An intuitionistic fuzzy graph’s variation coefficient measure with application to selecting a reliable alliance partner

Abstract

Group decision-making (GDM) is crucial in various components of graph theory, management science, and operations research. In particular, in an intuitionistic fuzzy group decision-making problem, the experts communicate their preferences using intuitionistic fuzzy preference relations (IFPRs). This approach is a way that decision-makers rank or select the most desirable alternatives by gathering criteria-based information to estimate the best alternatives using a wider range of knowledge and experience. This article proposes a new statistical measure in a fuzzy environment when the data is ambiguous or unreliable to solve a decision-making problem. This study uses the variation coefficient measure combined with intuitionistic fuzzy graphs (IFG) and Laplacian energy (LE) to solve a GDM problem that utilizes intuitionistic fuzzy preference relations (IFPRs) to select a reliable alliance partner. Initially, the Laplacian energy determines the weight of individual standards, and the obtained weight average further estimates the overall criterion weight vector. We establish the authority criteria weights using the variation coefficient measure and then ultimately rank the alternatives for each criterion using the same measure. We examine four distinct companies Alpha, Beta, Delta, and Zeta to conduct a realistic GDM to choose which alliance partner would be ideal. We successfully implemented the suggested technique, determining that Alpha satisfies company standards and is ranked first among other companies. Moreover, this technique is useful for all kinds of Intuitionistic fuzzy group decision-making problems to select optimal ones.

Introduction

Decision-making is the process of choosing the most favourable course to follow among a range of options¹, determining the optimal method for solving problems that fulfil the specific criteria, which can happen in everyday situations. Decisions are typically made by a group of specialists rather than individuals due to the complicated nature of the real-time environment and the absence of precise information². Selecting the optimal choice from a range of alternatives is a frequent undertaking process in the decision-making is a challenging task. The fuzzy set theory uses the group decision-making (GDM) technique to examine multiple existing research papers for problem-solving abilities. Individuals' knowledge, experiences, and insights all contribute to an improved understanding of the situation, while GDM uses a variety of aspects to solve challenging circumstances. Compared to individual decision-making, group decision-making (GDM) can provide better informed and flexible judgments. It can also help to identify and alleviate hidden biases and hazards that individuals may encounter. Furthermore, Liu³ suggested the non-additive fuzzy methods and peer interaction methods with log sigmoid techniques ensure the real-world scenarios for solving the GDM problem. Lu³ constructed the inter-valued Intuitionistic fuzzy multiplicative preference relations with the group best–worst method to enhance the group decision-making process. Mandal⁴ presented multi-criteria decision-making with inter-valued Intuitionistic fuzzy systems using a Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) approach to finding the rank criteria in the use case of the selection supervisor scenario. With the help of Statistical analysis can measure the degree of uncertainty linked to both individual and collective decisions. This analysis offers significant information regarding the dependability and resilience of the GDM solution.

The hybridization of intuitionistic fuzzy graphs and graph theory ensures optimal solutions in real-world scenarios. In this context, intuitionistic fuzzy graph models provide a more comprehensive view of the selection scenario than fuzzy graph models because they consider ambiguity and inconsistency in decisions made by various individuals or locations. Statistical methods can be utilized to analyse the uncertainty linked to IFG models and make statistically sound conclusions regarding the decision outcomes⁵. Graph theory plays a pivotal role in modern fields such as data science, artificial intelligence, and machine learning. It enhances the visualization of decision possibilities as nodes and their associations as edges, resulting in a transparent and short pictorial representation of the decision-making problem. Graph theory models can be combined with statistical techniques to examine patterns and connections in the decision data. This enables the detection of statistically significant correlations and patterns that can provide valuable insights for decision-making⁶. By incorporating graph theory and Intuitionistic fuzzy graphs (IFGs), those making decisions can utilize illustrations, analysis of information, and statistical methodologies to make well-informed and adapted assessments under the conditions of uncertainty. The suggested method is capable of effectively managing complicated decision issues that involve several criteria, ambiguous choices, and insufficient information. For example, Construct an IFG model wherein the nodes relate to symptoms and the edges signify possible associations with diseases. Applying statistical analysis to a framework can help to identify probable diagnoses by considering the advantage of patients’ data and expert judgments, which may vary in terms of certainty. An effective framework for making judgments in ambiguous circumstances is developed by fusing statistical techniques with graph theory and intuitionistic fuzzy graphs. To help with decision-making, this approach enables the analysis of complex relationships, the combination of several alternatives, and the creation of statistically valid outcomes.

The various existing research works are carried out dealing with a wide range of intricate phenomena with limited data, the introduction of fuzzy set theory by Zadeh in 1965 was imperative. Zadeh’s fuzzy set theory, as illustrated by his works⁷, has proven to be an effective method for identifying situations that involve uncertain or imprecise data. Fuzzy sets address those situations by assigning a degree to which an object belongs to a set. To address the uncertainty and unpredictability around the concept of the grade of membership. Atanassov⁸ proposed the introduction of a new grade in the fuzzy set theory known as the grade of non-membership. Furthermore, in 1975, the author innovatively established a fuzzy graph theory as an expansion of Euler's graph theory. This innovative study was the first person to deal with the uncertain connections between fuzzy sets and establish the structure for a fuzzy graph. Rosenfeld⁹ presents a discussion of the attributes of FG and suggests fuzzy equivalents for some important concepts in graph theory. Anjali¹⁰ first established the notion of energy in a fuzzy graph (FG) and provided a precise definition of the adjacency matrix of an FG. Parvathi¹¹ emphasizes the properties of FGs and introduces a new definition for the intuitionistic fuzzy graph. Deepa¹² provided an outline of an IFAM and described the energy of an IFG based on its adjacent matrix. Karthik and Sharief Basha¹³ extended the concept of the Laplacian energy (LE) of an intuitionistic fuzzy graph (IFG) by building upon the basic concept of the LE of a fuzzy graph (FG). Gutman¹⁴ proposed the utilization of a LE-based approach to assess the weights of criteria in decision-making matrices. In addition, he proposed the method to combine the discovered weights to acquire the collective LE weights. Alolaiyan² identified the shortcomings of the existing similarity measures and proposed an improved one within the framework of interval-valued IFSs. Muthukumar¹⁵ proposed a weighted similarity measure on intuitionistic fuzzy soft sets and discussed some of their basic properties. Tarannum¹⁶ proposed using the correlation coefficient and intuitionistic fuzzy sets in the medical field to classify Corona patients, to prioritize them for better treatment. Wei¹⁷ introduced correlation, the correlation coefficient of interval-valued intuitionistic fuzzy sets, and the operational principles of interval-valued intuitionistic fuzzy numbers. Intuitionistic fuzzy set theory has been applied in various domains, such as artificial intelligence in the medical field Jithendra^18,19,20, decision-making problems involving different similarity measures^{21,22,23,24,25,26}, and banking issues. Akram^27,28,29 introduced several innovative ideas on extended architectures and their applications in different types of fuzzy graphs.

Some statistical measures have been developed in recent years for fuzzy graph applications in decision-making. These measures have been proposed by Naveen and Basha^30,31,32, Jenifer and Helen³³, Rajagopal and Basha³⁴, Yagiz and Gokceoglu³⁵. It has been shown through intuitionistic fuzzy analysis that statistical knowledge and its practical uses have grown significantly in recent years. We are inspired to consider IFGs and their uses through this sort of fuzzy graph research. In this paper, we offer a method for resolving GDM problems when the IFG solely determines the alternatives and the weights (loads) of the criteria are entirely mysterious. We compute the relative weights resolute by every decision matrix using the LE metric to handle the vague information requirements. We collectively add all of the obtained LE weights (loads) to fulfil the overall weight (load) vector requirement. The variation grade for each grade of the alternatives is calculated after the variation coefficient measure has been used to rank the alternatives, and the most acceptable alternatives are then chosen by the decision-makers.

Using various existing research studies to incorporate opinions and demands from decision-makers may pose difficulties. Furthermore, the method of allocating appropriate weights to each condition may exhibit bias and have a significant impact on the final result. The decision-making process may become more complex and difficult to navigate due to the many factors and alternatives, thus hindering the identification of the most appropriate response. Updating IFG models efficiently to accommodate dynamic conditions and promptly adjusting desires in real-time continues to be an important challenge. Moreover, the issues of computational effectiveness and flexibility might be essential, especially when dealing with decision problems with fuzzy (ambiguous) data. This fuzzy graph research motivates us to explore IFGs and their applications. This research’s novel idea is the development of new statistical coefficients within an intuitionistic framework, specifically designed to address decision-making difficulties. Finding new techniques like current similarity measures in an intuitionistic fuzzy environment is another important part of this research. It also aims to present a methodical mathematical approach for solving decision-maker problems using newly created statistical measures in an intuitionistic fuzzy environment. In recent years, intuitionistic fuzzy set analysis has demonstrated a significant increase in statistical understanding as well as practical applications. Although there have been notable advancements in this field, numerous obstacles persist. Current research focuses on extending statistical coefficients from IFS to IFGs, which expands the ability to use the techniques (similarity measures) discussed in general. The study examines the various research efforts that address the gaps highlighted, such as how the statistical coefficients of IFGS have improved in real-life decision-making situations.

Objectives:

• 1. Intuitionistic fuzzy set theory introduces numerous similarity measures to solve decision-making problems, but applying the statistical coefficient (variation coefficient) in this field is a novel idea.

• 2. Apart from various similarity methods, the main objective of this work is to extend statistical coefficients from intuitionistic fuzzy sets to intuitionistic fuzzy graphs in decision-making problems to address real-life applications.

• 3. The main objective of this work is to find the Laplacian energy and correlation coefficient of intuitionistic fuzzy graphs to solve group decision-making problems supported by intuitionistic fuzzy preference relations.

• 4. The presented work has been successfully implemented for choosing the best alliance partner for a software company.

The rest of the content of this paper is divided into the following sections: section “Preliminaries” contains multiple explanations of fuzzy-related and statistical metrics. The procedure for calculating the suggested variation coefficient measure is outlined and additionally, presents the recommended steps (procedure I and II) in the format of a flowchart in section “Group decision-making built on IFGs Laplacian energy and variation coefficient”. Section “Real world application” presents the practical application of GDMP, whereas Section “Conclusion” outlines its intended development.

Preliminaries

Here, we briefly define some fuzzy key definitions and discuss some useful statistical metrics.

Definition 2.1

⁷(Zadeh, 1965)

A fuzzy set $A$ is defined as an assemblage of order pairs $\left(X, M\right)$, where $X$ is (a non-empty set) called the universe of discourse and the function $M = \mu_{A} \left( x \right):X \to \left[ {0,1} \right]$ is an MF of the fuzzy set $A=\left(X, M\right)$. Also, the value $M(x)$ is called the grade of MF of $x$ in $\left(X, M\right)$, for each $x\in X$.

$${\text{i.e.,}}\quad A = \left\{ {\left( {X,\mu_{A} \left( x \right)} \right)/x \in X} \right\}$$

(1)

Definition 2.2

⁸(Atanassov, 1983)

An IFS $A$ in $E$ is well-defined as an object in the form

$$A=\left\{\frac{x,{\mu }_{A}\left(x\right), {\vartheta }_{A}\left(x\right)}{x\in E}\right\}$$

(2)

with the condition $0\le {\mu }_{A}\left(x\right)+ {\vartheta }_{A}\left(x\right)\le 1$, where the functions ${\mu }_{A}\left(x\right):E\to [\text{0,1}]$ and ${\vartheta }_{A}\left(x\right): E\to [\text{0,1}]$ define the grade of membership and non-membership of the element $x\in E$ respectively, $\forall x\in E.$

Definition 2.3

⁹(Rosenfeld, 1975)

A FG ${G}_{i}=(V,\sigma ,\nu )$ is a nonempty set $V$ composed with a pair of functions $\sigma :V\to [0, 1]$ and $\nu :E\to [0, 1]$ such that for all x, y in $V$ satisfies $\nu \left(x, y\right)\le \text{min}(\sigma \left(x\right),\sigma \left(y\right))$, where $\sigma$ is called the fuzzy vertex set of G and $\nu$ is the fuzzy edge set of G, respectively.

Definition 2.4

³⁶(Atanassov, 2003)

An IFG ${G}_{i}=(V,E,\mu ,\vartheta )$ is an FG, where ‘V’ is the set of nodes and ‘E’ is the set of edges with an FMF $(\mu )$ well-defined on $VXV$ and a FNMF $(\vartheta )$, satisfying the conditions

• (i) $0\le {\mu }_{ij}+{\vartheta }_{ij}\le 1$

• (ii) $0\le {\mu }_{ij},{\vartheta }_{ij},{\pi }_{ij}\le 1$, where ${\pi }_{ij}=1-{(\mu }_{ij}+{\vartheta }_{ij})$.

Definition 2.5

¹¹(Parvathi, 2006)

An intuitionistic fuzzy adjacency matrix of an intuitionistic fuzzy graph is defined as the adjacency matrix of the corresponding intuitionistic fuzzy graph. That is for an intuitionistic fuzzy graph ${G}_{i}=(V,E,\mu ,\vartheta )$, an intuitionistic fuzzy adjacency matrix is defined by $A\left({G}_{i}\right)=\left[{a}_{ij}\right],$ where ${a}_{ij}=\left[{\mu }_{ij} ,{\vartheta }_{ij}\right].$ The adjacency matrix of an intuitionistic fuzzy graph can be written as two matrices one containing the entries as membership values and the other containing non-membership values.

$${\text{i}}.{\text{e}}.~\quad A\left( {G_{i} } \right)~ = \left[ {A_{\mu } \left( {G_{i} } \right),~\;A_{\vartheta } \left( {G_{i} } \right)} \right]$$

(3)

From Fig. 1, the adjacency matrix is given by

$$A\left({G}_{i}\right)=\left[\begin{array}{cccc}(0, 0)& (0.6, 0.2)& (0.5, 0.2)& (0.3, 0.6)\\ (0.6, 0.2)& (0, 0)& (0.5, 0.1)& (0.8, 0.1)\\ (0.5, 0.2)& (0.5, 0.1)& (0, 0)& (0.3, 0.5)\\ (0.3, 0.6)& (0.8, 0.1)& (0.3, 0.5)& (0, 0)\end{array}\right]$$

where, ${A}_{\mu }\left({G}_{i}\right)=\left[\begin{array}{cccc}0& 0.6& 0.5& 0.3\\ 0.6& 0& 0.5& 0.8\\ 0.5& 0.5& 0& 0.3\\ 0.3& 0.8& 0.3& 0\end{array}\right]$ and ${A}_{\vartheta }\left({G}_{i}\right)=\left[\begin{array}{cccc}0& 0.2& 0.2& 0.6\\ 0.2& 0& 0.1& 0.1\\ 0.2& 0.1& 0& 0.5\\ 0.6& 0.1& 0.5& 0\end{array}\right]$

Figure 1

Intuitionistic fuzzy graph $\left({G}_{i}\right)$.

Definition 2.6

³⁷(Chung, 2003)

Let G = (V, E) be an intuitionistic fuzzy graph on n vertices. The degree matrix $D({G}_{i})=\left[{d}_{ij}\right]$ of G is an $n\times n$ diagonal matrix defined as:

$$D({G}_{i})=\left[{d}_{ij}\right]=\left\{ \begin{array}{ll}deg\left({v}_{i}\right),&\quad if\, i = j \\ 0,& \quad otherwise\end{array} \right.$$

where, the degree $deg\left({v}_{i}\right)$ of a vertex counts the number of times an edge terminates at that vertex.

Definition 2.7

¹¹(Gutman, 2005)

Let $A\left({G}_{i}\right)$ be an adjacency matrix and $D\left({G}_{i}\right)=\left[{d}_{ij}\right]$ be a degree matrix of ${G}_{i}=(V,E,\mu ,\vartheta )$. The matrix

$$L\left({G}_{i}\right)=D\left({G}_{i}\right)-A\left({G}_{i}\right)$$

(4)

is defined as an intuitionistic fuzzy Laplacian matrix of IFG.

Using $A\left({G}_{i}\right)$ in Definition 2.5, the Laplacian matrix is

$$L\left({G}_{i}\right)=\left[\begin{array}{cccc}(1.4, 1)& (-0.6, -0.2)& (-0.5, -0.2)& (-0.3, -0.6)\\ (-0.6, -0.2)& (0.9, 0.4)& (-0.5, -0.1)& (-0.8, -0.1)\\ (-0.5,-0.2)& (-0.5, -0.1)& (1.3, 0.8)& (-0.3, 0.5)\\ (-0.3, -0.6)& (-0.8, -0.1)& (-0.3, -0.5)& (1.4, -0.2)\end{array}\right]$$

where, $L\left[{ A}_{\mu }\left({G}_{i}\right)\right]=\left[\begin{array}{cccc}1.4& -0.6& -0.5& -0.3\\ -0.6& 0.9& -0.5& -0.8\\ -0.5& -0.5& 1.3& -0.3\\ -0.3& -0.8& -0.3& 1.4\end{array}\right]$ and $L\left[{A}_{\vartheta }\left({G}_{i}\right)\right]=\left[\begin{array}{cccc}1& -0.2& -0.2& -0.6\\ -0.2& 0.4& -0.1& -0.1\\ -0.2& -0.1& 0.8& 0.5\\ -0.6& -0.1& -0.5& -1.2\end{array}\right]$

Definition 2.8

³⁸(Basha, 2015)

Consider an IFG ${G}_{i}=(V,E,\upmu ,\vartheta )$, in which ${\lambda }_{i}$, ${\theta }_{i}$ are the characteristic roots of the IFAM $A({G}_{i})$. Then the LE of the IFG is given by

$$LE\left({G}_{i}\right)=\left(LE\left({A}_{\mu }\left({G}_{i}\right)\right), LE\left({A}_{\vartheta }\left({G}_{i}\right)\right)\right),\quad \forall i=\text{1,2},3,\dots$$

(5)

Here, ${A}_{\mu }\left({G}_{i}\right)$ and ${A}_{\vartheta }\left({G}_{i}\right)$ signifies the two matrices (one contains membership and other contains non membership elements) of an IFAM $(A\left({G}_{i}\right))$ respectively, where as ${\lambda }_{i}$,${\theta }_{i}$ signifies the characteristic roots of ${A}_{\mu }\left({G}_{i}\right), {A}_{\vartheta }\left({G}_{i}\right)$. The LE of an IFG's matrix with ${A}_{\mu }\left({G}_{i}\right)$ and ${A}_{\vartheta }\left({G}_{i}\right)$ are specified by the equations

$$\left.\begin{array}{c}LE\left({A}_{\mu }\left({G}_{i}\right)\right)=\sum_{i=1}^{n}\left|{\lambda }_{i}-\frac{2\sum_{1\le i\le j\le n}\mu \left({u}_{i},{u}_{j}\right)}{n}\right|\\ LE\left({A}_{\vartheta }\left({G}_{i}\right)\right)=\sum_{i=1}^{n}\left|{\theta }_{i}-\frac{2\sum_{1\le i\le j\le n}\vartheta \left({v}_{i},{v}_{j}\right)}{n}\right|\end{array}\right\}$$

(6)

Eigenvalues (${\lambda }_{i})$ of $L\left[{ A}_{\mu }\left({G}_{i}\right)\right]=\{\text{0.0,1.5520,1.8506,2.5975}\}$

Eigenvalues (${\theta }_{i})$ of $L\left[{A}_{\vartheta }\left({G}_{i}\right)\right]=\{\text{0.0,0.5210,1.0900,1.7890}\}$

Using Fig. 1 and Definition 2.6, the Laplacian energy of an intuitionistic fuzzy graph ${G}_{i}$ is

$$LE\left({G}_{i}\right)=\left(LE\left({A}_{\mu }\left({G}_{i}\right)\right), LE\left({A}_{\vartheta }\left({G}_{i}\right)\right)\right)=\left[3.0001, 2.358\right]$$

Definition 2.9

${V}_{C}$ measure of IFGs:

Let $X=({x}_{1},{x}_{2},\dots , {x}_{n}$) and $Y=({y}_{1},{y}_{2},\dots ,{y}_{n})$ denotes the entries of two matrices of order n that include membership and non-membership elements, all of which are positive and are in the range [0, 1]. Then

(i) The Jaccard index³⁹ of two variables X and Y is defined as:

$$J\left(X,Y\right)=\frac{XY}{\Vert {X}_{i}^{2}\Vert +\Vert {Y}_{i}^{2}\Vert -XY}=\frac{\sum_{i=1}^{n}{x}_{i}{y}_{i}}{\sum_{i=1}^{n}{x}_{i}^{2}+\sum_{i=1}^{n}{y}_{i}^{2}-\sum_{i=1}^{n}{x}_{i}{y}_{i}}$$

(7)

where $XY=\sum_{i=1}^{n}{x}_{i}{y}_{i}$ is the inner product of membership elements of X and Y, $\Vert {X}_{i}^{2}\Vert =\sum_{i=1}^{n}{x}_{i}^{2}$ and $\Vert {Y}_{i}^{2}\Vert =\sum_{i=1}^{n}{y}_{i}^{2}$ are the Euclidean norm of X and Y.

(ii) The novel dice similarity measure⁴⁰ is well-defined as

$$D\left(X,Y\right)=\frac{2XY}{\Vert {X}_{i}^{2}\Vert +\Vert {Y}_{i}^{2}\Vert }=\frac{2\sum_{i=1}^{n}{x}_{i}{y}_{i}}{\sum_{i=1}^{n}{x}_{i}^{2}+\sum_{i=1}^{n}{y}_{i}^{2}}$$

(8)

(iii) The cosine similarity measure⁴¹ of the angle between X and Y is defined as

$$C\left(X,Y\right)=\frac{XY}{\Vert {X}_{i}^{2}\Vert \Vert {Y}_{i}^{2}\Vert }=\frac{\sum_{i=1}^{n}{x}_{i}{y}_{i}}{\sqrt{\sum_{i=1}^{n}{x}_{i}^{2}}\sqrt{\sum_{i=1}^{n}{y}_{i}^{2}}}$$

(9)

Therefore, the variation coefficient measure is given by ${V}_{C}\left(X, Y\right)=\alpha \left(\text{dice similarity measure}\right)+\left(1-\alpha \right)\text{cosine similarity measure}$

$$\left.\begin{array}{c}{V}_{C}\left(X, Y\right)=\alpha \frac{2XY}{\Vert {X}_{i}^{2}\Vert +\Vert {Y}_{i}^{2}\Vert }+(1-\alpha )\frac{XY}{\Vert {X}_{i}^{2}\Vert \Vert {Y}_{i}^{2}\Vert }\\ (or)\\ {V}_{C}\left(X, Y\right)= \alpha \frac{2\sum_{i=1}^{n}{x}_{i}{y}_{i}}{\sum_{i=1}^{n}{x}_{i}^{2}+\sum_{i=1}^{n}{y}_{i}^{2}}+(1-\alpha )\frac{\sum_{i=1}^{n}{x}_{i}{y}_{i}}{\sqrt{\sum_{i=1}^{n}{x}_{i}^{2}}\sqrt{\sum_{i=1}^{n}{y}_{i}^{2}}}\end{array}\right\}$$

(10)

where, $\alpha \in [\text{0,1}]$ is an intuitionistic fuzzy parameter.

In the fuzzy environment, the variation coefficient of two IFGs ${G}_{l}$ and ${G}_{p}$ is defined as

$${V}_{C}\left({G}_{l},{G}_{p}\right)=\alpha \frac{2\sum_{i=1}^{n}\left[{\mu }_{{G}_{l}}\left({x}_{i}\right){\mu }_{{G}_{p}}\left({y}_{i}\right)+{\vartheta }_{{G}_{l}}\left({x}_{i}\right){\vartheta }_{{G}_{p}}\left({y}_{i}\right)\right]}{\sum_{i=1}^{n}\left[{\mu }_{{G}_{l}}^{2}\left({x}_{i}\right)+{\vartheta }_{{G}_{l}}^{2}\left({x}_{i}\right)\right]+\sum_{i=1}^{n}\left[{\mu }_{{G}_{p}}^{2}\left({y}_{i}\right)+{\vartheta }_{{G}_{p}}^{2}\left({y}_{i}\right)\right]}+\left(1-\alpha \right)\frac{\sum_{i=1}^{n}\left[{\mu }_{{G}_{l}}\left({x}_{i}\right){\mu }_{{G}_{p}}\left({y}_{i}\right)+{\vartheta }_{{G}_{l}}\left({x}_{i}\right){\vartheta }_{{G}_{p}}\left({y}_{i}\right)\right]}{\sqrt{\sum_{i=1}^{n}\left[{\mu }_{{G}_{l}}^{2}\left({x}_{i}\right)+{\vartheta }_{{G}_{l}}^{2}\left({x}_{i}\right)\right]}\sqrt{\sum_{i=1}^{n}\left[{\mu }_{{G}_{p}}^{2}\left({y}_{i}\right)+{\vartheta }_{{G}_{p}}^{2}\left({y}_{i}\right)\right]}}$$

(11)

The function $V\left({G}_{l},{G}_{p}\right)$ also satisfies the following conditions:

• (a) $0\le {V}_{C}\left({G}_{l},{G}_{p}\right)\le 1$

• (b) ${V}_{C}\left({G}_{l},{G}_{p}\right)= {V}_{C}\left({G}_{p},{G}_{l}\right)$

• (c) ${V}_{C}\left({G}_{l},{G}_{p}\right)= 1, if\, {G}_{l}={G}_{p}$.

Group decision-making built on IFGs Laplacian energy and variation coefficient

Considering the seeming universality of GDM approaches, it might be challenging to identify the approach that will be most effective in solving a given issue. These types of problems have been addressed by several similarity metrics, but employing statistical measurements is a novel strategy.Here, we came up with a statistical measure called the variation coefficient measure together with LE to solve GDM issues using the following algorithm.

Algorithm

In order to identify GDMP based on IFPRs, let $w=\left({w}_{1},{w}_{2},\dots ,{w}_{m}\right)$ be a subjective weighting vector of authorities, where ${w}_{p}>0, p= {1,2},\dots ,m.,$ and $\sum_{p=1}^{m}{w}_{p}=1.$

Procedure-I

Step-(i): Find the LEs separately for membership and non-membership matrices of ${G}_{i}$, using the equations

$$\left.\begin{array}{c}LE\left({G}_{\upmu \left(i\right)}\right)=\sum_{i=1}^{n}\left|{\lambda }_{i}-\frac{2\sum_{1\le i\le j\le n}\mu \left({x}_{i},{x}_{j}\right)}{n}\right|\\ LE\left({G}_{\vartheta (i)}\right)=\sum_{i=1}^{n}\left|{\theta }_{i}-\frac{2\sum_{1\le i\le j\le n}\vartheta \left({y}_{i},{y}_{j}\right)}{n}\right|\end{array}\right\}$$

(12)

Step-(ii): Use the obtained LEs of the authorities ${e}_{p}$, find out the weight ${w}_{p}^{a}$ using the equation

$${w}_{p}^{a}=\left({\left({w}_{\mu }\right)}_{p},{\left({w}_{\vartheta }\right)}_{p}\right)=\left[\frac{LE\left({\left({G}_{\mu }\right)}_{p}\right)}{\sum_{i=1}^{m}LE\left({\left({G}_{\mu }\right)}_{i}\right)}, \frac{LE\left({\left({G}_{\vartheta }\right)}_{p}\right)}{\sum_{i=1}^{m}LE\left({\left({G}_{\vartheta }\right)}_{i}\right)}\right]$$

(13)

Step-(iii): Estimate the variation coefficient $V\left({G}_{l},{G}_{p}\right)$ between ${G}_{l}$ and ${G}_{p}$ for $p\ne l$ using the equation

$${V}_{c}\left({G}_{l},{G}_{p}\right)=\alpha \frac{2\sum_{i=1}^{n}\left[{\mu }_{{G}_{l}}\left({x}_{i}\right){\mu }_{{G}_{p}}\left({y}_{i}\right)+{\vartheta }_{{G}_{l}}\left({x}_{i}\right){\vartheta }_{{G}_{p}}\left({y}_{i}\right)\right]}{\sum_{i=1}^{n}\left[{\mu }_{{G}_{l}}^{2}\left({x}_{i}\right)+{\vartheta }_{{G}_{l}}^{2}\left({x}_{i}\right)\right]+\sum_{i=1}^{n}\left[{\mu }_{{G}_{p}}^{2}\left({y}_{i}\right)+{\vartheta }_{{G}_{p}}^{2}\left({y}_{i}\right)\right]}+\left(1-\alpha \right)\frac{\sum_{i=1}^{n}\left[{\mu }_{{G}_{l}}\left({x}_{i}\right){\mu }_{{G}_{p}}\left({y}_{i}\right)+{\vartheta }_{{G}_{l}}\left({x}_{i}\right){\vartheta }_{{G}_{p}}\left({y}_{i}\right)\right]}{\sqrt{\sum_{i=1}^{n}\left[{\mu }_{{G}_{l}}^{2}\left({x}_{i}\right)+{\vartheta }_{{G}_{l}}^{2}\left({x}_{i}\right)\right]}\sqrt{\sum_{i=1}^{n}\left[{\mu }_{{G}_{p}}^{2}\left({y}_{i}\right)+{\vartheta }_{{G}_{p}}^{2}\left({y}_{i}\right)\right]}}$$

(14)

Find the collective (average) variation coefficient grade $V\left({G}_{p}\right)$ using the equation

$${V}_{c}\left({G}_{p}\right)=\frac{1}{m-1}\sum_{l=1, p\ne l}^{n}{V}_{c}\left({G}_{p},{G}_{l}\right), p=\text{1,2},\dots ,m.$$

(15)

Step-(iv): Figure out the weight ${w}_{p}^{b}$ obtained by ${V}_{c}\left({G}_{p}\right)$ of the authority ${e}_{p}$ by using the equation

$${w}_{p}^{b}=\frac{{V}_{c}\left({G}_{p}\right)}{\sum_{i=1}^{m}{V}_{c}\left({G}_{i}\right)} , p=\text{1,2},\dots ,m.$$

(16)

Step-(v): Find out the authority ${e}_{p}$’s objective weight ${w}_{p}^{2}$ using the equation

$${w}_{p}^{2}=\eta {w}_{p}^{a}+\left(1-\eta \right){w}_{p}^{b} , p=\text{1,2},\dots ,m$$

(17)

where, $\eta \in [\text{0,1}]$ is an intuitionistic fuzzy parameter.

Step-(vi): Include the subjective weight ${w}_{p}^{a}$ and objective weight ${w}_{p}^{2}$ into the weight ${w}_{p}$ of the authority ${e}_{p}$ using the equation

$${w}_{p}=\gamma {w}_{p}^{a}+\left(1-\gamma \right){w}_{p}^{2} , p=\text{1,2},\dots ,m$$

(18)

where, $\gamma \in [\text{0,1}]$ is an intuitionistic fuzzy parameter.

Step-(vii): Calculate ${\mathcalligra{t}}_{i}^{\left(p\right)}$ using the formula,

$${\mathcalligra{t}}_{i}^{\left(p\right)}=\frac{1}{n}\sum_{j=1}^{n}{\mathcalligra{t}}_{ij}^{\left(p\right)} , \text{where } i=\text{1,2},\dots ,n \text { and } p= {1,2},\dots ,m.,$$

(19)

Step-(viii): Find out all values of ${\mathcalligra{t}}_{i}$(total intuitionistic fuzzy value) using the equation

$${\mathcalligra{t}}_{i}= \sum_{p=1}^{m}{w}_{p}{\mathcalligra{t}}_{i}^{\left(p\right)}, \forall i=\text{1,2},\dots ,n$$

(20)

Step-(ix): Estimate the score function through the equation

$${V}_{c}\left({\mathcalligra{t}}_{i}\right)={u}_{i}-{\vartheta }_{i},\forall i=\text{1,2},\dots ,n$$

(21)

The superior alternative ${\mathcalligra{t}}_{i}$ gives a greater value of ${V}_{c}\left({\mathcalligra{t}}_{i}\right)$, and this determines how the alternatives are ranked in order in a group.

Procedure II

Step-(x): Figure out the loyal IFPR as $D = ({x}_{ij}{)}_{n\times n}$ and find the matrix as follows:

$$D{=x}_{ij}=\left({\mu }_{ij},{\vartheta }_{ij}\right)=\left(\sum_{l=1}^{m}{w}_{l}{\mu }_{ij}^{(l)},\sum_{l=1}^{m}{w}_{l}{\vartheta }_{ij}^{(l)},\right),i,j=\text{1,2},\dots ,n$$

(22)

Step-(xi): For every alternative x_i, estimate the variation coefficient value ${V}_{c}\left({D}^{i},{D}^{+}\right)$ among Dⁱ and D⁺ and the variation coefficient value ${V}_{c}\left({D}^{i},{D}^{-}\right)$ between Dⁱ and D⁻ by the following equations

$${V}_{c}\left({D}^{i},{D}^{+}\right)=\frac{1}{n}\left\{2\alpha \sum_{j=1}^{n}\frac{{\mu }_{ij}}{{\mu }_{ij}^{2}+{\vartheta }_{ij}^{2}}+(1-\alpha )\sum_{j=1}^{n}\frac{{\mu }_{ij}}{\sqrt{{\mu }_{ij}^{2}+{\vartheta }_{ij}^{2}}}\right\}$$

(23)

And

$${V}_{c}\left({D}^{i},{D}^{-}\right)=\frac{1}{n}\left\{2\alpha \sum_{j=1}^{n}\frac{{\vartheta }_{ij}}{{\mu }_{ij}^{2}+{\vartheta }_{ij}^{2}}+(1-\alpha )\sum_{j=1}^{n}\frac{{\vartheta }_{ij}}{\sqrt{{\mu }_{ij}^{2}+{\vartheta }_{ij}^{2}}}\right\}$$

(24)

Step-(xii): For each alternative x_i, determine its estimation value by the equation

$${V}_{c}\left({x}_{i}\right)=\frac{{V}_{c}\left({D}^{i},{D}^{+}\right)}{{V}_{c}\left({D}^{i},{D}^{+}\right)+{V}_{c}\left({D}^{i},{D}^{-}\right)}$$

(25)

The larger the quantity of $V$(x_i), the higher the alternate x_i. Then we incorporated the rank of the substitutes. The subsequent two scenarios are supposed to illustrate how to obtain the included weights assists to rank the alternatives.

Flow chart

The VCM and LE-based GDMP working algorithm is shown in the following flow chart.

Real world application

Decision-making implementation

We apply the idea of IFGs to GDMP in this section. To show $how$ the idea of IFGs works in a vague situation based on IFPRs, the problem of how to choose an alliance partner for a software company is solved.For instance, the IFPR framework explains how to use an algorithm to choose an alliance partner for a software business.

Choosing the most acceptable alliance partner for Huawei software company

An alliance is created when two or more organizations or individuals decide to collaborate without compromising their independence. While maintaining their distinct identities, the firms in the alliance cooperate to accomplish a common purpose. In line with the corporate objectives, aims, and strategies, the alliance partner is in charge of creating and executing business and strategic connections with organizations.

Huawei is one of China's top five software businesses and it is a multinational telecommunications corporation that creates and exports smartphones, laptops, and communications devices. For 2020, this company projected annual sales of $136.7 billion. To build a completely connected, intelligent world, their business is committed to providing digital technology to every person, house, and organization. With operations in more than 170 nations and 197,000 employees, this corporation serves more than three billion people globally. Additionally, it offers a broad range of commercial offerings, such as industrial solutions, software platforms, and services related to product engineering. Through partnerships, open innovation, leadership and employee competence development, and continual organizational and process improvement, it is dedicated to becoming a global leader in IT solutions and services.

Currently, Huawei intends to form a strong alliance with a multinational corporation to boost its operations and competitiveness in the global market. Following extensive negotiations, five global businesses (alternatives) have shown an interest in forming a strategic alliance with Huawei. They are Alpha (${V}_{1}$), Beta (${V}_{2}$), Zeta (${V}_{3}$) and Delta (${V}_{4}$).A team of three decision-makers ${e}_{k}$ ($k=\text{1,2},3)$ from Huawei’s company operations outsource division, operation department managers, and engineering department managers were formed to pick the desired strategic alliance partner based on the following three criteria.

• 1. Reliable and experienced in the corresponding company.

• 2. Financially stable and willing to invest in the company.

• 3. Competent to bring a new business.

The following IFPRs are used to compare the alternatives using the criteria, and decision-makers use these results to express their own judgments.

The IFAM of $IFG$(${G}_{1})$ is

$$A\left({G}_{1}\right)=\left[\begin{array}{cccc}(0.0 , 0.0)& (0.5 , 0.3)& (0.2, 0.1)& (0.6 , 0.35)\\ (0.3 , 0.5)& (0.0 , 0.0)& (0.3 , 0.2)& (0.4 , 0.25)\\ (0.1 , 0.2)& (0.2 , 0.3)& (0.0 , 0.0)& (0.3 , 0.15)\\ (0.35 , 0.6)& (0.25 , 0.4)& (0.15 ,0.3)& (0.0 , 0.0)\end{array}\right]$$

The IFAM of $IFG$(${G}_{2})$ is

$$A\left({G}_{2}\right)=\left[\begin{array}{cccc}(0.0 , 0.0)& (0.6 , 0.35)& (0.4 , 0.25)& (0.3 , 0.2)\\ (0.35 , 0.6)& (0.0 , 0.0)& (0.2 , 0.1)& (0.5 , 0.3)\\ (0.25 , 0.4)& (0.1 , 0.2)& (0.0 , 0.0)& (0.2 , 0.1)\\ (0.2 , 0.3)& (0.3 , 0.5)& (0.1 , 0.2)& (0.0 , 0.0)\end{array}\right]$$

The IFAM of $IFG$(${G}_{3})$ is

$$A\left({G}_{3}\right)=\left[\begin{array}{cccc}(0.0 , 0.0)& (0.2 , 0.1)& (0.3 , 0.15)& (0.4 , 0.25)\\ (0.1 , 0.2)& (0.0 , 0.0)& (0.3 , 0.2)& (0.5 , 0.3)\\ (0.15 , 0.3)& (0.2 , 0.3)& (0.0 , 0.0)& (0.6 , 0.35)\\ (0.25 , 0.4)& (0.3 , 0.5)& (0.35 ,0.6)& (0.0 , 0.0)\end{array}\right]$$

Algorithm

Procedure-I

Step-(i): By formula (12), we compute the Laplacian energies of ${G}_{i} ,\forall i=\text{1,2},3.$

From Fig. 2 and A(G₁), we get $LE\left({G}_{1}\right)=(2.0480 , 2.0413)$,

Figure 2

GDM approach grounded on IFGs LE and Variation coefficient.

From Fig. 3 and A(G₂), we get $LE\left({G}_{2}\right)=(1.7745 , 2.1455)$,

Figure 3

$IFG$(${G}_{1})-$ IFPR for reliability and experience in the corresponding industry.

From Fig. 4 and A(G₃), we get $LE\left({G}_{3}\right)=(2.1941 , 1.8250)$.

Figure 4

$IFG$(${G}_{2})-$ IFPR for financially stable and willing to invest in the company.

Step-(ii): By the formula (13), we get the weights of ${G}_{i}$ determined with Laplacian energies.

$${w}_{1}^{a}=(0.3404 , 0.3395),$$

$${w}_{2}^{a}=\left(0.2949 , 0.3569\right),$$

$${w}_{3}^{a}=\left(0.3647 , 0.3036\right).$$

Step-(iii): Through (14) formula, we have

$${V}_{c}\left({G}_{1},{G}_{2}\right)=0.9112,$$

$${V}_{c}\left({G}_{1},{G}_{3}\right)=0.8748,$$

$${V}_{c}\left({G}_{2},{G}_{3}\right)=0.8098.$$

From Eq. (15), we get

$$\begin{aligned} {V}_{c}\left({G}_{1}\right)=0.8930,\\ {V}_{c}\left({G}_{2}\right)&=0.8604,\\ {V}_{c}\left({G}_{3}\right)&=0.8422. \end{aligned}$$

Step-(iv): By Eq. (16), we get

$$\begin{aligned} {w}_{1}^{b}=0.3440,\\{w}_{2}^{b}&=0.3315,\\ {w}_{3}^{b}&=0.3245. \end{aligned}$$

Step-(v): Taking $\eta =0.5$ in Eq. (17), we get

$$\begin{aligned} {w}_{1,\mu }^{2}&=0.3422 , {w}_{1,v}^{2}=0.3418,\\ {w}_{2,\mu }^{2}&=0.3132 , {w}_{2,v}^{2}=0.3442, {w}_{3,\mu }^{2}&=0.3446 , {w}_{3,v}^{2}=0.3141.\end{aligned}$$

So, the weights of authorities are

$$\begin{aligned} {w}_{1}^{2}=\left(0.3422 , 0.3418\right),\\ {w}_{2}^{2}&=\left(0.3132 , 0.3442\right),\\ {w}_{3}^{2}&=\left(0.3446 , 0.3141\right).\end{aligned}$$

Step-(vi): Taking $\gamma =0.5$ in Eq. (18), we get

$$\begin{aligned} {w}_{1,\mu }&=0.3413, w_{1,\nu }=0.3407,\\ {w}_{2,\mu }&=0.3041, {w}_{2,\nu }=0.3506,\\ {w}_{3,\mu }&=0.3547, {w}_{3,\nu }=0.3089 \end{aligned}$$

So, the impartial weights are

$$\begin{aligned}{w}_{1}&=(0.3423 , 0.3407),\\ {w}_{2}&=(0.3041 , 0.3506),\\ {w}_{3}&=\left(0.3547 , 0.3089\right). \end{aligned}$$

Step-(vii): By Eq. (19), we have ${\mathcalligra{t}}_{i}^{\left(p\right)}=\frac{1}{n}\sum_{j=1}^{n}{\mathcalligra{t}}_{ij}^{\left(p\right)} , i=\text{1,2},\dots ,m \text{and }p=\text{1,2},\dots ,m.$

Then from Fig. 2 and A(G₁) we get

$$\begin{aligned} {\mathcalligra{t}}_{1}^{\left(1\right)}&=\left(0.4333 , 0.2500\right), {\mathcalligra{t}}_{2}^{\left(1\right)}=\left(0.3333 , 0.3167\right),\\ {\mathcalligra{t}}_{3}^{\left(1\right)}&=(0.2000 , 0.2167), {\mathcalligra{t}}_{4}^{\left(1\right)}=(0.2500 , 0.4333). \end{aligned}$$

From Fig. 3 and A(G₂) we get

$$\begin{aligned} {\mathcalligra{t}}_{1}^{\left(2\right)}&=(0.4333 , 0.2667), {\mathcalligra{t}}_{2}^{\left(2\right)}=(0.3500 , 0.3333),\\ {\mathcalligra{t}}_{3}^{\left(2\right)}=(0.1833 , 0.2333), {\mathcalligra{t}}_{4}^{\left(2\right)}=(0.2000 , 0.3333) \end{aligned}$$

From Fig. 4 and A(G₃) we get

$$\begin{aligned} {\mathcalligra{t}}_{1}^{\left(3\right)}&=\left(0.3000 , 0.1667\right), {\mathcalligra{t}}_{2}^{\left(3\right)}=\left(0.3000 , 0.2333\right),\\ {\mathcalligra{t}}_{3}^{\left(3\right)}=(0.3167 , 0.3167), {\mathcalligra{t}}_{4}^{\left(3\right)}=\left(0.3000 , 0.5000\right)\end{aligned}$$

Step-(viii): From Eq. (20), we get

$$\begin{aligned} {\mathcalligra{t}}_{1,u}&=0.3861, {\mathcalligra{t}}_{1,\nu }=0.2302, {\mathcalligra{t}}_{2,u}=0.3266, {\mathcalligra{t}}_{2,v}=0.2968\\ {\mathcalligra{t}}_{3,u}&=0.2363, {\mathcalligra{t}}_{3,v}=0.2535, {\mathcalligra{t}}_{4,u}=0.2526, {\mathcalligra{t}}_{4,v}=0.4189.\end{aligned}$$

Therefore $\begin{aligned} {\mathcalligra{t}}_{1}&=(0.3861 , 0.2302), {\mathcalligra{t}}_{2}=(0.3266 , 0.2968)\\ {\mathcalligra{t}}_{3}&=(0.2363 , 0.2535), {\mathcalligra{t}}_{4}=\left(0.2526 , 0.4189\right)\end{aligned}$.

Step-(ix): By Eq. (21), we get

$$\begin{aligned} {V}_{c}\left({\mathcalligra{t}}_{1}\right)&=0.1559, {V}_{c}\left({\mathcalligra{t}}_{2}\right)=0.0298,\\ {V}_{c}\left({\mathcalligra{t}}_{3}\right)&=-0.0172, {V}_{c}\left({\mathcalligra{t}}_{4}\right)=-0.1663. \end{aligned}$$

Therefore

$${V}_{c}\left({\mathcalligra{t}}_{1}\right)>{V}_{c}\left({\mathcalligra{t}}_{2}\right)>{V}_{c}\left({\mathcalligra{t}}_{3}\right)>{V}_{c}\left({\mathcalligra{t}}_{4}\right)$$

and hence

$${\mathcalligra{t}}_{1}>{\mathcalligra{t}}_{2}>{\mathcalligra{t}}_{3}>{\mathcalligra{t}}_{4}.$$

Procedure II

In this scenario, we highlight the position outcome potential using our comparable variation coefficient approach. Through (22) in method II, we form the group IFPR as follows.

As of the matrices A(G₁), A(G₂) and A(G₃) we get

$$D=\left[\begin{array}{cccc}(\text{0,0})& \left(0.4246, 0.2558\right)& (0.2965, 0.1681)& (0.4385, 0.2666)\\ (0.2446, 0.4425)& (\text{0,0})& (0.2699, 0.1650)& (0.4663, 0.2830)\\ (0.1635, 0.3011)& (0.1698, 0.2650)& (\text{0,0})& (0.3763, 0.1943)\\ (0.2693, 0.4332)& (0.2832, 0.4660)& (0.2059, 0.3577)& (\text{0,0})\end{array}\right]$$

By using the Eqs. (23) and (24), we achieve

$$\begin{gathered} V_{C} \left( {D^{1} , \, D^{ + } } \right) \, = { 1}.{8}0{88},\quad V_{C} \left( {D^{2} , \, D^{ + } } \right) \, = { 1}.{5792}, \hfill \\ V_{C} \left( {D^{3} , \, D^{ + } } \right) \, = { 1}.{5391},\quad V\left( {D^{4} , \, D^{ + } } \right) \, = \, 0.{9923}, \hfill \\ \end{gathered}$$

and

$$\begin{gathered} V_{C} (D^{1} ,D^{ - } ) = { 1}.{3396},\quad V_{C} (D^{2} ,D^{ - } ) = { 1}.{8511}, \hfill \\ V_{C} (D^{3} ,D^{ - } ) = {2}.{5941},\quad V_{C} (D^{4} ,D^{ - } ) = {2}.{3161}. \hfill \\ \end{gathered}$$

Next, for each choice ${x}_{i},(i= 1, 2, 3, 4)$, Eq. (25) delivers the computation standards as

$${V}_{C}({x}_{1}) = 0.6292, {V}_{C}({x}_{2}) = 0.5447, {V}_{C}({x}_{3}) = 0.4537, {V}_{C}({x}_{4}) = 0.3749.$$

Since ${V}_{C}({x}_{1})$>${V}_{C}({x}_{2})$>${V}_{C}({x}_{3})$>${V}_{C}({x}_{4})$, we have ${x}_{1}>{x}_{2}>{x}_{3}>{x}_{4}$.

As a result, a committee of decision-makers from Huawei’s company discovered that the company Alpha $({V}_{1})$ fulfilled the company's criteria (requirements) and formed a strong alliance with Huawei multinational corporation (Fig. 5).

Figure 5

$IFG$(${G}_{3})-$ IFPR for the ability to bring a new business.

Accordingly, Table 1 presents the weight function and possible alternatives using Xu's functioning algorithm procedure-I, whereas Table 2 presents its ranking orders. Also, Tables 3 and 4 exhibit the variation coefficient measures and their ranking orders utilizing Xu's working algorithm procedure II. We acknowledge that for all values of $\eta \in [\text{0,1}]$ and $\gamma \in \left[\text{0,1}\right]$, the ranking order of the two methods stays the same.

Table 1 Weights of the alternatives for various values of “$\eta$ and $\gamma$” using procedure I.

Table 2 The ranking order of the alternatives by procedure I.

Table 3 The variation coefficient measure values using procedure II.

Table 4 The ranking order of the alternatives by procedure II.

Table 1 represents the objective weights $({w}_{p}^{2}$), subjective weights ${(w}_{p})$ and the total intuitionistic fuzzy values $({\mathcalligra{t}}_{i})$ for different values of $\eta and\gamma$.

Table 2 represents the values of the alternatives (${V}_{C}\left({\mathcalligra{t}}_{1}\right)$, ${V}_{C}\left({\mathcalligra{t}}_{2}\right)$, ${V}_{C}\left({\mathcalligra{t}}_{3}\right)$, ${V}_{C}\left({\mathcalligra{t}}_{4}\right)$) and the ranking order of the alternatives for different values of $\gamma$ using procedure-I and the experts team identified that ${t}_{1}$ ranked first among others.

Table 3 represents the variation coefficient value ${V}_{c}\left({D}^{i},{D}^{+}\right)$ among Dⁱ and D⁺ and the variation coefficient value ${V}_{c}\left({D}^{i},{D}^{-}\right)$ between Dⁱ and D⁻ for different values of $\alpha$.

Table 4 shows the values and ranking order of the alternatives for different alpha values using procedure II. It also helps to identify the best alternatives among others.

In the aforementioned results (Tables 1, 2, 3 and 4), procedures I and II investigate the significance of alternatives and determine the best alternatives by finding the objective weights, subjective weights, total intuitionistic fuzzy values, and variation coefficient values. After comparing the ranking orders of the two working procedures, we found that company Alpha $\left({V}_{1}\right)$ met the company's criteria and ranked first, followed by company Beta $\left({V}_{2}\right)$ in second place, company Delta $\left({V}_{4}\right)$ in third place, and company Zeta $\left({V}_{3}\right)$ in last place.

After conducting a general evaluation, the decision makers team identified that the rankings produced by the two working methodologies were identical. The proposed approach stands out as the best one despite the fact that the ranking order remains the same when compared to the different similarity measures due to an aspect termed "time consumption.". Our technique deviates from earlier similarity measures¹³ in terms of time consumption. This method yields results a little bit faster than previous mentioned method.

Conclusion

The various entropy and similarity estimation methods have been employed to resolve recent GDM problems utilizing intuitionistic fuzzy proofs. This study introduces the variation coefficient measure as a statistical technique for intuitionistic fuzzy graph preferences and highlights the significance of Laplacian energy in this scenario. Also utilize Xu’s approach to turn the intuitionistic preference connections that closely resemble substitutes into internal weights for the authorities, thus extracting information. The suggested methodology utilizes two working techniques to establish a ranking order and facilitate a comprehensive comparison of preferences (ranks). The variation coefficient and Laplacian energy were successfully applied to pick alliance partners, and their implementation will facilitate the assignment of replacement rankings. We successfully implemented the presented work to choose the best alliance partner for a software company and found that Alpha $({V}_{1})$ met the company's criteria, forming a strong alliance with a multinational software company. In the future, the authors will extend the work to statistical metrics such as correlation coefficient, regression coefficient, and association coefficients can be applied to additional forms of IFG in GDM techniques. The research study offers various advantages in the scenario of decision-making strategies, but also, has some limitations while applying the various fuzzy graphs. This concept can be extended to several forms of intuitionistic fuzzy graphs, such as Pythagorean fuzzy graphs, Hesitant fuzzy graphs, and Fermatian fuzzy graphs. This approach allows for the examination of various components within these graphs in future research work.