Multiple criteria decision making method based on neutrosophic hesitant fuzzy Heronian mean aggregation operators

Abstract

In this paper, we first extend the Heronian mean (HM), which can capture the correlations of the aggregated arguments, to aggregate the neutrosophic hesitant fuzzy set (NHFS) which is the generalization of neutrosophic sets (NSs) and hesitant fuzzy sets (HFSs). And then we propose some new Heronian mean operators for NHFS, including the neutrosophic hesitant fuzzy improved generalized weighted Heronian mean (NHFIGWHM) operator and the neutrosophic hesitant fuzzy improved generalized geometric weighted Heronian mean (NHFIGGWHM) operator. At the same time, some properties about these operators are investigated, such as idempotency, monotonicity, boundedness and so on. Furthermore, an approach for multiple criteria decision making (MCDM) based these operators is proposed. Finally, an example is given to illustrate the application and effectiveness of the developed approach.

Keywords

Neutrosophic hesitant fuzzy set Heronian mean operators multiple criteria decision making

1 Introduction

Multiple criteria decision making (MCDM) has been a focus in the politics, economic, military, management and the other fields in recent years. But in real life, the decision making information is often incomplete, indeterminate and inconsistent, and how to denote the decision information is the primary task of decision making. Under this circumstance, some methods for expressing the fuzzy information have been developed, such as fuzzy set (FS) [1], intuitionistic fuzzy set (IFS) [2, 3], and interval-valued intuitionistic fuzzy sets (IVIFSs) [4]. Although the theories of FSs and IFSs have been widely applied, they cannot handle the indeterminate and inconsistent information in different real-world problems. To solve this type of decision-making problems, Smarandache [5] proposed the neutrosophic set (NS) by adding an independent indeterminacy-membership function to IFS. Obviously, NS is the generalization of the present fuzzy sets including the fuzzy set, intuitionistic fuzzy set, paraconsistent set, paradoxist set and so on. So far, many great achievements have been made [6 –14]. For instance, Wang et al. [6] proposed a single valued neutrosophic set (SVNS) with some instances. Ye [7, 8] represented the correlation coefficient and the cross-entropy measure of SVNS. In addition, Ye [9] also proposed the concept of simplified neutrosophic sets (SNSs). Wang et al. [10] introduced the concept of interval neutrosophic sets (INSs) and provided the set-theoretic operators of INSs. In many cases, it is difficult for researchers to give consistent results or values because of differences in their knowledge backgrounds and thoughts. For example, the membership grade may be different values given by different persons even to the same element and set, one may want to give 0.5, while another person gives 0.6, or someone gives 0.7. To deal with the complicated cases, Torra and Narukawa [15], Torra [16] put forward a new concept of hesitant fuzzy sets (HFSs). As a generalization of fuzzy sets, HFSs use several possible values of an element to instead of the membership degree, which is an important method to represent indefinite information. Currently, the HFSs have attracted a great deal of attention [17 –20]. However, it cannot process indeterminate and inconsistent information, while the NS can easily denote uncertainty, incomplete and inconsistent information. Obviously, each of them has its advantages and disadvantages. So, we further propose the neutrosophic hesitant fuzzy sets (NHFSs) with combining the HFS and NS, which extend truth-membership degree, indeterminacy-membership degree, and falsity-membership degree of an element to a fixed set to HFS, i.e., which may have a few different values.

Heronian mean operator is a very important operator which considers the interrelationships of the aggregated arguments. Beliakov [21] had firstly proposed Heronian mean as an aggregation operator. Then Skora [22, 23] further proposed the generalized Heronian method. Liu [24] introduced Heronian mean operator and Heronian OWA operator respectively, which were similar to Bonferroni mean operator proposed by Yager [25] and Beliakov [26], then Heronian mean operator has been extended to intuitionistic uncertain linguistic information [27], neutrosophic uncertain linguistic information [28], interval-valued trapezoidal fuzzy information [29] and two-dimensional uncertain linguistic information [30]. Especially, Yu and Wu [31] compared Heronian mean operator with Bonferroni mean operator, the BM operator can reflect the relationship between any two criteria. However, it ignores the correlation between the criteria C_i and itself. Furthermore, the correlation between C_i and C_j (i ≠ j) is equal to the correlation between C_j and C_i (i ≠ j), which results in redundancy. Although the HM operator owns similar structure to BM, it can solve the stated two problems of the BM operator effectively.

Because HM has the some advantages comparing with BM mean, and it can take into account the interrelationships between the attributes in MCDM problems, while the NHFSs can also consider the advantages of HFS and NS, and can express the uncertain information, so it is necessary to extend the HM to NHFSs, and propose some decision making methods based on the extended HM which is the focus of this paper.

2 Preliminaries

2.1 The neutrosophic hesitant fuzzy set

Definition 1. [32]. Let X be a universe of discourse, with a generic element in X denoted by x. A single valued neutrosophic set A in X is denoted by: $A = {x (T_{A} (x), I_{A} (x), F_{A} (x)) | x \in X},$ (1) where the functions T_A (x), I_A (x) and F_A (x) denote the truth-membership, the indeterminacy-membership and the falsity-membership of the element x ∈ X to the set A respectively. For each point x in X, we have T_A (x) , I_A (x) , F_A (x) ∈ [0, 1], and 0 ≤ T_A (x) + I_A (x) + F_A (x) ≤3.

For convenience, we can use x = (T_x, I_x, F_x) to represent an element x in SVNS, and the element x is called a single valued neutrosophic number (SVNN).

Definition 2. [16]. Let X be a non-empty fixed set, a HFS A on X is defined in terms of a function h_A (x) that when applied to X returns a subset of [0,1], which can be expressed by the following mathematical symbol: $A = {〈 x, h_{A} (x) 〉 | x \in X},$ (2) where h_A (x) is a set of some values in [0, 1], representing the possible membership degrees of the element x ∈ X to A, we call h_A (x) a hesitant fuzzy element (HFE), denoted by h, which reads h = {γ|γ ∈ h} .

In the following, we will present the neutrosophic hesitant fuzzy set based on the combination of neutrosophic set with hesitant fuzzy set.

Definition 3. [33] Let X be a non-empty fixed set, a neutrosophic hesitant fuzzy set (NHFS) on X is expressed by: $N = {〈 x, \tilde{t} (x), \tilde{i} (x), \tilde{f} (x) 〉 x \in X},$ (3) where $\tilde{t} (x) = {\tilde{γ} | \tilde{γ} \in \tilde{t} (x)},$ $\tilde{i} (x) = {\tilde{δ} | \tilde{δ} \in \tilde{i} (x)},$ and $\tilde{f} (x) = {\tilde{η} | \tilde{η} \in \tilde{f} (x)}$ are three sets with some values in interval [0, 1], which represent the possible truth-membership hesitant degrees, indeterminacy-membership hesitant degrees, and falsity-membership hesitant degrees of the element x ∈ X to the set N, and satisfies the limits: $\tilde{γ} \in [0, 1]$ , $\tilde{δ} \in [0, 1]$ , $\tilde{η} \in [0, 1]$ and $0 \leq sup {\tilde{γ}}^{+} + sup {\tilde{δ}}^{+} + sup {\tilde{η}}^{+} \leq 3$ , where ${\tilde{γ}}^{+} = \cup_{\tilde{γ} \in \tilde{t} (x)} max {\tilde{γ}}$ , ${\tilde{δ}}^{+} = \cup_{\tilde{δ} \in \tilde{i} (x)} max {\tilde{δ}}$ , and ${\tilde{η}}^{+} = \cup_{\tilde{η} \in \tilde{f} (x)} max {\tilde{η}}$ for x ∈ X.

The $\tilde{n} = {\tilde{t} (x), \tilde{i} (x), \tilde{f} (x)}$ is called a neutrosophic hesitant fuzzy element (NHFE) which is denoted by the symbol $\tilde{n} = {\tilde{t}, \tilde{i}, \tilde{f}}$ .

Then, some basic operations of NHFEs are defined as follows:

Definition 4. [33] Let ${\tilde{n}}_{1} = {{\tilde{t}}_{1}, {\tilde{i}}_{1}, {\tilde{f}}_{1}}$ and ${\tilde{n}}_{2} = {{\tilde{t}}_{2}, {\tilde{i}}_{2}, {\tilde{f}}_{2}}$ be two NHFEs, and k > 0, the operations can be defined as follows: $(1) {\tilde{n}}_{1} \oplus {\tilde{n}}_{2} = {{\tilde{t}}_{1} \oplus {\tilde{t}}_{2}, {\tilde{i}}_{1} \otimes {\tilde{i}}_{2}, {\tilde{f}}_{1} \otimes {\tilde{f}}_{2}} = \underset{\begin{matrix} {\tilde{γ}}_{1} \in {\tilde{t}}_{1}, {\tilde{δ}}_{1} \in {\tilde{i}}_{1}, {\tilde{η}}_{1} \in {\tilde{f}}_{1}, \\ {\tilde{γ}}_{2} \in {\tilde{t}}_{2}, {\tilde{δ}}_{2} \in {\tilde{i}}_{2}, {\tilde{η}}_{2} \in {\tilde{f}}_{2} \end{matrix}}{\cup} {γ_{1} + γ_{2} - γ_{1} γ_{2}, δ_{1} δ_{2}, η_{1} η_{2}};$ (4) $(2) {\tilde{n}}_{1} \otimes {\tilde{n}}_{2} = {{\tilde{t}}_{1} \otimes {\tilde{t}}_{2}, {\tilde{i}}_{1} \oplus {\tilde{i}}_{2}, {\tilde{f}}_{1} \oplus {\tilde{f}}_{2}} = \underset{\begin{matrix} {\tilde{γ}}_{1} \in {\tilde{t}}_{1}, {\tilde{δ}}_{1} \in {\tilde{i}}_{1}, {\tilde{η}}_{1} \in {\tilde{f}}_{1}, \\ {\tilde{γ}}_{2} \in {\tilde{t}}_{2}, {\tilde{δ}}_{2} \in {\tilde{i}}_{2}, {\tilde{η}}_{2} \in {\tilde{f}}_{2} \end{matrix}}{\cup} {γ_{1} γ_{2}, δ_{1} + δ_{2} - δ_{1} δ_{2}, η_{1} + η_{2} - η_{1} η_{2}};$ (5) $(3) k {\tilde{n}}_{1} = \underset{{\tilde{γ}}_{1} \in {\tilde{t}}_{1}, {\tilde{δ}}_{1} \in {\tilde{i}}_{1}, {\tilde{η}}_{1} \in {\tilde{f}}_{1}}{\cup} {1 - {(1 - γ_{1})}^{k}, δ_{1}^{k}, η_{1}^{k}};$ (6) $(4) {\tilde{n}}_{1}^{k} = \underset{{\tilde{γ}}_{1} \in {\tilde{t}}_{1}, {\tilde{δ}}_{1} \in {\tilde{i}}_{1}, {\tilde{η}}_{1} \in {\tilde{f}}_{1}}{\cup} {γ_{1}^{k}, 1 - {(1 - δ_{1})}^{k}, 1 - (1 - η_{1})^{k}}$ (7)

Definition 5. For any an NHFE $\tilde{n}$ ,

$\begin{matrix} s (\tilde{n}) \\ = [\frac{1}{l} \sum_{i = 1}^{l} {\tilde{γ}}_{i} + \frac{1}{p} \sum_{i = 1}^{p} (1 - {\tilde{δ}}_{i}) + \frac{1}{q} \sum_{i = 1}^{q} (1 - {\tilde{η}}_{i})] / 3 \end{matrix}$ (8) is called the score function of $\tilde{n}$ , where l, p, q are the numbers of the values in $\tilde{γ}, \tilde{δ}, \tilde{η}$ , respectively. Obviously, $s (\tilde{n})$ is a value belonged to [0, 1].

Suppose ${\tilde{n}}_{1} = {{\tilde{t}}_{1}, {\tilde{i}}_{1}, {\tilde{f}}_{1}}$ and ${\tilde{n}}_{2} = {{\tilde{t}}_{2}, {\tilde{i}}_{2}, {\tilde{f}}_{2}}$ are any two NHFEs, the comparison method of NHFEs is expressed as follows [20, 21]:

If S (n₁) > S (n₂), then ${\tilde{n}}_{1} > {\tilde{n}}_{2}$ ;

If S (n₁) < S (n₂), then ${\tilde{n}}_{1} < {\tilde{n}}_{2}$ ;

If S (n₁) = S (n₂), then ${\tilde{n}}_{1} = {\tilde{n}}_{2}$ .

2.2 Some Heronian mean (HM) operators

Heronian mean (HM) operator, which could capture the interrelationships of the aggregated arguments, was defined as follows:

Definition 6. [34] A HM operator of dimension n is a mapping HM : Iⁿ → I. If: $HM (a_{1}, a_{2}, \dots, a_{n}) = \frac{2}{n (n + 1)} \sum_{i = 1}^{n} \sum_{j = i}^{n} \sqrt{a_{i} a_{j}},$ (9) where I = [0, 1], then HM is called the Heronian mean (HM) operator.

The generalized weighted Heronian mean (GWHM) operator proposed by Yu and Wu [31] is shown as follows:

Definition 7. [31] Let p, q ≥ 0, and x_i (i = 1, 2, ⋯ , n) be a set of nonnegative numbers, $W = {(w_{1}, w_{2}, \dots, w_{n})}^{T}$ is the weight vector of x_i (i = 1, 2, ⋯ , n), and satisfies w_i ≥ 0, $\sum_{i = 1}^{n} w_{i} = 1$ , if

$\begin{matrix} {GWHM}^{p, q} (a_{1}, a_{2}, \dots, a_{n}) \\ = {(\frac{2}{n (n + 1)} \sum_{i = 1}^{n} \sum_{j = i}^{n} (w_{i} a_{i})^{p} (w_{j} a_{j})^{q})}^{\frac{1}{p + q}} \end{matrix}$ (10) then GWHM^p,q is called a generalized weighted Heronian mean (GWHM) operator. Obviously, GWHM^p,q operator has not the property of idempotency. It seems to be counterintuitive. Liu [24, 30] proposed an improved generalized weighted Heronian mean (IGWHM) operator to overcome this shortcoming.

Definition 8. [24, 30] Let p, q ≥ 0, and a_i (i = 1, 2, ⋯ , n) be a set of nonnegative numbers, W = (w₁, w₂, ⋯ , w_n) ^T is the weight vector of a_i (i = 1, 2, ⋯ , n), and satisfies $W = {(w_{1}, w_{2}, \dots, w_{n})}^{T}$ , if

$\begin{matrix} {IGWHM}^{p, q} (a_{1}, a_{2}, \dots, a_{n}) \\ = \frac{{(\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j} a_{i}^{p} a_{j}^{q})}^{1 / p + q}}{{(\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j})}^{1 / p + q}} \end{matrix}$ (11) then IGWHM^p,q is called the improved generalized weighted Heronian mean (IGWHM) operator. The IGWHM^p,q operator has the properties of idempotency, monotonicity and boundedness [24, 30].

Further, Liu [24, 30] proposed the improved generalized geometric weighted Heronian mean (NGGWHM) operator.

Definition 9. [24, 30] Let p, q ≥ 0, and a_i (i = 1, 2, ⋯ , n) be a set of nonnegative numbers, W = (w₁, w₂, ⋯ , w_n) ^T is the weight vector of a_i (i = 1, 2, ⋯ , n), and satisfies $w_{i} \geq 0, \sum_{i = 1}^{n} w_{i} = 1$ , if

$\begin{matrix} {IGGWHM}^{p, q} (a_{1}, a_{2}, \dots, a_{n}) \\ = \frac{1}{p + q} \prod_{i = 1}^{n} {\prod_{j = i}^{n} ({pa}_{i} + {qa}_{j})}^{\frac{2 (n + 1 - i)}{n (n + 1)} \frac{w_{j}}{\sum_{k = i}^{n} w_{k}}}, \end{matrix}$ (12) then IGGWHM^p,q is called the improved generalized geometric weighted Heronian mean (IGGWHM) operator.

The IGGWHM has the properties of reducibility, idempotency, monotonicity and boundedness [24, 30].

3 Some Heronian mean operators based on neutrosophic hesitant fuzzy numbers

As mentioned above, because the IGWHM and IGGWHM operators can only aggregate the crisp numbers, and cannot aggregate the NHFSs. In this section, we will extend the IGWHM and IGGWHM operators to aggregate the NHFEs, and a neutrosophic hesitant fuzzy number improved generalized weighted Heronian mean (NHFIGWHM) operator and a neutrosophic hesitant fuzzy number improved generalized geometric weighted Heronian mean (NHFIGGWHM) operator are proposed as follows.

3.1 The NHFIGWHM operator

Definition 10. Let p, q ≥ 0 and ${\tilde{n}}_{j} = ({\tilde{t}}_{j}, {\tilde{i}}_{j}, {\tilde{f}}_{j})$ (j = 1, 2, ⋯ , n) be a set of the NHFEs with the weight vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_i ≥ 0, $\sum_{i = 1}^{n} w_{i} = 1$ , then a NHFIGWHM operator of dimension n is a mapping NHFIGWHM: Ωⁿ → Ω, and has

$\begin{matrix} {NHFIGWHM}^{p, q} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = {(\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}} \oplus_{i = 1}^{n} \oplus_{j = i}^{n} w_{i} w_{j} {\tilde{n}}_{i}^{p} \otimes {\tilde{n}}_{j}^{q})}^{1 / p + q}, \end{matrix}$ (13) where Ω is the set of all NHFEs.

Based on the operational rules of the NHFEs, we get the following theorems.

Theorem 1. Let p, q ≥ 0 and ${\tilde{n}}_{i} = (γ_{i}, δ_{i}, η_{i})$ (i = 1, 2, ⋯ , n) be the set of the NHFEs, then the aggregated result from Definition 10 is still a NHFE, and even

$\begin{matrix} {NHFIGWHM}^{p, q} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = ({(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{γ}}_{j} \in {\tilde{t}}_{j}}{\cup} {(1 - γ_{i}^{p} γ_{j}^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}, \\ 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}}{\cup} {(1 - (1 - δ_{i})^{p} (1 - δ_{j})^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}, \\ 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{η}}_{i} \in {\tilde{f}}_{i}, {\tilde{η}}_{j} \in {\tilde{f}}_{j}}{\cup} {(1 - (1 - η_{i})^{p} (1 - η_{j})^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}) \end{matrix}$ (14)

Proof. Since $\begin{array}{l} {\tilde{n}}_{i}^{p} = \underset{{\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{η}}_{i} \in {\tilde{f}}_{i}}{\cup} (γ_{i}^{p}, 1 - {(1 - δ_{i})}^{p}, 1 - {(1 - η_{i})}^{p}), \\ {\tilde{n}}_{j}^{q} = \underset{{\tilde{γ}}_{j} \in {\tilde{t}}_{j}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}, {\tilde{η}}_{j} \in {\tilde{f}}_{j}}{\cup} (γ_{j}^{q}, 1 - {(1 - δ_{j})}^{q}, 1 - {(1 - η_{j})}^{q}), \\ {\tilde{n}}_{i}^{p} \otimes {\tilde{n}}_{j}^{q} = \underset{\begin{matrix} {\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{η}}_{i} \in {\tilde{f}}_{i}, \\ {\tilde{γ}}_{j} \in {\tilde{t}}_{j}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}, {\tilde{η}}_{j} \in {\tilde{f}}_{j} \end{matrix}}{\cup} \\ (γ_{i}^{p} γ_{j}^{q}, 1 - {(1 - δ_{i})}^{p} {(1 - δ_{j})}^{q}, 1 - {(1 - η_{i})}^{p} {(1 - η_{j})}^{q}), \end{array}$ and $\begin{array}{l} w_{i} w_{j} {\tilde{n}}_{i}^{p} \otimes {\tilde{n}}_{j}^{q} = \underset{\begin{array}{l} {\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{η}}_{i} \in {\tilde{f}}_{i}, \\ {\tilde{γ}}_{j} \in {\tilde{t}}_{j}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}, {\tilde{η}}_{j} \in {\tilde{f}}_{j} \end{array}}{\cup} (1 - {(1 - γ_{i}^{p} γ_{j}^{q})}^{w_{i} w_{j}}, \\ {(1 - {(1 - δ_{i})}^{p} {(1 - δ_{j})}^{q})}^{w_{i} w_{j}}, {(1 - {(1 - η_{i})}^{p} {(1 - η_{j})}^{q})}^{w_{i} w_{j}}) \end{array}$ then $\begin{matrix} \oplus_{i = 1}^{n} \oplus_{j = i}^{n} (w_{i} w_{j} {\tilde{n}}_{i}^{p} \otimes {\tilde{n}}_{j}^{q}) = (1 - \prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{γ}}_{j} \in {\tilde{t}}_{j}}{\cup} {(1 - γ_{i}^{p} γ_{j}^{q})}^{w_{i} w_{j}}, \\ \prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}}{\cup} {(1 - (1 - δ_{i})^{p} (1 - δ_{j})^{q})}^{w_{i} w_{j}}, \\ \prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{η}}_{i} \in {\tilde{f}}_{i}, {\tilde{η}}_{j} \in {\tilde{f}}_{j}}{\cup} {(1 - (1 - η_{i})^{p} (1 - η_{j})^{q})}^{w_{i} w_{j}}), \end{matrix}$ further $\begin{matrix} \frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}} \oplus_{i = 1}^{n} \oplus_{j = i}^{n} (w_{i} w_{j} {\tilde{n}}_{i}^{p} \otimes {\tilde{n}}_{j}^{q}) \\ = (1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{γ}}_{j} \in {\tilde{t}}_{j}}{\cup} {(1 - γ_{i}^{p} γ_{j}^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}}, \\ {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}}{\cup} {(1 - (1 - δ_{i})^{p} (1 - δ_{j})^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}}, \\ {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{η}}_{i} \in {\tilde{f}}_{i}, {\tilde{η}}_{j} \in {\tilde{f}}_{j}}{\cup} {(1 - (1 - η_{i})^{p} (1 - η_{j})^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}}) \end{matrix}$

and $\begin{array}{l} {(\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}} \oplus_{i = 1}^{n} \oplus_{j = i}^{n} (w_{i} w_{j} {\tilde{n}}_{i}^{p} \otimes {\tilde{n}}_{j}^{q}))}^{1 / p + q} \\ = ({(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{γ}}_{j} \in {\tilde{t}}_{j}}{\cup} {(1 - γ_{i}^{p} γ_{j}^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}, \\ 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{\begin{matrix} {\tilde{δ}}_{i} \in {\tilde{i}}_{i}, \\ {\tilde{δ}}_{j} \in {\tilde{i}}_{j} \end{matrix}}{\cup} {(1 - {(1 - δ_{i})}^{p} {(1 - δ_{j})}^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}, \\ 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{\begin{matrix} {\tilde{η}}_{i} \in {\tilde{f}}_{i}, \\ {\tilde{η}}_{j} \in {\tilde{f}}_{j} \end{matrix}}{\cup} {(1 - {(1 - η_{i})}^{p} {(1 - η_{j})}^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}) \end{array}$

So $\begin{array}{l} N H F I G W H M^{p, q} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = ({(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{γ}}_{j} \in {\tilde{t}}_{j}}{\cup} {(1 - γ_{i}^{p} γ_{j}^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}, \\ 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{\begin{matrix} {\tilde{δ}}_{i} \in {\tilde{i}}_{i}, \\ {\tilde{δ}}_{j} \in {\tilde{i}}_{j} \end{matrix}}{\cup} {(1 - {(1 - δ_{i})}^{p} {(1 - δ_{j})}^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}, \\ 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{\begin{matrix} {\tilde{η}}_{i} \in {\tilde{f}}_{i}, \\ {\tilde{η}}_{j} \in {\tilde{f}}_{j} \end{matrix}}{\cup} {(1 - {(1 - η_{i})}^{p} {(1 - η_{j})}^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}) \end{array}$

Moreover, the NHFIGWHM operator also has the following properties:

(1) Theorem 2 (Idempotency). Let ${\tilde{n}}_{j} = \tilde{n} = {\tilde{t}, \tilde{i}, \tilde{f}}$ , where $\tilde{t} = γ$ , $\tilde{i} = δ$ and $\tilde{f} = η$ , then

$\begin{matrix} {NHFIGWHM}^{p, q} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = \tilde{n} = {\tilde{t}, \tilde{i}, \tilde{f}} . \end{matrix}$ (15)

Proof. Since ${\tilde{n}}_{j} = \tilde{n} = {\tilde{γ}, \tilde{δ}, \tilde{η}}$ , for all j, we have $\begin{array}{l} N H F I G W H M^{p, q} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = ({(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{\begin{array}{l} {\tilde{γ}}_{i} \in {\tilde{t}}_{i} \\ {\tilde{γ}}_{j} \in {\tilde{t}}_{j} \end{array}}{\cup} {(1 - γ_{}^{p} γ_{}^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}, \\ 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{\begin{array}{l} {\tilde{δ}}_{i} \in {\tilde{i}}_{i}, \\ {\tilde{δ}}_{j} \in {\tilde{i}}_{j} \end{array}}{\cup} {(1 - {(1 - δ_{})}^{p} {(1 - δ_{})}^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}, \\ 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{\begin{array}{l} {\tilde{η}}_{i} \in {\tilde{f}}_{i}, \\ {\tilde{η}}_{j} \in {\tilde{f}}_{j} \end{array}}{\cup} {(1 - {(1 - η_{})}^{p} {(1 - η_{})}^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}) \\ = ({(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{γ}}_{i} \in {\tilde{t}}_{i} {\tilde{γ}}_{j} \in {\tilde{t}}_{j}}{\cup} {(1 - γ_{}^{p + q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}, \\ 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{\begin{array}{l} {\tilde{δ}}_{i} \in {\tilde{i}}_{i}, \\ {\tilde{δ}}_{j} \in {\tilde{i}}_{j} \end{array}}{\cup} {(1 - {(1 - δ_{})}^{p + q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}, \\ 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{\begin{array}{l} {\tilde{η}}_{i} \in {\tilde{f}}_{i}, \\ {\tilde{η}}_{j} \in {\tilde{f}}_{j} \end{array}}{\cup} {(1 - {(1 - η_{})}^{p + q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}) \\ = \underset{\begin{array}{l} {\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{η}}_{i} \in {\tilde{f}}_{i}, \\ {\tilde{γ}}_{j} \in {\tilde{t}}_{j}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}, {\tilde{η}}_{j} \in {\tilde{f}}_{j} \end{array}}{\cup} ({(1 - {({(1 - γ_{}^{p + q})}^{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}, \\ 1 - {(1 - {({(1 - {(1 - δ_{})}^{p + q})}^{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}, \\ 1 - {(1 - {({(1 - {(1 - η_{})}^{p + q})}^{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}) \\ = \underset{\begin{array}{l} {\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{η}}_{i} \in {\tilde{f}}_{i}, \\ {\tilde{γ}}_{j} \in {\tilde{t}}_{j}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}, {\tilde{η}}_{j} \in {\tilde{f}}_{j} \end{array}}{\cup} ({(1 - (1 - γ_{}^{p + q}))}^{\frac{1}{p + q}}, \\ 1 - {(1 - (1 - {(1 - δ)}^{p + q}))}^{\frac{1}{p + q}}, 1 - {(1 - (1 - {(1 - η)}^{p + q}))}^{\frac{1}{p + q}}) \\ = \underset{\begin{array}{l} {\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{η}}_{i} \in {\tilde{f}}_{i}, \\ {\tilde{γ}}_{j} \in {\tilde{t}}_{j}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}, {\tilde{η}}_{j} \in {\tilde{f}}_{j} \end{array}}{\cup} ({(γ_{}^{p + q})}^{\frac{1}{p + q}}, \\ 1 - {({(1 - δ)}^{p + q})}^{\frac{1}{p + q}}, 1 - {({(1 - η)}^{p + q})}^{\frac{1}{p + q}}) \\ = \underset{{\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{η}}_{i} \in {\tilde{f}}_{i}, {\tilde{γ}}_{j} \in {\tilde{t}}_{j}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}, {\tilde{η}}_{j} \in {\tilde{f}}_{j}}{\cup} (γ, δ, η) = \tilde{n} = {\tilde{t}, \tilde{i}, \tilde{f}}, \end{array}$

(2) Theorem 3 (monotonicity). Let ${\tilde{n}}_{j}$ (j = 1, 2, ⋯ , n) and ${\tilde{n}}_{j}^{'} (j = 1, 2, \dots, n)$ be two sets of NHFEs, and suppose $γ_{j} \geq γ_{j}^{'}$ , $δ_{j} \leq δ_{j}^{'}$ and $η_{j} \leq η_{j}^{'}$ for all j, then

$\begin{matrix} {NHFIGWHM}^{p, q} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ \geq {NHFIGWHM}^{p, q} ({\tilde{n}}_{1}^{'}, {\tilde{n}}_{2}^{'}, \dots, {\tilde{n}}_{n}^{'}) . \end{matrix}$ (16)

Proof. (1) Since $γ_{j} \geq γ_{j}^{'}$ and p, q > 0, w_j ≥ 0 for all j, then $\begin{matrix} γ_{i}^{p} γ_{j}^{q} \geq γ_{i}^{' p} γ_{j}^{' q}, 1 - γ_{i}^{p} γ_{j}^{q} \leq 1 - γ_{i}^{' p} γ_{j}^{' q}, \\ \prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{γ}}_{j} \in {\tilde{t}}_{j}}{\cup} {(1 - γ_{i}^{p} γ_{j}^{q})}^{w_{i} w_{j}} \\ \leq \prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{γ}}_{j} \in {\tilde{t}}_{j}}{\cup} {(1 - γ_{i}^{' p} γ_{j}^{' q})}^{w_{i} w_{j}}, \\ 1 - \prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{γ}}_{j} \in {\tilde{t}}_{j}}{\cup} {(1 - γ_{i}^{p} γ_{j}^{q})}^{w_{i} w_{j}} \\ \geq 1 - \prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{γ}}_{j} \in {\tilde{t}}_{j}}{\cup} {(1 - γ_{i}^{' p} γ_{j}^{' q})}^{w_{i} w_{j}}, \end{matrix}$

so ${(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{γ}}_{j} \in {\tilde{t}}_{j}}{\cup} {(1 - γ_{i}^{p} γ_{j}^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}} \geq {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{γ}}_{j} \in {\tilde{t}}_{j}}{\cup} {(1 - γ_{i}^{' p} γ_{j}^{' q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}} .$

(2) Since $δ_{j} \leq δ_{j}^{'}$ and p, q > 0, w_j ≥ 0 for all j, then $\begin{matrix} (1 - δ_{i})^{p} \geq (1 - δ_{i}^{'})^{p} and \\ (1 - δ_{j})^{q} \geq (1 - δ_{j}^{'})^{q}, \end{matrix}$ then $\begin{matrix} (1 - δ_{i})^{p} (1 - δ_{j})^{q} \geq (1 - δ_{i}^{'})^{p} (1 - δ_{j}^{'})^{q}, \\ 1 - (1 - δ_{i})^{p} (1 - δ_{j})^{q} \leq 1 - (1 - δ_{i}^{'})^{p} (1 - δ_{j}^{'})^{q}, \end{matrix}$ $\begin{matrix} \prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}}{\cup} {(1 - (1 - δ_{i})^{p} (1 - δ_{j})^{q})}^{w_{i} w_{j}} \\ \leq \prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}}{\cup} {(1 - (1 - δ_{i}^{'})^{p} (1 - δ_{j}^{'})^{q})}^{w_{i} w_{j}}, \end{matrix}$

further, $\begin{matrix} {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}}{\cup} {(1 - (1 - δ_{i})^{p} (1 - δ_{j})^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}} \\ \leq {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}}{\cup} {(1 - (1 - δ_{i}^{'})^{p} (1 - δ_{j}^{'})^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}}, \\ 1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}}{\cup} {(1 - (1 - δ_{i})^{p} (1 - δ_{j})^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}} \\ \geq 1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}}{\cup} {(1 - (1 - δ_{i}^{'})^{p} (1 - δ_{j}^{'})^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}}, \end{matrix}$ and $\begin{array}{l} {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{\begin{matrix} {\tilde{δ}}_{i} \in {\tilde{i}}_{i}, \\ {\tilde{δ}}_{j} \in {\tilde{i}}_{j} \end{matrix}}{\cup} {(1 - {(1 - δ_{i})}^{p} {(1 - δ_{j})}^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}} \\ \geq {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}}{\cup} {(1 - {(1 - {δ^{'}}_{i})}^{p} {(1 - {δ^{'}}_{j})}^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}}, \end{array}$

so $\begin{array}{l} 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{\begin{matrix} {\tilde{δ}}_{i} \in {\tilde{i}}_{i}, \\ {\tilde{δ}}_{j} \in {\tilde{i}}_{j} \end{matrix}}{\cup} {(1 - {(1 - δ_{i})}^{p} {(1 - δ_{j})}^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}} \\ \leq 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}}{\cup} {(1 - {(1 - {δ^{'}}_{i})}^{p} {(1 - {δ^{'}}_{j})}^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}} . \end{array}$

(3) Similar to the (2), we have

$\begin{matrix} 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{η}}_{i} \in {\tilde{f}}_{i}, {\tilde{η}}_{j} \in {\tilde{f}}_{j}}{\cup} {(1 - (1 - η_{i})^{p} (1 - η_{j})^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}} \\ \leq 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{η}}_{i} \in {\tilde{f}}_{i}, {\tilde{η}}_{j} \in {\tilde{f}}_{j}}{\cup} {(1 - (1 - η_{i}^{'})^{p} (1 - η_{j}^{'})^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p + q}} . \end{matrix}$

According to (1–3), we can get $\begin{matrix} {NHFIGWHM}^{p, q} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ \geq {NHFIGWHM}^{p, q} ({\tilde{n}}_{1}^{'}, {\tilde{n}}_{2}^{'}, \dots, {\tilde{n}}_{n}^{'}) \end{matrix}$

(3) Theorem 4 (Boundedness). The NHFIGWHM operator locates between the max and min operators:

$\begin{matrix} min ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ \leq NHFIGWHM ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ \leq max ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \end{matrix}$ (17)

Proof. Let $m = min ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots {\tilde{n}}_{n})$ , $M = max ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots {\tilde{n}}_{n})$ , since $m \leq {\tilde{n}}_{j} \leq M$ , we can get the following result according to Theorem 3: $\begin{matrix} m & = & min ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ \leq & NHFIGWHM ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \end{matrix}$

and $\begin{matrix} NHFIGWHM ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ \leq max ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) = M \end{matrix}$

then $\begin{matrix} m & = & min ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ \leq & NHFIGWHM ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ \leq & max ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) = M . \end{matrix}$

In the following, we will discuss some specials of the NHFIGWHM operator in regard to the parameters p and q.

(1) When p = 0, then

$\begin{matrix} {NHFIGWHM}^{0, q} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = ({(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{γ}}_{j} \in {\tilde{t}}_{j}}{\cup} {(1 - γ_{j}^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{q}}, \\ 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{δ}}_{j} \in {\tilde{i}}_{j}}{\cup} {(1 - (1 - δ_{j})^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{q}}, \\ 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{η}}_{j} \in {\tilde{f}}_{j}}{\cup} {(1 - (1 - η_{j})^{q})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{q}}) . \end{matrix}$ (18)

(2) When q = 0, then

$\begin{matrix} {NHFIGWHM}^{p, 0} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = ({(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{γ}}_{i} \in {\tilde{t}}_{i}}{\cup} {(1 - γ_{i}^{p})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p}}, \\ 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{δ}}_{i} \in {\tilde{i}}_{i}}{\cup} {(1 - (1 - δ_{i})^{p})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p}}, \\ 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{η}}_{i} \in {\tilde{f}}_{i}}{\cup} {(1 - (1 - η_{i})^{p})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{p}}) . \end{matrix}$ (19)

(3) When p = q = 1, then $\begin{array}{l} N H F I G W H M^{1, 1} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = ({(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{γ}}_{j} \in {\tilde{t}}_{j}}{\cup} {(1 - γ_{i}^{} γ_{j}^{})}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{2}}, \\ 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{\begin{matrix} {\tilde{δ}}_{i} \in {\tilde{i}}_{i}, \\ {\tilde{δ}}_{j} \in {\tilde{i}}_{j} \end{matrix}}{\cup} {(1 - (1 - δ_{i}) (1 - δ_{j}))}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{2}}, \\ 1 - {(1 - {(\prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{\begin{matrix} {\tilde{η}}_{i} \in {\tilde{f}}_{i}, \\ {\tilde{η}}_{j} \in {\tilde{f}}_{j} \end{matrix}}{\cup} {(1 - (1 - η_{i}) (1 - η_{j}))}^{w_{i} w_{j}})}^{\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}}})}^{\frac{1}{2}}) . \end{array}$

3.2 The NHFIGGWHM operator

Definition 11. Let p, q ≥ 0 and ${\tilde{n}}_{j} = ({\tilde{t}}_{j}, {\tilde{i}}_{j}, {\tilde{f}}_{j})$ (j = 1, 2, ⋯ , n) be a set of the NHFEs with the weight vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_i ≥ 0, $\sum_{i = 1}^{n} w_{i} = 1$ , then a NHFIGGWHM operator of dimension n is a mapping NHFIGGWHM: Ωⁿ → Ω, and has

$\begin{matrix} {NHFIGGWHM}^{p, q} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = \frac{1}{p + q} \otimes_{i = 1}^{n} \otimes_{j = i}^{n} {(p {\tilde{n}}_{i} \oplus q {\tilde{n}}_{j})}^{\frac{2 (n + 1 - i)}{n (n + 1)} \frac{w_{j}}{\sum_{k = i}^{n} w_{k}}}, \end{matrix}$ (21) where Ω is the set of all the NHFEs.

Based on the operational rules of the NHFEs, we get the following theorems.

Theorem 5. Let p, q ≥ 0 and ${\tilde{n}}_{i} = (γ_{i}, δ_{i}, η_{i})$ (i = 1, 2, ⋯ , n) be the collection of NHFEs, then aggregated value by (21) is can be denoted as: $\begin{array}{l} N H F I G G W H M^{p, q} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = (1 - {(1 - \prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{\begin{matrix} {\tilde{γ}}_{i} \in {\tilde{t}}_{i}, \\ {\tilde{γ}}_{j} \in {\tilde{t}}_{j} \end{matrix}}{\cup} {(1 - {(1 - γ_{i})}^{p} {(1 - γ_{j})}^{q})}^{\frac{2 (n + 1 - i)}{n (n + 1)} \frac{w_{j}}{\sum_{k = i}^{n} w_{k}}})}^{\frac{1}{p + q}}, \\ {(1 - \prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}}{\cup} {(1 - δ_{i}^{p} δ_{j}^{q})}^{\frac{2 (n + 1 - i)}{n (n + 1)} \frac{w_{j}}{\sum_{k = i}^{n} w_{k}}})}^{\frac{1}{p + q}}, \\ {(1 - \prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{η}}_{i} \in {\tilde{f}}_{i}, {\tilde{η}}_{j} \in {\tilde{f}}_{j}}{\cup} {(1 - η_{i}^{p} η_{j}^{q})}^{\frac{2 (n + 1 - i)}{n (n + 1)} \frac{w_{j}}{\sum_{k = i}^{n} w_{k}}})}^{\frac{1}{p + q}}) \end{array}$

The proof of this theorem is similar to Theorem 1, it is omitted here.

Similar to Theorems 2–4, the NHFIGGWHM operator has the following properties.

(1) Theorem 6 (Reducibility).

$\begin{matrix} Let W = {(\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})}^{T} then \\ {NFHIGGWHM}^{p, q} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = {NFHGGHM}^{p, q} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) . \end{matrix}$ (23)

(2) Theorem 7 (Idempotency). Let ${\tilde{n}}_{j} = \tilde{n} = {\tilde{t}, \tilde{i}, \tilde{f}}$ , where $\tilde{t} = γ$ , $\tilde{i} = δ$ and $\tilde{f} = η$ , then ${NHFIGGWHM}^{p, q} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) = \tilde{n} .$ (24)

(3) Theorem 8 (monotonicity). Let ${\tilde{n}}_{j} (j = 1, 2, \dots, n)$ and ${\tilde{n}}_{j}^{'} (j = 1, 2, \dots, n)$ be two sets of NHFEs, and suppose $γ_{j} \geq γ_{j}^{'}$ , $δ_{j} \leq δ_{j}^{'}$ and $η_{j} \leq η_{j}^{'}$ for all j, then

$\begin{matrix} {NHFIGGWHM}^{p, q} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ \geq {NHFIGGWHM}^{p, q} ({\tilde{n}}_{1}^{'}, {\tilde{n}}_{2}^{'}, \dots, {\tilde{n}}_{n}^{'}) \end{matrix}$ (25)

(3) Theorem 9 (Boundedness). The NHFIGGWHM operator locates between the max and min operators:

$\begin{matrix} min ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ \leq NHFIGGWHM ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ \leq max ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \end{matrix}$ (26)

In the following, we will discuss some specials of the NHFIGGWHM in regard to the parameters p and q.

(1) When p = 0, then $\begin{matrix} {NHFIGGWHM}^{0, q} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = (1 - {(1 - \prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{γ}}_{j} \in {\tilde{t}}_{j}}{\cup} {(1 - (1 - γ_{j})^{q})}^{\frac{2 (n + 1 - i)}{n (n + 1)} \frac{w_{j}}{\sum_{k = i}^{n} w_{k}}})}^{\frac{1}{q}}, \\ {(1 - \prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{δ}}_{j} \in {\tilde{i}}_{j}}{\cup} {(1 - δ_{j}^{q})}^{\frac{2 (n + 1 - i)}{n (n + 1)} \frac{w_{j}}{\sum_{k = i}^{n} w_{k}}})}^{\frac{1}{q}}, \\ {(1 - \prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{η}}_{j} \in {\tilde{f}}_{j}}{\cup} {(1 - η_{j}^{q})}^{\frac{2 (n + 1 - i)}{n (n + 1)} \frac{w_{j}}{\sum_{k = i}^{n} w_{k}}})}^{\frac{1}{q}}) . \end{matrix}$ (27)

(2) When q = 0, then

$\begin{matrix} {NHFIGGWHM}^{p, 0} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = (1 - {(1 - \prod_{i = 1}^{n} \underset{{\tilde{γ}}_{i} \in {\tilde{t}}_{i}, {\tilde{γ}}_{j} \in {\tilde{t}}_{j}}{\cup} {(1 - (1 - γ_{i})^{p})}^{\frac{2 (n + 1 - i)}{n (n + 1)}})}^{\frac{1}{p}}, \\ {(1 - \prod_{i = 1}^{n} \underset{{\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}}{\cup} {(1 - δ_{i}^{p})}^{\frac{2 (n + 1 - i)}{n (n + 1)}})}^{\frac{1}{p}}, \\ {(1 - \prod_{i = 1}^{n} \underset{{\tilde{η}}_{i} \in {\tilde{f}}_{i}, {\tilde{η}}_{j} \in {\tilde{f}}_{j}}{\cup} {(1 - η_{i}^{p})}^{\frac{2 (n + 1 - i)}{n (n + 1)}})}^{\frac{1}{p}}) . \end{matrix}$ (28)

Obviously, when q = 0, NHFIGGWHM^p,0 does not have any relationship with w.

In addition, the parameters p and q don’t have the property of interchangeability.

(3) When p = q = 1, then $\begin{array}{l} N H F I G W G H M^{1, 1} ({\tilde{n}}_{1}, {\tilde{n}}_{2}, \dots, {\tilde{n}}_{n}) \\ = (1 - {(1 - \prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{\begin{matrix} {\tilde{γ}}_{i} \in {\tilde{t}}_{i}, \\ {\tilde{γ}}_{j} \in {\tilde{t}}_{j} \end{matrix}}{\cup} {(1 - (1 - γ_{i}) (1 - γ_{j}))}^{\frac{2 (n + 1 - i)}{n (n + 1)} \frac{w_{j}}{\sum_{k = i}^{n} w_{k}}})}^{\frac{1}{2}}, \\ {(1 - \prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{δ}}_{i} \in {\tilde{i}}_{i}, {\tilde{δ}}_{j} \in {\tilde{i}}_{j}}{\cup} {(1 - δ_{i} δ_{j})}^{\frac{2 (n + 1 - i)}{n (n + 1)} \frac{w_{j}}{\sum_{k = i}^{n} w_{k}}})}^{\frac{1}{2}}, \\ {(1 - \prod_{i = 1}^{n} \prod_{j = i}^{n} \underset{{\tilde{η}}_{i} \in {\tilde{f}}_{i}, {\tilde{η}}_{j} \in {\tilde{f}}_{j}}{\cup} {(1 - η_{i}^{} η_{j}^{})}^{\frac{2 (n + 1 - i)}{n (n + 1)} \frac{w_{j}}{\sum_{k = i}^{n} w_{k}}})}^{\frac{1}{2}}) . \end{array}$ (29)

4 A decision-making method based on the NHFIGWHM or NHFIGGWHM operators

In this section, we propose a MCDM method based on the NHFIGWHM or NHFIGGWHM operators.

For a multiple criteria decision making problem, let A = {A₁, A₂, ⋯ , A_m} be a collection of m alternatives, C = {C₁, C₂, ⋯ , C_m} be a collection of n criteria, which weight vector is w = (w₁, w₂, ⋯ w_n) ^T satisfying w_j ∈ [0, 1], $\sum_{j = 1}^{n} w_{j} = 1$ . Suppose that ${\tilde{n}}_{ij} = ({\tilde{t}}_{ij}, {\tilde{i}}_{ij}, {\tilde{f}}_{ij})$ is the evaluation value of the criteria C_j with respect to the alternative A_i which is expressed in the form of the NHFEs, where ${\tilde{t}}_{ij} = {{\tilde{γ}}_{ij} | {\tilde{γ}}_{ij} \in {\tilde{t}}_{ij} .}, {\tilde{i}}_{ij} = {{\tilde{δ}}_{ij} | {\tilde{δ}}_{ij} \in {\tilde{i}}_{ij} .}$ and ${\tilde{f}}_{ij} = {{\tilde{η}}_{ij} | {\tilde{η}}_{ij} \in {\tilde{f}}_{ij} .}$ are three collections of some values in interval [0, 1], which represent the possible truth-membership hesitant degrees, indeterminacy-membership hesitant degrees, and falsity-membership hesitant degrees, and satisfies following limits: $\tilde{γ} \in [0, 1]$ , $\tilde{δ} \in [0, 1]$ , $\tilde{η} \in [0, 1]$ , and $0 \leq sup {\tilde{γ}}^{+} + sup {\tilde{δ}}^{+} + sup {\tilde{η}}^{+} \leq 3$ , where ${\tilde{γ}}^{+} = \underset{{\tilde{γ}}_{ij} \in {\tilde{t}}_{ij}}{\cup} max {γ_{ij}}$ , ${\tilde{δ}}^{+} = \underset{{\tilde{δ}}_{ij} \in {\tilde{i}}_{ij}}{\cup} max {δ_{ij}},$ ${\tilde{η}}^{+} = \underset{{\tilde{η}}_{ij} \in {\tilde{f}}_{ij}}{\cup} max {η_{ij}}$ .

Then we need rank the alternatives.

The steps of the proposed method are shown as follows:

Step 1. Utilize the NHFIGWHM operator

$\begin{matrix} {\tilde{I}}_{k} & = & {NHFIGWHM}^{p, q} ({\tilde{n}}_{k 1}, {\tilde{n}}_{k 2}, \dots, {\tilde{n}}_{kn}) \\ = & {(\frac{1}{\sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j}} \sum_{i = 1}^{n} \sum_{j = i}^{n} w_{i} w_{j} {\tilde{n}}_{ki}^{p} {\tilde{n}}_{kj}^{q})}^{1 / - p + q} \end{matrix}$ (30) or NHFIGGWHM operator

$\begin{matrix} {\tilde{I}}_{k} & = & {NHFIGWHM}^{p, q} ({\tilde{n}}_{k 1}, {\tilde{n}}_{k 2}, \dots, {\tilde{n}}_{kn}) \\ = & \frac{1}{p + q} \otimes_{i = 1}^{n} \otimes_{j = i}^{n} {(p {\tilde{n}}_{ki} \oplus q {\tilde{n}}_{kj})}^{\frac{2 (n + 1 - i)}{n (n + 1)} \frac{w_{j}}{\sum_{x = i}^{n} w_{x}}} \end{matrix}$ (31) to derive the overall preference values ${\tilde{I}}_{k} (k = 1, 2, \dots, m)$ .

Step 2. Utilize the score function defined by (8) to calculate the ranking values. We can use an average method to improve the score function as follows.

$\begin{matrix} s_{i} = s ({\tilde{I}}_{i}) \\ = [\frac{1}{l_{i}} \sum_{i = 1}^{l_{i}} γ_{i} + \frac{1}{p_{i}} \sum_{i = 1}^{p_{i}} (1 - δ_{i}) + \frac{1}{q_{i}} \sum_{i = 1}^{q_{i}} (1 - η_{i})] / 3 \end{matrix}$ (32) where l_i, p_i and q_i are the numbers of values in ${\tilde{t}}_{i}, {\tilde{i}}_{i}, {\tilde{f}}_{i}$ .

Step 3. Rank all the alternatives A_i (i = 1, 2, ⋯ , m) and obtain the best one(s).

Step 4. End.

5 A numerical example

In this part, we cite an example [24, 30] to illustrate the application of NHFIGWHM or NHFIGGWHM operators in MCDM problems.

Suppose that one investment company intends to select an enterprise from the following four alternatives to invest. The four enterprises are marked by A_i (i = 1, 2, 3.4), and they are measured by three criteria: (1) C₁(the risk index); (2) C₂(the growth index); (3) C₃(environmental impact index), and the evaluation values are represented by NHFEs and their weight is w = (0.35, 0.25, 0.4) ^T. The decision matrix R is shown in the Table 1.

5.1 The evaluation steps by the NHFIGWHM operator

Step 1. Calculate the comprehensive evaluation values I_i (i = 1, 2, 3, 4) of each alternative by the NHFIGWHM operators based on Equation (30) (Suppose p = q = 1): $\begin{matrix} {\tilde{I}}_{1} & = & {{0.3853, 0.4134, 0.4498}, {0.2211}, \\ {0.4135, 0.4765}} \\ {\tilde{I}}_{2} & = & {{0.7}, {0.1731}, {0.2211, 0.2553}}; \\ {\tilde{I}}_{3} & = & {{0.5360, 0.6}, {0.2849, 0.3330}, \\ {0.3217}}; \\ {\tilde{I}}_{4} & = & {{0.6694}, {0.1662}, {0.2, 0.2385}} . \end{matrix}$

Step 2. Calculate the score function according to Equation (32): $\begin{matrix} s_{1} & = & 0.5834; s_{2} = 0.7629; s_{3} = 0.6351; \\ s_{4} & = & 0.7613 . \end{matrix}$

So, s₂ > s₄ > s₃ > s₁.

Step 3. Rank all the alternatives A_j (j = 1, 2, 3, 4) in accordance with the values of s_j (j = 1, 2, 3, 4), we can get A₂ ≻ A₄ ≻ A₃ ≻ A₁ and A₂ is the best option.

Step 4. End.

Similarly, we use the NHFIGGWHM operator to solve this problem, when p = q = 1, we can get the same ranking result A₂ ≻ A₄ ≻ A₃ ≻ A₁.

5.2 The influence on the decision making of this example by changing the parameters p, q

When the parameter p, q adopt different values, we may get different ranking results, and the corresponding score values with the ranking of alternatives are shown in Table 2.

From the Table 2, we know different parameters p, q may influence the ranking results in the NHFIGWHM operator. If we let the parameter p fixed, different values and rankings of the alternatives can be obtained as the parameter q is changed, we can find that:

(1) when p = 0, and

when q ∈ [0, 10.16), the ranking of the four alternatives is A₂ ≻ A₄ ≻ A₃ ≻ A₁;

when q ∈ [10.16, 99.49), the ranking of the four alternatives is A₄ ≻ A₂ ≻ A₃ ≻ A₁;

when q ∈ [99.49, +), the ranking of the four alternatives is A₄ ≻ A₂ ≻ A₁ ≻ A₃.

(2) when p = 1, and

when q ∈ [0, 0.89), the ranking of the four alternatives is A₄ ≻ A₂ ≻ A₃ ≻ A₁;

when q ∈ [0.89, 7.88), the ranking of the four alternatives is A₂ ≻ A₄ ≻ A₃ ≻ A₁;

when q ∈ [7.88, +), the ranking of the four alternatives is A₄ ≻ A₂ ≻ A₃ ≻ A₁.

If we let the parameter q fixed, different values and rankings of the alternatives can be obtained as the parameter p is changed, we can find that:

(3) when q = 0, and

when p ∈ [0, 99.49), the ranking of the four alternatives is A₄ ≻ A₂ ≻ A₃ ≻ A₁;

when q = 0, p ∈ [99.49, +), the ranking of the four alternatives is A₄ ≻ A₂ ≻ A₁ ≻ A₃.

(4) when q = 1, and when p ∈ [0, 1.11), the ranking of the four alternatives is A₂ ≻ A₄ ≻ A₃ ≻ A₁;

when p ∈ [1.11, 99.49), the ranking of the four alternatives is A₄ ≻ A₂ ≻ A₃ ≻ A₁;

when p ∈ [99.49, +), the ranking of the four alternatives is A₄ ≻ A₂ ≻ A₁ ≻ A₃.

From above calculation, we find that the best alternative is A₄ or A₂ and their score function values are very close, so the decision makers could assign any values of the parameters p, q. However, in practical applications, we generally adopt the values of the two parameters as p = q = 1, which is not only easy and intuitive but also fully capture the correlations between criteria.

5.3 Comparison analysis

In order to verify the effectiveness of the proposed approach, a comparison analysis is given based on the same illustrative example here. We can compare with two existing methods. One is the method that was introduced by Şahin and Liu [36], which established correlation coefficients under the neutrosophic hesitant fuzzy environment. The ranking of four alternatives is A₂ ≻ A₄ ≻ A₃ ≻ A₁, and the best alternative is A₂, while the worst alternative is always A₁. Another method was introduced by Liu [35], which is based on the generalized Hybrid weighted average operator of interval Neutrosophic hesitant set. The ranking of four alternatives is also A₂ ≻ A₄ ≻ A₃ ≻ A₁, and the best alternative is A₂, while the worst alternative is always A₁. It can be seen that the results of the proposed approach is same as methods of Şahin and Liu [36] and Liu [35]. Therefore, the proposed method is feasible and effective.

6 Conclusion

The HFS can make the membership function of one element to a set expressed by several possible values, however, it cannot deal with the indeterminate and inconsistent information, and the NS can easily express incomplete, uncertainty, and inconsistent information. Therefore, we combine the HFS and NS to propose the neutrosophic hesitant fuzzy sets (NHFSs), which includes truth-membership degree, indeterminacy-membership degree, and falsity-membership degree of an element to a fixed set to HFS. Then, we propose a score function and give a comparison method of NHFSs, and extend the Heronian mean operators to the NHFEs, including neutrosophic hesitant fuzzy improved generalized weighted Heronian mean (NHFIGWHM) operator, and neutrosophic hesitant fuzzy improved generalized weighted geometric Heronian mean (NHFIGGWHM) operator, and discuss their properties. Furthermore, we give the decision making method for the MCDM problems with neutrosophic hesitant fuzzy information, and give the specific decision steps. A significant feature of the proposed method is that it can deal with many kinds of fuzzy information. In the future, we will apply the improved operators to broader application fields, such as supply chain management, science-technology assessment and the performance evaluation.

Footnotes

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Nos. 71471172 and 71271124), the Special Funds of Taishan Scholars Project of Shandong Province, National Soft Science Project of China (2014GXQ4D192), and Shandong Provincial Social Science Planning Project (No. 15BGLJ06).

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