1 Introduction

Real-world problems involve some optimization problems that are often constrained, and most of these problems are considered multi-objective optimization problems (MOOPs). No single solution exists for a MOOP; instead, different solutions generate trade-offs for different objectives. Furthermore, MOOPs arise naturally in most fields, and solving them has been a challenging problem for researchers [1,2,3]. Evolutionary computation (EC) methods have been identified as more effective methods to tackle the challenges that arise from the MOOPs, for which the form of the Pareto-optimal front (discontinuity, nonconvexity, etc.) is not important [4, 5]. Moreover, most multi-objective evolutionary algorithms (MOEAs) use the dominance concept [6,7,8,9,10,11].

To solve the constrained optimization of all real-world problems, constrained evolutionary algorithm optimization (CEAO) implements an evolutionary algorithm (EA) combined with a constraint-handling technique (CHT). In work by [12], an infeasible individual will be divided into different categories based on their distances to the feasible region, and ranking will be conducted according to the classes. The authors of [13] introduced an approach that assigns high and low priorities to constraints and objective functions, respectively. The authors of [14] proposed a CHT that only considers the inequality constraints, wherein the algorithm uses tournament selection that has better convergence properties in comparison to the proportionate selection operator [15]. However, the latter algorithm employs niche count for all populations, which may increase the complexity of the computation.

The authors of [16] introduced a novel approach that ignores any solution that violates any of the assigned constraints. The authors of  [17] first proposed the use of a genetic algorithm (GA) population-based approach plus a controlled mutation operator to keep diversity among feasible solutions. The work of [18] proposed a CHT where three different non-dominated rankings of the population are performed using objective function values, different constraints, and a combination of all objective functions and values. Although the technique can handle infeasible solutions carefully and maintain diversity in the population, the algorithm performs poorly in choosing parameter values and is computationally expensive. The authors of [19] developed an EA based on the nondominated sorting concept that uses the min–max formulation for constraint handling.

The authors of [20] ran the simulation of the non-dominated sorting genetic algorithm II (NSGA-II) on a seven-constrained nonlinear problem, which exhibited better performance than Ray-Tai-Seow’s algorithm. The authors of [21] conducted an overview and analysis of the most popular CHTs using EAs along with pros and cons. The authors of [22] combined a penalty function and multi-objective optimization technique, in which the ranking scheme is borrowed from the latter technique. The authors of [23] suggested two approaches, namely Objective Exchange Genetic Algorithm and Objective Switching Genetic Algorithm, for solving constrained MOOPs. A new partial order relation from the constraint MOOPs was proposed by [24], under which the Pareto optimum set satisfies the constraints. The authors of [25] introduced the Blended Space EA, which uses a rank obtained by blending an individual’s rank in the objective space to check dominance.

The authors of [26] introduced a two-phase algorithm, which separates the objective function and constraints. The authors of [27] introduced a MOO-based EA (Cai and Wang method), abbreviated as CW, in addition to three other models for constrained optimization. In the proposed approach, the simplex crossover was used to enrich the exploitation and exploration abilities. The authors of [28] proposed an EA based on an evolutionary strategy for constrained MOOPs. The method uses a min–max formulation for constraint handling in which feasible individuals and infeasible individuals evolve toward Pareto optimality and feasibility, respectively. The authors of [29] suggested Pareto Descent Repair (PDR) to search for feasible solutions. The authors of [30] proposed an adaptive tradeoff model for constrained evolutionary optimization to address three main issues: evaluating an infeasible solution in case the population contains only infeasible solutions,achieving a balance between feasible and infeasible individuals when the population contains both solutions; and selecting the feasible solution in case the population possesses only feasible solutions.

The authors of [31] suggested a heuristic hybrid of particle swarm optimization (PSO) and ant colony optimization for the optimum design of trusses, which showed to handle the problem-specific constraints using a fly-back mechanism. The work of [32] suggested an infeasibility-driven EA (IDEA), which can retain a proportion of infeasible solutions among the population members and preserve diversity compared to NSGAII. The authors of [33] investigated an EA solution for approximate Karush–Kuhn–Tucker (KKT) conditions of smooth problems. The results of some test problems indicate that EA’s operators lead the search process to a point close to the KKT point. The authors of [34] discussed the most critical techniques, many of which were previously proposed [21, 35]. The previous work also addressed some state-of-the-art constrained handling techniques, including feasibility rules based on GA [17], epsilon-constrained method [36], penalty functions [37, 38], and ensemble of constraint-handling methods [39, 40]. The authors of  [41] introduced an evolutionary scheme for handling boundary constraints, combined it with differential evolution (DE) and compared the proposed method with other boundary constraints handling techniques. The results indicated the proposed approach is much better than the existing methods.

The authors of [42] developed a water cycle algorithm, inspired by observations of the water cycle process that could be applied to a number of constraint optimization problems. The authors of [43] introduced a population-based algorithm based on the mine blast explosion concept and then applied the proposed approach to some constraint optimization problems in comparison to other well-known optimizers. The authors of [44] used a constraint consensus method that helps an infeasible individual to move towards the feasible region and then combined the method with a memetic algorithm. The research conducted by [45] developed a feasible-guiding strategy to guide the evolution of individuals, in which a revised objective function technique with a feasible guiding strategy based on NSGA-II is introduced to handle constrained MOOPs. The study proposed by [46] proposed a class of constraint handling strategies in which infeasible individuals are repaired when they are considered in the search space and explicitly preserve the feasibility of the solutions.

The authors of [47] used a hybrid of PSO and GA to improve the balance between exploration and exploitation by using genetic operators, namely crossover and mutation in PSO. A few years later, the authors of [48] extended the parameter-less CHT so as to provide a balance between the feasible and infeasible solutions in a GA population. The authors of [49] proposed a new approach, known as the boundary update (BU) technique, which is able to handle constraints implicitly by updating variable bounds. The BU approach was tested on several constrained optimization problems and found to be very efficient. The method proposed by [49] possesses the potential to couple with MOEA.

It is noteworthy to mention that the majority of the mentioned studies focused on CHTs for single-objective optimization with little attention to multi-objective optimization. This is attributed to the fact that most constraint handling methods developed for single-objective optimization could also be modified for multi-objective optimization[34]. The main contribution of this work is reviewing the most state-of-the-art constraint handling techniques in population-based optimization for single- and multi-objective optimization problems, and then the study addresses the bibliometric analysis, with a focus on multi-objective, in the field.

To attain a better understanding of the research field and to provide new insights from relevant publications, this work aimed to answer the following questions:

  • RQ1: What are CHTs, and how are they important?

  • RQ2: What are the disadvantages of the different CHTs?

  • RQ3: What are the key subjects and keywords regarding constraint handling techniques?

  • RQ4: Could we extract the most active journals, researchers, and countries in the field?

  • RQ5: What are the basic statistics of constraint handling techniques for multi-objective population-bassed algorithms?

  • RQ6: What are the most active countries and affiliations in the field?

  • RQ7: What are the gaps found in literature and future trajectory in the area?

The reminder of the study is as follows. Section 2 describes the research methodology. Section 3 presents the CHTs in EAs. Section 4 describes the other approaches. Section 5 addresses the benchmark test problems. Section 6 discusses the scientometric analysis. Section 7 provides a summary of the study along with recommendations for future research. Concluding remarks are offered in the last section.

2 Research Methodology

The research methodology in this work was divided into several stages (Fig. 1). First of all, documents from databases were gathered from databases, namely, Scopus and Web of Science (WOS). For this aim, the authors used special keywords, namely (TITLE-ABS-KEY (constrained AND multi AND objective AND evolutionary AND optimization) OR TITLE-ABS-KEY (constraint AND handling AND multi AND objective AND evolutionary AND optimization) OR TITLE-ABS-KEY (constrained AND multi AND objective AND swarm AND optimization) OR TITLE-ABS-KEY (constraint AND handling AND multi AND objective AND swarm AND optimization) to find the related articles published as of May 4, 2021. Supplementary A and B present the data extracted from Scopus and WOS, respectively. Since some of the articles were duplicates, they were identified and removed from the library in the next stage using Mendeley as a powerful reference manager. Also, some research questions for this study were designed. An overview, along with a general illustration of CHTs is provided in the next stage. A social network analysis, including co-occurrence, co-authorship, citation, and citation network analyses, is then conducted using VOSviewer [50, 51] and RStudio. Also, some interesting analytical features, such as number of pages and authors per article, were conducted in this stage. The last section required preparing the findings, identifying important gaps, and determining future research directions.

Fig. 1
figure 1

Research Procedure

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3 Constraint Handling Methods in Evolutionary Algorithms (RQ1)

Almost all real-world problems are considered constraint problems. A general form of a constrained multi-objective optimization problem (CMOOP) is described as follows (Eqs. 1, 2, 3):

$${\text{Maximize }}\left( {{\text{Minimize}}} \right)\,{\text{F}}\left( {\text{x}} \right) = \left( {f_{1} \left( x \right), \ldots ,f_{t} \left( x \right)} \right)$$
(1)

s.t.

$${h}_{i}\left(x\right)=0 i=1,\dots .,n$$
(2)
$${g}_{j}\left(x\right)\le 0 j=1,\dots ,m$$
(3)

where F(x) is the objective vector; and t, n, and m are the number of objective function, equality, and inequality constraints, respectively. There is no single solution for a MOOP that simultaneously optimizes each objective, instead, there exists a number of Pareto optimal solutions. A Pareto front of possible solutions is called optimal or nondominated if improving anyone's objective further would lead to a decrease in other objectives. According to previous surveys [21, 35], a simple taxonomy of the constraint handling methods in nature-inspired optimization algorithms is as follows:

  • - Penalty functions methods

  • - Decoderss

  • - Special operators

  • - Separation techniquess

The first and fourth techniques are discussed in detail later in the paper. As an example of decoders, [52] proposed a homomorphous mapping (HM) method between an n-dimensional cube and feasible space. The feasible region can be mapped onto a sample space where a population-based algorithm could run a comparative performance [52,53,54,55]. However, this method requires high computational costs. A special operator is used to preserve the feasibility of a solution or move within a special region [56,57,58]. Nevertheless, this method is hindered by the initialization of feasible solutions in the initial population, which is challenging with highly-constrained optimization problems. In addition, The authors of [59] presented a taxonomy of CHTs in MOEA as follows (Fig. 2):

  • Penalty functions

  • Separation method

  • Retaining the infeasible solutions

  • Hybrid methods

Fig. 2
figure 2

Taxonomy of different constraint handling methods in MOEA

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Generally, penalty function techniques are one of the most simple CHTs. There are several types of penalty functions used with EAs, the most important ones include [60]:

  • Death penalty

  • Dynamic penalty

  • Static penalty

  • Adaptive penalty

  • Stochastic ranking

Details regarding the penalty function methods will be discussed in the next section.

3.1 Penalty Function Approach

The penalty function method is one of the easiest and most common ways to handle constraints in multi-objective evolutionary algorithms.

From a mathematical point of view, two types of penalty functions could be considered as follows:

  • Interior methods

  • Exterior methods

In the first type of penalty function, interior methods, the penalty factor is selected such that the value will be small away from the constraint boundaries and need an initial feasible solution [59], whereas exterior methossds do not need an initial feasible solution [4]. Also, it should be noted that some of the infeasible solutions should be retained in the populations so that they are able to converge to a solution, which lies in the boundary between the feasible and infeasible regions [61].

The penalty function method ignores any infeasible solution [62]. First, all constraints should be normalized, and for each solution, the constraint violations are calculated as follows (Eq. 4):

$$w_{j} \left( {x^{i} } \right) = \left\{ {\begin{array}{*{20}l} {\left| {\overline{g}\left( {x^{i} } \right)} \right|,} \hfill & {if\,\overline{g}\left( {x^{i} } \right) < 0} \hfill \\ {0,} \hfill & {otherwise} \hfill \\ \end{array} } \right.,$$
(4)

where \(\overline{g }({x}^{i})\) refer to the normalized values for a given constraint \({\overline{g} }_{j}\left({x}^{i}\right)\ge 0, j=1,\dots ,J\). Once the violations for the constraints are calculated, the values are added to determine the overall violation as follows (Equation. 5):

$$\Omega \left( {x^{i} } \right) = j = \sum\limits_{j = 1}^{J} {w_{j} \left( {x^{i} } \right)}$$
(5)

Also, a penalty parameter is multiplied by the sum of constraint violations and then added to the objective function values. If a proper penalty parameter is selected, MOEAs will work well; otherwise, a set of infeasible solutions or poor solutions distribution is possible.

3.1.1 Static Penalty Functions

In the static penalty proposed by [63], the penalty coefficient increases as a higher level of violation is reached. In fact, penalty functions do not change, a static penalty function is suggested, and several levels of violation are introduced in which the static penalty parameter is changed in case higher levels of violation are achieved [64]. In the static penalty function, the expanded objective function is (Equation 6):

$$\varphi \left( {\text{x}} \right) = {\text{f}}\left( {\text{x}} \right) + \mathop \sum \limits_{{{\text{j}} = 1}}^{{\text{p}}} {\text{C}}_{{{\text{kj}}}} {\text{G}}_{{\text{j}}}$$
(6)

where \({G}_{j}={max\left\{0,{g}_{j}(x)\right\}}^{\beta }\); and k = 1,…,l where l presents the number of violation levels.

3.1.2 Dynamic Penalty Functions

In this category, functions are changed based on the iteration number. The authors of [65] proposed the following dynamic penalty function, in which the penalty increases when the iteration number increases.

Dynamic multi-objective optimization problems (DMOOPs) involve the simultaneous optimization of different objectives subject to a number of given constraints, where the objective functions, constraints, and/or dimensions of the objective space could change over time. EAs have acquired great attention among researchers for solving the above-mentioned problems.s

3.1.3 Adaptive Penalty Functions

In this category, infeasible individuals are penalized according to the feedback taken from the search process. The authors of [66, 67] proposed a CHT based on the adaptive penalty function and distance measure, which both change as the objective function value and constraint violations of an individual varies.

Penalty-based constraint handing for multi-objective optimization is similar to single-objective problems in which a penalty factor is added to all the objectives. The authors of [67] proposed a self-adaptive penalty function suitable for solving constraint multi-objective optimization problems using evolutionary algorithms. In the self-adaptive penalty function method, the amount of penalty added to infeasible individuals are identified by tracking the number of feasible individuals. Also, the method uses improved objective values instead of the original objective function values [68].

3.1.4 Annealing-Based Penalty Functions

The authors of [69] introduced a multiplicative penalty function based on simulated annealing. In this type of penalty function, the temperature is decreased when the iteration number increases, which leads to an increased penalty.

3.1.5 Co-Evolutionary-Based Penalty Functions

The authors of [21] proposed a co-evolutionary approach in which the population is partitioned into two subpopulations. The first population evolves solutions, and the second population evolves penalty factors. In this approach, the penalty function considers information taken from the amount of constraint violations and a number of violations.

There are other types of penalty function methods, and Table 1 presents a summary and critique of the techniques for constraint handling.

Table 1 The important CHTs (penalty function)-RQ2
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3.2 Separation of Objective Function and Constraints

Unlike the penalty function technique, another approach exists that separates the values of objective functions and constraints in the nature-inspired algorithms (NIAs) [70], which is known as the separation of objective function and constraints. The authors of [71] initially proposed the idea of dividing the search space into two phases. In the first phase, feasible solutions are found, and optimizing the objective function is considered in the second phase.

Representative methods of this type of CHT are as follows:

  • Constraint dominance principle (CDP)

  • Epsilon CHT

  • Feasibility rules

The next section provides further details about this type of constraint handling method.

3.2.1 Constraint Dominance Principle

Three feasibility rules are applied to compare the two solutions in the constraint dominance principle (CDP). If \({x}^{1}\) is feasible and \({x}^{2}\) is infeasible, then \({x}^{1}\) would be better than \({x}^{2}\). If both solutions are infeasible, then the solution with a smaller constraint violation is better. If both are feasible, then the one dominating the other is better. The authors of [20] adopted CDP to handle constraints in NSGAII (NSGAII-CDP), in which the population is divided into feasible and infeasible sub-populations. NSGAII-CDP first selects offspring from the feasible solutions and then selects solutions from the infeasible solutions. The authors of [72] also adopted CDP to handle constraints in the MOEA/D framework.

3.2.2 ε-Constrained (EC) Method

The basic principle of the ε-constrained method, first introduced by [73], is similar to the superiority of feasible solution (SF) proposed by [74] (Equations. 12–13). The epsilon value is updated until the parameter k reaches the control generations \({T}_{c}\). The authors of [75] embedded the epsilon CHT in MOEA/to set the epsilon value adaptively to r comparison. Also, the violation threshold is based on the constraint type, the feasible space size, and the search outcome. In the method proposed by [75], the infeasible solutions with violations less than threshold are identified (Eqs. 7–8).

$$\varepsilon \left( 0 \right) = V\left( {x_{\theta } } \right)$$
(7)
$$\varepsilon \left( k \right) = \left\{ {\begin{array}{*{20}l} {\varepsilon \left( 0 \right)\left( {1 - \frac{k}{{T_{c} }}} \right)^{cp} ,} \hfill & {0 < k < T_{c} } \hfill \\ {0,} \hfill & {k > T_{c} } \hfill \\ \end{array} } \right.$$
(8)

where \({x}_{\theta }\) presents the top \(\theta\) th individual at initialization; and the cp parameter is selected between [2,10, 68].

3.2.3 Feasibility Rules

The popularity of this method depends on its ability to be coupled to a range of algorithms without announcing new parameters (factors) [34].

The feasibility rules proposed by [17] are simple, could be integrated into a variety of algorithms without adding new parameters, and thus, are largely used in the research field. The authors of [76, 77] developed feasibility rules for the selection process, which have been adopted by different evolutionary algorithms such as DE, PSO, and GA. According to the number of feasible solutions, the search space could be divided into three phases as follows [68]:

  • No feasible solution is found.

  • There exists at least one feasible solution.

  • Integrating the parent–offspring population has more feasible solutions than the size of the next generation population.

The feasibility rules used in multi-objective optimization, also known as the superiority of feasible solution (SF), are addressed as follows [68] (Equation 9):

$$fitness_{m} \left( x \right) = \left\{ {\begin{array}{*{20}l} {f_{m} \left( x \right)} \hfill & {if\, x\, is \,feasible} \hfill \\ {f_{worst}^{m} + v\left( x \right)} \hfill & {otherwise} \hfill \\ \end{array} } \right.$$
(9)

where \({f}_{worst}^{m}\) and v(x) show the mth objective value of the worst feasible solution and the overall constraint violation, respectively.

3.2.3.1 Feasibility Rules in Differential Evolution (DE)

Although the feasibility rules introduced by [17] have also been widely used by other researchers in DE [78], [79] [77, 80,81,82,83], [84] [30], they have been rarely used in multi-objective differential evolution. Particularly, [85] used Pareto dominance in constrained space instead of the sum of constraint violations. Later, the authors of [79] adopted the Pareto dominance in Generalized Differential Evolution (GDE), but encountered difficulties when there exist more than three constraints and/or objective functions.

The authors of [78] proposed a scheme for partitioning the objective space using the conflict information for multi-objective optimization. The authors of [86] introduced an operational efficient model based on Data Envelopment Analysis (DEA) and introduced DE along with the feasibility rules to optimize the mentioned model. The authors of [87] proposed a combined constraint handling framework, known as CCHF, for solving constrained optimization problems, in which the features of two well-known CHTs (i.e. feasibility rules and multi-objective optimization) were addressed in three different situations (feasible situation, infeasible situation, and semi-feasible situation).

3.2.3.2 Feasibility Rules in PSO

The authors of [88] employed feasibility rules as a constraint handling technique to recognize the most competitive PSO variant when solving constrained numerical optimization problems (CNOPs). In the research by [88], local-best was identified to be better than global best PSO. The authors of [89, 90] adapted an artificial bee colony algorithm (ABC) to solve CNOPs by using feasibility rules by modifying the probability assignment for the roulette wheel selection. The authors of [91] compared different GA variants using the feasibility rules as the constraint handling method. In the work of [92], a hybrid version of PSO to solve constrained optimization problems was introduced and the authors found that the swarm at each generation is split into several sub-swarms. Also, the hybrid version applied the feasibility rules to compare particles in the swarm.

3.2.3.3 Feasibility Rules in GA

The authors of [93] proposed a GA with a new multi-parent crossover for solving constraint optimization problems, in which the feasibility rules were added to handle the constraints. The latter authors also solved constraint numerical optimization problems by using different GA variants along with feasibility rules and found that all GAs perform equally. In the work of [26] a two-phase framework for solving constraint optimization problems was introduced. Specifically, the first phase ignores the objective function and the genetic algorithm minimizes the violation of the solutions, while the second phase optimizes bi-objective functions, including the original objective function and constraints satisfaction. Moreover, feasibility rules is applied to assign fitness values to the individuals.

3.2.3.4 Feasibility Rules in Other Population-Based Algorithms

Feasibility rules have been adapted to other population-based algorithms, such as artificial immune systems [94,95,96,97,98,99], organizational evolutionary algorithm [100, 101], biogeography-based optimization [102], and bacterial foraging optimization [103].

3.3 Retaining Infeasible Solutions in the Population

Another CHT is used to retain the infeasible individuals in the population. In other words, a constraint multi-objective optimization problem with m objective is transformed to an optimization problem with m + 1 objectives, which could save the infeasible solution during the evolution process [32, 104, 105] proposed a constraint handling technique so that individuals with low Pareto rank and low constraint violation will be chosen.

3.4 Hybrid Methods

Hybrid methods combine several CHTs to handle constraints. The authors of [106] addressed four different hybrid methods in their work: (1) the ensemble of constraint-handling method [68] separates the population into three sub-populations; (2) the adaptive trade-off model [107, 108] contains two different CHTs, (3) in the push and pull search [109], the population is pushed to the unconstrained Pareto front (push) then the population is pulled back to the Pareto front (pull), (4) the two-phase framework (ToP) [110] solves a constraint multi-objective optimization problem by first converting the objective functions into a single objective function via the weighting method then, in the second phase, a constrained MOEA is adopted to attain the Pareto feasible solutions. The disadvantages for each category are summarized in Fig. 3.

Fig. 3
figure 3

The main disadvantages of CHTs in MOEA- RQ2

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The authors of [34] presented a state-of-the-art taxonomy of CHTs, which is illustrated in Figs. 4 and 5 presents the different state-of-the-art CHTs that have been used since 2000. As mentioned, the authors of [17] first applied feasibility rules to the genetic algorithm. The authors of [111] introduced stochastic ranking, which employs a user-defined parameter instead of using penalty factors and is able to control the infeasible solutions based on the sum of constraint violation and objective function values. In a study of [38], the epsilon-constraint method was proposed to transform the constraint optimization problem into an unconstrained one. The authors of [38] addressed a multi-constrained optimization problem based on the KS function. The authors of [112] proposed a boundary search approach inspired by the ant colony metaphor based on conducting a boundary search between a feasible and infeasible solution. The authors of [32] proposed an additional objective to solve a bi-objective optimization problem, where the first objective is the original problem and the second objective is the constraint violation measure. In the work of [39], a combination of four CHTs, namely feasibility rules, stochastic ranking, self-adaptive penalty function, and the epsilon-constraint method to solve constraint numerical optimization problems, was addressed.

Fig. 4
figure 4

State-of-the-art CHTs [34]

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Fig. 5
figure 5

Timeline of different state-of-the-art CHTs

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3.5 Stochastic Ranking

The authors of [111] proposed the stochastic ranking (SR) approach to balance between the objective and penalty functions stochastically. The method was tested using a strategy evolution on several benchmarks, and the results showed that the method is able to improve the search performance with a user-defined parameter without introducing complicated variation operators. SR has also been coupled with other population-based algorithms, such as ant colony optimization (ACO) [113, 114], differential evolutionary (DE) [115,116,117], and evolutionary programming (EP) [118].

3.6 Ensemble Techniques

Ensemble CHTs provide a new research platform to tackle constrained multi-objective optimization problems. Combining several CHTs could improve the capability of an approach compared with a single CHTs [34, 119]. For instance, [120] proposed a combination of four CHTs, namely nondominated sorting, constrained-domination principle, multiple constraint ranking, and dynamic penalty function, and incorporated the proposed technique into an MOEA based on NSGAII. Some other ensemble CHTs have been reported [39, 121, 122]. Although the ensemble CHT has a competitive performance, it suffers from being parameter-dependent.

3.7 Multi-Objective Concept

Based on the multi-objective optimization concept, a constraint single-objective optimization problem is transferred to an unconstrained multi-objective optimization problem [4]. The multi-objective version of the optimization problem possesses an extra objective function, which presents the sum of constraint violation [32, 32,123,124,125].

The authors of [126] presented a taxonomy for constraint handling strategies in multi-objective GA, which include:

  • Penalty function methods

  • Separation method

  • Special operators

  • Repair methods

Among these strategies, the penalty function method is not straightforward in multi-objective GA since the fitness assignment is based on the non-dominance rank of a solution rather than its objective function values [127]. Yet, the penalty function method is one of the most popular CHTs in constraint multi-objective optimization. Whenever the multi-objective function and constraint violation for each constraint are assessed, the sum of violations is added to each objective function value considering the multiplication of the penalty parameter [128, 129].The authors of [23] proposed two approaches, namely OEGADO and OSGADO. The OEGADO runs several GAs in parallel so that each GA optimizes one objective, whereas The OSGADO runs each objective sequentially with a common population for all objectives.

3.8 Repair Approaches

There are several techniques used as repair algorithms, in which the search space is reduced (since only feasible individuals are considered):

  • In the permutation encodings method, each solution of an EA population is simply signified as an ordered list [130, 131].

  • Repair procedures in binary representations, which could be shown as fixing the number of 1 s in binary representations and Hopfield networks [132, 133].

  • Repair methods in graphs are represented as spanning trees and repairing graphs [134, 135].

  • Repair methods in grouping GAs, are proper for scenarios where a number of items should be assigned to a set of groups [136, 137].

Pure EAs do not perform well in complex combinatorial problems with a high number of constraints [138, 139]. Single-solution-based algorithms (e.g. local search, simulated annealing) have good performance in exploitation, while population-based algorithms (e.g. swarm intelligence, EA are exploration-oriented. In these problems, the hybridization of population-based algorithms with single-based algorithms can improve the power of both exploration and exploitation [139,140,141]. A memetic algorithm is a hybridization of an EA and a local search (LS) approach that LS is applied to improve the quality of the fitness function. On the other hand, LS could be used as a CHT [139], i.e. the local repair algorithm only consider feasible individuals leading to reducing the search space. Repair methods could be applied to EAs in several ways, such as in permutation encodings [142], [131], in binary representation [143], and in graphs and trees [134–144]. Although repair algorithms have numerous advantages, some disadvantages do exist. For instance, repair algorithms are problem-specific and must be designed for a specific problem [145]. Table 2 shows a summary of the disadvantages of the state-of-art CHTs.

Table 2 A summary of disadvantages of the state-of-art CHTs- RQ2
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4 Other Approaches

Table 3 provides a summary of novel approaches proposed between 2020 and 2021 to tackle constrained multi-objective optimization problems. Based on Table 3, there are signs of a renewed interest in constrained multi-objective optimization, even the clear superior amount of research in constrained single-objective optimization.

Table 3 Novel approaches for constrained multi-objective optimization problems between 2020 and 2021
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As a general, a taxonomy of CHTs in MOEAs could be summarized in Fig. 6. The CHTs presented in Fig. 6 have been explained in details in previous sections. As it is mentioned earlier, most constraint handling techniques developed for single-objective optimization problems can be applied to MOOPs. It is worthy to note that among them, stochastic ranking [72] [146]–150], penalty function [23, 67], multi-objective method [32] [151] [127] [152], Epsilon constrained method [153–160], transforming method [106, 161, 100, 101, 162–166], feasibility rules (with a modification) [167, 168], hybrid methods [169–173], and repair operators [148, 149, 174–178] have been addressed to multi-objective optimization problems.

Fig. 6
figure 6

A generalized taxonomy of CHTS for multi-objective optimization problems

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5 Benchmark Test Problems

Many benchmark or test problems have been suggested to measure the evolutionary algorithms' performance. On the other hand, benchmark problems help researchers better understand an algorithm's strengths and weaknesses [179]. These test problems are classified as single-objective such as Rosenbrock [180], G01-G09 [181], Himmelblau’s problem [182], Welded Beam [183], Pressure Vessel [184], Speed Reducer [185], Corrugated bulkheads design [186], Heater exchanger [187], Multiple disk clutch brake [188],

Rolling element bearing [189], Car side design [190], Stepped beam design problem [191], multi-objective including BNH [192], OSY [193],, ZDT [194], BT [107, 108], Truss2D [195], and many-objective optimization problems, for example, C-DTLZ [196], WFG[197], DTLZ [198]. Among the above-mentioned test problems, some of them are still unconstrained. More details are suggested in a review paper in the field by [197].

6 Scientometric Analysis (RQ3- RQ6)

A scientometric analysis is conducted to scientifically measure and analyze the literature in a particular field of study and has attracted much attention from researchers 119–208]. To perform the analysis in this work, VOSviewer [51] and RStudio were used. The following sub-sections provide new insight into the scientometric analysis in the field.

6.1 Citation Statistics

Figure 7 displays the trend of published documents, which shows that the number of documents in the field significantly increased from 2003 until the end of 2021 (just above 470 documents).

Fig. 7
figure 7

Trend of published documents

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Figure 8 presents a combo chart of the number of documents vs. total citations. In 2002, the most citations was achieved (the paper entitled: “A fast and elitist multi-objective genetic algorithm: NSGA-II” [20] has received more than 43,000 citations, according to Scopus). It is apparent that the number of citations has increased dramatically according to the trend.

Fig. 8
figure 8

Combo chart of number of documents vs. total citations (Scopus)

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According to the WOS, the number of citations of the top articles in the field was analyzed and is presented in Supplementary C (Fig. 1). Of the 735 related documents in WOS, about 45,824 citations were identified from the related papers, with an average of 1992.35 citations per year and an average of 62.35 citations per item. [20, 209, 210] are the top 3 cited articles with 20,013, 2609, and 1591 citations in WOS, respectively.

6.2 Statistics Based on Document Types

Among the document types, including articles, proceedings papers, reviews, and other items indexed by WOS, a total of 735 publications on constraint handling multi-objective population-based optimization algorithms were found (Table 4). From the search, articles were the most popular document type, comprising a total of 522 articles (71.02% of 735 documents) with 2.60 authors per publication (APP). Also, articles as the document type had the highest CPP2021 of 84.10, followed by proceedings papers with TP of 220 (29.93% of contributions and APP = 2.13). Moreover, there is a significant difference between the TC2021 article and the proceedings paper.

Table 4 Citations analysis based on document type
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Figure 9 presents the distribution of documents based on different types, according to WOS. It is clear from the figure that conference papers have the most contributions before 2010, followed by articles. However, since 2010, articles have had the most contributions in the field. It is also interesting to note that book/book chapters have been published since 2000. However, most book/book chapters have been published after 2010.

Fig. 9
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Type of research outputs

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6.3 Publication Statistics Based on Journal

Table 5 presents the top 20 journals that have published the greatest number of constraint handling multi-objective population-based algorithms papers based on Scopus. Accordingly, Lecture Notes In Computer Science (117), Applied Soft Computing Journal (57), and Swarm and Evolutionary Computation (30) are most are most utilized, which predominate in the field of optimization and evolutionary computations.

Table 5 The top 20 sources that have published the greatest number of constraint handling multi-objective population-based algorithms (Scopus)
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A total of 735 articles were published in 399 journals, which are classified among the 51 WOS categories in SCI-EXPANDED. Table 6 lists the 10 most productive WOS categories. A total of 271 articles (36.87% of 735 articles) were published in the first category (Computer Science Artificial Intelligence), of which 83.39% were published in Engineering Electrical Electronic (23.26%) and Computer Science Theory Methods (23.26%). Comparing the top 10 categories, the highest CPP2021 of articles published in the Computer Science Theory Methods category is 190.011, which includes the paper entitled: “A fast and elitist multi-objective genetic algorithm: NSGA-II” by [20], and the highest APP for articles published in the Energy Fuels category is 2.97.

Table 6 The top 10 productive WOS categories
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Figure 10 provides a comparison of the development trends of the top four productive WOS categories, including “Computer Science Artificial Intelligence”, “Engineering Electrical Electronic”, “Computer Science Theory Methods”, and “Computer Science Interdisciplinary Applications”. Between 2001 and 2021, Computer Science Artificial Intelligence was the most predominant category and has possessed the highest number of publications since 2004, excluding the period between 2007 and 2008. The three other categories possess fluctuations between 2001 and 2021, and as of writing this paper, “Computer Science Theory Methods” and “Engineering Electrical Electronic” have the same TP of 171.

Fig. 10
figure 10

Comparison of the development trends of the top four productive WOS categories

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6.4 Publication Statistics by Countries

From Fig. 11, China, India, and the USA are the top three active countries in the field according to Scopus (respectively), while China, the USA, and India are the top 3 active territories in the field based on WOS, respectively. It is pertinent to mention that the USA is ranked second based on WOS, but India is ranked second, according to Scopus. Also, it can be seen that there is a significant difference between the first rank (China) and second rank (India) based on the number of publications indexed by Scopus. Moreover, Fig. 12 presents the collaboration among countries, where the links across the circles depict the collaborations, and the circles' size represents the countries' activities in the field. The green and yellow colors present the keywords that have been used recently, while the dark blue indicates those used earlier (around 2008).

Fig. 11
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Research output of top 10 most productive countries across all databases

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Fig. 12
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Country network visualization

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Figure 13 displays the growth rate of the top 5 active countries in comparison to the world. While China and the USA have smooth trends between 2000 and 2021, India, the UK, and Australia show some fluctuations. Between 2002 and 2003, India presented the highest growth rate, then the trend continued smoothly until 2014, when it increased until 2015. The trend for the UK shows two growths between 2002–2003 and 2007–2008. While the number of articles published by Australia is much less than the four other countries, there was a significant rise between 2015 and 2016.

Fig. 13
figure 13

Growth rate of published documents for the top 5 countries

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6.5 Statistics Based on the Subject Area

Figure 14 presents the distribution of articles based on the subject area. Computer science, Engineering, and Mathematics possess the most contributions, with 936, 619, and 580 published articles, respectively. Comparatively, Pharmacology, Medicine, and Economics own the least contributions, with 1, 4, and 6 published documents in the field, respectively.

Fig. 14
figure 14

Distribution based on the subject area (Scopus)

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6.6 Statistics Based on Authors

Figures 15 shows the top authors with the most publication according to Scopus (Supplementary C, Figure 2 presents statistics based on WOS). Kalyanmoy Deb from “Michigan State University (USA)”, Ray T. from “University of New South Wales (Australia)”, and Carlos A. Coello Coello from “Cinvestav-IPN (Mexico)” with 38, 32, and 28 publications are the top 3 authors in the field indexed by Scopus. WANG Y from “City University Hong Kong (Hong Kong)”, Carlos A. Coello Coello from “Cinvestav-IPN (Mexico)”, and Ray T. from “University of New South Wales (Australia)” are the top 3 authors in the area with 21, 20, and 17 documents (indexed by WOS), respectively. According to WOS, 1717 authors have worked on constraint multi-objective population-based optimization.

Fig. 15
figure 15

The most active authors in the field (Scopus)

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In total, 0.5241% of authors own more than 10 documents; 1.6307% possess between 5 and 10 documents; 4.5428% have between 3 and 5 papers; 11.7647% own 2 papers; and 81.5377% possess 1 document (Fig. 16). Figure 17 presents the collaboration among the authors, where links across the circles depict the collaborations, and the circles' size shows the authors activities in the field. In addition, the yellow color represents recent activity, and the dark blue color depicts the contributions prior to 2014.

Fig. 16
figure 16

Contribution of authors based on the number of documents

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Fig. 17
figure 17

Collaboration among the authors (overlay visualizatio

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6.7 Statistics on Keywords

Keywords indicate the basic parts of a certain field of research and could offer insight into the organization and knowledge provided in the articles. Figure 18 provides an overlay visualization of the co-occurrence analyses via a network map based on the Scopus database. Each node in the network represents a keyword, and the link between nodes indicates the co-occurrence of the keywords. The top keywords in Scopus include multi-objective optimization, Pareto optimal solution, evolutionary algorithm, artificial intelligence, machine design, stochastic systems, distributed power generation, and reliability. The color of each circle represents the identified cluster, and each circle's size illustrates the keywords' importance. Table 7 presents the top keywords of 1-word, 2-word, and 3-word lengths extracted from Scopus. Specifically, optimization, algorithm, and scheduling are the top 1-word length keywords indexed by Scopus; genetic algorithm, constraint handling, and constrained optimization are the top 2-word length keywords; and constraint-handling techniques, Particle Swarm Optimization (PSO), and multi-objective optimization are the top 3-word length keywords indexed by Scopus.

Fig. 18
figure 18

Network visualization of keywords

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Table 7 Top 1-,2-, and 3- word keywords used in the field
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6.8 Publication Statistics by Number of Pages (Pages Count)

As of writing this paper, May of 2021, approximately 22395.8 pages of documents on constraint handling multi-objective population-based algorithms were published, with an average of 13.0435 pages per paper. About 22.53931% of the articles possess between 10 and 15 pages; 12.17239% of the manuscripts are between 15 and 20 pages; 39.07979 % of the papers are between 5 and 10 pages; and 67.09377% of the manuscripts are between 5 and 20 pages. Figure 19 presents the distribution of the manuscripts based on page count.

Fig. 19
figure 19

Distribution of documents based on page count

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7 Summary and Future Research (RQ7)s

The paper presents an analysis and overview of CHTs applied to multi-objective population-based algorithms. The first part of the paper defines the main idea of CHTs, and the second part discusses a detailed scientometric analysis of the field. Some important technical points are extracted as follows:

  • In the death penalty method, no information is used from infeasible points.

  • The static penalty method is problem-dependent and may need several penalty parameters.

  • Dynamic penalty method may converge to either an infeasible or feasible solution that is far from the global optimum.

  • The main disadvantage of the annealing penalty method is its sensitivity to its factors' values.

  • Setting parameters in the adaptive penalty method is difficult, and the method needs the definitions of additional parameters.

  • The additional parameters may affect the fitness function evaluations in the self-adaptive penalty method.

  • The main difficulty in SGA is selecting the penalty factors for each sub-population.

  • If the population is completely infeasible, choose solutions with a smaller overall constraint violation.

  • Retaining a proportion of infeasible solutions in the population may enhance the convergence and diversity of the algorithm.

  • Two types of CHTs, namely repair methods and special genetic operators, focus only on the feasible space.

  • Feasible solutions could be used to repair infeasible solutions (repairing population).

  • According to the constraint dominance principle, the feasible solution is always preferred over the infeasible solution, which may cause loss of important information from infeasible individuals.

  • Retaining a huge number of infeasible solutions may cause low convergence speed.

  • Although special operators are known to be highly comparative CHT, their applicability is limited, which makes this technique difficult to run.

  • Decoder is an interesting CHTs, but it involves a high computational cost and, thus, is now rarely used.

  • Although the ensemble CHT has a competitive performance, the method is parameter-dependent.

  • Although the stochastic ranking method has been employed in several nature-inspired algorithms, it is not often used for the multi-objective version of the algorithms.

  • Epsilon constraint method has been known as a powerful CHT, however, in some cases, premature convergence has been reported, while other works report that the method relies on gradient-based mutation.

  • Using multi-objective concept as a CHT may require gradient calculation.

  • Recently, feasibility rules have been recognized as one of the most powerful CHTs, which are simple and flexible; however, one of the major disadvantages of this method is premature convergence since this technique favors feasible solutions.

As a future direction, the authors have identified the top 5 most-used keywords and research fields in the last three years (2019–2021) based on Scopus. Tables 8 and 9 show the mentioned keywords and research fields for this time period. It is obvious that multi-objective optimization, constraint optimization, and evolutionary algorithms are the most famous keywords in the last three years. It should be noted that CHTs for multi-objective optimization has not received much attention compared with single-objective optimization. It is suggested that researchers focus on such methods in future works. Also, the BU technique, which is able to handle constraints directly, possesses the potential to couple with a multi-objective evolutionary algorithm (MOEA) as well. Furthermore, it is suggested to focus on constraint handling techniques for many-objective optimization problems (with more than three objectives) as it is not received much attention. In addition, according to Tables 9 and, GA, DE, and PSO remain the top 3 algorithms, which are expected to be further explored in the future. Moreover, Engineering, Computer Science, and Mathematics have been the top 3 research fields in the last two years, and it is projected that research work will advance in these areas in the future. It is also recommended to review the applications of constrained multi-objective evolutionary algorithms in different sectors; including engineering design problems [212], scheduling optimization problems [176] [214,215,216], and resource optimization problems [217].

Table 8 Top 5 keywords in 2019 and 2021
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Table 9 Top 5 research fields in 2019 and 2021
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8 Discussion and Conclusion

Constraint population-based optimization involves using a population-based algorithm combined with a CHT to solve a constraint optimization problem. The first part of the paper defines the main idea of CHTs, and the second part discusses detailed scientometric analysis of the field. It is noteworthy that most of the mentioned studies in the literature focused on CHTs for single-objective optimization with little attention to multi-objective optimization. This paper presents an analysis and evaluation of the CHTs, focusing on multi-objective optimization population-based algorithms, which support evolutionary and swarm intelligence algorithms. To the best of our knowledge, this study is the first analysis of relevant journals evaluated over the most relevant journals, keywords, authors, and articles in this field. All related papers, including research articles, reviews, book/book chapters, conference papers, etc., were extracted and analyzed. Publication statistics by year, journal, country, affiliation, author, number of pages, number of authors, and keywords are discussed in this paper as follows:

  • According to WOS, 45,824 citations have been received by the related papers, which is an average of 1992.35 citations per year and an average of 62.35 citations per item in WOS.

  • Based on WOS, articles were the most popular document type, with a total of 522 articles (71.02%), and 2.60 authors per publication.

  • Articles as the document type had the highest CPP2021 of 84.10, followed by proceedings papers with TP of 220 (29.93% of contributions and APP = 2.13).

  • Conference papers have the most contributions before 2010 followed by articles. However, since 2010, articles have had the most contributions in the field.

  • A total of 271 articles (36.87% of the total), with 2.31 authors per publication (on average), were published in the Computer Science Artificial Intelligence category, according to WOS.

  • In total, 271 articles (36.87% of 735 articles) were published in the first category (Computer Science Artificial Intelligence), and a total of 83.39% were published in the first three categories: Engineering Electrical Electronic (23.26%) and Computer Science Theory Methods (23.26%).

  • The highest CPP2021 of articles published in Computer Science Theory Methods is 190.011, which includes the paper “A fast and elitist multi-objective genetic algorithm: NSGA-II” by [20], and the highest APP for articles published in ‘Energy fuels’ is 2.97.

  • “Computer Science Artificial Intelligence,” “Engineering Electrical Electronic,” “Computer Science Theory Methods,” and “Computer Science Interdisciplinary Applications” were the top 4 productive WOS categories in the field.

  • China, USA, and India were the top three active countries in the field, according to WOS.

  • Computer science, Engineering, and Mathematics have the most contributions, with 936, 619, and 580 published articles, respectively. Pharmacology, Medicine, and Economics own the least contributions, with 1, 4, and 6 published documents in the field, according to Scopus.

  • Kalyanmoy Deb from “Michigan State University (USA)”, Ray T. from “University of New South Wales (Australia)”, and Carlos A. Coello Coello from “Cinvestav-IPN (Mexico)” are the top 3 authors in the field with 38, 32, and 28 publications (indexed by Scopus), respectively. WANG Y from “City University Hong Kong (Hong Kong)”, Carlos A. Coello Coello from “Cinvestav-IPN (Mexico)”, and Ray T. from “University of New South Wales (Australia)” are the top 3 authors in the area with 21, 20, and 17 documents (indexed by WOS), respectively.

  • Almost 0.5241% of authors own more than 10 documents; 1.6307% possess between 5 and 10 documents; 4.5428% have between 3 and 5 papers; 11.7647% of authors own 2 papers; and 81.5377% of authors possess 1 document.

  • Approximately 22.53931% of the articles possess between 10 and 15 pages; 12.17239% of the manuscripts are between 15 and 20 pages; 39.07979% of the papers are between 5 and 10 pages; and 67.09377% of the manuscripts are between 5 and 20 pages.