Interval-valued intuitionistic fuzzy prioritized hybrid weighted aggregation operator and its application in decision making

Abstract

Interval-valued intuitionistic fuzzy numbers (IVIFNs) are usually utilized in multi-attribute decision making problems, providing an accurate description of incomplete and uncertain information. The existing IVIFN-based works mainly focus on either the prioritization relationship of attributes or both the importance and the ordered position of them. However, these aspects of attributes are all important in practical use and they should be simultaneously considered. To this end, this paper proposes an interval-valued intuitionistic fuzzy prioritized hybrid weighted aggregation (IVIF-PHWA) operator. The contribution of this paper is threefold: 1) A unit prioritized hybrid weighted aggregation (UPHWA) operator is proposed, which considers the prioritization relationship, the importance and the ordered position of attributes. 2) The UPHWA operator is extended into IVIF-PHWA operator, whose properties are all investigated. These properties illustrate that the IVIF-PHWA operator can better deal with the incomplete and imprecise information. 3) This work develops an IVIF-PHWA operator-based multi-attribute decision making method to solve the decision making problems in IVIF environment. Finally, a practical example is provided to validate the efficiency and effectiveness of the proposed approach.

Keywords

Interval-valued intuitionistic fuzzy set multi-attribute decision making hybrid weighted aggregation operator prioritization relationship

1 Introduction

The multi-attribute decision making (MADM) problem is one of the central topics in decision making. In MADM problems, it is usually to choose the optimal alternative from multiple alternatives according to some criteria [31]. The MADM approaches have been widely used in many fields, such as economy [3, 40], management [4,31,38 , 4,31,38] , military [15], engineering [26, 39], etc.

1.1 Literature review

By reviewing the literatures [5–14 , 21–24], both data type and aggregation operator are the key factors in MADM problems, which concern the accuracy of evaluation results.

Considering for the data type, crisp numbers and intuitionistic fuzzy numbers (IFNs) [41] have been widely investigated in many decision works [5,6,9–11 , 5,6,9–11]. However, sometimes the assessment information is more uncertain and imprecise and it is difficult for decision makers to express their subjective judgments with crisp numbers or IFNs. Atanassov and Gargov [1] proposed the interval-valued intuitionistic fuzzy sets (IVIFSs), which were widely used in various fuzzy environments. In the following decades, IVIFSs have generated many excellent results [1–4 , 28–36]. For instance, Xu [17] firstly defined interval-valued intuitionistic fuzzy numbers (IVIFNs) and some operational laws of them. Xu [2] introduced some relations and operations of interval-valued intuitionistic fuzzy numbers (IVIFNs) and defined some types of matrices, including interval-valued intuitionistic fuzzy (IVIF) matrix, IVIF similarity matrix and IVIF equivalence matrix. Zhang et al. [3] gave a definition of IVIF entropy and constructed a MADM model based on both the IVIF weighted averaging operator and its ranking functions. Li [4] constructed a pair of nonlinear fractional programming models to solve MADM problems with both ratings of alternatives on attributes and weights being expressed with IVIFSs.

There are two kinds of approaches with respect to the information aggregation operators. The first kind is a hot research topic and has received many excellent results [5–8,13–15 , 5–8,13–15]. For example, Yager [5] introduced the ordered weighted aggregation (OWA) operator, which provides a parameterized class of aggregation operators, including the minimum, maximum and average as special case. Xu et al. [6] proposed the hybrid weighted averaging (HWA) operator, which considers both the ordered position and the importance of the attributes. However, the HWA operator does not satisfy the properties of boundary and idempotency. In view of this, Lin and Jiang [8] introduced the hybrid weighted arithmetical averaging (HWAA) operator to improve the HWA operator. However, Wang [7] found that the HWAA operator did not satisfy monotonicity and might provide inconsistent output in decision making systems.

The second kind concentrates on the prioritization relationships among the attributes. For instance, Yager [9, 10] introduced prioritized aggregation (PA) operator and prioritized ordered weighted averaging (POWA) operator. Yu and Xu [11] proposed a prioritized intuitionistic fuzzy aggregation (PIFA) operator and applied it in MADM. Yu et al. [12] proposed the interval-valued intuitionistic fuzzy prioritized weighted average (IVIF-PWA) operator to solve group decision making problems. Chen [16] developed a prioritized aggregation operator-based approach to the MADM problems by using IVIFSs.

1.2 Limitations

Although lots of work on MADM is proposed, there are still some limitations, which are summarized as follows.

All of these methods mentioned above considers either the prioritization relationship of attributes or both the importance and ordered position of attributes. However, all aspects of the attributes are important and are needed to be referred in some practical decisions.

In addition, the HWA operator and its extending operators do not satisfy some basic properties, including boundary and idempotency. This is pointed out by Lin and Jiang [8] and Wang [7]. However, most importantly, these properties are very necessary in decision making.

1.3 Contributions

In view of these limitations, this paper proposes an interval-valued intuitionistic fuzzy prioritized hybrid weighted aggregation (IVIF-PHWA) operator to solve the MADM problems. Its evolvement is demonstrated in Fig. 1. It is worth noting that this work introduces some new aggregation operators derived from HWA operator. The main contributions are described as follows. 1) A unit hybrid weighted aggregation (UHWA) operator is proposed by extending the HWA operator. Furthermore, a unit prioritized hybrid weighted aggregation (UPHWA) operator is proposed to consider the prioritization relationship, importance and ordered position of attributes simultaneously. 2) The UPHWA is extended into intuitionistic fuzzy based and interval-valued intuitionistic based operators, named as IF-PHWA operator and IVIF-PHWA operator, respectively. The desirable properties of them are all investigated and proved in this work. Besides, an interval-valued intuitionistic fuzzy basic unit monotonic (IVIF-BUM) function is introduced in order to transform the derived interval-valued intuitionistic weights to crisp numbers. 3) An IVIF-PHWA operator-based model is built to solve the MADM problems in the IVIF environment.

The remainder of this paper is organized as follows. In Section 2, some basic concepts and notations are reviewed briefly. Section 3 proposes a new system of HWA operator, including UHWA, UPHWA and IF-PHWA operators. In Section 4, an IVIF-PHWA operator is introduced. And its weight vector can be calculated by the proposed IVIF-BUM function. Section 5 presents a MADM model based on the IVIF-PHWA operator. In Section 6, a numerical example is provided to validate the superiority of the proposed approach. The conclusion is summarized in Section 7.

2 Preliminary

Before presenting the new aggregation operators derived from the HWA operator, several relevant definitions of IVIFSs and aggregation operators are briefly reviewed in this section.

2.1 Interval-valued intuitionistic fuzzy sets and their basic operations

Definition 2.1. [1] Let X ={ x ₁, x ₂, ·· · , x _n } be a universe of discourse, then an interval-valued intuitionistic fuzzy set α in X is given by

$α = (μ_{α} (x), υ_{α} (x)), x \in X,$ (1) where μ _α (x) ⊆ [0, 1], υ _α (x) ⊆ [0, 1] and 0 ≤ (supμ _α (x) + sup υ _α (x)) ≤1, ∀ x ∈ X. μ _α (x) and υ _α (x) denote the membership degree and the non-membership degree of element x in the set α, respectively.

The pair (μ _α (x) , υ _α (x)) is defined as an interval-valued intuitionistic fuzzy number [17]. For convenience, an IVIFN is denoted by $([μ_{α}^{L}, μ_{α}^{H}],$ $[υ_{α}^{L}, υ_{α}^{H}])$ , where $[μ_{α}^{L}, μ_{α}^{H}] \subseteq [0, 1]$ , $[υ_{α}^{L}, υ_{α}^{H}] \subseteq [0, 1]$ and $0 \leq μ_{α}^{H} + υ_{α}^{H} \leq 1$ .

Let α and β be any two IVIFNs. The basic operations are defined in [17], including α ∩ β, α ∪ β, α ⊕ β,α ⊗ β, λα and α ^λ.

Definition 2.2. [18] Let $α = ([μ_{α}^{L}, μ_{α}^{H}], [υ_{α}^{L}, υ_{α}^{H}])$ be an IVIFN. The composite score function s is defined as

$\begin{matrix} s (α) = δ \cdot [μ_{α}^{L} \cdot (2 - μ_{α}^{L} - 2 υ_{α}^{H})] + \\ (1 - δ) \cdot [1 - υ_{α}^{L} \cdot (2 - υ_{α}^{L} - 2 μ_{α}^{H})], \end{matrix}$ (2) where s (α) ∈ [0, 1], and δ ∈ (0, 1) is a parameter that reflects the attitudes of decision makers.

Definition 2.3. [18] Let $α = ([μ_{α}^{L}, μ_{α}^{H}], [υ_{α}^{L}, υ_{α}^{H}])$ be an IVIFN. The composite accuracy function h, based on the measures of accuracy and not-hesitancy of IVIFN, is defined as

$\begin{matrix} h (α) = φ \cdot 2 μ_{α}^{L} \cdot υ_{α}^{L} + \\ (1 - φ) \cdot [1 - (1 - μ_{α}^{H} - υ_{α}^{H})^{2}], \end{matrix}$ (3) where h (α) ∈ [0, 1], and φ ∈ (0, 1) is a parameter that reflects the attitudes of decision makers.

2.2 Some typical aggregation operators

Definition 2.4. [6, 42] A hybrid weighted aggregation (HWA) operator of dimension n is a mapping HWA: R ⁿ → R, which has an associated weight vector w = (w ₁, w ₂, . . . , w _n) ^T, with w _j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ . The HWA operator is defined as

$HW A_{(w, λ)} (a_{1}, a_{2}, . . ., a_{n}) = \sum_{j = 1}^{n} w_{j} b_{j},$ (4) where b _j is the jth largest of the weighted arguments nλ _i a _i (i = 1, 2, . . . , n), λ = (λ ₁, λ ₂, . . . , λ _n) ^T is the weight vector of a _i (i = 1, 2, . . . , n), with λ _i ∈ [0, 1] and $\sum_{i = 1}^{n} λ_{i} = 1$ , and n is a balancing coefficient.

Especially, considering for a kind of MADM problems in which there exists a prioritization relationship over the attributes, Yager [9, 10] introduced several prioritized aggregation operators. In [10], Yager proposed the prioritized ordered weighted averaging operator.

Let a collection of attributes A = {a ₁, a ₂, . . . , a _n} be prioritized such that a _i > a _j if i < j(i, j = 1, 2, . . , n). Assume that a _i (x) ∈ [0, 1] is the degree of satisfaction to the ith attribute by the alternative x. The goal is to get an overall satisfaction a (x) of x to the multiple attributes as the OWA operator of the individual attribute satisfaction in such a way as to reflect the organization of the attributes with both the scope and priority relationships between them.

Definition 2.5. [10] A prioritized ordered averaging (POWA) operator of dimension n is a mapping POWA:R ⁿ → R. For an alternative x, the POWA operator is defined as

$a (x) = POWA (a_{1}, a_{2}, . . ., a_{n}) = \sum_{j = 1}^{n} w_{j} a_{θ (j)},$ (5) where θ is an index function, θ (j) is the index of the jth most satisfied attribute, a _θ(j) is the jth largest of a _i, w = (w ₁, w ₂, . . . , w _n) ^T is an associated weight vector, by using a basic unit monotonic (BUM) function.

2.3 Basic unit monotonic function

A basic unit monotonic (BUM) function proposed by Yager [19] is presented as follows.

Definition 2.6. A BUM function is a mapping g : [0, 1] → [0, 1], which satisfies the following properties

g (0) =0,

g (1) =1,

g (x) ≥ g (y), where x > y.

3 New aggregation operators derived from hybrid weighted aggregation

In this section, we propose some new aggregation operators derived from the HWA operator. Then the desirable properties of them are all investigated.

The HWA operator generalizes both the weighted arithmetical averaging (WAA) operator [37] and the OWA operator [5] and considers both the importance and the ordered position of attributes. However, HWA operator does not satisfy the properties of boundary and idempotency, which is pointed out by Lin and Jiang [8]. In order to overcome these drawbacks, a UHWA operator is firstly proposed and presented as follows.

3.1 Unit hybrid weighted aggregation (UHWA) operator

Definition 3.1. A UHWA operator of dimension n is a mapping UHWA:R ⁿ → R, which has an associated weight vector w = (w ₁, w ₂, . . . , w _n) ^T, with w _j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ , the UHWA operator is defined as

$UHW A_{(w, λ)} (a_{1}, a_{2}, . . ., a_{n}) = \sum_{j = 1}^{n} w_{j} b_{j},$ (6) where b _j is the jth largest of the weighted arguments qλ _i a _i (i = 1, 2, . . . , n), λ = (λ ₁, λ ₂, . . . , λ _n) ^T is the weight vector of a _i (i = 1, 2, . . . , n) with λ _i ∈ [0, 1] and $\sum_{i = 1}^{n} λ_{i} = 1$ , and q is a balancing coefficient, $q = \frac{1}{\max (λ_{i})}$ .

The balancing coefficient q is different from the balancing coefficient n of HWA operator. q is related to the largest weight of a _i (i = 1, 2, . . . , n), but n is related to the number of a _i. Specially, if every λ _i equal each other, (i.e., $λ_{i} = \frac{1}{n}$ (i = 1, 2, . . . , n)), then $q = \frac{1}{\max (λ_{i})} = n$ .

In order to prove the properties of UHWA operator conveniently, let b _j = qλ _θ(j) a _θ(j), where θ (j) is the index of the jth largest of the weighted arguments qλ _i a _i(i = 1, 2, . . . , n). With that, Equation. (6) can be written as

$\begin{matrix} UHW A_{(w, λ)} (a_{1}, a_{2}, . . ., a_{n}) \\ = \sum_{j = 1}^{n} w_{j} q λ_{θ (j)} a_{θ (j)} . \end{matrix}$ (7) The UHWA operator has the desirable properties, including quasi-idempotency, monotonicity and boundary.

3.1.1 Quasi-idempotency

Theorem 3.1. Let a _i (i = 1, 2, . . . , n) be a collection of equal values (i.e., a _i = a) and $λ_{i} = \frac{1}{n}$ (i = 1, 2, . . . , n), then

$UHW A_{(w, λ)} (a_{1}, a_{2}, . . ., a_{n}) = a .$ (8)

Proof. According to Definition 3.1, we have

$\begin{matrix} UHW A_{(w, λ)} (a_{1}, a_{2}, . . ., a_{n}) \\ = \sum_{j = 1}^{n} w_{j} q λ_{θ (j)} a_{θ (j)} \\ = a, \end{matrix}$ where $q = \frac{1}{\max (λ_{i})} = n$ .

3.1.2 Monotonicity

Theorem 3.2. Let a _i and $a_{i}^{'}$ (i = 1, 2, . . . , n) be two collections of values. Both of their weight vectors are λ = (λ ₁, λ ₂, . . . , λ _n) ^T. If $a_{i} \geq a_{i}^{'}$ , then

$UHWA (a_{1}, a_{2}, . . ., a_{n}) \geq UHWA (a_{1}^{'}, a_{2}^{'}, . . ., a_{n}^{'})$ (9)

Proof. Because $a_{i} - a_{i}^{'} \geq 0 (i = 1, 2, . . ., n)$ , we have

$\begin{matrix} UHWA (a_{1}, a_{2}, . . ., a_{n}) - UHWA (a_{1}^{'}, a_{2}^{'}, . . ., a_{n}^{'}) \\ = \sum_{j = 1}^{n} w_{j} q λ_{θ (j)} a_{θ (j)} - \sum_{j = 1}^{n} w_{j} q λ_{θ (j)} a_{θ (j)}^{'} \\ = \sum_{j = 1}^{n} w_{j} q λ_{θ (j)} (a_{θ (j)} - a_{θ (j)}^{'}) \geq 0 . \end{matrix}$

3.1.3 Boundary

Theorem 3.3. Let a _i (i = 1, 2, . . . , n) be a collection of values. Then

$\begin{matrix} UHWA \underset{n}{\underset{︸}{(a_{*}, . . ., a_{*})}} \leq UHWA (a_{1}, a_{2}, . . ., a_{n}) \\ \leq UHWA \underset{n}{\underset{︸}{(a^{*}, . . ., a^{*})}}, \end{matrix}$ (10) where a _* = min (a _i) and a ^* = max (a _i).

Proof. According to Theorem 3.2, we can get Theorem 3.3 easily.

An example is provided and derived from [7]. By this example, we can explain the difference between UHWA operator and HWA operator [6, 42].

Example 3.1. Let (a ₁, a ₂, a ₃, a ₄) = (1, 1, 1, 1) and w = λ = (1, 0, 0, 0) ^T, then HWA (a ₁, a ₂, a ₃, a ₄) is calculated by Xu [6, 42]

$\begin{matrix} HWA (a_{1}, a_{2}, a_{3}, a_{4}) = \sum_{j = 1}^{n} w_{j} b_{j} = 4 > \max {a_{i}}, \end{matrix}$ where n = 4, b ₁ = nλ ₁ a ₁ = 4, b ₂ = b ₃ = b ₄ = 0 and max {a _i} =1.

UHWA (a ₁, a ₂, a ₃, a ₄) is calculated using Equation. (6).

$\begin{matrix} UHWA (a_{1}, a_{2}, a_{3}, a_{4}) = \sum_{j = 1}^{n} w_{j} b_{j} = \max {a_{i}} \end{matrix}$ where max (λ _i) =1, $q = \frac{1}{\max (λ_{i})} = 1$ , b ₁ = qλ ₁ a ₁ = 1, b ₂ = b ₃ = b ₄ = 0 and max {a _i} =1.

In this example, HWA (a ₁, a ₂, a ₃, a ₄) > max {a _i}, but UHWA (a ₁, a ₂, a ₃, a ₄) = max {a _i}. It clearly concludes that HWA operator does not satisfy the boundary and the idempotent in some special cases. But the UHWA operator satisfies the properties which are investigated from Theorem 3.1 to Theorem 3.3. The reason for this phenomenon is because of the different balancing coefficients (i.e.,n and q). n is related to the number of a _i, while q is related to the greatest weight of a _i, which is easy to be obtained. In a word, q is more appropriate to be a balancing coefficient.

In MADM problems, not only both the importance and the ordered position of the attributes should be considered, but also the prioritization relationship of attributes should be analyzed. For this reason, a unit prioritized hybrid weighted aggregation (UPHWA) operator derived from UHWA operator is proposed.

3.2 Unit prioritized hybrid weighted aggregation (UPHWA) operator

A prioritization relationship of attributes is defined as: Let a collection of attributes A = {a ₁, a ₂, . . . , a _n} be prioritized such that a _i > a _j if i < j(i, j = 1, 2, . . , n). Assume that a _i (x) ∈ [0, 1] is the degree of satisfaction to the ith attribute by alternative x. The goal is to get an overall satisfaction a (x) of x to the multiple attributes as the UHWA operator of the individual attribute satisfaction in such a way as to reflect the organization of the attributes with all the importance, scope and priority relationships between them.

Definition 3.2. A unit prioritized hybrid weighted aggregation (UPHWA) operator of dimension n is a mapping UPHWA:R ⁿ → R. The weight vector w = (w ₁, w ₂, . . . , w _n) ^T, with w _j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ , is an associated weight vector by using a BUM function. For any alternative x, there is

$a (x) = UPHWA (a_{1}, a_{2}, . . ., a_{n}) = \sum_{j = 1}^{n} w_{j} c_{j},$ (11) where c _j is the jth largest of the weighted attributes qλ _i a _i, λ = (λ ₁, λ ₂, . . . , λ _n) ^T is the weight vector of a _i (i = 1, 2, . . . , n) with λ _i ∈ [0, 1] and $\sum_{i = 1}^{n} λ_{i} = 1$ , and q is the balancing coefficient, $q = \frac{1}{\max (λ_{i})}$ .

Furthermore, let c _j = qλ _θ(j) a _θ(j) and θ be an index function. θ (j) is the index of the jth largest of the weighted attributes qλ _i a _i, a _θ(j) is the jth most satisfied attribute, λ _θ(j) is the weight corresponding to a _θ(j). Then Equation. (11) can be written as

$\begin{matrix} UPHWA (a_{1}, a_{2}, . . ., a_{n}) \\ = \sum_{j = 1}^{n} w_{j} q λ_{θ (j)} a_{θ (j)} . \end{matrix}$ (12)

In addition, UPHWA operator also has the important properties, including quasi-idempotency, monotonicity and boundary. The proofs are omitted here.

Because the IFNs are powerful tools to deal with the MADM problems, the UPHWA operator is extended to the intuitionistic fuzzy prioritized hybrid weighted aggregation (IF-PHWA) operator.

3.3 Intuitionistic fuzzy prioritized hybrid weighted aggregation (IF-PHWA) operator

Definition 3.3. Let a collection of attributes A = {a ₁, a ₂, . . . , a _n} be prioritized such that a _i > a _j if i < j(i, j = 1, 2, . . , n), and a _i = (μ _i, υ _i) be an IFN, for all i. An intuitionistic fuzzy prioritized hybrid weighted aggregation (IF-PHWA) operator of dimension n is a mapping F: R ⁿ → R. The weight vector w = (w ₁, w ₂, . . . , w _n) ^T, with w _j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ , is an associated weight vector by using an IF-BUM function [4]. For an alternative x, there is

$F_{(p, w, λ)} (a_{1}, a_{2}, . . ., a_{n}) = \sum_{j = 1}^{n} w_{j} c_{j},$ (13) where c _j is the jth largest of the weighted attributes qλ _i a _i, λ = (λ ₁, λ ₂, . . . , λ _n) ^T is the weight vector of a _i (i = 1, 2, . . . , n) with λ _i ∈ [0, 1] and $\sum_{i = 1}^{n} λ_{i} = 1$ , and q is the balancing coefficient, $q = \frac{1}{\max (λ_{i})}$ .

Furthermore, let c _j = qλ _θ(j) a _θ(j). θ (j) is the index of the jth largest of the weighted attributes qλ _i a _i, a _θ(j) is the jth most satisfied attribute, λ _θ(j) is the weight corresponding to a _θ(j). The Equation. (13) can be written as

$F_{(p, w, λ)} (a_{1}, a_{2}, . . ., a_{n}) = \sum_{j = 1}^{n} w_{j} q λ_{θ (j)} a_{θ (j)} .$ (14)

In addition, the aggregated value of a _i = (μ _i, υ _i) (i = 1, 2, . . . , n), using IF-PHWA operator, is still an IFN. And IF-PHWA also has the properties including quasi-idempotency, monotonicity and boundary. For a compact representation, the proofs are also omitted here.

The IF-PHWA operator is applied in intuitionistic fuzzy environment. However, sometimes it is hard for decision makers to express their opinions with IFSs. Atanassov and Gargov [1] generalized the IFS by introducing the IVIFS, which represents the degree of membership and non-membership by closed subintervals of [0, 1]. This approach has effectively expanded the ability of the IFS to handle uncertain and imprecise information. Because of its ability to solve practical decision making problems, an interval-valued intuitionistic fuzzy prioritized hybrid weighted aggregation (IVIF-PHWA) operator is proposed, which is derived from UPHWA and IF-PHWA operators. Before proposing the IVIF-PHWA operator, an intuitionistic fuzzy basic unit monotonic (IVIF-BUM) function is defined, which is utilized to transform the interval-valued intuitionistic fuzzy weights to crisp numbers.

4 The proposed IVIF-PHWA operator

4.1 Interval-valued intuitionistic fuzzy basic unit monotonic (IVIF-BUM) function

Definition 4.1. Let A = {a ₁, a ₂, . . . , a _n} be a collection of IVIFNs, and S = {a _S(1), a _S(2), . . . , a _S(k)} (k ≤ n) is a subset of A. S (i) is the value of least satisfied attributes. The IVIF-BUM function is given as

$\begin{matrix} \tilde{f} (S) = \frac{1}{N} (\sum_{i = 1}^{k} μ_{a_{S (i)}}^{L} + \sum_{i = 1}^{k} μ_{a_{S (i)}}^{H} - \sum_{i = 1}^{k} υ_{a_{S (i)}}^{L} - \\ \sum_{i = 1}^{k} υ_{a_{S (i)}}^{H} + 2 k), \end{matrix}$ (15) where $N = \sum_{i = 1}^{n} μ_{a_{i}}^{L} + \sum_{i = 1}^{n} μ_{a_{i}}^{H} - \sum_{i = 1}^{n} υ_{a_{i}}^{L} - \sum_{i = 1}^{n} υ_{a_{i}}^{H} + 2 n$ .

An IVIF-BUM function is mapping $\tilde{f} : [0, 1] \to [0, 1]$ , which satisfies the following properties:

$\tilde{f} (0) = 0$ ,

$\tilde{f} (1) = 1$ ,

$\tilde{f} (x) \geq \tilde{f} (y)$ , where x > y.

Proof.

Let S = 0, it means that S is a Null. Then k = 0, $\tilde{f} (0) = 0$ using Equation. (15).

Let S = 1, it means that S is a Full. Then k = n, $\tilde{f} (1) = 1$ using Equation. (15).

Let S ₁ = x and S ₂ = y. Assume that there are k _x = k + m and k _y = k alternatives to satisfy the conditions, respectively. By using Equation. (15)

$\begin{matrix} \tilde{f} (S_{x}) = \frac{1}{N} (\sum_{i = 1}^{(k + m)} μ_{a_{S (i)}}^{L} + \sum_{i = 1}^{(k + m)} μ_{a_{S (i)}}^{H} - \\ \sum_{i = 1}^{(k + m)} υ_{a_{S (i)}}^{L} - \sum_{i = 1}^{(k + m)} υ_{a_{S (i)}}^{H} + 2 (k + m)), \end{matrix}$

$\begin{matrix} \tilde{f} (S_{y}) = \frac{1}{N} (\sum_{i = 1}^{k} μ_{a_{S (i)}}^{L} + \sum_{i = 1}^{k} μ_{a_{S (i)}}^{H} - \\ \sum_{i = 1}^{k} υ_{a_{S (i)}}^{L} - \sum_{i = 1}^{k} υ_{a_{S (i)}}^{H} + 2 k), \end{matrix}$

$\begin{matrix} \tilde{f} (S_{x}) - \tilde{f} (S_{y}) = \frac{1}{N} (\sum_{i = (k + 1)}^{(k + m)} μ_{a_{S (i)}}^{L} + \sum_{i = (k + 1)}^{(k + m)} μ_{a_{S (i)}}^{H} - \\ \sum_{i = (k + 1)}^{(k + m)} υ_{a_{S (i)}}^{L} - \sum_{i = (k + 1)}^{(k + m)} υ_{a_{S (i)}}^{H} + 2 (m)), \end{matrix}$ where $N = \sum_{i = 1}^{n} μ_{a_{i}}^{L} + \sum_{i = 1}^{n} μ_{a_{i}}^{H} - \sum_{i = 1}^{n} υ_{a_{i}}^{L} - \sum_{i = 1}^{n} υ_{a_{i}}^{H} + 2 n$ .

If a _S(i) = ([0, 0] , [1, 1]) , i = ((k + 1) , (k + 2) , . . . , (k + m)), and $\min (μ_{a_{i}}^{L} + μ_{a_{i}}^{H} - υ_{a_{i}}^{L} - υ_{a_{i}}^{H}) = - 2 m$ . There is

$\tilde{f} (S_{x}) - \tilde{f} (S_{y}) = 0$ , else, $\min (μ_{a_{i}}^{L} + μ_{a_{i}}^{H} - υ_{a_{i}}^{L} - υ_{a_{i}}^{H}) \geq - 2 m$ . There is

$\tilde{f} (S_{x}) - \tilde{f} (S_{y}) \geq 0$ .

With the IVIF-BUM function, the IVIF-PHWA operator is introduced as follows.

4.2 Interval-valued intuitionistic fuzzy prioritized hybrid weighted aggregation (IVIF-PHWA) operator

Definition 4.2. Let a collection of attributes A = {a ₁, a ₂, . . . , a _n} be prioritized such that a _i > a _j if i < j(i, j = 1, 2, . . , n), and $a_{i} = ([μ_{i}^{L}, μ_{i}^{H}], [υ_{i}^{L}, υ_{i}^{H}])$ be an IVIFN, for all i. An interval-valued intuitionistic fuzzy prioritized hybrid weighted aggregation (IVIF-PHWA) operator of dimension n is a mapping F:R ⁿ → R. The weight vector w = (w ₁, w ₂, . . . , w _n) ^T, with w _j ∈ [0, 1] and $\sum_{j = 1}^{n} w_{j} = 1$ , is an associated weight vector with the IVIF-BUM function. For an alternative x, there is

${\tilde{F}}_{(p, w, λ)} (a_{1}, a_{2}, . . ., a_{n}) = \sum_{j = 1}^{n} w_{j} c_{j},$ (16) where c _j is the jth largest of the weighted attributes qλ _i a _i, λ = (λ ₁, λ ₂, . . . , λ _n) ^T is the weight vector of a _i (i = 1, 2, . . . , n) with λ _i ∈ [0, 1] and $\sum_{i = 1}^{n} λ_{i} = 1$ , and q is the balancing coefficient, $q = \frac{1}{\max (λ_{i})}$ .

${\tilde{F}}_{(p, w, λ)} (a_{1}, a_{2}, . . ., a_{n}) = \sum_{j = 1}^{n} w_{j} q λ_{θ (j)} a_{θ (j)} .$ (17)

The IVIF-PHWA operator has the following desirable properties.

4.2.1 IVIFN consistency

Theorem 4.1. Let $a_{i} = ([μ_{a_{i}}^{L}, μ_{a_{i}}^{H}], [υ_{a_{i}}^{L}, υ_{a_{i}}^{H}]) (i = 1, 2, . . ., n)$ be a collection of the IVIFNs. Then the aggregated result is still an IVIFN, and

$\begin{matrix} {\tilde{F}}_{(w, p, λ)} (a_{1}, a_{2}, . . ., a_{n}) \\ = ([1 - \prod_{j = 1}^{n} {(1 - μ_{a_{θ (j)}}^{L})}^{w_{j} q λ_{θ (j)}}, 1 - \\ \prod_{j = 1}^{n} {(1 - μ_{a_{θ (j)}}^{H})}^{w_{j} q λ_{θ (j)}}], [\prod_{j = 1}^{n} {(υ_{a_{θ (j)}}^{L})}^{w_{j} q λ_{θ (j)}}, \\ \prod_{j = 1}^{n} {(υ_{a_{θ (j)}}^{H})}^{w_{j} q λ_{θ (j)}}]) . \end{matrix}$ (18)

Proof. The proof of this theorem is similar to the proof of Theorem 3.2 in [23].

4.2.2 Quasi-idempotency

Theorem 4.2. Let $a_{i} = ([μ_{a_{i}}^{L}, μ_{a_{i}}^{H}],$ $[υ_{a_{i}}^{L}, υ_{a_{i}}^{H}])$ (i = 1, 2, . . . , n) be equal(i.e., a _i = a) and $λ_{i} = \frac{1}{n}$ , then

${\tilde{F}}_{(w, p, λ)} (a_{1}, a_{2}, . . ., a_{n}) = a .$ (19) Proof. According to Definition 4.2, we have

$\begin{matrix} {\tilde{F}}_{(w, p, λ)} (a_{1}, a_{2}, . . ., a_{n}) \\ = \sum_{j = 1}^{n} w_{j} q λ_{θ (j)} a_{θ (j)} \\ = a . \end{matrix}$ where $q = \frac{1}{\max (λ_{i})} = n$ , $λ_{θ (j)} = \frac{1}{n}$ , a _θ(j) = a.

4.2.3 Monotonicity

Theorem 4.3. Let $a_{i} = ([μ_{a_{i}}^{L}, μ_{a_{i}}^{H}],$ $[υ_{a_{i}}^{L}, υ_{a_{i}}^{H}])$ and $a_{i}^{'} = ([μ_{a_{i}^{'}}^{L}, μ_{a_{i}^{'}}^{H}], [υ_{a_{i}^{'}}^{L}, υ_{a_{i}^{'}}^{H}])$ (i = 1, 2, . . . , n) be two collections of the IVIFNs. Both of their weight vectors are λ = (λ ₁, λ ₂, . . . , λ _n) ^T. If $μ_{a_{i}}^{L} \leq μ_{a_{i}^{'}}^{L}$ , $μ_{a_{i}}^{H} \leq μ_{a_{i}}^{H}$ , $υ_{a_{i}}^{L} \geq υ_{a_{i}^{'}}^{L}$ and $υ_{a_{i}}^{H} \geq υ_{a_{i}^{'}}^{H}$ (i = 1, 2, . . . , n), then ${\tilde{F}}_{(w, p, λ)} (a_{1}, a_{2}, . . ., a_{n}) \leq {\tilde{F}}_{(w, p, λ)} (a_{1}^{'}, a_{2}^{'}, . . ., a_{n}^{'}) .$ (20)

Proof. Because $μ_{a_{i}}^{L} \leq μ_{a_{i}^{'}}^{L}$ for all i, we know that

$1 - \prod_{j = 1}^{n} {(1 - μ_{c_{j}}^{L})}^{w_{j}} \leq 1 - \prod_{j = 1}^{n} {(1 - μ_{c_{j}^{'}}^{L})}^{w_{j}},$

In a similar way, we also have

$\begin{matrix} 1 - \prod_{j = 1}^{n} {(1 - μ_{c_{j}}^{H})}^{w_{j}} \leq 1 - \prod_{j = 1}^{n} {(1 - μ_{c_{j}^{'}}^{H})}^{w_{j}}, \\ \prod_{j = 1}^{n} {(υ_{c_{j}}^{L})}^{w_{j}} \geq \prod_{j = 1}^{n} {(υ_{c_{j}^{'}}^{L})}^{w_{j}}, \end{matrix}$ and

$\begin{matrix} \prod_{j = 1}^{n} {(υ_{c_{j}}^{H})}^{w_{j}} \geq \prod_{j = 1}^{n} {(υ_{c_{j}^{'}}^{H})}^{w_{j}}, \end{matrix}$ where c _j = qλ _θ(j) a _θ(j), $c_{j}^{'} = q λ_{θ (j)} a_{θ (j)}^{'}$ .

It concludes that ${\tilde{F}}_{(w, p, λ)} (a_{1}, a_{2}, . . ., a_{n}) \leq {\tilde{F}}_{(w, p, λ)} (a_{1}^{'}, a_{2}^{'}, . . ., a_{n}^{'})$ .

4.2.4 Boundary

Theorem 4.4. Let $a_{i} = ([μ_{a_{i}}^{L}, μ_{a_{i}}^{H}],$ $[υ_{a_{i}}^{L}, υ_{a_{i}}^{H}]) (i = 1, 2, . . ., n)$ be a collection of the IVIFNs. Then

$\begin{matrix} {\tilde{F}}_{(w, p, λ)} \underset{n}{\underset{︸}{(a_{*}, . . ., a_{*})}} \leq {\tilde{F}}_{(w, p, λ)} (a_{1}, a_{2}, . . ., a_{n}) \\ \leq {\tilde{F}}_{(w, p, λ)} \underset{n}{\underset{︸}{(a^{*}, . . ., a^{*})}} \end{matrix}$ (21) where $a_{*} = ([\min (μ_{a}^{L}), \min (μ_{a}^{H})], [\max (μ_{a}^{L}), \max (μ_{a}^{H})])$ , $a^{*} = ([\max (μ_{a}^{L}), \max (μ_{a}^{H})], [\min (μ_{a}^{L}), \min (μ_{a}^{H})])$ .

Proof. According to Theorem 4.3, we can get Theorem 4.4 easily.

5 A multi-attribute decision making model based on the proposed IVIF-PHWA operator

In this section, the IVIF-PHWA operator-based approach is introduced to solve a MADM problem in the IVIF environment.

For a MADM problem, let X = {x ₁, x ₂, . . . , x _m} be a collection of alternatives, U = {u ₁, u ₂, . . . , u _n} be a collection of attributes. Each alternative is assessed against each of the attribute. The assessment of alternative x _k against the attribute u _i is given by an IVIFN. The main decision process can be summarized in the following.

Step 1: Let A = {a ₁, a ₂, . . . , a _n} be a collection of attributes, in which there exists a prioritization relationship between the attributes, such that a _i > a _j if i < j(i, j = 1, 2, . . , n). This is a linear ordering with a ₁ having the highest priority.

Step 2: Calculate the balancing coefficient q and b _i = b _i (x _k) (i = 1, 2, . . . , n ; k = 1, 2, . . . , m).

$q = \frac{1}{\max (λ_{i})},$ (22)

$b_{i} = q λ_{i} a_{i} (x_{k}) .$ (23)

Step 3: Calculate the satisfactions of b _i.

${\begin{matrix} S_{0} = ([1, 1], [0, 0]) \\ S_{i} = b_{i}, (i = 1, 2, . . ., n) . \end{matrix}$ (24)

Step 4: The importance weights T _i are calculated.

${\begin{matrix} T_{1} = ([1, 1], [0, 0]) \\ T_{i} = \prod_{k = 1}^{i - 1} S_{k} = S_{i - 1} T_{i - 1}, (i = 2, 3, . . ., n) . \end{matrix}$ (25)

Step 5: Calculate the scores s (b _i) of b _i using Equation. (2), where δ is given according to the requirement.

Step 6: Rank the attribute satisfactions according to the descending order of s (b _i) and obtain θ (j) for a _i (x _k). There is

$c_{j} = q λ_{θ (j)} a_{θ (j)} (x_{k}) .$ (26)

Step 7: Let M = {T ₁, T ₂, . . . , T _n} be a set of T _i, M _j = {T _θ(1), T _θ(2), . . . , T _θ(l)} (l ≤ n) and M ₀ be a Null. Apply the IVIF-BUM function Equation (15) to obtain the IVIF-PHWA weights w _j

$w_{j} = \tilde{f} (M_{j}) - \tilde{f} (M_{j - 1}),$ (27) where j = 1, 2, . . . , n.

Step 8: Using Equation (16), the final aggregated values Z (x _k)(k = 1, 2, . . . , m) are obtained.

Step 9: Calculate the scores s (Z (x _k))(k = 1, 2, . . . , m) of Z (x _k) using Equation (2), where the δ is given according to the decision makers. Then rank all the alternatives with the ranking of s (Z (x _k)).

6 Practical example

The MADM plays an important role in a variety of fields, such as business investment, supplier selection, medical diagnosis, etc. The threat assessment of aerial targets is also an important application of MADM.

There six different aerial targets X = {x ₁, x ₂, x ₃, x ₄, x ₅, x ₆} are planned to be evaluated. Furthermore, there are five attributes for each target in this threat assessment. They are Target type, Flight time, Flight velocity, Short-cut route and Flight height (U = {u ₁, u ₂, u ₃, u ₄, u ₅}), and their weight vector is given as w = (0.2, 0.2, 0.19, 0.2, 0.21) ^T. The prioritization relationship of these attributes is u ₅ > u ₁ > u ₃ > u ₄ > u ₂. Suppose that the attribute values u _i (x _k) (i = 1, 2, . . . , 5 ; k = 1, 2, . . . , 6) of x _k with respect to u _i can be represented as IVIFNs in Table 1.

6.1 The MADM method based on IVIF-PHWA operator

The proposed method based on IVIF-PHWA operator is presented as follows.

Step 1: Let A = {a ₁, a ₂, a ₃, a ₄, a ₅} be a collection of attributes and a ₁ > a ₂ > a ₃ > a ₄ > a ₅. According to the analysis above, there are a ₁ = u ₅, a ₂ = u ₁, a ₃ = u ₃, a ₄ = u ₄, a ₅ = u ₂.

Step 2: Calculate the balancing coefficient q using Equation. Equation. (22) and b _i (i = 1, 2, . . . , 5) using Equation. (23).

Step 3: Using Equation. (24), compute the satisfactions of b _i (i = 1, 2, . . . , 5).S ₀ = ([1, 1] , [0, 0]), S ₁ = b ₁ (x _k), S ₂ = b ₂ (x _k), S ₃ = b ₃ (x _k), S ₄ = b ₄ (x _k), S ₅ = b ₅ (x _k), where k = 1, 2, . . . , 6.

Step 4: The importance weights T _i (x _k) (i = 1, 2, . . . , 5 ; k = 1, 2, . . . , 6) are calculated using Equation. (25).

Step 5: Calculate the scores of b _i (x _k) (i = 1, 2, . . . , 5 ; k = 1, 2, . . . , 6) using Equation. (2), where δ is set as 0.5. The results are shown in Table 2.

Step 6: According to the s (b _i), sort the attribute satisfactions and obtain θ (j) for a _i (x _k). Then, c _j are calculated using Equation. (26).

Step 7: Calculate the weights w _j for IVIF-PHWA operator using IVIF-BUM function via Equation. (15) and (27). The calculated results are shown in Table 3.

Step 8: According to the above results and the IVIF-PHWA operator Equation. (16), The aggregated values are Z (x ₁) = ([1, 1] , [0, 0]),Z (x ₂) = ([0.6406, 0.6883] , [0.2135, 0.2456]), Z (x ₃) = ([0.6537, 0.6926] , [0.2008, 0.2416]), Z (x ₄) = ([0.8023, 0.8364] , [0, 0.1326]), Z (x ₅) = ([0.8231, 1] , [0, 0]) and Z (x ₆) = ([0.4933, 0.5120] , [0.3789, 0.4108]).

Step 9: Calculate the scores of Z (x _k) , (k = 1, 2, . . . , 6) using Equation. (2), where the δ is 0.5. s (Z (x ₁)) =1, s (Z (x ₂)) =0.7343, s (Z (x ₃)) =0.7406, s (Z (x ₄)) =0.8741, s (Z (x ₅)) =0.9844, s (Z (x ₆)) =0.5559.

Then, the alternatives are ranked according to the obtained scores, x ₁ > x ₅ > x ₄ > x ₃ > x ₂ > x ₆.

6.2 Comparison validation

To validate the superiority of the proposed method, some comparative methods are provided. According to the data type, the comparative methods can be divided into two categories: 1) The method for complete ranking of incomplete interval information (RankI) [20], the IVIFPA operator-based approach [16], the interval-valued intuitionistic fuzzy number hybrid aggregation (IVIFHA) operator-based method [23], and 2) the method based on the proposed IF-PHWA operator and the intuitionistic fuzzy hybrid averaging (IFHA) operator-based approach [43].

6.2.1 Experiment on the different IVIFN-based operators

The IVIFNs in Table 1 are used in all of these comparative algorithms to compare the results on a common basis. Then, the ranking process of each comparative method is described as follows.

RankI [20]: Firstly, we utilize the score function to calculate the membership of attributes based on the data in Table 1. Then the fuzzy dominance relation between two targets is computed. Finally, we calculate the entire dominance degree of each target Z ₁ (x _k). The calculated results are shown as follows. Z ₁ (x ₁) =0.9167, Z ₁ (x ₂) =0.5167, Z ₁ (x ₃) =0.3833, Z ₁ (x ₄) =0.4167, Z ₁ (x ₅) =0.5167, Z ₁ (x ₆) =0.2500.

The ranking is x ₁ > x ₅ = x ₂ > x ₄ > x ₃ > x ₆.

IVIFPA [16]: Firstly, we acquire the synthesized value ${\tilde{Q}}_{k}^{m}$ of the alternative x _k (k = 1, 2, . . . , 6) in the mth (m = 1, 2, . . . , 5) priority class and calculate the IVIF weight ${\tilde{W}}_{k}^{m}$ . Furthermore, the IVIFPA operator is applied to obtain the synthetic evaluation of the alternative x _k as follows. Z ₂ (x ₁) = ([1, 1] , [0, 0]), Z ₂ (x ₂) = ([0.8199, 0.8729] , [0.0368, 0.0536]), Z ₂ (x ₃) = ([0.9230, 0.9481] , [0.0167, 0.0247]), Z ₂ (x ₄) = ([0.9920, 0.9952] , [0, 0.0023]), Z ₂ (x ₅) = ([0.9852, 1] , [0, 0]), Z ₂ (x ₆) = ([0.6852, 0.7126] , [0.1539, 0.1910]).

And then we calculate the overall evaluation value E _k of the alternative x _k using

$E = \frac{1}{4} (2 + μ_{α}^{L} + μ_{α}^{H} - υ_{α}^{L} - υ_{α}^{H}) .$ (28) E ₁ = 1, E ₂ = 0.9006, E ₃ = 0.9574, E ₄ = 0.9962, E ₅ = 0.9963, E ₆ = 0.7632.

The ranking is x ₁ > x ₅ ≈ x ₄ > x ₃ > x ₂ > x ₆.

IVIFHA [23]: Firstly, the weight vector w of the function IVIFHA is derived by the normal distribution based method [44], which is w = (0.1117, 0.2365, 0.3036, 0.2365, 0.1117) ^T. Furthermore, the IVIFHA operator is applied to compute the aggregated values of the alternative x _k (k = 1, 2, . . . , 6) as follows. Z ₃ (x ₁) = ([1, 1] , [0, 0]), Z ₃ (x ₂) = ([0.6570, 0.7115] , [0.1866, 0.2275]), Z ₃ (x ₃) = ([0.6213, 0.6590] , [0.2396, 0.2788]), Z ₃ (x ₄) = ([0.7255, 0.7799] , [0, 0.1848]), Z ₃ (x ₅) = ([0.8096, 1] , [0, 0]), Z ₃ (x ₆) = ([0.4746, 0.4974] , [0.3657, 0.4164]).

Finally, the scores of Z ₃ (x _k) (k = 1, 2, . . . , 6) are calculated according to [17]. s (Z ₃ (x ₁)) =1, s (Z ₃ (x ₂)) =0.4772, s (Z ₃ (x ₃)) =0.3810, s (Z ₃ (x ₄)) =0.6603, s (Z ₃ (x ₅)) =0.9048, s (Z ₃ (x ₆)) =0.0950.

The ranking is x ₁ > x ₅ > x ₄ > x ₂ > x ₃ > x ₆.

6.2.2 Performance analysis

The proposed method and the first three comparative methods (from RANKI to IVIFHA) are based on the IVIFNs in Table 1. For a clearer comparison, the rankings of them are listed in Table 4. From these rankings, it can be observed that the ranking of proposed method is similar to the ranking of IVIFPA while significantly different from the ones of RANKI and IVIFHA. The reasons can be described as follows: 1) Comparing with the proposed method, RANKI and IVIFHA do not consider the prioritized relationship of attributes. Therefore, they generate an inconsistent ranking result with the actual expectation. On the contrary, IVIFPA and the proposed method consider the prioritized relationship of attributes. However, according to IVIFPA, three aggregated values are approximately equal, i.e., E ₁ = 1, E ₅ = 0.9963, E ₄ = 0.9962. Because of these slightly different values, there are some confusion in the practical applications. In contrast, the three aggregated values obtained by the proposed method are significantly different. That is because that the proposed method not only takes the prioritized relationship into consideration, but also considers the importance and the ordered position of attributes.

6.2.3 Experiment on IVIFNs vs IFNs

In order to validate the superiority of IVIFN-based method in this work, we utilize IF-PHWA and IFHA to deal with this problem, which are based on the intuitionistic fuzzy numbers. For the IFN generation, we choose three subsets of IVIFNs, which are derived from the values in Table 1, including $a^{L} = (μ_{α}^{L}, υ_{α}^{L})$ , $a^{H} = (μ_{α}^{H}, υ_{α}^{H})$ and $a^{M} = (μ_{α}^{M}, υ_{α}^{M})$ , where $μ^{M} = \frac{(μ_{α}^{L} + μ_{α}^{H})}{2}$ and $υ^{M} = \frac{(υ_{α}^{L} + υ_{α}^{H})}{2}$ .

IF-PHWA: Similar to the steps of IVIF-PHWA operator-based method, the final aggregated values Z _4L (x _k), Z _4H (x _k) and Z _4M (x _k) of alternatives x _k (k = 1, 2, . . . , 6) are calculated using the IF-PHWA operator Equation. (13) as follows.

1) The aggregated values Z _4L (x _k) are Z _4L (x ₁) = (1, 0), Z _4L (x ₂) = (0.6403, 0.2137), Z _4L (x ₃) = (0.6529, 0.2015), Z _4L (x ₄) = (0.8023, 0), Z _4L (x ₅) = (0.8235, 0), Z _4L (x ₆) = (0.4925, 0.3794).

The scores of Z _4L (x _k) are calculated according to Chen and Tan [45]: s (Z _4L (x ₁)) =1, s (Z _4L (x ₂)) =0.4266, s (Z _4L (x ₃)) =0.4514, s (Z _4L (x ₄)) =0.8023, s (Z _4L (x ₅)) =0.8235 and s (Z _4L (x ₆)) =0.1131.

Then according to Xu and Yager [41], the ranking is x ₁ > x ₅ > x ₄ > x ₃ > x ₂ > x ₆.

2) The scores s (Z _4H (x _k)) are obtained in a similar way to s (Z _4L (x _k)). Then the ranking of alternatives is x ₁ = x ₅ > x ₄ > x ₃ > x ₂ > x ₆.

3) Similarly, according to the calculated s (Z _4M (x _k)), the ranking of alternatives is x ₁ > x ₅ > x ₄ > x ₃ > x ₂ > x ₆.

IFHA [43]: Firstly, the weight vector w of the function IFHA is derived by the normal distribution based method [44], which is w = (0.1117, 0.2365, 0.3036, 0.2365, 0.1117) ^T. Furthermore, the IFHA operator is applied to calculate the aggregated values of the alternative x _k (k = 1, 2, . . . , 6) as follows.

1) The aggregated values Z _5L (x _k) are Z _5L (x ₁) = (1, 0), Z _5L (x ₂) = (0.6553, 0.1877), Z _5L (x ₃) = (0.6215, 0.2397), Z _5L (x ₄) = (0.7235, 0), Z _5L (x ₅) = (0.8055, 0), Z _5L (x ₆) = (0.4762, 0.3628).

Similar to the s (Z _4L (x _k)), the scores of Z _5L (x _k) are calculated according to [45]: s (Z _5L (x ₁)) =1, s (Z _5L (x ₂)) =0.4676, s (V _L (x ₃)) =0.3819, s (Z _5L (x ₄)) =0.7235, s (Z _5L (x ₅)) =0.8055 and s (Z _5L (x ₆)) =0.1134.

Then according to [41], the ranking is x ₁ > x ₅ > x ₄ > x ₂ > x ₃ > x ₆.

2) The scores s (Z _5H (x _k)) are obtained in a similar way to s (Z _5L (x _k)). Then the ranking of alternatives is x ₁ = x ₅ > x ₄ > x ₂ > x ₃ > x ₆.

3) Similarly, according to the calculated s (Z _5M (x _k)), the ranking is x ₁ > x ₅ > x ₄ > x ₂ > x ₃ > x ₆.

For clearer demonstration, the ranking results of the proposed method, IF-PHWA and IFHA are listed in Table 5.

6.2.4 Performance analysis

Comparing with IF-PHWA, IFHA has a different ranking between x ₂ and x ₃, which is inconsistent with actual expectation. That is because IFHA does not take the prioritized relationship of attributes into consideration. In addition, the results of IF-PHWA and IFHA show that the tiny difference of data can bring different rankings. However, by comparing the results in Table 5, the proposed IVIF-PHWA operator is more suitable to describe the uncertain and imprecise information.

In summary, the IVIF-PHWA operator-based appr-oach makes decision making more precise and reasonable when addressing MADM problems in more universal interval-valued intuitionistic fuzzy environment.

7 Conclusion

In this paper, a unit prioritized hybrid weighted aggregation (UPHWA) operator is firstly proposed to overcome the shortcoming of the existing aggregation operators. Secondly, the UPHWA operator is extended to an IF-PHWA operator in order to adapt to the intuitionistic fuzzy environment and an IVIF-PHWA operator for the IVIF environment, respectively. The important properties of all the proposed operators are investigated and proved. Finally, a decision making method based on IVIF-PHWA operator is developed to solve the MADM problem in more universal IVIF environment. The effectiveness and practicality of the proposed method is validated by a practical example.

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