The Correlation Coefficient of Hesitancy Fuzzy Graphs in Decision Making

The hesitancy fuzzy graphs (HFGs), an extension of fuzzy graphs, are useful tools for dealing with ambiguity and uncertainty in issues involving decision-making (DM). This research implements a correlation coefficient measure (CCM) to assess the strength of the association between HFGs in this article since CCMs have a high capacity to process and interpret data. The CCM that is proposed between the HFGs has better qualities than the existing ones. It lowers restrictions on the hesitant fuzzy elements’ length and may be used to establish whether the HFGs are connected negatively or favorably. Additionally, a CCMbased attribute DM approach is built into a hesitant fuzzy environment. This article suggests the use of weighted correlation coefficient measures (WCCMs) using the CCM concept to quantify the correlation between two HFGs. The decisionmaking problems of hesitancy fuzzy preference relations (HFPRs) are considered. This research proposes a new technique for assessing the relative weights of experts based on the uncertainty of HFPRs and the correlation coefficient degree of each HFPR. This paper determines the ranking order of all alternatives and the best one by using the CCMs between each option and the ideal choice. In the meantime, the appropriate example is given to demonstrate the viability of the new strategies.


Introduction
When experts have a variety of opinions on an element, HFS, which allows the membership grade of an element to be represented by various potential values between 0 and 1, is capable of determining the membership grades. In group decision making (GDM) with anonymity, HFS outperforms regular fuzzy sets and their various expansions in many ways. As a result, it has attracted a lot of academics. Following the introduction of HFS, Torra et al. [1] provided several fundamental operations on HFSs, including complement, union, and intersection. The components of an HFS are referred to as hesitant fuzzy elements (HFEs) by Xia et al. [2], who also provide the addition and multiplication operations over HFEs. Following that, Liao et al. [3] developed the HFSs' subtraction and division operations. Different aggregation operators, such as the HFWA, HFWG, HFOWA, HFOWG, HFHA, and HFHG operators [2], were suggested to combine the hesitant fuzzy information in multiple-criteria decision-making based on the extension concept by Torra [4]. Later, Liao et al. [5,6] added several tentative fuzzy hybrid weighted aggregation operators that offer many benefits over the aforementioned operators. A few dynamic aggregation methods were also put out by Liao et al. [7] to combine several stages of hazy information. Liao et al. [8] established the HFPR and looked into its multiplicative consistency to better assist the DM process. Zhu et al. [9] developed a goal programming approach based on the HFPR and used it for GDM to generate a ranking from an HFPR. Some DM techniques, including the hesitant fuzzy VIKOR method [10], the hesitant fuzzy TOPSIS method [11], the hesitant fuzzy TODIM method [12], and the satisfaction degree-based interactive DM technique [13], have been developed for DM with hesitant fuzzy formation. All of these accomplishments support the notion of HFS and demonstrate how effective it is at handling ambiguity and uncertainty in the DM process. One of the most commonly used metrics in data analysis, pattern recognition, machine learning, decision-making, etc. is correlation. It gauges how smoothly two variables travel along a linear path. The CCM has been expanded into several ambiguous situations since it initially emerged in Karl Pearson's concept of statistics. Fuzzy correlation and CCMs [14][15][16][17][18][19] are classified into three types: hesitant fuzzy correlation and CCMs [20], intuitionistic fuzzy correlation and CCMs [21][22][23][24][25][26][27][28], and fuzzy correlation and CCMs [29][30][31][32][33][34][35][36].
Additionally, certain tentative fuzzy information measures have been created, including entropy measures [29], similarity measures (SMs) [30], and distance measures [31][32][33][34][35]. It is believed that the correlation is a kind of ambiguous link between the variables [36,37]. The statistical and engineering sciences depend heavily on it [38]. The CC has also been expanded to support various fuzzy settings [15]. Many CCMs of HFSs were put out by Chen et al. [39] and used in clustering analysis. The intervalvalued HFSs and the accompanying correlation coefficient formulae were created by Chen et al. [39], who also used a particular numerical example to show how they may be used to cluster data containing interval-valued reluctant fuzzy information. Xu and Xia analysed SMs of HFSs based on distance measures, while Xu et al. [30] investigated CCMs of HFSs. Meng et al. [40] developed some novel Shapley weighted correlation coefficients for hesitant fuzzy sets without accounting for the length of the hesitant fuzzy elements or the order of their potential values. Guidong Sun et al. additionally included two examples concerning medical diagnosis and cluster analysis to compare the enhanced weighted correlation coefficient with the current CCMs. These improved versions were offered in the context of mathematics and stochastic process principles to make them more understandable. A new CCM between HFSs that has more appealing features than the ones currently in use was reported by Liu et al. in [41].
Hesitancy fuzzy graphs (HFGs), a novel graph structure, and some fundamental ideas about it were presented by Pathinathan et al. [42] in 2015. Despite introducing the idea of hesitant fuzzy elements (HFEs) to the vertices and edges of the graph, they didn't do so. Instead of HFEs, they employ IF-values, and they represent these IF-values using triples that include the membership, hesitancy, and nonmembership degrees of vertices and edges. It can be shown that Pathinathan et al. [42] definitions have several structural characteristics in common with neutrosophic graphs [43][44][45]. Karaaslan [46] established the terms homomorphism, isomorphism, weak isomorphism, and co-weak isomorphism between two HFGs as well as created some new notions, including the Cartesian product of HFSs and HFPRs. Furthermore, certain current CCMs are shockingly ambiguous when the two things are the same. One of the most prevalent human activities is DM. The main challenge is determining the finest technique to order an alternative or choose the best option [47]. It is possible to apply HFSs to a wide range of DM problems. A DM problem involving multiple attributes and multiple people requires the optimal alternative. Reddy et al. [48] presented a concept of Laplacian energy (LE) of hesitancy fuzzy graphs (HFGs) in DM problems. In addition, the evaluation of reservoir schemes and satellite communication systems uses practical examples to illustrate their applicability.
The format of the article is as follows. Section 2 provides an overview of the literature relevant to the given work and identifies the research gap. Section 3 presents the fundamental concepts of HFG and the measures of correlation coefficients between the HFGs. Section 4 describes a method for calculating the expert scores as well as a working procedure and flow chart based on HFGS. Section 5 provides an illustrative example that verifies the working procedure. In the conclusions section, this research presents the results of this study.

Literature Review
Researchers have carried out several useful studies for Electric Vehicle Charging Stations (EVCSs). The literature presents a number of multi-criteria decision-making based location models based on different objectives. One of the most widely used decision making techniques is objective decision making.
Guo et al. [49] used the multi-criteria decision-making (MCDM) technique to take into consideration some subjective and crucial factors for choosing an EVCS location using the fuzzy TOPSIS technique. Shahraki et al. [50] developed an optimization model based on vehicle travel patterns to optimize public charging demand and determine where to locate public charging stations for maximum electric vehicle mileage. Considering spatial temporal constraints such as electric taxi range, charging time, and charging station capacity, Tu et al. [51] developed a spatial-temporal demand coverage technique for improving the placement of charging stations for electric taxis. A case study was presented by He et al. [52] in response to a request for information on the location of public electric vehicle (EV) charging stations in Beijing, China. Liu et al. [53] proposed an integrated MCDM approach to location planning of EVCS where the criteria weights are determined using the grey DEMATEL model by Liu et al. and the optimal location is assessed and chosen using the UL-MULTIMOORA technique. Ren et al. [54] proposed a hesitant fuzzy linguistic stepwise weight assessment ratio analysis with the weight aggregated sum product assessment (HFL-SWARA-WASPAS) model to handle this problem. The HFL-SWARA approach is employed to calculate the weights of criteria. Wei et al. [55] developed new probabilistic language weighted dice similarity measures (PLWDSM) and probabilistic linguistic-weighted generalized dice similarity measures (PLWDGDSM) (PLWGDSM).
The mentioned literature review highlights various challenges related to EVCS location selection studies. Previous research has largely treated assessment factors as independent when developing site selection models. However, in many real-world circumstances, criteria may have convoluted and interconnected ties. Therefore, the purpose of this study is to fill these gaps by extending Xu's method based on the correlation coefficient measures for the evaluation and selection of EVCSs. Further, the energy technique is utilized to determine the weights of criteria by considering their interactions.

Preliminaries
This section presents several topics relevant to HFGs: hesitancy fuzzy preference relations (HFPR), correlation coefficient (CC), and the energy of HFGs.
Definition 3.1. Suppose that X is the fixed set that has been introduced into the HFS. An HFS is defined on Y using a function that, when applied to Y, gives the subset of [0, 1] as shown in the following mathematical symbol: where the hesitant fuzzy element (HFE) h X Y ð Þ is defined as a collection of different values in the interval 0; 1 ½ , that indicates the various degree of membership to which the values yEY to the set X .

Definition 3.2.
Considering h is the HFE, the lower and upper limits of an HFE are described below: The covariance of HG 1 and HG 2 is defined as follows The correlation coefficients of HFG HG 1 and HG 2 are defined as follows Á represent the energy of hesitancy fuzzy membership, nonmembership, and hesitant element matrix.

Method for Calculating Expert Scores
The optimal ranking result is achieved by taking a pair-wise comparison of two alternatives and building an HFPR to avoid the influence of the limited ability of human thinking in the decision-making process. Thus, the most common way for decision makers to express their preferences is through the use of HFPRs. Assume that Y ¼ y 1 ; y 2 ; . . . ; y n f gis the set of replacements, and E ¼ e 1 ; e 2 ; . . . ; e m f gis the set of experts. The expert e m offers preference information to every couple of replacements and develops a hesitancy fuzzy preference relationship (HFPRS) where a l ij ¼ r l ij ; s l ij ; q l ij , 0 r l ij þ s l ij þ q l ij 1 for all i is having 1; 2; : : : ; n: Energy can determine the ambiguous HFGs indicator. Every HFPR A l seems to be an HFG in the adjacent matrix such that the ambiguous specificity is determined by energy. There are certain values between 0 and 1 in an HFPR. As a result of the complexity of the decision-making process, the lack of knowledge about the domain, and so on, people may have a set of possible values when determining the preference degree of one object over another. This type of preference information is ideally portrayed by HFEs, which enable elements to belong to a set represented by several possible values. A decisionmaking problem often involves predetermined weights for experts based on their social position, prestige, and competence recognized in specific domains. HFPRs may not always be taken into account, even when they reflect the expert's actual expertise, since even their judgement about the alternatives is measured during the problem-solving process. In the practical decision-making process, weights can play a crucial role in ranking alternatives. Therefore, how we allocate reasonable weights to the experts becomes crucial. The predefined significance weights of the experts are considered subjective weights in this paper. By comparing the HFPRs with the predefined weights, it may be possible to derive the objective weights of the experts more accurately by looking at their preferred replacements based on their HFPRs. It is usually anticipated that the uncertainty degree of the HFPR will be as small as possible at the time of the decision-making process for more certainty of the results to be obtained. As a result, the correlation coefficient degree CC A l ; A k À Á among the two HFPRs A l ; A k and the mean degree of correlation coefficients of A l to the others can be computed.
As per the mentioned analysis, to evaluate the objective weights of the experts, this paper designed the following working procedure.

Working Procedure
A working procedure is being developed for GDM valid problems, focusing on HFPRs.
Consider w ¼ w 1 ; w 2 ; w 3 ; . . . ; w m f g is a subjective scoring vector of experts for the decisionmaking issues based on HFPRs, where w m . 0, m ¼ 1; 2; 3; ::; l, and the total of all the scoring values of the experts is equal to one is written as P l i¼1 w i ¼ 1.
The average correlation coefficient degree CC M k ð Þ À Á of M k ð Þ to the others is calculated by Stage V. Compute the "objective" scores w 2 k of the expert e k is w 2 Stage VI. Compute the subjective and objective scores w 1 k and w 2 k of the expert e k is Stage II. Compute the values of r k ð Þ i equivalent to m experts in to a collection of HFVs of the replacements t i to another replacement is Stage III. Compute the score function of r i is where the score functions are constructed based on the higher value of the replacement, and then a ranking order is constructed.
Working procedure I consider both subjective and objective information when weighing the weights of experts. Based on the objective and subjective weight information, the decision-maker determines the value of c. A collective HFPR is then created by integrating the HFPRs.
Working Procedure-II Stage I. Compute the cooperative HFPR A l ¼ a ij À Á nÂn by Stage II. Compute the CCMs CC B i ; B þ ð Þbetween B i and B þ for every alternative t i is Stage III. Compute the CCMs CC M i ; M À ð Þbetween M i and M À for each replacement t i is Stage IV. For every replacement t i , compute the values of g t i ð Þ is The maximum value of g t i ð Þ is greater than the replacements t i . And we estimate the rank of the replacements. The working procedures I and II are given to illustrate how to achieve absorbed scores to classify replacements in the two following instances. The order of rankings of the replacements is now assigned. The above working procedure have seen in Fig. 1.

Application: Choosing a Prominent Location for Electric Vehicle Charging Stations
Electric vehicles are the workhorses of today's public transportation system, offering accessible transportation to everyone. They represent a crucial first step in introducing large electric vehicles to the larger transportation industry since they are an integral part of everyday life in many cities. A piece of technology that provides electricity to EVs is known as an EV charger. Its primary function is to maintain a vehicle's mobility by recharging the battery of an EV. Due to the availability of certain smart charging Figure 1: The measure of the correlation coefficient between hesitancy fuzzy graphs stations, you may never need to visit a service station if you need to charge regularly. The main advantage of electric vehicles is the improvement in air quality they may bring to urban areas. Additionally, EVs may reduce the pollutants that contribute to pollution and environmental change, enhancing public health and minimizing ecological harm. Because there are so many different kinds of EV charging stations available, some people may find it perplexing. So, to assist you in making the best decision, below is a list of EV charging stations.
Choose Trickle charge if you're EV needs a little more power to get through the day. A more moderate charge is delivered in this mode. It includes a typical three-prong, 220 V connector that is used to charge your car, making it ideal for charging smaller vehicle types. Because it is the simplest, AC charging is the most popular way to charge EVs. The charging point is directly plugged into your home's network before being connected by cable to your automobile and supplying power to the vehicle's battery. Direct current, which travels straight from the source to the car, is used by fast chargers. Fast chargers skip the converter, allowing the batteries to charge more quickly. A company wishes to launch smart charging stations for electric vehicles charging at some particular locations. This research took into account the four locations on the Dwight D. Eisenhower highway road from Denver to Kansas in the United States for launching EVCS such as A 1 ¼ Strasburg; A 2 ¼ Liman; A 3 ¼ Flaglers; and A 4 ¼ Burlington as shown in Figs. 2 and 3. Also, this research considered these locations as alternatives The selection of the finest location from the four alternatives for launching EVCSs is based on the criteria's as Economic Factors, Land values, Petrol stations. Here we consider C ¼ C 1 ¼ Economic Factors; C 2 ¼ Land values; C 3 ¼ Petrol stations f g with preference information provided in the form of HFPR R ¼ r ij À Á , for the particular constraints that appear in the matrices below, respectively. Suppose that in the GDM problem, there are four replacements t i ; i ¼ 1; 2; 3 ð Þand three experts e b ; b ¼ 1; 2; and 3 ð Þ . Assume that the scores for every expert are 0:5 respectively and build the HFPRs as shown below. In the first place, DMs rated the language for each of the criteria based on weighted significance scores (see Table 1).
Adjacency matrix from Fig. 4      Stage IV: By calculating the values of the scores w b k using Eq. (13), we get w b ¼ 0:3289; 0:3310; 0:3401 ð Þ Stage V: Compute the "objective" scores w 2 k of the expert e k . Suppose g ¼ 0:5, which means the objective weight is affected by half of the weight determined by the CCM. The objective weighting vector can be obtained by Eq. (14), we get