Research on Cooperative Task Assignment of Multi-Agent Track

Bolt Operation Robot Based on Optimized Multi-Objective Particle

Swarm Optimization

Yanni Shen

, Jianjun Meng

1,2,3,4,*

, Defang Lv

, Gaoyang Meng

and Zhicheng Wang

School of Mechanical Engineering, Lanzhou Jiaotong University, Lanzhou 730070, China

Institute of Mechanical and Electrical Technology, Lanzhou Jiaotong University, Lanzhou 730070, China

Gansu Engineering Technology Research Center of Logistics and Transportation Equipment Informatization,

Lanzhou 730070, China

Gansu Logistics and Transportation Equipment Industry Technology Center, Lanzhou 730070, China

Keywords: Multi-Agent, Track Bolt, Unmanned Vehicles, Task Assignment, Cooperative Control.

Abstract: To improve the efficiency of single-person, single-machine track bolt maintenance during railway skylight

periods, we propose a collaborative task assignment control method using a multi-agent track bolt operation

robot. A control decision model is developed with constraints on operation time and distance, aiming to

optimize both total collaborative operation distance and completion time. By incorporating equations for

robot speed and operation time, we derive the Pareto solution set for multi-agent task assignment. The

method’s effectiveness is verified through an enhanced particle evolution technique within the multi-

objective particle swarm optimization (MOPSO) algorithm, and its performance is compared with that of

standard MOPSO. Simulations in a real-world track bolt maintenance environment show that this approach

produces a higher-quality Pareto solution set for task assignment.

1 INTRODUCTION

As railway construction in China accelerates, daily

track maintenance has become increasingly essential.

The condition of track bolts, a critical component of

the track structure, directly impacts train safety and

stability. However, the current single-person

maintenance method during skylight periods is labor-

intensive, inefficient, and poses safety risks. Thus,

developing an efficient Multi-Agent Track Bolt

Operation Robot (MATBOR) is imperative. In

alignment with the "Digital Railway Planning"

initiative by China National Railway Group Co., Ltd.,

we aim to achieve comprehensive digitalization and

intelligence in railway operations, thereby enhancing

modernization efforts. Improving intelligent maintenance

equipment for track bolts is crucial.

https://orcid.org/0009-0006-7316-3806

https://orcid.org/0000-0001-6856-2577

https://orcid.org/0009-0004-6779-2521

https://orcid.org/0000-0002-2973-7976

https://orcid.org/0009-0008-1531-9547

This article addresses the collaborative task

assignment problem for Multi-Agent Track Bolt

Operation Robots (MATBOR), a typical NP-hard

challenge marked by high computational complexity

and long processing times (Li et al., 2022).

The Particle Swarm Optimization (PSO)

algorithm is recognized for its high efficiency,

simplicity, and quick convergence, making it ideal for

solving single-objective optimization problems.

However, it is not naturally suited for multi-objective

optimization. Consequently, improving PSO for

multi-objective tasks has become a significant

research focus. Many researchers have extended the

original algorithm and implemented various

improvements to enhance the performance of the

Multi-Objective Particle Swarm Optimization

(MOPSO) algorithm (Figueiredo et al., 2016; Lv et

al., 2016; Zhou et al., 2022; Sunet al., 2024; Wang and

248

Shen, Y., Meng, J., Lv, D., Meng, G., Wang and Z.

Research on Cooperative Task Assignment of Multi-Agent Track Bolt Operation Robot Based on Optimized Multi-Objective Particle Swarm Optimization.

DOI: 10.5220/0013632100004671

In Proceedings of the 7th International Conference on Environmental Science and Civil Engineering (ICESCE 2024), pages 248-256

ISBN: 978-989-758-764-1; ISSN: 3051-701X

Liu, 2016; Khan et al., 2016; Wang et al., 2021).

For instance, Liu et al. proposed a co-evolutionary

PSO algorithm that employs synthetic immune

principles, dividing the population into elite and

ordinary subpopulations, which co-evolve for

improved convergence and global search abilities

(Liu et al., 2013). Goh et al. introduced a

collaborative evolution paradigm that combines

competition and cooperation to simultaneously tackle

static and dynamic multi-objective problems (Goh

and Tan, 2009). Song et al. developed a collaborative

evolutionary PSO algorithm based on a bottleneck

objective instructional strategy, maintaining diversity

through distributed collaboration across multiple

populations (Song et al., 2020). Huang et al. proposed

a dual-phase multi-task allocation approach utilizing

Discrete Particle Swarm Optimization (TMA-DPSO),

which iteratively updates particle positions and

velocities to enhance solutions (Huang et al., 2022).

Lastly, Li et al. proposed a gBest strategy, utilizing a

newly defined virtual generation distance index, to

enhance search efficiency (Li et al., 2023).

Building on this foundation, This study introduces

a method for task assignment in MATBOR, utilizing

an optimized MOPSO algorithm. The key

innovations of this study include:

(1) A two-stage subsampling method is

implemented to improve the algorithm's convergence

speed and accuracy.

(2) To address issues such as high computational

complexity, limited diversity in Pareto optimal

solutions, and challenges in handling complex

constraints, this study employs a simple adaptive grid

method to optimize the multi-objective particle

swarm optimization algorithm, thereby enhancing its

efficiency.

(3) Applying a collaborative task assignment

method to MATBOR to enhance maintenance

efficiency during railway skylight periods.

2 METHODOLOGY

2.1 Collaborative Task Assignment

Model for Multi-Agent Rail Bolt

Operation Robot

2.1.1 Problem Description

The collaborative task assignment problem for a

multi-agent rail bolt operation robot system can be

described as follows: assuming there are m

MATBORs operating during a specific skylight

period, represented by the robot set

1234

{, , , , , }

Rrrrr r=  . if the range of the rail bolt

area to be serviced does not exceed

max

L and n bolts

need to be maintained, the task set T can be

represented as

1234

{, , , , , }

Ttttt t=  . Here,

represents the task assignment for the ith robot,

[1, ]im∈

;

represents the jth task to be assigned,

[1, ]

n∈

. The working time during the skylight

period must not exceed

max

T . As shown in Figure 1,

the central control center sends the specific areas and

kilometers that require maintenance to the signal

receiving station, which then relays these

requirements to the monitoring operation screen used

by the maintenance personnel and the MATBORs

during the window period. Once the MATBORs

begin working, they transmit the status of each

completed task and equipment information to the

signal receiving station in real-time. The maintenance

work during the skylight period is considered

successfully completed when all MATBORs have

finished their assigned tasks simultaneously.

Figure 1: Schematic Diagram of Task Assignment for

MATBOR.

2.1.2 Constraints

The MATBOR collaborative task assign-ment model

includes the following key constraints:

(1) Task Coordination Constraints

To ensure that no task is executed multiple times

or left unexecuted, task coordination constraints are

incorporated into the model. This requires that each

task must be executed exactly once and can only be

assigned to a sing-le robot, as expressed in equation

(1).



T∀∈

(1)

represents the assignment of tasks; n

represents total number of tasks.

Research on Cooperative Task Assignment of Multi-Agent Track Bolt Operation Robot Based on Optimized Multi-Objective Particle

Swarm Optimization

249

(2) Robot Coordination Constraints

To prevent errors in task execution, the model

includes constraints for robot coordination. These

constraints guarantee that each robot can undertake

only one task at a t-ime during the assignment process,

as illustrated in equation (2).



iR∀∈

(2)

m represents total number of multi-agent systems.

(3) Task Status

The variable representing whether the task

assigned to the robot

r is defined as follows: if the

task is assigned to the robot, then:

x =

; if the task

is not assigned to the robot, then:

x =

. As shown

in equation (3).

{0,1}

x ∈

iR∀∈

，

T∀∈

(3)

(4) Homework

All robots begin from the same starting point, and

the bolts are sequentially numbered 1, 2, 3... n starting

fro-m that origin.

(5) Operation Time and Distance Constraints

Given the time and distance limitations for

maintenance work during the skylight period, the

assigned MATBOR must not exceed the specified

time, and the total distance traveled by a single robot

must remain within the maximum allowable distance.

These constraints are express-ed in equations (4) and

(5).

imax

≤



(4)

imax

DL≤ (5)

Were:

()

ij ij

ijij

Ttx



−+⋅Δ

+⋅Δ





irtij

PPx

=−⋅



Where,

represents robot

completes task

assignment and running time;

represents robot

total travel distance;

max

represents robot

maximum running time;

max

represents robot

maximum driving distance;

and

represent

complete the starting and ending bolt numbers of the

task separately;

represents ground speed:

Vms=

;

represents operating speed:

0.45 /

Vms=

;

represents task j execution time;

represents initial position:

(, )

iii

rrr

xy=

;

represents task element coordinates:

(, )

jjj

ttt

xy=

2.1.3 Objective Function

To more effectively evaluate the task assignment

results for the rail bolt robot, this model uses two

objective functions: MATBOR task duration and

overall travel distance.

Task completion time refers to the duration

required to complete the final task in the maintenance

process, while total travel distance refers to the sum

of all distances traveled by the track bolt robots

during the skylight period. The corresponding

calculation formulas are given in equations (6) and

(7).

FmaxT



(6)



(7)

In equations (6) and (7),

indicates the total

time needed to finish the final task in the complete

maintenance process, which corresponds to the

maximum task completion time.

represents the

sum of the travel distances of all participating rail bolt

robots across all systems. Since the goal is for the rail

bolt robots to complete tasks as quickly as possible

while minimizing resource consumption during task

assignment, the model put forward in this paper

considers both optimization objectives: minimizing

and

simultaneously. Based on these two

objectives, the optimal objective function for

MATBOR collaborative task assignment is

formulated in equation (8).

[, ]

zminFF= (8)

The MATBOR collaborative task assignment

model involves both discrete and continuous

variables, which complicates the solution space and

makes it more difficult to search effectively.

Additionally, the model includes multiple complex

constraints, such as inequality and equality

constraints, further increasing the irregularity of the

solution space and the difficulty in finding feasible

solutions. To tackle the challenges of multi-objective

and multi-constraint collaborative task assignment in

MATBOR, this paper introduces MOPSO algorithm

that incorporates a quadratic sampling adaptive grid

to address the multi-MATBOR collaborative task

assignment challenge.

ICESCE 2024 - The International Conference on Environmental Science and Civil Engineering

250

2.2 Multi-Objective PSO Based on

Quadratic Sampling Adaptive Grid

2.2.1 Pareto Optimal Solution

Given the multiplicity and complexity of different

objective functions, it is usually impossible for all

objectives to reach their maximum or minimum

values simultaneously. As a result, multi-objective

optimization problems rarely have a single optimal

solution. However, practical problems require

decision-making to identify the best possible solution.

To address this, Pareto optimal solutions are utilized

to assess and balance conflicting objectives.

In multi-objective optimization, several

objectives are optimized simultaneously. A solution is

considered Pareto optimal if no objective can be

enhanced without negatively impacting another (Lu

et al., 2024). For two decision vectors

and

，if

dominates

, denoted as

y

, this means that

is no worse than

in all objectives, and

enhanced in at least one objective.

A decision vector

is considered a Pareto

optimal solution if no other vector in the objective

space can dominate it. The group of all these solutions

constitutes the Pareto optimal set (PS), while its

graphical depiction is referred to as the Pareto optimal

frontier (PF). According to the definition, enhancing

one objective in a Pareto optimal solution necessarily

diminishes at least one other. In multi-objective

optimization, this collection is commonly known as

the non-dominated solution set. The algorithm

proposed in this paper aims to identify the Pareto

optimal solution set, thereby enhancing the efficiency

and balance of MATBOR task assignment.

2.2.2 Secondary Sampling

The more particle samples selected in the state space,

the higher the approximation accuracy becomes (Liu,

2017). To address the issue where the weights of

certain particles may reduce the effective sample

space after multiple iterations, thereby affecting

estimation accuracy, the quadratic sampling method

has been introduced (Douc and Cappe, 2005). During

the resampling process, particles with higher weights

are duplicated, while those with lower weights are

discarded, ensuring the particle count remains

constant. Various subsampling methods exis (Li et al.,

2015).

2.2.3 Improved Multi-Objective Particle

Swarm Optimization Algorithm

The PSO algorithm, developed by Kennedy and

Eberhart in 1995, is a swarm intelligence technique

modeled after birds' food-searching behavior. It is

especially adept at addressing complex, nonlinear

continuous optimization challenges (Kennedy and

Eberhart, 1995). Over time, PSO has been enhanced

and adapted to tackle discrete problems as well. These

improvements have expanded its applicability,

enabling it to effectively address NP-hard problems,

combinatorial optimization, and multi-objective

optimization challenges, while also incorporating

global guidance techniques (Yan et al., 2015; Gao et

al., 2023; Lu et al., 2023). In PSO, particles adjust

their direction and velocity for the next iteration by

considering both their individual flight history and

shared information from the swarm, demonstrating

collective intelligence. The particle update equations

are provided in (9) and (10).

11 2 2

( 1) () ( () ()) ( () ())

iiii i

vt vt crpt xt cr gt xt

+= + − + −

(9)

(1) (1) (1)

iii

xt xt vt+= ++ + (10)

Equation (9) represents the velocity update

formula, while equation(10) is the position update

formula. In these equations:

(1)

vt+ is the new

velocity of particle i at time t+1;

()

vt is the velocity

of particle i at time t;

is the inertia weight, which

controls the influence of the particle's previous

velocity on its current velocity;

c and

c are

acceleration constants, representing the weights of

individual cognition and group cognition,

respectively;

r and

r are random numbers in the

range [0,1], used to maintain randomness;

()

t i s

the global best position, representing the optimal

position found by the entire particle swarm;

()

t is

the current position of particle i at time t and

(1)

xt+

is the new position of particle i at time t+1.

This paper introduces a two-stage subsampling

method to address the challenges of low solution

accuracy and slow search speed during the middle

and later phases of the PSO algorithm. In the first

stage, particles are sampled from the search space,

where those farther from the non-dominated solutions

are discarded, and those closer are retained and

replicated, enhancing convergence speed and

accuracy. However, this may reduce particle diversity.

To counter this, the second-stage sampling focuses on

non-dominated particles, discarding high-density

particles and replicating low-density ones to preserve

diversity. Since the grid is updated only when extreme

Research on Cooperative Task Assignment of Multi-Agent Track Bolt Operation Robot Based on Optimized Multi-Objective Particle

Swarm Optimization

251

boundary particles appear in the storage set, the mesh

size can sometimes grow too large during evolution,

affecting performance. In the second stage, the target

area is segmented into smaller regions via a grid, and

particle density in each region is used for estimation.

The grid size is adjusted adaptively based on particle

evolution. This leads to the development of MOPSO

algorithm that incorporates quadratic sampling and an

adaptive grid (QSAGMOPSO), as depicted in the

flowchart in Figure 2.

Figure 2: Flowchart of the Algorithm.

Step 1: Initialize the particle position data by

dividing each dimension evenly, using these divisions

as the initial coordinates in the decision space. To

expand the distribution range, each dimension is

divided according to its value range.

Step 2: Utilize the layered sampling method for

secondary sampling. In the first stage, particles with

higher weights are duplicated, while those with lower

weights are discarded, maintaining a constant number

of particles throughout the resampling process.

Step 3: Apply the secondary sampling method

using a layered approach. In the initial stage, particles

with greater weights are duplicated, while those with

lesser weights are removed. The overall number of

particles stays constant during the resampling process.

Step 4: Non-dominated solutions that meet the

criteria are stored in an external file. When the file

reaches its maximum capacity, new qualifying

particles are added by screening the existing ones,

ensuring the particle count remains constant.

Step 5: Determine the Pareto optimal solution set

by assessing the fitness value of each particle for

every objective. Analyze the dominance relationships

between the particles, and collect all non-dominated

solutions to form the current Pareto optimal set.

Step 6: Update each particle's velocity and

position by applying the corresponding update

formulas.

Step 7: Input the particle coordinates into each

objective function and compute the corresponding

function values.

Step 8: Utilize the adaptive grid method to

compute the density of each particle in the Pareto

optimal solution set. The search space is divided into

smaller regions, with higher particle density

indicating a greater number of particles within a grid.

Low-density particles are replicated to preserve

diversity.

Step 9: Using the historical data of each particle,

identify the personal best particle (pbest) and the

global best particle (gbest) based on their past fitness

levels.

3 RESULTS

3.1 Experimental Comparison

To assess the feasibility and effectiveness of the

QSAGMOPSO algorithm, benchmark problems

ZDT1, ZDT2, and ZDT3 were selected for testing,

with the details provided in Table 1. The population

size was set to 400, with 200 generations of iterations

and a file set size of 200. Each test was run 20 times,

as shown in Figure 3. In the figure, red dots represent

the true Pareto frontier, while green dots indicate the

Pareto frontier identified by the QSAGMOPSO

algorithm. A summary of the comparison results,

including the mean and standard deviation for the

ZDT functions, is provided in Table 1.

In the Table 1, variable range:

x≤≤

1, 2, 3, ,im= 

30m =

ICESCE 2024 - The International Conference on Environmental Science and Civil Engineering

252

Table 1: Benchmark Test Table.

Test

uestions

Objective Function

ZDT1



=−





−



ZDT2

()

fgx







=−









−



ZDT3

()

211

1/sin10

gfgf



=− −





−



As shown in Table 2, while various indicators can

be used to assess two Pareto frontiers, no single

measure is completely reliable. To address this, the

paper employs two additional quality indicators,

and

∈+

, to compare algorithm performance. These

indicators evaluate how closely the solutions align

with the true Pareto front. Both

and

∈+

, range

from 0 to 1, where higher

(or lower

∈+

) values

indicate a better alignment with the true Pareto front.

The results suggest that the proposed QSAGMOPSO

algorithm outperforms the MOPSO algorithm.

Table 2: Performance Comparison Results of MOPSO and

QSAGMOPSO on Benchmark Functions.

Problem MOPSO QSAGMOPSO

∈+

ZDT1 Mean 0.8907 0.1564 0.9093 0.1471

Std. 0.0971 0.2790 0.0400 0.1806

ZDT2 Mean 0.6683 0.6500 0.8694 0.4520

Std. 0.4927 0.2472 0.2668 0.2286

ZDT3 Mean 0.7420 0.3695 0.8122 0.1222

Std. 0.8101 0.4553 0.8534 0.1813

Figure 3: Pareto front of ZDT1, ZDT2, ZDT3.

3.2 Task Assignment Coding

Since the particle swarm optimization algorithm uses

real number encoding, it cannot be directly applied to

the discrete task assignment problem in MATBOR.

Therefore, this paper adopts a method of truncating

decimal places and retaining integer digits to decode

the results obtained by the algorithm, effectively

transforming the problem from a continuous domain

into a discrete one. Assuming there are m rail bolt

robots that need to perform n tasks during the skylight

period, the assignment plan is expressed as an n-bit

array

123

[, , , , ]

tt t t . Each

in the array is a

randomly generated number falling within the range

[1, ]m

,and the integer part of

represents the id of

the robot assigned to perform the jth task. Tasks with

Research on Cooperative Task Assignment of Multi-Agent Track Bolt Operation Robot Based on Optimized Multi-Objective Particle

Swarm Optimization

253

the same integer part are executed by the same rail

bolt robot, while the decimal part is rounded during

the encoding process.

For example, if there are three track bolt robots

tasked with performing 300 jobs during the skylight

period, the task assignment code arrangement is

shown in Table 3.

Table 3: Example of Task Assignment Scheme.

Particle numbe



66 67



230 231



300

Robot codin

3 2 1

In this task assignment scheme, bolts numbered 1

to 66 are executed by Robot 3, bolts numbered 67 to

230 are handled by Robot 2, and bolts numbered 231

to 300 are assigned to Robot 1.

3.3 Example Simulation

A simulation was conducted in Matlab to address the

task assignment problem for multi-agent rail bolt

operation robots. The resulting comparison of the

algorithm's performance is illustrated in Figure 4 and

Figure 5.

From Figure 4 and Figure 5, it can be observed

that both algorithms were run 20 times across

different task quantity scenarios. Two multi-objective

optimization algorithms were used to record all

Pareto frontiers obtained in each instance. To further

evaluate the Pareto solution sets, two metrics were

introduced: Average Ratio (AR) and Spacing Metric

(SP). The results of these evaluations are presented in

Tables 4 and 5.

Table 4: Comparison of AR Values for Pareto Solution Sets

of Two Algorithms.

Number of

tasks

Number of

robots

MOPSO

QSAGMOPS

300

3 0.15 0.63

5 0.37 0.77

10 0.69 0.83

400

3 0.35 0.71

5 0.57 0.85

10 0.79 0.97

Table 5: Comparison of SP Values for Pareto Solution Sets

of Two Algorithms.

Number of

tasks

Number of

robots

MOPSO

QSAGMOPS

300

3 313.41 236.94

5 302.43 259.13

10 222.92 209.24

400

3 378.59 357.91

5 347.45 323.25

10 328.21 305.59

According to Tables 4 and 5, Table 4 compares the

AR values of the two algorithms. In the instance tests,

the AR values for the proposed QSAGMOPSO

algorithm were consistently higher than those for the

MOPSO algorithm, indicating that the

QSAGMOPSO algorithm produces higher-quality

solutions. Table 5 presents the average SP values for

each scenario. Across all scenarios, the proposed

QSAGMOPSO algorithm consistently achieved the

lowest average SP values, reflecting better

distribution and uniformity of solutions along the

Pareto front.

Figure 4: Comparison chart showing the completion of the

same task(300 pieces) by different numbers of robots. a:3

robots; b:5 robots; c:10 robots.

ICESCE 2024 - The International Conference on Environmental Science and Civil Engineering

254

Figure 5: Comparison chart showing the completion of the

same task(400 pieces) by different numbers of robots. a:3

robots; b:5 robots; c:10 robots.

4 CONCLUSION

In this article, we address a previously unexplored

problem in multi-robot task assignment: the

collaborative task assignment of multi-agent rail bolt

operation robots. By introducing the concept of

collaborative control from multi-agent systems into

railway engineering, we redefine the fully automated

rail bolt robot as an intelligent agent capable of

independent decision-making. With a single

instruction from a staff member, the robot can

efficiently complete the heavy maintenance and

engineering tasks of track bolts during the skylight

period.

We further improved the classical multi-objective

particle swarm optimization (MOPSO) algorithm by

integrating quadratic sampling and a straightforward

adaptive grid partitioning method. These

enhancements tackle the problems of slow

convergence and getting stuck in local optima during

the later stages of the traditional PSO algorithm. The

simulation results demonstrate that the task

assignment method proposed in this paper greatly

enhances task efficiency.

In future research, we will refine the constraints

based on real-world engineering requirements and

further develop the task assignment model to meet

broader practical needs.

ACKNOWLEDGEMENTS

The authors acknowledge the support of Lanzhou

Jiaotong University for this study.

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