A VIKOR-based framework to optimize the location of fast-charging stations with proportional hesitant fuzzy information

Abstract

In this study, a multi-criteria group decision making (MCGDM) framework is constructed for electric vehicle fast-charging station (EVFCS) selection using a proportional hesitant fuzzy set (PHFS) that can describe two aspects of information: the possible membership degrees in the hesitant fuzzy elements and associated proportion representing statistical information from different groups. A newly extended distance measure for PHFSs is introduced and an extended maximizing deviation method is constructed to obtain criteria weights objectively. Accordingly, an integrated PHFS-VIKOR (VlseKriterijum-ska Optimizacija I Kompromisno Resenje) method embedded with a new distance measure and extended maximizing deviation method is presented. With increasing concerns about range anxiety, it is essential to seek an optimal location for EVFCS considering efficient utilization of resources and long-term development of socio-economy under proportional hesitant fuzzy environment. Lastly, an illustration with sensitivity analysis and comparative analyses is provided to demonstrate the validity and robustness of our proposal.

Keywords

Multi-criteria group decision making proportional hesitant fuzzy set distance measure VIKOR maximizing deviation method

1 Introduction

Compared with conventional internal combustion engine vehicles, electric vehicles (EVs) have many advantages such as energy security, economic growth and environment protection. The vigorous promotion of EVs can facilitate a country’s sustainable development. However, users have to face the practical problem whether EVs is running out of power before reaching its destination or finding a charging station. The basic and strong construction of electric vehicle fast-charging stations (EVFCSs) can handle with it.

Many literatures have investigated the distribution of electric vehicle charging stations (EVCSs), which are divided into three categories. The first category examines strategic planning aspects regarding the layout of EVCSs such as location, network design and the capacity of stations for strategic planning [1 –3]. The second category aims to construct a well-functioning operational system that, at least, ensures a minimum quality of service and satisfies the maximum of users’ demands to reduce the operational costs based on network data analysis [4] and statistics for operational planning including EVCS availability [5] and the forecasting user demands [6]. Note that the above studies related to the site selection of EVCS are theoretical rather than practical. In practice, aspects such as convenience of transportation or possibility of further capability expansion shall be concerned for EVCS selection problem, which is an essential multi-criteria decision making (MCDM) problem. In recent related studies, some researchers have proposed many MCDM methods [7 –12] from the perspective of decision management.

Taking the hesitation during the process of decision making into account, hesitant fuzzy set (HFS) [13] was introduced to model the uncertainty and hesitation by assigning the degree of membership of an element. Although HFSs serve as successful information representation models for individuals, they largely fail to model group assessments where the assessments are collected from the subset of individual HFS inputs. Information loss or distortion is inevitable, since the objective proportions of expert subgroups with the same individual assessment cannot be reflected well. To enhance the modelling ability of high order uncertainties, with probability interpretations of HFS, probabilistic hesitant fuzzy set [14] has been proposed. The probability information for each membership degree is subjectively determined by the experts themselves. Moreover, probabilistic hesitant fuzzy set only reflects the certainty of individual decision maker (DM) for hesitation. A new challenge emerges in relation to ensuring the accuracy and reliability of the proportional information provided. To deal with this issue, in group decision making (GDM), weighted hesitant fuzzy set [15] and proportional hesitant fuzzy set [16], focused on the process of obtaining proportional information objectively and statistically. The collective proportional information indicates the decision making group has a high preference for the associated membership degree while the meaning for a small one is just converse. Analogously, in combination with the fuzzy linguistic approach, modelling a group assessment set objectively, the proportional hesitant fuzzy linguistic term set (PHFLTS) [17] is proposed, which clearly indicates that for a group of experts, their individual opinions can be dealt in the use of two approaches: treating them individually by hesitant fuzzy linguistic term set (HFLTS) possibility distributions [18] or treating them as a whole by PHFLTS [19 –21]. Further, the approach of the probabilistic hesitant linguistic term set [22 –31] and that of linguistic assessment distribution are subjective ways of obtaining the importance of each element in a hesitant linguistic term set.

In this paper, we focus on GDM problem and allow DMs to provide their evaluation information anonymously. The proportional information is collected statistically from a group of experts opinions in terms of proportional hesitant fuzzy set. PHFSs offer a brand new perspective for understanding the collective decision matrix construction paradigm, for which not only improve the quality of evaluations by including objective statistical information but also provide a simple solution to the gathering of individual opinions. Different stakeholders, such as EV manufacturers, charging station infrastructure, charging station operators and the central government, shall cooperate and foster the development of the EV market in a group manner to reinforce the democracy and rationality of the decision making result, which can extract the real case scenarios [32] to highlight competitive advantages for location selection. For instance, three groups provide their evaluation values between 0 and 1 on the alternation over certain criteria. Suppose that five professionals provide the value 0.6, ten professionals provide the value 0.7 and five professionals provide the value 0.9, which can be denoted by {0.6, 0.7, 0.9} using hesitant fuzzy element. Note that the proportional information elaborating on the characteristic of group DMs are loss. HFS can be viewed as a special case of PHFS as it is embedded with the same probabilities for each element in it. Precisely and statically, the above instance can be expressed by {0.6 (0.25) , 0.7 (0.5) , 0.9 (0.25)} utilizing the proportional hesitant fuzzy element (PHFE). Recently, a series of MCGDM methods based on PHFS were investigated, such as the operation rule and aggregation operator-based MCGDM methods [33, 34], measure-based MCGDM methods [35, 36], consensus-based MCGDM methods [37 –39], TOPSIS (Technique for Ordering Preference Similarity)-based interactive MCGDM method [40, 41] and outranking-based MCGDM method [42]. Existing research has paid increasing attentions to address these two critical issues simultaneously.

The VIKOR [43] (VIsekriterijumska optimizacija i KOmpromisno Resenje) method is a new well-known developed MCGDM technique, which introduces the multi-criteria ranking index based on the particular measure of “closeness” to the ideal solution. For ranking and selecting from a set of alternatives, the VIKOR method shall determine compromise options by maximizing group utility for the “majority” and minimizing individual regret for the “opponent”. Some researchers have extended the VIKOR method to various uncertain environment such as Pythagorean fuzzy environment [44], intuitionistic fuzzy environment [45], q-rung orthopair fuzzy environment [46]. Ş. Emeç and G. Akkaya [47] combined the stochastic analytic hierarchy process and fuzzy VIKOR approach for the warehouse location selection problem. Several useful applications have been reported for the VIKOR methods [48 –51] to manage other real-life problems in recent years. Note that there is no literature investigating the extension of VIKOR to accommodate the circumstance of PHFSs. Most of the decision-making methods with respect to PHFS are based on various aggregation operators so that the discrepancy of alternatives is almost neglected. Determining an optimal option that simultaneously satisfies all the criteria is difficult. Thus, obtaining a good compromise solution using the VIKOR method is preferred. Some major difficulties and challenges exist to extend the VIKOR method to the MCGDM method with PHFSs: (1) The distance of PHFSs must be defined to measure the closeness of two PHFSs; (2) To avoid subjective randomness, the criteria weights should be determined objectively for MCGDM with PHFSs. (3) It is impractical that the above research is based on the hypothesis that an optimal solution must exist. The solution is usually a set of non-inferior solutions, or a compromise solution, which is the closest to the ideal and an agreement established by mutual concessions because there may be no solution satisfying all criteria simultaneously when criteria are non-commensurable and conflicting.

Hence, from the above analysis, MCGDM method based on the distance measure and compromise solution should be explored. Additionally, motivated by the VIKOR method, this study proposes an integrated PHFS-VIKOR approach based on the new distance measure with proportional hesitant fuzzy information to manage MCGDM problems with unknown weight information on the criteria.

The main motivation and contributions of this study are summarized as follows.

(1) The fuzzy and random information is considered and integrated into the VIKOR method to render all information from different groups.

(2) Maximizing the deviation method makes full use of the objective proportional hesitant fuzzy information of each characteristic established to determine the optimal criteria weights where information on criteria weights is incompletely unknown.

(3) The proposed PHFS-VIKOR method involves the parameter ν, which can be regarded as a behaviour preference measure to help the DMs obtain the compromise solution by taking an appropriate value of ν based on his/her preference.

(4) The proposed PHFS-VIKOR method focuses on the compromise solution, which is the closest to the ideal solution that realizes an agreement established by mutual concessions in addressing the location selection for EVFCSs, which can reach a compromise proposal when criteria are non-comparable or conflicting.

The remainder of this paper is organized as follows. Section 2 contains the concept of PHFSs and the ideal variable and nadir variable of discrete stochastic variables. Additionally, a brief introduction to the classic VIKOR method is given. In Section 3, a novel distance and maximum deviation method for PHFSs are given. Subsequently, an integrated PHFS-VIKOR method is proposed. In Section 4, a case study and together with sensitivity analysis about selecting an optimal EVFCS location is given. In Section 5, comparisons and discussion in three aspects are conducted to demonstrate the advantage of the proposed method. Finally, Section 6 presents the conclusion.

2 Preliminary

2.1 PHFS concept

Definition 1. [16] Let X be a reference set, a proportional hesitant fuzzy set (PHFS) E on X is represented by the following mathematical notation: $E = {〈 x, H (x) | P (x) 〉 | x \in X},$ (1) where

(1) H (x) = {h₁, h₂, . . . , h_v, . . . , h_ϕ} is a set of values [0,1] that h_v (v = 1, 2, . . . , ϕ) denotes the whole possible membership degrees of the element x ∈ X to the set E.

(2) P (x) = {p₁, p₂, . . . , p_v, . . . , p_ϕ} is a set of values [0,1] where p_v (v = 1, 2, . . . , ϕ) denotes the proportion of membership degree and $\sum_{v = 1}^{ϕ} p_{v} (x) = 1$ .

For convenience, given x ∈ X, H (x) |P (x) is called a proportional hesitant fuzzy element (PHFE), indicated by $\begin{matrix} \forall x \in X, \\ H (x) | P (x) = {h_{v} (x) | p_{v} (x), v = 1, 2, . . ., ϕ}, \end{matrix}$ where p_v (x) is the proportion of membership degree h_v (x). Formally, PHFE can be regarded as a discrete random variable. Therefore, in this paper, based on probability theory, we deal with the uncertainty of hesitant fuzzy information. A large proportion indicates that the decision-making group has a high preference for the associated membership degree while the meaning for a small one is the converse. Conveniently, we replace H (x) |P (x) with H|P which can be represented by the two vectors: ∀v = 1, 2, . . . , ϕ, P = (p₁, . . . , p_v, . . . , p_ϕ) and H = (h₁, . . . , h_v, . . . , h_ϕ) as demonstrated in Table 1.

Table 1

Possibility distribution of PHFE

H	h ₁	...	h _v	...	h _ϕ
P	p ₁	...	p _v	...	p _ϕ

Remark 1. For a clear discussion in the remainder of the paper, we make some clear presuppositions:

(1) The whole underlying values h_v (x) of PHFE comprise the basic event. For x ∈ X, the proportion of h_v (x) satisfies $\sum_{v = 1}^{ϕ} p_{v} (x) = 1$ and p_v (x) ⩾0.

(2) Given a PHFE of H|P, its term h_v|p_v are lined up based on the value of h_v in an ascending order, specifically, h₁ < . . . < h_v < . . . < h_ϕ;

(3) The number of underlying membership degrees of the element x ∈ X is equal to a constant ϕ, and arbitrary PHFEs share the same domain H = (h₁, . . . , h_v, . . . , h_ϕ).

(4) Let p_v (ξ) denote the possibility when PHFE ξ takes the value h_v with p_v (x) ⩾0 for any h_v ∈ H. If $p_{v} (ξ) = \frac{1}{ϕ}$ for each v, the discrete stochastic variable PHFE degenerates to a hesitant fuzzy element.

Definition 2. Let ξ₁ and ξ₂ be two discrete stochastic variables with the possibility distribution P (ξ₁) and P (ξ₂) that the alternative carries an assessment value provided by a group of DMs. The distance between ξ₁ and ξ₂, denoted by d (ξ₁, ξ₂), is defined as follows: $d (ξ_{1}, ξ_{2}) = \sqrt{\frac{1}{2} [P_{1} - P_{2}] \cdot B \cdot [P_{1} - P_{2}]^{T}},$ (2) where $P_{1} - P_{2} = (p_{1}^{σ (1)} - p_{2}^{σ (1)}, p_{1}^{σ (2)} - p_{2}^{σ (2)}, . . ., p_{1}^{σ (ϕ)} - p_{2}^{σ (ϕ)})$ and $p_{1}^{σ (i)}$ represents the σ (i)-th element of vector P₁. Additionally, for each element b_ij of the matrix B = (b_ij) _ϕ×ϕ is defined as b_ij = (h^σ(ϕ)) ² - (h^σ(i) - h^σ(j)) ², ∀i, j = 1, 2, . . . , ϕ. The stochastic variables ξ₁ and ξ₂ have the same underlying value span; that is, the underlying value h^σ(j) is associated with the proportion $p_{1}^{σ (i)}$ or $p_{2}^{σ (i)}$ .

2.2 Ideal/nadir variants and their properties

Definition 3. [52] Let ξ₁, ξ₂, . . . , ξ_n be n discrete stochastic variables, with n ⩾ 2. The ideal variant ξ⁺ has a possibility distribution P (ξ⁺) defined as follows: $\begin{matrix} p_{1} (ξ^{+} = h_{1}) = 0, \\ . . ., \\ p_{k - 1} (ξ^{+} = h_{k - 1}) = 0, \\ p_{k} (ξ^{+} = h_{k}) = 1 - \sum_{v = k + 1}^{ϕ} p_{v} (ξ^{+}), \\ p_{k + 1} (ξ^{+} = h_{k + 1}) = \max_{u} {p_{k + 1} (ξ_{u}) | u = 1, 2, . . ., n}, \\ . . ., \\ p_{ϕ} (ξ^{+} = h_{ϕ}) = \max_{u} {p_{ϕ} (ξ_{u}) | u = 1, 2, . . ., n} . \end{matrix}$ (3)

such that $\sum_{v = 1}^{ϕ} p_{v} (ξ^{+}) = 1$ .

Definition 4. [52] Let ξ₁, ξ₂, . . . , ξ_n be discrete stochastic variables, with n ⩾ 2. The nadir variant ξ^- has a possibility distribution P (ξ^-) defined as follows: $\begin{matrix} p_{1} (ξ^{-} = h_{1}) = \max_{u} {p_{1} (ξ_{u}) | u = 1, 2, . . ., n}, \\ . . ., \\ p_{t - 1} (ξ^{-} = h_{t - 1}) = \max_{u} {p_{t - 1} (ξ_{u}) | u = 1, 2, . . ., n}, \\ p_{t} (ξ^{-} = h_{t}) = 1 - \sum_{v = 1}^{t - 1} p_{v} (ξ^{-}), \\ p_{t + 1} (ξ^{-} = h_{t + 1}) = 0, \\ . . ., \\ p_{ϕ} (ξ^{-} = h_{ϕ}) = 0 . \end{matrix}$ (4)

such that $\sum_{v = 1}^{ϕ} p_{v} (ξ^{-}) = 1$ .

Example 1. The domain that the condition values of PHFE can take is H = (h₁, . . . , h_v, . . . , h_ϕ), ∀v = 1, 2, . . . , ϕ, h_v ∈ [0, 1] and the possibility distributions of three discrete stochastic variables ξ₁ ξ₂ and ξ₃ are presented in Table 2. Each discrete stochastic variable may take on ϕ = 9 possible positive values. From the possibility distributions of these three discrete stochastic variables, the possibility distributions of the ideal and nadir variables ξ⁺ and ξ^- are determined by Formulae (3) and (4). The results are also presented in Table 2. The graphical exposition of the possibility distributions of ξ₁ ξ₂ and ξ₃, as well as those of ξ⁺ and ξ^-, are illustrated in Fig. 1.

Table 2

Possibility distributions of PHFEs ξ₁ ξ₂ and ξ₃, as well as those of ξ⁺ and ξ^-

PH	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
ξ ₁		1/9			2/9	2/9	4/9
ξ ₂	1/9	2/9	4/9			2/9
ξ ₃			2/9			1/3		1/3	1/9
ξ ⁺						1/9	4/9	3/9	1/9
ξ ^-	1/9	2/9	4/9		2/9

Fig. 1

Possibility distributions of ξ₁ ξ₂ and ξ₃, as well as those of ξ⁺ and ξ^-.

2.3 The classical VIKOR method

To solve the MCGDM problems, Opricovic and Tzeng [53, 54] introduced the VIKOR method. {A₁, A₂, . . . , A_m} is the set of m alternatives, and {C₁, C₂, . . . , C_n} is the set of n conflicting criteria. Let (w₁, w₂, . . . , w_n) ^T be the weight vector of the criteria, which can express the interactive importance of criteria w_j^, which are derived by the maximizing deviation method in this paper. Let f_ij be the rating functions, which provide the evaluation value of the alternative A_i with respect to the criterion C_j. The VIKOR method uses the following Lp-metric: $\begin{matrix} L_{p, i} = {\sum_{j = 1}^{n} {[\frac{w_{j} (f_{j}^{*} - f_{ij})}{(f_{j}^{*} - f_{j}^{-})}]}^{p}}^{\frac{1}{p}}, \\ 0 < p < \infty, i = 1, 2, . . ., m \end{matrix}$ (5) where $f_{j}^{*} = max_{i} f_{ij}$ and $f_{j}^{-} = min_{i} f_{ij}$ ∀i = 1, 2, . . . , m are the best and worst values of all criteria functions, respectively. The necessary steps of the VIKOR method are depicted as follows:

Step 1. Determine the alternatives, evaluation criteria and form of a group of DMs. The finite set of alternatives is denoted by A = {A₁, A₂, . . . , A_m}, and the set of criteria is denoted by C = {C₁, C₂, . . . , C_n}. Suppose that there are k DMs.

Step 2. Determine each criteria weight. The maximizing deviation method is used in this study to calculate each criteria weight.

Step 3. Compute the group utility values S_i and individual regret value R_i with respect to alternative A_i using the following formula: $S_{i} = L_{1, i} = \sum_{j = 1}^{n} \frac{w_{j} (f_{j}^{*} - f_{ij})}{(f_{j}^{*} - f_{j}^{-})}, i = 1, 2, . . ., m$ (6) $R_{i} = L_{\infty, i} = max_{j} \frac{w_{j} (f_{j}^{*} - f_{ij})}{(f_{j}^{*} - f_{j}^{-})}, i = 1, 2, . . ., m$ (7) where S_I represents the distance of the i-th alternative to the ideal solution, indicating the best combination, and R_i represents the distance of the i-th alternative to the nadir solution, indicating the worst combination, and w_j is the weight of the j-th criteria.

Step 4. Compute the value Q_i (i = 1, 2, . . . , m) using the following formula: $\begin{matrix} Q_{i} = v [\frac{S_{i} - S^{*}}{S^{-} - S^{*}}] + (1 - v) [\frac{R_{i} - R^{*}}{R^{-} - R^{*}}], \\ \forall i = 1, 2, . . ., m, \end{matrix}$ (8) where $S^{*} = min_{i} S_{i}$ , $S^{-} = max_{i} S_{i}$ and $R^{*} = min_{i} R_{I}$ , and $R^{-} = max_{i} R_{I}$ , i = 1, 2, . . . , m. The parameter v ∈ [0, 1] represents the weight for the strategy of the maximum group utility, and 1–v is the weight of the individual regret. The final ranking result is stable with a decision-making strategy, which conforms to the majority rule if v > 0.5, or the consensus rule if v = 0.5, or the veto rule if v < 0.5.

Step 5. Three ranking lists will be obtained by the sorting values S_i, R_i, Q_i (i = 1, 2, . . . , m).

Step 6. Determine the alternative A₁ as a compromise solution if it is ranked first by Q_i and satisfies the following two conditions (i = 1, 2, . . . , m):

Condition 1. Acceptable advantage: Q (A₂) - Q (A₁) ⩾ DQ, where the alternative A₂ is ranked second by Q_I. DQ = 1/ - (m - 1), and m is the number of alternatives.

Condition 2. Acceptable stability in decision making: The alternative A₁ must also be ranked first by S_i and/or R_i (i = 1, 2, . . . , m). However, if the alternative A₁ could not satisfy Conditions 1 or 2, a set of alternatives will be regarded as the compromise solution. We can obtain the set of alternatives using the following rules:

(1) The compromise solution comprises the alternatives A₁ and A₂ if the alternative A₁ cannot satisfy Condition 2.

(2) The compromise solution comprises the alternatives A₁, A₂, . . . , A_m if the alternative A₁ cannot satisfy Condition 1, and the alternative A_m is determined by the relation Q (A_m) - Q (A₁) < DQ.

The resulting compromise solution provides a balance between the maximum group utility of the “majority” (represented by min S_i) and the minimum individual regret of the “opponent” (represented by min R_i). It should be mentioned that, although the VIKOR method has numerous advantages, the performance ratings f_ij are generally quantified using crisp values. However, under many circumstances, such as those where DMs cannot express their preferences accurately, crisp data are inadequate to model real-life situations. That is, given that human judgements including preferences are often vague, it is both difficult and inaccurate to rate them as exact numerical values. Therefore, we employ the PHFE capturing the DM evaluation values and define the Manhattan Lp- metric of PHFE as well as present the extended VIKOR method.

3 The extended VIKOR method for the decision-making problem within proportional hesitant fuzzy information

3.1 Identification of the ideal and nadir solutions

Because the ideal and nadir solutions are used as reference points in ranking the alternatives, appropriately defining them and identifying them are important steps in the proposed method. Let

$ξ^{+} = (ξ_{1}^{+}, ξ_{2}^{+}, . . ., ξ_{n}^{+}), ξ_{j}^{+} = {h_{v} | p_{v} (ξ_{j}^{+}) | v = 1, 2, . . ., ϕ}$

and

$ξ^{-} = (ξ_{1}^{-}, ξ_{2}^{-}, . . ., ξ_{n}^{-}), ξ_{j}^{-} = {h_{v} | p_{v} (ξ_{j}^{-}) | v = 1, 2, . . ., ϕ},$

∀j = 1, 2, . . . , n, be the ideal and nadir verities under all criteria, respectively. Let p_v (ξ⁺) and p_v (ξ^-), ∀v = 1, 2, . . . , ϕ, denote the probabilities that PHFE $ξ_{j}^{+}$ and $ξ_{j}^{-}$ are equal to h_v where p_v (ξ⁺) and p_v (ξ^-) are determined based on Definitions 3 and 4, respectively.

3.2 A metric measuring the distance between two PHFSs

The VIKOR method is a distance-based decision method. Therefore, the definition of the distance between PHFSs is vital. Inspired by the above distance formulae, we define the distance between two PHFSs as follows.

Definition 5. For any two PHFSs E₁ and E₂, The distance between E₁ and E₂, denoted by d (E₁, E₂), is defined as follows: $d (E_{1}, E_{2}) = \frac{1}{n} \sum_{i = 1}^{n} \sqrt{\frac{1}{2} \cdot B \cdot [P_{1 i} - P_{2 i}]^{T}}$ (9) where $P_{1 i} - P_{2 i} = (p_{1 i}^{σ (1)} - p_{2 i}^{σ (1)}, p_{1 i}^{σ (2)} - p_{2 i}^{σ (2)}, . . ., p_{1 i}^{σ (ϕ)} - p_{2 i}^{σ (ϕ)})$ and $p_{1 i}^{σ (j)}$ , ∀j = 1, 2, . . . , ϕ, is the possibility associated with the status value h_σ(j), and each element b_ij of the matrix B = (b_ij) _ϕ×ϕ is defined as b_ij = (h^σ(ϕ)) ² - (h^σ(i) - h^σ(j)) ².

Obviously, B is a symmetric, positive definite matrix and satisfies $\frac{1}{2} [P_{1 i} - P_{2 i}] \cdot B \cdot [P_{1 i} - P_{2 i}]^{T} ⩾ 0$ . It can be seen that the smaller the difference between h^σ(i) and h^σ(j) is, the larger the value of b_ij is. Each b_ij has the following properties:

0 ⩽ b_ij ⩽ (h_σ(ϕ)) ²;

b_ij = b_ji;

b_ii = (h_σ(ϕ)) ² = max {b_ij|i, j = 1, 2, . . . , ϕ};

b_{i
₁
j
₁} > b_{i
₂
j
₂}, if j₂ < j₁ < i₁, j₂ > j₁ > i₁,

i₂ < i₁ < j₁ or i₂ > i₁ > j₁.

The distance measure d (ξ₁, ξ₂) between PHFE ξ₁ and ξ₂ has all the properties of a metric such as non-negativity, symmetry and triangle inequality. These properties are stated formally below.

Property 1(Non-negativity). For any two discrete stochastic variables ξ₁ and ξ₂, 0 ⩽ d (ξ₁, ξ₂) ⩽1 and d (ξ₁, ξ₂) =0 if and only if P₁ = P₂.

Property 2 (Symmetry). For any two discrete stochastic variables ξ₁ and ξ₂, d (ξ₁, ξ₂) = d (ξ₂, ξ₁).

Property 3 (Triangle inequality). For any discrete stochastic variables ξ₁, ξ₂ and ξ₃, d (ξ₁, ξ₂) + d (ξ₂, ξ₃) ⩾ d (ξ₁, ξ₃).

Example 2. Let E₁ and E₂ be any two PHFSs. $E_{1} = {〈 x_{1}, 0.2 | 0.5, 0.3 | 0.3, 0.8 | 0.2 〉,$ $〈 x_{2}, 0.5 | 0.6, 0.7 | 0.4 〉} .$ $E_{2} = {〈 x_{1}, 0.2 | 0.5, 0.4 | 0.5 〉, 〈 x_{2}, 0.5 | 0.6, 0.6 | 0.4 〉}$

According to Definition 2, P₁₁ - P₂₁ = (0, 0, 0.3, - 0.5, 0, 0, 0, 0.2, 0), P₁₂ - P₂₂ = (0, 0, 0, 0, 0, - 0.4, 0.4, 0, 0) and the matrix B is $B = (\begin{matrix} 0.81 & 0.80 & 0.77 & 0.72 & 0.65 & 0.56 & 0.45 & 0.32 & 0.17 \\ 0.80 & 0.81 & 0.80 & 0.77 & 0.72 & 0.65 & 0.56 & 0.45 & 0.32 \\ 0.77 & 0.80 & 0.81 & 0.80 & 0.77 & 0.72 & 0.65 & 0.56 & 0.45 \\ 0.72 & 0.77 & 0.80 & 0.81 & 0.80 & 0.77 & 0.72 & 0.65 & 0.56 \\ 0.65 & 0.72 & 0.77 & 0.80 & 0.81 & 0.80 & 0.77 & 0.72 & 0.65 \\ 0.56 & 0.65 & 0.72 & 0.77 & 0.80 & 0.81 & 0.80 & 0.77 & 0.72 \\ 0.45 & 0.56 & 0.65 & 0.72 & 0.77 & 0.80 & 0.81 & 0.80 & 0.77 \\ 0.32 & 0.45 & 0.56 & 0.65 & 0.72 & 0.77 & 0.80 & 0.81 & 0.80 \\ 0.17 & 0.32 & 0.45 & 0.56 & 0.65 & 0.72 & 0.77 & 0.8 & 0.81 \end{matrix}) .$

Accordingly, the distance between two PHFSs is d (E₁, E₂)= 0.045.

Definition 6. The proportional hesitant fuzzy Manhattan L p-metric of PHFSs is in terms of the following form: $\begin{matrix} L_{p, i} = {\sum_{j = 1}^{n} {[\frac{w_{j} d (ξ_{j}^{+}, ξ_{ij})}{d (ξ_{j}^{+}, ξ_{j}^{-})}]}^{p}}^{\frac{1}{p}}, \\ 0 < p < \infty, i = 1, 2, . . ., m \end{matrix}$ (10) where w_j (j = 1, 2, . . . , n) are the corresponding criteria weights, and satisfy 0 ⩽ w_j ⩽ 1 (j = 1, 2, . . . , n) and $\sum_{j = 1}^{n} w_{j} = 1$ . Additionally, $d (ξ_{j}^{+}, ξ_{ij}) = \sqrt{\frac{1}{2} [P (ξ_{j}^{+}) - P (ξ_{ij})] B {[P (ξ_{j}^{+}) - P (ξ_{ij})]}^{T}}$ , and $d (ξ_{j}^{+}, ξ_{j}^{-}) = \sqrt{\frac{1}{2} [P (ξ_{j}^{+}) - P (ξ_{j}^{-})] B {[P (ξ_{j}^{+}) - P (ξ_{j}^{-})]}^{T}}$ , ∀i = 1, 2, . . . , m, j = 1, 2, . . . , n .

3.3 Maximizing the deviation method to determine the criteria weights

Maximizing the deviation method has been widely applied to determine the criteria weights in MCGDM problems. For a criterion, if the deviation degree among the evaluation values of all alternatives is relatively large, then the criterion is relatively important in the group decision-making problem; otherwise, the criterion is relatively unimportant. In this paper, evaluation values in a group decision making problem are in the form of PHFEs, which is difficult to measure the deviation degree directly. Hence, we here construct an optimization model based on the maximizing deviation method to determine the optimal relative criteria weights under proportional hesitant fuzzy environment. For the criterion C_j ∈ C, the deviation of the alternative A_i to the other alternative can be expressed as follows:

$\begin{matrix} d (ξ_{ij}, ξ_{kj}) \\ = \sqrt{\frac{1}{2} [P (ξ_{ij}) - P (ξ_{kj})] B {[P (ξ_{ij}) - P (ξ_{kj})]}^{T}}, \end{matrix}$ (11)

$\begin{matrix} d (ξ_{ij}) & = \sum_{k = 1}^{m} d (ξ_{ij}, ξ_{kj}) \\ = \sum_{k = 1}^{m} \sqrt{\frac{1}{2} [P (ξ_{ij}) - P (ξ_{kj})] B {[P (ξ_{ij}) - P (ξ_{kj})]}^{T}}, \end{matrix}$ (12)

where i = 1, 2, . . . , m, j = 1, 2, . . . , n, k = 1, 2, . . . , m, k ≠ I. By summing all the deviations of the alternative A_i with respect to the criterion C_j, we can obtain the deviation with respect to the criterion C_j

$\begin{matrix} D_{j} (ξ_{ij}) & = \sum_{j = 1}^{n} d (ξ_{ij}) \\ = \sum_{j = 1}^{n} \sum_{k = 1}^{m} \sqrt{\frac{1}{2} [P (ξ_{ij}) - P (ξ_{kj})] B {[P (ξ_{ij}) - P (ξ_{kj})]}^{T}}, \end{matrix}$ (13)

where i = 1, 2, . . . , m, j = 1, 2, . . . , n, k = 1, 2, . . . , m, k ≠ I.

The importance of a criterion is determined by the deviation degree. The larger the deviation is, the more important the criterion is. Thus, the sum of the deviation of all criteria should be as large as possible, and we have the following programming model:

$\begin{matrix} D (w) = & \sum_{j = 1}^{n} D_{j} (ξ_{ij}) w_{j} = \sum_{j = 1}^{n} \sum_{j = 1}^{n} \sum_{k = 1}^{m} \\ \sqrt{\frac{1}{2} [P (ξ_{ij}) - P (ξ_{kj})] B {[P (ξ_{ij}) - P (ξ_{kj})]}^{T}} \cdot w_{j}, \end{matrix}$ (14)

where i = 1, 2, . . . , m, j = 1, 2, . . . , n, k = 1, 2, . . . , m, k ≠ I.

Based on the above analysis, we can construct a non-linear programming model to select the weight vector w, which maximizes all deviation values for all the criteria as follows: $(M - 1) {\begin{matrix} max D (w) = \sum_{j = 1}^{n} D_{j} (ξ_{ij}) w_{j} \\ s . t . w_{j} ⩾ 0, j = 1, 2, . . ., n, \sum_{j = 1}^{n} w_{j}^{2} = 1 \end{matrix}$ (15)

To solve the above model, we let

$\begin{matrix} L (w, λ) \\ = \sum_{j = 1}^{n} \sum_{j = 1}^{n} \sum_{k = 1}^{m} \sqrt{\frac{1}{2} [P (ξ_{ij}) - P (ξ_{kj})] B {[P (ξ_{ij}) - P (ξ_{kj})]}^{T}} \\ \cdot w_{j} + \frac{λ}{2} (\sum_{j = 1}^{n} w_{j}^{2} - 1), \end{matrix}$ (16)

which indicates the Lagrange function of the constrained optimization problem (M-1) where λ is a real number denoting the Lagrange multiplier variable. The partial derivatives of L are then computed as follows:

${\begin{matrix} \frac{\partial L}{\partial w_{j}} = \sum_{j = 1}^{n} \sum_{k = 1}^{m} \sqrt{\frac{1}{2} [P (ξ_{ij}) - P (ξ_{kj})] B {[P (ξ_{ij}) - P (ξ_{kj})]}^{T}} + λ w_{j} = 0 \\ \frac{\partial L}{\partial λ} = \frac{1}{2} (\sum_{j = 1}^{n} w_{j}^{2} - 1) = 0 \end{matrix}$ (17)

We obtain the exact solution as follows:

${\begin{matrix} w_{j} = - \frac{\sum_{j = 1}^{n} \sum_{k = 1}^{m} \sqrt{\frac{1}{2} [P (ξ_{ij}) - P (ξ_{kj})] B {[P (ξ_{ij}) - P (ξ_{kj})]}^{T}}}{λ}, \\ λ = - \sqrt{\sum_{j = 1}^{n} {(\sum_{j = 1}^{n} \sum_{k = 1}^{m} \sqrt{\frac{1}{2} [P (ξ_{ij}) - P (ξ_{kj})] B {[P (ξ_{ij}) - P (ξ_{kj})]}^{T}})}^{2}}, \forall j = 1, 2, . . ., n . \end{matrix}$ (18)

For convenience, we denote $(w_{1}^{*}, w_{2}^{*}, . . ., w_{n}^{*})$ as the solution result from (M-1).

3.4 Ranking of the alternatives based on the extended VIKOR method

Step 1. Compute the proportional hesitant fuzzy group utility measure and proportional hesitant fuzzy individual regret measure, respectively, as follows: $S_{i} = \sum_{j = 1}^{n} \frac{w_{j}^{*} d (ξ_{j}^{+}, ξ_{ij})}{d (ξ_{j}^{+}, ξ_{j}^{-})},$ (19) $R_{i} = max_{j} \frac{w_{j}^{*} d (ξ_{j}^{+}, ξ_{ij})}{d (ξ_{j}^{+}, ξ_{j}^{-})},$ (20) for each alternative where $S^{*} = min_{i} S_{i}$ , and $S^{-} = max_{i} S_{i}, R^{*} = min_{i} R_{i}, R^{-} = max R_{i}, i = 1, 2, . . ., m$ .

Step 2. Rank all alternatives in increasing order of the proportional hesitant fuzzy compromise measure Q_i, computed using the following formula: $\begin{matrix} Q_{i} = v [\frac{S_{i} - S^{*}}{S^{-} - S^{*}}] + (1 - v) [\frac{R_{i} - R^{*}}{R^{-} - R^{*}}], \\ i = 1, 2, . . ., m \end{matrix}$ (21)

4 Case study

In the 21st century, we are facing dual energy and environmental issues; thus, exploring a new frame of sustainable transport development is extremely urgent. Experts in the automobile industry field have been actively trying to replace gasoline power with battery power for engines to solve the urgent problem. It turns out that EVs are a new type of clean, efficient and eco-friendly energy vehicle. Hence, improving the electric vehicle charging facility network construction, convenience for users, cost of user charging and EVFCS construction, will directly affect the adoption of EVs. Therefore, studies on the optimal location of electric vehicle chargers are of great importance.

To highlight the effectiveness of the integrated PHFS-VIKOR method, an application of the optimal selection of EVFCS is given. Because the PHFSs are considered a useful tool to manage the hesitation and indeterminacy in the decision data, we use PHFEs to collect information from different groups of DMs considering various parameters.

4.1 Implementation of the proposed approach

For investments in charging stations, a reasonable location will increase the utilization rate and reduce the costs of construction and operation. Based on previous analyses [55, 56], an evaluation index system comprising 6 criteria based on the literature review was constructed in Fig. 2. Assume that a company would like to select an optimal location to build a new electric vehicle charger. The committee includes 5 stakeholders from different fields such as nearby residents, EV manufacturers, the charging station infrastructure, charging station operators and the central government. Additionally, the evaluation is collected as PHFEs shown as Tables 3, 4, 5 and 6.

Fig. 2

Evaluation index system in a MCGDM problem in selecting an optimal site for Evs.

Table 3

Ratings of the location A₁ by DMs under all criteria

Criteria	DM1	DM2	DM3	DM4	DM5	Collective opinion
C ₁	{0.5}	{0.5}	{0.4}	{0.4}	{0.5}	{0.4 (2/5) , 0.5 (3/5)}
C ₂	{0.4}	{0.3}	{0.3}	{0.3}	{0.3}	{0.3 (4/5) , 0.4 (1/5)}
C ₃	{0.4}	{0.4}	{0.4}	{0.4}	{0.4}	{0.4 (1)}
C ₄	{0.6}	{0.4}	{0.6}	{0.4}	{0.4}	{0.4 (3/5) , 0.6 (2/5)}
C ₅	{0.4}	{0.3}	{0.4}	{0.5}	{0.5}	{0.3 (1/5) , 0.4 (2/5) , 0.5 (2/5)}
C ₆	{0.4}	{0.4}	{0.6}	{0.6}	{0.5}	{0.4 (2/5) , 0.5 (1/5) , 0.6 (2/5)}

Table 4

Ratings of the location A₂ by DMs under all criteria

Criteria	DM1	DM2	DM3	DM4	DM5	Collective opinion
C ₁	{0.7}	{0.6}	{0.5}	{0.6}	{0.6}	{0.5 (1/5) , 0.6 (3/5) , 0.7 (1/5)}
C ₂	{0.4}	{0.6}	{0.5}	{0.5}	{0.6}	{0.4 (1/5) , 0.5 (2/5) , 0.6 (2/5)}
C ₃	{0.7}	{0.8}	{0.7}	{0.7}	{0.7}	{0.7 (4/5) , 0.8 (1/5)}
C ₄	{0.7}	{0.5}	{0.5}	{0.5}	{0.5}	{0.5 (4/5) , 0.7 (1/5)}
C ₅	{0.6}	{0.6}	{0.6}	{0.6}	{0.6}	{0.6 (1)}
C ₆	{0.8}	{0.9}	{0.6}	{0.7}	{0.6}	{0.6 (2/5) , 0.7 (1/5) , 0.8 (1/5) , 0.9 (1/5)}

Table 5

Ratings of the location A₃ by DMs under all criteria

Criteria	DM1	DM2	DM3	DM4	DM5	Collective opinion
C ₁	{0.4}	{0.4}	{0.3}	{0.3}	{0.3}	{0.3 (3/5) , 0.4 (2/5)}
C ₂	{0.7}	{0.7}	{0.8}	{0.7}	{0.7}	{0.7 (4/5) , 0.8 (1/5)}
C ₃	{0.4}	{0.4}	{0.6}	{0.3}	{0.3}	{0.3 (2/5) , 0.4 (2/5) , 0.6 (1/5)}
C ₄	{0.5}	{0.5}	{0.7}	{0.7}	{0.7}	{0.5 (2/5) , 0.7 (3/5)}
C ₅	{0.6}	{0.6}	{0.6}	{0.6}	{0.5}	{0.5 (1/5) , 0.6 (4/5)}
C ₆	{0.5}	{0.5}	{0.5}	{0.4}	{0.4}	{0.4 (2/5) , 0.5 (3/5)}

Table 6

Ratings of the location A₄ by DMs under all criteria

Criteria	DM1	DM2	DM3	DM4	DM5	Collective opinion
C ₁	{0.8}	{0.8}	{0.7}	{0.6}	{0.7}	{0.6 (1/5) , 0.7 (2/5) , 0.8 (2/5)}
C ₂	{0.9}	{0.8}	{0.9}	{0.8}	{0.6}	{0.6 (1/5) , 0.8 (2/5) , 0.9 (2/5)}
C ₃	{0.8}	{0.6}	{0.6}	{0.6}	{0.8}	{0.6 (3/5) , 0.8 (2/5)}
C ₄	{0.6}	{0.5}	{0.8}	{0.8}	{0.8}	{0.5 (1/5) , 0.6 (1/5) , 0.8 (3/5)}
C ₅	{0.7}	{0.9}	{0.9}	{0.7}	{0.8}	{0.7 (2/5) , 0.8 (1/5) , 0.9 (2/5)}
C ₆	{0.6}	{0.5}	{0.6}	{0.6}	{0.6}	{0.5 (1/5) , 0.6 (4/5)}

According to subsection 3.1, possibility distributions of the components of the ideal solution ξ⁺ and nadir solution ξ^- are identified and are shown as Tables 7 and 8.

Table 7

Possibility distributions of the components of the ideal solution ξ⁺

Ideal solution	Membership degree
	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
$ξ_{1}^{+}$						1/5	2/5	2/5
$ξ_{2}^{+}$							1/5	2/5	2/5
$ξ_{3}^{+}$						2/5	1/5	2/5
$ξ_{4}^{+}$							3/5	2/5
$ξ_{5}^{+}$							2/5	1/5	2/5
$ξ_{6}^{+}$						2/5	1/5	1/5	1/5

Table 8

Possibility distributions of the components of the nadir solution ξ^-

Nadir solution	Membership degree
	0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
$ξ_{1}^{-}$			3/5	2/5
$ξ_{2}^{-}$			4/5	1/5
$ξ_{3}^{-}$			2/5	3/5
$ξ_{4}^{-}$				3/5	2/5
$ξ_{5}^{-}$			1/5	2/5	2/5
$ξ_{6}^{-}$				2/5	3/5

Thereafter, we obtained the deviation of alternative A_i based on the criterion C_j by formula (14), shown as Table 9.

Table 9

Deviation values

	C ₁	C ₂	C ₃	C ₄	C ₅	C ₆
A ₁	0.48	1.02	0.46	0.42	0.28	0.58
A ₂	0.48	0.74	0.46	0.3	0.48	0.46
A ₃	0.64	0.74	0.46	0.3	0.28	0.7
A ₄	0.88	0.90	0.66	0.46	0.32	0.46

To obtain the weight vector of the criteria, we solve the model (M-1) (15) and the weight vector of the criteria is derived as w = (0.4208, 0.4441, 0.2980, 0.3857, 0.5026, 0.3681) ^T by MATLAB.

The performances of four locations are ranked using combination weights and the integrated PHFS-VIKOR method. Based on the values of Q_i, the ranking of all alternatives in descending order are A₂ ≻ A₄ ≻ A₃ ≻ A₁ when v = 0.5. The best alternative is found to be A₂, and the compromise solution is A₄, because Q₄ - Q₂ = 0.1262 < 1/(4 - 1) =0.33. Based on the above results, this proposed model can easily evaluate and select a best alternative given a v value. In the next subsection, to examine the rationality and stability of the proposed framework and analysis results, the discriminatory ability and sensitivity analysis of the Q_i value are presented.

4.2 Sensitivity analysis

In the extended decision making framework, there are some modelling parameters. To determine whether the modelling parameters (the weight for the strategy of the maximum group utility in this study) will have an observable effect on the final results of the decision-making process, sensitivity analysis must be executed.

Given the feature of the maximum group utility on the rankings of the alternatives, one experiment is conducted to observe how the parameter v validates the obtained results. Generally, the maximum group utility value of v is taken as 0.5. Actually, v can fetch any value from 0 to 1. The related results according to the value of v are illustrated in Table 10, which shows that the ranking of four alternatives are not influenced by the v value, indicating that the priorities of these alternatives are the same in terms of both maximum group utility and minimum individual regret so that the obtained results of the proposed approach are robust and reliable. In detail, the ranking of A₄ increases with the increase of v, revealing that A₄ has a higher level of risk when one DM focuses on maximum group utility. It was reported that the ranking of A₂ is in the top when the v value is small, indicating that the ranking will rise as the importance of the minimum individual regret increases. Therefore, the proposed method can obtain various rankings when taking different values depending on the specific applications in the optimal selection problem.

Table 10
S_i, R_i and Q(v) _i values obtained by the weight v and compromise solutions

A ₁ A ₂ A ₃ A ₄ Ranking Compromise Solutions

S_i 2.15 0.94 1.60 0.34 A₄ ≻ A₂ ≻ A₃ ≻ A₁ A ₄

R_i 0.31 0 0.37 0.2 A₂ ≻ A₄ ≻ A₁ ≻ A₃ A ₂

0 0.85 0 1 0.54 A₂ ≻ A₄ ≻ A₁ ≻ A₃ A ₂

0.1 0.85 0.03 0.97 0.49 A₂ ≻ A₄ ≻ A₁ ≻ A₃ A ₂

0.2 0.86 0.07 0.93 0.44 A₂ ≻ A₄ ≻ A₁ ≻ A₃ A ₂

0.3 0.87 0.11 0.90 0.40 A₂ ≻ A₄ ≻ A₁ ≻ A₃ A₂, A₄

0.4 0.88 0.14 0.87 0.35 A₂ ≻ A₄ ≻ A₃ ≻ A₁ A₂, A₄

Q(v) _i 0.5 0.88 0.18 0.83 0.30 A₂ ≻ A₄ ≻ A₃ ≻ A₁ A₂, A₄

0.6 0.89 0.21 0.80 0.26 A₂ ≻ A₄ ≻ A₃ ≻ A₁ A₂, A₄

0.7 0.90 0.25 0.77 0.21 A₄ ≻ A₂ ≻ A₃ ≻ A₁ A₄, A₂

0.8 0.91 0.29 0.73 0.16 A₄ ≻ A₂ ≻ A₃ ≻ A₁ A₄, A₂

0.9 0.91 0.32 0.70 0.12 A₄ ≻ A₂ ≻ A₃ ≻ A₁ A₄, A₂

1 0.92 0.36 0.67 0.07 A₄ ≻ A₂ ≻ A₃ ≻ A₁ A₄, A₂

		A ₁	A ₂	A ₃	A ₄	Ranking	Compromise Solutions
S_i		2.15	0.94	1.60	0.34	A₄ ≻ A₂ ≻ A₃ ≻ A₁	A ₄
R_i		0.31	0	0.37	0.2	A₂ ≻ A₄ ≻ A₁ ≻ A₃	A ₂
	0	0.85	0	1	0.54	A₂ ≻ A₄ ≻ A₁ ≻ A₃	A ₂
	0.1	0.85	0.03	0.97	0.49	A₂ ≻ A₄ ≻ A₁ ≻ A₃	A ₂
	0.2	0.86	0.07	0.93	0.44	A₂ ≻ A₄ ≻ A₁ ≻ A₃	A ₂
	0.3	0.87	0.11	0.90	0.40	A₂ ≻ A₄ ≻ A₁ ≻ A₃	A₂, A₄
	0.4	0.88	0.14	0.87	0.35	A₂ ≻ A₄ ≻ A₃ ≻ A₁	A₂, A₄
Q(v) _i	0.5	0.88	0.18	0.83	0.30	A₂ ≻ A₄ ≻ A₃ ≻ A₁	A₂, A₄
	0.6	0.89	0.21	0.80	0.26	A₂ ≻ A₄ ≻ A₃ ≻ A₁	A₂, A₄
	0.7	0.90	0.25	0.77	0.21	A₄ ≻ A₂ ≻ A₃ ≻ A₁	A₄, A₂
	0.8	0.91	0.29	0.73	0.16	A₄ ≻ A₂ ≻ A₃ ≻ A₁	A₄, A₂
	0.9	0.91	0.32	0.70	0.12	A₄ ≻ A₂ ≻ A₃ ≻ A₁	A₄, A₂
	1	0.92	0.36	0.67	0.07	A₄ ≻ A₂ ≻ A₃ ≻ A₁	A₄, A₂

5 Comparisons and discussion

This section aims to verify the feasibility of the proposed methods compared with several relevant methods.

5.1 Comparison analysis between HFSs and PHFSs

To demonstrate the capability of PHFSs, a numerical example [57] assumed with the equal possibility distribution is conducted using the proposed method.

Case 1: The civil aviation administration of Taiwan (CAAT) appraised and elected the best airline in Taiwan to improve the service quality of each domestic airline. Thus, the CAAT constructs a committee to investigate the four major domestic airlines. For a better comparison, the collective opinions of DMs with the assumption that the proportion of the whole underlying membership is identical are presented in Table 11. The weight vector of criteria is w = (0 . 1, 0 . 2, 0 . 4, 0 . 3) ^T as described previously [57]. According to the proposed approach in Section 3, the S_i, R_i and Q(v) _i values obtained by the weight v and compromise solutions are demonstrated in Table 12.

Table 11
Proportional hesitant fuzzy decision matrix adapted previously [57] with the identical proportion

UNI Air (B₁) Trans Asia(B₂) Mandarin(B₃) Daily Air(B₄)

C ₁ {0.6 (1/3) , 0.7 (1/3) , 0.9 (1/3)} {0.7 (1/3) , 0.8 (1/3) , 0.9 (1/3)} {0.5 (1/3) , 0.6 (1/3) , 0.8 (1/3)} {0.6 (1/2) , 0.9 (1/2)}

C ₂ {0.6 (1/2) , 0.8 (1/2)} {0.5 (1/3) , 0.8 (1/3) , 0.9 (1/3)} {0.6 (1/3) , 0.7 (1/3) , 0.9 (1/3)} {0.7 (1/2) , 0.9 (1/2)}

C ₃ {0.3 (1/3) , 0.6 (1/3) , 0.9 (1/3)} {0.4 (1/2) , 0.8 (1/2)} {0.3 (1/3) , 0.5 (1/3) , 0.7 (1/3)} {0.2 (1/3) , 0.4 (1/3) , 0.7 (1/3)}

C ₄ {0.4 (1/3) , 0.5 (1/3) , 0.9 (1/3)} {0.5 (1/3) , 0.6 (1/3) , 0.7 (1/3)} {0.5 (1/2) , 0.7 (1/2)} {0.4 (1/2) , 0.5 (1/2)}

	UNI Air (B₁)	Trans Asia(B₂)	Mandarin(B₃)	Daily Air(B₄)
C ₁	{0.6 (1/3) , 0.7 (1/3) , 0.9 (1/3)}	{0.7 (1/3) , 0.8 (1/3) , 0.9 (1/3)}	{0.5 (1/3) , 0.6 (1/3) , 0.8 (1/3)}	{0.6 (1/2) , 0.9 (1/2)}
C ₂	{0.6 (1/2) , 0.8 (1/2)}	{0.5 (1/3) , 0.8 (1/3) , 0.9 (1/3)}	{0.6 (1/3) , 0.7 (1/3) , 0.9 (1/3)}	{0.7 (1/2) , 0.9 (1/2)}
C ₃	{0.3 (1/3) , 0.6 (1/3) , 0.9 (1/3)}	{0.4 (1/2) , 0.8 (1/2)}	{0.3 (1/3) , 0.5 (1/3) , 0.7 (1/3)}	{0.2 (1/3) , 0.4 (1/3) , 0.7 (1/3)}
C ₄	{0.4 (1/3) , 0.5 (1/3) , 0.9 (1/3)}	{0.5 (1/3) , 0.6 (1/3) , 0.7 (1/3)}	{0.5 (1/2) , 0.7 (1/2)}	{0.4 (1/2) , 0.5 (1/2)}

Table 12

S_i, R_i and Q(v) _i values and ranking order using the proposed VIKOR method

		UNI Air	Trans	Mandarin	Daily	Ranking	Compromise Solutions
		(B₁)	Asia(B₂)	(B₃)	Air(B₄)
S_i		0.47	0.42	0.56	0.67	B₂ ≻ B₁ ≻ B₃ ≻ B₄	B₂, B₁
R_i		0.15	0.15	0.15	0.3	B₁ ∼ B₂ ∼ B₃ ≻ B₄	B₁, B₂, B₃
	0	0.41	0.41	0.41	0.82	B₁ ∼ B₂ ∼ B₃ ≻ B₄	B₁, B₂, B₃
	0.1	0.38	0.38	0.38	0.76	B₂ ≻ B₁ ≻ B₃ ≻ B₄	B₂, B₁
	0.2	0.35	0.35	0.36	0.70	B₂ ≻ B₁ ≻ B₃ ≻ B₄	B₂, B₁
	0.3	0.32	0.32	0.34	0.64	B₂ ≻ B₁ ≻ B₃ ≻ B₄	B₂, B₁
	0.4	0.30	0.29	0.32	0.58	B₂ ≻ B₁ ≻ B₃ ≻ B₄	B₂, B₁
Q(v)_i	0.5	0.27	0.26	0.29	0.52	B₂ ≻ B₁ ≻ B₃ ≻ B₄	B₂, B₁
	0.6	0.24	0.23	0.27	0.46	B₂ ≻ B₁ ≻ B₃ ≻ B₄	B₂, B₁
	0.7	0.22	0.20	0.25	0.40	B₂ ≻ B₁ ≻ B₃ ≻ B₄	B₂, B₁
	0.8	0.19	0.17	0.22	0.34	B₂ ≻ B₁ ≻ B₃ ≻ B₄	B₂, B₁
	0.9	0.16	0.14	0.20	0.29	B₂ ≻ B₁ ≻ B₃ ≻ B₄	B₂, B₁
	1	0.13	0.11	0.18	0.23	B₂ ≻ B₁ ≻ B₃ ≻ B₄	B₂, B₁

As described previously [57], Trans Asia (B₂) is the best alternative whose service quality has the highest satisfaction degree without compromise solutions. In this paper, we utilized the proposed approach and obtained the consistent result that Trans Asia (B₂) is one of the best airlines in terms of service quality. Along with the change in the weight v, however, it appears to be a compromise alternative to UNI Air (B₁). The primary reason for the slight difference between the two results is the information collection method. In the former method, the evaluation values from a group of DMs are collected by HFEs, leading to proportional information loss. Additionally, the new metric measuring the distance between the collective evaluations and ideal/nadir solution not only considers the possibility distribution, but also the values of the stochastic variables.

Case 2 In Case 1, it is assumed that the underlying membership in each alternative captured by PHFE is equal. However, the expert in the committee practically holds different views, not always with the same ratio in one PHFE. For a better comparison, the collective opinions of the DMs, with the assumption that the proportion of the whole underlying membership in each PHFE is selected randomly for the experiment, are presented in Table 13. According to the proposed approach in Section 3, the S_i, R_i and Q(v) _i values obtained by weight v and the compromise solutions are demonstrated in Table 14.

Table 13

Proportional hesitant fuzzy decision matrix with the assumed possibility distribution

	UNI Air(B₁)	Trans Asia(B₂)	Mandarin(B₃)	Daily Air(B₄)
C ₁	{0.6 (4/7) , 0.7 (1/7) , 0.9 (2/7)}	{0.7 (5/7) , 0.8 (1/7) , 0.9 (1/7)}	{0.5 (1/7) , 0.6 (1/7) , 0.8 (5/7)}	{0.6 (6/7) , 0.9 (1/7)}
C ₂	{0.6 (4/7) , 0.8 (3/7)}	{0.5 (4/7) , 0.8 (1/7) , 0.9 (2/7)}	{0.6 (1/7) , 0.7 (1/7) , 0.9 (5/7)}	{0.7 (5/7) , 0.9 (2/7)}
C ₃	{0.3 (1/7) , 0.6 (3/7) , 0.9 (3/7)}	{0.4 (5/7) , 0.8 (2/7)}	{0.3 (1/7) , 0.5 (2/7) , 0.7 (4/7)}	{0.2 (1/7) , 0.4 (5/7) , 0.7 (1/7)}
C ₄	{0.4 (4/7) , 0.5 (2/7) , 0.9 (1/7)}	{0.5 (3/7) , 0.6 (3/7) , 0.7 (1/7)}	{0.5 (3/7) , 0.7 (4/7)}	{0.4 (3/7) , 0.5 (4/7)}

Table 14

S_i, R_i and Q(v) _i values and ranking order using the proposed VIKOR method

		UNI Air	Trans	Mandarin	Daily	Ranking	Compromise Solutions
		(B₁)	Asia(B₂)	(B₃)	Air(B₄)
S_i		0.51	0.58	0.37	0.78	B₃ ≻ B₁ ≻ B₂ ≻ B₄	B₃, B₁
R_i		0.23	0.15	0.10	0.28	B₃ ≻ B₂ ≻ B₁ ≻ B₄	B₃, B₂
	0	0.63	0.41	0.27	0.77	B₃ ≻ B₂ ≻ B₁ ≻ B₄	B₃, B₂
	0.1	0.59	0.39	0.25	0.72	B₃ ≻ B₂ ≻ B₁ ≻ B₄	B₃, B₂
	0.2	0.54	0.36	0.23	0.67	B₃ ≻ B₂ ≻ B₁ ≻ B₄	B₃, B₂
	0.3	0.49	0.34	0.22	0.62	B₃ ≻ B₂ ≻ B₁ ≻ B₄	B₃, B₂
	0.4	0.44	0.32	0.20	0.57	B₃ ≻ B₂ ≻ B₁ ≻ B₄	B₃, B₂
Q(v)_i	0.5	0.39	0.30	0.18	0.52	B₃ ≻ B₂ ≻ B₁ ≻ B₄	B₃, B₂
	06	0.35	0.27	0.16	0.47	B₃ ≻ B₂ ≻ B₁ ≻ B₄	B₃, B₂
	0.7	0.30	0.25	0.14	0.43	B₃ ≻ B₂ ≻ B₁ ≻ B₄	B₃, B₂
	0.8	0.25	0.23	0.12	0.38	B₃ ≻ B₂ ≻ B₁ ≻ B₄	B₃, B₂
	0.9	0.20	0.21	0.10	0.33	B₃ ≻ B₁ ≻ B₂ ≻ B₄	B₃, B₁
	1	0.15	0.18	0.09	0.28	B₃ ≻ B₁ ≻ B₂ ≻ B₄	B₃, B₁

From the Table 14, the best airline is Mandarin (B₃) with the compromise solution UNI Air (B₁), a finding that is evidently different from the solution result in Case 1, which results from the loss of information relative to the proportion. Analogous to Case 1, Daily Air remains the worst airline with respect to service quality.

In the context of electric vehicle charging station selection using the same data, we demonstrate the advantages of the proposed methodology over TOPSIS. On the other hand, to illustrate the advantage of PHFSs, we utilized PHFSs and HFSs (a special case of PHFSs) to cope with one example where every membership in each PHFE is distributed with equal proportion while the other is distributed randomly. The comparative results indicate that loss of information can cause large differences in ranking, adversely impacting decision making.

5.2 Comparison analysis using aggregation operator-based MCGDM method

Considering that the aggregation operator is one of the most widely used tools in MCGDM methods, the second comparative method is an aggregation operator, namely, the proportional hesitant fuzzy weighted averaging (PHFWA) operator proposed by Zhang et al. [34] using the same data as in our case study. The score and deviation degree results of all alternatives are demonstrated in Table 15.

Table 15
Score and deviation degree based on Zhang et al. [34]

A ₁ A ₂ A ₃ A ₄

Score 0.747 0.905 0.857 0.962

Deviation degree 2.285e-05 8.70e-06 7.378e-06 1.373e-06

	A ₁	A ₂	A ₃	A ₄
Score	0.747	0.905	0.857	0.962
Deviation degree	2.285e-05	8.70e-06	7.378e-06	1.373e-06

Based on the proposed comparison method [34], the ranking order is A₄ ≻ A₂ ≻ A₃ ≻ A₁, which is aligned with our proposed method when parameter v is increased. However, the calculation of aggregation is complicated. Additionally, the discrimination among the alternatives is not obvious. Furthermore, the DM behaviour is not considered. The results validate the feasibility of the proposed VIKOR method. The results can also prove the robustness of the proposed methods to some extent.

5.3 Comparison analysis using TOPSIS

To demonstrate the effectiveness of the proposed extended VIKOR, it was compared to the extended TOPSIS method. Computational results for the extended TOPSIS based on the proportional hesitant fuzzy information with the same data from Tables 3, 4, 5, and 6 are given in Table 16, according to the relative closeness coefficient of each alternative.

Table 16
Closeness degree of the alternative A_i

A ₁ A ₂ A ₃ A ₄

$d_{i}^{+}$ 0.81 0.38 0.57 0.10

$d_{i}^{-}$ 0.09 0.53 0.34 0.80

C _I 0.10 0.59 0.37 0.89

	A ₁	A ₂	A ₃	A ₄
$d_{i}^{+}$	0.81	0.38	0.57	0.10
$d_{i}^{-}$	0.09	0.53	0.34	0.80
C _I	0.10	0.59	0.37	0.89

Clearly, the larger is the distance $d_{i}^{-}$ and the smaller is the distance $d_{i}^{+}$ , the higher is the closeness coefficient C_I of the alternative. Therefore, we can determine the ranking order of all alternatives and select the best one according to C_I. From the above analysis, we can obtain the ranking A₄ ≻ A₂ ≻ A₃ ≻ A₁. Obviously, TOPSIS determines the optimal solution associated with the shortest distance from the ideal solution and the farthest distance from the negative ideal solution. The shortcoming of this method is that it does not consider the importance of these distances and cannot effectively capture the DM’s behaviour.

By comparing Tables 10 and 16, it can be seen that the ranking of four alternatives solved from the extended VIKOR method was not in accordance with the extended TOPSIS completely. After using the extend VIKOR method, the ranking results vary with the parameter v, in which A₂ is the first rank and A₁ or A₃ is the fourth rank when v ∈ [0, 0.6] while A₄ is the first rank and A₁ is the fourth rank when v ∈ [0.7, 1] with respect to six selected criteria in the optimal location selection problem. After using the extended TOPSIS method, the ranking is A₄ ≻ A₂ ≻ A₃ ≻ A₁. The result obtained via the TOPSIS technique always satisfies the closest to the ideal variety and the farthest from the nadir variety. By contrast, the proposed VIKOR method sets the weight v and 1-v to justify all possible combinations. The slight difference between the two extended methods based on proportional hesitant fuzzy information is that TOPSIS does not consider the minority bias but represents the majority opinions.

6 Conclusions

In this paper, a VIKOR-based framework to optimize the location of fast-charging stations has been developed using proportional hesitant fuzzy information. PHFSs have been utilized to describe the random and fuzziness information in MCGDM. A new distance measure has been proposed to overcome the limitations of the extant proportional hesitant fuzzy distance measure. Moreover, the maximum deviation method has been proposed based on the defined proportional hesitant fuzzy distance to calculate the objective criteria weights. Finally, the PHFS-VIKOR method has been developed to determine the ranking order of locations. Additionally, a case study, sensitivity analysis and comparative analyses have been conducted to prove the practicality, feasibility and robustness of the proposed framework. In the future, we will focus on the operation of PHFS to alleviate the information loss and investigate the nature of PHFS in the form of continuous randomness. Based on the above, we will construct new MCGDM methods to satisfy the various realistic scenarios.

Conflict of interest

All authors declare that there is no conflict of interests regarding the publication of this paper.

Footnotes

Acknowledgments

The authors thank the editors and anonymous reviewers for their helpful comments and suggestions. This work is supported by the National Natural Science Foundation of China (No. 71901226), and the Scientific Research Project of Hunan Provincial Department of Education of China (No. 17B289).

References

Niu

, Zhang

and Wang

, Hierarchical power control strategy on small-scale electric vehicle fast charging station, Journal of Cleaner Production 199 (2018), 1043–1049.

, Kockelman

K.M.

and Perrine

K.A.

, Optimal locations of US fast charging stations for long-distance trip completion by battery electric vehicles, Journal of Cleaner Production 214 (2019), 452–461.

Yuan

, Song

, Shao

, Sun

and Wu

, A charging strategy with the price stimulus considering the queue of charging station and ev fast charging demand, Energy Procedia 145 (2018), 400–405.

Cai

, Jia

, Chiu

A.S.F.

, Hu

and Xu

, Siting public electric vehicle charging stations in Beijing using big-data informed travel patterns of the taxi fleet, Transportation Research Part D: Transport and Environment 33 (2014), 39–46.

Kim

, Son

, Lee

J.M.

and Ha

H.T.

, Scheduling and performance analysis under a stochastic model for electric vehicle charging stations, Omega 66 (2017), 278–289.

Yang

, Sun

, Deng

, Zhao

and Zhou

, Optimal sizing of PEV fast charging stations with Markovian demand characterization, IEEE Transactions on Smart Grid 10 (2018), 4457–4466.

, Ju

, Gonzalez

E.D.R.S.

, Giannakis

and Wang

, Study of site selection of electric vehicle charging station based on extended GRP method under picture fuzzy environment, Comput Ind Eng 135 (2019), 1271–1285.

, Zhong

, Yao

and Wu

, An interval type-2 fuzzy analysis towards electric vehicle charging station allocation from a sustainable perspective, Sustainable Cities And Society 40 (2018), 335–351.

Liu

, Yang

, Zhou

and Tian

, An integrated multi-criteria decision making approach to location planning of electric vehicle charging stations, IEEE Transactions on Intelligent Transportation Systems 20 (2018), 362–373.

10.

Erbaş

, Kabak

, Özceylan

and Çetinkaya

, Optimal siting of electric vehicle charging stations: A GIS-based fuzzy multi-criteria decision analysis, Energy 163 (2018), 1017–1031.

11.

Wang

, Wang

X.-k.

, Peng

J.-j.

and Wang

J.-q.

, The differences in hotel selection among various types of travellers: A comparative analysis with a useful bounded rationality behavioural decision support model, Tourism Management 76 (2020).

12.

Tian

, Peng

J.-j.

, Zhang

W.-y.

, Zhang

and Wang

J.-q.

, Tourism environmental impact assessment based on improved AHP and picture fuzzy PROMETHEE II methods, Technological and Economic Development of Economy (2019), 1–24.

13.

Torra

, Hesitant fuzzy sets, International Journal of Intelligent Systems 25 (2010), 529–539.

14.

Z. B, Decision method for research and application based on preference relation., Southeast University, Nanjing, 2014.

15.

Zhang

and Wu

, Weighted hesitant fuzzy sets and their application to multi-criteria decision making, Journal of Advances in Mathematics and Computer Science (2014), 1091–1123.

16.

Xiong

, Chen

and Chin

, A novel MAGDM approach with proportional hesitant fuzzy sets, International Journal of Computational Intelligence Systems 11 (2018), 256–271.

17.

Chen

Z.-S.

, Chin

K.-S.

, Li

Y.-L.

and Yang

, Proportional hesitant fuzzy linguistic term set for multiple criteria group decision making, Information Sciences 357 (2016), 61–87.

18.

Chen

Z.-S.

, Chin

K.-S.

, Mu

N.-Y.

, Xiong

S.-H.

, Chang

J.-P.

and Yang

, Generating HFLTS possibility distribution with an embedded assessing attitude, Information Sciences 394–395 (2017), 141–166.

19.

Chen

Z.-S.

, Chin

K.-S.

, Martinez

and Tsui

K.-L.

, Customizing semantics for individuals with attitudinal HFLTS possibility distributions, IEEE Transactions on Fuzzy Systems 26 (2018), 3452–3466.

20.

Chen

Z.-S.

, Yang

, Wang

X.-J.

, Chin

K.-S.

and Tsui

K.-L.

, Fostering linguistic decision-making under uncertainty: A proportional interval type-2 hesitant fuzzy TOPSIS approach based on Hamacher aggregation operators and andness optimization models, Information Sciences 500 (2019), 229–258.

21.

Chen

Z.S.

, Li

, Kong

W.T.

and Chin

K.S.

, Evaluation and selection of hazmat transportation alternatives: A PHFLTS- and TOPSIS-integrated multi-perspective approach, International Journal of Environmental Research and Public Health 16 (2019).

22.

Pang

, Wang

and Xu

, Probabilistic linguistic term sets in multi-attribute group decision making, Information Sciences 369 (2016), 128–143.

23.

Tian

, Nie

and Wang

, Probabilistic linguistic multicriteria decision-making based on evidential reasoning and combined ranking methods considering decisionmakers’ psychological preferences, Journal of the Operational Research Society 10.1080/01605682.2019.1632752 (2019).

24.

Liu

and Teng

, Probabilistic linguistic TODIM method for selecting products through online product reviews, Information Sciences 485 (2019), 441–455.

25.

Liu

and Li

, An extended MULTIMOORA method for probabilistic linguistic multi-criteria group decision-making based on prospect theory, Comput Ind Eng 136 (2019), 528–545.

26.

Liu

and Li

, A novel decision-making method based on probabilistic linguistic information, Cognitive Computation (2019), 1–13.

27.

Liu

, Li

and Teng

, Bidirectional projection method for probabilistic linguistic multi-criteria group decision-making based on power average operator, International Journal of Fuzzy Systems 10.1007/s40815-019-00705-y (2019).

28.

Liu

and Teng

, Some Muirhead mean operators for probabilistic linguistic term sets and their applications to multiple attribute decision making, Applied Soft Computing 68 (2018), 396–431.

29.

Liu

and Li

, The PROMTHEE II method based on probabilistic linguistic information and their application to decision making, Informatica 29 (2018), 303–320.

30.

Luo

, Zhang

, Wang

and Li

, Group decision-making approach for evaluating the sustainability of constructed wetlands with probabilistic linguistic preference relations, Journal of the Operational Research Society 10.1080/01605682.2018.1510806 (2019).

31.

Xiao

and Wang

, Multistage decision support framework for sites selection of solar power plants with probabilistic linguistic information, Journal of Cleaner Production 230 (2019), 1396–1409.

32.

Dey

, Bairagi

, Sarkar

and Sanyal

S.K.

, Group heterogeneity in multi member decision making model with an application to warehouse location selection in a supply chain, Comput Ind Eng 105 (2017), 101–122.

33.

Garg

and Kaur

, Algorithm for probabilistic dual hesitant fuzzy multi-criteria decision-making based on aggregation operators with new distance measures, Mathematics 6 (2018), 280.

34.

Zhang

, Xu

and He

, Operations and integrations of probabilistic hesitant fuzzy information in decision making, Information Fusion 38 (2017), 1–11.

35.

Song

, Xu

and Zhao

, A novel comparison of probabilistic hesitant fuzzy elements in multi-criteria decision making, Symmetry 10 (2018), 177.

36.

Zhang

, Wang

and Wang

, Multi-criteria group decision-making method based on TODIM with probabilistic interval-valued hesitant fuzzy information, Expert Systems 36 (2019).

37.

Tian

, Xu

and Fujita

, Sequential funding the venture project or not? A prospect consensus process with probabilistic hesitant fuzzy preference information, Knowledge-Based Systems 161 (2018), 172–184.

38.

Zhou

and Xu

, Group consistency and group decision making under uncertain probabilistic hesitant fuzzy preference environment, Information Sciences 414 (2017), 276–288.

39.

and Wang

, Multi-criteria decision-making with probabilistic hesitant fuzzy information based on expected multiplicative consistency, Neural Computing and Applications 10.1007/s00521-018-3753-1 (2018).

40.

, Liu

X.-D.

, Wang

Z.-W.

and Zhang

S.-T.

, Dynamic emergency decision-making method with probabilistic hesitant fuzzy information based on GM (1, 1) and TOPSIS, IEEE Access 7 (2018), 7054–7066.

41.

Ding

, Xu

and Zhao

, An interactive approach to probabilistic hesitant fuzzy multi-attribute group decision making with incomplete weight information, Journal of Intelligent & Fuzzy Systems 32 (2017), 2523–2536.

42.

and Wang

J.Q.

, Multi-criteria outranking methods with hesitant probabilistic fuzzy sets, Cognitive Computation 9 (2017), 611–625.

43.

Opricovic

, Multicriteria optimization of civil engineering systems, Faculty of Civil Engineering, Belgrade 2 (1998), 5–21.

44.

Chen

, Remoteness index-based Pythagorean fuzzy VIKOR methods with a generalized distance measure for multiple criteria decision analysis, Information Fusion 41 (2018), 129–150.

45.

Mousavi

S.M.

, Vahdani

and Behzadi

S.S.

, Designing a model of intuitionistic fuzzy VIKOR in multi-attribute group decision-making problems, Iranian Journal of Fuzzy Systems 13 (2016), 45–65.

46.

, Li

, Liao

, Zavadskas

E.K.

, Al-Barakati

, Barnawi

, Taylan

and Herrera-Viedma

, Hospitality brand management by a score-based q-rung orthopair fuzzy VIKOR method integrated with the best worst method, Economic Research-Ekonomska Istraživanja 32 (2019), 3266–3295.

47.

Emeç

Ş.

and Akkaya

, Stochastic AHP and fuzzy VIKOR approach for warehouse location selection problem, Journal of Enterprise Information Management 31 (2018), 950–962.

48.

Z.B.

, Ahmad

and Xu

J.p.

, A group decision making framework based on fuzzy VIKOR approach for machine tool selection with linguistic information, Applied Soft Computing 42 (2016), 314–324.

49.

Liu

H.C.

, You

J.X.

, You

X.Y.

and Shan

M.M.

, A novel approach for failure mode and effects analysis using combination weighting and fuzzy VIKOR method, Applied Soft Computing 28 (2015), 579–588.

50.

Wang

, Zhang

H.Y.

, Wang

J.Q.

and Li

, Picture fuzzy normalized projection-based VIKOR method for the risk evaluation of construction project, Applied Soft Computing 64 (2018), 216–226.

51.

Shen

K.W.

and Wang

J.Q.

, Z-VIKOR method based on a new weighted comprehensive distance measure of Z-number and its application, IEEE Transactions on Fuzzy Systems 10.1109/tfuzz.2018.2816581 (2018), 1–1.

52.

Jiang

, Liang

and Sun

, A method for discrete stochastic MADM problems based on the ideal and nadir solutions, Comput Ind Eng 87 (2015), 114–125.

53.

Opricovic

and Tzeng

G.H.

, Multicriteria planning of post earthquake sustainable reconstruction, Computer Aided Civil and Infrastructure Engineering 17 (2002), 211–220.

54.

Opricovic

and Tzeng

G.-H.

, Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS, European Journal of Operational Research 156 (2004), 445–455.

55.

, Yang

, Zhang

, Chen

and Wang

, Optimal site selection of electric vehicle charging stations based on a cloud model and the PROMETHEE method, Energies 9 (2016), 157.

56.

Guo

and Zhao

, Optimal site selection of electric vehicle charging station by using fuzzy TOPSIS based on sustainability perspective, Applied Energy 158 (2015), 390–402.

57.

Liao

and Xu

, A VIKOR-based method for hesitant fuzzy multi-criteria decision making, Fuzzy Optimization and Decision Making 12 (2013), 373–392.

A VIKOR-based framework to optimize the location of fast-charging stations with proportional hesitant fuzzy information

Abstract

Keywords

1 Introduction

2 Preliminary

2.1 PHFS concept

3.1 Identification of the ideal and nadir solutions

3.2 A metric measuring the distance between two PHFSs

4.1 Implementation of the proposed approach

5.1 Comparison analysis between HFSs and PHFSs

Table 15 Score and deviation degree based on Zhang et al. [34] A 1 A 2 A 3 A 4 Score 0.747 0.905 0.857 0.962 Deviation degree 2.285e-05 8.70e-06 7.378e-06 1.373e-06

Table 16 Closeness degree of the alternative A i A 1 A 2 A 3 A 4 d i + 0.81 0.38 0.57 0.10 d i - 0.09 0.53 0.34 0.80 C I 0.10 0.59 0.37 0.89

Conflict of interest

Footnotes

Acknowledgments

References

Table 15
Score and deviation degree based on Zhang et al. [34]

A ₁ A ₂ A ₃ A ₄

Score 0.747 0.905 0.857 0.962

Deviation degree 2.285e-05 8.70e-06 7.378e-06 1.373e-06

Table 16
Closeness degree of the alternative A_i

A ₁ A ₂ A ₃ A ₄

$d_{i}^{+}$ 0.81 0.38 0.57 0.10

$d_{i}^{-}$ 0.09 0.53 0.34 0.80

C _I 0.10 0.59 0.37 0.89