Intuitionistic fuzzy power geometric Heronian mean operators and their application to multiple attribute decision making

Abstract

This paper investigates intuitionistic fuzzy information aggration problem with the interrelationship among the input values and the “singular point” (i.e. the input value was either too large or too small) which canot be solved by most existing aggregation operators. To accomplish this, this paper combines the geometric Heronian mean (GHM) operator with the power geometric (PG) operator under intuitionistic fuzzy environment. Then, the intuitionistic fuzzy power GHM (IFPGHM) operator and the weighted intuitionistic fuzzy power GHM (WIFPGHM) operator are presented. The new operators capture not only the correlations between the input arguments but also the relative closeness of decision making information such that they can better solve the intuitionistic fuzzy information aggregation problem with diversified connections between arguments. The desirable properties of these new extensions of GHM operator and their special cases are investigated. Finally, based on the WIFPGHM operator, we present an approach to multiple attribute decision making and illustrate that approach with a practical example.

Keywords

Intuitionistic fuzzy number multiple attribute decision making Geometric heronian mean operator power geometric operator

1 Introduction

Multiple attribute decision making (MADM), as one branch of decision making science, has long been a concern of academic researchers for many years. In fact, MADM has a variety of applications in many domains, which range from inventory management, financial risk management, designing online learning systems and evaluating of renewable power sources, to disaster evaluation of the floods [1 –5]. Due to the complexity of decision making objects and the fuzzy characteristics of human thinking, it is difficult to evaluate the attribute characteristics of alternatives with precise values in most cases. To overcome this problem, fuzzy theory is introduced as a means to better express decision making information.

The fuzzy set (FS), initially introduced by Zadeh [6], has been widely used to express decision making information in fuzzy or uncertain circumstances. Several scholars argue that accurately representing complex decision making information using FS is impossible, because the sets only have memberships. For example, a survey conducted by the board of directors of a company showed that 51 percent of shareholders are in favor of passing a restructuring plan, while 37 percent are opposed. Expressing the results of this survey using FS is difficult. Because of this, Atanassov further put forward the intuitionistic fuzzy set (IFS) based on FS [7]. The IFS has three components: a membership degree function, a nonmembership degree function and a hesitancy degree function.This makes IFS more flexible and efficient than FS, particularly when expressing uncertainty information in decision making. By using IFS, the results of the survey above can be shown by membership 0.51 and nonmembership 0.37. Therefore, many scholars have focused on the study of intuitionistic fuzzy decision making problems [8 –10].

In order to aggregate the intuitionistic fuzzy individual evaluation values into a collective one, a large number of aggregate operators have been introduced during the past few years, such as intuitionistic fuzzy weighted geometric operator [11], intuitionistic fuzzy weighted averaging operator [11], intuitionistic fuzzy hybrid averaging operator [12], and intuitionistic fuzzy ordered weighted distance operator [13]. Su et al. presented the induced generalized intuitionistic fuzzy ordered weighted averaging operator by using induced variables [14]. Liao et al. introduced the generalized intuitionistic fuzzy hybrid weighted geometric operator and applied it to MADM [15]. Zeng et al. developed the intuitionistic fuzzy generalized probabilistic ordered weighted averaging operator [16]. Garg presented some intuitionistic fuzzy Hamacher interaction weighted operators to solve the MADM problems [17].

However, operators as mentioned above assume that input values are independent. This assumption overlooks the interconnectedness between attributes in decision making information infusing. Therefore, the information aggregation methods based on these operators do not have high reliability and veracity in decision making. The Heronian mean (HM) operator has the desirable characteristic of capturing correlations between aggregated arguments. Beliakov et al. theoretically proved that HM operator meets the conditions of an aggregation operator [18]. The HM operator have been widely applied to the fuzzy MADM problem in recent years. Zang et al. designed a method for MADM with interval-valued dual hesitant fuzzy information based on HM operator [20]. Han et al. proposed an unbalanced linguistic generalized Heronian mean aggregation operator to solve MADM problem [19]. Liu et al. proposed the intuitionistic fuzzy Archimedean HM operator based on the Archimedean t-conorm and t-norm [22]. Peng et al. introduced the Frank HM operator to aggregate linguistic intuitionistic decision making information [21].To extend and develop HM operator, Yu defined the geometric Heronian mean (GHM) operator based on the geometric averaging operator [23].

Operating from a different perspective, in order to take the relative closeness of decision making information into consideration, and pay close attention to the optimization of the whole in decision making, Yager originally proposed the power average (PA) operator [24]. Liu et al. combined the PA operator with Schweizer–Sklar operations under intuitionistic fuzzy environment and presented the interval-valued intuitionistic fuzzy Schweizer–Sklar PA operator [25]. Wen et al. proposed intuitionistic fuzzy evidential PA aggregation operator based on the D-S evidence theory [26]. Wen et al. presented intuitionistic fuzzy entropy weighted PA aggregation operator and developed a method to solve multiple attribute group decision making problems [27]. Rani et al. presented some PA operators to solve the complex intuitionistic fuzzy MADM problems [28]. Xu and Yager further put forward the power geometric(PG) operator [29]. In recent years, many researchers have devoted themselves to studying the PG operator in different fuzzy environments [30 –34].

A new problem arises when considering these operators only form a single factor, as this may lead to one-sided decision making conclusions. Therefore, it is necessary to design some new information aggregating operators, which can simultaneously take multiple factors into account in the decision making process at the same time, if required. For example, when faced with a risky decision making problem (such as selecting the location of nuclear power plants or selecting aircraft engine suppliers), people often choose robust decision making methods. They hope to mine the correlation between the economic and social attribute variables, while avoiding the impact of “singular point” on the decision making results. Thus, the aim of this paper is to solve MADM problems in which the attribute variables are interrelated and the evaluation values have the singular points. To do this, we first propose the intuitionistic fuzzy power GHM (IFPGHM) operator and the weighted intuitionistic fuzzy power GHM (WIFPGHM) operator. And then, apply them to deal with MADM prpblems in intuitionistic fuzzy enviroment.

The motivation of this paper is reposed on the following facts:

The existing intuitionistic fuzzy information aggregation operators are mainly proposed from a one-dimensional perspective. There was less research about multi-objective information aggregation with intuitionistic fuzzy information.

The GHM operator can capture the relationship between the input arguments. In addition, the GHM operator has two flexible parameters, and we could select the appropriate parameters to meet the different requirements in information infusing process. Until now, no operator exists that is designed based on the GHM and PG operators.

At present, the power Bonferroni mean (PBM) operator is the main way used to solve the above MADM problems, which was introduced initially by He et al. based on the BM and PA operators [35]. However, Liu et al. recently pointed out that the BM operator has the defect of computing complexity and overusing interactive computation to capture correlation in information infusing process, while the GHM operator can effectively avoid these situations [36].

In this paper, we intend to achieve the following contributions:

We introduce some new operators to reduce the influence of singular value and deeply mine the relationship among aggregated arguments in the information infusing process, i.e., the IFPGHM and WIFPGHM operators.

We develop a multi-objective MADM method on the basis of the proposed operator under the intuitionistic fuzzy environment.

Some comparative analyses between the proposed operators and method and other existing operators and methods are conducted based on the derived results. The advantages of the proposed operators and decision making method are highlighted.

The rest of this paper is organized as follows: Section 2 reviews some fundamental concepts, such as the intuitionistic fuzzy number (IFN), the comparison law of IFNs, the GHM and PG operators. In Sections 3, we design some new operators and investigate their properties. A new method that can be used to solve MADM problems with intuitionistic fuzzy information is presented in Section 4. In Section 5, the validity and feasibility of this proposed technique is explained through a numerical example. In Section 6, a comparative analysis with other MADM methods is provided. Finally, conclusions and directions for future research are presented in Section 7.

2 Preliminaries

2.1 Intuitionistic fuzzy values

Atanassov first proposed the concept of IFS as a flexible and efficient tool for expressing the fuzzy and uncertain information [7].

Definition 1. [7] Let X be a fixed set. An intuitionistic fuzzy set on X can be defined as $A = {< x, μ_{A} (x), υ_{A} (x), π_{A} (x) > | x \in X}$ where μ_A (x) : X → [0, 1] and υ_A (x) : X → [0, 1] satisfy the condition 0 ≤ μ_A (x) + υ_A (x) ≤1, ∀x ∈ X, and μ_A (x) and υ_A (x) represent the membership degree and the non-membership degree of x to A. π_A (x) indicates the degree of hesitance of the element x in the set A, and π_A (x) =1 - μ_A (x) - υ_A (x).

To facilitate information modeling in decision making, Xu called the pair α = (μ_α, υ_α) an IFN with the conditions μ_α ∈ [0, 1], υ_α ∈ [0, 1], μ_α + υ_α ∈ [0, 1], and defined the following operation laws [11].

Definition 2. [11] Let α_i = (μ_{α
_i}, υ_{α
_i}) (i = 1, 2), and α = (μ_α, υ_α) be three IFNs, then

α₁⊕ α₂ = (μ_{α
₁} + μ_{α
₁} - μ_{α
₁}μ_{α
₂}, υ_{α
₁}υ_{α
₂}) ;

α₁⊗ α₂ = (μ_{α
₁}μ_{α
₂}, υ_{α
₁} + υ_{α
₂} - υ_{α
₁}υ_{α
₂}) ;

$λ α = (1 - (1 - μ_{α})^{λ}, υ_{α}^{λ}, λ > 0;$

$α^{λ} = (υ_{α}^{λ}, 1 - (1 - υ_{α})^{λ}), λ > 0$ .

To compare the IFNs, Xu [11] further gave the following method:

Definition 3. [11] Let α_i = (μ_{α
_i}, υ_{α
_i}) (i = 1, 2), and α = (μ_α, υ_α) be three IFNs. S_α = μ_α - υ_α is called the score function of α, and h_α = μ_α + υ_α is called the accuracy degree function of α. Based on S_α and h_α, the ordering method for two IFNs is defined as

If S_{α
₁} > S_{α
₂}, then α₁ > α₂;

If S_{α
₁} < S_{α
₂}, then α₁ < α₂;

If S_{α
₁} = S_{α
₂}, then

if h_α
₁ = h_α
₂, then α₁ = α₂ ;

if h_α
₁ > h_α
₂, then α₁> α₂ ;

if h_α
₁ < h_α
₂, then α₁ < α₂.

There is also other comparison rules of IFNs in the literature. For example, Szmidt et al. proposed a comparison rule between IFNs based on the reliability of information [37]. Zhang et al. introduced a novel approach for ranking IFNs by using the similarity measure and the accuracy degree of IFNs [38]. Jafarian et al. presented a valuation-based method for ranking IFNs by mapping the IFNs to the crisp values [39]. Lakshmana et al. proposed a total ordering rule on IFNs by using the upper lower dense sequence [40]. However, the comparison rule proposed by Xu et al., as shown in Definition 3, is simple and convenient. So it is widely used in decision making.

2.2 Geometric Heronian mean and power geometric operator

The HM operator is a common tool used to capture the interrelationships between decision variables. Yu proposed the GHM operator by generalizing the HM and geometric average operators [23].

Definition 4. [23] Let p, q, a_i (i = 1, 2, …, n) are non-negative real numbers. If ${GHM}^{p, q} (a_{1}, a_{2}, \dots, a_{n}) = \frac{1}{p + q} \prod_{i = 1, j = i}^{n} ({pa}_{i} + {qa}_{j})^{\frac{2}{n (n + 1)}}$ then GHM^p,q is called geometric Heronian mean (GHM) operator.

To reduce the effect of the outliers, Yager originally proposed a PA operator [24]. Xu and Yager further put forward the PG operator [29].

Definition 5. [29] Let a_i (i = 1, 2, …, n) are real numbers. The power geometric (PG) operator is defined as $PG (a_{1}, a_{2}, \dots, a_{n}) = \prod_{i = 1}^{n} a_{i}^{\frac{1 + T (a_{i})}{\sum_{i = 1}^{n} (1 + T (a_{i}))}}$ where $T (a_{i}) = \sum_{j = 1, j \neq i}^{n} Sup (a_{i}, a_{j})$ , Sup (a_i, a_j) is the support for Sup (a_i, a_j)≥ and Sup (a_i, a_j)≥, with the conditions:

Sup (a_i, a_j) ∈ [0, 1];

Sup (a_i, a_j) = Sup (a_i, a_j);

If d (a_i, a_j) ≤ d (a_l, a_k), then Sup (a_i, a_j) ≥ Sup (a_l, a_k), where d (a_i, a_j) is the distance between a_i and a_j.

Remark 1. The support (i.e., Sup) measure is essentially a similarity index. Therefore, if d ∈ [0, 1], we can let Sup (a_i, a_j) =1 - d (a_i, a_j).

3 Intuitionistic fuzzy power GHM

In this section, we combine the HM and PG operators, and propose some new operators who capture not only the correlations between the aggregated arguments but also the relative closeness of the input values.

Definition 6. Let p ≥ 0, q ≥ 0 and p + q ≠ 0, and α_i = (μ_i, υ_i) (i = 1, 2, …, n) are IFNs. If

${IFPGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) = \frac{1}{p + q} \otimes_{i = 1, j = i}^{n} {(p α_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}} \oplus q α_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{\frac{2}{n (n + 1)}}$ (1)

then IFPGHM^p,q is called the intuitionistic fuzzy power GHM (IFPGHM) operator, where $T (α_{i}) = \sum_{j = 1, j \neq i}^{n} Sup (α_{i}, α_{j})$

Sup (α_i, α_j) is the support for α_i and α_j, with the following conditions:

Sup (α_i, α_j)∈ [0, 1] ;

Sup (α_i, α_j) = Sup (α_j, α_i) ;

If d (α_i, α_j) ≤ d (α_l, α_k), then Sup (α_i, α_j) ≥ Sup (α_l, α_k).

Theorem 1. Let p ≥ 0, q ≥ 0 and p + q ≠ 0, and α_i = (v_i, υ_i) (i = 1, 2, …, n) are IFNs, then the aggregated value by the IFPGHM operator is also an intuitionistic fuzzy number and

$\begin{matrix} {IFPGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) = (1 - {(1 - \prod_{i = 1, j = i}^{n} {(1 - {(1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{p} {(1 - u_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{q})}^{\frac{2}{n (n + 1)}})}^{\frac{1}{p + q}}, \end{matrix} {(1 - \prod_{i = 1, j = i}^{n} {(1 - {(1 - (1 - υ_{i})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{p} {(1 - (1 - υ_{j})^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{q})}^{\frac{2}{n (n + 1)}})}^{\frac{1}{p + q}}) .$ (2)Proof: Since $α_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}} = (u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}}, 1 - (1 - υ_{i})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}}), α_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}} = (u_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}}, 1 - (1 - υ_{j})^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})$ then by the operational laws of IFNs, we have $p α_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}} = (1 - (1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p}, (1 - (1 - υ_{i})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p})$ $q α_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}} = (1 - (1 - u_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q}, (1 - (1 - υ_{j})^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q})$ The above imply that $\begin{matrix} p α_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}} \oplus q α_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}} = & (1 - (1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p} + 1 - (1 - u_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q} - (1 - (1 - u_{i})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p}) (1 - (1 - u_{j})^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q}, & (1 - (1 - υ_{i})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p} (1 - (1 - υ_{j})^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q}) \end{matrix}$ $\begin{matrix} = (1 - (1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p} (1 - u_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q}, (1 - (1 - υ_{i})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p} (1 - (1 - υ_{j})^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q}) \end{matrix}$ and ${(p α_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}} \oplus q α_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{\frac{2}{n (n + 1)}} = ({(1 - (1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p} (1 - u_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q})}^{\frac{2}{n (n + 1)}},$ $(1 - (1 - υ_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p} (1 - (1 - υ_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q})^{\frac{2}{n (n + 1)}}) .$ Hence ${IFPGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) = \frac{1}{p + q} \otimes_{i = 1, j = i}^{n} {(p α_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}} \oplus q α_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{\frac{2}{n (n + 1)}}$ $= \frac{1}{p + q} \otimes_{i = 1, j = i}^{n} ({(1 - (1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p} (1 - u_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q})}^{\frac{2}{n (n + 1)}}, {(1 - (1 - υ_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p} (1 - (1 - υ_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q})}^{\frac{2}{n (n + 1)}})$ $= \frac{1}{p + q} (\prod_{i = 1, j = i}^{n} {(1 - (1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p} (1 - u_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q})}^{\frac{2}{n (n + 1)}},$ $1 - \prod_{i = 1, j = i}^{n} {(1 - (1 - (1 - υ_{i})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p} (1 - (1 - υ_{j})^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q})}^{\frac{2}{n (n + 1)}})$ $= (1 - {(1 - \prod_{i = 1, j = 1}^{n} {(1 - {(1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{p} {(1 - u_{j}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{q})}^{\frac{2}{n (n + 1)}})}^{\frac{1}{p + q}},$ ${(1 - \prod_{i = 1, j = 1}^{n} {(1 - {(1 - (1 - υ_{i})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{p} {(1 - (1 - υ_{j})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{q})}^{\frac{2}{n (n + 1)}})}^{\frac{1}{p + q}}) .$ By u_i ∈ [0, 1], υ_i ∈ [0, 1], u_i + υ_i ∈ [0, 1], i = 1, 2, ⋯ , n, we get $0 \leq 1 - {(1 - \prod_{i = 1, j = 1}^{n} {(1 - {(1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{p} {(1 - u_{j}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{q})}^{\frac{2}{n (n + 1)}})}^{\frac{1}{p + q}} \leq 1$ and $0 \leq {(1 - \prod_{i = 1, j = 1}^{n} {(1 - {(1 - (1 - υ_{i})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{p} {(1 - (1 - υ_{j})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{q})}^{\frac{2}{n (n + 1)}})}^{\frac{1}{p + q}} \leq 1$ Furthermore, $1 - υ_{i} \geq u_{i} \Rightarrow (1 - υ_{i})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}} \geq u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}}$ $\Rightarrow 1 - (1 - υ_{i})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}} \leq 1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}}$ $\Rightarrow (1 - (1 - υ_{i})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p} \leq (1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p}$ $\Rightarrow (1 - (1 - υ_{i})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p} (1 - (1 - υ_{j})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q} \leq (1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p} (1 - u_{j}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q}$ $\Rightarrow 1 - (1 - (1 - υ_{i})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p} (1 - (1 - υ_{j})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q} \geq 1 - (1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p} (1 - u_{j}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q}$ $\begin{matrix} \Rightarrow \prod_{i = 1, j = i}^{n} (1 - (1 - (1 - υ_{i})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p} (1 - (1 - υ_{j})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q}) \geq & \prod_{i = 1, j = i}^{n} (1 - (1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{p} (1 - u_{j}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})^{q}) \end{matrix}$ $\begin{matrix} {(1 - \prod_{i = 1, j = i}^{n} {(1 - {(1 - (1 - υ_{i})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{p} {(1 - (1 - υ_{j})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{q})}^{\frac{2}{n * (n + 1)}})}^{\frac{1}{p + q}} & \leq {(1 - \prod_{i = 1, j = i}^{n} {(1 - {(1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{p} {(1 - u_{j}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{q})}^{\frac{2}{n (n + 1)}})}^{\frac{1}{p + q}} \end{matrix}$ $\begin{matrix} 1 - {(1 - \prod_{i = 1, j = i}^{n} {(1 - {(1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{p} {(1 - u_{j}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{q})}^{\frac{2}{n * (n + 1)}})}^{\frac{1}{p + q}} & + {(1 - \prod_{i = 1, j = i}^{n} {(1 - {(1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{p} {(1 - u_{j}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{q})}^{\frac{2}{n (n + 1)}})}^{\frac{1}{p + q}} \leq 1 \end{matrix}$ The theorem is proved.

Remark 2. If $\frac{1 + T (α_{i})}{\sum_{t = 1}^{n} (1 + T (α_{t}))} = \frac{1}{n}$ (i = 1, 2, ⋯ , n), then $\begin{matrix} {IFPGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) & = \frac{1}{p + q} \otimes_{i = 1, j = i}^{n} {(p α_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}} \oplus q α_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{\frac{2}{n (n + 1)}} & = \frac{1}{p + q} \otimes_{i = 1, j = i}^{n} {(p α_{i} \oplus q α_{j})}^{\frac{2}{n (n + 1)}} & = {IFGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) \end{matrix}$ which we call the intuitionistic fuzzy GHM (IFGHM) operator [23].

Remark 3. If q → 0, then $\begin{matrix} {IFPGHM}^{p, q} (α_{1}, \dots, α_{n}) & = (1 - {(1 - \prod_{i = 1}^{n} {(1 - {(1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{p})}^{\frac{2}{n (n + 1)}})}^{\frac{1}{p}}, \end{matrix}$ $\begin{matrix} {(1 - \prod_{i = 1}^{n} {(1 - {(1 - (1 - υ_{i})^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{p})}^{\frac{2}{n (n + 1)}})}^{\frac{1}{p}}) . \end{matrix}$ which we call the generalized intuitionistic fuzzy PG operator.

Property 1. Let α_i (i = 1, 2, ⋯ , n) be a collection of IFNs, if all α_i are equal to α, then ${IFGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) = α .$ (3)

Proof: $\begin{matrix} {IFPGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) & = \frac{1}{p + q} \otimes_{i = 1, j = i}^{n} {(p α_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}} \oplus q α_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{\frac{2}{n (n + 1)}} & = \frac{1}{p + q} \otimes_{i = 1, j = i}^{n} {(p α^{\frac{n (1 + T (α))}{\sum_{t = 1}^{n} (1 + T (α))}} \oplus q α^{\frac{n (1 + T (α))}{\sum_{t = 1}^{n} (1 + T (α))}})}^{\frac{2}{n (n + 1)}} \end{matrix}$ $\begin{matrix} = \frac{1}{p + q} \otimes_{i = 1, j = i}^{n} {(p α \oplus q α)}^{\frac{2}{n (n + 1)}} & = \frac{1}{p + q} \otimes_{i = 1, j = i}^{n} (p + q) α^{\frac{2}{n (n + 1)}} = α . □ \end{matrix}$

Property 2. Let α_i (i = 1, 2, ⋯ , n) be a collection of IFNs, if $α_{1}^{'}, α_{2}^{'}, \dots, α_{n}^{'}$ is any permutation of α₁, α₂, ⋯ , α_n, then

${IFPGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) = {IFPGHM}^{p, q} (α_{1}^{'}, α_{2}^{'}, \dots, α_{n}^{'})$ (4)

Proof: $\begin{array}{l} {IFPGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) \\ = \frac{1}{p + q} \otimes_{i = 1, j = i}^{n} {(p α_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}} \oplus q α_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{\frac{2}{n (n + 1)}} \\ = \frac{1}{p + q} \otimes_{i = 1, j = i}^{n} {(p α_{i}^{'}^{\frac{n (1 + T (α_{i}^{'}))}{\sum_{t = 1}^{n} (1 + T (α_{t}^{'}))}} \oplus q α_{j}^{'}^{\frac{n (1 + T (α_{j}^{'}))}{\sum_{t = 1}^{n} (1 + T (α_{t}^{'}))}})}^{\frac{2}{n (n + 1)}} \\ = {IFPGHM}^{p, q} (α_{1}^{'}, α_{2}^{'}, \dots, α_{n}^{'}) \end{array}$

Property 3. Let α_i (i = 1, 2, ⋯ , n) be a collection of IFNs, $β_{i} = α_{i}^{\frac{n (1 + T (α_{t}))}{\sum_{i = t}^{n} (1 + T (α_{t}))}}$ , β⁺ = max {β_i}, β^- = min {β_i}, then $β^{-} \leq {IFPGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) \leq β^{+}$ (5) .

Proof: $\begin{matrix} {IFPGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) \\ = \frac{1}{p + q} \otimes_{i = 1, j = i}^{n} {(p α_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}} \oplus q α_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{\frac{2}{n (n + 1)}} \\ \leq \frac{1}{p + q} \otimes_{i = 1, j = i}^{n} {(p β^{+} \oplus q β^{+})}^{\frac{2}{n (n + 1)}} \\ = \frac{1}{p + q} \otimes_{i = 1, j = i}^{n} {((p + q) β^{+})}^{\frac{2}{n (n + 1)}} = β^{+} . \end{matrix}$

Similarly, β^- ≤ IFPGHM^p,q (α₁, α₂, ⋯ , α_n).

Therefore, we obtain $β^{-} \leq {IFPGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) \leq β^{+} .$

Example 1. Let α₁ = (0.4, 0.3), α₂ = (0.5, 0.2), α₃ = (0.2, 0.5), α₄ = (0.1, 0.6). Without loss of generality we assume $T (α_{i}) = \sum_{j = 1, j \neq i}^{4} Sup (α_{i}, α_{j}) = \sum_{j = 1, j \neq i}^{4} (1 - d (α_{i}, α_{j})),$

(i = 1, 2, ⋯ , n), where $d (α_{i}, α_{j}) = \sqrt{\frac{1}{2} ((u_{i} - u_{j})^{2} + (υ_{i} - υ_{j})^{2} + (π_{i} - π_{j})^{2})} .$

First, we utilize the IFPGHM operator (p = 1, q = 1) to aggregate these IFNs, as described in the following: $\begin{matrix} {(d (α_{i}, α_{j}))}_{4 \times 4} = (\begin{matrix} 0 & 0.1 & 0.2 & 0.3 \\ 0.1 & 0 & 0.3 & 0.4 \\ 0.2 & 0.3 & 0 & 0.1 \\ 0.3 & 0.4 & 0.1 & 0 \end{matrix}), \\ {(Supp (α_{i}, α_{j}))}_{4 \times 4} = (\begin{matrix} 1 & 0.9 & 0.8 & 0.7 \\ 0.9 & 1 & 0.7 & 0.6 \\ 0.8 & 0.7 & 1 & 0.9 \\ 0.7 & 0.6 & 0.9 & 1 \end{matrix}) . \end{matrix}$

This shows that ${(T (α_{i}))}_{4 \times 1} = (0.727, 0.667, 0.727, 0.667) .$

Then $\begin{matrix} {IFPGHM}^{1, 1} (α_{1}, α_{2}, α_{3}, α_{4}) \\ = \frac{1}{2} \otimes_{i = 1, j = i}^{4} {(α_{i}^{\frac{4 (1 + T (α_{i}))}{\sum_{t = 1}^{4} (1 + T (α_{t}))}} \oplus q α_{j}^{\frac{4 (1 + T (α_{j}))}{\sum_{t = 1}^{4} (1 + T (α_{t}))}})}^{\frac{1}{10}} = (0.389, 0.327) . \end{matrix}$

Utilizing the intuitionistic fuzzy geometric (IFG) operator [42], then we can get $IFG (α_{1}, α_{2}, α_{3}, α_{4}) = (\prod_{i = 1}^{4} u_{i}^{\frac{1}{4}}, 1 - \prod_{i = 1}^{4} υ_{i}^{\frac{1}{4}}) = (0.251, 0.421) .$

Utilizing the intuitionistic fuzzy PG (IFPG) operator [41], then we can get $\begin{matrix} IFG (α_{1}, α_{2}, α_{3}, α_{4}) = \prod_{i = 1}^{4} α_{i}^{\frac{1 + T (α_{i})}{\sum_{i = 1}^{4} (1 + T (α_{i}))}} \\ = (\prod_{i = 1}^{4} u_{i}^{\frac{1 + T (α_{i})}{\sum_{i = 1}^{4} (1 + T (α_{i}))}}, 1 - \prod_{i = 1}^{n} υ_{i}^{\frac{1 + T (α_{i})}{\sum_{i = 1}^{4} (1 + T (α_{i}))}}) = (0.383, 0.317) . \end{matrix}$

Utilizing the IFGHM operator [23], then we can get $\begin{matrix} {IFGHM}^{1, 1} (α_{1}, α_{2}, α_{3}, α_{4}) \\ = (1 - {(1 - \prod_{i = 1, j = i}^{4} {(1 - (1 - u_{i}) (1 - u_{j}))}^{\frac{1}{10}})}^{\frac{1}{2}}, \end{matrix}$ $\begin{matrix} {(1 - \prod_{i = 1, j = i}^{4} {(1 - (1 - (1 - υ_{i})) (1 - (1 - υ_{j})))}^{\frac{1}{10}})}^{\frac{1}{2}}) \\ = (0.267, 0.425) . \end{matrix}$

Table 1

Comparison with other geometric operators

Operator	Are the correlations between the aggregated arguments captured?	Is the relative closeness of the fused values considered?	Does the operator contain parameters?	Aggregation value of example 1	Distance to the IFG	Distance to the IFPG
IFG	No	No	No	(0.251, 0.421)	0	0.120
IFPG	No	Yes	No	(0.383, 0.317)	0.120	0
IFGHM	Yes	No	Yes	(0.267, 0.425)	0.018	0.112
IFPGBM	Yes	Yes	Yes	(0.416, 0.298)	0.149	0.029
IFPGHM	Yes	Yes	Yes	(0.389, 0.327)	0.122	0.014

Fig.1

Scores of the IFPGHM.

Utilizing the intuitionistic fuzzy power geometric BM (IFPGBM) operator [35], then we can get $\begin{matrix} {IFPGBM}^{1, 1} (α_{1}, α_{2}, α_{3}, α_{4}) \\ = \frac{1}{2} \otimes_{i, j = 1, j = i}^{4} {(α_{i}^{\frac{4 (1 + T (α_{i}))}{\sum_{t = 1}^{4} (1 + T (α_{t}))}} \oplus q α_{j}^{\frac{4 (1 + T (α_{j}))}{\sum_{t = 1}^{4} (1 + T (α_{t}))}})}^{\frac{1}{12}} \\ = (0.416, 0.298) . \end{matrix}$

As can be seen in Table 1, only the IFPGHM and IFPGBM operators satisfy all of the conditions. The IFGHM, IFPGBM and IFPGHM operators have two alterable parameters. When taking some unique parameter values, they can degenerate into other aggregation operators, which makes the information infusing process more flexible and better capable of meeting the needs of different users. Furthermore, the IFPGHM operator combines the advantages of the IFPG and IFGHM operators, so it can be regarded as an expansion of them. Compared with the IFPGHM operator, the IFPGBM operator has the defect of providing potentially redundant information, and cannot capture the relationship between the attribute value and itself [36]. In addition, from Table 1,we can see that the aggregation result of the IFPGHM operator is closer to that of the IFG and IFPG operators than the IFPGBM operator. This shows that the IFPGHM operator better retains the mathematical structure of the IFG and IFPG operators.Therefore, the IFPGHM operator is more suitable for MADM problems with fused information.

Figure 1 clearly demonstrates that the IFPGHM operator is not necessarily monotonic. If we let p = q = m, then Fig. 2 reveals that the scores obtained by the IFPGHM operator decrease as the parameters increase. This finding indicates that parameter m can reflect the risk preference of the decision maker. A decision makers in a negative mood could use the IFPGHM operator and choose a larger value for the parameter m, while a decision makers with a positive attitude could use the IFPGHM operator and choose a smaller value for the parameter m.

The hypothetical condition that all the arguments have the equal importance in Definition 6 does not hold in many situations. In many decision making cases, we should consider the attributes’ degrees of importance. If we assume that each attribute has a weight representing its importance, then we propose the intuitionistic fuzzy weighted power GHM operator, as described below.

Definition 7. Let p ≥ 0, q ≥ 0 and p + q ≠ 0, and α_i(i = 1, 2, ⋯ , n) are IFNs with the weight w_i (1, 2, ⋯ , n),w_i ∈ [0, 1], $\sum_{i = 1}^{n} w_{i} = 1$ . If

${WIFPGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) = \frac{1}{p + q} \otimes_{i = 1, j = i}^{n} {(p α_{i}^{\frac{{nw}_{i} (1 + T (α_{i}))}{\sum_{t = 1}^{n} w_{t} (1 + T (α_{t}))}} \oplus q α_{j}^{\frac{{nw}_{j} (1 + T (α_{j}))}{\sum_{t = 1}^{n} w_{t} (1 + T (α_{t}))}})}^{\frac{2}{n (n + 1)}}$ (6) then WIFPGHM^p,q is called the weighted intuitionistic fuzzy power GHM(WIFPGHM) operator, where $T (α_{i}) = \sum_{j = 1, j \neq i}^{n} w_{j} Sup (α_{i}, α_{j})$

Fig.2

Scores of the IFPGHM with p = q.

The detailed computational formula of the WIFPGHM operator can be expressed as follows.

Theorem 2. Let p ≥ 0, q ≥ 0 and p + q ≠ 0, and α_i = (u_i, υ_i) (i = 1, 2, ⋯ , n) are IFNs with the weighting vector $w = (w_{1}, w_{2}, \dots, w_{l})^{T}, w_{i} \in [0, 1], \sum_{i = 1}^{n} w_{i} = 1,$ then the aggregated value by the WIFPGHM operator is also an intuitionistic fuzzy number and

${WIFPGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) = (1 - {(1 - \prod_{i = 1}^{n} {(1 - {(1 - u_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{p} {(1 - u_{j}^{\frac{{nw}_{j} (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{q})}^{\frac{2}{n (n + 1)}})}^{\frac{1}{p + q}}, ({(1 - \prod_{i = 1}^{n} {(1 - {(1 - v_{i}^{\frac{{nw}_{t} (1 + T (α_{i}))}{\sum_{t = 1}^{n} w_{t} (1 + T (α_{t}))}})}^{p} {(1 - v_{j}^{\frac{{nw}_{j} (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{q})}^{\frac{2}{n (n + 1)}})}^{\frac{1}{p + q}}) .$ (7)

Remark 1. If $\frac{1 + T (α_{i})}{\sum_{t = 1}^{n} (1 + T (α_{t}))} = \frac{1}{n} (i = 1, \dots, n)$ , then $\begin{matrix} {WIFPGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) & = \frac{1}{p + q} \otimes_{i = 1, j = i}^{n} {(p α_{i}^{\frac{{nw}_{i} (1 + T (α_{i}))}{\sum_{t = 1}^{n} w_{t} (1 + T (α_{t}))}} \oplus q α_{j}^{\frac{{nw}_{j} (1 + T (α_{j}))}{\sum_{t = 1}^{n} w_{t} (1 + T (α_{t}))}})}^{\frac{2}{n (n + 1)}} \end{matrix}$ $\begin{matrix} = \frac{1}{p + q} \otimes_{i = 1, j = i}^{n} (p α_{i}^{w_{i}} \oplus q α_{j}^{w_{j}})^{\frac{2}{n (n + 1)}} & = {IFWGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) \end{matrix}$ which we call the intuitionistic fuzzy weighted GHM operator [23].

Remark 2. If $w_{t} = \frac{1}{n}, (i = 1, 2, \dots, n)$ , then $\begin{matrix} {WIFPGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) & = \frac{1}{p + q} \otimes_{i = 1, j = i}^{n} {(p α_{i}^{\frac{n (1 + T (α_{i}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}} \oplus q α_{j}^{\frac{n (1 + T (α_{j}))}{\sum_{t = 1}^{n} (1 + T (α_{t}))}})}^{\frac{2}{n (n + 1)}} \\ = {IFWGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) \end{matrix}$

That is to say the WIFPGHM operator reduces to the IFPGHM operator.

Theorem 2 and Properties 4-6 can be proved along the same lines as the proof of Theorem 1 and Properties 1-3, therefore, we omit the details.

Property 4. Let α_i (i = 1, 2, ⋯ , n) be a collection of IFNs, if all α_i are equal to α, then ${WIFPGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) = α .$ (8)

Property 5. Let α_i (i = 1, 2, ⋯ , n) be a collection of IFNs, if $α_{1}^{'}, α_{2}^{'}, \dots, α_{n}^{'}$ is any permutation of α₁, α₂, ⋯ , α_n, then

$\begin{matrix} {WIFPGHM}^{p, q} (α_{1}, α_{2}, \dots, α_{n}) \\ = {WIFPGHM}^{p, q} (α_{1}^{'}, α_{2}^{'}, \dots, α_{n}^{'}) \end{matrix}$ (9)

Property 6. Let α_i (i = 1, 2, ⋯ , n) be a collection of IFNs with the weighting vector w = (w₁, w₂, ⋯ , w_l) ^T $w_{i} \in [0, 1], \sum_{i = 1}^{n} w_{i} = 1,$ $β_{i} = α^{\frac{{nw}_{i} (1 + T (α i))}{\sum_{t = 1}^{n} wt (1 + T (α_{t}))}}$ , $β^{+} = max_{i} {β_{i}}$ , $β^{-} = min_{i} {β_{i}}$ , then $β^{-} \leq {WIFPGHM}^{p, q} (α_{1}, α_{2}, . . ., α_{n}) \leq β^{+} .$ (10)

4 An approach to MADM with the WIFPGHM

In this section, we consider the MADM problems in which the evaluation of attributes takes the form of intuitionistic fuzzy information.

Let X = {x₁, x₂, ⋯ , x_n} be a set of n alternatives, and C = {c₁, c₂, ⋯ , c_m} be a set of attributes with the weight vector w = (w₁, w₂, ⋯ , w_l) ^T, satisfying, $w_{i} \in [0, 1], \sum_{i = 1}^{n} w_{i} = 1$ . Suppose that α_ij = (u_{α
_ij}, υ_{α
_ij}) is the intuitionistic fuzzy evaluated value of the attribute g_j ∈ G on the alternative x_i ∈ X, which constructs the intuitionistic fuzzy decision matrix H = (α_ij) _n×m.

To rank the alternatives, we apply WIFPGHM operator to develop an approach to MADM under intuitionistic fuzzy environment, which involves the following steps:

Step 1. Transform intuitionistic fuzzy decision matrix H = (α_ij) _n×m into the normalized decision matrix $\bar{H} = ({\bar{α}}_{ij})_{n \times m}$ , by using the following conversion formula:

${\bar{α}}_{ij} = {\begin{matrix} @ c @ α_{ij}, for benefit criteria g_{j} \\ N (α_{ij}), for cost criteria g_{j} \end{matrix}$ where N (α_ij) is the complement of α_ij defined as N (α_ij) = (υ_αij, u_αij), i = 1, 2, ⋯ , n, j = 1, 2, ⋯ , m.

Step 2. Calculate $Sup ({\bar{α}}_{li}, {\bar{α}}_{lj}) = 1 - d ({\bar{α}}_{li}, {\bar{α}}_{lj})$ , i, j = 1, 2, ⋯ , m ; l = 1, 2, ⋯ , n, where

$d ({\bar{α}}_{li}, {\bar{α}}_{lj}) = \sqrt{\frac{1}{2} (({\bar{u}}_{li} - {\bar{u}}_{lj})^{2} + ({\bar{υ}}_{li} - {\bar{υ}}_{lj})^{2} + ({\bar{π}}_{li} - {\bar{π}}_{lj})^{2})}$ is the Euclidean distance defined by Eulalia S. and Janusz K. [41]. Then, comprise the support degree matrix S (l) = (Sup (α_li, α_lj)) _m×m, l = 1, 2, ⋯ , n.

Step 3. Calculate the weighted support T (α_li) of intuitionistic fuzzy value α_li using other intuitionistic fuzzy values α_lj. Then, utilize T (α_li) to calculate the weighted support exponent η_li. $T (α_{li}) =, \sum_{j = 1, j \neq i}^{m} w_{j} Sup ({\bar{α}}_{li}, {\bar{α}}_{lj}), η_{li} = \frac{{mw}_{i} (1 + T ({\bar{α}}_{li}))}{\sum_{t = 1}^{m} w_{t} (1 + T ({\bar{α}}_{lt}))}$

Step 4. Calculate the comprehensive evaluation value α_i of each alternative x_i(i = 1, 2, ⋯ , n) by the WIFPGHM:

$\begin{matrix} α_{i} = WIFPGHM (\bar{α_{i 1}}, \bar{α_{i 2}, . . ., \bar{α_{im}}}) \\ = (1 - {(1 - \prod_{k = 1, j = k}^{m} {(1 - {(1 - u_{{\bar{α}}_{ik}}^{\frac{{mw}_{k} (1 + T ({\bar{α}}_{ik}))}{\sum_{t = 1}^{m} w_{t} (1 + T ({\bar{α}}_{it}))}})}^{p} {(1 - u_{{\bar{α}}_{ij}}^{\frac{{mw}_{k} (1 + T ({\bar{α}}_{ij}))}{\sum_{t = 1}^{m} w_{t} (1 + T ({\bar{α}}_{it}))}})}^{q})}^{\frac{2}{m (m + 1)}})}^{\frac{1}{p + q}}, \end{matrix} {(1 - \prod_{k = 1, j = k}^{m} {(1 - {(1 - (1 - υ_{{\bar{α}}_{ik}})^{\frac{{mw}_{k} (1 + T ({\bar{α}}_{ik}))}{\sum_{t = 1}^{m} w_{t} (1 + T ({\bar{α}}_{it}))}})}^{p} {(1 - (1 - υ_{{\bar{α}}_{ij}})^{\frac{{mw}_{k} (1 + T ({\bar{α}}_{ij}))}{\sum_{t = 1}^{m} w_{t} (1 + T ({\bar{α}}_{it}))}})}^{q})}^{\frac{2}{m (m + 1)}})}^{\frac{1}{p + q}})$ (11)

Step 5. Calculate the score function value and accuracy degree function value of each α_i according to Definition 3, respectively.

Step 6. Rank all the alternatives x_i (i = 1, 2, ⋯ , n) and select the best one(s) according to Step 5.

5 A numerical example

In this section, we will use a practical example to demonstrate the application of the proposed method.

Example 2. (adapted from [43]): A company wants to increase profits and reduce operational risks through the purchase and merging of a high-quality business. There are five possible takeover targets available in the market.

x₁ is a new energy car company;

x₂ is a restaurant management company;

x₃ is a mobile phone software company;

x₄ is an office supply company;

x₅ is a household appliances company.

The purchasing company must consider the following four attributes (whose weighting vector is w = (0.2, 0.1, 0.3, 0.4) ^T):

c₁ is the risk factor;

c₂ is the growth factor;

c₃ is the influence power to society;

c₄ is the environmental impact factor.

All α_ij are contained in the intuitionistic fuzzy decision matrix H = (α_ij) _5×4 (see Table 2).

Table 2
Intuitionistic fuzzy decision matrix

c ₁ c ₂ x _i c ₄

x ₁ (0.4, 0.5) (0.5, 0.4) (0.2, 0.7) (0.2, 0.5)

x ₂ (0.6, 0.4) (0.6, 0.3) (0.6, 0.3) (0.3, 0.6)

x ₃ (0.5, 0.5) (0.4, 0.5) (0.4, 0.4) (0.5, 0.4)

x ₄ (0.7, 0.2) (0.5, 0.4) (0.2, 0.5) (0.3, 0.7)

x ₅ (0.5, 0.3) (0.3, 0.4) (0.6, 0.2) (0.4, 0.4)

	c ₁	c ₂	x _i	c ₄
x ₁	(0.4, 0.5)	(0.5, 0.4)	(0.2, 0.7)	(0.2, 0.5)
x ₂	(0.6, 0.4)	(0.6, 0.3)	(0.6, 0.3)	(0.3, 0.6)
x ₃	(0.5, 0.5)	(0.4, 0.5)	(0.4, 0.4)	(0.5, 0.4)
x ₄	(0.7, 0.2)	(0.5, 0.4)	(0.2, 0.5)	(0.3, 0.7)
x ₅	(0.5, 0.3)	(0.3, 0.4)	(0.6, 0.2)	(0.4, 0.4)

Step 1. Since c₁ and c₄ are cost type attributes that are different from the other attributes, H = (α_ij) _5×4 is transformed into the normalized intuitionistic fuzzy decision matrix $\bar{H} = ({\bar{α}}_{ij})_{5 \times 4}$ (see Table 3).

Table 3

Normalized intuitionistic fuzzy decision matrix

	c ₁	c ₂	c ₃	c ₄
x ₁	(0.5, 0.4)	(0.5, 0.4)	(0.2, 0.7)	(0.5, 0.2)
x ₂	(0.4, 0.6)	(0.6, 0.3)	(0.6, 0.3)	(0.6, 0.3)
x ₃	(0.5, 0.5)	(0.4, 0.5)	(0.4, 0.4)	(0.4, 0.5)
x ₄	(0.2, 0.7)	(0.5, 0.4)	(0.2, 0.5)	(0.7, 0.3)
x ₅	(0.3, 0.5)	(0.3, 0.4)	(0.6, 0.2)	(0.4, 0.4)

Step 2. Calculate $Sup ({\bar{α}}_{li}, {\bar{α}}_{lj})$ , and then comprise the support degree matrix S (l) = (Sup (α_li, α_lj)) _4×4, l = 1, 2, ⋯ , 5. $\begin{matrix} S (1) = (\begin{matrix} 1 & 1 & 0.7 & 0.8 \\ 1 & 1 & 0.7 & 0.8 \\ 0.7 & 0.7 & 1 & 0.56 \\ 0.8 & 0.8 & 0.56 & 1 \end{matrix}), \\ S (2) = (\begin{matrix} 1 & 0.74 & 0.74 & 0.74 \\ 0.74 & 1 & 1 & 1 \\ 0.74 & 1 & 1 & 1 \\ 0.74 & 1 & 1 & 1 \end{matrix}), \\ S (3) = (\begin{matrix} 1 & 0.9 & 0.83 & 0.9 \\ 0.9 & 1 & 0.9 & 1 \\ 0.83 & 0.9 & 1 & 0.9 \\ 0.9 & 1 & 0.9 & 1 \end{matrix}), \\ S (4) = (\begin{matrix} 1 & 0.7 & 0.8 & 0.54 \\ 0.7 & 1 & 0.74 & 0.83 \\ 0.8 & 0.74 & 1 & 0.56 \\ 0.54 & 0.83 & 0.56 & 1 \end{matrix}), \\ S (5) = (\begin{matrix} 1 & 0.9 & 0.7 & 0.9 \\ 0.9 & 1 & 0.74 & 0.9 \\ 0.7 & 0.74 & 1 & 0.8 \\ 0.9 & 0.9 & 0.8 & 1 \end{matrix}) . \end{matrix}$

Table 4

The comprehensive evaluation values obtained using the WIFPGHM

p, q	x ₁	x ₂	x ₃	x ₄	x ₅
p = 0.01, q = 0	α₁ = (0.6217, 0.2532)	α₂ = (0.7414, 0.2137)	α₃ = (0.6694, 0.2555)	α₄ = (0.5219, 0.3197)	α₅ = (0.6277, 0.1926)
p = 0, q = 0.01	α₁ = (0.5188, 0.2533)	α₂ = (0.6858, 0.2280)	α₃ = (0.4992, 0.3779)	α₄ = (0.5975, 0.2877)	α₅ = (0.5097, 0.2665)
p = 1, q = 0	α₁ = (0.4668, 0.4017)	α₂ = (0.6087, 0.3413)	α₃ = (0.5222, 0.3897)	α₄ = (0.3792, 0.4602)	α₅ = (0.4860, 0.3170)
p = 0, q = 1	α₁ = (0.3184, 0.5096)	α₂ = (0.4176, 0.5263)	α₃ = (0.2934, 0.6349)	α₄ = (0.3778, 0.5456)	α₅ = (0.2886, 0.5847)
p = 5, q = 1	α₁ = (0.3261, 0.5523)	α₂ = (0.5314, 0.4008)	α₃ = (0.3822, 0.5172)	α₄ = (0.3135, 0.5126)	α₅ = (0.3931, 0.4344)
p = 1, q = 5	α₁ = (0.2978, 0.5914)	α₂ = (0.4384, 0.4961)	α₃ = (0.2985, 0.6194)	α₄ = (0.2997, 0.5564)	α₅ = (0.2966, 0.5475)
p = 1, q = 1	α₁ = (0.4107, 0.4238)	α₂ = (0.5628, 0.3637)	α₃ = (0.4342, 0.4665)	α₄ = (0.4013, 0.4603)	α₅ = (0.4281, 0.3729)
p = 2, q = 2	α₁ = (0.3664, 0.4837)	α₂ = (0.5365, 0.3871)	α₃ = (0.3954, 0.5027)	α₄ = (0.3601, 0.4846)	α₅ = (0.4008, 0.4116)
p = 6, q = 6	α₁ = (0.2755, 0.6219)	α₂ = (0.4787, 0.4275)	α₃ = (0.3043, 0.6035)	α₄ = (0.2697, 0.5257)	α₅ = (0.3127, 0.5110)
p = 10, q = 10	α₁ = (0.239, 0.67)	α₂ = (0.4503, 0.4433)	α₃ = (0.2651, 0.6472)	α₃ = (0.2292, 0.5401)	α₅ = (0.2677, 0.5491)

Table 5

The orderings of the alternatives

p, q	Score	Ordering
p = 0.01, q = 0	s = (0.3684, 0.5276, 0.4139, 0.2021, 0.4351)	x₂ ≻ x₅ ≻ x₃ ≻ x₁ ≻ x₄
p = 0, q = 0.01	s = (0.2655, 0.4578, 0.1213, 0.3098, 0.2432)	x₂ ≻ x₄ ≻ x₁ ≻ x₅ ≻ x₃
p = 1, q = 0	s = (0.0651, 0.2673, 0.1326, - 0.081, 0.169)	x₂ ≻ x₅ ≻ x₃ ≻ x₁ ≻ x₄
p = 0, q = 1	s = (-0.1912, - 0.1088, - 0.3415, - 0.1679, - 0.2961)	x₂ ≻ x₄ ≻ x₁ ≻ x₅ ≻ x₃
p = 5, q = 1	s = (-0.2263, 0.1306, - 0.135, - 0.1991, - 0.0413)	x₂ ≻ x₅ ≻ x₃ ≻ x₄ ≻ x₁
p = 1, q = 5	s = (-0.2936, - 0.0576, - 0.321, - 0.2567, - 0.251)	x₂ ≻ x₅ ≻ x₄ ≻ x₃ ≻ x₁
p = 1, q = 1	s = (-0.0131, 0.1991, - 0.0323, - 0.0589, 0.0552)	x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄
p = 2, q = 2	s = (-0.1173, 0.1494, - 0.1073, - 0.1246, - 0.0108)	x₂ ≻ x₅ ≻ x₃ ≻ x₁ ≻ x₄
p = 6, q = 6	s = (-0.3464, 0.0512, - 0.2992, - 0.256, - 0.1983)	x₂ ≻ x₅ ≻ x₄ ≻ x₃ ≻ x₁
p = 10, q = 10	s = (-0.4311, 0.007, - 0.382, - 0.3109, - 0.2815)	x₂ ≻ x₅ ≻ x₄ ≻ x₃ ≻ x₁

Step 3. Calculate the weighted support T (α_li) of intuitionistic fuzzy value α_li using other intuitionistic fuzzy values α_lj. Then, utilize T (α_li) to calculate the weighted support exponent η_li, i = 1, 2, 3, 4 ; l = 1, ⋯ , 5. $\begin{matrix} T = (\begin{matrix} 0.5 & 0.25 & 0.5892 & 0.8656 \\ 0.4413 & 0.2735 & 0.8206 & 1.0942 \\ 0.5254 & 0.28 & 0.788 & 1.12 \\ 0.4083 & 0.2262 & 0.6299 & 0.7731 \\ 0.5 & 0.2535 & 0.6706 & 1.04 \end{matrix}), \end{matrix}$ $\begin{matrix} η = (\begin{matrix} 0.7281 & 0.3034 & 1.1572 & 1.8113 \\ 0.6407 & 0.2831 & 1.2141 & 1.8620 \\ 0.6714 & 0.2817 & 1.1806 & 1.8663 \\ 0.7031 & 0.3061 & 1.2205 & 1.7703 \\ 0.6886 & 0.2878 & 1.1505 & 1.8731 \end{matrix}) . \end{matrix}$

Step 4. Calculate the comprehensive evaluation value of c₃ (i = 1, 2, ⋯ , 5) by the WIFPGHM operator. The results are shown in Table 4.

Step 5. Rank the alternatives x_i (i = 1, 2, ⋯ , 5) according to α_i (i = 1, 2, ⋯ , n) by the binary relation of IFNs described in Definition 3. The orderings of the alternatives are shown in Table 5.

As can be seen in Table 5, the ordering of the alternatives varies with the parameters p and q. All the results also show that x₂ is the optimal alternative. If one of the parameters is fixed, different alternative values can be obtained as the other parameter changes, as shown in Figs. 3–6.

Fig.3

Scores of the WIFPGHM with p = 1, q ∈ [1, 10].

Fig.4

Scores of the WIFPGHM with q = 1, p ∈ [1, 10].

Fig.5

Scores of the WIFPGHM with p = 2, q ∈ [1, 10].

Fig.6

Scores of the WIFPGHM with q = 2, p ∈ [1, 10].

From Figs. 3–6, one can find that the score function values gained by the WIFPGHM operator decrease with the increase of p(or q) for the same input values. Therefore, the parameters should be chosen based on the risk preference types of experts during the decision making process, as discussed in detail in Section 3. In view of the computational complexity and recognition capability of the decision making modeling, we would not recommend using parameters that are too large or too small in the WIFPGHM operator. For example, if p → q, then the WIFPGHM operator degenerates to the generalized intuitionistic fuzzy PG operator. This does not capture the correlation among the aggregated arguments. For this reason, taking p = q=1 is recommended, as this not only makes the aggregation calculations simple but also represents the correlation and the relative closeness of individual arguments.

6 Comparative analysis with other methods

In this section, the proposed method is compared with other related methods, which were designed based on an aggregation operator to solve intuitionistic fuzzy MADM problems with attribute weights that were completely known. We now consider the MADM method based on the intuitionistic fuzzy geometric weighted HM (IFGWHM) operator developed by Yu [23], the MADM method based on the weighted intuitionistic fuzzy PG averaging (WIFPGA) operator developed by Zhang [41], and the decision making method based on the weighted intuitionistic fuzzy geometric BM (WIFGBM) operator developed by Xia et al. [42]. Comparative analysis results are given in Tables 6 and 7.

Table 6
Comparison with other method

Ref [21] Ref [31] Ref [34] Proposed method

Operator IFWGHM WIFPGA WIFGBM WIFPGHM

Example 2 p = q = 1 x₂ ≻ x₁ ≻ x₅ ≻ x₃ ≻ x₄ x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄ x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄ x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄

	Ref [21]	Ref [31]	Ref [34]	Proposed method
Operator	IFWGHM	WIFPGA	WIFGBM	WIFPGHM
Example 2 p = q = 1	x₂ ≻ x₁ ≻ x₅ ≻ x₃ ≻ x₄	x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄	x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄	x₂ ≻ x₅ ≻ x₁ ≻ x₃ ≻ x₄

Table 7

Comprehensive evaluation values (p = q = 1)

Operator	x ₁	x ₂	x ₃	x ₄	x ₅
IFWGHM	α₁ = (0.4026, 0.4596)	α₂ = (0.5188, 0.4121)	α₃ = (0.4005, 0.5104)	α₄ = (0.3600, 0.5090)	α₅ = (0.3512, 0.4248)
WIFGBM	α₁ = (0.8056, 0.1212)	α₂ = (0.8671, 0.1057)	α₃ = (0.8151, 0.1407)	α₄ = (0.7962, 0.1418)	α₅ = (0.8104, 0.1027)
WIFPGA	α₁ = (0.0244, 0.9168)	α₂ = (0.0984, 0.8766)	α₃ = (0.0305, 0.9283)	α₄ = (0.0155, 0.9385)	α₅ = (0.0278, 0.8642)
WIFPGHM	α₁ = (0.4123, 0.4208)	α₂ = (0.5627, 0.3639)	α₃ = (0.4343, 0.4662)	α₄ = (0.4013, 0.4597)	α₅ = (0.4280, 0.3731)

As can be seen in Table 6, the major differences between the four methods come from how the diversity in the aggregation operators. Although the operators are all based on the geometric averaging operator, they have different emphases and angles in the decision making process. The IFWGHM and WIFGBM operators can capture the correlations among the aggregated arguments, but they are likely to be affected by a singular point [35]. The WIFPGA operator takes the relative closeness of decision making information into consideration, and plays attention to using global optimization technique in decision making. The WIFPGHM operator takes into consideration not only the correlations between the aggregated arguments but also the relative closeness of the fused values, which is the most advanced development of the IFWGHM and WIFPGA operators. According to Table 7, we can find that the comprehensive evaluation values derived from the WIFPGHM operator is an approximation of the values calculated by the IFWGHM operator. This finding shows that the WIFPGHM operator preserves the characteristics of the IFWGHM operator.

Generally, the method proposed in this paper has the following advantages:1) For the use of IFWGHM operator, the proposed method can solve the problem that attribute information is not robust and has correlation characteristics. 2) The proposed method can set the parameters according to the decision maker’s requirements, which reflects flexibility in the decision making process. 3) The interactive operation between attributes is not repeated, which causes the calculation efficiency of the decision model to be better than the BM operator-based method. Due to the complicated and dynamic nature of decision making circumstances, and the diversified decision making process, the design of a generally applicable method for fuzzy information aggregation is a tough task. In addition, the existing fuzzy information aggregation operators have their own strengths and weaknesses. Furthermore, none of existing operators can always perform better than the others in any decision making environment. Therefore, in order to ensure the decision results are reliable and credible, we should rationally select a operator based on the situation as regards the problem and needs.

7 Conclusion

This paper analyzes the relative strength of the GHM and PG operators. Then, the power GHM operators under intuitionistic fuzzy conditions are developed, known as the intuitionnistic fuzzy power GHM (IFPGHM) operator and the weighted intuitionistic fuzzy power GHM (WIFPGHM) operator. Their desirable characteristics are analyzed in this paper, and some commonly used special cases concerning parameters p and q are also provided. Based on the WIFPGHM operator,we also developed a detailed calculation procedure to solve the intuitionistic fuzzy MADM problem. The application and efficiency of the new procedure is illustrated with practical examples.

The designed power GHM operators with parameters p and q can provide more choices for the user, because people can select reasonable parameter values, according to the decision making situations. This is an interesting topic and is worthy of being further study in the future. Besides, it is necessary and significant to take the proposed aggregation operators as a means to solve the real MADM problems such as supplier selection, investment strategy selection, data mining, and website quality evaluation.

Footnotes

Acknowledgments

This work was supported by the National Natur-al Science Foundation of China(Grant No.111712), the Natural Science Foundation of Anhui Prov-ince (Grant No. 1908085QG306), the Ministry of Ed-ucation Humanities and Social Sciences Research Youth Fund Project(18YJCZH216) and the Research on Bidding for Henan Government Decision making (2018B461).

References

Kaur

, Garg

Multi-attribute decision-making based on Bonferroni mean operators under cubic intuitionistic fuzzy set environment, Entropy 20(1) (2018), 65.

Liao

H.C.

, Mi

X.M.

, Xu

Z.S.

, Herrera

Intuitionistic Fuzzy Analytic Network Process, IEEE Transactions on Fuzzy Systems 26(5) (2018), 2578–2590.

Garg

, Kumar

, Garg

MADM-based parametric selection and ranking of E-learning websites using fuzzy COPRAS, IEEE Transactions on Education 62(1) (2018), 11–18.

D.J.

, Liao

H.C.

Visualization and quantitative research on intuitionistic fuzzy studies, Journal of Intelligent & Fuzzy Systems 30(6) (2016), 3653–3663.

Garg

, Arora

Bonferroni mean aggregation operators under intuitionistic fuzzy soft set environment and their applications to decision-making, Journal of the Operational Research Society 69(11) (2018), 1711–1724.

Zadeh

L.A.

Fuzzy sets, Information & Control 8(3) (1965), 338–353.

Atanassov

K.T.

Intuitionistic fuzzy sets, Physica-Verlag HD Press, 1999.

Garg

, Kumar

Linguistic interval-valued Atanassov intuitionistic fuzzy sets and their applications to group decision-making problems, IEEE Transactions on Fuzzy Systems, (2019), DOI: 10.1109/TFUZZ.2019.2897961.

Sirbiladze

, Sikharulidze

Extensions of Probability Intuitionistic Fuzzy Aggregation Operators in Fuzzy MCDM/MADM, International Journal of Information Technology & Decision Making 17(02) (2018), 621–655.

10.

Garg

, Kumar

A novel exponential distance and its based TOPSIS method for interval-valued intuitionistic fuzzy sets using connection number of SPA theory, (2018), DOI: 10.1007/s10462-018-9668-5.

11.

Z.S.

Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems 14(6) (2008), 1179–1187.

12.

Zhao

, Xu

Z.S.

, Ni

M.F.

, Liu

S.S.

Generalized aggregation operators for intuitionistic fuzzy sets, International Journal of Intelligent Systems 25(1) (2010), 1–30.

13.

Zeng

, Su

Intuitionistic fuzzy ordered weighted distance operator, Knowledge-Based Systems 24(8) (2011), 1224–1232.

14.

Z.X.

, Xia

G.P.

, Chen

M.Y.

, Wang

Induced generalized intuitionistic fuzzy OWA operator for multi-attribute group decision making, Expert Systems with Applications 39(2) (2012), 1902–1910.

15.

Liao

H.C.

, Xu

Z.S.

Intuitionistic Fuzzy Hybrid Weighted Aggregation Operators, International Journal of Intelligent Systems 29(11) (2014), 971–993.

16.

Zeng

, Su

, Zhang

Intuitionistic fuzzy generalized probabilistic ordered weighted averaging operator and its application to group decision making, Technological and Economic Development of Economy 22(2) (2015), 1–17.

17.

Garg

Intuitionistic Fuzzy Hamacher Aggregation Operators with Entropy Weight and Their Applications to Multi-criteria Decision-Making Problems, Iranian Journal of Science and Technology, Transactions of Electrical Engineering, (2019), doi: 10.1007/s40998-018-0167-0.

18.

Beliakov

, Pradera

, Calvo

Aggregation functions, Heidelberg: Springer Press, 2007.

19.

Han

, Chen

H.Y.

, Zhu

J.M.

, Liu

J.P.

An approach to linguistic multiple attribute decision-making based on unbalanced linguistic generalized heronian mean aggregation operator, Computational Intelligence and Neuroscience 2018 (2018), 1–25.

20.

Zang

, Zhao

, Li

Interval-Valued Dual Hesitant Fuzzy Heronian Mean Aggregation Operators and their Application to Multi-Attribute Decision Making, International Journal of Computational Intelligence and Applications 17(1) (2018), 1–20.

21.

Peng

H.G.

, Wang

J.Q.

, Cheng

P.F.

A linguistic intuitionistic multi-criteria decision-making method based on the Frank Heronian mean operator and its application in evaluating coal mine safety, International Journal of Machine Learning and Cybernetics 9(6) (2018), 1053–1068.

22.

Liu

P.D.

, Chen

S.M.

Group Decision Making Based on Heronian Aggregation Operators of Intuitionistic Fuzzy Numbers, IEEE Trans Cybern 99 (2017), 2514–2530.

23.

Intuitionistic fuzzy geometric Heronian mean aggregation operators, Applied Soft Computing 13(2) (2013), 1235–1246.

24.

Yager

R.R.

The power average operator, Systems Man & Cybernetics Part A Systems & Humans IEEE Transactions on 31(6) (2001), 724–731.

25.

Jiang

, Wei

B.Y.

Intuitionistic fuzzy evidential power aggregation operator and its application in multiple criteria decision-making, International Journal of Systems Science 49(3) (2018), 582–594.

26.

Liu

P.D.

, Wang

Some interval-valued intuitionistic fuzzy Schweizer– Sklar power aggregation operators and their application to supplier selection, International Journal of Systems Science 49(6) (2018), 1188–1211.

27.

Jiang

, Wei

B.Y.

, Liu

, Zheng

H.Q.

Intuitionistic Fuzzy Power Aggregation Operator Based on Entropy and Its Application in Decision Making, International Journal of Intelligent Systems 33(1) (2018), 49–67.

28.

Wan

S.P.

, Dong

J.Y.

Power geometric operators of trapezoidal intuitionistic fuzzy numbers and application to multi-attribute group decision making, Applied Soft Computing 29 (2015), 153–168.

29.

Z.S.

, Yager

R.R.

Power-Geometric Operators and Their Use in Group Decision Making, IEEE Transactions on Fuzzy Systems 18(1) (2010), 94–105.

30.

Dimple

, Harish

Complex intuitionistic fuzzy power aggregation operators and their applications in multicriteria decision-making, Expert Systems, (2018), 10.1111/exsy.12325.

31.

Wei

G.W.

, Lu

Pythagorean fuzzy power aggregation operators in multiple attribute decision making, International Journal of Intelligent Systems 33(1) (2018), 169–186.

32.

Liu

P.D.

, Liu

The neutrosophic number generalized weighted power averaging operator and its application in multiple attribute group decision making, International Journal of Machine Learning and Cybernetics 9(2) (2018), 347–358.

33.

Jiang

, Wei

B.Y.

, Liu

, Li

X.Y.

, Zheng

H.Q.

Intuitionistic Fuzzy Power Aggregation Operator Based on Entropy and Its Application in Decision Making, Interna-tional Journal of Intelligent Systems 33(1) (2018), 49–67.

34.

Shi

M.H.

, Xiao

Hesitant fuzzy linguistic aggregation operators and their application to multiple attribute group decision making, Fuzzy Systems and Mathematics 31(5) (2017), 68–79.

35.

Y.D.

, He

, Jin

, Chen

H.Y.

Intuitionistic Fuzzy Power Geometric Bonferroni Means and Their Application to Multiple Attribute Group Decision Making, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 23(2) (2015), 285–315.

36.

Liu

, Tao

Z.F.

, Chen

H.Y.

, Zhou

L.G.

A magdm method based on 2-tuple linguistic heronian mean and new operational laws, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 24(4) (2016), 593–627.

37.

Szmidt

, Kacprzyk

Amount of Information and Its Reliability in the Ranking of Atanassov’s Intuitionistic Fuzzy Alternatives// Recent Advances in Decision Making. Berlin: Springer, 2009.

38.

Zhang

X.M.

, Xu

Z.S.

A new method for ranking intuitionistic fuzzy values and its application in multi-attribute decision making, Fuzzy Optimization and Decision Making 11(2) (2012), 135–146.

39.

Jafarian

, Rezvani

M.A.

A valuation-based method for ranking the intuitionistic fuzzy numbers, Journal of Intelligent & Fuzzy Systems 24(1) (2013), 133–144.

40.

Lakshmana

G.N.V.

, Jeevaraj

, Sivaraman

Complete Ranking of Intuitionistic Fuzzy Numbers, Fuzzy Information and Engineering 8(2) (2016), 237–254.

41.

Zhang

Z.M.

Generalized Atanassov’s intuitionistic fuzzy power geometric operators and their application to multiple attribute group decision making, Information Fusion 14(4) (2013), 460–486.

42.

Xia

M.M.

, Xu

Z.S.

, Zhu

Geometric Bonferroni means with their application in multi-criteria decision making, Knowledge-Based Systems 40 (2013), 88–100.

43.

Szmidt

, Kacprzyk

Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems 114(3) (2000), 505–518.

Intuitionistic fuzzy power geometric Heronian mean operators and their application to multiple attribute decision making

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Intuitionistic fuzzy values

2.2 Geometric Heronian mean and power geometric operator

3 Intuitionistic fuzzy power GHM

Table 6 Comparison with other method Ref [21] Ref [31] Ref [34] Proposed method Operator IFWGHM WIFPGA WIFGBM WIFPGHM Example 2 p = q = 1 x2 ≻ x1 ≻ x5 ≻ x3 ≻ x4 x2 ≻ x5 ≻ x1 ≻ x3 ≻ x4 x2 ≻ x5 ≻ x1 ≻ x3 ≻ x4 x2 ≻ x5 ≻ x1 ≻ x3 ≻ x4

Footnotes

Acknowledgments

References