A new metaphor-less simple algorithm based on Rao algorithms: a Fully Informed Search Algorithm (FISA)

Optimization algorithms

Optimization algorithms can be broadly classified into two types: exact methods and approximate methods. Exact methods are capable of guaranteeing an optimal solution, but they may require significant computational resources and time. In contrast, approximate methods focus on finding good solutions in a reasonable amount of time. Heuristic algorithms are a popular type of approximate methods that are designed to quickly generate high-quality solutions to a wide range of problems. The effectiveness of heuristic algorithms depends on the nature of the problem being solved (Gogna & Tayal, 2013).

Metaheuristic methods

Metaheuristics refer to methods that guide the search process and are often inspired by nature. Unlike heuristic algorithms, due to their problem-independent nature, these algorithms can be utilized to optimize a diverse range of problems. These methods are among the most important and promising research in the optimization domain.

The general principles of metaheuristic methods are as follows:

• Employing a given number of repetitive efforts

• Employing one or more agents (particles, neurons, ants, chromosomes, etc.)

• Operation (in multi-factor mode with a cooperation-competition mechanism)

• Creating methods of self-change and self-transformation

Nature has two great tactics:

1. Rewarding strong personal characteristics and punishing weaker ones.

2. Introducing random mutations, which can lead to the birth of new individuals.

Recently, many optimization techniques have been proposed that operate on the basis of natural behaviors and social behaviors. In the initialization stage, these algorithms randomly generate solutions and in the later stages, they rely on natural processes to produce better answers.

Some of the popular and widely used types of metaheuristics are Particle swarm optimization algorithm (PSO) (Kennedy & Eberhart, 1995), differential evolution (DE) (Storn & Price, 1995), genetic algorithm (GA) (Whitley, 1994), firefly algorithm (Yang, 2009), ant colony optimization (Dorigo & Di Caro), bat algorithm (Yang, 2010), teaching–learning-based optimization (TLBO) (Rao, Savsani & Vakharia, 2011), grey wolf optimizer (GWO) (Mirjalili, Mirjalili & Lewis, 2014), artificial bee colony algorithm (ABC) (Karaboga & Basturk, 2007), imperialist competitive algorithm (ICA) (Atashpaz-Gargari & Lucas, 2007), moth-flame optimization algorithm (MFO) (Mirjalili, 2015), gravitational search algorithm (GSA) (Rashedi, Nezamabadi-pour & Saryazdi, 2009), shuffled frog-leaping algorithm (SFLA) (Eusuff, Lansey & Pasha, 2006), whale optimization algorithm (WOA) (Mirjalili & Lewis, 2016), etc.

Now the question is why all these new optimization algorithms, either modified or combined, are needed. The main reason for this is the inability to determine with certainty which optimization or metaheuristic algorithm is appropriate for resolving a problem, and only through the comparison of the outcomes can it be asserted which algorithm provides a superior approach. In addition, based on (Mirjalili, Mirjalili & Lewis, 2014), an algorithm may perform well for some groups of functions but not for some other groups. Therefore, the motivation to modify algorithms or introduce new algorithms has been very high in recent years (Mirjalili & Lewis, 2016).

In 2020, Rao (2020) suggested three effective and powerful straightforward algorithms for optimization problems without the use of metaphors. These methods use the most and least optimal solutions in each iteration and the casual interrelations between possible solutions. Additionally, these methods need no control parameter other than the population size and the number of iterations.

Rao algorithms have been successfully used by researchers in the short time since they have been introduced, some of which include the engineering design optimization (Rao & Pawar, 2020a; Rao & Pawar, 2020b; Rao & Pawar, 2022; Rao & Pawar, 2023), weight optimization of reinforced concrete cantilever retaining wall (Kalemci & Ikizler, 2020), cropping pattern under a constrained environment (Kumar & Yadav, 2019), construction scheduling (Yılmaz & Dede, 2023), photovoltaic cell parameter estimation (Premkumar et al., 2020; Wang et al., 2020), optimization of energy systems (Rao et al., 2022), and optimal power flow (Sahay, Upputuri & Kumar, 2023).

A significant proportion of optimization problems encountered in practical applications contain shifted functions, for which the performance of Rao algorithms may not be much optimal, as demonstrated in the simulation section. This article introduces a new algorithm, called Fully Informed Search Algorithm (FISA), which is based on Rao algorithms and address this drawback. FISA not only outperforms the original Rao algorithms in optimizing shifted functions but also preserves their simplicity and requires no control parameters. The suggested algorithm’s performance is evaluated by optimizing benchmark problems with shifted functions and real-world problems. The results demonstrate that FISA outperforms not only the original Rao algorithms but also other state-of-the-art methods, indicating its superior performance.

The article continues in four sections: the formulation of the suggested and Rao algorithms is presented in the following section. Reporting the outcomes of simulations and presenting and discussing the findings are done in next section. Lastly, the conclusions are presented.

Rao algorithms

The basic formulations of Rao algorithms, which are very simple algorithms without control parameters, rely on the difference vectors obtained by subtracting the position (location) of the worst individual from the location of the finest individual in the present iteration. This ensures that the population always moves towards a better solution. These algorithms consist of three distinct movements (position update) vectors for updating the position which are defined as follows (Rao, 2020):

Rao-1 algorithm: (1)${\displaystyle X_{i,j}^{\mathrm{new}}=X_{i,j}^{\mathrm{Iter}}+r_{1,j}\left(X_{\mathrm{best},j}^{\mathrm{Iter}}-X_{\mathrm{worst},j}^{Iter}\right).}$

Rao-2 algorithm: (2)${\displaystyle \left\{\begin{array}{c}\mathrm{if}f\left(X_i^{\mathrm{Iter}}\right)<f\left(X_k^{\mathrm{Iter}}\right)\hfill \\ X_{i,j}^{\mathrm{new}}=X_{i,j}^{\mathrm{Iter}}+r_{1,j}\left(X_{\mathrm{best},j}^{\mathrm{Iter}}-X_{\mathrm{worst},j}^{\mathrm{Iter}}\right)+r_{2,j}\left(\left|X_{i,j}^{\mathrm{Iter}}\right|-\left|X_{k,j}^{\mathrm{Iter}}\right|\right);\hfill \\ \mathrm{else}\hfill \\ X_{i,j}^{\mathrm{new}}=X_{i,j}^{\mathrm{Iter}}+r_{1,j}\left(X_{\mathrm{best},j}^{\mathrm{Iter}}-X_{\mathrm{worst},j}^{Iter}\right)+r_{2,j}\left(\left|X_{k,j}^{\mathrm{Iter}}\right|-\left|X_{i,j}^{\mathrm{Iter}}\right|\right).\hfill \end{array}\right.}$

Rao-3 algorithm: (3)${\displaystyle \left\{\begin{array}{c}\mathrm{if}f\left(X_i^{\mathrm{Iter}}\right)<f\left(X_k^{\mathrm{Iter}}\right)\hfill \\ X_{i,j}^{\mathrm{new}}=X_{i,j}^{\mathrm{Iter}}+r_{1,j}\left(X_{\mathrm{best},j}^{\mathrm{Iter}}-\left|X_{\mathrm{worst},j}^{\mathrm{Iter}}\right|\right)+r_{2,j}\left(\left|X_{i,j}^{\mathrm{Iter}}\right|-X_{k,j}^{\mathrm{Iter}}\right);\hfill \\ \mathrm{else}\hfill \\ X_{i,j}^{\mathrm{new}}=X_{i,j}^{\mathrm{Iter}}+r_{1,j}\left(X_{\mathrm{best},j}^{\mathrm{Iter}}-\left|X_{\mathrm{worst},j}^{\mathrm{Iter}}\right|\right)+r_{2,j}\left(\left|X_{k,j}^{\mathrm{Iter}}\right|-X_{i,j}^{\mathrm{Iter}}\right).\hfill \end{array}\right.}$

In the above equations, $X_i^{\mathrm{Iter}}$ represents the ith solution’s location in the present iteration Iter; j(changing from 1 to D) represents the jth dimension of each solution; $X_{\mathrm{best}}^{\mathrm{Iter}}$ and $X_{\mathrm{worst}}^{\mathrm{Iter}}$ represent the position of the highest and lowest performing members of the population during the present iteration, in that order; r₁ and r₂ are two randomly selected values between 0 and 1 with the dimension of D; $X_k^{\mathrm{Iter}}$ represents the position of the kth solution, which is indiscriminately chosen; and $f\left(.\right)$ represents the numerical output of the function being optimized of the corresponding solution in the present iteration. The location of the ith solution in the next iteration is obtained using Eq. (4): (4)${\displaystyle \left\{\begin{array}{c}{X}_i^{\mathrm{Iter}+1}=X_i^{\mathrm{new}}\mathrm{if}f\left(X_i^{\mathrm{new}}\right)\le f\left(X_i^{\mathrm{Iter}}\right)\hfill \\ X_i^{\mathrm{Iter}+1}=X_i^{\mathrm{Iter}}\mathrm{else}.\hfill \end{array}\right.}$

The proposed Fully Informed Search Algorithm (FISA)

The performance of Rao algorithms in optimizing shifted functions, which may be the case for many real-world problems, may be suboptimal. This will be demonstrated in the simulation section. Therefore, we propose a new algorithm, the Fully Informed Search Algorithm (FISA), which is based on Rao algorithms and is designed to address this issue. FISA improves the optimization of shifted functions while retaining the simplicity and absence of control parameters of the original algorithms. Similar to Rao algorithms, FISA moves the population towards better solutions. In summary, FISA can be summarized as follows: (5)${\displaystyle X_{i,j}^{\mathrm{new}}=X_{i,j}^{\mathrm{Iter}}+r_{1,j}\left(M{X}_{\mathrm{best},j}^{\mathrm{Iter}}-X_{i,j}^{\mathrm{Iter}}\right)+r_{2,j}\left(X_{i,j}^{\mathrm{Iter}}-M{X}_{\mathrm{worst},j}^{\mathrm{Iter}}\right).}$

In fact, in FISA, each member moves away from the mean position of the individuals within the population that have worse fitness values and approaches the mean position of the individuals that have better fitness values than the associated member. Then, the position of each member is updated using Eq. (4). In Eq. (5), the values of $M{X}_{\mathrm{best},j}^{\mathrm{Iter}}$ and $M{X}_{\mathrm{worst},j}^{\mathrm{Iter}}$ in each iteration are calculated using Eqs. (6) and (7), respectively: (6)${\displaystyle M{X}_{\mathrm{best}}^{\mathrm{Iter}}=\frac{X_{\mathrm{best}}^{\mathrm{Iter}}+\sum _{l\in Bi}{X}_l^{\mathrm{Iter}}}{\text{length}\left(Bi\right)+1}}$ (7)${\displaystyle M{X}_{\mathrm{worst}}^{\mathrm{Iter}}=\frac{X_{\mathrm{worst}}^{\mathrm{Iter}}+\sum _{l\in Wi}{X}_l^{\mathrm{Iter}}}{\text{length}\left(Wi\right)+1}}$

where B_i and W_i are the set of population members that have a better and worse fitness value than the i ^th member in iteration Iter, respectively, and length(.) represents the count of the individuals in the set.

The flowchart of FISA is shown in Fig. 1.

FISA for solving CEC2005 problems

To assess the effectiveness of the suggested algorithm in comparison to the original Rao algorithms, we have chosen 14 real-world shifted functions with 30 dimensions (including unimodal, multimodal, and expanded multimodal functions), numbered in the order introduced in CEC 2005 (Liu et al., 2013), whose data were extracted from Suganthan et al. (2005). These functions have been successfully utilized in many articles (Ghasemi, Aghaei & Hadipour, 2017; Birogul, 2019; Ghasemi et al., 2019; Ghasemi et al., 2022a; Ghasemi et al., 2022b; Ghasemi et al., 2023; Akbari, Rahimnejad & Gadsden, 2021; Premkumar et al., 2021; Zou et al., 2022). The total count of function evaluations (NFE) during the execution of each algorithm is considered 300,000 based on Liu et al. (2013); accordingly, the number of population members of each algorithm during the optimization is 30; therefore, the convergence curves, in this study, have been extracted for 10,000 iterations. Furthermore, to acquire the optimal solution for each function, each algorithm has been executed independently for 25 runs. A summary of the results comprising the average value, standard deviation, and rank for each algorithm amongst the investigated algorithms is given in Table 1. In this table, N_b and N_w represent the total instances where the corresponding algorithm has the best or the worst result among the studied algorithms, respectively. M_R also represents the mean ranking of each algorithm for 14 test functions.

After a concise investigation of Table 1, it can be observed that the FISA algorithm noticeably outperformed the original Rao algorithms, particularly for functions F2, F4, and F6. The proposed algorithm surpassed the Rao-2 and Rao-3 algorithms for all investigated functions, except for the F8 test function where it achieved the same performance as the Rao algorithms. Although the FISA algorithm showed slightly lower performance than the Rao-1 algorithm for test functions F1, F7, and F9, these results were close, and FISA outperformed the Rao-1 algorithm for the remaining 10 test functions. These findings indicate that the proposed algorithm has a strong capability to achieve optimal solutions for practical problems. Additionally, the convergence behaviors of different algorithms for solving the selected functions are presented in Fig. 2, which provides evidence of the superior convergence behavior of the suggested algorithm.

FISA for solving CEC2014 problems

In the second part of demonstrating the efficacy of the suggested method, FISA, in comparison with the RAO algorithms, 30 test functions from CEC 2014 Test Functions, with dimension 30 are selected (Askari & Younas, 2021; Suganthan et al., 2005; Liu & Nishi, 2022; Meng et al., 2022; Band et al., 2022), whose data were extracted from (Suganthan et al., 2005). The optimal value for all functions is 0. The population number was selected as 30 and the stopping criterion was selected as 10,000 iterations for all algorithms; so that the NFE for each algorithm is equal to 300,000. The experiment is conducted by running each algorithm independently 25 times for every function. The summary of the results comprising the Mean value, standard deviation, and best value for each of the investigated algorithms is given in Table 2.

After reviewing the results presented in Table 2, taking into account the average value, standard deviation, and also the best optimal value of 25 runs, we can see that the suggested algorithm has a significant advantage over the Rao algorithms. The last row of the table shows the number of test functions in which each algorithm achieved their best solution, denoted as Nb. It is evident that the proposed method outperformed the other algorithms by obtaining the best solution in 23 out of 30 test functions. This indicates the strength and efficiency of the suggested technique as a new optimization algorithm. The convergence characteristics of algorithms for some selected test functions are displayed in Fig. 3.

Pressure vessel optimal design

The goal of the problem is to minimize the overall costs of a pressure vessel, comprising material, forming, and welding expenses. As depicted in Fig. 4, this problem involves four design variables: shell thickness (denoted as x₁ or T _s), head thickness (denoted as x₂ or T_h), inner radius (denoted as x₃ or R), and cylindrical section length (denoted as x₄ or L). While x₃ and x₄ are continuous variables, x₁ and x₂ are discrete variables represented as integer multiples of 0.0625 in.

The problem’s objective function is nonlinear and it has both a linear and a nonlinear inequality constraint, which are illustrated below (Askarzadeh, 2016):

Minimize: (8)${\displaystyle f\left(X\right)=0.6224x_1{x}_3{x}_4+1.7781x_2{x}_3^2+3.1661x_1^2{x}_4+19.84x_1^2{x}_3}$

subject to: (9)${\displaystyle g_1\left(X\right)=-x_1+0.0193x_3\le 0,}$ (10)${\displaystyle g_2\left(X\right)=-x_2+0.00954x_3\le 0,}$ (11)${\displaystyle g_3\left(X\right)=-\pi x_3^2{x}_4-\frac{4}{3}\pi x_3^3+1,296,000\le 0,}$ (12)${\displaystyle g_4\left(X\right)=x_4-240\le 0,}$

0 ≤ x_i ≤ 100, i = 1, 2

10 ≤ x_i ≤ 200, i = 3, 4.

Table 3 compares the outcomes achieved by the proposed algorithm for the problem and other widely used standard algorithms, including quantum-inspired PSO (QPSO) and Gaussian QPSO (G-QPSO) (Coelho, Dos Santos Coelho & Coelho, 2010), ABC (Akay & Karaboga, 2012), a GA equipped with a constraint-handling via dominance-based tournament selection (GA4) (Coello Coello et al., 2002), co-evolutionary PSO (CPSO) (He & Wang, 2007), co-evolutionary DE (CDE) (Huang, Wang & He, 2007), unified PSO (UPSO) (Parsopoulos & Vrahatis, 2005), Crow search algorithm (CSA) (Askarzadeh, 2016), hybridizing a genetic algorithm with an artificial immune system (HAIS-GA) (Bernardino et al., 2008), bacterial foraging optimization algorithm (BFOA) (Mezura-Montes & Hernández-Ocana, 2008), evolution strategies (ES) (Mezura-Montes & Coello, 2008), A modification of the T-Cell algorithm (Aragón, Esquivel & Coello, 2010), a GA enhanced with a self-adaptive penalty method (GA3) (Coello Coello, 2000), queuing search (QS) algorithm (Zhang et al., 2018), and a GA equipped with a constraint-handling via automatic dynamic penalization (ADP) method (BIANCA) (Montemurro, Vincenti & Vannucci, 2013). Furthermore, the best solution found for the problem using the suggested technique was shown in Table 4. Tables 3 and 4 demonstrate that FISA outperforms other algorithms with the smallest value for the best solution, and ranking first for the worst solution and average value. This confirms that FISA performs well and reliably in tackling the pressure vessel optimal design problem.

Tension/compression spring optimal design

According to Fig. 5, the goal of the problem includes reducing the tension/compression spring weight while satisfying four inequality limitations (one linear and three nonlinear). The problem comprises three continuous design variables: wire diameter (denoted as d or x₁), mean coil diameter (denoted as D or x₂), and the number of active coils (denoted as P or x₃) (Askarzadeh, 2016).

Minimize: (13)${\displaystyle f\left(X\right)=\left(x_3+2\right)x_2{x}_1^2}$

subject to: (14)${\displaystyle g_1\left(X\right)=1-\frac{x_2^3{x}_3}{71,785x_1^4}\le 0,}$ (15)${\displaystyle g_2\left(X\right)=\frac{4x_2^2-x_1{x}_2}{12,566\left(x_1^3{x}_2-x_1^4\right)}+\frac{1}{5,108x_1^2}-1\le 0,}$ (16)${\displaystyle g_3\left(X\right)=1-\frac{140.45x_1}{x_2^2{x}_3}\le 0,}$ (17)${\displaystyle g_4\left(X\right)=\frac{x_1+x_2}{1.5}-1\le 0.}$

0.05 ≤ x₁ ≤ 2, 0.25 ≤ x₂ ≤ 1.3, 2 ≤ x₃ ≤ 15.

Table 5 describes the outcomes of the proposed algorithm for the problem in comparison with other standard and well-known algorithms including BFOA (Mezura-Montes & Hernández-Ocana, 2008), T-Cell (Aragón, Esquivel & Coello, 2010), CDE (Huang, Wang & He, 2007), a cultural algorithm (CA) (Coello Coello & Becerra, 2004), CPSO (He & Wang, 2007), GA4 (Coello Coello et al., 2002), GA3 (Coello Coello, 2000), TEO (Kaveh & Dadras, 2017), G-QPSO (Coelho, Dos Santos Coelho & Coelho, 2010), SBO (Ray & Liew, 2003), evolutionary algorithms ((l + k)-ES) (Mezura-Montes & Coello, 2005), grey wolf optimizer (GWO) (Mirjalili, Mirjalili & Lewis, 2014), UPSO (Parsopoulos & Vrahatis, 2005), QS (Zhang et al., 2018), SDO (Zhao, Wang & Zhang, 2019), BIANCA (Montemurro, Vincenti & Vannucci, 2013), and water cycle algorithm (WCA) (Eskandar et al., 2012). Furthermore, Table 6 shows the optimal solution obtained for the problem using the suggested algorithm. The tables demonstrate that FISA outperforms all other algorithms in terms of the best value, with the worst solution and average value being the smallest. This suggests that FISA is more effective than other competitive optimizers in solving this problem.

Welded beam optimal design

The goal of the problem is the minimization of the cost of a welded beam. The problem has four continuous decision variables, as depicted in Fig. 6, namely x₁(h), x₂(l), x₃(t), and x₄(b), along with two linear and five nonlinear inequality limitations (Askarzadeh, 2016).

Minimize: (18)${\displaystyle f\left(X\right)=1.10471x_2{x}_1^2+0.04811x_3{x}_4\left(14+x_2\right)}$

Subject to: (19)${\displaystyle g_1\left(X\right)=\tau \left(x\right)-{\tau }_{\max }\le 0,}$ (20)${\displaystyle g_2\left(X\right)=\sigma \left(x\right)-{\sigma }_{\max }\le 0,}$ (21)${\displaystyle g_3\left(X\right)=x_1-x_4\le 0,}$ (22)${\displaystyle g_4\left(X\right)=0.10471x_1^2+0.04811x_3{x}_4\left(14+x_2\right)-5\le 0.}$ (23)${\displaystyle g_5\left(X\right)=0.125-x_1\le 0,}$ (24)${\displaystyle g_6\left(X\right)=\delta \left(x\right)-{\delta }_{\max }\le 0,}$ (25)${\displaystyle g_7\left(X\right)=P-P_c\left(x\right)\le 0,}$

Where (26)${\displaystyle \tau \left(x\right)=\sqrt{{\left({\tau }^{\prime }\right)}^2+2{\tau }^{\prime }\tau ″\frac{x_2}{2R}+{\left(\tau ″\right)}^2}}$

(27-28)${\displaystyle {\tau }^{\prime }=\frac{P}{\sqrt{2}{x}_1{x}_2},\tau ″=\frac{MR}{J},}$

(29-31)${\displaystyle M=P\left(L+\frac{x_2}{2}\right),R=\sqrt{\frac{x_2^2}{4}+{\left(\frac{x_1+x_3}{2}\right)}^2},\delta \left(x\right)=\frac{4P{L}^3}{E{x}_3^3{x}_4}}$

(32-33)${\displaystyle J=2\left[\sqrt{2}{x}_1{x}_2\left\{\frac{x_2^2}{12}+{\left(\frac{x_1+x_3}{2}\right)}^2\right\}\right],\sigma \left(x\right)=\frac{6PL}{x_4{x}_3^2},}$

(34)${\displaystyle P_c\left(x\right)=\frac{4.013E\sqrt{\frac{x_4^6{x}_3^2}{36}}}{L^2}\left(1-\frac{x_3}{2L}\sqrt{\frac{E}{4G}}\right),}$

P = 6,000 lb; L = 14 in; E = 30e6 psi G = 12e6 psi, τ_max =13,000 psi, σ_max =30,000 psi

δ_max = 0.25 in, 0.1 ≤ x₁ ≤ 2, 0.1 ≤ x₂ ≤ 10, 0.1 ≤ x₃ ≤ 10, 0.1 ≤ x₄ ≤ 2.

The performance of the suggested method for the problem is presented in Table 7, where it is compared to other widely used algorithms, including a cooperative PSO with stochastic movements (EPSO) (Ngo, Sadollah & Kim, 2016), BFOA (Mezura-Montes & Hernández-Ocana, 2008), T-Cell (Aragón, Esquivel & Coello, 2010), CDE (Huang, Wang & He, 2007), a hybrid real-parameter GA (HSA-GA) (Hwang & He, 2006), CPSO (He & Wang, 2007),TEO (Kaveh & Dadras, 2017), SBO (Ray & Liew, 2003), Derivative-Free Filter Simulated Annealing Method (FSA) (Hedar & Fukushima, 2006), GA4 (Coello Coello et al., 2002), (l + k)-ES (Mezura-Montes & Coello, 2005), GWO (Mirjalili, Mirjalili & Lewis, 2014), SFO (Shadravan, Naji & Bardsiri, 2019), HGSO (Hashim et al., 2019), UPSO (Parsopoulos & Vrahatis, 2005), WCA (Eskandar et al., 2012), BIANCA (Montemurro, Vincenti & Vannucci, 2013). Furthermore, Table 8 displays the optimal solution obtained by the suggested algorithm for the problem. Table 7 and 8 demonstrate that FISA is capable of discovering the most effective optimization solution. The statistical findings also indicate that FISA surpasses other methods and can more effectively handle the constrained engineering problems.