Flood algorithm: a novel metaheuristic algorithm for optimization problems

Terms and conditions

The terms and definitions used in this study are given in Table 1.

Introduction

Optimization can be defined as the attempt to achieve the best result by making choices on the inputs of an objective function within the framework of certain criteria. It is a frequently encountered problem in almost every field of science and real-world problems. Solving complex optimization problems (e.g., the NP-complete problems) with conventional gradient-based methods is a time-consuming process (Chen, Cai & Wang, 2018). The complexity of the problem makes it impossible to search for all possible solutions or combinations in an acceptable time frame (Yang, 2010b).

Metaheuristic algorithms are an impressive area of research that has made significant advances in solving challenging optimization problems (Dokeroglu et al., 2019). They use random operators, trial-and-error processes and random scanning of the problem solving space to generate efficient solutions to optimization problems (Dehghani et al., 2023). The optimization process in meta-heuristic algorithms starts by generating a certain number of random feasible solutions in the problem space. In an iterative process, candidate solutions are updated and improved according to the algorithm instructions. After the algorithm is fully implemented, the best solution among the candidate solutions is presented as the solution to the problem (Dehghani et al., 2020).

Simple concepts, easy implementation, no need for a derivation process, efficiency in high-dimensional problems, efficiency in non-linear and non-convex environments are some of the advantages that lead to the popularity and widespread use of metaheuristic algorithms (Cavazzuti, 2013). These advantages and their ability to solve problems in a wide variety of domains without knowing the details and definitions of the problems, and to provide near-optimal solutions, gives them an advantage over traditional techniques (Rajpurohit et al., 2017). Therefore, in recent decades, a number of metaheuristic algorithms have been proposed and successfully applied to solve complex optimization problems in various scientific domains and real-world problems. According to the number of related studies, well-known and most used algorithms are: Genetic Algorithm (GA) (Holland, 1992), Simulated Annealing (SA) (Kirkpatrick, Gelatt & Vecchi, 1983), and Particle Swarm Optimization (PSO) (Eberhart & Kennedy, 1995). There are numerous studies in the literature where these algorithms, improved versions, or hybrid versions of these algorithms are proposed or applied to a problem.

Although metaheuristic algorithms show strong optimization ability in solving nonlinear global optimization problems, some of them fall into local optima when faced with different optimization problems (Yang, 2010b). Moreover, a metaheuristic may perform well on one set of optimization problems but poorly on another (Wolpert & Macready, 1997).

These fundamental gaps encourage the discovery and development of new metaheuristic algorithms with satisfactory performance. For this reason, there is an increasing amount of work in the literature to propose new metaheuristic algorithms. In line with these motivations, this article proposes a new metaheuristic algorithm called the flood algorithm.

The contributions of this study can be expressed as follows:

• A new optimization algorithm called Flood Algorithm (FA) is developed to model the flow of flood water.

• Fifteen standard benchmark functions, eight of them from the CEC2022 test suite, and three engineering design problems were used to evaluate the performance of FA in solving optimization problems.

• The performance of FA in real-world applications is tested by solving an exam seating problem, which is a permutation problem and consists of eleven sessions.

• The performance of FA is validated against the three well-known metaheuristic algorithms.

The main advantage of the proposed FA approach for global optimization problems is that it has a single basic parameter and thus can be easily adapted to different optimization problems. The second advantage of FA is its highly effective efficiency in dealing with high-dimensional optimization problems. The third advantage of the proposed method is its strong performance in handling real-world optimization applications.

The rest of the article is organized as follows: “Literature Review” section presents a literature review about metaheuristic algorithms. “Flood Algorithm” section describes the proposed flood algorithm. “Flood Algorithm Validation” section presents the application and results of the proposed algorithm and three other algorithms on benchmark functions and a real-world problem and compares the results, and “Conclusion” section concludes this article and provides suggestions for future work.

Literature review

Metaheuristic algorithms are inspired by nature and mimic biological and physical processes such as the behavior of animals, insects, birds, living things, physical laws, biological sciences, human activities, rules of the games, and any other evolution-based process to solve optimization problems (Tanhaeean, Tavakkoli-Moghaddam & Akbari, 2022; Dehghani et al., 2023). For example, the GA (Holland, 1992) mimics evolutionary processes in nature, PSO (Eberhart & Kennedy, 1995) mimics swarm movement of birds or fish while the SA (Kirkpatrick, Gelatt & Vecchi, 1983) mimics physical metal annealing processes. Based on the development of various metaheuristic algorithms in the last decades, Metaheuristic algorithms can be divided into three main categories: Evolutionary Algorithms (EA), which follow the natural evolutionary process found in nature. Swarm Intelligence Algorithms (SI), which include swarm-based techniques that mimic the social behavior of groups of insects or animals. And other metaheuristic algorithms that mimic principles of physics, chemistry, gaming and human behavior (Azizi, Talatahari & Gandomi, 2023; Hashim et al., 2022; Ayyarao et al., 2022).

EA are based on the mechanism of natural selection. They mimic the way natural evolution and genetic mechanisms works. GA and Differential Evolution (Storn & Price, 1997) are among the most widely used evolutionary algorithms designed based on the modeling of the reproductive process, natural selection, Darwin’s theory of evolution, and the use of random operators of selection, crossover, and mutation (Dehghani et al., 2023). Some other algorithms in this group are: Genetic Programming (Koza, 1992), Evolution Strategy (Bäck, 1995), Covariance Matrix Adaptation Evolutionary Strategy (Hansen & Ostermeier, 2001), Wild Horse Optimizer (Naruei & Keynia, 2022) and, Biogeography-Based Optimization (Simon, 2008).

SI algorithms have been developed inspired by natural swarming phenomena, the collective and self-organizing behavior of birds, fishes and, other living things in nature. Some of the known meta-heuristic algorithms are: PSO (Eberhart & Kennedy, 1995), Ant Colony Optimization (Dorigo, Maniezzo & Colorni, 1996), Firefy Algorithm (Yang, 2010a), Gray Wolf Optimization (Mirjalili, Mirjalili & Lewis, 2014), Whale Optimization Algorithm (Mirjalili & Lewis, 2016), White Shark Optimizer (Braik et al., 2022), Harris Hawks Optimizer (Heidari et al., 2019), War Strategy Optimization Algorithm (Ayyarao et al., 2022), Shrimp and Goby Association Search Algorithm (Sang-To et al., 2023), Orchard Algorithm (Kaveh, Mesgari & Saeidian, 2023), Gannet Optimization Algorithm (Pan et al., 2022), Dung Beetle Optimizer (Xue & Shen, 2023) and Honey Badger Algorithm (Hashim et al., 2022).

The third group algorithms are developed based on mathematical modeling of various events, concepts, laws, and forces in physics, chemistry, games, and human behavior. Some of the well-known metaheuristic algorithms are: SA (Kirkpatrick, Gelatt & Vecchi, 1983), Gravitational Search Algorithm (Rashedi, Nezamabadi-Pour & Saryazdi, 2009), Group Teaching Optimization Algorithm (Zhang & Jin, 2020), Henry Gas Solubility Optimization (Hashim et al., 2019), Fireworks Algorithm (Tan & Zhu, 2010), Chaos Game Optimization (Talatahari & Azizi, 2021), Harmony Search Algorithm (Yang, 2009), Hunger Games Search (Yang et al., 2021), Boxing Match Algorithm (Tanhaeean, Tavakkoli-Moghaddam & Akbari, 2022), Atomic Orbital Search (Azizi, 2021).

From another perspective, metaheuristics can be classified into two groups: trajectory-based metaheuristics and population-based metaheuristics. The main difference between these two types of methods is based on the number of solutions used at each step of the algorithm. Trajectory-based algorithms use a single agent or solution that moves through the design space or search space. Population-based algorithms, on the other hand, use a large number of agents or solutions at each search step. Population-based algorithms are much more popular in the literature than trajectory-based algorithms. For example, EA and SI are all population-based algorithms. The most common examples of trajectory-based algorithms are: SA (Kirkpatrick, Gelatt & Vecchi, 1983), Tabu Search (Glover, 1989), Iterated Tabu Search (Misevicius, Lenkevicius & Rubliauskas, 2006), Guided Local Search (Davenport et al., 1994) and Variable neighbourhood Search (Mladenović & Hansen, 1997).

Flood algorithm

In this section, a new metaheuristic algorithm called the flood algorithm is discussed in detail. The FA is a metaheuristic algorithm for optimization problems that can be used to approach global optimization in a large search space within an acceptable time frame. Metaheuristic algorithms are divided into two groups according to their search mechanisms: trajectory-based algorithms, which work on a single solution at each iteration, and population-based algorithms, which work on a collection of solutions. The FA is a trajectory-based algorithm. It starts with a random solution and searches for a global optimum by manipulating that solution at each iteration.

Mathematical model

This section describes the mathematical formulation of the proposed FA. Since the FA includes both exploration and exploitation processes, it can be considered a global optimization algorithm. The pseudo-code of the FA including the initialization, search and evaluation, and the termination steps is presented in Algorithm 1, and the flowchart is presented in Fig. 1. Mathematically, the steps of the proposed algorithm are detailed as follows.

Algorithm 1 :

Pseudo-code of the flood algorithm.

$m\leftarrow \mathrm{m}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{m}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{s}\mathrm{t}\mathrm{o}{}^{ˇ}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{y}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$

$v_{init}\leftarrow 1,v_{min}\leftarrow 0.1,v\leftarrow v_{init}$

$s\leftarrow s_0$                         ⊳ initial state

$s_{best}\leftarrow s_0$

while $v>v_{min}$ do

for k = 1 to m do

$s_{new}\leftarrow \mathrm{g}\mathrm{e}\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{o}\mathrm{m}\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{s}$

if $f(s_{new})<f(s)$ then      ⊳ fitness value comparison

$v\leftarrow v\ast (f(s)/f(s_{new}))$   ⊳ speed update

$s\leftarrow s_{new}$

if $f(s)<f(s_{best})$ then

$s_{best}\leftarrow s$

break

$neighbours[k]\leftarrow s_{new}$

$v\leftarrow v\ast (f(s)/f(min(neighbours)))$      ⊳ speed update

$s\leftarrow min(neighbours)$

DOI: 10.7717/peerj-cs.2278/table-12

Initialization

The FA is a trajectory-based algorithm starting from a single starting point $X_{init}$ as defined in Eq. (1) and with a starting speed $V_{init}$ of 1.

(1) $$\begin{array}{rl}& X_{init}:[x_1,...,x_i,...,x_n],i=1,2,...,n,\\ & n\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{d}\mathrm{e}\mathrm{c}\mathrm{i}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e}\mathrm{s}\end{array}$$

This point represents a candidate solution to the problem for which the values of the decision variables are randomly determined using Eq. (2) and represents the location where the flood starts.

(2) $$X_{init}:x_i=l{b}_i+r\times (u{b}_i-l{b}_i)$$where $x_i$ is the value of the $i$th decision variable, $r$ is a random real number in the range [0, 1], and $l{b}_i$ and $u{b}_i$ are the lower and upper bounds of the $i$th decision variable, respectively. After evaluating the fitness of the starting point, the FA runs in iterations until the termination condition is met. The starting point $X_{init}$ is used as the current solution $S_{cur}$ and starting speed $X_{init}$ is used as the current speed of the first iteration with Eq. (3).

(3) $$\begin{array}{c}{S}_{cur}=X_{init}\\ V_{cur}=V_{init}\end{array}$$

Finding a neighbour solution

The search process of the FA involves finding a certain number of neighbours of the current solution and checking their fitness values in each iteration. A neighbour solution is generated by randomly updating the value of one of the variables of the current solution within the bounds of that variable. Suppose the optimization problem has $n$ variables. The current solution is $X=x_1,...,x_k,...,x_n$ and the variable $x_k$ (whose lower and upper bounds are $l{b}_k$ and $u{b}_k$) is selected for variation. The new value of the selected variable is randomly determined by using Eq. (4) and the neighbour solution is obtained as seen in Eq. (5).

(4) $$x_k^{\mathrm{\prime }}=l{b}_k+r\times (u{b}_k-l{b}_k)$$

(5) $$S_{new}:[x_1,...,x_k^{\mathrm{\prime }},...,x_n]$$

Search process

The search mechanism of the FA is based on the two rules of flood movement. It is designed to search the solution space by scanning a predetermined number of neighbour points as defined in Eqs. (4) and (5) of the current solution in each iteration. The current solution $S_{cur}$ is used as the reference point in each iteration and updated at the end of each iteration. When a neighbour with a better (smaller) fitness value than the current solution is found in an iteration as seen in Fig. 2A, it stops scanning the neighbours and continues searching the solution space over that point (modeling the flow of water towards lower elevations) in the next iteration as seen in Eq. (6).

(6) $$\begin{array}{r}{S}_{cur}=S_i,f({\mathbf{S}}_{\mathbf{i}})<f({\mathbf{S}}_{\mathbf{c}\mathbf{u}\mathbf{r}})\\ \\ S_i\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{f}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{t}\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{b}\mathrm{e}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{i}\mathrm{t}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{s}\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\end{array}$$

If a given number of neighbours are scanned and no neighbour with a better fitness value is found as seen in Fig. 2B, in the next iteration it continues to search the solution space over the point with the best fitness value among the neighbours (modeling the flow of water by overflowing from the weakest point of the obstacles it encounters) as defined in Eq. (7).

(7) $$\begin{array}{l}{S}_{cur}=S_i,S_i\in \{S_1\dots S_m\}\wedge f({\mathbf{S}}_{\mathbf{i}}):min\{f({\mathbf{S}}_{\mathbf{1}})\dots f({\mathbf{S}}_{\mathbf{m}})\}\\ m\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{m}\mathrm{a}\mathrm{x}\mathrm{i}\mathrm{m}\mathrm{u}\mathrm{m}\mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{o}\mathrm{f}\mathrm{n}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{b}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{s}\end{array}$$

In both cases, the speed is updated in proportion to the ratio of the fitness values of the current solution and the new solution which is selected as the current solution for the next iteration as seen in Eq. (8).

(8) $$\begin{array}{r}{V}_{cur}^{t+1}=V_{cur}^t\times \frac{f({\mathbf{S}}_{\mathbf{c}\mathbf{u}\mathbf{r}})}{f({\mathbf{S}}_{\mathbf{n}\mathbf{e}\mathbf{w}})},\\ t\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{s}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{c}\mathrm{u}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\end{array}$$

The speed increases when the new solution has a better fitness value and decreases when the new solution has a worse fitness value. Also; as part of the search mechanism, the FA maintains the best solution ( $S_{best}$) found so far by repeatedly updating it during the search process.

Termination

The termination process of FA mimics the stopping of a flood by decreasing its speed in areas where the slope decreases or there is a reverse slope. The FA starts the search from a randomly chosen starting point with an initial speed ( $v_{init}$) of 1. It maintains the speed $v$ by repeatedly updating it at each iteration based on the ratio of the fitness value of the current solution $f(S_{cur})$ and the fitness value of the solution selected for the next iteration $f(S_{n}ew)$ as defined in Eq. (8). If the fitness value of the solution selected for the next iteration is better than the current solution, the speed is increased, and if it is worse, the speed is decreased. The algorithm stops when the speed falls below the minimum speed ( $v_{min}$) of 0.1.

The most commonly used stopping conditions for metaheuristics in the literature are the maximum number of iterations and the maximum number of consecutive iterations without improving the current solution (Corominas, 2023). These conditions can be easily applied to the FA as well as to many other metaheuristics. In particular, the maximum number of iterations is a fair way to compare metaheuristics. Therefore, the maximum number of iterations is used as a stopping condition in the FA to compare it with other algorithms.

Flood algorithm validation

To determine the performance of a metaheuristic algorithm, a sufficient number of experiments and case studies should be performed. As part of the verification of the FA, 15 benchmark functions, three engineering design problems and a real-world problem of preparation of an exam seating plan were used. In order to verify the search capability of the FA, GA, SA, and PSO algorithm are executed on the same benchmark function set, engineering design problems and the real-world problem. The results obtained are compared to the FA. These algorithms were chosen because they are the most widely used algorithms in the literature on optimization problems as shown in Fig. 3.

Benchmark functions

In this subsection, FA is investigated on a set of 15 benchmark functions, eight of which are from the CEC2022 test suite. The range, dimension, type, and formulation of functions are listed in Table 2. The list contains unimodal, multimodal, separable, and non-separable functions with different dimensions. The main goal is to evaluate the performance of the FA by using the difficulty of the different benchmark problems.

Table 2:

Benchmark functions used in experiments.

Function	Range	D	Type	Formulation
Ackley	−32, 32	30	MN	$f(\mathbf{x})=-20exp(-0.2{\sum }_{i=1}^D{x}_i^2)-exp(1/d{\sum }_{i=1}^D\cos 2\pi x_i)+exp(1)+20$
Beale	−4.5, 4.5	2	UN	$f(\mathbf{x})=(1.5-x_1+x_1{x}_2{)}^2+(2.25-x_1+x_1{x}_2^2{)}^2+(2.625-x_1+x_1{x}_2^3{)}^2$
Easom	−100, 100	2	UN	$f(\mathbf{x})=-\cos x_1\cos x_{2}exp(-(x_1-\pi {)}^2-(x_2-\pi {)}^2)$
Michalewicz	0, $\pi$	10	MS	$f(\mathbf{x})=-{\sum }_{i=1}^D\sin x_i{\sin }^{20}i{x}_i^2/\pi$
Quartic	−1.28, 1.28	30	US	$f(\mathbf{x})={\sum }_{i=1}^{D}i{x}_i^4+Rand$
Rastrigin	−5.12, 5.12	30	MS	$f(x)=10D+{\sum }_{i=1}^D[x_i^2-10\cos 2\pi x_i]$
Shubert	−10, 10	2	MN	$f(\mathbf{x})={\sum }_{i=1}^{5}i\cos ((i+1)x_1+i){\sum }_{i=1}^{5}i\cos ((i+1)x_2+i)$
Six hump camel back	−5, 5	2	MN	$f(\mathbf{x})=(4-2.1x_1^2+(x_1^4/3))x_{1}32+x_1{x}_2+(-4+4x_2^2)x_2^2$
Sphere	−100, 100	30	US	$f(\mathbf{x})={\sum }_{i=1}^d{x}_i^2$
Bent Cigar	−100, 100	30	UN	$f(\mathbf{x})=x_1^2+{10}^6{\sum }_{i=2}^n{x}_i^2$
HGBat	−15, 15	30	MN	$f(\mathbf{x})={\left|{\left({\sum }_{i=1}^D{x}_i^2\right)}^2-{\left({\sum }_{i=1}^D{x}_i\right)}^2\right|}^{\frac{1}{2}}+\frac{0.5{\sum }_{i=1}^D{x}_i^2+{\sum }_{i=1}^D{x}_i}{D}+0.5$
Griewank	−100, 100	30	MN	$f(\mathbf{x})=\frac{1}{4,000}{\sum }_{i=1}^n{x}_i^2-\prod _{i=1}^n\cos \left(\frac{x_i}{\sqrt{i}}\right)+1$
Rosenbrock	−10, 10	30	MN	$f(\mathbf{x})={\sum }_{i=1}^{D-1}\left(100{(x_{i+1}-x_i^2)}^2+{(x_i-1)}^2\right)$
Levy	−10, 10	30	MN	$f(\mathbf{x})={\sin }^2(\pi x_1)+{\sum }_{i=1}^{n-1}{(x_i-1)}^2[1+10{\sin }^2(\pi x_{i+1})]+(x_D-1{)}^2$
Zakharov	−5, 10	30	UN	$f(\mathbf{x})={\sum }_{i=1}^n{x}_i^2+{\left(\frac{1}{2}{\sum }_{i=1}^{n}i{x}_i\right)}^2+\left(\frac{1}{2}{\sum }_{i=1}^{n}i{x}_i\right)$

DOI: 10.7717/peerj-cs.2278/table-2

Note:

D, dimension; M, multimodal; U, unimodal; S, separable; N, non-seperable.

Experimantal results

For benchmark functions, the FA, GA, SA, and PSO algorithm were run 100 times on a 64-bit computer with 16 GB RAM and Intel Core i5 CPU, with the parameter values given in Table 3. To ensure the fairness of the experiment, the population-based GA and PSO algorithm were run for 100 generations/cycle on a population of 100 individuals which means 10,000 function evaluations, and the trajectory-based FA and SA were run for 10,000 steps which also means 10,000 function evaluations. The algorithms were compared in terms of fitness value and convergence performance on benchmark functions.

Table 3:

Parameter values of algorithms.

Algorithm	Parameter	Value
FA	Neighbour count	30
SA	$k_{max}$	10,000
PSO	c1	2
	c2	2
	Inertia weight	0.8
GA	Crossover rate	0.8
	Mutation rate	0.05

DOI: 10.7717/peerj-cs.2278/table-3

The minimum, maximum, average, and standard deviation of fitness values obtained for each algorithm are given in Table 4. When the results are analyzed, it is seen that the FA produces better results than the other algorithms in 31 out of 60 different comparison areas (especially on multi-dimensional functions), including best, worst, average, and standard deviation for each benchmark function. In addition to these comparisons, the Wilcoxon Ranksum Test (Wilcoxon, 1992), a non-parametric statistical test, was used to evaluate the results statistically. The ranks and p-values (for statistically significance level at $\alpha$ = 0.05) of the results obtained by the FA against the results obtained by the other three algorithms for 15 benchmark functions are presented in Table 5. According to the Wicoxon Ranksum Test results, the FA outperformed the GA in 12 out of 15 functions, the PSO in 11 out of 15 functions, and the SA in nine out of 15 functions.

Table 4:

Comparative results of FA with GA, PSO, and SA (bolded numbers represent the best values).

Function		FA	GA	PSO	SA
Ackley	Best	0.630264721	0.030177472	6.037359303	0.615644664
	Worst	1.081308917	9.852174721	15.00098476	1.890181228
	Mean	0.840619763	4.827023494	9.887241081	1.21125692
	StdDev	0.102621848	2.458913203	1.485300744	0.284202246
Beale	Best	8.1218E−09	1.50587E−05	5.32812E−15	2.57522E−06
	Worst	0.763954554	0.017210748	0.735040539	9.542461754
	Mean	0.243985461	0.004337761	0.007350405	0.646968955
	StdDev	0.357440055	0.003958047	0.073504054	1.522015766
Easom	Best	−0.999984745	−0.999733691	−1	−0.999989072
	Worst	−8.0573E−05	−6.48809E−05	−0.999999997	−5.71945E−09
	Mean	−0.559012012	−0.486018129	−1	−0.895178564
	StdDev	0.497941702	0.336835486	4.53634E−10	0.295969316
Michalewicz	Best	−9.655901223	−5.42823829	−9.035137456	−9.657388029
	Worst	−9.628734725	−3.154691032	−5.024521985	−9.590041807
	Mean	−9.644992901	−4.157626983	−6.958815078	−9.641043957
	StdDev	0.004907447	0.387775307	0.805336095	0.013817348
Quatric	Best	7.11854E−07	1.04821E−06	0.033055868	0.000210782
	Worst	1.43489E−05	3.265309166	1.620159254	0.001263889
	Mean	6.29445E−06	0.41820625	0.36219887	0.000661741
	StdDev	2.80352E-06	0.719218	0.287628592	0.000220706
Rastrigin	Best	0.924309115	0.044107217	70.95627643	1.041741143
	Worst	2.358124187	38.83575726	197.9559435	5.304766346
	Mean	1.526722464	13.23619498	116.5297821	2.621956417
	StdDev	0.317304598	8.777635217	24.5573117	0.834267081
Shubert	Best	−186.7308897	−186.7248143	−186.7309088	−186.7309027
	Worst	−186.7163675	−177.2558732	−184.7264808	−186.6764707
	Mean	−186.7289436	−185.1291998	−186.7108645	−186.7263416
	StdDev	0.002344217	1.682359271	0.200442803	0.007494791
Sixhump camelback	Best	−1.031628452	−1.031470545	−1.031628453	−1.031628351
	Worst	−1.031566413	−0.980048867	−1.031628452	−1.031261734
	Mean	−1.031617878	−1.022595193	−1.031628452	−1.031582382
	StdDev	1.18116E−05	0.009334517	4.09659E−12	6.57447E−05
Sphere	Best	1.88159E-05	0.005840427	622.3560198	1.803937595
	Worst	0.007377774	4,378.884757	8,637.558451	5.282826775
	Mean	0.001233648	798.8495789	2,180.258882	3.533097216
	StdDev	0.00141772	901.084707	1142.749793	0.734131596
Bent Cigar	Best	1,662,639.001	52,208.38673	306,655,833.9	1,852,577.25
	Worst	11,315,660.86	3,904,504,464	3,968,514,125	12,610,546.22
	Mean	5,132,787.392	718,352,211.1	1,836,472,254	5,456,679.957
	StdDev	1,991,141.08	705,159,229.7	790,638,188.1	2,221,453.951
HGBat	Best	0.273985563	0.580532757	11.37543708	0.309354292
	Worst	1.490868581	121.6640925	115.6925306	1.248709457
	Mean	0.680847783	24.20371437	45.2520308	0.622166672
	StdDev	0.355691216	25.01670369	20.00026061	0.292947893
Griewank	Best	0.070946968	0.010918879	1.181959635	0.171126239
	Worst	0.444453243	2.303547801	2.288771959	0.736621856
	Mean	0.234341341	0.717726596	1.511488535	0.387217392
	StdDev	0.069850661	0.415042013	0.228033621	0.092241659
Rosenbrock	Best	4.984522855	29.01762542	75.22963067	7.060738999
	Worst	141.1286931	373.1991982	236.5369623	141.876423
	Mean	57.07679761	69.06191623	133.5588939	62.7984998
	StdDev	31.50056937	50.00424247	35.83975214	37.44949298
Levy	Best	0.030788386	2.710117694	2.552523385	0.008846786
	Worst	0.141272967	9.097557631	29.71448051	0.057414966
	Mean	0.069551636	4.171587424	10.68035128	0.026914693
	StdDev	0.022433369	1.281362725	5.429156246	0.010534223
Zakharov	Best	155.825532	0.046912753	236.0268778	166.4557122
	Worst	538.0403138	81.9534772	945.6585919	565.4200336
	Mean	332.1240272	26.33974406	494.4301908	339.3193892
	StdDev	75.9756031	17.76458461	147.9450299	76.08398062

DOI: 10.7717/peerj-cs.2278/table-4

Table 5:

Wilcoxon rank-sum test results.

	FA vs. GA			FA vs. PSO			FA vs. SA
Function	R+	R−	p-value	R+	R−	p-value	R+	R−	p-value
Ackley	14,370	5,730	4.86E−26	15,050	5,050	2.56E−34	13,972	6,128	9.54E−22
Beale	11,832	8,268	1.34E−05	5,118	14,982	1.95E−33	12,693	7,407	1.07E−10
Easom	10,705	9,395	1.10E−01	5,050	15,050	2.56E−34	9,571	10,529	2.42E−01
Michalewicz	15,050	5,050	2.56E−34	15,050	5,050	2.56E−34	10,259	9,841	6.10E−01
Quatric	14,771	5,329	8.89E−31	15,050	5,050	2.56E−34	15,050	5,050	2.56E−34
Rastrigin	14,621	5,479	5.88E−29	15,050	5,050	2.56E−34	14,095	6,005	4.97E−23
Shubert	15,045	5,055	2.98E−34	5,150	14,950	5.02E−33	11,162	8,938	6.60E−03
Six hump camel back	15,050	5,050	2.56E−34	5,050	15,050	2.56E−34	12,607	7,493	4.20E−10
Sphere	15,048	5,052	2.72E−34	15,050	5,050	2.56E−34	15,050	5,050	2.56E−34
Bent Cigar	14,758	5,342	1.29E−30	15,050	5,050	2.56E−34	10,475	9,625	3.00E−01
HGBat	14,609	5,491	8.18E−29	15,050	5,050	2.56E−34	10,056	10,044	9.89E−01
Griewank	13,985	6,115	7.01E−22	15,050	5,050	2.56E−34	14,241	5,859	1.33E−24
Rosenbrock	10,656	9,444	1.39E−01	14,743	5,357	1.96E−30	10,338	9,762	4.82E−01
Levy	15,050	5,050	2.56E−34	15,050	5,050	2.56E−34	5,250	14,850	9.26E−32
Zakharov	5,050	15,050	2.56E−34	13,394	6,706	3.10E−16	10,298	9,802	5.45E−01

DOI: 10.7717/peerj-cs.2278/table-5

Note:

Bold values represent the values better than the level of significance $\alpha$ = 0.05.

The convergence comparison graphs of the algorithms are given in Fig. 4. These graphs clearly show the better final result as well as the better convergence trend of the proposed FA compared to GA, SA, and PSO.

Another investigation on the benchmark functions was the search patterns of the flood algorithm. The search patterns of the flood algorithm on the solution space of the benchmark functions are given in Figs. 5 and 6. In the graphs in Figs. 5 and 6, the number of neighbours, which is the only parameter of the FA that needs to be determined in advance, is considered as 30 and the benchmark functions are considered as two-dimensional. It can be seen that the flood algorithm has a successful search propagation around the local and global minimum points of the functions. When the effect of the number of neighbours on the search propagation is examined, it is seen that as the number of neighbours increases, as expected the search propagation becomes sharper towards the regions with minimum points, as seen in Fig. 7. Increasing the number of neighbours leads to better results, but increases the execution time of the algorithm at the same rate. The graphs in Fig. 7 show the results of the analysis on two benchmark functions (Rastrigin and Michalewicz) for three different numbers of neighbours (10, 30 and 100).

Conclusion

Optimization problems are problems that we encounter in many different fields such as engineering, finance, and health. It is often not feasible to produce solutions to these problems in polynomial time using deterministic algorithms, due to reasons such as the size of the solution set increasing exponentially according to the number of variables in the objective function and/or the solution space not being convex. In recent decades, many studies have been carried out proposing metaheuristic algorithms that produce intuition-based solutions to solve these optimization problems within an acceptable time frame. The works are largely inspired by natural phenomena. The applications of these highly flexible algorithms in the modeling of complex optimization problems in the engineering world are increasing. In this article, a new meta-heuristic optimization algorithm, called the Flood Algorithm, is proposed. FA is a trajectory-based algorithm inspired by the movement of flood waters on the ground. By scanning the neighbouring points of a randomly determined starting point, when it finds a neighbour with a better fitness value than itself, it passes through that point (modeling the flow of water towards lower points), and when a predetermined number of neighbours are scanned and a neighbour with a better fitness value cannot be found, it passes between neighbours. It is designed to scan the solution space over the point with the best fitness value (modeling the passage of water overflowing from the weakest point of the obstacles it encounters). Experimental results show that the proposed algorithm has stronger search and convergence ability than GA, SA and PSO in most of the comparisons. Currently, the authors are working to compare FA with a larger number of metaheuristic algorithms on a larger pool of problems. The authors look forward to seeing applications of FA on different real-world problems in the future.

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