Multi-robot task allocation based on an automatic clustering strategy employing an enhanced dynamic distributed PSO

Shaymaa M. Jawad Alzubairi; Alexander Petunin; Amjad Jaleel Humaidi

doi:10.1556/1848.2025.00935

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International Review of Applied Sciences and Engineering

Volume/Issue:: Accepted Manuscript / Online First

Multi-robot task allocation based on an automatic clustering strategy employing an enhanced dynamic distributed PSO

Authors:

Shaymaa M. Jawad Alzubairi

Shaymaa M. Jawad Alzubairi Ural Federal University, Ekaterinburg, Russia

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Shaymaaalzubairi77@gmail.com

Alexander Petunin

Alexander Petunin Ural Federal University, Ekaterinburg, Russia

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a.a.petunin@urfu.ru

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Amjad Jaleel Humaidi

Amjad Jaleel Humaidi University of Technology, Baghdad, Iraq

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amjad.j.humaidi@uotechnology.edu.iq https://orcid.org/0000-0002-9071-1329

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Online Publication Date:: 27 Feb 2025

Publication Date:: 27 Feb 2025

Article Category:: Research Article

DOI:: https://doi.org/10.1556/1848.2025.00935

Keywords:: automatic clustering; multi-robot task allocation MRTA; MRS; mobile robot; particle swarm optimization

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Abstract

Systems based on mobile multirobots have gained considerable attention in the past two decades because of their efficacy and flexibility in various real-world applications. An essential component of these systems is multi-robot task allocation (MRTA), which concerns allocating tasks to mobile robots in an efficient manner. The effectiveness of MRTA is influenced by the size of the search space and computational time, and both increase substantially as the number of tasks and robots involved increases. This study introduces an effective solution to the MRTA problem by employing a two-stage approach. First, nearby tasks are automatically grouped into clusters by using an enhanced dynamic distributed particle swarm optimization algorithm. Second, mobile robots are assigned to the closest clusters. To demonstrate the effectiveness of this approach. Simulations are conducted to compare the proposed method with particle swarm optimization and differential evolution approaches. Numerical results confirm that the proposed approach exhibits highly competitive performance in terms of clustering cost, clustering time, and overall time (clustering and assigning time). This approach is advantageous for real-world applications involving numerous robots and targets.

Abstract

1 Introduction

For the last two decades, multi-robot systems (MRS) have been on the robotics community's agenda. The primary motivation for this interest is the considerable advantages and potential that MRS offers compared with single-robot systems. The main benefits of using a team of robots include improving system reliability, resolving task complexity, increasing performance, and simplifying the design process [1, 2]. MRS has been utilized in various fields for numerous applications, including surveillance [3], search and rescue [4, 5], agriculture [6–8], pick-up and delivery [9], and healthcare [10–12].

Assigning a group of robots to a set of tasks in a way that maximizes the system's overall performance while taking into account a number of constraints is one of the most difficult MRS challenges. The multi-robot task allocating problem is the name given to this issue [13, 14]. The MRTA problem is defined as: given a collection of tasks and robots capable of executing them, the objective is to assign the tasks to suitable robots in order to optimize a defined criterion, such as minimizing the total completion time. In recent years, extensive research has been conducted to address this problem. The main solutions can be grouped into two main approaches, namely, market-based and optimization approaches, depending on whether the MRS architecture is centralized or decentralized [15, 16], as shown in Fig. 1. Market-based approaches [17–19], which are typically used with decentralized architectures rely on direct communication between robots to negotiate. Optimization-based (swarm) approaches are usually applied in centralized systems, where cooperative behavior emerges from simple interactions between robots by communicating with the control unit [20].

Fig. 1.

Central and decentralized control architectures

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00935

The aforementioned approaches can be further classified as learning-based [21, 22], behavior-based [23, 24], and clustering [25, 26]. This study focuses on clustering strategies that involve grouping nearby tasks into clusters and allocating robots to these clusters. Many clustering algorithms are widely used, but most of them require a certain number of tasks and clusters, a requirement that is often impossible to meet in real-world scenarios. This limitation has led to research on algorithms that do not require a predefined number of tasks and clusters. Such algorithms, known as automatic clustering approaches, determine clusters' numbers on the basis of cluster validity measures. Identifying the ideal number of clusters and tasks per cluster remains a crucial challenge in dynamic clustering-based task allocation [27].

Recent advancements in heuristics and metaheuristics have contributed to the development of automatic clustering methods [28, 29], An ideal cluster number can be automatically determined using a modified multi-objective genetic algorithm (GA)-based clustering framework even in the absence of sample label information. For instance, a multi-objective differential evolution approach to dynamic clustering without previous knowledge of the cluster number was improved in Reference [30] so that only a few features from all of the available features in the data are used. The cuckoo search algorithm was used in Reference [31] to determine the optimal grouping of N objects into K clusters. Reference [32] introduced a unique clustering methodology based on ant colony optimization that uses two objective functions to assess the quality of different clustering solutions, without requiring a priori information.

In Reference [25], classical and automatic clustering problems were solved using the swarm-based Emperor Penguin Colony algorithm, which does not require previous knowledge to classify data. In Reference [33], the optimal cluster number and the centroid value were determined using an autonomous clustering algorithm called AC-Mean ABC, which is based on the mean artificial bee colony.

Amongst clustering methods, particle swarm optimization (PSO) shows considerable promise for continuous problems, such as clustering [34]. A novel particle representation method for choosing the ideal number of clusters from a variety of options is provided by the multi-elitist PSO model presented in Reference [35]. Additionally, a dynamic distributed PSO algorithm has been created by researchers especially for robotic swarms' autonomous task clustering [36]. For the automatic clustering problem, a dynamic distributed double-guided PSO was modified in Reference [26] to reduce the search space in two stages. The crazy firefly algorithm and the variable-step-size firefly algorithm were hybridized separately with a normal PSO algorithm in Reference [37] in order to solve the clustering problem's velocity problems.

Despite the advancement of various clustering algorithms, many of them still face challenges in achieving the desired levels of automation, quality, and efficiency. Automatically determining the optimal number of clusters and the number of tasks per cluster in large datasets remains a challenge.

This study proposed a novel approach to the automatic clustering problem. The method is called enhanced dynamic distributed particle swarm optimization (ED²PSO), which can automatically and efficiently handle the problem of an unknown number of clusters and tasks per cluster and optimally cluster a huge as well as middle and small number of tasks in competitive time.

The remaining sections of the paper are arranged as follows: Section 2 outlines the MRTA problem. Section 3 provides background information. Section 4 presents the proposed approach. Section 5 shows the experiments and results, and Section 6 contains the conclusions and future work directions.

2 MRTA problem

As the research on MRS progresses, the question “which robot can execute which task” has become increasingly prominent. This question has directed substantial focus towards the task allocation problem in MRS. MRTA aims to optimal allocating tasks to robots to achieve the best overall team performance [16].

2.1 Problem description

The MRTA problem in this study is that many independent targets dispersed randomly throughout an area full of obstacles must be instantly inspected by identical mobile robots. The following assumptions were made.

MRS operates under centralized control.
All robots are homogeneous.
All tasks are homogeneous (static intendent targets).
Each target (SR) requires only one robot for inspection.
A robot can inspect only one target at a time, but it can handle multiple targets sequentially (MT).
All targets must be inspected.
The completion time of all targets needs to be as short as possible.

MRS in this work is centralized control that distributes identical tasks (independent targets) amongst identical robots (TA). The mission is completed when all tasks are successfully distributed to robots, as shown in Fig. 2.

Fig. 2.

MRS centralized control with independent targets

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00935

Given the homogeneity of robots and tasks, the clustering strategy is the best solution. This strategy reduces the robot team's average overall completion mission time [38, 39] through a two-stage methodology. Firstly, nearby tasks (targets) are grouped into clusters, as indicated in Fig. 3. Secondly, each robot is allocated to the nearest cluster.

Fig. 3.

Clustering nearby independent targets and assigning robots to clusters

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00935

In the aforementioned methodology, the MRTA problem is transformed into an automatic clustering problem, where the goal is to group nearby targets into clusters, and then to an assignment problem, where each robot needs to be assigned to the closest group.

2.2 Formula of the MRTA problem

In this paper, the following is a description of the MRTA problem model. Let $T = {T_{1}, T_{2}, \dots \dots ., T_{N}}$ represent the set of identifiers' targets (Cartesian floating points) that will be allocated to the swarm's robots. Let $R = {R_{1}, R_{2}, \dots \dots ., R_{M}}$ represent the set of identifiers' robots (Cartesian floating points) in the robots' swarm. The MRTA problem comprises the following subproblems:

Part I (clustering problem): grouping all the nearby targets into clusters.
Part II (assignment problem): allocating robots R to the clusters to ensure cost-effective utilization of all robots.

Part I: The clustering problem can be defined as partitioning the set of targets, T, into CL clusters based on distance.

C L = {{C L}_{1}, {C L}_{2}, \dots \dots, {C L}_{K}}

has the following properties.

{C L}_{K} = R_{M}

(1)

\cup_{i = 1}^{K} {C L}_{i} = T

(2)

{C L}_{i} \cap {C L}_{j} = \emptyset, i \neq j, i = 1, 2, . . K, j = 1, 2, . . K

(3)

{C L}_{i} \neq \emptyset, i = 1, 2, \dots . . K

(4)

The following is a formulation of the automatic clustering problem:

\min_{C L \in θ} ƒ (C L)

(5)

subject to Eq. (1– 4), where

θ

is the set of feasible solutions containing all possible clustering of T targets into CL clusters. ƒ is the fitness function. Within-cluster sum of squares (WCSS) is utilized to measure the tightness of the clusters Eq (12).

Part II: The allocation problem can be defined as allocating R robot to the nearest valid clusters CL. Swarm allocation is presented as follows:

A = {A_{1}, A_{2}, \dots \dots, A_{R_{M}}}

, where the allocation has the following expression:

a_{r} = \min \sum_{i = 1}^{R_{M}} \sum_{j = 1}^{{C L}_{K}} c_{i j} x_{i j} .

(6)

Let

x_{i j}

be a binary variable, such that

x_{i j} = 1

if a robot is allocated to the closest cluster; otherwise,

x_{i j} = 0

c_{i j}

is the cost of assigning robot

i

to cluster

j

and calculated by the Euclidian equation of distance between the cluster centroid and robot location Eq (13). It is subject to

\sum_{j \in C L} x_{i j} \leq 1, \forall i \in R

(7)

\sum_{i \in R} x_{i j} \leq 1, \forall j \in C L

(8)

Once every robot has been allocated to the proper clusters, the task allocation process is complete [36, 38].

3 Background

An overview of PSO algorithms is given in this section. In 1995, Berhart and Kennedy developed PSO, a stochastic optimization technique [40, 41]. It's a population-based optimization algorithm where the population is known as a swarm. The swarm is made up of several individuals, or particles, each representing a potential solution. The principle of the algorithm involves moving these particles to find the optimal solution. $PSO$ algorithm is initialized with a set of random solutions (particles). Each particle $i$ contains:

Its current position $x_{i}$ .
Its current velocity $v_{i}$ .
Its best position (Pbesti), which is associated with the best fitness value the particle has achieved thus far.
The global best position (Gbest) associated with the best fitness value found amongst all particles.

All particles modify their direction in the space in each iteration in order to move toward their Pbest_i and Gbest locations. After determining the Pbest_i and the Gbest values, the particle updates their velocity and position in accordance with the equations

v_{i + 1} = w * v_{i} + c_{1} * r a n d * ({P b e s t}_{i} - x_{i}) + c_{2} * r a n d * (G b e s t - x_{i})

(9)

x_{i + 1} = x_{i} + v_{i + 1}

(10)

where

w

is the coefficient of inertia that steadily lowers the velocity over time,

r a n d

is a random number between 0 and 1 and

c_{1}

and

c_{2}

are the coefficients of acceleration. The stopping condition is typically the iterations' maximum number for the algorithm to execute. The stopping condition depends on the particular problem that is being optimized, similar to other parameters.

4 The proposed approach

The proposed approach is described in this section, addressing the automatic clustering problem by using ED²PSO [36, 42] and the allocation of robots to these clusters.

4.1 Concept of dynamic distributed PSO

While swarm intelligence methods are appealing for MRTA problems, they have two notable drawbacks: these approaches may occasionally become stuck in local optimum and can exhibit slow progress in terms of value of the fitness function, leading to slow convergence to the target areas.

To address these issues, Ayari and Bouamama [36, 42] introduced a modified PSO referred to as dynamic distributed PSO (D²PSO) by adding the following parameters: local optima detector (LOD) for personal best (LOD_Pbest) and LOD for global best (LOD_Gbest). These detectors monitor the successive iterations' number in which Pbest and Gbest do not exhibit any improvement. When particles fail to enhance their Pbest, they cease to contribute effectively to finding the global optimal solution, indicating that they have reached a saturation point and require an external thrust to regain effectiveness. This thrust is provided by D²PSO, which diversifies the search space by steering particles toward possibly better and unexplored regions. Similarly, if the global best (Gbest) does not improve for a certain iterations' number, then it might be trapped in local optimum, potentially misleading others by drawing them towards this suboptimal area. D²PSO addresses this issue by pushing trapped particles away from local optimum, thus enhancing the overall search process.

Although D²PSO offers a practical improvement over classical PSO by addressing some of the latter's drawbacks, it does not differentiate between particles that have failed to improve their Pbest, because they are stuck in a local optimum and must be redirected to explore the search space further and those that have truly found the best solution. Similarly, the method assumes that if the Gbest value remains unchanged, then the algorithm is trapped in a local optimum. This assumption overlooks the possibility that these values might actually represent the best solution, a case that would warrant stopping the algorithm early.

The proposed method, ED²PSO, solved the problem of D²PSO by adding tests to check the Pbest values to see whether they were trapped in the local optimum or reached the best solution. It also enhanced overall performance by reaching the best solution in a less iterations' number. The proposed method, ED²PSO, utilizes the two detectors of D²PSO and one of its equations [36] but employs them in a novel way, as we will see in the next paragraph.

4.2 Concept of ED²PSO

LOD is a parameter used throughout the optimization process and is updated locally. Each particle in the swarm has its own LOD_Pbest. If the Pbest value of a particle remains unchanged in an iteration, then its LOD_Pbest will increase by one. This operation continues for a certain number of iterations. Gbest has LOD_Gbest, which also increases when the Gbest value is unchanged.

Parameters S_p and S_g are given at the initialization stage. S_p is the threshold at which the maximum number of iterations a particle's Pbest can remain unchanged, and S_g is the threshold at which the maximum number of iterations Gbest can remain unchanged. When the Pbest of a particle does not improve in S_p iterations (i.e. LOD_Pbest = S_p), the algorithm checks if its value matches Gbest. If it does, this improvement indicates that it is the best solution amongst all particles and may be the required solution, and the iteration continues. Otherwise, the particle is considered saturated and requires external thrust and restructuring to Pbest_temp_i as follows:

{P b e s t_t e m p}_{i} = {P b e s t}_{i_{1}} + \frac{i_{1}}{i_{1} + i_{2}} * ({G b e s t_{h i s t}}_{i_{2}} - {P b e s t}_{i}),

(11)

where

i_{1} = r a n d o m (1, M)

M = (p o p u l a t i o n s i z e)

i_{2} = r a n d o m (1, s i z e (G b e s t_h i s t))

and

G b e s t_h i s t

represents the historical values of Gbest. If the Gbest value does not change for a predefined number of successive iterations, then the current value might be the best solution, and the algorithm should stop without completing all iterations. The proposed method prevents the Pbest of all particles from becoming trapped in local optima, thereby ensuring that Gbest also avoids falling into local optima.

4.3 Fitness function

A)Part I (Clustering fitness function):

The primary goal of data clustering is to minimize the similarity amongst instances within a single cluster while maximizing the dissimilarity between different clusters. In this work, the goal is to group nearby targets into clusters and maximize the distance between the clusters. The proposed algorithm utilized within-cluster sum of squares (WCSS) [43]. WCSS calculates the total squared distance between each cluster centroid and all of the cluster's objects. Well-separated and compact clusters are indicated by lower WCSS values. The WCSS for a given cluster can be computed using the formula.

W C S S = \sum_{i = 1}^{K} \sum_{j = 1}^{N_{i}} {(t_{i j} - c_{i})}^{2},

(12)

where K is the clusters' number,

N_{i}

is the targets' number in cluster

i

t_{i j}

is the j-th target in cluster

i

and

c_{i}

is the centroid of cluster

i

B)Part II (Assignment fitness function)

To assign each robot to the nearest cluster. The proposed approach utilized the Euclidian equation to calculate the distance between cluster centroid and robot location.

c_{i j} = \sqrt{{({x_{C L}}_{j} - {x_{R}}_{i})}^{2}} + {({y_{C L}}_{j} - {y_{R}}_{i})}^{2} .

(13)

Where

{x_{C L}}_{j}

and

{x_{R}}_{i}

are the X coordinates of cluster

{C L}_{j}

and robot

R_{i}

, respectively. Similarly,

{y_{C L}}_{j}

and

{y_{R}}_{i}

are the Y coordinates of cluster

{C L}_{j}

and robot

R_{i}

, respectively.

4.4 Flowchart of ED2PSO: automatic clustering

In the beginning, the targets' number (T), their locations, the robots' number (R), and the PSO parameters must be specified. The LOD parameters (S_p and S_g) should be initialized, and the initial particles must be generated. In this approach, each particle represents a different decomposition for distributing targets into clusters. The cost value of each particle is determined using the selected fitness function (WCSS), as shown in (Eq. (12)) in order to assess the clustering.

In each iteration, Pbest for all particles is calculated. If any particle's Pbest value remains unchanged, then its LOD_Pbest increases by 1. Then, LOD_Pbest should be checked to see if it has reached the predefined threshold (S_p). If so, the Pbest of that specific particle should be compared with the Gbest value. If the values match, then the solution of the particle might be the best solution, and the algorithm continues till its maximum number. If they do not match, then the particle is falling in a local optimum, and its Pbest is restructured to Pbest_temp_i (Eq. (11)).

Gbest must be calculated before updating the velocities and positions of particles and comparing its value with the historical values from previous iterations. If the value does not change, then its LOD_Gbest increases by 1. Afterwards, LOD_Gbest is checked. If it has reached the predefined threshold (S_g), then the algorithm might have found the solution, so the simulation can be stopped without completing all the iterations. Otherwise, the algorithm continues, updating the particles' positions and velocities. The approach repeats till the maximum number of iterations. Then, it returns the clusters' number (K), the positions of their centroids, and the targets' number in each cluster.

This approach transforms the MRTA problem into an MTSP problem, which is then divided into several TSP problems. The complete approach is illustrated in the flowchart in Fig. 4, which shows in detail the application of ED²PSO to address the problem.

Fig. 4.

Flowchart of automatic task clustering using ${ED}^{2}$ PSO

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00935

4.5 Assigning robots to clusters

Following the distribution of targets into clusters, robots need to be optimally assigned to the clusters based on the shortest distance. Each cluster is defined by its number of targets and centroid; the cluster's centroid is the location where the total distances between all targets within the cluster are the smallest. Robot-to-cluster distance is determined by the robot-to-nearest cluster's centroid Eq. (13). The steps for this assignment are shown in Table 1.

Table 1.

Steps of assigning robots to the closest clusters

1:	Initialize an empty list to store assignments.
2:	For each robot, calculate the distance to each cluster centroid.
3:	Sort the robots and centroids based on their distances in ascending order.
4:	Whilst there are still unassigned robots or targets
	Select the robot with the shortest distance.
	Select the target with the shortest distance.
	Add the assignment of this robot to this target to list of assignments.
	Remove the robot and targets from their respective lists.
5:	Return the list of assignments.

5 Experiments and results

In this section, a comparative analysis of PSO, differential evolution (DE), and the proposed approach for automatically allocating independent targets to multiple robots is presented. The comparative analysis is supported by numerical results. The simulations were conducted in four cases, each with varying numbers of targets and robots, as outlined in Table 2.

Table 2.

Number of targets and robots for all cases

	Case I	Case II	Case III	Case IV
Targets Number	10	100	500	1,000
Robots Number	3	3	4	5

The simulation experiments were conducted on a laptop with a Core (TM) i7-11800H CPU and 16 GB RAM by using MATLAB software. The probabilistic roadmap algorithm was employed to distribute targets randomly across the workspace for each simulation run. The locations of the robots varied in each run.

The efficiency of the ED²PSO was evaluated by comparing clustering time, the overall total time (clustering and assignment), and the Davies-Bouldin (DB) index to demonstrate clustering performance. The DB index [44] is determined the average of within-cluster objects distances to between-clusters distances, as follows:

D B = \frac{1}{K} \sum_{k = 1}^{K} \max_{k \neq k^{'}} \frac{S_{n} (c_{k}) + S_{n} (c_{k^{'}})}{S (c_{k}, c_{k^{'}})}

(14)

where K is the quantity of clusters,

S_{n}

is the average of all targets distance in cluster

C_{k}

to their centroid cluster

c_{k}

and

S (c_{k}, c_{k^{'}})

is the distance between cluster centroids

c_{k}

and

c_{k^{'}}

. A small ratio indicates that clusters are well-separated and compact, leading to a low DB-index, which signifies high clustering quality.

The following parameters were used: damping ratio $w_{d a m p}$ = 1; personal learning coefficient $c h 1 = c h i \times p h i 1$ ; global learning coefficient $c h 2 = c h i \times p h i 2$ , where $p h i 1 = 2.05$ , $p h i 2 = 2.05$ , $p h i = p h i 1 + p h i 2$ and $c h i = 2 / (p h i - 2 + \sqrt{{p h i}^{2} - 4 * p h i})$ ; inertia weight $w = c h i$ ; $S_{p}$ = 10; $S_{g}$ = 50; and number of iterations = 200 times.

Table 3 presents the numerical results of target clustering using ED²PSO, DE, and PSO for the four cases in Table 2. The results in Table 3 confirm that the proposed approach outperformed DE and GA in terms of clustering time and the overall time (clustering and assignment). Additionally, the performance of the ED²PSO algorithm measured by the DB index was comparable to that of DE and better than that of PSO and positively influenced the quality of the automatic clustering problem.

Table 3.

The automatic task clustering's numerical results using ${E D}^{2}$ PSO, PSO, and DE

	Case I			Case II			Case III			Case IV
	ED²PSO	PSO	DE	ED²PSO	PSO	DE	ED²PSO	PSO	DE	ED²PSO	PSO	DE
Average clustering time (s)	0.5220	4.1140	1.3238	0.4389	3.0514	0.7590	0.5953	4.2500	0.8446	0.7871	4.7853	1.5106
Average overall total time (s)	1.3886	4.1907	2.1886	1.1679	3.1013	1.3725	1.3453	4.3114	1.4488	1.5682	4.8564	2.4269
Average DB index values	0.6140	0.6830	0.6268	0.7900	0.8032	0.8097	0.7704	0.7674	0.7766	0.8375	0.8556	0.8618

The optimization algorithms produced different numerical results in each run, even with the same parameters and workspace. Therefore, each algorithm was run 20 times per case to obtain accurate results and verify the proposed approach. The average values were calculated. Table 3 presents average numerical results of automatic target clustering of the four cases in Table 2 by using ED2PSO, DE, and PSO. The results in Table 3 confirm that the proposed approach outperformed DE and GA in terms of clustering time and the overall time (clustering and assignment). Additionally, the performance of the ED²PSO algorithm measured by the DB index was comparable to that of DE and better than that of PSO.

As shown in Figs 5 –7 and the results in Table 3, ED²PSO demonstrated superior efficacy in automatic task or target clustering compared with the other algorithms. This superiority was evident in terms of clustering time, the overall total time and the DB index value across the different cases that ranged from a small number of targets and robots to a large number.

Fig. 5.

Graph showing the clustering time required by ED²PSO, PSO and DE algorithms

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00935

Fig. 6.

Graph depicting the overall total time for ED²PSO, PSO and DE algorithms

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00935

Fig. 7.

Graph depicting the DB index value for ED²PSO, PSO and DE algorithms

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00935

The images in Figs 8 –11 illustrate the execution of the proposed approach on a 2D real-like factory map. The map's dimension is 100 × 100 m. The same numbers of targets and robots as those in the four cases in Table 3 were used. Each case involved a varying number of targets and robots in diverse locations. In each case, the clusters were automatically created in accordance with the number of robots. Each cluster contained the closest targets to each other. The centroid points were calculated and created based on the targets' locations in each cluster; they are represented by small circles with different colors and numbers in Figs 8 –11. The robots are represented by large circles. As shown, each robot has the same number and color of the cluster centroid point allocated to it.

Fig. 8.

Image depicting the execution of the proposed algorithm for Case 1 (3 robots and 10 targets)

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00935

Fig. 9.

Image depicting the execution of the proposed algorithm for Case 2 (3 robots and 100 targets)

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00935

Fig. 10.

Image depicting the execution of the proposed algorithm for Case 3 (4 robots and 500 targets)

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00935

Fig. 11.

Image depicting the execution of the proposed algorithm for Case 4 (5 robots and 1,000 targets)

Citation: International Review of Applied Sciences and Engineering 2025; 10.1556/1848.2025.00935

At the conclusion of the ED²PSO algorithm's execution, clusters equivalent to the number of robots were created automatically and successfully, and a centroid for each cluster was specified. Then, each robot was assigned to the closest cluster centroid.

Other optimization techniques like Bee Algorithm, Cuckoo search algorithm, sparrow search algorithm, Whale Optimization Algorithm, Sea Lion Optimization Approach can be used to conduct a comparison study to the proposed PSO algorithm [45–51]. One may suggest to implement the proposed method in real-time environment using FPGA technology or other embedded system hardware design [52–60].

6 Conclusion

This paper addresses the MRTA problem, where a group of robots is required to be allocated to a set of predetermined targets optimally. A two-phase approach has been proposed, which efficiently can handle a small or large quantity of targets and robots. In Phase I, nearby targets are automatically and efficiently grouped into clusters by employing the ED2PSO algorithm. In Phase II, each robot is assigned to the nearest cluster optimally.

In this work, an enhancement has been done to the D2PSO algorithm. Although the algorithm effectively improves diversification through LODPbest and LODGbest operators, which enhance convergence by escaping local optimum, it does not distinguish between local optimum and the best solution. The ED2PSO addresses this issue by performing tests to distinguish the local optimum from the best solution. The numerical simulations conducted in this work revealed the efficacy of the proposed approach in addressing the MRTA problem in general and the efficacy of the ED2PSO algorithm in addressing the clustering problem in competitive time.

For future work, the proposed approach can be adapted to take into consideration obstacles and robot trajectories while clustering targets and assigning them to robots. It can be improved to be used in an unknown environment or in the presence of dynamic obstacles or targets.

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International Review of Applied Sciences and Engineering

Print ISSN:: 2062-0810

Online ISSN:: 2063-4269

Editorial Board

Mohammad Nazir AHMAD, Institute of Visual Informatics, Universiti Kebangsaan Malaysia, Malaysia

Murat BAKIROV, Center for Materials and Lifetime Management Ltd., Moscow, Russia

Nicolae BALC, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Umberto BERARDI, Toronto Metropolitan University, Toronto, Canada

Ildikó BODNÁR, University of Debrecen, Debrecen, Hungary

Sándor BODZÁS, University of Debrecen, Debrecen, Hungary

Fatih Mehmet BOTSALI, Selçuk University, Konya, Turkey

Samuel BRUNNER, Empa Swiss Federal Laboratories for Materials Science and Technology, Dübendorf, Switzerland

István BUDAI, University of Debrecen, Debrecen, Hungary

Constantin BUNGAU, University of Oradea, Oradea, Romania

Shanshan CAI, Huazhong University of Science and Technology, Wuhan, China

Michele De CARLI, University of Padua, Padua, Italy

Robert CERNY, Czech Technical University in Prague, Prague, Czech Republic

Erdem CUCE, Recep Tayyip Erdogan University, Rize, Turkey

György CSOMÓS, University of Debrecen, Debrecen, Hungary

Tamás CSOKNYAI, Budapest University of Technology and Economics, Budapest, Hungary

Anna FORMICA, IASI National Research Council, Rome, Italy

Alexandru GACSADI, University of Oradea, Oradea, Romania

Eugen Ioan GERGELY, University of Oradea, Oradea, Romania

Janez GRUM, University of Ljubljana, Ljubljana, Slovenia

Géza HUSI, University of Debrecen, Debrecen, Hungary

Ghaleb A. HUSSEINI, American University of Sharjah, Sharjah, United Arab Emirates

Nikolay IVANOV, Peter the Great St. Petersburg Polytechnic University, St. Petersburg, Russia

Antal JÁRAI, Eötvös Loránd University, Budapest, Hungary

Gudni JÓHANNESSON, The National Energy Authority of Iceland, Reykjavik, Iceland

László KAJTÁR, Budapest University of Technology and Economics, Budapest, Hungary

Ferenc KALMÁR, University of Debrecen, Debrecen, Hungary

Tünde KALMÁR, University of Debrecen, Debrecen, Hungary

Milos KALOUSEK, Brno University of Technology, Brno, Czech Republik

Jan KOCI, Czech Technical University in Prague, Prague, Czech Republic

Vaclav KOCI, Czech Technical University in Prague, Prague, Czech Republic

Imre KOCSIS, University of Debrecen, Debrecen, Hungary

Imre KOVÁCS, University of Debrecen, Debrecen, Hungary

Angela Daniela LA ROSA, Norwegian University of Science and Technology, Trondheim, Norway

Éva LOVRA, Univeqrsity of Debrecen, Debrecen, Hungary

Elena LUCCHI, Eurac Research, Institute for Renewable Energy, Bolzano, Italy

Tamás MANKOVITS, University of Debrecen, Debrecen, Hungary

Igor MEDVED, Slovak Technical University in Bratislava, Bratislava, Slovakia

Ligia MOGA, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Marco MOLINARI, Royal Institute of Technology, Stockholm, Sweden

Henrieta MORAVCIKOVA, Slovak Academy of Sciences, Bratislava, Slovakia

Phalguni MUKHOPHADYAYA, University of Victoria, Victoria, Canada

Balázs NAGY, Budapest University of Technology and Economics, Budapest, Hungary

Husam S. NAJM, Rutgers University, New Brunswick, USA

Jozsef NYERS, Subotica Tech College of Applied Sciences, Subotica, Serbia

Bjarne W. OLESEN, Technical University of Denmark, Lyngby, Denmark

Stefan ONIGA, North University of Baia Mare, Baia Mare, Romania

Joaquim Norberto PIRES, Universidade de Coimbra, Coimbra, Portugal

László POKORÁDI, Óbuda University, Budapest, Hungary

Roman RABENSEIFER, Slovak University of Technology in Bratislava, Bratislava, Slovak Republik

Mohammad H. A. SALAH, Hashemite University, Zarqua, Jordan

Dietrich SCHMIDT, Fraunhofer Institute for Wind Energy and Energy System Technology IWES, Kassel, Germany

Lorand SZABÓ, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Csaba SZÁSZ, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

Ioan SZÁVA, Transylvania University of Brasov, Brasov, Romania

Péter SZEMES, University of Debrecen, Debrecen, Hungary

Edit SZŰCS, University of Debrecen, Debrecen, Hungary

Radu TARCA, University of Oradea, Oradea, Romania

Zsolt TIBA, University of Debrecen, Debrecen, Hungary

László TÓTH, University of Debrecen, Debrecen, Hungary

László TÖRÖK, University of Debrecen, Debrecen, Hungary

Anton TRNIK, Constantine the Philosopher University in Nitra, Nitra, Slovakia

Ibrahim UZMAY, Erciyes University, Kayseri, Turkey

Andrea VALLATI, Sapienza University, Rome, Italy

Tibor VESSELÉNYI, University of Oradea, Oradea, Romania

Nalinaksh S. VYAS, Indian Institute of Technology, Kanpur, India

Deborah WHITE, The University of Adelaide, Adelaide, Australia

International Review of Applied Sciences and Engineering
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2023
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International Review of Applied Sciences and Engineering
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International Review of Applied Sciences and Engineering
Language	English
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