Normal intuitionistic fuzzy Bonferroni mean operators and their applications to multiple attribute group decision making

Abstract

Normal intuitionistic fuzzy numbers (NIFNs), which express their membership degree and non-membership degree as normal fuzzy numbers, can better character normal distribution phenomena existing in the real world. In this paper, we investigate the multiple attribute decision making problems with normal intuitionistic fuzzy information. Firstly, we introduce some operational laws, score function and accuracy function of NIFNs. Then, motivated by the ideal of Bonferroni mean, which can capture the interrelationship between input arguments, we develop some normal intuitionistic fuzzy aggregation operators, including the normal intuitionistic fuzzy Bonferroni mean (NIFBM) operator, the normal intuitionistic fuzzy weighted Bonferroni mean (NIFWBM) operator, the normal intuitionistic fuzzy geometric Bonferroni mean (NIFGBM) operator and the normal intuitionistic fuzzy geometric weighted Bonferroni mean (NIFGWBM) operator; and discuss some desirable properties of these operators. Furthermore, based on these operators, we propose a new approach for the multiple attribute group decision making under normal intuitionistic fuzzy environment. Finally, we give a numerical example to illustrate the effectiveness and feasibility of the developedapproach.

Keywords

Multiple attribute group decision making normal intuitionistic fuzzy numbers Bonferroni mean normal intuitionistic fuzzy Bonferroni mean operator normal intuitionistic fuzzy geometric Bonferroni mean operator

1 Introduction

Since the fuzzy set (FS) theory was originally introduced in 1965 by Zadeh [1], it has been widely used in various fields of modern society, such as MADM, information retrieval, machine learning, etc. However, FS theory only assigns to each element a membership degree. Hence, it is difficult for FS to comprehensively describe the uncertainty and vagueness of the objective elements in the real world [2]. In 1986, Atanassov [3] further proposed the notion of intuitionistic fuzzy set (IFS), which is characterized by a membership degree and a non-membership degree at the same time. In recent years, IFS has become a research hot topic in decision making fields and received more and more attention. Xu [4], Xu and Yager [5] developed some intuitionistic fuzzy information aggregation operators to solve the multiple attributes decision making problems, respectively.

In 1989, Based on the IFS theory, Atanassov and Gargov [6] further introduced the notion of interval-valued intuitionistic fuzzy (IVIF) sets and defined some basic operational laws. The prominent characteristics of IVIF is that both the membership degree and the non-membership degree are expressed as interval numbers. Yu [7] developed the generalized interval-valued intuitionistic fuzzy weighted geometric (GIIFWG) and generalized interval-valued intuitionistic fuzzy ordered weighted geometric (GIIFOWG) operators, and discussed their properties and special cased in detail. Liu and Yuan [8] originally proposed the concept of fuzzy number intuitionistic fuzzy set (FNIFS) in which the membership degree and the non-membership degree are both represented by triangular fuzzy numbers, and they defined some basic operations and discussed the relationships among the FNIFS, the IFS and the IVFS. In 2006, Shu et al. [9] proposed the definition of triangular intuitionistic fuzzy numbers (TIFNs) and some basic operations, and then they used them to implement the fault-tree analysis. Lin et al. [10] proposed some prioritized aggregation operators for aggregating fuzzy number intuitionistic fuzzy information, and developed an approach to deal with fuzzy number intuitionistic fuzzy MADM problems. Wang [11] originally introduced the concepts of trapezoidal intuitionistic fuzzy numbers (TrIFNs) and interval-valued trapezoidal intuitionistic fuzzy numbers (IVTrIFNs), which can be regarded as the extension of TIFNs. In 2010, based on the IFS and linguistic set, Wang and Li [12] defined intuitionistic linguistic set (ILS) as well as their operations, expected values, score function and accuracy function, and proposed some intuitionistic linguistic fuzzy aggregation operators. Liu and Zhang [13] proposed the notion of intuitionistic uncertain linguistic number (IULN) which is the generalization of IFNs and ULNs, as well as some operational laws and comparison criteria.

From the aforementioned research results, we can see that some scholars have extended IFS from different angles, and proposed IVIFNs, FNIFNs, TrIFNs, TIFNs, ILNs, IULNs, and so on. The existing research results have extended the domains of IFNs from discrete sets to continuous sets, and can effectively represent the information of decision making problems than traditional IFNs. In the reality of social life, it is worth noting that a lot of economic and social phenomena conform to normal distribution, such as random measurement error, average annual rainfall in a region, etc. In 1996, Yang and Ko [14] presented the concept of normal fuzzy numbers (NFNs) which are very suitable to depict these normal distribution information. In 2013, Wang et al. [15, 16] introduced the definition of normal intuitionistic fuzzy numbers (NIFNs) in which the membership degree and the non-membership degree are expressed as NFNs, and defined their operations, score function. Further, they proposed some normal intuitionistic fuzzy aggregation operators to aggregate normal intuitionistic fuzzy information. However, these normal intuitionistic fuzzy aggregation operators only emphasize the importance of each data or ordered position and they cannot describe the interrelationships of individual data. Therefore, it is necessary to pay more attention to this issue.

The Bonferroni mean (BM) was originally introduced by Bonferroni [17] in 1950. The desirable characteristic of the BM is its capability to capture the interrelationship between input arguments [18]. In 2009, Yager [19] further studied the BM operator and proposed some generalizations of BM which can effectively enhance its modeling capability. Recently, BM has received more and more attention from researchers. However, the drawback of the traditional BM operator is that it can only aggregate the input arguments which are expressed as crisp numbers. Some scholars studied the BM operator to accommodate the situations where the attribute values take the form of other types of domains, such as IFNs [18], IVIFNs [20], HFNs [21], uncertain linguistic variables [22], 2-tuple linguistic variables [23]. However, so far, no scholars study the BM operators to accommodate the normal intuitionistic fuzzy information.

From the above analysis, we noticed that there is no method proposed for aggregating NIFNs and considering the interrelationship between input normal intuitionistic fuzzy arguments at the same time. Therefore, it is necessary to pay more attention to this issue. In Section 2, we introduce some basic concepts and operational laws related to the NIFNs. In Section 3, we develop the normal intuitionistic fuzzy aggregation operators to aggregate NIFNs, and discuss some desirable properties of these developed operators. In Section 4, we extend the geometric Bonferroni mean with intuitionistic normal fuzzy information, and propose the normal intuitionistic fuzzy geometric Bonferroni mean operator and its weighted version. In Section 5, we utilize these operators to develop a multiple attribute decision procedure with normal intuitionistic fuzzy information, and give the decision making steps. In Section 6, a practical example is provided to verify the developed approach and to demonstrate its practicality and effectiveness. In Section 7, we conclude this paper and give some remarks

2 Preliminaries

In this section, we introduce some basic concepts and basic operational laws related to NFNs and NIFNs, and give the definition of Bonferroni mean operator.

2.1 Normal fuzzy numbers

Definition 1. [14] Let R is a real number set, and x, a, σ ∈ R. A normal fuzzy number (NFN) can be denoted as A = (a, σ) when its membership function is expressed as: $A (x) = e^{- {(\frac{(x - a)}{σ})}^{2}} (σ > 0)$ (1)

And the set of NFNs can be denoted as N.

Definition 2. [24] Let A = (α, σ) , B = (β, φ) be two NFNs, then the operations between A and B are defined as follows: $(1) λ A = λ (α, σ) = (λ α, λ σ), λ > 0$ (2) $(2) A + B = (α, σ) + (β, φ) = (α + β, σ + φ)$ (3)

2.2 Normal intuitionistic fuzzy numbers

Definition 3. [15] Let X be an ordinary finite nonempty set, A =< (a, σ) , u _A, υ _A > is a normal intuitionistic fuzzy number (NIFN), which is character by a membership function: $μ_{A} (x) = μ_{A} e^{- {(\frac{x - α}{σ})}^{2}}, x \in X$ (4) and a non-membership function: $υ_{A} (x) = 1 - (1 - υ_{A}) e^{- {(\frac{x - α}{σ})}^{2}}, x \in X$ (5) μ _A and υ _A indicate the membership degree and the non-membership degree of the element to the NFN (a, σ), respectively, satisfying 0 ≤ u _A ≤ 1, 0 ≤ υ _A ≤ 1, 0 ≤ u _A + υ _A ≤ 1. Compared with the classical IFNs, the universe of discourse of NIFNs expand from discrete to continuous, and NIFNs can effectively describe a large number of normal distribution in the socio-economic environment. Besides, compared with NFNs, NIFNs add the non-membership function which expresses the degree that the alternatives do not belong to (a, σ), as a result, NFNs can describe the uncertainty and fuzziness of objects more detailed and comprehensively.

Let A = 〈 (α, σ) , μ, υ〉, A ₁ = 〈 (α ₁, σ ₁) , μ ₁, υ ₁〉and A ₂ = 〈 (α ₂, σ ₂) , μ ₂, υ ₂〉 be any three NIFNs, then the operation laws can be defined as follows:

$\begin{matrix} (1) A_{1} \oplus A_{2} = 〈 (α_{1} + α_{2}, σ_{1} + σ_{2}), \\ 1 - (1 - μ_{1}) (1 - μ_{2}), υ_{1} υ_{2} 〉 \end{matrix}$ (6)

$\begin{matrix} (2) A_{1} \otimes A_{2} & = & 〈 (α_{1} α_{2}, α_{1} α_{2} \sqrt{\frac{σ_{1}^{2}}{α_{1}^{2}} + \frac{σ_{2}^{2}}{α_{2}^{2}}}), \\ μ_{1} μ_{2}, 1 - (1 - υ_{1}) (1 - υ_{2}) 〉 \end{matrix}$ (7) $(3) λ A = 〈 (λ α, λ σ), 1 - {(1 - u)}^{λ}, υ^{λ} 〉, λ \geq 0$ (8) $(4) A^{λ} = 〈 (α^{λ}, λ^{\frac{1}{2}} α^{λ - 1} σ), μ^{λ}, 1 - {(1 - υ)}^{λ} 〉, λ \geq 0$ (9)

Definition 4. [15] Let A =〈 (α, σ) , μ, υ 〉 be any NIFN, then its score function and accuracy function are respectively defined as follows: $S_{1} (A) = α (μ - υ), S_{2} (A) = σ (μ - υ) .$ (10) $H_{1} (A) = α (μ + υ), H_{2} (A) = σ (μ + υ) .$ (11)

To rank any two NIFNs A _i = 〈 (α _i, σ _i) , μ _i, υ _i〉(i = 1, 2), Wang and Li [15] introduced the following method:

If S ₁ (A ₁) > S ₁ (A ₂), then, A ₁> A ₂.

If S ₁ (A ₁) =S ₁ (A ₂) and H ₁ (A ₁) > H ₁ (A ₂), then, A ₁> A ₂.

If S ₁ (A ₁) = S ₁ (A ₂) and H ₁ (A ₁) = H ₁ (A ₂),

then:

If S ₂ (A ₁) < S ₂ (A ₂), then, A ₁> A ₂;

If S ₂ (A ₁) =S ₂ (A ₂) and H ₂ (A ₁) < H ₂ (A ₂),then, A ₁> A ₂;

If S ₂ (A ₁) =S ₂ (A ₂) and H ₂ (A ₁) =H ₂ (A ₂), then, A ₁=A ₂.

2.3 Bonferroni mean

The BM was originally introduced by Bonferroni [17], which was defined as follows.

Definition 5. [17] Let p, q ≥ 0, and a _i (i = 1, 2, ·· · , n) be a collection of nonnegative real numbers, then the aggregation function: ${BM}^{p, q} (α_{1}, α_{2}, \cdot \cdot \cdot, α_{n}) = {(\frac{1}{n (n - 1)} \sum_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} a_{i}^{p} a_{j}^{q})}^{\frac{1}{p + q}}$ (12) is called a Bonferroni mean(BM) operator.

However, the classical BM operator can only aggregate the input arguments which are described by nonnegative real numbers, but are invalid if the aggregation information is given in other forms, such as NIFNs. We shall extend the classical BM operator to accommodate the NIFNs and propose the following definitions.

3 Normal intuitionistic fuzzy Bonferroni mean operator and its weighted form

In this section, based on the Definition of NIFNs and BM, we propose the normal intuitionistic fuzzy Bonferroni mean (NIFBM) operator and its weighted form, and discuss their desirable properties.

Definition 6. Let A _i = 〈 (α _i, σ _i) , μ _i, σ _i 〉 (i = 1, 2, …, n) be a collection of NIFNs, and p, q ≥ 0, then we call:

$\begin{matrix} {NIFBM}^{p, q} (A_{1}, A_{2}, \cdot \cdot \cdot, A_{n}) \\ = {(\frac{1}{n (n - 1)} \sum_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} (A_{i}^{p} \otimes A_{j}^{q}))}^{\frac{1}{p + q}} \end{matrix}$ (13) the normal intuitionistic fuzzy Bonferroni mean (NIFBM) operator.

Theorem 1. Let A _i = 〈 (α _i, σ _i) , μ _i, σ _i 〉 (i = 1, 2, …, n) be a collection of NIFNs, and p, q ≥ 0, then the aggregated result by Equation (13) is also a NIFN, and ${NIFBM}^{p, q} (A_{1}, A_{2}, \cdot \cdot \cdot, A_{n}) = < (α, σ), μ, υ >$ (14) where $\begin{matrix} α = (\frac{1}{n (n - 1)} {\sum_{i, j = 1}}_{i \neq j}^{n} a_{i}^{p} a_{j}^{q}) \frac{1}{p + q} \\ σ = \frac{1}{\sqrt{p + q}} {(\frac{\overset{n}{\underset{i \neq j}{\sum_{i, j = 1}}} a_{i}^{p} a_{j}^{q}}{n (n - 1)})}^{\frac{1}{p + q} - 1} \\ (\frac{{\sum_{i, j = 1}}_{i \neq j}^{n} a_{i}^{p} a_{j}^{q} \sqrt{p {(\frac{σ_{i}}{a_{i}})}^{2} + q {(\frac{σ_{j}}{a_{j}})}^{2}}}{n (n - 1)}) \end{matrix}$ $\begin{matrix} μ = {(1 - {\prod_{i, j = 1}}_{i \neq j}^{n} {(1 - μ_{i}^{p} μ_{j}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}} \\ υ = 1 - {(1 - {\prod_{i, j = 1}}_{i \neq j}^{n} {(1 - {(1 - υ_{i})}^{p} {(1 - υ_{j})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}} \end{matrix}$

In the following, let us discuss some properties of the NIFBM ^p,q operator.

(1) Idempotency: if all A _i = A =〈 (α, σ) , μ, υ 〉 for all i (i = 1, 2, ·· · , n), then:

$\begin{matrix} {NIFBM}^{p, q} (A_{1}, A_{2}, \cdot \cdot \cdot, A_{n}) \\ = {NIFBM}^{p, q} (A, A, \cdot \cdot \cdot, A) = A \end{matrix}$ (15)

(2) Monotonicity: Let A _i =〈 (α _i, σ _i) , μ _i, υ _i 〉 and B _i = 〈 (α _j, σ _j) , μ _j, υ _j 〉 (i, j = 1, 2, ·· · , n) be two collections of NIFNs, if A _i ≤ B _i for all i, then ${NIFBM}^{p, q} (A_{1}, \cdot \cdot \cdot, A_{n}) \leq {NIFBM}^{p, q} (B_{1}, \cdot \cdot \cdot, B_{n})$ (16)

(3) Commutativity: Let B _i =〈 (α _j, σ _j) , μ _j, υ _j 〉 be any permutation of A _i =〈 (α _i, σ _i) , μ _i, υ _i 〉, then:

$\begin{matrix} {NIFBM}^{p, q} (A_{1}, A_{2} \cdot \cdot \cdot, A_{n}) \\ = {NIFBM}^{p, q} (B_{1}, B_{2}, \cdot \cdot \cdot, B_{n}) \end{matrix}$ (17)

(4) Boundedness: Let A _i =〈 (α _i, σ _i) , μ _i, υ _i 〉 be a set of NIFNs, and then $min_{i} (A_{i}) \leq {NIFBM}^{p, q} (A_{1}, A_{2}, \cdot \cdot \cdot, A_{n}) \leq \max_{i} (A_{i})$ (18)

From the aforementioned analysis, we can see that only the input arguments and their interrelationships are considered in the NIFBM ^p,q operator, but the importance of each input argument is not involved in the NIFBM ^p,q operator. Nevertheless, in many practical situations, the weights of the attributes should be taken into account. To solve this problem, we shall define the weighted form of the NIFBM ^p,q operator.

Definition 7. Let A _i = 〈 (α _i, σ _i) , μ _i, υ _i〉 (i = 1, 2, …, n) be a collection of NIFNs, and NIFWBM : Ω ⁿ → Ω, if: $\begin{matrix} {NIFWBM}_{w}^{p, q} (A_{1}, A_{2}, \cdot \cdot \cdot, A_{n}) \\ = {(\frac{1}{n (n - 1)} \underset{n}{\underset{i \neq j}{\sum_{i, j = 1}}} {(ω_{i} A_{i})}^{p} \otimes {(ω_{j} A_{j})}^{q})}^{\frac{1}{p + q}} \end{matrix}$ (19) where ω = (ω ₁, ·· · , ω _n) ^T is the weight vector of A _i (i = 1,2, … ,n), ω _i ∈ [0, 1], $\sum_{i = 1}^{n} ω_{i} = 1 .$ Then, ${NIFWBM}_{w}^{p, q}$ is called a normal intuitionistic fuzzy weighted Bonferroni mean operator.

Theorem 2. Let A _i = 〈 (α _i, σ _i) , μ _i, υ _i 〉 (i = 1, 2, …, n) be a collection of NIFNs, and p, q ≥ 0, then the aggregated result by Equation (19) is also a NIFN, and

$\begin{matrix} {NIFWBM}_{w}^{p, q} (A_{1}, A_{2}, \cdot \cdot \cdot, A_{n}) \\ = 〈 (α_{ω}, σ_{ω}), μ_{ω}, υ_{ω} 〉 \end{matrix}$ (20) where: $\begin{matrix} α_{ω} = {(\frac{1}{n (n - 1)} \underset{n}{\underset{i \neq j}{\sum_{i, j = 1}}} (ω_{i} a_{i})^{p} (ω_{j} a_{j})^{q})}^{\frac{1}{p + q}} \\ σ_{ω} = {(\frac{1}{p + q})}^{\frac{1}{2}} {(\frac{\overset{n}{\underset{i \neq j}{\sum_{i, j = 1}}} ω_{i} a_{i})^{p} (ω_{j} a_{j})^{q}}{n (n - 1)})}^{\frac{1}{p + q} - 1} \\ (\frac{{\sum_{i, j = 1}}_{i \neq j}^{n} (ω_{i} a_{i})^{p} {(ω_{j} a_{j})}^{q} \sqrt{p {(\frac{σ_{i}}{a_{i}})}^{2} + q {(\frac{σ_{j}}{a_{j}})}^{2}}}{n (n - 1)}) \end{matrix}$

$\begin{matrix} μ_{ω} = {(1 - \underset{n}{\underset{i \neq j}{\prod_{i, j = 1}}} {(1 - {(1 - {(1 - μ_{i})}^{ω_{i}})}^{p} {(1 - {(1 - μ_{j})}^{ω_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}} \end{matrix}$

$\begin{matrix} υ_{ω} = 1 - {(1 - \underset{n}{\underset{i \neq j}{\prod_{i, j = 1}}} {(1 - {(1 - υ_{i}^{ω_{i}})}^{p} {(1 - υ_{j}^{ω_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}} \end{matrix}$

Similar to Theorem 1, it can be proved by using mathematical induction on n that Equation (20)holds.

It is easy to prove that the ${NIFWBM}_{w}^{p, q}$ operator has the properties of monotonicity, but it has not the property of idempotency, commutativity and boundedness.

4 Normal intuitionistic fuzzy Bonferroni mean operator and its weighted form

In this section, we shall extend the GBM operator to accommodate the situations where the individual arguments are NIFNs, and propose the normal intuitionistic fuzzy geometric Bonferroni mean operator and its weighted form, and study their desirable properties.

Definition 8. [25] Let a _i (i = 1, 2, ·· · , n) be a collection of non-negative numbers, p, q ≥ 0 and p, q cannot take the value 0 at the same time, if ${GB}^{p, q} = \frac{1}{p + q} ({\prod_{i, j = 1}}_{i \neq j}^{n} {({pa}_{i} + {qa}_{j})}^{\frac{1}{n (n - 1)}})$ (21)

Then we call GBM ^p,q the geometric Bonferroni mean (GBM).

Definition 9. Let A _i (i = 1, 2, ·· · , n) be a collection of NIFNs, p, q ≥ 0 and p, q cannot take the value 0 at the same time, if ${NIFGBM}^{p, q} = \frac{1}{p + q} ({\prod_{i, j = 1}}_{i \neq j}^{n} {({pA}_{i} \oplus {qA}_{j})}^{\frac{1}{n (n - 1)}})$ (22) then we call NIFGBM ^p,q the normal intuitionistic fuzzy geometric Bonferroni mean (NIFGBM).

Theorem 3. Let A _i = 〈 (α _i, σ _i) , μ _i, υ _i〉 (i = 1, 2, …, n) be a collection of NIFNs, and p, q ≥ 0, then the aggregated value by Equation (22) is also a NIFN, and

$\begin{matrix} {NIFGBM}^{p, q} (A_{1}, A_{2}, \cdot \cdot \cdot, A_{n}) \\ = 〈 (α_{η}, σ_{η}), μ_{η}, υ_{η} 〉 \end{matrix}$ (23) where $\begin{matrix} α_{η} = \frac{1}{p + q} {\prod_{i, j = 1}}_{i \neq j}^{n} {(p α_{i} + q α_{j})}^{\frac{1}{n (n - 1)}} \\ σ_{η} = \frac{1}{p + q} ({\prod_{i, j = 1}}_{i \neq j}^{n} {(p α_{i} + q α_{j})}^{\frac{1}{n (n - 1)}} \\ \sqrt{\frac{{\sum_{i, j = 1}}_{i \neq j}^{n} {(\frac{p σ_{i} + q σ_{j}}{p α_{i} + q α_{j}})}^{2}}{n (n - 1)}}) \\ μ_{η} = 1 - {({\prod_{i, j = 1}}_{i \neq j}^{n} {(1 - {(1 - μ_{i})}^{p} {(1 - μ_{j})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}} \\ υ_{η} = {(1 - {\prod_{i, j = 1}}_{i \neq j}^{n} {(1 - υ_{i}^{p} υ_{j}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}} \end{matrix}$

Similar to Theorem 1, it can be proved by using mathematical induction on n that Equation (23)holds.

In the following, let us confider some desirable properties of the NIFGBM ^p,q operator.

(1) Idempotency: if all A _i = A =〈 (α, σ) , μ, υ 〉for all i (i = 1, 2, ·· · , n), then

$\begin{matrix} {NIFGBM}^{p, q} (A_{1}, A_{2} \cdot \cdot \cdot, A_{n}) \\ = {NIFGBM}^{p, q} (A, \cdot \cdot \cdot, A) = A \end{matrix}$ (24)

(2) Monotonicity: let A _i =〈 (α _i, σ _i) , μ _i, υ _i 〉 and $B_{i} = 〈 ({\tilde{α}}_{i}, {\tilde{σ}}_{i}), {\tilde{μ}}_{i}, {\tilde{υ}}_{i} 〉 (i = 1, 2, \cdot \cdot \cdot, n)$ be two collections of NIFNs, if A _i ≤ B _i for all i, then

$\begin{matrix} {NIFGBM}^{p, q} (A_{1}, A_{2}, \cdot \cdot \cdot, A_{n}) \\ \leq {NIFGBM}^{p, q} (B_{1}, B_{2}, \cdot \cdot \cdot, B_{n}) \end{matrix}$ (25)

(3) Commutativity: Let $B_{i} = 〈 ({\tilde{α}}_{i}, {\tilde{σ}}_{i}), {\tilde{μ}}_{i}, {\tilde{υ}}_{i} 〉$ be any permutation of A _i =〈 (α _i, σ _i) , μ _i, υ _i 〉, then

$\begin{matrix} {NIFGBM}^{p, q} (A_{1}, A_{2}, \cdot \cdot \cdot, A_{n}) \\ = {NIFGBM}^{p, q} (B_{1}, B_{2}, \cdot \cdot \cdot, B_{n}) \end{matrix}$ (26)

(4) Boundedness: Let A _i = 〈 (α _i, σ _i) , μ _i, υ _i〉 (i = 1, 2, …, n), be a set of NIFNs, and then:

$\begin{matrix} min_{i} (A_{i}) \leq {NIFGBM}^{p, q} (A_{1}, A_{2}, \cdot \cdot \cdot, A_{n}) \\ \leq \max_{i} (A_{i}) \end{matrix}$ (27)

Similarly, we define the weighted form of NIFGBM ^p,q in the following.

Definition 10. Let A _i = 〈 (α _i, σ _i) , μ _i, υ _i〉 (i = 1, 2, …, n) be a collection of NIFNs, and NIFWGBM : Ω ⁿ → Ω, if: $\begin{matrix} {NIFWGBM}_{w}^{p, q} (A_{1}, A_{2}, \cdot \cdot \cdot, A_{n}) = \\ \frac{1}{p + q} ({\prod_{i, j = 1}}_{i \neq j}^{n} {({pA}_{i}^{ω_{i}} \oplus {qA}_{j}^{ω_{j}})}^{\frac{1}{n (n - 1)}}) \end{matrix}$ (28) where ω = (ω ₁, ω ₂, ·· · , ω _n) ^T is the weight vector of A _i (i = 1, 2, ·· · , n), ω _i ∈ [0, 1], $\sum_{i = 1}^{n} ω_{i} = 1 .$ Then, NIFWGBM is called the normal intuitionistic fuzzy weighted geometric Bonferroni mean operator.

Theorem 4. Let A _i = 〈 (α _i, σ _i) , μ _i, υ _i〉 (i = 1, 2, …, n) be a collection of NIFNs, and p, q ≥ 0, ω = (ω ₁, ω ₂, ·· · , ω _n) ^T is the weight vector of A _i, ω _i ∈ [0, 1], $\sum_{i = 1}^{n} ω_{i} = 1$ , then the aggregated value by Equation (28) is also a NIFN, and

$\begin{matrix} {NIFWGBM}_{ω}^{p, q} (A_{1}, A_{2}, \cdot \cdot \cdot, A_{n}) \\ = 〈 (α_{ω}, σ_{ω}), μ_{ω}, υ_{ω} 〉 \end{matrix}$ (29) where $\begin{matrix} α_{ω} = \frac{1}{p + q} {\prod_{i, j = 1}}_{i \neq j}^{n} {(p α_{i}^{ω_{i}} + q α_{j}^{ω_{j}})}^{\frac{1}{n (n - 1)}} \\ σ_{ω} = \frac{1}{p + q} {\prod_{i, j = 1}}_{i \neq j}^{n} ({(p α_{i}^{ω_{i}} + q α_{j}^{ω_{j}})}^{\frac{1}{n (n - 1)}} \end{matrix}$ $\begin{matrix} \sqrt{{\sum_{i, j = 1}}_{i \neq j}^{n} \frac{1}{n (n - 1)} {(\frac{p ω_{i}^{\frac{1}{2}} α_{i}^{ω_{i} - 1} σ_{i} + q ω_{j}^{\frac{1}{2}} α_{j}^{ω_{j} - 1} σ_{j}}{p α_{i}^{ω_{i}} + q α_{j}^{ω_{j}}})}^{2}}) \end{matrix}$ $\begin{matrix} μ_{ω} = 1 - {(1 - {\prod_{i, j = 1}}_{i \neq j}^{n} {(1 - {(1 - μ_{i}^{ω_{i}})}^{p} {(1 - μ_{j}^{ω_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}} \end{matrix}$ $\begin{matrix} υ_{ω} = {(1 - {\prod_{i, j = 1}}_{i \neq j}^{n} {(1 - {(1 - {(1 - υ_{i})}^{ω_{i}})}^{p} {(1 - {(1 - υ_{j})}^{ω_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}} \end{matrix}$

Similar to Theorem 1, it can be proved by using mathematical induction on n that Equation (29) holds.

It is easy to prove that the ${NIFWGBM}_{ω}^{p, q}$ operator has the properties of monotonicity, but it has not the property of idempotency, commutativity and boundedness.

5 An approach to multiple attribute decision making with normal intuitionistic fuzzy information

In this section, we utilize the NIFWBM (or NIFWGBM) operator to develop an approach for multiple attribute group decision making under normal intuitionistic fuzzy environment.

A multiple attribute group decision making problem under the normal intuitionistic fuzzy environment can be represented as follows: Let A = {A ₁, A ₂, ⋯ , A _m} be a set of alternatives, and C ={ C ₁, C ₂, ⋯ , C _n } be the set of attributes, whose weight vector is ω = (ω ₁, ω ₂, ⋯ , ω _n) ^T, with ω _i ∈ [0, 1] and $\sum_{i = 1}^{n} ω_{i} = 1 (i = 1, 2, \dots, n)$ . Let D = {D ₁, D ₂, ⋯ , D _t} be a set of decision makers, and λ = (λ ₁, λ ₂, ⋯ , λ _t) is the decision makers weight, with λ _j ≥ 0 (j = 1,2, … ,t), $\sum_{j = 1}^{t} λ_{j} = 1$ . Suppose $E^{k} = [e_{ij}^{k}]_{m \times n}$ (k = 1,2, … ,t) is the decision matrix given by the decision maker D _k for alternatives A _i (i = 1, 2, …, m) with respect to the attribute C _j (j = 1, 2, ·· · , n), where $e_{ij}^{k}$ takes the form of NIFNs represented by $e_{ij}^{k} = 〈 (α_{ij}^{k}, σ_{ij}^{k}), μ_{ij}^{k}, υ_{ij}^{k} 〉$ .

In the following, we apply the NIFWBM (or NIFWGBM) operator to MAGDM with normal intuitionistic fuzzy information, which involves the following steps:

Step 1.Normalization. In general, there are mainly two different attribute types in multiple attribute decision making, they are benefit attributes (the bigger the performance values the better) and cost attributes (the smaller the performance values the better). Hence, we need to transform the decision matrix $E^{k} = [e_{ij}^{k}]_{m \times n} (k = 1, 2, \dots, t)$ $(e_{ij}^{k} =$ $〈 (α_{ij}^{k}, σ_{ij}^{k}), μ_{ij}^{k}, υ_{ij}^{k} 〉)$ into the normalization matrix $R_{Nor}^{k} = [r_{Norij}^{k}]_{m \times n}$ $(r_{Norij}^{k} = 〈 (α_{Norij}^{k}, σ_{Norij}^{k})$ , $μ_{Norij}^{k}, υ_{Norij}^{k} 〉)$ (i = 1,2, …m; j = 1,2,…, n) by Wang’s method [16], where

$\begin{matrix} r_{Norij}^{k} = 〈 (\frac{α_{ij}}{{max}_{i} (α_{ij})}, \frac{σ_{ij}}{{max}_{i} (σ_{ij})} \frac{σ_{ij}}{α_{ij}}), μ_{ij}^{k}, υ_{ij}^{k} 〉, \\ For benefit attribute of C_{j} \end{matrix}$ (30)

$\begin{matrix} r_{Norij}^{k} = 〈 (\frac{{min}_{i} (α_{ij})}{α_{ij}}, \frac{σ_{ij}}{{max}_{i} (σ_{ij})} \frac{σ_{ij}}{α_{ij}}), μ_{ij}^{k}, υ_{ij}^{k} 〉, \\ For cost attribute of C_{j} \end{matrix}$ (31)

Step 2. Utilize the NIFWBM operator Equation (20) or the NIFWGBM operator Equation (29) to aggregate all of the normal intuitionistic fuzzy decision matrices $R_{Nor}^{k} = [r_{Norij}^{k}]_{m \times n}$ given by all decision makers into the collective normal intuitionistic fuzzy decision matrix R _Col = [r _Colij] _m×n.

Step 3. Utilize the NIFWBM operator Equation (20) or the NIFWGBM operator Equation (29) to aggregate all of the preference values r _Colij in the ith line of R _Col, and get the collective overall preference value θ _i (i = 1, 2, ⋯ , m) of alternative A _i (i = 1, 2, ⋯ , m).

Step 4. Rank θ _i (i = 1, 2, ⋯ , m) in descending order using the ranking method of NIFNs defined in Definition 4, and obtain the priority of the alternatives A _i (i = 1, 2, ⋯ , m) according to θ _i (i = 1, 2, ⋯ , m).

Step 5. End.

6 Numerical example

In this subsection, we utilize a multiple attribute decision making problem adapted from [26] to illustrate the application of the proposed approach. Stock market investors face a real problem is how to choose the stock with the value of investment. Especially in China, with the continuous development of the stock market, investors pay more attention to the stock investment value analysis. Therefore, the establishment of an effective stock value evaluation method is of great significance. However, due to most of the financial indicators are approximately obey normal distribution, different types of IFNs, such as TIFNs, TrIFNs, ILNs, are not suitable to express the stock investment value information. However, the NIFNs can effectively describe the phenomenon of normal distribution and evaluate the stock investment value information. In the following, we utilize the above MAGDM with normal intuitionistic fuzzy information to evaluate the stock alternatives.

Suppose an investment institution want to select a best stock from several stocks coming from the same industry. After preliminary screening, four stocks (alternatives) denoted as {A ₁, A ₂, A ₃, A ₄}, are taken into consideration. In order to evaluate the stock investment value, we utilized the method proposed in [26] to extract the four key financial attributes described as follows(Suppose that the weight vector of four attributes is ω = (0 . 15, 0 . 25, 0 . 32, 0 . 28) ^T):

C₁: Earnings per share;

C₂: Net asset value per share;

C₃: Undistributed profits per share;

C₄: Equity ratio.

It is obvious that the above four attributes are all benefit attributes. The four possible alternatives {A ₁, A ₂, A ₃, A ₄} are evaluated by three decision makers DM _i (i = 1, 2, 3) (whose weight vector is λ = {0.4, 0.32, 0.28}) by using NIFNs under the above four attributes. In the following, we utilize the proposed method in Section 5 to select the best alternative.

6.1 Decision making steps

To obtain the ranking of four stock investment value, the following steps are involved:

Step 1. Data transformation. Given a set of time series values of a certain financial indicators of a company from 2006 to 2014, which are obtained from some related websites or stock software, denoted by {x ₁, x ₂, …, x _n}, it can be transformed into a NFN <α, σ> by using the following method: $α = \frac{\sum_{i = 1}^{n} x_{i}}{n}, σ = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} (x_{i} - α)^{2}}$ (32)

Then, by utilizing the Equation (32), three decision makers can evaluate the four possible alternatives under the above four attributes C _j (j = 1, 2, 3, 4). Three normal intuitionistic fuzzy decision matrices $E^{k} = [e_{ij}^{k}]_{4 \times 4}$ (k = 1, 2, 3) are shown in Tables 1–3.

Step 2. Normalization. By using the Equation (32), we can transform the normal intuitionistic fuzzy decision matrix into the normalized decision matrix $R_{Nor}^{k} = [r_{Norij}^{k}]_{4 \times 4}$ (see Tables 4–6).

Step 3. Utilize the NIFWGB ^p,q operator (without the loss of generality, here, let p = q = 1) and the decision information given in matrix to aggregate all the normalized decision matrices $R_{Nor}^{k} = [r_{Norij}^{k}]_{4 \times 4}$ (k = 1, 2, 3) into a collective decision matrix R _Col = [r _Colij] _4×4 (see Table 7).

Step 4. Utilize the NIFWGB ^p,q operator (suppose p = q = 1) to aggregate all the preferences values in the ith line of R _Col = [r ^Colij] _4×4 and derive the collective overall preference values θ _i (i = 1, 2, 3, 4) of alternative A _i (i = 1, 2, 3, 4). We can get: $\begin{matrix} θ_{1} = 〈 (0.5006, 0.0261), 0.9662, 0.0145 〉 \\ θ_{2} = 〈 (0.4881, 0.0229), 0.9672, 0.0125 〉 \\ θ_{3} = 〈 (0.5084, 0.0267), 0.9655, 0.0155 〉 \\ θ_{4} = 〈 (0.4921, 0.028), 0.9699, 0.0148 〉 \end{matrix}$

According to Definition 4, we calculate the scores of all the alternatives: $\begin{matrix} S_{1} (θ_{1}) = 0 . 4764, S_{1} (θ_{2}) = 0 . 4661 \\ S_{3} (θ_{3}) = 0 . 483, S_{4} (θ_{4}) = 0 . 47 \end{matrix}$

By using the ranking method of NIFNs defined in Definition 4, we can derive the θ _i (i = 1, 2, 3, 4) in descending order: $θ_{3} > θ_{1} > θ_{4} > θ_{2}$

Step 5. Rank all the alternatives A _i (i = 1, 2, 3, 4) as follows: $A_{3} > A_{1} > A_{4} > A_{2}$

Then, the best alternative is A ₃

6.2 Comparison analyses and discussions

To verify the feasibility and effectiveness of the proposed method, the proposed method are compared with the TOPSIS method based on TIFNs [27] and the normal intuitionistic fuzzy decision making method proposed in [28], respectively. Meanwhile, the advantages of the proposed method are also highlighted at the end of this section.

6.2.1 A comparison analysis with the existing MAGDM method using triangular intuitionistic fuzzy numbers (TIFNs)

To illustrate the advantages of the proposed method, the TOPSIS method with TIFNs proposed in [27] is applied to solve the aforementioned example by transforming the NIFNs into the TIFNs.

Firstly, the method proposed by Zhang et al. [15] are utilized to transform a NIFN 〈 (α, σ) , μ, υ〉 into a TIFN denoted by $〈 (\tilde{α}, \tilde{b}, \tilde{c}), \tilde{μ}, \tilde{υ} 〉$ , where $\tilde{α} = α - 3 σ$ , $\tilde{b} = α, \tilde{c} = α + 3 σ, \tilde{μ} = μ, υ = \tilde{υ}$ . Then, the decision information of each expert can be transformed into TIFNs denoted by r _ij =< (a _ij, b _ij, c _ij) , μ, υ >, and by using the formula ${\tilde{r}}_{ij} = < (\frac{{\tilde{a}}_{ij}}{{\tilde{c}}_{j}^{+}}, \frac{{\tilde{b}}_{ij}}{{\tilde{c}}_{j}^{+}}, \frac{{\tilde{c}}_{ij}}{{\tilde{c}}_{j}^{+}}), \tilde{μ}, \tilde{υ} > (j \in B)$ , where B denotes benefit attributes, we can normalize a TIFN decision matrix into a normalized decision matrix and obtain the collective decision matrix by using the TIFAA operator. Further, the collective overall preference values θ _i (i = 1, 2, 3, 4) of alternative A _i (i = 1, 2, 3, 4) can be derived as follows. $\begin{matrix} θ_{1} & = & < (- 0.2545, 0.2449, 0.7443), 0.5, 0.3 > \\ θ_{2} & = & < (- 0.1713, 0.2167, 0.7075), 0.55, 0.25 > \\ θ_{3} & = & < ((- 0.4196, 0.2724, 0.9645), 0.55, 0.25 > \\ θ_{4} & = & < (- 0.2235, 0.2269, 0.6074), 0.25, 0.3 > \end{matrix}$

By using the distance measure for TIFNs defined in [27], the distance between all the alternatives and the positive-ideal solution and the negative-ideal solution can be calculated as follows: $\begin{matrix} d_{1}^{+} = 0.07, d_{1}^{-} = 0.073, d_{2}^{+} = 0.06, d_{2}^{-} = 0.084 \\ d_{3}^{+} = 0.04, d_{3}^{-} = 0.103, d_{4}^{+} = 0.026, d_{4}^{-} = 0.118 \end{matrix}$

Hence, the relative closeness coefficient C _i (i = 1, 2, 3, 4) of each alternative A _i (i = 1, 2, 3, 4) to the positive-ideal solution can be calculated as follows: $C_{1} = 0.4906, C_{2} = 0.4167, C_{3} = 0.2809, C_{4} = 0.821$

According to the relative closeness coefficients C _i (i = 1, 2, 3, 4), the ranking is A ₃ ≻ A ₂ ≻ A ₁ ≻ A ₄. Thus, the most desirable alternative is A ₃. It is obvious that the ranking obtained by the proposed method are completely different from Zhang’s method proposed in [27], although the best alternative is still A ₃. The main reason is that TIFNs cannot well express normal distribution phenomenon, which widely exists in natural phenomena, social phenomena and production activities, just as mentioned in the Introduction part. However, NIFNs, as an extension of NFNs, are more suitable to depict the normal distribution information than other IFNs. Especially, according to the Central Limit Theorem, we know that a large number of random phenomena approximately obey normal distribution. Thus, NIFNs are of much more realistic senses than TIFNs, and the proposed method is more reasonable and reliable to aggregate the evaluation information conforming to normal distribution. In addition, this proposed method can capture the interrelationship between input normal intuitionistic fuzzy arguments at the same time. As a result, compared with TIFNs and the corresponding method proposed in [27], NIFNs along with the proposed method are more practical.

6.2.2 A comparison analysis with the existing MAGDM method using NIFNs

Wang and Li [28] proposed a method to aggregate attribute values by using normal intuitionistic fuzzy aggregation operators and rank the alternatives by comparing the relative closeness of alternatives to the positive-ideal solution and the negative-ideal solution. We can utilize the normal intuitionistic fuzzy arithmetic average (NIFAA) operator to aggregate all of the individual normal intuitionistic fuzzy decision matrices and obtain the collective overall preference values θ _i (i = 1, 2, 3, 4) of alternative A _i (i = 1, 2, 3, 4), and by using the distance equation defined in [28], we can calculate the distance from the positive ideal solution A ⁺ and the negative ideal solution A ^- for each alternative A _i (i = 1, 2, 3, 4), respectively. $\begin{matrix} d_{1}^{+} = 0.7828, d_{1}^{-} = 0.5427, \\ d_{2}^{+} = 0.7043, d_{2}^{-} = 0.5461 \\ d_{3}^{+} = 0.6236, d_{3}^{-} = 0.5685, \\ d_{4}^{+} = 0.8060, d_{4}^{-} = 0.5207 \end{matrix}$

Furthermore, the relative closeness coefficient C _i (i = 1, 2, 3, 4) of each alternative A _i (i = 1, 2, 3, 4) to the positive-ideal solution A ⁺ can be calculate as follows: $\begin{matrix} C_{1} = 0.5906, C_{2} = 0.5633, \\ C_{3} = 0.5231, C_{4} = 0.6075 \end{matrix}$

According to the relative closeness coefficients C _i (i = 1, 2, 3, 4), the ranking is A ₃ ≻ A ₂ ≻ A ₁ ≻ A ₄. Thus, the most desirable alternative is A ₃. Noted that the best alternative is still A ₃. However, the other alternatives ranking result is completely different from the proposed method. Although both the proposed method and the method proposed in [28] can deal with NIFNs, the proposed method can also capture the interrelationship between input arguments.

According to the abovementioned comparison analyses, the proposed method has the following advantages:

First, compared with other fuzzy numbers such as IFNs, TIFNs and so on, NIFNs can better express the phenomenon of normal distribution, which is a widespread phenomenon in practical life. Just as Li and Liu mentioned in [29] that the fuzzy concepts characterized by the normal membership function are more suitable to the inherent subjective nature of human think. Although the representation of NIFNs looks more complex and requires more computation than other IFNs, we can easily reduce the complexity and amount of computation by utilizing programming software, such as MATLAB software.

Second, the normal intuitionistic fuzzy Bonferroni mean operators and the proposed method can not only accommodate normal intuitionistic fuzzy information but also consider the interrelationship between the individual arguments, which are more important for decision makers to evaluate some complex decision-making problems under the normal intuitionistic fuzzy environment in which the interrelationship of individual arguments should be considered.

7 Conclusion

An NIFF is the generalization of an IFN and a NFN, which can better describe normal distribution phenomena in the real world. In this paper, we investigate the MAGDM problems in which input arguments take the form of NIFNs. Firstly, we introduced the definition, operational laws and score function of NIFNs. Then, motivated by the ideal of Bonferroni mean, we proposed some normal intuitionistic fuzzy aggregation operators. The desirable characteristic of these operators is that they can not only accommodate the normal intuitionistic fuzzy information but also consider the interrelationship between then input arguments. We study some desirable properties of these aggregation operators. Based on these operators, we further proposed an approach to multiple attribute decision making with normal intuitionistic fuzzy information. Finally, an illustrative example has been given to verify the proposed method and demonstrate its practicality and effectiveness by the comparison with other methods. In the future, we shall continue working in the extension and application of the normal intuitionistic fuzzy multiple attribute decision making to otherdomains.

Footnotes

Acknowledgments

The authors thank the editors and anonymous reviewers for their helpful comments and suggestions. This paper was supported by the National Natural Science Foundation of China (No. 71271124, 71471172), the Humanities and Social Sciences Research Project of Ministry of Education of China (No. 13YJC630104), the Shandong Provincial Natural Science Foundation (No. ZR2013GQ011).

References

Zadeh

1965

Fuzzy Sets

Information and Control 8 338 353

Šostaks

2011

Mathematics in the context of fuzzy sets: Basic ideas, concepts, and some remarks on the history and recent trends of development

Mathematical Modelling and Analysis 16 173 198

Atanassov

1986

Intuitionistic fuzzy sets

Fuzzy sets and Systems 20 87 96

2007

Intuitionistic fuzzy aggregation operators

IEEE Transations on Fuzzy Systems 15 1179 1187

Yager

2006

Some geometric aggregation operators based on intuitionistic fuzzy sets

International Journal of General Systems 35 417 433

Atanassov

Gargov

1989

Interval valued intuitionistic fuzzy sets

Fuzzy Sets and Systems 31 343 349

2013

Decision making based on generalized geometric operator under interval-valued intuitionistic fuzzy environment

Journal of Intelligent and Fuzzy Systems 25 471 480

Liu

Yuan

2007

Fuzzy number intuitionistic fuzzy set

Fuzzy Systems and Mathematics 21 88 91

Shu

Cheng

Chang

2006

Using intuitionistic fuzzy sets for fault-tree analysis on printed circuit board assembly

Microelectronics Reliability 46 2139 2148

10.

Lin

Zhao

Wei

2013

Fuzzy number intuitionistic fuzzy prioritized operators and their application to multiple attribute decision making

Journal of Intelligent and Fuzzy Systems 24 879 888

11.

Wang

2008

Overview on fuzzy multi-criteria decision-making approach

Control and Decision 23 601 606

12.

Liu

2012

A competency evaluation method of human resources managers based on multi-granularity linguistic variables and VIKOR method

Technological and Economic Development of Economy 18 696 710

13.

Liu

Zhang

2012

Intuitionistic uncertain linguistic aggregation operators and their application to group decision making

Systems Engineering—Theory & Practice 32 2704 2711

14.

Yang

1996

On a class of fuzzy c-numbers clustering procedures for fuzzy data

Fuzzy Sets and Systems 84 49 60

15.

Wang

Zhang

2012

Multi-criteria decision-making method based on induced intuitionistic normal fuzzy related aggregation operators

International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 20 559 578

16.

Wang

Zhou

Zhang

Chen

2014

Multi-criteria decision-making method based on normal intuitionistic fuzzy-induced generalized aggregation operator

TOP 1103 1122

17.

Bonferroni

1950

Sulle medie multiple di potenze

Bolletino Matematica Italiana 5 267 270

18.

Yager

2011

Intuitionistic fuzzy Bonferroni means

IEEE Transactions on Systems, Man and Cybernetics 41 568 578

19.

Yager

2009

On generalized Bonferroni mean operators for multi-criteria aggregation

International Journal of Approximate Reasoning 50 1279 1286

20.

Chen

2011

A multi-criteria decision making procedure based on interval-valued intuitionistic fuzzy bonferroni means

Journal of Systems Science and Systems Engineering 20 217 228

21.

Zhu

Xia

2012

Hesitant fuzzy geometric Bonferroni means

Information Sciences 205 72 85

22.

Wei

Zhao

Lin

Wang

2013

Uncertain linguistic Bonferroni mean operators and their application to multipleattribute decision making

Applied Mathematical Modelling 37 5277 5285

23.

Zhang

2014

2-tuple linguistic bonferroni mean operators and their application to multiple attribute group decision making

British Journal of Mathematics & Computer Science 4 1567 1614

24.

2001

Regression prediction for fuzzy time series

Appl Math A J Chin Univ(Ser. A) 16 455 461

25.

Xia

Zhu

2013

Geometric Bonferroni means with their application in multi-criteriadecision making

Knowledge-Based Systems 40 88 100

26.

Zhang

2005

A new method for stock investment value analysis based on the attibute synthetic evaluation system and vague sets

Chinese Journal of Management Science 13 15 21

27.

Zhang

Nan

2012

TOPSIS for MADM with triangular intuitionistic fuzzy numbers

Operations Research And Management Science 21 96 101

28.

Wang

2013

Multi-criteria decision-making method based on intuitionistic normal fuzzy aggregation operators

Systems Engineering—Theory & Practice 33 1501 1508

29.

Liu

2004

Study on the universality of the normal cloud model

Engineering Science 6 28 34