1. Introduction
In many applications, such as in engineering, businesses and industrial designs, optimization is extremely important. Many researchers ask a common question: there are so many optimization algorithms, so what is the best?
It is a simple question, but unfortunately, there is no simple answer. We cannot answer this question simply for several reasons. One reason is that the complexity and diversity of problems in the real world often make it easier to solve some problems, whereas others can be extremely difficult. Consequently, a single method is unlikely to solve all types of problems. Another reason is due to the so-called no free lunch (NFL) theorem, which reads that no universal algorithm exists for all problems [
1].
This theorem states that: in the search for an extremity of an objective function, if any algorithm A surpasses another algorithm B, then algorithm B surpasses other objective functions. In general, the NFL theorem applies to the scenario of either deterministic or stochastic parameters, where the objective or cost function can be defined using a set of continuous (or discrete or mixed) parameters [
1].
The aim of every optimization solution involves at least some of the following: to reduce energy and costs and to maximize profit, output, performance and efficiency. Metaheuristic algorithms are currently turning out to be incredible techniques for solving numerous complex issues using streamlining [
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13]. By far, most heuristic and metaheuristic algorithms were inspired by the behavior of biological system frameworks or potential frameworks in nature. There are several different types of algorithms; some of the popular and modern algorithms are discussed below.
The beginning of the study of metaheuristic algorithms was developed in the thesis of [
2] in 1992, where he proposed an algorithm that was inspired by an ant colony (AC). This search technique was inspired by the swarm intelligence of social ants using a pheromone as a chemical messenger.
In 1995, ref. [
3] proposed an algorithm called particle swarm optimization (PSO), which was created based on the swarm behavior of birds and fish. The multiple-particle swarm moves in the search space, starting from some initial random guess. The swarm gives information about the current best solution. Then, to focus on the high-quality solutions, they share the global best solution.
In 2006, ref. [
4] presented a novel numerical algorithm for stochastic optimization that is based on weeds. In a simple but successful optimization algorithm known as invasive weed optimization (IWO), the robustness, adaptation and randomness of the colonizing weeds is tested.
In 2009, ref. [
5] proposed an algorithm called the firefly algorithm (FA), which is based on the behavior of firefly flashlights in the summer night’s sky. Then, they provided a comparison study between the FA and PSO algorithms.
In 2010, ref. [
6] proposed an algorithm called the cuckoo search algorithm (CS), which is based on some cuckoo species that engage in brood parasitism.
In 2010, ref. [
7] proposed a novel nature-inspired metaheuristic algorithm called the bat algorithm (BA). This algorithm is based on microbats’ behavior, which involves using echolocation for finding certain directions.
In 2012, ref. [
8] proposed an algorithm called the flower pollination algorithm (FPA). Using another domain for metaheuristic inspiration, this algorithm is based on the pollination behavior of plants and types of pollination spreads in fields.
In 2013, ref. [
9] presented research on complex system modeling and calculations using a novel biologically inspired approach known as the root growth algorithm (RG). This general model of optimization gleaned ideas from the behaviors of root growth in soil.
In 2016, ref. [
10] proposed a novel algorithm for nature-based metaheuristic optimization named the whale optimization algorithm (WOA), which mimics humpback whales’ social behavior. The bubble-net hunting strategy was the inspiration of the algorithm.
The bio-inspired computational technique for geometrically optimized joints of a compact coplanar waveguide (CPW)-fueled microstrip antenna with a defective ground structure was introduced in 2017 by [
11] and is known as the adaptive bacterial foraging optimization (ABFO). The ABFO was compared with the original (BFO) technique, PSO, the invasive weed optimization technique, and the artificial bee colony (ABC) to check its adequacy.
In 2018, ref. [
12] proposed a modern model for optimization that was inspired by nature called the squirrel search algorithm (SSA). This optimization algorithm imitates the southern flying squirrel’s complex foraging behavior and its effective method of transport known as gliding. The proposed algorithm mathematically modeled this behavior to realize the optimization process. In terms of the efficiency of the proposed SSA, the statistical analysis, convergence rate analysis, Wilcoxon test and ANOVA were evaluated with respect to the classic and modern CEC 2014 benchmarks. The performance of the SSA over other popular optimization techniques with regard to the optimization accuracy and convergence rate was demonstrated in an exhaustive comparative analysis.
In 2019, ref. [
13] proposed a new modified algorithm with a high calculation speed and simplified camel-based structural optimization (modified CA). The results showed that the modified camel algorithm is preferable when compared with particle swarm optimization (PSO) and the crow searching algorithm (CSA).
In 2020, the smart flower optimization algorithm (SFOA) was proposed in [
14] to provide a new kind of nature-inspired optimization algorithm. There were two modes for the proposed algorithm: sunny and cloudy or snowy, depending on weather conditions. For the testing of SFOA’s efficiency using statistical analysis and Wilcoxon’s test, a collection of 15 benchmarking features in the CEC 2015 was used. For the design of a system of adaptive IIR to adapt to an unknown system, SFOA was used.
Each of these algorithms has specific weaknesses and focal points. For example, simulations annealing can almost guarantee that the ideal arrangement is found if the cooling process is moderate and the simulation is sufficiently long [
15].
In this study, another metaheuristic technique was proposed, known as the moss rose optimization algorithm (MROA), which is based on the flowering of this type of plant. In addition to other factors, such as day length and daytime temperature, the ability of the moss rose to flower at some time of day depends mainly on light factors. In the rest of the paper, the moss rose optimization algorithm for optimizing the plant’s floral diameters is proposed. A brief overview of the algorithm is then given, and a comparison with other algorithms is provided to show that the proposed algorithm works correctly.
2. Moss Rose Optimization Algorithm
In this part, the proposed plant algorithm’s inspiration is considered. Then, the mathematical model of the algorithm is presented.
2.1. Inspiration
The inspiration for the proposed approach was the flowering behavior of the moss rose. The scientific name of this plant is Portulaca grandiflora Hook. This plant is classified as a long-day plant. The most important factor that activates the flowering is exposure to red light. The light interacts with photoreceptors (phytochromes) to make a special protein called florigen.
A phytochrome is a multiform pigment that is capable of absorbing red light. These pigments absorb light in a very narrow and specific spectral range. The phytochromes are the first step in providing information about the red phytochrome (Pr) and far-red phytochrome (Pfr) levels in a signaling system, leading to developmental changes in gene expression. There are two convertible forms of the phytochrome molecule: first, Pr absorbs red light at a 660 nm wavelength, and second, Pfr absorbs 730 nm long-red light. The light from the Sun contains more red light than far-red light. Pr is biologically inactive; when red photons are present, it is transformed into the active form, Pfr. When far-red photons are available, Pfr is transformed back into Pr. In other words, if Pfr is present, there will be biological reactions affecting the phytochromes. The reactions cannot occur if most of the Pfr has been replaced by Pr [
16], as shown in
Figure 1a.
The phytochromes are located in the cytoplasm but enter the core to allow light-response genes to be transcribed. It can pass into the nucleus when Pr is converted to Pfr. Once inside the nucleus, Pfr binds to other proteins to form a transcription complex, which leads to the production of light-controlled genes. This operation is shown in
Figure 1b [
16].
From the information above regarding the biological mechanism, we see that light is the flowering signal of moss roses. Light activates photoreceptors and triggers signal cascades in plant cells of apical or lateral meristems [
17].
2.2. Mathematical Model
The moss rose optimization is a population-based optimization algorithm. For the proposed algorithm, the search space can be modified based on the mechanisms of flowering in the plant. In this algorithm, it is assumed that each moss rose has the ability to produce flowers in a dimensional search space.
Since the moss rose algorithm depends on the population, a population of a flowering cluster can be represented in a matrix as follows:
where
M denotes the number of flowers in a plant and
Dim denotes the number of variables (dimension). In the proposed algorithm, the diameter of each flower of a moss rose leads to a random solution to the optimization problem. Each flower has a fitness value that depends on the fitness function value of the optimization problem that represents the diameter of its flower. A better fitness value represents the diameter of a larger flower. New flower diameters (solutions) allow the algorithm to predict and prepare to complete their flowering during the same day in the decision space based on internal mechanisms. The definitions of the properties of the MROA are represented in
Table 1.
The mathematical model that was used to simulate the flowering mechanisms of a moss rose is presented in the following equations:
Create random variables that represent the flowering diameters:
where
—the diameters of roses between max and min values.
Generate the flowering age parameter, which depends on the current diameter and the maximum diameter that the flower will reach. The equation for the
is
where
—maximum flower diameter and
—total hours in a day.
Generate the phytochrome parameter, which depends on the max flowering diameter, the number of hours the flowers are open, a random time in the morning and the minimum flowering diameter:
where
BF—biological opening factor (controls the clock time of flowering; the value is 3),
SF—scaling factor (to ensure the maximum reaction of the rose to light effect; the value is 2.7),
—maximum number of hours the flowers are open and
—minimum flower diameter.
Calculate the new
fd according to the following equation:
where Pr—red photon wavelength (nm) and
—best flowering diameter, which is calculated based on the fitness function.
Any flower on the plant can update its diameter according to the random sunlight receptors in response to the composition of the ‘phytochrome’ in the apical meristem without the other. Therefore, the same concept can be extended with a dimensional search space by changing the exponential morning clock function ‘clock’, which is chosen randomly between 7 a.m. and 3 p.m.
In order to have a clear visualization of the work of the algorithm, the basic work of the two equations mentioned in steps 2 and 3 involves adding the values that control the improvement of the random variable. As for step 4, this is the basic equation for updating the randomly imposed value and it contains the values of the variables that were generated in the previous two steps.
The moss rose will close its flowers during the night and for several hours of the day, and its flowers will bloom during the rest of the day. This cyclic prototype permits a moss rose to be repositioned around another solution. This can guarantee that the intensification of the space is characterized by two arrangements. To diversify the inquiry space, the flowers should have the option to look outside the local space of the best arrangements they are comparing. This can be achieved by changing the scope of the phytochrome during a 24 h day/night cycle, as shown in
Figure 2.
In stochastic algorithms, a calculation should have the option to adjust between intensification and diversification stages to compute the promising regions of the pursuit space and eventually join to produce the global optimum. For adjusting the intensification and diversification, the flowering diameter of the roses was decreased during successive iterations. It is shown in
Figure 1 and
Figure 2 that the MROA investigates the pursuit space when the phytochrome is enacted and the organic clock of the moss rose works regularly.
- 5.
The MROA finalizes the optimization process when evaluating the maximum number of fitness functions or obtaining the accuracy of the global optimum. The pseudocode of the MROA is illustrated in
Figure 3. The flowchart of the MROA is shown in
Figure 4
3. Computational Results of Benchmark Functions
To check the efficiency of the proposed MROA, a total of 18 benchmark functions were implemented. Two types of functions composed these benchmarks:
- (a)
Unimodal functions for benchmarking;
- (b)
Functions for multimodal benchmarks.
The experiments were carried out using the R2019a version of MATLAB software and Windows 10 Pro N on a laptop with an i7 Core 2.4 GHz processor, a 256 Gb SSD and 8 Gb RAM.
Table 2 lists the global/local benchmarking functions’ names, measurements, attributes and the associated values. Details of the 18 benchmark functions are given in [
18,
19].
Three well-known search algorithms, namely, the crow searching algorithm (CSA) [
20], the modified camel algorithm (MCA) [
17] and particle swarm optimization (PSO) [
3], were used as tests of MROA’s benchmarking functions. The maximum population size and the iteration values were set to 50 and 5000, respectively, for all algorithms to ensure that the comparison was fair. All algorithms in
Table 3 had the default parameters. For the proposed MROA, the default parameter values of maximum diameter, minimum diameter, Pr and daylight hours were set to 4, 0.5, 660 nm and [
7,
15], respectively. Each algorithm was run 20 times for every function to evaluate the algorithms using the benchmark functions. The results of the benchmark functions are shown in
Table 4. For each algorithm, the minimum value of the functions is called ‘Min. Value’, the average solution for a given function after 20 implementations is called ‘Mean’, the standard function deviation for each algorithm is ‘Std. Dev.’ and a bold number represents the best mean for each benchmark function for every single algorithm. The algorithms were ranked according to the smallest mean solution. It is easy to detect that MROA produced much better results than the compared algorithms based on the results of the mean values illustrated in
Table 4.
It can be concluded from
Table 4 that the MROA algorithm found the best solution (global optimum) for all benchmark functions. The multimodal benchmark functions illustrated the second part of the functions testing, which contained seven functions. Given the difficulty of solving these functions, this means that the global searchability test is hard for any algorithm [
19]. It is shown in
Table 4 that the MROA ranked first for all functions (Colville Function, Easom Function, Quartic Function, Quartic with Noise Function, Schwefel 2.36 Function, Griewank Function and Schaffer 6 Function) based on the mean of each solutions. These results show that the MROA provided better results than the other algorithms.
As displayed above, the benchmark functions were classified into two parts. The first part (unimodal) contained eleven functions, where MROA had the best mean values for eight of the eleven unimodal functions, (bold numbers in the
Table 4), the eight unimodal functions are: Acklay2 Function, Dixon and Price Function, Matyas Function, Powell Sum Function, Elliptic Function, Rosenbrock Function, Step 2 Function and Sum Square Function, but had the second-best mean values for three functions (Beale Function, Leon Function and Schwefel 2.22 Function).
5. Conclusions
In this study, a new algorithm known as the moss rose optimization algorithm (MROA), which was inspired by the flowering process of the moss rose, was suggested. During the day, a moss rose opens toward the sunlight. For the flowering of plants, proteins, such as florigen, play a major role. Florigen is released to allow for plant flowering with various diameters at different points of the light cycle. The flowers are opened several hours a day and are then closed in preparation for the next day to re-adjust the diameters. The main structure of the algorithm is as follows:
Generate random variables.
Identify the values of the variables that produced the best solution to the calculated fitness function.
Generate a new variable representing the lifetime of the variable.
Generating a second variable that represents the extent of the influence of natural factors upon updating the variable data.
Update the variables and compare the results with the previous results to obtain convergence toward a better outcome.
The proposed MROA was tested for extremely demanding modern requirements. The results of the benchmark function tests showed that the proposed algorithm was better than other algorithms, especially for multimodal functions, as illustrated in
Table 4. In addition, these tests showed that the MROA balanced the exploitation and exploration stages with the results well (especially for multimodal functions) in comparison to other algorithms, namely, the crow search algorithm (CSA), the modified camel algorithm (MCA) and particle swarm optimization (PSO).
MROA’s success was also confirmed by optimizing restricted selected engineering applications, namely, an anti-jamming smart antenna as an online problem and the computation of microstrip antenna dimensions for a 2.4 GHz frequency as an offline problem.
For the online problem, the algorithm perfectly removed jamming signals by nullifying them. The advantage of this algorithm in this application came from the algorithm speed when finding the results.
For the offline problem, the algorithm found the optimal dimensions that operated at a target resonant frequency despite the complexity of the system; the other algorithms did not give accurate results for this application.