Intuitionistic fuzzy normalized weighted geometric Bonferroni means with reducibility and boundedness and application in decision making 1

Abstract

Bonferroni mean (BM) is an important aggregation operator in decision making. The desirable characteristic of the BM is that it can capture the interrelationship between the aggregation arguments or the individual attributes. The optimized weighted geometric Bonferroni mean (OWGBM) and the generalized optimized weighted geometric Bonferroni mean (GOWGBM) proposed by Jin et al in 2016 are the extensions of the BM. However, the OWGBM and the GOWGBM have neither the reducibility nor the boundedness, which will lead to the illogical and unreasonable aggregation results and might make the wrong decision. To overcome these existing drawbacks, based on the normalized weighted Bonferroni mean (NWBM) and the GOWGBM, we propose the normalized weighted geometric Bonferroni mean (NWGBM) and the generalized normalized weighted geometric Bonferroni mean (GNWGBM), which can not only capture the interrelationship between the aggregation arguments, but also have the reducibility and the boundedness. Further, we extend the NWGBM and the GNWGBM to the intuitionistic fuzzy decision environment respectively, and develop the intuitionistic fuzzy normalized weighted geometric Bonferroni mean (IFNWGBM) and the generalized intuitionistic fuzzy normalized weighted geometric Bonferroni mean (GIFNWGBM). Subsequently, we prove some properties of these operators. Moreover, we present a new intuitionistic fuzzy decision method based on the IFNWGBM and the GIFNWGBM. Two application examples and comparisons with other existing methods are used to verify the validity of the proposed method.

Keywords

Intuitionistic fuzzy number Bonferroni mean geometric Bonferroni mean decision making

1 Introduction

The Bonferroni mean (BM) originally was defined by Bonferroni [1] and then was generalized by Yager [2]. The prominent characteristic of the BM is that it can reflect the interrelationship of the individual attribute but not consider the importance of each attribute. In recent years, the BM has been extended and generalized to many other forms, such as the generalized Bonferroni mean (GBM) [3], the uncertain Bonferroni mean (UBM) [4], the intuitionistic fuzzy Bonferroni mean (IFBM) and the weighted intuitionistic fuzzy Bonferroni mean (IFWBM) [5], the intuitionistic fuzzy geometric Bonferroni mean (GBM) and the weighted intuitionistic fuzzy geometric Bonferroni mean (IFWGBM) [6,7], the interval-valued intuitionistic fuzzy Bonferroni mean (IVIFBM) and the weighted interval-valued intuitionistic fuzzy Bonferroni mean (IVIFWBM) [8], the generalized intuitionistic fuzzy weighted Bonferroni mean (GIFWBM) and the generalized intuitionistic fuzzy weighted geometric Bonferroni mean (GIFWGBM) [9]. However, according to the research by Zhou [10], the BM, the UBM, the IFBM and the IFGBM ignore the weight vector of the aggregation arguments. Although the IFWBM, the IFWGBM, the IVIFWBM and the GIFWBM considered this issue, we cannot respectively obtain the IFBM, the IFGBM, the IVIFBM and the GIFBM, when all the weights of the aggregated arguments are the same, that is, these operators have not reducibility. To overcome this problem, Xia [9] proposed the revised BM and GWBM, which can just reflect the correlation-ship between the individual attribute and all attributes, but change the structure of the BM. Therefore, Zhou [10] proposed the normalized weighted Bonferroni mean (NWBM) and the intuitionistic fuzzy normalized weighted Bonferroni mean (IFNWBM), which can deal with the above issue successfully. Afterward, Sun [11] introduced the hesitant fuzzy normalized weighted geometric Bonferroni mean (HFNWGBM). Since then, many scholars extended the NWBM into various decision making environments, and proposed some different types of aggregation operators including uncertain linguistic BM [12], intuitionistic uncertain linguistic weighted BM [13], linguistic neutrosophic number BM [14], bipolar neutrosophic Frank Choquet BM [15], Dombi-normalized weighted BM [16], Pythagorean fuzzy geometric BM [17], etc.

Based on the NWBM [10] and the GBM [3], Jin [18] developed the optimized weighted geometric Bonferroni mean (OWGBM) and the generalized optimized weighted geometric Bonferroni mean (GOWGBM) in 2016, whose characteristics were to reflect the preference and interrelationship of the aggregation arguments. Then, Jin extended these operators to intuitionistic fuzzy decision environments. However, by scrutinizing the algebraic structures of the OWGBM and GOWGBM proposed by Jin [18], there are still three issues which remain to be further addressed.

(1) The reducibility is one of the most important properties of the weighted aggregation operator. However, the OWGBM and GOWGBM proposed by Jin [18] have no the reducibility, which illustrates that when all the weights of the aggregation arguments are the same, the OWGBM and GOWGBM are not reduced to the GBM and the GGBM. Thus, the OWGBM and the GOWGBM are not extensions of the GBM and the GGBM.

(2) The boundedness is one of the most important properties of mean type aggregation operator. The OWGBM and the GOWGBM proposed by Jin [18] have no the boundedness, which implies that the aggregation result obtained by the OWGBM and the GOWGBM is out of the range of the aggregated arguments, and is not consistent with the fact that BM is a mean type aggregation operator.

(3) Unfortunately, the IFOWGBM and the GIFOWGBM proposed by Jin [18] are extensions of the OWGBM and the GOWGBM, which inevitably have neither the reducibility nor the boundedness, and the decision results based them are also unreasonable.

To bridge the abovementioned drawbacks, which are exactly the original motivation of this paper, we will propose the normalized weighted geometric Bonferroni mean (NWGBM) and the generalized normalized weighted geometric Bonferroni mean (GNWGBM), which have better algebraic structures than the OWGBM and GOWGBM proposed by Jin [18], and have these following properties, such as reducibility, idempotency, monotonicity and boundedness. Furthermore, we will extend the NWGBM and the GNWGBM to intuitionistic fuzzy decision environment, and propose the intuitionistic fuzzy normalized weighted geometric Bonferroni mean (IFNWGBM) and the generalized intuitionistic fuzzy normalized weighted geometric Bonferroni mean (GIFNWGBM). And we also give the mathematical expressions of these operators and study their desirable properties. Then, we will proposed the decision method based on the IFNWGBM and the GIFNWGBM and application examples are used to illustrate the valid of the proposed method.

The main contributions of this paper can be summarized as follows:

(1) The reducibility and boundedness of the OWGBM and GOWGBM proposed by Jin [18] are analyzed. Let the weights of attributes be the same, we found that the OWGBM and the GOWGBM have no the reducibility. Meanwhile, it is also found from a simple computational example that the OWGBM and the GOWGBM have no the boundedness.

(2) We propose the normal weighted geometric Bonferroni mean and generalized normal weighted geometric Bonferroni mean with reducibility and boundedness, which have better algebraic structures, fix the drawbacks of the OWGBM and the GOWGBM mentioned above and thus effectively generalize the BM.

(3) The NWGBM and the GNWGBM under intuitionistic fuzzy environment are proposed, and their computational formulas and some properties are given.

(4) A new decision method based on the IFNWGBM and the GIFNWGBM is developed. Two application examples and comparisons with other existing methods are used to illustrate the effectiveness of the proposed operators.

The remainder of this paper is organized as follows: Section 2 briefly reviews some basic concepts of BM and its extensions, and explains that the OWGBM and the GOWGBM have neither the reducibility nor the boundedness. In section 3, we propose the normalized weighted geometric Bonferroni mean (NWGBM) and the generalized normalized weighted geometric Bonferroni mean (GNWGBM), and study some properties of these operators. Section 4 proposes the intuitionistic fuzzy normalized weighted geometric Bonferroni mean (IFNWGBM) and the generalized intuitionistic fuzzy normalized weighted geometric Bonferroni mean (GIFNWGBM), and their mathematical expressions are also given correspondingly. A new decision method based on IFNWGBM and GIFNWGBM is proposed in section 5. Furthermore, two practical examples are provided in section 6 to demonstrate the applications of these operators. Section 7 ends with the conclusions of this study.

2 Some basic concepts

For this paper to be as self-contained as possible, we will review some basic concepts needed throughout, including the BM and some extensions under classic and intuitionistic fuzzy decision environment.

2.1 BM and its extensions

Definition 2.1 [1] Let p, q ≥ 0 and a_i (i = 1, 2, ⋯ , n) be a collection of nonnegative numbers. If $BM (a_{1}, a_{2}, \dots, a_{n}) = (\frac{1}{n (n - 1)} \sum_{i, j = 1, i \neq j}^{n} a_{i}^{p} a_{j}^{q})^{\frac{1}{p + q}}$ (1) then BM is called the Bonferroni mean (BM).

Based on the geometric mean and the BM, the geometric Bonferroni mean (GBM) was introduced by Xia [6].

Definition 2.2 [1] Let p, q ≥ 0 and a_i (i = 1, 2, ⋯ , n) be a collection of nonnegative numbers. If

$\begin{matrix} GBM & (a_{1}, a_{2}, \dots, a_{n}) \\ = \frac{1}{p + q} (\prod_{i, j = 1, i \neq j} (p a_{i} + q a_{j})^{\frac{1}{n (n - 1)}}) \end{matrix}$ (2) then GBM is called the geometric Bonferroni mean (GBM).

Xia [9] also defined the generalized weighted geometric Bonferroni mean (GWGBM).

Definition 2.3 [9] Let p, q, r ≥ 0 and a_i (i = 1, 2, ⋯ , n) be a collection of nonnegative numbers, with the weight vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_i > 0, i = 1, 2, ⋯ , n and $\sum_{i = 1}^{n} w_{i} = 1$ . If $\begin{matrix} GWGBM (a_{1}, a_{2}, \dots, a_{n}) \\ = \frac{1}{p + q + r} (\prod_{i, j, k = 1}^{n} (p a_{i} + q a_{j} + r a_{k})^{w_{i} w_{j} w_{k}}) \end{matrix}$ (3) then GWGBM is called the generalized weighted geometric Bonferroni mean (GWGBM).

Based on the GBM, NWBM and GNWBM, Jin [18] defined the optimized weighted geometric Bonferroni mean and generalized optimized weighted geometric Bonferroni mean.

Definition 2.4 [18] Let p, q ≥ 0 and a_i (i = 1, 2, ⋯ , n) be a collection of nonnegative numbers, with the weight vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_i > 0, i = 1, 2, ⋯ , n and $\sum_{i = 1}^{n} w_{i} = 1$ . If $\begin{matrix} OWGBM (a_{1}, a_{2}, \dots, a_{n}) \\ = \frac{1}{p + q} \prod_{i, j = 1, i \neq j}^{n} (p a_{i}^{w_{i}} + q a_{j}^{\frac{w_{j}}{1 - w_{i}}}) \end{matrix}$ (4) then OWGBM is called the optimized weighted geometric Bonferroni mean (OWGBM).

Definition 2.5 [18] Let p, q, r ≥ 0 and a_i (i = 1, 2, ⋯ , n) be a collection of nonnegative numbers, with the weight vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_i > 0, i = 1, 2, ⋯ , n and $\sum_{i = 1}^{n} w_{i} = 1$ . If $\begin{matrix} GOWGBM (a_{1}, a_{2}, \dots, a_{n}) \\ = \frac{1}{p + q + r} \prod_{\binom{i, j, k = 1,}{i \neq j \neq k}}^{n} (p a_{i}^{w_{i}} + q a_{j}^{\frac{w_{j}}{1 - w_{i}}} + r a_{k}^{\frac{w_{k}}{1 - w_{i} - w_{j}}}) \end{matrix}$ (5) then GOWGBM is called the generalized optimized weighted geometric Bonferroni mean (GOWGBM).

2.2 Some concepts of IFS

Definition 2.6 [19] Let a set X be fixed. An intuitionistic fuzzy set (IFS) A on X is defined as follows: $A = {< x, μ_{A} (x), ν_{A} (x) > | x \in X >}$ (6) where μ_A (x) ∈ [0, 1] and ν_A (x) ∈ [0, 1] satisfy 0 ≤ μ_A (x) ≤1, 0 ≤ ν_A (x) ≤1 for all x ∈ X, and μ_A (x) , ν_A (x) are the membership and nonmembership degrees of the element x ∈ X to A, respectively.

Xu and Yager [20,21] called the pair <μ_A (x) , ν_A (x)> in A an intuitionistic fuzzy number (IFN), and for convenience, it was expressed as α =< μ_α, ν_α > with the conditions μ_α ∈ [0, 1] , ν_α ∈ [0, 1], and 0 ≤ μ_α + ν_α ≤ 1.

Xu [20,21] defined some basic operations on IFNs.

Definition 2.7 [20, 21] Let α =< μ_α, ν_α > , β = < μ_β, ν_β > be two IFNs, then the four basic operations are defined as follows: $(1) α \oplus β = < μ_{α} + μ_{β} - μ_{α} μ_{β}, ν_{α} ν_{β} >;$ (7) $(2) α ⨂ β = < μ_{α} μ_{β}, ν_{α} + ν_{β} - ν_{α} ν_{β} >;$ (8) $(3) λ α = < 1 - (1 - μ_{α})^{λ}, ν_{α}^{λ} >, λ > 0;$ (9) $(4) α^{λ} = < μ_{α}^{λ}, 1 - (1 - ν_{α})^{λ} >, λ > 0 .$ (10)

To effectively rank IFNs, Chen and Tan [22] introduce the score function s (α) = μ_α - ν_α, and Hong and Choi [23] proposed the accuracy function s (α) = μ_α + ν_α. Based on the score function and the accuracy function, Xu and Yager [21] gave the comparison method of the IFNs.

Definition 2.8 [21] Let α =< μ_α, ν_α > and β =< μ_β, ν_β > be two IFNs, then

(1) If s (α) ≥ s (β), then α is larger than β, denoted by α ≻ β;

(2) If s (α) = s (β), then

(a) If h (α) ≥ h (β), then α is larger than β, denoted by α ≻ β;

(b) If h (α) = h (β), then α and β represent the same information, i.e. μ_α = μ_β and ν_α = ν_β, denoted by α = β.

3 NWGBM and GNWGBM

In this section, we first analyze the drawbacks of the OWGBM and the GOWGBM proposed by Jin [18]. Then, based on the NWBM and the GNWBM, we will propose the normal weighted geometric Bonferroni mean and generalized normal weighted geometric Bonferroni mean with reducibility and boundedness, which can overcome the drawbacks of the OWGBM and the GOWGBM.

3.1 The drawbacks of the OWGBM and the GOWGBM

The OWGBM and the GOWGBM indeed can reflect the interrelationship between aggregation arguments, but have no the reducibility and boundedness.

(1) The OWGBM and the GOWGBM have no the reducibility

Let $w = (\frac{1}{n}, \frac{1}{n}, \dots, \frac{1}{n})^{T}$ , then we have

$OWGBM (a_{1}, a_{2}, \dots, a_{n}) = \frac{1}{p + q} \prod_{i, j = 1, i \neq j}^{n} (p a_{i}^{\frac{1}{n}} + q a_{j}^{\frac{1}{n - 1}}) \neq \frac{1}{p + q} \prod_{i, j = 1, i \neq j}^{n} (p a_{i} + q a_{j})^{\frac{1}{n (n - 1)}} = GBM (a_{1}, a_{2}, \dots, a_{n})$

and

$GOWGBM (a_{1}, a_{2}, \dots, a_{n}) = \frac{1}{p + q + r} \prod_{i, j, k = 1, i \neq j \neq k}^{n} (p a_{i}^{\frac{1}{n}} + q a_{j}^{\frac{1}{n - 1}} + r a_{k}^{\frac{1}{n - 2}}) \neq \frac{1}{p + q + r} \prod_{i, j, k = 1, i \neq j \neq k}^{n} (p a_{i} + q a_{j} + r a_{k})^{\frac{1}{n (n - 1) (n - 2)}} = GGBM (a_{1}, a_{2}, \dots, a_{n}) .$

Obviously, we can see from the above analysis that the OWGBM and the GOWGBM are not reduced to the GBM and the GGBM, that is, the OWGBM and the GOWGBM have no the reducibility.

(2) The OWGBM and the GOWGBM have no the boundedness

Example 3.1. Let a₁ = 3, a₂ = 4, a₃ = 5 be three crisp numbers with the weight vector w = (0.3, 0.4, 0.3) ^T. Then, if p = q = 1, we have OWGBM (3, 4, 5) =873.18. And if p = q = r = 1, we have GOWGBM (3, 4, 5) =67343.90.

Example 3.1 indeed illustrated that the OWGBM and the GOWGBM have no the boundedness.

3.2 NWGBM

Definition 3.2 Let p, q ≥ 0 and a_i (i = 1, 2, ⋯ , n) be a collection of nonnegative numbers, with the weight vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_i > 0, i = 1, 2, ⋯ , n and $\sum_{i = 1}^{n} w_{i} = 1$ . If $\begin{matrix} NWGBM (a_{1}, a_{2}, \dots, a_{n}) \\ = \frac{1}{p + q} \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (p a_{i} + q a_{j})^{\frac{w_{i} w_{j}}{1 - w_{i}}} \end{matrix}$ (11) then NWGBM is called the normalized weighted geometric Bonferroni mean (NWGBM).

Obviously, the NWGBM has the following properties.

Theorem 3.3 Let p, q ≥ 0 and a_i (i = 1, 2, ⋯ , n) be a collection of nonnegative numbers, with the weight vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_i > 0, i = 1, 2, ⋯ , n and $\sum_{i = 1}^{n} w_{i} = 1$ .

(1) (Reducibility) If $w_{i} = \frac{1}{n} (i = 1, 2, \dots, n)$ , then NWGBM (a₁, a₂, ⋯ , a_n) = GBM (a₁, a₂, ⋯ , a_n).

(2) (Idempotency) If a_i = a (i = 1, 2, ⋯ , n), then NWGBM (a₁, a₂, ⋯ , a_n) = a.

(3) (Monotonicity) Let b_i (i = 1, 2, ⋯ , n) be a collection of nonnegative numbers. If a_i ≤ b_i, for all i, then NWGBM (a₁, a₂, ⋯ , a_n) ≤ NWGBM (b₁, b₂, ⋯ , b_n).

(4) (Boundedness) $min_{i} {a_{i}} \leq NWGBM (a_{1}, a_{2}, \dots, a_{n}) \leq max_{i} {a_{i}}$ .

Proof. The proofs are straightforward.

3.3 GNWGBM

Definition 3.4 Let p, q, r ≥ 0 and a_i (i = 1, 2, ⋯ , n) be a collection of nonnegative numbers, with the weight vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_i > 0, i = 1, 2, ⋯ , n and $\sum_{i = 1}^{n} w_{i} = 1$ . If $\begin{matrix} GNWGBM (a_{1}, a_{2}, \dots, a_{n}) \\ = \frac{1}{p + q + r} \prod_{\binom{i, j = 1,}{i \neq j \neq k}}^{n} (p a_{i} + q a_{j} + r a_{k})^{\frac{w_{i} w_{j} w_{k}}{(1 - w_{i}) (1 - w_{i} - w_{j})}} \end{matrix}$ (12) then GNWGBM is called the generalized normalized weighted geometric Bonferroni mean (GNWGBM).

Obviously, the GNWGBM has the following properties.

Theorem 3.5 Let p, q, r ≥ 0 and a_i (i = 1, 2, ⋯ , n) be a collection of nonnegative numbers, with the weight vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_i > 0, i = 1, 2, ⋯ , n and $\sum_{i = 1}^{n} w_{i} = 1$ .

(1) (Reducibility) If $w_{i} = \frac{1}{n} (i = 1, 2, \dots, n)$ , then GNWGBM (a₁, a₂, ⋯ , a_n) = GGBM (a₁, a₂, ⋯ , a_n).

(2) (Idempotency) If a_i = a (i = 1, 2, ⋯ , n), then GNWGBM (a₁, a₂, ⋯ , a_n) = a.

(3) (Monotonicity) Let b_i (i = 1, 2, ⋯ , n) be a collection of nonnegative numbers. If a_i ≤ b_i, for all i, then GNWGBM (a₁, a₂, ⋯ , s_n) ≤ GNWGBM (b₁, b₂, ⋯ , b_n).

(4) (Boundedness) $min_{i} {a_{i}} \leq GNWGBM (a_{1}, a_{2}, \dots, a_{n}) \leq max_{i} {a_{i}}$ .

Proof. The proofs are straightforward.

Example 3.6. Let a₁ = 3, a₂ = 4, a₃ = 5 be three crisp numbers with the weight vector w = (0.3, 0.4, 0.3) ^T. Then, if p = q = 1, we have NWGBM (3, 4, 5) =3.9767. And if p = q = r = 1, we have GNWGBM (3, 4, 5) =4.

From Theorem 3.3, Theorem 3.5 and Example 3.6, we have these conclusions that the NWGBM and the GNWGBM have the reducibility and the boundedness, which have overcome the existing drawbacks of the OWGBM and the GOWGBM.

4 IFNWGBM and GIFNWGBM

Xu [19, 20] presented the basic aggregation operators for intuitionistic fuzzy information. Based on these aggregation operators and the operations for IFNs, we will propose the intuitionistic fuzzy normalized weighted geometric Bonferroni mean and generalized intuitionistic fuzzy normalized weighted geometric Bonferroni mean.

4.1 IFNWGBM

Definition 4.1 Let p, q ≥ 0 and α_i = < μ_{α_i}, ν_{α_i} > (i = 1, 2, ⋯ , n) be a collection of IFNs, with the weight vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_i > 0, i = 1, 2, ⋯ , n and $\sum_{i = 1}^{n} w_{i} = 1$ . If $\begin{matrix} IFNWGBM (α_{1}, α_{2}, \dots, α_{n}) \\ = \frac{1}{p + q} ⨂_{\binom{i, j = 1,}{i \neq j}}^{n} (p α_{i} oplus q α_{j})^{\frac{w_{i} w_{j}}{1 - w_{i}}} \end{matrix}$ (13) then IFNWGBM is called the intuitionistic fuzzy normalized weighted geometric Bonferroni mean (IFNWGBM).

Theorem 4.2 Let p, q ≥ 0 and α_i = < μ_{α_i}, ν_{α_i} > (i = 1, 2, ⋯ , n) be a collection of IFNs, with the weight vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_i > 0, i = 1, 2, ⋯ , n and $\sum_{i = 1}^{n} w_{i} = 1$ , then

$\begin{matrix} IFNWGBM (α_{1}, α_{2}, \dots, α_{n}) = < 1 - \\ (1 - \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - (1 - μ_{α_{i}})^{p} (1 - μ_{α_{j}})^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}})^{\frac{1}{p + q}}, \\ (1 - \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - ν_{α_{i}}^{p} ν_{α_{j}}^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}})^{\frac{1}{p + q}} > . \end{matrix}$ (14)

Proof. By the operational laws described in Definition 2.7, we get $p α_{i} = < 1 - (1 - μ_{α_{i}})^{p}, ν_{α_{i}}^{q} >$ , $q α_{j} = < 1 - (1 - μ_{α_{j}})^{p}, ν_{α_{j}}^{q} >$ , and then $p α_{i} ⨂ q α_{j} = < 1 - (1 - μ_{α_{i}})^{p} (1 - μ_{α_{j}})^{q}, ν_{α_{i}}^{p} ν_{α_{j}}^{q} > .$

Further, we get that

$(p α_{i} oplus q α_{j})^{\frac{w_{i} w_{j}}{1 - w_{i}}} = < (1 - (1 - μ_{α_{i}})^{p} (1 - μ_{α_{j}})^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}}, 1 - (ν_{α_{i}}^{p} ν_{α_{j}}^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}} >$

and

$⨂_{i, j = 1, i \neq j}^{n} (p α_{i} oplus q α_{j})^{\frac{w_{i} w_{j}}{1 - w_{i}}} = < \prod_{i, j = 1, i \neq j}^{n} (1 - (1 - μ_{α_{i}})^{p} (1 - μ_{α_{j}})^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}}, 1 - \prod_{i, j = 1, i \neq j}^{n} (1 - ν_{α_{i}}^{p} ν_{α_{j}}^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}} > .$

So, we have

$IFNWGBM (α_{1}, α_{2}, \dots, α_{n}) = \frac{1}{p + q} ⨂_{\binom{i, j = 1,}{i \neq j}}^{n} (p α_{i} oplus q α_{j})^{\frac{w_{i} w_{j}}{1 - w_{i}}} = < 1 - (\prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - (1 - μ_{α_{i}})^{p} (1 - μ_{α_{j}})^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}})^{\frac{1}{p + q}}, (1 - \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - ν_{α_{i}}^{p} ν_{α_{j}}^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}})^{\frac{1}{p + q}} > .$

Theorem 4.3 Let p, q ≥ 0 and α_i = < μ_{α_i}, ν_{α_i} > (i = 1, 2, ⋯ , n) be a collection of IFNs, with the weight vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_i > 0, i = 1, 2, ⋯ , n and $\sum_{i = 1}^{n} w_{i} = 1$ .

(1) (Reducibility) If $w_{i} = \frac{1}{n} (i = 1, 2, \dots, n)$ , then IFNWGBM (α₁, α₂, ⋯ , α_n) = IFGBM (α₁, α₂, ⋯ , α_n).

(2) (Idempotency) If α_i = α = < μ_α, ν_α > (i = 1, 2, ⋯ , n), then IFNWGBM (α₁, α₂, ⋯ , α_n) = α.

(3) (Monotonicity) Let β_i = < μ_{β_i}, ν_{β_i} > (i = 1, 2, ⋯ , n) be a collection of IFNs. If α_i ≤ β_i, i.e. μ_{α_i} ≤ μ_{β_i}, ν_{α_i} ≥ ν_{β_i} for all i, then IFNWGBM (α₁, α₂, ⋯ , α_n) ≤ IFNWGBM (α₁, α₂, ⋯ , α_n).

(4) (Boundedness) α^- ≤ IFNWGBM (α₁, α₂, ⋯ , α_n) ≤ α⁺, where $α^{-} = < min_{i} {μ_{α_{i}}}, max_{i} {ν_{α_{i}}} >$ and $α^{+} = < max_{i} {μ_{α_{i}}}, min_{i} {ν_{α_{i}}} >$ .

Proof. (1) Since $w_{i} = \frac{1}{n}$ , we get

$IFNWGBM (α_{1}, α_{2}, \dots, α_{n}) = < 1 - (\prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - (1 - μ_{α_{i}})^{p} (1 - μ_{α_{j}})^{q})^{\frac{1}{n (n - 1)}})^{\frac{1}{p + q}}, (1 - \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - ν_{α_{i}}^{p} ν_{α_{j}}^{q})^{\frac{1}{n (n - 1)}})^{\frac{1}{p + q}} > = IFGBM (α_{1}, α_{2}, \dots, α_{n}) .$

That is, when $w_{i} = \frac{1}{n}$ , the IFNWGBM is reduced to the IFGBM [6].

(2) Since α_i = α, based on Theorem 4.2, we get

$IFNWGBM (α_{1}, α_{2}, \dots, α_{n}) = < 1 - (1 - \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - (1 - μ_{α})^{p} (1 - μ_{α})^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}})^{\frac{1}{p + q}}, (1 - \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - ν_{α}^{p} ν_{α}^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}})^{\frac{1}{p + q}} > = < 1 - (1 - (1 - (1 - μ_{α})^{p + q})^{\sum_{\binom{i, j = 1,}{i \neq j}}^{n} \frac{w_{i} w_{j}}{1 - w_{i}}})^{\frac{1}{p + q}}, (1 - (1 - ν_{α}^{p + q})^{\sum_{\binom{i, j = 1,}{i \neq j}}^{n} \frac{w_{i} w_{j}}{1 - w_{i}}})^{\frac{1}{p + q}} > = < 1 - (1 - (1 - (1 - μ_{α})^{p + q}))^{\frac{1}{p + q}}, (1 - (1 - ν_{α}^{p + q}))^{\frac{1}{p + q}} > = < 1 - ((1 - μ_{α})^{p + q})^{\frac{1}{p + q}}, (1 - (1 - ν_{α}^{p + q}))^{\frac{1}{p + q}} > = < μ_{α}, ν_{α} > = α .$

(3) Let IFNWGBM (α₁, α₂, ⋯ , α_n) = < μ_α, ν_α > and IFNWGBM (β₁, β₂, ⋯ , β_n) = < μ_β, ν_β >.

Since p, q > 0, μ_{α_i} ≤ μ_{β_i} and ν_{α_i} ≥ μ_{β_i} for all i, we can get that

(1 - μ_{α_i}) ^p ≥ (1 - μ_{β_i}) ^p, (1 - ν_{α_i}) ^q ≥ (1 - μ_{β_i}) ^q

and

(1 - μ_{α_i}) ^p (1 - ν_{α_i}) ^q ≥ (1 - μ_{β_i}) ^p (1 - μ_{β_i}) ^q.

Then we have that

1 - (1 - μ_{α_i}) ^p (1 - ν_{α_i}) ^q ≤ 1 - (1 - μ_{β_i}) ^p (1 - μ_{β_i}) ^q

and

$\prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - (1 - μ_{α_{i}})^{p} (1 - ν_{α_{i}})^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}} \leq \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - (1 - μ_{β_{i}})^{p} (1 - μ_{β_{i}})^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}}$ .

Further, we can obtain that

$(1 - \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - (1 - μ_{α_{i}})^{p} (1 - ν_{α_{i}})^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}})^{\frac{1}{p + q}} \geq (\prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - (1 - μ_{β_{i}})^{p} (1 - μ_{β_{i}})^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}})^{\frac{1}{p + q}}$

and

$1 - (1 - \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - (1 - μ_{α_{i}})^{p} (1 - ν_{α_{i}})^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}})^{\frac{1}{p + q}} \leq 1 - (\prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - (1 - μ_{β_{i}})^{p} (1 - μ_{β_{i}})^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}})^{\frac{1}{p + q}},$ i.e. μ_α ≤ μ_β.

Similarly, since p, q > 0, ν_{α_i} ≥ ν_{β_i} for all i, we can get $ν_{α_{i}}^{p} ν_{α_{j}}^{q} \geq ν_{β_{i}}^{p} ν_{β_{j}}^{q}$ , and hence we have $(1 - ν_{α_{i}}^{p} ν_{α_{j}}^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}} \leq (1 - ν_{β_{i}}^{p} ν_{β_{j}}^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}} .$

Moreover,we get that

$1 - \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - ν_{α_{i}}^{p} ν_{α_{j}}^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}} \geq 1 - \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - ν_{β_{i}}^{p} ν_{β_{j}}^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}}$

and

$(1 - \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - ν_{α_{i}}^{p} ν_{α_{j}}^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}})^{\frac{1}{p + q}} \geq (1 - \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - ν_{β_{i}}^{p} ν_{β_{j}}^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}})^{\frac{1}{p + q}},$ i.e. ν_α ≥ ν_β.

So, we have that μ_α - ν_α ≤ μ_β - ν_β. That is IFNWGBM (α₁, α₂, ⋯ , α_n) ≤ IFNWGBM (α₁, α₂, ⋯ , α_n) .

(4) Let IFNWGBM (α₁, α₂, ⋯ , α_n) = < μ_α, ν_α >.

Since $min_{i} {μ_{α_{i}}} \leq μ_{α_{i}} \leq max_{i} {μ_{α_{i}}}$ , for all i, then we get

$(1 - min_{i} {μ_{α_{i}}})^{p} \geq (1 - μ_{α_{i}})^{p} \geq (1 - max_{i} {μ_{α_{i}}})^{p}$

and

$(1 - min_{i} {μ_{α_{i}}})^{q} \geq (1 - μ_{α_{j}})^{q} \geq (1 - max_{i} {μ_{α_{i}}})^{q} .$

Further, we have

$(1 - min_{i} {μ_{α_{i}}})^{p + q} \geq (1 - μ_{α_{i}})^{p} (1 - μ_{α_{j}})^{q} \geq (1 - max_{i} {μ_{α_{i}}})^{p + q}$

and

$\prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - (1 - min_{i} {μ_{α_{i}}})^{p + q})^{\frac{w_{i} w_{j}}{1 - w_{i}}} \leq \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - (1 - μ_{α_{i}})^{p} (1 - μ_{α_{j}})^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}} \leq \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - (1 - max_{i} {μ_{α_{i}}})^{p + q})^{\frac{w_{i} w_{j}}{1 - w_{i}}} .$

Then, we obtain

$1 - (1 - min_{i} {μ_{α_{i}}})^{p + q} \leq \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - (1 - μ_{α_{i}})^{p} (1 - μ_{α_{j}})^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}} \leq 1 - (1 - max_{i} {μ_{α_{i}}})^{p + q} .$

Thus, we have

$1 - min_{i} {μ_{α_{i}}} \geq (1 - \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - (1 - μ_{α_{i}})^{p} (1 - μ_{α_{j}})^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}})^{\frac{1}{p + q}} \geq 1 - max_{i} {μ_{α_{i}}}$

and

$min_{i} {μ_{α_{i}}} \leq 1 - (1 - \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - (1 - μ_{α_{i}})^{p} (1 - μ_{α_{j}})^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}})^{\frac{1}{p + q}} \leq max_{i} {μ_{α_{i}}} .$

i.e. $min_{i} {μ_{α_{i}}} \leq μ_{α} \leq max_{i} {μ_{α_{i}}} .$

Additionally, since $min_{i} {ν_{α_{i}}} \leq ν_{α_{i}} \leq max_{i} {ν_{α_{i}}}$ for all i, then we obtain that

$(min_{i} {ν_{α_{i}}})^{p + q} \leq ν_{α_{i}}^{p} ν_{α_{j}}^{q} \leq (max_{i} {ν_{α_{i}}})^{p + q}$

and

$(1 - (min_{i} {ν_{α_{i}}})^{p + q})^{\frac{w_{i} w_{j}}{1 - w_{i}}} \geq (1 - ν_{α_{i}}^{p} ν_{α_{j}}^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}} \geq (1 - (max_{i} {ν_{α_{i}}})^{p + q})^{\frac{w_{i} w_{j}}{1 - w_{i}}}$ .

Hence, we have that

$\prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - (min_{i} {ν_{α_{i}}})^{p + q})^{\frac{w_{i} w_{j}}{1 - w_{i}}} \geq \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - ν_{α_{i}}^{p} ν_{α_{j}}^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}} \geq \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - (max_{i} {ν_{α_{i}}})^{p + q})^{\frac{w_{i} w_{j}}{1 - w_{i}}}$ .

i.e. $1 - (min_{i} {ν_{α_{i}}})^{p + q} \geq \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - ν_{α_{i}}^{p} ν_{α_{j}}^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}} \geq 1 - (max_{i} {ν_{α_{i}}})^{p + q}$ .

Then, we get that

$min_{i} {ν_{α_{i}}} \leq (1 - \prod_{\binom{i, j = 1,}{i \neq j}}^{n} (1 - ν_{α_{i}}^{p} ν_{α_{j}}^{q})^{\frac{w_{i} w_{j}}{1 - w_{i}}})^{\frac{1}{p + q}} \leq max_{i} {ν_{α_{i}}} .$

i.e. $min_{i} {ν_{α_{i}}} \leq ν_{α} \leq max_{i} {ν_{α_{i}}} .$

Thus, we have $min_{i} {μ_{α_{i}}} - max_{i} {ν_{α_{i}}} \leq μ_{α} - ν_{α} \leq max_{i} {μ_{α_{i}}} - min_{i} {ν_{α_{i}}}$ .

So, we have α^- ≤ IFNWGBM (α₁, α₂, ⋯ , α_n) ≤ α⁺ .

4.2 GIFNWGBM

Definition 4.4 Let p, q, r ≥ 0 and α_i = < μ_{α_i}, ν_{α_i} > (i = 1, 2, ⋯ , n) be a collection of IFNs, with the weight vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_i > 0, i = 1, 2, ⋯ , n and $\sum_{i = 1}^{n} w_{i} = 1$ . If

$\begin{matrix} GIFNWGBM (α_{1}, α_{2}, \dots, α_{n}) = \\ \frac{1}{p + q + r} ⨂_{\binom{i, j, j = 1,}{i \neq j \neq k}}^{n} (p α_{i} oplus q α_{j} oplus r α_{k})^{\frac{w_{i} w_{j} w_{k}}{(1 - w_{i}) (1 - w_{i} - w_{j})}} \end{matrix}$ (15) then GIFNWGBM is called the generalized intuitionistic fuzzy normalized weighted geometric Bonferroni mean (GIFNWGBM).

Theorem 4.5 Let p, q, r ≥ 0 and α_i = < μ_{α_i}, ν_{α_i} > (i = 1, 2, ⋯ , n) be a collection of IFNs, with the weight vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_i > 0, i = 1, 2, ⋯ , n and $\sum_{i = 1}^{n} w_{i} = 1$ , then

$\begin{matrix} GIFNWGBM (α_{1}, α_{2}, \dots, α_{n}) \\ = < 1 - (1 - \prod_{\binom{i, j, k = 1,}{i \neq j \neq k}}^{n} (1 - (1 - μ_{α_{i}})^{p} (1 - μ_{α_{j}})^{q} \\ (1 - μ_{α_{k}})^{r})^{\frac{w_{i} w_{j} w_{k}}{(1 - w_{i}) (1 - w_{i} - w_{j})}})^{\frac{1}{p + q + r}}, \\ (1 - \prod_{\binom{i, j, k = 1,}{i \neq j \neq k}}^{n} (1 - ν_{α_{i}}^{p} ν_{α_{j}}^{q} ν_{α_{k}}^{r})^{\frac{w_{i} w_{j} w_{k}}{(1 - w_{i}) (1 - w_{i} - w_{j})}})^{\frac{1}{p + q + r}} > . \end{matrix}$ (16)

Proof. The proof of Theorem 4.5 is similar to Theorem 4.2, and we do not duplicate here.

Theorem 4.6 Let p, q, r ≥ 0 and α_i = < μ_{α_i}, ν_{α_i} > (i = 1, 2, ⋯ , n) be a collection of IFNs, with the weight vector w = (w₁, w₂, ⋯ , w_n) ^T such that w_i > 0, i = 1, 2, ⋯ , n and $\sum_{i = 1}^{n} w_{i} = 1$ .

(1) (Reducibility) If $w_{i} = \frac{1}{n} (i = 1, 2, \dots, n)$ , then GIFNWGBM (α₁, α₂, ⋯ , α_n) = GIFGBM (α₁, α₂, ⋯ , α_n).

(2) (Idempotency) If α_i = α = < μ_α, ν_α > (i = 1, 2, ⋯ , n),then GIFNWGBM (α₁, α₂, ⋯ , α_n) = α.

(3) (Monotonicity) Let β_i = < μ_{β_i}, ν_{β_i} > (i = 1, 2, ⋯ , n) be a collection of IFNs. If α_i ≤ β_i, i.e. μ_{α_i} ≤ μ_{β_i}, ν_{α_i} ≥ ν_{β_i} for all i,then GIFNWGBM (α₁, α₂, ⋯ , α_n) ≤ GIFNWGBM (α₁, α₂, ⋯ , α_n).

(4) (Boundedness) α^- ≤ GIFNWGBM (α₁, α₂, ⋯ , α_n) ≤ α⁺, where $α^{-} = < min_{i} {μ_{α_{i}}}, max_{i} {ν_{α_{i}}} >$ and $α^{+} = < max_{i} {μ_{α_{i}}}, min_{i} {ν_{α_{i}}} >$ .

Proof. The proof of Theorem 4.6 is similar to Theorem 4.3, it is omitted here.

5 Intuitionistic fuzzy decision method based on the IFNWGBM and the GIFNWGBM

In this section, we will present an intuitionistic fuzzy decision method based on the IFNWGBM and the GIFNWGBM.

For a multiple attribute decision making problem, let X = {x₁, x₂, ⋯ , x_m} be a set of m alternatives, and C = {c₁, c₂, ⋯ , c_n} be a set of n attributes, whose weight vector is w = (w₁, w₂, ⋯ , w_n) ^T such that w_i ≥ 0, i = 1, 2, ⋯ , n and $\sum_{i = 1}^{n} w_{i} = 1$ . Assume that the performance of the alternative x_i with respect to the attribute c_j is represented by an IFN α_ij =< μ_{α_ij}, ν_{α_ij} >, where μ_{α_ij} indicates the degree that the alternative x_i satisfies the attribute c_j, and ν_{α_ij} indicates the degree that the alternative x_i does not satisfy the attribute c_j, such that 0 ≤ μ_{α_ij}, ν_{α_ij} ≤ 1, μ_{α_ij} + ν_{α_ij} ≤ 1 . The intuitionistic fuzzy decision making matrix evaluated by experts is A = (α_ij) _mn, where α_ij =< μ_{α_ij}, ν_{α_ij} >. In the following, we give the concrete step of intuitionistic fuzzy decision making based on the IFNWGBM and the GIFNWGBM.

Step 1 The experts or the decision makers evaluate the alternatives with respect to the attributes to form the intuitionistic fuzzy decision making matrix A = (α_ij) _mn.

Step 2 If all the attributes are of the same type, then the attribute values do not need normalization. Whereas there are, generally, benefit attributes and cost attributes in multiple attribute decision making, in such case, we may transform the attribute values of the cost type into the attribute values of the benefit type. Then, the intuitionistic fuzzy decision making matrix A = (α_ij) _mn can be transformed into the matrix B = (β_ij) _mn, where

$β = {\begin{matrix} α_{ij}, & for benefit attribute c_{j} \\ {\bar{α}}_{ij}, & for cost attribute c_{j} \end{matrix}$ (17) and ${\bar{α}}_{ij}$ is the complement of α_ij such that ${\bar{α}}_{ij} = < ν_{α_{ij}}, μ_{α_{ij}} >$ (see [20]).

Step 3 Aggregate all attribute values β_ij (j = 1, 2, ⋯ , n) to the comprehensive attribute value corresponding to the alternative x_i by IFNWGBM or the GIFNWGBM is shown as follows: $z_{i} = IFNWGBM (β_{i 1}, β_{i 2}, \dots, β_{in})$ (18) or $z_{i} = GIFNWGBM (β_{i 1}, β_{i 2}, \dots, β_{in}) .$ (19)

Step 4 Calculate the score value s (z_i) of the comprehensive attribute value z_i corresponding to the alternative x_i. If there is no difference between two score values s (z_i) and s (z_j), then we need to calculate the accuracy values h (z_i) and h (z_j) of the comprehensive attribute value z_i and z_j.

Step 5 Rank all the alternatives x_i (i = 1, 2, ⋯ , n) according to the score value s (z_i) and the accuracy value h (z_i) of z_i (i = 1, 2, ⋯ , n), and then select the best alternative.

6 Application examples

In this section, we will use two application examples to illustrate the proposed decision method based on the IFNWGBM and the GIFNWGBM. Comparative analysis are used to verify the advantages of the proposed decision method.

6.1 Example 6.1

6.1.1 Air-conditioning systems in the library Installation [5]

In ancient times, the development of culture has led to the emergence of books, and the rapid increase of books has produced an early library whose main function is to preserve books. Thus, the library is the product of the history of human civilization to a certain stage. The excellent cultural and scientific achievements of mankind are preserved by the early library. In modern times, the function of the library has converted from the previous preservation of books to reading, borrowing, etc. The library plays an irreplaceable role in the continuous progress of mankind and the sustainable development of society. At present, more and more local governments are establishing public libraries.

Now a city is planning to build a large, comprehensive and modern public library. One of the problems facing the city development commissioner is to determine what kind of air-conditioning systems should be installed in the library. The contractor offers five feasible alternative x_i (i = 1, 2, 3, 4, 5), which might be adapted to the physical structure of the library. Suppose that three attributes: (1) c₁ : economic, (2) c₂ :functional, (3) c₃ : operational, are taken into consideration in the installation problem, the weight vector of the attribute c_j (j = 1, 2, 3) is w = (0.3, 0.5, 0.2) ^T, which are evaluated by experts. Assume that the characteristics of the alternatives x_i (i = 1, 2, 3, 4, 5) with respect to the attribute c_j (j = 1, 2, 3) are represented by the IFNs α_ij =< μ_{α_ij}, ν_{α_ij} >, and all α_ij are contained in the intuitionistic fuzzy matrix A = (α_ij) _mn (see Table 1).

Table 1
Intuitionistic fuzzy information matrix A

c ₁ c ₂ c ₃

x ₁ <0.3, 0.4> <0.7, 0.2> <0.5, 0.3>

x ₂ <0.5, 0.2> <0.4, 0.1> <0.7, 0.1>

x ₃ <0.4, 0.5> <0.7, 0.2> <0.4, 0.4>

x ₄ <0.2, 0.6> <0.8, 0.1> <0.8, 0.2>

x ₅ <0.9, 0.1> <0.6, 0.3> <0.2, 0.5>

	c ₁	c ₂	c ₃
x ₁	<0.3, 0.4>	<0.7, 0.2>	<0.5, 0.3>
x ₂	<0.5, 0.2>	<0.4, 0.1>	<0.7, 0.1>
x ₃	<0.4, 0.5>	<0.7, 0.2>	<0.4, 0.4>
x ₄	<0.2, 0.6>	<0.8, 0.1>	<0.8, 0.2>
x ₅	<0.9, 0.1>	<0.6, 0.3>	<0.2, 0.5>

Step 1 The decision making matrix is given by the decision maker (see Table 1).

Step 2 All the attributes are the same type, so the attribute values are not need normalization.

Step 3 Utilize the IFNWGBM (p = q = 1) to aggregate all the attribute values α_ij (j = 1, 2, 3) of the ith line, and get the comprehensive attribute value z_i corresponding to the alternative x_i:

z₁ = <0.5353, 0.2831 > , z₂ = <0.5104, 0.1295 > , z₃ = <0.5395, 0.3325 > , z₄ = <0.6497, 0.2392 > , z₅ = <0.6158, 0.2718 > .

Step 4 Calculate the score function value s (z_i) of the comprehensive attribute value z_i corresponding to the alternative x_i:

s (z₁) =0.2522, s (z₂) =0.3809, s (z₃) =0.2070, s (z₄) =0.4105, s (z₅) =0.3440 .

Step 5 Since s (z₄) > s (z₂) > s (z₅) > s (z₁) > s (z₃), the ranking of the alternatives is x₄ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₃. Thus, x₄ is the best option.

6.1.2 Analysis about the parameters p and q

Further, we discuss the influence of the parameters p and q on decision making results based on the IFNWGBM, and the ranking results with the different values are shown in Table 2.

Table 2
Score values obtained by the IFNWGBM and the ranking of alternatives

p, q x ₁ x ₂ x ₃ x ₄ x ₅ Ranking

0.1,0.1 0.2605 0.3886 0.2185 0.4405 0.4160 x₄ ≻ x₅ ≻ x₂ ≻ x₁ ≻ x₃

0.1,0.2 0.2434 0.3903 0.1982 0.4090 0.3987 x₄ ≻ x₅ ≻ x₂ ≻ x₁ ≻ x₃

0.2,0.1 0.2662 0.3820 0.2265 0.4376 0.4053 x₄ ≻ x₅ ≻ x₂ ≻ x₁ ≻ x₃

0.5,0.5 0.2573 0.3854 0.2141 0.4280 0.3870 x₄ ≻ x₅ ≻ x₂ ≻ x₁ ≻ x₃

0.5,0.8 0.2446 0.3853 0.1990 0.4001 0.3616 x₄ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₃

0.8,0.5 0.2596 0.3800 0.2174 0.4201 0.3722 x₄ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₃

0.8,1 0.2488 0.3827 0.2033 0.4055 0.3477 x₄ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₃

1,0.8 0.2558 0.3802 0.2118 0.4151 0.3541 x₄ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₃

1,1 0.2522 0.3809 0.2070 0.4105 0.3440 x₄ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₃

1,3 0.1920 0.3611 0.1403 0.2268 0.1875 x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃

3,1 0.2172 0.3523 0.1695 0.2594 0.2309 x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃

2,2 0.2395 0.3713 0.1879 0.3782 0.2623 x₄ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₃

3,3 0.2248 0.3622 0.1641 0.3544 0.2064 x₂ ≻ x₄ ≻ x₁ ≻ x₅ ≻ x₃

5,5 0.1947 0.3483 0.1157 0.3241 0.1482 x₂ ≻ x₄ ≻ x₁ ≻ x₅ ≻ x₃

5,7 0.1747 0.3405 0.0929 0.2728 0.1100 x₂ ≻ x₄ ≻ x₁ ≻ x₅ ≻ x₃

7,5 0.1755 0.3392 0.0935 0.2777 0.1287 x₂ ≻ x₄ ≻ x₁ ≻ x₅ ≻ x₃

p, q	x ₁	x ₂	x ₃	x ₄	x ₅	Ranking
0.1,0.1	0.2605	0.3886	0.2185	0.4405	0.4160	x₄ ≻ x₅ ≻ x₂ ≻ x₁ ≻ x₃
0.1,0.2	0.2434	0.3903	0.1982	0.4090	0.3987	x₄ ≻ x₅ ≻ x₂ ≻ x₁ ≻ x₃
0.2,0.1	0.2662	0.3820	0.2265	0.4376	0.4053	x₄ ≻ x₅ ≻ x₂ ≻ x₁ ≻ x₃
0.5,0.5	0.2573	0.3854	0.2141	0.4280	0.3870	x₄ ≻ x₅ ≻ x₂ ≻ x₁ ≻ x₃
0.5,0.8	0.2446	0.3853	0.1990	0.4001	0.3616	x₄ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₃
0.8,0.5	0.2596	0.3800	0.2174	0.4201	0.3722	x₄ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₃
0.8,1	0.2488	0.3827	0.2033	0.4055	0.3477	x₄ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₃
1,0.8	0.2558	0.3802	0.2118	0.4151	0.3541	x₄ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₃
1,1	0.2522	0.3809	0.2070	0.4105	0.3440	x₄ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₃
1,3	0.1920	0.3611	0.1403	0.2268	0.1875	x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃
3,1	0.2172	0.3523	0.1695	0.2594	0.2309	x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃
2,2	0.2395	0.3713	0.1879	0.3782	0.2623	x₄ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₃
3,3	0.2248	0.3622	0.1641	0.3544	0.2064	x₂ ≻ x₄ ≻ x₁ ≻ x₅ ≻ x₃
5,5	0.1947	0.3483	0.1157	0.3241	0.1482	x₂ ≻ x₄ ≻ x₁ ≻ x₅ ≻ x₃
5,7	0.1747	0.3405	0.0929	0.2728	0.1100	x₂ ≻ x₄ ≻ x₁ ≻ x₅ ≻ x₃
7,5	0.1755	0.3392	0.0935	0.2777	0.1287	x₂ ≻ x₄ ≻ x₁ ≻ x₅ ≻ x₃

From Table 2, we can easily see that the values of p and q in the IFNWGBM exert a great influence upon the ranking results of the alternatives. When the parameters p and q take smaller values, the ranking of the alternatives is x₄ ≻ x₅ ≻ x₂ ≻ x₁ ≻ x₃ or x₄ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₃. When the parameters p and q take bigger values, the ranking of the alternatives is x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃ or x₂ ≻ x₄ ≻ x₁ ≻ x₅ ≻ x₃. Therefore, the IFNWGBM is considerably flexible by using the parameters p and q, and the decision maker can select the parameters p and q according to the certain situation.

Furthermore, it is observed from Table 2 that the score values of each alternative decrease as the parameters p and q increase, so the parameters p and q can reflect the attitudinal character of the decision maker in the decision making approach. Therefore, in decision making process, if the decision maker is pessimistic, the higher values can be assigned to the parameters p and q, and the score values of alternatives will decrease. If the decision maker is optimistic, the lower values can be assigned to the parameters p and q, and the score values of alternatives will increase.

Obviously, the bigger the parameters p and q, the more the calculation effort we needed. When p = q = 1, the IFNWGBM is not only simple but also can fully capture the interrelationship of the individual arguments, and therefore, in decision making we usually take the values of the parameters p and q as p = q = 1.

6.1.3 Sensitivity analysis about the weight vector

It is well-known that weights of attributes are very important in multi-attribute decision making. To observe how the different weights of attributes affect the ranking of the alternatives, we calculate the comprehensive attribute values and score values of the alternatives under p = q = 1. For convenience, the score values and the ranking of the alternatives are tabulated in Table 3.

Table 3
The ranking of the alternatives under the different weight vector

Score values of the alternatives

Weight vector x ₁ x ₂ x ₃ x ₄ x ₅ Ranking

(0.1,0.5,0.4) 0.3062 0.4439 0.2384 0.5340 0.1433 x₄ ≻ x₂ ≻ x₁ ≻ x₃ ≻ x₅

(0.2,0.5,0.3) 0.2708 0.4127 0.2126 0.4602 0.2401 x₄ ≻ x₂ ≻ x₁ ≻ x₃ ≻ x₅

(0.4,0.5,0.1) 0.2476 0.3476 0.2207 0.3761 0.4053 x₄ ≻ x₅ ≻ x₂ ≻ x₁ ≻ x₃

(0.3,0.1,0.6) 0.1286 0.4553 0.0318 0.3244 0.3651 x₂ ≻ x₅ ≻ x₄ ≻ x₁ ≻ x₃

(0.3,0.2,0.5) 0.1747 0.4396 0.0894 0.3687 0.3155 x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃

(0.3,0.3,0.4) 0.2088 0.4227 0.1353 0.3959 0.3002 x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃

(0.3,0.4,0.3) 0.2342 0.4037 0.1738 0.4097 0.3091 x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃

(0.3,0.5,0.2) 0.2522 0.3809 0.2070 0.4105 0.3440 x₄ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₃

(0.3,0.6,0.1) 0.2618 0.3518 0.2358 0.3950 0.4227 x₅ ≻ x₄ ≻ x₂ ≻ x₁ ≻ x₃

(0.6,0.1,0.3) 0.1152 0.4274 0.0259 0.2937 0.4531 x₅ ≻ x₂ ≻ x₄ ≻ x₁ ≻ x₃

(0.6,0.2,0.2) 0.1579 0.3731 0.0898 0.3206 0.4198 x₅ ≻ x₂ ≻ x₄ ≻ x₁ ≻ x₃

(0.6,0.3,0.1) 0.2162 0.3503 0.1798 0.3456 0.5191 x₅ ≻ x₂ ≻ x₄ ≻ x₁ ≻ x₃

Score values of the alternatives
(0.1,0.5,0.4)	0.3062	0.4439	0.2384	0.5340	0.1433	x₄ ≻ x₂ ≻ x₁ ≻ x₃ ≻ x₅
(0.2,0.5,0.3)	0.2708	0.4127	0.2126	0.4602	0.2401	x₄ ≻ x₂ ≻ x₁ ≻ x₃ ≻ x₅
(0.4,0.5,0.1)	0.2476	0.3476	0.2207	0.3761	0.4053	x₄ ≻ x₅ ≻ x₂ ≻ x₁ ≻ x₃
(0.3,0.1,0.6)	0.1286	0.4553	0.0318	0.3244	0.3651	x₂ ≻ x₅ ≻ x₄ ≻ x₁ ≻ x₃
(0.3,0.2,0.5)	0.1747	0.4396	0.0894	0.3687	0.3155	x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃
(0.3,0.3,0.4)	0.2088	0.4227	0.1353	0.3959	0.3002	x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃
(0.3,0.4,0.3)	0.2342	0.4037	0.1738	0.4097	0.3091	x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃
(0.3,0.5,0.2)	0.2522	0.3809	0.2070	0.4105	0.3440	x₄ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₃
(0.3,0.6,0.1)	0.2618	0.3518	0.2358	0.3950	0.4227	x₅ ≻ x₄ ≻ x₂ ≻ x₁ ≻ x₃
(0.6,0.1,0.3)	0.1152	0.4274	0.0259	0.2937	0.4531	x₅ ≻ x₂ ≻ x₄ ≻ x₁ ≻ x₃
(0.6,0.2,0.2)	0.1579	0.3731	0.0898	0.3206	0.4198	x₅ ≻ x₂ ≻ x₄ ≻ x₁ ≻ x₃
(0.6,0.3,0.1)	0.2162	0.3503	0.1798	0.3456	0.5191	x₅ ≻ x₂ ≻ x₄ ≻ x₁ ≻ x₃

As seen in Table 3, when the weight w₂ of the attribute c₂ are larger, i.e. w₂ = 0.5 or 0.6, the best alternative is x₄ or x₅. When the weight w₁ of the attribute c₁ are larger, i.e. w₁ = 0.6, the best alternative is x₅. And when the weight w₃ of the attribute c₃ are larger, i.e. w₃ = 0.5 or 0.6, the best alternative is x₂. This example illustrates that the different weights of attributes do affect the ranking of the alternatives. In addition, this example also demonstrates the stability of the IFNWGBM operator. It can be seen from Table 3 that when the weight vector is(0.1, 0.5, 0.4) and (0.2, 0.5, 0.3), the worst alternative is x₅, and in other cases, the alternative x₃ is always the worst one, and x₁ is always the second-worst one. In account of the very importance of the weight vector for the ranking of the alternatives in decision making process, we should try our best to guarantee that the weight vector is objective and prepare well for the information aggregation by using the IFNWGBM.

6.1.4 Comparisons with other methods

To further prove the effectiveness of the proposed method, in this subsection, we compare the proposed method with the other existing methods based on the same illustrative example, including IFBM defined by Xu [5], WIFGBM proposed by Xia [6] and IFOWGBM proposed by Jin [18]. For convenience, let p = q = 1, and comparative analysis results are listed in Table 4.

Table 4
Score values obtained by the IFNWGBM and the ranking of alternatives

methods operator Reducibility Boundedness Weight vector Ranking the best alternative

[5] IFWBM No Yes known x₂ ≻ x₅ ≻ x₄ ≻ x₁ ≻ x₃ x ₂

[6] IFWGBM No Yes known x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃ x ₂

[18] IFOWGBM No No known x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃ x ₂

Proposed method IFNWGBM Yes Yes known x₄ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₃ x ₄

methods	operator	Reducibility	Boundedness	Weight vector	Ranking	the best alternative
[5]	IFWBM	No	Yes	known	x₂ ≻ x₅ ≻ x₄ ≻ x₁ ≻ x₃	x ₂
[6]	IFWGBM	No	Yes	known	x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃	x ₂
[18]	IFOWGBM	No	No	known	x₂ ≻ x₄ ≻ x₅ ≻ x₁ ≻ x₃	x ₂
Proposed method	IFNWGBM	Yes	Yes	known	x₄ ≻ x₂ ≻ x₅ ≻ x₁ ≻ x₃	x ₄

As can be seen from Table 4, the first difference in the above four methods is in the use of the aggregation operators. The IFWBM used by Xu [5] is based on arithmetic mean, but the other three aggregation operators are based on geometric mean. The second difference is that the proposed aggregation operator has the reducibility, while IFWBM by Xu [5], WIFGBM by Xia [6] and IFOWGBM by Jin [18] have no this property. The third difference is that IFOWGBM by Jin [18] has no boundedness, and the other three aggregation operators have this property. Thus, there are significant differences in the ranking of the alternatives between these aggregation operators. The advantages of the four methods are that they can not only consider the importance of each attribute but also capture the interrelationship of the individual attribute. However, since the IFWBM, WIFGBM and IFOWGBM have no the reducibility, and the IFOWGBM has no the boundedness, therefore the proposed method can produce a reasonable and effective ranking result.

6.2 Example 6.2

6.2.1 Introduction of teacher in a Chinese university [24]

Management School in a Chinese University wants to introduce a teacher from four alternatives. A set of four attributes are considered: C = (c₁, c₂, c₃, c₄) ^T = {morality, research capability, teaching skill, education background }, whose weight vector is w = (w₁, w₂, w₃, w₄) ^T = (0.2, 0.3, 0.3, 0.2) ^T which are given by experts. The experts evaluate four alternatives x_i (i = 1, 2, 3, 4) in relation to the attributes with C = (c₁, c₂, c₃, c₄) ^T. The evaluation information on the four alternative x_i (i = 1, 2, 3, 4) under the attributes C = (c₁, c₂, c₃, c₄) ^T are represented by the IFNs α_ij =< μ_{α_ij}, ν_{α_ij} >, and form the intuitionistic fuzzy matrix A = (α_ij) _4×4, which contains all α_ij (see Table 5).

Table 5
Intuitionistic fuzzy information matrix A

c ₁ c ₂ c ₃ c ₄

x ₁ <0.9, 0.0> <0.5, 0.4> <0.8, 0.1> <0.5, 0.4>

x ₂ <0.7, 0.2> <0.9, 0.0> <0.6, 0.3> <0.8, 0.1>

x ₃ <0.4, 0.5> <0.7, 0.2> <0.9, 0.0> <0.7, 0.0>

x ₄ <0.6, 0.3> <0.6, 0.3> <0.7, 0.2> <0.9, 0.0>

In the next,we use the proposed IFNWGBM to aggregate the intuitionistic fuzzy values of the alternative x_i, and the comprehensive attribute value z_i corresponding to the alternative x_i under p = q = 1 are obtained.

z₁ = <0.6816, 0.2099 > , z₂ = <0.7611, 0.1298 > , z₃ = <0.7073, 0.1829 > , z₄ = <0.6960, 0.2005 > .

Then, by using the score function, we also obtain that the score function values of the comprehensive attribute value z_i are respectively

s (z₁) =0.4717, s (z₂) =0.6318, s (z₃) =0.5244, s (z₄) =0.4955 .

Since s (z₂) > s (z₃) > s (z₄) > s (z₁), the ranking order of the alternatives is x₂ ≻ x₃ ≻ x₄ ≻ x₁ and the alternative x₂ is the best one.

6.2.2 Comparisons with other methods

At present, there are two types of aggregation operators that can capture the interrelationship between the individual arguments or the attributes. One type is to reflect the interrelationship between the individual arguments directly by using the arithmetic mean or geometric mean and some of their extensions, such as Bonferroni mean, Heronian mean, etc. The other is to represent the interrelationship between the attributes by using non-additive measure, of which the Choquet integral based on the fuzzy measure is a typical representation. In the following, To illustrate the advantages of the explored IFNWGBM, we compare the proposed method with the other existing operators, including the intuitionistic fuzzy geometric Heronian mean (IFGHM) proposed by Yu [24], the IFWBM put forward by Xu and Yager [5] and the intuitionistic fuzzy Choquet integral operator (IFCI) defined by Xu [25] and Tan [26].

(1) If we use the IFGHM proposed by Yu [24] to aggregate the intuitionistic fuzzy information of the alternatives, we can compute the comprehensive attribute value z_i corresponding to the alternative x_i under p = q = 1.

z₁ = <0.8951, 0.0721 > , z₂ = <0.9259, 0.0449 > , z₃ = <0.9049, 0.0631 > , z₄ = <0.9074, 0.0611 > .

The by using the score function of IFN, we can calculate that the score of the comprehensive attribute value z_i are respectively

s (z₁) =0.8230, s (z₂) =0.8811, s (z₃) =0.8418, s (z₄) =0.8463 .

Since s (z₂) > s (z₄) > s (z₃) > s (z₁), the ranking of the alternatives is x₂ ≻ x₄ ≻ x₃ ≻ x₁. Thus, x₂ is the best alternative.

However, it is noticed from the calculated results of the example that there is an issue with the IFGHM put forward by Yu [24]. First, there is a small difference between the comprehensive attribute values z₃ =<0.9049, 0.0631 > and z₄ =<0.9074, 0.0611 >, which result in the little distinction degree. Obviously, the absolute difference between the membership degree of z₃ and z₄ is only equal to 0.9074 - 0.9049 = 0.0025, and the absolute difference between the non-membership degree of z₃ and z₄ is only equal to 0.0631 - 0.0611 = 0.0020. Furthermore, we also see from the score function values that the absolute difference between the score function value of the alternative x₃ and x₄ is only equal to 0.8463 - 0.8418 = 0.0045. The above discussion about the difference between the alternative x₃ and x₄ illustrates that the IFGHM put forward by Yu [24] has a poor distinguishing power.

(2) If we apply the IFWBM proposed by Xu and Yager [5], the comprehensive attribute value z_i corresponding to the alternative x_i under p = q = 1 are respectively.

z₁ = <0.2610, 0.5431 > , z₂ = <0.3011, 0.5357 > , z₃ = <0.2672, 0.5837 > , z₄ = <0.2671, 0.5495 > .

Then, by using the score function, we can obtain that the score function values of the comprehensive attribute value z_i are respectively

s (z₁) = -0.2821, s (z₂) = -0.2347, s (z₃) = -0.3165, s (z₄) = -0.2824 .

Since s (z₂) > s (z₁) > s (z₄) > s (z₃), the ranking order of the alternatives is x₂ ≻ x₁ ≻ x₄ ≻ x₃ and so x₂ is the best alternative.

However, it is noticed that there are three problems with the IFWBM defined by Xu and Yager [5]. the first problem is that the IFWBM has no the reducibility, which means that the IFWBM is not the generalization of IFBM and the aggregation results by the operator may lead to a reasonable decision. The second one is that there are still small differences in the comprehensive attribute values and the score functions. For example, from z₁ =<0.2610, 0.5431 > , z₂ = <0.3011, 0.5357 > and z₄ =<0.2671, 0.5495 >, we can find that there is almost indifference between the membership degree of z₃ and z₄, and is also almost indifference between the non-membership degree of z₁ and z₄. At the same time, the difference between the score function value of the alternative x₁ and x₄ is only equal to 0.2824 - 0.2821 = 0.0003. The above discussion about the differences between the alternatives x₁, x₃ and x₄ illustrate that the distinguishing power of the IFWBM defined by Xu and Yager [5] is poor.The last one is that the membership degree of each aggregation result obtained by the IFWBM is bigger than the non-membership degree, which is not consistent with the initial information of the experts. This shows that the aggregation results obtained by the IFWBM are very strange and interesting, which are worthy to be studied further.

(3) It is noted that the IFCI defined by Xu [25] and Tan et al [26] can be used in decision making only if the additive property of the attributes has been met. Therefore, if we utilize the IFCI, we will find that we could use the IFCI directly unless the IFCI was reduced the intuitionistic fuzzy weighted average operator (IFWA) [20]. Because the aggregation result through the IFWA cannot reflect the interrelationship between the aggregation arguments or the individual attributes, it is meaningless to compare the aggregation results used by the IFWA [25, 26] with those used by the IFWBM proposed by Xu and Yager [5], the IFGHM proposed by Yu [24] and the proposed IFNWGBM.

For the convenience of comparing the merits and demerits between these operators, some results obtained by them are listed in Table 6.

Table 6
Comparisons with other operators

Operators Difference between maximum membership degree and minimum membership degree Difference between maximum nonmembership degree and minimum nonmembership degree Difference between score values of the best alternative and the worst one Interrelationship between the aggregation arguments Ranking

IFGHM 0.0344 0.0272 0.0581 Yes x₂ ≻ x₄ ≻ x₃ ≻ x₁

IFWBM 0.0401 0.0480 0.0818 No x₂ ≻ x₁ ≻ x₄ ≻ x₃

IFCI not available not available not available Yes not available

Proposed method 0.0795 0.0801 0.1601 Yes x₂ ≻ x₃ ≻ x₄ ≻ x₁

Operators	Difference between maximum membership degree and minimum membership degree	Difference between maximum nonmembership degree and minimum nonmembership degree	Difference between score values of the best alternative and the worst one	Interrelationship between the aggregation arguments	Ranking
IFGHM	0.0344	0.0272	0.0581	Yes	x₂ ≻ x₄ ≻ x₃ ≻ x₁
IFWBM	0.0401	0.0480	0.0818	No	x₂ ≻ x₁ ≻ x₄ ≻ x₃
IFCI	not available	not available	not available	Yes	not available
Proposed method	0.0795	0.0801	0.1601	Yes	x₂ ≻ x₃ ≻ x₄ ≻ x₁

It can be seen from the results in Table 6 that the difference between the comprehensive attribute values of the alternatives by the proposed method is relatively larger than the other two operators, and so is the difference between the score function values of the alternative. This illustrates that the proposed IFNWGBM has a stronger distinguishing power than the IFWBM proposed by Xu and Yager [5] and the IFGHM put forward by Yu [24]. In addition, the proposed IFNWGBM has the reducibility, which can not only take the weights of attributes into account but also reflect the interrelationship between the aggregation arguments. In a word, these above comparisons and analyses showed that the decision result obtained by the proposed IFNWGBM is more reasonable and valid.

6.3 Some advantages of the proposed operators

Through the above two application examples, we can see that compared with the previous operators proposed by Xu and Yager [5], Yu [24], Xu [25] and Tan et al [26], our developed operators have some advantages:

(1) The proposed operators have the reducibility and boundedness. The reducibility means that the proposed operators do not change the mathematical structure of Bonferroni mean, and guarantees that the proposed operators are the generalization of the OWGBM and the GOWGBM. Meanwhile the boundedness ensures that the proposed operators are the averaging aggregation operator.

(2) The proposed operators have the flexibility by the parameters p and q, and the decision makers can select the parameters p and q according to the attitudinal character of the decision makers in actual decision making.

(3) When the weight of the attribute occurs in a relatively small change, the best alternative and the worst one remain unchanged. So, the proposed operators have, to some degree, the stability.

(4) The proposed operators have a strong distinguishing power, which can better distinguish the difference between the alternatives, and make the ranking of the alternatives more reasonable.

One coin has two sides. The proposed operators have two disadvantages:

(1) The computation load in the process of information aggregation is much heavier, especially when the parameters p and q take some large values.

(2) When the attributes do not fit the additive property, the proposed operators cannot be used to aggregate the decision information.

7 Conclusions

In order to overcome the shortcomings of the OWGBM and the GOWGBM, we have proposed the NWGBM and GNWGBM. Then, we have developed the IFNWGBM and GIFNWGBM, which are suited for intuitionistic fuzzy decision environment. Two examples are used to show the applications of these operators and comparisons with other methods. The chief achievements of this paper can be summarized as follows:

(1) In the classic decision environment, we have defined the NWGBM and the GNWGBM, which can not only capture the interrelationship between input arguments, but also have the reducibility and boundedness. Thus, the proposed operators have overcome the drawbacks of the OWGBM and the GOWGBM.

(2) In the intuitionistic fuzzy decision environment, we have developed the IFNWGBM and the GIFNWGBM. And we have investigated their desirable properties, such as the reducibility, idempotency, monotonicity and boundedness.

(3) By applying these operators to two application examples and comparisons with other operators, we have modified the practicability and validity of the proposed operators.

Although we have developed the NWGBM, the GNWGBM and their extensions under intuitionist fuzzy decision environments, there are still some problems that need study in the future.

(1) How to choose the proper parameters p and q in the real decision making is worth continuing to study.

(2) There are certain relationships and connections between the BM, its extension and the aggregation operators based on the Choquet integral. How to describe them is a critical problem.

(3) We will study how to take the applications of these operators to deal with the decision making problems in real world.

(4) In addition, considering the advantages of the NWGBM and the GNWGBM, we will also extend then to other decision making environments such as Pythagorean fuzzy sets (PHFs) [27], Pythagorean hesitant fuzzy sets (PHFSs) [28] and generalized orthopair fuzzy sets (GOFSs) [29] and so on.

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Intuitionistic fuzzy normalized weighted geometric Bonferroni means with reducibility and boundedness and application in decision making 1

Abstract

Keywords

1 Introduction

2 Some basic concepts

2.1 BM and its extensions

3.1 The drawbacks of the OWGBM and the GOWGBM

3.2 NWGBM

4.1 IFNWGBM

6.1 Example 6.1

6.1.1 Air-conditioning systems in the library Installation [5]

Table 1 Intuitionistic fuzzy information matrix A c 1 c 2 c 3 x 1 <0.3, 0.4> <0.7, 0.2> <0.5, 0.3> x 2 <0.5, 0.2> <0.4, 0.1> <0.7, 0.1> x 3 <0.4, 0.5> <0.7, 0.2> <0.4, 0.4> x 4 <0.2, 0.6> <0.8, 0.1> <0.8, 0.2> x 5 <0.9, 0.1> <0.6, 0.3> <0.2, 0.5>

6.2.1 Introduction of teacher in a Chinese university [24]

Table 5 Intuitionistic fuzzy information matrix A c 1 c 2 c 3 c 4 x 1 <0.9, 0.0> <0.5, 0.4> <0.8, 0.1> <0.5, 0.4> x 2 <0.7, 0.2> <0.9, 0.0> <0.6, 0.3> <0.8, 0.1> x 3 <0.4, 0.5> <0.7, 0.2> <0.9, 0.0> <0.7, 0.0> x 4 <0.6, 0.3> <0.6, 0.3> <0.7, 0.2> <0.9, 0.0>

7 Conclusions

References

Table 1
Intuitionistic fuzzy information matrix A

c ₁ c ₂ c ₃

x ₁ <0.3, 0.4> <0.7, 0.2> <0.5, 0.3>

x ₂ <0.5, 0.2> <0.4, 0.1> <0.7, 0.1>

x ₃ <0.4, 0.5> <0.7, 0.2> <0.4, 0.4>

x ₄ <0.2, 0.6> <0.8, 0.1> <0.8, 0.2>

x ₅ <0.9, 0.1> <0.6, 0.3> <0.2, 0.5>

Table 5
Intuitionistic fuzzy information matrix A

c ₁ c ₂ c ₃ c ₄

x ₁ <0.9, 0.0> <0.5, 0.4> <0.8, 0.1> <0.5, 0.4>

x ₂ <0.7, 0.2> <0.9, 0.0> <0.6, 0.3> <0.8, 0.1>

x ₃ <0.4, 0.5> <0.7, 0.2> <0.9, 0.0> <0.7, 0.0>

x ₄ <0.6, 0.3> <0.6, 0.3> <0.7, 0.2> <0.9, 0.0>