An Improved Multi-Objective Artificial Physics Optimization Algorithm Based on Multi-Strategy Fusion

The elements of the population of the artificial physics optimization (APO) algorithm are assigned mass, velocity, and displacement attributes. It is a new type of heuristic algorithm, but it still has defects such as low efficiency of non-dominated solution selection and unbalanced search capacity of global and local. This paper introduces the mechanism of improved fast non-dominant sorting and partition-guided individual evolution into the APO algorithm to overcome this imperfection. An improved multi-objective artificial physics optimization algorithm based on fast non-dominated sorting (IFNS-MOAPO) is proposed, which is the result of integrating numerous strategies into the APO algorithm. Firstly, an improved fast non-dominated sorting strategy is introduced. This strategy can increase the efficiency of selecting non-dominated solutions and decrease the running time of the algorithm. Secondly, the mechanism of individual evolution guided by partition is proposed. For individuals in infeasible and feasible domains, different mass functions and virtual force calculation rules are adopted to update the algorithm iteratively to boost the convergence performance. To verify the comprehensive performance of the IFNS-MOAPO algorithm, eleven benchmark test problems are selected for simulation experiments and compared with five algorithms in terms of runtime duration, Pareto front plots, and metric values. The results show that the IFNS-MOAPO algorithm has good distribution and can converge to the true Pareto front quickly. It is a useful tool for solving constrained multi-objective optimization problems (CMOPs).

research of MOAs plays an irreplaceable role both in theory and practice.
To solve MOPs, scholars have done a lot of research on algorithms. From the beginning of the traditional optimization algorithm, and then to the intelligent optimization algorithm, are the condensation of wisdom. Traditional optimization algorithms are generally aimed at structural problems, most of which fall into the category of convex optimization [9]. The essence of the algorithm is to transform multiple objective functions into a single objective function based on some rules and then solve the MOP by a singleobjective optimization method [10], [11]. Although the traditional optimization algorithm has a fast convergence speed and a clear termination criterion, it can only find one of the Pareto solution sets of the optimization problem at a time, and the solution results are strongly dependent on the initial values. The emergence of the intelligent algorithm breaks the situation that MOP cannot be solved effectively. And it can better reflect the characteristics of MOP. Well-known intelligent optimization algorithms include evolutionary algorithms (EA) [12] and particle swarm optimization (PSO) algorithms [13]. EA are population-based heuristic search methods that can be solved without prior knowledge.
Due to the various advantages of intelligent algorithms, there has been a recent research boom in the related academic community. As far as multi-objective optimization itself is concerned, representative MOEAs are binary differential evolution with self-learning for multi-objective feature selection [14], a decomposition-based archiving approach for multiobjective evolutionary optimization [15]. These algorithms have high robustness, high nonlinearity, parallelism and wide applicability, and can find globally optimal solutions. At the same time, MOEAs also have some drawbacks, such as high computational cost, low operational efficiency, and poor diversity of solution sets. To overcome these problems, researchers proposed many improved MOEAs [18], [19]. Meanwhile, other meta-heuristic algorithms are also actively sought to solve MOP. PSO algorithm is one of them. The key points of the study of PSO are as follows: the maintenance of population diversity and distributivity, the selection of nondominant solutions, and the pruning of the archive set. The multi-objective particle swarm optimization (MOPSO) algorithm has been widely used to solve MOPs. However, it is prone to fall into the local optimum when solving discrete problems. Therefore, most of the improved algorithms are now based on certain strategies to avoid premature algorithms, and the inclusion of some strategies is accompanied by an increase in algorithm running time [20], [21]. The APO algorithm has some similarities with the PSO algorithm, and its performance can be compared with or even surpasses the classical optimization algorithms [22], [23]. The core of the APO algorithm contains the selection of the nondominated set, the calculation of the mass function, and the virtual force. Although the RMOAPO algorithm reflects the concept of Pareto domination, it is more complicated to determine the individual number based on individual order value, neighborhood radius and congestion distance, which increases the computational complexity [24]. The APO algorithm is a new type of algorithm, which is one of the means to solve optimization problems, especially for SOPs, showing powerful solving ability. Based on this, many scholars have tried to apply it to solve MOPs. However, the APO algorithm suffers from the defects of low efficiency of non-dominated solution selection, poor convergence, and distribution in solving MOPs. To address the shortcomings of the traditional APO algorithm with low efficiency of non-dominated solution selection, an improved fast non-dominated sorting strategy is introduced to save the time cost of the algorithm. In the iterative solution of the traditional APO algorithm, it is necessary to first synthesize multiple objectives into one objective, and then find the optimal individual and the worst individual in the population before the final calculation of individual quality, which does not fully reflect the concept of Pareto domination and cannot fully reflect the characteristics of multi-objective optimization [25], [26]. This paper reconstructs the quality function based on individual partitioning and establishes a direct mapping between the calculation of individual quality and individual ordinal values and constraint violation degrees, which fully reflects the characteristics of multi-objective optimization. Also, the mass function and virtual force correction rules distinguish between constrained and unconstrained search according to feasible and infeasible solutions. The simulation results show that the improved algorithm proposed in this paper can accelerate the convergence speed, enhance the diversity of the population, and effectively solve the MOP.
The remainder of this paper is structured as follows. Section 2 introduces the basic definition of constrained multi-objective optimization problems (CMOPs). Section 3 gives a brief introduction to the traditional APO algorithm. Section 4 presents the design of key elements and computational steps of the multi-objective artificial physics optimization algorithm based on improved fast nondominated sorting (IFNS-MOAPO). In section 5, a comparison experiment with other algorithms is conducted to verify the comprehensive performance of the IFNS-MOAPO algorithm. Finally, Section 6 presents the conclusions.

II. MATHEMATICAL EXPRESSION OF CMOP
The APO algorithm is widely used in practical problems. In these practical problems, optimization models are often constructed with constraints and more than one objective function. In this paper, the mathematical expression of CMOP is defined as: where k is the number of conflicting objectives, m and s are the respective numbers of inequality constraints and equality VOLUME 10, 2022 constraints. x ∈ R n is the n-dimensional vector of decision variables. U i and L i are the upper bound and lower bound of the decision variable x i , i = 1, 2, · · · , n, respectively.

III. APO ALGORITHMS
The APO algorithm is inspired by Newton's force law. Individuals in the population adjust their motion according to the total virtual force and inertia of other individuals to obtain a satisfactory solution. In addition, the individuals in the algorithm are given mass properties, which are constantly updated according to the optimal individual, the worst individual, and their fitness value. As the mass changes, the virtual force of the individual also changes. The rules for calculating the velocity of an individual in the APO algorithm are related to the virtual force and mass of the individual.
As the virtual force of the individual changes, the velocity changes accordingly, and thus the position of the individual changes accordingly. The APO algorithm is described as follows: Definition 1: The mass function of the individual i.
where f (x best ) denotes the function value of the best individual, f (x worst ) denotes the function value of the worst individual, and f (x i ) denotes the function value of the individual i.
According to Definition 1, the calculation of individual mass requires the realization of knowing three values. These three values are the fitness value of the best individual, the fitness value of the worst individual, and the fitness value of the individuals in the population, respectively. However, for MOPs, the best and worst individuals cannot be found due to conflicting objective functions.
Definition 2: The virtual force exerted on individual i via individual j in kth dimension.
where G is the ''gravitational constant'' and x j,k − x i,k is the distance from individual i to individual j in the kth dimension. It can be seen from Definition 2. that the virtual force calculation rules of the traditional APO algorithm do not distinguish individuals in the feasible and infeasible domains. Considering that the test object of this paper is the CMOP, it is necessary to make corrections to the virtual force calculation rules based on individual partitions.
where ω is the inertia weight; λ is a random variable generated within (0, 1) with normal distribution. Definition 5: The position update of the individual i at generation t + 1.
After calculating the total force, velocity, and position of the individual iare updated by using (5) and (6), respectively.

IV. PROPOSED ALGORITHM: IFNS-MOAPO
In this paper, the IFNS-MOAPO algorithm is proposed. Improvements are made based on the APO algorithm as follows: 1) To address the problem of low efficiency of the non-dominated solution selection method of the basic APO algorithm, ameliorated non-dominated sorting strategy is introduced to decrease the running time cost of the algorithm. And each individual is assigned a different ordinal number in combination with the congestion distance. 2) The individual mass function is reconstructed according to the different individual partitions to fully reflect the characteristics of multi-objective optimization. 3) More specific and detailed virtual force calculation rules are constructed according to the different regions of individuals. Through the above improvements, the running efficiency, convergence, and distribution of the algorithm are increased, so that the algorithm will evolve in a better direction, and eventually, the global optimal solution set is found.

A. CONSTRUCTING NON-DOMINATED SETS
The traditional APO algorithm uses Deb's non-dominated sorting method to construct the non-dominated set, which requires all individuals in the population to be compared with each other to generate non-dominated individuals. Therefore, this construction method does not have good structural efficiency.
The improved fast non-dominated sort converts the strategy of comparing one individual with other individuals into a strategy of comparing two individuals with other individuals at the same time, where the two individuals are either unrelated or the second one dominates the first one, and any individual dominated by one of the two individuals is removed, and only the individual not dominated by both individuals is retained. The worst situation of the improved fast nondominated sort is that the second comparison individual is not found. And then the improved fast non-dominated sort degenerates. But even it is theoretically more efficient than Deb's non-dominated sort.
Let P denote the evolutionary population, x and y are individuals in the population. The definition of individual non-dominant relationships is as follows.
Definition 6: For any individual x and y, x, y ∈ P, we say that x and y are irrelevant if there is no dominant relationship between x and y.
Definition 7: For the certain individual x ∈ P, if there is no y ∈ P such that y x, we say that x is the non-dominant individual of the set P. Definition 8: We often say x d y that the relationship d is not transitive, if x y or x and y are irrelevant.
The IFNS-MOAPO algorithm performs the selection of non-dominated solutions after the population initialization is completed. The selection method is a quicksort algorithm for improved construction of non-dominated sets. After executing the above algorithms, the population reaches the state of stratification based on the level of non-inferior solutions. Then an ordinal number is assigned to each level so that individuals in the same layer have the same non-dominated order. In the following discussion, we use i rank to denote the non-dominated ordinal value of an individual i. For the set F 1 , all individuals are endowed with non-dominated ordinal values i rank = 1. For the set F 2 , individuals are assigned the non-dominated ordinal values i rank = 2, and so on. Until all individuals in the same stratum in the whole population have the same non-dominated i rank for the entire population. For the individuals in the same stratum, their strengths and weaknesses need to be defined in a certain way. To achieve selective sorting of individuals with the same i rank , crowding and crowding distance comparison operators are introduced. The result is that the individuals in the population can be infinitely close to the true front and uniformly distributed throughout the Pareto domain.

B. PARTITION RECONFIGURATION MASS FUNCTION
The mass function of the traditional MOAPO algorithm is calculated by knowing the objective function values of the best and worst individuals in the population, but the MOP has multiple objective functions, so each sub-objective function needs to be combined into a single objective function through certain strategies. To this end, the mass function is reconstructed in such a way that the mass of individuals in the feasible domain is limited to (1,2), and the mass function of the individual in the infeasible domain is limited to (0, 1]. The mass function in the IFNS-MOAPO algorithm is modified as: where N is the number of individuals in the population, and rank(i) is a natural number between 1 ∼ N .
where CV (x i ) is the constraint violation degree value of individual i and CV (x) is the overall constraint violation.

C. VIRTUAL FORCE CALCULATION RULES FOR PARTITION RECONSTRUCTION
The discussion in this paper is directed to CMOP, which has constraints that limit the range of the optimal solution. The constraints divide the whole search space into feasible design regions and infeasible design regions. The points that satisfy the constraints will fall in the feasible domain, and the points that do not satisfy the constraints will fall in the infeasible domain. The rules for calculating the virtual forces in the APO algorithm should satisfy that individuals with good adaptation values have a gravitational effect on individuals with poor adaptation values. The rules for the virtual are modified based on the difference in the area where the two individuals are located and are discussed in the following four cases.
When both individuals are in the infeasibility domain, the rule for calculating the virtual force is the magnitude of the constraint violation degree value. The individual with a small constraint violation degree acts as a gravitational force on the individual with a large constraint violation degree, and conversely, the individual with a large constraint violation degree acts as a repulsive force on the individual with a small constraint violation degree.

D. ALGORITHM DESCRIPTION
The effectiveness of MOAPO algorithms lies in the fact that an individual in the population is subject to virtual forces from other individuals and constantly adjusts its position to move in a more optimal direction. The constraint processing technique simply uses the constraint holding method to pull individuals that are not in the feasible domain to the boundary of the feasible domain. It will fall into local search. The constraint violation value is introduced in the improved algorithm to quantify the degree to which a solution violates a constraint and to establish a direct mapping between it and the design of the mass function. The constraint violation CV (x) of a solution x is calculated as: where P m (x) is defined as the mth inequality constraint and q s (x) is defined as the sth equality constraint. when x is in the feasible design region, CV (x) = 0, a solution x is called a feasible solution. When x is in the feasible design region, CV (x) > 0, x is an infeasible solution.
We use the IFNS-MOAPO algorithm to solve MOPS. Figure 1 shows the design idea of the algorithm.
We use the IFNS-MOAPO algorithm to solve MOPS. The following algorithm is described in pseudocode form.

V. EXPERIMENTAL STUDIES
The ability of the improved algorithm to deal with problems can be judged by the comprehensive performance of the improved algorithm on the test function. In the following sections, we first list the names of the test functions. The specific values of the key parameters in the six algorithms are also given. Next, we plot the comparison of the frontier obtained by the six algorithms with the true Pareto. The time complexity of IFNS-MOAPO is analyzed. Then, at the end of this section, we quantitatively analyze the distributivity and convergence of each algorithm by specific metric values.

A. EXPERIMENTAL ENVIRONMENT CONFIGURATION AND BENCHMARK TEST PROBLEMS
The experimental environment of this paper is configured as 11th Gen Intel(R) Core (TM) i5-1135G7 CPU @ 2.50GHz 4.90GHz, 16.00GB, Windows 10 flagship 64-bit OS. The software environment is MATLAB R2020b (64-bit), and some algorithms implementations rely on the EMO platform.
To verify the effectiveness of IFNS-MOAPO algorithm, the MOPs with constraints are selected as the standard test set [27], [28], as shown in The comparison algorithms in this paper are NSGA-II [29], CMOPSO [30], MOAPO [25], MOEA/D-DAE [31], CCMO [32], respectively. The reasons for choosing these comparison algorithms are as follows: (i) NSGA-II is an algorithm based  4: Improved fast non-dominated sort stratification 5: Calculate the individual's crowding distance 6: Assign each individual an ordinal number rank(i) 7: while (t < Max iterations ) do 8: for each individual from the archive do 9: calculate the constraint violation value for each individual by (13). 10: if i ∈ feasible domain, then 11: calculate the individual's mass with (7). 12: else 13: calculate the individual's mass with (8). 14: end if 15: select the corresponding virtual force calculation rules to calculate the force receives from other individuals in the kth dimension by (9)-(12). 16: calculation of the combined virtual forces by (4). 17: update the speed and position of each individual by (5) and (6). 18: end do 19: t = t + 1 20: update Pareto front by the algorithm. 21: end while 22: return Pareto Front (PF) 23: end procedure on non-dominated sorting, and it is used as a comparison to test the effectiveness of the improved non-dominated sorting method in the IFNS-MOAPO algorithm to save time costs. (ii) The reason for choosing MOPSO algorithm as the comparison object is that the essence of traditional APO algorithm is similar to that of particle swarm algorithm. (iii) Compare with the traditional MOAPO algorithm and analyze the effectiveness of the improved strategy in the IFNS-MOAPO algorithm. (iv) MOEA/D-DAE and CCMO have excellent performance and are relatively advanced algorithms in recent years. Compared with these two algorithms, the comprehensive performance of the improved algorithm can be better evaluated. The other parameters for each compared algorithm are set according to the original paper. The specific values of the key parameters in the six algorithms are shown in TABLE 2.

B. EXPERIMENTAL RESULTS OF THE FINAL NON-DOMINATED FRONT
The comparison between the Pareto front obtained by the algorithm and the true front can directly reflect the degree of approximation to the real front and the advantages and disadvantages of the distribution of the algorithm. Figure 2 shows the comparison plot of the Pareto front obtained by the six algorithms with the true Pareto front. For each test function, there are six plots: the one on the right is for IFNS-MOAPO, and the others are for NSGA-II, CMOPSO, MOAPO, MOEA/D-DAE, and CCMO.
As Figure 2 shows, the improved algorithm proposed in this paper and the other six algorithms can find the non-inferior solution set of the test function. The improved algorithm in this paper can not only find feasible individuals but also these individuals are very close to or even coincide with the true Pareto frontier, which meets our expectations.

C. ALGORITHM EFFICIENCY
The algorithm efficiency can be analyzed from two aspects. From the theoretical aspect, we can analyze the algorithm's time complexity. From the application aspect, we can evaluate the running time taken by the algorithm. In this paper, the time complexity of non-dominated solution selection is changed from O(kN 2 ) to O(kN log N ) after the improvement of the traditional MOAPO algorithm [33,34].  The algorithm runtime result statistics in Table 3

D. PERFORMANCE METRICS
The convergence and distribution of the improved algorithm can be illustrated by scalar values. The common performance evaluation indexes include GD, SP, HV, and IGD. HV and IGD are comprehensive performance evaluation indexes. The calculation of the HV metric strictly follows the Pareto dominance principle, the larger the HV value, the better the algorithm performance. the IGD metric is the reverse mapping of the GD metric. the smaller the value of the IGD metric, the better the performance of the algorithm. The calculation of the HV index value does not need to know the Pareto optimal surface. HV and IGD are defined as: where S is the nondominant set, v i denotes the supervolume of reference points and non-dominant individuals, λ is the Lebesgue measure. n is the number of vectors in the frontier after running the algorithm. min¯i ∈P j −ī denotes the minimum Euclidean distance from the pointj on the Pareto optimal surface to the individualī in the optimal solution set P.

E. EXPERIMENTAL RESULTS ON TEST PROBLEMS
The IFNS-MOAPO algorithm is a random search algorithm. Due to the randomness of the algorithm, the results of a single experiment are not representative. Therefore, to avoid the errors caused by the random search of the algorithm ,  TABLE 4 and TABLE 5 show the comparison results of the mean and standard deviation (mean vs. std) of HV metrics and IGD metrics between the IFNS-MOAPO algorithm and NSGA-II, CMOPSO, MOAPO, MOEA/D-DAE, CCMO after 30 consecutive independent runs on MOP1-7, respectively, where the best results are shown in bold.       TABLE 6 TABLE 4 and TABLE 7 show the comparison results of the mean and standard deviation (mean vs. std) of HV metrics and IGD metrics between the IFNS-MOAPO algorithm and NSGA-II, CMOPSO, MOAPO, MOEA/D-DAE, CCMO after 30 consecutive independent runs on VNT series test functions, respectively, where the best results are shown in bold.   7 shows that the mean values of IGD metrics of the IFNS-MOAPO algorithm on the VNT series test functions are lower than those of the comparison algorithm, which fully illustrates that the IFNS-MOAPO algorithm also has good distributivity and convergence on the three-objective test functions.
The conventional MOAPO algorithm has the problem of partitioning ambiguity in the calculation rules of mass function and virtual force, which leads to the inferiority of HV and IGD index values to the IFNS-MOAPO algorithm on most of the test functions selected in this paper; the CMOPSO also has the inferior performance to the IFNS-MOAPO algorithm on the test functions due to its tendency to fall into local optimum.
The box plots can visually reflect the dispersion of the sample data as well as the symmetry and tail weight of the overall distribution. Figure 3 and Figure    From Figure 3, we can observe that IFNS-MOAPO has superiority for the HV statistic followed by CCMO. For instance, IFNS-MOAPO shows higher HV statistical results on most of the seven MOP test problems. Compared with the MOAPO algorithm, the HV statistics of IFNS-MOAPO are significantly improved, indicating that the strategy of partitioning to reconstruct the mass function and virtual force rules is effective.
From Figure 4, we can observe that IFNS-MOAPO has superiority for the IGD statistic followed by CCMO and MOEA/D-DAE. The CMOPSO algorithm shows high IGD statistics on MOP1, MOP6, and MOP7 test problems with poor distribution. The effectiveness of the improved strategy in this paper is further illustrated by comparing the IGD statistics results with those of the traditional MOAPO algorithm. Establishing a direct mapping of the mass function to the individual ordinal number and reconstructing the virtual force calculation rules according to the region can significantly improve the distributivity and convergence of the IFNS-MOAPO algorithm.

F. APPLICATION ON REAL-WORLD ENGINEERING OPTIMIZATION
The disc brake design problem is used to further test the effectiveness of the algorithm in this paper [35]. The problem has two objectives, which are the weight of the brake and the stopping time. The problem contains four variables, x 1 and x 2 denote the diameters of the inner and outer rings of the brake, x 3 denotes the engagement force, and x 4 denotes the number of friction surfaces. The constraints of the problem are the minimum distance between radii, the maximum length of the brake, pressure, temperature, and torque limits, respectively. The specific optimization problem and constraints are shown in Eq. (16).
Each algorithm was run 30 times independently on this problem, with the number of function evaluations set to 100000 and population size of 100. We used HV to evaluate the performance of the IFNS-MOAPO, and the results after 30 runs are summarized in TABLE 8. The Pareto front plots obtained for the six algorithms are given in Figure 5.
As TABLE 8 shows, the result of IFNS-MOAPO is significantly better than the other five algorithms on this realworld problem in terms of HV metric. CCMO achieves the second-best result followed by NSGAII and MOEADDAE. CMOPSO and MOAPO obtain the worst results.
As Figure 5 shows, NSGA-II, MOAPO, MOEADDAE, and CCMO demonstrate worse performance on distribution compared to IFNS-MOAPO because the individual distribution at the ends of their approximate PFs is sparse. All points of CMOPSO cannot converge to the PF, so CMOPSO is not valid for this practical engineering problem. Therefore, IFNS-MOAPO achieves the best performance concerning convergence and distribution compared to the other peer algorithms.

VI. CONCLUSION AND FUTURE STUDIES
In this paper, we propose a new multi-objective optimization algorithm, namely IFNS-MOAPO, for the CMOPs. The improving fast non-dominated sorting strategy, the conception of partition reconstruction mass function, and the conception of partition reconstruction virtual force calculation rules have made the IFNS-MOAPO algorithm more effective in solving CMOPs. By comparing IFNS-MOAPO with other optimization algorithms (NSGA-II, CMOPSO, MOAPO, MOEA/D-DAE, and CCMO), we demonstrate that our IFNS-MOAPO is efficient in dealing with CMOPs. The main findings of the paper can be summarized as follows.
(1) On the basis of the traditional APO algorithm, the improved fast non-dominated ranking strategy can efficiently reduce the run time of our algorithm. For most of the test functions, the IFNS-MOAPO algorithm has the fastest running time, although this advantage is very small. (2) The mass function of partition reconstruction and the virtual force calculation method make full use of the characteristics of the feasible and infeasible domains in the CMOPs. The improved strategy speeds up the convergence of the algorithm and enhances the diversity of populations.
The IFNS-MOAPO algorithm is strongly influenced by the parameters, especially the problem of adjusting the gravitational factor and inertial weights. In our future work, we will investigate the parameter setting problem and the computational complexity of the algorithm. In addition, this paper is only a theoretical analysis of the IFNS-MOAPO algorithm. In the future, we will use the IFNS-MOAPO algorithm to solve MOPs in practical engineering, such as engineering vehicle suspension parameters, robot path planning, and logistics allocation.