An Amalgamated Kruskal’s Algorithm to ascertain the shortest route in the futuristic smart world

Abstract

As the globe enters a new era, web applications will become indispensable to managing business. Businesses can easily grow, become simpler, and accomplish their objective much faster by employing web applications. Creating a web application in cloud computing allows for the more affordable leveraging of cloud-based services. This makes it easier to avoid setting up and maintaining several servers. To get around cloud computing’s built-in restrictions such as scalability, security, and bandwidth limitations, the future smart world of cloud computing will be coupled with LiFi connectivity. Beyond creating the web application, it is important to promote this web application among the network of users as quickly and effectively as possible. This manuscript proposes a strategy to address these challenges. There are two primary components to this MCDM technique. The first step is to model the problem as a graph and weigh the edges by employing the Hamacher aggregation operator. The second step involves using a fresh iteration of Kruskal’s technique in conjunction with this approach to discover a Minimum Spanning Tree as a resolution. This manuscript adds to the literature by solving real-world Minimum Spanning Tree problems by combining existing algorithms with MCDM techniques. This technique is demonstrated for marketing a web application(created via cloud service) in a future smart world using LiFi technology.

Keywords

Cloud computing LiFi technology Kruskal’s technique minimum spanning tree Hamacher aggregation operator

1 Introduction

The digital world is consistently evolving and the actions we engage in it change every day. The creation of web applications is an essential part of contemporary technology because it allows companies and organizations to make their products and information available online, opening up new opportunities for development, collaboration, and innovation. Because technology is erratic, our work remains essential for success in this cutthroat environment. Creating a web application in cloud computing allows for the more affordable leveraging of cloud-based services. Businesses need web applications that can stand out in this fiercely competitive market, and those applications can only be successful if they are created utilizing unique modern technologies. Considering this, the future smart world will be equipped with LiFi technology.

LiFi is a type of wireless communication that connects devices by using light to convey information and coordinates. Li-Fi is a light-based communication technology that can send data quickly over the visible, ultraviolet, and infrared spectrums. Currently, the only lights that can be utilized to transmit data in visible light are LED lamps. This technology has similarities to Wi-Fi. The main technological distinction is that Li-Fi transmits data by modulating light intensity instead of Wi-Fi, which induces an electric tension in an antenna to transmit data. Li-Fi can operate in places that would ordinarily be sensitive to electromagnetic interference(such as military facilities, hospitals, and airplane cabins). This particular LiFi function protects against interference from possible hackers on the sensitive network. Connecting rural areas with major cities and hubs is essential for the economy to grow rapidly. Two Gujarati villages now have access to high-speed internet to a firm from Ahmedabad’s LiFi-based technology, marking the first step towards the creation of smart villages in the state. The first smart villages in India are Akrund and Navanagar, which are located in Gujarat’s Aravalli district and boast LiFi-based internet connectivity. Schools, hospitals, post offices, and government offices in these two cities have access to a faster and safer internet connection. The state of technology is improving daily. Future will witness a LiFi revolution in the smart world [20].

Beyond creating the web application using cloud service in the future smart world, it is important to promote this web application among the network of users quickly and most effectively. A challenge in marketing a web application developed through cloud computing employing LiFi across the user network is often considered to as the Minimum Spanning Tree $(\vec{MST})$ problem. The most widely investigated problem in combinatorial optimization is the $\vec{MST}$ problem, which can be addressed using a variety of algorithms [11]. Kruskal’s [3] is a prominent MST algorithm. Several investigations [4, 10] in the literature use this technique to solve real-world issues. Certain criteria can be established when looking into the scores of links in marketing a web application. Methods for Multi Criteria Decision-Making(MCDM) are adequate for locating and assessing properties. Fuzzy MCDM methods, which are frequently employed in criterion weighting and alternative ranking, are applicable at this point in the literature.

Zadeh [26] created a structure for Fuzzy set $(\vec{FS})$ in which each element has a Membership Score $(\vec{MS})$ . Due to multiple uncertainties, there are some issues that we face daily that $\vec{FS}$ theory fails to address. To get rid of these ambiguities, Atanassov [2] designed a structure of Intuitionistic Fuzzy set $(\vec{IFS})$ , where each element is given $\vec{MS}$ and Non-Membership $(\vec{NMS})$ subject to the constraint 0 ≤ $\vec{MS}$ + $\vec{NMS}$ ≤ 1. In addition to the existence theory, Yager [25] developed the q-Rung Orthopair Fuzzy set $(q - \vec{ROFS})$ with the constraint that ≤ ${\vec{MS}}^{q}$ + ${\vec{NMS}}^{q}$ ≤ 1. With the formulation of reference parameters by Riaz and Hashmi [18], a structure of Linear Diophantine Fuzzy set $(\vec{LDFS})$ emerged to expand the decision-maker’s independence when handling real-world situations.

The limitations of the aforementioned theories, render them ineffective for addressing problems involving parametrization tools. Molodtsov [15] devised the deductive theory of the Soft set $(\vec{SS})$ to provide mathematical tools that are not subject to the aforementioned restriction.

Conventional complex analysis can be useful in many areas of Mathematics. In order to resolve the Problems of complex analysis in a fuzzy environment, Ramot et al. [17] devised the structure of Complex Fuzzy set $(\vec{CFS})$ , in which each element is given a Complex-Valued Membership Score $(\vec{CVMS})$ (i.e.), $Ω_{\vec{A}} (s_{p}) e^{i 2 π w_{Ω_{\vec{A}}} (s_{p})}$ where $Ω_{\vec{A}} (s_{p})$ depicts Amplitude Term $(\vec{AT})$ and $w_{Ω_{\vec{A}}} (s_{p})$ depicts Phase Term $(\vec{PT})$ subject to the constraint that $Ω_{\vec{A}} (s_{p}), w_{Ω_{\vec{A}}} (s_{p})$ ∈ [0,1]. Huseyin and Kamaci [6] developed Complex Linear Diophantine Fuzzy set $(\vec{CLDFS})$ structure to address many of the real-life problems in a 2D frame as a refinement to the existing theory. Several researchers discussed some of the novel distance measures for the $\vec{FS}$ extension theory [8, 21]. In addition to that, there was a development of aggregation operators for the theories of $\vec{FS}$ extensions. to discard information that is unclear or unreliable while making decisions [7, 12]. Furthermore, many researchers worked together on the hybridization and extension of $\vec{FS}$ [1, 13, 14, 23, 24].

Upon reviewing the previously mentioned literature, the following research gaps become apparent:

1.
It has been noted with great attention that the hybridization of Kruskal’s algorithm with several $\vec{FS}$ extensions to account for data uncertainty in experimental settings has already been the subject of attempts to optimize and find solutions to the $\vec{SP}$ problem. However, none of the $\vec{FS}$ extensions utilizing the parametrization tool have been hybridized with Kruskal’s algorithm. Therefore, it is imperative to investigate the suitability of mathematical modeling of fuzzy set extension involving parametrization tools in the hybridization of Kruskal’s algorithm to confirm their efficacy in obtaining the desired optimal path.
2.
In addition, there is a dearth of suitable methods that can handle the operator perspective of the relative weights given to various process performances involving uncertainty and ambiguity in the experimental dataset.
This manuscript makes the following contributions to address the aforementioned research gaps: 1.
To address Minimum Spanning Tree problems in practice, Kruskal’s algorithm is integrated with $\vec{CLDFSS}$ to account for data uncertainty in experimental settings.
2.
Extending the Hamacher operations to combine the $\vec{CLDFS}$ information is a significant portion of the work.
3.
We have used Hamacher aggregation operators with $\vec{CLDFS}$ information to create a few models for the MCDM situation.
Novelty of the suggested methodology:

Despite the abundance of papers on $\vec{SP}$ problems in the literature, none address the issue in a $\vec{CLDFS}$ setting. The following is a summary of some novel contributions made by the suggested study: 1.
Using the $\vec{CVMS}$ , $\vec{CVNMS}$ , and $\vec{CV}$ reference parameters, the suggested method applies $\vec{CLDFSS}$ theory to capture uncertainty and open up possibilities for incorporating the $\vec{AS}$ into decision-making. The earlier fuzzy methods utilized in the literature to handle the $\vec{SP}$ problem are mostly overshadowed by this characteristic of the proposed study.
2.
To the fullest extent of our understanding, no prior work has addressed Kruskal’s algorithm in a $\vec{CLDFS}$ setting. Being the first of its kind, our study positioned our suggested model as a trailblazer and an exemplar in this field.
3.
We put out a model with the intended objectives to solve a $\vec{MST}$ problem of marketing a web application(created via cloud service) in a future smart world using LiFi technology by combining existing algorithms with MCDM techniques.
4.
The suggested method helps decision makers obtain optimal results by employing a parametrization tool to get precise results.
5.
The suggested model is a simple approach that is simpler to solve computationally when compared to previous studies.
This manuscript adds to the literature by solving real-world $\vec{MST}$ problems of marketing a web application(created via cloud service) in a future smart world using LiFi technology by combining existing algorithms with MCDM techniques. The structure of the Article is outlined in Table 1:

Table 1
Structure of the article

Segment 2 Concepts of $\vec{CLDFSS}$ together with the fundamentals on graphs and its extensions are examined.

Segment 3 The conceptualization of Complex Linear Diophantine Fuzzy Soft Hamacher Weighted averaging $(\vec{CLDFSHWA})$ operator and the

practical Hamacher operational laws are investigated.

Segment 4 Conceptual framework for locating the shortest path $(\vec{SP})$ connecting each vertex of the weighted graph is developed.

Segment 5 Discusses the suggested MCDM technique combining Kruskal’s algorithm and the Hamacher aggregation operator through an actual case study.

Segment 6 Sensitivity Analysis is performed.

Segment 7 Manuscript is compiled with future developments.

2 Preliminaries

In this segment, we scrutinize the core concepts of $\vec{CLDFSS}$ and their operational definitions, which are extremely beneficial for the investigated works. Also, we examine the fundamentals of graphs and their extensions which provide structure in assessing performances. Furthermore, in every investigation, the Universal Set $(\vec{US})$ is indicated using the symbol $\vec{ℶ}$ and the Attribute Set $(\vec{AS})$ is indicated using the symbol $\vec{R}$ .

2.1 Elementary concepts

Definition 2.1. [15] The Soft set $\vec{K}$ over $\vec{ℶ}$ is an arrangement of specific type $\vec{K} = {(r, \vec{J} (r)) / r \in \vec{R}, \vec{J} (r) \in P (\vec{K})}$

where $P (\vec{K})$ is the power set of $\vec{K}$ .

Definition 2.2. [17] The Complex Fuzzy Set $\vec{A}$ over $\vec{ℶ}$ is a configuration of particular form $\vec{A} = {(s_{p}, 〈 Ω_{\vec{A}} (s_{p}) e^{i v_{Ω_{\vec{A}}} (s_{p})} 〉)$ : s _p ∈ $\vec{ℶ}$ }

where (for $Ω_{\vec{A}} (s_{p})$ ∈ [0, 1] and $v_{Ω_{\vec{A}}} (s_{p})$ ∈ [0, 2π]) symbolizes the Complex-valued Membership Score

$(\vec{CVMS})$ of s _p ∈ $\vec{ℶ}$ . The Complex Fuzzy set can also be identified as $\vec{A} = {(s_{p}, 〈 Ω_{\vec{A}} (s_{p}) e^{i 2 π w_{Ω_{\vec{A}}} (s_{p})} 〉)$ : s _p ∈ $\vec{ℶ}$ }

where (for $Ω_{\vec{A}} (s_{p})$ ∈ [0, 1] and $w_{Ω_{\vec{A}}} (s_{p})$ ∈ [0, 1]) symbolizes the $\vec{CVMS}$ of s _p ∈ $\vec{ℶ}$ .

Definition 2.3. [6] The Complex Linear Diophantine Fuzzy set $(\vec{CLDFS})$ $\vec{T}$ over $\vec{ℶ}$ is an arrangement of specific type $\vec{T}$ = ${(s_{p}, 〈 Ω_{\vec{T}} (s_{p}) e^{i 2 π (w_{Ω_{\vec{T}}} (s_{p}))}, ϒ_{\vec{T}} (s_{p}) e^{i 2 π (w_{ϒ_{\vec{T}}} (s_{p}))} 〉$ ,

$〈 ϱ_{\vec{T}}^{p} e^{i 2 π (w_{ϱ_{\vec{T}}^{p}})}, ϑ_{\vec{T}}^{p} e^{i 2 π (w_{ϑ_{\vec{T}}^{p}})} 〉)$ : s _p ∈ $\vec{ℶ}}$

corresponding to the conditions $0 \leq ϱ_{\vec{T}}^{p} + ϑ_{\vec{T}}^{p} \leq 1$ , $0 \leq ϱ_{\vec{T}}^{p} Ω_{\vec{T}} (s_{p}) + ϑ_{\vec{T}}^{p} ϒ_{\vec{T}} (s_{p}) \leq 1$ and $0 \leq w_{ϱ_{\vec{T}}^{p}} + w_{ϑ_{\vec{T}}^{p}} \leq 1$ , $0 \leq w_{ϱ_{\vec{T}}^{p}} w_{Ω_{\vec{T}}} (s_{p}) + w_{ϑ_{\vec{T}}^{p}} w_{ϒ_{\vec{T}}} (s_{p}) \leq 1$ , where $Ω_{\vec{T}} (s_{p}) e^{i 2 π (w_{Ω_{\vec{T}}} (s_{p}))}$ , $ϒ_{\vec{T}} (s_{p}) e^{i 2 π (w_{ϒ_{\vec{T}}} (s_{p}))}$ , $ϱ_{\vec{T}}^{p} e^{i 2 π (w_{ϱ_{\vec{T}}^{p}})}$ , $ϑ_{\vec{T}}^{p} e^{i 2 π (w_{ϑ_{\vec{T}}^{p}})}$ symbolizes the $\vec{CVMS}$ , Complex-valued Non Membership Score $(\vec{CVNMS})$ and Complex-valued reference parameters $(\vec{CVRP} s)$ of s _p ∈ $\vec{ℶ}$ respectively. The Complex Linear Diophantine Fuzzy Number $(\vec{CLDFN})$ is symbolized as $\vec{T} = (〈 (Ω_{\vec{T}},$ $w_{Ω_{\vec{T}}}), (ϒ_{\vec{T}}, w_{ϒ_{\vec{T}}}) 〉, 〈 (ϱ_{\vec{T}}, w_{ϱ_{\vec{T}}}), (ϑ_{\vec{T}}, w_{ϑ_{\vec{T}}}) 〉)$

Definition 2.4. [6] The Score function for a $\vec{CLDFN}$ $\vec{C}$ = $(〈 (Ω_{\vec{C}}, w_{Ω_{\vec{C}}}), (ϒ_{\vec{C}}, w_{ϒ_{\vec{C}}}) 〉, 〈 (ϱ_{\vec{C}}, w_{ϱ_{\vec{C}}}), (ϑ_{\vec{C}}, w_{ϑ_{\vec{C}}}) 〉)$ is an configuration of particular form $\vec{S} (\vec{C}) = \frac{1}{4} [(Ω_{\vec{C}} - ϒ_{\vec{C}}) + (w_{Ω_{\vec{C}}} - w_{ϒ_{\vec{C}}}) + (ϱ_{\vec{C}} - ϑ_{\vec{C}}) + (w_{ϱ_{\vec{C}}} - w_{ϑ_{\vec{C}}})]$ , where $\vec{S} (\vec{C})$ ∈ [-1,1].

Definition 2.5. [22] Let $\vec{ℶ}$ = {s ₁, s ₂, . . . , s _n} be the $\vec{US}$ . Then a Complex Linear Diophantine Fuzzy Soft set $(\vec{CLDFSS})$ is a pair $(\vec{J}, \vec{R})$ over $\vec{ℶ}$ and $\vec{J}$ is a mapping identified as $\vec{J}$ : $\vec{R}$ → $\vec{CLDFSU} (\vec{ℶ})$ such that $\vec{J} (r) = Φ$ , if r $\notin \vec{R}$ in which $\vec{CLDFSU} (\vec{ℶ})$ consists of the collection of all $\vec{CLDFS}$ . The $\vec{CLDFSS}$ is described and specified as $〈 \vec{J}, \vec{R} 〉$ = ${〈 r, \vec{J} (r) 〉 / r \in \vec{R}, \vec{J} (r) \in \vec{CLDFSU} (\vec{ℶ})}$

(i.e.), $\vec{J} (r)$ = ${(s_{p}, 〈 Ω_{\vec{J} (r)} (s_{p}) e^{i 2 π (w_{Ω_{\vec{J} (r)}} (s_{p}))},$

$\begin{matrix} ϒ_{\vec{J} (r)} (s_{p}) e^{i 2 π (w_{ϒ_{\vec{J} (r)}} (s_{p}))} 〉, \\ 〈 ϱ_{\vec{J} (r)}^{p} e^{i 2 π (w_{ϱ_{\vec{J} (r)}^{p}})}, ϑ_{\vec{J} (r)}^{p} e^{i 2 π (w_{ϑ_{\vec{J} (r)}^{p}})} 〉) : s_{p} \in \vec{ℶ}} . \end{matrix}$

The Complex Linear Diophantine Fuzzy Soft Number $(\vec{CLDFSN})$ establishes a certain form $\vec{J} (r) = (〈 (Ω_{\vec{J} (r)}, w_{Ω_{\vec{J} (r)}}), (ϒ_{\vec{J} (r)}, w_{ϒ_{\vec{J} (r)}}) 〉$ ,

$〈 (ϱ_{\vec{J} (r)}, w_{ϱ_{\vec{J} (r)}}), (ϑ_{\vec{J} (r)}, w_{ϑ_{\vec{J} (r)}}) 〉)$

2.2 Fundamental definitions on graphs

Definition 2.6. Pursuant to crisp graph theory [19], an ordered triple $(\vec{V (G)}, \vec{E (G)}, \vec{Ψ_{G}})$ comprising of a non-empty set $\vec{V (G)}$ of vertices, a set $\vec{E (G)}$ of edges, and an incidence function $\vec{Ψ_{G}}$ constitute a Graph $\vec{G}$ . A graph $\vec{H}$ is said to be a Subgraph of $\vec{G}$ if $\vec{V (H)} \subseteq \vec{V (G)}$ and $\vec{E (H)} \subseteq \vec{E (G)}$ , and $\vec{Ψ_{H}}$ is the restriction of $\vec{Ψ_{G}}$ . A subgraph $\vec{H}$ is referred to as a Spanning Subgraph if $\vec{V (H)}$ = $\vec{V (G)}$ . A Walk $\vec{W}$ in $\vec{G}$ is an alternating sequence of vertices and edges as its terms that is finite and non-null. If the edges of Walk $\vec{W}$ are distinct, then $\vec{W}$ is referred to as a Trial. If the vertices of the Trial $\vec{W}$ are distinct then $\vec{W}$ is referred to as a Path. A Cycle is a closed trial with a distinct origin and internal vertices. A connected acyclic graph is Tree. A Spanning Tree of $\vec{G}$ is a spanning subgraph of $\vec{G}$ that is a tree. The Minimum Spanning Tree has all the characteristics of a spanning tree with the additional restriction of having the lowest weights possible across all conceivable spanning trees. Pursuant to fuzzy graph theory, a pair $(\vec{η}, \vec{ρ})$ is said to be a Fuzzy Graph [16] symbolized by $\vec{G^{'}}$ = $(\vec{V}, \vec{E})$ of $\vec{G}$ = $(\vec{V}, \vec{E})$ where $\vec{η}$ : $\vec{V}$ → [0,1] is a fuzzy subset of a nonempty set $\vec{V}$ and $\vec{ρ}$ : $\vec{V \times V}$ → [0,1] is a symmetric fuzzy relation on $\vec{η}$ in conjunction with the condition that $\vec{ρ} (a, b)$ = $\vec{ρ} (ab)$ = $\vec{η} (a)$ ∧ $\vec{η} (b)$ ∀ a, b ∈ $\vec{V}$ .

Definition 2.7. [5] Following are the definitions of the $\vec{T}$ -norm and ${\vec{T}}^{*}$ -conorm that Hamacher established. $\vec{T} (s_{1}, s_{2})$ = $\frac{s_{1} s_{2}}{ς + (1 - ς) (s_{1} + s_{2} - s_{1} s_{2})}$ ,ς>0

${\vec{T}}^{*} (s_{1}, s_{2})$ = $\frac{s_{1} + s_{2} - s_{1} s_{2} - (1 - ς) s_{1} s_{2}}{1 - (1 - ς) s_{1} s_{2}}$ ,ς>0

3 Aggregation operator

The aggregation operators are effective models for aggregating and summing together a finite set of numerical information into a single numerical information. Hamacher aggregation operators are more adaptable and powerful to ascertain the correlations between any number of attributes [9]. This section aims to develop the conceptualization of Complex Linear Diophantine Fuzzy Soft Hamacher Weighted averaging $(\vec{CLDFSHWA})$ operator and investigate the practical Hamacher operational laws.

3.1 Operational laws based on Hamacher norm

Definition 3.1. Let

$\vec{J} = (〈 (Ω_{\vec{J}}, w_{Ω_{\vec{J}}}), (Υ_{\vec{J}}, w_{Υ_{\vec{J}}}) 〉, 〈 (ϱ_{\vec{J}}, w_{ϱ_{\vec{J}}}), (ϑ_{\vec{J}}, w_{ϑ_{\vec{J}}}) 〉),$ ${\vec{J}}_{11} = (〈 (Ω_{{\vec{J}}_{11}}, w_{Ω_{{\vec{J}}_{11}}}), (Υ_{{\vec{J}}_{11}}, w_{Υ_{{\vec{J}}_{11}}}) 〉, 〈 (ϱ_{{\vec{J}}_{11}}, w_{ϱ_{{\vec{J}}_{11}}}), (ϑ_{{\vec{J}}_{11}}, w_{ϑ_{{\vec{J}}_{11}}}) 〉),$ and ${\vec{J}}_{12} = (〈 (Ω_{{\vec{J}}_{12}}, w_{Ω_{{\vec{J}}_{12}}}), (Υ_{{\vec{J}}_{12}}, w_{Υ_{{\vec{J}}_{12}}}) 〉, 〈 (ϱ_{{\vec{J}}_{12}}, w_{ϱ_{{\vec{J}}_{12}}}), (ϑ_{{\vec{J}}_{12}}, w_{ϑ_{{\vec{J}}_{12}}}) 〉)$ be three $\vec{C ℒ D ℱ S N}$ and $ς > 0$ then the following definitions are Operational laws based on Hamacher $\vec{T}$ -norm and ${\vec{T}}^{*}$ -conorm:

{\vec{J}}_{11} \oplus {\vec{J}}_{12} = {\begin{cases} 〈 (\frac{Ω_{{\vec{J}}_{11}} + Ω_{{\vec{J}}_{12}} - Ω_{{\vec{J}}_{11}} Ω_{{\vec{J}}_{12}} - (1 - ς) Ω_{{\vec{J}}_{11}} Ω_{{\vec{J}}_{12}}}{1 - (1 - ς) Ω_{{\vec{J}}_{11}} Ω_{{\vec{J}}_{12}}}, \\ \frac{w_{Ω_{{\vec{J}}_{11}}} + w_{Ω_{{\vec{J}}_{12}}} - w_{Ω_{{\vec{J}}_{11}}} w_{Ω_{{\vec{J}}_{12}}} - (1 - ς) w_{Ω_{{\vec{J}}_{11}}} w_{Ω_{{\vec{J}}_{12}}}}{1 - (1 - ς) w_{Ω_{{\vec{J}}_{11}}} w_{Ω_{{\vec{J}}_{12}}}}), \\ (\frac{Υ_{{\vec{J}}_{11}} Υ_{{\vec{J}}_{12}}}{ς + (1 - ς) (Υ_{{\vec{J}}_{11}} + Υ_{{\vec{J}}_{12}} - Υ_{{\vec{J}}_{11}} Υ_{{\vec{J}}_{12}})}, \\ \frac{w_{Υ_{{\vec{J}}_{11}}} w_{Υ_{{\vec{J}}_{12}}}}{ς + (1 - ς) (w_{Υ_{{\vec{J}}_{11}}} + w_{Υ_{{\vec{J}}_{12}}} - w_{Υ_{{\vec{J}}_{11}}} w_{Υ_{{\vec{J}}_{12}}})}) 〉, \\ 〈 (\frac{ϱ_{{\vec{J}}_{11}} + ϱ_{{\vec{J}}_{12}} - ϱ_{{\vec{J}}_{11}} ϱ_{{\vec{J}}_{12}} - (1 - ς) ϱ_{{\vec{J}}_{11}} ϱ_{{\vec{J}}_{12}}}{1 - (1 - ς) ϱ_{{\vec{J}}_{11}} ϱ_{{\vec{J}}_{12}}}, \\ \frac{w_{ϱ_{{\vec{J}}_{11}}} + w_{ϱ_{{\vec{J}}_{12}}} - w_{ϱ_{{\vec{J}}_{11}}} w_{ϱ_{{\vec{J}}_{12}}} - (1 - ς) w_{ϱ_{{\vec{J}}_{11}}} w_{ϱ_{{\vec{J}}_{12}}}}{1 - (1 - ς) w_{ϱ_{{\vec{J}}_{11}}} w_{ϱ_{{\vec{J}}_{12}}}}), \\ (\frac{ϑ_{{\vec{J}}_{11}} ϑ_{{\vec{J}}_{12}}}{ς + (1 - ς) (ϑ_{{\vec{J}}_{11}} + ϑ_{{\vec{J}}_{12}} - ϑ_{{\vec{J}}_{11}} ϑ_{{\vec{J}}_{12}})}, \\ \frac{w_{ϑ_{{\vec{J}}_{11}}} w_{ϑ_{{\vec{J}}_{12}}}}{ς + (1 - ς) (w_{ϑ_{{\vec{J}}_{11}}} + w_{ϑ_{{\vec{J}}_{12}}} - w_{ϑ_{{\vec{J}}_{11}}} w_{ϑ_{{\vec{J}}_{12}}})}) 〉 \end{cases}}

{\vec{J}}_{11} \otimes {\vec{J}}_{12} = {\begin{cases} 〈 (\frac{Ω_{{\vec{J}}_{11}} Ω_{{\vec{J}}_{12}}}{ς + (1 - ς) (Ω_{{\vec{J}}_{11}} + Ω_{{\vec{J}}_{12}} - Ω_{{\vec{J}}_{11}} Ω_{{\vec{J}}_{12}})}, \\ \frac{w_{Ω_{{\vec{J}}_{11}}} w_{Ω_{{\vec{J}}_{12}}}}{ς + (1 - ς) (w_{Ω_{{\vec{J}}_{11}}} + w_{Ω_{{\vec{J}}_{12}}} - w_{Ω_{{\vec{J}}_{11}}} w_{Ω_{{\vec{J}}_{12}}})}), \\ (\frac{Υ_{{\vec{J}}_{11}} + Υ_{{\vec{J}}_{12}} - Υ_{{\vec{J}}_{11}} Υ_{{\vec{J}}_{12}} - (1 - ς) Υ_{{\vec{J}}_{11}} Υ_{{\vec{J}}_{12}}}{1 - (1 - ς) Υ_{{\vec{J}}_{11}} Υ_{{\vec{J}}_{12}}}, \\ \frac{w_{Υ_{{\vec{J}}_{11}}} + w_{Υ_{{\vec{J}}_{12}}} - w_{Υ_{{\vec{J}}_{11}}} w_{Υ_{{\vec{J}}_{12}}} - (1 - ς) w_{Υ_{{\vec{J}}_{11}}} w_{Υ_{{\vec{J}}_{12}}}}{1 - (1 - ς) w_{Υ_{{\vec{J}}_{11}}} w_{Υ_{{\vec{J}}_{12}}}}) 〉, \\ 〈 (\frac{ϱ_{{\vec{J}}_{11}} ϱ_{{\vec{J}}_{12}}}{ς + (1 - ς) (ϱ_{{\vec{J}}_{11}} + ϱ_{{\vec{J}}_{12}} - ϱ_{{\vec{J}}_{11}} ϱ_{{\vec{J}}_{12}})}, \\ \frac{w_{ϱ_{{\vec{J}}_{11}}} w_{ϱ_{{\vec{J}}_{12}}}}{ς + (1 - ς) (w_{ϱ_{{\vec{J}}_{11}}} + w_{ϱ_{{\vec{J}}_{12}}} - w_{ϱ_{{\vec{J}}_{11}}} w_{ϱ_{{\vec{J}}_{12}}})}), \\ (\frac{ϑ_{{\vec{J}}_{11}} + ϑ_{{\vec{J}}_{12}} - ϑ_{{\vec{J}}_{11}} ϑ_{{\vec{J}}_{12}} - (1 - ς) ϑ_{{\vec{J}}_{11}} ϑ_{{\vec{J}}_{12}}}{1 - (1 - ς) ϑ_{{\vec{J}}_{11}} ϑ_{{\vec{J}}_{12}}}, \\ \frac{w_{ϑ_{{\vec{J}}_{11}}} + w_{ϑ_{{\vec{J}}_{12}}} - w_{ϑ_{{\vec{J}}_{11}}} w_{ϑ_{{\vec{J}}_{12}}} - (1 - ς) w_{ϑ_{{\vec{J}}_{11}}} w_{ϑ_{{\vec{J}}_{12}}}}{1 - (1 - ς) w_{ϑ_{{\vec{J}}_{11}}} w_{ϑ_{{\vec{J}}_{12}}}}) 〉 \end{cases}}

κ \vec{J} = {\begin{cases} 〈 (\frac{{(1 + (ς - 1) Ω_{\vec{J}})}^{κ} - {(1 - Ω_{\vec{J}})}^{κ}}{{(1 + (ς - 1) Ω_{\vec{J}})}^{κ} + (ς - 1) {(1 - Ω_{\vec{J}})}^{κ}}, \\ \frac{{(1 + (ς - 1) w_{Ω_{\vec{J}}})}^{κ} - {(1 - w_{Ω_{\vec{J}}})}^{κ}}{{(1 + (ς - 1) w_{Ω_{\vec{J}}})}^{κ} + (ς - 1) {(1 - w_{Ω_{\vec{J}}})}^{κ}}), \\ (\frac{ς {(Υ_{\vec{J}})}^{κ}}{{(1 + (ς - 1) (1 - Υ_{\vec{J}}))}^{κ} + (ς - 1) {(Υ_{\vec{J}})}^{κ}}, \\ \frac{ς {(w_{Υ_{\vec{J}}})}^{κ}}{{(1 + (ς - 1) (1 - w_{Υ_{\vec{J}}}))}^{κ} + (ς - 1) {(w_{Υ_{\vec{J}}})}^{κ}}) 〉, \\ 〈 (\frac{{(1 + (ς - 1) ϱ_{\vec{J}})}^{κ} - {(1 - ϱ_{\vec{J}})}^{κ}}{{(1 + (ς - 1) ϱ_{\vec{J}})}^{κ} + (ς - 1) {(1 - ϱ_{\vec{J}})}^{κ}}, \\ \frac{{(1 + (ς - 1) w_{ϱ_{\vec{J}}})}^{κ} - {(1 - w_{ϱ_{\vec{J}}})}^{κ}}{{(1 + (ς - 1) w_{ϱ_{\vec{J}}})}^{κ} + (ς - 1) {(1 - w_{ϱ_{\vec{J}}})}^{κ}}), \\ (\frac{ς {(ϑ_{\vec{J}})}^{κ}}{{(1 + (ς - 1) (1 - ϑ_{\vec{J}}))}^{κ} + (ς - 1) {(ϑ_{\vec{J}})}^{κ}}, \\ \frac{ς {(w_{ϑ_{\vec{J}}})}^{κ}}{{(1 + (ς - 1) (1 - w_{ϑ_{\vec{J}}}))}^{κ} + (ς - 1) {(w_{ϑ_{\vec{J}}})}^{κ}}) 〉 \end{cases}}_{κ > 0}

{\vec{J}}^{κ} = {\begin{cases} 〈 (\frac{ς {(Ω_{\vec{J}})}^{κ}}{{(1 + (ς - 1) (1 - Ω_{\vec{J}}))}^{κ} + (ς - 1) {(Ω_{\vec{J}})}^{κ}}, \\ \frac{ς {(w_{Ω_{\vec{J}}})}^{κ}}{{(1 + (ς - 1) (1 - w_{Ω_{\vec{J}}}))}^{κ} + (ς - 1) {(w_{Ω_{\vec{J}}})}^{κ}}), \\ (\frac{{(1 + (ς - 1) Υ_{\vec{J}})}^{κ} - {(1 - Υ_{\vec{J}})}^{κ}}{{(1 + (ς - 1) Υ_{\vec{J}})}^{κ} + (ς - 1) {(1 - Υ_{\vec{J}})}^{κ}}, \\ \frac{{(1 + (ς - 1) w_{Υ_{\vec{J}}})}^{κ} - {(1 - w_{Υ_{\vec{J}}})}^{κ}}{{(1 + (ς - 1) w_{Υ_{\vec{J}}})}^{κ} + (ς - 1) {(1 - w_{Υ_{\vec{J}}})}^{κ}}) 〉, \\ 〈 (\frac{ς {(ϱ_{\vec{J}})}^{κ}}{{(1 + (ς - 1) (1 - ϱ_{\vec{J}}))}^{κ} + (ς - 1) {(ϱ_{\vec{J}})}^{κ}}, \\ \frac{ς {(w_{ϱ_{\vec{J}}})}^{κ}}{{(1 + (ς - 1) (1 - w_{ϱ_{\vec{J}}}))}^{κ} + (ς - 1) {(w_{ϱ_{\vec{J}}})}^{κ}}), \\ (\frac{{(1 + (ς - 1) ϑ_{\vec{J}})}^{κ} - {(1 - ϑ_{\vec{J}})}^{κ}}{{(1 + (ς - 1) ϑ_{\vec{J}})}^{κ} + (ς - 1) {(1 - ϑ_{\vec{J}})}^{κ}}, \\ \frac{{(1 + (ς - 1) w_{ϑ_{\vec{J}}})}^{κ} - {(1 - w_{ϑ_{\vec{J}}})}^{κ}}{{(1 + (ς - 1) w_{ϑ_{\vec{J}}})}^{κ} + (ς - 1) {(1 - w_{ϑ_{\vec{J}}})}^{κ}}) 〉 \end{cases}}_{κ > 0}

Definition 3.2. In relation to the collective $\vec{CLDFSN} s$ ${\vec{J}}_{pq} = (〈 (Ω_{{\vec{J}}_{pq}}, w_{Ω_{{\vec{J}}_{pq}}}), (ϒ_{{\vec{J}}_{pq}}, w_{ϒ_{{\vec{J}}_{pq}}}) 〉$ , $〈 (ϱ_{{\vec{J}}_{pq}}, w_{ϱ_{{\vec{J}}_{pq}}}), (ϑ_{{\vec{J}}_{pq}}, w_{ϑ_{{\vec{J}}_{pq}}}) 〉)$ , p=1,2,...,n, q=1,2,...,m, the $\vec{CLDFSHWA}$ operator is characterized and designated as $\vec{CLDFSHWA} ({\vec{J}}_{11}, {\vec{J}}_{12}, . . ., {\vec{J}}_{nm})$ = ${oplus}_{q = 1}^{m} ξ_{q} ({oplus}_{p = 1}^{n} δ_{p} {\vec{J}}_{pq})$

where ξ, δ > 0 is a weight vector, $\sum_{q = 1}^{m} ξ_{q} = 1$ and $\sum_{p = 1}^{n} δ_{p} = 1$ .

Theorem 3.3. The aggregated value to the collective $\vec{CLDFSN} s$ obtained by using the $\vec{CLDFSHWA}$ operator remains a $\vec{CLDFSN}$ , which is designated as

$\vec{CLDFSHWA} ({\vec{J}}_{11}, {\vec{J}}_{12}, . . ., {\vec{J}}_{nm})$ = $〈 (\frac{\prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 + (ς - 1) Ω_{{\vec{J}}_{pq}})^{δ_{p}})^{ξ_{q}} - \prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 - Ω_{{\vec{J}}_{pq}})^{δ_{p}})^{ξ_{q}}}{\prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 + (ς - 1) Ω_{{\vec{J}}_{pq}})^{δ_{p}})^{ξ_{q}} + (ς - 1) \prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 - Ω_{{\vec{J}}_{pq}})^{δ_{p}})^{ξ_{q}}}, \frac{\prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 + (ς - 1) w_{Ω_{{\vec{J}}_{pq}}})^{δ_{p}})^{ξ_{q}} - \prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 - w_{Ω_{{\vec{J}}_{pq}}})^{δ_{p}})^{ξ_{q}}}{\prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 + (ς - 1) w_{Ω_{{\vec{J}}_{pq}}})^{δ_{p}})^{ξ_{q}} + (ς - 1) \prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 - w_{Ω_{{\vec{J}}_{pq}}})^{δ_{p}})^{ξ_{q}}}), (\frac{ς \prod_{q = 1}^{m} (\prod_{p = 1}^{n} (ϒ_{{\vec{J}}_{pq}})^{δ_{p}})^{ξ_{q}}}{\prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 + (ς - 1) (1 - ϒ_{{\vec{J}}_{pq}}))^{δ_{p}})^{ξ_{q}} + (ς - 1) \prod_{q = 1}^{m} (\prod_{p = 1}^{n} (ϒ_{{\vec{J}}_{pq}})^{δ_{p}})^{ξ_{q}}}, \frac{ς \prod_{q = 1}^{m} (\prod_{p = 1}^{n} (w_{ϒ_{{\vec{J}}_{pq}}})^{δ_{p}})^{ξ_{q}}}{\prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 + (ς - 1) (1 - w_{ϒ_{{\vec{J}}_{pq}}}))^{δ_{p}})^{ξ_{q}} + (ς - 1) \prod_{q = 1}^{m} (\prod_{p = 1}^{n} (w_{ϒ_{{\vec{J}}_{pq}}})^{δ_{p}})^{ξ_{q}}}) 〉, 〈 (\frac{\prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 + (ς - 1) ϱ_{{\vec{J}}_{pq}})^{δ_{p}})^{ξ_{q}} - \prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 - ϱ_{{\vec{J}}_{pq}})^{δ_{p}})^{ξ_{q}}}{\prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 + (ς - 1) ϱ_{{\vec{J}}_{pq}})^{δ_{p}})^{ξ_{q}} + (ς - 1) \prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 - ϱ_{{\vec{J}}_{pq}})^{δ_{p}})^{ξ_{q}}}, \frac{\prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 + (ς - 1) w_{ϱ_{{\vec{J}}_{pq}}})^{δ_{p}})^{ξ_{q}} - \prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 - w_{ϱ_{{\vec{J}}_{pq}}})^{δ_{p}})^{ξ_{q}}}{\prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 + (ς - 1) w_{ϱ_{{\vec{J}}_{pq}}})^{δ_{p}})^{ξ_{q}} + (ς - 1) \prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 - w_{ϱ_{{\vec{J}}_{pq}}})^{δ_{p}})^{ξ_{q}}}), (\frac{ς \prod_{q = 1}^{m} (\prod_{p = 1}^{n} (ϑ_{{\vec{J}}_{pq}})^{δ_{p}})^{ξ_{q}}}{\prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 + (ς - 1) (1 - ϑ_{{\vec{J}}_{pq}}))^{δ_{p}})^{ξ_{q}} + (ς - 1) \prod_{q = 1}^{m} (\prod_{p = 1}^{n} (ϑ_{{\vec{J}}_{pq}})^{δ_{p}})^{ξ_{q}}}, \frac{ς \prod_{q = 1}^{m} (\prod_{p = 1}^{n} (w_{ϑ_{{\vec{J}}_{pq}}})^{δ_{p}})^{ξ_{q}}}{\prod_{q = 1}^{m} (\prod_{p = 1}^{n} (1 + (ς - 1) (1 - w_{ϑ_{{\vec{J}}_{pq}}}))^{δ_{p}})^{ξ_{q}} + (ς - 1) \prod_{q = 1}^{m} (\prod_{p = 1}^{n} (w_{ϑ_{{\vec{J}}_{pq}}})^{δ_{p}})^{ξ_{q}}}) 〉$

——–(1) where ξ, δ > 0 is a weight vector, $\sum_{q = 1}^{m} ξ_{q} = 1$ and $\sum_{p = 1}^{n} δ_{p} = 1$ .

The $\vec{CLDFSHWA}$ operator possesses the following properties, which are easily demonstrable.

Theorem 3.4. (Idempotency) If the entire ${\vec{J}}_{pq}$ is same, (i.e)., for every p,q ${\vec{J}}_{pq}$ = $\vec{J}$ , then $\vec{CLDFSHWA} ({\vec{J}}_{11}, {\vec{J}}_{12}, . . ., {\vec{J}}_{nm})$ = $\vec{J}$

Theorem 3.5. (Monotonicity) Let ${\vec{J}}_{pq}$ and ${\vec{J^{'}}}_{pq}$ be two $\vec{CLDFSN}$ , if ${\vec{J}}_{pq}$ ≤ ${\vec{J^{'}}}_{pq}$ for all values of p,q, then $\vec{CLDFSHWA} ({\vec{J}}_{11}, {\vec{J}}_{12}, . . ., {\vec{J}}_{nm})$ ≤

$\vec{CLDFSHWA} ({\vec{J^{'}}}_{11}, {\vec{J^{'}}}_{12}, . . ., {\vec{J^{'}}}_{nm})$

Theorem 3.6. (Boundedness) Let ${\vec{J}}_{pq}$ be the collection of $\vec{CLDFSN}$ and let

${\vec{J}}^{-} = (〈 (_{q}^{m i n}_{p}^{m i n} {Ω_{{\vec{J}}_{p q}}},_{q}^{m i n}_{p}^{m i n} {w_{Ω_{{\vec{J}}_{p q}}}}), (_{q}^{m a x}_{p}^{m a x} {Υ_{{\vec{J}}_{p q}}},_{q}^{m a x}_{p}^{m a x} {w_{Υ_{{\vec{J}}_{p q}}}}) 〉, 〈 (_{q}^{m i n}_{p}^{m i n} {ϱ_{{\vec{J}}_{p q}}},_{q}^{m i n}_{p}^{m i n} {w_{ϱ_{{\vec{J}}_{p q}}}}), (_{q}^{m a x}_{p}^{m a x} {ϑ_{{\vec{J}}_{p q}}},_{q}^{m a x}_{p}^{m a x} {w_{ϑ_{{\vec{J}}_{p q}}}}) 〉),$

${\vec{J}}^{+} = (〈 (_{q}^{m a x}_{p}^{m a x} {Ω_{{\vec{J}}_{p q}}},_{q}^{m a x}_{p}^{m a x} {w_{Ω_{{\vec{J}}_{p q}}}}), (_{q}^{m i n}_{p}^{m i n} {Υ_{{\vec{J}}_{p q}}},_{q}^{m i n}_{p}^{m i n} {w_{Υ_{{\vec{J}}_{p q}}}}) 〉, 〈 (_{q}^{m a x}_{p}^{m a x} {ϱ_{{\vec{J}}_{p q}}},_{q}^{m a x}_{p}^{m a x} {w_{ϱ_{{\vec{J}}_{p q}}}}), (_{q}^{m i n}_{p}^{m i n} {ϑ_{{\vec{J}}_{p q}}},_{q}^{m i n}_{p}^{m i n} {w_{ϑ_{{\vec{J}}_{p q}}}}) 〉)$ ,

then ${\vec{J}}^{-}$ ≤ $\vec{CLDFSHWA} ({\vec{J}}_{11}, {\vec{J}}_{12}, . . ., {\vec{J}}_{nm})$ ≤ ${\vec{J}}^{+}$

4 Conceptual framework

This segment presents a conceptual framework for locating the shortest path $(\vec{SP})$ connecting each vertex of the weighted graph. There are two primary components to this algorithm. The first step is to model the problem as a graph and weigh the edges using $\vec{CLDF}$ values from $\vec{CLDFS}$ values by employing Hamacher aggregation operator. The second step involves using a fresh iteration of Kruskal’s algorithm to solve a $\vec{CLDF}$ $\vec{MST}$ . The intriguing part is that visualizing a problem as a graph and determining the weights of alternatives(the graph’s edges) employing a $\vec{CLDF}$ via the Hamacher aggregation operator and additionally, utilizing Kruskal’s technique in conjunction with this approach to discover a $\vec{MST}$ as a resolution. Figure 1 depicts an in-depth flowchart of this framework that includes the basic steps of an algorithm.

Fig. 1

Flowchart of the Conceptual framework.

Below are the steps of an $\vec{CLDFSS}$ MCDM method that uses the Hamacher aggregation operator to determine the weights of alternatives.

$\vec{STEP}$ 1: Define the problem.

$\vec{STEP}$ 2: Decide on the experts in accordance with the problem, and let $\vec{B}$ = {b ₁, b ₂, . . . , b _z} be the collective group of experts.

$\vec{STEP}$ 3: Create an unweighted graph that depict the problem. Let $\vec{G}$ = $(\vec{V (G)}, \vec{E (G)})$ be an undirected graph in which vertices represents the points and edges represents the alternatives. Let $\vec{ℶ} = {s_{1}, s_{2}, . . ., s_{n}}$ be the collective group of alternatives.

$\vec{STEP}$ 4: Examine the attribute and attribute weights in conjunction with the experts. Let $\vec{R}$ signifies the collective attribute set and ξ signifies the collective weight vector of the attribute set with ξ _q ∈ [0, 1] and $\sum_{q = 1}^{m} ξ_{q} = 1$

$\vec{STEP}$ 5: Normalize the information set of the experts to remove the influence of various decision values (benefits and costs). The information set does not need to be normalized if the attributes are all of the same type. Suppose not, then the information set needs to be normalized. (i.e.),There are no modifications in the information set if the attribute is of the benefit type, but if it is of the cost type, the $\vec{MS}$ and $\vec{NMS}$ switch positions.

$\vec{STEP}$ 6: Aggregate all the $\vec{CLDFS}$ values by using eqn(1).

$\vec{STEP}$ 7: Analyze on the $\vec{CLDF}$ weights of alternatives(edges), which is obtained in $\vec{STEP}$ 6 by using the $\vec{CLDFSHWA}$ operator.

The $\vec{CLDF}$ adaptation of Kruskal’s algorithm is preferred for processing $\vec{MST}$ $\vec{CLDF}$ environment. This algorithm constructs a tree in order to determine the $\vec{SP}$ between each graph vertex. Table 2 displays the pseudocode of $\vec{CLDF}$ kruskal’s algorithm. The $\vec{CLDF}$ Kruskal’s algorithm has the following steps:

$\vec{STEP}$ 8: Transform the unweighted graph from $\vec{STEP}$ 3 into the $\vec{CLDF}$ weighted graph with the help of values derived in $\vec{STEP}$ 6.

$\vec{STEP}$ 9: Compute the Score function for all $\vec{CLDFN}$ with the aid of Definition 2.4.

$\vec{STEP}$ 10: Schedule each edges(alternatives) in an ascending sequence of scores.

$\vec{STEP}$ 11: Begin with the lowest score first and start developing the $\vec{MST}$ .

$\vec{STEP}$ 12: Gravitate towards the subsequent lowest-scoring

(unused) edge that fails to generate a cycle to the formation of $\vec{MST}$ . If the insertion of the subsequent lowest-scoring (unused) edge fails to generate a cycle, then include it in the $\vec{MST}$ . If not, don’t insert it.

$\vec{STEP}$ 13: Proceed with $\vec{STEP}$ 12 until the $\vec{MST}$ contains $(\vec{V (G)} - 1)$ edges.

Table 2

Pseudocode algorithm

ALGORITHM:	$\vec{MST}$ Kruskal"s algorithm
INPUT:	A graph(undirected) $\vec{G}$ = $(\vec{V}, \vec{E (G)})$ , collection of $\vec{CLDF}$ edge weights.
OUTPUT:	A tree $\vec{T}$ = $(\vec{V}, \vec{E (T)})$ .
1:	Compute the score of each edge
2:	Organize the edges of $\vec{E (G)}$ in ascending sequence by scores
3:	for each edge e _z ∈ $\vec{E (G)}$ taken in ascending sequence of scores
4:	if e _z fails to generate a circuit with the edges in $\vec{E (T)}$
5:	set $\vec{E (T)}$ ← $\vec{E (T)}$ ∪ {e _z}
6:	return $\vec{E (T)}$

5 Integration of application

This segment is intended to employ the suggested $\vec{CLDF}$ MCDM combining Kruskal’s algorithm and the Hamacher aggregation operator through an actual case study. In accordance with the suggested algorithm described in the preceding segment, there are two primary components to this algorithm. The first step is to model the problem as a graph and weigh the edges using $\vec{CLDF}$ values from $\vec{CLDFS}$ values by employing Hamacher aggregation operator. The second step involves using a fresh iteration of Kruskal’s algorithm to solve a $\vec{CLDF}$ $\vec{MST}$ .

5.1 Case study

The Case Study deals with the implementation of $\vec{CLDF}$ MCDM with the $\vec{CLDFSHWA}$ operator and kruskal’s algorithm for marketing a web application(created via cloud service) in a future smart world using LiFi technology. The problem is solved sequentially according to the suggested approach described in the previous section.

5.1.1 Cloud computing architecture

A service that allows users to access computing resources(servers, software, networking, databases, storage, analytics, and intelligence) over the internet is called Cloud Computing. It offers quicker innovation, adaptable resources, and economies of scale.

Cloud Computing uses two different types of models such as

1.
Deployment Models
2.
Service Models
Cloud Deployment Model: Cloud Computing operates as a cloud-based computing system with an offer to select a deployment model. It mainly addresses what are the different ways that a person can deploy a rented server on the internet based on how much data needs to be stored and who has access to the infrastructure.

There are three kinds of deployment models on the cloud. 1.
Public Cloud: Public Cloud is characterized as computing services that are made available to each individual who desires to access or buy them through the open Internet by third-party suppliers. Free or on-demand sales options are available, allowing users to pay only for the CPU cycles, storage, or bandwidth they use.
2.
Private Cloud: A private cloud is a cloud computing system that is confined to a single entity. A self-service portal allows you to provision underlying compute resources like storage and CPU on demand. These resources are present in any cloud infrastructure. In a private cloud, all resources are segregated under the management of a single company.
3.
Hybrid Cloud: A hybrid cloud is a mixed computing environment in which the applications are executed utilizing an amalgam of computing, storage, and services from both private and public clouds.
Cloud Service Model: Service Model deals with what kind of service a cloud provider offers. There are three different kinds of services which are offered by cloud platforms. 1.
Infrastructure as a Service(IaaS) means granting users full access to the server’s operating system. It offers network components, hardware, storage, servers, and space for data centers. it might also contain software.
2.
In Platform as a Service(PaaS), one is not given access to everything. Instead, access is granted to the dashboard level, where users upload data and the cloud provider processes the rest.
3.
Software as a Service(SaaS) is a software distribution paradigm in which a cloud provider hosts applications and makes them online accessible to users.

5.1.2 LiFi: A route to a brand-new form of communication

LiFi(light fidelity) is a bidirectional wireless system that transfers data using infrared or LED light. It initially emerged in 2011. As compared to wifi, which uses radio frequency, LiFi technology simply requires a light source with a chip to broadcast an internet signal by light waves.

Benefits of the LiFi technology:

1. High data speed. 2. High level of security.

3. Low power. 4. Conserving RF spectrum.

5. Less hazardous to humans. 6. Electricity saving.

Lifi Revolution in a Future Smart World

This segment gains insight into the potential of LiFi technology in numerous fields. Here are some of the fields that will benefit from LiFi technology in the smart world of the future.

1.
Offices - Fast, secure, and reliable connectivity for modern offices with LiFi technology: LiFi technology provides quick, safe, and stable connectivity for modern offices. There is a need for dependable connectivity in the continuously changing modern business environment. LiFi technology provides steady, quick, and secure network access, making it the perfect choice for companies looking for a dependable network that won’t disappoint them.
2.
Education - Empowering education with faster, more secure connectivity: High-speed and secure wireless networks are more important as the educational sector develops rapidly. However, constraints and issues about data privacy and security are becoming more prevalent for conventional wireless networks like WiFi, 4G, and 5G. LiFi, a technology that employs light to carry data, presents a workable alternative for schools that need high-speed, safe, and versatile wireless connectivity. LiFi is a more secure solution for schools that value data privacy because it only functions within line-of-sight of the light source, unlike conventional wireless networks. Additionally, LiFi allows quick and dependable connectivity, enabling students and teachers to work without any delays with a variety of online applications.
3.
Government and Defence - Revolutionize data transfer and security in the public sector and Military with LiFi technology: Real-time data sharing and communication are essential for success in government and defense organizations. However, due to interference with sensitive equipment or explosive items in particular cyberspaces, radio-based technologies like Wi-Fi and 4G/5G are restricted. Here is where LiFi technology is useful. This enables quick field data link creation in humanitarian help and disaster scenarios where there is a paucity of infrastructure and acquiring permission for RF spectrum usage will require some time. LiFi gives an additional level of physical security. LiFi is a light-based internet connection, therefore anything that tries to intercept, jam, or track it from outside the infrared cone would fail. This makes it the perfect option for command posts where military personnel can access dedicated USB keys to stay online safely and move around effortlessly between LiFi coverage regions.
4.
Industry and Transportation - Optimize the transportation and industrial processes with LiFi technology: Reliable and secure wireless communication is essential in today’s fast-paced industrial and transportation industries. The restrictions and potential security threats that traditional wireless networks like WiFi and Bluetooth frequently experience make them a lesser choice for demanding industrial situations. LiFi can provide real-time machine-to-machine connectivity in industrial environments, ensuring precise and effective data flow. In manufacturing operations. This can be especially crucial, where prompt communication may boost productivity. In order to save energy and clutter, LiFi can also provide a solution for industrial lighting systems by fusing the functionality of illumination and data connection. LiFi can offer dependable communication systems for automobiles and transportation infrastructure in the transportation sector. LiFi can facilitate quick and secure data transfer between automobiles, benefiting both road safety and traffic flow. LiFi also offers fast data transfer rates and improved security features, making it a viable choice for in-flight communication systems in the aviation sector.
5.
Healthcare - Safe and dependable healthcare communication with LiFi technology: LiFi’s quick data transmission rates and minimal latency make it possible for healthcare workers to communicate in real time. When making snap decisions in an emergency, this can be extremely important. LiFi reduces energy use and physical clutter by merging the functions of illumination and data transmission without the need for separate systems.
6.
Aerospace - Revolutionize aircraft communication with High-speed, secure LiFi technology: The potential of LiFi to permit real-time communication between ground control and spacecraft is a benefit in the aerospace industry. LiFi can offer dependable communication systems that allow pilots and operators to receive and distribute crucial data in real time with its high data transfer speeds and minimal latency. This is crucial in circumstances where speedy decision-making is necessary.
A Sequential Algorithmic Procedure

$\vec{STEP}$ 1: All data pertaining to the environment, objects, and people is connected to networks in the modern era of the Internet of Things(IoT). Future IoT systems will require more dynamicity, heterogeneity, and scalability due to the increasing human-centricity of IoT applications. The IoT and cloud computing are both developing technologies that have already been incorporated into our daily lives. To satisfy present and future needs, a new IT paradigm is being formed by the complimentary elements of the IoT and cloud. Parallelly hacking also takes place in a wide range and results in various problems, including a disruption in customer service, a loss of financial and personal data, lost revenue, a loss of reputation, and even the closure of a corporation.

Hacking cannot permeate daily life in the modern era of LiFi technology, since LiFi employs the infrared and visible light spectrum as its connecting medium rather than radio frequencies, it is currently one of the safest technologies for data transmission. Any waves that cannot pass through physical obstructions render the network invisible to everyone outside the LED lamp’s light beam.

In the smart future of LiFi technology, let’s say suppose a person develops a web application using cloud computing and uploads it to the Internet of LiFi. Several distinct strategies can be employed to promote this web application and increase users. The person seeks to connect with people in order to market the web application in the quickest way possible. The shortest linked path in a testing group of six users will be determined.

$\vec{STEP}$ 2: Through the process of sector study, the problem’s decision-makers are narrowed down to three experts and let $\vec{B}$ = {b ₁, b ₂, b ₃} be the collective group of experts with weight vector {0.3, 0.4, 0.3}.

$\vec{STEP}$ 3: The problem is depicted by an unweighted, undirected graph $\vec{G}$ = $(\vec{V (G)}, \vec{E (G)})$ in which vertices represent the users and the links between users are set as edges. Let $\vec{ℶ} = {s_{1}, s_{2}, . . ., s_{8}}$ be the collective group of alternatives.

$\vec{STEP}$ 4: Let $\vec{R} = {r_{1}, r_{2}, r_{3}}$ signifies the collective attribute set as r ₁: Social reinforcement(negative feedback), r ₂: Human heterogeneity, and r ₃: Attentional decline with the weight vectors {0.2, 0.3, 0.5}.

The characteristics of $\vec{AS}$ are outlined below: 1.
Social reinforcement(negative feedback) influences an individual’s adoption of new knowledge to some degree whenever they are presented with it. Social reinforcement, or negative feedback, causes people to become less attentive to information after they have been exposed to it for a while, which decreases the amount of information that spreads.
2.
Human heterogeneity may have an impact on the dissemination of information. Furthermore, the level of acceptance of the knowledge may differ from person to person during the initial interaction. Thus, human variation may potentially have a significant impact on the dissemination of information.
3.
The Attentional decline of a person undoubtedly influences the entire spreading actions. As a point of fact, the simultaneous dissemination of multiple forms of information is actually a highly intricate process. Thus, Human heterogeneity, Social reinforcement(negative feedback), and Attentional decline may potentially have a significant impact on the spreading of information.
Since the scores are complex-valued, the $\vec{AT}$ assesses the truthfulness and falsehood scores and $\vec{PT}$ assesses the truthfulness and falsehood scores based on the span of time between two users since they first encounter

The attributes are laid out in the following manner. 1.
The attribute "Social reinforcement(negative feedback)" renders it apparent the alternative is "more" or "less".
2.
The attribute "Human heterogeneity" renders it apparent the alternative is "high" or "low".
3.
The attribute "Attentional decline" renders it apparent the alternative is "more" or "less".
The following is an exemplified description.

Attributes The specifications for $\vec{CLDFSS}$

"Social reinforcement (negative feedback)" $(〈 \vec{CVMS}, \vec{CVNMS} 〉, 〈 more, less 〉)$

"Human heterogeneity" $(〈 \vec{CVMS}, \vec{CVNMS} 〉, 〈 high, low 〉)$

"Attentional decline" $(〈 \vec{CVMS}, \vec{CVNMS} 〉, 〈 more, less 〉)$

As an illustration, Expert b ₁ estimates that the characteristic "Human heterogeneity" has a score

(〈(0.7, 0.9) , (0.3, 0.1) 〉, 〈(0.8, 0.8) , (0.2, 0.2) 〉). According to this score, s ₁ has a truthfulness score of 0.7, a falsehood score of 0.3, and a truthfulness score of 0.9 based on the span of time between two users since they first encounter, a falsehood score of 0.1 based on the span of time between two users since they first encounter. The pair 〈(0.8, 0.8) , (0.2, 0.2) 〉 is interpreted as a truthfulness and falsehood score for the reference parameter, enabling to comprehend that s ₁ should be 0.8 score high, 0.2 score low, and 0.8 score high based on the span of time between two users since they first encounter, and 0.2 score low based on the span of time between two users since they first encounter. In a similar way, all other data are discussed.

$\vec{STEP}$ 5: There are no modifications in the information set since all the attributes are of the benefit type. The values of three decision makers based on the attributes are given in Tables 3–5.

Table 3
Values of the decision maker b ₁

$\vec{ℶ}$ r ₁ r ₂ r ₃

s ₁ (〈(0.6, 0.7) , (0.4, 0.3) 〉, (〈(0.7, 0.9) , (0.3, 0.1) 〉, (〈(0.6, 0.8) , (0.4, 0.2) 〉

〈(0.7, 0.7) , (0.3, 0.3) 〉) (〈(0.8, 0.8) , (0.2, 0.2) 〉 (〈(0.7, 0.7) , (0.3, 0.3) 〉

s ₂ (〈(0.7, 0.7) , (0.3, 0.2) 〉, (〈(0.7, 0.6) , (0.3, 0.4) 〉, (〈(0.7, 0.9) , (0.3, 0.1) 〉

〈(0.7, 0.8) , (0.2, 0.2) 〉) (〈(0.6, 0.7) , (0.4, 0.3) 〉 (〈(0.8, 0.9) , (0.1, 0.1) 〉

s ₃ (〈(0.5, 0.6) , (0.4, 0.4) 〉, (〈(0.8, 0.7) , (0.2, 0.3) 〉, (〈(0.7, 0.8) , (0.3, 0.2) 〉

〈(0.6, 0.6) , (0.4, 0.4) 〉) (〈(0.8, 0.8) , (0.2, 0.2) 〉 (〈(0.7, 0.8) , (0.2, 0.2) 〉

s ₄ (〈(0.5, 0.7) , (0.4, 0.3) 〉, (〈(0.7, 0.7) , (0.2, 0.2) 〉, (〈(0.5, 0.6) , (0.5, 0.4) 〉

〈(0.6, 0.6) , (0.3, 0.3) 〉) (〈(0.7, 0.8) , (0.2, 0.3) 〉 (〈(0.6, 0.7) , (0.4, 0.3) 〉

s ₅ (〈(0.9, 0.8) , (0.1, 0.2) 〉, (〈(0.9, 0.9) , (0.1, 0.1) 〉, (〈(0.7, 0.6) , (0.3, 0.4) 〉

〈(0.9, 0.8) , (0.1, 0.2) 〉) (〈(0.8, 0.9) , (0.1, 0.1) 〉 (〈(0.7, 0.7) , (0.3, 0.3) 〉

s ₆ (〈(0.8, 0.7) , (0.2, 0.3) 〉, (〈(0.8, 0.6) , (0.2, 0.3) 〉, (〈(0.9, 0.7) , (0.1, 0.3) 〉

〈(0.7, 0.7) , (0.2, 0.2) 〉) (〈(0.8, 0.8) , (0.2, 0.2) 〉 (〈(0.7, 0.8) , (0.3, 0.2) 〉

s ₇ (〈(0.8, 0.9) , (0.2, 0.1) 〉, (〈(0.7, 0.8) , (0.3, 0.1) 〉, (〈(0.8, 0.6) , (0.2, 0.4) 〉

〈(0.7, 0.8) , (0.3, 0.2) 〉) (〈(0.8, 0.8) , (0.2, 0.2) 〉 (〈(0.6, 0.7) , (0.4, 0.3) 〉

s ₈ (〈(0.6, 0.6) , (0.4, 0.3) 〉, (〈(0.7, 0.6) , (0.3, 0.4) 〉, (〈(0.6, 0.5) , (0.4, 0.5) 〉

〈(0.6, 0.7) , (0.3, 0.3) 〉) (〈(0.7, 0.8) , (0.3, 0.2) 〉 (〈(0.7, 0.7) , (0.3, 0.3) 〉

Table 4
Values of the decision maker b ₂

$\vec{ℶ}$ r ₁ r ₂ r ₃

s ₁ (〈(0.7, 0.8) , (0.3, 0.2) 〉, (〈(0.8, 0.9) , (0.2, 0.1) 〉, (〈(0.7, 0.8) , (0.3, 0.2) 〉

〈(0.8, 0.8) , (0.2, 0.2) 〉) (〈(0.8, 0.8) , (0.2, 0.1) 〉 (〈(0.8, 0.7) , (0.2, 0.3) 〉

s ₂ (〈(0.8, 0.8) , (0.2, 0.1) 〉, (〈(0.8, 0.7) , (0.2, 0.3) 〉, (〈(0.8, 0.8) , (0.2, 0.2) 〉

〈(0.8, 0.9) , (0.1, 0.1) 〉) (〈(0.7, 0.8) , (0.3, 0.2) 〉 (〈(0.7, 0.8) , (0.3, 0.2) 〉

s ₃ (〈(0.6, 0.7) , (0.4, 0.3) 〉, (〈(0.7, 0.8) , (0.3, 0.2) 〉, (〈(0.8, 0.8) , (0.2, 0.2) 〉

〈(0.7, 0.7) , (0.3, 0.3) 〉) (〈(0.7, 0.7) , (0.2, 0.3) 〉 (〈(0.8, 0.9) , (0.1, 0.1) 〉

s ₄ (〈(0.6, 0.8) , (0.3, 0.2) 〉, (〈(0.8, 0.8) , (0.2, 0.2) 〉, (〈(0.6, 0.7) , (0.4, 0.3) 〉

〈(0.7, 0.7) , (0.2, 0.2) 〉) (〈(0.8, 0.9) , (0.1, 0.1) 〉 (〈(0.7, 0.8) , (0.3, 0.2) 〉

s ₅ (〈(0.9, 0.9) , (0.1, 0.1) 〉, (〈(0.9, 0.8) , (0.1, 0.2) 〉, (〈(0.8, 0.7) , (0.2, 0.3) 〉

〈(0.9, 0.8) , (0.1, 0.2) 〉) (〈(0.8, 0.9) , (0.1, 0.1) 〉 (〈(0.8, 0.8) , (0.2, 0.2) 〉

s ₆ (〈(0.9, 0.8) , (0.1, 0.2) 〉, (〈(0.9, 0.7) , (0.1, 0.2) 〉, (〈(0.9, 0.7) , (0.1, 0.3) 〉

〈(0.8, 0.8) , (0.2, 0.2) 〉) (〈(0.9, 0.9) , (0.1, 0.1) 〉 (〈(0.8, 0.7) , (0.2, 0.3) 〉

s ₇ (〈(0.9, 0.8) , (0.1, 0.2) 〉, (〈(0.8, 0.9) , (0.2, 0.1) 〉, (〈(0.9, 0.7) , (0.1, 0.2) 〉

〈(0.7, 0.8) , (0.3, 0.2) 〉) (〈(0.8, 0.8) , (0.2, 0.2) 〉 (〈(0.7, 0.8) , (0.3, 0.2) 〉

s ₈ (〈(0.7, 0.7) , (0.3, 0.3) 〉, (〈(0.8, 0.6) , (0.2, 0.4) 〉, (〈(0.7, 0.6) , (0.3, 0.4) 〉

〈(0.7, 0.8) , (0.3, 0.2) 〉) (〈(0.6, 0.7) , (0.4, 0.3) 〉 (〈(0.8, 0.6) , (0.2, 0.4) 〉

Table 5
Values of the decision maker b ₃

$\vec{ℶ}$ r ₁ r ₂ r ₃

s ₁ (〈(0.5, 0.6) , (0.4, 0.4) 〉, (〈(0.6, 0.6) , (0.4, 0.4) 〉, (〈(0.6, 0.8) , (0.4, 0.2) 〉

〈(0.6, 0.7) , (0.4, 0.3) 〉) (〈(0.7, 0.7) , (0.3, 0.3) 〉 (〈(0.8, 0.7) , (0.2, 0.3) 〉

s ₂ (〈(0.9, 0.9) , (0.1, 0.1) 〉, (〈(0.8, 0.9) , (0.2, 0.1) 〉, (〈(0.9, 0.7) , (0.1, 0.2) 〉

〈(0.8, 0.9) , (0.2, 0.1) 〉) (〈(0.8, 0.8) , (0.2, 0.2) 〉 (〈(0.9, 0.9) , (0.1, 0.1) 〉

s ₃ (〈(0.7, 0.9) , (0.3, 0.1) 〉, (〈(0.8, 0.7) , (0.2, 0.3) 〉, (〈(0.8, 0.6) , (0.2, 0.4) 〉

〈(0.8, 0.8) , (0.2, 0.2) 〉) (〈(0.8, 0.9) , (0.1, 0.1) 〉 (〈(0.6, 0.7) , (0.4, 0.3) 〉

s ₄ (〈(0.6, 0.7) , (0.4, 0.3) 〉, (〈(0.7, 0.7) , (0.3, 0.2) 〉, (〈(0.7, 0.8) , (0.3, 0.2) 〉

〈(0.7, 0.8) , (0.3, 0.2) 〉) (〈(0.8, 0.8) , (0.2, 0.2) 〉 (〈(0.7, 0.7) , (0.3, 0.3) 〉

s ₅ (〈(0.8, 0.8) , (0.2, 0.2) 〉, (〈(0.9, 0.8) , (0.1, 0.2) 〉, (〈(0.8, 0.7) , (0.2, 0.3) 〉

〈(0.9, 0.8) , (0.1, 0.2) 〉) (〈(0.7, 0.8) , (0.3, 0.2) 〉 (〈(0.7, 0.8) , (0.3, 0.2) 〉

s ₆ (〈(0.8, 0.9) , (0.2, 0.1) 〉, (〈(0.7, 0.8) , (0.3, 0.2) 〉, (〈(0.8, 0.7) , (0.2, 0.3) 〉

〈(0.9, 0.8) , (0.1, 0.2) 〉) (〈(0.6, 0.6) , (0.4, 0.3) 〉 (〈(0.7, 0.8) , (0.3, 0.2) 〉

s ₇ (〈(0.8, 0.7) , (0.2, 0.3) 〉, (〈(0.6, 0.8) , (0.4, 0.2) 〉, (〈(0.6, 0.8) , (0.4, 0.2) 〉

〈(0.9, 0.7) , (0.1, 0.3) 〉) (〈(0.7, 0.7) , (0.3, 0.3) 〉 (〈(0.8, 0.9) , (0.2, 0.1) 〉

s ₈ (〈(0.7, 0.6) , (0.3, 0.4) 〉, (〈(0.7, 0.9) , (0.3, 0.1) 〉, (〈(0.7, 0.9) , (0.3, 0.1) 〉

〈(0.8, 0.7) , (0.2, 0.3) 〉) (〈(0.9, 0.9) , (0.1, 0.1) 〉 (〈(0.8, 0.8) , (0.2, 0.2) 〉

$\vec{STEP}$ 6: Aggregate all the $\vec{CLDFS}$ values of three experts based on three attributes by using eqn(1) with ς = 2. The aggregated table is given in Table 6

Table 6
Aggregated table

$\vec{ℶ}$ Aggregated Values

s ₁ (〈(0.6627, 0.8013) , (0.3325, 0.1987) 〉,

〈(0.7636, 0.7326) , (0.2363, 0.2473) 〉)

s ₂ (〈(0.8033, 0.7997) , (0.1967, 0.1737) 〉,

〈(0.7740, 0.8461) , (0.1889, 0.1539) 〉)

s ₃ (〈(0.7429, 0.7512) , (0.2533, 0.2488) 〉,

〈(0.7318, 0.8011) , (0.1983, 0.1989) 〉)

s ₄ (〈(0.6464, 0.7269) , (0.3291, 0.2540) 〉,

〈(0.7061, 0.7744) , (0.2496, 0.2224) 〉)

s ₅ (〈(0.8421, 0.7700) , (0.1579, 0.2299) 〉,

〈(0.7944, 0.8152) , (0.1784, 0.1848) 〉)

s ₆ (〈(0.8573, 0.7293) , (0.1427, 0.2510) 〉,

〈(0.7803, 0.7806) , (0.2143, 0.2082) 〉)

s ₇ (〈(0.7951, 0.7798) , (0.2049, 0.1908) 〉,

〈(0.7446, 0.7952) , (0.2554, 0.2048) 〉)

s ₈ (〈(0.6962, 0.7026) , (0.3045, 0.2921) 〉,

〈(0.7554, 0.7447) , (0.2402, 0.2553) 〉)

$\vec{STEP}$ 7: Analyze on the $\vec{CLDF}$ weights of alternatives(edges), which is obtained in $\vec{STEP}$ 6 by using the $\vec{CLDFSHWA}$ operator. $\vec{STEP}$ 8: The unweighted graph is transformed into the $\vec{CLDF}$ weighted graph with the help of values derived in $\vec{STEP}$ 6 and it is given in Fig. 2.

Fig. 2
$\vec{CLDF}$ weighted graph.

$\vec{STEP}$ 9: The Score function for all $\vec{CLDFN}$ with the aid of Definition 2.4. is given in Table 7.

Table 7
Aggregated table

Edges( $\vec{ℶ}$ ) Score Values

s ₁ 0.4864

s ₂ 0.6275

s ₃ 0.5319

s ₄ 0.4497

s ₅ 0.6176

s ₆ 0.5828

s ₇ 0.5647

s ₈ 0.4517

$\vec{STEP}$ 10: The edges(alternatives) in an ascending sequence of scores is scheduled as s ₄ < s ₈ < s ₁ < s ₃ < s ₇ < s ₆ < s ₅ < s ₂.

$\vec{STEP}$ 11: Edge s ₄ has the lowest scoring. The $\vec{MST}$ $\vec{T}$ = $(\vec{V (T)}, \vec{E (T)})$ therefore starts to develop from this edge as seen in Fig. 3.

Fig. 3
$\vec{MST}$ starts to develop from s ₄.

$\vec{STEP}$ 12: The unused edge with the next-lowest score value is s ₈. The edge s ₈ does not generate a cycle. so it is included to $\vec{ST}$ as shown in Fig. 4.

Fig. 4
$\vec{MST}$ develops with s ₈.

$\vec{STEP}$ 12: The process continues with $\vec{STEP}$ 12 until the $\vec{ST}$ has five edges.

$\vec{STEP}$ 12: The unused edge with the next-lowest score value is s ₁. The edge s ₁ does not generate a cycle. so it is included to $\vec{ST}$ as shown in Fig. 5. At present, the $\vec{ST}$ has three edges.

Fig. 5
$\vec{MST}$ develops with s ₁.

$\vec{STEP}$ 12: The unused edge with the next-lowest score value is s ₃. The edge s ₃ does not generate a cycle. so it is included to $\vec{ST}$ as shown in Fig. 6. At present, the $\vec{ST}$ has four edges.

Fig. 6
$\vec{MST}$ develops with s ₃.

$\vec{STEP}$ 12: The unused edge with the next-lowest score value is s ₇. The edge s ₇ does not generate a cycle. so it is included to $\vec{ST}$ as shown in Fig. 7. At present, the $\vec{ST}$ has five edges.

Fig. 7
$\vec{MST}$ develops with s ₇.

$\vec{STEP}$ 12: Since the unused edge with the next-lowest score values of s ₆, s ₅, and s ₂ generates cycle, they are not included in the $\vec{ST}$ as given in Fig. 8. Moreover, the $\vec{ST}$ has five edges, $\vec{STEP}$ 12 ends, and the algorithm is concluded.

Fig. 8
$\vec{MST}$ does not develop with s ₆,s ₅,s ₂.

The $\vec{MST}$ is the solution to the challenge of marketing a web application(created via cloud service) in a future smart world using LiFi technology, as shown in Fig. 9. The edges s ₄, s ₈, s ₁, s ₃, and s ₇ are chosen based on the attributes.

Fig. 9
$\vec{MST}$ solution to promote the web application (created via cloud computing) in the future smart world among users.
6 Sensitivity analysis

Attributes	The specifications for $\vec{CLDFSS}$
"Social reinforcement (negative feedback)"	$(〈 \vec{CVMS}, \vec{CVNMS} 〉, 〈 more, less 〉)$
"Human heterogeneity"	$(〈 \vec{CVMS}, \vec{CVNMS} 〉, 〈 high, low 〉)$
"Attentional decline"	$(〈 \vec{CVMS}, \vec{CVNMS} 〉, 〈 more, less 〉)$

$\vec{ℶ}$	r ₁	r ₂	r ₃
s ₁	(〈(0.6, 0.7) , (0.4, 0.3) 〉,	(〈(0.7, 0.9) , (0.3, 0.1) 〉,	(〈(0.6, 0.8) , (0.4, 0.2) 〉
	〈(0.7, 0.7) , (0.3, 0.3) 〉)	(〈(0.8, 0.8) , (0.2, 0.2) 〉	(〈(0.7, 0.7) , (0.3, 0.3) 〉
s ₂	(〈(0.7, 0.7) , (0.3, 0.2) 〉,	(〈(0.7, 0.6) , (0.3, 0.4) 〉,	(〈(0.7, 0.9) , (0.3, 0.1) 〉
	〈(0.7, 0.8) , (0.2, 0.2) 〉)	(〈(0.6, 0.7) , (0.4, 0.3) 〉	(〈(0.8, 0.9) , (0.1, 0.1) 〉
s ₃	(〈(0.5, 0.6) , (0.4, 0.4) 〉,	(〈(0.8, 0.7) , (0.2, 0.3) 〉,	(〈(0.7, 0.8) , (0.3, 0.2) 〉
	〈(0.6, 0.6) , (0.4, 0.4) 〉)	(〈(0.8, 0.8) , (0.2, 0.2) 〉	(〈(0.7, 0.8) , (0.2, 0.2) 〉
s ₄	(〈(0.5, 0.7) , (0.4, 0.3) 〉,	(〈(0.7, 0.7) , (0.2, 0.2) 〉,	(〈(0.5, 0.6) , (0.5, 0.4) 〉
	〈(0.6, 0.6) , (0.3, 0.3) 〉)	(〈(0.7, 0.8) , (0.2, 0.3) 〉	(〈(0.6, 0.7) , (0.4, 0.3) 〉
s ₅	(〈(0.9, 0.8) , (0.1, 0.2) 〉,	(〈(0.9, 0.9) , (0.1, 0.1) 〉,	(〈(0.7, 0.6) , (0.3, 0.4) 〉
	〈(0.9, 0.8) , (0.1, 0.2) 〉)	(〈(0.8, 0.9) , (0.1, 0.1) 〉	(〈(0.7, 0.7) , (0.3, 0.3) 〉
s ₆	(〈(0.8, 0.7) , (0.2, 0.3) 〉,	(〈(0.8, 0.6) , (0.2, 0.3) 〉,	(〈(0.9, 0.7) , (0.1, 0.3) 〉
	〈(0.7, 0.7) , (0.2, 0.2) 〉)	(〈(0.8, 0.8) , (0.2, 0.2) 〉	(〈(0.7, 0.8) , (0.3, 0.2) 〉
s ₇	(〈(0.8, 0.9) , (0.2, 0.1) 〉,	(〈(0.7, 0.8) , (0.3, 0.1) 〉,	(〈(0.8, 0.6) , (0.2, 0.4) 〉
	〈(0.7, 0.8) , (0.3, 0.2) 〉)	(〈(0.8, 0.8) , (0.2, 0.2) 〉	(〈(0.6, 0.7) , (0.4, 0.3) 〉
s ₈	(〈(0.6, 0.6) , (0.4, 0.3) 〉,	(〈(0.7, 0.6) , (0.3, 0.4) 〉,	(〈(0.6, 0.5) , (0.4, 0.5) 〉
	〈(0.6, 0.7) , (0.3, 0.3) 〉)	(〈(0.7, 0.8) , (0.3, 0.2) 〉	(〈(0.7, 0.7) , (0.3, 0.3) 〉

$\vec{ℶ}$	r ₁	r ₂	r ₃
s ₁	(〈(0.7, 0.8) , (0.3, 0.2) 〉,	(〈(0.8, 0.9) , (0.2, 0.1) 〉,	(〈(0.7, 0.8) , (0.3, 0.2) 〉
	〈(0.8, 0.8) , (0.2, 0.2) 〉)	(〈(0.8, 0.8) , (0.2, 0.1) 〉	(〈(0.8, 0.7) , (0.2, 0.3) 〉
s ₂	(〈(0.8, 0.8) , (0.2, 0.1) 〉,	(〈(0.8, 0.7) , (0.2, 0.3) 〉,	(〈(0.8, 0.8) , (0.2, 0.2) 〉
	〈(0.8, 0.9) , (0.1, 0.1) 〉)	(〈(0.7, 0.8) , (0.3, 0.2) 〉	(〈(0.7, 0.8) , (0.3, 0.2) 〉
s ₃	(〈(0.6, 0.7) , (0.4, 0.3) 〉,	(〈(0.7, 0.8) , (0.3, 0.2) 〉,	(〈(0.8, 0.8) , (0.2, 0.2) 〉
	〈(0.7, 0.7) , (0.3, 0.3) 〉)	(〈(0.7, 0.7) , (0.2, 0.3) 〉	(〈(0.8, 0.9) , (0.1, 0.1) 〉
s ₄	(〈(0.6, 0.8) , (0.3, 0.2) 〉,	(〈(0.8, 0.8) , (0.2, 0.2) 〉,	(〈(0.6, 0.7) , (0.4, 0.3) 〉
	〈(0.7, 0.7) , (0.2, 0.2) 〉)	(〈(0.8, 0.9) , (0.1, 0.1) 〉	(〈(0.7, 0.8) , (0.3, 0.2) 〉
s ₅	(〈(0.9, 0.9) , (0.1, 0.1) 〉,	(〈(0.9, 0.8) , (0.1, 0.2) 〉,	(〈(0.8, 0.7) , (0.2, 0.3) 〉
	〈(0.9, 0.8) , (0.1, 0.2) 〉)	(〈(0.8, 0.9) , (0.1, 0.1) 〉	(〈(0.8, 0.8) , (0.2, 0.2) 〉
s ₆	(〈(0.9, 0.8) , (0.1, 0.2) 〉,	(〈(0.9, 0.7) , (0.1, 0.2) 〉,	(〈(0.9, 0.7) , (0.1, 0.3) 〉
	〈(0.8, 0.8) , (0.2, 0.2) 〉)	(〈(0.9, 0.9) , (0.1, 0.1) 〉	(〈(0.8, 0.7) , (0.2, 0.3) 〉
s ₇	(〈(0.9, 0.8) , (0.1, 0.2) 〉,	(〈(0.8, 0.9) , (0.2, 0.1) 〉,	(〈(0.9, 0.7) , (0.1, 0.2) 〉
	〈(0.7, 0.8) , (0.3, 0.2) 〉)	(〈(0.8, 0.8) , (0.2, 0.2) 〉	(〈(0.7, 0.8) , (0.3, 0.2) 〉
s ₈	(〈(0.7, 0.7) , (0.3, 0.3) 〉,	(〈(0.8, 0.6) , (0.2, 0.4) 〉,	(〈(0.7, 0.6) , (0.3, 0.4) 〉
	〈(0.7, 0.8) , (0.3, 0.2) 〉)	(〈(0.6, 0.7) , (0.4, 0.3) 〉	(〈(0.8, 0.6) , (0.2, 0.4) 〉

$\vec{ℶ}$	r ₁	r ₂	r ₃
s ₁	(〈(0.5, 0.6) , (0.4, 0.4) 〉,	(〈(0.6, 0.6) , (0.4, 0.4) 〉,	(〈(0.6, 0.8) , (0.4, 0.2) 〉
	〈(0.6, 0.7) , (0.4, 0.3) 〉)	(〈(0.7, 0.7) , (0.3, 0.3) 〉	(〈(0.8, 0.7) , (0.2, 0.3) 〉
s ₂	(〈(0.9, 0.9) , (0.1, 0.1) 〉,	(〈(0.8, 0.9) , (0.2, 0.1) 〉,	(〈(0.9, 0.7) , (0.1, 0.2) 〉
	〈(0.8, 0.9) , (0.2, 0.1) 〉)	(〈(0.8, 0.8) , (0.2, 0.2) 〉	(〈(0.9, 0.9) , (0.1, 0.1) 〉
s ₃	(〈(0.7, 0.9) , (0.3, 0.1) 〉,	(〈(0.8, 0.7) , (0.2, 0.3) 〉,	(〈(0.8, 0.6) , (0.2, 0.4) 〉
	〈(0.8, 0.8) , (0.2, 0.2) 〉)	(〈(0.8, 0.9) , (0.1, 0.1) 〉	(〈(0.6, 0.7) , (0.4, 0.3) 〉
s ₄	(〈(0.6, 0.7) , (0.4, 0.3) 〉,	(〈(0.7, 0.7) , (0.3, 0.2) 〉,	(〈(0.7, 0.8) , (0.3, 0.2) 〉
	〈(0.7, 0.8) , (0.3, 0.2) 〉)	(〈(0.8, 0.8) , (0.2, 0.2) 〉	(〈(0.7, 0.7) , (0.3, 0.3) 〉
s ₅	(〈(0.8, 0.8) , (0.2, 0.2) 〉,	(〈(0.9, 0.8) , (0.1, 0.2) 〉,	(〈(0.8, 0.7) , (0.2, 0.3) 〉
	〈(0.9, 0.8) , (0.1, 0.2) 〉)	(〈(0.7, 0.8) , (0.3, 0.2) 〉	(〈(0.7, 0.8) , (0.3, 0.2) 〉
s ₆	(〈(0.8, 0.9) , (0.2, 0.1) 〉,	(〈(0.7, 0.8) , (0.3, 0.2) 〉,	(〈(0.8, 0.7) , (0.2, 0.3) 〉
	〈(0.9, 0.8) , (0.1, 0.2) 〉)	(〈(0.6, 0.6) , (0.4, 0.3) 〉	(〈(0.7, 0.8) , (0.3, 0.2) 〉
s ₇	(〈(0.8, 0.7) , (0.2, 0.3) 〉,	(〈(0.6, 0.8) , (0.4, 0.2) 〉,	(〈(0.6, 0.8) , (0.4, 0.2) 〉
	〈(0.9, 0.7) , (0.1, 0.3) 〉)	(〈(0.7, 0.7) , (0.3, 0.3) 〉	(〈(0.8, 0.9) , (0.2, 0.1) 〉
s ₈	(〈(0.7, 0.6) , (0.3, 0.4) 〉,	(〈(0.7, 0.9) , (0.3, 0.1) 〉,	(〈(0.7, 0.9) , (0.3, 0.1) 〉
	〈(0.8, 0.7) , (0.2, 0.3) 〉)	(〈(0.9, 0.9) , (0.1, 0.1) 〉	(〈(0.8, 0.8) , (0.2, 0.2) 〉

$\vec{ℶ}$	Aggregated Values
s ₁	(〈(0.6627, 0.8013) , (0.3325, 0.1987) 〉,
	〈(0.7636, 0.7326) , (0.2363, 0.2473) 〉)
s ₂	(〈(0.8033, 0.7997) , (0.1967, 0.1737) 〉,
	〈(0.7740, 0.8461) , (0.1889, 0.1539) 〉)
s ₃	(〈(0.7429, 0.7512) , (0.2533, 0.2488) 〉,
	〈(0.7318, 0.8011) , (0.1983, 0.1989) 〉)
s ₄	(〈(0.6464, 0.7269) , (0.3291, 0.2540) 〉,
	〈(0.7061, 0.7744) , (0.2496, 0.2224) 〉)
s ₅	(〈(0.8421, 0.7700) , (0.1579, 0.2299) 〉,
	〈(0.7944, 0.8152) , (0.1784, 0.1848) 〉)
s ₆	(〈(0.8573, 0.7293) , (0.1427, 0.2510) 〉,
	〈(0.7803, 0.7806) , (0.2143, 0.2082) 〉)
s ₇	(〈(0.7951, 0.7798) , (0.2049, 0.1908) 〉,
	〈(0.7446, 0.7952) , (0.2554, 0.2048) 〉)
s ₈	(〈(0.6962, 0.7026) , (0.3045, 0.2921) 〉,
	〈(0.7554, 0.7447) , (0.2402, 0.2553) 〉)

Edges( $\vec{ℶ}$ )	Score Values
s ₁	0.4864
s ₂	0.6275
s ₃	0.5319
s ₄	0.4497
s ₅	0.6176
s ₆	0.5828
s ₇	0.5647
s ₈	0.4517

This segment examines the impact of variations in ς. ς = 2 has been established in the case study. Table 8 demonstrates the rank as ς increases. It demonstrates unequivocally that, in the case of the $\vec{CLDFSHWA}$ operator, changing ς has no impact on ranking. The $\vec{MST}$ still has the same ascending sequence of scores.

Table 8
Ranking results with increasing value of ς

ς $\vec{CLDFSHWA}$ operator

0.5 s ₄ < s ₈ < s ₁ < s ₃ < s ₇ < s ₆ < s ₅ < s ₂

1 s ₄ < s ₈ < s ₁ < s ₃ < s ₇ < s ₆ < s ₅ < s ₂

1.5 s ₄ < s ₈ < s ₁ < s ₃ < s ₇ < s ₆ < s ₅ < s ₂

2 s ₄ < s ₈ < s ₁ < s ₃ < s ₇ < s ₆ < s ₅ < s ₂

ς	$\vec{CLDFSHWA}$ operator
0.5	s ₄ < s ₈ < s ₁ < s ₃ < s ₇ < s ₆ < s ₅ < s ₂
1	s ₄ < s ₈ < s ₁ < s ₃ < s ₇ < s ₆ < s ₅ < s ₂
1.5	s ₄ < s ₈ < s ₁ < s ₃ < s ₇ < s ₆ < s ₅ < s ₂
2	s ₄ < s ₈ < s ₁ < s ₃ < s ₇ < s ₆ < s ₅ < s ₂

The outcomes obtained through the $\vec{CLDFSHWA}$ operator appear to be stable Since in the instance of the $\vec{CLDFSHWA}$ operator, altering of ς has no impact on ranking.

7 Conclusion

This work is the first to combine the $\vec{CLDFS}$ MCDM method and Kruskal’s algorithm with the Hamacher aggregation operator. It was motivated by the dearth of hybrid fuzzy approaches to graph problems. For discrete numbers, the traditional graph techniques work well. In real life, circumstances rely on several uncontrollable factors. $\vec{FS}$ extensions should therefore be incorporated into graph algorithms, and this work adds to the body of knowledge by putting out a hybrid strategy for handling fuzzy $\vec{MST}$ issues. The primary contribution of this endeavor is outlined below.

The analyzed study uses $\vec{CLDFSS}$ to deal with insufficient, ambiguous, and contradictory data by incorporating $\vec{MS}$ , $\vec{NMS}$ , and reference parameter regarding the $\vec{AS}$ .

This study is initiated due to the need for more hybrid fuzzy techniques to handle graph-oriented problems. It is the initial attempt to combine the Kruskal’s algorithm, Hamacher operator, and $\vec{CLDFSS}$ . For precise numbers, the traditional graph techniques are appropriate. Scenarios in the real world depend on several uncontrollable factors. As a result, $\vec{CLDFSS}$ should be considered in graph algorithm and this study adds to the body of knowledge by suggesting a hybrid method for addressing $\vec{CLDFSS}$ $\vec{MST}$ problems.

The proposed $\vec{MST}$ problem creates avenues for a lot of people to promote a web application(created via cloud computing) in the shortest way possible in the future smart world of LiFi technology.

For further study, this hybrid approach can be used to study various $\vec{MST}$ problems, such as LAN networks, Tour operations, Computer networks, Transportation networks, Computer networks, etc. in the future smart world of LiFi technology. The $\vec{CLDFSS}$ can be integrated with other MCDM techniques and different operators other than the Hamacher Aggregation operator can be used to rebuild this algorithm.

Acknowledgment

The article has been written with the joint financial support of RUSA-Phase 2.0 grant sanctioned vide letter No.F 24-51/2014-U, Policy (TN Multi-Gen), Dept. of Edn. Govt. of India, Dt. 09.10.2018, DST-PURSE 2nd Phase programme vide letter No. SR/PURSE Phase 2/38 (G) Dt. 21.02.2017 and DST (FIST - level I) 657876570 vide letter No.SR/FIST/MS-I/2018/17Dt. 20.12.2018.

References

Abdulazeez Alkouri,

Abdul Razak Salleh, Complex intuitionistic fuzzy sets, in: Ch. 2nd International conference on Fundamental and Applied Sciences (2012), 464–470.

Atanassov

K.T.

, Intuitionistic fuzzy sets, Fuzzy Sets andSystems 20(1) (1986), 87–96.

Broutin

Devroye

McLeish

, Note on the Structure ofKruskal’s Algorithm, Algorithmica 56 (2010), 141–159.

Huang

, An Improved Kruskal Algorithm-Two Branch KruskalAlgorithm, Chinese Scientific Papers Online (2011), 1–13.

Hamacher

, Uber logische verknupfungen unscharfer Anssagen undderen Zugehorige Bewertungs-Funktion, Progr. Cybern. Syst.Res. 3 (1978), 276–288.

Huseyin

Complex linear diophantine fuzzy sets and their cosine measures with applications, Complex and Intelligent System (2021).

Kannan

Garg

Jayakumar

Hananf

Aty

Haleem

, Linear diophantine multi-fuzzy aggregation operators and itsapplication in digital transformation, Journal of Intelligent& Fuzzy Systems 45(2) (2023), 3097–3107.

Kumar Tanuj Bajaj Rakesh, , On complex intuitionistic fuzzy softsets with distance measures and entropies, Journal ofMathematics (2014), 1–12.

Izatmand

Mahmood

Ali Aslam

Chinram,

, Generalizedhamacher aggregation operators based on linear diophantine uncertainlinguistic setting and their applications in decision-makingproblems, IEEE Access 9 (2021), 126748–126764.

10.

Liang

, Application of minimum spanning tree in network design, Journal of Suzhou Education Institute 2 (2008), 150–153.

11.

Xia

Wang

, Research and Improvement of KruskalAlgorithm, Journal of Computer and Communications 5(2017), 63–69.

12.

Liu

Mahmood

Ali

, Complex q-rung orthopair fuzzyaggregation operators and their applications in multiple-attributegroup decision making, Information 11(5) (2020).

13.

Maji

P.K.

Biswas

Roy

A.R.

, Intuitionistic fuzzy soft sets, Journal of Fuzzy Mathematics 9(3) (2001), 677–692.

14.

Murat Kirisci Necip Simsek, , Decision making method related toPythagorean Fuzzy Soft Sets with infectious diseases application, Journal of King Saud University - Computer and InformationSciences 8 (2022), 5968–5978.

15.

Molodtsov

, Soft set theory - first results, Computers andMathematics with Applications 37 (1999), 19–31.

16.

Mordeson

J.N.

Mathew

Malik

, Fuzzy graph theory withapplications to human trafficking, Springer InternationalPublishing 365 (2018).

17.

Ramot

Milo

Friedman

Kandel

, Complex fuzzy sets, IEEE Transactions on Fuzzy Systems 10(2) (2002), 171–186.

18.

Riaz

Hashmi

M.R.

, Linear diophantine fuzzy set and itsapplications towards multi-attribute decision making problems, Journal of Intelligent and Fuzzy Systems 37(4) (2019), 5417–5439.

19.

Ruohonen

Graph theory, Translation by Janne Tamminen, Kung-Chung Lee and Robert Piche (2013).

20.

Sharma

P.K.

Ryu

J.H.

Park

K.Y.

, et al. Li-Fi based on securitycloud framework for future IT environment, Human-CentricComputing and Information Science 8(23) (2018).

21.

Ullah

Mahmood

Ali

Jan

, On some distance measuresof complex Pythagorean fuzzy sets and their applications in patternrecognition, Complex and Intelligent Systems 6(2020), 15–27.

22.

Vimala Jayakumar, Ashma Banu, Muhammad Haris Saeed, Hamed Alsulami, Aftab Hussain Muhammad Saeed, , Development of complex lineardiophantine fuzzy soft set in determining a suitable agri-drone forspraying fertilizers and pesticides, IEEE Access 11(2023), 9031–9041.

23.

Vimala

Mahalakshmi

Rahman

A.U.

Saeed

A customized TOPSIS method to rank the best airlines to fly during COVID-19 pandemic with q-rung orthopair multi-fuzzy soft information, Soft Computing (2023), Doi:https://doi.org/10.1007/s00500-023-08976-2.

24.

Yager

R.R.

, Pythagorean fuzzy subsets, Edmonton, Canada, IEEE, IFSA World 871 Congressand NAFIPS Annual Meeting 2013 Joint (2013), 57–61.

25.

Yager

R.R.

, Generalized orthopair fuzzy sets, IEEE Transactionson Fuzzy Systems 25(5) (2017), 1222–1230.

26.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

Segment 2	Concepts of $\vec{CLDFSS}$ together with the fundamentals on graphs and its extensions are examined.
Segment 3	The conceptualization of Complex Linear Diophantine Fuzzy Soft Hamacher Weighted averaging $(\vec{CLDFSHWA})$ operator and the
	practical Hamacher operational laws are investigated.
Segment 4	Conceptual framework for locating the shortest path $(\vec{SP})$ connecting each vertex of the weighted graph is developed.
Segment 5	Discusses the suggested MCDM technique combining Kruskal’s algorithm and the Hamacher aggregation operator through an actual case study.
Segment 6	Sensitivity Analysis is performed.
Segment 7	Manuscript is compiled with future developments.

An Amalgamated Kruskal’s Algorithm to ascertain the shortest route in the futuristic smart world

Abstract

Keywords

1 Introduction

2.1 Elementary concepts

2.2 Fundamental definitions on graphs

3 Aggregation operator

3.1 Operational laws based on Hamacher norm

4 Conceptual framework

5.1 Case study

5.1.1 Cloud computing architecture

Table 8 Ranking results with increasing value of ς ς CLDFSHWA → operator 0.5 s 4 < s 8 < s 1 < s 3 < s 7 < s 6 < s 5 < s 2 1 s 4 < s 8 < s 1 < s 3 < s 7 < s 6 < s 5 < s 2 1.5 s 4 < s 8 < s 1 < s 3 < s 7 < s 6 < s 5 < s 2 2 s 4 < s 8 < s 1 < s 3 < s 7 < s 6 < s 5 < s 2

Acknowledgment

References