Generalized Shapley probability neutrosophic hesitant fuzzy Choquet aggregation operators and their applications

Abstract

The probabilistic neutrosophic hesitant fuzzy numbers are considered to be effective tools for dealing with such decision problems when both subjective and objective uncertainties exist simultaneously. However, the existing methods for dealing with real-life decision-making in the context is based on the assumption that the relationships between all criteria are independent and irrelevant. It is worth noting that this assumption is not sufficient. In fact, there may be interrelationships between attributes. In order to consider the correlation between factors from a more global perspective, the generalized Shapley probabilistic neutrosophic hesitant fuzzy Choquet averaging (GS-PNHFCA) operator and the generalized Shapley probabilistic hesitant fuzzy Choquet geometric (GS-PNHFCG) operator are investigated. Next, in order to find the optimal weight vector about DMs and criteria, a model is constructed by the maximizing score deviation (MSD) method. In addition, based on the integrated operators and built models, an algorithm for solving the MCGDM problem of PNHFN is designed. The effectiveness and practicability of the algorithm is proved by comparison with existingresults.

Keywords

Probabilistic neutrosophic hesitant fuzzy set multi-criteria group decision-making (MCGDM)Choquet integral shapley function aggregation operator

1 Introduction

With the increase of human activities and the diversification of information, the MCGDM issue has become a hot issue [1 , 32]. Due to the limitations of the decision maker’s cognitive level, a large amount of decision information is depicted by fuzzy sets. Naturally, fuzzy sets have also been extensively investigated in the relevant fields of algebra [11 , 55]. With the advent of the information age, due to the cognitive uncertainty and cognitive inconsistency of decision makers, a large amount of decision information cannot be accurately represented. Different types of fuzzy set (FS) theory [2, 50] depict the information only from one aspect of cognitive inconsistency. In order to describe the MCGDM information from three aspects: epistemic certainty, epistemic indetermination and epistemic inconsistency, as an extension of FS, the notion of neutrosophic set (NS) was introduced [38]. The advantage of NS is that it can explain fuzzy information from three independent aspects: truth membership information (T_A (x) ∈ [0, 1]), indeterminacy membership information (I_A (x) ∈ [0, 1]) and falsity membership information (F_A (x) ∈ [0, 1]). By using NS, all the epistemic information can be described by decision maker (DM). If 0 ≤ T_A (x) + I_A (x) + F_A (x) ≤1, the NS reduces a intuitionistic FS. If 1 < T_A (x) + I_A (x) + F_A (x) ≤3, this phenomenon indicates that the decision maker lacks confidence for the alternatives and lead to overlapping of preference information. For instance a voting question, voters have three options: support, opposition, abstention. This type of information is counted in terms of support and is also included in the aspect of abstention. On the other hand, some of the abstentions may have a tendency to opposition. In the end, the result will be greater than 1. Therefore, the NS is better at describing fuzzy information in real life, such as medical diagnosis [25, 47], the decision-making problems [18 , 48], etc. It also began to be studied in the field of algebra [35 , 57].

Additionally, in actual situations, decision makers often show hesitation. For better describe the actual situations, Ye [49] first defined single-valued neutrosophic hesitant fuzzy set (SVNHFS) with truth membership hesitant degrees (TMHDs), indeterminacy membership hesitant degrees (IMHDs) and falsity membership hesitant degrees (FMHDs), separately. Each type of hesitation membership may contains multiple degrees. For convenience, in this paper, SVNHFS is simply expressed as neutrosophic hesitant fuzzy set (NHFS). As far as current research is concerned, the application of NHFS is more flexible than NS. For example, Ji et al. [10] proposed a improved projection-based TODIM method with NHFSs and applied to the personnel assignment problems. According to the neutrosophic linguistic situations, Wang et.al. [42] investigated a battery of Maclaurin symmetric mean (MSM) aggregation operators. In [49], Ye structured the similarity (distance) measures of NHFS. Other researchers have made a lot of contributions [19,20 , 28].

But in many practical situations, there is a key factor in the existing methods that are not reflected. The main reason is that these methods are based on the premise that there is no relationship between various alternatives. Obviously, this is unreasonable in real situations. When there is a correlation between attributes, the Choquet integral can help us to consider this situations [3 , 43].

When the information about probability is considered part of a fuzzy data, it is easy to be lost after the fuzzy operations. Since of the lack of knowledge of real-world knowledge, it is difficult to fully acquire the probability distribution of information. Peng et al. [29] described a probabilistic fuzzy linguistic set (PFLS) through a new perspective. Then, probabilistic hesitant fuzzy set [45], probabilistic dual hesitant fuzzy set [8] and probabilistic interval intuitionistic hesitant fuzzy set [51] were investigated. This pattern can be applied to NHFS to reduce the problem of data loss. Peng et al. [30] and Shao [36] introduced probabilistic multi-valued neutrosophic set (PMVNS) and probabilistic neutrosophic hesitant fuzzy set (PNHFS), respectively. Peng et al. proposed the convex combination operation and aggregation operator under PMVNS. Shao et al. researched PNHFS’ operation laws and basic properties. Simultaneously, probabilistic neutrosophic hesitant fuzzy averaging (geometric) operators under PNHFS were introduced. Under the probabilistic neutrosophic hesitant fuzzy environment, Shao [37] introduced PNHF Choquet averaging (PNHFCA) and PNHF Choquet geometric (PNHFCG)operators.

By observing the methods mentioned above, all factors are aggregated based on the independent character. Obviously, those factors may be interrelationship in real situations. Nevertheless, the interrelationships between the factors are caught by the Choquet integral. But the approach only describes the interrelationship between two nearby factors. For better results, more hidden information can be obtained by considering the relationship between the combinations or their ordered positions globally. Thence, the generalized Shapley PNHFCA (GS-PNHFCA) and generalized Shapley PNHFCG (GS-PNHFCG) operators are proposed. If the weight information of DMs and the criteria may be partially unknown, by applying the maximizing score deviation (MSD) approach, a new model to obtain the whole weight information are established. The MSD approach is inspired by [9, 44]. Finally, through a global perspective, the correlation between attributes is considered and an algorithm for handling MCGDM problems is given with PNHFNs.

This paper is structured as follows: In Section 2, the conceptions and operations of the PNHFNs are described. In Section 3, the GS-PNHFCA and GS-PNHFCG operators are established, respectively. In Section 4, a model based on the MSD method to determine the optimal measures are given, an algorithm for MCGDM problems with PNHFNs are signed. In Section 5, a numerical example is produced to test the given method. In Section 6, the conclusions and next study directions are indicated.

2 Preliminaries

Definition 2.1. ([48]) Suppose X is a finite reference set. A NHFS on X is described by the following formula: $\begin{matrix} N = {〈 x, \tilde{t} (x), \tilde{i} (x), \tilde{f} (x) 〉 | x \in X} \end{matrix}$ in which $\tilde{t} (x)$ , $\tilde{i} (x)$ and $\tilde{f} (x) \in P$ . $P$ is the power set of [0, 1]. $\tilde{t} (x)$ , $\tilde{i} (x)$ and $\tilde{f} (x)$ depict the truth-membership hesitant function, indeterminacy-membership hesitant function and falsity-membership hesitant function of x ∈ X, respectively. Hold the following restrictions: 0≤α, β, γ ≤ 1 and 0 ≤ α⁺ + β⁺ + γ⁺ + ≤3, where $α \in \tilde{t} (x), β \in \tilde{i} (x), γ \in \tilde{f} (x)$ , α⁺ = max {α}, β⁺ =max {β}, γ⁺ = max {γ} for x ∈ X.

Definition 2.2. ([30, 36]) Suppose X is a finite reference set. A PNHFS on X is depicted by the following formula:

$\begin{matrix} N = & {〈 x, T (x) | P^{T} (x), I (x) | P^{I} (x), F (x) | P^{F} (x) 〉 \\ | x \in X} . \end{matrix}$ (1)

The T (x) |P^T (x), I (x) |P^I (x) and F (x) |P^F (x) are three factors of N, where T (x), I (x) and F (x) are called the possible truth-membership hesitant degree, indeterminacy-membership hesitant degree and falsity-membership hesitant degree to x, respectively. P^T (x), P^I (x) and P^F (x) are the corresponding probabilistic information for these three types of hesitant degrees. The conditions that need to be held: $\begin{matrix} α_{a}, β_{b}, γ_{c} \in [0, 1], 0 \leq α^{+} + β^{+} + γ^{+} \leq 3; \\ P_{a}^{T}, P_{b}^{I}, P_{c}^{F} \in [0, 1]; \sum_{a = 1}^{# T} P_{a}^{T} \leq 1, \\ \sum_{b = 1}^{# I} P_{b}^{I} \leq 1, \sum_{c = 1}^{# F} P_{c}^{F} \leq 1 . \end{matrix}$ where α_a ∈ T (x), β_b ∈ I (x), γ_c ∈ F (x). α⁺ = max {α_a}, β⁺ = max {β_b}, γ⁺ = max {γ_c}, $P_{a}^{T} \in P^{T}$ , $P_{b}^{I} \in P^{I}$ , $P_{c}^{F} \in P^{F}$ . The symbols # T, # I and # F are the cardinal numbers of elements in the components T (x) |P^T (x), I (x) |P^I (x) and F (x) |P^F (x), respectively.

Generally, A probabilistic neutrosophic hesitant fuzzy number (PNHFN) is described by the symbol: $\begin{matrix} N = {T | P^{T}, I | P^{I}, F | P^{F}} . \end{matrix}$

Definition 2.3. ([30]) Suppose X is a finite reference set, N is a PNHFN, N holds the following conditions:

$\tilde{N} = 〈 T (x) | {\tilde{P}}^{T} (x), I (x) | {\tilde{P}}^{I} (x), F (x) | {\tilde{P}}^{F} (x) 〉$ (2) where ${\tilde{P}}_{a}^{T} = \frac{P_{a}^{T}}{\sum P_{a}^{T}}$ , ${\tilde{P}}_{b}^{I} = \frac{P_{b}^{I}}{\sum P_{b}^{I}}$ , ${\tilde{P}}_{c}^{F} = \frac{P_{c}^{F}}{\sum P_{c}^{F}}$ .

Then $\tilde{N}$ is called a normalized PNHFN.

Definition 2.4. ([36]) Suppose N₁ = {T₁|P^{T
₁}, I₁|P^{I
₁}, F₁|P^{F
₁}} and N₂ = {T₂|P^{T
₂}, I₂|P^{I
₂}, F₂|P^{F
₂}} are two PNHFNs, then the basic operations of PNHFNs as follows:

$(N_{1})^{c} = ⋃_{α_{1} \in T_{1}, β_{1} \in I_{1}, γ_{1} \in F_{1}} {γ_{1} | P_{1}^{F_{1}}, 1 - β_{1} | P_{1}^{I_{1}},$ $α_{1} | P_{1}^{T_{1}}}$ ,

$(N_{1})^{λ} = ⋃_{α_{1} \in T_{1}, β_{1} \in I_{1}, γ_{1} \in F_{1}} {{(α_{1})^{λ} | P_{1}^{T_{1}}}, {1 - (1 - β_{1})^{λ} | P_{1}^{I_{1}}}, {1 - (1 - γ_{1})^{λ} | P_{1}^{F_{1}}}}$ ,

$λ (N_{1}) = ⋃_{α_{1} \in T_{1}, β_{1} \in I_{1}, γ_{1} \in F_{1}} {{1 - (1 - λ_{1})^{λ} | P_{1}^{T_{1}}},$ ${(β_{1})^{λ} | P_{1}^{I_{1}}}, {(γ_{1})^{λ} | P_{1}^{F_{1}}}}$ ,

N₁ ⊕ N₂ = ⋃ _{substackα₁∈T₁,β₁∈I₁,γ₁∈F₁,η₂∈T₂,θ₂∈I₂,μ₂∈F₂} {{α₁ + η₂ - α₂η₂| $P_{1}^{T_{1}} P_{2}^{T_{2}}}, {β_{1} θ_{2} | P_{1}^{I_{1}} P_{2}^{I_{2}}}, {γ_{1} μ_{2} | P_{1}^{F_{1}} P_{2}^{F_{2}}}}$ ,

$N_{1} \otimes N_{2} = ⋃_{α_{1} \in T_{1}, β_{1} \in I_{1}, γ_{1} \in F_{1}, η_{2} \in T_{2}, θ_{2} \in I_{2}, μ_{2} \in F_{2}} {{α_{1} η_{2} | P_{1}^{T_{1}} P_{2}^{T_{2}}},$ ${β_{1} + θ_{2} - β_{1} θ_{2} | P_{1}^{I_{1}} P_{2}^{I_{2}}}, {γ_{1} + μ_{2} - γ_{1} μ_{2} |$ $P_{1}^{F_{1}} P_{2}^{F_{2}}}}$ ,

where

P_{1}^{T_{1}}

;

P_{1}^{I_{1}}

and

P_{1}^{F_{1}}

are hesitant probabilities of α₁ ∈ T₁, β₁ ∈ I₁ and γ₁ ∈ F₁, respectively.

P_{2}^{T_{2}}

;

P_{2}^{I_{2}}

and

P_{1}^{F_{2}}

are corresponding hesitant probabilities of η₂ ∈ T₂, θ₂ ∈ I₂ and μ₂ ∈ F₂.

Fuzzy measures have been extensively studied in actual situations [5 , 46].

Definition 2.5. ([39]) Let X be a reference set, $P (X)$ be the power set of X. A fuzzy measure μ on X is a function μ: P (X) → [0, 1] which holds:

μ (∅) =0, μ (X) =1;

If A ⊆ B, then μ (A) ≤ μ (B), $\forall A, B \subseteq P (X)$ ;

Let X = {x₁, x₂, ⋯ , x_n}. $\forall A, B \in P (X)$ , A∩ B = ∅. If fuzzy measure μ holds the following requisition, μ (A ∪ B) = μ (A) + μ (B) + λμ (A) μ (B), λ ∈ (-1, ∞)

then μ is called a λ-fuzzy measure.

Theorem 2.1. ([39]) Let X = {x₁, x₂, ⋯ , x_n} be a reference set, then λ-fuzzy measure μ holds: $μ (X) = {\begin{matrix} \frac{1}{λ} (\prod_{i = 1}^{n} [1 + λ μ (x_{i})] - 1) if λ \neq 0 \\ \sum_{i = 1}^{n} μ (x_{i}) if λ = 0 \end{matrix}$ (3)Since μ (X) =1, parameter λ can be determined by $\begin{matrix} λ + 1 = \prod_{i = 1}^{n} (1 + λ μ (x_{i})) . \end{matrix}$

Definition 2.6. (([6])) Suppose μ is a fuzzy measure of X, X = {x₁, x₂, ⋯ , x_n}, f is a measurable, real-valued and nonnegative mapping on X. The formula of Choquet integral with respect to fuzzy measure μ is described by: $\begin{matrix} \int fd μ = \sum_{i = 1}^{n} f (x_{(i)}) [μ (A_{(i)}) - μ (A_{(i + 1)})] \end{matrix}$ where {x₍₁₎, x₍₂₎, ⋯ , x_(n)} is a permutation of X such that f (x₍₁₎) ≤ f (x₍₂₎) ≤ ⋯ ≤ f (x_(n)), A_(i) = {x_(i), x_(i+1), ⋯ , x_n} and A_(n+1) = 0,.

Definition 2.7. ([22]) Let P (X) be the power set of X = {x₁, x₂, ⋯ , x_n}, μ be a fuzzy measure, then generalized Shapley function can be constructed, $ψ_{S} (μ, X) = \sum_{T \subseteq X ∖ S} \frac{(n - s - t)! t!}{(n - s + 1)!} (μ (S \cup T) - μ (T)), \forall S \in P (X)$ . which μ is depicted as a fuzzy measure, parameter n, s, t stands for corresponding cardinal number of X, S, T. Based on the above definition, we can understand that this generalized Shapley function can summarize the impact of each alliance in the global.

3 The generalized Shapley probabilistic neutrosophic hesitant fuzzy Choquet aggregation operators

The relationship between the indicators reflected by the PNHF information is described from a holistic perspective. The GS-PNHFCA and GS-PNHFCG operators are proposed in this section.

3.1 The comparison method of PNHFNs

When dealing with MCGDM issues with PNHFNs, it is necessary to compare PNHFNs. Thus, in this subsection, the score function and deviation function of PNHFN are introduced. Then introduced an approach of ranking PNHFNs.

Definition 3.1. Suppose N = {T|P^T, I|P^I, F|P^F} is a PNHFN, then a score function of the PNHFN is marked as:

$S (N) = \sum_{a = 1}^{# T} α_{a} P_{a}^{T} + \sum_{b = 1}^{# I} (1 - β_{b}) P_{b}^{I} - \sum_{c = 1}^{# F} γ_{c} P_{c}^{F}$ (4)

Definition 3.2. Suppose N = {T|P^T, I|P^I, F|P^F} is a PNHFN, then we can define a deviation function D (N) as: $\begin{matrix} D (N) = & \sum_{a = 1}^{# T} (α_{a} - S (N))^{2} \cdot P_{a}^{T} + \\ \sum_{b = 1}^{# I} (1 - β_{b} - S (N))^{2} \cdot P_{b}^{I} + \end{matrix}$ (5) $\sum_{c = 1}^{# F} (γ_{c} - S (N))^{2} \cdot P_{c}^{F}$ (6)

The distance from the score valued in the PNHFN N is described by the deviation function. Thus, the deviation value is called a consistency indicator of the PNHFN N. The higher value of D (N), the lower consistency of N. Based on the Definition 3.1 and 3.2, a method of ordering two PNHFNs is developed.

Definition 3.3. For PNHFNs N₁ and N₂, the arrangement approach for N₁ and N₂ as follows:

If S (N₁) > S (N₂), then the PNHFN N₁ is superior to N₂;

If S (N₁) = S (N₂) , D (N₁) > D (N₂), then the PNHFN N₁ is inferior to N₂;

If S (N₁) = S (N₂) , D (N₁) = D (N₂), then the PNHFN N₁ is equal to N₂.

3.2 GS-PNHFCA operator based on the PNHFS

By Definition 2.3 and Choquet integral, we construct a generalized Shapley probabilistic neutrosophic hesitant fuzzy Choquet averaging (GS-PNHFCA) operator and a generalized Shapley probabilistic neutrosophic hesitant fuzzy Choquet geometric (GS-PNHFCG) operator under the probabilistic neutrosophic hesitant fuzzy environment. Meanwhile, some basic properties are summarized.

Definition 3.4. Suppose N_i = {T_i|P^{T
_i}, I_i|P^{I
_i}, F_i|P^{F
_i}} (i = 1, 2, ⋯ , n) is a PNHFN, μ is a fuzzy measure on X. Then, the GS-PNHFCA operator is described as: $\begin{matrix} GS - PNHFCA (N_{1}, N_{2}, \dots, N_{n}) \\ = \oplus_{i = 1}^{n} [ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)] N_{(i)} \end{matrix}$ (7) where $P_{(i)}^{T_{(i)}}$ , $P_{(i)}^{I_{(i)}}$ and $P_{(i)}^{F_{(i)}}$ are corresponding probability values of α_(i), β_(i) and γ_(i). (·) is a permutation on X such that N₍₁₎ ≤ N₍₂₎ ≤ ⋯ ≤ N_(n), N_(n+1) = 0. ψ_{A
_(i)} (μ, A) is the generalized Shapley value of N_(i) about μ, A_(i) = {x_(i), x_(i+1), ⋯ , x_(n)}.

Theorem 3.1. Suppose N_i (i = 1, 2, ⋯ , n) is a PNHFN, μ is a fuzzy measure on N = {N₁, N₂, ⋯, N_n}. Then the GS-PNHFCA operator PNHFCA (N₁, N₂, ⋯ , N_n) is still a PNHFN, $\begin{matrix} GS - PNHFCA (N_{1}, N_{2}, \dots, N_{n}) \\ = \oplus_{i = 1}^{n} [ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)] N_{(i)} \\ = ⋃_{α_{(i)} \in T_{(i)}, β_{(i)} \in I_{(i)}, γ_{(i)} \in F_{(i)}} { \\ {1 - \prod_{i = 1}^{n} (1 - α_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} | \prod_{i = 1}^{n} P_{α_{(i)}}^{T_{(i)}}}, \\ {\prod_{i = 1}^{n} (β_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} | \prod_{i = 1}^{n} P_{β_{(i)}}^{I_{(i)}}}, \\ {\prod_{i = 1}^{n} (γ_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} | \prod_{i = 1}^{n} P_{γ_{(i)}}^{F_{(i)}}}} . \end{matrix}$ (8)

Proof. By Definition 2.7, ∀i ∈ {1, 2, ⋯ , n}, $\begin{matrix} ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A) \geq 0; \\ \sum_{i = 1}^{n} (ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)) = 1 . \end{matrix}$ Mathematical induction can be utilized to solve this problem. Obviously, when n = 1, the Theorem 3.1 is certainly true.

(1) When n = 2, according to the operation laws of PNHFN, we have $\begin{matrix} GS - PNHFCA (N_{1}, N_{2}) = \\ (ψ_{A_{(1)}} (μ, A) - ψ_{A_{(2)}} (μ, A)) N_{(1)} \oplus \\ (ψ_{A_{(2)}} (μ, A) - ψ_{A_{(3)}} (μ, A)) N_{(2)} \\ = ⋃_{α_{(1)} \in T_{(1)}, β_{(1)} \in I_{(1)}, γ_{(1)} \in F_{(1)}} { \\ {1 - (1 - α_{(1)})^{ψ_{A_{(1)}} (μ, A) - ψ_{A_{(2)}} (μ, A)} (μ, A)) N_{(1)} | P_{α_{(1)}}^{T_{(1)}}}, \\ {(β_{(1)})^{ψ_{A_{(1)}} (μ, A) - ψ_{A_{(2)}} (μ, A)} | P_{β_{(1)}}^{I_{(1)}}}, \\ {(γ_{(1)})^{ψ_{A_{(1)}} (μ, A) - ψ_{A_{(2)}} (μ, A)} (μ, A)) N_{(1)} | P_{γ_{(1)}}^{F_{(1)}}}} \\ \oplus ⋃_{α_{(2)} \in T_{(2)}, β_{(2)} \in I_{(2)}, γ_{(2)} \in F_{(2)}} { \\ {1 - (1 - α_{(2)})^{ψ_{A_{(2)}} (μ, A) - ψ_{A_{(3)}} (μ, A)} | P_{α_{(2)}}^{T_{(2)}}}, \\ {(β_{(2)})^{ψ_{A_{(2)}} (μ, A) - ψ_{A_{(3)}} (μ, A)} | P_{β_{(2)}}^{I_{(2)}}}, \\ {(γ_{(2)})^{ψ_{A_{(2)}} (μ, A) - ψ_{A_{(3)}} (μ, A)} | P_{γ_{(2)}}^{F_{(2)}}}} \\ = ⋃_{α_{(i)} \in T_{(i)}, β_{(i)} \in I_{(i)}, γ_{(i)} \in F_{(i)}} { \\ {1 - \prod_{i = 1}^{2} (1 - α_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} | \prod_{i = 1}^{2} P_{α_{(1)}}^{T_{(i)}}}, \\ {\prod_{i = 1}^{2} (β_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} | \prod_{i = 1}^{2} P_{β_{(i)}}^{I_{(i)}}}, \\ {\prod_{i = 1}^{2} (γ_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} | \prod_{i = 1}^{2} P_{γ_{(i)}}^{F_{(i)}}} \end{matrix}$ Thus, Eq. (7) is true.

(3) Suppose Eq. (7) holds for n = k, then $\begin{matrix} GS - PNHFCA (N_{1}, N_{2}, \dots, N_{k}) \\ = \oplus_{i = 1}^{k} (ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)) N_{(i)} \\ = ⋃_{α_{(i)} \in T_{(i)}, β_{(i)} \in I_{(i)}, γ_{(i)} \in F_{(i)}} { \\ {1 - \prod_{i = 1}^{k} (1 - α_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} | \prod_{i = 1}^{k} P_{α_{(i)}}^{T_{(i)}}}, \\ {\prod_{i = 1}^{k} (β_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} | \prod_{i = 1}^{k} P_{β_{(i)}}^{I_{(i)}}}, \\ {\prod_{i = 1}^{k} (γ_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} | \prod_{i = 1}^{k} P_{γ_{(i)}}^{F_{(i)}}}} . \end{matrix}$

Therefore, when n = k + 1, $\begin{matrix} GS - PNHFCA (N_{1}, N_{2}, \dots, N_{k}, N_{k + 1}) \\ = ⋃_{α_{(i)} \in T_{(i)}, β_{(i)} \in I_{(i)}, γ_{(i)} \in F_{(i)}} { \\ {1 - \prod_{i = 1}^{k} (1 - α_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} | \prod_{i = 1}^{k} P_{α_{(i)}}^{T_{(i)}}}, \\ {\prod_{i = 1}^{k} (β_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} | \prod_{i = 1}^{k} P_{β_{(i)}}^{I_{(i)}}}, \\ {\prod_{i = 1}^{k} (γ_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} | \prod_{(i) = 1}^{k} P_{γ (i)}^{F_{(i)}}}} \\ \oplus ⋃_{α_{(k + 1)} \in T_{(k + 1)}, β_{(k + 1)} \in I_{(k + 1)}, γ_{(k + 1)} \in F_{(k + 1)}} { \\ {1 - (1 - α_{(k + 1)})^{ψ_{A_{(k + 1)}} (μ, A) - ψ_{A_{(k + 2)}} (μ, A)} | P_{α (k + 1)}^{T_{(k + 1)}}}, \\ {(β_{(k + 1)})^{ψ_{A_{(k + 1)}} (μ, A) - ψ_{A_{(k + 2)}} (μ, A)} | P_{β_{(k + 1)}}^{I_{(k + 1)}}}, \\ {(γ_{(k + 1)})^{ψ_{A_{(k + 1)}} (μ, A) - ψ_{A_{(k + 2)}} (μ, A)} | P_{γ_{(k + 1)}}^{F_{(k + 1)}}}} \\ = ⋃_{α_{(i)} \in T_{(i)}, β_{(i)} \in I_{(i)}, γ_{(i)} \in F_{(i)}} { \\ {1 - \prod_{i = 1}^{k + 1} (1 - α_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} | \prod_{i = 1}^{k + 1} P_{α_{(i)}}^{T_{(i)}}}, \end{matrix}$ $\begin{matrix} {\prod_{i = 1}^{k + 1} (β_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} | \prod_{i = 1}^{k + 1} P_{β_{(i)}}^{I_{(i)}}}, \\ {\prod_{i = 1}^{k + 1} (γ_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} | \prod_{i = 1}^{k + 1} P_{γ_{(i)}}^{F_{(i)}}}} \end{matrix}$ Thus, Eq. (8) holds for n = k + 1. Generally, ∀n, Eq. (8) is right. Next, $\begin{matrix} 1 - \prod_{i = 1}^{n} (1 - α_{(n)})^{ψ_{A_{(n)}} (μ, A) - ψ_{A_{(n + 1)}} (μ, A)} \in [0, 1] \\ \prod_{i = 1}^{n} (β_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(n + 1)}} (μ, A)} \in [0, 1] \\ \prod_{i = 1}^{n} (γ_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} \in [0, 1] \\ \prod_{i = 1}^{n} P_{α_{(i)}}^{T_{(i)}}, \prod_{i = 1}^{n} P_{β_{(i)}}^{I_{(i)}}, \prod_{i = 1}^{n} P_{γ_{(i)}}^{F_{(i)}} \in [0, 1] . \end{matrix}$ In conclusion, the Theorem 3.1 is right.

Theorem 3.2. (Idempotency) Suppose N_i = {{α|P₁} , {β|P₂} , {γ] |P₃}}, i = 1, 2, ⋯ , n, μ is a fuzzy measure of N = {N₁, N₂, ⋯ , N_n} and P₁, P₂, P₃ ∈ {0, 1}, then

$\begin{matrix} GS - PNHFCA (N_{1}, N_{2}, \dots, N_{n}) \\ = {{α | P_{1}}, {β | P_{2}}, {γ | P_{3}}} . \end{matrix}$ (9)

Proof. By Theorem 3.1, $\begin{matrix} \prod_{i = 1}^{n} P_{j} = P_{j} (j = 1, 2, 3) . \end{matrix}$ Simultaneously, it holds $\begin{matrix} GS - PNHFCA (N_{1}, N_{2}, \dots, N_{n}) \\ = {{1 - \prod_{i = 1}^{n} (1 - α)^{ψ_{A_{(n)}} (μ, A) - ψ_{A_{(n + 1)}} (μ, A)} | \prod P_{1}}, \\ {\prod_{i = 1}^{n} (β)^{ψ_{A_{(n)}} (μ, A) - ψ_{A_{(n + 1)}} (μ, A)} | \prod P_{2}}, \\ {\prod_{i = 1}^{n} (γ)^{ψ_{A_{(n)}} (μ, A) - ψ_{A_{(n + 1)}} (μ, A)} | \prod P_{3}}} \\ = {{1 - (1 - α)^{\sum (ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A))} | P_{1}}, \\ {(β)^{\sum (ψ_{A_{(n)}} (μ, A) - ψ_{A_{(n + 1)}} (μ, A))} | P_{2}}, \\ {(γ)^{\sum (ψ_{A_{(n)}} (μ, A) - ψ_{A_{(n + 1)}} (μ, A))} | P_{3}}} \end{matrix}$ According to the definition of fuzzy measure, $\begin{matrix} \sum (ψ_{A_{(n)}} (μ, A) - ψ_{A_{(n + 1)}} (μ, A)) = 1 . \end{matrix}$ Eventually, $\begin{matrix} GS - PNHFCA (N_{1}, N_{2}, \dots, N_{n}) \\ = {{α | P_{1}}, {β | P_{2}}, {γ | P_{3}}} . \end{matrix}$

Theorem 3.3. (Commutativity) Suppose μ is a fuzzy measure of X = {x₁, x₂, ⋯ , x_n}. By swapping the locations of the elements in the collection {N₁, N₂, ⋯ , N_n}, an new set ${N_{1}^{'}, N_{2}^{'}, \dots, N_{n}^{'}}$ is organized. Then $\begin{matrix} GS - PNHFCA (N_{1}, N_{2}, \dots, N_{n}) \\ = GS - PNHFCA (N_{1}^{'}, N_{2}^{'}, \dots, N_{n}^{'}) . \end{matrix}$

Proof. By Definition 3.4, it is easy to illustrate this conclusion.

Theorem 3.4. (Monotonicity) Suppose N_i = {{α_i| $P_{α_{i}}^{T_{i}}}, {β_{i} | P_{β_{i}}^{I_{i}}}, {γ_{i} | P_{γ_{i}}^{F_{i}}}}$ and ${\tilde{N}}_{i} = {{{\tilde{α}}_{i} | P_{{\tilde{α}}_{i}}^{{\tilde{T}}_{i}}}, {{\tilde{β}}_{i} |$ $P_{{\tilde{β}}_{i}}^{{\tilde{I}}_{i}}}$ , ${{\tilde{γ}}_{i} | P_{{\tilde{γ}}_{i}}^{{\tilde{F}}_{i}}}}$ (i = 1, 2, ⋯ , n) are two PNHFNs. {(i)} is an array, satisfies N₍₁₎ ≤ N₍₂₎ ≤ ⋯ ≤ N_(n) and ${\tilde{N}}_{(1)} \leq {\tilde{N}}_{(2)} \leq \dots \leq {\tilde{N}}_{(n)}$ . For any factors in $N, \tilde{N}$ , holds $α_{(i)} \leq {\tilde{α}}_{(i)}, β_{(i)} \geq {\tilde{β}}_{(i)}, γ_{(i)} \geq {\tilde{γ}}_{(i)},$ $P_{α_{(i)}}^{T_{(i)}} = P_{{\tilde{α}}_{(i)}}^{{\tilde{T}}_{(i)}}, P_{β_{(i)}}^{I_{(i)}} = P_{{\tilde{β}}_{(i)}}^{{\tilde{I}}_{(i)}}, P_{γ_{(i)}}^{F_{(i)}} = P_{{\tilde{γ}}_{(i)}}^{{\tilde{F}}_{(i)}} .$

Then $\begin{matrix} GS - PNHFCA (N_{1}, N_{2}, \dots, N_{n}) \\ \leq GS - PNHFCA ({\tilde{N}}_{1}, {\tilde{N}}_{2}, \dots, {\tilde{N}}_{n}) . \end{matrix}$

Proof. By Definition 3.4, for every aggregated element, there are $(1 - \prod (1 - α_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)}) \prod P_{α_{(i)}}^{T_{(i)}} \leq (1 - \prod (1 - {\tilde{α}}_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)}) \prod P_{{\tilde{α}}_{(i)}}^{{\tilde{T}}_{(i)}}$

$(1 - \prod (β_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)}) \prod P_{β_{(i)}}^{I_{(i)}} \leq (1 - \prod ({\tilde{β}}_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)}) \prod P_{{\tilde{β}}_{(i)}}^{{\tilde{I}}_{(i)}}$

$\prod (γ_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} \prod P_{γ_{(i)}}^{F_{(i)}} \geq \prod ({\tilde{γ}}_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}}} \prod P_{{\tilde{γ}}_{(i)}}^{{\tilde{F}}_{(i)}} .$

then according to Definition 3.1 and 3.3, the conclusion $\begin{matrix} GS - PNHFCA (N_{1}, N_{2}, \dots, N_{n}) \\ \leq GS - NHFCA ({\tilde{N}}_{1}, {\tilde{N}}_{2}, \dots, {\tilde{N}}_{n}) . \end{matrix}$ is correct.

Theorem 3.5. (Boundedness) Suppose N_i = {{α_i| $P_{α_{i}}^{T_{i}}}, {β_{i} | P_{β_{i}}^{I_{i}}}, {γ_{i} | P_{γ_{i}}^{F_{i}}}}$ is a PNHFN (i = 1, 2, ⋯ , n), μ is a fuzzy measure on X. If $\begin{matrix} N^{-} = & {{\min {α_{i}} | \min {P_{α_{i}}^{T_{i}}}}, \\ {\max {β_{i}} | \max {P_{β_{i}}^{I_{i}}}}, \\ {\max {γ_{i}} | \max {P_{γ_{i}}^{F_{i}}}}}, \\ N^{+} = & {{\max {α_{i}} | \max {P_{α_{i}}^{T_{i}}}}, \\ {\min {β_{i}} | \min {P_{β_{i}}^{I_{i}}}}, \\ {\min {γ_{i}} | \min {P_{γ_{i}}^{F_{i}}}}} . \end{matrix}$ Then $\begin{matrix} GS - PNHFCA (N^{-}) \\ \leq GS - PNHFCA (N_{1}, N_{2}, \dots, N_{n}) \\ \leq GS - PNHFCA (N^{+}) \end{matrix}$

Proof. ∀N_i, $\begin{matrix} \min {α_{i}} \leq α_{(i)} \leq \max {α_{i}}, \\ \min {β_{i}} \leq β_{(i)} \leq \max {β_{i}}, \\ \min {γ_{i}} \leq γ_{(i)} \leq \max {γ_{i}}; \\ \min {P_{α_{i}}^{T_{i}}} \leq P_{α_{(i)}}^{T_{(i)}} \leq \max {P_{α_{i}}^{T_{i}}}, \\ \min {P_{β_{i}}^{I_{i}}} \leq P_{β_{(i)}}^{I_{(i)}} \leq \max {P_{β_{i}}^{I_{i}}}, \\ \min {P_{γ_{i}}^{F_{i}}} \leq P_{γ_{(i)}}^{F_{(i)}} \leq \max {P_{γ_{i}}^{F_{i}}} . \end{matrix}$ Thus,

$\begin{matrix} 1 - \prod (1 - α_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} \\ \geq & 1 - \prod (1 - \min {α_{i}})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} \\ = & 1 - (1 - \min {α_{i}})^{\sum (ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A))} \\ = & \min {α_{i}} = \min {α_{(i)}}, \\ \prod (β_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} \\ \leq & \prod (\max {β_{i}})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} \\ = & (\max {β_{i}})^{\sum (ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A))} \\ = & \max {β_{i}} = \max {β_{(i)}}, \\ \prod (γ_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} \\ \leq & \prod (\max {γ_{i}})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} \\ = & (\max {γ_{i}})^{\sum (ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A))} \\ = & \max {γ_{i}} = \max {γ_{(i)}} . \end{matrix}$

For the probabilities, we have $\begin{matrix} \prod \min {P_{α_{i}}^{T_{i}}} \leq \prod P_{α_{(i)}}^{T_{(i)}}, \\ \prod P_{β_{(i)}}^{I_{(i)}} \leq \prod \max {P_{β_{i}}^{I_{i}}}, \\ \prod P_{γ_{(i)}}^{F_{(i)}} \leq \prod \max {P_{γ_{i}}^{F_{i}}} . \end{matrix}$

Thus, about GS - PNHFCA (N^-), we know, $\begin{matrix} GS - PNHFCA (N^{-}) \\ = & ⋃ {{\min {α_{i}} | \prod \min {P_{α_{i}}^{T_{i}}}}, \end{matrix}$ $\begin{matrix} {\max {β_{i}} | \prod \max {P_{β_{i}}^{I_{i}}}}, {\max {γ_{i}} | \prod \max {P_{γ_{i}}^{F_{i}}}}} . \end{matrix}$ By Definition 3.1 and 3.3, $\begin{matrix} GS - PNHFCA (N^{-}) \\ \leq GS - PNHFCA (N_{1}, N_{2}, \dots, N_{n}) . \end{matrix}$ Similarly, the following inequality is illustrated. $\begin{matrix} GS - PNHFCA (N_{1}, N_{2}, \dots, N_{n}) \\ \leq GS - PNHFCA (N^{+}) . \end{matrix}$

3.3 GS-PNHFCG operator based on the PNHFS

Definition 3.5. Suppose N_i = {T_i|P^{T
_i}, I_i|P^{I
_i}, F_i|P^{F
_i}} (i = 1, 2, ⋯ , n) is a PNHFN, μ is a fuzzy measure on X. Then the GS-PNHFCG operator is defined as: $\begin{matrix} GS - PNHFCG (N_{1}, N_{2}, \dots, N_{n}) \\ = & \otimes_{i = 1}^{n} (ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)) N_{(i)} \end{matrix}$ (10) where $P_{(i)}^{T_{(i)}}$ , $P_{(i)}^{I_{(i)}}$ and $P_{(i)}^{F_{(i)}}$ are corresponding probability values of α_(i), β_(i) and γ_(i). (·) is a permutation on X such that N₍₁₎ ≤ N₍₂₎ ≤ ⋯ ≤ N_(n), N_(n+1) = 0. ψ_{A
_(i)} (μ, A) is the generalized Shapley value of N_(i) about μ, A_(i) = {x_(i), x_(i+1), ⋯ , x_(n)}.

Theorem 3.6. Suppose N_i (i = 1, 2, ⋯ , n) are n PNHFNs, μ is a fuzzy measure on X. Then the integrated value by GS-PNHFCG operator GS - PNHFCOG (N₁, N₂, ⋯ , N_n) is still a PNHFN, and $\begin{matrix} GS - PNHFCG (N_{1}, N_{2}, \dots, N_{n}) \\ = & \otimes_{i = 1}^{n} (ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)) N_{(i)} \\ = & ⋃_{α_{(i)} \in T_{(i)}, β_{(i)} \in I_{(i)}, γ_{(i)} \in F_{(i)}} { \\ {\prod_{i = 1}^{n} (α_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} | \prod_{i = 1}^{n} P_{β_{(i)}}^{T_{(i)}}}, \\ {1 - \prod_{i = 1}^{n} (1 - β_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} | \prod_{i = 1}^{n} P_{β_{(i)}}^{I_{(i)}}}, \\ {1 - \prod_{i = 1}^{n} (1 - γ_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} | \prod_{i = 1}^{n} P_{γ_{(i)}}^{F_{(} i)}}}, \end{matrix}$ (11)

Proof. Proof process similar to Theorem 3.1.

Since proof processes are similarity to the properties of GS-PNHFCA operator, thus the next proof processes are omitted:

Theorem 3.7.

(Idempotency) Suppose N_i = {{α|P₁} , {β|P₂} , {γ] |P₃}} is a normalized PNHFN (i = 1, 2, ⋯ , n), μ is a fuzzy measure of X, then

$\begin{matrix} GS - PNHFCG (N_{1}, N_{2}, \dots, N_{X}) \\ = & {{α | P_{1}}, {β | P_{2}}, {γ | P_{3}}} . \end{matrix}$ (12)

(Commutativity) Suppose A = {N₁, N₂, ⋯ , N_n} is a PNHFN, B = {N_λ(1), N_λ(2), ⋯ , N_λ(n)} is a new collection obtained after the elements in A = {N₁, N₂, ⋯ , N_n} are rearranged. $\begin{matrix} PNHFCOG (N_{1}, N_{2}, \dots, N_{n}) \\ = & PNHFCOG {N_{λ (1)}, N_{λ (2)}, \dots, N_{λ (n)}} . \end{matrix}$

(Monotonicity) Suppose $N_{i} = {{α_{i} | P_{α_{i}}^{T_{i}}}, {β_{i} | P_{β_{i}}^{I_{i}}}, {γ_{i} | P_{γ_{i}}^{F_{i}}}}$ and ${\tilde{N}}_{i} = {{{\tilde{α}}_{i} | P_{{\tilde{α}}_{i}}^{{\tilde{T}}_{i}}}, {{\tilde{β}}_{i} | P_{{\tilde{β}}_{i}}^{{\tilde{I}}_{i}}}$ , ${{\tilde{γ}}_{i} | P_{{\tilde{γ}}_{i}}^{{\tilde{F}}_{i}}}}$ (i = 1, 2, ⋯ , n) are two PNHFNs. {(i)} is a prerequisite such that N₍₁₎ ≤ N₍₂₎ ≤ ⋯ ≤ N_(n) and ${\tilde{N}}_{(1)} \leq {\tilde{N}}_{(2)} \leq \dots \leq {\tilde{N}}_{(n)}$ . ∀N_(i) and $\forall {\tilde{N}}_{(i)}$ , $α_{(i)} \leq {\tilde{α}}_{(i)}$ , $β_{(i)} \geq {\tilde{β}}_{(i)}$ , $γ_{(i)} \geq {\tilde{γ}}_{(i)}$ ; $P_{α_{(i)}}^{T_{(i)}} \leq P_{{\tilde{α}}_{(i)}}^{{\tilde{T}}_{(i)}}$ , $P_{β_{(i)}}^{I_{(i)}} \leq P_{{\tilde{β}}_{(i)}}^{{\tilde{I}}_{(i)}}$ , $P_{γ_{(i)}}^{F_{(i)}} \geq P_{{\tilde{γ}}_{(i)}}^{{\tilde{F}}_{(i)}}$ .

Then $\begin{matrix} GS - PNHFCG {N_{1}, N_{2}, \dots, N_{n}} \\ \leq GS - PNHFCG {{\tilde{N}}_{1}, {\tilde{N}}_{2}, \dots, {\tilde{N}}_{n}} . \end{matrix}$

(Boundedness)

Suppose $N_{i} = {{α_{i} | P_{α_{i}}^{T_{i}}}, {β_{i} | P_{β_{i}}^{I_{i}}}, {γ_{i} |$ $P_{γ_{i}}^{F_{i}}}}$ is a PNHFN (i = 1, 2, ⋯ , n), μ is a fuzzy measure of X. If $\begin{matrix} N^{-} = & {{\min {α_{i}} | \min {P_{α_{i}}^{T_{i}}}}, \\ {\max {β_{i}} | \max {P_{β_{i}}^{I_{i}}}}, \\ {\max {γ_{i}} | \max {P_{γ_{i}}^{F_{i}}}}}, \\ N^{+} = & {{\max {α_{i}} | \max {P_{α_{i}}^{T_{i}}}}, \\ {\min {β_{i}} | \min {P_{β_{i}}^{I_{i}}}}, \\ {\min {γ_{i}} | \min {P_{γ_{i}}^{F_{i}}}}} . \end{matrix}$ Then $\begin{matrix} GS - PNHFCA (N^{-}) \\ \leq & GS - PNHFCA (N_{1}, N_{2}, \dots, N_{n}) \\ \leq & GS - PNHFCA (N^{+}) \end{matrix}$

Lemma 3.1. If x_i ≥ 0, w_i ≥ 0, i = 1, 2, ⋯ , n and $\sum_{i = 1}^{n} w_{i} = 1$ , then $\begin{matrix} \prod_{i = 1}^{n} (x_{i})^{w_{i}} \leq \sum_{i = 1}^{N} w_{i} x_{i} . \end{matrix}$

Theorem 3.8. Suppose $N_{i} = {{α_{i} | P_{α_{i}}^{T_{i}}}, {β_{i} | P_{β_{i}}^{I_{i}}},$ ${γ_{i} | P_{γ_{i}}^{F_{i}}}}$ (i = 1, 2, ⋯ , n) is a PNHFN, μ is a fuzzy measure, then $\begin{matrix} GS - PNHFCG (N_{1}, N_{2}, \dots, N_{n}) \\ \leq & GS - PNHFCA (N_{1}, N_{2}, \dots, N_{n}) . \end{matrix}$ (13)

Proof. Based on Lemma 3.1, Definition 3.4 and 3.5, the following relationship is obtained $\begin{matrix} \prod (α_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} \\ \leq & \sum (ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)) α_{(i)} \\ = & 1 - \sum (ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)) (1 - α_{(i)}) \\ \leq & 1 - \prod (1 - α_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} . \end{matrix}$ Thus, $\begin{matrix} \prod (α_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} \prod P_{α_{(i)}}^{T_{(i)}} \\ \leq & (1 - \prod (1 - α_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)}) \prod P_{α_{(i)}}^{T_{(i)}} . \end{matrix}$ Similarity, we have $\begin{matrix} (1 - \prod (β_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)}) \prod P_{β_{(i)}}^{I_{(i)}} \\ \geq & \prod (1 - β_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} P_{β_{(i)}}^{I_{(i)}}, \\ \prod (γ_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)} \prod P_{γ_{(i)}}^{F_{(i)}} \\ \leq & (1 - \prod (1 - γ_{(i)})^{ψ_{A_{(i)}} (μ, A) - ψ_{A_{(i + 1)}} (μ, A)}) P_{γ_{(i)}}^{F_{(i)}} . \end{matrix}$ Thus, by Eqs. (3) and Definition 3.3, $\begin{matrix} PNHFCOG (N_{1}, N_{2}, \dots, N_{n}) \\ \leq & PNHFCOA (N_{1}, N_{2}, \dots, N_{n}) . \end{matrix}$

4 A MCGDM method with PNHFNs

In this section, by the maximizing deviation approach, a MSD approach is structured under the probabilistic neutrosophic hesitant fuzzy environment. Suppose Z = {z₁, z₂, ⋯ , z_m} is the group of alternatives, C = {C₁, C₂, ⋯ , C_n} is a collection of criteria. A decision matrix is structured by decision makers d, d = 1, 2, ⋯ , q, denoted by $D_{d} = [N_{ij}^{d}]_{m \times n}$ , where $N_{ij} = {{α_{i, j}^{d} | P_{α_{i, j}}^{d}},$ ${β_{i, j}^{d} | P_{β_{i, j}}^{d}}, {γ_{i, j}^{d} | P_{γ_{i, j}}^{d}}}$ , (i = 1, 2, ⋯ , m ; j = 1, 2, ⋯, n ; d = 1, 2, ⋯ , q) is a PNHFN. The decision matrix describes the attribute information that the attributes z_i satisfy the criteria C_j.

4.1 The model based on the maximizing score deviation (MSD) method to determine the decision makers’ weights

Since the maximizing deviation approach is utility for obtaining weight information, we further propose an MSD approach to determine the optimal measure information about the decision makers and criteria.

For some reasons, the weight information of all decision makers is incomplete, a projection model is proposed to obtain the optimal fuzzy measure for every criterion C_j (Building model 1): $\max \sum_{i = 1}^{m} \sum_{d = 1}^{q} \sum_{k = 1}^{q} | s_{ij}^{d} - s_{ij}^{k} | θ_{d} (μ^{j}, E);$ (14) $s . t . {\begin{matrix} μ^{j} (\emptyset) = 0, μ^{j} (E) = 1 \\ μ^{j} (S) \leq μ^{j} (Y), \forall S, Y \subseteq E, S \subseteq Y \\ μ^{j} (d) \in W_{d}, d = 1, 2, \dots, q \end{matrix}$ (15) where θ_d (μ_j, E) is a generalized shapley value of decision maker d about fuzzy measure μ^j, W_d is the value range of d.

Next, a model is structured to adjust the fuzzy measure on the criterion C_j. According to the MSD method, if all attribute weights are incomplete, the programming model is instituted to find the optimal fuzzy measure for attribute C_j (Buildingmodel 2): $\max \sum_{j = 1}^{n} \sum_{i = 1}^{m} \sum_{e = 1}^{m} | s_{ij} - s_{ej} | θ_{C_{j}} (ν, C);$ (16) $s . t . {\begin{matrix} ν (\emptyset) = 0, ν (C) = 1 \\ ν (S) \leq ν (Y), \forall S, Y \subseteq C, S \subseteq Y \\ ν (C_{j}) \in W_{C_{j}}, j = 1, 2, \dots, n \end{matrix}$ (17) where θ_{C
_j} (ν, C) is the generalized Shapley value of the attribute C_j about fuzzy measure ν, W_{C
_j} is the value range of C_j.

4.2 The decision step

Through the above models and the GS-PNHFCA (GS-PNHFCG) operator, a procedure for resolving the MCGDM problem with PNHFNs is proposed as follows:

Normalize the probabilistic neutrosophic hesitant fuzzy information;

Determine the optimal fuzzy measure on decision maker e_i about attribute C_j by model 1;

According to Definition 2.7 and the GS-PNHFCA or GS-PNHFCG operator to obtain the whole PNHF matrix;

Utilize model 2 and to determine the optimal fuzzy measure about attribute C_j;

Utilize the the GS-PNHFCA or GS-PNHFCG operator to obtain the overall PNHFNs of the alternative z_i;

Calculate the score value and deviation value of every alternative z_i by Definition 3.1 and 3.2;

Rank and select the optimal alternative. The higher the score value, the better the alternative. When the score value is the same, the smaller the deviation value, the better the alternative.

End.

5 An illustrative example and analysis

In this part, an illustrative example is given to describe the practicability of the proposed method.

This example is adapted from Peng et al. [30]. In modern society, Choosing the suitable Logistics company plays an increasingly important role in the company’s development. Suppose that four possible logistics suppliers {a₁, a₂, a₃, a₄} are selected to determine the best one for development of ABC Machnery Manufacturing Co., Ltd Four attributes are considered as the main objects: C₁: The equipment system; C₂: The ability of manage; C₃: Safety; C₄: After-sales service. The incompletely weight vector about the attributes is given by W_C = {[0.2, 0.4] , [0.3, 0.4] , [0.1, 0.3] , [0.1, 0.3]}. A professional team of three experts, described by {e₁, e₂, e₃}. The evaluation values are shown by the PNHFNs in Tables 1, 2 and 3. Suppose the experts’ weight vectors are known: W_{e
₁} = {[0.2, 0.5] , [0.4, 0.5] , [0.1, 0.2] , [0.2, 0.3]}, W_{e
₂} = {[0.2, 0.3] , [0.2, 0.3] , [0.3, 0.4] , [0.3, 0.5]}, W_{e
₃} = {[0.2, 0.4] , [0.2, 0.3] , [0.3, 0.4] , [0.3, 0.4]}.

Table 1
PNHF decision matrix of e₁

C₁ C₂

z₁ {{0.4|0.6, 0.6|0.22} , {0.4|0.6} , {0.3|0.4, 0.4|0.5}} {{0.3|0.4, 0.6|0.4} , {0.5|0.5, 0.6|0.4} , {0.3|0.4}}

z₂ {{0.5|0.4, 0.6|0.3} , {0.4|0.2, 0.6|0.5} , {0.3|0.4}} {{0.6|0.5} , {0.4|0.3, 0.6|0.5} , {0.4|0.6, 0.6|0.3}}

z₃ {{0.5|0.7} , {0.4|0.3, 0.5|0.4} , {0.4|0.3, 0.6|0.5}} {{0.4|0.5, 0.6|0.5} , {0.5|0.6} , {0.4|0.4, 0.5|0.4}}

z₄ {{0.5|0.3} , {0.2|0.1, 0.4|0.5, 0.6|0.2} , {0.5|0.7}} {{0.6|0.5} , {0.4|0.5, 0.6|0.5} , {0.5|0.3, 0.6|0.5}

C₃ C₄

z₁ {{0.7|0.5, 0.8|0.5} , {0.3|0.5, 0.4|0.4} , {0.5|0.6}} {{0.5|0.4, 0.7|0.6} , {0.3|0.5, 0.5|0.4} , {0.5|0.4}}

z₂ {{0.7|0.3, 0.8|0.5} , {0.4|0.6} , {0.4|0.5, 0.6|0.4}} {{0.6|0.4, 0.8|0.4} , {0.4|0.2, 0.6|0.5} , {0.5|0.3}}

z₃ {{0.6|0.5} , {0.4|0.5, 0.5|0.3} , {0.4|0.5, 0.6|0.4}} {{0.6|0.5} , {0.5|0.4, 0.6|0.4} , {0.5|0.6, 0.6|0.4}}

z₄ {{0.6|0.3, 0.8|0.5} , {0.4|0.6} , {0.5|0.3, 0.6|0.5}} {{0.6|0.5, 0.8|0.4} , {0.4|0.6} , {0.4|0.5, 0.5|0.4}

	C₁	C₂
z₁	{{0.4\|0.6, 0.6\|0.22} , {0.4\|0.6} , {0.3\|0.4, 0.4\|0.5}}	{{0.3\|0.4, 0.6\|0.4} , {0.5\|0.5, 0.6\|0.4} , {0.3\|0.4}}
z₂	{{0.5\|0.4, 0.6\|0.3} , {0.4\|0.2, 0.6\|0.5} , {0.3\|0.4}}	{{0.6\|0.5} , {0.4\|0.3, 0.6\|0.5} , {0.4\|0.6, 0.6\|0.3}}
z₃	{{0.5\|0.7} , {0.4\|0.3, 0.5\|0.4} , {0.4\|0.3, 0.6\|0.5}}	{{0.4\|0.5, 0.6\|0.5} , {0.5\|0.6} , {0.4\|0.4, 0.5\|0.4}}
z₄	{{0.5\|0.3} , {0.2\|0.1, 0.4\|0.5, 0.6\|0.2} , {0.5\|0.7}}	{{0.6\|0.5} , {0.4\|0.5, 0.6\|0.5} , {0.5\|0.3, 0.6\|0.5}
	C₃	C₄
z₁	{{0.7\|0.5, 0.8\|0.5} , {0.3\|0.5, 0.4\|0.4} , {0.5\|0.6}}	{{0.5\|0.4, 0.7\|0.6} , {0.3\|0.5, 0.5\|0.4} , {0.5\|0.4}}
z₂	{{0.7\|0.3, 0.8\|0.5} , {0.4\|0.6} , {0.4\|0.5, 0.6\|0.4}}	{{0.6\|0.4, 0.8\|0.4} , {0.4\|0.2, 0.6\|0.5} , {0.5\|0.3}}
z₃	{{0.6\|0.5} , {0.4\|0.5, 0.5\|0.3} , {0.4\|0.5, 0.6\|0.4}}	{{0.6\|0.5} , {0.5\|0.4, 0.6\|0.4} , {0.5\|0.6, 0.6\|0.4}}
z₄	{{0.6\|0.3, 0.8\|0.5} , {0.4\|0.6} , {0.5\|0.3, 0.6\|0.5}}	{{0.6\|0.5, 0.8\|0.4} , {0.4\|0.6} , {0.4\|0.5, 0.5\|0.4}

Table 2

PNHF decision matrix of e₂

	C₁	C₂
z₁	{{0.6\|0.5} , {0.4\|0.2, 0.6\|0.6} , {0.4\|0.6, 0.6\|0.2}}	{{0.5\|0.4, 0.7\|0.4} , {0.6\|0.4} , {0.4\|0.6, 0.5\|0.4}}
z₂	{{0.3\|0.4} , {0.5\|0.4} , {0.2\|0.2, 0.4\|0.5, 0.6\|0.3}}	{{0.5\|0.6} , {0.6\|0.4} , {0.5\|0.3, 0.6\|0.4}}
z₃	{{0.4\|0.6, 0.6\|0.2} , {0.6\|0.3} , {0.5\|0.4, 0.6\|0.5}}	{{0.6\|0.4, 0.8\|0.4} , {0.5\|0.3, 0.7\|0.5} , {0.5\|0.4}}
z₄	{{0.5\|0.4, 0.6\|0.4} , {0.5\|0.3} , {0.3\|0.4, 0.6\|0.5}}	{{0.7\|0.5} , {0.5\|0.6, 0.6\|0.3} , {0.5\|0.6}
	C₃	C₄
z₁	{{0.5\|0.3, 0.6\|0.5} , {0.4\|0.4, 0.6\|0.6} , {0.3\|0.6}}	{{0.6\|0.6} , {0.3\|0.5} , {0.4\|0.4, 0.5\|0.3, 0.6\|0.3}}
z₂	{{0.5\|0.4, 0.6\|0.3} , {0.5\|0.6, 0.6\|0.3} , {0.5\|0.5}}	{{0.5\|0.6, 0.6\|0.4} , {0.4\|0.5, 0.6\|0.3} , {0.3\|0.4}}
z₃	{{0.5\|0.4, 0.6\|0.5} , {0.5\|0.4, 0.7\|0.5} , {0.5\|0.8}}	{{0.4\|0.6, 0.7\|0.4} , {0.3\|0.4, 0.4\|0.6} , {0.5\|0.5}}
z₄	{{0.5\|0.6} , {0.5\|0.5} , {0.4\|0.2, 0.6\|0.5, 0.7\|0.3}}	{{0.5\|0.5, 0.7\|0.5} , {0.5\|0.4} , {0.4\|0.6, 0.6\|0.3}

Table 3

PNHF decision matrix of e₃

	C₁	C₂
z₁	{{0.5\|0.4, 0.7\|0.4} , {0.5\|0.6, 0.7\|0.4} , {0.6\|0.3}}	{{0.4\|0.6} , {0.6\|0.4, 0.7\|0.3} , {0.5\|0.3, 0.7\|0.6}}
z₂	{{0.5\|0.4, 0.6\|0.4} , {0.5\|0.4} , {0.4\|0.6, 0.6\|0.4}}	{{0.4\|0.5, 0.5\|0.3} , {0.5\|0.5, 0.6\|0.4} , {0.4\|0.6}}
z₃	{{0.3\|0.4, 0.5\|0.4} , {0.4\|0.4, 0.5\|0.6} , {0.3\|0.4}}	{{0.6\|0.4} , {0.5\|0.3} , {0.4\|0.5, 0.5\|0.4}}
z₄	{{0.4\|0.4, 0.5\|0.6} , {0.4\|0.4} , {0.5\|0.4, 0.6\|0.2}}	{{0.5\|0.6, 0.7\|0.4} , {0.4\|0.3} , {0.3\|0.4, 0.5\|0.5}
	C₃	C₄
z₁	{{0.4\|0.5} , {0.3\|0.4} , {0.3\|0.2, 0.5\|0.5, 0.6\|0.3}}	{{0.3\|0.4, 0.5\|0.6} , {0.7\|0.6} , {0.5\|0.4, 0.6\|0.6}}
z₂	{{0.7\|0.4} , {0.4\|0.6, 0.7\|0.3} , {0.4\|0.5, 0.5\|0.5}}	{{0.4\|0.6} , {0.4\|0.5} , {0.4\|0.3, 0.5\|0.5, 0.6\|0.2}}
z₃	{{0.4\|0.5, 0.6\|0.5} , {0.4\|0.5} , {0.6\|0.4, 0.8\|0.5}}	{{0.4\|0.4, 0.5\|0.4} , {0.5\|0.6} , {0.3\|0.5, 0.5\|0.5}}
z₄	{{0.6\|0.5, 0.8\|0.5} , {0.4\|0.6, 0.6\|0.4} , {0.5\|0.3}}	{{0.8\|0.5} , {0.4\|0.5, 0.6\|0.3, 0.8\|0.2} , {0.3\|0.4}

5.1 The steps of the proposed method

In order to evaluate these four logistics providers, the main steps can be summarized as follows,

Step 1. Normalize the PNHF Information.

The decision matrixes are normalized by Eq. (2). Because of space constraints, the normalized evaluation information are not listed.

Step 2. Determine the optimal fuzzy measure on the general managers set E about attribute C₁ by Model 14.

\begin{matrix} \max & \frac{(133 * μ^{1} (e_{2}))}{3000} - \frac{(161 * μ^{1} (e_{1}))}{3000} + \frac{(7 * μ^{1} (e_{3}))}{750} \\ - & \frac{(7 * μ^{1} (e_{1}, e_{2}))}{750} - \frac{(133 * μ^{1} (e_{1}, e_{3}))}{3000} \\ + & \frac{(161 * μ^{1} (e_{2}, e_{3}))}{3000} + 797 / 600; \\ s . t . & {\begin{matrix} μ^{1} (\emptyset) = 0, μ^{1} (e_{1}, e_{2}, e_{3}) = 1 \\ μ^{1} (S) \leq μ^{1} (Y), \forall S, Y \subseteq {e_{1}, e_{2}, e_{3}}, S \subseteq Y \\ μ^{1} (e_{1}) \subseteq [0.2, 0.5], μ^{1} (e_{2}) \subseteq [0.2, 0.3], \\ μ^{1} (e_{3}) \subseteq [0.2, 0.4] . \end{matrix} \end{matrix}

With MATLAB software, related results are obtained.

\begin{matrix} μ^{1} (e 1) = μ^{1} (e 3) = μ^{1} (e 1, e 3) = 0.2, \\ μ^{1} (e 2) = μ^{1} (e 1, e 2) = 0.3, \\ μ^{1} (e 2, e 3) = μ^{1} (e 1, e 2, e 3) = 1 . \end{matrix}

Similarly, utilize the Model 14, we can get the optimal fuzzy measure about attribute C_j (j = 2, 3, 4).

$\begin{matrix} μ^{2} (e 1) = μ^{2} (e 2) = μ^{2} (e 1, e 2) = 0.2, \\ μ^{2} (e 3) = 0.4, \\ μ^{2} (e 1, e 3) = μ^{2} (e 2, e 3) = μ^{3} (e 1, e 2, e 3) = 1 . \\ μ^{3} (e 1) = μ^{3} (e 1, e 3) = 0.5, \\ μ^{3} (e 2) = μ^{3} (e 3) = μ^{3} (e 2, e 3) = 0.2, \\ μ^{3} (e 1, e 2) = μ^{3} (e 1, e 2, e 3) = 1 . \\ μ^{4} (e 1) = μ^{4} (e 2) = μ^{4} (e 1, e 2) = 0.2, \\ μ^{4} (e 3) = μ^{4} (e 1, e 3) = 0.4, \\ μ^{4} (e 2, e 3) = μ^{4} (e 1, e 2, e 3) = 1 . \end{matrix}$

Step 3. By Eq. 2 and GS-PNHFCA operator, the whole PNHF decision matrix R = [N_ij] _4×4 can be obtained (See Table 4)

Table 4
The whole PNHF decision matrix R

C₁

z₁ {{0.5456|0.3750, 0.5577|0.125, 0.6389|0.3750, 0.6486|0.1250} , {0.4423|0.15, 0.5380|0.45, 0.5145|0.1, 0.6259|0.3},

{0.4709|0.33, 0.4801|0.42, 0.5729|0.11, 0.5840|0.14}}

z₂ {{0.3984|0.285, 0.4072|0.215, 0.4477|0.285, 0.4558|0.215} , {0.4926|0.29, 0.5061|0.71} ,

{0.268|0.12, 0.3131|0.08, 0.3924|0.30, 0.4584|0.2, 0.4904|0.18, 0.5729|0.12}}

z₃ {{0.3641|0.3750, 0.4473|0.3750, 0.4912|0.1250, 0.5577|0.1250},

{0.4999|0.1720, 0.5486|0.2580, 0.5073|0.2280, 0.5527|0.3420} ,

{0.4011|0.1650, 0.4066|0.2570, 0.4435|0.21, 0.4495|0.35}}

z₄ {{0.4572|0.2, 0.5164|0.2, 0.50|0.3, 0.5544|0.3} , {0.4386|0.125, 0.4489|0.625, 0.455|0.25} ,

{0.3840|0.2948, 0.5494|0.3752, 0.4168|0.1452, 0.5964|0.1848}}

C₂

z₁ {{0.4142|0.25, 0.4762|0.25, 0.4974|0.25, 0.5506|0.25} , {0.5785|0.3192, 0.6|0.2508, 0.6429|0.2408, 0.6481|0.1892},

{0.4222|0.198, 0.4514|0.132, 0.4996|0.402, 0.5432|0.268}}

z₂ {{0.4567|0.63, 0.5218|0.37} , {0.487|0.2128, 0.5281|0.3472, 0.5533|0.1672, 0.60|0.2728} ,

{0.409|0.2881, 0.4436|0.1419, 0.4166|0.3819, 0.4517|0.1881}}

z₃ {{0.5834|0.25, 0.6617|0.25, 0.6|0.25, 0.6751|0.25} , {0.50|0.375, 0.5531|0.625} ,

{0.4277|0.28, 0.489|0.22, 0.4373|0.28, 0.50|0.22}}

z₄ {{0.5805|0.6, 0.6912|0.4} , {0.4277|0.335, 0.4517|0.165, 0.4454|0.335, 0.4704|0.165} ,

{0.368|0.165, 0.50|0.21, 0.3748|0.275, 0.5092|0.35}}

C₃

z₁ {{0.6248|0.1875, 0.7106|0.1875, 0.6512|0.3125, 0.7283|0.3125} ,

{0.3255|0.224, 0.3887|0.176, 0.3651|0.336, 0.436|0.246},

{0.4111|0.2, 0.4326|0.5, 0.4406|0.3}}

z₂ {{0.6413|0.2138, 0.7206|0.3562, 0.6682|0.1613, 0.7416|0.2687},

{0.4325|0.4489, 0.4406|0.2211, 0.461|0.2211, 0.4697|0.1089} ,

{0.4325|0.28, 0.5554|0.22, 0.4357|0.28, 0.5595|0.22}}

z₃ {{0.5563|0.22, 0.5834|0.28, 0.5739|0.22, 0.6|0.28} , {0.4261|0.275, 0.489|0.165, 0.4687|0.35, 0.5379|0.21} ,

{0.4437|0.2464, 0.5698|0.1936, 0.4567|0.3136, 0.5864|0.2464}}

z₄ {{0.4305|0.1875, 0.5844|0.3125, 0.4435|0.1875, 0.5839|0.3125} , {0.4325|0.6, 0.4384|0.4} ,

{0.4624|0.075, 0.5157|0.125, 0.5329|0.1875, 0.5964|0.3125, 0.5625|0.1125, 0.6294|0.1875}}

C₄

z₁ {{0.4362|0.16, 0.4457|0.24, 0.5393|0.24, 0.5471|0.36} , {0.4988|0.56, 0.5073|0.44},

{0.4607|0.16, 0.50|0.12, 0.5346|0.12, 0.514|0.24, 0.5578|0.18, 0.5964|0.18}}

z₂ {{0.4463|0.25, 0.4898|0.3, 0.459|0.25, 0.5015|0.3} , {0.40|0.1812, 0.4641|0.1081, 0.4054|0.4437, 0.4704|0.2662} ,

{0.3626|0.1875, 0.4146|0.3125, 0.4625|0.125}}

z₃ {{0.4238|0.3, 0.5532|0.2, 0.4772|0.3, 0.5946|0.2} , {0.4146|0.2, 0.4607|0.3, 0.4222|0.2, 0.4692|0.3} ,

{0.3808|0.3, 0.50|0.3, 0.3878|0.2, 0.5092|0.2}}

z₄ {{0.7047|0.28, 0.7115|0.22, 0.7593|0.28, 0.7648|0.22} , {0.4373|0.5, 0.5503|0.3, 0.6478|0.2} ,

{0.3398|0.3752, 0.3424|0.2948, 0.3997|0.1848, 0.4026|0.1452}}

	C₁
z₁	{{0.5456\|0.3750, 0.5577\|0.125, 0.6389\|0.3750, 0.6486\|0.1250} , {0.4423\|0.15, 0.5380\|0.45, 0.5145\|0.1, 0.6259\|0.3},
	{0.4709\|0.33, 0.4801\|0.42, 0.5729\|0.11, 0.5840\|0.14}}
z₂	{{0.3984\|0.285, 0.4072\|0.215, 0.4477\|0.285, 0.4558\|0.215} , {0.4926\|0.29, 0.5061\|0.71} ,
	{0.268\|0.12, 0.3131\|0.08, 0.3924\|0.30, 0.4584\|0.2, 0.4904\|0.18, 0.5729\|0.12}}
z₃	{{0.3641\|0.3750, 0.4473\|0.3750, 0.4912\|0.1250, 0.5577\|0.1250},
	{0.4999\|0.1720, 0.5486\|0.2580, 0.5073\|0.2280, 0.5527\|0.3420} ,
	{0.4011\|0.1650, 0.4066\|0.2570, 0.4435\|0.21, 0.4495\|0.35}}
z₄	{{0.4572\|0.2, 0.5164\|0.2, 0.50\|0.3, 0.5544\|0.3} , {0.4386\|0.125, 0.4489\|0.625, 0.455\|0.25} ,
	{0.3840\|0.2948, 0.5494\|0.3752, 0.4168\|0.1452, 0.5964\|0.1848}}
	C₂
z₁	{{0.4142\|0.25, 0.4762\|0.25, 0.4974\|0.25, 0.5506\|0.25} , {0.5785\|0.3192, 0.6\|0.2508, 0.6429\|0.2408, 0.6481\|0.1892},
	{0.4222\|0.198, 0.4514\|0.132, 0.4996\|0.402, 0.5432\|0.268}}
z₂	{{0.4567\|0.63, 0.5218\|0.37} , {0.487\|0.2128, 0.5281\|0.3472, 0.5533\|0.1672, 0.60\|0.2728} ,
	{0.409\|0.2881, 0.4436\|0.1419, 0.4166\|0.3819, 0.4517\|0.1881}}
z₃	{{0.5834\|0.25, 0.6617\|0.25, 0.6\|0.25, 0.6751\|0.25} , {0.50\|0.375, 0.5531\|0.625} ,
	{0.4277\|0.28, 0.489\|0.22, 0.4373\|0.28, 0.50\|0.22}}
z₄	{{0.5805\|0.6, 0.6912\|0.4} , {0.4277\|0.335, 0.4517\|0.165, 0.4454\|0.335, 0.4704\|0.165} ,
	{0.368\|0.165, 0.50\|0.21, 0.3748\|0.275, 0.5092\|0.35}}
	C₃
z₁	{{0.6248\|0.1875, 0.7106\|0.1875, 0.6512\|0.3125, 0.7283\|0.3125} ,
	{0.3255\|0.224, 0.3887\|0.176, 0.3651\|0.336, 0.436\|0.246},
	{0.4111\|0.2, 0.4326\|0.5, 0.4406\|0.3}}
z₂	{{0.6413\|0.2138, 0.7206\|0.3562, 0.6682\|0.1613, 0.7416\|0.2687},
	{0.4325\|0.4489, 0.4406\|0.2211, 0.461\|0.2211, 0.4697\|0.1089} ,
	{0.4325\|0.28, 0.5554\|0.22, 0.4357\|0.28, 0.5595\|0.22}}
z₃	{{0.5563\|0.22, 0.5834\|0.28, 0.5739\|0.22, 0.6\|0.28} , {0.4261\|0.275, 0.489\|0.165, 0.4687\|0.35, 0.5379\|0.21} ,
	{0.4437\|0.2464, 0.5698\|0.1936, 0.4567\|0.3136, 0.5864\|0.2464}}
z₄	{{0.4305\|0.1875, 0.5844\|0.3125, 0.4435\|0.1875, 0.5839\|0.3125} , {0.4325\|0.6, 0.4384\|0.4} ,
	{0.4624\|0.075, 0.5157\|0.125, 0.5329\|0.1875, 0.5964\|0.3125, 0.5625\|0.1125, 0.6294\|0.1875}}
	C₄
z₁	{{0.4362\|0.16, 0.4457\|0.24, 0.5393\|0.24, 0.5471\|0.36} , {0.4988\|0.56, 0.5073\|0.44},
	{0.4607\|0.16, 0.50\|0.12, 0.5346\|0.12, 0.514\|0.24, 0.5578\|0.18, 0.5964\|0.18}}
z₂	{{0.4463\|0.25, 0.4898\|0.3, 0.459\|0.25, 0.5015\|0.3} , {0.40\|0.1812, 0.4641\|0.1081, 0.4054\|0.4437, 0.4704\|0.2662} ,
	{0.3626\|0.1875, 0.4146\|0.3125, 0.4625\|0.125}}
z₃	{{0.4238\|0.3, 0.5532\|0.2, 0.4772\|0.3, 0.5946\|0.2} , {0.4146\|0.2, 0.4607\|0.3, 0.4222\|0.2, 0.4692\|0.3} ,
	{0.3808\|0.3, 0.50\|0.3, 0.3878\|0.2, 0.5092\|0.2}}
z₄	{{0.7047\|0.28, 0.7115\|0.22, 0.7593\|0.28, 0.7648\|0.22} , {0.4373\|0.5, 0.5503\|0.3, 0.6478\|0.2} ,
	{0.3398\|0.3752, 0.3424\|0.2948, 0.3997\|0.1848, 0.4026\|0.1452}}

Step 4. By the following formula to determine the optimal measure about attribute C_j (j = 1, 2, 3, 4).

\begin{matrix} \max & 0.1076 ν (C_{2}) - 0.4672 ν (C_{1}) + 0.1048 ν (C_{3}) \\ + 0.2548 ν (C_{4}) - 0.1798 ν (C_{1}, C_{2}) \\ - 0.1812 ν (C_{1}, C_{3}) - 0.1062 ν (C_{1}, C_{4}) \\ + 0.1062 ν (C_{2}, C_{3}) + 0.1812 ν (C_{2}, C_{4}) \\ + 0.1798 ν (C_{3}, C_{4}) - 0.2548 ν (C_{1}, C_{2}, C_{3}) \\ - 0.1048 ν (C_{1}, C_{2}, C_{4}) - 0.1076 ν (C_{1}, C_{3}, C_{4}) \\ + 0.4672 ν (C_{2}, C_{3}, C_{4}) + 2.03 \\ s . t . & {\begin{matrix} ν (\emptyset) = 0, ν (C_{1}, C_{2}, C_{3}, C_{4}) = 1 \\ ν (S) \leq ν (Y), \\ \forall S, Y \subseteq {C_{1}, C_{2}, C_{3}, C_{4}}, S \subseteq Y \\ ν (C_{1}) \subseteq [0.2, 0.4], ν (C_{2}) \subseteq [0.3, 0.4], \\ ν (C_{3}) \subseteq [0.1, 0.3, ν (C_{4}) \subseteq [0.1, 0.3] . \end{matrix} \end{matrix}

This model is solved by Matlab software, $\begin{matrix} ν (C_{1}) = ν (C_{3}) = ν (C 1, C_{3}) = 0.2; \\ ν (C_{2}) = ν (C_{4}) = ν (C_{1}, C_{2}) = ν (C_{1}, C_{4}) \\ = & ν (C_{2}, C_{3}) = ν (C_{1}, C_{2}, C_{3}) = 0.3; \\ ν (C_{2}, C_{4}) = ν (C_{3}, C_{4}) = ν (C_{1}, C_{2}, C_{4}) \\ = & ν (C_{1}, C_{3}, C_{4}) = ν (C_{2}, C_{3}, C_{4}) = 1; \end{matrix}$

Step 5. By Eq. (2.7) and the GS-PNHFCA operator to integrate the PNHFN N_i of the alternative z_i (i = 1, 2, 3, 4). The PNHFN N_i is omitted, because the space is limited.

Step 6. Calculate the score value of PNHFN N_i,

\begin{matrix} N_{1} = 0.4886, N_{2} = 0.8804, \\ N_{3} = 0.6192, N_{4} = 0.7993 . \end{matrix}

Step 7. Comparison method based on PNHFNs, all alternatives z_i (i = 1, 2, 3, 4) are ranked in descending order as follows, the most suitable alternative is chosen.

\begin{matrix} z_{2} > z_{4} > z_{3} > z_{1} . \end{matrix}

Then we obtain the optimal alternative is the z₂

Similarly, when the GS-PNHFCG operator is utilized to select the optimal alternative, the ranking is $\begin{matrix} z_{2} > z_{4} > z_{3} > z_{1} . \end{matrix}$ Thus, we know z₂ is the best logistics company.

5.2 Comparison analysis and discussion

The following comparisons are made utilizing existing methods, based on the same problem background. Peng et.al combines TOPSIS-based QUALIFLEX method and aggregation operator to rank the alternatives, the result is z₄ > z₂ > z₁ > z₃. Thus, the optimal alternative is z₄. See Table 5.

Table 5
Rankings results captured using different methods

Method Sort of Results Optimal Alternative Worst Alternative

TOPSIS - based QUALIFLEX method [30] z₄ > z₂ > z₁ > z₃ z ₄ z₃

GS - PNHFCA operator z₂ > z₄ > z₃ > z₁ z ₂ z₁

GS - PNHFCG operator z₂ > z₄ > z₃ > z₁ z ₂ z₁

Method	Sort of Results	Optimal Alternative	Worst Alternative
TOPSIS - based QUALIFLEX method [30]	z₄ > z₂ > z₁ > z₃	z ₄	z₃
GS - PNHFCA operator	z₂ > z₄ > z₃ > z₁	z ₂	z₁
GS - PNHFCG operator	z₂ > z₄ > z₃ > z₁	z ₂	z₁

The optimal option by our methods are different to those obtained by the Peng’s method. The result shows that alternative z₄ is the best choice. There are two motivations about this situation. Firstly, the interaction relationships between the decision makers and attributes are reflected in our method. Secondly, incomplete measurement information for decision makers and attributes affects the final result. In general, Peng’s method only deal with the probabilistic neutrosophic hesitant fuzzy information with independent criteria, unknown weight factors. When the decision makers’ and alternatives’ weight information are partly realized, our method can actually operate those probabilistic neutrosophic hesitant fuzzy information.

6 Conclusions

The PNHFS is integrated of a NHFS and a PHFS. The generalized Shapley Choquet integral (GSCI) operator describes the interrelation among elements. We applied the GSCI operator to PNHFS, and structured the GS-PNHFCA and GS-PNHFCG operators. Through these two operators, the importance of the alliance or its orderly position is first considered comprehensively, and then the correlation between the coalitions or their ordered positions is summarized globally. In the same context, the weight information of the criteria and estimator is partially know. Since the MSD method is a effective approach of receiving whole weight information, thus the model was structured to get the optimal fuzzy measure information. Next, through the GS-PNHFCA and GS-PNHFCG operators and MSD method, we structured a method to deal with the MCGDM problems under the probabilistic neutrosophic hesitant fuzzyenvironment.

In the next study, the proposed method will be applied to some practical issues in other fields, like emergency response, risk evaluation. Simultaneously, the complexity of the situation will be exponential growth when the fuzzy measure is described on the power set. Then, in order to reduce the complexity of calculating the fuzzy measure, some different types of fuzzy measures are considered, including λ- fuzzy measure, p- symmetry measure. Further, the single valued (interval) neutrosophic Choquet integral can be investigated.

7 Compliance with ethical standards

Acknowledgments This work was supported by National Natural Science Foundation of China (Grant No. 61573240).

Author contributions All authors have contributed equally to this paper.

Conflicts of interest The authors declare no conflict of interest.

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	C₁	C₂
z₁	{{0.6\|0.5} , {0.4\|0.2, 0.6\|0.6} , {0.4\|0.6, 0.6\|0.2}}	{{0.5\|0.4, 0.7\|0.4} , {0.6\|0.4} , {0.4\|0.6, 0.5\|0.4}}
z₂	{{0.3\|0.4} , {0.5\|0.4} , {0.2\|0.2, 0.4\|0.5, 0.6\|0.3}}	{{0.5\|0.6} , {0.6\|0.4} , {0.5\|0.3, 0.6\|0.4}}
z₃	{{0.4\|0.6, 0.6\|0.2} , {0.6\|0.3} , {0.5\|0.4, 0.6\|0.5}}	{{0.6\|0.4, 0.8\|0.4} , {0.5\|0.3, 0.7\|0.5} , {0.5\|0.4}}
z₄	{{0.5\|0.4, 0.6\|0.4} , {0.5\|0.3} , {0.3\|0.4, 0.6\|0.5}}	{{0.7\|0.5} , {0.5\|0.6, 0.6\|0.3} , {0.5\|0.6}
	C₃	C₄
z₁	{{0.5\|0.3, 0.6\|0.5} , {0.4\|0.4, 0.6\|0.6} , {0.3\|0.6}}	{{0.6\|0.6} , {0.3\|0.5} , {0.4\|0.4, 0.5\|0.3, 0.6\|0.3}}
z₂	{{0.5\|0.4, 0.6\|0.3} , {0.5\|0.6, 0.6\|0.3} , {0.5\|0.5}}	{{0.5\|0.6, 0.6\|0.4} , {0.4\|0.5, 0.6\|0.3} , {0.3\|0.4}}
z₃	{{0.5\|0.4, 0.6\|0.5} , {0.5\|0.4, 0.7\|0.5} , {0.5\|0.8}}	{{0.4\|0.6, 0.7\|0.4} , {0.3\|0.4, 0.4\|0.6} , {0.5\|0.5}}
z₄	{{0.5\|0.6} , {0.5\|0.5} , {0.4\|0.2, 0.6\|0.5, 0.7\|0.3}}	{{0.5\|0.5, 0.7\|0.5} , {0.5\|0.4} , {0.4\|0.6, 0.6\|0.3}

	C₁	C₂
z₁	{{0.5\|0.4, 0.7\|0.4} , {0.5\|0.6, 0.7\|0.4} , {0.6\|0.3}}	{{0.4\|0.6} , {0.6\|0.4, 0.7\|0.3} , {0.5\|0.3, 0.7\|0.6}}
z₂	{{0.5\|0.4, 0.6\|0.4} , {0.5\|0.4} , {0.4\|0.6, 0.6\|0.4}}	{{0.4\|0.5, 0.5\|0.3} , {0.5\|0.5, 0.6\|0.4} , {0.4\|0.6}}
z₃	{{0.3\|0.4, 0.5\|0.4} , {0.4\|0.4, 0.5\|0.6} , {0.3\|0.4}}	{{0.6\|0.4} , {0.5\|0.3} , {0.4\|0.5, 0.5\|0.4}}
z₄	{{0.4\|0.4, 0.5\|0.6} , {0.4\|0.4} , {0.5\|0.4, 0.6\|0.2}}	{{0.5\|0.6, 0.7\|0.4} , {0.4\|0.3} , {0.3\|0.4, 0.5\|0.5}
	C₃	C₄
z₁	{{0.4\|0.5} , {0.3\|0.4} , {0.3\|0.2, 0.5\|0.5, 0.6\|0.3}}	{{0.3\|0.4, 0.5\|0.6} , {0.7\|0.6} , {0.5\|0.4, 0.6\|0.6}}
z₂	{{0.7\|0.4} , {0.4\|0.6, 0.7\|0.3} , {0.4\|0.5, 0.5\|0.5}}	{{0.4\|0.6} , {0.4\|0.5} , {0.4\|0.3, 0.5\|0.5, 0.6\|0.2}}
z₃	{{0.4\|0.5, 0.6\|0.5} , {0.4\|0.5} , {0.6\|0.4, 0.8\|0.5}}	{{0.4\|0.4, 0.5\|0.4} , {0.5\|0.6} , {0.3\|0.5, 0.5\|0.5}}
z₄	{{0.6\|0.5, 0.8\|0.5} , {0.4\|0.6, 0.6\|0.4} , {0.5\|0.3}}	{{0.8\|0.5} , {0.4\|0.5, 0.6\|0.3, 0.8\|0.2} , {0.3\|0.4}

Generalized Shapley probability neutrosophic hesitant fuzzy Choquet aggregation operators and their applications

Abstract

Keywords

1 Introduction

2 Preliminaries

3.1 The comparison method of PNHFNs

4.1 The model based on the maximizing score deviation (MSD) method to determine the decision makers’ weights

5 An illustrative example and analysis

Table 5 Rankings results captured using different methods Method Sort of Results Optimal Alternative Worst Alternative TOPSIS - based QUALIFLEX method [30] z4 > z2 > z1 > z3 z 4 z3 GS - PNHFCA operator z2 > z4 > z3 > z1 z 2 z1 GS - PNHFCG operator z2 > z4 > z3 > z1 z 2 z1

7 Compliance with ethical standards

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