This paper presents a novel bio-inspired optimization algorithm called Coronavirus Optimization Algorithm (COVIDOA). COVIDOA is an evolutionary search strategy that mimics the mechanism of Coronavirus when hijacking human cells. COVIDOA is inspired by the frameshifting technique used by the Coronavirus for replication. The proposed algorithm is tested using 20 standard benchmark optimization functions with different parameter values. Besides, we utilized 5 IEEE Congress of Evolutionary Computation (CEC) benchmark test functions (CECC06, 2019 Competition) and 5 CEC 2011 real-world problems to prove the proposed algorithm's efficiency. The proposed algorithm is compared to eight of the most popular and recent meta-heuristic algorithms from the state-of-the-art in terms of best cost, the average cost (AVG), corresponding standard deviation (STD), and convergence speed. The results demonstrate that COVIDOA is superior to most existing meta-heuristics.
Research Article
COVIDOA: A Novel Evolutionary Optimization Algorithm Based on Coronavirus Replication Lifecycle
https://doi.org/10.21203/rs.3.rs-1592094/v1
This work is licensed under a CC BY 4.0 License
Version 1
posted
You are reading this latest preprint version
Coronavirus
optimization
frameshifting
best cost
convergence
evolutionary algorithm
Nature is full of principles and mechanisms that inspire scientists to develop complex computational problems (Darwish, 2018). Researchers developed various renature-inspired algorithms such as Genetic Algorithm (GA) (Holland 1992) and Differential Evolution (DE) (Rocca et al., 2011) over the years. These algorithms are based on the theory of natural evolution. Another group of Algorithms mimics the behavior of birds, animals, insects, plants, or fish such as Particle Swarm Optimization (PSO) (Kennedy & Eberhart, 1995), Artificial Bee Colony (ABC) (Karaboga, 2005), Chicken Swarm Optimization (CSO) (Meng et al., 2014), Flower Pollination Algorithm (FPA) (Yang, 2012), Grey Wolf Optimization (GWO) (Mirjalili et al., 2014), Whale Optimization Algorithm (WOA) (Mirjalili & Lewis, 2016), Cuckoo Search (CS) (Yang & Deb, 2009), Bird Mating Optimizer (Askarzadeh, 2014), Social Spider Optimization (SSO) (James & Li, 2015), Krill Herd (Gandomi & Alavi, 2015), and Seagull optimization algorithm (SOA) (Dhiman & Kumar, 2019). Other algorithms based on physical phenomena such as Water Cycle Algorithm (WCA) (Eskandar et al., 2014), Central Force Optimization (CFO) (Formato, 2007), Gravitational Search Algorithm (GSA) (Rashedi et al., 2009), Water Wave Optimization (WWO) (Zheng, 2015), and Gradient-based optimizer (GBO) (Ahmadianfar et al., 2020). Many other optimization algorithms are proposed by (Martínez-Álvarez et al., 2003; Das et al., 2009; Brooks & Morgan, 2009; Rashedi et al., 2010; Birbil & Fang, 2003; Pál, 2006; Kaveh & Talatahari, 2010; Shah-Hosseini, 2009; Hsiao et al., 2005; Sacco & Oliveira, 2005; Moscato et al., 2004; Rechenberg, 1989; Simon, 2008; Baluja, 1994; Nordin et al., 1998; Brameier & Banzhaf, 2007; O'Neill & Ryan, 2001; Andre et al., 2001 ).
Generally speaking, optimization algorithms are classified into three categories: swarm-based, physics-based, and evolutionary algorithms. Swarm-based algorithms such as ABC, PSO, CSO, and CS, mimic how a group of agents would behave with each other and their environment (Ab Wahab et al., 2015). Physics-based algorithms are based on a mathematical idea or physical process such as CFO and GSA, based on Newton's gravitational law (Alatas & Can, 2015). On the other hand, evolutionary algorithms are search methods inspired by biological evolution mechanisms, such as reproduction and mutation (Yu & Gen, 2010). The most popular evolutionary algorithm is GA, which is inspired by Darwin's theory of biological evolution. As mentioned in (Fogel, 1997), evolutionary algorithms have some advantages over other types of optimization algorithms, such as:
(1) They are conceptually simple: all evolutionary algorithms have similar necessary steps: initialization, fitness evaluation, selection, crossover, and mutation.
(2) In evolutionary algorithms, the individuals with the highest fitness are selected for reproduction, leading to new individuals' production closer to the optimum solution.
(3) Broad applicability: researchers can apply evolutionary algorithms to any problem formulated in the form of an optimization function. A list of the most popular nature-inspired algorithms shown in Fig. 1.
Since 2020, the world suffers from the pandemic of coronavirus disease 2019 (COVID-19). Researchers worldwide are doing their best to understand this novel virus's mechanism and find an effective therapy for this disease (Kang et al., 2020). More than one researcher discussed the mechanism of the novel Coronavirus from different perspectives in the optimization field. (Martínez-Álvarez et al., 2020) proposed a bioinspired metaheuristic algorithm based on the propagation model of Coronavirus, and the experimental results showed quite remarkable performance of the algorithm. (Al-Betal et al., 2020) proposed an optimization algorithm based on herd immunity's effect in tackling the covid pandemic. The comparative analysis showed that the proposed algorithm yields very competitive results compared to other well-established methods. Another algorithm proposed in (Hosseini et al., 2020) models the coronavirus distribution process as an optimization problem to minimize the number of COVID-19 infected countries and slow down the epidemic.
Once the virus is inside the human body, the most severe problem is replication and transcription, in which new copies of the virus are created and target new healthy cells (Milewska et al., 2020). This paper presents a novel evolutionary optimization algorithm named Coronavirus Disease Optimization Algorithm (COVIDOA).
COVIDOA mimics the attacking behavior of Coronavirus to human cells. It is worth mentioning that almost all kinds of viruses have the same general steps for replication: entry, uncoating, replication, assembly, and virion release. However, replication between viruses is greatly varied depending on the genes involved in them (Domingo et al., 1996).
In addition to the advantages of evolutionary algorithms, COVIDOA has several advantages when compared with other similar mechanisms:
-
Based on the virus's novelty and the lack of research on its various aspects, the reported numerical data about coronavirus lifecycle may not be accurate. Therefore, the proposed algorithm parameters, such as the number of virus particles in each generation and the number of viral proteins generated by each particle, haven't been set fixed values. These reasons give the researchers flexibility to use the extendable values for the controlling parameters that most fit according to their problem.
-
As mentioned in (Bar-On et al., 2020), the mutation rate of Coronavirus is 1×10− 6, which is very low; however, the mutation rate in the proposed algorithm is set a larger value in the range [0.1 0,001], which helps in exploring new promising regions and avoid getting stuck in a local minimum.
-
This study simulates a different virus replication technique known as the frameshifting technique (Kelly et al., 2020; Brian & Baric, 2020; Kelly et al., 2020). The virus's frameshifting technique is used to create more copies of itself and leads to large-scale changes to polypeptide length and chemical composition. It is considered the most harmful for the molecular evolution of human cell proteins resulting in a non-functional protein that often disrupts the biochemical processes of a cell (Rapley & Whitehouse, 2015). Applying the frameshifting technique in the proposed algorithm helps update solutions so that the solutions in each generation will not become too similar, which would allow the algorithm to converge to the global minimum.
The rest of the paper is structured as follows. Section 2 describes the inspiration and mathematical model of the proposed algorithm (COVIDOA). Experiments using test benchmark functions and the obtained results are discussed in Sections 3. Finally, the conclusion and future work of this study is presented in Section 4.
In this section, the inspiration and mathematical models of COVIDOA are presented.
2.1. Inspiration
The new Coronavirus disease (COVID-19) is an infectious respiratory disease caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). SARS-CoV-2 belongs to the Coronaviruses' family, named for crown-like spikes on their surface [51], as shown in Fig. 2.
The Coronavirus consists of a set of genetic instructions inside an oily membrane. These instructions are encoded in 30000 letters of Ribonucleic Acid (RNA) - a, c, g, and u - then read by the infected cell and translated into many types of virus proteins (Bar-On et al., 2020). Human-to-human transmission of SARS-CoV-2 occurs primarily via respiratory droplets from coughs and sneezes. Complications may include acute respiratory distress syndrome (ARDS), multi-organ failure, septic shock, and death (Kelly et al., 2020). Like other Coronaviruses, SARS-CoV-2 (COID-19) has four structural proteins, including the spike (S), envelope (E), and membrane (M) that constitute the viral coat, and the nucleocapsid (N) protein, which encapsulates the viral RNA (Brian & Baric, 2020).
What the virus wants to do is make more copies of itself and send it out to damage more human healthy cells; the virus can do this through a set of stages as follows:
2.1.1. Virus entry and uncoating
For replication, Coronavirus needs to use the human cell's protein-making machinery. So, it first needs to gain entry into the cell. The virus contains a set of proteins around it called spike (S) proteins; it uses its spike proteins as a key to get inside a human cell (Tu et al., 2020; Bergmann & Silverman, 2020). One spike of the virus binds to a protein called angiotensin-converting enzyme 2 (ACE-2) (Shang et al., 2020) on the surface of some human cells, as shown in Fig. 3. Coronavirus has a sort of membrane around it that hides its genetic material from the outside world; human cells have the same membrane that hides their material from the outside world. So, when those two things come together, the virus must find a way to get inside the host cell (Schoeman & Fielding, 2019). Once inside, all of the structural proteins are removed, and the virus contents, which are the genomic RNA, will be released into the host cell cytoplasm. This process is referred to as virus uncoating (Yamauchi & Greber 2016), as shown in Fig. 4.
2.1.2. Virus Replication
Suppose the virus is getting fused in the cell membrane. In the next step, its small genetic material needs to hijack big cellular machinery. It will be a tedious task if the virus has few proteins to hijack the cell. The virus genome starts to find something in the host cell called a ribosome (Ziebuhr, 2020); a ribosome turns the virus RNA into many virus proteins through the ribosomal frameshifting technique (Ivanov & Atkins, 2015), as shown in Fig. 5.
2.1.2.1. Ribosomal frameshifting during genome translation
Ribosomal frameshifting is also known as translational frameshifting, a biological phenomenon that occurs during translation (Ivanov & Atkins, 2015). This phenomenon results from multiple unique proteins from a single messenger RNA (mRNA) molecule (Cobb & Matthew, 2015). To understand this, we need to understand translation and frameshifting separately. The translation is when the mRNA molecule provides information to ribosomes, leading to protein molecules' formation (Ivanov & Atkins, 2015; Jacks et al., 1988). At the same time, frameshifting is a process when a specific reading frame of RNA molecule shifts to another reading frame to provide a new protein sequence (Tu et al., 2020; Shang et al., 2020; Yamauchi & Greber 2016).
In Fig. 6, we have a protein sequence ATG-CCT-GCA-TCA. If we read the sequence from ATG-CCT…, this will be a specific reading frame, and if we read the sequence starting from TGC-CTG…, this will be another reading frame. In simple words, with the same RNA sequence, we can have two unique reading frames depending on how we read the sequence.
On the mRNA strand, proteins are translated by reading tri-nucleotides known as codons (e.g., ATG). Each codon is translated into single amino acid (Moghal et al., 2014). Thus, shifting any number (not divisible by 3) of nucleotides in the reading frame will result in subsequent codons being read differently (Sharma et al., 2020).
In a translating ribosome, a frameshifting can result in either a nonsense mutation (Tu et al., 2020; Sharma et al., 2020) or creating a new protein after the frameshift. The most common types of frameshifting are − 1 frameshifting and + 1 frameshifting (Pascual, 2020).
A. -1 Frameshifting
In − 1 frameshifting, the ribosome slips back one nucleotide (RNA letter) and continues translation in the − 1 frame, as shown in Fig. 7(a).
B. + 1 Frameshifting
The ribosome starts translation from the + 1 frame when 0 is the initial position, as shown in Fig. 7(b). Because of shifting, the sequences are read differently and translated into different proteins.
2.1.2.2. Synthesis of both genomic and subgenomic RNA species
The ribosomal frameshifting technique results in two types of RNAs, genomic RNA and subgenomic RNAs. Genomic RNA is produced through the replication process and becomes the genome of the new virus particle. At the same time, Subgenomic RNAs are translated into many structural proteins (S: spike protein, E: envelope protein, M: membrane protein, and N: nucleocapsid protein). The genomic RNA and subgenomic RNAs are combined to form a viral particle (Pascual, 2020; Khan et al., 2020). Finally, the new virion is released trying to hijack new healthy cells, Fig. 8.
2.1.3. Virus Mutation
As coronaviruses spread from person to person, they randomly accumulate more mutations to escape from the immune system (Khan et al., 2020). Mutations involve changing one or more letters that represent the virus genome. As mentioned in (Bar-On et al., 2020), Coronavirus has lower mutation rates (≈ 10 − 6 per site per cycle) in comparison to influenza (≈ 3×10 − 5 per site per cycle).
2.2. Mathematical model of COVIDOA
In this section, the mathematical model of COVIDOA is provided. COVIDOA are summarized in the following steps:
-
Initialization: population of solutions is randomly initialized, and the cost is evaluated for each solution. The solutions are then ordered ascendingly according to the fitness function, and the first solution is considered the best solution.
-
Virus replication phase through frameshifting technique: for each solution in the population, a parent is selected using roulette wheel selection (Lipowski & Lipowska, 2012) then,
a. Frameshifting technique is applied to produce several proteins from the selected parent as follows:
b. For each protein:
i. If the + 1 frameshifting technique is used, the parent solution's values are shifted to the right direction by 1, and the value in the first position is set a random value in the range [minVal maxVal] as follows.
$${\text{S}}_{\text{k}}\left(1\right)=\text{r}\text{a}\text{n}\text{d}(\text{m}\text{i}\text{n}\text{V}\text{a}\text{l},\text{m}\text{a}\text{x}\text{V}\text{a}\text{l}),$$
1
$${\text{S}}_{\text{k}}(2:\text{D})= \text{P}(1:\text{D}-1),$$
2
Where minVal and maxVal are the minima and maximum values for the variables in each solution.
ii. If the − 1frameshifting technique is used, the parent solution values are shifted backward by 1, and the value in the last position is to set a random value in the range [minVal, maxVal].
$${\text{S}}_{\text{k}}\left(\text{D}\right)=\text{r}\text{a}\text{n}\text{d}(\text{m}\text{i}\text{n}\text{V}\text{a}\text{l},\text{m}\text{a}\text{x}\text{V}\text{a}\text{l}),$$
3
$${\text{S}}_{\text{k}}(1:\text{D}-1)= \text{P}(2:\text{D}),$$
4
The symbol Sk refers to the kth generated protein, P is the parent solution, D is the problem dimension (number of variables in each solution). The result of frameshifting represents a new protein sequence.
c. New virion formation: a uniform crossover is applied to the generated sub proteins to produce a new virion (new solution).
3. Mutation: a mutation operator is applied to the solution created in the previous step to generate a new mutated solution as follows:
Zi=\(\left\{\begin{array}{c}r if rand\left(\text{0,1}\right)<MR \\ {\text{x}}_{\text{i}} otherwise \end{array}\right.\) (5)
X is the solution before mutation. Z is the mutated solution, xi and Zi are the ith element in the old and new solutions, respectively, i = 1, …, D, and r is random value in the range [minVal, maxVal]. MR is the mutation rate.
4. The objective function is evaluated for the new solution, and the population is updated for the next generation (the solutions with the highest fitness remain, and the others are removed).
5. Repeat steps (2–4) for the new population until termination criteria are achieved. For example, the maximum number of iterations is reached.
6. Output the best solution found so far.
The flowchart of the proposed algorithm is shown in Fig. 9.
2.3. parameters of the proposed algorithm:
The parameters of the proposed algorithm are suggested as follows:
-
Max_Iter: maximum number of iterations.
-
PopNo: number of solutions in the population.
-
MinVal and MaxVal: minimum and maximum value of variables in a solution.
-
D: problem dimension (number of variables in each solution).
-
CostFunction: objective function;
-
MR: Mutation Rate, MR is set a value in the range [0.1,0.001].
The pseudocode of the proposed algorithm is as follows:
-
Shifting: a number that represents the type of frameshifting used. For example, shiftingNo = 1 means that the + 1 frameshifting technique is used. We noticed that the + 1 frameshifting technique yields the best results.
-
numOfProtiens: number of proteins generated during virus replication in the proposed algorithm, numOfProteins is 2.
3.1. Benchmark functions
To test the efficiency of the proposed COVIDOA, we utilized 30 benchmark functions. The first 20 are classical standard benchmark test functions (http://benchmarkfcns.xyz). We selected 5 functions from IEEE CEC 2019 Competition (https://www.mathworks.com/matlabcentral/fileexchange/72123-cec-06-2019-matlab implementation), while the remaining 5 are selected from CEC 2011 Competition on Testing Evolutionary Algorithms on Real World Optimization Problems (Das & Suganthan, 2011) as follows:
I. Classical benchmark problems
Twenty standard optimization functions from the literature are discussed and used for testing the efficiency of the proposed algorithm. These functions are classified into 4 groups: unimodal, multimodal, fixed-dimension, and n-dimensional functions (Hussain et al., 2019; Jamil & Yang, 2013). In fixed-dimension problems, the number of design variables (problem dimension) is fixed, while the other n-dimension problems use any design variables. A multimodal function has multiple (at least locally optimum) solutions instead of a unimodal function with a single optimum solution (Hussain et al., 2019). As described in Table 1 in the appendix, the chosen optimization functions are described in terms of the function name, formula, problem dimension (D), range of possible values, the global optimum, and the group of benchmark functions to which it belongs.
II. IEEE CEC 2019 benchmark problems
In addition to the classical benchmark functions, 5 CEC benchmark functions are utilized for evaluation. These are a group of modern test functions known as "The 100-Digit Challenge" and intended to be used in single objective numerical optimization IEEE competitions (Abdullah & Ahmed, 2019). As shown in Table 2 in the appendix, these functions are described in terms of problem dimension, range of possible values, and the global optimum (https://www.mathworks.com/).
III. CEC 2011 Real World Problems
For further evaluation, COVIDOA applied to 5 real-world optimization problems. These are bound-constrained real-world optimization problems selected from the CEC 2011 Competition on Testing Evolutionary Algorithms on Real-World Optimization. These problems are as follows (Das & Suganthan, 2011):
Lennard-Jones Potential Problem.
Transmission Network Expansion Planning (TNEP) problem.
Tersoff Potential Function Minimization Problem for model Si(B).
Tersoff Potential Function Minimization Problem for model Si(C).
Spread spectrum radar polyphase problem.
The detailed description of these real-world problems is discussed in the 2011 IEEE Congress on Evolutionary Computation (IEEE-CEC 2011) (Das & Suganthan, 2011).
3.2. Experimental results
COVIDOA is implemented in MATLAB R2016a software. COVIOA is utilized to solve the previously mentioned test problems. The results are compared with 8 well-known and recent optimization algorithms: GA (Holland, 1992), DE (Rocca et al., 2011), PSO (Kennedy & Eberhart, 1995), FPA (Yang, 2012), GWO (Mirjalili et al., 214), WOA (Mirjalili & Lewis, 2016), SOA (Dhiman & Kumar, 2019), and CHO (Al-Betar et al., 2020). We selected this group of algorithms for many reasons:
-
Most of them are recent and published in reputable sources.
-
All of them have high performance in single-objective optimization on various benchmark functions.
-
Their MATLAB implementations are publicly available on the MATLAB website (https://www.mathworks.com/).
-
Some of them are evolutionary algorithms, such as GA and DE, which are the same category as COVIDOA.
CHIO algorithm simulates Coronavirus, as is COVIDOA, but each one has its inspiration.
In optimization algorithms, the obtained results change at each run due to the random process. The commonly used number of runs is 30, which would give acceptable statistical precision. So, the proposed algorithm and the state-of-the-art algorithms are run 30 times.
The proposed and state-of-the-art algorithms use Max_Iter = 500 and PopNo = 1000 for the classical benchmark functions. The comparison is made in terms of optimum cost, average cost, standard deviation (STD), and convergence speed. The authors downloaded the source code of the state-of-the-art optimization algorithms from the MATLAB website.
Tables (1–4) show the results of the best cost, average cost, standard deviation, and convergence speed, respectively, for the 20 classical benchmark functions. Table 1 and the pie chart in Fig. 10 show that the proposed algorithm reaches the optimum global cost in 18 of 20 problems and gets very close to the global optimum in the 2 remaining problems. Table 2 and the chart in Fig. 11 proved the COVIDOA algorithm's efficiency in terms of the average cost. It reaches the minimum average cost in 17 from 20 problems and the second minimum average cost in 3. The third criterion is STD, which shows how the cost values are far from the average cost. Low STD values mean that the cost values over the iterations are clustered closely around the average cost. Table 3 and Fig. 12 show that the COVIDOA algorithm reaches the minimum STD values in 17 from 20 problems, the second minimum in 2, and the third minimum in 2, which means that the results of COVIDOA are more reliable than the other algorithms with higher STD values.
Compared with the recently proposed algorithm, CHIO, which simulates the effect of herd immunity in tackling covid pandemic, COVIDOA is the best. As shown in tables (1–4) and figures (10–15), CHIO reaches the minimum optimum cost in 7 benchmark functions only from 25; in contrast, COVIDOA reaches the minimum optimum cost in 21 from 25 test functions. This indicates that COVIDOA has robust exploration capabilities in comparison with CHIO.
Compared with PSO, GWO, and WOA, COVIDOA is superior according to the best cost, average cost, and STD values in most of the test problems. It has a higher convergence speed as it reaches the global minimum after the first few numbers of iterations, as is the case in functions (F3, F8, F7, F15, and F16).
| Problem | Algorithm | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| No | Name | GA (Holland, 1992) | DE (Rocca et al., 2011) | PSO (Kennedy & Eberhart, 1995) | FPA (Yang, 2012) | GWO (Mirjalili et al., 214) | WOA (Mirjalili & Lewis, 2016) | CHIO (Al-Betar et al., 2020) | SOA (Dhiman & Kumar, 2019) | Proposed COVIDOA |
| 1 | Dixon-price function | 0.66667 | 0.40228 | 0.6667 | 4.9183 | 1 | 0.6667 | 1.694 | 0.6667 | 0.27378 |
| 2 | Happy Cat Function | 0.1386 | 0.014702 | 0.24166 | 231.478 | 0.0122 | 1.4353 | 0.2691 | 0.005142 | 0.0023146 |
| 3 | Crosslegtable Function | -0.08493 | -0.084778 | -0.07981 | -0.0006630 | -3.869e-04 | -0.0016362 | -2.606e-04 | -2.4310e-04 | -1 |
| 4 | Eggholder Function | -4886.18 | -7445.3819 | -5858.46 | -6292.2901 | -6.006 e + 03 | -6319.4385 | -6385 | -5441.7 | -7825.143 |
| 5 | Stybtang Function | -566.287 | -626.6587 | -626.658 | -530.9072 | -626.086 | -555.9751 | -619.1 | -605.2622 | -626.621 |
| 6 | Schwefel function | -837.965 | -837.9529 | -837.965 | -837.9657 | -837.965 | -837.9658 | -837.9548 | -837.9658 | -837.9658 |
| 7 | Keane Function | -0.67367 | -0.67367 | -0.67367 | -0.67367 | -0.6736 | -0.67367 | -0.6737 | -0.67367 | -0.67367 |
| 8 | Trid Function | -2 | -2 | -2 | -2 | -2 | -2 | -2.0000 | -2 | -2 |
| 9 | Schaffern4fcn Function | 0.2926 | 0.29258 | 0.29258 | 0.29258 | 0.2926 | 0.29258 | 0.2926 | 0.29258 | 0.29258 |
| 10 | Branin Function | 0.39789 | 0.39789 | 0.39789 | 0.39789 | 0.39789 | 0.39789 | 0.4071 | 0.39789 | 0.39789 |
| 11 | Wolfe Function | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
| 12 | Zettl Function | -0.0037 | -0.0037 | -0.0037 | -0.0037 | -0.0038 | -0.0038 | -0.0037 | -0.0038 | -0.0038 |
| 13 | Alpine N. 2 Function | -14320.0 | -23700.87 | -14320.08 | -8649.361 | -2369 | -23700.7978 | -1.7386 | -14277 | -23563.73 |
| 14 | Cross-in-Tray Function | -2.0626 | -2.0626 | -2.0626 | -2.0626 | -2.0626 | -2.0626 | -2.0626 | -2.0626 | -2.0626 |
| 15 | McCormick Function | -1.9105 | -1.9105 | -1.9105 | -1.9105 | -1.9105 | -1.9105 | -1.9105 | -1.9105 | -1.9105 |
| 16 | Gramacy & Lee Function | -2.8739 | -2.8739 | -2.8739 | -2.8739 | -2.8739 | -2.8739 | -2.8739 | -2.8739 | -2.8739 |
| 17 | Testtubeholder Function | -10.8723 | -10.8723 | -10.8723 | -10.8723 | -10.8723 | -10.8723 | -10.8723 | -10.8723 | -10.8723 |
| 18 | Shubert Function | -186.7309 | -186.7309 | -186.7309 | -186.7309 | -186.7309 | -186.7309 | -186.7082 | -186.7309 | -186.7309 |
| 19 | Price 2 Function | 0.9 | 0.9 | 0.9 | 0.9004 | 0.9 | 0.9 | 0.9001 | 0.9 | 0.9 |
| 20 | Dejong5 | 0.998 | 0.998 | 0.998 | 0.998 | 0.998 | 0.998 | 0.9980 | 0.998 | 0.998 |
The curves in Figs. 14 and 15 represent the relationship between the iterations and the corresponding best cost for the classical test functions. The obtained results using the selected test problems are divided into two groups and displayed in Figs. 14 and 15. Figure 14 represents the test problems for which the COVIDOA algorithm outperforms the other algorithms. In contrast, Fig. 15 shows the results of test problems in which the COVIDOA algorithm has a performance very close to the others.
| Problem | Algorithm | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| No | Name | GA (Holland, 1992) | DE (Rocca et al., 2011) | PSO (Kennedy & Eberhart, 1995) | FPA (Yang, 2012) | GWO (Mirjalili et al., 214) | WOA (Mirjalili & Lewis, 2016) | CHIO (Al-Betar et al., 2020) | SOA (Dhiman & Kumar, 2019) | Proposed COVIDOA |
| 1 | Dixon-price function | 15.3545 | 126.5770 | 6.3509 | 1.0998e + 03 | 46.6686 | 30.0319 | 1.0734e + 03 | 9.897e + 03 | 5.23636 |
| 2 | Happy Cat Function | 0.6517 | 0.0445 | 0.2636 | 371.4819 | 0.0802 | 20.4486 | 0.2930 | 0.0477 | 0.0137 |
| 3 | Crosslegtable Function | -0.0683 | -0.0427 | -0.0427 | -0.7909 | -5.1528e-04 | -2.6865e-04 | -2.182e-04 | -0.0047 | -0.8980 |
| 4 | Eggholder Function | -4.70e + 03 | -6.75e + 03 | -5.628e + 03 | -5.681e + 03 | -5.2816e + 03 | -6.2799e + 03 | -5.679e + 03 | -4.262e + 03 | -7.23e + 03 |
| 5 | Stybtang Function | -393.6128 | -619.9509 | -619.2246 | -475.8865 | -577.2454 | -552.6846 | -572.8967 | -594.1131 | -622.7337 |
| 6 | Schwefel function | -835.3788 | -821.9348 | -837.8732 | -837.5112 | -837.5351 | -837.9275 | -835.5825 | -837.6662 | -837.9367 |
| 7 | Keane Function | -0.673659 | -0.673667 | -0.673667 | -0.67359 | -0.673661 | -0.673633 | -0.6736 | -0.673519 | -0.673667 |
| 8 | Trid Function | -1.9999 | -1.9999 | -1.9999 | -2 | -1.9999 | -1.9999 | -1.9996 | -1.9993 | -2 |
| 9 | Schaffern4fcn Function | 0.2928 | 0.2930 | 0.2926 | 0.2930 | 0.2927 | 0.2928 | 0.2961 | 0.2947 | 0.2927 |
| 10 | Branin Function | 0.3980 | 0.3982 | 0.3979 | 0.3984 | 0.3987 | 0.3984 | 0.4673 | 0.4205 | 0.3981 |
| 11 | Wolfe Function | 0.0144 | 1.7214e-04 | 8.5733e-05 | 0 | 3.3785e-04 | 1.4367e-04 | 0.0055 | 3.7236e-04 | 0 |
| 12 | Zettl Function | -0.0038 | -0.0038 | -0.0038 | -0.0036 | -0.0038 | -0.0038 | -0.0028 | -0.0036 | -0.0038 |
| 13 | Alpine N. 2 Function | -1.32e + 04 | -2.114e + 04 | -1.402e + 04 | -5.826e + 03 | -1.2565e + 0 | -2.1515e + 04 | -9.569e + 03 | -2.014e + 03 | -2.18e + 04 |
| 14 | Cross-in-Tray Function | -2.0626 | -2.0626 | -2.0626 | -2.0626 | -2.0626 | -2.0626 | -2.0626 | -2.0626 | -2.0626 |
| 15 | McCormick Function | -1.9105 | -1.9105 | -1.9105 | -1.9105 | -1.9105 | -1.9105 | -1.9103 | -1.9105 | -1.9105 |
| 16 | Gramacy & Lee Function | -2.87389 | -2.87384 | -2.87389 | -2.87389 | -2.87389 | -2.87389 | -2.8739 | -2.87385 | -2.87389 |
| 17 | Testtubeholder Function | -10.8718 | -10.8720 | -10.8721 | -10.8718 | -10.8721 | -10.8717 | -10.8697 | -10.8638 | -10.8721 |
| 18 | Shubert Function | -186.6132 | -186.6495 | -186.6853 | -186.4929 | -186.6285 | -186.6954 | -186.4249 | -186.2621 | -186.7009 |
| 19 | Price 2 Function | 0.90037 | 0.900945 | 0.900233 | 0.902144 | 0.9006 | 0.90033 | 0.9031 | 0.91701 | 0.90004 |
| 20 | Dejong5 | 1.0115 | 1.0065 | 0.9987 | 1.0218 | 1.0122 | 1.0100 | 1.1783 | 1.2333 | 0.9980 |
Additionally, to prove the results' statistical significance, the test results of the 20 classical benchmark functions are compared using Wilcoxon rank-sum test at the %5 significance level (Derrac et al., 2011). A small p-value (typically ≤ 0.05) indicates strong evidence against the null hypothesis (Szucs & Ioannidis, 2017). A null hypothesis is a type of hypothesis used in statistics that assumes there is no significant difference between the two methods' averages values. Table 5 introduces the p values computed by Wilcoxon rank-sum test that compares the COIDOA with eight well-known metaheuristic algorithms for the 20 classical benchmark functions. We observed from Table 5 that all p values are less than a 5% significance level for all comparative algorithms, strong evidence against the null hypothesis. Therefore, we conclude that the COVIDOA is better than all other comparative algorithms.
| Problem | Algorithm | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| No | Name | GA (Holland, 1992) | DE (Rocca et al., 2011) | PSO (Kennedy & Eberhart, 1995) | FPA (Yang, 2012) | GWO (Mirjalili et al., 214) | WOA (Mirjalili & Lewis, 2016) | CHIO (Al-Betar et al., 2020) | SOA (Dhiman & Kumar, 2019) | Proposed COVIDOA |
| 1 | Dixon-price function | 269.8620 | 1.1364e + 03 | 62.4047 | 3.545e + 03 | 894.3655 | 451.6185 | 4.0605e + 03 | 7.220e + 03 | 52.7791 |
| 2 | Happy Cat Function | 0.1139 | 78.6462 | 46.1026 | 109.0887 | 0.0522 | 0.0406 | 0.0390 | 0.2955 | 0.0294 |
| 3 | Crosslegtable Function | 0.0321 | 0.0392 | 0.0357 | 1.557e-04 | 4.8352e-05 | 2.212e-04 | 3.1194e-05 | 2.6201e-05 | 2.8268e-05 |
| 4 | Eggholder Function | 380.0907 | 878.5967 | 426.2801 | 574.4192 | 450.2399 | 236.1962 | 701.8892 | 979.6934 | 425.3867 |
| 5 | Stybtang Function | 36.0487 | 42.5232 | 13.4428 | 45.6925 | 26 | 21.2045 | 49.1493 | 50.9105 | 18.6986 |
| 6 | Schwef Function | 3.5750 | 0.3603 | 0.2791 | 1.9761 | 0.6225 | 2.0103 | 3.2773 | 6.3983 | 0.1706 |
| 7 | Keane Function | 9.276e-05 | 4.1333e-06 | 3.492e-06 | 0.0011 | 1.242e-04 | 7.286e-04 | 7.1552e-05 | 0.0033 | 6.6663e-08 |
| 8 | Trid Function | 0.0015 | 9.1279e-04 | 8.566e-05 | 3.721e-04 | 5.869e-04 | 0.0023 | 7.1631e-04 | 0.0023 | 1.9900e-05 |
| 9 | Schaffern4fcn Function | 9.484e-04 | 0.0033 | 0.0469 | 0.0016 | 6.954e-04 | 0.0031 | 0.0050 | 0.0041 | 5.6588e-04 |
| 10 | Branin Function | 0.0016 | 0.0013 | 3.902-04 | 0.0035 | 0.0063 | 0.0023 | 0.4256 | 0.0246 | 3.6041e-04 |
| 11 | Wolfe Function | 0.0393 | 0.0027 | 0.0019 | 0 | 0.0076 | 0.0032 | 0.0303 | 0.0083 | 0 |
| 12 | Zettl Function | 1.696e-04 | 2.3912e-04 | 6.959e-05 | 0.0011 | 8.467e-04 | 1.704e-04 | 0.0034 | 0.0015 | 1.1646e-04 |
| 13 | Alpine N. 2 Function | 5.807e + 03 | 2.1124e + 04 | 1.247e + 03 | 2.4308e + 03 | 6.94e + 034 | 3.901e + 03 | 5.3061e + 03 | 2.3812e + 03 | 1.7739e + 03 |
| 14 | Cross-in-Tray Function | 3.718e-05 | 2.8873e-05 | 4.8930e-06 | 3.0880e-04 | 3.903e-05 | 3.9988e-05 | 1.4093e-04 | 0.0012 | 4.5473e-06 |
| 15 | McCormick Function | 6.450e-05 | 1.3749e-04 | 1.361e-06 | 8.3451e-05 | 0.0013 | 7.382e-04 | 0.0013 | 0.0041 | 2.6101e-07 |
| 16 | Gramacy & Lee Function | 4.198e-04 | 0.0010 | 1.856e-05 | 7.1457e-06 | 4.162e-04 | 6.207e-05 | 4.8450e-05 | 4.3901e-04 | 4.1554e-08 |
| 17 | Testtubeholder Function | 0.0034 | 0.0018 | 0.0015 | 0.0065 | 0.0038 | 0.0058 | 0.0103 | 0.0207 | 0.0021 |
| 18 | Shubert Function | 0.6984 | 0.5832 | 0.4346 | 1.2720 | 1.2720 | 0.3873 | 0.7193 | 0.7625 | 0.2339 |
| 19 | Price 2 Function | 0.0034 | 0.0068 | 0.0045 | 0.0063 | 0.0054 | 0.0050 | 0.0096 | 0.0375 | 1.4613e-04 |
| 20 | Dejong5 | 0.1339 | 0.1025 | 8.7297e-05 | 0.1555 | 0.1185 | 0.1910 | 0.4304 | 0.7947 | 3.4653e-05 |
| Problem | Algorithms | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
| No | Name | GA (Holland, 1992) | DE (Rocca et al., 2011) | PSO (Kennedy & Eberhart, 1995) | FPA (Yang, 2012) | GWO (Mirjalili et al., 214) | WOA (Mirjalili & Lewis, 2016) | CHIO (Al-Betar et al., 2020) | SOA (Dhiman & Kumar, 2019) | Proposed COVIDOA |
| 1 | Dixon-price function | Moderate | Moderate | Moderate | slow | Moderate | Moderate | Moderate | slow | Moderate |
| 2 | Happy Cat Function | Moderate | Slow | Moderate | slow | slow | slow | slow | slow | Moderate |
| 3 | Crosslegtable Function | Moderate | Moderate | Moderate | slow | slow | slow | slow | slow | Fast |
| 4 | Eggholder Function | Slow | Slow | Slow | slow | slow | slow | Slow | slow | Moderate |
| 5 | Stybtang Function | Slow | Slow | Slow | slow | slow | fast | Fast | slow | Fast |
| 6 | Schwef Function | Fast | Fast | Fast | fast | fast | fast | Moderate | fast | Fast |
| 7 | Keane Function | Fast | Fast | Fast | fast | fast | fast | Fast | fast | Fast |
| 8 | Trid Function | Fast | Fast | Fast | fast | fast | fast | fast | slow | Fast |
| 9 | Schaffern4fcnFunction | Fast | Fast | Fast | fast | fast | fast | fast | moderate | Fast |
| 13 | Alpine N. 2 Function | Slow | Moderate | Slow | slow | moderate | slow | moderate | Slow | Moderate |
| 14 | Cross-in-Tray Function | Fast | Fast | Fast | fast | fast | fast | fast | slow | Fast |
| 15 | McCormick Function | Fast | Fast | Fast | fast | fast | fast | fast | fast | Fast |
| 18 | Shubert Function | Fast | Fast | Fast | fast | fast | fast | fast | fast | Fast |
| 19 | Price 2 Function | Fast | Fast | Fast | moderate | fast | fast | fast | fast | Fast |
| 20 | Dejong5 | Fast | Fast | Fast | fast | fast | fast | fast | moderate | Fast |
CEC benchmark functions, COVIDOA, and state-of-the-art algorithms search for the optimum cost for 250 iterations with 1000 solutions in each generation. The results of the best cost, average cost, and STD values are discussed in Table 6, and the convergence curves are shown in Fig. 16. COVIDOA is superior to the other algorithms in CEC01, CEC06, and CEC01. The CEC03 problem reaches the minimum best cost and the second minimum average cost ad STD value. In the case of CEC07, however, it is not the best; it achieves excellent results compared to GA, FPA, GWO, WOA, SOA, and CHIO algorithms.
All test results for the CEC benchmark functions compared using the Wilcoxon rank-sum test to prove their statistical significance. Table 7 shows the p values computed by Wilcoxon rank-sum test that compares the COIDOA with other well-known algorithms for CEC benchmark functions. It is evident from Table 7 that all p values are less than 5% which proves the statistical significance of COVIDOA.
| Problem | Algorithm | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| No | Name | COVIDOA vs. GA | COVIDOA vs. DE | COVIDOA vs. PSO | COVIDOA vs. FPA | COVIDOA vs. GWO | COVIDOA vs. WOA | COVIDOA vs. CHIO | COVIDOA vs. SOA |
| 1 | Dixon-price function | 2.2242e-06 | 8.0835e-24 | 1.3497e-09 | 1.6207e-129 | 6.6181e-12 | 1.0616e-13 | 4.0517e-134 | 2.2667e-102 |
| 2 | Happy Cat Function | 2.3444e-41 | 2.9609e-142 | 4.6130e-59 | 7.0570e-151 | 7.8828e-78 | 6.2314e-153 | 3.3083e-158 | 2.3994e-168 |
| 3 | Crosslegtable Function | 3.8478e-147 | 2.6216e-149 | 1.2983e-131 | 5.9989e-71 | 8.8060e-112 | 3.0525e-168 | 1.7277e-168 | 2.1944e-5 |
| 4 | Eggholder Function | 1.5930e-08 | 3.5431e-35 | 4.7752e-101 | 6.9415e-107 | 4.2077e-102 | 1.2085e-97 | 7.0318e-99 | 1.8854e-169 |
| 5 | Stybtang Function | 3.5988e-89 | 3.8579e-92 | 9.8348e-156 | 3.5551e-160 | 1.3910e-150 | 4.2362e-151 | 4.8158e-148 | 7.4638e-172 |
| 6 | Schwefel function | 1.3895e-123 | 1.9569e-164 | 4.7482e-132 | 2.6478e-24 | 2.7706e-42 | 1.3129e-155 | 1.9594e-121 | 7.4398e-142 |
| 7 | Keane Function | 2.1931e-145 | 4.5728e-141 | 7.8435e-146 | 7.4084e-141 | 1.8957e-138 | 2.5242e-139 | 4.3658e-155 | 1.3654e-145 |
| 8 | Trid Function | 2.3005e-04 | 1.2665e-07 | 5.4793e-15 | 3.9880e-12 | 3.6942e-132 | 2.7804e-129 | 2.0510e-140 | 7.0287e-143 |
| 9 | Schaffern4fcn Function | 8.6497e-151 | 1.4164e-133 | 1.2696e-139 | 3.4423e-57 | 8.8837e-158 | 1.5795e-160 | 5.5992e-53 | 1.0205e-04 |
| 10 | Branin Function | 1.4628e-170 | 1.5300e-166 | 9.9148e-147 | 3.9973e-56 | 4.0798e-31 | 1.5096e-163 | 3.7413e-04 | 1.4814e-99 |
| 11 | Wolfe Function | 8.6069e-11 | 1.6745e-18 | 1.3438e-25 | 1.3438e-25 | 8.1128e-25 | 8.2198e-25 | 3.2408e-05 | 8.2198e-25 |
| 12 | Zettl Function | 2.4618e-47 | 3.7395e-48 | 7.9714e-46 | 4.2188e-51 | 8.4116e-43 | 2.6398e-43 | 6.3415e-64 | 2.3241e-60 |
| 13 | Alpine N. 2 Function | 1.1328e-63 | 2.1170e-87 | 4.9124e-160 | 9.4075e-169 | 2.3117e-98 | 5.4748e-96 | 5.4748e-96 | 3.0303e-170 |
| 14 | Cross-in-Tray Function | 2.9415e-190 | 6.3621e-190 | 9.4571e-165 | 2.9144e-118 | 1.6057e-167 | 4.4898e-185 | 8.7420e-20 | 7.1009e-30 |
| 15 | McCormick Function | 4.8145e-208 | 7.9734e-199 | 2.4157e-205 | 2.7981e-185 | 1.7211e-193 | 3.8569e-208 | 1.5483e-54 | 1.6457e-188 |
| 16 | Gramacy & Lee Function | 5.0302e-214 | 1.1779e-213 | 1.7334e-200 | 2.6517e-189 | 2.1106e-212 | 1.5681e-192 | 2.3659e-191 | 3.2889e-191 |
| 17 | Testtubeholder Function | 1.3355e-161 | 2.0663e-138 | 1.4054e-120 | 6.2333e-26 | 2.0588e-151 | 1.2910e-163 | 1.4419e-48 | 9.7121e-15 |
| 18 | Shubert Function | 7.8405e-182 | 4.1448e-96 | 2.8226e-121 | 3.6690e-18 | 5.0688e-103 | 1.7861e-161 | 4.6324e-105 | 1.9701e-164 |
| 19 | Price 2 Function | 5.8287e-19 | 2.4336e-06 | 1.7689e-07 | 2.1156e-119 | 1.6710e-31 | 1.1040e-24 | 3.8123e-70 | 3.1442e-18 |
| 20 | Dejong5 | 1.7349e-183 | 6.1675e-188 | 2.6969e-179 | 7.4328e-178 | 3.8529e-175 | 5.8155e-177 | 4.8973e-178 | 3.7030e-182 |
To test the impact of changing parameter values on the performance of OVIDOA, we used 9 different scenarios by changing the values of the parameters MR (Mutation Rate) and numOfProtiens. We utilized the values of 0.1, 0.01, ad 0.001 for MR, 2, 4, and 6 for numOfProtiens which produces 9 scenarios, as shown in Table 8. The results of each scenario on the selected 5 IEEE CEC benchmark problems are presented in Table 9. The best-obtained results are highlighted in bold. We noticed that scenario 1 (MR = 0.1 and numOfProtiens = 2) has better results, followed by scenario 4. The common between these two scenarios is MR = 0.1 which represents a higher mutation rate. This comparison shows that higher MR values are better for improving the performance of the proposed algorithm.
| Problem | Metric | Algorithm | ||||||||
| GA (Holland, 1992) | DE (Rocca et al., 2011) | PSO (Kennedy & Eberhart, 1995) | FPA (Yang, 2012) | GWO (Mirjalili et al., 214) | WOA (Mirjalili & Lewis, 2016) | CHIO (Al-Betar et al., 2020) | SOA (Dhiman & Kumar, 2019) | Proposed COVIDOA | ||
| CEC01 | Best | 4.79e + 07 | 8.067e + 09 | 2.130e + 08 | 2.525e + 09 | 6.58e + 06 | 4.585e + 09 | 7.011e + 06 | 7.35e + 10 | 1.25e + 06 |
| AVG | 7.767e + 09 | 3.648e + 10 | 4.108e + 09 | 3.4008e + 10 | 4.260e + 09 | 1.623e + 10 | 2.2465e + 11 | 1.294e + 11 | 1.044e + 09 | |
| STD | 3.649e + 10 | 3.729e + 10 | 1.394e + 10 | 8.531e + 10 | 4.8333e + 10 | 5.6522e + 10 | 1.3991e + 11 | 1.755e + 11 | 6.249e + 09 | |
| CEC03 | Best | 12.7024 | 12.7024 | 12.7024 | 12.7024 | 12.7024 | 12.7024 | 12.7024 | 12.7024 | 12.7024 |
| AVG | 12.7024 | 12.7025 | 12.7024 | 12.7026 | 12.7024 | 12.7024 | 12.7025 | 12.7028 | 12.7025 | |
| STD | 1.8779e-04 | 2.3779e-04 | 4.8999e-05 | 3.7993e-04 | 1.8226e-04 | 1.0041e-04 | 2.5001e-04 | 5.063e-04 | 9.8359e-05 | |
| CEC06 | Best | 10.0164 | 7.7598 | 8.5145 | 9.4978 | 9.2790 | 7.7528 | 9.3672 | 8.0529 | 7.6402 |
| AVG | 10.7198 | 8.7656 | 9.7656 | 9.7070 | 9.5928 | 8.6969 | 9.6018 | 9.2519 | 8.6512 | |
| STD | 0.6542 | 8.6156 | 0.7421 | 0.5951 | 0.5048 | 1.2646 | 0.6372 | 0.8336 | 0.4291 | |
| ECE07 | Best | 296.0888 | 165.6218 | 242.9147 | 176.8028 | 305.1 | 546.7268 | 277.5 | 317.7 | 276.0837 |
| AVG | 409.9065 | 265.9382 | 388.3867 | 334.7019 | 461.5165 | 570.3746 | 316.4750 | 566.4644 | 376.4779 | |
| STD | 231.6249 | 186.9450 | 266.6071 | 159.2303 | 168.3036 | 176.3487 | 119.4227 | 170.8287 | 163.8042 | |
| CEC10 | Best | 20.1179 | 20.0925 | 20.1074 | 20.3277 | 20.3589 | 20.0006 | 20.2471 | 20.1112 | 19.4927 |
| AVG | 20.4208 | 20.1859 | 20.2848 | 20.3669 | 20.3789 | 20.0226 | 20.3975 | 20.2414 | 19.4976 | |
| STD | 0.0823 | 0.1245 | 0.1128 | 0.0686 | 0.0697 | 0.0863 | 0.0933 | 0.1412 | 0.0574 | |
For testing COVIDOA on CEC real-world problems, we obtain our results over 500 iterations. The proposed and state-of-the-art algorithms were run 25 independent times as suggested by IEEE-CEC 2011 Competition (Das & Suganthan, 2011). Table 10 and Fig. 17 show the obtained results for the selected CEC real-world problems. The proposed algorithm achieves the optimum best cost, average cost, and STD values for all 5 selected problems.
Although the general steps of COVIDOA and other evolutionary algorithms, such as GA and DE, are very similar, COVIDOA is superior to them, as shown in tables (1–10). This progress is caused by the additional step proposed in the replication phase of COVIDOA, the frameshifting technique. Adding frameshifting technique in the replication process helps COVIDOA update solutions in each generation, helping to get to global optimum rapidly faster.
| Problem | Algorithm | |||||||
|---|---|---|---|---|---|---|---|---|
| COVIDOA vs. GA | COVIDOA vs. DE | COVIDOA vs. PSO | COVIDOA vs. FPA | COVIDOA vs. GWO | COVIDOA vs. WOA | COVIDOA vs. CHIO | COVIDOA vs. SOA | |
| CEC01 | 2.0762e-19 | 3.9935e-44 | 4.0173e-28 | 1.2177e-28 | 7.4696e-24 | 3.5076e-25 | 9.3679e-33 | 9.7806e-73 |
| CEC03 | 4.3959e-10 | 2.0317e-06 | 2.8802e-14 | 6.8629e-04 | 1.6530e-18 | 2.6432e-19 | 2.8370e-17 | 1.8324e-08 |
| CEC06 | 9.1167e-05 | 7.2701e-19 | 1.7786e-05 | 4.3378e-31 | 7.0423e-26 | 2.0190e-23 | 1.8914e-35 | 2.6879e-18 |
| ECE07 | 2.8384e-16 | 3.5116e-19 | 5.2696e-12 | 3.7006e-13 | 2.8596e-09 | 6.4990e-32 | 6.5814e-21 | 6.1956e-26 |
| CEC10 | 6.4014e-13 | 1.8025e-19 | 1.0889e-28 | 5.0020e-28 | 3.9464e-26 | 8.7411e-19 | 4.3551e-32 | 1.4533e-20 |
| Scenario | Parameters | |
|---|---|---|
| MR | numOfProtiens | |
| 1 | 0.1 | 2 |
| 2 | 0.01 | 2 |
| 3 | 0.001 | 2 |
| 4 | 0.1 | 4 |
| 5 | 0.01 | 4 |
| 6 | 0.001 | 4 |
| 7 | 0.1 | 6 |
| 8 | 0.01 | 6 |
| 9 | 0.001 | 6 |
3.3. Explorations and exploitation capabilities of COVIDOA
It is essential to test the efficiency of the proposed algorithm. In other words, it is necessary to test its exploration and exploitation capabilities. In exploration, the algorithm searches for new solutions in new regions, while exploitation means using existing solutions and improving them, so their fitness will improve (Epitropakis et al., 2008). Mutation and crossover steps in COVIDOA are used to create new solutions; so, they are methods to explore the problem space. On the other hand, selecting an existing parent virus and applying the frameshifting technique to produce new children represents the exploitation of the current solution features. Unimodal test functions F1, F7, F10, and F13 can evaluate the exploitation feature because they have only one global optimum solution. Multimodal functions F2, F3, F4, and F5 can help assess the exploration capability of COVIDOA because they have many optimum solutions.
| Problem | Metric | Scenario 1 | Scenario 2 | Scenario 3 | Scenario 4 | Scenario 5 | Scenario 6 | Scenario 7 | Scenario 8 | Scenario 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| CEC01 | Best | 8.41 e + 07 | 1.93e + 08 | 4.84 e + 08 | 7.22e + 06 | 8.94e + 08 | 4.36e + 08 | 6.64e + 08 | 6.31e + 08 | 6.10e + 08 |
| AVG | 6.90e + 09 | 7.34e + 09 | 5.21e + 09 | 6.15e + 08 | 4.82e + 09 | 4.47e + 09 | 1.03e + 10 | 9.49e + 09 | 6.44e + 09 | |
| STD | 1.75e + 10 | 2.38e + 10 | 1.51e + 10 | 1.99e + 10 | 1.80e + 10 | 1.20e + 10 | 3.22e + 10 | 4.06e + 10 | 1.79e + 10 | |
| CEC03 | Best | 12.7024 | 12.7024 | 12.7024 | 12.7025 | 12.7025 | 12.7025 | 12.7025 | 12.7025 | 12.7025 |
| AVG | 12.7025 | 12.7025 | 12.7025 | 12.7025 | 12.7026 | 12.7026 | 12.7025 | 12.7026 | 12.7026 | |
| STD | 2.08e-04 | 8.34e-05 | 1.26e-04 | 2.91e-04 | 3.03e-04 | 2.17e-04 | 1.60e-04 | 1.14e-04 | 2.29e-04 | |
| CEC06 | Best | 7.6402 | 9.0038 | 9.3928 | 9.0148 | 8.344 | 7.7169 | 8.8126 | 8.7189 | 9.1483 |
| AVG | 8.6512 | 9.7509 | 9.9594 | 9.4252 | 8.9131 | 9.2187 | 8.8800 | 9.2262 | 9.6000 | |
| STD | 0.4291 | 1.0948 | 0.9469 | 0.5378 | 0.8895 | 1.2194 | 0.4741 | 0.7438 | 0.5159 | |
| ECE07 | Best | 429.593 | 467.8152 | 525.5403 | 276.0837 | 388.5537 | 508.3128 | 455.5922 | 560.6439 | 404.9701 |
| AVG | 478.6812 | 600.8125 | 699.5776 | 493.5525 | 460.4821 | 640.0468 | 468.0766 | 719.8709 | 521.5527 | |
| STD | 135.4222 | 139.3537 | 101.7200 | 202.0033 | 137.9164 | 157.4621 | 87.5660 | 145.9606 | 218.8682 | |
| CEC10 | Best | 19.3901 | 20.241 | 20.3035 | 20.301 | 20.104 | 20.2851 | 20.3317 | 20.2618 | 20.2906 |
| AVG | 20.1491 | 20.3671 | 20.3797 | 20.3672 | 20.2833 | 20.3194 | 20.3450 | 20.3261 | 20.3269 | |
| STD | 0.2631 | 0.0939 | 0.0591 | 0.0478 | 0.1631 | 0.0615 | 0.0531 | 0.1012 | 0.0788 |
3.4. Convergence of COVIDOA
In Table 4, the convergence speed of COVIDOA and the other algorithms for the classical benchmark functions are classified into three groups: Fast, Moderate, and Low, where algorithms that reach the minimum cost in the first 100 iterations classified as fast convergence algorithms, those that get the minimum cost from iteration 100 to 300 are moderate convergence algorithms, and the others classified as slow convergence algorithms. As shown in Table 5 and Fig. 13, the proposed algorithm has fast convergence in the majority (16 from 20) of the test problems and moderate in the others. In contrast, other state-of-the-art algorithms may have slow convergence of the test problems.
Overall results reveal that COVIDOA reaches the minimum best cost, average cost, and standard deviation in most test problems. Additionally, it has high exploration and exploitation capabilities and a very high convergence speed during iterations.
| Problem | Meric | Algorithms | ||||||||
| GA (Holland, 1992) | DE (Rocca et al., 2011) | PSO (Kennedy & Eberhart, 1995) | FPA (Yang, 2012) | GWO (Mirjalili et al., 214) | WOA (Mirjalili & Lewis, 2016) | CHIO (Al-Betar et al., 2020) | SOA (Dhiman & Kumar, 2019) | COVIDOA | ||
| Lennard-Jones Potential Problem | Best | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 | -1 |
| AVG | -0.9984 | -0.9949 | -0.9999 | -0.9968 | -0.9967 | -0.9996 | -0.9998 | -0.9953 | -1 | |
| STD | 0.0177 | 0.0461 | 0.0019 | 0.0409 | 0.0516 | 0.0087 | 5.7843e-04 | 0.0549 | 4.8460e-07 | |
| Spread spectrum radar polyphase problem. | Best | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 |
| AVG | 0.5004 | 0.5 | 0.5 | 0.5014 | 0.5003 | 0.5007 | 0.5 | 0.5002 | 0.5 | |
| STD | 0.0056 | 0 | 0 | 0.0096 | 0.0040 | 0.0094 | 0 | 0.0018 | 0 | |
| Tersoff Potential Function Minimization Problem for model Si(B). | Best | -2.6237 | -2.6237 | -2.6237 | -2.6237 | -2.6237 | -2.6237 | -2.6237 | -2.6237 | -2.6237 |
| AVG | -2.6237 | -2.6237 | -2.6237 | -2.6235 | -2.6236 | -2.6237 | -2.6228 | -2.6236 | -2.6237 | |
| STD | 1.3471e-05 | 1.5074e-04 | 3.3914e-05 | 0.0036 | 0.0016 | 2.5855e-04 | 0.0145 | 2.3704e-04 | 2.7394e-07 | |
| Tersoff Potential Function Minimization Problem for model Si(C). | Best | -2.666 | -2.666 | -2.666 | -2.666 | -2.6660 | -2.6660 | -2.6660 | -2.666 | -2.666 |
| AVG | -2.6660 | -2.6660 | -2.6660 | -2.6660 | -2.6656 | -2.6660 | -2.6654 | -2.6660 | -2.6660 | |
| STD | 1.8503e-05 | 5.0644es-04 | 6.2625e-06 | 2.8149e-05 | 0.0068 | 7.0142e-05 | 0.0088 | 2.0518e-06 | 5.8070e-07 | |
| Transmission Network Expansion Planning (TNEP) problem | Best | 435 | 435 | 435 | 435 | 435 | 442 | 442 | 435 | |
| AVG | 435.0640 | 435.4125 | 435.0740 | 440.3519 | 441.4249 | 442.4413 | 478.2145 | 435.6794 | 435 | |
| STD | 0.7134 | 2.5585 | 1.6547 | 32.7886 | 21.9935 | 9.8685 | 13.7593 | 3.7615 | 0 | |
A novel evolutionary optimization algorithm (COVIDOA) inspired by the replication lifecycle of SARS-CoV-2 is presented. The proposed COVIDOA was tested by solving 20 classical benchmark problems and 5 CEC benchmark test functions. The proposed COVIDOA is compared with the state-of-the-art nature-inspired optimization algorithms in terms of best cost, average cost, standard deviation, and convergence speed. The proposed algorithm implemented using MATLAB R2016a software, and the source code of the state-of-the-art algorithms and the benchmark problems are downloaded from the mathworks.com website. The experimental results proved that the proposed algorithm outperforms the state-of-the-art optimization algorithms in most test problems and has very close results to other algorithms in the rest of the test problems. COVIDOA has very high exploitation and exploration capabilities and very high convergence speed compared to other metaheuristics.
Future work may include the implementation of COVIDOA in solving large-scale problems in different fields.
Conflict of Interest:
The authors declare that they have no conflict of interest.
No fund is received by authors.
Ethical approval:
This article does not contain any studies with human participants performed by any authors.
- Ab Wahab MN, Nefti-Meziani S, Atyabi A, "A Comprehensive Review of Swarm Optimization Algorithms," PLoS ONE, 10(5), 2015.
- Abdullah, J. M., & Ahmed, T. (2019). Fitness dependent optimizer: inspired by the bee swarming reproductive process. IEEE Access, 7, 43473-43486.
- Alatas, Bilal & Can, Umit., "Physics based Metaheuristic Optimization Algorithms for Global Optimization," 2015.
- Al-Betar, M. A., Alyasseri, Z. A. A., Awadallah, M. A., & Doush, I. A. (2020). Coronavirus herd immunity optimizer (CHIO). Neural Computing and Applications, 1-32.
- Al-Betar, M. A., Alyasseri, Z. A. A., Awadallah, M. A., & Doush, I. A. (2020). Coronavirus herd immunity optimizer (CHIO). Neural Computing and Applications, 1-32.
- Andre, J., Siarry, P., & Dognon, T., "An improvement of the standard genetic algorithm fighting premature convergence in continuous optimization," Advances in engineering software, 32(1), 49-60, 2001.
- Askarzadeh, A., "Bird mating optimizer: an optimization algorithm inspired by bird mating strategies," Communications in Nonlinear Science and Numerical Simulation, 19(4), 1213-1228, 2014.
- Baluja, S., "Population-based incremental learning. a method for integrating genetic search-based function optimization and competitive learning", Carnegie-Mellon Univ Pittsburgh Pa Dept Of Computer Science, 1994.
- Bar-On, Y. M., Flamholz, A., Phillips, R., & Milo, R., "SARS-CoV-2 (COVID-19) by the numbers". Science Forum, Elife, 2020.
- Bergmann, C. C., & Silverman, R. H., "COVID-19: Coronavirus replication, pathogenesis, and therapeutic strategies," Cleveland Clinic journal of medicine., 2020.
- Birbil, Ş. İ., & Fang, S. C., "An electromagnetism-like mechanism for global optimization," Journal of global optimization, 25(3), 263-282, 2003.
- Brameier, M. F., & Banzhaf, W., "Linear genetic programming," Springer Science & Business Media, 2007.
- Brian, D. A., & Baric, R. S., "Coronavirus genome structure and replication," Coronavirus replication and reverse genetics," 1-30, 2020.
- Brooks, S. P., & Morgan, B. J., "Optimization using simulated annealing," Journal of the Royal Statistical Society: Series D (The Statistician), 44(2), 241-257. 1995.
- Cobb, Matthew, "Who discovered messenger RNA?". Current Biology. 25 (13): 526-532, 2015.
- Darwish, A., "Bio-inspired computing: Algorithms review, deep analysis, and the scope of applications," Future Computing and Informatics Journal, 3(2), 231-246, 2018.
- Das S, Suganthan PN (2010) Problem definitions and evaluation criteria for cec 2011 competition on testing evolutionary algorithms on real world optimization problems. Jadavpur University, Nanyang Technological University, Kolkata, pp 341–359
- Das, S., Biswas, A., Dasgupta, S., & Abraham, A., "Bacterial foraging optimization algorithm: theoretical foundations, analysis, and applications," Foundations of computational intelligence, 3, 23-55, 2009.
- Derrac J, García S, Molina D, Herrera F (2011) A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms. Swarm Evol Comput 1(1):3–18.
- Dhiman, G., & Kumar, V., "Seagull optimization algorithm: Theory and its applications for large-scale industrial engineering problems," Knowledge-Based Systems, 165, 169-196, 2019.
- Domingo, E., Escarmís, C., Sevilla, N., Moya, A., Elena, S. F., Quer, J., ... & Holland, J. J. (1996). Basic concepts in RNA virus evolution. The FASEB Journal, 10(8), 859-864.
- E. Rashedi, H. Nezamabadi-Pour, and S. Saryazdi, "GSA: a gravitational search algorithm," Information Sciences, vol. 179, no. 13, pp. 2232–2248, 2009.
- Epitropakis, Michael & Plagianakos, Vassilis & Vrahatis, Michael, "Balancing the exploration and exploitation capabilities of the Differential Evolution Algorithm," 2686-2693, 2008.
- Eskandar, H., Sadollah, A., Bahreininejad, A., & Hamdi, M., "Water cycle algorithm–A novel metaheuristic optimization method for solving constrained engineering optimization problems," Computers & Structures, 110, 151-166, 2014.
- Fogel, D. B., "The Advantages of Evolutionary Computation," BCEC, 1-11, 1997.
- Gandomi, A. H., & Alavi, A. H., "Krill herd: a new bio-inspired optimization algorithm," Communications in nonlinear science and numerical simulation, 17(12), 4831-4845, 2015.
- Holland JH., "Genetic algorithms," Sci Am, 267, 66–72, 1992.
- Hosseini, E., Ghafoor, K. Z., Sadiq, A. S., Guizani, M., & Emrouznejad, A. (2020). Covid-19 optimizer algorithm, modeling and controlling of coronavirus distribution process. IEEE Journal of Biomedical and Health Informatics, 24(10), 2765-2775.
- Hsiao, Y. T., Chuang, C. L., Jiang, J. A., & Chien, C. C., "A novel optimization algorithm: space gravitational optimization," In 2005 IEEE international conference on systems, man and cybernetics, 3, 2323-2328, 2005.
- http://benchmarkfcns.xyz
- https://commons.wikimedia.org/wiki/File:3D_medical_animation_corona_virus.jpg
- https://time.com/5839932/how-remdesivir-works-coronavirus/
- https://www.mathworks.com/
- https://www.mathworks.com/matlabcentral/fileexchange/72123-cec-06-2019-matlab implementation
- https://www.microscope.com/coronavirus-under-an-electron-microscope/
- Hussain, K., Salleh, M. N. M., Cheng, S., & Shi, Y., "Metaheuristic research: a comprehensive survey," Artificial Intelligence Review, 52(4), 2191-2233, 2019.
- Ivanov IP, Atkins JF., "Ribosomal frameshifting in decoding antizyme mRNAs from yeast and protists to humans: close to 300 cases reveal remarkable diversity despite underlying conservation". Nucleic Acids Research. 35 (6): 1842–1858, 2015.
- Jacks T, Madhani HD, Masiarz FR, Varmus HE, "Signals for ribosomal frameshifting in the Rous sarcoma virus gag-pol region," cell. 55 (3): 447–458, 1988.
- James, J. Q., & Li, V. O., "A social spider algorithm for global optimization," Applied Soft Computing, 30, 614-627, 2015.
- Kang, S., Peng, W., Zhu, Y., Lu, S., Zhou, M., Lin, W., ... & Deng, M., "Recent Progress in understanding 2019 novel coronavirus associated with human respiratory disease: detection, mechanism and treatment", International Journal of Antimicrobial Agents, 55(5), 2020.
- Karaboga, D., "An Idea Based on Honey Bee Swarm For Numerical Optimization," S2CID 8215393, 200, 1-10, 2005.
- Kaveh, A., & Talatahari, S., "A novel heuristic optimization method: charged system search," Acta Mechanica, 213(3), 267-289, 2010.
- Kelly, J. A., Olson, A. N., Neupane, K., Munshi, S., San Emeterio, J., Pollack, L., ... & Dinman, J. D., "Structural and functional conservation of the programmed− 1 ribosomal frameshift signal of SARS coronavirus 2 (SARS-CoV-2)", Journal of Biological Chemistry, 295(31), 10741-10748, 2020.
- Kelly, J. A., Olson, A. N., Neupane, K., Munshi, S., San Emeterio, J., Pollack, L., ... & Dinman, J. D., "Structural and functional conservation of the programmed− 1 ribosomal frameshift signal of SARS coronavirus 2 (SARS-CoV-2)", Journal of Biological Chemistry, 295(31), 10741-10748, 2020.
- Kennedy, J., & Eberhart, R., "Particle swarm optimization," In Proceedings of ICNN'95-International Conference on Neural Networks, 4, 1942-1948, 1995.
- Khan, M. I., Khan, Z. A., Baig, M. H., Ahmad, I., Farouk, A. E., Song, Y. G., & Dong, J. J., "Comparative genome analysis of novel coronavirus (SARS-CoV-2) from different geographical locations and the effect of mutations on major target proteins", An in-silico insight. PLoS One, 15(9), 2020.
- Lipowski, A., & Lipowska, D., "Roulette-wheel selection via stochastic acceptance," Physica A: Statistical Mechanics and its Applications, 391(6), 2193-2196, 2012.
- Martínez-Álvarez, F., Asencio-Cortés, G., Torres, J. F., Gutiérrez-Avilés, D., Melgar-García, L., Pérez-Chacón, R., ... & Troncoso, A., “Coronavirus Optimization Algorithm: A bioinspired metaheuristic based on the COVID-19 propagation model”, arXiv preprint arXiv:2003.13633.
- Meng, X., Liu, Y., Gao, X., & Zhang, H., "A new bio-inspired algorithm: chicken swarm optimization," International conference in swarm intelligence, 86-94, 2014.
- Milewska, A., Kula-Pacurar, A., Wadas, J., Suder, A., Szczepanski, A., Dabrowska, A., ... & Rajfur, Z., “Replication of SARS-CoV-2 in human respiratory epithelium”, Journal of Virology, 94(15), 2020.
- Mirjalili, S., & Lewis, A., "The whale optimization algorithm," Advances in engineering software, 95, 51-67, 2016.
- Mirjalili, S., Mirjalili, S. M., & Lewis, A., "Grey wolf optimizer," Advances in engineering software, 69, 46-61, 2014.
- Moghal A, Mohler K, Ibba M . "Mistranslation of the genetic code" FEBS Letters. 588 (23), 2014.
- Momin Jamil and Xin-She Yang, "A literature survey of benchmark functions for global optimization problems,' Int. Journal of Mathematical Modelling and Numerical Optimisation, 4(2), 150 –194, 2013.
- Moscato, P., Cotta, C., & Mendes, A., "Memetic algorithms," New optimization techniques in engineering," 53-85, 2004.
- Napthine S, Ling R, Finch LK, Jones JD, Bell S, Brierley I, Firth AE (June 2017). "Protein-directed ribosomal frameshifting temporally regulates gene expression"—nature Communications.
- Nordin, P., Keller, R. E., & Francone, F. D., “Genetic programming”, W. Banzhaf (Ed.). Springer, 1998.
- O'Neill, M., & Ryan, C., "Grammatical evolution," IEEE Transactions on Evolutionary Computation, 5(4), 349-358, 2001.
- Pál, K. F., "Hysteretic optimization, faster and simpler," Physica A: Statistical Mechanics and its Applications, 360(2), 525-533, 2006.
- Pascual, M. R. (2020), "Coronavirus SARS-CoV-2: Analysis of subgenomic mRNA transcription, 3CLpro and PL2pro protease cleavage sites and protein synthesis", arXiv preprint arXiv, 2020.
- R. A. Formato, "Central force optimization: a new metaheuristic with applications in applied electromagnetics," Progress in Electromagnetics Research, vol. 77, pp. 425–491, 2007.
- Rapley, R., & Whitehouse, D. (Eds.). (2015). Molecular biology and biotechnology. Royal Society of Chemistry.
- Rashedi, E., Nezamabadi-Pour, H., & Saryazdi, "BGSA: binary gravitational search algorithm," Natural Computing, 9(3), 727-745, 2010.
- Rechenberg, I., "Evolution strategy: Nature's way of optimization. In Optimization: Methods and applications", possibilities and limitations, 106-126, 1989.
- Rocca, P.; Oliveri, G.; Massa, A., "Differential Evolution as Applied to Electromagnetics," IEEE Antennas and Propagation Magazine, 53 (1), 38-49, 2011.
- Sacco, W. F., & Oliveira, C. R. E. A., "A new stochastic optimization algorithm based on a particle collision metaheuristic," Proceedings of 6th WCSMO, 2005.
- Schoeman, D., Fielding, B.C., "Coronavirus envelope protein, "current knowledge," 16(1), 1-22, 2019.
- Shah-Hosseini, H., "The intelligent water drops algorithm: a nature-inspired swarm-based optimization algorithm," International Journal of Bio-inspired Computation, 1(1-2), 71-79, 2009.
- Shang, J., Ye, G., Shi, K., Wan, Y., Luo, C., Aihara, H., ... & Li, F., "Structural basis of receptor recognition by SARS-CoV-2", Nature, 581(7807), 221-224, 2020.
- Sharma, Jyoti; Keeling, Kim M.; Rowe, Steven M., "Pharmacological approaches for targeting cystic fibrosis nonsense mutations," European Journal of Medicinal Chemistry, 2020.
- Simon, D., "Biogeography-based optimization", IEEE transactions on evolutionary computation, 12(6), 702-713, 2008.
- Szucs, D., & Ioannidis, J. (2017). When null hypothesis significance testing is unsuitable for research: a reassessment. Frontiers in human neuroscience, 11, 390.
- Terpos, E., Ntanasis‐Stathopoulos, I., Elalamy, I., Kastritis, E., Sergentanis, T. N., Politou, M., ... & Dimopoulos, "Hematological findings and complications of COVID‐19", American journal of hematology, 2020.
- Tu, Y. F., Chien, C. S., Yarmishyn, A. A., Lin, Y. Y., Luo, Y. H., Lin, Y. T., ... & Wang, M. L., "A review of SARS-CoV-2 and the ongoing clinical trials". International journal of molecular sciences, 21(7), 2020.
- Wang, S., Zhang, Y., Liu, S., Peng, H., Mackey, V., & Sun, L., "Coronaviruses and the associated potential therapeutics for the viral infections," J Infect Dis Ther, 8(417), 2020.
- Yamauchi, Y., & Greber, U. F., "Principles of virus uncoating," cues and the snooker ball. Traffic, 17(6), 569-592, 2016.
- Yang, X. S., "Flower pollination algorithm for global optimization," In International conference on unconventional computing and natural computation," 240-249, 2012.
- Yang, X. S., & Deb, S., "Cuckoo search via Lévy flights," In 2009 World Congress on Nature & biologically inspired computing (NaBIC), 210-214, 2009.
- Yu, X., & Gen, M., "Introduction to evolutionary algorithms," Springer Science & Business Media, 2010.
- Zheng, Y. J., "Water wave optimization: a new nature-inspired metaheuristic," Computers & Operations Research, 55, 1-11, 2015.
- Ziebuhr, J., "The coronavirus replicase: insights into a sophisticated enzyme machinery." The Nidoviruses, Springer, Boston, MA., 3-11, 2020.
- Supp.docx
Appendix A and B