Fuzzy-Based Clustering of Wireless Sensor Networks for Multiple Mobile Agent Itinerary Planning

International Journal of Computer Networks & Communications (IJCNC) vol 16, No 6, November 2024
DOI: 10.5121/ijcnc.2024.16604 57
FUZZY-BASED CLUSTERING OF WIRELESS SENSOR
NETWORKS FOR MULTIPLE MOBILE AGENT
ITINERARY PLANNING
Nidhi Kashyap1
, Shuchita Upadhyaya1
,Monika Poriye*1
, Sachin Lalar2
1
Department of Computer Science and Applications, Kurukshetra University,
Kurukshetra, India
4
Department of Engineering and Technology, Gurugram University Gurugram
ABSTRACT
Mobile agent (MA) technology exhibits remarkable efficiency when integrated into Wireless Sensor
Networks (WSNs) for information processing tasks. MAs reduce network overhead by executing processing
code locally on nodes and selectively transmitting significant data to designated remote sensor nodes,
thereby enhancing data fusion and acquisition while minimizing energy depletion. However, in large-scale
networks, relying on a single MA leads to significant delays, necessitating the use of multiple MAs to
operate asynchronously and minimize latency. The challenge lies in effectively grouping nodes to ensure
MAs reach their intended destinations.
To address this challenge, this paper introduces a novel approach, the Adaptive FCM Clustering
Algorithm (AFCM), a fuzzy-based clustering algorithm designed for addressing network partitioning
challenges in Multiple Mobile Agent Itinerary Planning (MIP). A systematic analysis of the existing
literature examines various MIP algorithms, emphasizing their strengths and uncovering potential
research gaps. AFCM is specifically developed to create disjoint and load-balanced partitions tailored for
multi-mobile agent itinerary planning. A Methodical analysis with three traditional clustering algorithms
is conducted. The correctness of the Adaptive Fuzzy C-Means (AFCM) algorithm is demonstrated through
a detailed manual application on a wireless network comprising 15 nodes.
KEYWORDS
Clustering, Itinerary planning, Mobile agent, Routing, Wireless sensor networks.
1. INTRODUCTION
Mobile agents are software agents capable of autonomously migrating with their processing code
and data state to perform specified data processing tasks for remote users [1], [2]. They can
resume execution even after disconnection and process data at designated nodes. This flexibility
allows them to efficiently utilize network bandwidth, conserve energy, and minimize latency [3],
[4], [5], [6]. However, the deployment of mobile agents is only necessary when dealing with
substantial amounts of data transmission. In traditional Wireless Sensor Networks (WSNs),
deploying numerous sensor nodes in close proximity often leads to redundant sensed data.
Transmitting this redundant data individually consumes significant energy and bandwidth [7], [8],
[9], [10], [11]. In contrast, mobile agents migrate to each node, process and accumulate reduced
data in their payload, and perform aggregation with previously accumulated and newly retrieved
data [12], [13], [14]. By delivering only processed and aggregated information to the intended
node, mobile agents enable accurate decision-making based on significant information.

58
Despite its numerous benefits, mobile agent (MA) technology also presents challenges, with
mobile agent routing being a prominent issue [15], [16], [17], [18]. Routing mobile agents involves
determining their optimal itinerary, which includes the sequence of migration and the group of
nodes to be visited. The itinerary must be carefully planned to ensure that the collaborative
system performs better than the traditional system. The problem is divided into three steps to
address this:
 dividing the network into appropriate and disjoint clusters
 creating a group of source nodes to be visited within a single itinerary
 deciding the visiting order for completing their tasks.
This research paper addresses the initial step of the stated problem, which involves partitioning
the network into disjoint domains.To address this challenge, the study introduces an algorithm
that partitions a Wireless Sensor network into distinct, non-overlapping domains, thereby
improving the efficiency of the mobile agent system.
The proposed algorithm introduces a novel approach for autonomously determining the
cardinality for network segmentation. It dynamically selects the optimal number of domains
(clusters) and effectively partitions the network into non-overlapping, disjoint segments.
Additionally, the algorithm adeptly resolves the challenge of assigning nodes equidistant from
two centroids to the correct domain, ensuring precise network segmentation.
2. LITERATURE REVIEW
This research investigates the use of the mobile agent paradigm for Wireless Sensor Networks
(WSNs) in communicating data. The mobile agent paradigm offers advantages such as efficient
resource utilization, reduced network bandwidth usage, improved scalability, and offline stability.
However, the effective operation of mobile agents requires careful planning of their migration
path to avoid energy consumption and delays. Path determination can be achieved through static
or dynamic itinerary planning. Static planning is suitable for known node sequences in physical
data monitoring, while dynamic planning is used for target tracking with mobile and evolving
targets. Table 1 presents a summary of the literature reviewed in this study, outlining the main
findings and methodologies explored.

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Table 1: Clustering Techniques in Wireless Sensor Networks: A Review of Relevant Studies
Algorithms with
sources
Strategy used Research gap and strength
of the proposed algorithm
Description of the
algorithm
GIGM-MIP [19],
Aloui et al. [20],
SMIP [21],
Daramola et al. [22]
k-means and
x-means
The challenge is to identify
k value
Consider the proximity of
source nodes to associate
respective CH(Cluster
Head).
MST-MIP [3], TBID
[12], MINDS [23],
NOID [24], ILS [25],
Disjoint MIP [26],
CBID [27], SNOID
[28]
Tree-
structured
Consume more energy and
takea long time to traverse
as MA has to migrate each
node twice by following
reverse mapping
Single MA is dedicated to
each stemmed branch
CSA-MIP [4],Kuila et
al. [29], Wu et al.
[30], Rajagopalan et
al. [31],GA-MIP [32]
GA based Not effective for time-
critical applications
Select the sequence of
source nodes for the gene
array randomly.
CL-MIP [33], EMIP
[34], OM-MIP [35],
MAEF [36]
VCL based Centers are chosen on
density base
Partitions are in
circularsectorzones
SGMIP [35].
,
DSGMIP [37],
Bendjima et al. [38]
Directionality
based
Difficult to identify angle θ Distribute sensor nodes in
concentric zones,
originating with VCL
lines
AG-MIP [39],
SLMADA [40]
Angle based Outperforms when almost
nodes are in the same
direction. Still an issue to
determine angle θ
Partitions the network into
concentric sector zones,
using two beelines with
angle θ
BM-FPA [41],
MFGSA [42]
Evolutionary
technique
Good enough, but due to the
collaboration of a number
of techniques, it becomes
very complex
The results obtained from
fuzzy-based membership
are carefully incorporated
into a PSO-based
clustering technique,
which is iteratively
executed to determine the
global optimal results.
RA-MDP [43] k-mediods Difficult to identify k value CHs are chosen using
angle gap-based strategy
The review of literature, detailed in Table 1, highlights different algorithms employed for
network partitioning in multiple Mobile Agent Itinerary Planning (MIP), each with its unique
advantages and limitations. Building on these observations, this study presents a new approach
with the Adaptive FCM Clustering Algorithm, designed to generate disjoint and balanced
partitions tailored for multi-mobile agent itinerary planning.
This research paper addresses the challenges of clustering in context to Mobile Agent routing in
Wireless Sensor Networks, specifically focusing on issues related to overlapping and load
balancing. The paper introduces the proposed AFCM algorithm and compares it with three
traditional clustering algorithms. To validate the algorithm, a manual demonstration is provided
using a 15-node wireless network. Additionally, the algorithm was implemented in Python to
further verify its accuracy, and the results supported its correctness. The conclusions and findings
of these investigations are discussed in the concluding section of the paper.

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3. ENHANCING MULTI-MOBILE AGENT ITINERARY PLANNING:
INTRODUCING AN ADAPTIVE FUZZY C-MEANS ALGORITHM FOR
NETWORK CLUSTERING
After conducting a review of existing literature on multi-mobile agent itinerary planning (MIP), it
was observed that network clustering plays a crucial role in achieving efficient MIP. Various
algorithms, including k-means, x-means, tree-based, genetic algorithm-based, center location-
based, and directional-based approaches, have been proposed to partition the network and
optimize MIP.
The main drawback of the k-means clustering algorithm is its requirement to specify the number
of clusters beforehand. Variants such as x-means and Fuzzy c-Means (FCM) are built upon the k-
means framework. In x-means clustering, an initial minimum number of centroidsis assumed, and
clusters are subsequently adjusted to achieve optimal configuration. FCM algorithms, on the
other hand, start with a fixed number of clusters and iteratively update centroids while assigning
data points to clusters based on membership values to optimize results. Although x-means, k-
means, and FCM share some characteristics, FCM is distinguished as a prominent soft clustering
method, allowing data points to belong to multiple clusters simultaneously according to their
membership degrees.
However, Fuzzy c-Means (FCM) encounters challenges when a sensor node has equal
membership values for multiple destination itineraries, potentially resulting in imbalanced
domains within multi-mobile agent itinerary planning (MIP) systems. To address this issue, this
paper proposes a revised version of the FCM algorithm, termed Adaptive Fuzzy C-Means
(AFCM), designed to improve network clustering in MIP applications.
The AFCM algorithm addresses the issue of imbalanced domains by associating each sensor node
with the Domain Initial (DI) that has the lower expected load. This method helps to balance the
domains, thereby enhancing the efficiency of multi-mobile agent itinerary planning (MIP)
systems. The research will assess the AFCM algorithm’s capability to manage cases where sensor
nodes have equal membership values for multiple DIs and evaluate its effectiveness in creating
balanced domains.
By overcoming the limitations of existing clustering algorithms, particularly Fuzzy C-Means
(FCM), the proposed study aims to improve the performance and efficacy of MIP systems. The
evaluation of the AFCM algorithm's performance in network clustering for MIP will contribute to
the advancement of more effective itinerary planning techniques in multi-agent systems.
3.1. Distinguishing Clustering Algorithms: Exploring AFCM's Motivation and
Uncovering Algorithmic Differentiations
Table 2 offers a detailed comparison of the k-means, x-means, and fuzzy c-means clustering
algorithms. This analysis aims to clarify the reasons for introducing the Advanced Fuzzy C-
Means (AFCM) algorithm. By highlighting the unique characteristics and limitations of each
algorithm, the table helps to identify the specific strengths and weaknesses that drive the
development of AFCM as a novel approach in clustering techniques.

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Table 2: Comparison between k-means, x-means, and fuzzy c-means clustering algorithms
Aspect k-means x-means fuzzy c-means
Determining the
Number of
Clusters
Requires specifying the
number of clusters (k) in
advance.
Begins with initial
centroids and adjusts the
number of clusters
dynamically.
Requires specifying
the number of
clusters ahead of
time.
Centroid
Computation
Computes centroids as
the mean of data points
in each cluster.
Identifies optimal sub-
clusters by decomposing
clusters, leading to new
centroids.
Iteratively updates
centroids based on
the weighted values
of data points.
Membership
Assignment
Assigns data points to
the nearest centroid,
resulting in hard
assignments.
Uses a likelihood-based
criterion for probabilistic
assignment to sub-clusters.
Assigns membership
values indicating the
degree of belonging
to each cluster.
Handling
Overlapping and
Noise
Ineffective at handling
overlapping or noisy
data.
Manages overlapping
clusters to some extent by
decomposing them into
sub-clusters.
Effectively manages
overlapping and
noisy data.
Flexibility and
Adaptability
Fixed number of clusters;
lacks adaptability.
Flexible, as it dynamically
determines the number of
clusters during clustering.
Flexible in both the
number of clusters
and the degree of
membership.
Performance
Performs well with well-
separated and spherical
clusters.
Offers enhanced
performance by
automatically determining
the optimal number of
clusters.
Robust performance
on datasets with
overlapping or non-
spherical clusters.
Ease of Mobile
Agent Itinerary
Planning
Less suitable due to fixed
cluster number and
inability to handle
overlapping.
More suitable as it adjusts
cluster numbers
dynamically, aiding
itinerary planning.
Highly suitable due
to its flexibility in
handling overlapping
clusters and varying
degrees of
membership.
This table offers a succinct overview of the primary distinctions among the algorithms. It is
important to note that their performance can vary based on the particular dataset and the nature of
the clustering task.
3.1.1. Description of the Adaptive Fuzzy C-Means (AFCM) Clustering Algorithm for
Sensor Networks
This section provides a detailed, point-by-point description of the Adaptive Fuzzy C-Means
(AFCM) Clustering Algorithm, tailored for use in Sensor Networks.
1. AFCM approach: The Adaptive Fuzzy C-Means (AFCM) Clustering Algorithm for
Sensor Networks operates without the need to specify the number of clusters (k) in
advance.

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2. Selection of processing node (PE): AFCM selects a centralized processing node (PE)
within the network.
3. Determining k: The number of Sensor Nodes (SNs) in the vicinity of the PE becomes the
value of k. These SNs are referred to as domain initials (DI).
4. Assignment of Remaining Sensor Nodes (SNs): The remaining sensor nodes are
allocated to the domain initials (DIi) within the set DI. For each remaining sensor node,
the membership value (µi, j) is computed relative to each domain initial (DIi) in its
vicinity.
5. Sum of membership values: The sum of membership values (∑ µi, j) of each remaining
SN with respect to each DIi is calculated in advance.
6. Calculation of Sum of membership values: The total sum of membership values (∑ µi, j)
for each remaining sensor node (SN) with respect to each domain initial (DIi) is
computed beforehand.
7. Assignment to Domain Initials (DIi): Each sensor node is allocated to the domain initial
(DIi) that has the highest membership value. After each assignment, the sum of
membership values (∑ µi, j) is recalculated.
8. Finalizing Assignments: The assignment process continues until all sensor nodes in the
network are allocated to a designated domain initial (DIi). For nodes that have identical
membership values across multiple DIs, they are assigned to the DIi with the lowest total
sum of membership values (∑ µi, j).
9. Threshold Comparison: The updated sum of membership values (∑ µi, j) is compared
against a predefined threshold, which represents the minimum value required for
deploying a single Mobile Agent (MA). Domain initials (DIi) with ∑ µi, j values falling
below this threshold are excluded from further consideration.
10. Domain Reduction: To minimize the number of domain initials (DIs) or DIi, existing
domain initials and their assigned nodes are consolidated. This merging process
facilitates the creation of non-overlapping, load-balanced domains, ensuring a more
efficient distribution of sensor nodes.
3.1.2. Distinctive Features of the Adaptive Fuzzy C-Means (AFCM) Algorithm in
Comparison to k-means, x-Means, and Fuzzy c-Means (FCM)
1. Requirement for Cluster Number Specification:
 k-means: Requires an explicit specification of the number of clusters prior to
execution, which can limit its adaptability to diverse data distributions.
 x-Means: Addresses the limitation of fixed cluster numbers by iteratively adjusting
and refining the number of centroids based on data characteristics.
 Fuzzy c-Means (FCM): Does not necessitate a fixed number of clusters from the
outset, but still relies on initial estimates that can influence clustering outcomes.
 Adaptive Fuzzy c-Means (AFCM): Advances beyond these methods by determining
the number of clusters dynamically, based on the proximity of nodes to a central
processing node, thus eliminating the need for pre-specified cluster numbers.
2. Cluster Initialization:
 k-means: Initializes clusters either randomly or through a predefined method, which
may not always align with the data distribution.
 x-Means: Starts with an initial cluster configuration and iteratively adjusts the cluster
count to enhance data fit.

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 Fuzzy c-Means (FCM): Initiates with centroids and iteratively refines them based on
membership values, though initial cluster estimates still play a role.
 AFCM: Uniquely establishes initial clusters (domain initials) using nodes in
proximity to the processing node, thus aligning initial clusters more closely with
local data characteristics.
3. Cluster Assignment Method:
 k-means: Assigns nodes to the nearest centroid, potentially leading to imbalanced
clusters if the initial number of clusters is suboptimal.
 x-Means: Iteratively refines cluster assignments as the number of clusters is adjusted,
improving alignment with data distribution.
 Fuzzy c-Means (FCM): Utilizes fuzzy membership values to assign nodes to clusters,
allowing for partial membership but not necessarily optimizing cluster count.
 AFCM: Assigns nodes to clusters based on the highest membership value in an
iterative manner, ensuring effective and balanced allocation of all nodes.
4. Clustering Optimization:
 k-means: Does not perform dynamic optimization of cluster numbers after
initialization, which can lead to inefficiencies if the fixed number of clusters is not
ideal.
 x-Means: Enhances clustering by iteratively optimizing the number of clusters,
adapting based on data fit.
 Fuzzy c-Means (FCM): Focuses on centroid optimization according to membership
values but does not dynamically adjust the number of clusters.
 AFCM: Provides a sophisticated optimization approach by calculating the sum of
membership values in advance. It discards clusters with low membership sums and
redistributes nodes to remaining clusters, resulting in non-overlapping and load-
balanced domains.
5. Handling Low Membership Clusters:
 k-means: Fixed clusters remain unchanged regardless of membership distribution,
which may not address low membership issues.
 x-Means: Adapts the number of clusters based on data fit but does not specifically
address low membership clusters.
 Fuzzy c-Means (FCM): Concentrates on refining centroid positions without
dynamically adjusting or eliminating low membership clusters.
 AFCM: Actively manages clusters with low membership sums by eliminating them
and reassigning their nodes, thereby optimizing clustering efficiency and balance.
In summary, the AFCM algorithm offers significant improvements over k-means, x-Means, and
FCM by dynamically determining the number of clusters based on node proximity, optimizing
cluster assignments, and achieving more balanced and efficient clustering in sensor networks.
Table 3 gives a comprehensive comparison that includes all aspects for k-means, x-Means,
Fuzzy c-Means (FCM), and Adaptive Fuzzy c-Means (AFCM)

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Table 3: Comprehensive Comparison of k-means, x-Means, Fuzzy c-Means (FCM), and Adaptive
Fuzzy c-Means (AFCM): Key Aspects and Distinctive Features
Aspect k-means x-Means Fuzzy c-Means
(FCM)
Adaptive Fuzzy c-
Means (AFCM)
Pre-
determined
Number of
Clusters
Requires pre-
determination
of the number
of clusters (k).
Adjusts the
number of clusters
iteratively,
refining k based
on data
distribution.
Does not require a
fixed number of
clusters but may
need initial
estimates.
Does not require
pre-determined
number of clusters;
determines
dynamically based
on node proximity.
Dynamic
Adjustment
of Clusters
No dynamic
adjustment;
clusters are
fixed after
initialization.
Dynamically
adjusts the
number of clusters
by evaluating and
splitting clusters.
Adjusts cluster
centroids based on
membership values
but not the number
of clusters.
Dynamically adjusts
the number of
clusters based on
nodes' proximity to
the processing node.
Cluster
Initialization
Clusters are
initialized
randomly or
using a
predefined
method.
Initializes clusters
and then iterates
to refine the
number based on
fit.
Initializes centroids
and adjusts
iteratively based on
membership values.
Initial clusters are
formed from nodes
within the
processing node's
vicinity.
Cluster
Assignment
Nodes are
assigned to the
nearest cluster
centroid.
Nodes are
assigned
iteratively as
clusters are
refined.
Nodes are assigned
based on fuzzy
membership values
to each cluster.
Nodes are assigned
to clusters based on
highest membership
value iteratively.
Optimization No optimization
of cluster count;
fixed after
initialization.
Optimizes cluster
count iteratively,
adding or
removing clusters
as needed.
Optimizes cluster
centroids based on
membership values
but does not adjust
cluster count
dynamically.
Optimizes clustering
by calculating
membership values
and dropping
clusters below a
threshold.
Cluster
Efficiency
May result in
imbalanced
clusters if the
number of
clusters is not
optimal.
Improves cluster
balance by
dynamically
adjusting cluster
count.
Handles overlapping
data but may not
balance clusters as
dynamically.
Achieves non-
overlapping, load-
balanced clusters by
dynamically
adjusting and
optimizing
assignments.
Handling of
Low
Membership
Clusters
Not applicable;
clusters are
fixed.
Not specifically
addressed; adjusts
cluster count
based on data fit.
Does not
dynamically drop
clusters; focuses on
centroid
adjustments.
Drops clusters with
low membership
sums and reassigns
nodes to existing
clusters to balance
load.
3.1.3. Comparative Analysis of Fuzzy c-Means (FCM) and Adaptive Fuzzy c-Means
(AFCM) Algorithms: Advancements, Adaptability, and Efficiency
Comparing the Fuzzy c-Means (FCM) and Adaptive Fuzzy c-Means (AFCM) algorithms is
crucial for understanding their respective strengths and advancements in clustering research.
While FCM requires pre-determined cluster numbers and operates with static cluster adjustments,
AFCM introduces dynamic cluster determination based on node proximity, significantly
enhancing its adaptability to varying data distributions. This dynamic approach allows AFCM to
optimize clustering through iterative membership calculations and cluster reduction, leading to

65
more balanced and efficient clustering outcomes compared to FCM. Additionally, evaluating
these algorithms helps identify their suitability for different applications, particularly in scenarios
requiring dynamic adjustments and efficient resource management. Overall, this comparison
highlights AFCM's practical advantages in handling dynamic and resource-sensitive clustering
tasks more effectively than FCM.
Table 4 offers a concise yet thorough comparison between the Fuzzy c-Means (FCM) and
Adaptive Fuzzy c-Means (AFCM) algorithms, highlighting their key differences and
demonstrating the advantages of AFCM over FCM.
Table 4: Distinctions between Fuzzy C-Means (FCM) and Adaptive Fuzzy C-Means (AFCM) algorithms
Distinction Issues Fuzzy c-Means (FCM) Adaptive Fuzzy c-Means
(AFCM)
Approach Standard FCM algorithm Adaptive FCM algorithm
Pre-determined number of
clusters (k)
Required Not required
Selection of processing node
(PE)
Not applicable Centralized processing node
Determining k Not applicable Number of Sensor Nodes
(SNs) in the vicinity of the PE
Assigning remaining SNs Based on membership
values
Based on membership values
within the vicinity of each
domain initial (DIi)
Sum of membership values Not calculated in advance Calculated in advance
Assignment to DIi Maximum membership
value
Maximum membership value
with consideration of the
lowest sum of membership
values (∑ µi, j)
Assignment completion Based on membership
values
Based on membership values
and ∑ µi, j values
Comparing with threshold value Not applicable Comparison of updated ∑ µi, j
values with a threshold value
Reducing domains Not applicable Merging domain initials and
assigned nodes for load-
balanced domains
These distinctions highlight the key differences between the FCM and AFCM algorithms,
emphasizing the adaptive nature of AFCM, which does not require pre-determining the number
of clusters (k) and incorporates additional steps for processing node selection, assignment
completion, and domain reduction to achieve load-balanced domains in sensor networks.
4. ADAPTIVE FUZZY-BASED CLUSTERING ALGORITHM (AFCM): A
COMPREHENSIVE DESCRIPTION
The proposed algorithm addresses critical challenges in clustering, such as load balancing and
cluster overlapping. Unlike traditional set theories, fuzzy-based partitioning captures the degree
of belongingness of each sensor node (SN) to the network. Equation 1 illustrates that the degree
of membership (µ) represents the similarity between nodes. Sensor nodes are grouped based on
the domain initial (DI) with the highest µi,j value, where µi,j∈ [0, 1]. Nodes near the center have
higher µ values, while those near the boundary have lower µ values. The algorithm utilizes
membership functions to map distances to degrees of membership (µi,j).

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4.1. Objective and Methodology: Associating Source Nodes with Domain Initials
In a collaborative system, the processing element (PE) assigns mobile agents (MAs) to specific
starting points known as domain initials (DIs), where DIi represents the candidate’s starting point
for an MA's journey. The set DI consists of source nodes that are within the transmission range of
the PE.
The objective of the proposed algorithm is to associate source nodes with their corresponding
DIs. This association ensures that for every source node i in the set SN (∀ i∈ N), there exists a DIj
in the set DI such that i is within the transmission range of j i.e. jrange. Any source nodes that are
not directly associated with a DI are placed in the set NDA (not directly associated). The
algorithm employs fuzzy-based clustering to determine the degree of association µi,j, calculated
using equation 2. The clustering process follows the constraints specified in equations 3 and 4,
ensuring exhaustive partitioning and disjointness among the DIs.
µ𝒊, 𝒋 =
[
𝟏 𝒊𝒇 𝒊 = 𝒋
𝒅𝒊, 𝒋
−𝟐
𝒎−𝟏
∑ 𝒅𝒍,𝒋
−𝟐
𝒎−𝟏
|𝑫𝑰|
𝒍=𝟏
𝑶𝒕𝒉𝒆𝒓𝒘𝒊𝒔𝒆
]
∀𝒊 ∈ 𝑵, 𝒍 ≠ 𝒋 (1)
µ𝒊, 𝒋 =
𝒅𝒊,𝒋
−𝟐
𝒎−𝟏
∑ 𝒅𝒊, 𝒋
−𝟐
𝒎−𝟏
|𝑫𝑰|
𝒍=𝟏
(2)
Here,
|DI| is the cardinality of the set containing source nodes within the vicinity of the processing
element (PE).
‘m’ The balancing exponent that determines the level of crispness or fuzziness in the clustering
process.
• The value of ‘m’ can be adjusted adaptively.
• A lower value of ‘m’ results in sharper boundaries between clusters, making them more
distinct.
• Conversely, a higher value of ‘m’ leads to softer boundaries between clusters, allowing
for more overlap and uncertainty.
Equation 3 ensures that the total degree of association between each source node and the domain
initials within its vicinity always adds up to one. This constraint guarantees the disjunction of any
two domains, meaning that no source node is connected to more than one domain initial.
∑ µ𝒊, 𝒋 = 𝟏, ∀𝒋 ∈ {𝟏, 𝟐, 𝟑, … . . , 𝒏}
|𝑫𝑰|
𝒊=𝟏
(3)
In Equation 4, it is ensured that the sum of the degree of membership values of each source node
which comes under it’s vicinity (irange)with respect to each domain initial is always non-zero. This
constraint guarantees that each domain initial has an expected assigned load from each source
node in the network, preventing any null assignments.
∑ µ𝒊, 𝒋
𝒏
𝒊=𝟏
> 𝟎, ∀𝒋 ∈ {𝟏, 𝟐, 𝟑 … . , |𝑫𝑰|}, ∃ 𝒋 ∶ 𝒅(𝒊 , 𝒋) ≤ 𝒊𝒓𝒂𝒏𝒈𝒆 (4)

67
In this section, a concise description of the proposed partitioning algorithm for wireless sensor
networks is provided. Once the domain initials (DIs) have been established, the subsequent step
involves allocating the remaining nodes to their corresponding domain initials (DIi).
4.2. Problem Statement
Clustering of the wireless sensor network into disjoint and equally loaded appropriate number of
domains.
Given:
1. SN = {SN∈ i = 1, 2,….., n} – Documented set of n sensor nodes.
2. D = {dij, where, i∈ 1 to n; j ∈1 to n; i ≠ j} – Given by Table 2: Spatial distance of each
sensor node in illustrated wireless sensor network, where dij is the Euclidean distance,
described as the whole number.
3. The transmission range of each sensor node is taken as 5.
To find out:
Exhaustive partition of a set of source nodes (SN) into c number of domains.
Steps for proposed AFCM algorithm
1. Given: Set of n number of sensor nodes. SN = {1, 2, 3, …….n};
2. N = {SN– x: x is any centrally located node in the network};/*set of nodes that are to be
associated with the domains. This set contains all sensor nodes except the processing
element.
3. Set Level=0; Choose Llevel = {i: d ( PE, i) ≤ PErange;/* MA will be dispatched by PE by
choosing the nodes which directly come under PE’s vicinity. These nodes are termed
domain initials.
4. Update Nlevel = {N - Llevel};
5. Determine subsets of Llevel, DI = {∃𝑗 : d ( i, j) ≤ jrange: ∀ i∈ Nlevel,∀ j ∈Llevel;/* Set of
domain initials to which source node may be connected.
6. NDA = {∃ i ∈ Nlevel, : d ( i, j ) >jrange,∀j ∈Llevel}; /* Set of nodes i∈ N, which are not
directly connected to Llevel.
If ( NDA ≠ ∅ ) then
Nlevel = Nlevel - NDA;
7. Find µ𝑖, 𝑗 =
𝑑𝑖,𝑗
−2
𝑚−1
∑ 𝑑𝑖,𝑗
−2
𝑚−1
|𝐷𝐼|
𝑙=1
, ∀𝑖 ∈Nlevelw.r.t. each DI such that d (i, j) ≤ irange∀𝑖 ∈ Nlevel, ∀𝑗 ∈
Llevel;
8. Calculate the estimated load of each DI by adding µ values of each DI w.r.t. nodes that
come under their vicinity;
9. Associate the nodes with the DIi
10. If (NDA ≠ ∅) then
i) Llevel = Nlevel,
ii) Level = Level +1,
iii) Nlevel= NDA,
iv) Go to step 5 and repeat until NDA = ∅;
11. Check if there is any domain having data to be carried by the mobile agent less than the
MA’s threshold value then less loaded domain needs to be dropped out by following step
11 else go to step 13;

68
12. Identify each node of this domain (to be dropped out) with their corresponding Llevel and
assign identified nodes to their respective same level by considering membership values;
13. Stop;
In some domains, the amount of data that needs to be carried by the Mobile Agent (MA)
associated with the domain's nodes may be less than the MA's threshold value. The threshold
value represents the minimum data quantity for which a dedicated MA should be deployed. In
such cases, these domains should be excluded or dropped out. Consequently, all relevant nodes
will be associated with their respective domain, denoted by j∈Li, where the membership degree
(µ) is maximum. Similarly, the remaining nodes that were previously associated with the dropped
domain (j) should be connected to the remaining domains (j).
4.3. Evaluating Efficiency of Fuzzy-Based Clustering Algorithm for Wireless Sensor
Network Partitioning: An Illustrative Example
In Figure 1, a specific instance of a wireless sensor network comprising 15 nodes is depicted. The
figure also demonstrates the spatial separation between nodes within the network.
Figure 1. Representation of a Wireless Sensor Network with 15 Nodes
The Fuzzy C-means (FCM) clustering algorithm, initially applied to wired networks as detailed
in [44], is here re-examined in the context of wireless sensor networks (WSNs) using the same
illustrative network. This study shifts focus to explore fuzzy-based clustering in WSNs, as
presented in Table 5, which outlines the range matrix for the discussed network. The objective is
to demonstrate the algorithm’s effectiveness in partitioning both wired and wireless networks
while ensuring that clusters maintain non-overlapping domains. The research highlights the
significant differences between clustering wired and wireless networks due to their distinct
characteristics: wired networks have fixed, stable topologies that optimize static connections,
whereas WSNs feature dynamic, irregular topologies with mobile nodes, necessitating adaptable
clustering strategies to address issues such as energy constraints and communication variability.
By comparing clustering outcomes in WSNs with those from wired networks, this study
emphasizes the need for tailored approaches that effectively manage the unique challenges of
each network type.
The distinction between applying the proposed clustering algorithm to wired versus wireless
networks lies in how proximity is considered. In wireless networks, clusters are formed based on
the wireless range between the processing elements and the centroids of the nodes selected for
clustering. This approach accounts for the variable communication range and signal strength
11
12
13
4
15
5
3
2
1
6
7
8
9
10
14

69
inherent to wireless environments. In contrast, in wired networks, clustering is based on the
physical proximity of nodes to each other rather than their proximity to a centroid. This reflects
the fixed and stable nature of wired connections, where mutual proximity is the primary factor in
cluster formation. The application of the proposed algorithm is illustrated as follows:
Table 5: Spatial distance of each sensor node in wireless sensor network[39]
Node 8, selected as the Processing Node (PE) due to its central location, serves as the basis for
the proposed fuzzy-based clustering algorithm. With the chosen PE, nodes 4, 5, 7, 9, and 14 will
be grouped together as elements of the set DI. Table 6 provides a list of domains to which each
source node may be connected. However, node 10 does not have any domain (DIi) within its
transmission range. Therefore, node 10 will be categorized as a member of the set NDA (Not
Directly Associated) in this context.Table 7presents the degree of membership (belongingness) of
each source node (SNi) to its corresponding Domain Initial (DI).
Table6: Set of domain initials to which node may be connected
Node SNi
Expected domain initials (DI) to which
SNinode may be connected
1 {4, 5}
2 {4, 5}
3 {4}
6 {4, 7}
10 {}
11 {7}
12 {7}
13 {7, 9, 14}
15 {9, 14}
Table 7: Membership values (µ) of each sensor node for the current set of domain initials
di,j 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
1 0 3 6 3 2 8 8 6 8 11 11 12 12 10 12
2 3 0 3 2 3 7 8 6 8 12 12 12 12 10 12
3 6 3 0 3 6 4 6 6 7 10 10 10 10 8 10
4 3 2 3 0 3 5 6 4 6 10 10 10 9 8 10
5 2 3 6 3 0 8 6 4 6 10 10 10 10 8 10
6 8 7 4 5 8 0 5 8 9 5 5 5 7 9 11
7 8 8 6 6 6 5 0 3 4 6 4 4 4 4 6
8 6 6 6 4 4 8 3 0 2 9 7 6 6 4 6
9 8 8 7 6 6 9 4 2 0 10 8 6 4 2 4
10 11 12
1
0
10 10 5 6 9
1
0
0 2 4 8 10 12
11 11 12
1
0
10 10 5 4 7 8 2 0 2 6 8 10
12 12 12
1
0
10 10 5 4 6 6 4 2 0 4 6 8
13 12 12
1
0
9 10 7 4 6 4 8 6 4 0 2 4
14 10 10 8 8 8 9 4 4 2 10 8 6 2 0 2
15 12 12
1
0
10 10 11 6 6 4 12 10 8 4 2 0

70
SNi
Membership
value (µ)with DI4
Membership value
(µ)with DI5
Membership
value (µ)with
DI7
Membership
value
(µ)with DI9
Membership
value (µ)with
DI14
1 2/5 = 0.4 3/5 = 0.6 -- -- --
2 3/5 = 0.6 2/5 = 0.4 -- -- --
3 1 -- -- -- --
6 1/2 = 0.5 -- 1/2 = 0.5 -- --
10 -- -- -- -- --
11 -- -- 1 -- --
12 -- -- 1 -- --
13 -- -- 1/4 = 0.25 1/4 = 0.25 1/2 = 0.5
15 -- -- -- 1/3 = 0.333 2/3 = 0.666
In this scenario, the expected assigned load for each domain initial can be calculated by summing
the membership values of each domain initial with respect to each sensor node. For the given
example, the expected assigned load for domain initials 4, 5, 7, 9, and 14 are 2.5, 1, 2.75, 0.583,
and 1.166, respectively.
Table 8: Association of succeeding sensor nodes to their respective domain initial
Sensor
nodes with
feasible DIs
DI4 (carried
load (2.5), set
of related
nodes)
DI5 (carried
load (1), set of
related nodes)
DI7 (carried
load (2.75), set
of related
nodes)
DI9 (carried
load (0.583),
set of related
nodes)
DI14(carried
load (1.166), set
of related
nodes)
3 2.5, {3} 1,{} 2.75, {} 0.583, {} 1.166, {}
11 2.5, {3} 1, {} 2.75, { 11, 12} 0.583, {} 1.166, {}
12 2.5, {3} 1, {} 2.75, { 11, 12} 0.583, {} 1.166, {}
1 1.9, {3} 1, {1} 2.75, { 11, 12} 0.583, {} 1.166, {}
2 1.9, {2, 3} 0.6, {1} 2.75, { 11, 12} 0.583, {} 1.166, {}
6 1.9, {2, 3, 6} 0.6, {1} 2.25, { 11, 12} 0.583, {} 1.166, {}
15 1.9, {2, 3, 6} 0.6, {1} 2.25, {11, 12} 0.25, {} 1.166, {15}
13 1.9, {2, 3, 6} 0.6, {1} 2, {11, 12} 0 1.166, {13, 15}
3 2.5, {3} 1,{} 2.75, {} 0.583, {} 1.166, {}
Table 8 presents the nodes in ascending order of the cardinality of set DI (as shown in Table 6),
ensuring that nodes with smaller cardinality sets are associated first. In the assignment process,
each node is associated with the DIi that has the highest membership value, as indicated in Table
7. However, if multiple DIi have the same membership value for a node, the node is connected to
the DIi with the lower assigned load. This approach ensures that all nodes within the vicinity of
DIi are appropriately associated.
In contrast, the set NDA contains one element, node 10, which needs to be connected to the
source nodes that are already linked to DI in the previous iteration. In this iteration, the elements
of set DI are replaced by the elements of set N, i.e., 1, 2, 3, 6, 11, 12, 13, and 15, while the
elements of set N are replaced by the element of set NDA, which is 10. Table 9 displays the
degree of belongingness of node 10 with respect to the new set DI.
Table 9: Membership values (µ) of the set NDA for the new set DI
SNi
µw.r.t.
DI1
µw.r.t.
DI2
µw.r.t.
DI3
µw.r.t. DI6
µw.r.t.
DI11
µw.r.t.
DI12
µw.r.t.
DI13
µw.r.t.
DI15
10 -- -- -- 0.211 0.526 0.263 -- --
Since node 10 comes under the vicinity of 6, 11, and 12 nodes directly. Degree of belongingness

71
of the node w.r.t. the relatively new data sets 6, 11, and 12 are 0.211, 0.526, and 0.263
respectively. So, consequently, node 10 will be associated with 11 (already node 11 has been
connected to node 7). Thus, node 10 would be connected to node 4 indirectly. Further, load
assigned to DIs 4, 5, 7, 9, and 14 become 1.9, 0.6, 2.526, 0, and 1.166 respectively.
Therefore, none of the sensor nodes is assigned to the domain initial DI9 and the load assigned to
DI5 is less than the MA threshold value. Thus, both domains need to be dropped out. It is
required that nodes associated with the domains DI5 and DI9 should be associated with some
other domain initials. DI9 contains a single element, 9 and DI5 contains two elements nodes 1
and 5. Thus, the set of nodes {1, 5, 9} needs to be dropped out. Similarly, this set of nodes will
be connected to the respective DI.
5. EVALUATIVE ASSESSMENT
 In this paper, the modification of the proposed algorithm produced three distinct
domains: {1, 2, 3, 4, 6}, {7, 10, 11, 12}, and {5, 9, 13, 14, 15}, with node 8 chosen as the
Processing Element.
 In contrast, applying the FCM algorithm to general/wired networks, as described in [46],
resulted in a different partitioning of the instance network into three domains: {1, 2, 3, 4,
5}, {6, 7, 10, 11, 12}, and {9, 13, 14, 15}.
 Both networks were structured with an equal number of domains and an identical number
of nodes per domain, though the specific nodes within each domain varied.
 The loads for the wired networks were determined as minL(wired), minL(wired) + 0.2,
and minL(wired) + 0.4, where minL(wired) represents the minimum load.
 The loads for the wireless networks were minL(wireless), minL(wireless) + 0.374, and
minL(wireless) + 0.306, with the minimum load for wireless networks denoted as
minL(wireless).
 In both cases, the domains maintained the same cardinality of 4, 5, and 5 nodes; however,
the composition of nodes within each cluster differed between the two algorithms.
6. CONCLUSION
The research paper introduces the Adaptive Fuzzy c-Means (AFCM) algorithm to address the
limitations of existing clustering methods like k-means, x-means, and Fuzzy c-Means (FCM) in
multi-mobile agent itinerary planning (MIP). The AFCM algorithm enhances network clustering
by effectively managing sensor nodes with equal membership values and ensuring balanced, non-
overlapping domains. This improvement optimizes the efficiency of MIP systems, particularly
under varying constraints. The algorithm demonstrated success in partitioning networks into
well-balanced domains and suggested an appropriate number of mobile agents for optimized
performance.
Future research can be focused on exploring the AFCM algorithm's performance under diverse
network constraints and in larger-scale systems. Additionally, integrating AFCM with real-world
MIP applications could further validate its effectiveness and scalability.
Compliance with Ethical Standards
The authors have no conflicts of interest to declare. All the co-authors have seen and agree with
the contents of the manuscript and there is no financial interest to report.

72
Each author have made substantial contributions to the conception or design of the work; or the
acquisition, analysis, or interpretation of data.
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