Power average operators of linguistic intuitionistic fuzzy numbers and their application to multiple-attribute decision making

Abstract

Linguistic intuitionistic fuzzy number (LIFN) is a special intuitionistic fuzzy number which can more easily describe the uncertainty and the vagueness information existing in the real world, and the power average (PA) operator can relieve some influences of unreasonable attribute values given by biased decision makers. In this paper, we will extend the PA operator to the LIFNs and propose some new operators, and develop some new decision making methods. Firstly, we introduce the definition, properties, score function, and operational rules of the LIFNs. Then, we propose some linguistic intuitionistic fuzzy power operators, such as linguistic intuitionistic fuzzy power averaging (LIFPA) operator, linguistic intuitionistic fuzzy weighted power averaging (LIFWPA) operator, linguistic intuitionistic fuzzy power geometric (LIFPG) operator, linguistic intuitionistic fuzzy weighted power geometric (LIFWPG) operator, linguistic intuitionistic fuzzy generalized power averaging (LIFGPA) operator, linguistic intuitionistic fuzzy generalized weighted power averaging (LIFGWPA) operator. At the same time, we study some effective properties of these operators. Then, three methods based on the LIFWPA operator, LIFWPG operator and LIFGWPA operator for multi-attribute decision making are proposed. Finally, we use an illustrative example to demonstrate the practicality and effectiveness of the proposed methods.

Keywords

Multiple attribute decision making linguistic intuitionistic fuzzy numbers power average operators

1 Introduction

Multiple attribute decision making (MADM) has an extensive use in real decision making. Because of the complexity of decision making problems, it is difficult to describe the attributes by crisp numbers. In order to easily character the attributes, Zadeh [39] defined of the fuzzy set (FS), Atanassov [1] defined the intuitionistic fuzzy set (IFS) which is with membership degree and non-membership degree. Due to its capability of accommodating hesitation in human decision processes, IFSs have been widely applied to decision making process. Zhang and Xu [43] proposed a new ranking method for intuitionistic fuzzy numbers (IFNs). Nguyen [25] developed new similarity degree of IFNs. Guo and Song [6] proposed the information entropy of IFNs based on the amount of knowledge. At the same time, some famous decision making methods have also been extended to process the IFNs, such as TOPSIS method [10], GRA method [24], VIKOR method [26], TODIM method [11], DEMATEL method [5] and so on. Yu [38] discussed the research status of IFS by using a citation network analysis. In real decision making, there exists a great deal of qualitative information, which is expressed by the linguistic information [14 , 23]. Sine Zadeh [40 –42] firstly defined the linguistic variable (LV), there are many researches on the LVs. Herrera and Verdegay [8] developed the operational rules for the LVs. Liu and Jin [15] defined the intuitionistic uncertain linguistic variable.

As above analysis, an IFN is characterized by real-valued membership and non-membership degree defined on interval [0, 1], and the hesitancy degree can be deduced based on the membership and non-membership degrees. However, due to the vagueness and complexity of the decision making environment, the crisp values are insufficient or inadequate to express the membership and non-membership functions in IFN, a feasible solution is to represent such membership degree and non-membership degree by linguistic variables, So, Chen and Liu [3] defined the concept called linguistic intuitionistic fuzzy numbers (LIFNs). The linguistic intuitionistic fuzzy number (LIFN) is in the form of γ = (S_α, S_β) which follows the membership degree and non-membership degree by linguistic variables based on the given linguistic term set. The advantages of both linguistic term sets and IFNs are combined by LIFNs. Because the membership and non-membership degrees are expressed by linguistic variables, LIFNs can easier express the fuzzy information than IFNs.

The information aggregation operators have gotten more and more attentions and have also become an important research topic [12–16 , 37]. Some aggregation operators were developed, such as arithmetic and geometric weighted aggregation operators for IFNs [31, 34], generalized aggregation operators of IFNs [44], neutral averaging operators of IFNs [7]. However, these operators cannot consider the relationship between the attributes. Further, Yager [36] presented a power-average (PA) operator by power weighting vectors which depend on the input data. Then some new extended PA operators were developed, such as UPOWG (uncertain power ordered weighted geometric) operator [35], 2-tuple linguistic PA (2TLPA) operator [28], intuitionistic fuzzy PA (IFPA) operator [33], generalized IFPA (GIFPA) operator [45], linguistic weighted PA (LWPA) operator [29], and so on.

Because PA operator can consider the relationship between the attributes by the power weighting, which can relieve the some influences of unreasonable data given by biased decision makers, and linguistic intuitionistic fuzzy numbers can be flexible for decision-makers to exactly quantify their viewpoints with linguistic variables. However, the existing PA operators cannot process the linguistic intuitionistic fuzzy numbers, so there is an important significance to study the new PA operators for the linguistic intuitionistic fuzzy numbers. In this paper, we will propose the power averaging operator under linguistic intuitionistic fuzzy information environment, and then propose some new methods for multi-attribute decision making problems and discuss some cases and properties of them.

2 Preliminaries

2.1 The intuitionistic fuzzy numbers and the linguistic term set

Atanassov [1] defined intuitionistic fuzzy set (IFS), which is shown as follows:

Definition 1. [1] Suppose X is a universe of discourse, an intuitionistic fuzzy set in X is defined as A ={ 〈 x, μ_A (x) , v_A (x) 〉 |x ∈ X }, where the functions μ_A : X → [0, 1] and v_A : X → [0, 1] are the degree of membership and the degree of non-membership of the element ∀_x ∈ X to A, also 0 ≤ μ_A (x) + v_A (x) ≤ 1. π_A (x) = 1 - μ_A (x) - v_A (x) is called the degree of indeterminacy or hesitation of x to A.

For an IFS A and a given x, Xu and Yager [34] called the pair (μ_A(x), v_A(x)) an intuitionistic fuzzy number (IFN).

In real decision making, there are great deals of qualitative information which can be easily expressed by linguistic variables [4 , 40]. In order to describe them, it is necessary to set the appropriate linguistic term set. Suppose that S = {s_i|i = 0, 1, …, t} is a linguistic term set with odd cardinality, where t is a positive integer.

For example, a set of seven linguistic terms S could be given as follows [36]: $\begin{matrix} S & = & {s_{0} = none, s_{1} = very low, s_{2} = low, \\ s_{3} = medium, s_{4} = high, \\ s_{5} = very high, s_{6} = perfect} . \end{matrix}$

In general, in order to minimize the loss of information, the above linguistic term set S is extended to a continuous linguistic set $\bar{S}$ which is explained by Xu [30]. It is omitted here.

2.2 Linguistic intuitionistic fuzzy numbers

Definition 2. [3] Let $s_{α}, s_{β} \in \bar{S}$ and γ = (s_α , s_β), if α + β ≤ t, then we can call γ the linguistic intuitionistic fuzzy numbers.

Remark 1. If α ∈ [0, t], then (s_α, neg (s_α)) is also a LIFN, where neg (s_α) = s_t-α.

Remark 2. The uncertain linguistic variables [30] can be converted to the LIFNs. Let $\tilde{S} = [s_{α}, s_{β}]$ be an uncertain linguistic variable (ULV), where α, β ∈ [0, t] and α ≤ β, then $\tilde{S} = [s_{α}, s_{β}]$ can be expressed by a LIFN (s_α , s_t-β).

For convenience, suppose Γ_[0,t] is the set of all LIFNs.

Definition 3. [3] Let γ = (s_α, s_β) , γ₁ = (s_α1, s_β1) , and γ₂ = (s_α2, s_β2) ∈ Γ_[0,t], λ > 0, then the operational laws of the LIFNs are defined as follows: $\begin{matrix} γ_{1} \oplus γ_{2} = (s_{α 1}, s_{β 1}) \oplus (s_{α 2}, s_{β 2}) \\ = (s_{α 1 + α 2 - \frac{α 1 α 2}{t}}, s_{\frac{β 1 β 2}{t}}); \end{matrix}$ (1) $\begin{matrix} γ_{1} \otimes γ_{2} = (s_{α 1}, s_{β 1}) \otimes (s_{α 2}, s_{β 2}) \\ = (s_{\frac{α 1 α 2}{t}}, s_{β 1 + β 2 - \frac{β 1 β 2}{t}}); \end{matrix}$ (2) $λ γ = λ (s_{α}, s_{β}) = (s_{t - t {(1 - \frac{α}{t})}^{λ}}, s_{t {(\frac{β}{t})}^{λ}});$ (3) $γ^{λ} = {(s_{α}, s_{β})}^{λ} = (s_{t {(\frac{α}{t})}^{λ}}, s_{t - t {(1 - \frac{β}{t})}^{λ}}) .$ (4)

In the following, we will introduce the comparison method of two LIFNs [3].

Definition 4. [3] Let γ = (s_α,s_β) ∈ Γ_[0,t], and suppose $Ls (γ) = α - β;$ (5) $Lh (γ) = α + β .$ (6)

Then we call Ls (γ) and Lh (γ) the score function and the accuracy function of the LIFN γ, respectively.

Definition 5. [3] Let γ₁ = (s_α1, _β1) , γ₂ = (s_α2 , s_β2) ∈ Γ_[0,t], then

If Ls (γ₁) < Ls (γ₂), then γ₁ is smaller than γ₂, denoted by γ₁ ≺ γ₂;

If Ls (γ₁) = Ls (γ₂), then

if Lh (γ₁) < Lh (γ₂), then γ₁ is smaller than γ₂, denoted by γ₁ ≺ γ₂;

if Lh (γ₁) = Lh (γ₂), then γ₁ and γ₂ have the same information, denoted by γ₁ = γ₂;

Obviously, we have (s₀, s_t) ≤ (s_α, s_β) ≤ (s_t, s₀) for any (s_α, s_β) ∈ Γ_[0,t]. For any two LIFNs γ₁ = (s_{α
₁}, s_{β
₁}), γ₂ = (s_{α
₂}, s_{β
₂}) ∈ Γ_[0,t]. If α₁ ≥ α₂ and β₁ ≤ β₂, then γ₁ ≥ γ₂.

Definition 6. Let γ₁ = (s_{α
₁}, s_{β
₁}) , γ₂ = (s_{α
₂} , s_{β
₂}) ∈ Γ_[0,t] be two LIFNs, then the Euclidean distance between γ₁ and γ₂ is defined as follows: $d ({\tilde{γ}}_{1}, {\tilde{γ}}_{2}) = \frac{1}{2 t} (| α_{1} - α_{2} | + | β_{1} - β_{2} |) .$ (7)

2.3 The Power Aggregation (PA) operator

Definition 7. [32] For real numbers γ_i (i = 1, 2, …, n), the power average operator is defined as following: $PA (γ_{1}, γ_{2}, \dots, γ_{n}) = \frac{\sum_{i = 1}^{n} (1 + T (γ_{i})) \cdot γ_{i}}{\sum_{i = 1}^{n} (1 + T (γ_{i}))}$ (8) where $T (γ_{i}) \sum_{\begin{array}{l} j = 1 \\ j \neq i \end{array}}^{n} S u p (γ_{i}, γ_{j})$ (9) and sup(γ_i, γ_j) means the degree to which γ_j supports γ_i. It satisfies the following rules: Sup (γ_i, γ_j) = Sup (γ_j, γ_i) , Sup (γ_i, γ_j) ∈ [0, 1] and Sup (γ_i, γ_j) ≥ Sup (γ_m, γ_n), if |γ_i - γ_j| ≤ |γ_m - γ_n|.

3 Linguistic intuitionistic fuzzy aggregation operators

In Section 2, we have introduced the concept of linguistic intuitionistic fuzzy numbers where membership and non-membership are represented as linguistic terms. In addition, the PA operator can take the relationship between the attributes. So it is necessary to extend the PA to the LIFNs. In this section, we will develop some kinds of linguistic intuitionistic fuzzy power aggregation operators.

3.1 Linguistic intuitionistic fuzzy power averaging operator

Definition 8. Let γ_i = (s_{α_i}, s_{β_i})(i = 1, 2, ⋯ , n) be a collection of the LIFNs, then we can define LIFPA (linguistic intuitionistic fuzzy power aggregation) operator as follows:

$\begin{matrix} LIFPA (γ_{1}, γ_{2}, \dots, γ_{n}) \\ = \frac{\sum_{i = 1}^{n} (1 + T (γ_{i})) γ_{i}}{\sum_{i = 1}^{n} (1 + T (γ_{i}))} \end{matrix}$ (10) where $T (γ_{i}) \sum_{\begin{array}{l} j = 1 \\ j \neq i \end{array}}^{n} S u p (γ_{i}, γ_{j})$ (11) and Sup (γ_i, γ_j) means the support for γ_i from γ_j, Sup (γ_i, γ_j) = 1 - d (γ_i, γ_j). it satisfies the following rules.

Sup (γ_i, γ_j) = Sup (γ_j, γ_i) , Sup (γ_i, γ_j) ∈ [0, 1],and Sup (γ_i, γ_j) ≥ Sup (γ_m, γ_n), if |γ_i - γ_j| ≤ |γ_m - γ_n|.

Theorem 1. Let γ_i = (s_{α_i}, s_{β_i}) (i = 1, 2, …, n) be a collection of the LIFNs, then, the result aggregated from Definition 8 is still a LIFN, and even

$\begin{matrix} LIFPA (γ_{1}, γ_{2}, \dots, γ_{n}) \\ = (s_{t - t (\frac{\prod_{i = 1}^{n} [(t - α_{i}) {(1 - \frac{α_{i}}{t})}^{T (γ_{i})}]}{t^{n}})}^{1 / \sum_{i = 1}^{n} (1 + T (γ_{i}))}, s_{t (\frac{\prod_{i = 1}^{n} [β_{i} {(\frac{β_{i}}{t})}^{T (γ_{i})}]}{t^{n}})}^{1 / \sum_{i = 1}^{n} (1 + T (γ_{i}))}) \end{matrix}$ (12)

Proof.

when n = 1, it’s obvious that the Equation (12) is right.

Suppose when n = k, the Equation (12) is right, i.e.,

$\begin{matrix} LIFPA (γ_{1}, γ_{2}, \dots, γ_{k}) \\ = (s_{t - t (\frac{\prod_{i = 1}^{k} [(t - α_{i}) {(1 - \frac{α_{i}}{t})}^{T (γ_{i})}]}{t^{k}})}^{1 / \sum_{i = 1}^{k} (1 + T (γ_{i}))}, s_{t (\frac{\prod_{i = 1}^{k} [β_{i} {(\frac{β_{i}}{t})}^{T (γ_{i})}]}{t^{k}})}^{1 / \sum_{i = 1}^{k} (1 + T (γ_{i}))}) \end{matrix}$ (13)

Then when n = k + 1, we have $\begin{matrix} L I F P A (γ_{1}, γ_{2}, ..., γ_{k}, γ_{k + 1}) = \frac{\sum_{i = 1}^{k + 1} (1 + T (γ_{i})) γ_{i}}{\sum_{i = 1}^{k + 1} (1 + T (γ_{i}))} \\ = [\frac{\sum_{i = 1}^{k} (1 + T (γ_{i})) γ_{i}}{\sum_{i = 1}^{k} (1 + T (γ_{i}))}] & \oplus [\frac{(1 + T (γ_{k + 1})) γ_{k + 1}}{1 + T (γ_{k + 1})}] \\ = (s_{t - t (\frac{\prod_{i = 1}^{k} [(t - α_{i}) {(1 - \frac{α_{i}}{t})}^{T (γ_{i})}]}{t^{k}})}^{1 / \sum_{i = 1}^{k} (1 + T (γ_{i}))}, s_{t (\frac{\prod_{i = 1}^{k} [β_{i} {(\frac{β_{i}}{t})}^{T (γ_{i})}]}{t^{k}})}^{1 / \sum_{i = 1}^{k} (1 + T (γ_{i}))}) \\ \oplus (s_{t - t (1 - \frac{α_{k + 1}}{t})}^{1 + T (γ_{k + 1})}, s_{t (\frac{β_{k + 1}}{t})}^{1 + T (γ_{k + 1})}) \\ = (s_{t - t (\frac{\prod_{i = 1}^{k + 1} [(t - α_{i}) {(1 - \frac{α_{i}}{t})}^{T (γ_{i})}]}{t^{k + 1}})}^{1 / \sum_{i = 1}^{k + 1} (1 + T (γ_{i}))}, s_{t (\frac{\prod_{i = 1}^{k + 1} [β_{i} {(\frac{β_{i}}{t})}^{T (γ_{i})}]}{t^{k + 1}})}^{1 / \sum_{i = 1}^{k + 1} (1 + T (γ_{i}))}) \end{matrix}$

So, when n = k + 1, the formula (12) is right.

According to (i) and (ii), we can get the formula (12) holds for any n.

In the following, we will discuss some properties of LIFPA operator as follows:

Theorem 2. Let γ_i = (s_{α_i}, s_{β_i})(i = 1, 2, ⋯ , n) be a collection of the LIFNs, if Sup (γ_k, γ_j) = c, then the LIFPA operator will reduce to the linguistic intuitionistic fuzzy arithmetic averaging operator (LIFAA) shown as follows: $LIFPA (γ_{1}, γ_{2}, \dots, γ_{n}) = \frac{1}{n} \sum_{i = 1}^{n} γ_{i}$ (14)

Proof. Since all Sup (γ_k, γ_j) = c, k ≠ j, then $\begin{matrix} LIFPA (γ_{1}, γ_{2}, \dots, γ_{n}) \\ = \frac{(1 + (n - 1) c) γ_{1} + (1 + (n - 1) c) γ_{2} + \dots + (1 + (n - 1) c) γ_{n}}{\sum_{i = 1}^{n} 1 + (n - 1) c} \\ = \frac{(1 + (n - 1) c) (γ_{1} + γ_{2} + \dots + γ_{n})}{\sum_{i = 1}^{n} 1 + (n - 1) c} = \frac{1}{n} \sum_{i = 1}^{n} γ_{i} \end{matrix}$

Theorem 3. (Idempotency). Let γ_i = (s_{α_i}, s_{β_i}) (i = 1, 2, …, n) be a collection of the LIFNs, and for all γ_i = (s_α, s_β) = γ, i = (1, 2, …, n), then $LIFPA (γ_{1}, γ_{2}, \dots, γ_{n}) = γ$ (15)

Proof. Since all γ_i = (s_α, s_β) = γ we have $\begin{matrix} L I F P A (γ_{1}, γ_{2}, ..., γ_{n}) = \frac{\sum_{i = 1}^{n} (1 + T (γ_{i})) γ_{i}}{\sum_{i = 1}^{n} (1 + T (γ_{i}))} \\ = (s_{t - t (\frac{\prod_{i = 1}^{n} [(t - α_{i}) {(1 - \frac{α_{i}}{t})}^{T (γ_{i})}]}{t^{n}})}^{1 / \sum_{i = 1}^{n} (1 + T (γ_{i}))}, s_{t (\frac{\prod_{i = 1}^{n} [β_{i} {(\frac{β_{i}}{t})}^{T (γ_{i})}]}{t^{n}})}^{1 / \sum_{i = 1}^{n} (1 + T (γ_{i}))}) \\ = (s_{t - t (\frac{{((t - α) {(1 - \frac{α}{t})}^{T (γ)})}^{n}}{t^{n}})}^{1 / n (1 + T (γ))}, s_{t (\frac{{(β {(\frac{β}{t})}^{T (γ)})}^{n}}{t^{n}})}^{1 / n (1 + T (γ))}) \\ = (s_{α}, s_{β}) = γ \\ = (s_{t - t (\frac{{((t - α) {(1 - \frac{α}{t})}^{T (γ)})}^{n}}{t^{n}})}^{1 / n (1 + T (γ))}, s_{t (\frac{{(β {(\frac{β}{t})}^{T (γ)})}^{n}}{t^{n}})}^{1 / n (1 + T (γ))}) \\ = (s_{α}, s_{β}) = γ \end{matrix}$

Theorem 4. (Boundedness). Let γ_i = (s_{α_i}, s_{β_i}) (i = 1, 2, ⋯ , n) be a collection of the LIFNs, suppose $γ_{max} = (s_{max_{i} α_{i}}, s_{min_{i} β_{i}})$ , $γ_{min} = (s_{min_{i} α_{i}}, s_{max_{i} β_{i}})$ , then the LIFPA operator satisfies: $γ_{min} \leq LIFPA (γ_{1}, γ_{2}, \dots, γ_{n}) \leq γ_{max}$ (16)

Proof. Since s_minα ≤ s_αi ≤ s_maxα, s_minβ ≤ s_βi ≤ s_maxβ for all i. And suppose $w_{i} = \frac{1 + T (γ_{i})}{\sum_{i = 1}^{n} 1 + T (γ_{i})}$

we can get $\begin{matrix} LIFPA (γ_{1}, γ_{2}, \dots, γ_{n}) = \frac{\sum_{i = 1}^{n} (1 + T (γ_{i})) γ_{i}}{\sum_{i = 1}^{n} (1 + T (γ_{i}))} \\ = \sum_{i = 1}^{n} w_{i} γ_{i} = \sum_{i = 1}^{n} w_{i} (s_{α i}, s_{β i}), \end{matrix}$ since $\begin{matrix} w_{i} s_{min a} \leq w_{i} s_{ai} \leq w_{i} s_{max a}, \\ w_{i} s_{min β} \leq w_{i} s_{β i} \leq w_{i} s_{max β} \end{matrix}$ then $\begin{matrix} \sum_{i = 1}^{n} w_{i} s_{min α} \leq \sum_{i = 1}^{n} w_{i} s_{α i} \\ \leq \sum_{i = 1}^{n} w_{i} s_{max α}, \sum_{i = 1}^{n} w_{i} s_{min β} \\ \leq \sum_{i = 1}^{n} w_{i} s_{β i} \leq \sum_{i = 1}^{n} w_{i} s_{max β} \end{matrix}$ since [γ_min = (s_minα, s_maxβ)] ≤ [γ_i = (s_αi, s_βi)] ≤ [γ_max = (s_maxa, s_minβ)], we can obtain $\begin{matrix} LIFPA (γ_{min}, γ_{min}, \dots, γ_{min}) \\ \leq LIFPA (γ_{1}, γ_{2}, \dots, γ_{n}) \\ \leq LIFPA (γ_{max}, γ_{max}, \dots, γ_{max}) \end{matrix}$ according to the Theorem 3, we can get $\begin{matrix} LIFPA (γ_{min}, γ_{min}, \dots, γ_{min}) = γ_{min} \\ LIFPA (γ_{max}, γ_{max}, \dots, γ_{max}) = γ_{max} \end{matrix}$ so, we can get γ_min ≤ LIFPA (γ₁, γ₂, …, γ_n) ≤ γ_max.

Theorem 5. (Commutativity). Let (γ₁, γ₂, …, γ_n) and $(γ_{1}^{'}, γ_{2}^{'}, \dots, γ_{n}^{'})$ be two collections of the LIFNs, if $(γ_{1}^{'}, γ_{2}^{'}, \dots, γ_{n}^{'})$ is any permutation of (γ₁, γ₂, …, γ_n), then $LIFPA (γ_{1}^{'}, γ_{2}^{'}, \dots, γ_{n}^{'}) = LIFPA (γ_{1}, γ_{2}, \dots, γ_{n})$ (17)

Proof. Let $\begin{matrix} LIFPA (γ_{1}, γ_{2}, \dots, γ_{n}) = \frac{\sum_{i = 1}^{n} (1 + T (γ_{i})) γ_{i}}{\sum_{i = 1}^{n} (1 + T (γ_{i}))}, \\ LIFPA (γ_{1}^{'}, γ_{2}^{'}, \dots, γ_{n}^{'}) = \frac{\sum_{i = 1}^{n} (1 + T (γ_{i}^{'})) γ_{i}^{'}}{\sum_{i = 1}^{n} (1 + T (γ_{i}^{'}))} \end{matrix}$

Since $(γ_{1}^{'}, γ_{2}^{'}, \dots, γ_{n}^{'})$ is any permutation of (γ₁ , γ₂, …, γ_n), then we have $\sum_{i = 1}^{n} (1 + T (γ_{i})) γ_{i} = \sum_{i = 1}^{n} (1 + T (γ_{i}^{'})) γ_{i}^{'}$ and $\sum_{i = 1}^{n} (1 + T (γ_{i})) = \sum_{i = 1}^{n} (1 + T (γ_{i}^{'}))$ so, $LIFPA (γ_{1}^{'}, γ_{2}^{'}, \dots, γ_{n}^{'}) = LIFPA (γ_{1}, γ_{2}, \dots, γ_{n}) .$

In Equations (10) and (12), we think all aggregated attributes are of equivalent importance. However, in real decision making, the weights of the attributes should also be taken into consideration. In the following, we will propose another power averaging operator which is called the linguistic intuitionistic fuzzy weighted power averaging (LIFWPA) operator.

Definition 9. Let (γ₁, γ₂, …, γ_n) be a collection of LIFNs, the linguistic intuitionistic fuzzy weighted power averaging (LIFWPA) operator isdefined as $LIFWPA (γ_{1}, γ_{2}, \dots, γ_{n}) = \frac{\sum_{i = 1}^{n} (1 + T (γ_{i})) w_{i} γ_{i}}{\sum_{i = 1}^{n} w_{i} (1 + T (γ_{i}))}$ (18) where $T (γ_{i}) = \sum_{i = 1, j \neq i}^{n} Sup (γ_{i}, γ_{j})$ , and w = (w₁, w₂, … , w_n) ^T be the weight vector of LIFNs (γ₁, γ₂, …, γ_n), 0 ≤ w_i ≤ 1 (i = 1, 2, …, n) and $\sum_{i = 1}^{n} w_{i} = 1 .$

Theorem 6. Let γ_i = (s_{α_i}, s_{β_i})(i = 1, 2 … , n) be a collection of the LIFNs, then, the result aggregated from Definition 9 is still a LIFN, and even

$L I F W P A (γ_{1}, γ_{2}, ..., γ_{n}) = \frac{\sum_{i = 1}^{n} (1 + T (γ_{i})) w_{i} γ_{i}}{\sum_{i = 1}^{n} w_{i} (1 + T (γ_{i}))} = (s_{{_{t - [t (\prod_{i = 1}^{n} {(1 - \frac{α_{i}}{t})}^{(1 + T (γ_{i}) ω_{i})})]}}^{1 / \sum_{i = 1}^{n} (1 + T (γ_{i})) ω_{i}}}, s_{_{t {(\prod_{i = 1}^{n} [{(\frac{β_{i}}{t})}^{(1 + T (γ_{i})) ω_{i}}])}^{^{1 / \sum_{i = 1}^{n} (1 + T (γ_{i})) ω_{i}}}}})$ (19)

The LIFWPA operator has the following properties:

Theorem 7. Let γ_i = (s_{α_i}, s_{β_i})(i = 1, 2, …, n) be a collection of the LIFNs, if Sup (γ_i, γ_j) = c, c ∈ [0, 1] , i ≠ j, then the LIFWPA operator will reduce to the linguistic intuitionistic fuzzy weighted averaging operator as follows:

$LIFWPA (γ_{1}, γ_{2}, \dots, γ_{n}) = \frac{1}{n} \sum_{i = 1}^{n} w_{i} γ_{i}$ (20)

Theorem 8. (Boundedness). Let γ_i = (s_{α_i}, s_{β_i}) (i = 1, 2, …, n) be a collection of the LIFNs, suppose $γ_{max} = (s_{max_{i} α_{i}}, s_{min_{i} β_{i}})$ , $γ_{min} = (s_{min_{i} α_{i}}, s_{max_{i} β_{i}})$ , then the LIFWPA operator satisfies: $γ_{min} \leq LIFWPA (γ_{1}, γ_{2}, \dots, γ_{n}) \leq γ_{max}$ (21)

Theorem 9. (Idempotency) Let (γ₁, γ₂, …, γ_n) be a collection of LIFNs, if all γ_i = (i = 1, 2, …, n) are equal, i.e. γ_i = γ (i = 1, 2, …, n) for all i, then $LIFWPA (γ_{1}, γ_{2}, \dots, γ_{n}) = γ$ (22)

3.2 The linguistic intuitionistic fuzzy Power geometric operators

Definition 10. Let (γ₁, γ₂, …, γ_n) be a collection of LIFNs. The linguistic intuitionistic fuzzy power geometric (LIFPG) operator is defined as $LIFPG (γ_{1}, γ_{2}, \dots, γ_{n}) = \prod_{i = 1}^{n} γ_{i}^{\frac{1 + T (γ_{i})}{\sum_{i = 1}^{n} (1 + T (γ_{i}))}}$ (23) where $T (γ_{i}) = \sum_{j = 1, j \neq i}^{n} Sup (γ_{i}, γ_{j})$ . Obviously, the LIFPG operator is a nonlinear weighted geometric aggregation operator, and the weight $\frac{1 + T (γ_{i})}{\sum_{i = 1}^{n} (1 + T (γ_{i}))}$ of the argument γ_j (j = 1, 2, …, n) depends on all the input values γ_j (j = 1, 2, …, n).

Theorem 10. Let γ_i = (s_{α_i}, s_{β_i})(i = 1, 2, …, n) be a collection of the LIFNs, the result aggregated from Definition 11 is still a LIFN, and even

$\begin{matrix} LIFPG (γ_{1}, γ_{2}, \dots, γ_{n}) = \prod_{i = 1}^{n} γ_{i}^{\frac{1 + T (γ_{i})}{\sum_{i = 1}^{n} (1 + T (γ_{i}))}} \\ = [s_{t \prod_{i = 1}^{n} {(\frac{α_{i}}{t})}^{1 + T (γ_{i)} / - \sum_{i = 1}^{n} 1 + T (γ_{i})}}, s_{t - t \prod_{i = 1}^{n} {(1 - \frac{β_{i}}{t})}^{1 + T (γ_{i}) / - \sum_{i = 1}^{n} 1 + T (γ_{i})}}] \end{matrix}$ (24)

The LIFPG operator has some properties as follows:

Theorem 11. Let γ_i = (s_{α_i}, s_{β_i}) (i = 1, 2, …, n) be a collection of the LIFNs, if Sup (γ_i, γ_j) = c, c ∈ [0, 1] , i ≠ j, then the LIFPG operator will reduce to the linguistic intuitionistic fuzzy geometric operator as follows: $LIFPG (γ_{1}, γ_{2}, \dots, γ_{n}) = {(\prod_{i = 1}^{n} γ_{i})}^{\frac{1}{n}}$ (25)

Theorem 12. (Commutativity). Let (γ₁, γ₂, …, γ_n) and $(γ_{1}^{'}, γ_{2}^{'}, \dots, γ_{n}^{'})$ be two collections of the LIFNs, if $(γ_{1}^{'}, γ_{2}^{'}, \dots, γ_{n}^{'})$ is any permutation of (γ₁, γ₂, …, γ_n) then

$\begin{matrix} LIFPG (γ_{1}^{'}, γ_{2}^{'}, \dots, γ_{n}^{'}) \\ = LIFPG (γ_{1}, γ_{2}, \dots, γ_{n}) \end{matrix}$ (26)

Theorem 13. (Idempotency). Let (γ₁, γ₂, …, γ_n) be a collection of LIFNs, if all γ_i (i = 1, 2, …, n) are equal, i.e. γ_i = γ for all i, then $LIFPG (γ_{1}, γ_{2}, \dots, γ_{n}) = γ$ (27)

Theorem 14. (Boundedness). Let γ_i = (s_{α_i}, s_{β_i}) (i = 1, 2 ⋯ , n) be a collection of the LIFNs, suppose $γ_{max} = (s_{max_{i} α_{i}}, s_{min_{i} β_{i}})$ , $γ_{min} = (s_{min_{i} α_{i}}, s_{max_{i} β_{i}})$ , then the LIFPG operator satisfies: $γ_{min} \leq LIFPG (γ_{1}, γ_{2}, \dots, γ_{n}) \leq γ_{max}$ (28)

Similar to the LIFPA operator, the LIFPG operator also neglects their own weights. The next, we will extend the LIFPG operator to the linguistic intuitionistic fuzzy weighted power geometric (LIFWPG) operator which can consider the weights of the aggregated arguments.

Definition 11. Let (γ₁, γ₂, …, γ_n) be a collection of LIFNs, and then the linguistic intuitionistic fuzzy weighted power geometric (LIFWPG) operator is defined as $LIFWPG (γ_{1}, γ_{2}, \dots γ_{n}) = \prod_{i = 1}^{n} γ_{i}^{\frac{w_{i} (1 + T (γ_{i}))}{\sum_{i = 1}^{n} w_{i} (1 + T (γ_{i}))}}$ (29) where $T (γ_{i}) = \sum_{j = 1, j \neq i}^{n} sup (γ_{i}, γ_{j})$ , and w = (w₁, w₂, … , w_n) ^T be the weight vector of LIFNs (γ₁, γ₂, …, γ_n), 0 ≤ w_i ≤ 1 (i = 1, 2, …, n) and $\sum_{i = 1}^{n} w_{i} = 1$ .

Theorem 15. Let γ_i = (s_{α_i}, s_{β_i}) (i = 1, 2 ⋯ , n) be a collection of LIFNs, then, the result aggregated from Definition 11 is still a LIFN, and even

$\begin{matrix} LIFWPG (γ_{1}, γ_{2}, \dots, γ_{n}) \\ = [s_{t \prod_{i = 1}^{n} {(\frac{α_{i}}{t})}^{(1 + T (γ_{i)}) w_{i} / - \sum_{i - 1}^{n} (1 + T (γ_{i})) w_{i}}}, s_{t - t \prod_{i = 1}^{n} {(1 - \frac{β_{i}}{t})}^{(1 + T (γ_{i})) w_{i} / - \sum_{i = 1}^{n} (1 + T (γ_{i})) w_{i}}}] \end{matrix}$ (30)

The LIFWPG operator has the idempotency, commutativity and boundedness.

3.3 The linguistic intuitionistic fuzzy generalized power operators

Definition 12. Let (γ₁, γ₂, …, γ_n) be a collection of LIFNs, and then the linguistic intuitionistic fuzzy generalized power average (LIFGPA) operator is defined as

$\begin{matrix} LIFGPA (γ_{1}, γ_{2}, \dots, γ_{n}) \\ = {(\frac{\sum_{i = 1}^{n} (1 + T (γ_{i})) γ_{i}^{λ}}{\sum_{i = 1}^{n} (1 + T (γ_{i}))})}^{\frac{1}{λ}} \end{matrix}$ (31) where $T (γ_{i}) = \sum_{j = 1, j \neq i}^{n} Sup (γ_{i}, γ_{j})$ , and λ is a parameter such that λ ∈ (0, + ∞).

Theorem 16. Let γ_i = (s_{α_i}, s_{β_i})(i = 1, 2, …, n) be a collection of the LIFNs, then, the result aggregated from Definition 12 is still a LIFN, and even $L I F G P A (γ_{1}, γ_{2}, ..., γ_{n}) = (s_{t (1 - {(\prod_{i = 1}^{n} {(1 - {(\frac{α_{i}}{t})}^{λ})}^{1 + T (γ_{i})})}^{\frac{1}{\sum_{i = 1}^{n} (1 + T (γ_{i}))}})}^{1 / λ}, s_{t - t (1 - {(\prod_{i = 1}^{n} {(1 - {(1 - \frac{β_{i}}{t})}^{λ})}^{1 + T (γ_{i})})}^{\frac{1}{\sum_{i = 1}^{n} (1 + T (γ_{i}))}})}^{1 / λ})$ (32)

The LIFGPA operator has the following properties.

Theorem 17. Let γ_i = (s_{α_i}, s_{β_i}) (i = 1, 2, …, n) be a collection of the LIFNs, if sup(γ_i, γ_j) = c, then the LIFGPA operator will reduce to the linguistic intuitionistic fuzzy generalized arithmetic averaging operator as follows: $LIFGPA (γ_{1}, γ_{2}, \dots, γ_{n}) = {(\frac{1}{n} \sum_{i = 1}^{n} γ_{i}^{λ})}^{1 / - λ}$ (33)

Theorem 18. (Commutativity). Let (γ₁, γ₂, …, γ_n) and $(γ_{1}^{'}, γ_{2}^{'}, \dots, γ_{n}^{'})$ be two collections of the LIFNs, if $(γ_{1}^{'}, γ_{2}^{'}, \dots, γ_{n}^{'})$ is any permutation of (γ₁, γ₂, …, γ_n). Then

$\begin{matrix} LIFGPA (γ_{1}^{'}, γ_{2}^{'}, \dots, γ_{n}^{'}) \\ = LIFGPA (γ_{1}, γ_{2}, \dots, γ_{n}) \end{matrix}$ (34)

Theorem 19. (Idempotency). Let (γ₁, γ₂, …, γ_n) be a collection of LIFNs, if all γ_i (i = 1, 2, …, n) are equal, i.e. γ_i = γ, for all i, then $LIFGPA (γ_{1}, γ_{2}, \dots, γ_{n}) = γ$ (35)

Theorem 20. (Boundedness). Let γ_i = (s_{α_i}, s_{β_i}) be a collection of the LIFNs, suppose $γ_{max} = (s_{max_{i} α_{i}}, s_{min_{i} β_{i}})$ , $γ_{min} = (s_{min_{i} α_{i}}, s_{max_{i} β_{i}})$ , then the LIFGPA operator satisfies: $γ_{min} \leq LIFGPA (γ_{1}, γ_{2}, \dots, γ_{n}) \leq γ_{max}$ (36)

In the following, we will discuss some cases of the LIFGPA operator.

when λ → 0, $LIFGPA (γ_{1}, γ_{2}, \dots, γ_{n}) = \prod_{i = 1}^{n} γ_{i}^{\frac{1 + T (γ_{i})}{\sum_{i = 1}^{n} (1 + T (γ_{i}))}}$

when λ = 1, $LIFGPA (γ_{1}, γ_{2}, \dots, γ_{n}) = \frac{\sum_{i = 1}^{n} (1 + T (γ_{i})) γ_{i}}{\sum_{i = 1}^{n} (1 + T (γ_{i}))}$

Since the LIFGPA operator neglects the importance of their own weights. Here, we will introduce another power averaging operator which is called the linguistic intuitionistic fuzzy generalized weighted power averaging (LIFGWPA) operator to overcome the shortcoming.

Definition 13. Let γ_i = (s_αi, s_βi) (i = 1, 2 ⋯ , n) be a collection of the LIFNs, and then the linguistic intuitionistic fuzzy generalized weighted power average (LIFGWPA) operator is defined as

$\begin{matrix} LIFGWPA (γ_{1}, γ_{2}, \dots, γ_{n}) \\ = {(\frac{\sum_{i = 1}^{n} w_{i} (1 + T (γ_{i})) γ_{i}^{λ}}{\sum_{i = 1}^{n} w_{i} (1 + T (γ_{i}))})}^{1 / - λ} \end{matrix}$ (37) where $T (γ_{i}) = \sum_{j = 1, j \neq i}^{n} Sup (γ_{i}, γ_{j})$ , and w = (w₁, w₂, …, w_n) ^T is the weight vector of LIFNs (γ₁, γ₂, …, γ_n), 0 ≤ w_i ≤ 1 (i = 1, 2, …, n) and $\sum_{i = 1}^{n} w_{i} = 1$ . λ is a parameter such that λ ∈ (0, + ∞)

Theorem 21. Let γ_i = (s_{α_i}, s_{β_i})(i = 1, 2, …, n) be a collection of LIFNs, then, the result aggregated from Definition 13 is still an LIFN, and even $L I F G W P A (γ_{1}, γ_{2}, ..., γ_{n}) = = (s_{t (1 - {(\prod_{i = 1}^{n} {(1 - {(\frac{α_{i}}{t})}^{λ})}^{(1 + T (γ_{i})) w_{i}})}^{\frac{1}{\sum_{i = 1}^{n} (1 + T (γ_{i})) w_{i}}})}^{1 / λ}, s_{t - t (1 - {(\prod_{i = 1}^{n} {(1 - {(1 - \frac{β_{i}}{t})}^{λ})}^{(1 + T (γ_{i})) w_{i}})}^{\frac{1}{\sum_{i = 1}^{n} (1 + T (γ_{i})) w_{i}}})}^{1 / λ})$ (38)

In the following, we will discuss some properties of the LIFGWPA operator. It can be proved that the LIFGWPA operator has the following properties.

Theorem 22. Let γ_i = (s_{α_i}, s_{β_i})(i = 1, 2, …, n) be a collection of LIFNs, if Sup (γ_i, γ_j) = c, then the generalized weighted power averaging operator of LIFNs reduces to the linguistic intuitionistic fuzzy generalized weighted averaging (LIFGWA) operator as follows: $LIFGWPA (γ_{1}, γ_{2}, \dots, γ_{n}) = {(\sum_{i = 1}^{n} w_{i} γ_{i}^{λ})}^{1 / - λ}$ (39)

Theorem 23. (Idempotency). Let {γ₁, γ₂, … , γ_n } be a collection of LIFNs, if all γ_i (i = 1, 2, …, n) are equal, i.e. γ_i = γ, for all i, then $LIFGWPA (γ_{1}, γ_{2}, \dots, γ_{n}) = γ$ (40)

Theorem 24. (Boundedness). Let γ_i = (s_{α_i}, s_{β_i}) (i = 1, 2, …, n) be a collection of the LIFNs, suppose $γ_{max} = (s_{max_{i} α_{i}}, s_{min_{i} β_{i}}), γ_{min} = (s_{min_{i} α_{i}}, s_{max_{i} β_{i}})$ , then the LIFGPA operator satisfies: $γ_{min} \leq LIFGWPA (γ_{1}, γ_{2}, \dots, γ_{n}) \leq γ_{max}$ (41)

In the following, we will discuss some cases of the LIFGWPA operator.

when λ → 0, $LIFGWPA (γ_{1}, γ_{2}, \dots, γ_{n}) = \prod_{i = 1}^{n} γ_{i}^{\frac{(1 + T (γ_{i})) w_{i}}{\sum_{i = 1}^{n} (1 + T (γ_{i})) w_{i}}}$

when λ = 1, $LIFGWPA (γ_{1}, γ_{2}, \dots, γ_{n}) = \frac{\sum_{i = 1}^{n} w_{i} (1 + T (γ_{i})) γ_{i}}{\sum_{i = 1}^{n} w_{i} (1 + T (γ_{i}))}$

4 Some MADM methods based on the proposed operators

In this section, we will use the LIFWPA, LIFWPG and LIFGWPA operators to solve the MADM problems in which the attribute values take the form of LIFNs.

Let X ={ x₁, x₂, …, x_m } be a discrete set of alternatives, C ={ c₁, c₂, …, c_n } be a set of attributes, and ω = (ω₁, ω₂, …, ω_n) ^T be the weighting vector of the attributes, where ω_j ∈ [0, 1] , j = 1, 2, …, n . $\sum_{j = 1}^{n} ω_{j} = 1$ . For each alternative x_i ∈ X, the decision maker gives his/her preference value γ_ij with respect to attribute c_j ∈ C. where γ_ij = (s_{α
_ij}, s_{β
_ij}) takes the form of LIFNs. The goal of the decision making is to give the ranking of all alternatives.

Approach I. Firstly, we give the steps of decision making based on the LIFWPA operator as follows.

Step 1. Normalize the decision matrix.

Since there are two types of attributes, i.e. benefit type and cost type, we can transform the cost attribute values to benefit type, and the transformed decision matrix is expressed by $\tilde{R} = {[{\tilde{r}}_{ij}]}_{m \times n}$ , (i = 1, 2, ⋯ , m, j = 1, 2, ⋯ , n, where ${\tilde{γ}}_{ij} = {\begin{matrix} (s_{α_{ij}}, s_{β_{ij}}) for benefit attribute C_{j} \\ (s_{β_{ij}}, s_{α_{ij}}) for cost attribute C_{j} \end{matrix}$ (42)

Step 2. Calculate the supports

$\begin{matrix} Sup ({\tilde{γ}}_{ij}, {\tilde{γ}}_{ij}) = 1 - d ({\tilde{γ}}_{ij}, {\tilde{γ}}_{ij}), \\ i = 1, 2, \dots, m; j, l = 1, 2, \dots, n . l \neq j . \end{matrix}$ (43) where $d ({\tilde{γ}}_{ij}, {\tilde{γ}}_{il})$ is Euclidean distance between two LIFNs ${\tilde{γ}}_{ij}$ and ${\tilde{γ}}_{il}$ .

Step 3. Calculate the support $T ({\tilde{γ}}_{ij})$ and the weights ϖ_ij, we have $\begin{array}{l} T ({\tilde{γ}}_{i j}) \sum_{\begin{array}{l} l = 1 \\ j \neq l \end{array}}^{n} S u p ({\tilde{γ}}_{i j}, {\tilde{γ}}_{i l}) \\ i = 1, 2, . . . , m . j = 1, 2, . . . , n . \end{array}$ (44) $\begin{array}{l} ϖ_{i j} = \frac{ω_{j} (1 + T ({\tilde{γ}}_{i j}))}{\sum_{k = 1}^{l} ω_{j} (1 + T ({\tilde{γ}}_{i j}))}, \\ i = 1, 2, \dots, m, j = 1, 2, \dots n . \end{array}$ (45) and

$\begin{matrix} ϖ_{ij} & = & \frac{ω_{j} (1 + T ({\tilde{γ}}_{ij}))}{\sum_{k = 1}^{l} ω_{j} (1 + T ({\tilde{γ}}_{ij}))}, \\ i = 1, 2, \dots, m, j = 1, 2, \dots n . \end{matrix}$ (46) where ϖ_ij ≥ 0, and $\sum_{j = 1}^{n} ϖ_{j} = 1$ .

Step 4. Calculate the comprehensive evaluation value of each alternative, we have ${\tilde{γ}}_{i} = LIFWPA ({\tilde{γ}}_{i 1}, {\tilde{γ}}_{i 2}, \dots, {\tilde{γ}}_{in})$ where i = 1, 2, ⋯ , m.

Step 5. Calculate the score function by Equation (5).

Step 6. Rank all the alternatives {x₁, x₂, … , x_m } and select the best one(s). The bigger the score function is, the better the alternative is.

Approach II. Then, we use the LIFWPG operator to solve the MADM problems above-mentioned as follows.

Steps 1–3. Same as Steps 1–3 in Approach I.

Step 4. Calculate the comprehensive evaluation value by the LIFWPG operator, we have ${\tilde{γ}}_{i} = LIFWPG ({\tilde{γ}}_{i 1}, {\tilde{γ}}_{i 2}, \dots, {\tilde{γ}}_{in})$ where i = 1, 2, ⋯ , m.

Steps 5–6. Same as Steps 5–6 in Approach I.

Approach III. Finally, we will adopt the LIFGWPA operator to solve the MADM problems above-mentioned as follows.

Steps 1–3. Same as Steps 1–3 in Approach I.

Step 4. Calculate the comprehensive evaluation value by the LIFGWPA operator, we have $γ_{i} = LIFGWPA ({\tilde{γ}}_{i 1}, {\tilde{γ}}_{i 2}, \dots, {\tilde{γ}}_{in})$ where i = 1, 2, ⋯ , m.

Steps 5–6. Same as Steps 5–6 in Approach I.

5 An application example

In this section, we will give an example to explain the proposed method. Suppose there is a decision making problem of selecting the best global supplier is describe as follows.

A manufacturing company desires to select a best global supplier for its most critical parts used in assembling process. Suppose that X ={ x₁, x₂, x₃, x₄ } is a set of four potential global suppliers (i.e., alternatives) under consideration and C ={ c₁, c₂, c₃, c₄, c₅ } is a set of attributes, where (c₁, c₂, c₃, c₄, c₅) stand for “overall cost of the production”, “quality of the production”, “service performance of supplier”, “supplier’s profile”, “risk factor”, respectively. And the attribute weight ω = (0.3, 0.3, 0.2, 0.1, 0.1) ^T. The four alternatives x_i (1, …, 4) are to be evaluated using the LIFNs based on the linguistic term set: $\begin{matrix} S & = & {s_{0} = extremely poor, s_{1} = very poor, \\ s_{2} = poor, s_{3} = slightly poor, s_{4} = fair, \\ s_{5} = slightly good, s_{6} = good, \\ s_{7} = very good, s_{8} = extremely good} \end{matrix}$

Then we can construct the decision matrix R = (γ_ij) _4×5 shown in Table 1.

5.1 Procedure of decision making based on the LIFWPA operator

Step 1. Normalize the decision matrix, we get $\tilde{R} = [\begin{matrix} (s_{1}, s_{7}) (s_{6}, s_{2}) (s_{4}, s_{3}) (s_{7}, s_{1}) (s_{2}, s_{5}) \\ (s_{2}, s_{6}) (s_{5}, s_{2}) (s_{6}, s_{1}) (s_{6}, s_{2}) (s_{1}, s_{7}) \\ (s_{1}, s_{6}) (s_{5}, s_{3}) (s_{7}, s_{1}) (s_{75}, s_{1}) (s_{4}, s_{3}) \\ (s_{2}, s_{5}) (s_{7}, s_{1}) (s_{4}, s_{3}) (s_{6}, s_{1}) (s_{4}, s_{4}) \end{matrix}]$

Step 2. Calculate the supports $Sup ({\tilde{γ}}_{ij}, {\tilde{γ}}_{il})$ we can get(S_ij,il = Sup_ij,il) $\begin{matrix} S_{11, 12} = S_{12, 11} = 0.3750, S_{11, 13} = S_{13, 11} = 0.5625, \\ S_{11, 14} = S_{14, 11} = 0.2500, S_{11, 15} = S_{15, 11} = 0.8125, \\ S_{12, 13} = S_{13, 12} = 0.8125, S_{12, 14} = S_{14, 12} = 0.8750, \\ S_{12, 15} = S_{15, 12} = 0.5625, S_{13, 14} = S_{14, 13} = 0.6875, \\ S_{13, 15} = S_{15, 13} = 0.7500, S_{14, 15} = S_{15, 14} = 0.4375, \\ S_{21, 22} = S_{22, 21} = 0.5625, S_{21, 23} = S_{23, 21} = 0.4375, \\ S_{21, 24} = S_{24, 21} = 0.5000, S_{21, 25} = S_{25, 21} = 0.8750, \\ S_{22, 23} = S_{23, 22} = 0.8750, S_{22, 24} = S_{24, 22} = 0.9375, \\ S_{22, 25} = S_{25, 22} = 0.4375, S_{23, 24} = S_{24, 23} = 0.9375, \\ S_{23, 25} = S_{25, 23} = 0.3125, S_{24, 25} = S_{25, 24} = 0.3750, \\ S_{31, 32} = S_{32, 31} = 0.5625, S_{31, 33} = S_{33, 31} = 0.3125, \\ S_{31, 34} = S_{34, 31} = 0.4375, S_{31, 35} = S_{35, 31} = 0.6250, \\ S_{32, 33} = S_{33, 32} = 0.7500, S_{32, 34} = S_{34, 32} = 0.8750, \\ S_{32, 35} = S_{35, 32} = 0.9375, S_{33, 34} = S_{34, 33} = 0.8750, \\ S_{33, 35} = S_{35, 33} = 0.6875, S_{34, 35} = S_{35, 34} = 0.8125, \\ S_{41, 42} = S_{42, 41} = 0.4375, S_{41, 43} = S_{43, 41} = 0.7500, \\ S_{41, 44} = S_{44, 41} = 0.5000, S_{41, 45} = S_{45, 41} = 0.8125, \\ S_{42, 43} = S_{43, 42} = 0.6875, S_{42, 44} = S_{44, 42} = 0.9375, \\ S_{42, 45} = S_{45, 42} = 0.6250, S_{43, 44} = S_{44, 43} = 0.7500, \\ S_{43, 45} = S_{45, 43} = 0.9375, S_{44, 45} = S_{45, 44} = 0.6875 . \end{matrix}$

Step 3. Calculate $T ({\tilde{γ}}_{ij}) (i, = 1, 2, 3, 4 . j = 1, 2, 3, 4, 5)$ by formula (41), and we get $\begin{matrix} T_{11} = 2.0000, T_{12} = 2.6250, T_{13} = 2.8125, \\ T_{14} = 2.2500, T_{15} = 2.5625, T_{21} = 2.3750, \\ T_{22} = 2.8125, T_{23} = 2.5625, T_{24} = 2.7500, \\ T_{25} = 2.0000, T_{31} = 1.9375, T_{32} = 3.1250, \\ T_{33} = 2.6250, T_{34} = 3.0000, T_{35} = 3.0625, \\ T_{41} = 2.5000, T_{42} = 2.6875, T_{43} = 3.1250, \\ T_{44} = 2.8750, T_{45} = 3.0625 . \end{matrix}$

Then calculate the weights ω_k (k = 1, 2, …, n), and we get $\begin{matrix} ϖ_{11} = 0.2623, ϖ_{12} = 0.3169, ϖ_{13} = 0.2222, \\ ϖ_{14} = 0.0947, ϖ_{15} = 0.1038, ϖ_{21} = 0.2857, \\ ϖ_{22} = 0.3228, ϖ_{23} = 0.2011, ϖ_{24} = 0.1058, \\ ϖ_{25} = 0.0847, ϖ_{31} = 0.2414, ϖ_{32} = 0.3390, \\ ϖ_{33} = 0.1986, ϖ_{34} = 0.1096, ϖ_{35} = 0.1113, \\ ϖ_{41} = 0.2781, ϖ_{42} = 0.2930, ϖ_{43} = 0.2185, \\ ϖ_{44} = 0.1026, ϖ_{45} = 0.1076 \end{matrix}$

Step 4. Utilize the LIFPWA operator to calculate the comprehensive evaluation value of each alternative, and we get $\begin{matrix} {\tilde{γ}}_{1} & = & (s_{4.5986}, s_{3.1309}), {\tilde{γ}}_{2} = (s_{4.5307}, s_{2.6478}), \\ {\tilde{γ}}_{3} & = & (s_{4.9444}, s_{2.5278}), {\tilde{γ}}_{4} = (s_{5.2222}, s_{2.3093}) \end{matrix}$

Step 5. Calculate the score function, we get $\begin{matrix} Ls ({\tilde{γ}}_{1}) & = & 1.4677, Ls ({\tilde{γ}}_{2}) = 1.8829, \\ Ls ({\tilde{γ}}_{3}) & = & 2.4166, Ls ({\tilde{γ}}_{4}) = 2.9129 . \end{matrix}$

Step 6. Rank all the alternatives and select the best one by the score function, we get $x_{4} ≻ x_{3} ≻ x_{2} ≻ x_{1}$

So the best choice is x₄.

5.2 Procedure of decision making based on the LIFWPG operator

Steps 1–3. Same as the above steps 1–3.

Step 4. Utilize the LIFGPWA operator to calculate the comprehensive evaluation value of each alternative, we can get $\begin{matrix} {\tilde{γ}}_{1} = (s_{3.1025}, s_{4.5995}), {\tilde{γ}}_{2} = (s_{3.5514}, s_{4.1148}), \\ {\tilde{γ}}_{3} = (s_{3.5355}, s_{3.5545}), {\tilde{γ}}_{4} = (s_{4.0516}, s_{3.1621}) \end{matrix}$

Step 5. Calculate the score index, we get $\begin{matrix} Ls ({\tilde{γ}}_{1}) = - 1.4970, Ls ({\tilde{γ}}_{2}) = - 0.5634, \\ Ls ({\tilde{γ}}_{3}) = - 0.0189, Ls ({\tilde{γ}}_{4}) = 0.8895 \end{matrix}$

Step 6. Rank all the alternatives and select the best one by the score index, we get $x_{4} ≻ x_{3} ≻ x_{2} ≻ x_{1}$

So the best choice is x₄.

5.3 Procedure of decision making based on the LIFGWPA operator

Steps 1–3. Same as the above steps 1–3.

Step 4. Utilize the LIFGWPA operator to calculate the comprehensive evaluation value of each alternative (suppose λ = 2), we get.

$\begin{matrix} {\tilde{γ}}_{1} = (s_{4.9160}, s_{2.8711}), {\tilde{γ}}_{2} = (s_{4.7440}, s_{2.4222}), \\ {\tilde{γ}}_{3} = (s_{5.1899}, s_{2.3640}), {\tilde{γ}}_{4} = (s_{5.4437}, s_{2.1748}) \end{matrix}$

Step 5. Calculate the score index, we get

$\begin{matrix} Ls ({\tilde{γ}}_{1}) = 2.0449, Ls ({\tilde{γ}}_{2}) = 2.3218, \\ Ls ({\tilde{γ}}_{3}) = 2.8259, Ls ({\tilde{γ}}_{4}) = 3.2689 \end{matrix}$

Step 6. Rank all the alternatives and select the best one by the score index, we get

$x_{4} ≻ x_{3} ≻ x_{2} ≻ x_{1} .$

Obviously, there are the same ranking results for these three methods.

In addition, in order to demonstrate the influence of the parameter λ on ranking results, we use the different values λ in LIFGWPA operator in step 4 to rank the alternatives. The ranking results are shown in Table 2.

As we can see from Table 2, the ordering of the alternatives may be different for the different value λ in LIFGWPA operator.

When 0 < λ ≤ 6, the ordering of the alternatives is x₄ ≻ x₃ ≻ x₂ ≻ x₁ and the best alternative is x₄.

When 7 ≤ λ, the ordering of the alternatives is x₄ ≻ x₃ ≻ x₁ ≻ x₂ and the best alternative is x₄.

5.4 Comparing with the existing methods

In order to verify the validness of the methods in this paper, we use the method proposed by Chen [3] to rank this example, and get $\begin{matrix} {\tilde{γ}}_{1} = (s_{4.5161}, s_{3.2297}), {\tilde{γ}}_{2} = (s_{4.4404}, s_{2.7439}), \\ {\tilde{γ}}_{3} = (s_{4.8042}, s_{2.6564}), {\tilde{γ}}_{4} = (s_{5.2192}, s_{2.3191}) \end{matrix}$

Then we calculate the score function, and get $\begin{matrix} Ls ({\tilde{γ}}_{1}) = 1.2864, Ls ({\tilde{γ}}_{2}) = 1.6965, \\ Ls ({\tilde{γ}}_{3}) = 2.1478, Ls ({\tilde{γ}}_{4}) = 2.9001 \end{matrix}$

So we can get the ranking result x₄ ≻ x₃ ≻ x₂ ≻ x₁ and the best choice is x₄.

Obviously, there are the same ranking result and the same best alternative from these methods. Comparing with the proposed methods in this paper, the method proposed by Chen [3] is based on the LIFHA operator, it is only a special case of the LIFWPA operator and LIFGWPA operator proposed in this paper and it doesn’t consider the relationship between the attributes by the power weighting. Obviously, the linguistic intuitionistic fuzzy power operators in this paper can relieve the some influences of unreasonable data given by biased decision makers, the fundamental aspect of these operators is that the weight of each input argument depends on the other input arguments and allows argument values to support each other. In the LIFGPA operator, when we change the value of λ, the result is different, and it provides the more flexible features as λ is assigned different values.

6 Conclusion

In this paper, we explore some Power aggregation operators based on LIFNs and develop some methods to deal with the MADM problems in which the attribute values take the form of LIFNs. Firstly, we proposed the linguistic intuitionistic fuzzy power averaging (LIFPA) operator, the linguistic intuitionistic fuzzy weighted power averaging (LIFWPA) operator, the linguistic intuitionistic fuzzy power geometric (LIFPG) operator, the linguistic intuitionistic fuzzy weighted power geometric (LIFWPG) operator, the linguistic intuitionistic fuzzy generalized power averaging (LIFGPA) operator and linguistic intuitionistic fuzzy generalized weighted power averaging (LIFGWPA) operator. Then, we propose some methods for multi-criteria decision making based on the LIFWPA operator, LIFWPG operator and LIFGWPA operator, and describe the operational processes in detail. Finally, an application example is given to describe the developed approach and to verify its practicality and effectiveness. The advantages of these methods are that they can consider the relationship between the attributes so as to relieve the some influences of unreasonable data given by biased decision makers. In the further research, it is necessary and meaningful to study some new aggregation operators based on the linguistic intuitionistic fuzzy numbers because the linguistic intuitionistic fuzzy numbers are widely applied to each domain in the real world, or extend the proposed methods to interval valued intuitionistic hesitant fuzzy set [2] and interval neutrosophic hesitant set [19] and so on.

Footnotes

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (Nos. 71471172 and 71271124), the Special Funds of Taishan Scholars Project of Shandong Province, National Soft Science Project of China (2014GXQ4D192), Shandong Provincial Social Science Planning Project (No. 15BGLJ06), the Teaching Reform Research Project of Undergraduate Colleges and Universities in Shandong Province (2015Z057).

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