**1. Introduction**

Electric vehicles play a very momentous role in addressing fossil fuel pollution and they are capable of making a paradigm shift in the entire transportation sector. Transportation is a significant contributor to urban air pollution and the reduction of urban city emission is the need of the hour. Electric vehicles make the world more liveable and provide a pollution-free mode of transportation in urban areas. The high level of pollution is degrading the environment and it has made the concept of sustainable development a fairy-tale phenomenon [1,2]. Sustainable consumption needs to be adopted by conducting timely environmental and sustainability assessments to prevent any large-scale ecological disaster from happening. This is where electrical vehicles come into the picture, with sustainability, environment-protecting and pocket-friendly being a few of their rewards. Electric vehicles use a minimum of one electric motor or traction motor for propulsion. They maybe self-contained with a generator or battery for converting the fuel into electricity or they may be power-drivenvia a collector scheme by using electricity from off-vehicle sources [3]. The problem of the energy crisis in the world can be tackled in the future by using this option. Ever-rising gas prices force people to look for alternative modes of

**Citation:** Ghosh, A.; Ghorui, N.; Mondal, S.P.; Kumari, S.; Mondal, B.K.; Das, A.; Gupta, M.S. Application of Hexagonal Fuzzy MCDM Methodology for Site Selection of Electric Vehicle Charging Station. *Mathematics* **2021**, *9*, 393. https://doi.org/10.3390/math9040393

Academic Editor: Michael Voskoglou

Received: 28 December 2020 Accepted: 9 February 2021 Published: 16 February 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

energy supply, or to even switch over to walking or availing public transportation. The other reason why electrical vehicles are preferred over fossil-fueled vehicles is due to their eco-friendly nature and the cheaper cost of driving them [4,5]. Filling up an electric car costs less than filling up a full tank. Internal combustion engines emit more CO2 emissions than electric vehicles [6–8].

Moreover, for the promotion of charging stations, we need to focus on two facets: the power of the government and the role of the market. The government influences taxation policies and subsidy policies directly on the stations, thus affecting the income of the charging station. On the other hand, from the market viewpoint, the key is the demand and supply relationship, i.e., the consumers who demand the electric vehicles and the supply of electric vehicle charging and the electric vehicle charging stations [9].Electric vehicles can usually be charged in three ways i.e., by using inductive charging, conductive charging and the battery replacement method [7].The inductive method of charging works through an electromagnetic transmission keeping no contact between the charging station and the electric vehicle. The conductive method has a battery connected by a cable which is directly plugged into an electricity provider, whereas the battery replacement method replaces the discharged batteries with new batteries in a charging station, keeping in mind the internal connections and the dimensions of the batteries, resulting in it being the least used method. The charging station operators mostly use conductive charging since it is more efficient and cheaper [10].The benefits of electric vehicles outweigh the problems. They are energy efficient since they convert a higher rate of electrical energy from the grid to power, compared to gasoline vehicles. They do not make noise or pollute the air since the electric motors have a quiet and smooth operation, requiring low maintenance. Their use of electricity reduces the need for fossil fuels [11]. The major problem with electric vehicles is their shorter driving range. The time taken to recharge the vehicle is also usually long and the battery packs are heavy and expensive [12]. Establishing an economic, efficient and convenient electric vehicle charging station will speed up the purchase of electric vehicles for consumers, thereby developing the sector and it viceversa will hinder electric vehicle adoption leading to lesser incentives for investing in its development [13]. Selecting a suitable site for electric vehicle charging stations is of utmost importance since it has a direct linkage to the operation efficiency and the service quality of the charging stations throughout its entire life cycle [14–16]. Electric vehicle charging station (EVCS) site selection requires a comprehensive analysis of social, economic, environmental, operations and urban planning for which several prospective and potential alternative locations were considered with respect to a range of criteria. Hence, location selection is seen as a multiple-criteria decision-making (MCDM) problem. The technique for order of preference by similarity to ideal solution (TOPSIS) is an aggregation of the MCDM method, having the benefits of computational efficiency and the ability to measure in simple mathematical form the relative performance of each alternative criterion [17–23]. However, due to information deficiency and vagueness along with human qualitative judgements, some criteria fail to be measured by a crisp value and can only be a fuzzy value [24] such as the development planning of road networks and petrol stations. Fuzzy theory effectively tackles this issue [25].In today's times, finding a suitable EVCS site requires a multi-criteria approach as well as accuracy and reliability in the maps [26]. The efficacy of the final decision depends on the quality of the data which are used to produce criteria maps. The geographic information system-based multi-criteria decision analysis (GIS-based MCDA) method converts spatial as well as non-spatial data into information with respect to the decision-maker's own judgement [27]. To promote e-vehicle usage, longterm infrastructure installation is the need of the hour. For long-term infrastructure creation, detailed study of the criteria and in-depth analysis is required. This paper addresses the problem of charging infrastructure creation for e-vehicles in a city environment. This model can be replicated in different cities for a pollution-free mode of transport. Widespread usage of e-vehicles will make the environment greener.

#### *1.1. Literature Review*

The construction of an electric vehicle charging station is the turning point in its life cycle. Selecting an appropriate place for setting up the charging station and determining its capacity will benefit all the stakeholders and endorse sustainable development of the entire industry. Some studies have been shown this with respect to economy and technology. Lee et al. [28] examined the price competition among EVCSs using renewable power generators by employing game theory with relevant physical constraints. Rivera et al. [29] put forward a novel architecture for plug-in electric vehicles: a DC charging station using a grid-tied neutral point clamped converter. Wang et al. [30] utilized threephase uncontrolled rectification chargers to study the harmonic amplification of EVCSs. Ding, Hu and Song [31] studied the energy storage system as a potential supplement of an electric bus fast charging station by employing mixed integer nonlinear programming for valuing the energy storage system. Fan, Sainbayar and Ren [32] calculated the effect of limiting electric vehicles' full state of charge to total charged energy and the revenue of EVCS. Capasso and Veneri [33] built a DC fast charging architecture for plug-in hybrid vehicles as well as fully electric vehicles by integrating the fleets of hybrid/road electric vehicles with renewable energy sources. Li et al. [34] studied the control of EVCSs and the management of energy and fit them in a dynamic price framework by developing a realtime simulation system for evaluating how the EVCS meets the charging and discharging requirements for grid-to-vehicle, vehicle-to-grid and vehicle-to-vehicle. Nansai et al. [35] conducted the life-cycle analysis on EVCSs in three phases of installation, transportation and production of electric vehicle charging equipment and then compared the carbon dioxide, carbon monoxide, sulphur oxides and nitrogen oxide emissions of electric vehicles and gasoline vehicles.

Research also focuses on the electric vehicle's size, its placement and the fields in which it is used. Khalkhali et al. [36] determined the optimal location and size of plug-in hybrid EVCSs using data envelopment analysis by maximizing the benefit of the distribution system management. Frade et al. [15] studied the location of EVCSs in Portugal by employing a maximal covering model for defining the capacity and number of EVCSs. Liu [37] investigated the nascent electric vehicle market in Beijing and then formulated an assignment model for different charging infrastructure assignment strategies. Wirges, Linder and Kessler [38] predicted several scenarios for charging infrastructure development until 2020 in Germany and also formulated a dynamic spatial model for the allocation of the EVCSs in the country. Wang et al. [39] developed an EVCS location model considering the electricity consumed along the roads and the oil sales, after which the EVCS quantity and layout were calculated. He et al. [40] developed an equilibrium modeling framework for deciding the optimal allocation of the charging stations in a metropolitan area. Liu et al. [41] proposed a two-step screening method for determining the optimal EVCS site, taking environmental factors and service radius into consideration.

There have been a large number of researches which have employed a type of decisionmaking method. Pashajavid and Golkar [42] put forward a scenario optimization algorithm by using multivariate stochastic modeling methodology for load demand by allocating the charging station of plug-in electric vehicles and also used a particle swarm optimization algorithm to minimize voltage deviation and energy loss in the distribution system. Chen, Kockelman and Khan [43] determined the optimal charging station location assignment in Seattle by developing a mixed-integer programming model which minimized the station access cost of the electric vehicle users and took trip attributes, parking demand, population density and local job as constraints. Sathaye and Kelley [44] used a continuous facility location model considering the demand uncertainty for finding the optimal location of the public-funded electric vehicle infrastructure on the highway corridors. Wang et al. [45] determined the optimal size and location of EVCSs with respect to smart grids by proposing a multi-objective EVCS planning method, maximizing the electric vehicle traffic flow under the constraint of battery capacity and the final optimal solution data-envelopment analysis was obtained and employed. Dong, Liu and Lin [46]

formulated an activity-based assessment method for the evaluation of the feasibility of electric vehicles taking the heterogeneous travelling population and subsequently applying a genetic algorithm for determining the sub optimal location. Xu et al. [47] identified the candidate centralized charging station using a proposed mathematical model with the minimum total transportation distance. You and Hsieh [48] used the mixed-integer programming model to determinine the best location of the EVCS which would maximize the number of people who could complete round-trip itineraries, along with developing an efficient hybrid genetic algorithm which would obtain a compromised solution in a reasonable time. Lee et al. [49] first collected users' charging and traveling behaviors along with the batteries' state of charge and then proposed a location model of the rapid EVCS by using a probabilistic distribution function for the remaining fuel range. Baouche et al. [50] formulated an integer linear optimization model, taking the electric vehicle's input consumption as the optimal model. Sadeghi-Barzani [16] postulated a mixed-integer non-linear optimization approach for the determination of the optimal size and place of fast charging stations. Yao et al. [51] devised a multi-objective collaborative planning strategy to tackle the planning issue in electric vehicle charging systems and integrated power distributions by using a decomposition-based multi-objective evolutionary algorithm model and equilibrium-based traffic assignment model. Ma et al. [52] used an agent-based model to optimize the sketch of the initial EVCS. Chung and Kwon [53] devised a multi-period optimization model to perform the EVCS planning, basing it on the real traffic flow data of the Korean Expressway network in 2011. Lam, Leung and Chu [54] used four solution methods i.e., the greedy approach, chemical reaction optimization, iterative mixed-integer linear programming and effective mixed-integer linear programming for the EVCS place problem by checking them against practical and artificial cases. Cai et al. [55] found the relation between public charging infrastructure development and the travel patterns mined from big-data taken using large-scale trajectory data in Beijing.

Multi-criteria decision-making was used in transhipment site selection [56], shopping mall site selection [57], railway station site selection [58], waste management site selection [59] and macro-site selection of solar/wind hybrid power station [60].

The major drawback of the aforementioned studies which use multi-criteria decisionmaking approaches is the lack of incorporation between evaluation criteria and spatial data, since selecting a suitable site for an EVCS is a spatial decision predicament which involves conflicting, incommensurate and multiple evaluation criteria and a large set of evaluation criteria. Two unique parts of research i.e., multi-criteria decision-making and GIS can aid each other to overcome the intersections between evaluation criteria and spatial data. GIS integrates the spatially-referenced data in a problem-solving environment whereas multi-criteria decision-making refers to procedures and techniques used in structural decision problems by evaluating, prioritizing and designing the alternative decisions [61]. GIS-based multi-criteria decision-making techniques are usually used for spatial decision problems such as freight village site selection [62] wind power plant site selection [63] and refugee camp site selection [64]. Even though there have been multiple studies on the GIS-based multi-criteria decision-making approach on spatial decision problems as mentioned above, still there is a gap in the selection of EVCS sites using GIS.

#### *1.2. Objectives of the Study*

The present research has the following objectives:


#### *1.3. Novelties of the Study*

Several researchers have explored fuzzy numbers with MCDM techniques AHP, TOPSIS, and COPRAS. Hardly any research has been done using hexagonal fuzzy numbers

under MCDM methodology. The HFN defuzzification formulae has been developed and utilized. Distance measured between two HFN is also defined. The formulae has been developed to calculate the hexagonal fuzzy weight of factors and sub-factors. A technique has been developed to incorporate more than one decision maker's opinion into a single comprehensive value in terms of HFN. Two different ranking methods, fuzzy AHP-TOPSIS and fuzzy AHP-COPRAS has been used in this research. GIS software has been used for distance measurement and graphical presentation of the selected sites.

#### *1.4. Structure of the Paper*

The remainder of the paper is organized in the following way: Section 2 depicts the concept of fuzzy numbers, HFN, its properties, distance measure, and defuzzification formulae. It also includes the MCDM technique AHP, fuzzy TOPSIS and fuzzy COPRAS. Section 3 contains the numerical application and description of factors and sub-factors. Section 4 represents the comparison and sensitivity analysis, respectively. The results and discussion covered in Sections 5 and 6 describes the future scope and conclusion.

The structured framework of the study represented in Figure 1 describes the sequential steps followed in this paper. After the initial selection of factors and sub-factors, their weights are calculated by AHP and FAHP. Subsequently, FTOPSIS and FCOPRAS MCDM tools have been applied for the selection of preferred locations followed by comparative and sensitivity analysis, overall results and discussions.

**Figure 1.** Structural framework of the study. FTOPSIS: fuzzy technique for order of preference by similarity to ideal solution; FCOPRAS: fuzzy complex proportional assessment; MCDM: multiple-criteria decision-making.

#### **2. Preliminaries**

#### *2.1. Fuzzy Set*

The fuzzy set theory was developed by the author [25] to deal with the impreciseness of real-life issues [65–73].

**.**

**Definition 1.** *A set <sup>T</sup>*ˆ*, defined as <sup>T</sup>*<sup>ˆ</sup> <sup>=</sup> {(*τ*, *<sup>μ</sup>T*ˆ(*τ*) : *<sup>τ</sup>* <sup>∈</sup> *<sup>T</sup>*ˆ, *<sup>μ</sup>T*ˆ(*τ*) <sup>∈</sup> (0, 1)}, *where <sup>μ</sup>T*ˆ(*τ*) *represents the membership function of T*ˆ *which takes value from zero to one. In real life situations, where the information is vague and uncertain, fuzzy logic can be efficiently used to deal with these problems.*

**Definition 2.** *Hexagonal fuzzy number (HFN) A number αHFN* = {(*h*1, *h*2, *h*3, *h*4, *h*5, *h*6), *μα*(*x*)} *is defined as HFN if it satisfies the following properties:*


The following Figure 2 represents the membership function of symmetric HFN.

$$\mu\_a(x) = \begin{cases} 0, & for | x \le h\_1 \\ \frac{0.5(x - h\_1)}{h\_2 - h\_1}, & for | h\_1 \le x \le h\_2 \\ 0.5 + \frac{0.5(x - h\_2)}{h\_3 - h\_2}, & for | h\_2 \le x \le h\_3 \\ 1, & for | h\_3 \le x \le h\_4 \\ 1 - \frac{0.5(x - h\_4)}{h\_5 - h\_4}, & for | h\_4 \le x \le h\_5 \\ 0.5 - \frac{0.5(x - h\_5)}{h\_6 - h\_5}, & for | h\_5 \le x \le h\_6 \\ 0, & for | x \ge h\_6 \end{cases} \tag{1}$$

(**a**)

**Figure 2.** (**a**) Representation of the membership function of linear hexagonal fuzzy number (HFN). (**b**) Hexagonal fuzzy number as union of different regions.

Where *h*1, *h*2, *h*3, *h*4, *h*<sup>5</sup> *and h*<sup>6</sup> are real numbers such that *h*<sup>1</sup> ≤ *h*<sup>2</sup> ≤ *h*<sup>3</sup> ≤ *h*<sup>4</sup> ≤ *h*<sup>5</sup> ≤ *h*6. Note 2.1: Hexagonal fuzzy numbers (HFN) capture the hesitancy and uncertainty in broader aspects compared to triangular fuzzy numbers (TFN), trapezoidal fuzzy numbers (TrFN), and pentagonal fuzzy numbers (PFN) as the latter undertakes three, four, and five numbers, respectively, to represent the impreciseness and ambiguity of the decision maker (DM). If we consider the linguistic term corresponding to TFN then it is represented

numerically as (low, medium, high) where the medium value corresponds to the best possible chances of the quantity. For TrFN, the numerical presentation is expressed as (very low, low, high, very high) and the maximum possibility lies in a range of (low, high). Further, if we take PFN, it considers the numerical behavior as (very low, low, medium, high, very high), here the middle value denotes the maximum possibility, which is 1. HFN undertakes the highest level of distribution in the numerical form as (very very low, very low, low, high, very high, very very high), i.e., the maximum possible spread can be accommodated in HFN. With respect to a developing country in an unplanned city, many of the attributes under consideration have a wider range of linguistic representation which cannot be captured using TFN, TrFN and PFN. For example, the consumption level in a city with a heterogeneous earning pattern can be better captured using HFN with a wider range. The hesitancy of decision makers in such an environment requires a broader range of values to depict uncertainty; HFN enables this.

The hesitancy of decision makers (DMs) can be better captured in HFN. Different levels of hesitancy in linguistic terms corresponding to different HFN. It is easy to understand and analyze that the linguistic terms "weakly important" represented in HFN as (1.1, 1.2, 1.3, 1.4, 1.5, 1.6), "absolutely important" represented in HFN as (4.6, 4.8, 5, 5.2, 5.4, 5.7) and so on. These range from lower values of HFN to higher values. Similarly, the inverse of these HFN, specifically, the opposites of the linguistic terms can be obtained by using Equation (7).

#### *2.2. Arithmetic Operations of Linear Symmetric HFN*

Let *U* = (*u*1, *u*2, *u*3, *u*4, *u*5, *u*6) and *V* = (*v*1, *v*2, *v*3, *v*4, *v*5, *v*6) be two HFN, then their general arithmetic operations can be defined in the following way:

1. Addition:

$$(\mathcal{U} + V) = (\boldsymbol{u}\_1 + \boldsymbol{v}\_1, \boldsymbol{u}\_2 + \boldsymbol{v}\_2, \boldsymbol{u}\_3 + \boldsymbol{v}\_3, \boldsymbol{u}\_4 + \boldsymbol{v}\_4, \boldsymbol{u}\_5 + \boldsymbol{v}\_5, \boldsymbol{u}\_6 + \boldsymbol{v}\_6) \tag{2}$$

2. Subtraction:

$$(\mathcal{U} - V) = (\mathfrak{u}\_1 - \mathfrak{v}\_{6\prime}, \mathfrak{u}\_2 - \mathfrak{v}\_{5\prime}, \mathfrak{u}\_3 - \mathfrak{v}\_{4\prime}, \mathfrak{u}\_4 - \mathfrak{v}\_{3\prime}, \mathfrak{u}\_5 - \mathfrak{v}\_{2\prime}, \mathfrak{u}\_6 - \mathfrak{v}\_1) \tag{3}$$

3. Multiplication:

$$(\!(\!L\times V) = (\!u\_1\upsilon\_1, \;u\_2\upsilon\_2, \;u\_3\upsilon\_3, \;u\_4\upsilon\_4, \;u\_5\upsilon\_5, \;u\_6\upsilon\_6) \tag{4}$$

4. Scalar Multiplication:

$$
\mathfrak{a}\mathfrak{U} = (\mathfrak{a}\mathfrak{u}\_1, \mathfrak{a}\mathfrak{u}\_2, \mathfrak{a}\mathfrak{u}\_3, \mathfrak{a}\mathfrak{u}\_4, \mathfrak{a}\mathfrak{u}\_5, \mathfrak{a}\mathfrak{u}\_6) \tag{5}
$$

5. Division:

$$
\left(\frac{\mathcal{U}}{V}\right) = \left(\frac{u\_1}{v\_6}, \frac{u\_2}{v\_5}, \frac{u\_3}{v\_4}, \frac{u\_4}{v\_3}, \frac{u\_5}{v\_2}, \frac{u\_6}{v\_1}\right) \tag{6}
$$

6. Inverse:

$$
\Delta U^- = \left(\frac{1}{\mu\_6}, \frac{1}{\mu\_5}, \frac{1}{\mu\_4}, \frac{1}{\mu\_3}, \frac{1}{\mu\_2}, \frac{1}{\mu\_1}\right) \tag{7}
$$

#### *2.3. Distance Measure of Two HFN*

**Definition 3.** *Let A* \**<sup>d</sup>* <sup>=</sup> (*a*1, *<sup>a</sup>*2, *<sup>a</sup>*3, *<sup>a</sup>*4, *<sup>a</sup>*5, *<sup>a</sup>*6) *and <sup>B</sup>* \**<sup>d</sup>* <sup>=</sup> (*b*1, *<sup>b</sup>*2, *<sup>b</sup>*3, *<sup>b</sup>*4, *<sup>b</sup>*5, *<sup>b</sup>*6) *be two HFNs, then the distance between the two HFNs can be determined as:*

$$\begin{aligned} d\left(\overrightarrow{A}\_d, \overrightarrow{B}\_d\right) &= \\ \sqrt{1/6\left[\left(a\_1 - b\_1\right)^2 + \left(a\_2 - b\_2\right)^2 + \left(a\_3 - b\_3\right)^2 + \left(a\_4 - b\_4\right)^2 + \left(a\_5 - b\_5\right)^2 + \left(a\_6 - b\_6\right)^2\right]} \quad \text{(8)} \end{aligned}$$

Example 1. Let *U* = (1.2, 1.3, 1.5, 1.6, 1.8, 2) and *V* = (1, 1.3, 1.7, 2, 2.1, 2.3) be two HFNs then, their distance

$$\begin{cases} d\left(l, \ V\right) \\ = \sqrt{1/6 \left[ \left(1.2 - 1\right)^2 + \left(1.3 - 1.3\right)^2 + \left(1.5 - 1.7\right)^2 + \left(1.6 - 2\right)^2 + \left(1.8 - 2.1\right)^2 + \left(2 - 2.3\right)^2 \right]} \\ = 0.26 \end{cases}$$

#### *2.4. Centroid-Based Method for the Defuzzification of Hexagonal Fuzzy Numbers*

A HFN can be considered as the union of two triangles and two trapeziums, e.g., ΔADH, ΔBCG, ABCD and CDEF together form a HFN.

Further, the trapezium is a union of two triangles and one rectangle. Applying a centroid-based method to triangles and rectangles, and finally summing them, we obtain the centroid of the HFN. Since, the defuzzified value should remain within the range of a HFN, the formulae given below provides the required defuzzified value. Derivation of the HFN is executed in the following way:

(i). Centroid of

$$
\Delta BCG = \left(\frac{h\_4 + h\_5 + h\_6}{3}, \frac{0.5}{3}\right) \tag{9}
$$

(ii). Centroid of

$$
\Delta ADH = \left(\frac{h\_1 + h\_2 + h\_3}{3}, \frac{0.5}{3}\right) \tag{10}
$$

	- (a) Centroid of

$$
\Delta ADI = \left(\frac{h\_2 + 2h\_3}{3}, \frac{1}{3}\right) \tag{11}
$$

(b) Centroid of

$$
\Delta BCD = \left(\frac{2h\_4 + h\_5}{3}, \frac{1}{3}\right) \tag{12}
$$

(c) Centroid of Rectangle

$$ABIf = \left(\frac{h\_3 + h\_4}{2}, \frac{0.5}{2}\right) \tag{13}$$

$$\mathbb{C}\_{ABCD} \left( \frac{2h\_2 + 7h\_3 + 7h\_4 + 2h\_5}{18}, \frac{11}{2} \right) \tag{14}$$

(iv). Centroid of Trapezium CDEF is calculated in the similar order and we obtain:

$$\mathbb{C}\_{CDEF} = \left(\frac{2h\_2 + 7h\_3 + 7h\_4 + 2h\_5}{18}, \frac{25}{12}\right) \tag{15}$$

The defuzzified value is determined by summing Equations (9), (10), (14) and (15) and dividing the denominator by the sum of the quantities of the numerator.

$$\mathbb{C}\_{HGCEFD} = \left(\frac{3h\_1 + 3h\_2 + 10h\_3 + 10h\_4 + 5h\_5 + 3h\_6}{34}, \frac{10}{3}\right) \tag{16}$$

Example 2. Let *U* = (1.5, 1.6, 1.7, 1.9, 2, 2.1), then the defuzzified value of

$$\mathcal{U} = \frac{4.5 + 4.8 + 17 + 19 + 10 + 6.3}{34} = 1.8 \tag{17}$$

#### *2.5. Determination of Hexagonal Fuzzy Weights of Factors and Sub-Factors*

We have extended the methodology developed by Buckley [74] for TFNs in the context of determining hexagonal fuzzy weight.

Step 1. The geometric mean value of the HFN is obtained using:

$$k\_{\mathbf{c}} = \left(\prod\_{d=1}^{j} y\_{cd}\right)\_{\prime} c = 1,2,...,i.$$

Step 2. Summation of each *kc*

Step 3. To calculate the inverse of each *kc* and arrange it in increasing order. Step 4. To find the hexagonal fuzzy weight of factors and sub-factors using the following equation:

$$w\_c = k\_c \ast \left(k\_1 + k\_2 + \dots + k\_i\right)^{-1} \tag{18}$$

Step 5. The global hexagonal fuzzy weight of sub-factors are computed by the product of factor weight with the respective sub-factor fuzzy weight.

#### *2.6. Fuzzy Analytic Hierarchy Process (FAHP)*

The AHP was introduced by Satty [75]. It is used widely for the evaluation of factor and sub-factor weights. AHP helps in solving the real-life situations with a scientific approach. The comparison of factors and sub-factors, thereby giving preference in linguistic terms can be considered as a hesitant task for DMs, thus HFN with AHP methodology captures the vagueness of the problem. The determination of factors' and sub-factors' weights are important for ranking the electric vehicle charging station. AHP works with a problem hierarchy where a comparison matrix is constructed to represent subjective judgments regarding criteria and sub-criteria. In this work, FAHP is taken instead of AHP, keeping in mind the fuzzy setting represents the uncertainties of the decision experts. The FAHP concept with fuzzy logic allows the DMs in the evaluation of reliable results. The steps of FAHP are given below.

#### Step 1. Construction of a comparison matrix in terms of HFN by a group of decision experts.

Let a group of '*H*' decision-makers assigned for the comparison of factors and subfactors. Let each DM express their preference in the pairwise comparison of factors and sub-factors. Thus, '*h*' set of matrices are obtained, *Th* = {*tcdh*}.

Where *tcdh* <sup>=</sup> (*<sup>m</sup>cdh*, *<sup>n</sup>cdh*, *ocdh*, *<sup>p</sup>cdh*, *<sup>q</sup>cdh*) denotes the HFN of *<sup>c</sup>* factor to *<sup>d</sup>* factor as expressed by the '*h*' DM.

$$\begin{cases} \begin{aligned} \tilde{m}\_{cd} &= \min\_{h=1,2,\dots,H} \tilde{m}\_{cdh} \\ \tilde{n}\_{cd} &= \min\_{h=1,2,\dots,H} \tilde{n}\_{cdh} \\ \tilde{o}\_{cd} &= \sqrt[H]{\prod\_{h=1}^{H} \tilde{o}\_{cdh}} \\ \tilde{p}\_{cd} &= \sqrt[H]{\prod\_{h=1}^{H} \tilde{p}\_{cdh}} \\ \tilde{q}\_{cd} &= \max\_{h=1,2,\dots,H} \tilde{q}\_{cdh} \\ \tilde{r}\_{cd} &= \max\_{h=1,2,\dots,H} \tilde{r}\_{cdh} \end{aligned} \tag{19}$$

Step 2. Defuzzification of HFN:

A HFN can be defuzzified by using the centroid-based method used in this paper. Thus using Equation (16), the HFN is transformed to a crisp value.

Step 3. Normalization of the defuzzified matrix:

$$N\_{cd} = \frac{M\_{cd}}{\sum\_{c=1}^{i} M\_{cd}}, \text{ where } c = 1, 2, \dots, i; \ d = 1, 2, \dots, j;\tag{20}$$

Step 4. Estimation of factors' and sub-factors' weights:

$$E = \frac{N^{th} 
rootvalue}{\sum N^{th} 
root} \tag{21}$$

Step 5. To test the Consistence Index (*C*.*I*) of the matrix:

$$\mathbf{f}\left(\mathbf{C}.I\right) = \frac{\mathbf{x}\_{\text{max}} - \bar{f}}{\bar{f} - 1} \tag{22}$$

where *j* denotes the size of the matrix.

Step 6. Determination of Consistence Ratio (*C.R*):

$$\text{C.R} = \frac{\text{C.I}}{\text{R.I}} \tag{23}$$

where *R.I* is stand for Random Index, and its value differs with the size of the matrix "*n*".

The assessment of *C.R* ≤ 0.1 is acceptable and indicates that the weights obtained are justified. Thus further evaluation is not essential.

#### *2.7. Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) and (FTOPSIS)*

The TOPSIS MCDM tool is an extensively used technique, developed by Hwang and Yoon [76] to rank the alternatives, thus giving an idea as to which choice is most preferred. The TOPSIS method is considered to be a distance measure method in which the optimal alternative obtained is farthest away from the negative ideal solution (NIS) and nearest to the positive ideal solution (PIS). The linguistic human decisions can be reflected suitably with Fuzzy TOPSIS (FTOPSIS). The approach is useful in handling the complexity of the situation involving several factors and their sub-factors. In this research, for the selection of the best site to construct an electric vehicle charging station, it is dependent on multiple conflicting factors and sub-factors, thus the MCDM method FTOPSIS introduced by Sodhi and Prabhakar [77] is one of the most helpful and reliable methods. The fuzzy logic extends our goal to obtain more sensitive results in this regard. The steps of FTOPSIS are described below.

Step 1: Construction of the decision matrix by the help of decision experts in terms of linguistic terms. The linguistic terms are then converted to a HFN. Step 2: To evaluate the normalized HFN fuzzy decision matrix:

$$\begin{array}{l}\text{N}\dot{\text{D}}=\left[n\_{\text{g}}\right]\_{\text{st}'}\text{g}=\left[1,2,\ldots,\text{s}\right]\text{h}=\left[1,2,\ldots,\text{t}\right] \\\text{N}\_{\text{g}\text{h}}=\left(\frac{a\_{1\text{g}}\text{h}}{a\_{\text{g}}^{\text{a}}},\frac{a\_{2\text{g}}\text{h}}{a\_{\text{g}}^{\text{a}}},\frac{a\_{3\text{g}}\text{h}}{a\_{\text{g}}^{\text{a}}},\frac{a\_{4\text{g}}\text{h}}{a\_{\text{g}}^{\text{a}}},\frac{a\_{5\text{g}}\text{h}}{a\_{\text{g}}^{\text{a}}},\frac{a\_{6\text{g}}\text{h}}{a\_{\text{g}}^{\text{a}}}\right)\text{d}\in\text{B.A},\ a\_{6}^{\*}=\text{max}a\_{\text{g}}\text{h} \\\text{N}\_{\text{g}\text{h}}=\left(\frac{a\_{\text{h}}^{\text{a}}}{a\_{\text{g}}^{\text{a}}},\frac{a\_{\text{g}}^{\text{a}}}{a\_{\text{g}}^{\text{a}}},\frac{a\_{\text{h}}^{\text{a}}}{a\_{\text{g}}^{\text{a}}},\frac{a\_{\text{h}}^{\text{a}}}{a\_{\text{g}}^{\text{a}}},\frac{a\_{\text{h}}^{\text{a}}}{a\_{\text{g}}^{\text{a}}},\frac{a\_{\text{h}}^{\text{a}}}{a\_{\text{g}}^{\text{a}}}\right)\text{d}\in\text{N}.\text{B.A},a\_{\text{h}}^{\*}=\text{min}a\_{1\text{g}}\text{h}\end{array} \tag{24}$$

where B.A and N.B.A signifies the benefit attributes and non-benefit attributes, respectively. Step 3: To evaluate the weighted fuzzy normalized matrix, the sub-factors' fuzzy weights are multiplied with the normalized fuzzy value:

$$\text{WN} = [P\_{\text{g}}\mathbf{h}]\_{\text{st}}\mathbf{g} = \mathbf{1}, \mathbf{2}, \dots, \mathbf{s} \text{ :} \mathbf{h} = \mathbf{1}, \mathbf{2}, \dots, \mathbf{t} \tag{25}$$

where

$$P\_{\mathbb{S}^h} = N\_{\mathbb{S}^h} \times \hat{W}\_{\mathbb{H}^\times} \text{ g} = 1, \, 2, \ldots, s; h = 1, \, 2, \ldots, t \tag{26}$$

Step 4: Calculate the fuzzy positive ideal solution (FPIS) (*PIS*+) and fuzzy negative ideal solution (FNIS) (*NIS*−), where *h*<sup>+</sup> *<sup>g</sup>* denotes the maximum value of *hgh* and *h*<sup>−</sup> *<sup>g</sup>* denotes the minimum value of *hgh*:

$$\begin{aligned} PIS^{+} &= \left\{ a\_1^{+}, a\_2^{+}, \dots, a\_t^{+} \right\} = \left\{ \left( \max a\_{\text{gh}} \middle| h \in M\_{\text{B}} \right), \left( \min a\_{\text{gh}} \middle| h \in M\_{\text{NB}} \right) \right\} \\ NIS^{-} &= \left\{ a\_1^{-}, a\_2^{-}, \dots, a\_t^{-} \right\} = \left\{ \left( \min a\_{\text{gh}} \middle| h \in M\_{\text{B}} \right), \left( \max \text{h}\_{\text{gh}} \middle| h \in M\_{\text{NB}} \right) \right\} \end{aligned} \tag{27}$$

where *MB* denotes the benefit attributes and *MNB* denotes the non-benefit attributes. Step 5: Calculation of the distance measure of all alternatives from the PIS and NIS. The two Euclidean distances for individual alternatives can be calculated as follows:

$$\begin{aligned} L\_{\mathcal{S}}^{+} &= \sum\_{h=1}^{t} \mathbf{d}(P\_{\mathfrak{g}h}, h\_{\mathfrak{g}}^{+}) \; , \; \mathcal{g} = 1 \; \; 2 \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \;$$

where d(., .) denotes the Euclidean distance between two fuzzy numbers. Step 6: Determination of the relative closeness to the ideal alternatives:

$$R\_{\mathcal{S}} = \frac{L\_{\mathcal{S}}^{-}}{L\_{\mathcal{S}}^{-} + L\_{\mathcal{S}}^{+}} \text{ , } \mathcal{g} = 1 \text{ , 2 , ... , < \ s \tag{29}$$

Step 7: Rank the alternatives:

The alternatives are ranked based on the score obtained by *Rg*. The larger value of *Rg* signifies the better alternatives.

#### *2.8. Fuzzy COPRAS Methodology*

The complex proportional assessment (COPRAS) method was first introduced by Zavadskas, Kaklauskas and Sarka [78]. Fuzzy COPRAS is an extended approach of the COPRAS technique, widely used for decision-making problems. It uses the stepwise ranking and evaluation procedure for the alternatives with reference to significance and utility degree. A few applications of the COPRAS method are in economics, construction and property management. An extension of the COPRAS method is Fuzzy COPRAS, which is frequently used in decision-making problems. Ghose et al. [79] used a hybrid fuzzy COPRAS method for selecting the optimal material to be used for a solar car. They took into consideration 19 materials which had 14 different properties. Using a sensitivity analysis, the robustness of the model was checked. The reason for using the fuzzy-based MCDM technique was that it helps decision makers to get over the problems of ambiguous data. Tolga and Durak [80] used the fuzzy COPRAS technique in the air cargo sector for evaluating the potential capability. 18 criteria were chosen for selecting the best out of the present six alternatives. The steps of the COPRAS method are illustrated below:

Step 1. Decision matrix is constructed in terms of HFN, the alternatives are given linguistic terms by the decision experts with respect to the criteria.

Step 2. Normalized decision matrix is formulated using Equation (1), in the similar way, we constructed for TOPSIS normalized matrix.

Step 3. Weighted normalized matrix is constructed by multiplying the criteria weights with fuzzy normalized matrix using Equation (19).

Step 4. Aggregation of beneficial *B*<sup>+</sup> *<sup>g</sup>* and non-beneficial indices *NB*<sup>−</sup> *<sup>g</sup>* for each alternative are evaluated.

$$B\_{\mathcal{G}}^{+} = \{ \sum\_{h=1}^{m} N \mathcal{W}^{a\_1}, \sum\_{h=1}^{m} N \mathcal{W}^{a\_2}, \sum\_{h=1}^{m} N \mathcal{W}^{a\_3}, \sum\_{h=1}^{m} N \mathcal{W}^{a\_4}, \sum\_{h=1}^{m} N \mathcal{W}^{a\_5}, \sum\_{h=1}^{m} N \mathcal{W}^{a\_6}, \sum\_{h=1}^{m} N \mathcal{W}^{a\_8} \} \tag{30}$$

$$N B\_{\mathcal{X}}^{-} = \left\{ \sum\_{h=m+1}^{l} N W^{4\_1}, \sum\_{h=m+1}^{l} N W^{4\_2}, \sum\_{h=m+1}^{l} N W^{4\_3}, \sum\_{h=m+1}^{l} N W^{4\_4}, \sum\_{h=m+1}^{l} N W^{4\_5}, \sum\_{h=m+1}^{l} N W^{4\_6} \right\} \tag{31}$$

where *h* = 1, 2, ... , m represents the benefit attribute of the alternatives and *h* = *m* + 1, *m* + 2, . . . , *t* represents the non-benefit attributes of the alternative.

Step 5. Finally, the aggregated beneficial and non-beneficial indices are defuzzified using the Equation (9) and *R*+*<sup>g</sup>* and *R*−*<sup>g</sup>* are determined.

Step 6. Calculation of *Rg* using the following formulae:

$$R\_{\mathcal{J}} = R\_{+\mathcal{J}} + \frac{R\_{-\min} \sum\_{\mathcal{S}=1}^{l} R\_{-\mathcal{S}}}{R\_{-\mathcal{S}} \sum\_{\mathcal{S}=1}^{l} \frac{R\_{-\min}}{R\_{-\mathcal{S}}}} \tag{32}$$

Step 7. Ranking of the alternatives are done using the formulae:

$$R = \frac{R\_{\circ}}{R\_{\text{max}}} \ast 100\% \tag{33}$$

where *Rg* represents the g-thdefuzzified value and *Rmax* represents the maximum defuzzified value of individual alternative.

#### **3. Hexagonal Fuzzy MCDM Methodology for Site Selection of Electric Vehicle Charging Station (Numerical Application)**

*3.1. The Factors and Sub Factors Taken in This Research Have Been Explained in the Following Way* 3.1.1. Economic Factors (*C*1)

The prosperity of a nation is comprehended by the state of its economy. The assessment of the economic factors reveals the feasibility of the undertaken study. The factors considered are:


• Public facilities (*C*15): Kinay et al. [84] studied multi-criteria chance-constrained capacitated single source discrete facility location problems. Public facilities refer to schools, colleges, grocery stores, shopping malls, bus stops and the other everyday amenities which are used by commoners on a mass scale. In the case of a charging station being built near a location with a large density of public facilities, it will act as a boon since money will frequently change hands and thereby develop the area.
