initial parameters of the search and thus rectifies the local entrapment of molds. It is seen that some algorithms lack global search capability. For the efficient performance of any algorithm, there should a proper balance between its local and global search capabilities. In the proposed research, to incorporate these local and global search requirements, no composite operations are involved. Second, the efficacy of any algorithm is judged by the simulation time required to simulate a particular objective function. The experimental section of the paper reveals that the suggested method has improved solution efficacy to a greater extend. Lastly, during the cumulative run process, as the simulation successive progresses, premature convergence “local area stagnation” is the most common problem of many algorithms. This local area stagnation problem is omitted in the proposed method by utilizing the properties of ergodicity. The basic feature of Chaotic SMA is to enhance the optimization process by improving local search capabilities. Twenty-three benchmark problems including uni-modal, multi-modal and fixed dimensions are tested using the suggested chaotic slime algorithm.

4 Test benchmark functions

The developed Chaotic SMA algorithm has been simulated on Intel Core TM, i5-3320 M CPU@2.60 GHz system. These standard benchmark function are characterized by their objective fitness in parameter space within a particular dimension (Dim), range, and frequency ($f_{\min }$). In the entire work, F1–F7 represent as uni-modal test functions (U-Modal), F8–F13 are multi-modal test functions (M-Modal) and F14–F23 are fixed dimension (FD) functions. The effectiveness of the proposed Chaotic SMA optimization technique is examined by referring benchmark functions are taken [91, 92]. The characteristics of benchmark functions differ from each other. Some functions show better performance in exploring local search while a few functions are found to excellent in determining global optima. Table 4 illustrates equations of F1–F7 with their name, dimension, range, and frequency. On similar grounds, Tables 5 and 6 explore the details of multi-modal and fixed dimensions functions.

Table 4 Standard uni-modal benchmark

Table 5 Multimodal test function (M-modal)

Table 6 Fixed dimension test function

In the whole research study, 30 search agents are taken into considerations and the proposed algorithm is simulated for maximum iterations of 500. Figures 3, 4, and 5 illustrate the 3D view along with their objective space representing convergence for F1–F7, F8–F13 and F14–F23, respectively. It is clear from the various comparative results that the newly developed chaotic strategy appreciably increases convergence rate and thus improves its ability to easily escape from local minima entrapment.

Fig. 3

Three-dimensional view of F1–F7 along with convergence curve for SMA and CSMA

Fig. 4

Three-dimensional view of F8–F13 along with convergence curve for SMA and CSMA

Fig. 5

Three-dimensional view of F14–F23 along with convergence curve for SMA and CSMA

5 Results of proposed algorithm

In this section, test results for benchmark functions are discussed with their average, best, worst, median, standard deviation, and p value have been taken into account. The stochastic complexity of the proposed algorithm is justified and analyzed by running the algorithm for 30 trial checks and 500 iterations. To analyze the feasibility of the solution, Wilcoxon sum test has been taken into account. The parameter setting for the proposed CSMA method is illustrated in Table 7. On similar grounds, results are compared with other universally validated systems.

Table 7 Parameter setting for the proposed method

5.1 Testing of uni-modal functions (U-modal)

The search process for the best position depends upon the capability of the search agents to reach closer to origin. During the search process by various agents, there may be the possibility of getting entrap far or nearby and accordingly defined in terms of exploration and exploitation. Exploration comes under the global search process and exploitation falls under the local search category. The statistical outcomes of U-Modal (F1–F7) have a few pick points with increased convergence validates the effectiveness of the proposed algorithm. Figure 6 illustrates a comparison between Chaotic SMA and different methods. It can be seen from the convergence curves that the suggested algorithm converges to optima much earlier.

Fig. 6

Convergence curve for U-Modal test function showing comparison of CSMA with other algorithms

To check the appropriateness of the proposed algorithm, each test function is simulated with SMA and CSMA. Table 8 illustrates statistical outcomes of the uni-modal benchmark function in terms of average, standard deviation, the best value, worst value, median value, and p value. In the search space, there are some regions of global optima whereas some regions are stagnated to local optima. The global search process determines the exploration phase whereas in the local search process, exploitation phase is evaluated. The performance of any algorithm is judged by its ability to attain the maxima or minima with less computation time. Table 9 shows the computational time in terms of best, mean, and worst time. Table 10 shows the comparison of the CSMA method with other techniques such as PSO [93], GWO [15], GSA [94], BA [95], FA [96], GA [97], BDA [98], BPSO [99], MFO [16], MVO [6], BGSA [100], SMS [101], FPA [102], DE [103], ALO [104], and WOA [105] in terms of average (AVG) and standard deviation (SD).

Table 8 Test results for U-modal function using CSMA

Table 9 Simulation time for uni-modal benchmark problems using CSMA

Table 10 Comparative results of U-Modal test function

The characteristics of benchmark functions vary from each other. These test functions have different exploration and exploitation search capacities. Griewank, levy, Ackley, Rastrigin, Schwefel functions, sphere function, explore many local minima points while sum square function, Zakharov are applicable to explore global minima points. In this regard, test assessment of seven uni-modal benchmark functions (F1–F7) are analyzed. The test results for each function in terms of average and standard deviations over 30 independent trial runs and 500 iterations are recorded. The scalability assessment is carried out to investigate the impact of chaos on the solutions of SMA. The statistical results shown in Table 10 reveals an appreciable gap between CSMA and other methodologies. As it can be seen in Table 10, that by introducing a chaotic function, the exploration and exploitation phase of SMA has been improved. The results of CSMA when compared with PSO, GWO, DE, FA, GA, BBO, MFO, SCA, and SSA show remarkable performance in dealing with F3, F4, and F7 test functions. As per the convergence curves in Fig. 6, it is observed that the optimality of results is increased with higher efficiency. On the other hand, the former method are found to be subjected to premature convergence. Further, to verify the effectiveness of the proposed method, independent trial runs for each benchmark function are illustrated in Fig. 7. Comparative analysis revealed that the sinusoidal chaotic function facilitates to explore the local search phase more intensively.

Fig. 7

Trial runs of Uni-modal benchmark function

5.2 Testing of multi-modal test functions (M-modal)

Figure 8 illustrates comparison between Chaotic SMA and other methods for multi-modal benchmark (F8–F13).The multi-modal function is tested for 30 introductory attempts and 500 iterations and results are shown in Table 11. Simulation time for M-Modal utilizing CSMA appears in Table 12. Table 13 shows compared results with other meta-heuristics search algorithms such as PSO [93], GWO [15], GSA [94], BA [95], FA [96], GA [97], BDA [98], BPSO [99], MFO [16], MVO [6], BGSA [100], SMS [101], FPA [102], DE [103], ALO [104], and WOA [105] in terms of average value and std. deviation. It can be seen Fig. 8 that test outcomes of M-Modal (F8–F13) have some pick points with increased convergence using CSMA, justifies the effectiveness of algorithm in solving multi-modal test functions.

Fig. 8

Convergence curve for M-Modal test function showing comparison of CSMA with other algorithms

Table 11 Testing of multi-modal using CSMA

Table 12 Simulation time for M-modal using CSMA

Table 13 Comparison of multi-modal test function

From the statistical data analysis illustrated in Table 13, it is observed that the optimality of multi-modal test functions is marginally improved by implementing the sinusoidal chaotic function. According to the best and standard deviation results, CSMA shows better performance for almost all seven test functions. As can be seen from convergence curves shown in Fig. 8, CSMA gives optimal convergences except for a few of the test functions. It can be understood from the convergence comparison that CSMA converges faster and seizes the run as soon as it reaches the stop criterion. As per comparative curves shown in Fig. 8, it is observed that the suggested algorithm shows superior performance in dealing with F9, F10 and F11 and comparative operation in the case of F8, F12 and F13. Trial run accuracy matrices shown in Fig. 9 reveal that proposed CSMA significantly searches the local and global space more intensively.

Fig. 9

Trial runs of M-Modal functions

5.3 Testing of fixed dimension function (F-modal)

Fixed dimension test functions (F14–F23) are tested for 30 trial runs and 500 iterations as shown in Fig. 11. Simulation results for FD test function using CSMA are shown in Table 14. Table 15 illustrates simulation results of F-Modal. It can be seen that results for F-Modal have many pick points with better convergence. Simulation time for F-Modal benchmark problems using CSMA is shown in Table 15. Table 16 illustrates CSMA results compared with others variants such as GWO [15], PSO [10], GSA [4], DE [1], and FEP [91] in terms of average (AVG) and standard deviation (SD).

Table 14 Simulation results for fixed dimension test function using CSMA

Table 15 Simulation time for fixed dimension using CSMA

Table 16 Comparison of FD test function with other methods

The statistical data analysis illustrated in Table 16 shows that solution accuracy of fixed dimension test functions is appreciably improved by implementing the sinusoidal chaotic function. Rendering to the best and standard deviation results, CSMA shows enhanced performance for nearly all ten test functions. As can be seen from convergence curves shown in Fig. 10, CSMA gives optimal convergences except for a few of the test functions. It can be implicit from the convergence evaluation that CSMA converges faster and seizes the run as soon as it reaches the halt criterion. As per comparative curves shown in Fig. 8, it is observed that the suggested algorithm shows superior performance in dealing with F14, F18, F19 and F20 and comparative operation in the case of F15, F21, F22 and F23. Trial runs shown in Fig. 11 reveal that proposed CSMA significantly searches the local and global space for finding the optimal solution.

Fig. 10

Convergence curve for FD test function showing comparison of CSMA with other algorithms

Fig. 11

Trial run of FD benchmark functions

6 Multi-disciplinary engineering design problems

In this section, ten real-world design problems are tested which includes “3-bar truss problem, pressure –vessel design, compression design, welded beam, cantilever beam design, gear train design problem, speed reducer problem, Belleville spring problem, rolling element problem and multidisc clutch brake problem” [112]. Each design problem is simulated with CSMA algorithm. The abbreviations for various multidisciplinary engineering functions (EFs) has been shown in Table 17. The comparison of the engineering design problem with their average, standard deviation, best, worst and p value has been elucidated in Table 18 and simulation time for best, mean and average values are shown in Table 19.

Table 17 Abbreviations for 10 types of design problems

Table 18 Engineering design problems by CSMA

Table 19 Computation time for engineering function (EF) using CSMA

6.1 Three-bar truss design problem

The proposed chaotic SMA method is applied for solving problem of truss design as shown in Fig. 12 [113, 114]. Truss design problem depends on two variables and three parameters. The main focus of truss design problem is to minimize weight. The various constraints involved in truss bar design problem are warping, deflection and stress. These constraints are optimized to achieve the desired objective. The mathematical modeling of 3-bar truss are illustrated through Eqs. (9.1) to (9.1d) subject to various constraints. The solutions of CSMA were compared with existing methods and illustrated in Table 20. It is seen that the suggested method appreciably improves the objective of cost minimization. The design problem is modeled as shown below:

Fig. 12

Truss engineering design

Table 20 Comparative analysis of CSMA results with other methods for 3-bar truss problem

Consider,

$$ \vec{y} = \left[ {y_{1} ,y_{2} } \right] = \left[ {A_{1} ,A_{2} } \right]. $$

(9.1)

Minimize,

$$ f\left( {\vec{y}} \right) = \left( {2\sqrt 2 y_{1} + y_{2} } \right) * l. $$

(9.1a)

Subjected to:

$$ g_{1} \left( {\vec{y}} \right) = \frac{{\sqrt 2 y_{1} + y_{2} }}{{\sqrt 2 y_{1}^{2} + 2y_{1} y_{2} }}P - \sigma \le 0 $$

(9.1b)

$$ g_{2} \left( {\vec{y}} \right) = \frac{{y_{2} }}{{\sqrt 2 y_{1}^{2} + 2y_{1} y_{2} }}P - \sigma \le 0 $$

(9.1c)

$$ g_{3} \left( {\vec{y}} \right) = \frac{1}{{\sqrt 2 y_{2} + y_{1} }}P - \sigma \le 0. $$

(9.1d)

6.2 Pressure vessel engineering problem

The design specification for this kind of engineering problem as illustrated in Fig. 13 [113, 114] is selected for reference. The chaotic SMA is applied to diminish the expense which includes the material cost and welding cost to form the vessel in cylindrical form. The four variables used to design the pressure vessel are: (i) shell thickness (T_s), (ii) head thickness (T_h), (iii) length of cylindrical unit (L_h), (iv) without head thickness (R). These four variables are modeled as y₁–y₄. The numerical formulation of this kind of problem is shown in Eqs. (9.2) through (9.2e). Table 21 shows the result assessment of suggested CSMA with some recent algorithms. From the comparative analysis, it is found that CSMA effectively reduces the expense of design by controlling the design variables:

Fig. 13

Pressure vessel engineering design

Table 21 Comparative analysis CSMA results with other methods

Consider:

$$ \vec{y} = \left[ {y_{1} y_{2} y_{3} y_{4} } \right] = \left[ {T_{{\text{s}}} T_{{\text{h}}} RL_{{\text{h}}} } \right]. $$

(9.2)

Minimize;

$$ f(\vec{y}) = 0.6224y_{1} y_{3} y_{4} + 1.7781y_{2} y_{3}^{2} + 3.1661y_{1}^{2} y_{4} + 19.84y_{1}^{2} y_{3} $$

(9.2a)

Subject to:

$$ g_{1} \left( {\vec{y}} \right) = - y_{1} + 0.0193y_{3} \le 0 $$

(9.2b)

$$ g_{2} \left( {\vec{y}} \right) = y_{3} + 0.00954y_{3} \le 0 $$

(9.2c)

$$ g_{3} \left( {\vec{y}} \right) = - \pi y_{3}^{2} y_{4} - \frac{4}{3}\pi y_{3}^{3} + 1296000 \le 0 $$

(9.2d)

$$ g_{4} \left( {\vec{y}} \right) = y_{4} - 240 \le 0. $$

(9.2e)

Variable range $0 \le \,y_{1} \, \le \,99$, $0 \le \,y_{2} \, \le \,99$, $10 \le \,y_{3} \, \le \,200$, $10 \le \,y_{4} \, \le \,200$, $0 \le \,y_{1} \, \le \,99$.

6.3 Compression spring engineering design

Compression spring design concerned with mechanical engineering [113, 114] is shown in Fig. 14. The main intention of this type of problem is minimization of the spring weight. There are three design variables: (1) no. of active coils (N_c), (2) wire diameter (d_r), and (3) mean coil diameter (D_m). The formulation is shown in the Eqs. (9.3) through (9.5f). The proposed method is applied to solve compression spring design problem and results are compared with other methods shown in Table 22 for validation. It is clearly seen from the analysis that CSMA method is efficient for reducing spring weight marginally.

Fig. 14

Compression spring engineering design

Table 22 Comparison of CSMA with other methods

Consider

$$ \vec{y} = \left[ {y_{1} y_{2} y_{3} } \right] = \left[ {d_{{\text{r}}} D_{{\text{m}}} N_{{\text{c}}} } \right], $$

(9.3)

Minimize

$$ f(\vec{y}) = \left( {y_{3} + 2} \right)y_{2} y_{1}^{2} , $$

(9.3a)

Subject to:

$$ g_{1} \left( {\vec{y}} \right) = 1 - \frac{{y_{2}^{3} y_{3} }}{{71785y_{1}^{4} }} \le 0, $$

(9.3b)

$$ g_{2} \left( {\vec{y}} \right) = \frac{{4y_{2}^{2} - y_{1} y_{2} }}{{12566\left( {y_{2} y_{1}^{3} - y_{1}^{4} } \right)}} + \frac{1}{{5108y_{1}^{2} }} \le 0, $$

(9.3c)

$$ g_{2} \left( {\vec{y}} \right) = \frac{{4y_{2}^{2} - y_{1} y_{2} }}{{12566\left( {y_{2} y_{1}^{3} - y_{1}^{4} } \right)}} + \frac{1}{{5108y_{1}^{2} }} \le 0, $$

(9.3d)

$$ g_{3} \left( {\vec{y}} \right) = 1 - \frac{{140.45y_{1} }}{{y_{2}^{2} y_{3} }} \le 0, $$

(9.3e)

$$ g_{4} \left( {\vec{y}} \right) = \frac{{y_{1} + y_{2} }}{1.5} - 1 \le 0. $$

(9.3f)

Variable range 0.005 ≤ $y_{1}$ ≤ 2.00, 0.25 ≤ $y_{2}$ ≤ 1.30, 2.00 ≤ $y_{3}$ ≤ 15.0.

6.4 Welded beam design

In welded beam design, welding is carried out by fusing different sections by molten metal as shown in Fig. 15 [113, 114]. The main focus is on minimization of the making cost of the welded beam. The four variables are: (1) bar thickness (b) is specified by y₁, (2) bar length (l) is specified by y₂, (3) weld thickness (h) is specified by y₃, and (4) the bar height (h) is specified by y₄ which is subject to constraints, such as Buckling of bar (P_c), side constraints, End deflection of beam (d), bending stress of the beam (h) and stress of shear (s). The equations of the above-mentioned design problem are noted by Eqs. (9.4) through (9.4n). The results are compared with other methods as shown in Table 23. The comparative analysis shows that proposed method is competent for handling beam design problem more precisely.

Fig. 15

Welded beam engineering design

Table 23 Comparative analysis of welded beam design with other methods

Consider

$$ \vec{y} = \left[ {y_{1} y_{2} y_{3} y_{4} } \right] = \left[ {hltb} \right]. $$

(9.4)

Minimize

$$ f(\vec{y}) = 1.10471y_{1}^{2} y_{2} + 0.04811y_{3} y_{4} \left( {14.0 + y_{2} } \right) $$

(9.4a)

Subject to

$$ g_{1} (\vec{y}) = \tau (\vec{y}) - \tau_{{{\text{maxi}}}} \le 0, $$

(9.4b)

$$ g_{2} (\vec{y}) = \sigma (\vec{y}) - \sigma_{{{\text{maxi}}}} \le 0, $$

(9.4c)

$$ g_{3} (\vec{y}) = \delta (\vec{y}) - \delta_{{\text{maxi}}} \le 0, $$

(9.4d)

$$ g_{4} (\vec{y}) = y_{1} - y_{4} \le 0, $$

(9.4e)

$$ g_{5} (\vec{y}) = P_{i} - P_{{\text{c}}} (\vec{y}) \le 0, $$

(9.4f)

$$ g_{6} (\vec{y}) = 0.125 - y_{1} \le 0, $$

(9.4g)

$$ g_{7} (\vec{y}) = 1.10471y_{1}^{2} + 0.04811y_{3} y_{4} (14.0 + y_{2} ) - 5.0 \le 0. $$

(9.4h)

Variable range 0.1 ≤ $y_{1}$ ≤ 2, 0.1 ≤ $y_{2}$ ≤ 10, 0.1 ≤ $y_{3}$ ≤ 10, 0.1 ≤ $y_{4}$ ≤ 2,

where

$$ \tau (\vec{y}) = \sqrt {(\tau^{/} )^{2} + 2\tau^{/} \tau^{//} \frac{{y_{2} }}{2R} + (\tau^{//} )^{2} ,} $$

(9.4i)

$$ \tau^{/} = \frac{{P_{i} }}{{\sqrt 2 y_{1} y_{2} }},\tau^{//} = \frac{MR}{J},M = P_{i} \left( {L + \frac{{y_{2} }}{2}} \right), $$

(9.4j)

$$ R = \sqrt {\frac{{y_{2}^{2} }}{4} + \left( {\frac{{y_{1} + y_{3} }}{2}} \right)^{2} } , $$

(9.4k)

$$ J = 2\left\{ {\sqrt 2 y_{1} y_{2} \left[ {\frac{{y_{2}^{2} }}{4} + \left( {\frac{{y_{1} + y_{3} }}{2}} \right)^{2} } \right]} \right\}, $$

(9.4l)

$$ \sigma (\vec{y}) = \frac{{6P_{i} L}}{{y_{4} y_{3}^{2} }},\quad \delta (\vec{y}) = \frac{{6P_{i} L^{3} }}{{Ey_{2}^{2} y_{4} }}, $$

(9.4m)

$$ P_{{\text{c}}} (\vec{y}) = \frac{4.013E {\frac{\sqrt{y_{3}^{2}y_{4}^{6}}}{36}} }{L^{2}} \left(1-{\frac{y_{3}}{2L}} {\sqrt{\frac{E}{4G}}}\right)$$

(9.4n)

$$ \begin{gathered} P_{i} = 6000lb,L = 14\;{\text{in}},\delta_{\max i} = 0.25\;{\text{in}},E = 30 \times 1^{6} \;{\text{psi}},G = 12 \times 10^{6} \;{\text{psi}}, \hfill \\ \tau_{{\text{maxi}}} = 13600\;{\text{psi}},\sigma_{{\text{maxi}}} = 3000\;{\text{psi}}{.} \hfill \\ \end{gathered} $$

6.5 Cantilever beam design

This is civil engineering problem in which main focus is minimization of beam weight as shown in Fig. 16. In beam design, there are five elements $l_{1} ,l_{2} ,l_{3} ,l_{4} \;{\text{and}}\;l_{5}$ [114]. The main aim is minimization of the weight of the beam shown in Eq. (9.5). Taking care that displacement of vertical constraint not to disturb during finishing process of the beam for final optimal solution shown by Eq. (9.5a) to Eq. (9.5b). The results shown in Table 24 validate that CSMA algorithm efficiently reduces the weight of the beam. The formulation of design is given below:

Fig. 16

Cantilever beam engineering design

Table 24 Comparative analysis of beam problem with other methods

Consider

$$ \vec{x} = \left[ {l_{1} l_{2} l_{3} l_{4} l_{5} } \right]. $$

(9.5)

Minimize

$$ \begin{gathered} f(\vec{x}) = 0.6224(l_{1} + l_{2} + l_{3} + l_{4} + l_{5} ), \hfill \\ \hfill \\ \end{gathered} $$

(9.5a)

Subject to

$$ g(\vec{x}) = \frac{61}{{l_{1}^{3} }} + \frac{37}{{l_{2}^{3} }} + \frac{19}{{l_{3}^{3} }} + \frac{7}{{l_{4}^{3} }} + \frac{1}{{l_{5}^{3} }} \le 1. $$

(9.5b)

6.6 Gear train design

In this method, the four variables $g_{1} ,g_{2} ,g_{3} ,\,{\text{and}}\,g_{4}$ are reformed to diminish the scalar value and teeth ratio as shown in Fig. 17 [113]. Teeth on each gear are the decision variables in designing process. The gear train design problem is formulated through Eq. (9.6a) to Eq. (9.6b). The simulation results shown in Table 25 reveal that CSMA method gives comparison of results with other methods. From the assessment of test results, it is seen that proposed method effectively evaluates the gear train ratio.

Fig. 17

Gear train problem

Table 25 Comparison of gear train problem with other methods

Let us consider:

$$ \vec{g} = \left[ {g_{1} g_{2} g_{3} g_{4} } \right] = \left[ {M_{A} M_{B} M_{C} M_{D} } \right] $$

(9.6a)

Minimizing

$$ f(\vec{g}) = \left( {\frac{1}{6.931} - \frac{{g_{3} g_{4} }}{{g_{1} g_{4} }}} \right)^{2} . $$

(9.6b)

Subject to: $12 \le g_{1} ,g_{2} ,g_{3} ,g_{4} \le 60$.

6.7 Speed reducer engineering design problem

The fundamental problem is to limit the heaviness of the speed reducer. This type of design problem consist of seven variables as shown in Fig. 18 [113]. The seven variables are face width $(x_{1} )$, teeth module $(x_{2} )$, pinion teeth $(x_{3} )$, first shaft length $(x_{4} )$, second shaft length $(x_{5} )$, the first shaft diameter $(x_{6} )$ and second shaft diameter $(x_{7} )$. The results shown in Table 26 shows that optimum fitness has been improved to some extend from previous evaluation. The equations are formulated as given below:

Fig. 18

Speed reducer engineering design problem

Table 26 Comparison of speed reducer problem with other methods

Minimizing

$$ \begin{aligned} f(\vec{x}) & = 0.7854x_{1} x_{2} (3.3333x_{3}^{2} + 14.9334x_{3} - 43.0934) - 1.508x_{1} (x_{6}^{2} + x_{7}^{2} ) + 7.4777(x_{6}^{3} + x_{7}^{3} ) \\ & \quad + 0.7854(x_{4} x_{6}^{2} + x_{5} x_{7}^{2} ) \\ \end{aligned} $$

Subject to:

$$ g_{1} (\overrightarrow {x} ) = \frac{27}{{x_{1} x_{2}^{2} x_{3} }} - 1 \le 0 $$

(9.7a)

$$ g_{2} (\overrightarrow {x} ) = \frac{397.5}{{x_{1} x_{2}^{2} x_{3}^{2} }} - 1 \le 0 $$

(9.7b)

$$ g_{3} (\overrightarrow {x} ) = \frac{{1.93x_{4}^{3} }}{{x_{2} x_{3} x_{6}^{4} }} - 1 \le 0 $$

(9.7c)

$$ g_{4} (\overrightarrow {x} ) = \frac{{1.93x_{5}^{3} }}{{x_{2} x_{3} x_{7}^{4} }} - 1 \le 0 $$

(9.7d)

$$ g_{5} (\overrightarrow {x} ) = \frac{1}{{110x_{6}^{3} }}\sqrt {\left( {\frac{{745.0x_{4} }}{{x_{2} x_{3} }}} \right)}^{2} + 16.9 \times 10^{6} - 1 \le 0 $$

(9.7e)

$$ g_{6} (\overrightarrow {x} ) = \frac{1}{{85x_{7}^{3} }}\sqrt {\left( {\frac{{745.0x_{5} }}{{x_{2} x_{3} }}} \right)}^{2} + 157.5 \times 10^{6} - 1 \le 0 $$

(9.7f)

$$ g_{7} (\vec{x}) = \frac{{x_{2} x_{3} }}{40} - 1 \le 0 $$

(9.7g)

$$ g_{8} (\vec{x}) = \frac{{5x_{2} }}{{x_{1} }} - 1 \le 0 $$

(9.7h)

$$ g_{9} (\overrightarrow {x} ) = \frac{{x_{1} }}{{12x_{2} }} - 1 \le 0 $$

(9.7i)

$$ g_{10} (\overrightarrow {x} ) = \frac{{1.5x_{6} + 1.9}}{{12x_{2} }} - 1 \le 0 $$

(9.7j)

$$ g_{11} (\overrightarrow {x} ) = \frac{{1.1x_{7} + 1.9}}{{x_{5} }} - 1 \le 0 $$

(9.7k)

where $2.6 \le x_{1} \le 3.6,0.7 \le x_{2} \le 0.8,17 \le x_{3} \le 28,7.3 \le x_{4} \le 8.3,7.8 \le x_{5} \le 8.3,2.9 \le x_{6} \le 3.9\;{\text{and}}\;5 \le x_{7} \le 5.5.$

6.8 Belleville spring design

This type of design problem is shown in Fig. 19 [113]. In this method, one of the design parameter is selected according to variable ratio. The focus of this method is to minimize weight within certain constraints. The designed variables are internal diameter of the spring (DIMI), external diameter of the spring (DIME), spring height (SH) and spring thickness (ST). The results are compared with other methods as shown in Table 27. The assessment of CHHO with other methods reveals that present method is effective in solving spring design problem more precisely. The mathematical formulation for spring design are given below:

Fig. 19

Belleville spring engineering design

Table 27 Comparative analysis of Belleville spring design variables with other methods

Minimizing

$$ f(x) = 0.07075\pi ({\text{DIM}}_{{\text{E}}}^{2} - {\text{DIM}}_{{\text{I}}}^{2} )t $$

(9.8a)

Subject to:

$$ b_{1} (x) = G - \frac{{4P\lambda_{\max } }}{{(1 - \delta^{2} )\alpha {\text{DIM}}_{{\text{E}}} }}\left[ {\delta (S_{{\text{H}}} - \frac{\lambda \max }{2}) + \mu t} \right] \ge 0 $$

(9.8b)

$$ b_{2} (x) = \left( {\frac{{4P\lambda_{\max } }}{{(1 - \delta^{2} )\alpha {\text{DIM}}_{{\text{E}}} }}\left[ {\left( {S_{{\text{H}}} - \frac{\lambda }{2}} \right)(S_{{\text{H}}} - \lambda )t + t^{3} } \right]} \right)_{{\lambda_{\max } }} - P_{{{\text{MAX}}}} \ge 0 $$

(9.8c)

$$ b_{3} (x) = \lambda_{1} - \lambda_{\max } \ge 0 $$

(9.8d)

$$ b_{4} (x) = H - S_{{\text{H}}} - t \ge 0 $$

(9.8e)

$$ b_{5} (x) = {\text{DIM}}_{{{\text{MAX}}}} - {\text{DIM}}_{{\text{E}}} \ge 0 $$

(9.8f)

$$ b_{6} (x) = {\text{DIM}}_{{\text{E}}} - {\text{DIM}}_{{\text{I}}} \ge 0 $$

(9.8g)

$$ b_{7} (x) = 0.3 - \frac{{S_{{\text{H}}} }}{{{\text{DIM}}_{{\text{E}}} - {\text{DIM}}_{{\text{I}}} }} \ge 0 $$

(9.8h)

where $\alpha = \frac{6}{\pi \ln J}\left( {\frac{J - 1}{J}} \right)^{2}$

$$ \delta = \frac{6}{\pi \ln J}\left( {\frac{J - 1}{{\ln J}} - 1} \right) $$

$$ \mu = \frac{6}{\pi \ln J}\left( {\frac{J - 1}{2}} \right) $$

$$ P_{{{\text{MAX}}}} = 5400lb $$

$$ \begin{gathered} P = 30e6\;{\text{psi}},\lambda_{\max } = 0.2\;{\text{in}},\delta = 0.3,G = 200\;{\text{Kpsi}}, \hfill \\ H = 2\;{\text{in}},{\text{DIM}}_{{{\text{MAX}}}} = 12.01\;{\text{in}},J = \frac{{{\text{DIM}}_{{\text{E}}} }}{{{\text{DIM}}_{{\text{I}}} }},\lambda_{1} = f(a)a,a = \frac{{S_{{\text{H}}} }}{t}. \hfill \\ \end{gathered} $$

6.9 Rolling element bearing design

The major aspect of this kind of design is to improve the dynamic load carrying capacity of rolling bearing element as illustrated in Fig. 20 [89]. There are ten parameters which decides the optimum design of bearing for improving the load bearing power. Out of these ten variables only five variables are of much consideration. These major variables are (1) diameter of the ball (DIMB), (2) diameter pitch (DIMP), (3) ball numbers (N_b), (4) outer curvature coefficient and (5) inner curvature coefficient. Rest of five variables only affect indirectly to the internal portion of the geometry. The proposed algorithm is applied to solve rolling design problem and outputs are compared with other methods as illustrated in Table 28. From the result assessment, it can be seen that CSMA gives analogous outcomes. The design equations are formulated through the following equations:

Fig. 20

Rolling element bearing problem

Table 28 Comparative analysis of rolling element design variables

Maximizing

$$ C_{{\text{D}}} = f_{{\text{c}}} N^{2/3} {\text{DIM}}_{{\text{B}}}^{1.8} $$

(9.9a)

$$ {\text{if}}\,\,{\text{DIM}} \le 25.4\;{\text{mm}} $$

$$ C_{{\text{D}}} = 3.647f_{{\text{c}}} N^{2/3} {\text{DIM}}_{{\text{B}}}^{1.4} $$

(9.9b)

$$ {\text{if}}\,\,{\text{DIM}} \ge 25.4\;{\text{mm}} $$

Subjected to:

$$ r_{1} (x) = \frac{{\theta_{0} }}{{2\sin^{ - 1} \left( {\frac{{{\text{DIM}}_{{\text{B}}} }}{{{\text{DIM}}_{{{\text{MAX}}}} }}} \right)}} - N + 1 \ge 0 $$

(9.9c)

$$ r_{2} (x) = 2{\text{DIM}}_{{\text{B}}} - K_{{{\text{DIM}}_{{{\text{MIN}}}} }} ({\text{DIM}} - \dim ) \ge 0 $$

(9.9d)

$$ r_{3} (x) = K_{{{\text{DIM}}_{{{\text{MAX}}}} }} ({\text{DIM}} - \dim ) \ge 0 $$

(9.9e)

$$ r_{4} (x) = \beta B_{W} - {\text{DIM}}_{{\text{B}}} \le 0 $$

(9.9f)

$$ r_{4} (x) = {\text{DIM}}_{{{\text{MAX}}}} - 0.5({\text{DIM}} + \dim ) \ge 0 $$

(9.9g)

$$ r_{5} (x) = {\text{DIM}}_{{{\text{MAX}}}} - 0.5({\text{DIM}} + \dim ) \ge 0 $$

(9.9h)

$$ r_{6} (x) = (0.5 + {\text{re}})({\text{DIM}} + \dim ) \ge 0 $$

(9.9i)

$$ r_{7} (x) = 0.5({\text{DIM}} - {\text{DIM}}_{{{\text{MAX}}}} - {\text{DIM}}_{{\text{B}}} ) - \alpha {\text{DIM}}_{{\text{B}}} \ge 0 $$

(9.9j)

$$ r_{8} (x) = f_{I} \ge 0.515 $$

(9.9k)

$$ r_{9} (x) = f_{0} \ge 0.515, $$

(9.9l)

where

$$ f_{{\text{c}}} = 37.91\left[ {1 + \left\{ {1.04\left( {\frac{1 - \varepsilon }{{1 + \varepsilon }}} \right)^{1.72} \left( {\frac{{f_{{\text{I}}} \left( {2f_{0} - 1} \right)}}{{f_{0} \left( {2f_{{\text{I}}} - 1} \right)}}} \right)^{0.41} } \right\}^{10/3} } \right]^{ - 0.3} \times \left[ {\frac{{\varepsilon^{0.3} \left( {1 - \varepsilon } \right)^{1.39} }}{{\left( {1 + \varepsilon } \right)^{1/3} }}} \right]\left[ {\frac{{2f_{{\text{I}}} }}{{2f_{{\text{I}}} - 1}}} \right]^{0.41} $$

$$ \theta_{0} = 2\pi - 2\cos^{ - 1} \left( {\frac{{\left[ {\left\{ {({\text{DIM}} - \dim )/2 - 3(t/4)} \right\}^{2} + \left( {{\text{DIM}}/2 - t/4 - {\text{DIM}}_{{\text{B}}} } \right)^{2} - \left\{ {\dim /2 + t/4} \right\}^{2} } \right]}}{{2\left\{ {({\text{DIM}} - \dim )/2 - 3(t/4)} \right\}\left\{ {D/2 - t/4 - {\text{DIM}}_{{\text{B}}} } \right\}}}} \right) $$

$$ \varepsilon = \frac{{{\text{DIM}}_{{\text{B}}} }}{{{\text{DIM}}_{{{\text{MAX}}}} }},\quad f_{{\text{I}}} = \frac{{R_{{\text{I}}} }}{{{\text{DIM}}_{{\text{B}}} }},\quad f_{0} = \frac{{R_{0} }}{{{\text{DIM}}_{{\text{B}}} }},\quad t = {\text{DIM}} - \dim - 2{\text{DIM}}_{{\text{B}}} $$

$$ {\text{DIM}} = 160,\quad \dim = 90,\quad B_{W} = 30,\quad R_{{\text{I}}} = R_{0} = 11.033 $$

$\begin{gathered} 0.5\left( {{\text{DIM}} + \dim } \right) \le {\text{DIM}}_{{{\text{MAX}}}} \le 0.6\left( {{\text{DIM}} + \dim } \right),0.15\left( {{\text{DIM}} - \dim } \right) \le {\text{DIM}}_{{\text{B}}} \le 0.45\left( {{\text{DIM}} - \dim } \right),4 \le N \le 50 \hfill \\ 0.515 \le f_{{\text{I}}} \quad {\text{and}}\quad f_{0} \le 0.6 \hfill \\ 0.4 \le K_{{{\text{DIM}}_{{{\text{MIN}}}} }} \le 0.5,\quad 0.6 \le K_{{{\text{DIM}}_{{{\text{MIN}}}} }} \le 0.7,\quad 0.3 \le {\text{re}} \le 0.1,\quad 0.02 \le {\text{re}} \le 0.1,\quad 0.6 \le \beta \le 0.85. \hfill \\ \end{gathered}$

6.10 Multidisc-clutch design

Brake design is one of the most crucial problem in engineering and is shown in Fig. 21 [143]. The clutch-design problem is mainly fabricated to minimize the overall weight. The five design variables are inner surface radius (R_in), outer surface radius (R_o), thickness of disc’s (Th), actuating force (Fac) and count of friction surface (S_f). In Table 29, results are compared and observed that optimum fitness is found to better than other methods. The equations for Multidisc brake problem are given below:

Fig. 21

Multidisc clutch break design

Table 29 Comparative analysis of multiple-disc clutch brake problem with other methods

Minimizing

$$ f\left( {R_{{{\text{in}}}} ,R_{{\text{o}}} ,S_{{\text{f}}} ,{\text{Th}}} \right) = \pi {\text{Th}}\gamma \left( {R_{0}^{2} - R_{{{\text{in}}}}^{2} } \right)\left( {S_{{\text{f}}} + 1} \right) $$

(9.10a)

where $R_{{{\text{in}}}} \in 60,61,62 \ldots 80;\quad R_{{\text{o}}} \in 90,91, \ldots 110;\quad {\text{Th}} \in 1,1.5,2,2.5,3;\quad F_{{{\text{ac}}}} \in 600,610,620,1000;\quad S_{{\text{f}}} \in 2,3,4,5,6,7,8,9.$

Subject to:

$$ m_{1} = R_{0} - R_{{{\text{in}}}} - \Delta R \ge 0 $$

(9.10b)

$$ m_{2} = L_{{{\text{MAX}}}} - (S_{{\text{f}}} + 1)({\text{Th}} + \alpha ) \ge 0 $$

(9.10c)

$$ m_{3} = {\text{PM}}_{{{\text{MAX}}}} - {\text{PM}}_{\pi } \ge 0 $$

(9.10d)

$$ m_{4} = {\text{PM}}_{{{\text{MAX}}}} Y_{{{\text{MAX}}}} + {\text{PM}}_{\pi } Y_{{{\text{SR}}}} \ge 0 $$

(9.10e)

$$ m_{5} = Y_{{{\text{SR}}_{{{\text{MAX}}}} }} - Y_{{{\text{SR}}}} \ge 0 $$

(9.10f)

$$ m_{6} = t_{{{\text{MAX}}}} - t \ge 0 $$

(9.10g)

$$ m_{7} = {\text{DC}}_{{\text{h}}} - {\text{DC}}_{{\text{f}}} \ge 0 $$

(9.10h)

$$ m_{8} = t \ge 0 $$

(9.10i)

where

${\text{PM}}_{\pi } = \frac{{F_{{{\text{ac}}}} }}{{\Pi \left( {R_{0}^{2} - R_{{{\text{in}}}}^{2} } \right)}}$

$$ Y_{{{\text{SR}}}} = \frac{{2\pi n\left( {R_{0}^{3} - R_{{{\text{in}}}}^{3} } \right)}}{{90\left( {R_{0}^{2} - R_{{{\text{in}}}}^{2} } \right)}} $$

$$ t = \frac{{i_{x} \pi n}}{{30\left( {{\text{DC}}_{{\text{h}}} + {\text{DC}}_{{\text{f}}} } \right)}} $$

To check the effectiveness of chaotic SMA, algorithm is tested for 30 trial runs. The algorithm is tested with respect for best value, worst value, p value and standard deviation along with the trial runs. Furthermore, a comparative analysis with recent optimization methods is provided for justifying the validity of tested results for each of the standard function’s and design problems. Figure 22a and b shows 30 trial runs and for 10 multidisciplinary engineering problems to check the optimality of the algorithm.

Fig. 22

Trial run test for engineering design problems

7 Conclusion

In the proposed research, the exploitation phase of the classical SMA has been enhanced by incorporating sinusoidal chaotic function. The resultant chaotic SMA has been applied to 23 standard benchmark problems. In the set of experiments, CSMA was compared with basic SMA. The test results of benchmark functions are also compared with other algorithms in terms of mean and standard deviation. To check the soundness of the proposed algorithm, results of CSMA has been compared with others recently developed and well-known classical optimizers such as PSO, DE, SSA, MVO, GWO, DE, MFO, SCA, CS, TSA, PSO-DE, GA, HS, Ray and Sain, MBA, ACO, and MMA. Experimental results suggest that chaotic strategy enables SMA to improve the exploitation phase with better convergence. Simulation results shows that the developed chaotic SMA algorithm outperforms on almost all benchmark functions. Furthermore, the CSMA is applied to solve 10 real-world engineering design problems. Each design problem with a specific objective function has been simulated by implementing CSMA. Engineering design problems are mostly analyzed in terms of weight minimization and reduced manufacturing cost. The comparative analysis reveals that the proposed method effectively explores the search space to optimize objective fitness and proofs that CSMA can demonstrate good results not only on unrestricted issues but also on restricted issues. It is seen that the resultant chaotic slime mold algorithm is capable of giving more optimistic and convergent results. Thus, the proposed CSMA may be a good choice for solving numerical optimization problems.

8 Limitation and future scope

The proposed optimizer is giving a powerful and optimal solution depending on the type of chaotic strategy opted, if chaotic strategies are not properly decided, it may lead to insignificant results also. Therefore, it is recommended that proper selection of chaotic strategy is required for significant results, so that the exploitation phase of the existing algorithm can be explored in a most significant way. In the future, the proposed chaotic variant may be significantly used to solve the various engineering and design optimization problems including power system optimization problems such as Economic Load Dispatch, Generation Scheduling problem, Unit commitment problem, and Automatic generation control and load frequency issues of realistic power system and in more deeper sense, it may be applied to solve power system dispatch and unit commitment problems considering electric and hybrid electric vehicles including uncertainty of wind and solar power.