A novel multi-agent simulation based particle swarm optimization algorithm.

Recently, there has been considerable research on combining multi-agent simulation and particle swarm optimization in practice. However, most existing studies are limited to specific engineering fields or problems without summarizing a general and universal combination framework. Moreover, particle swarm optimization can be less effective in complex problems due to its weakness in balancing exploration and exploitation. Yet, it is not common to combine multi-agent simulation with improved versions of the algorithm. Therefore, this paper proposes an improved particle swarm optimization algorithm, introducing a multi-level structure and a competition mechanism to enhance exploration while balancing exploitation. The performance of the algorithm is tested by a set of comparison experiments. The results have verified its capability of converging to high-quality solutions at a fast rate while holding the swarm diversity. Further, a problem-independent simulation-optimization approach is proposed, which integrates the improved algorithm into multi-agent systems, aiming to simulate realistic scenarios dynamically and solve related optimization problems simultaneously. The approach is implemented in a response planning system to find optimal arrangements for response operations after the Sanchi oil spill accident. Results of the case study suggest that compared with the commonly-used shortest distance selection method, the proposed approach significantly shortens the overall response time, improves response efficiency, and mitigates environmental pollution.

Introduction

The merging of optimization and simulation technologies has seen rapid growth recently. The optimization of simulation models refers to finding the best values of some decision variables for a system where the performance is evaluated based on the output of a simulation model of this system [1]. However, due to the complexity, analytical simulation methods may be unsuitable for large-scale systems in manufacturing, supply chain management, financial management, etc. Agent-based modeling (ABM) is a bottom-up modeling method with high autonomy and interactivity, which is particularly suitable for complex system research [2]. Given the advantages of ABM, researchers have been solving optimization problems by integrating multi-agent simulation (MAS) with a wide range of methods, including game theory, reinforcement learning, and swarm intelligence (SI). Applications can be found in various fields such as intelligent transport systems, efficient electric grids, and smart buildings. Among the methods mentioned above, SI has been proved to be robust and efficient for optimization problems. As a classic category of SI, particle swarm optimization (PSO) has the advantage of simple parameter setting, fast convergence rate, high stability, and strong scalability, making it an excellent option for complex systems.

However, despite a high convergence rate, the standard formation of PSO (SPSO) [3] is likely to stagnate at a local optimum, which reduces the solution accuracy. Thus, it is necessary to modify the algorithm for the application in complex problems where stable optimal solutions are indispensable. Different approaches have been considered to improve SPSO, among which setting parameters is especially representative [4]. Changing coefficients is a typical way of parameter setting. The performance of SPSO is affected by the values of its coefficients, i.e., the acceleration coefficients and the inertia weight. Thus, several attempts were made to tune the values of these parameters, such as the random weight method (RPSO) [5], the constriction factor approach (CPSO) [6], and the fuzzy adaptive approach (FAIPSO) [7]. Apart from the coefficients, the population size of the swarm is also a vital parameter. The basic idea for adjusting population size is to concentrate the individuals in the most promising area during the exploitation phase of the algorithm. De Oca et al. [8] conducted this idea by introducing incremental social learning (IPSO). On the other hand, as SPSO came from a simplified simulation of bird clustering behaviors, some researchers tried to improve it by mimicking other bird activities simultaneously. Neshat et al. [9] modified SPSO based on the predation phenomenon in bird swarms (PPSO). Besides these approaches, Li et al. [10] decoupled exploration and exploitation to make the algorithm more effective in large-scale optimization.

In MAS, agents can be either homogeneous or heterogeneous. Homogeneous agents are usually treated as particles when PSO is applied [11]. Yang et al. [12] presented a multi-agent-based model which simulated human behaviors in a multi-exit evacuation environment, where individuals were homogeneous agents and PSO was introduced to simulate their movement. Ali et al. [13] proposed a collective motion and self-organization control of a swarm of 10 unmanned aerial vehicles, which were divided into two clusters of five agents each. PSO was adopted to provide the best agents of the cluster. While homogeneous systems often simulate certain collective activities of agents with the same properties, heterogeneous systems are more practical for realistic scenarios due to the diversity of agents. Kanaga and Valarmathi [14] proposed a multi-agent model using PSO to solve patient scheduling problems, using a PSO Agent to perform the algorithm. Yu et al. [15] designed a Web services selection model, using PSO to select the optimal Web services combination in E-business. The model was realized by MAS. Thiel et al. [16] simulated Hanoi dwellers’ choices between traditional markets and a hypothetical new one, aiming to find the optimal location for the ready-to-build local supermarket by using PSO to maximize sales volume. Mahad et al. [17] proposed an intelligent MAS approach to optimize power supply in community-based multi-microgrids systems, where various layers of autonomous and intelligent agents took decisions based on the PSO method.

Although there are many studies on improving SPSO, there are not so many attempts to combine proposed algorithms with MAS and apply them to practical problems. In fact, utilizing improved algorithms in simulation-optimization systems is necessary because complex tasks require high-performance optimizers, and SPSO may not meet this requirement. Besides, existing studies on the combination of MAS and PSO may vary in engineering fields, yet they all designed the system based on specific problems of their own research field without summarizing a general and universal framework, which is field-independent and can be implemented in various settings. Therefore, this paper focuses on a more in-depth exploration of improving PSO and integrating it into MAS to solve practical problems. The main contributions of this paper are as follows:

Proposing an improved particle swarm optimization algorithm (CoPSO) according to the biological nature of birds. It has a multi-level structure and a competition mechanism, aiming to improve performance by enhancing exploration while balancing exploitation.

Proposing a general simulation-optimization approach (MAS-CoPSO) to integrate CoPSO into MAS. It is supposed to achieve a very close combination between simulation and optimization, and the application should not be limited to certain engineering fields and specific problems.

Conducting a series of experiments to substantiate the validity of CoPSO.

Implementing MAS-CoPSO in a case study of the Sanchi oil spill accident to solve the response planning problem.

Methods and materials

This section elaborates on the methodologies used in this study. Firstly, a brief introduction to MAS technology is given. Detailed descriptions of the proposed CoPSO algorithm and the MAS-CoPSO approach follow afterward.

Multi-agent simulation

The fundamental element in MAS is the agents, which can be software or physical entities. Although different agents exhibit distinct behaviors, they share some common properties. For instance, they all have a certain degree of autonomy that enables them to work even without human intervention, they can communicate and interact with each other, and they are capable of perceiving and reacting to the changes in the environment as well as determining the proper behaviors to achieve the final goal [18]. In a simulation system, agents can be either homogeneous or heterogeneous depending on the specific problem background. Heterogeneous agents can deal with diverse scenarios and complex tasks, making it more suitable for complicated realistic systems [11]. Based on the aim of proposing a general framework for the combination of MAS and CoPSO, this paper mainly focuses on heterogeneous systems due to their universality.

Improved particle swarm optimization algorithm

PSO mimics the behavior of bird swarms. The fundamental idea is to assume a potential solution to the optimization problem as a bird without quality and volume, i.e., a particle. Particles are described by their positions and velocities. Generally, the position of a particle represents a specific solution while the velocity represents the searching direction and scope, which leads the particle to fly through the solution space. Every particle will get a fitness value by calculating the objective function. It will also adjust the value according to the experience of itself and the neighbors.

Supposing the search space is D-dimensional, the ith particle is represented as , where , d ∈ [1, D], and are the lower and upper bounds of the dth dimension, respectively. The velocity of the ith particle is represented as , where , d ∈ [1, D], V and V are the minimum velocity and maximum velocity specified by the user, respectively. In SPSO [3], during each iteration, particle i updates its position and velocity as follows: where w is the inertia weight, r1 and r2 are random values between 0 and 1, c1 and c2 are acceleration constants which control how the particle moves, is the best previous position of the ith particle, and gbest = (gbest1, gbest2, …, gbest) is the best position among all the particles in the swarm.

PSO encourages exploration during the early stage of the iterations while encouraging exploitation in the latter stage. However, Angeline [19] has pointed out that the balance between exploration and exploitation is subtle and difficult to control in PSO, leading to stagnation or premature convergence. Inspired by the features of bird flocks, this paper proposes CoPSO to improve the performance of PSO, introducing a multi-level structure and a competition mechanism.

Multi-level structure

According to the biological nature of birds, their ability to sing and create a complete song is learned from their parents. Experiments have shown that if a bird is reared in silence, it can only scream. In addition, birds’ singing skills are not set in stone. Basically, as they grow up, their expertise in singing also gets improved [20]. Based on this phenomenon, CoPSO has a multi-level structure, dividing the particles in a swarm into different singing levels. It is stipulated that the closer a particle is to a possible global optimum, the higher its level. The level of the ith particle is denoted by l, which will increase if the particle doesn’t update its personal best position pbest in μ continuous iterations. Since birds of different ages learn to sing at different rates, particles of different levels have different acceleration constants.

Fig 1 illustrates how the particles’ levels change during the iterations in two-dimensional unimodal problems. At the early stage of the algorithm, particles are like newborn birds with little knowledge about singing. They will explore the search space and try to find promising areas. Once found, they might temporarily stop updating their personal best positions, leading to an upgrade. At the latter stage of the algorithm, some particles can be very close to the global optimum. Consequently, they will find it difficult to locate better positions, and their levels are also relatively higher.

Fig 1

Variations of particles’ levels during the iterations in two-dimensional unimodal problems.

The points represent the particles. (a) The early stage. (b) The latter stage.

Competition mechanism

While a particle’s high level is likely to come from finding the global optimum, there are exceptions. Note that falling into local optima can also stop particles from updating their personal best positions. Since particles’ levels are linked to their ages (the higher level and the more skillful in singing, the older), the idea of the competition mechanism inside bird swarms is utilized to improve the algorithm. The survival resources of swarms and the lifespan of birds are both limited. Therefore, a bird swarm is constantly renewed by the death of old birds and the born of new ones. If a particle’s level is too high, it becomes too old to possess the resources of the swarm. The population size is set as a constant, once a particle’s level reaches the upper bound L, it will be replaced by a randomly generated new one. The new particle has the lowest level since it’s a newborn.

Fig 2 depicts how the competition mechanism functions in two-dimensional multimodal problems. If particles prematurely converge to a local optimum, they will upgrade around it. It can be seen from Fig 2a that few particle gets close to the global optimum. However, once the particles in the local area become too high-leveled, they will be replaced by new birds. As these new particles are randomly generated, they have the potential to explore unknown search space and finally locate the global optimum, which is shown in Fig 2b.

Fig 2

Competition mechanism in two-dimensional multimodal problems.

The points represent the particles. (a) Particles are trapped in the local optimum. (b) Particles jump out of it.

Since particles tend to move closer to the global optimum and oscillate around it as the algorithm runs, in practice, the value of μ will be increased as the level rises, which allows an elaborate control over the balance between exploration and exploitation. The pseudo-code of CoPSO is shown in Algorithm 1.

Algorithm 1: Pseudo-code of CoPSO

Input: The value of L, the value of μ, and the maximum number of iterations I

Output: The final solution gbest and its fitness value f(gbest)

1 Initialize gbest;

2 for each particle i do

3  Generate x and v randomly;

4  pbest = x;

5  if f(x) is better than f(gbest) then

6   gbest = x;

7  end

8 end

9 while I is not met do

10  for each particle i do

11   Update v according to Eq (1);

12   Update x according to Eq (2);

13   if f(x) is better than f(pbest) then

14    pbest = x;

15   end

16   if f(x) is better than f(gbest) then

17    gbest = x;

18   end

19   if pbest hasn’t changed in μ continuous iterations then

20    l+ = 1;

21   end

22   if l == L then

23    Regenerate x and v randomly;

24    l = 0;

25   end

26  end

27 end

Problem-independent simulation-optimization approach

The architecture of the proposed MAS-CoPSO approach is shown in Fig 3, where the MAS module and the CoPSO module are the main components. For the optimization problem brought up by the user (e.g., decision making, scheduling, route planning, etc.), a suitable simulation time step should be chosen first according to specific backgrounds. In every time step, MAS is supposed to simulate the problem scene, during which the agents’ behaviors might change the values of related variables and parameters. Simulation statistics are then put into the optimization module, whose objective can be maximizing the efficiency or minimizing the cost of certain activities. CoPSO algorithm is supposed to optimize the objective function while satisfying all the constraints. The optimization result will then play as the input of MAS, instructing the simulation process in the next time step. This iteration will go on until meeting the stop criteria set by the user. The outcome of MAS-CoPSO is the final solution to the problem. For complex systems, the solution may not be strictly optimal (actually it is intractable to find the strict global optimum for these problems). However, MAS-CoPSO can compromise between the solution quality and computational cost, making it possible to get a satisfying result in a reasonable period of time.

Fig 3

Architecture of the MAS-CoPSO approach.

Successful simulation of the problem scene relies on a proper setup of the environment (which reflects the background of the problem) and an appropriate design of all the agents. In MAS, agents have autonomous behaviors and complex interactions, and the relationship between them can be various (e.g., cooperation, competition, control, etc.). Value changes of related variables and parameters link simulation and optimization together. For CoPSO, simulation results will influence the calculation of the objective function’s fitness value and the expression of constraints. Optimization results of CoPSO will function as the instructions for simulation in the next time step, i.e., how to adjust agents’ behaviors and tune the values of variables, parameters, and environmental settings. These two modules compose a cohesive framework through consistent information exchange and close integration.

Experiments and analysis of the optimization algorithm

To substantiate the validity of the proposed CoPSO algorithm, this study conducts numerical comparison experiments and computational complexity analysis. Exhaustive descriptions and discussions are given in this section.

Comparison experiments

The performance of CoPSO is compared with a set of classic algorithms:

SPSO [3]: the standard formation of PSO, introduced in detail in the previous section.

RPSO [5]: PSO with a random inertia weight factor designed for tracking dynamic systems.

CPSO [6]: PSO with a new constriction factor related to the original acceleration coefficients.

IPSO [8]: PSO with an increasing population size, i.e., whenever the algorithm can not find a satisfactory solution, add a new particle to the population.

Benchmarks and parameter setting

Five well-known benchmark functions commonly used in literature [21] are applied to evaluate the performance of CoPSO, both in terms of solution quality and convergence rate. They are non-linear minimization problems that present different difficulties to the optimizers. Table 1 lists the functions, the problem dimension n, the global minimum fitness value f, and the search space ranges (which are also the initial ranges in this research).

Table 1

Benchmark functions used in this study.

	Function	n	f min	Search range
Sphere function	f_1(x)=∑i=1nx_i^2	30	0	[−100, 100]n
Schwefel’s function	f_2(x)=∑i=1n|x_i|+∏i=1n|x_i|	30	0	[−10, 10]n
Quartic function	f_3=∑i=1nix_i^4+rand[0,1)	30	0	[−1.28, 1.28]n
Rosenbrock function	f_4=∑i=1n100×(x_i+1-x_i^2)^2+(1-x_i)^2	30	0	[−100, 100]n
Griewank function	f_5=14000∑i=1nx_i^2-∏i=1ncos(x_ii)+1	30	0	[−600, 600]n

The problem dimension n, the global optimum f, and the search range are listed in the table.

The main parameters are set based on the recommendation in [22]. In CPSO, the maximal velocity V and the minimal velocity V are 100,000 and -100,000, respectively (since it is believed that V and V are not even needed in CPSO [6]). For other algorithms, V is 10% of the search space’s upper bound while V is 10% of the lower one. The acceleration constants c1 and c2 are both 2.05 for CPSO. For SPSO, RPSO, and IPSO, the figure is 2.0. For CoPSO, L is 3, the initial value of μ is 5 (added by 1 for each level up), and the gap between the acceleration constants of each level is 0.2 (c1 and c2 are the same with a maximum value of 2.0). A fixed inertia weight value of 0.6 is adopted for SPSO, IPSO, and CoPSO. The maximum iteration number is 1000 for all algorithms, meaning that the maximum particle number is 1000 in IPSO. In other cases, the population size is 80. A total of 50 repeating runs are conducted for each experiment. In addition, as recommended by Yang et al. [23], Wilcoxon rank sum tests are conducted for the statistical analysis between CoPSO and the other algorithms with a significance level of 0.05.

Results and discussion

Table 2 summarizes all the experimental results of 50 independent runs. Wilcoxon rank sum tests further verify the significance of these numerical results. The numbers in bold represent the comparatively best values. In general, CoPSO greatly outperforms SPSO, RPSO, CPSO, and IPSO on most metrics. For the exceptions, it still shows equivalent performances to the comparison algorithms. These results indicate that the multi-level structure and the competition mechanism can improve the solution quality of CoPSO.

Table 2

Experimental results for all algorithms on benchmark functions.

		CoPSO	SPSO	RPSO	CPSO	IPSO
f 1	Best	2.8549E-65	5.5264E-09	1.5411E-35	1.5978E-23	3.0563E-11
	Worst	9.7607E-57	5.7243E-06	6.5542E-29	1.2931E-19	7.2027E-09
	Median	1.0691E-61	8.5793E-08	5.3866E-33	9.3553E-22	3.6676E-10
	Mean	3.0003E-58	4.1779E-07	1.3600E-30	7.2911E-21	7.5476E-10
	Std	1.4938E-57	9.5508E-07	9.2625E-30	2.1853E-20	1.2102E-09
	p-value	-	7.0661E-18	7.0661E-18	7.0661E-18	7.0661E-18
f 2	Best	4.9143E-37	3.2218E-05	1.2864E-21	5.8821E-11	3.5853E-06
	Worst	8.5886E-15	9.8248E+00	6.4473E-13	6.8001E-07	2.1356E+02
	Median	1.3265E-31	3.1720E-04	1.0877E-19	2.6652E-09	9.8882E-05
	Mean	3.8317E-16	3.9115E-01	1.6602E-14	3.7949E-08	7.5323E+00
	Std	1.4994E-15	1.6359E+00	9.2441E-14	1.1971E-07	3.3733E+01
	p-value	-	7.0661E-18	5.1559E-11	7.0661E-18	7.0661E-18
f 3	Best	3.0297E-03	2.3906E-02	8.9131E-03	2.7188E-03	7.4989E-03
	Worst	1.7855E-02	1.3631E-01	5.5517E-02	2.1025E-02	3.4737E-02
	Median	8.6013E-03	7.1187E-02	4.0110E-03	1.0032E-02	1.7729E-02
	Mean	8.7629E-03	7.3491E-02	2.5168E-02	1.0271E-02	1.8165E-02
	Std	3.4998E-03	24612E-02	9.1510E-03	4.6240E-03	6.8623E-03
	p-value	-	7.0661E-18	3.7961E-13	1.0448E-01	3.9592E-12
f 4	Best	3.4746E-03	1.4929E+01	6.4345E+00	7.6875E+00	4.0219E+00
	Worst	8.1844E+01	4.6145E+02	1.5713E+02	9.5701E+02	1.7156E+02
	Median	1.3684E+01	7.2457E+01	2.4572E+01	2.5592E+01	2.6071E+01
	Mean	1.7385E+01	6.7829E+01	4.4824E+01	8.7070E+01	4.4996E+01
	Std	1.8940E+01	6.8713E+01	3.3475E+01	1.5140E+01	3.6439E+01
	p-value	-	3.8499E-14	6.9808E-10	3.7706E-12	3.5908E-12
f 5	Best	0	3.3119E-10	0	0	9.4358E-13
	Worst	2.9459E-02	5.1369E-02	4.4293E-02	6.6471E-02	5.3964E-02
	Median	7.3960E-03	7.3961E-03	7.3960E-03	7.3960E-03	9.8573E-03
	Mean	7.1907E-03	9.0566E-03	9.6510E-03	1.0782E-02	1.1233E-02
	Std	8.2867E-03	1.1201E-02	1.1035E-02	1.3882E-02	1.0670E-02
	p-value	-	1.0732E-02	3.8092E-01	3.7739E-01	8.9401E-04

Best, Worst, Median, Mean, and Std are the best, worst, median, average, and the standard deviation of the final results in 50 independent runs, p-value is the statistical result obtained by Wilcoxon rank sum test with a significance level of 0.05.

It is believed that CoPSO can preserve excellent exploration and exploitation abilities while keeping a good balance between these two. In other words, CoPSO is able to jump out of local optima and fully exploit the promising areas at a fast rate. To substantiate this hypothesis, swarm diversity is used to measure an algorithm’s exploration ability. Supposing the size of swarm S is N, the problem dimension is D, and x represents the position of particle i. As recommended by Yang et al. [23], the diversity of swarm S is computed as follows: where D(S) is the swarm diversity of S and is the average position of the swarm. Further, the converging speed can reflect whether an algorithm can compromise between exploration and exploitation properly. During the iterations, average swarm diversities and average global best fitness values (of 50 runs, in logarithmic form) for each algorithm on five benchmarks are recorded in Fig 4.

Fig 4

Average swarm diversities and average global best fitness values of all algorithms at each iteration.

Note that the vertical axes are in logarithmic form. (a) The results on f1. (b) The results on f2. (c) The results on f3. (d) The results on f4. (e) The results on f5.

Generally, CoPSO converges faster with better solutions than all other algorithms in most cases. As for swarm diversity, CoPSO holds a similar tendency on all benchmarks, i.e., decreasing rapidly in the early stage of the evolution and staying at a relatively high value (or even the highest) in the latter stage. This can be explained by the structure of CoPSO. Particles of different levels are in different regions of the search space. Those of lower levels are more likely to be far away from the global best position while those of higher levels can be very close to it. CoPSO treats different levels differently by enhancing exploration for lower levels and promoting exploitation for higher ones. Thus, the algorithm has more potential to locate promising areas in the early stage and then converges to them very quickly, causing the swarm diversity to decline. However, particles may not upgrade because they find a global optimum, but because they are trapped in local optima. In this case, the competition mechanism will force the swarm to escape from local areas, which increases the diversity simultaneously.

Despite the overall similarity, the behaviors of CoPSO differ slightly on different kinds of test functions. f1 is a simple unimodal function with only one global minimum, making it possible to achieve fast convergence. Although all algorithms converge exponentially to the optima, CoPSO greatly surpasses the others due to better exploitation of the promising areas at the beginning and high-intensity exploration in the latter stage. For f2, CoPSO has a competitive performance compared to RPSO and even shows better potential in the end as a result of higher swarm diversity. f3 is a noisy quartic function. f4 is a classic optimization problem with a global minimum inside a long, narrow, parabolic-shaped valley. f5 is a multimodal function where the number of local optima increases exponentially as the problem dimension increases. For all these complicated benchmarks, CoPSO always keeps an excellent balance between exploration and exploitation, which helps it to achieve the fastest converging rate or the shortest stagnation at local optima. The other algorithms lack the adjusting ability of CoPSO. For instance, IPSO puts too much emphasis on exploration (holding the highest swarm diversity on most benchmarks) while RPSO and CPSO are too focused on exploitation, which degenerates their overall performances.

Computational complexity analysis

According to Yang et al. [23], given a fixed number of fitness evaluations, the computational complexity of an evolutionary algorithm is generally calculated by analyzing the extra cost in each generation without considering the cost of function evaluations, which is problem-dependent. As CoPSO inherits the simple structure of SPSO, its computational complexity will be analyzed by comparison with SPSO.

Assuming the population size is N and the problem dimension is D, updating particles’ states takes O(N × D) time in each iteration for SPSO. By comparison, adding line 19 to line 25 in Algorithm 1 costs extra time for CoPSO. Specifically, particles tend to update pbest frequently at the early stage of the iterations, so it only costs O(N) extra time to check l and update related parameters. As l increases, the algorithm may have to regenerate some new particles, which costs O(N × D) extra time in the worst case. However, this situation is only possible in certain generations. Besides, the value increase of μ also controls the time cost of this part. As for the space complexity, since CoPSO needs to store the information of l and the time particles stay at the same pbest, it requires O(N) extra memory than SPSO (which takes O(N × D) space to save particles’ positions and velocities). To conclude, with proper parameter setting, the computational complexity of CoPSO can be controlled within an acceptable range.

Table 3 lists the average computing time for a single run (in 50 repeating experiments) with the same parameters as the numerical experiments. SPSO, RPSO, and CPSO take nearly the same time on different functions due to their similar algorithmic structures. However, IPSO has a growing population, dramatically increasing fitness evaluations at the latter stage of the iterations. Consequently, the computational cost of IPSO is the highest. Based on the complexity analysis above, CoPSO takes acceptable extra time in each iteration compared to SPSO, verified by the experimental result that CoPSO costs slightly higher than SPSO on the benchmarks. In conclusion, CoPSO surpasses the other algorithms in solution quality while remaining computationally efficacious.

Table 3

Average runtime for all algorithms on benchmark functions (in second).

	CoPSO	SPSO	RPSO	CPSO	IPSO
f 1	9.3840e-02	9.2220e-02	9.0920e-02	8.9540e-02	5.3714e-01
f 2	9.6400e-02	9.2880e-02	9.3700e-02	9.0220e-02	5.6548e-01
f 3	1.5344e-01	1.5248e-01	1.5138e-01	1.5044e-01	9.3878e-01
f 4	1.0150e-01	9.8220e-02	9.9000e-02	9.8580e-02	5.7976e-01
f 5	1.5624e-01	1.5144e-01	1.5274e-01	1.5212e-01	9.3274e-01

Case study of the simulation-optimization approach

To demonstrate that MAS-CoPSO can be used in practical applications, a case study of a real marine oil spill accident is carried out. This section presents the accident background, the implementation of MAS-CoPSO in a response planning system, and the mathematical formulation of the optimization problem. Further, an in-depth analysis of the system’s performance is given.

Background

Frequent marine oil spills have posed severe tests to the environment. Reducing the loss caused by oil spills has become an important issue. The capability of the MAS-CoPSO approach is tested through a case study of the Sanchi oil spill accident, which is caused by a collision between Panamanian oil tanker Sanchi and Chinese bulk carrier Changfeng Crystal. The accident happened on January 6, 2018, and the collision site is about 160 nautical miles east of the Yangtze River Estuary. Fig 5 illustrates the approximate location where the accident happened and the areas around it. Tons of condensate oil carried by Sanchi leaked into the ocean, causing significant economic loss and environmental pollution. Researchers have pointed out that oil tankers are transporting around 90% of all the oil around the world [24]. Furthermore, ship collision is the primary cause of many catastrophic marine oil spills, which suggests the representativeness of the case study.

Fig 5

Geographical map of the approximate collision site and surrounding areas.

Map services and data available from the U.S. Geological Survey.

Zhong and You [25] have concluded that after spill accidents, oil leaked into the ocean forms scattered slicks around the spill site and undergoes various physical and chemical changes due to environmental forces. The most dominant processes are spreading, evaporation, emulsification, and dispersion. These processes will dramatically change the volume, area, thickness, and viscosity of those slicks, which will affect the efficiency of oil spill response operations as a consequence.

There are some equipped response teams berthed at docks near the spill site. They will conduct a wide range of actions to clean up the ocean, among which booms, skimmers, chemical dispersants, and in situ burning are the most frequently used methods. Booms are generally the first equipment deployed after an oil spill and are often used to protect shorelines, divert oil to certain areas, or concentrate oil for further recovery and burning. Skimmers are adopted to recover oil or oil-water mixtures from the water surface. The characters of the slick will affect the working efficiency of skimmers. Chemical dispersants can reduce the oil-water interfacial tension, which should be initialized as soon as possible. In situ burning indicates controlled burning of the oil at or near the site. In this case, it is assumed that the spraying of dispersants and the concentration of leaked oil have been completed quickly after the accident. Besides, in situ burning will emit toxic substances and waste the leaked oil. Thus, the primary response operation is recovery.

This study focuses on supporting response planning after the accident. A simulation-optimization system is built based on the proposed MAS-CoPSO method. The overall objective is to clean up the polluted sea area in the most efficient way, i.e., as soon as possible. The system is supposed to realize interactive spill simulation and response optimization while considering the influences of the oceanic environment. It should also give the most reasonable allocations for available resources and the schedule of response teams’ actions.

Implementation

The implementation of MAS-CoPSO in the system is described in Fig 6, which is an instantiation of Fig 3. The MAS module and the CoPSO module work collectively, composing a dynamic system for marine oil spill response planning. The CoPSO algorithm is encapsulated as a function to be called before each time step. Meanwhile, agents of the simulation module consistently update their behaviors and interact with each other, realizing the oil spill fate modeling and response operation modeling. The agents must follow specific rules in reality. For instance, the property changes of oil slicks are calculated by weathering models considering spreading, evaporation, emulsification, and dispersion. Skimmers and ships of response teams should obey the rules for oil recovery. The performance of a response team can be affected by the actions of other teams and the characteristics of oil slicks. For example, when a skimmer works on a slick to collect oil, the volume and area of the slick, the evaporated and dispersed oil rates, the viscosity and the water content of the oil, will all be affected. If more than one team are instructed to head for the same slick, they will collaborate.

Fig 6

Implementation of the MAS-CoPSO approach in the case study.

To complete the global objective of minimizing response time while cleaning up the spill site to a large extent (the stop criteria, only a small portion of leaked oil remained in the ocean), CoPSO is utilized to maximize the decreased volume of oil in each simulation time step. Meanwhile, MAS should simulate equipment location, response process, oil spill fate, and transportation. The system can be further updated to suit different purposes and requirements.

Mathematical formulation of the optimization problem

Since the weathering process dramatically affects oil properties, it is necessary to simulate it as accurately as possible. Furthermore, the objective function and constraints of the optimization problem can be derived from the model.

Simulation of the weathering process

First of all, the initial area of a slick, A0 (m2), can be calculated by the equation shown as follows [26]: where ρ (g ⋅ cm−3) is the density of seawater, ρ0 (g ⋅ cm−3) is the density of oil, ν (0.801 × 10−6m2 ⋅ s−1 under 30°C) is the kinematic viscosity of seawater, V0 (m3) is the inital volume of the slick, g (m ⋅ s−1) is the acceleration of gravity, k2 and k3 are constants with values of 1.21 and 1.53, respectively.

The area changing rate of the slick due to spreading can be modeled as follows [27]: where A (m2) is the current surface area of the slick, t (s) refers to the time since the accident happened, V (m3) is the current volume of the slick, K1 (s−1) is a dominant physicochemical parameter of the oil with a default value of 150.

Evaporation is assumed as the most influential environmental effect, causing a loss of up to 20-50% of the spill. The rate that oil evaporates from the sea surface is modeled by the following equation [28]: where F (%) is the current volume fraction of the oil that has been evaporated with an initial value of F( = 0, T (K) is the oil temperature, A and B are empirical constants with fixed values of 6.3 and 10.3, respectively. K (m ⋅ s−1) is the mass transfer coefficient for evaporation which can be calculated as follows [29]: where v (m ⋅ s−1) is the wind speed. T and T are the initial boiling point and the gradient of the oil distillation curve, respectively. Their values can be calculated through functions of the oil API (American Petroleum Institute) degree as follows:

Emulsification can also influence the slick characteristics dramatically, especially for the oil viscosity. The initial oil viscosity μ0 (%) can be calculated using the following equation [29]: where AC (%) is the asphaltene content of the parent oil, which is a constant. As the viscosity changes over time, its changing rate is given by [30]: where μ (%) is the current viscosity, C3 is a constant for the final water content, C4 is an oil-dependent constant, Y (%) is the fractional water content in the emulsion, it changes over time, which can be computed with the following equation [27]: where K is an empirical constant between 1 × 10−6 and 2 × 10−6, Y has an initial value of Y( = 0.

Finally, dispersion is also a vital factor in the loss of oil. The volume of oil naturally dispersed changes over time, which can be modeled by [27]: where V (m3) is the total dispersed oil volume until now with an initial value of V( = 0 and ζ (cP) is the oil-water interfacial tension.

Objective function and constraints

In the response process, the optimization problem is formulated on the basis of each simulation time step. Suppose that the ith response team is recovering oil on the kth slick during time step T, define ORR (m3 ⋅ hr−1) as the oil recovery rate of team i (i.e., the amount of oil that the team can recover per hour) and ST (m) as the current oil thickness of slick k. ORR is determined by the properties of the team’s skimmer and ST: where α and β are empirical coefficients obtained from experimental tests, reflecting the features of the skimmer. ST can be calculated by: where V (m3) and A (m2) are the current oil volume and the area of the kth slick, respectively. A can be calculated according to Eqs (3) and (4), V can be computed as follows: where V (m3) is the remaining oil volume of this slick at the end of the last time step (i.e., the initial oil volume of the current time step), EV (m3) is the volume loss caused by evaporation, DV (m3) is the volume loss caused by dispersion, and SV (m3) is the volume recovered by team i. Notice that SV may not exist, which indicates that no response team is working on slick k right now. The system can get the value of EV, DV, and SV according to Eqs (5), (8) and (9), respectively. During the simulation, for simplification, a short T should be chosen and the system will use the value of V for the calculations regarding V.

The aim of the system is to clean up the polluted sea area as soon as possible. Therefore, based on the proposed method, given a fixed simulation time step, the oil volume loss will be maximized during each step, which is composed of the evaporation loss, the dispersion loss and the recovered volume. Suppose there are m oil slicks and n response teams during T, V represents the total oil volume loss in this period. F and μ refer to the evaporated oil fraction and the oil viscosity of slick k, respectively. The objective function and constraints are given by: s.t.

Performance of the system

To examine the efficiency of MAS-CoPSO, its performance is compared with the shortest distance selection approach (SDS). Ye et al. [31] have suggested that SDS is a simple strategy commonly used in marine oil spill emergency response, which indicates a process that allows a response team to choose the nearest oil slick as the target for oil recovery. After meeting the cleaning requirements, they will choose the second nearest oil slicks to continue.

Therefore, when conducting SDS, the distances between ships and slicks are the judgment criteria for response planning. When initializing the simulation process, each response team (corresponding to a certain response agent) is assigned with the closest oil slick (corresponding to a certain spill agent) to clean. Before each time step, instead of calling the CoPSO module, SDS will check whether the slick is cleaned up, if the oil volume is below the set threshold, the ship will head to the closest one in the remaining slicks, otherwise, it will stay at the current location until finishing the cleaning task. Compared to MAS-CoPSO, SDS is more straightforward, which explains why it is commonly used in real accidents. However, SDS makes plans without considering the oceanic forces and the interactions between agents, which according to previous discussions, can dramatically influence the overall efficiency. Thus, comparative experiments are conducted for further analysis.

Simulation settings

Chen et al. [32] introduced the process and consequences of the accident in detail. Based on their description, it is assumed that the spilled oil was split into ten slicks within the East Sea with a total volume of 113,000 tons. Table 4 lists the oil volumes and locations (described by the distance and direction from the approximate collision site) of these slicks. The ship-mounted skimmers belonging to five different response teams are the only available nearby cleanup means to be applied. Assume that all the teams have enough storage space for recovered oil. As the allocation process needs specific transportation time, the speed of ships is set at 30 km ⋅ h−1. Skimmers of different teams have different recovering efficiencies, which are affected by α and β in Eq (9). Table 5 lists the values of the empirical coefficients for different teams. Fig 7 illustrates the simulation process. The red cross represents the collision site, the black spills represent the oil slicks (the size reflects the oil volume of the corresponding slick), and the ships represent the response teams carrying the recovery task. The position and size of a slick will change over time. A ship can either be heading to a slick or working on it.

Table 4

Oil volumes and site locations of ten slicks formed after the accident.

Slick	Oil volume (ton)	Location (from the collision site)
		Distance (km)	Direction
1	9318.66	14	Northwest
2	6593.79	12	Northwest
3	15020.19	11	North
4	8344.60	4	Southwest
5	6258.39	2	North
6	13907.59	2	Southwest
7	16689.20	13	Southeast
8	5562.99	2	Southwest
9	7253.58	15	Southeast
10	24051.01	0	None

Table 5

Empirical coefficients for the calculations of five ship-mounted skimmers’ oil recovery rates.

Types of skimmers	Empirical coefficients
	α	β
SK1(TeamA)	0.01437	0.05602
SK2(TeamB)	-0.00791	0.84975
SK3(TeamC)	-0.01591	1.54975
SK4(TeamD)	0.02372	0.03583
SK5(TeamE)	-0.01026	1.27589

Fig 7

Illustration of the simulation process.

As mentioned above, spreading, evaporation, emulsification, and dispersion are the main oceanic effects. Table 6 lists the inputs for the modeling of these processes, including environmental parameters and the characteristics of the leaked oil (condensate oil). As [26] suggested, the drifting speed of oil slicks is about 2.5-4.5% of the wind speed, so it is set at 0.03 m ⋅ s−1.

Table 6

Parameters for the simulation of weathering processes.

Parameter	Value	Unit
Sea water temperature (T)	278.15	K
Wind speed (vwind)	10	m ⋅ s−1
Sea water density (ρw)	1.02	g ⋅ cm−3
Interface tension (ζt)	2.30 × 103	cP
Oil density (ρ0)	0.77	g ⋅ cm−3
Oil temperature (TK)	288.15	K
Oil API	52.27	None
Oil AC	1.21	%

The model is built in AnyLogic®. 50 repeating runs are carried out with 50 iterations per time step (T is one hour). Once the volume of the remaining oil is reduced to 10% of the original figure, the simulation will stop.

Table 7 presents the simulation results of MAS-CoPSO and SDS with the same stop criteria and time step. The operation time for achieving an oil recovery rate of 90% is 129.82 hours based on the optimal outcomes given by the MAS-CoPSO system, which is only 74.2% of the figure for SDS. Shorter response time means higher operation efficiency, less volume of wasted oil, and slighter damage to the environment. The results also indicate that total recovered oil holds a proportion of 65.54% in MAS-CoPSO, which is approximately 1.2 times higher than the number for SDS.

Table 7

Simulation results of MAS-CoPSO and SDS.

	MAS-CoPSO	SDS
Operation time (hr)	129.82	174.96
Recovered oil (Team A) (ton)	4133.35	2203.70
Recovered oil (Team B) (ton)	14658.40	13329.95
Recovered oil (Team C) (ton)	27268.71	20088.75
Recovered oil (Team D) (ton)	5598.22	5726.52
Recovered oil (Team E) (ton)	22401.52	20258.68
Total recovered oil (%)	65.54	54.52
Evaporated oil (%)	24.13	35.34
Dispersed oil (%)	0.52	0.51
Remaining oil (%)	9.81	9.63

The operation time (hr), the oil volume recovered by each team (ton), the proportion of each oil loss type (%), and the proportion of remaining oil after the response process (%) are listed in the table.

Fig 8 illustrates the variations of remaining oil volume for each slick in MAS-CoPSO and SDS scenarios during the entire simulation process. The optimization outcomes of MAS-CoPSO are highly related to the thickness of the oil slicks, and the system intends to keep a balanced volume level for each slick by optimizing the time for allocation and the response efficiency (see Fig 8a). However, oil thickness does not affect the decision-making of the SDS model, so it will not try to reduce the volume steadily. As a result of different strategies, the curves of MAS-CoPSO are much smoother than those of SDS. In addition, the SDS method might ignore slicks too far from the response teams (e.g., Slick 1), leading to long-time exposure of the oil, emitting more harmful substances (see Fig 8b).

Fig 8

Variations of remaining oil volume for each slick in MAS-CoPSO and SDS.

(a) The results of MAS-CoPSO. (b) The results of SDS.

Fig 9 depicts the variations of accumulated recovered volume for each team in two approaches. MAS-CoPSO tries to find optimal schedules for all teams in every time step, so the curves climbs stably (see Fig 9a). Meanwhile, SDS always cleans up one slick before moving to another, leading to discontinuous curves, which also hinders the increase of the overall efficiency (see Fig 9b).

Fig 9

Variations of accumulated recovered volume for each team in MAS-CoPSO and SDS.

(a) The results of MAS-CoPSO. (b) The results of SDS.

The above results show that the response planning system based on MAS-CoPSO outperforms the commonly used SDS strategy in terms of time consumption, oil recovery rate, and environmental impact. Therefore, oil spill response teams can schedule their operations according to the simulation-optimization outcomes of this system rather than relying on the inefficient SDS approach. Moreover, complex problems and high-intensity interactions can enhance the advantages of MAS-CoPSO, indicating the system’s ability to function in scenarios with higher oil volumes and more response teams. Even though the case study is simplified and focuses on the oil recovery process, the system has the potential to comprehensively support multiple cleanup techniques concerning booms, chemical dispersants, and in situ burning. Besides, modeling of the weathering process can be easily enriched by considering more complicated oceanic forces such as dissolution, photo-oxidation, sedimentation, and biodegradation [25]. Hydrodynamic simulation of oil spill trajectories can also be considered. In addition, the application range of the MAS-CoPSO-based system can be further enlarged by handling uncertainties and risk assessments. With these extensions, oil spill response teams can get a practical simulation-optimization tool to support their planning process.

Conclusion

An improved particle swarm optimization algorithm called CoPSO is proposed in this paper. Inspired by the features of bird flocks, a multi-level design and a competition mechanism are introduced into CoPSO to balance the exploration and exploitation more effectively, increase solution accuracy, achieve fast convergence, and avoid stagnation at local optima. The performance of CoPSO is tested on a series of well-known benchmark functions. Comparisons are made between CoPSO, SPSO, and three classic variants of it. Experimental results have verified its capability.

This paper also proposes a problem-independent simulation-optimization approach called MAS-CoPSO to combine CoPSO with MAS. MAS-CoPSO can achieve a dynamic simulation of various complex systems and give satisfying solutions to related optimization problems. In MAS-CoPSO, the relationship between the simulation module and the optimization module is highly cohesive. The effectiveness of MAS-CoPSO is substantiated through a response planning system for the Sanchi oil spill accident. The system simulates the accident scene with consideration of the oil slicks’ physicochemical evolution. Case study results show that the system can optimize response device allocation, operation scheduling, and time consumption, causing slighter damage to the environment.

MAS-CoPSO can also be applied in other scenarios to solve different optimization problems depending on the users’ requirements. For future studies, we will consider the combination with other methods, such as fuzzy control and neural network.

20 Jun 2022

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Reviewer #1: The contribution of this paper is good and I am happy to endorse its acceptance at some point. However, there are several major and minor comments to address. I have listed them as follows:

• First off, please clearly state the gap targeted in this paper at the end of introduction and list down the hypotheses

• In terms of research method and design, there is not much in the paper.

• The comparative algorithms in the experiments are not properly acknowledged and cited

• I also suggest adding some figures to better articular the content as the paper looks very dry at the moment.

• Analysis of the results is missing in the paper. There is a big gap between the results and conclusion. There should be the result analysis between these two sections. After comparing the numerical methods, you have to be able to analyse the results and relate them to their structures. It would be interesting to have your thoughts on why the method works that way? Such analyses would be the core of your work where you prove your understanding of the reason behind the results. You can also link the findings to the hypotheses of the paper. Long story short, this paper requires a very deep analysis from different perspectives

• There is no statistical test to judge about the significance of the numerical method’s results. Without such a statistical test, the conclusion cannot be supported

• There is no discussion on the cost effectiveness of the proposed method. What is the computational complexity? What is the runtime? Please include such discussions. You can also use the big oh notation to show the computation complexity.

• Some mathematical notations and Lemma presentations are not rigorous enough to correctly understand the contents of the paper. The authors are requested to recheck all the definition of variables and further clarify these equations.

Reviewer #2: This paper proposes a hybrid particle swarm optimisation method for multi-agent simulation case study. The paper needs some revisions and after applying them can be published.

1. please clear what are the main research gaps in Abstract.

2. The CoPSO algorithm is not clear and can be described more.

3. The major novelties of this work should be listed in the introduction.

4. Next section after introduction should be methods and materials.

5. In the section of case study, please describe the technical details of benchmarks only not methods details.

6. It can be helpful to have a better understanding of CoPSO with adding some relevant references and adaptive PSO as well such as a) FAIPSO: fuzzy adaptive informed particle swarm optimization. Neural Computing and Applications. 2013 Dec;23(1):95-116. b) "A new kind of PSO: predator particle swarm optimization." International Journal on Smart Sensing and Intelligent Systems 5, no. 2 c) An adaptive particle swarm optimizer with decoupled exploration and exploitation for large scale optimization." Swarm and Evolutionary Computation 60 (2021): 100789.

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16 Aug 2022

Dear Editors and Reviewers

Thank you for your careful review and constructive comments regarding our manuscript. These comments are all valuable and helpful for improving our paper. All the authors have seriously discussed about all the comments. We have tried our best to revise the manuscript in accordance with the comments and highlighted in yellow all the amends in our revised manuscript. The point-by-point replies are listed below. We sincerely appreciate your help.

Reply to Reviewer 1 Comments

Comment 1

First off, please clearly state the gap targeted in this paper at the end of introduction and list down the hypotheses.

-Response & Revision:

Thank you very much for the valuable feedback on our Introduction. We have modified the end of our Introduction as follows:

We can see from the literature that although there are many studies on improving SPSO, there are not so many attempts to combine proposed algorithms with MAS and apply them to practical problems. In fact, utilizing improved algorithms in simulation-optimization systems is necessary because complex tasks require high-performance optimizers, and SPSO may not meet this requirement. Besides, existing studies on the combination of MAS and PSO may vary in engineering fields, yet they all designed the system based on specific problems of their own research field without summarizing a general and universal framework, which is field-independent and can be implemented in various settings. Therefore, this paper focuses on a more in-depth exploration of improving PSO and integrating it into MAS to solve practical problems. The main contributions of this paper are as follows:

1. We proposed an improved particle swarm optimization algorithm called CoPSO according to the biological nature of birds. CoPSO has a multi-level structure and a competition mechanism, aiming to improve performance by enhancing exploration while balancing exploitation.

2. We proposed a general approach to integrate CoPSO into MAS called MAS-CoPSO. MAS-CoPSO is supposed to achieve a very close combination between simulation and optimization, and the application should not be limited to certain engineering fields and specific problems.

3. We conducted a series of comparison experiments to substantiate the validity of CoPSO.

4. We implemented MAS-CoPSO in a case study of the Sanchi oil spill accident to solve the response planning problem.

Comment 2

In terms of research method and design, there is not much in the paper.

-Response & Revision:

We appreciate a lot for your comments on the formation of the paper. To address this problem, we have adjusted the structure of the paper by adding section Methods and materials after Introduction. There are three subsections in Methods and materials, namely, Multi-agent Simulation, The proposed CoPSO algorithm, and The MAS-CoPSO approach. These three parts have described the research method and design of the paper in detail.

Comment 3

The comparative algorithms in the experiments are not properly acknowledged and cited.

-Response & Revision:

Thank you for pointing out this problem. We have four comparative algorithms, namely, the standard formation of PSO (SPSO) [1], the random weight method (RPSO) [2], the constriction factor approach (CPSO) [3], and the incremental particle swarm optimizer (IPSO) [4]. We briefly mentioned these algorithms in Introduction without proper acknowledgments and citations in the experiments. For modification, we have cited these algorithms and described their contents in section Experiments and analysis of CoPSO.

Comment 4

I also suggest adding some figures to better articular the content as the paper looks very dry at the moment.

-Response & Revision:

Thank you for your constructive suggestion. Firstly, we have added Fig 1 and Fig 2 (the mark numbers subject to the revised manuscript) for a better description of CoPSO. They are both two-dimensional schematic diagrams. Fig 1 illustrates how the particles’ levels change during the iterations. Fig 2 depicts how the competition mechanism functions in the algorithm. We have also added Fig 5 in Case study of MAS-CoPSO to enrich the background description. Fig 5 is a geographical map illustrating the approximate location where the Sanchi oil spill accident happened. In addition, we have modified some original figures. Fig 4 in the revised manuscript contains subgraphs of benchmark functions’ converging behaviors and their swarm diversity variations. We have also redrawn Fig 7 to depict the simulation process of the case study more clearly. Fig 8 and Fig 9 are modified for better comparison.

Comment 5

Analysis of the results is missing in the paper. There is a big gap between the results and conclusion. There should be the result analysis between these two sections. After comparing the numerical methods, you have to be able to analyze the results and relate them to their structures. It would be interesting to have your thoughts on why the method works that way? Such analyses would be the core of your work where you prove your understanding of the reason behind the results. You can also link the findings to the hypotheses of the paper. Long story short, this paper requires a very deep analysis from different perspectives.

-Response & Revision:

Thank you for pointing out this problem. As we have adjusted the formation of the paper, modifications for this comment can be found in section Experiments and analysis of CoPSO. Considering some exceptional results may hugely affect the “Mean” performance and conceal the algorithms’ properties, we have added the comparison item “Median” in Table 2. Further, we have also introduced the concept of swarm diversity [5] to analyze the algorithms’ exploration abilities. After comparing the diversity evolution and the convergence behavior of each algorithm on all the benchmarks in Fig 4, we have discussed the results based on the architectures of the algorithms and concluded that the multi-level structure and the competition mechanism did improve the performance of CoPSO by enhancing exploration while balancing exploitation. A detailed analysis can be found in Results and discussion of Comparison experiments.

Comment 6

There is no statistical test to judge about the significance of the numerical method’s results. Without such a statistical test, the conclusion cannot be supported.

-Response & Revision:

Thank you for this feedback. For modification, as recommended by Yang et al. [5], we have conducted Wilcoxon rank sum test for the statistical analysis between CoPSO and the other algorithms with a significance level of 0.05. The p-values were listed in Table 2. Results of the statistical test have supported our conclusion. A detailed discussion can be found in Results and discussion of Comparison experiments.

Comment 7

There is no discussion on the cost effectiveness of the proposed method. What is the computational complexity? What is the runtime? Please include such discussions. You can also use the big oh notation to show the computation complexity.

-Response & Revision:

Thank you for this comment. We have added subsection Computational complexity analysis in Experiments and analysis of CoPSO to make up for this part. According to Yang et al. [5], given a fixed number of fitness evaluations, the computational complexity of an evolutionary algorithm is generally calculated by analyzing the extra cost in each generation without considering the cost of function evaluations, which is problem-dependent. As CoPSO inherits the simple structure of SPSO, we have analyzed the computational complexity (in terms of time and space, using the big oh notation) of CoPSO by comparing it with SPSO. We have also compared the runtime of all the algorithms and analyzed the results listed in Table 3.

Comment 8

Some mathematical notations and Lemma presentations are not rigorous enough to correctly understand the contents of the paper. The authors are requested to recheck all the definition of variables and further clarify these equations.

-Response & Revision:

Thank you for pointing out this problem. We have rechecked the paper and modified the following ambiguous mathematical notations and lemma presentations (the mark numbers of the tables and equations subject to the revised manuscript):

As the position and velocity of a particle (for example, particle i) are supposed to be vectors, we have modified the presentation of these two variables to the bold form x_i=(x_i^1,x_i^2,…,x_i^D) and v_i=(v_i^1,v_i^2,…,v_i^D), where the superscripts indicate dimensions. We have also modified the personal best position of particle i and the global best position to 〖pbest〗_i=(〖pbest〗_i^1,〖pbest〗_i^2,…,〖pbest〗_i^D) and gbest=(〖gbest〗^1,〖gbest〗^2,…,〖gbest〗^D), respectively. For a more concise expression, we have modified the denotation of the level of particle i to l_i with an upper bound of L_max. Finally, we have further clarified Eq (1), Eq (2), and Algorithm 1.

We have added the definitions of n (the problem dimension) and f_min (the global optimum) in the note of Table 1.

We have added the units of ρ_ω and ρ_0 in Eq (3).

We have supplemented the definitions of A, t and V in Eq (4).

We have supplemented the definitions of F_E and added the calculation of T_O and T_G in Eq (5).

We have added the unit of μ_0 and the explanation of AC in Eq (6), as well as the unit of μ and the calculation of Y_W in Eq (7).

We have supplemented the definitions of V_D and added the unit of ζ_t in Eq (8). We have also modified the presentation of Eq (8) to avoid the misunderstanding of variable A and variable V.

We have supplemented the definitions of 〖ORR〗_i in Eq (9) and 〖ST〗_k in Eq (10). We have also added the explanation of parameter α and β in Eq (9), as well as the calculation of A_k in Eq (10).

We have supplemented the definitions and the calculations of V_k, V_k0, 〖EV〗_k, DV_k, and 〖SV〗_i in Eq (11).

We have added the definitions of V_step in Eq (12), 〖F_E〗_k in Eq (14), and μ_k in Eq (16). We have also modified Eq (13), Eq (14), and Eq (16) for better presentation.

We have stated our objective in text form for a better description of Eqs (12)~(18) (The aim of our system is to clean up the polluted sea area as soon as possible. Therefore, based on our method, given a fixed simulation time step, we will maximize the oil volume loss during each step, which is composed of the evaporation loss, the dispersion loss and the recovered volume.).

We have added the meaning of each result item in the note of Table 7.

References in replies:

[1] Kennedy J, Eberhart R. Particle swarm optimization. In: Proceedings of ICNN’95-international conference on neural networks. vol. 4. IEEE; 1995. p. 1942–1948.

[2] Eberhart RC, Shi Y. Tracking and optimizing dynamic systems with particle swarms. In: Proceedings of the 2001 congress on evolutionary computation (IEEE Cat. No. 01TH8546). vol. 1. IEEE; 2001. p. 94–100.

[3] Clerc M, Kennedy J. The particle swarm-explosion, stability, and convergence in a multidimensional complex space. IEEE transactions on Evolutionary Computation. 2002;6(1):58–73.

[4] De Oca MAM, Stutzle T, Van den Enden K, Dorigo M. Incremental social learning in particle swarms. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics). 2010;41(2):368–384.

[5] Yang Q, Chen WN, Da Deng J, Li Y, Gu T, Zhang J. A level-based learning swarm optimizer for large-scale optimization. IEEE Transactions on Evolutionary Computation. 2017;22(4):578–594.

Reply to Reviewer 2 Comments

Comment 1

Please clear what are the main research gaps in Abstract.

-Response & Revision:

Thank you very much for pointing out the problems in our Abstract. For modification, we have stated the main research gaps and the contributions of our paper as follows:

Recently, there have been considerable researches on combining multi-agent simulation and particle swarm optimization in practice. However, most existing studies are limited to specific engineering fields or problems without summarizing a general and universal combination framework. Moreover, particle swarm optimization can be less effective in complex problems due to its weakness in balancing exploration and exploitation. Yet, it is not common to integrate MAS with improved versions of the algorithm. Therefore, this paper proposed an improved particle swarm optimization algorithm called CoPSO, introducing a multi-level structure and a competition mechanism to enhance exploration while balancing exploitation. Further, we integrated CoPSO into MAS by a problem-independent approach called MAS-CoPSO, aiming to simulate realistic scenarios dynamically and solve related optimization problems simultaneously.

Comment 2

The CoPSO algorithm is not clear and can be described more.

-Response & Revision:

Thank you for this feedback. We have modified subsection Improved particle swarm optimization algorithm with competition mechanism to address this problem, a detailed description can be found there. Firstly, we have introduced the standard formation of PSO and stated our motivation, i.e., improving the performance of PSO by enhancing exploration while balancing exploitation. Inspired by the biological nature of birds, we have proposed a multi-level structure and a competition mechanism in CoPSO. We divide particles into different levels and stipulate that the closer a particle is to a possible global optimum, the higher its level. If a particle doesn't update its personal best position in μ continuous iterations, it will upgrade. Particles of different levels learn at different acceleration constants. Fig 1 is a two-dimensional schematic diagram of the variations of particles’ levels during the iterations.

However, a higher level doesn’t always a mean closer distance to the global optimum, it can also indicate that the particle has fallen into local optima. The basic idea of the competition mechanism is to replace particles of extremely high levels with randomly generated new ones. Thus, the algorithm gets the potential to escape from local optima and explore the unknown search space. Fig 2 illustrates how the competition mechanism functions, particles in Fig 2a are trapped in local areas, once their levels get too high, we will regenerate new particles to replace them. From Fig 2b we can see that these new particles have located the right promising area.

The pseudo-code of CoPSO is shown in Algorithm 1.

Comment 3

The major novelties of this work should be listed in the introduction.

-Response & Revision:

Thank you for pointing out this problem. We have listed the novelties of the paper in the end of our Introduction as follows:

The main contributions of this paper are as follows:

1. We proposed an improved particle swarm optimization algorithm called CoPSO according to the biological nature of birds. CoPSO has a multi-level structure and a competition mechanism, aiming to improve performance by enhancing exploration while balancing exploitation.

2. We proposed a general approach to integrate CoPSO into MAS called MAS-CoPSO. MAS-CoPSO is supposed to achieve a very close combination between simulation and optimization, and the application should not be limited to certain engineering fields and specific problems.

3. We conducted a series of comparison experiments to substantiate the validity of CoPSO.

4. We implemented MAS-CoPSO in a case study of the Sanchi oil spill accident to solve the response planning problem.

Comment 4

Next section after introduction should be methods and materials.

-Response & Revision:

Thank you for your constructive comments on the formation of the paper. We have added section Methods and materials after Introduction. There are three subsections in Methods and materials, namely, Multi-agent Simulation, The proposed CoPSO algorithm, and The MAS-CoPSO approach. These three parts have described the research method and design of the paper in detail.

Comment 5

In the section of case study, please describe the technical details of benchmarks only not methods details.

-Response & Revision:

Thank you for the valuable feedback. Based on the modeling of an oil spill accident, we conducted MAS-CoPSO for response planning and used the shortest distance selection approach (SDS) as a contrast. SDS is commonly used in real oil spill response emergencies [1], which makes the comparison reasonable. We have added a more detailed description of the procedure of SDS in subsection The performance of the MAS-CoPSO-based system. Instead of calling the CoPSO module, SDS used the distances between ships and oil slicks as the measure for planning, which was the main difference between these two approaches. When initializing the simulation process, we assigned each response team with the closest oil slick to clean. Before each time step, we would check whether the slick was cleaned up, if the oil volume was below the set threshold, the ship would head to the closest one in the remaining slicks, otherwise, it would stay at the current location until completing the recovery.

Comment 6

It can be helpful to have a better understanding of CoPSO with adding some relevant references and adaptive PSO as well such as a) FAIPSO: fuzzy adaptive informed particle swarm optimization. Neural Computing and Applications. 2013 Dec;23(1):95-116. b) "A new kind of PSO: predator particle swarm optimization." International Journal on Smart Sensing and Intelligent Systems 5, no. 2 c) An adaptive particle swarm optimizer with decoupled exploration and exploitation for large scale optimization." Swarm and Evolutionary Computation 60 (2021): 100789.

-Response & Revision:

Thank you for providing these papers. We have cited it in our Introduction.

References in replies:

[1] Ye X, Chen B, Li P, Jing L, Zeng G. A simulation-based multi-agent particle swarm optimization approach for supporting dynamic decision making in marine oil spill responses. Ocean & Coastal Management. 2019;172:128–136.

[2] Neshat M. FAIPSO: fuzzy adaptive informed particle swarm optimization. Neural Computing and Applications. 2013;23(1):95–116.

[3] Neshat M, Sargolzaei M, Masoumi A, Najaran A. A new kind of PSO: predator particle swarm optimization. International Journal on Smart Sensing and Intelligent Systems. 2012;5(2).

[4] Li D, Guo W, Lerch A, Li Y, Wang L, Wu Q. An adaptive particle swarm optimizer with decoupled exploration and exploitation for large scale optimization. Swarm and Evolutionary Computation. 2021;60:100789.

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29 Aug 2022

8 Sep 2022

Dear Reviewers:

Thank you for your careful review and constructive comments regarding our manuscript. These comments are all valuable and helpful for improving our paper. All the authors have seriously discussed about all the comments. We have tried our best to revise the manuscript in accordance with the comments and highlighted in yellow all the amends in our revised manuscript. The point-by-point replies are listed below. We sincerely appreciate your help.

Reply to Reviewer 1 Comments

Comment 1

The results of your comparative study should be discussed in-depth and with more insightful comments on the behavior of your algorithm on various case studies. Discussing results should not mean reading out the tables and figures once again.

-Response & Revision:

Thank you very much for the valuable feedback. We have modified the discussion of the comparative study results in the revised manuscript as follows:

Generally, CoPSO converges faster with better solutions than all other algorithms in most cases. As for swarm diversity, CoPSO holds a similar tendency on all benchmarks, i.e., decreasing rapidly in the early stage of the evolution and staying at a relatively high value (or even the highest) in the latter stage. This can be explained by the structure of CoPSO. Particles of different levels are in different regions of the search space. Those of lower levels are more likely to be far away from the global best position while those of higher levels can be very close to it. CoPSO treats different levels differently by enhancing exploration for lower levels and promoting exploitation for higher ones. Thus, the algorithm has more potential to locate promising areas in the early stage and then converges to them very quickly, causing the swarm diversity to decline. However, particles may not upgrade because they find a global optimum, but because they are trapped in local optima. In this case, the competition mechanism will force the swarm to escape from local areas, which increases the diversity simultaneously.

Despite the overall similarity, the behaviors of CoPSO differ slightly on different kinds of test functions. f_1 is a simple unimodal function with only one global minimum, making it possible to achieve fast convergence. Although all algorithms converge exponentially to the optima, CoPSO greatly surpasses the others due to better exploitation of the promising areas at the beginning and high-intensity exploration in the latter stage. For f_2, CoPSO has a competitive performance compared to RPSO and even shows better potential in the end as a result of higher swarm diversity. f_3 is a noisy quartic function. f_4 is a classic optimization problem with a global minimum inside a long, narrow, parabolic-shaped valley. f_5 is a multimodal function where the number of local optima increases exponentially as the problem dimension increases. For all these complicated benchmarks, CoPSO always keeps an excellent balance between exploration and exploitation, which helps it to achieve the fastest converging rate or the shortest stagnation at local optima. The other algorithms lack the adjusting ability of CoPSO. For instance, IPSO puts too much emphasis on exploration (holding the highest swarm diversity on most benchmarks) while RPSO and CPSO are too focused on exploitation, which degenerates their overall performances.

Comment 2

Avoid lumping references as in [x, y] and all other. Instead summarize the main contribution of each referenced paper in a separate sentence. For scientific and research papers, it is not necessary to give several references that say exactly the same. Anyway, that would be strange, since then what is innovative scientific contribution of referenced papers? For each thesis state only one reference.

-Response & Revision:

Thank you for pointing out this problem. We have checked the references and removed items containing the same information as those kept in the revised manuscript (i.e., references 21, 23, 25, 26, and 27 of the original version).

Comment 3

Avoid using first person.

-Response & Revision:

Thank you for pointing out this problem. We have carefully checked the whole manuscript and modified related content.

Comment 4

Avoid using abbreviations and acronyms in title, abstract, headings and highlights.

-Response & Revision:

Thank you for your constructive suggestion. We have carefully checked these parts and modified related expressions to avoid using abbreviations and acronyms.

Comment 5

Please avoid having heading after heading with nothing in between, either merge your headings or provide a small paragraph in between.

-Response & Revision:

Thank you for pointing out this problem. We have noticed that this problem exists at the beginning of some sections and subsections. For modification, we have provided brief summaries in between. Please review the rebuttal letter or the revised manuscript for details.

Comment 6

The first time you use an acronym in the text, please write the full name and the acronym in parenthesis. Do not use acronyms in the title, abstract, chapter headings and highlights.

-Response & Revision:

Thank you for this feedback. We have carefully checked the manuscript and modified related content.

Comment 7

The results should be further elaborated to show how they could be used for the real applications.-Response & Revision:

Thank you for this comment. Our method has the potential to be used for real applications mainly because: 1) case study results indicate its superiority over SDS (which is commonly used in real situations); 2) it can still function in more complex scenarios; 3) it can be easily extended according to the users’ requirements and achieve a comprehensive simulation of reality. We have further discussed the results in the revised manuscript as follows:

The above results show that the response planning system based on MAS-CoPSO outperforms the commonly used SDS strategy in terms of time consumption, oil recovery rate, and environmental impact. Therefore, oil spill response teams can schedule their operations according to the simulation-optimization outcomes of this system rather than relying on the inefficient SDS approach. Moreover, complex problems and high-intensity interactions can enhance the advantages of MAS-CoPSO, indicating the system's ability to function in scenarios with higher oil volumes and more response teams. Even though the case study is simplified and focuses on the oil recovery process, the system has the potential to comprehensively support multiple cleanup techniques concerning booms, chemical dispersants, and in situ burning. Besides, modeling of the weathering process can be easily enriched by considering more complicated oceanic forces such as dissolution, photo-oxidation, sedimentation, and biodegradation [25]. Hydrodynamic simulation of oil spill trajectories can also be considered. In addition, the application range of the MAS-CoPSO-based system can be further enlarged by handling uncertainties and risk assessments. With these extensions, oil spill response teams can get a practical simulation-optimization tool to support their planning process.

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12 Sep 2022

12 Sep 2022

Dear Editors and Reviewers:

Thank you for your careful review and constructive comments regarding our manuscript. These comments are all valuable and helpful for improving our paper. All the authors have seriously discussed about all the comments. We have tried our best to revise the manuscript in accordance with the comments. The point-by-point replies are listed below. We sincerely appreciate your help.

Reply to Reviewer 1 Comments

Comment 1

Are all the images used in this work copyrights free? If not, have the authors obtained proper copyrights permission to re-use them? Please kindly clarify, and this is just to ensure all the figures are fine to be published in this work.

-Response & Revision:

Thank you very much for the comment. We have carefully checked the images used in this work. All the images are copyrights free.

Comment 2

Also, the list of references should be carefully checked to ensure consistency with between all references and their compliances with the journal policy on referencing.

-Response & Revision:

Thank you for the suggestion. We have checked the reference list according to the manuscript content and the journal policy. The reference list is complete and correct. None of the cited papers have been retracted.

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26 Sep 2022

A novel multi-agent simulation based particle swarm optimization algorithm

PONE-D-22-15746R3

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3 Oct 2022

PONE-D-22-15746R3

A novel multi-agent simulation based particle swarm optimization algorithm

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