1 Introduction

Nature has provided humanity with methodologies and approaches for solving highly complex problems [44]. The GA is a well-known nature-inspired algorithm in the optimization field [19], where scientists mimic the evolution of natural biological organisms to find high-quality solutions to optimize problems [20]. The amazing success of the GA in solving various optimization problems motivated researchers to develop additional heuristics such as ACO [14], DE [43], CSA [23], and PSO [24]. Such algorithms usually search for the best solution by generating all solutions to the problem, evaluating them, and then updating them until a maximum number of iterations is reached. Despite the efficiency of these algorithms, they face many difficulties when solving some real problems, such as multiple objectives [33, 34], many objectives [22], constraints [3], uncertainty [7], and local optima [27], which requires a modification of these metaheuristics to handle such difficulties.

Optimization techniques are divided into single- and multi-objective [10]. The first type achieves the optimal solution by comparing the utilized objective function. In the case of multi-objective optimization, the goal is to find the POSs [36, 37]. Solving multi-objective problems is classified into two main groups: Priori and Posteriori [17]. In the a priori method, sufficient information must be provided before making any decision. This information can aggregate all the objectives into a single one by defining the single-objective function as a weighted sum of the normalized costs associated with each objective [35]. In such a case, a single-objective algorithm can be suitable, as it is straightforward and does not require any modification of the basic algorithm. However, it has disadvantages, such as consulting with the decision-maker to determine the preferred weights. A posteriori method aims to find a set of POSs. Then, apply the decision-making [40]. It can find all the POSs in one run. This approach maintains the formulation of the multi-objective problem and does not need to aggregate the different objectives or determine a set of weights. NSGA [12], MOPSO [8], MOEA/D [50], PAES [28], MOSMA [21], MOJS [6], MOMPA [18], MOVPA [39], and MOSOA [13] are examples of the well-regarded multi-objective optimization algorithms. These algorithms suffer from limitations such as slow convergence and inefficiency in solving complex higher-dimensional problems or when the number of objectives increases. Interested readers are referred to [36,37,38], and [30] for more algorithms.

COVIDOA is one of the most recently proposed swarm intelligence metaheuristics [25, 26]. It is inspired by Coronavirus particles' behavior inside the human body. A coronavirus particle goes through a replication life cycle within a human host cell, creating more copies of itself that can eventually infect and damage more healthy human cells. The key stages of the Coronavirus replication lifecycle include virus attachment to human cells through spike protein [16], virus entry and uncoating [31], virus replication through the ribosomal frameshifting technique [4], virus mutation [5], and new virion assembly and release [32]. COVIDOA has been employed in various optimization problems and has shown superior performance [25, 25, 26, 26] due to its excellent exploration and exploitation capabilities and high convergence speed.

This work presents a novel multi-objective COIVDOA (MOCOVIDOA) which generalizes the original COVIDOA for solving optimization problems with up to three objective functions. The proposed algorithm utilizes the archive concept to store and retrieve the POSs. The roulette wheel selection selects a non-dominated solution from the archive. The reasons for developing the multi-objective version of COVIDOA are as follows:

  • Engineering design problems inherently have multiple conflicting objectives with multiple constraints. Single-objective techniques are usually failed to optimize all objectives.

  • The NFL theorem [13] demonstrates that no single optimization method can find the optimum solution for all optimization problems. This theory motivated us to develop new approaches that may perform better in solving complex optimization problems.

  • The outstanding performance of COVIDOA and BCOVIDOA in single-objective optimization [25, 25, 26, 26] motivated the authors to derive the MOCOVIDOA to solve multi-objective real-world problems.

  • Most of the current approaches are suitable only for unconstrained problems. The COVIDOA is proposed to solve both constrained and unconstrained problems.

This paper is organized as follows: An overview of multi-objective optimization and a concise description of the single-objective COVIDOA are presented in Sect. 2. The proposed MOCOVIDOA is introduced in Sect. 3. We discussed the benchmark and engineering test problems and the obtained results in Sect. 4. Finally, the conclusion is presented in Sect. 5.

2 Preliminaries

2.1 Multi-objective optimization

Multi-objective optimization involves optimizing a mathematical problem with multiple objective functions, where optimizing these functions simultaneously is the main challenge. A multi-objective optimization can be represented as a minimization optimization problem [33, 34]:

$$Minimize: F\left(\overrightarrow{t}\right)=\{{f}_{1}\left(t\right),{ f}_{2}\left(t\right), ..., {f}_{o}\left(t\right)\}$$
(1)
$$Subject~ to:{ g}_{i}\left(t\right)\ge 0, 1\le i\le p$$
(2)
$${h}_{i}\left(t\right)=0, 1\le i\le q$$
(3)
$${L}_{i}\le {t}_{i}\le {U}_{i}, 1\le i\le D$$
(4)

where o is the number of objective functions, p is the number of inequality constraints, q is the number of equality constraints, D is the problem dimension, \({g}_{i}\) is the ith inequality constraint, \({h}_{i}\) is the ith equality constraint, and \({L}_{i}\) and \({U}_{i}\) are the lower and upper bounds of the ith variable.

It should be noted that, in single-objective optimization, it is easy to find the optimal solution to the problem because of a single objective. In contrast, multi-objective optimization requires finding solutions representing the best trade-off among the different objectives [11].

To compare solutions for multiple objectives, relational operators are not suitable due to the existence of multiple conflicting objectives; instead, the dominance operator [49] is utilized as follows:

In a minimization problem, a solution x1 is said to dominate another solution x2 (denoted as\({x}_{1}\prec {x}_{2}\)) if and only if:

$$\forall i \in \left\{1, \dots , o\right\}, {f}_{i}\left({x}_{1}\right)\le {f}_{i}\left({x}_{2}\right), and$$
(5)
$$\exists i\in \left\{1, \dots , o\right\}, {f}_{i}\left({x}_{1}\right)<{f}_{i}\left({x}_{2}\right).$$
(6)

For the maximization problem, a solution x1 dominates x2 if:

$$\forall i \in \left\{1, \dots , o\right\}, {f}_{i}\left({x}_{1}\right)\ge {f}_{i}\left({x}_{2}\right)\quad{and}$$
(7)
$$\exists i\in \left\{1, \dots , o\right\}, {f}_{i}\left({x}_{1}\right)>{f}_{i}\left({x}_{2}\right).$$
(8)

A solution is better than another if it has the same fitness value as the other solution in all objectives and better fitness in at least one objective. A solution is called Pareto optimal if there is no other solution that dominates it as follows:

$$\nexists \overrightarrow{y}\in X\left|\overrightarrow{y} \right.\prec \overrightarrow{x}$$
(9)

Each multi-objective function has a set of non-dominated solutions called the Pareto optimal set, defined as follows (see Fig. 1):

Fig. 1
figure 1

Search space and Objective space in multi-objective optimization

Full size image
$${P}_{s}:=\{\overrightarrow{x}, \overrightarrow{y} \in X\left|\nexists \overrightarrow{y}\right.\prec \overrightarrow{x}\}$$
(10)

The set of Objective functions values for the Pareto optimal set is called the Pareto front as follows:

$${P}_{f}:=\{ f\left(\overrightarrow{x}\right)\left|\overrightarrow{x}\right. \in {P}_{s}\}$$
(11)

The general steps of almost all multi-objective optimization algorithms in the evolutionary computation field are similar. They begin the optimization process with an initial random population of solutions. Then they compare these solutions using the dominance operator to find the set of POSs and keep them in an archive. In the next iterations, each algorithm tries to improve the quality of the archive solutions to get the closest possible approximation to the true Pareto set [29].

There are two main conflicting objectives of all posterior multi-objective algorithms; convergence and coverage. Convergence refers to improving the quality of the non-dominated solutions obtained during the optimization process to improve their accuracy compared to the true POSs. Coverage means improving the distribution of the non-dominated solutions to cover the entire Pareto optimal front. Focusing on only one of the two objectives will negatively affect the other, so an effective algorithm must balance these two criteria well [49].

2.2 Coronavirus disease optimization algorithm

COVIDOA is a recent evolutionary metaheuristic inspired by the replication mechanism of Coronavirus when hijacking human body cells [25, 26]. The replication process of Coronavirus is divided into four main stages as follows (see Fig. 2 as well):

Fig. 2
figure 2

The replication lifecycle of Coronavirus

Full size image

2.2.1 Virus entry and uncoating

When a human is infected with COVID, the Coronavirus particles bind to the human cell via one of its structural proteins, spike protein. After getting inside the human cell, the virus contents are released.

2.2.2 Virus replication

The virus aims to make more copies to hijack as many as possible human healthy cells. The virus uses what is referred to as the ribosomal frameshifting technique for replication [4]. Frameshifting moves the reading frame of a virus protein sequence to another reading frame, leading to the creation of many different protein sequences. The newly generated viral proteins are then merged to form new virus particles. There are many types of frameshifting techniques; however, the most popular is + 1 frameshifting as follows [45]:

  •  + 1 frameshifting technique

The elements of the parent virus particle (parent solution) are moved in the right direction in 1 step. As a result of + 1 frameshifting, the first element is lost. In the proposed algorithm, the first element is set to a random value in the range [Lb, Ub] as follows:

$$S_k (1)=\mathrm{rand}(\mathrm{Lb},\mathrm{Ub}),$$
(12)
$$S_k (2:\mathrm{D})=\mathrm{ P}(1:\mathrm{D}-1),$$
(13)

where Lb and Ub are the lower and upper bounds for the variables in each solution, P is the parent sequence, and \({\mathrm{S}}_{\mathrm{k}}\) is the generated viral protein number k.

2.2.3 Virus mutation

Coronavirus uses the mutation technique to resist the human immune system. In the COVID algorithm, the mutation is applied to the previously created virus particle (solution) to produce a new one as follows:

$$ Zi = \left\{ {\begin{array}{*{20}l} r \hfill & {if\ {\text{ }}r\ and\ \left( {0,1} \right)< MR} \hfill \\ {X_{i} } \hfill & {otherwise} \hfill \\ \end{array} } \right. $$
(14)

MR refers to the mutation rate. Where 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, where D is the problem dimension, r is a random value in the range [Lb, Ub]. Coronavirus has a very low mutation rate (1 × 10–6), as mentioned in [2], however, using such a low mutation rate will limit the algorithm's ability to explore the search space. Thus, the mutation rate in the proposed algorithm is set to a larger value in the range [0.005, 0.5], which helps explore new promising regions of the search space.

2.2.4 New virion release

The newly created virus particle leaves the infected cell targeting new healthy cells. In the proposed algorithm, if the fitness of the new solution is better than the parent solution fitness, the parent solution is replaced by the new one; otherwise, the parent solution remains as follows:

$${P}_{i}^{t}=\left\{\begin{array}{l}{Z}_{i}^{t}\quad if f({Z}_{i}^{t})f({P}_{i}^{t})\\ {P}_{i}^{t}\quad otherwise\end{array}\right.$$
(15)

where \({P}_{i}^{t}\) is the ith parent (parent particle) in the tth iteration, and \({Z}_{i}^{t}\) is the ith generated child (new virion) in the tth iteration.

The pseudocode of the COVID algorithm is shown in Fig. 3.

Fig. 3
figure 3

The pseudo-code of the COVIDOA algorithm

Full size image

As an evolutionary algorithm, COVIDOA has many advantages, such as:

  1. 1.

    Its concept is easy to understand.

  2. 2.

    It searches from a population of points, not a single point.

  3. 3.

    It can handle complex problems and parallelism.

  4. 4.

    It is robust to the local minima/maxima problem.

Despite these advantages, COVIDOA suffers from limitations such as its high time complexity compared with other algorithms due to mutation, crossover, and frameshifting processes. Important parameters such as mutation rate, crossover probability, and frameshifting number will make it difficult for the algorithm to converge if they are not chosen appropriately.

3 Multi-objective coronavirus optimization algorithm (MOCOVIDOA)

The search mechanism of MOCOVIDOA is the same as in COVIDOA, which is based on the replication mechanism of Coronavirus. MOCOVIDOA uses the dominance operator to compare solutions considering multiple criteria (objective functions). All the POSs obtained so far are stored in an archive. A controller is used to determine which solution should be kept in the archive and which should be removed, which is done similarly to MOPSO [8]. The archive controller works according to the following rules:

  • If the archive is empty, the current solution should be accepted.

  • If another solution dominates the archive, the particular solution should be removed.

  • The particular solution should be stored in the archive if the external population does not dominate a solution.

  • If new elements dominate solutions in the archive, they should be removed.

Two approaches must be done to improve the coverage of solutions in the Pareto optimal front; solutions from the least crowded regions in the archive must be selected to contribute to the improvement of others, and solutions with many neighbors should be thrown away from the archive. A leader selection mechanism is utilized to perform these two approaches as follows:

MOVOIDOA selects a solution from the least populated regions in the archive using the roulette wheel selection. The probability of selecting a solution is defined as follows:

$${P}_{i}=\frac{c}{{N}_{i}}$$
(16)

where c is a constant greater than 1, and \({N}_{i}\) represents the number of solutions in the vicinity of the ith solution.

The undesired solutions are those with many neighbors. The undesired solutions should be removed when the archive becomes full. The probability of selecting a solution for removal from the archive is defined as follows:

$${P}_{i}=\frac{{N}_{i}}{c}$$
(17)

In MOCOVIDOA, Eq. (15) should be replaced by the following equation because of multiple objectives.

$${P}_{i}^{t}={Z}_{i}^{t}~ if~ f({Z}_{i}^{t})\prec f({P}_{i}^{t})$$
(18)

where \({P}_{i}^{t}\) is the ith parent (parent particle) in the tth iteration, and \({Z}_{i}^{t}\) is the ith generated child (new virion) in the tth iteration.

The rest of the operators used in MOCOVIDOA are the same as those used in COVIDOA.

The computational complexity of MOCOVIDOA is estimated as follows:

  1. 1.

    The MOCOVIDOA requires \(O\left(m \times n \right)\) for calculating the fitness functions of search agents, where m represents the number of objective functions, and n is the population size.

  2. 2.

    The MOCOVIDOA requires \(O\left(m \times {n}_{\mathrm{archive}}\right)\) time to update the archive where \({n}_{archive}\) refers to the number of non-dominated solutions in the archive.

  3. 3.

    Steps 1 and 2 are repeated until the maximum number of iterations (\(Max\_iter\)) is reached.

Hence the time complexity of MOCOVIDOA is \(O\left(Max\_iter\times m \times n \times {n}_{archive}\right)\), which is more expensive than the complexity of the original single-objective COVIDOA (\(O\left(\mathrm{Max}\_\mathrm{iter}\times m \times n \times {n}_{\mathrm{archive}}\right)\)) due to an additional archive of solutions and multiple objective functions.

The computational complexity of the proposed MOCOVIDOA is the same as that of the well-known multi-objective algorithms: MOPSO, NSGA-II, SPEA2, and PEAS. However, it is better than the complexity of other algorithms such as NSGA and SPEA, O(mn3). According to the space complexity, MOCOVIDOA requires the same space as other algorithms, such as MOPSO, but it needs more space compared to some algorithms, such as NSGA-II, due to creating the archive.

The pseudocode of the proposed MOCOVIDOA is presented in Fig. 4, and the flowchart is illustrated in Fig. 5.

Fig. 4
figure 4

The pseudo-code of the MOCOVIDOA algorithm

Full size image
Fig. 5
figure 5

Flowchart of the proposed MOCOVIDOA

Full size image

4 Experimental results and discussion

This section presents the performance of the proposed algorithm in solving 27 multi-objective problems with diverse characteristics and the comparisons with the state-of-the-art algorithms according to various evaluation metrics.

4.1 Experimental setup

All the experiments were run on a laptop with the following specifications: Intel(R) Core(TM) i7-1065G7 processor, RAM of 8.0 GB size, and Windows 10 Ultimate 64-bit operating system. All the algorithms are developed using MATLAB R2016a development environment.

Twenty-seven benchmark case studies are used to evaluate the performance of MOCOVIDOA, as follows:

  • Six unconstrained ZDT test suites, including ZDT1-ZDT4, ZDT6, and ZDT1, with linear Pareto front (Zitzler et al., 2002).

  • Ten unconstrained CEC-2009 functions, including UF1-UF10 [51].

  • Five constrained test functions include BNH, TNK, CONSTR, SRN, and OSY [47].

  • Six real-world constrained engineering design problems include welded beam, speed reducer, four-bar truss, welded beam, and gear train design (Dept, 1991, [15, 42]. The details of these cases studied can be found in Appendices A, B, and C.

The reasons for selecting these benchmark problems for testing are as follows:

  • The selected test functions have diverse characteristics, such as different Pareto optimal fronts (concave, convex, linear, and separated).

  • Solving constrained real-world problems is more challenging.

The results are compared to five well-regarded and recent multi-objective approaches, including the multi-objective slime mold algorithm (MOSMA) [21], multi-objective ant lion optimizer (MOALO) [36, 37], multi-objective multi verse optimizer (MOMVO) [36, 37], multi-objective particle swarm optimization (MOPSO) [8], and multi-objective evolutionary algorithm based on decomposition (MOEA/D) [50] to verify the superiority of MOCOVIDOA. According to their original works, the parameters associated with these algorithms have remained the same.

All algorithms were run 30 times, and the best results are reported in the tables below. We have utilized 200 iterations and 100 search agents for all algorithms and a size of 100. The parameters of MOCOVIDOA are set as follows: mutation rate (MR) = 0.05, number of viral proteins (\(\mathrm{num\, of\, Protiens}\)) = 2, and the shifting number = 1. These parameter values yield the best results during the experiments.

4.2 Evaluation metrics

Five metrics are utilized to evaluate the performance of MOCOVIDOA: GD, IGD, Hypervolume, and Delta P are used for measuring convergence of solutions, whereas SP and MS are used for measuring coverage as follows:

  • Generational distance (GD):

    $$GD=\frac{\sqrt{\sum_{i=1}^{no}{d}_{i}^{2}}}{n}$$
    (16)

Where no refers to the numerical solutions in the obtained Pareto optimal set, and di indicates the Euclidean distance between the ith obtained Pareto optimal solution and the closest true Pareto optimal solution in the reference set.

  • Inverse generational distance (IGD):

    $$IGD=\frac{\sqrt{\sum_{i=1}^{nt}{({d}_{i}^{^{\prime}})}^{2}}}{n}$$
    (17)

Where nt refers to the numerical solutions in the true Pareto optimal set, \({d}_{i}^{^{\prime}}\) Indicates the Euclidean distance between the ith true Pareto optimal solution and the closest Pareto optimal solution obtained in the reference set.

The GD and IGD metrics are used to quantify the convergence of solutions. In other words, they are used to measure to what extent the obtained POSs are close to the true POSs.

  • Spacing (SP):

    $$SP\triangleq \sqrt{\frac{1}{no-1}\sum_{i=1}^{no}({\overline{d }- {d}_{i})}^{2}}$$
    (18)

Where \(\overline{d }\) is the average of all \({d}_{i}\). SP is used to measure the coverage of solutions where lower SP values indicate better coverage.

  • Maximum spread (MS):

    $$MS=\sqrt{\sum_{i=1}^{o}\mathrm{max}(d({a}_{i} , {b}_{i}))}$$
    (19)

The o refers to the number of s used to measure the coverage of the obtained POSs.

  • Hyper volume (HV):

    $$HV=\mathrm{volume} (\bigcup_{i=1}^{A}{V}_{i})$$
    (20)

HV represents the volume covered by the non-dominated solutions (A). Larger HV values indicate better convergence and diversity. The hypervolume \({\mathrm{V}}_{\mathrm{i}}\) of the ith solution is calculated for a reference point which can be calculated by creating a vector of the worst objective function values.

  • Delta P (\(\Delta \mathrm{P}\)):

    $$\Delta P=max (mean \left(IGD\right), mean (GD))$$
    (21)

4.3 Results and discussion

4.3.1 Results of unconstrained test functions

The utilized unconstrained test functions comprise two well-known benchmark test suits, ZDT and CEC2009 (Zitzler et al., 2021, [5]. In the ZDT test suit, MOCOVIDOA is used to calculate the POSs for the 6 cases (ZDT1, ZDT2, ZDT3, ZDT4, ZDT6, and ZDT1 with linear Pareto front). The results of MOCOVIDOA are compared to five well-known approaches according to the previously mentioned evaluation metrics.

The results in Table 1 show the superiority of COVIDOA over its peers in most cases, and it achieves the best results in 30 of 36 test cases. However, in the remaining 6 cases, other algorithms slightly outperform it, such as MOSMA, MOALO, and MOMVO.

Table 1 Results for ZDT test suit
Full size table

The best Pareto fronts obtained from running all the ZDT test suit algorithms are shown in Figs. 6, 7, 8, 9, 10, 11. The figure shows high convergence and coverage of the Pareto optimum solutions obtained by the proposed algorithm since the obtained Pareto front is almost identical to the True Pareto front for all cases. The Pareto fronts of MOSMA and MOPSO are the closest to MOCOVIDOA, while MOMVO, MOALO, and MOEA/D show the lowest convergence and coverage. The comparison clearly shows that the Pareto front of MOCOVIDOA is better than other algorithms in all cases.

Fig. 6
figure 6

Obtained Pareto fronts for the ZDT1 test function

Full size image
Fig. 7
figure 7

Obtained Pareto fronts for the ZDT2 test function

Full size image
Fig. 8
figure 8

Obtained Pareto fronts for the ZDT3 test function

Full size image
Fig. 9
figure 9

Obtained Pareto fronts for the ZDT4 test function

Full size image
Fig. 10
figure 10

Obtained Pareto fronts for the ZDT6 test function

Full size image
Fig. 11
figure 11

Obtained Pareto fronts for the ZDT1LP test function

Full size image

Additionally, we conducted the Wilcoxon rank sum test to prove the superiority of the proposed algorithm. The p values obtained from comparing MOCOVIDOA against other algorithms are shown in Table 2.

Table 2 The P values obtained from the rank-sum test on ZDT1-ZDT6
Full size table

Smaller p values are strong evidence against the null hypothesis [46]. The null hypothesis is a type of hypothesis used in statistics that assumes no significant difference between the two methods' average values. All the p values shown in the table are less than 0.05 except the value obtained from comparing the performance of MOCOVIDOA with MOSMA for the ZDT1 test function, which shows that MOCOVIDOA is superior to MOSMA, MOPSO, MOMVO, MOALO, MOEA/D algorithms but has no significant difference with MOSMA for ZDT1 test function.

The CEC2009 test suit is divided into two-objective functions (UF1-UF7) and three-objective functions (UF8-UF10). As shown in Table 3, in the two objective functions, COVIDOA achieves the best results in all evaluation metrics for UF2 and UF4. For the remaining functions, MOCOVIDOA is the best for all metrics in UF1 except the spacing metric, for which MOMVO is the best. In UF3 and UF5 functions, MOEA/D achieves promising results in HV and \(\Delta P\) metrics. In UF5 and UF7, MOSMA is superior in terms of HV metric. According to the three-objective functions (UF8-UF10), MOCOVID achieves the best results for UF9. However, MOSMA and MOEA/D achieve the best values of spacing metrics for UF8 and UF10, respectively. The CEC2009 Pareto fronts obtained from running MOCOVIDOA are presented in Fig. 12.

Table 3 Results for CEC2009 test suit
Full size table
Fig. 12
figure 12

MOCOVIDOA Pareto front for CEC2009 test suit

Full size image

4.3.2 Results of constrained test functions

The second group of test functions includes benchmark functions with multiple constraints. A static penalty approach is utilized in the proposed algorithm to handle constraints. The penalty function aims to convert the constrained problem into an unconstrained one by using an artificial penalty for violating the constraint. After calculating the penalty for violating the constraints, this value is added to the fitness function value f(x) in the case of minimization problems or subtracted from f(x) in the case of maximization problems [48]. The five convergence and coverage metrics mentioned above are utilized to compare all algorithms' performance. Table 4 shows the results of solving the test functions: CONSTR, TNK, BNH, OSY, and SRN. The table shows that the MOCOVIDOA is superior in 21 of 30 cases. In the remaining cases, MOSMA, MOEA/D, and MOMVO slightly outperform MOCOVIDOA, especially in terms of SP and MS metrics.

Table 4 Results for TNK, CONSTR, BNH, SRN, and OSY functions
Full size table

The POSs obtained by MOCOVIDOA are presented in Fig. 13; it is obvious from the figure that the constrained test functions have different Pareto fronts, such as the concave front of the CONSTR and the wave-shaped front of the TNK function. The results showed that the proposed MOCOVIDOA could successfully approximate all these fronts. Additionally, the coverage of MOCOVIDOA is clear as the obtained Pareto fronts are well distributed along the true Pareto front.

Fig. 13
figure 13

MOCOVIDOA Pareto front for constrained problems: CONSTR, BNH, TNK, SRN, and OSY

Full size image

Finally, the superiority of MOCOVIDOA is supported by utilizing a set of popular real-world engineering design problems. The numerical results for all algorithms are shown in Table 5. Note that, the bold values in the tables indicate the best obtained results. As in the preceding test functions, the proposed MOCOVIDOA shows strong convergence and coverage properties compared to other algorithms. It outperforms other algorithms' peers in 26 of 36 cases. The MOCOVIDOA obtained Pareto fronts for engineering design problems are shown in Fig. 14.

Table 5 Results of different engineering design problems
Full size table
Fig. 14
figure 14

MOCOVIDOA Pareto front for engineering design problems

Full size image

The superior coverage of MOCOVIDOA is due to the COVID selection and archive mechanism, which includes exploring the uncovered areas of the search space by selecting solutions from less-populated neighborhoods from the archive and discarding solutions in the most crowded regions.

The proposed MOCOVIDOA uses the same updating mechanism as COVIDOA, based on the frameshifting technique for replication and mutation. Using the Frameshifting technique and mutation process with an appropriate mutation rate helps the proposed algorithm find the most promising region of the search space and converge to the set of POSs.

4.3.3 Sensitivity analysis

This section introduces a study about the sensitivity of MOCOVIDOA performance to different parameters as follows:

  1. A.

    A maximum number of iterations: We ran the proposed MOCOVIDOA using 100, 200, and 500 iterations to assess the impact of the number of iterations on MOCOVIDOA performance. The GD results of all cases displayed in Table 6 indicate that a maximum number of iterations of 200 yields the best results in most cases.

  2. B.

    Population size: A population size of 100, 200, and 500 is used to test the sensitivity of MOCOVIDOA to the number of solutions in the population. According to Table 7, the performance of MOCOVIDOA is better with the rise in population size. The results of 200 and 500 population sizes are close to each other; however, a population size of 200 would be preferable to reduce the algorithm's running time.

  3. C.

    Mutation rate (MR): The mutation rate greatly impacts the performance of MOCOVIDOA. We used three values for MR as follows: 0.5, 0.05, and 0.005, and we display the results in Table 8. It is shown from the table that an MR of 0.05 achieves the best GD values, which indicates that the value of MR should not be very high or very low to make a good balance between exploration and exploitation properties.

Table 6 Sensitivity analysis of a maximum number of iterations
Full size table
Table 7 Sensitivity analysis of population size
Full size table
Table 8 Sensitivity analysis of mutation rate (MR)
Full size table

5 Conclusion

This paper proposed the multi-objective version of the recently proposed COVIDOA called MOCOVIDOA, which aims to solve optimization problems with up to three objectives. It maintains the search mechanism of COVIDOA and an archiving concept and selection mechanism. The solutions in MOCOVIDOA are evaluated by using the Pareto optimal dominance operator. The proposed algorithm tests 27 test functions, including 16 unconstrained, five constrained, and six real-world engineering design problems. The performance is evaluated using different metrics, including IGD, GD, SP, HV, MS, and \(\Delta P\), presenting the best Pareto front obtained by the proposed algorithm. The superiority of the proposed algorithm is demonstrated by comparing it to five well-regarded multi-objective techniques: MOSMA, MOALO, MOPSO MOMVO, and MOEA/D. The results showed that the proposed MOCOVIDOA outperforms the other techniques in most test cases and its high convergence and coverage capabilities. Using test functions with different Pareto fronts demonstrates the ability of MOCOVIDOA to find different shapes of Pareto fronts. Finally, the results of solving different engineering problems demonstrate the capability of MOCOVIDOA to solve real-world multi-objective problems with multiple conflicting constraints.