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Article

Multiple-Attribute Decision Making Based on Interval-Valued Intuitionistic Fuzzy Generalized Weighted Heronian Mean

1
School of Economics and Management, North China Electric Power University, Beijing 102206, China
2
Department of Mathematics and Physics, North China Electric Power University, Baoding 071003, China
*
Author to whom correspondence should be addressed.
Information 2022, 13(3), 138; https://doi.org/10.3390/info13030138
Submission received: 22 December 2021 / Revised: 26 February 2022 / Accepted: 1 March 2022 / Published: 7 March 2022

Abstract

:
Due to the complexity and uncertainty of objective things, interval-valued intuitionistic fuzzy (I-VIF) numbers are often used to describe the attribute values in multiple-attribute decision making (MADM). Sometimes, there are correlations between the attributes. In order to make the decision-making result more objective and reasonable, it is often necessary to take the correlation factors into account. Therefore, the study of MADM based on the correlations between attributes in the I-VIF environment has important theoretical and practical significance. Thus, in this paper, we propose new operators (AOs) for I-VIF information that are able to reflect the completeness of the information, attribute relevance, and the risk preference of decision makers (DMs). Firstly, we propose some new AOs for I-VIF information, including I-VIF generalized Heronian mean (I-VIFGHM), I-VIF generalized weighted Heronian mean (I-VIFGWHM), and I-VIF three-parameter generalized weighted Heronian mean (I-VIFTPGWHM). The properties of the obtained operators, including their idempotency, monotonicity, and boundedness are studied. Furthermore, an MADM method based on the I-VIFGWHM operator is provided. Finally, an example is provided to explain the rationality and feasibility of the proposed method.

1. Introduction

MADM is an important branch of modern decision-making theory [1]. Its theory and method are widely used in many fields, such as those related to the economy, management, engineering, and the military [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. Effective attribute value integration is a core problem of MADM. Scholars have developed many AOs [22,23,24,25,26,27,28]. The generalized mean (GM) was proposed as a connective operator by Dyckhoff and Pedrycz [28]. This operator makes it easy to model the compensation degree, naturally including the minimum and maximum operators along with the arithmetic and geometric means as special examples. The GM can not only express the preferences of the DMs, but it can also take the decision information from the perspective of the whole into account. Therefore, scholars have paid attention to GM research.
With the increasing complexity of the social and economic environment, the decision-making problems that are experienced in various fields of people’s lives and during production are becoming more and more complex. It is often difficult for DMs to provide evaluation information in the form of an accurate value. Zadel put forward the fuzzy set concept, which states the uncertainty and fuzziness of things through membership. Subsequently, Atanassov developed the fuzzy set and put forward the concept of the intuitionistic fuzzy set, which describes the uncertainty and fuzziness of things through the non-membership degree and membership degree. Because of the complicacy and nondeterminacy of things, it is sometimes difficult to express the non-membership degree and membership degree of intuitionistic fuzzy sets using real values, and it is more suitable to express them in the form of interval-valued numbers [27]. Therefore, Atanassov and Gargov [29] extended the intuitionistic fuzzy set to the I-VIF set, and then the concepts of the I-VIF number, score function of the I-VIF number, and exact function of the I-VIF number were put forward [30]. With the introduction of I-VIF number ranking method [30], the I-VIF information MADM method has become more and more important. Garg [31] presented a new generalized improved score function and an I-VIF set-based method to solve the MCDM problem. Wei [32] proposed two new entropy measures based on the cosine function for intuitionistic fuzzy sets and I-VIF sets, which were applied to solve multi-criteria fuzzy group decision-making problems.
At present, many AOs assume that attributes are mutually independent. In real decisions, however, attributes are often interrelated. Therefore, more and more experts and scholars study the MADM with interrelated attributes. The Heronian mean AO is an important AO that considers the correlations between attributes. Wu [33] produced some Dombi Heronian mean AOs with I-VIF numbers and two MADM methods based on the I-VIF-weighted Dombi Heronian mean AO and interval-valued intuitionistic-weighted Dombi geometric Heronian mean AO. Yu [34] proposed a generalized I-VIF Heronian mean and a MCDM method based on this operator. Zang [35] proposed the interval-valued dual hesitant fuzzy Heronian mean AO and the interval-valued dual hesitant fuzzy geometric Heronian mean AO.
Therefore, the study of MADM based on the correlations between attributes in the I-VIF environment has important theoretical and practical significance. Thus, in this paper, we propose some new AOs for I-VIF information, including the I-VIFGHM, I-VIFGWHM, and I-VIFTPGWHM. They have the virtues of both the Heronian mean and the GM and can reflect the completeness of information, attribute relevance, and the risk preference of DMs.
The rest of the article is described below. In Section 2, some basic notions are introduced. In Section 3, some new AOs for I-VIF information are proposed, including the I-VIFGHM, I-VIFGWHM, and I-VIFTPGWHM. The properties of the obtained operators are studied, including their idempotency, monotonicity and boundedness. In Section 4, the MADM method based on the I-VIFGWHM operator is provided, and using an example, the proposed method is compared with the existing MADM methods. In Section 5, we provide a summary and prospects for future work.

2. Preliminaries

Definition 1. 
[29] Let Z be a nonempty set, and E ¯ = { z , θ ¯ E ¯ z , ϑ ¯ E ¯ z | z Z } be the I-VIF set, where θ ¯ E ¯ z , ϑ ¯ E ¯ z 0 , 1 , and s u p θ ¯ E ¯ z + s u p ϑ ¯ E ¯ z 1 , z Z .
If sup θ ¯ E ¯ z = inf θ ¯ E ¯ z and s u p ϑ ¯ E ¯ z = i n f ϑ ¯ E ¯ z , then the I-VIF sets degenerate into intuitionistic fuzzy sets.
Definition 2. 
[30] Let Z be a nonempty set, E ¯ = { z , θ ¯ E ¯ z , ϑ ¯ E ¯ z | z Z } be a I-VIF set, and θ ¯ E ¯ z , ϑ ¯ E ¯ z be the I-VIF number and be abbreviated as χ , δ , η , κ , where χ , δ , η , κ 0 , 1 and δ + κ 1 .
Definition 3. 
[30] Let α = χ , δ , η , κ , α ¯ 1 = χ 1 , δ 1 , η 1 , κ 1 , and α ¯ 2 = χ 2 , δ 2 , η 2 , κ 2 be three I-VIF numbers, λ > 0 , creating the following algorithms:
(1)
α ¯ 1 α ¯ 2 = χ 1 + χ 2 χ 1 χ 2 , δ 1 + δ 2 δ 1 δ 2 , η 1 η 2 , κ 1 κ 2
(2)
α ¯ 1 α ¯ 2 = χ 1 χ 2 , δ 1 δ 2 , η 1 + η 2 η 1 η 2 , κ 1 + κ 2 κ 1 κ 2
(3)
λ α ¯ = 1 ( 1 χ ) λ , 1 ( 1 δ ) λ , η λ , κ λ
(4)
α ¯ λ = χ λ , δ λ , 1 ( 1 η ) λ , 1 ( 1 κ ) λ
Theorem 1. 
[30] Let α = χ , δ , η , κ , α ¯ 1 = χ 1 , δ 1 , η 1 , κ 1 , and α ¯ 2 = ( χ 2 , δ 2 , η 2 , κ 2 be three I-VIF numbers, and let all of the operation results in Definition 3 continue to be I-VIF numbers.
Theorem 2. 
[30] Let α = χ , δ , η , κ , α ¯ 1 = χ 1 , δ 1 , η 1 , κ 1 , and α ¯ 2 = χ 2 , δ 2 , η 2 , κ 2 be three I-VIF numbers, λ , λ 1 , λ 2 0 , meaning that the following algorithms hold:
(1)
Commutative law α ¯ 1 α ¯ 2 = α ¯ 2 α ¯ 1 , α ¯ 1 α ¯ 2 = α ¯ 2 α ¯ 1 ;
(2)
Distributive law λ α ¯ 1 α ¯ 2 = λ α ¯ 1 λ α ¯ 2 , α ¯ 1 α ¯ 2 λ = α ¯ 1 λ α ¯ 2 λ ;
(3)
Associative law λ 1 α ¯ λ 2 α ¯ = λ 1 + λ 2 α ¯ , α ¯ λ 1 α ¯ λ 2 = α ¯ λ 1 + λ 2 .
To rank the I-VIF numbers, Xu [30] introduced the score function s α ¯ = ( χ δ + η κ ) / 2 and exact function h α ¯ = ( χ + δ + η + κ ) / 2 to calculate the score and accuracy of the I-VIF number α ¯ = ( [ χ , δ ] , [ η , κ ] ) , resulting in the order relationship between two I-VIF numbers, α ¯ 1 and α ¯ 2 :
Definition 4. 
[30] Let any two I-VIF numbers be α ¯ 1 and α ¯ 2 , and then
(1)
If s α ¯ 1 < s α ¯ 2 , then α ¯ 1 < α ¯ 2 ;
(2)
If s α ¯ 1 = s α ¯ 2 , then
(i)
If h α ¯ 1 < h α ¯ 2 , then α ¯ 1 < α ¯ 2 ;
(ii)
If h α ¯ 1 = h α ¯ 2 , then α ¯ 1 α ¯ 2 .

3. Some New Aggregation Operators for I-VIF Information

In this section, we propose some new AOs for I-VIF information, including the I-VIFGHM, I-VIFGWHM, and I-VIFTPGWHM.
Definition 5. 
Let z 1 , z 2 , , z n be a set of nonnegative real numbers, where s , t and, t 0 , w i i = 1 , 2 , , n is the weight of z i , and w i 0 , i = 1 n w i = 1 . Then
GWHM s , t ( z 1 , z 2 , , z n ) = 1 λ i , j = 1 j = i n w i z i s + w j z j s t / s 1 / t
is the generalized weighted Heronian mean, where λ = i , j = 1 , j = i n w i + w j t / s .
Definition 6. 
Let α ¯ i = ϕ i , f i , g i , h i , i = 1 , 2 , , n be a set of I-VIF numbers, θ , ϑ > 0 , s , t , and s , t 0 . Then,
I - VIFGHM s , t ( α ¯ 1 , α ¯ 2 , , α ¯ n ) = 2 n ( n + 1 ) i , j = 1 j = i n θ θ + ϑ α ¯ i s ϑ θ + ϑ α ¯ j s t / s 1 / t
is the I-VIFGHM.
Theorem 3. 
Let α ¯ i = ϕ i , f i , g i , h i , i = 1 , 2 , , n be a set of I-VIF numbers. If θ , ϑ > 0 ,   s , t , and s , t 0 ,   then the result aggregated by the I - VIFGHM operator is still an I-VIF number, and
I - VIFGH M s , t ( α ¯ 1 , α ¯ 2 , , α ¯ n ) = 2 n ( n + 1 ) n i , j = 1 j = i θ θ + ϑ α ¯ i s ϑ θ + ϑ α ¯ j s t / s 1 t = 1 i , j = 1 j = i n 1 1 1 ϕ i s θ / ( θ + ϑ ) 1 ϕ j s ϑ / ( θ + ϑ ) t / s 2 / n ( n + 1 ) 1 / t , 1 i , j = 1 j = i n 1 1 1 f i s θ / ( θ + ϑ ) 1 f j s ϑ / ( θ + ϑ ) t / s 2 / n ( n + 1 ) 1 / t 1 1 i , j = 1 j = i n 1 1 1 1 g i s θ / ( θ + ϑ ) 1 1 g j s ϑ / ( θ + ϑ ) t / s 2 / n ( n + 1 ) 1 / t , 1 1 i , j = 1 j = i n 1 1 1 1 h i s θ / ( θ + ϑ ) 1 1 h j s ϑ / ( θ + ϑ ) t / s 2 / n ( n + 1 ) 1 / t .
Proof. 
Using the operation laws of the I-VIF number, we obtain
α ¯ j s = ϕ j s , f j s , 1 1 g j s , 1 1 h j s ,   α ¯ i s = ϕ i s , f i s , 1 1 g i s , 1 1 h i s ,
θ θ + ϑ α ¯ i s = 1 1 ϕ i s θ / ( θ + ϑ ) , 1 1 f i s θ / ( θ + ϑ ) , 1 1 g i s θ / ( θ + ϑ ) , 1 1 h i s θ / ( θ + ϑ )
ϑ θ + ϑ α ¯ j s = 1 1 ϕ j s ϑ / ( θ + ϑ ) , 1 1 f j s ϑ / ( θ + ϑ ) , 1 1 g j s ϑ / ( θ + ϑ ) , 1 1 h j s ϑ / ( θ + ϑ )
θ θ + ϑ α ¯ i s ϑ θ + ϑ α ¯ j s = 1 1 ϕ i s θ / ( θ + ϑ ) 1 ϕ j s ϑ / ( θ + ϑ ) , 1 1 f i s θ / ( θ + ϑ ) 1 f j s ϑ / ( θ + ϑ ) , 1 1 g i s θ / ( θ + ϑ ) 1 1 g j s ϑ / ( θ + ϑ ) , 1 1 h i s θ / ( θ + ϑ ) 1 1 h j s ϑ / ( θ + ϑ )
then
θ θ + ϑ α ¯ i s ϑ θ + ϑ α ¯ j s t / s = 1 1 ϕ i s θ / ( θ + ϑ ) 1 ϕ j s ϑ / ( θ + ϑ ) t / s , 1 1 f i s θ / ( θ + ϑ ) 1 f j s ϑ / ( θ + ϑ ) t / s , 1 1 1 1 g i s θ / ( θ + ϑ ) 1 1 g j s ϑ / ( θ + ϑ ) t / s , 1 1 1 1 h i s θ / ( θ + ϑ ) 1 1 h j s ϑ / ( θ + ϑ ) t / s
and
n i , j = 1 j = i θ θ + ϑ α ¯ i s ϑ θ + ϑ α ¯ j s t / s = 1 i , j = 1 j = i n 1 1 1 ϕ i s θ / ( θ + ϑ ) 1 ϕ j s ϑ / ( θ + ϑ ) t / s , 1 i , j = 1 j = i n 1 1 1 f i s θ / ( θ + ϑ ) 1 f j s ϑ / ( θ + ϑ ) t / s , i , j = 1 j = i n 1 1 1 1 g i s θ / ( θ + ϑ ) 1 1 g j s ϑ / ( θ + ϑ ) t / s , i , j = 1 j = i n 1 1 1 1 h i s θ / ( θ + ϑ ) 1 1 h j s ϑ / ( θ + ϑ ) t / s
that is,
2 n ( n + 1 ) i , j = 1 j = i n θ θ + ϑ α ¯ i s ϑ θ + ϑ α ¯ j s t / s
= 1 i , j = 1 j = i n 1 1 1 ϕ i s θ / ( θ + ϑ ) 1 ϕ j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) , 1 i , j = 1 j = i n 1 1 1 f i s θ / ( θ + ϑ ) 1 f j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) , i , j = 1 j = i n 1 1 1 1 g i s θ / ( θ + ϑ ) 1 1 g j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) , i , j = 1 j = i n 1 1 1 1 h i s θ / ( θ + ϑ ) 1 1 h j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 )
and then,
2 n ( n + 1 ) n i , j = 1 j = i θ θ + ϑ α ¯ i s ϑ θ + ϑ α ¯ j s t / s 1 / t = 1 i , j = 1 j = i n 1 1 1 ϕ i s θ / ( θ + ϑ ) 1 ϕ j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) 1 / t , 1 i , j = 1 j = i n 1 1 1 f i s θ / ( θ + ϑ ) 1 f j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) 1 / t 1 1 i , j = 1 j = i n 1 1 1 1 g i s θ / ( θ + ϑ ) 1 1 g j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) 1 / t , 1 1 i , j = 1 j = i n 1 1 1 1 h i s θ / ( θ + ϑ ) 1 1 h j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) 1 / t
Since
0 1 i , j = 1 j = i n 1 1 1 f i s θ / ( θ + ϑ ) 1 f j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) 1 / t 1 ,
0 1 i , j = 1 j = i n 1 1 1 ϕ i s θ / ( θ + ϑ ) 1 ϕ j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) 1 / t 1 ,
0 1 1 i , j = 1 j = i n 1 1 1 1 g i s θ / ( θ + ϑ ) 1 1 g j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) 1 / t 1 ,
0 1 1 i , j = 1 j = i n 1 1 1 1 h i s θ / ( θ + ϑ ) 1 1 h j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) 1 / t 1 .
and, for all i = 1 , 2 , , n , ϕ i , f i , g i , h i 0 , 1 , f i + h i 1 , we obtain
0 1 i , j = 1 j = i n 1 1 1 f i s θ / ( θ + ϑ ) 1 f j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) 1 / t
+ 1 1 i , j = 1 j = i n 1 1 1 1 h i s θ / ( θ + ϑ ) 1 1 h j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) 1 / t 1 i , j = 1 j = i n 1 1 1 1 h i s θ / ( θ + ϑ ) 1 1 h j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) 1 / t + 1 1 i , j = 1 j = i n 1 1 1 1 h i s θ / ( θ + ϑ ) 1 1 h j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) 1 / t = 1 .
The proof is complete. □
Next, we will determine some properties of the I - VIFGHM operator.
Property 1. 
(Idempotency) Suppose that α ¯ i = α ¯ = ϕ , f , g , h and θ , ϑ > 0 ,   s , t , s , t 0 . Then,
I - VIFGHM s , t ( α ¯ 1 , α ¯ 2 , , α ¯ n ) = I - VIFGHM s , t ( α ¯ , α ¯ , , α ¯ ) = 2 n ( n + 1 ) i , j = 1 j = i n θ θ + ϑ α ¯ s ϑ θ + ϑ α ¯ s t / s 1 t = 2 n ( n + 1 ) i , j = 1 j = i n α ¯ t θ θ + ϑ ϑ θ + ϑ t / s 1 t = α ¯ .
Property 2. 
(Monotonicity) Let α ¯ i = ϕ α i , f α i , g α i , h α i , β ¯ i = ϕ β i , f β i , g β i , h β i i = 1 , 2 , , n be two sets of I-VIFnumbers,where ϕ α i ϕ β i ,   f α i f β i ,   g α i g β i ,   h α i h β i , i = 1 , 2 , , n .   I f   θ , ϑ > 0 ,   s , t , and s , t > 0 ;   then,
I - VIFGHM s , t ( α ¯ 1 , α ¯ 2 , , α ¯ n ) I - VIFGHM s , t ( β ¯ 1 , β ¯ 2 , , β ¯ n ) .
Proof. 
On the one hand, if 0 ϕ α i ϕ β i 1 , i = 1 , 2 , , n , s > 0 , then ϕ α i s ϕ β i s , and since 0 θ θ + ϑ ,   ϑ θ + ϑ 1 , we obtain
1 ϕ α i s 1 ϕ β i s 0 ,   1 ϕ α j s 1 ϕ β j s 0 ,   1 ϕ α i s θ / ( θ + ϑ ) 1 ϕ β i s θ / ( θ + ϑ ) 0 ,
1 ϕ α j s ϑ / ( θ + ϑ ) 1 ϕ β j s ϑ / ( θ + ϑ ) 0 ;
1 1 ϕ α i s θ / ( θ + ϑ ) 1 ϕ α j s ϑ / ( θ + ϑ ) 1 ϕ β i s θ / ( θ + ϑ ) 1 ϕ β j s ϑ / ( θ + ϑ ) 0 ,
0 1 1 ϕ α i s θ / ( θ + ϑ ) 1 ϕ α j s ϑ / ( θ + ϑ ) 1 1 ϕ β i s θ / ( θ + ϑ ) 1 ϕ β j s ϑ / ( θ + ϑ ) 1 ,
1 1 ϕ α i s θ / ( θ + ϑ ) 1 ϕ α j s ϑ / ( θ + ϑ ) t / s 1 1 ϕ β i s θ / ( θ + ϑ ) 1 ϕ β j s ϑ / ( θ + ϑ ) t / s ,
1 1 1 ϕ α i s θ / ( θ + ϑ ) 1 ϕ α j s ϑ / ( θ + ϑ ) t / s 1 1 1 ϕ β i s θ / ( θ + ϑ ) 1 ϕ β j s ϑ / ( θ + ϑ ) t / s ,
i , j = 1 , j = i n 1 1 1 ϕ α i s θ / ( θ + ϑ ) 1 ϕ α j s ϑ / ( θ + ϑ ) t / s i , j = 1 , j = i n 1 1 1 ϕ β i s θ / ( θ + ϑ ) 1 ϕ β j s ϑ / ( θ + ϑ ) t / s
then
i , j = 1 j = i n 1 1 1 ϕ α i s θ / ( θ + ϑ ) 1 ϕ α j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) i , j = 1 j = i n 1 1 1 ϕ β i s θ / ( θ + ϑ ) 1 ϕ β j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) ,
Therefore,
1 i , j = 1 j = i n 1 1 1 ϕ α i s θ / ( θ + ϑ ) 1 ϕ α j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) 1 i , j = 1 j = i n 1 1 1 ϕ β i s θ / ( θ + ϑ ) 1 ϕ β j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) .
resulting in
1 i , j = 1 j = i n 1 1 1 ϕ α i s θ / ( θ + ϑ ) 1 ϕ α j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) 1 / t 1 i , j = 1 j = i n 1 1 1 ϕ β i s θ / ( θ + ϑ ) 1 ϕ β j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) 1 / t
Similarly, it can be determined that
1 i , j = 1 j = i n 1 1 1 f α i s θ / ( θ + ϑ ) 1 f α j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) 1 / t 1 i , j = 1 j = i n 1 1 1 f β i s θ / ( θ + ϑ ) 1 f β j s ϑ / ( θ + ϑ ) t / s 2 n ( n + 1 ) 1 / t
On the other hand, for all i = 1 , 2 , , n since 1 g α i g β i 0 , then 1 g α j 1 g β j , and since θ , ϑ , s , t > 0 , then 1 g α i s 1 g β i s , 1 1 g α i s 1 1 g β i s , 1 1 g α i s θ θ + ϑ 1 1 g β i s θ θ + ϑ ,
1 1 g α i s θ θ + ϑ 1 1 g α j s ϑ θ + ϑ 1 1 g β i s θ θ + ϑ 1 1 g β j s ϑ θ + ϑ ,
1 1 1 g α i s θ θ + ϑ 1 1 g α j s ϑ θ + ϑ 1 1 1 g β i s θ θ + ϑ 1 1 g β j s ϑ θ + ϑ ,
1 1 1 g α i s θ θ + ϑ 1 1 g α j s ϑ θ + ϑ t / s 1 1 1 g β i s θ θ + ϑ 1 1 g β j s ϑ θ + ϑ t / s ,
1 1 1 1 g α i s θ θ + ϑ 1 1 g α j s ϑ θ + ϑ t / s 1 1 1 1 g β i s θ θ + ϑ 1 1 g β j s ϑ θ + ϑ t / s ,
i , j = 1 j = i n 1 1 1 1 g α i s θ θ + ϑ 1 1 g α j s ϑ θ + ϑ t / s i , j = 1 j = i n 1 1 1 1 g β i s θ θ + ϑ 1 1 g β j s ϑ θ + ϑ t / s ,
i , j = 1 j = i n 1 1 1 1 g α i s θ θ + ϑ 1 1 g α j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) i , j = 1 j = i n 1 1 1 1 g β i s θ θ + ϑ 1 1 g β j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) ,
1 i , j = 1 j = i n 1 1 1 1 g α i s θ θ + ϑ 1 1 g α j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 i , j = 1 j = i n 1 1 1 1 g β i s θ θ + ϑ 1 1 g β j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) ,
1 i , j = 1 j = i n 1 1 1 1 g α i s θ θ + ϑ 1 1 g α j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t 1 i , j = 1 j = i n 1 1 1 1 g β i s θ θ + ϑ 1 1 g β j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t ,
1 1 i , j = 1 j = i n 1 1 1 1 g α i s θ θ + ϑ 1 1 g α j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t 1 1 i , j = 1 j = i n 1 1 1 1 g β i s θ θ + ϑ 1 1 g β j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t .
Similarly, it can be determined that
1 1 i , j = 1 j = i n 1 1 1 1 h α i s θ θ + ϑ 1 1 h α j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t 1 1 i , j = 1 j = i n 1 1 1 1 h β i s θ θ + ϑ 1 1 h β j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t
According to (3), (4), (5), and (6), we have
1 i , j = 1 j = i n 1 1 1 ϕ α i s θ θ + ϑ 1 ϕ α j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t 1 1 i , j = 1 j = i n 1 1 1 1 g α i s θ θ + ϑ 1 1 g α j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t 1 i , j = 1 j = i n 1 1 1 ϕ β i s θ θ + ϑ 1 ϕ β j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t 1 1 i , j = 1 j = i n 1 1 1 1 g β i s θ θ + ϑ 1 1 g β j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t
and
1 i , j = 1 j = i n 1 1 1 f α i s θ θ + ϑ 1 f α j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t 1 1 i , j = 1 j = i n 1 1 1 1 h α i s θ θ + ϑ 1 1 h α j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t
1 i , j = 1 j = i n 1 1 1 f β i s θ θ + ϑ 1 f β j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t
1 1 i , j = 1 j = i n 1 1 1 1 h β i s θ θ + ϑ 1 1 h β j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t  
Let α ¯ = I - VIFGWHM s , t ( α ¯ 1 , α ¯ 2 , , α ¯ n ) , β ¯ = I - VIFGWHM s , t ( β ¯ 1 , β ¯ 2 , , β ¯ n ) , and use s α and s β to represent the scores of α and β respectively. Then,
2 s α = 1 i , j = 1 j = i n 1 1 1 ϕ α i s θ θ + ϑ 1 ϕ α j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t 1 1 i , j = 1 j = i n 1 1 1 1 g α i s θ θ + ϑ 1 1 g α j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t , + 1 i , j = 1 j = i n 1 1 1 f α i s θ θ + ϑ 1 f α j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t 1 1 i , j = 1 j = i n 1 1 1 1 h α i s θ θ + ϑ 1 1 h α j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t
2 s β = 1 i , j = 1 j = i n 1 1 1 ϕ β i s θ θ + ϑ 1 ϕ β j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t 1 1 i , j = 1 j = i n 1 1 1 1 g β i s θ θ + ϑ 1 1 g β j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t + 1 i , j = 1 j = i n 1 1 1 f β i s θ θ + ϑ 1 f β j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t 1 1 i , j = 1 j = i n 1 1 1 1 h β i s θ θ + ϑ 1 1 h β j s ϑ θ + ϑ t / s 2 / n ( n + 1 ) 1 / t
From (7) and (8), we can determine that s α s β . Then,
I - VIFGHM s , t ( α 1 , α 2 , , α n ) I - VIFGHM s , t ( β 1 , β 2 , , β n ) ,
That is, the monotonicity is valid.
The proof is complete. □
Property 3. 
(Boundedness) Let α ¯ i = [ ϕ α i , f α i ] , [ g α i , h α i ] ( i = 1 , 2 , , n ) be a set of I-VIF numbers, let
ζ + = max i ϕ α i , max i f α i , min i g α i , min i h α i ,
and let
ζ = min i ϕ α i , min i f α i , max i g α i , max i h α i ,
then
ζ I VIFGHM s , t α ¯ 1   ,   α ¯ 2   ,   , α ¯ n   ζ + .
Proof. 
Using Property 1, we obtain
I VIFGHM s , t ζ ,   ζ ,   , ζ = ζ
and I VIFGHM s , t ζ + ,   ζ ,   , ζ + = ζ + , Then, using Property 2, we have
I VIFGHM s , t ζ ,   ζ ,   , ζ I VIFGHM s , t α ¯ 1   ,   α ¯ 2   ,   , α ¯ n   I VIFGHM s , t ζ + ,   ζ ,   , ζ +
Therefore, Property 3 can be proven. □
Next, the I - VIFGWHM operator is presented.
Definition 7. 
Let α ¯ i = ϕ i , f i , g i , h i , i = 1 , 2 , , n be a set of I-VIF numbers.The weight w i 0 should satisfy i = 1 n w i = 1 ,and let λ = i = 1 , j = i n ( w i + w j ) t / s . If s , t > 0 , then
I - VIFGWHM s , t ( α ¯ 1 , α ¯ 2 , , α ¯ n ) = 1 λ i , j = 1 j = i n w i α ¯ i s w j α ¯ j s t / s 1 t
is the I - VIFGWHM .
Theorem 4. 
Let α ¯ i = ϕ i , f i , g i , h i ( i = 1 , 2 , , n )be a set of I-VIF numbers andthe weight w i 0 satisfy i = 1 n w i = 1 . If s , t > 0 , then the result aggregated by the I - VIFGWHM operator is still an I-VIF number, and
I - VIFGWHM s , t ( α ¯ 1 , α ¯ 2 , , α ¯ n ) = 1 λ i , j = 1 j = i n w i α ¯ i s w j α ¯ j s t / s 1 t
= 1 i , j = 1 j = i n 1 1 1 ϕ i s w i 1 ϕ j s w j t / s 1 / λ 1 / t , 1 i , j = 1 j = i n 1 1 1 f i s w i 1 f j s w j t / s 1 / λ 1 / t 1 1 i , j = 1 j = i n 1 1 1 1 g i s w i 1 1 g j s w j t / s 1 / λ 1 / t , 1 1 i , j = 1 j = i n 1 1 1 1 h i s w i 1 1 h j s w j t / s 1 / λ 1 / t .
Proof. 
By the operation laws of I-VIF number, we achieve
α ¯ j s = ϕ α j s , f α j s , 1 1 g α j s , 1 1 h α j s ,   α ¯ i s = ϕ α i s , f α i s , 1 1 g α i s , 1 1 h α i s ,
w i α ¯ i s = 1 1 ϕ α i s w i , 1 1 f α i s w i , 1 1 g α i s w i , 1 1 h α i s w i ,
w j α ¯ j s = 1 1 ϕ α j s w j , 1 1 f α j s w j , 1 1 g α j s w j , 1 1 h α j s w j ,
w i α ¯ i s w j α ¯ j s = 1 1 ϕ α i s w i 1 ϕ α j s w j , 1 1 f α i s w i 1 f α j s w j , 1 1 g α i s w i 1 1 g α j s w j , 1 1 h α i s w i 1 1 h α j s w j ,
then
w i α ¯ i s w j α ¯ j s t / s = 1 1 ϕ α i s w i 1 ϕ α j s w j t / s , 1 1 f α i s w i 1 f α j s w j t / s , 1 1 1 1 g α i s w i 1 1 g α j s w j t / s , 1 1 1 1 h α i s w i 1 1 h α j s w j t / s ,
and
n i , j = 1 j = i w i α ¯ i s w j α ¯ j s t / s = 1 i , j = 1 j = i n 1 1 1 ϕ α i s w i 1 ϕ α j s w j t / s , 1 i , j = 1 j = i n 1 1 1 f α i s w i 1 f α j s w j t / s , i , j = 1 j = i n 1 1 1 1 g α i s w i 1 1 g α j s w j t / s , i , j = 1 j = i n 1 1 1 1 h α i s w i 1 1 h α j s w j t / s ,
that is,
1 λ n i , j = 1 j = i w i α ¯ i s w j α ¯ j s t / s = 1 i , j = 1 j = i n 1 1 1 ϕ α i s w i 1 ϕ α j s w j t / s 1 / λ , 1 i , j = 1 j = i n 1 1 1 f α i s w i 1 f α j s w j t / s 1 / λ , i , j = 1 j = i n 1 1 1 1 g α i s w i 1 1 g α j s w j t / s 1 / λ , i , j = 1 j = i n 1 1 1 1 h α i s w i 1 1 h α j s w j t / s 1 / λ ,
and
1 λ n i , j = 1 j = i w i α ¯ i s w j α ¯ j s t / s 1 t = 1 i , j = 1 j = i n 1 1 1 ϕ α i s w i 1 ϕ α j s w j t / s 1 / λ 1 / t , 1 i , j = 1 j = i n 1 1 1 f α i s w i 1 f α j s w j t / s 1 / λ 1 / t 1 1 i , j = 1 j = i n 1 1 1 1 g α i s w i 1 1 g α j s w j t / s 1 / λ 1 / t , 1 1 i , j = 1 j = i n 1 1 1 1 h α i s w i 1 1 h α j s w j t / s 1 / λ 1 / t .
Since
0 1 i , j = 1 j = i n 1 1 1 ϕ α i s w i 1 ϕ α j s w j t / s 1 / λ 1 / t 1 ,
0 1 i , j = 1 j = i n 1 1 1 f α i s w i 1 f α j s w j t / s 1 / λ 1 / t 1 ,
0 1 1 i , j = 1 j = i n 1 1 1 1 g α i s w i 1 1 g α j s w j t / s 1 / λ 1 / t 1 ,
0 1 1 i , j = 1 j = i n 1 1 1 1 h α i s w i 1 1 h α j s w j t / s 1 / λ 1 / t 1
and for all i = 1 , 2 , , n , μ α i + ν α i 1 , we obtain
0 1 i , j = 1 j = i n 1 1 1 f α i s w i 1 f α j s w j t / s 1 / λ 1 / t
+ 1 1 i , j = 1 j = i n 1 1 1 1 h α i s w i 1 1 h α j s w j t / s 1 / λ 1 / t 1 i , j = 1 j = i n 1 1 1 1 h α i s w i 1 1 h α j s w j t / s 1 / λ 1 / t + 1 1 i , j = 1 j = i n 1 1 1 1 h α i s w i 1 1 h α j s w j t / s 1 / λ 1 / t = 1 .
The proof is complete. □
Next, we will present some of the properties of the I - VIFGWHM operator.
Property 4. 
(Idempotency) For i = 1 , 2 , , n , if α ¯ i = α ¯ = ϕ , f , g , h , then
I - VIFGWHM s , t ( α ¯ 1 , α ¯ 2 , , α ¯ n ) = I - VIFGWHM s , t ( α ¯ , α ¯ , , α ¯ ) = 1 λ i , j = 1 j = i n w i α ¯ s w j α ¯ s t / s 1 t = 1 λ i , j = 1 j = i n α ¯ t w i w j t / s 1 t = α ¯ t λ i , j = 1 j = i n w i w j t / s 1 t = α ¯ .
Property 5. 
(Monotonicity) Suppose that α ¯ i = ϕ α i , f α i , g α i , h α i and β ¯ i = ϕ β i , f β i , g β i , h β i ( i = 1 , 2 , , n ) are two sets of I-VIF numbers, where ϕ α i ϕ β i , f α i f β i , and g α i g β i , h α i h β i . Then
I - VIFGWHM s , t ( α ¯ 1 , α ¯ 2 , , α ¯ n ) I - VIFGWHM s , t ( β ¯ 1 , β ¯ 2 , , β ¯ n ) .
Proof. 
On the one hand, since 0 ϕ α i ϕ β i 1 , i = 1 , 2 , , n , s > 0 , then ϕ α i s ϕ β i s , and since 0 w i 1 , we achieve
1 ϕ α i s 1 ϕ β i s 0 , 1 ϕ α j s 1 ϕ β j s 0 ,   1 ϕ α i s w i 1 ϕ β i s w i 0 ,   1 ϕ α j s w j 1 ϕ β j s w j 0 ;
1 1 ϕ α i s w i 1 ϕ α j s w j 1 ϕ β i s w i 1 ϕ β j s w j 0 ,
0 1 1 ϕ α i s w i 1 ϕ α j s w j 1 1 ϕ β i s w i 1 ϕ β j s w j 1 ,
1 1 ϕ α i s w i 1 ϕ α j s w j t / s 1 1 ϕ β i s w i 1 ϕ β j s w j t / s ,
1 1 1 ϕ α i s w i 1 ϕ α j s w j t / s 1 1 1 ϕ β i s w i 1 ϕ β j s w j t / s ,
i = 1 j = i n 1 1 1 ϕ α i s w i 1 ϕ α j s w j t / s i = 1 j = i n 1 1 1 ϕ β i s w i 1 ϕ β j s w j t / s ,
Then,
i , j = 1 j = i n 1 1 1 ϕ α i s w i 1 ϕ α j s w j t / s 1 / λ i , j = 1 j = i n 1 1 1 ϕ β i s w i 1 ϕ β j s w j t / s 1 / λ ,
Therefore,
1 i , j = 1 j = i n 1 1 1 ϕ α i s w i 1 ϕ α j s w j t / s 1 / λ 1 i , j = 1 j = i n 1 1 1 ϕ β i s w i 1 ϕ β j s w j t / s 1 / λ .
As such, we have
1 i , j = 1 j = i n 1 1 1 ϕ α i s w i 1 ϕ α j s w j t / s 1 / λ 1 / t 1