Some intuitionistic linguistic dependent Bonferroni mean operators and application in group decision-making

Abstract

The dependent ordered weighted averaging (DOWA) operator can relieve the influence of unfair data from the aggregated arguments, and Bonferroni mean (BM) operator can capture the interrelationship of the aggregated arguments. In order to making fully use of the advantages of these two types of operators, we combine the DOWA with the BM operator in intuitionistic linguistic setting, and propose the intuitionistic linguistic dependent Bonferroni mean (ILDBM) operator and the intuitionistic linguistic dependent geometric Bonferroni mean (ILDGBM) operator. Simultaneously, several properties of these novel operators are discussed. Moreover, a method based on these operators is developed to solve the multi-attribute group decision making (MAGDM) problems with intuitionistic linguistic information. The advantages of the proposed method are (1) it can consider the interrelationship between any two attribute values; (2) it can relieve the influence of unfair attribute values given by some biased decision makers. Finally, an application example is represented to illustrate the practicality and validity of the developed method by comparing with the existing methods.

Keywords

Multiple attribute group decision making intuitionistic linguistic set dependent ordered weighted averaging operator Bonferroni mean intuitionistic linguistic dependent geometric Bonferroni mean (ILDGBM) operator

1 Introduction

The concept of intuitionistic fuzzy set (IFS) was presented by Atanassov [1], which is characterized by three important parameters: membership function, hesitancy function and non-membership function. Thus it is an effective tool to handle uncertainty and vagueness, and the researches on multi-attribute decision-making (MADM) problems with intuitionistic fuzzy information have received more and more attention. Wei [2] proposed a modified GRA method to solve MADM problems with intuitionistic fuzzy information. Zeng and Xiao [3], and Yue [4] developed an extended TOPSIS technique to find out the best alternative under intuitionistic fuzzy environment. Montajabiha [5] proposed a new extended PROMETHE II method for intuitionistic fuzzy MAGDM problems. Further, some new extensions of IFS have been studied and applied to decision makings [6, 7].

However, intuitionistic fuzzy set may not always be adequate to express the uncertain and fuzzy information in practical MADM problems, especially for qualitative aspects, while it is easy to provide the evaluation values by the means of linguistic variables. For example, when the moral character of students, the computer performance and so on are evaluated, they are easy to be expressed by the linguistic variables such as “very poor”, “poor”, “fair”, “good”, “very good”. So far, the research on the MAGDM problems based on linguistic variables has made many achievements. Liu and Zhang [8] developed an extended ELECTRE method to select supplier of a supply chain. Wu et al. [9] proposed a general MAGDM technique with linguistic information based on the VIKOR method to select an appropriate computer numerical control machine. Qin and Liu [10] proposed extended Muirhead mean operators for 2-tuple linguistic information and applied to select the supplier.

By combining linguistic variables with IFS, the concept of intuitionistic linguistic set (ILS) was introduced by Wang and Li [11], which gave the information about the membership and non-membership of an element to a linguistic variable. Obviously, ILS is an efficient approximate technique to handle the uncertain and fuzzy information by integrating the advantages of linguistic variables and IFS. Later, Wang et al. [12] proposed intuitionistic linguistic ordered weighted geometric (ILOWG) operator and intuitionistic linguistic hybrid geometric (ILOHG) operator, and applied them to MAGDM problems. Liu and Wang [13] proposed the intuitionitic linguistic power generalized weighted average (ILPGWA) operator and the intuitionitic linguistic power generalized ordered weighted average (ILPGOWA) operator. Ju et al. [14] extended Maclaurin symmetric mean operator to ILS by considering the interrelationships among the multi-input arguments.

Information aggregation operators are the good tool to deal with the MADM problems [15 –17]. Among them, the DOWA operator, and the BM operator are two of the most common operators for aggregating information. The DOWA operator was proposed by Xu [18], which can relieve the influence of unfair data from the aggregated arguments. Then, the DOWA operator was extended to interval set (IS) [19], IFS [20], uncertain linguistic set (ULS) [21], 2-dimension linguistic set (2DLS) [22 –24], ILS [25] and so on. The BM operator was proposed by Bonferroni [26], which can capture the interrelationship of the aggregated arguments. Further, the BM operator was extended to interval type-2 fuzzy set (IT2FS) [27], IFS [28], ULS [29], 2-tuple linguistic set (2TLS) [30], interval-valued 2-tuple linguistic set (IV2TLS) [31], ILS [32] and so on.

Moreover, with increasing the complexity of decision problems in the real application, in order to get the best alternative for a MAGDM problem, we need to solve the following three problems simultaneously. 1) The attribute values can be easily described for complex information. Obviously, the ILS can be utilized to solve this problem; 2) The attribute values of alternatives given by decision makers may be unreasonable because of bias of decision makers, which may be unduly high or unduly low. The DOWA operator can be utilized to solve this problem; 3) The attribute values may not be independent and have the interrelationships, and then the BM operator can be utilized to solve this problem. To solve above problems simultaneously, therefore, it is necessary to combine the DOWA operator with the BM operator in intuitionistic linguistic environment. So the goal of this paper is to propose some new intuitionistic linguistic operators and apply them to solve the MAGDM problems, which not only relieve the influence of unfair data, but also capture the interrelationship of the aggregated arguments.

The remainder of this study is constructed as follows. Section 2 reviews some basic concepts of the ILS, DOWA operator and BM. Section 3 proposes the intuitionistic linguistic weighted Bonferroni mean operator and the intuitionistic linguistic weighted geometric Bonferroni mean operator. Section 4 proposes the intuitionistic linguistic dependent weighted Bonferroni mean operator and intuitionistic linguistic dependent weighted geometric Bonferroni mean operator. Section 5 gives approach to intuitionistic linguistic MAGDM based on the proposed new operators. Section 6 presents a numerical example to illustrate the validity of the proposed method. Section 7 provides the concluding remarks.

2 Preliminaries

2.1 The linguistic variable

Referring to that S = (s₀, s₁, …, s_l-1) is a limited and well-ordered term set, where s_i represents a possible value for a linguistic variable and l is an odd numerical value. In normal conditions, l is assigned as 3,5,7,9, etc. If l = 7, the set could be given as follows [33, 34]:

S = (s₀, s₁, s₂, s₃, s₄, s₅, s₆) = {very poor, poor, faintly poor, fair, faintly good, good, very good} in which s_i ≥ s_j (i ≥ j).

In order to keep all the given information, the discrete linguistic term set S = (s₀, s₁, s₂, …, s_l-1) is extended to a continuous linguistic set $\tilde{S} = {s_{α} | α \in R^{+}}$ [35].

About the characteristics and operational rules, please refer to references [33 –35].

2.2 The intuitionistic linguistic number

Definition 1. [11] An intuitionistic linguistic set A in X is defined as $A = {< x [s_{α (x)}, (u_{A (x)}, v_{A (x)})] > | x \in X}$ (1) where $s_{α (x)} \in \tilde{S}, u_{A} (x) : X \to [0, 1], v_{A} (x) : X \to [0, 1]$ , with the conditions 0 ≤ u_A (x) + v_A (x) ≤1, ∀x ∈ X. The numbers u_A (x) and v_A (x) denote the membership degree and the non-membership degree of the element x to the linguistic variable s_α(x), respectively. If π_A (x) =1 - u_A (x) - v_A (x) , ∀ x ∈ X then π_A (x) is called a hesitancy degree to the linguistic variable s_α(x).

Definition 2. [11] Suppose A = {< x [s_α(x), (u_A(x), v_A(x))] > |x ∈ X} is an intuitionistic linguistic set, then the triad group <s_α(x), (u_A (x) , v_A (x))> is called an intuitionistic linguistic number (ILN), and A can also be regarded as a collection of all ILNs. And it can be also denoted as: $A = < s_{α (x)}, (u_{A} (x), v_{A} (x)) >$ (2)

Let ${\tilde{a}}_{1} = < s_{α (a_{1})}, (u (a_{1}), v (a_{1})) >$ and ${\tilde{a}}_{2} = < s_{α (a_{2})}, (u (a_{2}), v (a_{2})) >$ be any two ILNs, λ ≥ 0, then we have [11]: $\begin{matrix} a . & {\tilde{a}}_{1} + {\tilde{a}}_{2} = < s_{α (a_{1}) + α (a_{2})}, (1 - (1 - u (a_{1})) \\ (1 - u (a_{2})), v (a_{1}) v (a_{2})) > \end{matrix}$ (3) $\begin{matrix} b . & {\tilde{a}}_{1} \otimes {\tilde{a}}_{2} = < s_{α (a_{1}) \times α (a_{2})}, (u (a_{1}) u (a_{2}), \\ v (a_{1}) + v (a_{2}) - v (a_{1}) v (a_{2})) > \end{matrix}$ (4) $c . λ {\tilde{a}}_{1} = < s_{λ \times α (a_{1})}, (1 - (1 - u (a_{1}))^{λ}, (v (a_{1}))^{λ}) >$ (5) $d . {\tilde{a}}_{1}^{λ} = < s_{(α (a_{1}))^{λ}}, ((u (a_{1}))^{λ}, 1 - (1 - v (a_{1}))^{λ}) >$ (6)

Obviously, the above operational results are still the ILNs.

Definition 3. [25] Suppose that ${\tilde{a}}_{1} = < s_{α (a_{1})}, (u (a_{1}), v (a_{1})) >$ is an ILN, then the score value $S ({\tilde{a}}_{1})$ of ${\tilde{a}}_{1}$ can be represented as: $S ({\tilde{a}}_{1}) = \frac{α (a_{1})}{l - 1} \times [u (a_{1}) + \frac{1}{2} (1 - u (a_{1}) - v (a_{1}))]$ (7)

Definition 4. [25] Suppose that ${\tilde{a}}_{1} = 〈 s_{α (a_{1})}, (u (a_{1}), v (a_{1})) 〉$ is an ILN, then the accurate function $H ({\tilde{a}}_{1})$ of ${\tilde{a}}_{1}$ can be represented as: $H ({\tilde{a}}_{1}) = \frac{α (a_{1})}{l - 1} \times (u (a_{1}) + v (a_{1}))$ (8)

Definition 5. [25] Let ${\tilde{a}}_{1} = 〈 s_{α (a_{1})}, (u (a_{1}), v (a_{1})) 〉$ and ${\tilde{a}}_{2} = 〈 s_{α (a_{2})}, (u (a_{2}), v (a_{2})) 〉$ be any two ILNs, then

if $S ({\tilde{a}}_{1}) > S ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1} > {\tilde{a}}_{2}$ ;

if $S ({\tilde{a}}_{1}) = S ({\tilde{a}}_{2})$ , then

if $H ({\tilde{a}}_{1}) > H ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1} > {\tilde{a}}_{2}$ ;

if $H ({\tilde{a}}_{1}) = H ({\tilde{a}}_{2})$ , then ${\tilde{a}}_{1} = {\tilde{a}}_{2}$ .

Definition 6. [25] Let ${\tilde{a}}_{1} = < s_{α (a_{1})}, (u (a_{1}), v (a_{1})) >$ and ${\tilde{a}}_{2} = < s_{α (a_{2})}, (u (a_{2}), v (a_{2})) >$ be any two ILNs, then the normalized Hamming distance between ${\tilde{a}}_{1}$ and ${\tilde{a}}_{2}$ is given as follows:

$\begin{matrix} d (a_{1}, a_{2}) \\ = \frac{1}{2 (l - 1)} (| (1 + u (a_{1}) - v (a_{1})) α (a_{1}) \\ - (1 + u (a_{2}) - v (a_{2})) α (a_{2}) |) \end{matrix}$ (9)

2.3 DOWA operator

Definition 7. [18] Let (a₁, a₂, ⋯ , a_n) be a collection of input arguments, and let av be the average value of these input arguments, $av = \frac{1}{n} \sum_{j = 1}^{n} a_{j}$ , and (σ (1) , σ (2) , …, σ (n)) is a permutation of (1, 2, …, n) such that a_σ(j-1) ≥ a_σ(j) for all j = 2, …, n, if

$\begin{matrix} s (a_{σ (j)}, av) & = & 1 - \frac{| a_{σ (j)} - av |}{\sum_{j = 1}^{n} | a_{j} - av |} \\ (j = 1, 2, \dots, n) \end{matrix}$ (10) then s (a_σ(j), av) is called the similarity degree of the j-th largest argument a_σ(j) and the average value av.

Let w = (w₁, w₂, …, w_n) ^T be the weight vector of the OWA operator, then we define the following [18]: $w_{j} = \frac{s (a_{σ (j)}, av)}{\sum_{j = 1}^{n} s (a_{σ (j)}, av)} (j = 1, 2, \dots, n)$ (11) where s (a_σ(j), av) is defined by formula (10). Obviously, we have w_j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1 .$ Since $\sum_{j = 1}^{n} s (a_{σ (j)}, av) = \sum_{j = 1}^{n} s (a_{j}, av)$ (12)

Then, formula (11) can be rewritten as $w_{j} = \frac{s (a_{σ (j)}, av)}{\sum_{j = 1}^{n} s (a_{j}, av)} (j = 1, 2, \dots, n)$ (13)

In this case, we have

$\begin{matrix} OWA (a_{1}, a_{2}, \dots, a_{n}) & = & \frac{\sum_{j = 1}^{n} s (a_{σ (j)}, av) a_{σ (j)}}{\sum_{j = 1}^{n} s (a_{σ (j)}, av)} \\ = & \frac{\sum_{j = 1}^{n} s (a_{j}, av) a_{j}}{\sum_{j = 1}^{n} s (a_{j}, av)} \end{matrix}$ (14) then, the OWA is called dependent OWA (DOWA) operator.

2.4 The Benferrroni mean and geometric Bonferroni mean operators

Definition 8. [26] Let a_i (i = 1, 2, ⋯ , n) be a collection of non-negative real numbers, and p, q ≥ 0. If

$\begin{matrix} {BM}^{p, q} (a_{1}, a_{2}, \dots, a_{n}) \\ = {(\frac{1}{n (n - 1)} \sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} a_{i}^{p} a_{j}^{q})}^{\frac{1}{p + q}} \end{matrix}$ (15) then, BM^p, q is called the Benferrroni mean (BM) operator.

Definition 9. [36, 37] Let a_i (i = 1, 2, ⋯ , n) be a collection of non-negative real number and p, q ≥ 0. If

$\begin{matrix} {GBM}^{p, q} (a_{1}, a_{2}, \dots, a_{n}) \\ = \frac{1}{p + q} {(\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({pa}_{i} + {qa}_{j}))}^{\frac{1}{n (n - 1)}} \end{matrix}$ (16) then, GBM^p,q is called the geometric Benferroni mean (GBM) operator.

3 The ILWBM operator and the ILWGBM operator

3.1 The ILWBM operator

Definition 10. [38] Let ${\tilde{s}}_{j} = [s_{α_{j}}, (u_{j}, v_{j})]$ (j = 1, 2, …, n) be a set of ILNs, and p, q ≥ 0, ILWBM : Ωⁿ → Ω . If

$\begin{matrix} {ILWBM}_{w}^{p, q} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) \\ = {(\frac{1}{n (n - 1)} (\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({({nw}_{i} {\tilde{s}}_{i})}^{p} \otimes {({nw}_{j} {\tilde{s}}_{j})}^{q})))}^{\frac{1}{p + q}} \end{matrix}$ (17) where Ω is the set of all ILNs, and w = (w₁, w₂, …, w_n) ^T is the weight vector of $({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n})$ with meeting w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ , n is a balance parameter. Then, ${ILWBM}_{w}^{p, q}$ is called the intuitionistic linguistic weighted Bonferroni mean (ILWBM) operator.

Theorem 1. Let ${\tilde{s}}_{j} = [s_{α_{j}}, (u_{j}, v_{j})] (j = 1, 2, \dots, n)$ be a set of ILNs, and p, q ≥ 0, then, the result aggregated from Definition 10 is still an ILN, and even $\begin{array}{l} I L W B M_{w}^{p, q} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, ..., {\tilde{s}}_{n}) = [s_{{(\frac{1}{n (n - 1)} \sum_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} {(n w_{i} α_{i})}^{p} \times {(n w_{j} α_{j})}^{q})}^{^{\frac{1}{p + q}}}}, \\ ({(1 - \prod_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} {(1 - {(1 - {(1 - u_{i})}^{n w_{i}})}^{p} \times {(1 - {(1 - u_{j})}^{n w_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}}, \\ 1 - {(1 - \prod_{\begin{array}{l} i, j = 1 \\ i \neq j \end{array}}^{n} {(1 - {(1 - v_{i}^{n w_{i}})}^{p} \times {(1 - v_{_{j}}^{n w_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}})] \end{array}$ (18)

Then some special cases of the ILWBM operator are given as follows.

1) If q = 0, the ILWBM operator reduces to an intuitionistic linguistic generalized weighted mean operator; it follows that $\begin{array}{l} I L W B M_{w}^{p, 0} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, ..., {\tilde{s}}_{n}) \\ = [s_{{(\frac{1}{n} \sum_{i = 1}^{n} {(n w_{i} α_{i})}^{p})}^{^{\frac{1}{p}}}}, ({(1 - \prod_{i = 1}^{n} {(1 - {(1 - {(1 - u_{i})}^{n w_{i}})}^{p})}^{\frac{1}{n}})}^{\frac{1}{p}}, \\ 1 - {(1 - \prod_{i = 1}^{n} {(1 - {(1 - v_{i}^{n w_{i}})}^{p})}^{\frac{1}{n}})}^{\frac{1}{p}})] \end{array}$

2) If p = 1 and q = 0, then ILWBM operator reduces to an intuitionistic linguistic arithmetic weighted average operator. $\begin{matrix} {ILWBM}_{w}^{1, 0} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) \\ = [s_{(\sum_{i = 1}^{n} w_{i} α_{i})}, ((1 - \prod_{i = 1}^{n} {(1 - u_{i})}^{w_{i}}), \\ \prod_{i = 1}^{n} (v_{i}^{w_{i}}))] \end{matrix}$

3) If p → 0 and q = 0, then ILWBM operator reduces to an intuitionistic linguistic geometric weighted mean operator. $\begin{matrix} lim_{p \to 0} {ILWBM}_{w}^{p, 0} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) \\ = [s_{\prod_{i = 1}^{n} {({nw}_{j} α_{j})}^{\frac{1}{n}}}, ({(\prod_{i = 1}^{n} {(1 - {(1 - u_{i})}^{{nw}_{i}})}^{\frac{1}{n}})}^{\frac{1}{p + q}}, \\ 1 - \prod_{i = 1}^{n} {(1 - ν_{i}^{{nw}_{i}})}^{\frac{1}{n}})] \end{matrix}$

3.2 The ILWGBM operator

Definition 11. Let ${\tilde{s}}_{j} = [s_{α_{j}}, (u_{j}, v_{j})]$ (j = 1, 2, …, n) be a set of ILNs, and p, q ≥ 0, ILWGBM : Ωⁿ → Ω. If

$\begin{matrix} {ILWGBM}_{w}^{p, q} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) \\ = \frac{1}{p + q} {(\otimes_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (p {({\tilde{s}}_{i})}^{{nw}_{i}} \oplus q {({\tilde{s}}_{j})}^{{nw}_{j}}))}^{\frac{1}{n (n - 1)}} \end{matrix}$ (19) where Ω is the set of all ILNs, and w = (w₁, w₂, …, w_n) ^T is the weight vector of $({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n})$ with w_i ∈ [0, 1] and $\sum_{i = 1}^{n} w_{i} = 1$ , n is a balance parameter. Then ${ILWGBM}_{w}^{p, q}$ is called the intuitionistic linguistic weighted geometric Bonferroni mean (ILWGBM) operator.

Theorem 2. Let ${\tilde{s}}_{j} = [s_{α_{j}}, (u_{j}, v_{j})] (j = 1, 2, \dots, n)$ be a set of ILNs, and p, q ≥ 0, then, the result aggregated from Definition 11 is still an ILN, and even

$\begin{matrix} {ILWGBM}_{w}^{p, q} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) \\ = [s_{\frac{1}{p + q} {(\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (p {(α_{i})}^{{nw}_{i}} + q {(α_{j})}^{{nw}_{j}}))}^{\frac{1}{n (n - 1)}}}, (1 - {(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - {(1 - u_{α_{i}}^{{nw}_{i}})}^{p} {(1 - u_{α_{j}}^{{nw}_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}}, \\ {(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - {(1 - {(1 - v_{α_{i}})}^{{nw}_{i}})}^{p} {(1 - {(1 - v_{α_{j}})}^{{nw}_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}})] \end{matrix}$ (20)

Proof. By formulas (5) and (6), we obtain

$\begin{matrix} {\tilde{s}}_{i}^{{nw}_{i}} & = & (s_{α_{i}^{{nw}_{i}}}, (u_{i}^{{nw}_{i}}, 1 - (1 - v_{i})^{{nw}_{i}})) \\ {\tilde{s}}_{j}^{{nw}_{j}} & = & (s_{α_{j}^{{nw}_{j}}}, (u_{j}^{{nw}_{j}}, 1 - (1 - v_{j})^{{nw}_{j}})) \end{matrix}$ (21)

$\begin{matrix} p ({\tilde{s}}_{i})^{{nw}_{i}} & = & (s_{p u_{α_{i}}^{m w_{i}}}, (1 - (1 - u_{i}^{nw i})^{p}, \\ (1 - (1 - v_{i})^{{nw}_{i}})^{p})) \\ q ({\tilde{s}}_{j})^{{nw}_{j}} & = & (s_{q α_{j}^{nw j}}, (1 - (1 - u_{j}^{nw j})^{q}, \\ (1 - (1 - v_{j})^{{nw}_{j}})^{q})) \end{matrix}$ (22)

Then by formulas (3) and (4), we have

$\begin{matrix} p ({\tilde{s}}_{i})^{{nw}_{i}} \oplus q ({\tilde{s}}_{j})^{{nw}_{j}} \\ = (s_{p α_{i}^{{nw}_{i}} + q α_{j}^{nw j}}, (1 - (1 - u_{i}^{nw i})^{p} (1 - u_{j}^{{nw}_{j}})^{q}, \\ (1 - (1 - v_{i})^{{nw}_{i}})^{p} (1 - (1 - v_{j})^{{nw}_{j}})^{q})) \end{matrix}$ (23)

To sum up, we first prove the following formula

$\begin{matrix} [\otimes_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (p ({\tilde{s}}_{i})^{{nw}_{i}} \oplus q ({\tilde{s}}_{j})^{{nw}_{j}})] \\ = [s_{\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (p α_{i}^{{nw}_{i}} + q α_{j}^{{nw}_{j}})}, (\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (1 - {(1 - u_{α_{i}}^{{nw}_{i}})}^{p} {(1 - u_{α_{j}}^{{nw}_{j}})}^{q}), \\ 1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (1 - {(1 - {(1 - v_{α_{i}})}^{{nw}_{i}})}^{p} {(1 - {(1 - v_{α_{j}})}^{{nw}_{j}})}^{q}))] \end{matrix}$ (24)

By mathematical induction on n, we have

1) for n = 2, we obtain

$\begin{matrix} [\otimes_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{2} (p {({\tilde{s}}_{i})}^{{nw}_{i}} \oplus q {({\tilde{s}}_{j})}^{{nw}_{j}})] \\ = [(p {({\tilde{s}}_{1})}^{2 w_{1}} \oplus q {({\tilde{s}}_{2})}^{2 w_{2}}) \\ \otimes (p {({\tilde{s}}_{2})}^{2 w_{2}} \oplus q {({\tilde{s}}_{1})}^{2 w_{1}})] \\ = [s_{(p α_{1}^{2 w_{1}} + q α_{2}^{2 w_{2}}) \times (p α_{2}^{2 w_{2}} + q α_{1}^{2 w_{1}})}, \\ = ((1 - (1 - u_{1}^{2 w_{1}})^{p} (1 - u_{2}^{2 w_{2}})^{q}) \\ (1 - (1 - u_{2}^{2 w_{2}})^{p} (1 - u_{1}^{2 w_{1}})^{q}), \\ 1 - (1 - (1 - (1 - v_{1})^{2 w_{1}})^{p} \\ (1 - (1 - v_{2})^{2 w_{2}})^{q}) \\ (1 - (1 - (1 - v_{2})^{2 w_{2}})^{p} (1 - (1 - v_{1})^{2 w_{1}})^{q}))] \end{matrix}$ (25)

2) Suppose (24) holds when n = r, i.e.,

$\begin{matrix} [\otimes_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{r} (p {({\tilde{s}}_{i})}^{{rw}_{i}} \oplus q {({\tilde{s}}_{j})}^{{rw}_{j}})] & = & [s_{\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{r} (p α_{i}^{{rw}_{i}} + q α_{j}^{{rw}_{j}})}, (\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{r} (1 - (1 - u_{i}^{{rw}_{i}})^{p} (1 - u_{j}^{{rw}_{j}})^{q}), \\ 1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{r} (1 - (1 - (1 - v_{i})^{{rw}_{i}})^{p} (1 - (1 - v_{j})^{{rw}_{j}})^{q}))] \end{matrix}$ (26)

then, when n = r + 1, by formulas (3–5), we can obtain

$\begin{matrix} \otimes_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{r + 1} (p {({\tilde{s}}_{i})}^{(r + 1) w_{i}} \oplus q {({\tilde{s}}_{j})}^{(r + 1) w_{j}}) = (\otimes_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{r} (p {({\tilde{s}}_{i})}^{(r + 1) w_{i}} \oplus q {({\tilde{s}}_{j})}^{(r + 1) w_{j}})) \\ \otimes (\otimes_{i = 1}^{r} (p {({\tilde{s}}_{i})}^{(r + 1) w_{i}} \oplus q {({\tilde{s}}_{r + 1})}^{(r + 1) w_{r + 1}})) \otimes (\otimes_{j = 1}^{r} (p {({\tilde{s}}_{r + 1})}^{(r + 1) w_{r + 1}} \oplus q {({\tilde{s}}_{j})}^{(r + 1) w_{j}})) \end{matrix}$ (27)

Further, we need to prove that

$\begin{array}{l} \otimes_{i = 1}^{r} (p {({\tilde{s}}_{i})}^{(r + 1) w_{i}} \oplus q {({\tilde{s}}_{r + 1})}^{(r + 1) w_{r + 1}}) \\ = [s_{\prod_{i = 1}^{r} (p α_{i}^{(r + 1) w_{i}} + q α_{r + 1}^{^{(r + 1) w_{r + 1}}})}, (\prod_{i = 1}^{r} (1 - {(1 - u_{i}^{(r + 1) w_{i}})}^{p} {(1 - u_{r + 1}^{(r + 1) w_{r + 1}})}^{q}), \\ 1 - \prod_{i = 1}^{r} (1 - {(1 - {(1 - v_{i})}^{(r + 1) w_{i}})}^{p} {(1 - {(1 - v_{r + 1})}^{(r + 1) w_{r + 1}})}^{q}))] \end{array}$ (28)

By mathematical induction on r, we have for r = 2, by formula (23), we obtain $\begin{matrix} p {({\tilde{s}}_{i})}^{(r + 1) w_{i}} \oplus q {({\tilde{s}}_{r + 1})}^{(r + 1) w_{r + 1}} = (s_{p α_{i}^{(2 + 1) w_{i}} + q α_{2 + 1}^{(2 + 1) w_{2 + 1}}}, (\prod_{i = 1}^{r} (1 - (1 - u_{i}^{(2 + 1) w_{i}})^{p} (1 - u_{2 + 1}^{(2 + 1) w_{2 + 1}})^{q}), \\ 1 - \prod_{i = 1}^{r} (1 - (1 - (1 - v_{i})^{(2 + 1) w_{i}})^{p} (1 - (1 - v_{2 + 1})^{(2 + 1) w_{2 + 1}})^{q}) i = 1, 2 \end{matrix}$ (29) $\begin{matrix} \otimes_{i = 1}^{r} (p {({\tilde{s}}_{i})}^{(r + 1) w_{i}} \oplus q {({\tilde{s}}_{r + 1})}^{(r + 1) w_{r + 1}}) \\ = (p {({\tilde{s}}_{1})}^{(2 + 1) w_{1}} \oplus q {({\tilde{s}}_{2 + 1})}^{(2 + 1) w_{2 + 1}}) \otimes (p {({\tilde{s}}_{2})}^{(2 + 1) w_{2}} \oplus q {({\tilde{s}}_{2 + 1})}^{(2 + 1) w_{2 + 1}}) \\ = [s_{(p α_{1}^{(2 + 1) w_{1}} + q α_{2 + 1}^{(2 + 1) w_{2 + 1}}) \times (p α_{2}^{(2 + 1) w_{2}} + q α_{2 + 1}^{(2 + 1) w_{2 + 1}})}, (\prod_{i = 1}^{2} (1 - (1 - u_{α_{i}}^{(2 + 1) w_{i}})^{p} (1 - u_{α_{2 + 1}}^{(2 + 1) w_{2 + 1}})^{q}), \\ 1 - \prod_{i = 1}^{2} (1 - (1 - (1 - v_{α_{i}})^{(2 + 1) w_{i}})^{p} (1 - (1 - v_{α_{2 + 1}})^{(2 + 1) w_{2 + 1}})^{q}))] \end{matrix}$ (30)

3) If (28) holds for r = r₀, i . e .,

$\begin{matrix} \otimes_{i = 1}^{r} (p {({\tilde{s}}_{i})}^{(r + 1) w_{i}} \oplus q {({\tilde{s}}_{r + 1})}^{(r + 1) w_{r + 1}}) \\ = [s_{\prod_{i = 1}^{r_{0}} (p α_{i}^{(r_{0} + 1) w_{i}} + q α_{r_{0} + 1}^{(r_{0} + 1) w_{r_{0} + 1}})}, (\prod_{i = 1}^{r_{0}} (1 - (1 - u_{α_{i}}^{(r_{0} + 1) w_{i}})^{p} (1 - u_{α_{r_{0} + 1}}^{(r_{0} + 1) w_{r_{0} + 1}})^{q}), \\ 1 - \prod_{i = 1}^{r_{0}} (1 - (1 - (1 - v_{α_{i}})^{(r_{0} + 1) w_{i}})^{p} (1 - (1 - v_{α_{r_{0} + 1}})^{(r_{0} + 1) w_{r_{0} + 1}})^{q})] \end{matrix}$ (31)

then, when r = r₀ + 1, by formulas (23) (3) (4), we obtain

$\begin{matrix} \otimes_{i = 1}^{r} (p {({\tilde{s}}_{i})}^{(r + 1) w_{i}} \oplus q {({\tilde{s}}_{r + 1})}^{(r + 1) w_{r + 1}}) (\otimes_{i = 1}^{r_{0} + 1} (p {({\tilde{s}}_{i})}^{(r_{0} + 2) w_{i}} \oplus q {({\tilde{s}}_{r_{0} + 2})}^{(r_{0} + 2) w_{r_{0} + 2}})) \\ \otimes (p {({\tilde{s}}_{r_{0} + 2})}^{(r_{0} + 2) w_{r_{0} + 2}} \oplus q {({\tilde{s}}_{r_{0} + 2})}^{(r_{0} + 2) w_{r_{0} + 2}}) \\ = [s_{\prod_{i = 1}^{r_{0} + 1} (p α_{i}^{(r_{0} + 2) w_{i}} + q α_{r_{0} + 2}^{(r_{0} + 2) w_{r_{0} + 2}})}, (\prod_{i = 1}^{r_{0} + 1} (1 - (1 - u_{α_{i}}^{(r_{0} + 2) w_{i}})^{p} (1 - u_{α_{r_{0} + 2}}^{(r_{0} + 2) w_{r_{0} + 2}})^{q}), \\ 1 - \prod_{i = 1}^{r_{0} + 1} (1 - (1 - (1 - v_{α_{i}})^{(r_{0} + 2) w_{i}})^{p} (1 - (1 - v_{α_{r_{0} + 2}})^{(r_{0} + 2) w_{r_{0} + 2}})^{q}))] \end{matrix}$ (32)

The above proofs prove that formula (28) holds for r = r₀ + 1. From above all, formula (28) holds for all r.

Similarly, it is easy to prove that

$\begin{matrix} \oplus_{j = 1}^{r} (p {\tilde{s}}_{r + 1}^{(r + 1) w_{r + 1}} \otimes q {\tilde{s}}_{j}^{(r + 1) w_{j}}) \\ = [s_{\prod_{j = 1}^{r} (p α_{r + 1}^{(r + 1) w_{r + 1}} + q α_{j}^{(r + 1) w_{j}})}, (\prod_{j = 1}^{r} (1 - (1 - u_{α_{r + 1}}^{(r + 1) w_{r + 1}})^{p} (1 - u_{α_{j}}^{(r + 1) w_{j}})^{q}), \\ 1 - \prod_{j = 1}^{r} (1 - (1 - (1 - v_{α_{r + 1}})^{(r + 1) w_{r + 1}})^{p} (1 - (1 - v_{α_{j}})^{(r + 1) w_{j}})^{q}))] \end{matrix}$ (33)

Thus, by formulas (26) (28) (33), we can transform formula (27) as

$\begin{matrix} \otimes_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{r + 1} (p {\tilde{s}}_{i}^{(r + 1) w_{i}} \oplus q {\tilde{s}}_{j}^{(r + 1) w_{j}}) \\ = [s_{\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{r + 1} (p α_{i}^{(r + 1) w_{i}} + q α_{j}^{(r + 1) w_{j}})}, ((\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{r} (1 - {(1 - u_{α_{i}}^{{nw}_{i}})}^{p} {(1 - u_{α_{j}}^{{nw}_{j}})}^{q}), \\ 1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{r} (1 - {(1 - {(1 - v_{α_{i}})}^{{nw}_{i}})}^{p} {(1 - {(1 - v_{α_{j}})}^{{nw}_{j}})}^{q})) \\ \oplus (\prod_{i = 1}^{r} (1 - (1 - u_{α_{i}}^{(r + 1) w_{i}})^{p} (1 - u_{α_{r + 1}}^{(r + 1) w_{r + 1}})^{q}), \\ 1 - \prod_{i = 1}^{r} (1 - (1 - (1 - v_{α_{i}})^{(r + 1) w_{i}})^{p} (1 - (1 - v_{α_{r + 1}})^{(r + 1) w_{r + 1}})^{q})) \end{matrix}$

$\begin{matrix} \oplus (\prod_{j = 1}^{r} (1 - (1 - u_{α_{r + 1}}^{(r + 1) w_{r + 1}})^{p} (1 - u_{α_{j}}^{(r + 1) w_{j}})^{q}), \\ 1 - \prod_{j = 1}^{r} (1 - (1 - (1 - v_{α_{r + 1}})^{(r + 1) w_{r + 1}})^{p} (1 - (1 - v_{α_{j}})^{(r + 1) w_{j}})^{q})))] \\ = [s_{\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{r + 1} (p α_{i}^{{nw}_{i}} + q α_{j}^{{nw}_{j}})}, (\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{r + 1} (1 - {(1 - u_{α_{i}}^{{nw}_{i}})}^{p} {(1 - u_{α_{j}}^{{nw}_{j}})}^{q}), \\ 1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{r + 1} (1 - {(1 - {(1 - v_{α_{i}})}^{{nw}_{i}})}^{p} {(1 - {(1 - v_{α_{j}})}^{{nw}_{j}})}^{q}))] \end{matrix}$ (34)

The formula (34) proves that (24) holds for n = r + 1. Thus, (24) holds for all of n.

Then, by formula (23) and formula (6), we obtain $\begin{matrix} {(\otimes_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (p {\tilde{s}}_{i}^{{nw}_{i}} \oplus q {\tilde{s}}_{j}^{{nw}_{j}}))}^{\frac{1}{n (n - 1)}} & = & [s_{{(\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (p α_{i}^{{nw}_{i}} + q α_{j}^{{nw}_{j}}))}^{\frac{1}{n (n - 1)}}}, ({(\prod_{i, j = 1}^{n} (1 - (1 - u_{α_{i}}^{{nw}_{i}})^{p} (1 - u_{α_{j}}^{{nw}_{j}})^{q})}^{\frac{1}{n (n - 1)}}, \\ 1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - {(1 - (1 - v_{α_{i}})^{{nw}_{i}})}^{p} {(1 - (1 - v_{α_{j}})^{{nw}_{j}})}^{q})}^{\frac{1}{n (n - 1)}})] \end{matrix}$ (35)

And then, by formulas (5) and (35), we obtain $\begin{matrix} {ILWGBM}_{w}^{p, q} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) = {[\frac{1}{p + q} (\otimes_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (p {\tilde{s}}_{i}^{{nw}_{i}} \oplus q {\tilde{s}}_{j}^{{nw}_{j}}))]}^{\frac{1}{n (n - 1)}} \\ = [s_{\frac{1}{p + q} {(\prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (p {(α_{i})}^{{nw}_{i}} + q {(α_{j})}^{{nw}_{j}}))}^{\frac{1}{n (n - 1)}}}, (1 - {(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - {(1 - u_{α_{i}}^{{nw}_{i}})}^{p} {(1 - u_{α_{j}}^{{nw}_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}}, \\ {(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - {(1 - {(1 - v_{α_{i}})}^{{nw}_{i}})}^{p} {(1 - {(1 - v_{α_{j}})}^{{nw}_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}})] \end{matrix}$ (36)

The above proofs can prove that Equation (20) holds. Furthermore, since $\begin{matrix} 0 \leq 1 - {(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - {(1 - u_{α_{i}}^{{nw}_{i}})}^{p} {(1 - u_{α_{j}}^{{nw}_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}} \leq 1 \\ 0 \leq {(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - {(1 - {(1 - v_{α_{i}})}^{{nw}_{i}})}^{p} {(1 - {(1 - v_{α_{j}})}^{{nw}_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}} \leq 1 \end{matrix}$ and by $0 \leq u_{A} (x) + v_{A} (x) \leq 1, \forall x \in X, and π_{A} (x) = 1 - u_{A} (x) - v_{A} (x)$ we obtain that $\begin{matrix} 1 - {(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - {(1 - u_{α_{i}}^{{nw}_{i}})}^{p} {(1 - u_{α_{j}}^{{nw}_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}} \\ + {(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - {(1 - {(1 - v_{α_{i}})}^{{nw}_{i}})}^{p} {(1 - {(1 - v_{α_{j}})}^{{nw}_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}} \\ \leq 1 + {(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - {(1 - {(1 - v_{α_{i}})}^{{nw}_{i}})}^{p} {(1 - {(1 - v_{α_{j}})}^{{nw}_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}} \\ - {(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - {(1 - {(1 - v_{α_{i}})}^{{nw}_{i}})}^{p} {(1 - {(1 - v_{α_{j}})}^{{nw}_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}} = 1 \end{matrix}$

By now, we complete the proof of Theorem 2.

In the following, we discuss the special cases of the ILWGBM operator.

1) if q → 0, the ILWGBM operator reduces to an intuitionistic linguistic generalized weighted mean operator; it follows that $\begin{matrix} {ILWGBM}_{w}^{p, 0} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) \\ = [s_{\frac{1}{p} \prod_{i = 1}^{n} {(p {(α_{i})}^{{nw}_{i}})}^{\frac{1}{n}}}, (1 - {(1 - \prod_{i = 1}^{n} {(1 - {(1 - u_{α_{i}}^{{nw}_{i}})}^{p})}^{\frac{1}{n}})}^{\frac{1}{p}}, {(1 - \prod_{i = 1}^{n} {(1 - {(1 - {(1 - v_{α_{i}})}^{{nw}_{i}})}^{p})}^{\frac{1}{n}})}^{\frac{1}{p}})] \\ = {ILWGBM}_{w}^{p, 0} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) \end{matrix}$

2) if p = 1 and q = 0, then ILWGBM operator reduces to an intuitionistic linguistic arithmetic weighted average operator. $\begin{matrix} {ILWGBM}_{w}^{1, 0} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) \\ = {(\otimes_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({({\tilde{s}}_{i})}^{{nw}_{i}}))}^{\frac{1}{n (n - 1)}} = \prod_{i = 1}^{n} {({\tilde{s}}_{i})}^{w_{i}} \\ = [s_{\prod_{i = 1}^{n} {(α_{i})}^{w_{i}}}, (\prod_{i = 1}^{n} u_{α_{i}}^{w_{i}}, 1 - \prod_{i = 1}^{n} {(1 - v_{α_{i}})}^{w_{i}})] \end{matrix}$

3) if p = 2 and q → 0, then ILWGBM operator reduces to an intuitionistic linguistic square geometric weighted mean operator. $\begin{matrix} {ILWGBM}_{w}^{2, 0} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) = \frac{1}{2} {(\otimes_{i = 1}^{n} (2 {({\tilde{s}}_{i})}^{{nw}_{i}}))}^{\frac{1}{n}} \\ = [s_{\frac{1}{2} \prod_{i = 1}^{n} {(2 {(α_{i})}^{{nw}_{i}})}^{\frac{1}{n}}}, (1 - {(1 - \prod_{i = 1}^{n} {(1 - {(1 - u_{α_{i}}^{{nw}_{i}})}^{2})}^{\frac{1}{n}})}^{\frac{1}{2}}, {(1 - \prod_{i = 1}^{n} {(1 - {(1 - {(1 - v_{α_{i}})}^{{nw}_{i}})}^{2})}^{\frac{1}{n}})}^{\frac{1}{2}})] \end{matrix}$

4 The ILDWBM operator and the ILDWGBM operator

In the above discussion, the ILWBM and ILWGBM operators consider the mutual relations of each arguments and the different importance of each arguments. Because decision makers have the individual predilection in real-life decision making problems, some individuals may give unduly high or unduly low preference values to their preferred or repugnant objects. So it is disadvantageous to select the optimal project. In order to release the influence of experts’ preference, we combine the dependent ordered weighted averaging operator with the Bonferrioni mean operator under intuitionistic linguistic environment, and propose the intuitionistic linguistic dependent weighted Bonferroni mean (ILDWBM) operator and the intuitionistic linguistic dependent weighted geometric Bonferroni mean (ILDWGBM) operator.

4.1 The ILDWBM operator

Definition 12. [25] Let ${\tilde{s}}_{j} = [s_{α_{j}}, (u (α_{j}), v (α_{j}))]$ (j = 1, 2, …, n) be a set of the intuitionistic linguistic numbers, then $\tilde{x} = [s_{α_{\tilde{x}}}, (u (\tilde{x}), v (\tilde{x}))]$ is defined as the mean of the intuitionistic linguistic set, where

$\begin{matrix} s_{α_{\tilde{x}}} = \frac{1}{n} \sum_{j = 1}^{n} s_{α_{j}}, u (\tilde{x}) = 1 - {(\prod_{j = 1}^{n} (1 - u (α_{j})))}^{\frac{1}{n}}, \\ v (\tilde{x}) = {(\prod_{j = 1}^{n} v (α_{j}))}^{\frac{1}{n}} \end{matrix}$ (37)

If $s ({\tilde{s}}_{j}, \tilde{x}) = 1 - \frac{d ({\tilde{s}}_{j}, \tilde{x})}{\sum_{j = 1}^{n} d ({\tilde{s}}_{j}, \tilde{x})}$ (38)

then, the $s ({\tilde{s}}_{j}, \tilde{x})$ is called the similarity degree between the argument ${\tilde{s}}_{j}$ and $\tilde{x}$ . If $ω_{j} = \frac{s ({\tilde{s}}_{j}, \tilde{x})}{\sum_{j = 1}^{n} s ({\tilde{s}}_{j}, \tilde{x})}$ (39) then, the ω_j is called the dependent weight of the input arguments ${\tilde{s}}_{j}$ .

Definition 13. Let ${\tilde{s}}_{j} = [s_{α_{j}}, (u (α_{j}), v (α_{j}))]$ be a set of the intuitionistic linguistic numbers, p, q ≥ 0, and ILDWBM : Ωⁿ → Ω. If

$\begin{matrix} {ILDWBM}_{ω}^{p, q} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) \\ = {(\frac{1}{n (n - 1)} (\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({(n ω_{i} {\tilde{s}}_{i})}^{p} \otimes {(n ω_{j} {\tilde{s}}_{j})}^{q})))}^{\frac{1}{p + q}} \end{matrix}$ (40) where Ω is the set of all linguistic numbers, ω = (ω₁, ω₂, ⋯ , ω_n) ^T is the weight vector defined by formula (39), $\sum_{i = 1}^{n} ω_{i} = 1$ . Then ${ILDWBM}_{ω}^{p, q}$ is called the intuitionistic linguistic dependent weighted Bonferroni mean (ILDWBM) operator.

Theorem 3. Let ${\tilde{s}}_{j} = [s_{α_{j}}, (u (α_{j}), v (α_{j}))]$ be a set of the intuitionistic linguistic numbers, p, q ≥ 0, then the aggregated result from Definition 13 is also an ILN, and $\begin{matrix} {ILDWBM}_{ω}^{p, q} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) \\ = [s_{{(\frac{1}{n (n - 1)} \sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(n ω_{i} α_{i})}^{p} \times {(n ω_{j} α_{j})}^{q})}^{\frac{1}{p + q}}}, ({(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - {(1 - {(1 - u_{α_{i}})}^{n ω_{i}})}^{p} \times {(1 - {(1 - u_{α_{j}})}^{n ω_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}}, \\ 1 - {(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - {(1 - v_{α_{i}}^{n ω_{i}})}^{p} {(1 - v_{α_{j}}^{n ω_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}})] \end{matrix}$ (41)

Similar to Theorem 2, it can be proved by using mathematical induction on n.

If the values of the parameters p and q change in the ILDWBM operator, then some special cases can be obtained as follows:

1) if q = 0, the ILDWBM operator reduces to an intuitionistic linguistic dependent generalized weighted mean operator; it follows that $\begin{matrix} {ILDWBM}_{w}^{p, 0} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) \\ = {(\frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {({nw}_{i} {\tilde{s}}_{i})}^{p})}^{\frac{1}{p}} = {(\frac{1}{n} \sum_{i = 1}^{n} {({nw}_{i} {\tilde{s}}_{i})}^{p})}^{\frac{1}{p}} \\ = [s_{{(\frac{1}{n} \sum_{i = 1}^{n} {({nw}_{i} α_{i})}^{p})}^{\frac{1}{p}}}, ({(1 - \prod_{i = 1}^{n} {(1 - {(1 - {(1 - u_{i})}^{{nw}_{i}})}^{p})}^{\frac{1}{n}})}^{\frac{1}{p}}, 1 - {(1 - \prod_{i = 1}^{n} {(1 - {(1 - v_{i}^{{nw}_{i}})}^{p})}^{\frac{1}{n}})}^{\frac{1}{p}})] \end{matrix}$

2) if p = 1 and q = 0, then ILDWBM operator reduces to an intuitionistic linguistic dependent arithmetic weighted average operator. $\begin{matrix} {ILDWBM}_{w}^{1, 0} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) = (\frac{1}{n (n - 1)} (\oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (({nw}_{i} {\tilde{s}}_{i}) \otimes {({nw}_{j} {\tilde{s}}_{j})}^{0}))) \\ = \frac{1}{n (n - 1)} \oplus_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} ({nw}_{i} {\tilde{s}}_{i}) = \sum_{i = 1}^{n} w_{i} {\tilde{s}}_{i} = [s_{(\sum_{i = 1}^{n} w_{i} α_{i})}, ((1 - \prod_{i = 1}^{n} {(1 - u_{i})}^{w_{i}}), \prod_{i = 1}^{n} (v_{i}^{w_{i}}))] \end{matrix}$

3) if p → 0 and q = 0, then ILDWBM operator reduces to an intuitionistic linguistic dependent geometric weighted mean operator.

$\begin{matrix} lim_{p \to 0} {ILDWBM}_{w}^{p, 0} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) = lim_{p \to 0} {(\frac{1}{n} \sum_{i = 1}^{n} {({nw}_{i} {\tilde{s}}_{i})}^{p})}^{\frac{1}{p}} = \prod_{i = 1}^{n} {({nw}_{i} {\tilde{s}}_{i})}^{\frac{1}{n}} \\ = [s_{\prod_{i = 1}^{n} {({nw}_{j} α_{j})}^{\frac{1}{n}}}, ({(\prod_{i = 1}^{n} {(1 - {(1 - u_{i})}^{{nw}_{i}})}^{\frac{1}{n}})}^{\frac{1}{p + q}}, 1 - \prod_{i = 1}^{n} {(1 - ν_{i}^{{nw}_{i}})}^{\frac{1}{n}})] \end{matrix}$

4.2 The ILDWGBM operator

Definition 14. Let ${\tilde{s}}_{j} = [s_{α_{j}}, (u (α_{j}), v (α_{j}))]$ be a set of the intuitionistic linguistic numbers, and p, q ≥ 0, ILDWGBM : Ωⁿ → Ω. If

$\begin{matrix} {ILDWGBM}_{ω}^{p, q} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) \\ = \frac{1}{p + q} {(\otimes_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (p {({\tilde{s}}_{i})}^{n ω_{i}} \oplus q {({\tilde{s}}_{j})}^{n ω_{j}}))}^{\frac{1}{n (n - 1)}} \end{matrix}$ (42) where Ω is the set of all linguistic numbers, ω = (ω₁, ω₂, …, ω_n) ^T is the weight vector defined by formula (39), n is a balance parameter. Then ${ILDWGBM}_{ω}^{p, q}$ is called the intuitionistic linguistic dependent weighted geometric Bonferroni mean (ILDWGBM) operator.

Theorem 4. Let ${\tilde{s}}_{j} = [s_{α_{j}}, (u (α_{j}), v (α_{j}))]$ be a set of the intuitionistic linguistic numbers, and p, q ≥ 0. Then the aggregated result from Definition 14 is an ILN, and

$\begin{matrix} {ILDWGBM}_{ω}^{p, q} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) \\ = [s_{\frac{1}{p + q} {(\sum_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} (p {(α_{i})}^{n ω_{i}} + q {(α_{j})}^{n ω_{j}}))}^{\frac{1}{n (n - 1)}}}, (1 - {(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - {(1 - u_{α_{i}}^{n ω_{i}})}^{p} {(1 - u_{α_{j}}^{n ω_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}}, \\ {(1 - \prod_{\begin{matrix} i, j = 1 \\ i \neq j \end{matrix}}^{n} {(1 - {(1 - {(1 - v_{α_{i}})}^{n ω_{i}})}^{p} {(1 - {(1 - v_{α_{j}})}^{n ω_{j}})}^{q})}^{\frac{1}{n (n - 1)}})}^{\frac{1}{p + q}})] \end{matrix}$ (43)

Similar to the Theorem 3, it can be proved by using mathematical induction on n.

If the values of the parameters p and q change in the ILDWGBM operator, then some special cases can be obtained as follows:

1) If q = 0, the ILDWGBM operator reduces to an intuitionistic linguistic dependent generalized weighted mean operator; it follows that $\begin{matrix} {ILDWGBM}_{ω}^{p, 0} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) = \frac{1}{p} {(\otimes_{i = 1}^{n} (p {({\tilde{s}}_{i})}^{n ω_{i}}))}^{\frac{1}{n}} \\ = [s_{\frac{1}{p} \prod_{i = 1}^{n} {(p {(α_{i})}^{{nw}_{i}})}^{\frac{1}{n}}}, (1 - {(1 - \prod_{i = 1}^{n} {(1 - {(1 - u_{α_{i}}^{{nw}_{i}})}^{p})}^{\frac{1}{n}})}^{\frac{1}{p}}, \\ {(1 - \prod_{i = 1}^{n} {(1 - {(1 - {(1 - v_{α_{i}})}^{{nw}_{i}})}^{p})}^{\frac{1}{n}})}^{\frac{1}{p}})] \end{matrix}$

2) If p = 1 and q = 0, then ILWGBM operator reduces to an intuitionistic linguistic arithmetic weighted average operator.

${ILDWGBM}_{w}^{1, 0} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) = \prod_{i = 1}^{n} {({\tilde{s}}_{i})}^{w_{i}} = [s_{\prod_{i = 1}^{n} {(α_{i})}^{w_{i}}}, (\prod_{i = 1}^{n} u_{α_{i}}^{w_{i}}, 1 - \prod_{i = 1}^{n} {(1 - v_{α_{i}})}^{w_{i}})]$

3) If p = 2 and q → 0, then ILDWGBM operator reduces to an intuitionistic linguistic square geometric weighted mean operator. $\begin{matrix} {ILDWGBM}_{w}^{2, 0} ({\tilde{s}}_{1}, {\tilde{s}}_{2}, \dots, {\tilde{s}}_{n}) = \frac{1}{2} {(\otimes_{i = 1}^{n} (2 {({\tilde{s}}_{i})}^{{nw}_{i}}))}^{\frac{1}{n}} \\ = [s_{\frac{1}{2} \prod_{i = 1}^{n} {(2 {(α_{i})}^{{nw}_{i}})}^{\frac{1}{n}}}, (1 - {(1 - \prod_{i = 1}^{n} {(1 - {(1 - u_{α_{i}}^{{nw}_{i}})}^{2})}^{\frac{1}{n}})}^{\frac{1}{2}}, {(1 - \prod_{i = 1}^{n} {(1 - {(1 - {(1 - v_{α_{i}})}^{{nw}_{i}})}^{2})}^{\frac{1}{n}})}^{\frac{1}{2}})] \end{matrix}$

5 A MCGDM method based on intuitionistic linguistic numbers

Consider a multi-criteria group decision-making problem under intuitionistic linguistic environment: let A = (A₁, A₂, …, A_m) be a set of alternatives, and C = (C₁, C₂, …, C_n) be a set of criteria, w = (w₁, w₂, …, w_n) be the weighted vector of the criteria c_j, which satisfies w_j ≥ 0 (j = 1, 2, …, n), $\sum_{j = 1}^{n} w_{j} = 1$ . Let D = (D₁, D₂, …, D_k) be the set of decision maker, ω = (ω₁, ω₂, …, ω_t) be the expert weight, where ω_k ≥ 0(k = 1, …, t) and $\sum_{k = 1}^{t} ω_{k} = 1$ . Supposed that $R^{k} = [r_{ij}^{k}]_{m \times n}$ is the evaluation matrix of expert k, where $r_{ij}^{k} = [s_{ij}^{k}, (u_{ij}^{k}, v_{ij}^{k})]$ , $s_{ij}^{k} \in S,$ $0 \leq u_{ij}^{k} \leq 1$ , $0 \leq v_{ij}^{k} \leq 1$ and $0 \leq μ_{ij}^{k} + v_{ij}^{k} \leq 1$ . Then, the goal of this MAGDM problem is to rank the alternatives.

In last section, we proposed two types of aggregation operators, i.e., the ILWBM operator and ILWGBM operator, and they have the different advantages. The ILWBM operator emphasizes on the impact of the overall data, and allows the complementarity among the various data; the ILWGBM operator emphasizes on the coordination among the various data, and does not allow a “short board” phenomenon. In general, for the decision makers with optimistic attributes, the LWBM operator will be adopted and ILWGBM operator is selected for the decision makers with pessimistic attributes.

In the following, the multi-criteria group decision making method with intuitionistic linguistic information is described as follows:

Step 1. Calculate the comprehensive evaluation values ${\tilde{r}}_{i}^{k}$ and ${\tilde{r}}_{i}^{k^{'}}$ of each alternative from every expert by the ILWBM or ILWGBM operator respectively.

${\tilde{r}}_{i}^{k} = ILWBM ({\tilde{r}}_{i 1}^{k}, {\tilde{r}}_{i 2}^{k}, \dots, {\tilde{r}}_{in}^{k})$ (44) or ${\tilde{r}}_{i}^{k^{'}} = ILWGBM ({\tilde{r}}_{i 1}^{k^{'}}, {\tilde{r}}_{i 2}^{k^{'}}, \dots, {\tilde{r}}_{in}^{k^{'}})$ (45)

Step 2. Calculate the dependent weights $ω_{i}^{k}$ and $ω_{i}^{k^{'}}$ vector of $r_{i}^{k}$ and $r_{i}^{k^{'}}$ as follows. $s ({\tilde{r}}_{i}^{k}, {\tilde{x}}_{i}) = 1 - \frac{d ({\tilde{r}}_{i}^{k}, {\tilde{x}}_{i})}{\sum_{k = 1}^{t} d ({\tilde{r}}_{i}^{k}, {\tilde{x}}_{i})}$ (46) where, ${\tilde{x}}_{i} = \frac{1}{t} \sum_{k = 1}^{t} {\tilde{r}}_{i}^{k}$ , $ω_{i}^{k} = \frac{s ({\tilde{r}}_{i}^{k}, {\tilde{x}}_{i})}{\sum_{k = 1}^{t} s ({\tilde{r}}_{i}^{k}, {\tilde{x}}_{i})}$

or $s ({\tilde{r}}_{i}^{k^{'}}, {\tilde{x}}_{i}^{'}) = 1 - \frac{d ({\tilde{r}}_{i}^{k^{'}}, {\tilde{x}}_{i}^{'})}{\sum_{k = 1}^{t} d ({\tilde{r}}_{i}^{k^{'}}, {\tilde{x}}_{i}^{'})}$ where, ${\tilde{x}}_{i} = \frac{1}{t} \sum_{k = 1}^{t} {\tilde{r}}_{i}^{k}$ , $ω_{i}^{k^{'}} = \frac{s ({\tilde{r}}_{i}^{k^{'}}, {\tilde{x}}_{i}^{'})}{\sum_{k = 1}^{t} s ({\tilde{r}}_{i}^{k^{'}}, {\tilde{x}}_{i}^{'})} (\sum_{k = 1}^{t} ω_{i}^{k} = 1)$

Step 3. Calculate the collective evaluation value by ILDWBM or ILDWGBM operator. ${\tilde{r}}_{i} = {ILDWBM}_{ω}^{p, q} ({\tilde{r}}_{i}^{1}, {\tilde{r}}_{i}^{2}, \dots, {\tilde{r}}_{i}^{t})$ (47) or ${\tilde{r}}_{i}^{'} = {ILDWGBM}_{ω}^{p, q} ({\tilde{r}}_{i}^{1^{'}}, {\tilde{r}}_{i}^{2^{'}}, \dots, {\tilde{r}}_{i}^{t^{'}})$ (48)

Step 4. Calculate the score value of each alternatives by Definition 3.

Step 5. Rank the results of Step 4 in descending order.

Step 6. Rank the alternatives according to the orderings of step 5 by Definition 5 and select the optimal project.

6 An illustrative example

There is an investment company which has a sum of money and wants to invest the money in the best option. There is a panel with four possible alternatives in which to invest the money:

A₁ is a car company;

A₂ is a computer company;

A₃ is a TV company;

A₄ is a food company.

There are the following four attributes that the investment company must refer to make a decision (suppose that the weight vector of four criteria is w = (0.32, 0.26, 0.18, 0.24):

C₁ is the risk analysis;

C₂ is the growth analysis;

C₃ is the sociopolitical impact analysis;

C₄ is the environmental impact analysis.

The following decision matrix ${\tilde{R}}^{k} = [{\tilde{r}}_{ij}^{k}]_{4 \times 4}$ (k = 1, 2, 3) listed in Tables 1–3 are given by three decision makers under the above four criteria.

Table 1
Decision matrix ${\tilde{R}}^{1}$

C ₁ C ₂ C ₃ C ₄

A ₁ <s₅, (0.2, 0.7)> <s₂, (0.4, 0.6)> <s₅, (0.5, 0.5)> <s₃, (0.2, 0.6)>

A ₂ <s₄, (0.4, 0.6)> <s₅, (0.4, 0.5)> <s₃, (0.1, 0.8)> <s₄, (0 .5, 0 . 5)>

A ₃ <s₃, (0.2, 0.7)> <s₄, (0.2, 0.7)> <s₅, (0 .3, 0.7)> <s₅, (0.2, 0.7)>

A ₄ <s₆, (0 .5, 0 . 4)> <s₂, (0.2, 0 . 8)> <s₃, (0.2, 0 . 6)> <s₃, (0 .3, 0 . 6)>

	C ₁	C ₂	C ₃	C ₄
A ₁	<s₅, (0.2, 0.7)>	<s₂, (0.4, 0.6)>	<s₅, (0.5, 0.5)>	<s₃, (0.2, 0.6)>
A ₂	<s₄, (0.4, 0.6)>	<s₅, (0.4, 0.5)>	<s₃, (0.1, 0.8)>	<s₄, (0 .5, 0 . 5)>
A ₃	<s₃, (0.2, 0.7)>	<s₄, (0.2, 0.7)>	<s₅, (0 .3, 0.7)>	<s₅, (0.2, 0.7)>
A ₄	<s₆, (0 .5, 0 . 4)>	<s₂, (0.2, 0 . 8)>	<s₃, (0.2, 0 . 6)>	<s₃, (0 .3, 0 . 6)>

Table 2

Decision matrix ${\tilde{R}}^{2}$

	C ₁	C ₂	C ₃	C ₄
A ₁	<s₄, (0 .1, 0.7)>	<s₃, (0 .2, 0 . 7)>	<s₃, (0 .2, 0 . 8)>	<s₆, (0 .4, 0 . 5)>
A ₂	<s₅, (0.4, 0 . 5)>	<s₃, (0 .3, 0 . 6)>	<s₄, (0 .2, 0 . 6)>	<s₃, (0 .2, 0 . 7)>
A ₃	<s₄, (0.2, 0 . 6)>	<s₄, (0.2, 0.7)>	<s₂, (0 .4, 0 . 6)>	<s₃, (0 .3, 0.7)>
A ₄	<s₅, (0 .3, 0 . 6)>	<s₄, (0 .4, 0 . 5)>	<s₂, (0 .3, 0 . 6)>	<s₄, (0 .2, 0 . 6)>

Table 3

Decision matrix ${\tilde{R}}^{3}$

	C ₁	C ₂	C ₃	C ₄
A ₁	<s₅, (0 .2, 0 . 6)>	<s₃, (0 .3, 0 . 7)>	<s₄, (0 .4, 0 . 5)>	<s₄, (0 .2, 0 . 7)>
A ₂	<s₄, (0 .3, 0 . 7)>	<s₅, (0 .3, 0 . 6)>	<s₂, (0 .1, 0 . 8)>	<s₃, (0 .1, 0 . 8)>
A ₃	<s₄, (0.2, 0 . 7)>	<s₅, (0 .3, 0 . 6)>	<s₁, (0 .1, 0 . 8)>	<s₄, (0 .2, 0.7)>
A ₄	<s₃, (0 .2, 0 . 7)>	<s₃, (0 .1, 0 . 7)>	<s₄, (0 .3, 0 . 6)>	<s₅, (0 .4, 0 . 5)>

The evaluation steps are given as follows:

Step 1. Calculate the comprehensive evaluation values ${\tilde{r}}_{i}^{k}$ and ${\tilde{r}}_{i}^{k^{'}}$ of each alternative from every expert. (For calculating convenience, let p = 0.5, q = 0.5 .).

The comprehensive evaluation values of each alternative by ILWBM operator as follow: $\begin{matrix} {\tilde{r}}_{1}^{1} = < s_{3 . 524}, (0.304, 0.610) >, \\ {\tilde{r}}_{1}^{2} = < s_{3 . 676}, (0.205, 0.685) >, \\ {\tilde{r}}_{1}^{3} = < s_{3 . 917}, (0.262, 0.635) > \\ {\tilde{r}}_{2}^{1} = < s_{3 . 937}, (0.335, 0.608) >, \\ {\tilde{r}}_{2}^{2} = < s_{3 . 159}, (0.267, 0.608) >, \\ {\tilde{r}}_{2}^{3} = < s_{3 . 393}, (0.186, 0.733) > \\ {\tilde{r}}_{3}^{1} = < s_{3 . 877}, (0.218, 0.704) >, \\ {\tilde{r}}_{3}^{2} = < s_{3 . 060}, (0.262, 0.657) >, \\ {\tilde{r}}_{3}^{3} = < s_{3 . 263}, (0.193, 0.706) > \\ {\tilde{r}}_{4}^{1} = < s_{3 . 306}, (0.286, 0.620) >, \\ {\tilde{r}}_{4}^{2} = < s_{4.441}, (0.292, 0.581) >, \\ {\tilde{r}}_{4}^{3} = < s_{3 . 611}, (0.229, 0.636) > \end{matrix}$

The comprehensive evaluation values of each alternative by ILWGBM operator as follow: $\begin{matrix} {\tilde{r}}_{1}^{1^{'}} = < s_{3 . 768}, (0.331, 0.592) >, \\ {\tilde{r}}_{1}^{2^{'}} = < s_{4.102}, (0.236, 0.669) >, \\ {\tilde{r}}_{1}^{3^{'}} = < s_{4.207}, (0.282, 0.620) > \\ {\tilde{r}}_{2}^{1^{'}} = < s_{4.222}, (0.350, 0.593) >, \\ {\tilde{r}}_{2}^{2^{'}} = < s_{3 . 952}, (0.280, 0.592) >, \\ {\tilde{r}}_{2}^{3^{'}} = < s_{3 . 798}, (0.198, 0.719) > \\ {\tilde{r}}_{3}^{1^{'}} = < s_{3 . 904}, (0.233, 0.692) >, \\ {\tilde{r}}_{3}^{2^{'}} = < s_{3 . 541}, (0.282, 0.643) >, \\ {\tilde{r}}_{3}^{3^{'}} = < s_{3 . 846}, (0.203, 0.693) > \\ {\tilde{r}}_{4}^{1^{'}} = < s_{3 . 816}, (0.307, 0.587) >, \\ {\tilde{r}}_{4}^{2^{'}} = < s_{4.179}, (0.308, 0.567) >, \\ {\tilde{r}}_{4}^{3^{'}} = < s_{3 . 626}, (0.256, 0.616) > \end{matrix}$

Step 2. Calculate the dependent weight $ω_{i}^{k}$ and $ω_{i}^{k^{'}}$ vector of $r_{i}^{k}$ and $r_{i}^{k^{'}}$ $\begin{matrix} ω_{i}^{k} & = & [\begin{matrix} 0.384 & 0.240 & 0.376 \\ 0.250 & 0.472 & 0.278 \\ 0.302 & 0.469 & 0.229 \\ 0.389 & 0.244 & 0.367 \end{matrix}], \\ ω_{i}^{k^{'}} & = & [\begin{matrix} 0.386 & 0.229 & 0.385 \\ 0.297 & 0.454 & 0.249 \\ 0.482 & 0.279 & 0.239 \\ 0.481 & 0.264 & 0.255 \end{matrix}] \end{matrix}$

Step 3. Calculate the collective evaluation value by ILDWBM or ILDWGBM operator respectively, shown in Table 4.

Table 4

The comprehensive evaluation values

ILDBM	ILDGBM
${\tilde{r}}_{1} = < s_{3.644}, (0.252, 0.649) >$	${\tilde{r}}_{1}^{'} = < s_{1.711}, (0.294, 0.616) >$
${\tilde{r}}_{2} = < s_{3.361}, (0.251, 0.663) >$	${\tilde{r}}_{2}^{'} = < s_{1.717}, (0.290, 0.622) >$
${\tilde{r}}_{3} = < s_{3.267}, (0.217, 0.699) >$	${\tilde{r}}_{3}^{'} = < s_{1.714}, (0.263, 0.652) >$
${\tilde{r}}_{4} = < s_{3.684}, (0.263, 0.619) >$	${\tilde{r}}_{4}^{'} = < s_{1.714}, (0.310, 0.571) >$

Step 4. Calculate the score value of each alternative by Definition 3.

The results by ILDWGBM operator are listed as follows: $\begin{matrix} S ({\tilde{r}}_{1}) = 0.183, S ({\tilde{r}}_{2}) = 0.165, \\ S ({\tilde{r}}_{3}) = 0.141, S ({\tilde{r}}_{4}) = 0.198 . \end{matrix}$

The results by ILDWBM operator are listed as follows: $\begin{matrix} S ({\tilde{r}}_{1}^{'}) = 0.097, S ({\tilde{r}}_{2}^{'}) = 0.096, \\ S ({\tilde{r}}_{3}^{'}) = 0.087, S ({\tilde{r}}_{4}^{'}) = 0.106 . \end{matrix}$

Step 5. From the above results, we can get the ordering of the score values under ILDWBM operator: $S_{({\tilde{r}}_{4})} > S_{({\tilde{r}}_{1})} > S_{({\tilde{r}}_{2})} > S_{({\tilde{r}}_{3})}$ or the ordering of the score values under ILDGBM operator as follows: $S_{({\tilde{r}}_{4})} > S_{({\tilde{r}}_{1})} > S_{({\tilde{r}}_{2})} > S_{({\tilde{r}}_{3})}$

Step 6. Rank all of the alternatives according to the ordering of Step 5 by Definition 5 and select the optimal project.

From the results, we can get the ordering of alternatives under ILDWBM or ILDWGBM operator is A₄ ≻ A₁ ≻ A₂ ≻ A₃, so the best alternative is A₄. In order to illustrate the influence of the parameters p, q on decision making result of this example, we use the different values of p, q to rank the alternatives. The ranking results are shown in Table 5.

Table 5

The orders of alternatives by ILDWBM and ILDWGBM operator

p and q	Order
	ILDWBM	ILDWGBM
p = 0.5, q = 0.5	A₄ ≻ A₁ ≻ A₂ ≻ A₃	A₄ ≻ A₁ ≻ A₂ ≻ A₃
p = 0.4, q = 0.6	A₄ ≻ A₁ ≻ A₂ ≻ A₃	A₄ ≻ A₁ ≻ A₂ ≻ A₃
p = 0.3, q = 0.7	A₄ ≻ A₁ ≻ A₂ ≻ A₃	A₄ ≻ A₁ ≻ A₂ ≻ A₃
p = 0.2, q = 0.8	A₄ ≻ A₁ ≻ A₂ ≻ A₃	A₄ ≻ A₁ ≻ A₂ ≻ A₃
p = 0.1, q = 0.9	A₄ ≻ A₁ ≻ A₂ ≻ A₃	A₄ ≻ A₁ ≻ A₂ ≻ A₃
p = 1, q = 0	A₄ ≻ A₂ ≻ A₁ ≻ A₃	A₄ ≻ A₂ ≻ A₁ ≻ A₃
p = 1, q = 1	A₄ ≻ A₁ ≻ A₂ ≻ A₃	A₄ ≻ A₁ ≻ A₂ ≻ A₃

From the above results, we get A₄ ≻ A₁ ≻ A₂ ≻ A₃ or A₄ ≻ A₁ ≻ A₂ ≻ A₃ by the ILDWBM operator and the ILDWGBM operator. From the Table 5, we can get that the score value of ${\tilde{r}}_{4}$ is higher than others obviously, and the score value of ${\tilde{r}}_{3}$ is lower than others at all times. This indicates that the approach of this paper is robust and available.

In order to further verify the effective of the developed method, we utilize this method to solve the illustrate example given by Liu and Wang [13], and we can get the same ranking result. Besides, the aggregation operator proposed by Liu and Wang [13] can only consider capture the interrelationship of the aggregated arguments. However, the proposed aggregation operators in this paper can capture the interrelationship of the aggregated intuitionistic linguistic arguments and relieve the influence of unfair data from the aggregated arguments. Thus, the proposed method can obtain much more information in the process of decision making.

7 Conclusion

In this paper, we propose some intuitionistic linguistic aggregation operators, such as the intuitionistic linguistic dependent weighted Bonferroni mean (ILDWBM) operator and the intuitionistic linguistic dependent weighted geometric Bonferroni mean (ILDWGBM) operator, and then we analyze some desirable properties and special cases of these proposed operators. Clearly, these proposed operators can take the advantages of dependent ordered weighted averaging (DOWA) operator and Bonferroni mean operator, namely, they can relieve the influence of unfair data from the aggregated intuitionistic linguistic arguments and capture the interrelationship of the aggregated arguments. Besides, we develop a novel decision making method based on these proposed operators to handle the intuitionistic linguistic MAGDM problem with interactive condition. The advantages of the proposed method are (1) it adopted the intuitionistic linguistic variables which can more easily express the uncertain information; (2) it can consider the interrelationship between any two attribute values; (3) it can relieve the influence of unfair attribute values given by some biased decision makers. Finally, an example is provided to illustrate the validity and advantages of the developed method by comparing with existing approach. In the further works, we shall apply proposed operators to granular computing and evaluations about resources and Environment [39 –46].

Funding

This paper is supported by the National Natural Science Foundation of China (Nos. 71471172 and 71271124), the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045), Shandong Provincial Social Science Planning Project (Nos. 15BGLJ06, 16CGLJ31 and 16CKJJ27), the Teaching Reform Research Project of Undergraduate Colleges and Universities in Shandong Province (2015Z057), and Key research and development program of Shandong Province (2016GNC110016).

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