A review of nature-inspired algorithms on single-objective optimization problems from 2019 to 2023

Abstract

The field of nature inspired algorithm (NIA) is a vital area of research that consistently aids in solving optimization problems. One of the metaheuristic algorithm classifications that has drawn attention from researchers in recent decades is NIA. It makes a significant contribution by addressing numerous large-scale problems and achieving the best results. This research aims to identify the optimal NIA for solving single-objective optimization problems. The NIA discovered between 2019 and 2023 is presented in this study with a brief description. About 83 distinct NIAs have been studied in this study in order to address the optimization issues. In order to accomplish this goal, we have taken into consideration eight real-world single-objective optimization problems: the 3-bar truss design problem, the rolling element bearing, the pressure vessel, the cantilever beam, the I beam, the design of a welded beam, and the design of a spring. Based on a comparative study and bibliographic analysis, we have determined that two algorithms—the flow direction algorithm, and prairie dog optimization—give us the best results and optimal solutions for all eight of the engineering problems listed. Lastly, some perspectives on the limitations, difficulties, and future course are provided. In addition to providing future research guidelines, this will assist the novice and emerging researcher in providing a more comprehensive perspective on advanced NIA.

1 Introduction

The primary objective of optimization is to identify the best solution among all feasible solutions. It is one of the most crucial tools for decision-making and physical system analysis. According to Hajipour et al. (2015), an optimization procedure is the process of determining the best values for specific system characteristics in order to finish the system design at the lowest possible cost. One way to achieve this is to convert the problem into a mathematical model, which includes an objective function, maximization or minimization and set of restrictions. The set of variables associated with the problems is the key role to find the best objective among the different available resources. In the literature, there are different categories associated with the nature of the objective problems such as continuous versus discrete optimization, unconstrained versus constraint optimization, single and multi-objective optimization, deterministic versus stochastic optimization, etc. Continuous optimization problems are those that involve continuous variables; discrete optimization problems are those that involve discrete variables; unconstrained optimization problems are those that do not have any constraints; constraint optimization problems are those that do. On the other hand, the optimization problems which contain only one objective are called single objective problems while which contains more than one objective is called many objective problems. Stochastic optimization refers for the use of randomness in the objective function or optimization process. The classification of the different types of optimization methods are summarized in Fig. 1 (Janga Reddy and Nagesh Kumar 2020).

Fig. 1

Classification of optimization methods

Combinatorial optimization approaches are divided into two groups namely as exact and approximate methods. An exact method is one that produces an optimal solution with precision, while an approximate method produces a solution that is close to the exact solution. The dynamic programming and branch and bound method falls in the category of exact method. Combinatorial optimization considers an optimal object selected from the finite set, where the feasible solution set is either discrete or may be brought down to a discrete set. On the other hand, there are two types of programming approaches for continuous optimization problems: linear and nonlinear. Artificial intelligence and machine learning challenges and applications in the actual world are typically discrete, unconstrained, or discrete in nature (Mou et al. 2023; Hajipour et al. 2014; Zhu et al. 2023). According to Reddy and Kumar (2012), these conventional techniques might not be able to discover a workable solution for big nonlinear models. A non-linear and approximate method is divided into two categories: local search, which finds the best solution from a particular area of the search space, and global search, which finds the best solution from the entire search space. The three components of the global search method are heuristic, metaheuristic and random search. Heuristic algorithms are population-based algorithms that draw inspiration from the phenomenon of biological or human intelligence. These algorithms will converge to optimal solution or near to optimal solution. In random search (RS) methods, functions may or may not be continuous and differentiable. Such optimization methods are also known as derivative-free methods. Local search technique is divided into gradient-based optimization and non-gradient-based optimization. Gradient based optimization presents a powerful method if the objective function of an optimization problem is differentiable and gradient information is consistent. If the objective function is differentiable but finding the derivative is difficult then derivative free methods are extremely useful optimization tools.

All the above listed algorithms (traditional algorithms) are suffered from a number of drawbacks, including convergent to local optima, an unidentified search space as well as premature convergence (Shen et al. 2023). Furthermore, they only provide a single-based solution (Hashim et al. 2019). In recent years, researchers have been paying more attention to metaheuristic algorithm which are inspired by nature to solve the complex optimization problems and to address these issues (Cao et al. 2021; Li et al. 2023).Metaheuristic algorithm has a great position in various field like science and engineering to solve different type of optimization problems (Chen et al. 2023). Metaheuristics are one of the well-known approaches for solving a number of complex real-world problem and multi-objective optimization problems (Zhang et al. 2008; Xu et al. 2022; Cao et al. 2023). The Greek terms “meta” and “heuristic” which mean “away from” and “to discover” respectively, are combined to form the metaheuristic (Muazu et al. 2022). The meta heuristics is a recurrent generation process carried out by collaboratively similar intelligent techniques to explore the exploration space, learning rules used to assemble information to reach effectively close to the optimal solutions. In many cases, the metaheuristics require less computing time to arrive the optimal or near to optimal solution that the other iterative techniques, optimization algorithm and heuristic methods (Cao et al. 2020a). Metaheuristic algorithm gives better outcomes as compared to traditional method for different type of problems by inspiring congestion and evolutionary behavior of organism in the real world (Gharehchopogh et al. 2023; Gharehchopogh and Ibrikci 2023).

Every metaheuristic NIA, based on multiple physical, biological and ethological issues (Suman and Kumar 2006). Typically, these methods are discretized into two categories: single solution-based algorithms and population-based metaheuristic. Single solution-based NIA employes single candidate solution and refines its finding through the local search. However, the outcome obtained from a single search solution could suffer from premature convergence and stuck in local optima. Tabu search and Simulated Annealing are the two well-known single-solution dependent metaheuristics. On the other hand, the population-based NIA algorithm starts from the group of population from the search space and find the optimal solution use exploration and exploitation phases. Such algorithms are classified into four major categories, namely based on bio-inspired, physical, chemical and social and human-thinking. Bio-inspired algorithms can be further subdivided into three categories: evolutionary algorithm, swarm intelligence and plant-based algorithm. These algorithms can be combined to increase the overall accuracy of the solution.

In literature, several kinds of nature-inspired algorithms were developed between 2019 and 2023 by the researchers to address the single-objective optimization problems. In this present work, we examined 83 such algorithms and discussed their features and problem-solving approach. To the best of the authors’ knowledge, no research article has been published that claims this specific algorithm is the most effective at solving the single objective optimization problems. In order to select the best algorithms among the 83 existing NIAs, we have examined all such algorithms and taken into account the engineering design optimization problems as a benchmark function to compare their performances.

The rest of the work is organized into six sections. An introduction is provided in Sects. 1 and 2, a description of all existing 83 NIAs is provided. Section 3 give the different structural engineering design problems and their solution by using existing NIAs. The four best algorithms are identified and listed in this section too. In Sect. 4, we discussed the performance of the four obtained best algorithms on the 13 new benchmark functions. In Sect. 5, we conduct an experiment and results are presented through bibliographic analysis to validate the result. Finally, conclusion and future research direction are listed in Sect. 6.

2 A cyclopedia of NIA

This section aims to provide summarizing information about the earlier surveys on NIA discovered between 2019 to 2023.The list of the NIA discovered in this period is reported in Table 1. Along with algorithm and author name few remarks are also given in Table 1 which gives the source of inspiration of all newly developed algorithms. Each algorithm has initially worked on some benchmark problems for validation then thereafter few engineering problems have been solved using these algorithms. In the last column, we also have mentioned a list of those engineering problems.

Table 1 List of NIA discovered from 2019 to 2023

Along with developing new algorithms, few researchers also have written review papers in between 2019 to 2023 and the details of them is mentioned in Table 2.

Table 2 Review papers on NIA from 2019 to 2023

A large number of new NIA algorithms have been developed by the researchers between 2019 and 2023. To demonstrate the superiority of their algorithms over others, researchers have employed benchmark functions and variety of engineering design problems. To the best of the authors’ knowledge, no research article has been published that claims this specific algorithm is the most effective at solving the single objective optimization problems. Thus, this paper will help the researchers to identify the importance of each algorithm and their applications to solve the various engineering design optimization problems. Table 1 shows that of the 83 NIA algorithms, 40 worked on WBD, 38 on TSD, 37 on PVD, 15 on CBD, 8 on IBD, 15 on REB, 24 on SRP and 18 on TBTD. We have considered these NIAs in this study, as most of the researchers have frequently used these algorithms to compare the performance of their newly developed algorithm.

3 Comparative analysis

This study’s main goal is to review the several algorithms that may be used to solve single-objective constrained problems and determine the best solution. In this section, we have attempted to identify the most effective and efficient algorithms for solving the optimization problems. For this purpose, we studied all newly developed algorithms from 2019 to 2023 and will report the optimal solution obtained from them. From the 83 research publications listed in Table 1, we identified eight common engineering problems, namely welded beam design (WBM), tension spring design (TSD), pressure-vessel design (PVD), cantilever beam design (CBD), I Beam Design (IBD), Rolling Element Bearing (REB), speed reducer problem (SRP) and Three Bar Truss Design (TBTD). We have reported the results of all existing algorithms on these eight problems and compare their performances. The optimal results are obtained by considering 30 independent runs of each algorithm and iteration number as 500 to 1000 as a stopping criteria.

3.1 Welded beam design (WBD)

The goal of this problem is to decrease the welded cost of a WBD. The constraints are as follow:

• 1. sheer stress ($\upmu$);

• 2. bending stress in the beam (φ);

• 3. bucking load on the block (${B}_{c}$);

• 4. end deflection of the beam ($\Upsilon$);

• 5. side constraints.

This problem includes four variables likebreadth of weld (${t}_{1}$), the length of the connected part of the block (${t}_{2}$) height of the block (${t}_{3}$) and the thickness of the block (${t}_{4}$). The mathematical formulation of this design problem is listed as below:

$$\mathrm{Consider }\quad \overrightarrow{ t} = \left[ {t}_{1}{t}_{2}{t}_{3 }{t}_{4} \right],$$

$$\mathrm{Minimize} \quad f \left(\overrightarrow{t} \right)= 1.10471 {t}_{1}^{2}{t}_{2} + 0.04811 {t}_{3}{t}_{4} \left(14.0+{t}_{2}\right),$$

Subject to

$${y}_{1}\left(\overrightarrow{t}\right)=\upmu \left(\overrightarrow{t}\right)-{\upmu }_{{\text{max}}}\le 0,$$

$${y}_{2}\left(\overrightarrow{t}\right)=\mathrm{\varphi }\left(t\right)-{\mathrm{\varphi }}_{{\text{max}}}\le 0,$$

$${y}_{3}\left(\overrightarrow{t}\right)=\Upsilon \left(\overrightarrow{t}\right)-{\Upsilon }_{{\text{max}}}\le 0,$$

$${y}_{4}\left(\overrightarrow{t}\right)={{\text{t}}}_{1}-{{\text{t}}}_{4}\le 0,$$

$${y}_{5}\left(\overrightarrow{t}\right)={\text{T}}-{{\text{B}}}_{{\text{c}}}\left(\overrightarrow{x}\right)\le 0,$$

$${y}_{6}\left(\overrightarrow{t}\right)=0.125-{{\text{t}}}_{1}\le 0,$$

$${y}_{7}\left(\overrightarrow{t}\right)=1.10471 {{\text{t}}}_{1}^{2}+0.04811 {{\text{t}}}_{3}{{\text{t}}}_{4}\left(14.0+{{\text{t}}}_{2}\right)- 5.0\le 0,$$

$$0.1\le {t}_{1}\le 2, 0.1\le {t}_{2}\le 10, 0.1\le {t}_{3}\le 10, 0.1\le {t}_{4}\le 2,$$

where $\upmu \left(\overrightarrow{t}\right)=\sqrt{{\left({\upmu }^{\prime}\right)}^{2}+2{\upmu }^{\prime}{\upmu }^{\prime\prime}\frac{{t}_{2}}{2V}+{{\upmu }^{\prime\prime}}^{2}}$

$${\upmu }^{\prime}=\frac{T}{\sqrt{2}{t}_{1}{t}_{2}}; \quad \Upsilon\left(\overrightarrow{t}\right)=\frac{4T{l}^{3}}{E{t}_{3}^{2}{t}_{4}}; \quad \varphi \left(\overrightarrow{t}\right)=\frac{6TX}{{{t}_{4}t}_{3}^{2}},$$

$${\upmu }^{\prime\prime}=\frac{UV}{W}; \quad U=T\left(X+\frac{{t}_{2}}{2}\right); \quad V=\sqrt{\frac{{{\text{t}}}_{2}^{2}}{12}+{\left(\frac{{t}_{1}+{t}_{3}}{2}\right)}^{2}},$$

$$W=2\left\{\sqrt{2}{t}_{1}{t}_{2}\left[\frac{{{\text{t}}}_{2}^{2}}{4}{\left(\frac{{t}_{1}+{t}_{3}}{2}\right)}^{2}\right]\right\}; \quad {B}_{c}\left(\overrightarrow{t}\right)=\frac{4.013E\sqrt{\frac{{{{\text{t}}}_{3}^{2}{\text{t}}}_{4}^{6}}{36}}}{{X}^{2}}\left(1-\frac{{t}_{3}}{2X}\sqrt{\frac{E}{4G}} \right),$$

$$T=6000 lb, X=14\, in., {\Upsilon}_{max}=0.25 \,in., E=30\times {10}^{6}\, psi.$$

Out of 83 research papers, 40 researchers have worked on WBD problem. Table 3 shows the comparative analysis of all 40 algorithms for ${t}_{1}$ (thickness of the weld), ${t}_{2}$ (length of the attached part of the bar), ${t}_{3}$ (height of the bar), ${t}_{4}$ (thickness of the bar) and found that AO method is giving minimum cost out of all NIA algorithms.

Table 3 Result of different algorithms for WBD

3.2 Tension/compression spring design (TSD) problem

The aim of this problem is to get the lowest weight of TSD. The constraints are described as lowest deflection/shear stress and surge frequency together with design variables as well as mean coin diameter (${t}_{2}$), wire diameter (${t}_{1}$) and numeral of active coils (${t}_{3}$).

The mathematical formulation of the problem is as follow:

$${\mathrm{Consider } \quad \overrightarrow{t} = [ {t}_{1}t}_{2}{t}_{3 }]= \left[\mathrm{w\, c\, a}\right],$$

$$\mathrm{Minimize } \quad f \left(\overrightarrow{t}\right)=\left({t}_{3} +2\right){t}_{2}{{\text{t}}}_{1}^{2}.$$

Subject to

$${y}_{1}\left(\overrightarrow{t}\right)=1-\frac{{{\text{t}}}_{2}^{3}{t}_{3 }}{71785{{\text{t}}}_{1}^{4 }}\le 0,$$

$${y}_{2}\left(\overrightarrow{t}\right)=\frac{4{{\text{t}}}_{2}^{2}-{{\text{t}}}_{1}{{\text{t}}}_{2}}{12566({t}_{2}{{\text{t}}}_{1}^{3}-{{\text{t}}}_{1}^{4})}+\frac{1}{5108{{\text{t}}}_{1}^{2}}-1\le 0,$$

$${y}_{3}\left(\overrightarrow{t}\right)=1-\frac{140.45}{{{\text{t}}}_{2}^{3}{t}_{3 }}\le 0,$$

$${y}_{4}\left(\overrightarrow{t}\right)=\frac{{t}_{1}+{t}_{2}}{1.5}-1\le 0.$$

Variable range

$$0.05\le {t}_{1}\le 2, 0.25\le {t}_{2}\le 1.30, 2\le {t}_{3}\le 15.$$

Out of 83 research papers, 38 researchers have worked on the same problem. We have compared the optimum cost of each 38 algorithms in Table 4 for coin diameter (${t}_{2}$), wire diameter (${t}_{1}$) and numeral of active coils (${t}_{3}$).The AO method shows the optimal weight out of all.

Table 4 Result of different algorithms on TSD

3.3 Pressure vessel design (PVD) problem

In PVD, we reduce the manufacturecost and it contains four constraints, four parameters and four variables ${t}_{1}$ to ${t}_{4}$:${S}_{t}$(${t}_{1}$, width of the shell), ${H}_{t}$(${t}_{2},$ width of the head), M (${t}_{3}$, internal radius), N (${t}_{4}$, length of the component without head).

The mathematical model of this problem is as follow:

$${\mathrm{Consider }\quad \overrightarrow{t} = [t}_{1}{,t}_{2},{t}_{3 },{t}_{4 }]=\left[{S}_{t}{,H}_{t},M,N \right],$$

$$\mathrm{Minimize } \quad f\left(\overrightarrow{t}\right)=0.6224{t}_{1}{t}_{3}{t}_{4 }+1.7781{t}_{2 }{{\text{t}}}_{2}^{3}+3.1661{{\text{t}}}_{1 }^{2}{t}_{4 }+19.84{{\text{t}}}_{1 }^{2}{t}_{3 }.$$

Subject to

$${y}_{1}\left(\overrightarrow{t}\right)=-{t}_{1}+0.0193 {t}_{3}\le 0,$$

$${y}_{2}\left(\overrightarrow{t}\right)=-{t}_{2}+0.00954 {t}_{3}\le 0,$$

$${y}_{3}\left(\overrightarrow{t}\right)=-\Pi {{\text{t}}}_{3}^{2}{t}_{4 }-\frac{4}{3}\Pi {{\text{t}}}_{3}^{3}+\mathrm{1,296,000}\le 0,$$

$${y}_{4}\left(\overrightarrow{t}\right)=-{t}_{4}-240\le 0,$$

Variable range

$$0\le {t}_{1}\le 99, 0\le {t}_{2}\le 99, 10\le {t}_{3}\le 200, 10\le {t}_{4}\le 200.$$

Out of 83 research papers 37 researchers have worked on the same problem. We have compared the optimum cost of each 37 algorithms in Table 5 for ${S}_{t}$ (width of the shell), ${H}_{t}$ (width of the head), M (internal radius), N (length of the component without head).The PDO method shows the optimal cost out of all.

Table 5 Performance of dissimilar algorithms for solving PVD

3.4 Cantilever beam design (CBD) problem

The cantilever beam is built from five components, each component contains a vacant cross section with stable thickness. There is outer force performing at the open end of the cantilever. The mass of the beam is to be reduced while the higher limit is assigning on the vertical displacement of the free end. The devise variables are the width or height, ${u}_{i}$ of the cross section of each component. The problem is formulated mathematically using classical beam theory are as follow:

$$\mathrm{Minimize \,fitness }= 0.0624\times {(u}_{1}+{u}_{2}+u+{u}_{4}+{u}_{5}).$$

Subject to

$$g\left(x\right)=\frac{61}{{u}_{1}^{3}}+\frac{37}{{u}_{2}^{3}}+\frac{19}{{u}_{3}^{3}}+\frac{7}{{u}_{4}^{3}}+\frac{1}{{u}_{5}^{3}}-1\le 0.$$

Variable ranges

$$0.01\le {u}_{1},{u}_{2},{u}_{3},{u}_{4},{u}_{5}\le 100.$$

Out of 83 research papers 15 researchers have worked on same problem. We have compared the optimum cost of each 15 algorithms in Table 6 for ${u}_{1}{,u}_{2},{u}_{3},{u}_{4},{u}_{5}$ (height of five hollow square blocks with constant thickness) and found PDO method shows the optimal cost out of all.

Table 6 Performance of different algorithms for solving problem of CBD

3.5 I beam design (IBD)

The objective of this problem is to minimize the vertical deflection of the beam depend on correlated parameters. The IBD isfocus to two devise constraints: stress and the load’s cross-sectional area. This problem contain four design variables are as follows: the width of the flange’s $(v),$ the height of component ($u)$, the thickness of the web’s $({W}_{t,})$, and the thickness of the flange’s ${(F}_{t})$.

The formulation of this design is as follow:

$$Minimize\,fitness=\frac{5000}{\frac{1}{12}{W}_{t,}{\left(v-2{F}_{t}\right)}^{3}+\frac{1}{6}u{F}_{t}^{3}+2u{F}_{t}{\left(\frac{v-{F}_{t}}{2}\right)}^{2}}.$$

Subject to:

$${y}_{1}\left(x\right)=2u{F}_{t}+{W}_{t,}{\left(v-2{F}_{t}\right)}^{3}\le 300,$$

$${y}_{2}\left(x\right)=\frac{180,000{x}_{1}}{{W}_{t}{\left(v-2{F}_{t}\right)}^{3}+2u{F}_{t}[4{F}_{t}^{2}+3v(v-{F}_{t})]}+\frac{15,000 {x}_{2}}{(v-2{F}_{t}){W}_{t}^{3}+2{F}_{t}{u}^{3}}\le 6.$$

The variables are subject to:

$$10\le \mathrm{ v }\le 80, 10 \le u\le 50, 0.9\le {W}_{t,}\le 5, 0.9\le {F}_{t}\le 5.$$

Out of 83 research papers 8 researchers have worked on the same problem. We have compared the optimum cost of each 8 algorithms in Table 7 for width of the flange’s $(v),$ the height of component ($u)$, the thickness of the web’s $({W}_{t,})$, and the thickness of the flange’s ${(F}_{t})$ and found that ROA shows the optimal weight.

Table 7 Performance of different algorithms on problem of IBD

3.6 Rolling element bearing (REB) problem

The aim of this engineering problem is to maximize the carrying capacity of dynamic load of REBin so far as possible. Outcomes of this difficulty included 10 decision variables like diameter of a ball (${B}_{b})$, diameter of a pitch (${B}_{m})$, total number of balls (n), curvature coefficients of inner (${X}_{i}$) and outer (${X}_{0})$ raceway, ${K}_{{G}_{min}}, {K}_{{G}_{max}}, \mu , \nu , \boldsymbol{\varphi }$. The mathematical expression of this difficulty given below:

$$\mathrm{Maximize } \quad Z=\left\{\begin{array}{l}{F}_{c}\times {n}^\frac{2}{3}\times {B}_{b}^{1.8 } ;\qquad\qquad\quad if {B}_{b}\le 25.4 \\ 3.647 \times {F}_{c} \times {n}^{2/3}\times {B}_{b}^{1.4} ;\quad otherwise\end{array}\right..$$

Subject to:

$${h}_{1}\left(\overrightarrow{x}\right)=\frac{{\theta }_{0}}{2{{\text{sin}}}^{-1}(\frac{{B}_{b}}{{B}_{m}})}-n+1\le 1,$$

$${h}_{2}\left(\overrightarrow{x}\right)=2{B}_{b}-{K}_{{G}_{min}}\left(G-g\right)\ge 0,$$

$${h}_{3}\left(\overrightarrow{x}\right)={K}_{{G}_{max}}\left(G-g\right)-2{B}_{b}\ge 0,$$

$${h}_{4}\left(\overrightarrow{x}\right)=\varphi {W}_{b}-{B}_{b}\le 0,$$

$${h}_{5}\left(\overrightarrow{x}\right)={B}_{m}-0.5\times \left(G+g\right)\ge 0,$$

$${h}_{6}\left(\overrightarrow{x}\right)={B}_{m}-(0.5+{\nu )}\times \left(G+g\right)-{B}_{m}\ge 0,$$

$${h}_{7}\left(\overrightarrow{x}\right)=0.5\left(G-{B}_{m}-{B}_{b}\right)-\mu {B}_{b}\ge 0,$$

$${h}_{8}\left(\overrightarrow{x}\right)={X}_{i}\ge 0.515,$$

$${h}_{9}\left(\overrightarrow{x}\right)={X}_{0}\ge 0.515,$$

where

$${F}_{c}=37.91{\left[1+{\left(\frac{1+\sigma }{1-\sigma }\right)}^{1.72}{{\left(\frac{{X}_{i}\left(2{X}_{0}-1\right)}{{X}_{o}(2{X}_{I}-1}\right)}^{0.41}}^{10/3}\right]}^{-0.3}\times \left[\frac{{\sigma }^{0.3}{(1-\sigma )}^{1.39}}{{(1+\sigma )}^{1/3}}\right]\times {\left[\frac{2{X}_{i}}{2{X}_{i}-1}\right]}^{0.41},$$

$$k=\left[{\left\{\frac{G-g}{2}-3\left(\frac{L}{4}\right)\right\}}^{2}+{\left\{\frac{G}{2}-\frac{L}{4}-{B}_{b}\right\}}^{2}-{\left\{\frac{g}{2}+\frac{L}{4}\right\}}^{2}\right],$$

$$l=2\left\{\frac{G-g}{2}-3\left(\frac{L}{4}\right)\right\}\times \left\{\frac{G}{2}-\frac{L}{4}-{B}_{b}\right\},$$

$${\theta }_{0}=2\Pi -{{\text{cos}}}^{-1}\left(\frac{k}{l}\right),$$

where ${W}_{b}=30$, $G=160$, $g=90$, ${u}_{i}={u}_{0}=11.033$, $\sigma =\frac{{B}_{b}}{{B}_{m}}$, ${X}_{i}=\frac{{u}_{i}}{{B}_{b}}$, ${X}_{o}=\frac{{u}_{0}}{{B}_{b}}$, $L=G-g-2{B}_{b}$, $0.15\le (G-g)\le {B}_{b}\le 0.45$, $4\le n\le 50$, $0.515\le {X}_{i},{X}_{0}\le 0.60$, $0.4\le {K}_{{G}_{min}}\le 0.5$, $0.6\le {K}_{{G}_{max}}\le 0.7$, $0.3\le\upmu \le 0.4$, $0.02\le \nu \le 0.1$, $0.6\le \varphi \le 0.85$.

Out of 83 research papers 15 researchers have worked on the above problem. We have compared the optimum cost of each 41 algorithms in Table 8 for diameter of a ball (${B}_{b})$, diameter of a pitch (${B}_{m})$, total number of balls (n), curvature coefficients of inner (${X}_{i}$) and outer (${X}_{0})$ raceway curvature coefficient, ${K}_{{G}_{min}}, {K}_{{G}_{max}}, \mu , \nu , \boldsymbol{\varphi }$. The ROA method shows the optimal cost out of all.

Table 8 Performance of different algorithms to solve REB

3.7 Speed reducer problem (SRP)

The main objective is to minimize the weight of speed reducer in so far as possible through subject to constraints:

• gear teeth under bending stress;

• surface stress;

• shafts transverse deflections;

• stresses in the shafts.

SRP contain seven design variables (${u}_{1}\text{ to }{ u}_{7}$) named as width of the face (${u}_{1}$), a set of teeth $({u}_{2}$), total number of pinion teeth (${u}_{3}$), initial shaft distance between bearing $({u}_{4}$), second shaft distance between bearing $({u}_{5}$), first shaft diameter $({u}_{6}$) and second shaft diameter $({u}_{7}$).

$$Minimize \quad z=0.7854{u}_{1}{u}_{2}^{2}\times \left(3.3333\times {u}_{3}^{2}+14.9334\times {u}_{3}-43.0934\right)-1.508\times \left({u}_{6}^{2}+{u}_{7}^{2}\right)+7.4777({u}_{6}^{3}+ {u}_{7}^{3}).$$

Subject to

$${t}_{1}\left(\overrightarrow{u}\right)=\frac{27}{\left({u}_{1}{u}_{2}^{2}\times {u}_{3}\right)}-1\le 0,$$

$${t}_{2}\left(\overrightarrow{u}\right)=\frac{397.5}{({u}_{1}{u}_{2}^{2}\times {u}_{3}^{2})}-1\le 0,$$

$${t}_{3}\left(\overrightarrow{u}\right)=\frac{1.93{u}_{4}^{3}}{({u}_{2}{u}_{3}\times {u}_{6}^{4})}-1\le 0,$$

$${t}_{4}\left(\overrightarrow{u}\right)=\frac{1.93{u}_{5}^{3}}{({u}_{2}{u}_{3}\times {u}_{7}^{4})}-1\le 0,$$

$${t}_{5}\left(\overrightarrow{u}\right)=\frac{1}{110\times {u}_{6}^{3}}\times \sqrt{\left(\frac{745{u}_{4}^{2}}{{u}_{2}{u}_{3}}\right)+16.9\times {10}^{6}}-1\le 0,$$

$${t}_{6}\left(\overrightarrow{u}\right)=\frac{1}{85\times {u}_{7}^{3}}\times \sqrt{\left(\frac{745{u}_{5}^{2}}{{u}_{2}{u}_{3}}\right)+157.5\times {10}^{6}}-1\le 0,$$

$${t}_{7}\left(\overrightarrow{u}\right)=\frac{{u}_{2}{u}_{3}}{40}-1\le 0,$$

$${t}_{8}\left(\overrightarrow{u}\right)=\frac{5{u}_{2}}{{u}_{1}}-1\le 0,$$

$${t}_{9}\left(\overrightarrow{u}\right)=\frac{{u}_{1}}{{12u}_{2}}-1\le 0,$$

$${t}_{10}\left(\overrightarrow{u}\right)=\frac{1.5{u}_{6}+1.9}{{u}_{4}}-1\le 0,$$

$${t}_{11}\left(\overrightarrow{u}\right)=\frac{1.1{u}_{7}+1.9}{{u}_{5}}-1\le 0,$$

where $2.6\le {u}_{1}\le 3.6$, $0.7\le {u}_{2}\le 0.8$, $17\le {u}_{3}\le 28$, $7.3\le {u}_{4}\le 8.3$, $7.8\le {u}_{5}\le 8.3$, $2.9\le {u}_{6}\le 3.9$, $5.0\le {u}_{7}\le 5.5$.

Out of 83 research papers, 24 researchers have worked on same problem. We have compared the optimum cost of each 24 algorithms in Table 9 for width of the face (${u}_{1}$), a set of teeth $({u}_{2}$), total number of pinion teeth (${u}_{3}$), initial shaft distance between bearing $({u}_{4}$), second shaft distance between bearing $({u}_{5}$), first shaft diameter $({u}_{6}$) and second shaft diameter $({u}_{7}$) and found FDA method best among all.

Table 9 Performance of several algorithms for solving the SRP

3.8 Three bar truss design (TBTD) problem

TBTD problem is considered as one of the essential engineering problems, which focuses to find the smallest value under constraints such as bending, stress and bucking. This difficulty contains two decision variables, together with the region of the first, second and third bar.

The third bar truss problem is expressed mathematically as follow:

$$Minimize \,Fitness\left(\overrightarrow{x}\right)=\left(2\sqrt{2}{x}_{{B}_{1}}+{x}_{{B}_{2}}\right)\times u.$$

Subject to

$${t}_{1}\left(\overrightarrow{x}\right)=\frac{\sqrt{2}{x}_{{B}_{1}}+{x}_{{B}_{2}}}{\sqrt{2}{{x}_{{B}_{1}}}^{2}+2{x}_{{B}_{1}}{x}_{{B}_{2}}}L-\sigma \le 0,$$

$${t}_{2}\left(\overrightarrow{x}\right)=\frac{{x}_{{A}_{2}}}{\sqrt{2}{{x}_{{B}_{1}}}^{2}+2{x}_{{B}_{1}}{x}_{{B}_{2}}}L-\sigma \le 0,$$

$${t}_{3}\left(\overrightarrow{x}\right)=\frac{ 1}{\sqrt{2}{x}_{{B}_{2}}+{x}_{{B}_{1}}}L-\sigma \le 0,$$

$$0\le {x}_{{B}_{1}}, {x}_{{B}_{2}}\le 1 ; u=100\text{ cm}, L=\frac{2KN}{{\text{cm}}^{2}}, \sigma =\frac{2KN}{{\text{cm}}^{2}}.$$

Out of 83 research papers 18 researchers have worked on the same problem. After comparing the results for ${x}_{{B}_{1}}$ (area of the first and third bar) and ${x}_{{B}_{2}}$ (area of the second bar) in Table 10 we found that PDO method performs better for the above-mentioned problem.

Table 10 Performance of different algorithms for solving the problem of TBTD

4 Result and discussion

NIA are stochastic investigation techniques, can move about to any convoluted search space and situate optimal (near optimal) solutions in suitable computational time. They can present solutions to every complex optimization difficulty that is not easily solved by the predictable nonlinear programming (NLP) techniques because of their nature that may involve point of discontinuities of the search domain, non-differentiable objective functions, unfocused advice and values of the function. The algorithm discovered over the last 5 years is presented in this study with their brief description. We have tried to find out some best NIA to solve real world single objective optimization problems. For this purpose, we have found eight engineering problems and take the common algorithms for each problem from Table 1 to compare the performance of these algorithms. Tables 3 and 4 show that AO gives best result for WBD and TSD problem. Tables 5, 6 and 10 shows that PDO gives best result for PVD, CBD and TBTD problem. Tables 7 and 8 shows ROA gives significantly better result for IBD and REB problem. Table 9 shows that FDA gives best result for SRP problem. A summary of these results can be seen in Table 11. A little bit more analysis has been done considering those 4 algorithms mentioned in Table 11. Since all these algorithms have also been solved using benchmark problems so their average and standard deviation are compared for solution of 13 benchmark problems mentioned in Table 12 (Rezaei et al. 2022).The run time has been taken as 30.

Table 11 Best algorithm for real life single objective engineering problems

Table 12 Unimodal and multimodal benchmark function

After comparing the results of Tables 13 and 14 we found that FDA and PDO give us comparatively better results for solving single objective optimization problem. The major difference of FDA with other algorithms is using precise strategies to assign some interval of investigation process to global search and remaining to local search. This rule is archived by defining an area radius that decreases from highest values to small values, and sink filling procedure that helps to prevent from local solutions. The PDO break search space of the problem into some coteries, and the optimization procedure is passed out in each of the coteries. The solutions are combined, and the best solution is selected.

Table 13 Comparison of ROA and AO for average and standard deviation for dimension (D) = 100 on 13 benchmark function

Table 14 Comparison of FDA, AO and PDO for average and standard deviation for D = 100 on 13 benchmark function

Table 12 shows some commonly used unimodal and multimodal benchmark functions which are used in all 83-researchpaper to check the efficiency of their algorithms.

5 Bibliographic analysis

From previous section we concluded that FDA and PDO are better as compare to other advanced NIA for solving single objective optimization problems. We also have done some bibliographic analysis to verify our results. The same can be seen in Table 15. Serial Number. 43, 47, 48 and 73 are the details of NIA algorithms ROA (IF 8.665, Citation 66) FDA (IF 7.18, Citation 63), AO (IF 7.18, Citation 744) and PDA (IF 5.10 and Citation 41). From this table also we can conclude that all four algorithms (ROA, FDA, AO and PDO) are reasonable have good citation and impact factor. Overall, our analysis suggests that out of 83 algorithms, FDA and PDO are better to solve single objective optimization problem.

Table 15 Detail of 83 review papers in terms of Author name, Journal name, Impact factor and citation

6 Conclusion and future work

This paper reviewed the recent developed the nature-inspired algorithm from the year 2019 to 2023. The goal of this study is to identify the best nature-inspired algorithms to tackle the constrained and other optimization problems, and for that we considered the 83 well-known algorithms that were found between 2019 and 2023. We have selected eight common benchmark engineering design optimization problems from 83 research publications and compared their solutions of all standard algorithms to these problems in order to access their performance. From the analysis, it is determined that AO provides the best solution for WBD and TSD problems, PDO provides the best solution for PVD, CBD and TBTD problems, ROA provides the best solution for IBD and REB problems, while FDA give best result for SRP problem. Further, to determine, which algorithm from AO, FDA, ROA and PDO is the best, more analysis is done by considering the benchmark CEC problems. From the analysis, it is concluded that FDA and PDA are more effective at solving the large-scale optimization problems among the available 83 different algorithms. In the future, we shall extend our research to analyze more on this issue and their application to diverse field such as object detection (Zhang et al. 2023), feature selection problem (Liu et al. 2023), neural network, control design problems (Govindan et al. 2023), resource-constrained (Xuemin et al. 2023). One may try to apply these algorithms on multi objective optimization problems (Cao et al. 2020b) and they may be clubbed with machine learning to improve the efficiency.