Consensus-based group decision-making methods with probabilistic dual hesitant fuzzy preference relations and their applications

Abstract

Probabilistic dual hesitant fuzzy sets (PDHFSs) have good flexibility and integrity in expressing fuzzy and uncertain information. However, some crucial problems related to PDHFSs remain unsolved, such as how to define probabilistic dual hesitant fuzzy preference relations (PDHFPRs) and solve group decision-making (GDM) problems with PDHFPRs. This paper establishes the concept of PDHFPRs and investigates consensus-based GDM methods with PDHFPRs. First, a new distance measure is proposed to quantify the difference between two PDHFPRs, which does not increase the virtual elements of membership and non-membership degrees, and can contain all distance combination of membership and non-membership elements. Therefore, the distance calculation results are not affected by the subjectivity of decision-makers (DMs). Second, the consensus measures for PDHFPRs are proposed, which are effective tool to measure the consensus level among DMs. Moreover, two consensus-based GDM methods are proposed, which can improve the group consensus level for PDHFPRs by changing the PDHFPR with the worst consensus level or modifying the weights of DMs. Finally, the proposed methods are applied to the location selection of large-scale industrial solid waste treatment facilities. The comparison with existing methods illustrates the validity and feasibility of the proposed methods.

Keywords

Group decision-making probabilistic dual hesitant fuzzy preference relations distance measure consensus

1 Introduction

Decision-making is the process by which people choose the most satisfactory scheme from several feasible schemes as well as an indispensable activity in human social life. With the development of society and the progress of science and technology, people are faced with increasingly complex decision-making problems, which are difficult to describe accurately. To address this situation, in 1965, Zadeh [19] proposed the definition of fuzzy sets (FSs). Later, FSs were widely applied in the fields of fuzzy diagnosis, fuzzy preference evaluation, fuzzy information aggregation, fuzzy estimation and so on [3 , 40]. However, with the rapid development of FSs, the concept of FSs was extended to intuitionistic FSs [25, 39], interval-valued intuitionistic FSs [6 , 31], hesitant FSs [1 , 38], type-2 FSs [21 , 37], hesitant fuzzy linguistic term sets [24], fuzzy multiple sets [2] and dual hesitant FSs [5, 26], to name but a few. Furthermore, the probabilistic dual hesitant FS (PDHFS) [42] considers hesitations of membership degree, non-membership degree and corresponding probabilities for these two types of degree, which is a valuable extension of FS.

In the process of decision-making, due to the complexity of decision-making environment, it is easier for decision-makers (DMs) to give the preference information by making pairwise comparisons for all the alternatives. As a tool to effectively express decision information, preference relations are widely used in group decision-making (GDM) scenarios, and a widely used preference relation is multiplicative preference relation (MPR) represented by Satty’s 1–9 scale. Vargas [20] used MPR to express the preference intensity of voters, and proposed eigenvector method of pairwise comparisons in voting. Cabrera et al. [8] verified the eigenvector method based on MPR by the ranking problem of tourist destinations. Up to now, extensions of MPR have been introduced. For example, Saaty and Vargas [32] used the interval MPR to express the preference value of DMs. Xia and Xu [22] introduced the definition of hesitant MPR and used it in GDM methods. Another widely used preference relation is called fuzzy preference relations (FPR). Different from MPR, the information scale of FPR is uniformly and symmetrically distributed in the interval [0,1] with 0.5 as the center. With the development of FSs, various forms of FPRs and their applications in GDM have been studied. Liao et al. [12] considered the hesitant fuzzy information of the DM and proposed the concept of hesitant FPR, whose membership degree consists of a set of possible values. Zhou and Xu [36] established probabilistic FPR by combining probabilistic information with FPR. Considering the hesitations of membership degree and non-membership degree, Zhao et al. [26] proposed dual hesitant FPR and investigated GDM methods with dual hesitant FPRs. However, all these FPRs did not comprehensively consider hesitations of membership degree and non-membership degree and their occurrence possibilities, and cannot truly describe the original decision-making information. For this reason, inspired by PDHFS, this paper proposes the concept of probabilistic dual hesitant FPR (PDHFPR), which can describe decision information by considering the possible degrees that one alternative is preferred and non-preferred to another and the probabilities of these degrees.

The group consensus level for preference relations is a key attribute to consider in GDM problems. Results satisfying the expected consensus level threshold can be accepted by most DMs. Li and Wang [18] studied the expected multiplicative consistency and group consensus of probabilistic hesitant FPRs (PHFPRs), constructed a programming model to adjust the individual consistency level and the group consensus of PHFPRs simultaneously, and verified the validity of the decision-making model through an example of green energy project selection. A measurement method was proposed to measure the similarity between type-1 FSs and type-2 FSs in López-Ortega et al. [27], which was used to quantify the consensus level in GDM. In order to solve the problem of how interval additive preference relation can reach the preset consensus threshold, Wu et al. [40] designed an optimization model with interval additive preference relations. Labella et al. [4] established a new consensus reaching method to solve GDM problems, which used a comparative language expression based on hesitant fuzzy linguistic term sets to express the preferences of DMs. Moreover, Saaty and Vargas [33] aggregated different MPRs by geometric mean method, and then derived a welfare function for the group preference. Later they [34] defined the geometric dispersion of group judgments (or consensus indicator) to measure the consensus degree between MPRs. Different from MPR, PDHFPR is an extended form of FPR which can express decision information more completely. However, the consensus of PDHFPRs in GDM has never been studied. Thus, it is urgent to study the consensus of PDHFPRs and use it to solve the GDM problem with PDHFPRs.

Since PDHFS was proposed in 2017, it has been widely used in GDM. Hao et al. [42] studied the basic properties and operators of PDHFSs and proposed a visualization method for Arctic geopolitical risk evaluation. Garg and Kaur [15] proposed the weighted probabilistic dual hesitant fuzzy Maclaurin symmetric mean averaging (WPDHFMSMA) operator and weighted probabilistic dual hesitant fuzzy Maclaurin symmetric mean geometric (WPDHFMSMG) operator. Based on these operators, a GDM model was proposed to solve the uncertainty problem in medical diagnosis. Garg and Kaur [14] proposed the correlation coefficient and weighted correlation coefficients of PDHFSs and established a novel algorithm for addressing multi-criteria decision-making problems. Using Einstein norm operations, Garg and Kaur [13] developed two distance measures and a maximum deviation method to calculate the weight vectors of attributes and then constructed an multi-criteria decision-making algorithm. Ren et al. [43] proposed a new decision making method combining AHP and VIKOR under probabilistic dual fuzzy information, and then applied it to the strategy selection problem in artificial intelligence.

However, the existing research was only limited to GDM with PDHFSs. Because of the structure of preference relation and the importance of consensus in GDM, it is necessary to propose the definition of PDHFPR and study how to solve GDM problems with PDHFPRs using consensus. Especially, Garg and Kaur [13] proposed two distance measures on PDHFSs, but one distance measure needs to add elements to membership degree and non-membership degree, and the other distance measure does not consider all distance combination of membership and non-membership elements. Therefore, it is necessary to propose a new distance measure to accurately and comprehensively reflect the differences between PDHFSs.

Based on the above analysis, this paper aims to design consensus-based group decision-making methods with PDHFPRs. Specifically, the advantages of our methods are summarized as follows:

Based on PDHFSs, this paper defines PDHFPRs, which require the DMs to give preference values through pairwise comparison of alternatives without directly evaluating alternatives, and are more appropriate for the decision of DMs in complicated environments.

Different from the distance measures in Garg and Kaur [13], this paper proposes a new distance measure, which does not increase the virtual elements of membership and non-membership degrees, and can contain all distance combination of membership and non-membership elements.

The consensus measures for PDHFPRs and two consensus-based GDM methods with PDHFPRs are proposed. According to the wishes of DMs, the group consensus level for PDHFPRs can reach the expected consensus level threshold by changing the PDHFPR or modifying the weights of DMs.

To do so, the remainder of this paper is arranged as follows. After reviewing some basic definitions of PDHFS, concepts such as PDHFPRs, the distance measure of PDHFPRs and the consensus measure for PDHFPRs are given in Section 3. Based on PDHFPRs, two GDM methods are constructed in Section 4. Section 5 presents an illustrative example and comparative study. Finally, this paper ends with conclusions in Section 6.

2 Preliminaries

Definition 1 [42]. If X is a finite set of alternatives, a PDHFS H on X is denoted as: $H = {〈 x, f (x) | p (x), g (x) | q (x) 〉 | x \in X} .$ (1)f (x) |p (x) and g (x) |q (x) are two sets of some possible elements, where f (x) and g (x) represent the hesitant fuzzy membership degrees and non-membership degrees to the set X of x, respectively. p (x) and q (x) are the corresponding probabilities for these two types of degrees. Additionally, $0 ⩽ γ, τ ⩽ 1, 0 ⩽ γ^{+} + τ^{+} ⩽ 1,$ (2) $0 ⩽ p_{i} ⩽ 1, \sum_{i = 1}^{# f} p_{i} = 1, 0 ⩽ q_{i} ⩽ 1, \sum_{i = 1}^{# g} q_{i} = 1 .$ (3) where γ ∈ f (x), γ⁺∈ f⁺ (x) = ⋃ _γ∈h(x) max { γ }, τ ∈ g (x), τ⁺∈ g⁺ (x) = ⋃ _τ∈g(x) max { τ }, p_i ∈ p (x) and q_i ∈ q (x). The symbols # f and # g are the total numbers of elements in the f (x) |p (x) and g (x) |q (x) , respectively.

Remark 1. h = (f (x) |p (x) , g (x) |q (x) ) represents the probabilistic dual hesitant element (PDHFE) in Hao et al. [42] and is abbreviated as h = (f|p, g|q ).

Definition 2 [42]. Let h = (f|p, g|q ), h¹ = (f₁|p_{f
₁}, g₁|q_{g
₁} ), h² = (f₂|p_{f
₂}, g₂|q_{g
₂} ) be three PDHFEs. Then $\begin{matrix} h^{1} \oplus h^{2} = ⋃ γ_{1} \in f_{1}, τ_{1} \in g_{1}, \\ γ_{2} \in f_{2}, τ_{2} \in g_{2} ({(γ_{1} + γ_{2} - γ_{1} γ_{2}) | p_{γ_{1}} p_{γ_{2}}}, {(τ_{1} τ_{2}) | q_{τ_{1}} q_{τ_{2}}}) . \end{matrix}$ $\begin{matrix} h^{1} \otimes h^{2} = ⋃_{γ_{1} \in f_{1}, τ_{1} \in g_{1}, γ_{2} \in f_{2}, τ_{2} \in g_{2}} ({(γ_{1} γ_{2}) | p_{γ_{1}} p_{γ_{2}}}, \\ {(τ_{1} + τ_{2} - τ_{1} τ_{2}) | q_{τ_{1}} q_{τ_{2}}}), \end{matrix}$ $λ h = \underset{γ \in f, τ \in g}{\cup} ({(1 - (1 - γ)^{λ}) | p_{γ}}, {τ^{λ} | q_{τ}}),$ $h^{λ} = \underset{γ \in f, τ \in g}{\cup} ({γ^{λ} | p_{γ}}, {(1 - (1 - τ)^{λ}) | q_{τ}}) .$ where λ ⩾ 0.

Based on the above operation rules, Hao et al. [42] defined the probabilistic dual hesitant fuzzy weighted averaging operator (PDHFWA).

Definition 3 [42]. If h^Λ = (f_Λ|p_{f
_Λ}, g_Λ|q_{g
_Λ} ) (Λ = 1, 2, ·· · , n) are PDHFE sequences and φ = (φ₁, φ₂, ·· · , φ_m) ^T is the corresponding weight vector, where φ_i ⩾ 0, φ₁ + φ₂ + ·· · + φ_n = 1, then $\begin{matrix} PDHFWA (h^{1}, h^{2}, \cdot \cdot \cdot, h^{n}) = \oplus_{i = 1}^{n} φ_{i} h^{i} = \underset{τ_{1} \in g_{1,} τ_{2} \in g_{2}, \cdot \cdot \cdot, τ_{n} \in g_{n}}{⋃_{γ_{1} \in f_{1}, γ_{2} \in f_{2}, \cdot \cdot \cdot, γ_{n} \in f_{n},}} \\ ({(1 - \prod_{i = 1}^{n} (1 - γ_{i})^{φ_{i}}) | \prod_{i = 1}^{n} p_{γ_{i}}}, {\prod_{j = 1}^{n} τ_{j}^{φ_{j}} | \prod_{j = 1}^{n} q_{τ_{j}}}) . \end{matrix}$

Definition 4 [42]. Let h = (f|p, g|q ) be a PDHFE, the score function of the PDHFE is denoted as: $s (h) = \sum_{i = 1, γ \in f}^{# f} γ_{i} \cdot p_{i} - \sum_{j = 1, τ \in g}^{# g} τ_{j} \cdot q_{j},$ (4) and the deviation function of the PDHFE is denoted as: $σ (h) = (\sum_{i = 1, γ \in f}^{# f} (γ_{i} - s (h))^{2} \cdot p_{i} + {\sum_{j = 1, τ \in g}^{# g} (τ_{j} - s (h))^{2} \cdot q_{j})}^{1 / 2},$ (5) where s (h) is the score function of h, the symbols # f and # g are the total numbers of elements in the f|p and g|q, respectively.

If h^Λ (Λ = 1, 2) are two arbitrary PDHFEs, s (h^Λ) and σ (h^Λ) (Λ = 1, 2) are the corresponding score function and deviation function, respectively, then,

If s (h¹) > s (h²), PDHFE h¹ is superior to h², denoted by h¹ ≻ h²; in contrast, if s (h¹) < s (h²), PDHFE h¹ is inferior to h², denoted by h¹ ≺ h².

If s (h¹) = s (h²), then

If σ (h¹) > σ (h²), the PDHFE h¹ is inferior to h², denoted by h¹ ≺ h²;

If σ (h¹) = σ (h²), the PDHFE h¹ is indifferent to h², denoted by h¹ ∼ h²;

If σ (h¹) < σ (h²), the PDHFE h¹ is superior to h², denoted by h¹ ≻ h².

3 PDHFPRs, distance measure of PDHFPRs and consensus measure for PDHFPRs

This section introduces a series of concepts, such as PDHFPRs, the distance measure of PDHFPRs and the consensus measure for PDHFPRs.

3.1 PDHFPRs

Definition 5. Let X ={ x₁, x₂, ·· · , x_n } be a finite set of alternatives. A PDHFPR D on X is defined by D = (d_ij) _n×n, where for i, j = 1, 2, ·· · , n, d_ij = (f_ij|p_ij , g_ij|q_ij) is a PDHFE. f_ij indicates the possible degrees to which x_i is preferred to x_j, p_ij is the corresponding probabilities of f_ij, g_ij indicates the possible degrees to which x_i is non-preferred to x_j and q_ij is the corresponding probabilities of g_ij. The symbols # f_ij and # g_ij are the total numbers of elements in f_ij|p_ij and g_ij|q_ij, respectively. In addition, the following conditions should be satisfied:

For i = 1, 2, ·· · , n, d_ii = ({ 0.5|1 } , { 0.5|1 });

For i, j = 1, 2, ·· · , n, i ≠ j,

If f_ij≠ ø , g_ji ≠ ø, then ${\begin{matrix} f_{ij, l} = g_{ji, l}, g_{ij, k} = f_{ji, k} \\ p_{ij, l} = q_{ji, l}, q_{ij, k} = p_{ji, k} \end{matrix},$ (6) for all l = 1, 2, ·· · , # f_ij - 1, k = 1, 2, ·· · , # g_ij - 1, where f_ij,l ∈ f_ij indicates the lth element in f_ij, p_ij,r is the corresponding probability of f_ij,r, $\sum_{l = 1}^{# f_{ij}} p_{ij, l} = 1$ , g_ij,k ∈ g_ij indicates the kth element in g_ij, q_ij,k is the corresponding probability of g_ij,k, $\sum_{k = 1}^{# g_{ij}} q_{ij, k} = 1$ , f_ij,l < f_ij,l+1 and g_ij,k < g_ij,k+1.

If f_ij ≠ ø (p_ij ≠ ø) , g_ji = ø (q_ji = ø), then ${\begin{matrix} f_{ij, l} + f_{ji, # h_{ij} - l + 1} = 1, \\ p_{ij, l} + p_{ji, # h_{ij} - l + 1} = 1, # f_{ij} = # f_{ji}, \end{matrix}$ (7) for all l = 1, 2, ·· · , # f_ij - 1, , where f_ij,l ∈ f_ij indicates the lth element in f_ij, p_ij,l is the corresponding probability of f_ij,l, $\sum_{l = 1}^{# f_{ij}} p_{ij, l} = 1$ and f_ij,l < f_ij,l+1.

If f_ij = ø (p_ij = ø) , g_ji ≠ ø (q_ji ≠ ø), then ${\begin{matrix} g_{ij, k} + g_{ji, # g_{ij} - k + 1} = 1, \\ q_{ij, r} + q_{ji, # g_{ij} - k + 1} = 1, # g_{ij} = # g_{ji}, \end{matrix}$ (8) for all k = 1, 2, ·· · , # g_ij - 1 where g_ij,k ∈ g_ij indicates the kth element in g_ij, q_ij,k is the corresponding probability of g_ij,k, $\sum_{k = 1}^{# g_{ij}} q_{ij, k} = 1$ and g_ij,k < g_ij,k+1.

Unless otherwise specified, all PDHFPR D = (d_ij) _n×n, satisfy f_ij≠ ø and g_ji≠ ø.

Example 1 The DM gives decision-making information for three alternatives {x₁, x₂, x₃}. $D = (\begin{matrix} ({0.5 | 1}, {0.5 | 1}) & \begin{matrix} ({0.36 | 0.3, 0.6 | 0.7}, \\ {0.2 | 0.25, 0.4 | 0.75}) \end{matrix} & \begin{matrix} ({0.4 | 0.25, 0.56 | 0.75}, \\ {0.2 | 0.8, 0.3 | 0.2}) \end{matrix} \\ \begin{matrix} ({0.2 | 0.25, 0.4 | 0.75}, \\ {0.36 | 0.3, 0.6 | 0.7}) \end{matrix} & ({0.5 | 1}, {0.5 | 1}) & \begin{matrix} ({0.5 | 0.4, 0.7 | 0.6}, \\ {0.1 | 0.55, 0.3 | 0.45}) \end{matrix} \\ \begin{matrix} ({0.2 | 0.8, 0.3 | 0.2}, \\ {0.4 | 0.25, 0.56 | 0.75}) \end{matrix} & \begin{matrix} ({0.1 | 0.55, 0.3 | 0.45}, \\ {0.5 | 0.4, 0.7 | 0.6}) \end{matrix} & ({0.5 | 1}, {0.5 | 1}) \end{matrix})$

According to Definition 5, D is a PDHFPR.

Definition 6 Let X ={ x₁, x₂, ·· · , x_n } be a finite set of alternatives. PDHFPRs D^Λ (Λ = 1, 2, ·· · , m) on X are defined by $D^{Λ} = (d_{ij}^{Λ})_{n \times n} (Λ = 1, 2, \cdot \cdot \cdot, m)$ , where $d_{ij}^{Λ} = (f_{ij}^{Λ} | p_{ij}^{Λ}, g_{ij}^{Λ} | q_{ij}^{Λ})$ is a PDHFE. φ = (φ₁, φ₂, ·· · , φ_m) ^T is the corresponding weight vector, where φ_i ⩾ 0 and φ₁ + φ₂ + ·· · + φ_m = 1, then $D = PDHFWA (D^{1}, D^{2}, \cdot \cdot \cdot, D^{m}) = \sum_{t = 1}^{m} φ_{t} \otimes D^{t} = (d_{ij})_{n \times n}$ is a PDHFPR, and for all i, j = 1, 2, ·· · , n, $\begin{matrix} d_{ij} = \underset{g_{ij, β^{2}}^{2} \in g_{ij}^{2}, \cdot \cdot \cdot, g_{ij, β^{m}}^{m} \in g_{ij}^{m}}{\underset{f_{ij, α^{m}}^{m} \in f_{ij}^{m}, g_{ij, β^{1}}^{1} \in g_{ij}^{1},}{⋃_{f_{ij, α^{1}}^{1} \in f_{ij}^{1}, f_{ij, α^{2}}^{2} \in f_{ij}^{2}, \cdot \cdot \cdot,}}} \\ ({{(1 - \prod_{Λ = 1}^{m} (1 - f_{ij, α^{Λ}}^{Λ}))}^{φ_{Λ}} | \prod_{Λ = 1}^{m} p_{ij, α^{Λ}}^{Λ}}, \\ {\prod_{Λ = 1}^{m} (g_{ij, β^{Λ}}^{Λ^{φ_{Λ}}} | \prod_{Λ = 1}^{m} q_{ij, β^{Λ}}^{Λ}}) . \end{matrix}$ (9) This conclusion can be proven according to Definition 3 and Definition 5.

3.2 The distance measure for PDHFPRs

Definition 7 Let X ={ x₁, x₂, ·· · , x_n } be a finite set of alternatives. Two PDHFPRs on X are defined as $D^{Λ} = (d_{ij}^{Λ})_{n \times n} (Λ = 1, 2)$ , where for i, j = 1, 2, ·· · , n, $d_{ij}^{1} = (f_{ij}^{1} | p_{ij}^{1}, g_{ij}^{1} | q_{ij}^{1}), d_{ij}^{2} = (f_{ij}^{2} | p_{ij}^{2}, g_{ij}^{2} | q_{ij}^{2})$ , the distance measure between D¹ and D² is given as $\begin{matrix} dis (D^{1}, D^{2}) \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} \sum_{j = 1}^{n} (\frac{1}{# f_{ij}^{1} # f_{ij}^{2}} \sum_{l = 1}^{# f_{ij}^{1}} \sum_{v = 1}^{# f_{ij}^{2}} | f_{ij, l}^{1} p_{ij, l}^{1} - f_{ij, v}^{2} p_{ij, v}^{2} | \\ + \frac{1}{# g_{ij}^{1} # g_{ij}^{2}} \sum_{k = 1}^{# g_{ij}^{1}} \sum_{o = 1}^{# g_{ij}^{2}} | g_{ij, k}^{2} q_{ij, k}^{2} - g_{ij, o}^{2} q_{ij, o}^{2} |), \end{matrix}$ (10) where for Λ = 1, 2, $f_{ij, l}^{Λ} \in f_{ij}^{Λ}$ indicates the lth element in $f_{ij}^{Λ}$ , $p_{ij, l}^{Λ}$ is the corresponding probability of $f_{ij, l}^{Λ}$ , $\sum_{l = 1}^{# f_{ij}^{Λ}} p_{ij, l}^{Λ} = 1$ , $g_{ij, k}^{Λ} \in g_{ij}^{Λ}$ indicates the kth element in $g_{ij}^{Λ}$ , $q_{ij, k}^{Λ}$ is the corresponding probability of $g_{ij, k}^{Λ}$ , and $\sum_{k = 1}^{# g_{ij}^{Λ}} q_{ij, k}^{Λ} = 1$ . The symbols $# f_{ij}^{Λ}$ and $# g_{ij}^{Λ}$ are the total numbers of elements in $f_{ij}^{Λ} | p_{ij}^{Λ}$ and $g_{ij}^{Λ} | q_{ij}^{Λ}$ , respectively.

The distance measure between PDHFPRs D¹ and D² in Equation (10) satisfies the following properties:

Theorem 1 Let $D^{Λ} = (d_{ij}^{Λ})_{n \times n} (Λ = 1, 2, 3)$ be three PDHFPRs, where for i, j = 1, 2, ·· · , n, $d_{ij}^{Λ} = (f_{ij}^{Λ} | p_{ij}^{Λ}, g_{ij}^{Λ} | q_{ij}^{Λ})$ is a PDHFE, $f_{ij, l}^{Λ} \in f_{ij}^{Λ}$ indicates the lth element in $f_{ij}^{Λ}$ , $p_{ij, l}^{Λ}$ is the corresponding probability of $f_{ij, l}^{Λ}$ , $\sum_{l = 1}^{# f_{ij}^{Λ}} p_{ij, l}^{Λ} = 1$ , $g_{ij, k}^{Λ} \in g_{ij}^{Λ}$ indicates the kth element in $g_{ij}^{Λ}$ , $q_{ij, k}^{Λ}$ is the corresponding probability of $g_{ij, k}^{Λ}$ , and $\sum_{k = 1}^{# g_{ij}^{Λ}} q_{ij, k}^{Λ} = 1$ . The symbols $# f_{ij}^{Λ}$ and $# g_{ij}^{Λ}$ are the total numbers of elements in $f_{ij}^{Λ} | p_{ij}^{Λ}$ and $g_{ij}^{Λ} | q_{ij}^{Λ}$ . The distance measure in Equation (10) satisfies the following properties:

dis (D¹, D²)⩾0 ;

dis (D¹, D²) = dis (D², D¹) ;

dis (D¹, D²) ⩽ dis (D¹, D³) + dis (D³, D²) .

Proof (i) and (ii) are clearly established.

(iii) $\begin{matrix} dis (D^{1}, D^{2}) \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} \sum_{j = 1}^{n} {\frac{1}{# f_{ij}^{1} # f_{ij}^{2}} \sum_{l = 1}^{# f_{ij}^{1}} \sum_{v = 1}^{# f_{ij}^{2}} | f_{ij, l}^{1} p_{ij, l}^{1} - f_{ij, v}^{2} p_{ij, v}^{2} | \\ + \frac{1}{# g_{ij}^{1} # g_{ij}^{2}} \sum_{k = 1}^{# g_{ij}^{1}} \sum_{l = 1}^{# g_{ij}^{2}} | g_{ij, k}^{2} q_{ij, k}^{2} - g_{ij, o}^{2} q_{ij, o}^{2} |} \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} \sum_{j = 1}^{n} {\frac{# f_{ij}^{3}}{# f_{ij}^{1} # f_{ij}^{2} # f_{ij}^{3}} \sum_{l = 1}^{# f_{ij}^{1}} \sum_{v = 1}^{# f_{ij}^{2}} | f_{ij, l}^{1} p_{ij, l}^{1} - f_{ij, v}^{2} p_{ij, v}^{2} | \\ + \frac{# g_{ij}^{3}}{# g_{ij}^{1} # g_{ij}^{2} # g_{ij}^{3}} \sum_{k = 1}^{# g_{ij}^{1}} \sum_{l = 1}^{# g_{ij}^{2}} | g_{ij, k}^{1} q_{ij, k}^{1} - g_{ij, o}^{2} q_{ij, o}^{2} |} \\ = \frac{1}{n^{2}} \sum_{i = 1}^{n} \sum_{j = 1}^{n} {\frac{1}{# f_{ij}^{1} # f_{ij}^{2} # f_{ij}^{3}} \sum_{s = 1}^{# f_{ij}^{3}} \sum_{l = 1}^{# f_{ij}^{1}} \sum_{v = 1}^{# f_{ij}^{2}} | f_{ij, l}^{1} p_{ij, l}^{1} - f_{ij, v}^{2} p_{ij, v}^{2} | \\ + \frac{1}{# g_{ij}^{1} # g_{ij}^{2} # g_{ij}^{3}} \sum_{t = 1}^{# g_{ij}^{3}} \sum_{k = 1}^{# g_{ij}^{1}} \sum_{l = 1}^{# g_{ij}^{2}} | g_{ij, k}^{1} q_{ij, k}^{1} - g_{ij, o}^{2} q_{ij, o}^{2} |} \\ ⩽ \frac{1}{n^{2}} \sum_{i = 1}^{n} \sum_{j = 1}^{n} {\frac{1}{# f_{ij}^{1} # f_{ij}^{2} # f_{ij}^{3}} (\sum_{s = 1}^{# f_{ij}^{3}} \sum_{l = 1}^{# f_{ij}^{1}} \sum_{v = 1}^{# f_{ij}^{2}} | f_{ij, l}^{1} p_{ij, l}^{1} - f_{ij, s}^{3} p_{ij, s}^{3} | \\ + \sum_{s = 1}^{# f_{ij}^{3}} \sum_{l = 1}^{# f_{ij}^{1}} \sum_{v = 1}^{# f_{ij}^{2}} | f_{ij, s}^{3} p_{ij, s}^{3} - f_{ij, v}^{2} p_{ij, v}^{2} |) \\ + \frac{1}{# g_{ij}^{1} # g_{ij}^{2} # g_{ij}^{3}} (\sum_{t = 1}^{# g_{ij}^{3}} \sum_{k = 1}^{# g_{ij}^{1}} \sum_{l = 1}^{# g_{ij}^{2}} | g_{ij, k}^{1} q_{ij, k}^{1} - g_{ij, t}^{3} q_{ij, t}^{3} | \\ + \sum_{t = 1}^{# g_{ij}^{3}} \sum_{k = 1}^{# g_{ij}^{1}} \sum_{l = 1}^{# g_{ij}^{2}} | g_{ij, t}^{3} q_{ij, t}^{3} - g_{ij, o}^{2} q_{ij, o}^{2} |)} \\ = dis (D^{1}, D^{3}) + dis (D^{3}, D^{2}) \end{matrix}$

Therefore, Theorem 1 has been proved.

3.3 The consensus measure for PDHFPRs

Definition 8 Let X ={ x₁, x₂, ·· · , x_n } be a finite set of alternatives. PDHFPR sequences on X are defined by $D^{Λ} = (d_{ij}^{Λ})_{n \times n} (i, j = 1, 2, \cdot \cdot \cdot, n, Λ = 1, 2, \cdot \cdot \cdot, m)$ , where $d_{ij}^{Λ} = (f_{ij}^{Λ} | p_{ij}^{Λ}, g_{ij}^{Λ} | q_{ij}^{Λ})$ is a PDHFE, $f_{ij, α^{Λ}}^{Λ} \in f_{ij}^{Λ}$ indicates the α^Λth element in $f_{ij}^{Λ}$ , $p_{ij, α^{Λ}}^{Λ}$ is the corresponding probability of $f_{ij, α^{Λ}}^{Λ}$ , $\sum_{α^{Λ} = 1}^{# f_{ij}^{Λ}} p_{ij, α^{Λ}}^{Λ} = 1$ , $g_{ij, β^{Λ}}^{Λ} \in g_{ij}^{Λ}$ indicates the β^Λth element in $g_{ij, β^{Λ}}^{Λ}$ , $q_{ij, β^{Λ}}^{Λ}$ is the corresponding probability of $g_{ij, β^{Λ}}^{Λ}$ and $\sum_{β^{Λ} = 1}^{# g_{ij}^{Λ}} q_{ij, β^{Λ}}^{Λ} = 1$ . The symbols $# f_{ij}^{Λ}$ and $# g_{ij}^{Λ}$ are the total numbers of elements in $f_{ij}^{Λ} | p_{ij}^{Λ}$ and $g_{ij}^{Λ} | q_{ij}^{Λ}$ . The individual consensus measure for PDHFPRs D^Λ is denoted as $\begin{matrix} ICM (D^{Λ}) \\ = 1 - \frac{1}{m - 1} \sum_{κ \neq c} dis (D^{Λ}, D^{c}) = 1 - \frac{1}{(m - 1) n^{2}} \cdot \\ \sum_{c \neq Λ} \sum_{i = 1}^{n} \sum_{j = 1}^{n} (\frac{1}{# f_{ij}^{Λ} # f_{ij}^{c}} \sum_{α^{Λ} = 1}^{# f_{ij}^{Λ}} \sum_{α^{c} = 1}^{# f_{ij}^{c}} | f_{ij, α^{Λ}}^{Λ} p_{ij, α^{Λ}}^{Λ} - f_{ij, α^{c}}^{c} p_{ij, α^{c}}^{c} | \\ + \frac{1}{# g_{ij}^{Λ} # g_{ij}^{c}} \sum_{β^{Λ} = 1}^{# g_{ij}^{Λ}} \sum_{β^{c} = 1}^{# g_{ij}^{c}} | g_{ij, β^{Λ}}^{Λ} q_{ij, β^{Λ}}^{Λ} - g_{ij, β^{c}}^{c} q_{ij, β^{c}}^{c} |) . \end{matrix}$ (11)

Definition 9 Let X ={ x₁, x₂, ·· · , x_n } be a finite set of alternatives. PDHFPR sequences on X are defined by $D^{Λ} = (d_{ij}^{Λ})_{n \times n} (Λ = 1, 2, \cdot \cdot \cdot, m),$ where for i, j = 1, 2, ·· · , n, $d_{ij}^{Λ} = (f_{ij}^{Λ} | p_{ij}^{Λ}, g_{ij}^{Λ} | q_{ij}^{Λ})$ is a PDHFE, $f_{ij, α^{Λ}}^{Λ} \in f_{ij}^{Λ}$ indicates the α^Λth element in $f_{ij}^{Λ}$ , $p_{ij, α^{Λ}}^{Λ}$ is the corresponding probability of $f_{ij, α^{Λ}}^{Λ}$ , $\sum_{α^{Λ} = 1}^{# f_{ij}^{Λ}} p_{ij, α^{Λ}}^{Λ} = 1$ , $g_{ij, β^{Λ}}^{Λ} \in g_{ij}^{Λ}$ indicates the β^Λth element in $g_{ij, β^{Λ}}^{Λ}$ , $q_{ij, β^{Λ}}^{Λ}$ is the corresponding probability of $g_{ij, β^{Λ}}^{Λ}$ and $\sum_{β^{Λ} = 1}^{# g_{ij}^{Λ}} q_{ij, β^{Λ}}^{Λ} = 1$ . The symbols $# f_{ij}^{Λ}$ and $# g_{ij}^{Λ}$ are the total numbers of elements in $f_{ij}^{Λ} | p_{ij}^{Λ}$ and $g_{ij}^{Λ} | q_{ij}^{Λ}$ . The group consensus measure GCM is denoted as $\begin{matrix} GCM \\ = \frac{1}{m} \sum_{Λ = 1}^{m} ICM (D^{Λ}) = \frac{1}{{mn}^{2} (m - 1)} \cdot \\ \sum_{Λ = 1}^{m} \sum_{c \neq Λ} \sum_{i = 1}^{n} \sum_{j = 1}^{n} (\frac{1}{# f_{ij}^{Λ} # f_{ij}^{c}} \sum_{α = 1}^{# f_{ij}^{Λ}} \sum_{n = 1}^{# f_{ij}^{c}} | f_{ij, α^{Λ}}^{Λ} p_{ij, α^{Λ}}^{Λ} - f_{ij, α^{c}}^{c} p_{ij, α^{c}}^{c} | \\ + \frac{1}{# g_{ij}^{Λ} # g_{ij}^{c}} \sum_{β = 1}^{# g_{ij}^{Λ}} \sum_{l = 1}^{# g_{ij}^{c}} | g_{ij, β^{Λ}}^{Λ} q_{ij, β^{Λ}}^{Λ} - g_{ij, β^{c}}^{c} q_{ij, β^{c}}^{c} |) . \end{matrix}$ (12)

Next, the consensus measure between the two row vectors of PDHFPRs is studied.

Definition 10 Let X ={ x₁, x₂, ·· · , x_n } be a finite set of alternatives. Two PDHFPRs on X are defined by $D^{Λ} = (d_{ij}^{Λ})_{n \times n} (Λ = 1, 2)$ , where for i, j = 1, 2, ·· · , n, $d_{ij}^{Λ} = (f_{ij}^{Λ} | p_{ij}^{Λ}, g_{ij}^{Λ} | q_{ij}^{Λ})$ is a PDHFE, $f_{ij, l}^{Λ} \in f_{ij}^{Λ}$ indicates the lth element in $f_{ij}^{Λ}$ , $p_{ij, l}^{Λ}$ is the corresponding probability of $f_{ij, l}^{Λ}$ , $\sum_{l = 1}^{# f_{ij}^{Λ}} p_{ij, l}^{Λ} = 1$ , $g_{ij, k}^{Λ} \in g_{ij}^{Λ}$ indicates the kth element in $g_{ij}^{Λ}$ , $q_{ij, k}^{Λ}$ is the corresponding probability of $g_{ij, k}^{Λ}$ and $\sum_{k = 1}^{# g_{ij}^{Λ}} q_{ij, k}^{Λ} = 1$ . The symbols $# f_{ij}^{Λ}$ and $# g_{ij}^{Λ}$ are the total numbers of elements in $f_{ij}^{Λ} | p_{ij}^{Λ}$ and $g_{ij}^{Λ} | q_{ij}^{Λ}$ . If $V_{i}^{1}$ and $V_{i}^{2}$ are the ith row vectors of PDHFPRs D¹ and D², then the consensus measure between $V_{i}^{1}$ and $V_{i}^{2}$ is as follows: $\begin{matrix} ICM (V_{i}^{1}, V_{i}^{2}) \\ = 1 - \frac{1}{n} \cdot \sum_{j = 1}^{n} (\frac{1}{# f_{ij}^{1} # f_{ij}^{2}} \sum_{l = 1}^{# f_{ij}^{1}} \sum_{v = 1}^{# f_{ij}^{2}} | f_{ij, l}^{1} p_{ij, l}^{1} - f_{ij, v}^{2} p_{ij, v}^{2} | \\ + \frac{1}{# g_{ij}^{1} # g_{ij}^{2}} \sum_{k = 1}^{# g_{ij}^{1}} \sum_{o = 1}^{# g_{ij}^{2}} | g_{ij, k}^{1} q_{ij, k}^{1} - g_{ij, o}^{2} q_{ij, o}^{2} |) . \end{matrix}$ (13)

According to the consensus measure between the row vectors of PDHFPRs, the definition of the consensus measure of alternative x_i is as follows.

Definition 11 Let X ={ x₁, x₂, ·· · , x_n } be a finite set of alternatives, E ={ e₁, e₂, ·· · , e_m } be DMs, and φ = (φ₁, φ₂, ·· · , φ_m) be the corresponding weight vector. $D^{Λ} = (d_{ij}^{Λ})_{n \times n} (Λ = 1, 2, \cdot \cdot \cdot, m)$ and $D = PDHFWA (D^{1}, D^{2}, \cdot \cdot \cdot, D^{m}) = \sum_{t = 1}^{m} φ_{t} \otimes D^{t} = (d_{ij})_{n \times n}$ are the individual PDHFPRs and the comprehensive PDHFPRs of DMs. If $V_{i}^{Λ}$ and V_i are the ith row vectors of PDHFPRs D^Λ and D, then the individual consensus measure of alternative x_i is as follows: ${ACM}_{i} = \sum_{Λ = 1}^{m} φ_{Λ} ICM (V_{i}^{Λ}, V_{i}) .$ (14)

The group consensus measure is as follows: $GACM = \frac{1}{n} \sum_{i = 1}^{n} {ACM}_{i} .$ (15)

4 GDM methods based on PDHFPRs

In this section, we propose two GDM methods with PDHFPRs. The two GDM methods modify the PDHFPRs provided by DMs or modify the weights of DMs to make the DMs’ preferences relations reach the expected consensus level threshold and then obtain reasonable decision-making results.

Let X ={ x₁, x₂, ·· · , x_n } be a finite set of alternatives, E ={ e₁, e₂, ·· · , e_m } be DMs, and φ = (φ₁, φ₂, ·· · , φ_m) be the corresponding weight vector. In view of the flexibility and integrity of PDHFSs in expressing fuzzy and uncertain information, PDHFPRs are used to represent the decision-making information provided by DMs, and PDHFPR sequences $D^{Λ} = (d_{ij}^{Λ})_{n \times n} (Λ = 1, 2, \cdot \cdot \cdot, m)$ are constructed, where for i, j = 1, 2, ·· · , n, $d_{ij}^{Λ} = (f_{ij}^{Λ} | p_{ij}^{Λ}, g_{ij}^{Λ} | q_{ij}^{Λ})$ is a PDHFE. γ₀ ⩾ 0 is the specified consensus level threshold, and θ is the adjustment coefficient.

In actual decision-making, the weights of the DMs affect the decision-making result. However, it is usually difficult to give the weight of the DM. Generally, the smaller the individual consensus measure for PDHFPRs is, the more reliable the information provided by the DM is, and the higher the weight that should be assigned to the DM. Algorithm 1 is developed to calculate the weights of the DMs using the individual consensus measure. Meanwhile, the preference relation with the worst individual consensus level is modified to reach the preset threshold.

5 Algorithm 1

Step 1: Decision initialization. Let $D^{(I, Λ)} = (d_{ij}^{(I, Λ)})_{n \times n} (i, j = 1, 2, \cdot \cdot \cdot, n, Λ = 1, 2, \cdot \cdot \cdot, m)$ , where $d_{ij}^{(I, Λ)} = (f_{ij}^{(I, Λ)} | p_{ij}^{(I, Λ)}, g_{ij}^{(I, Λ)} | q_{ij}^{(I, Λ)})$ is a PDHFE, $f_{ij, α^{(I, Λ)}}^{(I, Λ)} \in f_{ij}^{(I, Λ)}$ indicates the α^(I,Λ)th element in $f_{ij}^{(I, Λ)}$ , $p_{ij, α^{(I, Λ)}}^{(I, Λ)}$ is the corresponding probability of $f_{ij}^{(I, Λ)}$ , $\sum_{α^{(I, Λ)} = 1}^{# f_{ij}^{(I, Λ)}} p_{ij, α^{(I, Λ)}}^{(I, Λ)} = 1$ , $g_{ij, β^{(I, Λ)}}^{(I, Λ)} \in g_{ij}^{(I, Λ)}$ indicates the β^(I,Λ)th element in $g_{ij}^{(I, Λ)}$ and $q_{ij, β^{(I, Λ)}}^{(I, Λ)} \in g_{ij}^{(I, Λ)}$ is the corresponding probability of $g_{ij}^{(I, Λ)}$ and $\sum_{β^{(I, Λ)} = 1}^{# g_{ij}^{(I, Λ)}} q_{ij, β^{(I, Λ)}}^{(I, Λ)} = 1$ . In addition, when I = 0, D^(0,Λ) = D^Λ.

Step 2: Calculate the individual weights φ_Λ of DMs e_Λ by means of the following formula: $φ_{Λ} = φ_{Λ}^{'} / \sum_{Λ = 1}^{m} φ_{Λ}^{'}, φ_{Λ}^{'} = ICM (D^{Λ}), Λ = 1, 2, \cdot \cdot \cdot, m .$

Thus, $0 ⩽ φ_{Λ} ⩽ 1, \sum_{Λ = 1}^{m} φ_{Λ} = 1$ .

Step 3: Use Equation (11) and Equation (12) to calculate the individual consensus measure ICM (D^(I,Λ)) and the group consensus measure GCM^(I) of PDHFPRs D^(I,Λ).

Step 4: Check the consensus level for D^(I,Λ). If GCM^(I) ⩽ γ₀ or I ⩾ I_max, then go to Step 6; otherwise, go to the next step.

Step 5: Modify the PDHFPR D^(I,ψ) with the worst individual consensus level, where $ICM (D^{(I, ψ)}) = min_{κ} {ICM (D^{(I, Λ)})} .$ D^(I+1,Λ) = $(d_{ij}^{(I + 1, Λ)})_{n \times n} = (f_{ij}^{(I + 1, Λ)} | p_{ij}^{(I, Λ)}, g_{ij}^{(I + 1, Λ)} | q_{ij}^{(I, Λ)})_{n \times n} (Λ = 1, 2, . . ., m)$ where $f_{ij, l}^{(I + 1, Λ)} \in f_{ij}^{(I + 1, Λ)}$ indicates the lth element in $f_{ij}^{(I + 1, Λ)}$ and $g_{ij, k}^{(I + 1, Λ)} \in g_{ij}^{(I + 1, Λ)}$ indicates the kth element in $g_{ij}^{(I + 1, Λ)}$ . Moreover, $f_{ij, α^{(I + 1, Λ)}}^{(I + 1, Λ)} = {\begin{matrix} f_{ij, α^{(I, Λ)}}^{(I, Λ)}, Λ \neq ψ, \\ θ \cdot f_{ij, α^{(I, Λ)}}^{(I, Λ)} + (1 - θ) \cdot \bar{f_{ij, α^{(I)}}^{(I)}}, Λ = ψ, \end{matrix}$ (16) $g_{ij, β^{(I + 1, Λ)}}^{(I + 1, Λ)} = {\begin{matrix} g_{ij, β^{(I, Λ)}}^{(I, Λ)}, Λ \neq ψ, \\ θ \cdot g_{ij, β^{(I, Λ)}}^{(I, Λ)} + (1 - θ) \cdot \bar{g_{ij, β^{(I)}}^{(I)}}, Λ = ψ, \end{matrix}$ (17) where θ is the adjustment coefficient, $\bar{f_{ij, α^{(I)}}^{(I)}} = \frac{1}{m - 1} (\sum_{Λ = 1, Λ \neq ψ}^{m} \sum_{α^{(I, Λ)} = 1}^{# f_{ij}^{(I, Λ)}} f_{ij, α^{(I, Λ)}}^{(I, Λ)} \cdot p_{ij, α^{(I, Λ)}}^{(I, Λ)})$ and $\bar{g_{ij, β^{(I)}}^{(I)}} = \frac{1}{m - 1} (\sum_{Λ = 1, Λ \neq ψ}^{m} \sum_{β^{(I, Λ)} = 1}^{# g_{ij}^{(I, Λ)}} g_{ij, β^{(I, Λ)}}^{(I, Λ)} \cdot q_{ij, β^{(I, Λ)}}^{(I, Λ)})$ .

Let I = I + 1; proceed to Step 3.

Step 6: Use Equation (9) to aggregate individual PDHFPRs D^(I,Λ) (Λ = 1, 2, ·· · , m) to obtain the comprehensive PDHFPR $D^{(I)} = SPDHFWA (D^{(I, 1)}, D^{(I, 2)}, \cdot \cdot \cdot, D^{(I, m)}) = \sum_{t = 1}^{m} φ_{t} \otimes D^{(I, t)} = (d_{ij}^{(I)})_{n \times n}$ , and $d_{ij}^{(I)} = (f_{ij}^{I}, g_{ij}^{I})$ can be obtained by $d_{i j}^{(I)} = \underset{f_{i j, α^{(I, 1)}}^{(I, 1)} \in f_{i j}^{(I, 1)}, \cdot \cdot \cdot, f_{i j, α^{(I, m)}}^{(I, m)} \in f_{i j}^{(I, m)}, g_{i j, β^{(I, 1)}}^{(I, 1)} \in g_{i j}^{(I, 1)}, \cdot \cdot \cdot, g_{i j, β^{(I, m)}}^{(I, m)} \in g_{i j}^{(I, m)}}{\cup} ({(1 - \prod_{Λ = 1}^{m} {(1 - f_{i j, α^{(I, Λ)}}^{(I, Λ)})}^{φ_{Λ}} | \prod_{Λ = 1}^{m} p_{i j, α^{(I, Λ)}}^{(I, Λ)}}, {\prod_{Λ = 1}^{m} {(g_{i j, β^{(I, Λ)}}^{(I, Λ)})}^{φ_{Λ}} | \prod_{Λ = 1}^{m} q_{i j, β^{(I, Λ)}}^{(I, Λ)}}) .$ (18)

Step 7: Aggregate all the probabilistic dual hesitant fuzzy preference values $d_{ij}^{(I)} (i, j = 1, 2, \cdot \cdot \cdot, n)$ of alternative x_i to obtain the comprehensive probabilistic dual hesitant fuzzy preference values $d_{i}^{(I)}$ , where $\begin{matrix} d_{i}^{(I)} = ⋃ f_{ij, α^{I}}^{I} \in f_{ij}^{I}, \\ g_{ij, β^{I}}^{I} \in g_{ij}^{I} ({(1 - \prod_{j = 1}^{n} (1 - f_{ij, α^{I}}^{(I)})^{\frac{1}{n}}) | \prod_{j = 1}^{n} p_{ij, α^{I}}^{(I)}}, \\ {\prod_{j = 1}^{n} (g_{ij, β^{I}}^{(I)})^{\frac{1}{n}} | \prod_{j = 1}^{n} q_{ij, β^{I}}^{(I)}}) . \end{matrix}$

Step 8: Apply Definition 4 to calculate the score function of the comprehensive probabilistic dual hesitant fuzzy preference values $d_{i}^{(I)} (i = 1, 2, \cdot \cdot \cdot, n)$ and obtain the ranking of alternatives.

When the decision expert does not want to modify the given PDHFPR, Algorithm 2 modifies the weights of DMs to make the preference relations reach the expected consensus level threshold to ensure reasonable decision-making results.

6 Algorithm 2

Step 1: Decision initialization. Let $D^{(I, Λ)} = (d_{ij}^{(I, Λ)})_{n \times n} (i, j = 1, 2, \cdot \cdot \cdot, n, Λ = 1, 2, \cdot \cdot \cdot, m)$ and $φ^{(I)} = (φ_{1}^{(I)}, φ_{2}^{(I)}, \cdot \cdot \cdot, φ_{m}^{(I)})$ , where $d_{ij}^{(I, Λ)} = (f_{ij}^{(I, Λ)} | p_{ij}^{(I, Λ)},$ $g_{ij}^{(I, Λ)} | q_{ij}^{(I, Λ)})$ is a PDHFE, $f_{ij, l}^{(I, Λ)} \in f_{ij}^{(I, Λ)}$ indicates the lth element in $f_{ij}^{(I, Λ)}$ , $p_{ij}^{(I, Λ)}$ is the corresponding probability of $f_{ij}^{(I, Λ)}$ , $\sum_{l = 1}^{# f_{ij}^{(I, Λ)}} f_{ij, l}^{(I, Λ)} = 1$ , $g_{ij, k}^{(I, Λ)} \in g_{ij}^{(I, Λ)}$ indicates the kth element in $g_{ij}^{(I, Λ)}$ , $q_{ij}^{(I, Λ)}$ is the corresponding probability of $g_{ij}^{(I, Λ)}$ and $\sum_{k = 1}^{# g_{ij}^{(I, Λ)}} q_{ij, k}^{(I, Λ)} = 1$ . When I = 0, D^(0,κ) = D^κ.

Step 2: Use Equation (9) to aggregate all individual PDHFPRs D^(I,Λ) (Λ = 1, 2, ·· · , m) to obtain the comprehensive PDHFPR D^(I) = SPDHFWA (D^(I,1), D^(I,2), $\cdot \cdot \cdot, D^{(I, m)}) = \sum_{t = 1}^{m} φ_{t} \otimes D^{(I, t)} = (d_{ij}^{(I)})_{n \times n}$ , where $d_{ij}^{(I)} = (f_{ij}^{I}, g_{ij}^{I})$ can be obtained by $d_{i j}^{(I)} = \cup l f_{i j, α^{(I, 1)}}^{(I, 1)} \in f_{i j}^{(I, 1)}, f_{i j, α^{(I, 2)}}^{(I, 2)} \in f_{i j}^{(I, 2)}, \cdot \cdot \cdot, f_{i j, α^{(I, m)}}^{(I, m)} \in f_{i j}^{(I, m)}, g_{i j, β^{(I, 1)}}^{(I, 1)} \in g_{i j}^{(I, 1)}, g_{i j, β (I, 2)}^{(I, 2)} \in g_{i j}^{(I, 2)}, \cdot \cdot \cdot, g_{i j, β^{(I, m)}}^{(I, m)} \in g_{i j}^{(I, m)} ({(1 - \prod_{Λ = 1}^{m} {(1 - f_{i j, α^{(I, Λ)}}^{(I, Λ)})}^{φ_{Λ}} | \prod_{Λ = 1}^{m} p_{i j, α^{(I, Λ)}}^{(I, Λ)}}, {\prod_{Λ = 1}^{m} {(g_{i j, β^{(I, Λ)}}^{(I, Λ)})}^{φ_{Λ}} | \prod_{Λ = 1}^{m} q_{i j, β^{(I, Λ)}}^{(I, Λ)}}) .$ (19)

Step 3: Use Equation (14) and Equation (15) to calculate the consensus measure ${ACM}_{i}^{(I)} (i = 1, 2, \cdot \cdot \cdot, n)$ of alternative x_i and the group consensus measure GACM^(I).

Step 4: Check the consensus level for D^(I,Λ). If GACM^(I) ⩽ γ₀ or I ⩾ I_max, then go to Step 6; otherwise, go to the next step.

Step 5: Modify the weight vector $φ^{(I)} = (φ_{1}^{(I)}, φ_{2}^{(I)}, \cdot \cdot \cdot, φ_{m}^{(I)})$ . First, calculate the contribution $C_{Λ}^{(I)} (Λ = 1, 2, \cdot \cdot \cdot, m)$ of DM e_Λ. When DM e_Λ is not involved in GDM, the consensus measure ${ACM}_{\bar{Λ} i}^{(I)} (i = 1, 2, \cdot \cdot \cdot, n)$ of alternative x_i is: ${ACM}_{\bar{Λ} i}^{(I)} = \sum_{Λ = 1, Λ \neq k}^{m} φ_{Λ}^{(I)^{'}} ICM (V_{i}^{(I, Λ)}, V_{i}^{(I)^{'}})$ (20) where $φ_{Λ}^{(I)^{'}} = φ_{Λ}^{(I)} / \sum_{Λ = 1 Λ \neq k}^{m} φ_{Λ}^{(I)}$ , $V_{i}^{(I)^{'}}$ is the ith row vector of comprehensive PDHFPR D^I′ obtained by Equation (9) without considering DM e_Λ, and the contribution of DM e_Λ to alternative x_i is calculated by: $C_{Λ, i}^{(I)} = {ACM}_{i}^{(I)} - {ACM}_{\bar{Λ} i}^{(I)}$ (21)

The contribution of DM e_Λ to all alternatives is calculated as: $C_{Λ}^{(I)} = \sum_{i = 1}^{n} C_{Λ, i}^{(I)}$ .

At this time, the weight of DM e_Λ is adjusted, as follows: $φ_{Λ}^{(I + 1)} = π_{Λ}^{(I + 1)} / \sum_{Λ = 1}^{m} π_{Λ}^{(I + 1)}, π_{Λ}^{(I + 1)} = φ_{Λ}^{(I)} {(1 - \frac{C_{Λ}^{(I)}}{m})}^{ρ},$ where ρ is the adjustment coefficient.

Let I = I + 1; proceed to Step 2.

Step 6: Aggregate all the probabilistic dual hesitant fuzzy preference values $d_{ij}^{(I)} (i, j = 1, 2, \cdot \cdot \cdot, n)$ of alternative x_i to obtain the comprehensive probabilistic dual hesitant fuzzy preference values $d_{i}^{(I)}$ , where $\begin{matrix} d_{i}^{(I)} = ⋃ f_{ij, α^{I}}^{(I)} \in f_{ij}^{(I)}, \\ g_{ij, β^{I}}^{(I)} \in g_{ij}^{(I)} \\ ({(1 - \prod_{j = 1}^{n} (1 - f_{ij, α^{I}}^{(I)})^{\frac{1}{n}}) | \prod_{j = 1}^{n} p_{ij, α^{I}}^{(I)}}, \\ {\prod_{j = 1}^{n} (g_{ij, β^{I}}^{(I)})^{\frac{1}{n}} | \prod_{j = 1}^{n} q_{ij, β^{I}}^{(I)}}) . \end{matrix}$

Step 7: Apply Definition 4 to calculate the score function of the comprehensive probabilistic dual hesitant fuzzy preference values $d_{i}^{(I)} (i = 1, 2, \cdot \cdot \cdot, n)$ and obtain the ranking of alternatives.

The flowchart of the above GDM methods with PDHFPRs is shown in Fig. 1.

Fig. 1

GDM methods with PDHFPRs.

5 PDHFPRs, distance measure of PDHFPRs and consensus measure for PDHFPRs

We demonstrate the validity of our methods by analyzing the location selection of large-scale industrial solid waste treatment facilities and compare our methods with existing methods.

5.1 Application to analyze the location selection of large-scale industrial solid waste treatment facilities

The increase in harmful substances, such as PM2.5, PM10, NO₂ and SO₂, in the air has caused a deterioration of air quality, including haze. Frequent haze weather is a product of economic society and human activities and has a serious impact on people’s health and daily travel. How to protect the environment, reduce pollution, and achieve sustainable development are major issues faced by human beings. Industrial solid waste is a substantial component of pollutants, and the protection of the environment cannot be separated from the effective treatment of pollutants. Therefore, choosing the most suitable site to construct industrial solid waste treatment facilities is of great significance to the development of industry and society. To date, the theory of PDHFSs has not been applied to industrial solid waste treatment facilities site selection.

Anhui Province is located in eastern China. In recent years, with the rapid development of industry, the demand for industrial solid waste treatment facilities has increased. However, there are few large-scale solid waste treatment facilities in the province.

Therefore, the most suitable locations to build new large-scale industries must be determined. The probabilistic dual hesitant fuzzy environment can retain more uncertain decision-making information. This paper builds GDM methods with PDHFPRs to determine the ideal location to construct industrial solid waste treatment facilities in Anhui Province. According to the total amount of industrial solid waste, the number of industrial enterprises, the sum of distances between prefecture level cities and the number of prefecture level cities in the coverage area, experts E ={ e₁, e₂, e₃ } from three related fields made pairwise comparisons of four locations X = {x₁, x₂, x₃, x₄} and gave the PDHFPRs $D^{Λ} = (d_{ij}^{Λ})_{4 \times 4} (i, j = 1, 2, 3, 4, Λ = 1, 2, 3)$ , as demonstrated in Tables 1 –3, where $d_{ij}^{(Λ)} = (f_{ij}^{(Λ)} | p_{ij}^{(Λ)}, g_{ij}^{(Λ)} | q_{ij}^{(Λ)})$ is a PDHFE.

Table 1
Preference values given by DM e₁

D ¹ x ₁ x ₂ x ₃ x ₄

x ₁ ({ 0.5|1  } , { 0.5|1  }) $\begin{matrix} ({0.4 | 0.2, 0.1 | 0.8}, \\ {0.5 | 0.4, 0.3 | 0.1, 0.2 | 0.5}) \end{matrix}$ $\begin{matrix} ({0.3 | 0.45, 0.6 | 0.55}, \\ {0.4 | 1}) \end{matrix}$ $\begin{matrix} ({0.5 | 0.6, 0.3 | 0.4}, \\ {0.24 | 0.7, 0.14 | 0.3}) \end{matrix}$

x ₂ / ({ 0.5|1  } , { 0.5|1  }) ({ 0.2|1  } , { 0.6|0.35, 0.8|0.65   }) $({0.25 | \frac{1}{3}, 0.7 | \frac{2}{3}}, {0.3 | 1})$

x ₃ / / ({ 0.5|1  } , { 0.5|1  }) $\begin{matrix} ({0.7 | 0.1, 0.6 | 0.3, 0.2 | 0.6}, \\ {0.2 | 0.8, 0.1 | 0.2}) \end{matrix}$

x ₄ / / / ({ 0.5|1  } , { 0.5|1  })

Table 2

Preference values given by DM e₂

D ²	x ₁	x ₂	x ₃	x ₄
x ₁	({ 0.5\|1 } , { 0.5\|1 })	$\begin{matrix} ({0.3 \| 1}, \\ {0.6 \| 0.55, 0.5 \| 0.45}) \end{matrix}$	$\begin{matrix} ({0.5 \| 0.4, 0.6 \| 0.5, \\ 0.7 \| 0.1}, {0.3 \| 1}) \end{matrix}$	({ 0.7\|1 } , { 0.2\|1 })
x ₂	/	$\begin{matrix} ({0.5 \| 1}, \\ {0.5 \| 1}) \end{matrix}$	$\begin{matrix} ({0.45 \| 1}, \\ {0.35 \| 0.25, 0.5 \| 0.75}) \end{matrix}$	$({0.3 \| \frac{9}{20}, 0.4 \| \frac{11}{20}}, {0.3 \| 1})$
x ₃	/	/	({ 0.5\|1 } , { 0.5\|1 })	$\begin{matrix} ({0.6 \| 0.42, 0.7 \| 0.58}, \\ {0.2 \| 0.7, 0.3 \| 0.3}) \end{matrix}$
x ₄	/	/	/	({ 0.5\|1 } , { 0.5\|1 })

Table 3

Preference values given by DM e₃

D ³	x ₁	x ₂	x ₃	x ₄
x ₁	({ 0.5\|1 } , { 0.5\|1 })	$\begin{matrix} ({0.6 \| 1}, \\ {0.25 \| 0.82, 0.3 \| 0.18}) \end{matrix}$	$({0.4 \| \frac{1}{3}, 0.5 \| \frac{2}{3}}, {0.3 \| \frac{1}{6}, 0.45 \| \frac{5}{6}})$	$({0.2 \| \frac{4}{7}, 0.45 \| \frac{3}{7}}, {0.5 \| 1})$
x ₂	/	({ 0.5\|1 } , { 0.5\|1 })	$\begin{matrix} ({0.6 \| 0.75, 0.7 \| 0.25}, \\ {0.2 \| 1}) \end{matrix}$	$({0.3 \| 1}, {0.4 \| \frac{1}{6}, 0.5 \| \frac{5}{6}})$
x ₃	/	/	({ 0.5\|1 } , { 0.5\|1 })	$\begin{matrix} ({0.5 \| 0.6, 0.4 \| 0.4}, \\ {0.2 \| 0.8, 0.4 \| 0.2}) \end{matrix}$
x ₄	/	/	/	({ 0.5\|1 } , { 0.5\|1 })

This paper uses two GDM approaches with PDHFPRs based on consensus to address the above problems and select the location of large-scale industrial solid waste treatment facilities, where the specified consensus level threshold γ₀ = 0.75 and the adjustment coefficient θ = 0.4 .

8 Algorithm 1

Step 1: The DMs provided PDHFPRs D^Λ = D^(0,Λ) (Λ = 1, 2, 3), as demonstrated in Tables 1 –3.

Step 2: Calculate the individual weights φ_Λ (Λ = 1, 2, 3) of DMs e_Λ. First, the individual consensus measure for PDHFPRs D^Λ is calculated, as follows: $\begin{matrix} ICM (D^{1}) = 0 . 7277, ICM (D^{2}) = 0.7222, \\ ICM (D^{3}) = 0.7156 . \end{matrix}$

Thus, $φ_{1}^{'} = 0 . 7277, φ_{2}^{'} = 0 . 7222, φ_{3}^{'} = 0 . 7156,$ and $\sum_{Λ = 1}^{3} φ_{Λ}^{'} = 2.1655 .$

The weights of DMs e_Λ are obtained as follows: φ₁ = 0 . 3361, φ₂ = 0 . 3335, φ₃ = 0 . 3304 .

Step 3: Use Equation (12) to calculate the group consensus measure GCM⁽⁰⁾ for PDHFPRs D^Λ (Λ = 1, 2, 3), we obtain ${GCM}^{(0)} = \frac{0.7277 + 0.7222 + 0.7156}{3} = 0.7218 .$

Step 4: Check the consensus level for D^Λ (Λ = 1, 2, 3). We obtain GCM⁽⁰⁾ = 0.7218 < γ₀, which means that the PDHFPR D D^k with the worst individual consensus level needs to be modified.

Step 5: Use Equation (16) and Equation (17) to modify the PDHFPR D³ with the worst individual consensus level and obtain the modified PDHFPR D^(1,3) as demonstrated in Table 4.

Table 4
The modified PDHFPR D^(1,3)

D ^(1,3) x ₁ x ₂ x ₃ x ₄

x ₁ ({ 0.5|1  } , { 0.5|1  }) $\begin{matrix} ({0.378 | 1}, \\ {0.3655 | 0.82, 0.3855 | 0.18}) \end{matrix}$ $\begin{matrix} ({0.4705 | \frac{1}{3}, 0.5105 | \frac{2}{3}}, \\ {0.33 | \frac{1}{6}, 0.39 | \frac{5}{6}}) \end{matrix}$ $\begin{matrix} ({0.416 | \frac{4}{7}, 0.516 | \frac{3}{7}}, \\ {0.323 | 1}) \end{matrix}$

x ₂ / ({ 0.5|1  } , { 0.5|1  }) $\begin{matrix} ({0.435 | 0.75, 0.475 | 0.25}, \\ {0.43775 | 1}) \end{matrix}$ $({0.3915 | 1}, {0.34 | \frac{1}{6}, 0.38 | \frac{5}{6}})$

x ₃ / / ({ 0.5|1  } , { 0.5|1  }) $\begin{matrix} ({0.5084 | 0.6, 0.4684 | 0.4}, \\ {0.203 | 0.8, 0.283 | 0.2}) \end{matrix}$

x ₄ / / / ({ 0.5|1  } , { 0.5|1  })

D ^(1,3)	x ₁	x ₂	x ₃	x ₄
x ₁	({ 0.5\|1 } , { 0.5\|1 })	$\begin{matrix} ({0.378 \| 1}, \\ {0.3655 \| 0.82, 0.3855 \| 0.18}) \end{matrix}$	$\begin{matrix} ({0.4705 \| \frac{1}{3}, 0.5105 \| \frac{2}{3}}, \\ {0.33 \| \frac{1}{6}, 0.39 \| \frac{5}{6}}) \end{matrix}$	$\begin{matrix} ({0.416 \| \frac{4}{7}, 0.516 \| \frac{3}{7}}, \\ {0.323 \| 1}) \end{matrix}$
x ₂	/	({ 0.5\|1 } , { 0.5\|1 })	$\begin{matrix} ({0.435 \| 0.75, 0.475 \| 0.25}, \\ {0.43775 \| 1}) \end{matrix}$	$({0.3915 \| 1}, {0.34 \| \frac{1}{6}, 0.38 \| \frac{5}{6}})$
x ₃	/	/	({ 0.5\|1 } , { 0.5\|1 })	$\begin{matrix} ({0.5084 \| 0.6, 0.4684 \| 0.4}, \\ {0.203 \| 0.8, 0.283 \| 0.2}) \end{matrix}$
x ₄	/	/	/	({ 0.5\|1 } , { 0.5\|1 })

The individual consensus measure and the group consensus measure for PDHFPRs D^(1,Λ) (Λ = 1, 2, 3) can be calculated as follows: $\begin{matrix} ICM (D^{(1, 1)}) = 0.7533, ICM (D^{(1, 2)}) = 0.7423, \\ ICM (D^{(1, 3)}) = 0.7613, {GCM}^{(1)} = 0.7523 > γ_{0} . \end{matrix}$

Step 6: Use Equation (9) to aggregate all individual PDHFPRs D^(1,Λ) (Λ = 1, 2, 3) to obtain the comprehensive PDHFPRs D⁽¹⁾. Due to space limitations in this paper, only select elements of D⁽¹⁾ are listed: $\begin{matrix} d_{12}^{(1)} = ({0.3608 | 0.2, 0.2675 | 0.8}, {0.4791 | 0.1804, \\ 0.4508 | 0.1476, 0.4588 | 0 . 0324, 0.4035 | 0.0451, \\ 0.3313 | 0.1845, 0.4876 | 0.0396, 0.3797 | 0.0369, \\ 0.3584 | 0.0495, 0.3372 | 0.0405, 0.4107 | 0 . 0099, \\ 0.3864 | 0.0081, 0.3521 | 0.2255}), \end{matrix}$ $\begin{matrix} d_{13}^{(1)} = ({0.4294 | 0 . 06, 0.5617 | 0.0917, 0.6013 | 0.0183, \\ 0.5724 | 0.1833, 0.5188 | 0.015, 0.4703 | 0.075 \\ 0.5273 | 0.0733 0.5394 | 0.1467, 0.444 | 0.12, \\ 0.5311 | 0.03, 0.6115 | 0.03 67, 0.4839 | 0.15}, \\ {0.3410 | 0.16670.3604 | 0.8333}) . \end{matrix}$

Step 7: Aggregate all the probabilistic dual hesitant fuzzy preference values $d_{ij}^{(1)} (i, j = 1, 2, 3, 4)$ of alternative x_i to obtain the comprehensive probabilistic dual hesitant fuzzy preference values $d_{i}^{(1)}$ , where $\begin{matrix} d_{1}^{(1)} = ({0.4667 | 0.0041, 0.4748 | 0.0031, \cdot \cdot \cdot, \\ 0.4923 | 0.005}, {0.3777 | 0.021, \cdot \cdot \cdot, 0.3352 | 0.0101}), \end{matrix}$ $\begin{matrix} d_{2}^{(1)} = ({0.4213 | 0.0203, 0.4287 | 0.0248, \cdot \cdot \cdot, 0.4416 | 0.0037}, \\ {0.3995 | 0.0029, \cdot \cdot \cdot, 0.3649 | 0.325}), \end{matrix}$ $\begin{matrix} d_{3}^{(1)} = ({0.4852 | 0.0004, 0.4819 | 0.0002, \cdot \cdot \cdot, 0.4841 | 0.0566}, \\ {0.3556 | 0.0202, \cdot \cdot \cdot, 0.3919 | 0.0001}), \end{matrix}$ $\begin{matrix} d_{4}^{(1)} = ({0.3261 | 0.0523, 0.3311 | 0.0131, \cdot \cdot \cdot, 0.3205 | 0.003}, \\ {0.4814 | 0.0013, \cdot \cdot \cdot, 0.5126 | 0.0087}) . \end{matrix}$

Step 8: Calculate the score function of the comprehensive probabilistic dual hesitant fuzzy preference values $d_{i}^{(1)} (i = 1, 2, 3, 4)$ as follows: $\begin{matrix} s (d_{1}^{(1)}) = 0.1102, s (d_{2}^{(1)}) = 0.041, \\ s (d_{3}^{(1)}) = 0.1191, s (d_{4}^{(1)}) = - 0.1764 . \end{matrix}$

Finally, the ranking of alternatives is as follows: x₃ ≻ x₁ ≻ x₂ ≻ x₄.

The decision expert reaches the group consensus level threshold by modifying the given PDHFPRs in Algorithm 1. However, in real life, some DMs do not want to change the given preference relations. Algorithm 2 improves the group consensus level by modifying the weight of DMs and then solves the problem of selecting the location of large-scale industrial solid waste treatment facilities.

9 Algorithm 2

Step 1: The DMs gave PDHFPRs D^Λ = D^(0,Λ) (Λ = 1, 2, 3), as demonstrated in Tables 1 –3. To facilitate comparison with Algorithm 1, the initial weights of DMs are as follows: $φ = (0 . 3361, 0 . 3335, 0 . 3304) .$

Step 2: Use Equation (9) to aggregate all individual PDHFPRs D^Λ (Λ = 1, 2, 3) to obtain the comprehensive PDHFPR $D = PDHFWA (D^{1}, D^{2}, D^{3}) = \sum_{i = 1}^{3} φ_{i} \otimes D^{i} = (d_{ij})_{4 \times 4} .$

Due to space limitations in this paper, only some select elements of D are listed: $\begin{matrix} d_{12} = ({0.4475 | 0.2, 0.3669 | 0.8} {0.4226 | 0.1804, \\ 0.4488 | 0.0396, 0.3977 | 0.1476, 0.4223 | 0 . 0324, \\ 0.3559 | 0.0451, 0.2923 | 0.1845, 0.378 | 0 . 0099, \\ 0.3106 | 0.2255, 0.3349 | 0.0369, 0.3557 | 0.0081, \\ 0.3297 | 0.0495, 0.3104 | 0.0405}), \\ d_{13} = ({0.4054 | 0 . 06, 0.5073 | 0.0733, 0.4803 | 0.15, \\ 0.5427 | 0.0917, 0.448 | 0.075, 0.5694 | 0.1833, \\ 0.6088 | 0.03 67, 0.5361 | 0.1467, 0.4401 | 0.12, \\ 0.5278 | 0.03, 0.5845 | 0.0183, 0.4985 | 0.015}, \\ {0.3778 | 0.8333, 0.3305 | 0.1667}) . \end{matrix}$

Step 3: Use Equation (3.9) and Equation (3.10) to calculate the consensus measure ${ACM}_{i}^{(I)} (i = 1, 2, 3, 4)$ of alternative x_i (i = 1, 2, 3, 4) and the group consensus measure GACM⁽⁰⁾, $\begin{matrix} {ACM}_{1}^{(0)} = 0.7222, {ACM}_{2}^{(0)} = 0.7478, \\ {ACM}_{3}^{(0)} = 0.7585, {ACM}_{4}^{(0)} = 0.7534 . \end{matrix}$ $\begin{matrix} {GACM}^{0} = 1 / 4 * (0.7222 + 0.7478 + 0.7585 \\ + 0.7534) = 0.7455 . \end{matrix}$

Step 4: Check the consensus level of D^Λ (Λ = 1, 2, 3). We can obtain GACM⁰ = 0.7455 < γ₀, which means that the weight of DMs e_Λ needs to be modified.

Step 5: When DM e_Λ (Λ = 1, 2, 3) is not involved in decision making, the individual consensus ${ACM}_{\bar{Λ} i}^{(0)} (i = 1, 2, 3, 4)$ of alternative x_i is shown in Table 5.

Table 5
Individual consensus measures without considering one specific DM

The individual consensus measure x ₁ x ₂ x ₃ x ₄

${ACM}_{i}^{(0)}$ 0.7222 0.7478 0.7585 0.7534

${ACM}_{\bar{1} i}^{(0)}$ 0.752 0.7968 0.8054 0.7805

${ACM}_{\bar{2} i}^{(0)}$ 0.7573 0.7668 0.8090 0.7952

${ACM}_{\bar{3} i}^{(0)}$ 0.8034 0.8185 0.8168 0.8218

The individual consensus measure	x ₁	x ₂	x ₃	x ₄
${ACM}_{i}^{(0)}$	0.7222	0.7478	0.7585	0.7534
${ACM}_{\bar{1} i}^{(0)}$	0.752	0.7968	0.8054	0.7805
${ACM}_{\bar{2} i}^{(0)}$	0.7573	0.7668	0.8090	0.7952
${ACM}_{\bar{3} i}^{(0)}$	0.8034	0.8185	0.8168	0.8218

Then, the contribution of DM e_Λ to alternative x_i is calculated as follows: $\begin{matrix} C_{1, 1}^{(0)} = - 0 . 0298, C_{1, 2}^{(0)} = - 0 . 049, C_{1, 3}^{(0)} = - 0 . 0469, \\ C_{1, 4}^{(0)} = - 0 . 0271, C_{2, 1}^{(0)} = - 0 . 0351, C_{2, 2}^{(0)} = - 0 . 019, \\ C_{2, 3}^{(0)} = - 0.0505, C_{2, 4}^{(0)} = - 0.0418, C_{3, 1}^{(0)} = - 0 . 0812, \\ C_{3, 2}^{(0)} = - 0 . 0707, C_{3, 3}^{(0)} = - 0.0583, C_{3, 4}^{(0)} = - 0.0684 . \end{matrix}$

The contribution of DM e_Λ to all alternatives is calculated by: $C_{1}^{(0)} = - 0 . 1528, C_{2}^{(0)} = - 0 . 1464, C_{3}^{(0)} = - 0 . 2786 .$

By letting the adjustment coefficient ρ = 10, the following results are obtained: $π_{1}^{(1)} = 0 . 1181, π_{2}^{(1)} = 0 . 1226, π_{3}^{(1)} = 0 . 0470 .$

Meanwhile, the weight of DM e_k is adjusted as follows: $φ_{1}^{(1)} = 0 . 4105, φ_{2}^{(1)} = 0 . 4261, φ_{3}^{(1)} = 0 . 1634 .$

The individual consensus measure ${ACM}_{i}^{(1)}$ and the group consensus measure GACM⁽¹⁾ of alternative x_i can be calculated as follows: $\begin{matrix} {ACM}_{1}^{(1)} = 0.7309, {ACM}_{2}^{(1)} = 0.7541, \\ {ACM}_{3}^{(1)} = 0.756, {ACM}_{4}^{(1)} = 0.7552 . \end{matrix}$ $\begin{matrix} {GACM}^{1} = 1 / 4 * (0.7309 + 0.7541 + 0.756 \\ + 0.7552) = 0.7491 < γ_{0} . \end{matrix}$

Next, the weight of DM e_Λ is adjusted again as follows: $φ_{1}^{(2)} = 0 . 5356, φ_{2}^{(2)} = 0 . 3763, φ_{3}^{(2)} = 0 . 0881 .$

In this case, the individual consensus measure ${ACM}_{i}^{(2)}$ and the group consensus measure GACM⁽²⁾ of alternative x_i can be calculated as follows: $\begin{matrix} {ACM}_{1}^{(2)} = 0.7483, {ACM}_{2}^{(2)} = 0.7613, \\ {ACM}_{3}^{(2)} = 0.7574, {ACM}_{4}^{(2)} = 0.7673 . \end{matrix}$ $\begin{matrix} {GACM}^{2} = 1 / 4 * (0.7483 + 0.7613 + 0.7574 \\ + 0.7673) = 0.7585 > γ_{0} . \end{matrix}$

Step 6: Aggregate all the probabilistic dual hesitant fuzzy preference values $d_{ij}^{2} (i, j = 1, 2, 3, 4)$ of alternative x_i to obtain the comprehensive probabilistic dual hesitant fuzzy preference value $d_{i}^{2}$ , where $\begin{matrix} d_{1}^{2} = ({0.4677 | 0.0041, 0.4865 | 0.005, \cdot \cdot \cdot, 0.4721 | 0.0031}, \\ {0.381 | 0.021, \cdot \cdot \cdot, 0.3122 | 0.0101}), \\ d_{2}^{2} = ({0.4142 | 0.0203, 0.4226 | 0.0248, \cdot \cdot \cdot, 0.4454 | 0.0037}, \\ {0.4032 | 0.0029, \cdot \cdot \cdot, 0.3857 | 0.325}), \\ d_{3}^{2} = ({0.4988 | 0.0004, 0.4968 | 0.0002, \cdot \cdot \cdot, 0.4858 | 0.0566}, \\ {0.3413 | 0.0202, \cdot \cdot \cdot, 0.3535 | 0.000055}), \\ d_{4}^{2} = ({0.3225 | 0.0523, 0.3252 | 0.0131, \cdot \cdot \cdot, 0.3048 | 0.003}, \\ {0.4746 | 0.0013, \cdot \cdot \cdot, 0.509 | 0.0087}) . \end{matrix}$

Step 7: Calculate the score function of the comprehensive probabilistic dual hesitant fuzzy preference values $d_{i}^{(2)} (i = 1, 2, 3, 4)$ : $\begin{matrix} s (d_{1}^{(2)}) = 0.1185, s (d_{2}^{(2)}) = 0.0488, \\ s (d_{3}^{(2)}) = 0.1233, s (d_{4}^{(2)}) = - 0.1826 . \end{matrix}$

Finally, the ranking of alternatives is as follows: x₃ ≻ x₁ ≻ x₂ ≻ x₄.

5.2 Comparative analysis

With the help of the above examples to determine the location of large-scale industrial solid waste treatment facilities, this section compares our proposed GDM methods with the methods proposed by Garg and Kaur [15] and Li and Wang [17].

On the one hand, the decision-making method in Garg and Kaur [15] is applied to address the location problem of large-scale industrial solid waste treatment facilities as follows:

First, according to Algorithm 1 in [15], the individual PDHFPRs are aggregated to obtain a comprehensive PDHFPR: $d_{ii}^{l} = ({0.5 | 1}, {0.5 | 1}),$ $\begin{matrix} d_{12}^{*} = ({0.4 | \frac{1}{15}, 0.1 | \frac{4}{15}, 0.3 | \frac{1}{3}, 0.6 | \frac{1}{3}}, \\ {0.5 | \frac{17}{60}, 0.3 | \frac{7}{75}, 0.2 | \frac{1}{6}, 0.25 | \frac{41}{150}, 0.6 | \frac{11}{60}}), \end{matrix}$ $\begin{matrix} d_{13}^{*} = ({0.3 | 0.15, 0.6 | 0.35, 0.5 | \frac{16}{45}, 0.4 | \frac{1}{9}, 0.7 | \frac{1}{30}}, \\ {0.4 | \frac{1}{3}, 0.3 | \frac{7}{18}, 0.45 | \frac{5}{18}}), \\ d_{14}^{*} = ({0.5 | \frac{1}{5}, 0.3 | \frac{2}{15}, 0.7 | \frac{1}{3}, 0.2 | \frac{4}{21}, 0.45 | \frac{1}{7}}, \\ {0.24 | \frac{7}{30}, 0.14 | 0.1, 0.2 | \frac{1}{3}, 0.5 | \frac{1}{3}}), \end{matrix}$ $\begin{matrix} d_{23}^{*} = ({0.2 | \frac{1}{3}, 0.45 | \frac{1}{3}, 0.6 | 0.25, 0.7 | \frac{1}{12}}, \\ {0.6 | \frac{7}{60}, 0.8 | \frac{13}{60}, 0.35 | \frac{5}{60}, 0.5 | 0.25, 0.2 | \frac{1}{3}}), \\ d_{24}^{*} = ({0.25 | \frac{1}{9}, 0.7 | \frac{2}{9}, 0.3 | \frac{29}{60}, 0.4 | \frac{11}{60}}, \\ {0.3 | \frac{2}{3}, 0.4 | \frac{1}{18}, 0.5 | \frac{5}{18}}), \\ d_{34}^{*} = ({0.7 | \frac{17}{75}, 0.6 | 0.24, 0.2 | 0.2, 0.5 | 0.2, 0.4 | \frac{2}{15}}, \\ {0.2 | \frac{23}{30}, 0.1 | \frac{1}{15}, 0.3 | 0.1, 0.4 | \frac{1}{15}}) . \end{matrix}$

Second, let k = 4. Based on the WPDHFMSMA operator ((Equation (23) in [15]), the comprehensive dual hesitant fuzzy preference value $d_{i}^{* 1} (i = 1, 2, 3, 4)$ of alternative x_i is obtained as follows: $\begin{matrix} d_{1}^{* 1} = ({0.1268 | 0.002, 0.1085 | 0.0013, \cdot \cdot \cdot, 0.1852 | 0.0016}, \\ {0.8014 | 0.022, \cdot \cdot \cdot, 0.8469 | 0.017}), \end{matrix}$ $\begin{matrix} d_{2}^{* 1} = ({0.0988 | 0.0105, 0.1375 | 0.021, \cdot \cdot \cdot, 0.1785 | 0.0028}, \\ {0.8215 | 0.0052, \cdot \cdot \cdot, 0.8219 | 0.0309}), \\ d_{3}^{* 1} = ({0.1785 | 0.0088, 0.1682 | 0.0093, \cdot \cdot \cdot, 0.1095 | 0.0123}, \\ {0.7405 | 0.0383, \cdot \cdot \cdot, 0.8759 | 0.0002}), \end{matrix}$ $\begin{matrix} d_{4}^{* 1} = ({0.0836 | 0.1193, 0.0695 | 0.0104, \cdot \cdot \cdot, 0.1482 | 0.0062}, \\ {0.8414 | 0.005, \cdot \cdot \cdot, 0.8136 | 0.0035}) . \end{matrix}$

Similarly, based on the WPDHFMSMG operator (Equation (24) in [15]), the comprehensive dual hesitant fuzzy preference value $d_{i}^{* 2} (i = 1, 2, 3, 4)$ of alternative x_i is obtained as follows: $\begin{matrix} d_{1}^{* 2} = ({0.8084 | 0.002, 0.7834 | 0.0013, \cdot \cdot \cdot, 0.869 | 0.0016}, \\ {0.1191 | 0.022, \cdot \cdot \cdot, 0.1638 | 0.017}), \\ d_{2}^{* 2} = ({0.7774 | 0.0105, 0.8365 | 0.021, \cdot \cdot \cdot, 0.8649 | 0.0028}, \\ {0.1351 | 0.0052, \cdot \cdot \cdot, 0.1295 | 0.0309}), \\ d_{3}^{* 2} = ({0.8649 | 0.0088, 0.8529 | 0.0093, \cdot \cdot \cdot, 0.7898 | 0.0123}, \\ {0.0795 | 0.0383, \cdot \cdot \cdot, 0.1895 | 0.0002}), \\ d_{4}^{* 2} = ({0.7468 | 0.1193, 0.7285 | 0.0104, \cdot \cdot \cdot, 0.8305 | 0.0062}, \\ {0.1462 | 0.005, \cdot \cdot \cdot, 0.1335 | 0.0035}) . \end{matrix}$

Finally, according to Definition 4, the score function of the comprehensive dual hesitation fuzzy preference values $d_{i}^{* 1}, d_{i}^{* 2} (i = 1, 2, 3, 4)$ is calculated, and the ranking of potential locations for large-scale industrial solid waste treatment facilities is obtained, as shown in Table 6.

Table 6
Score function values and ranking results using different operators

Operators x ₁ x ₂ x ₃ x ₄ Ranking

WPDHFMSMA –0.655 –0.6803 –0.6634 –0.7352 x₁ ≻ x₃ ≻ x₂ ≻ x₄

WPDHFMSMG 0.7116 0.6887 0.7097 0.624 x₁ ≻ x₃ ≻ x₂ ≻ x₄

Operators	x ₁	x ₂	x ₃	x ₄	Ranking
WPDHFMSMA	–0.655	–0.6803	–0.6634	–0.7352	x₁ ≻ x₃ ≻ x₂ ≻ x₄
WPDHFMSMG	0.7116	0.6887	0.7097	0.624	x₁ ≻ x₃ ≻ x₂ ≻ x₄

On the other hand, to implement the decision-making method in Li and Wang [17], let the consistency index threshold CI₀ = 0.9. For fair comparison, the consensus level threshold is the same as that in Section 5.1 of this paper, i.e., GCI₀ = 0.75. The specific GDM steps are summarized as follows:

First, the expected additive consistent PHFPRs D^Λ′ (Λ = 1, 2, 3) are calculated using Equation (12) in [17].

To save space, only some select elements of D^1′ are listed: $\begin{matrix} d_{12}^{(1^{'})} = {0.5375 | 0.0388, 0.425 | 0.0776, 0.4875 | 0.0259, \\ 0.375 | 0 . 0517, 0.6125 | 0.0044, 0.5 | 0.0088, \\ 0.5625 | 0 . 0029, 0.4625 | 0.0992, 0.3875 | 0.0576, \\ 0.275 | 0.1152, 0.3375 | 0.0384, 0.225 | 0.0768, \\ 0.4125 | 0.0661, 0.3 | 0.1323, \\ 0.45 | 0.0059, 0.35 | 0.1984}, \end{matrix}$ $\begin{matrix} d_{13}^{(1^{'})} = {0.25 | 0.06507, 0.275 | 0.09221, 0.375 | 0.1644, \\ 0.2 | 0 . 04662, 0.225 | 0.05238, 0.325 | 0.12522, \\ 0.35 | 0 . 0638, 0.45 | 0.12276, 0.3 | 0.09924, \\ 0.4 | 0.08547, 0.175 | 0.00972, 0.125 | 0.00648, \\ 0.15 | 0.01944, 0.425 | 0.01089, \\ 0.525 | 0.02178, 0.475 | 0.01452} . \end{matrix}$

Second, Equation (15) in [17] is used to calculate the consistency index of individual PHFPRs D^Λ (Λ = 1, 2, 3): $CI (D^{1}) = 0.9832, CI (D^{2}) = 0.9666, CI (D^{3}) = 0.9876 .$

Therefore, all PHFPRs D^Λ (Λ = 1, 2, 3) satisfy the acceptable consistency level.

Then, Equation (20) in [17] is used to obtain the weights of DMs: $φ_{1} = 0 . 3347, φ_{2} = 0 . 3291, φ_{3} = 0 . 3362 .$

In addition, Equation (16) in [17] is used for information integration, and the group PHFPR $D^{(1)} = (d_{ij}^{(1)})_{4 \times 4}$ is obtained: $d_{ii}^{(1)} = {0.5 | 1}, d_{12}^{(1)} = {0.4343 | 0.2, 0.3339 | 0.8},$ $\begin{matrix} d_{13}^{(1)} = {0.3994 | 0.06, 0.5335 | 0.1467, 0.5328 | 0.0917, \\ 0.4331 | 0.12, 0.4324 | 0.075, 0.5664 | 0.1833, \\ 0.4653 | 0.015, 0.4989 | 0.03, 0.4999 | 0.0733, \\ 0.5657 | 0.0183, 0.5993 | 0.0367, 0.466 | 0.15}, \end{matrix}$ $\begin{matrix} d_{14}^{(1)} = {0.465 | 0.3429, 0.549 | 0.2571, \\ 0.398 | 0.2286, 0.4821 | 0.1714}, \\ d_{23}^{(1)} = {0.4168 | 0.75, 0.4504 | 0.25}, \\ d_{24}^{(1)} = {0.2833 | 0.15, 0.3162 | 0.1833, \\ 0.4334 | 0.3, 0.4668 | 0.3667}, \end{matrix}$ $\begin{matrix} d_{34}^{(1)} = {0.5999 | 0.0252, 0.5662 | 0.0168, 0.6328 | 0.0348, \\ 0.5991 | 0.0232, 0.5664 | 0.0756, 0.5328 | 0.0504, \\ 0.5993 | 0.1044, 0.5657 | 0.696, 0.4325 | 0.1512, \\ 0.4654 | 0.2088, 0.4318 | 0.1392, 0.3989 | 0.1008} . \end{matrix}$

Third, the consensus measure for PHFPRs D^Λ (Λ = 1, 2, 3) is obtained using Equation (17) in [17]: $GCI (D^{1}) = 0.9529, GCI (D^{2}) = 0.9732, GCI (D^{3}) = 0.9430 .$

Since the consensus measure GCI (D^Λ) > GCI₀ = 0.75 for Λ = 1, 2, 3, there is no need to increase the consensus level for PHFPRs D^Λ.

Finally, by applying Equation (4) in [17], the score matrix E (D⁽¹⁾) of the group PHFPR $D^{(1)} = (d_{ij}^{(1)})_{4 \times 4}$ is obtained: $E (D^{(1)}) = (\begin{matrix} {0.5000} & {0.3540} & {0.5001} & {0.4742} \\ {0.6460} & {0.5000} & {0.4252} & {0.4017} \\ {0.4999} & {0.5748} & {0.5000} & {0.4950} \\ {0.5258} & {0.5983} & {0.5050} & {0.5000} \end{matrix}) .$

Then, the corresponding weights of alternatives x_i (i = 1, 2, 3, 4) are obtained using Equation (21) in [17]: $ω_{1} = 0.2285, ω_{2} = 0.2466, ω_{3} = 0.2587, ω_{4} = 0.2661 .$

In this way, the locations of large-scale industrial solid waste treatment facilities are ranked as follows: $x_{4} ≻ x_{3} ≻ x_{2} ≻ x_{1} .$

In summary, Table 7 shows the ranking results of the locations of large industrial solid waste treatment facilities obtained using different GDM methods.

Table 7

The ranking results of large-scale industrial solid waste treatment facilities locations

Approach	The ranking results of the locations	The most suitable location of large-scale industrial solid waste treatment facilities
Algorithm 1	x₃ ≻ x₁ ≻ x₂ ≻ x₄	x ₃
Algorithm 2	x₃ ≻ x₁ ≻ x₂ ≻ x₄	x ₃
Garg and Kaur [15]’s method	x₁ ≻ x₃ ≻ x₂ ≻ x₄	x ₁
Li and Wang [17]’s method	x₄ ≻ x₃ ≻ x₂ ≻ x₁	x ₄

5.3 Discussion

The most suitable location of large-scale industrial solid waste treatment facilities obtained by our methods are different from those obtained using the GDM methods of Garg and Kaur [15] and Li and Wang [17] for the following reasons.

(1) The method proposed by Garg and Kaur [15] got support from WPDHFMSMA operator and WPDHFMSMG to determine the location of large-scale industrial solid waste treatment facilities, However, our methods use the consensus of preference relations to solve this problem which include contain the consensus test and improvement of the PDHFPRs, so our proposed methods are more reasonable. In addition, the individual PDHFPRs are aggregated to obtain a group PDHFPR using Algorithm 1 in Garg and Kaur [15], where $\begin{matrix} d_{23}^{*} = ({0.2 | \frac{1}{3}, 0.45 | \frac{1}{3}, 0.6 | 0.25, 0.7 | \frac{1}{12}}, \\ {0.6 | \frac{7}{60}, 0.8 | \frac{13}{60}, 0.35 | \frac{5}{60}, 0.5 | 0.25, 0.2 | \frac{1}{3}}), \end{matrix}$ max {0.2, 0.45, 0.6, 0.7} + max {0.6, 0.8, 0.35, 0.5, 0.2} >1. Therefore, the PDHFE $d_{23}^{*}$ does not meet the definition of PDHFS. In contrast, Algorithm 1 of this paper uses the consensus level for the PDHFPR to objectively determine the weights of the DMs, which can effectively address decision-making situations where the weights of DMs is unknown. Therefore, the GDM algorithm in this paper is more reasonable.

(2) When the decision-making method proposed by Li and Wang [17] is used to select the location of large-scale industrial solid waste treatment facilities, the original PDHFPRs are converted into the corresponding PHFPRs, which can easily result in distortion and a loss of information, thereby leading to inaccurate decision-making results. In addition, when determining the location of large-scale industrial solid waste treatment facilities, the original PDHFPRs or the weights of the DMs must be adjusted to reach the expected consensus level threshold when using our proposed methods. However, when using the method proposed by Li and Wang [17], the consensus level among DMs does not need to be improved, which indicates that the proposed methods have higher requirements for the consensus level among DMs. Therefore, the GDM algorithms in this paper are more reasonable.

(3) This paper proposes two consensus-based GDM methods with PDHFPRs. Algorithm 1 provides an adjustment parameter and automatically modifies the PDHFPR provided by the DM through to meet the expected consensus level threshold. It does not require DMs to spend much time interacting, which can greatly improve the efficiency of decision-making. When the DMs do not want to modify the PDHFPR, Algorithm 2 modifies the DMs’ weights to make the DMs reach the expected consensus level to obtain reasonable decision-making results.

6 Conclusion

Two consensus-based GDM methods with PDHFPRs have been proposed in this paper. Based on the definition of PDHFPRs, the distance measure between two PDHFPRs has been defined. Note that the numbers of elements of the two PDHFEs are not required to be equal, and all distance combination of membership and non-membership elements are considered in the given distance measure, thereby avoiding the loss of original information. To measure the consensus level for PDHFPRs, a series of concepts, such as the individual consensus measure for PDHFPRs, the group consensus measure for PDHFPRs, and the consensus measure between the row vectors of the PDHFPR have also been presented. Finally, two GDM methods based on PDHFPRs have been proposed. These two methods improve the consensus level among DMs by modifying the PDHFPRs provided by DMs or the weight of DMs. The simulation results have illustrated that the proposed methods can be used effectively to solve the problem of selecting the most suitable location of large-scale industrial solid waste treatment facilities.

In the future work, it is interesting to apply the proposed methods to other fields, such as home care service provider selection and the selection of evaluation indices for digital libraries. In addition, due to factors such as the lack of knowledge and the urgency of time of DMs, some elements are often missing in the construction process of PDHFPRs. Therefore, future research could focus on the GDM method with incomplete PDHFPRs.

Footnotes

Acknowledgments

The work was supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (No. 71521001), National Natural Science Foundation of China (No. 71901001, 61806068), and Training Program of the Major Research Plan of the National Science Foundation of China (No. 91546108), Natural Science Foundation of Anhui Province (No. 2008085QG333), Key Research Project of Humanities and Social Sciences in Colleges and Universities of Anhui Province (No. SK2020A0038).

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D ¹	x ₁	x ₂	x ₃	x ₄
x ₁	({ 0.5\|1 } , { 0.5\|1 })	$\begin{matrix} ({0.4 \| 0.2, 0.1 \| 0.8}, \\ {0.5 \| 0.4, 0.3 \| 0.1, 0.2 \| 0.5}) \end{matrix}$	$\begin{matrix} ({0.3 \| 0.45, 0.6 \| 0.55}, \\ {0.4 \| 1}) \end{matrix}$	$\begin{matrix} ({0.5 \| 0.6, 0.3 \| 0.4}, \\ {0.24 \| 0.7, 0.14 \| 0.3}) \end{matrix}$
x ₂	/	({ 0.5\|1 } , { 0.5\|1 })	({ 0.2\|1 } , { 0.6\|0.35, 0.8\|0.65 })	$({0.25 \| \frac{1}{3}, 0.7 \| \frac{2}{3}}, {0.3 \| 1})$
x ₃	/	/	({ 0.5\|1 } , { 0.5\|1 })	$\begin{matrix} ({0.7 \| 0.1, 0.6 \| 0.3, 0.2 \| 0.6}, \\ {0.2 \| 0.8, 0.1 \| 0.2}) \end{matrix}$
x ₄	/	/	/	({ 0.5\|1 } , { 0.5\|1 })

Consensus-based group decision-making methods with probabilistic dual hesitant fuzzy preference relations and their applications

Abstract

Keywords

1 Introduction

2 Preliminaries

3.1 PDHFPRs

5 Algorithm 1

5.1 Application to analyze the location selection of large-scale industrial solid waste treatment facilities

Table 5 Individual consensus measures without considering one specific DM The individual consensus measure x 1 x 2 x 3 x 4 ACM i ( 0 ) 0.7222 0.7478 0.7585 0.7534 ACM 1 - i ( 0 ) 0.752 0.7968 0.8054 0.7805 ACM 2 - i ( 0 ) 0.7573 0.7668 0.8090 0.7952 ACM 3 - i ( 0 ) 0.8034 0.8185 0.8168 0.8218

Table 6 Score function values and ranking results using different operators Operators x 1 x 2 x 3 x 4 Ranking WPDHFMSMA –0.655 –0.6803 –0.6634 –0.7352 x1 ≻ x3 ≻ x2 ≻ x4 WPDHFMSMG 0.7116 0.6887 0.7097 0.624 x1 ≻ x3 ≻ x2 ≻ x4

6 Conclusion

Footnotes

Acknowledgments

References

Table 6
Score function values and ranking results using different operators

Operators x ₁ x ₂ x ₃ x ₄ Ranking

WPDHFMSMA –0.655 –0.6803 –0.6634 –0.7352 x₁ ≻ x₃ ≻ x₂ ≻ x₄

WPDHFMSMG 0.7116 0.6887 0.7097 0.624 x₁ ≻ x₃ ≻ x₂ ≻ x₄