Introduction

In many practical engineering optimization applications, there are often many problems with multiple conflicting objectives and may be necessary to be optimized at the same time, which are called multi-objective optimization problems (MOPs) [1,2,3,4]. For example, in the mixed flow shop scheduling problem with consistent sublots [1], the maximum completion time and total quantity of sublots were minimized at the same time. In the economic dispatch [2], researchers proposed a multi-objective optimization model for the unified allocation of water resources and wastewater discharge permits, which took into account the maximization of economic benefits of unit wastewater discharge. In the controller design [3], the application of an adaptive controller in the speed regulation of the four-bar mechanism was studied. In [4], the multi-objective optimization problem of job scheduling in cloud data center was proposed. Generally speaking, there is no single optimal solution to these problems, but rather a set of trade-off solutions. If the performance of one objective is improved, the performance of another objective may be reduced. Then it is the main purpose of MOPs to find balanced solutions among several conflicting objectives.

In recent years, multi-objective evolutionary algorithms (MOEAs) have become the focus of a lot of research. They have been found to provide a very promising performance in solving different types of MOPs [5,6,7,8]. For example, particle swarm optimization (PSO) [5], genetic algorithm (GA) [6], differential algorithm (DE) [7], ant colony optimization (ACO) [8], etc. Based on the policy approaches they use, the existing MOEAs can be classified into the following three categories. The first category is Pareto-based MOEAs. Since the optimal result of MOPs is a set of Pareto optimal solutions, the Pareto dominance relationship naturally becomes the criterion of differentiation in the evolutionary process of the algorithms. The representative algorithms include nondominated sorting genetic algorithm II (NSGA-II) [9], improving the strength Pareto evolutionary algorithm (SPEA2) [10], grid-based evolutionary algorithm (GrEA) [11] and shift-based density estimation (SDE) [12] and so on. The second category is the MOEAs based on indicators. The principle is that the objectives of MOPs are aggregated through a scaling function to generate a single scalar value, and then a single performance indicator is used to guide the search in the evolutionary process. The representative algorithms include SMS-EMOA [13], TS_R2EA [14] and HypE [15]. The third category is MOEAs based on decomposition. The principle is to transform the multi-objective optimization problem into a series of single-objective optimization sub-problems, and then optimize the sub-problems simultaneously. The diversity of the population is maintained by specifying a set of well-distributed reference points (or directions) to guide its individuals to search for different directions simultaneously. The representative algorithms include MOEA/D [16], MMOPSO [17] and MOEA/D-LWS [18].

In MOEAs, PSO was proposed more than twenty years ago, and initially, PSO was mainly used to solve single-objective optimization problems [19]. Compared with other evolutionary algorithms, PSO has the characteristics of lower computational cost and higher convergence speed. The good results of PSO in solving standard operating procedures validate the effectiveness of its optimization scheme, especially in a large and complex environment. Later, this also prompted researchers to extend PSO from single objective to multi-objective to solve MOPs [20]. Due to its relatively low computational cost, multi-objective particle swarm optimization (MOPSO) is also an advantage in real-life applications, such as multi-objective load scheduling in microgrids [21], carpool optimization [22] and job-shop scheduling [23] and so on.

PSO also faces many problems in the process of expanding from single objective to multi-objective. The first is the selection strategies of the global best (gbest) and the personal best (pbest). In PSO, each particle is guided by gbest and pbest, and different selections will lead to different flight directions of a particle, so different selection strategies have an important impact on convergence and diversity. Therefore, the selection of gbest and pbest is an important factor in MOPSO. As the number of objectives of the problems we are facing now is increasing, the traditional MOPSO [24] can not effectively select gbest and pbest by using the roulette selection method and dominance relationship, respectively. Therefore, we need better selection strategies for gbest and pbest. Of course, many researchers have put forward many selection strategies at present. For example, Cui et al. [25] selected the parents from the convergence archive and diversity archive, respectively, and generated offspring from the parents through crossover and mutation. Then they selected the excellent ones as gbest. Han et al. [26] selected solutions with convergence and diversity as gbest by using the distribution entropy of solutions. Coello et al. [27] used decomposition to determine the global optimal solutions. Xiong et al. [28] adopted the sparsity method to judge the particle with large sparsity of the two solutions in the current population as the individual history optimal scheme for the next iteration. The second is the external elite archive maintenance. In the process of evolution, the external archive can be used as a repository for the gbests. Therefore, to obtain a well-distributed approximate Pareto front, an appropriate archiving strategy is necessary to ensure that those non-dominated solutions with higher performance are retained. This reserves sufficiently excellent candidates for the next selection of gbest of particles. The gbest can guide the particle swarm to approach the true Pareto front and improve the exploration ability of the algorithm. So far, there have been some excellent examples. For example, Cui et al. [25] introduced a two-archive mechanism and designed a convergence archive and diversity archive to balance convergence and diversity and produce more high-performance candidates of gbest. Yang et al. [29] proposed a new method to maintain the external archive based on the vector angle value of each solution, which can balance convergence and diversity. Of course, some researchers believe that the selection of pbest and gbest will cost a certain amount of computing resources. For example, Zhang et al. [30] combined the competition mechanism with MOPSO and proposed a competitive MOPSO algorithm. This method has no selection of pbest, gbest, and no use of external archive compared to MOPSO. The leaders were selected by pairwise competition of particles in the population, which reduces the computational cost.

Based on the above problems, we get two key problems in MOPSO. The first is the selection of group leaders, namely the gbest and pbest, to provide the right search direction for all particles. The second is the maintenance of external archive to preserve high-quality solutions to promote the evolution of the population. This helps accelerate convergence and also keeps the diversity of the population. Inspired by [31,32,33,34], we propose a two-stage maintenance and multi-strategy selection for multi-objective particle swarm optimization (TMMOPSO). In this algorithm, each particle will be guided by two leaders, namely gbest and pbest. The gbest is selected from the external archive by making appropriate use of the hyper-cone domain. The pbest is selected by aggregation index. If the previous generation is better, it means that the population is deteriorating. Therefore, to make sure that the evolution of the population, we use an information synthesis to make the particles evolve in the right direction. To improve the convergence speed and the exploration ability of the population, first of all, the particles after flight are hybrid pretreated, and then the excellent particles are crossed and mutated to generate excellent offspring. These more excellent offspring are entered into the external archive. Then the hypercube is used to adopt a two-stage maintenance strategy for the external archive. At present, the grid-based algorithms are also very competitive in solving MaOPs, and hypercubes are very effective for measuring the distribution of solutions according to their positions. Examples include ε-MOEA [35] and GrEA [11]. Compared with existing MOPSOs, the new aspects of our proposed TMMOPSO are as follows.

  1. (1)

    In TMMOPSO, the update strategies of gbest and pbest are designed. The update of gbest adopts the adaptive hybrid strategy, which can greatly improve the convergence of the population. In addition, an aggregation method is proposed to update pbest, which has a stronger advantage in higher dimensions than the traditional pbest update.

  2. (2)

    An archive updating strategy combining hypercube and two-stage maintenance is proposed. Firstly, the hypercubes are established in the external archive. Secondly, the particles in the hypercube with high density are deleted by the two-stage strategy. Finally, keep the good performance and remove the bad performance. Thereby accelerating the convergence speed and ensuring the diversity of the population.

  3. (3)

    To improve the optimization efficiency and accelerate the optimization process, an optimal disturbance scheme is also proposed. The simulated binary crossover (SBX) and polynomial mutation (PM) are applied to excellent particles to produce excellent offspring. The offspring entering the external archive can improve the quality of solutions in the external archive and better guide the evolution of the population.

The rest of this paper is planned as follows. The following section introduces the related work. “The proposed TMMOPSO” gives a detailed introduction to TMMOPSO, which describes the selection strategy of gbest and pbest, the maintenance mode of the external archive and the optimal disturbance strategy, respectively. In “Simulation experiment”, the experimental research is introduced, comparing TMMOPSO with the current five MOPSOs and five MOEAs. In addition, the advantages of TMMOPSO are also verified in “Simulation experiment”. Finally, “Conclusion” concludes the paper.

Related work

Multi-objective optimization problem

For the general minimization multi-objective optimization problem [31], the mathematical expression is as follows:

$$ \begin{aligned} & \min \;F\left( x \right) = \left( {f_{1} \left( x \right),f_{2} \left( x \right), \ldots ,f_{m} \left( x \right)} \right) \nonumber \\ & {\mathrm{s.t.}}\;g_{i} \left( x \right) \ge 0,i = 1,2, \ldots ,q \nonumber \\ & h_{j} \left( x \right) = 0,j = 1,2, \ldots ,r , \end{aligned} $$
(1)

where \(x = \left( {x_{1} ,x_{2} , \ldots ,x_{n} } \right)\) is a n-dimensional decision vector in the decision space \(X \subset R^{n}\), \(F\left( x \right)\) represents the objective vector of m-dimension, \(f_{i} \left( x \right)\) represents the objective function of minimizing the i-th dimension, \(g_{i} \left( x \right) \ge 0\left( {i = 1,2, \ldots ,q} \right)\) is an inequality constraint, and \(h_{j} \left( x \right) = 0\left( {j = 1,2, \ldots ,r} \right)\) is an equality constraint.

There are always mutually exclusive objectives in MOPs, so the solutions of MOPs cannot optimize all objectives, and because different solutions can not be compared among different objectives, solving the MOPs is to search a set of non-dominated solutions in the search space. This set of solutions is called the Pareto optimal set. The relevant definitions [24] are given below:

Definition 1

(Pareto dominance) Given two decision vectors \(x_{A}\) and \(x_{B}\), it is said that \(x_{A}\) dominates \(x_{B}\) (denoted by \(x_{A} \prec x_{B}\)), if and only if

$$ \begin{aligned} & \forall i \in \left\{ {1,2, \ldots ,m} \right\}:f_{i} \left( {x_{A} } \right) \le f_{i} \left( {x_{B} } \right) \wedge , \nonumber \\ & \exists j \in \left\{ {1,2, \ldots ,m} \right\}:f_{j} \left( {x_{A} } \right) < f_{j} \left( {x_{B} } \right). \end{aligned} $$
(2)

Definition 2

(Pareto optimal solution) If a solution \(x\) is called Pareto optimal solution, if and only if \(x\) is not dominated by any other solution.

Definition 3

(Pareto optimal set) For a given MOP, the set composed of all Pareto optimal solutions is called the Pareto optimal set (PS), namely:

$$ {\mathrm{PS}} = \left\{ {x^{*} \in X|\neg \exists x \in X,x \prec x^{*} } \right\}. $$
(3)

Definition 4

(Pareto front) For a given MOP, Pareto front (PF) is defined as follows:

$$ {\mathrm{PF}} = \left\{ {F\left( {x^{*} } \right) = \left( {f_{1} \left( {x^{*} } \right),f_{2} \left( {x^{*} } \right),...,f_{m} \left( {x^{*} } \right)} \right)|x^{*} \in {\mathrm{PS}}} \right\}. $$
(4)

Multi-objective particle swarm optimization

MOPSO [24] is an optimization algorithm that expands from a single objective to multiple objectives based on PSO. It also inherits the properties of PSO, such as the position and velocity of particles are still influenced by gbest and pbest. Therefore, the particles of MOPSO are updated in the following way:

$$ \begin{aligned} v_{i,d} \left( {t + 1} \right) & = \omega v_{i,d} \left( t \right) + c_{1} r_{1} \left( {{\mathrm{pbest}}_{i,d} \left( t \right) - x_{i,d} \left( t \right)} \right) \nonumber \\ & \quad + c_{2} r_{2} \left( {{\mathrm{gbest}}_{i,d} \left( t \right) - x_{i,d} \left( t \right)} \right), \end{aligned} $$
(5)
$$ x_{i,d} \left( {t + 1} \right) = x_{i,d} \left( t \right) + v_{i,d} \left( {t + 1} \right), $$
(6)

where \(d = 1,2, \ldots ,D\), D is the dimension of decision space, t is the number of iterations, \(\omega\) is the inertia weight, \(c_{1}\) and \(c_{2}\) are learning factors, which are acceleration coefficients used to control the influence of \({\mathrm{pbest}}_{i,d} \left( t \right)\) and \({\text{g}}best_{i,d} \left( t \right)\) on particle velocity, respectively, \(r_{1}\) and \(r_{2}\) are two random values uniformly generated in \(\left[ {0,1} \right]\). The right side of Eq. (5) is composed of the inheritance of the current speed, individual cognition, and social cognition.

The proposed TMMOPSO

The main challenge of MOPSO is to balance the convergence and diversity of the population. Compared with the original MOPSO, the key points of TMMOPSO are to put forward a new strategy of automatically selecting gbest, aggregation index to update pbest, and an external archive maintenance strategy combining adaptive hypercube with two stages. The particles with good convergence and diversity are retained, so that the population can approach the true PF better and faster. In addition, to improve the performance of the algorithm, an optimal perturbation strategy is developed to generate excellent offspring from excellent parents.

Adaptive selection strategy

In PSO, the potential direction of population evolution is significantly related to the selection strategies of gbest and pbest. Therefore, a suitable selection strategy can improve the ability of the population to converge to the true PF. In addition, considering that the population is prone to fall into local optimum when solving complex MOPs, the updated population may lack diversity. Therefore, an adaptive hybrid gbest selection strategy is proposed. However, the dominance relationship is not so important in high-dimensional problems. In many cases, it will only be a non-dominated relationship. To solve this problem, this paper puts forward a new strategy of selecting the pbest.

Adaptive hybrid selection strategy of gbest

In MOPSO, the gbest of each particle is determined from the PS by using roulette wheel selection. In this paper, the gbest is determined by the adaptive hyper-cone domain to ensure that the particles in this domain learn from this gbest. Firstly, the non-dominated solutions generated in the external archive are used as the candidates of gbest. After having the candidates, the inner radius and outer radius of the hyper-cone domain are determined according to these candidates. The inner radius and outer radius of the i-th hyper-cone are defined as:

$$ R_{i,in} = \frac{{y_{{\mathrm{in}},m} }}{{\sqrt {y_{{\mathrm{in}},1}^{2} + y_{{\mathrm{in}},2}^{2} + ... + y_{{\mathrm{in}},m - 1}^{2} } }}, $$
(7)
$$ R_{i,ou} = \frac{{y_{{\mathrm{ou}},m} }}{{\sqrt {y_{{\mathrm{ou}},1}^{2} + y_{{\mathrm{ou}},2}^{2} + ... + y_{{\mathrm{ou}},m - 1}^{2} } }}, $$
(8)
$$ y_{{{\text{in}}}} = \frac{{F\left( {x_{i - 1} } \right) + F\left( {x_{i} } \right)}}{2}, $$
(9)
$$ y_{{\mathrm{ou}}} = \frac{{F\left( {x_{i} } \right) + F\left( {x_{i + 1} } \right)}}{2}, $$
(10)

where m represents the number of objectives, \(y_{{{\text{in}}}} = \left( {y_{{\text{in,1}}} ,y_{{{\text{in,}}2}} ,...,y_{{{\text{in,}}m}} } \right)\), \(y_{{\mathrm{ou}}} = \left( {y_{{{\mathrm{ou}},{1}}} ,y_{{\mathrm{ou}},2} ,...,y_{{\mathrm{ou}},m} } \right)\), \(R_{i,{\mathrm{in}}}\) and \(R_{i,{\mathrm{ou}}}\) are the inner radius and outer radius, respectively, \(F\left( {x_{i} } \right)\) represents the fitness value of the i-th particle, \(i \in 1,2,...,s\), s is the number of solutions in the PS. When \(i = 1\) or \(i = s\), the hyper-cone domain can be determined by calculating only its outer radius and the inner radius, respectively. The hyper-cone domain is delineated according to the inner radius and outer radius (as shown in Fig. 1). The advantages of the hyper-cone domain are that, on the one hand, there is only one candidate of gbest in a hyper-cone domain, which is conducive to the particles finding its own gbest. On the other hand, with the progress of the search, the workspace should be narrowed and the solutions gradually approach the true PF. This can greatly improve the convergence of the population. The hyper-cone domain is just suitable for such a process and is therefore adopted. For this strategy, there are two other possible situations in the learning evolution process. Firstly, in the process of learning evolution, the particles may be unevenly distributed in some hyper-cone domains. In this case, the candidates of the gbest of particles are controlled by the circular strategy. The purpose of this method is to increase the distribution of the population. Secondly, the particles may be concentrated in certain hyper-cone domains (see Fig. 2), which will result in a waste of the resources of the gbest. In this case, the Euclidean distance is used to measure the distance between the particles and the non-dominated solutions. For each particle, the nearest non-dominated solution is selected as the gbest leader of the particle, which makes the limited resources be utilized to the maximum, ensures the good distribution of the population, and increases the exploration ability of the algorithm. The pseudo-code of the gbest update is shown in Algorithm 1.

Fig. 1
figure 1

Establish the hyper-cone, PS is the Pareto optimal set and X is the solution

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Fig. 2
figure 2

Fall into the local optimum, PS is the Pareto optimal set and X is the solution

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figure a

Selection strategy of pbest

In general, the traditional update strategy of pbest is that if the updated particle dominates the current pbest, then the updated particle will replace the pbest of the next iteration. Otherwise, the pbest will not be updated. Although this traditional updating method is simple to maintain the population diversity, it has little influence on the dominance relationship in high-dimensional MOPs, and most of the time is a non-dominated relationship, and the learning efficiency is relatively low.

In this paper, we propose a strategy to update pbest in an aggregated manner. In this strategy, each objective is equally aggregated into a scalar metric, which is used for evaluation update. This index is defined as:

$$ c\left( {x_{i} } \right) = \sum\limits_{j = 1}^{m} {f_{j}^{^{\prime}} \left( {x_{i} } \right)}, $$
(11)
$$ f_{j}^{^{\prime}} \left( {x{}_{i}} \right) = \frac{{f_{j}^{{}} \left( {x{}_{i}} \right) - z_{j}^{\min } }}{{z_{j}^{\max } - z_{j}^{\min } }}, $$
(12)

where m is the number of objectives, \(f_{j}^{^{\prime}} \left( {x{}_{i}} \right)\) is the standardization of each objective, \(z_{{}}^{\min } = \left( {z_{1}^{\min } ,z_{2}^{\min } ,...,z_{m}^{\min } } \right)\) is the ideal point, and \(z_{j}^{\min } = \min f_{j} \left( {x_{i} } \right)\), \(z_{j}^{\max } = \max f_{j} \left( {x_{i} } \right),j = 1,2,...,m\) is the lowest point. The update mode of the pbest is as follows:

$$ {\mathrm{pbest}}_{i} \left( t \right) = \left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {x_{i} \left( t \right),} & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {} & {} & {} \\ \end{array} } & {} \\ \end{array} } & {} \\ \end{array} \;\;\;\;c_{t} \left( {x_{i} } \right) < c_{t - 1} \left( {x_{i} } \right)} \\ \end{array} } \\ {\begin{array}{*{20}c} {x_{i} \left( t \right) + N\left( {0,1} \right) \cdot x_{i} \left( {t - 1} \right),} & {c_{t} \left( {x_{i} } \right) \ge c_{t - 1} \left( {x_{i} } \right)} \\ \end{array} } \\ \end{array} } \right., $$
(13)

where \({\mathrm{pbest}}_{i} \left( t \right)\) is the individual history optimum of the i-th particle of t-th generation, and N(0,1) is Gaussian distribution. If \(c_{t} \left( {x_{i} } \right) < c_{t - 1} \left( {x_{i} } \right)\), that is, this generation is smaller than the previous generation, it means that the particle has learned and evolved, choosing the current position as \({\mathrm{pbest}}_{i} \left( t \right)\). If the optimal weighted value of the individual history of the previous generation is less than or equal to that of the current generation, it indicates that the particle does not learn and evolve. Then the information synthesis of the particle is carried out, and the synthesis method is the information fusion of the last and this time. In this way, it is more helpful to select the preeminent pbest and improve the convergence speed of the algorithm. The pseudo-code of its update strategy is shown in Algorithm 2.

figure b

Two-stage maintenance strategy

The maintenance of the external archive is crucial in MOPSO, which stores non-dominated solutions and its maximum threshold is the population size. When the number of non-dominated solutions exceeds the set archive maximum threshold, we need to perform a maintenance on the archive so that the number of stores does not exceed its maximum threshold. What we need to do is to select the solutions with good convergence and diversity among the optimal solutions to retain, and remove the non-dominated solutions with poor performance. Here we use the adaptive hypercube (when there are two objectives: it is the adaptive grid), and the densest hypercube is processed by the two-stage strategy. Firstly, according to the range of non-dominated solutions, we establish hypercubes, the number of hypercubes used here is not fixed, using a linear decreasing strategy. The purpose is to ensure that the number of particles in the grid in which the non-dominated solutions are located is not one. If the number is one, reduce the number of grids so that the density is not one. Secondly, the grid with the highest density is deleted in a two-stage strategy. The two-stage strategy is the use of comprehensive index SDE [12] in the first stage and the use of convergence index Achievement Scalarizing Function (ASF) [36] in the second stage. The adaptive two-stage maintenance strategy balances convergence and diversity to a certain extent. The pseudo-code of this strategy is shown in Algorithm 3.

figure c

A comprehensive index SDE is adopted in the first stage. Firstly, the positions of the moving non-dominated solutions in the objective space are compared according to Eq. (14). Then, the displacement distances of the non-dominated solutions after moving are calculated using Eq. (15). Finally, the density \(D_{i}\) of the i-th estimated non-dominated solution is calculated according to Eq. (16).

$$ f_{q,j} = \left\{ {\begin{array}{*{20}c} {f_{i,j} ,} & {f_{q,j} < f_{i,j} } \\ {f_{q,j} ,} & {f_{q,j} \ge f_{i,j} } \\ \end{array} } \right.\begin{array}{*{20}c}& {} \\ \end{array}, $$
(14)
$$ d_{i,q} = \sum\limits_{j = 1}^{m} {\left| {f_{q,j} - f_{i,j} } \right|}, $$
(15)
$$ D_{i} = \min \left( {d_{i,1} ,d_{i,2} ,...,d_{i,q} } \right), $$
(16)

where \(i = 1,2,...,n\), \(n\) denotes the number of non-dominated solutions to estimate the current density, and \(j = 1,2,...,m\), \(m\) denotes the number of objectives, and \(q = 1,2,...,n - 1\) denotes the non-dominated solutions to be moved. \(d_{i,q}\) denotes the distance calculated between the i-th non-dominated solution and the non-dominated solution \(q\) after displacement. The important thing to note here is that the smaller the density estimate, the denser the particles and the worse the diversity and convergence.

The second stage uses ASF, which can evaluate the convergence of the solution. The smaller the ASF value, the better the convergence. Therefore, what we delete here is the solution with a larger ASF value, which fully guarantees the convergence of the population. ASF is defined as follows:

$$ {\mathrm{ASF}} \left( {x_{j} ,z^{\min } ,w} \right) = \mathop {\max }\limits_{i = 1}^{M} \left( {\frac{{f_{i} \left( {x_{j} } \right) - z_{i}^{\min } }}{{w_{i} }}} \right)\;, $$
(17)
$$ w_{i} = \frac{{f_{i} \left( {x_{j} } \right)}}{{\sum\nolimits_{s = 1}^{M} {f_{s} \left( {x_{j} } \right)} }}, $$
(18)

where \(x_{j}\) is the solution in the densest hypercube, \(j = 1,2,3,...,\left| R \right|,\) \(z^{\min }\) is the ideal point, M is the number of objectives and \(w_{i}\) is the i-th generation favorable weight vector. In addition, according to [36] in this paper, if \(w_{i} = 0\), then set \(w_{i} = 10^{ - 6}\).

The optimal perturbation

Recent references [25, 36] show that excellent parents can produce better offspring after the operation, which can improve the performance of the algorithm. Therefore, this paper proposes an optimal disturbance strategy using a turbulence operator. The purpose of optimal perturbation in this paper is to produce better offspring. Let the PS be closer to the true PF. The specific method is that in the process of particle iteration, the hybrid strategy is first used to pretreat the particles. The hybrid strategy is inspired by [37] and is defined as:

$$ {\mathrm{new}}p = \left\{ {\begin{array}{*{20}c} {p_{{}} + N\left( {0,1} \right) \cdot \left| {{\mathrm{pbest}} - {\mathrm{gbest}}} \right|\;,} & {x \ge \alpha } \\ {p_{{}} + C\left( {0,1} \right) \cdot \left| {{\mathrm{pbest}} - {\mathrm{gbest}}} \right|\;,} & {x < \alpha } \\ \end{array} } \right.\begin{array}{*{20}c} , & {} \\ \end{array} $$
(19)

where \(p\) represents the position of particles before pretreatment, \({\mathrm{new}}p\) represents the position of particles after pretreatment, \(C\left( {0,1} \right), N\left( {0,1} \right)\) is Cauchy factor and Gaussian factor, which are the probability density based on Cauchy distribution and Gaussian distribution of \(x\), respectively. \(\alpha = 2 - \tfrac{2t}{T}\). Secondly, by selecting excellent parents for crossover and mutation [38], excellent particles can become better, more excellent particles can be generated, and offspring of higher quality can be obtained, which has a great impact on the performance of the algorithm. It can produce non-dominated solutions with better performance so that the population is closer to the true PF. The specific pseudo-code is shown in Algorithm 4.

figure d

The complete algorithm of TMMOPSO

Algorithm 5 shows the pseudo-code of TMMOPSO. It first initializes the particle swarm (line 1), which includes the position and velocity of the particles and pbest. Then, The maximum number of iterations is performed on the main loop of the algorithm. Secondly, the velocity and position of the particles are updated (line 4). Then select the excellent particles from the updated particles for disturbance (line 5), and the perturbed particles are saved in the archive for archiving update. Finally, update gbest and pbest (lines 7 and 8). The basic process of the algorithm is similar to most MOPSOs, which is to output a set of feasible solutions generated by a new population after several iterations.

figure e

Simulation experiment

Test problems and comparison algorithms

To evaluate the performance of the proposed algorithm, the TMMOPSO proposed in this paper will be evaluated using three sets of benchmark suites: ZDT [39], UF [40] and DTLZ [41]. These include five two-objective test problems for the ZDT benchmark suite, seven two-objective test problems and three three-objective test problems for the UF benchmark suite, seven DTLZ test problems with three objectives and seven DTLZ test problems with five objectives. The involved benchmark problems have different complex features and PF shapes, which makes it difficult for the algorithm to obtain an approximate PF with good diversity and convergence. It is important to note that ZDT5 is a discrete optimization problem, so it is not included in the benchmark problems used in this paper. The DTLZ8 and DTLZ9 are two constrained optimization problems, therefore they are not included in the class either.

To evaluate the performance of TMMOPSO, we compare the proposed TMMOPSO with five MOPSOs: CMOPSO [30], NMPSO [42], MPSOD [43], dMOPSO [27] and SMPSO [44]. In addition, to further evaluate TMMOPSO, we will compare it with five MOEAs: DGEA [45], IDBEA [46], MOEA/D [16], NSGAII [9] and SPEAR [47]. These algorithms have been proved to be competitive in solving MOPs.

Algorithm performance indicators

In this paper, three widely used performance indicators are used to evaluate the performance of the algorithms: Inverted Generational Distance (IGD) [48], Hypervolume (HV) [49] and Generational Distance (GD) [50].

  1. (1)

    IGD is used to measure the distance between the Pareto optimal solutions obtained by the algorithm and the true PF, which can well test the convergence and diversity of the algorithm. Let \(P^{ * }\) represents a set of uniformly distributed solutions in the objective space along the PF. \(S\) is the Pareto optimal set obtained by the algorithm. The IGD is described as:

    $$ {\mathrm{IGD}}\left( {P^{ * } ,S} \right) = \frac{1}{{\left| {P^{ * } } \right|}}\sum\limits_{{x^{ * } \in P^{ * } }} {\min {\mathrm{dist}}\left( {x^{ * } ,S} \right)} , $$
    (20)

where \(\min {\mathrm{dist}}\left( {x^{ * } ,S} \right)\) is the minimum Euclidean distance between solution \(x^{ * }\) in \(P^{ * }\) and \(S\), The smaller the IGD value of the algorithm, the better its convergence and diversity. IGD indicator has been widely used as the performance indicator of MOPs.

  1. (2)

    The HV indicator is also a comprehensive performance indicator, which is the volume of the region enclosed by the PS obtained by the algorithm and uniformly sampled points in the objective space. This indicator can estimate the convergence and diversity of the solution set obtained by the algorithm. The larger the HV value of the algorithm, the better the convergence and diversity of the algorithm. Supposing \(Z^{r} = \left( {Z_{1}^{r} ,Z_{2}^{r} , \ldots ,Z_{m}^{r} } \right)\) is a reference point dominated by all Pareto optimal solutions in the objective space, then the calculation formula of HV is defined as:

    $$ {\mathrm{HV}}\left( S \right) = \delta \left( {\mathop \cup \limits_{{_{x \in S} }} \left[ {f_{1} \left( x \right),z_{1}^{r} } \right] \times \cdots \times \left[ {f_{m} \left( x \right),z_{m}^{r} } \right]} \right), $$
    (21)

    where \(S\) denotes the Pareto optimal solution set obtained in the algorithm, \(\delta\) denotes the Lebesgue measure.

  1. (3)

    The GD indicator measures the average distance from the Pareto optimal solutions to the nearest neighbors on the true PF, it is defined as:

    $$ {\mathrm{GD}}\left( {P^{ * } ,S} \right) = \frac{1}{{\left| {S^{{}} } \right|}}\sum\limits_{s \in S} {\mathop {{\text{min}}}\limits_{{x^{ * } \in P^{ * } }} {\mathrm{dist}}\left( {x^{ * } ,s} \right)}, $$
    (22)

    where \(\min {\mathrm{dist}}\left( {x^{ * } ,S} \right)\) is the minimum Euclidean distance between solution \(x^{ * }\) in \(P^{ * }\) and \(S\). The smaller the GD value, the better the algorithm performance.

Parameter settings

In this paper, the TMMOPSO is compared with five existing MOPSOs and five representative MOEAs. These algorithms have been proved to be highly competitive in solving many complicated MOPs. To ensure the fairness of comparison, the parameter settings of all algorithms are consistent with the original references, as shown in Table 1.

Table 1 Parameter settings of all algorithms
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The parameter settings of the other ten comparison algorithms on the three benchmark suites are shown in Table 2, where N represents the population size, M is the number of objectives, D is the dimension of decision space, and FEs is the maximum evaluation times of the algorithm in the running process. All algorithms are independently run 30 times on each test problem, and all the experimental data of the algorithms are implemented on Intel(R) Core(TM) i5-10300H CPU @ 2.50 GHz 2.50 GHz, Windows 10 system, MATLAB R2020b. The source codes of the comparison algorithms are provided by PlatEMO [51].

Table 2 Parameter settings of all test problems
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Comparisons with five MOPSOs

Tables 3, 4, and 5 show the mean and standard deviation of five MOPSOs: CMOPSO, NMPSO, MPSOD, dMOPSO, SMPSO, and TMMOPSO proposed in this paper, which run independently on three benchmark suites ZDT, UF and DTLZ for 30 times. Where "mean" and "std" respectively represent the mean and standard deviation, and the optimal results on IGD, HV and GD are shown in bold.

Table 3 IGD values of five MOPSOs and TMMOPSO on all test problems
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Table 4 HV values of five MOPSOs and TMMOPSO on all test problems
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Table 5 GD values of five MOPSOs and TMMOPSO on all test problems
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According to the results summarized in the last row of Table 3, TMMOPSO achieves the best IGD on 18 test problems of the three benchmark suites ZDT, UF, and DTLZ, and the best IGD indicator results are obtained on more than half of the test problems. It shows good performance. Based on the evaluation indicators in this paper, it can be concluded from Table 3 that there are four test problems on ZDT benchmark suite that perform better than the comparison algorithms CMOPSO, NMPSO, MPSOD, dMOPSO and SMPSO. Only on ZDT2, the performance is poor, but it is second only to CMOPSO, and the performance is the second best, which is also beyond the other four comparison algorithms. On six test problems of the UF series outperform the comparison algorithms, and it is the second best on UF2 and the third best on UF1, UF5 and UF6 after CMOPSO and NMPSO. The DTLZ series of benchmarks perform well on four three-objective and four five-objective test problems. Based on the above analysis, under the IGD indicator, TMMOPSO outperforms the comparison algorithms on most of the test problems and performs better on the ZDT benchmark suite.

The comparison results of the HV indicator of TMMOPSO and the other five MOPSOs are shown in Table 4. Just like the conclusion drawn from the comparison results of IGD, TMMOPSO performs best on most of the test problems out of a total of 29 benchmark problems and is superior to the other comparison algorithms in terms of convergence and diversity. Table 5 shows the comparison results of the GD indicator between the TMMOPSO algorithm and the other five MOPSOs. According to the above empirical results, it can be concluded that the proposed TMMOPSO is highly competitive with the existing MOPSOs, so it can achieve better performance in terms of convergence and diversity when solving MOPs.

To be able to see the optimal solutions and the ideal solutions more intuitively, we also give the corresponding PF graphs of the six algorithms on the three test problems of ZDT3, UF9 and DTLZ6. As shown in Figs. 3, 4 and 5, through observation, the solutions obtained by TMMOPSO more uniformly approximate the true PF in these test problems. However, the approximate PF obtained by other comparison algorithms cannot be approximated as well as that obtained by TMMOPSO. For example, in Fig. 3, TMMOPSO and CMOPSO are closer to the true PF of ZDT3, while none of the remaining algorithms are significantly close to the true PF. But only TMMOPSO maintains good distribution and convergence. In Fig. 4, comparing the distribution, TMMOPSO has a better distribution near the true PF of ZDT3 compared with other comparison algorithms. In Fig. 5, TMMOPSO and CMOPSO are closer to the true PF, but the distribution of TMMOPSO is better than CMOPSO. In addition, the NMPSO in Fig. 5 is also close to the true PF, but the number of NMPSO distributed nearby is not much.

Fig. 3
figure 3

PF plots of five MOPSOs and TMMOPSO on ZDT3 test problem

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Fig. 4
figure 4

PF plots of five MOPSOs and TMMOPSO on UF9 test problem

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Fig. 5
figure 5

PF plots of five MOPSOs and TMMOPSO on DTLZ6 test problem

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Figure 6 shows the convergence trajectories of the test problems ZDT3, UF9, and DTLZ6. Through observation, it is found that TMMOPSO achieves the optimal IGD value of the algorithm earlier than the comparison algorithms, which indicates that our proposed algorithm accelerates the convergence speed of the population. Figure 7 shows the box plots of IGD values of TMMOPSO and the comparison algorithms running 30 times independently on the ZDT and DTLZ of the three objectives benchmark suites, which depicts the discrete distribution of data of the algorithms on the ZDT and DTLZ of the three objectives benchmark suites. In the figures, the abscissas coordinates 1, 2, 3, 4, 5 and 6 represent CMOPSO, NMPSO, MPSOD, dMOPSO, SMPSO, and TMMOPSO, respectively. As can be seen from the figures, on most of the test problems, the data of TMMOPSO is more concentrated than the other comparison algorithms, which indicates that TMMOPSO has better stability than the selected comparison algorithms.

Fig. 6
figure 6

Convergence trajectories of five MOPSOs and TMMOPSO on ZDT3, UF9 and DTLZ6

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Fig. 7
figure 7

Box plots of IGD values of five MOPSOs and TMMOPSO on ZDT and DTLZ test suites

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Comparisons with five MOEAs

Tables 6, 7, and 8 show the mean and standard deviation of five MOEAs: DGEA, IDBEA, MOEA/D, NSGAII, SPEAR, and TMMOPSO proposed in this paper running independently on the three benchmark suites of ZDT, UF and DTLZ for 30 times as the algorithm evaluation criteria. The optimal results on IGD, HV, and GD are shown in bold.

Table 6 IGD values of five MOEAs and TMMOPSO on all test problems
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Table 7 HV values of five MOEAs and TMMOPSO on all test problems
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Table 8 GD values of five MOEAs and TMMOPSO on all test problems
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According to the results in the last row of Table 6, TMMOPSO achieves the best performance on 17 of the 29 test problems. Among them, TMMOPSO outperforms all five comparison algorithms on the ZDT test suite, six on the UF test suite are excellent, three on the DTLZ test suite with three objectives, and three on the DTLZ test suite with five objectives are excellent than the comparison algorithms. In addition, it is second only to NSGAII on the DTLZ5 with three objectives, and second only to MOEA/D on the DTLZ1 with five objectives.

A similar conclusion that this algorithm has superior performance can also be obtained from Table 7. As can be seen from Table 7, TMMOPSO obtained the best HV value on 17 out of 29 test problems. Table 8 shows the comparison results of the GD value of TMMOPSO and comparison algorithms. Relatively speaking, from the final results of the GD indicator, TMMOPSO ranks second in the number of best GD values obtained among the selected test suites, second only to MOEA/D. The overall implementation of TMMOPSO has good performance on the benchmark suites with both regular and irregular PF. To sum up, the proposed TMMOPSO has superior performance in both convergence accuracy and search ability compared with the five representative MOEAs on the test problems used under IGD and HV indicators.

Figure 8 shows the convergence speed of MOEAs on test problems of ZDT3, UF9 and DTLZ6 with three objectives. In the figures, the abscissa 1, 2, 3, 4, 5, and 6 represent DGEA, IDBEA, MOEA/D, NSGAII, SPEAR, and TMMOPSO, respectively. It can be seen from the figures that the convergence speed of the proposed algorithm TMMOPSO is faster than that of the comparison algorithms. Figure 9 shows the box plots of IGD values on the ZDT and DTLZ of the three-objective benchmark set. According to the characteristics of the box plots, it can be seen that the results of TMMOPSO are more stable than that of the comparison algorithms on most test problems after running independently 30 times.

Fig. 8
figure 8

Convergence trajectories of five MOEAs and TMMOPSO on ZDT3, UF9 and DTLZ6

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Fig. 9
figure 9

Box plots of IGD values of five MOEAs and TMMOPSO on ZDT and DTLZ test suites

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Friedman rank test

To obtain more comprehensive statistical results, we conducted the Friedman rank test [25, 52] on the results in Tables 3, 4, 5, 6, 7 and 8, and corresponding P values were obtained. The P values in the tables strongly reflect the significant differences between the proposed algorithm and the comparison algorithms. Table 9 shows the Friedman rank test rankings of IGD values of all algorithms on ZDT, UF and DTLZ benchmark suites. According to the results of the Friedman rank test, TMMOPSO proposed in this paper has obtained the first ranking on ZDT and UF benchmark suites. It finishes second and third on the DTLZ benchmark suites of three objectives and five objectives, respectively. However, from the overall results, TMMOPSO is ranked first, NSGAII is ranked second and NMPSO is ranked third. The Friedman rank test rankings of HV values on the three benchmark suites for all algorithms are shown in Table 10. As can be seen from the results in Table 10, TMMOPSO wins first place on the ZDT and UF benchmark suites, and second and third place respectively on the DTLZ benchmark suites of three objectives and five objectives. In the last column, TMMOPSO ranks first, followed by NMPSO. Table 11 shows the Friedman rank test results of GD values of all algorithms, ranking first in ZDT and third in total. In general, TMMOPSO has achieved excellent ranking under both IGD, HV and GD indicators, which also reflects the excellent performance of the algorithm.

Table 9 Friedman rank test results of IGD values for all comparison algorithms and TMMOPSO
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Table 10 Friedman rank test results of HV values for all comparison algorithms and TMMOPSO
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Table 11 Friedman rank test results of GD values for all comparison algorithms and TMMOPSO
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Conclusion

In this paper, a two-stage maintenance and multi-strategy selection for multi-objective particle swarm optimization is proposed. Firstly, the optimal particles are disturbed, and then the redundant solutions in the external archive are deleted using a two-stage strategy, which maintains the external archive, promotes the convergence speed, ensures the diversity, and optimizes the quality of the solutions. In addition, a strategy of updating pbest by aggregation index aiming at high-dimensional objective space problems is proposed. Next, the adaptive selection of gbest is put forward, the method uses an adaptive hyper-cone domain to determine each domain only a gbest, which makes the population closer to the true front and further accelerates the convergence speed of the population. Through the simulation experiments, TMMOPSO is compared with the existing five MOPSOs and five MOEAs on 29 standard test problems. The experimental results show that TMMOPSO has better performance than other algorithms on these standard benchmark suites.