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Article

Methods for Multiple Attribute Group Decision Making Based on Intuitionistic Fuzzy Dombi Hamy Mean Operators

by
Zengxian Li
,
Hui Gao
and
Guiwu Wei
*
School of Business, Sichuan Normal University, Chengdu 610101, China
*
Author to whom correspondence should be addressed.
Symmetry 2018, 10(11), 574; https://doi.org/10.3390/sym10110574
Submission received: 28 September 2018 / Revised: 24 October 2018 / Accepted: 27 October 2018 / Published: 1 November 2018
(This article belongs to the Special Issue Multi-Criteria Decision Aid methods in fuzzy decision problems)
Versions Notes

Abstract

:
In this paper, we extended the Hamy mean (HM) operator, the Dombi Hamy mean (DHM) operator, the Dombi dual Hamy mean (DDHM), with the intuitionistic fuzzy numbers (IFNs) to propose the intuitionistic fuzzy Dombi Hamy mean (IFDHM) operator, intuitionistic fuzzy weighted Dombi Hamy mean (IFWDHM) operator, intuitionistic fuzzy Dombi dual Hamy mean (IFDDHM) operator, and intuitionistic fuzzy weighted Dombi dual Hamy mean (IFWDDHM) operator. Following this, the multiple attribute group decision-making (MAGDM) methods are proposed with these operators. To conclude, we utilized an applicable example for the selection of a car supplier to prove the proposed methods.

1. Introduction

Multiple attribute decision making (MADM) is a key branch of decision theory. The definition of intuitionistic fuzzy sets (IFSs) [1,2] has been utilized to deal with uncertainty and imprecision. The introduction of intuitionistic fuzzy entropy by Burillo and Bustince [3] caught the attention of researchers. Xu [4,5] developed a number of aggregation operators with intuitionistic fuzzy numbers (IFNs.) Xu [6] defined the intuitionistic preference relations for multiple attribute group decision making (MAGDM). Li [7] proposed the Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) models for MADM. Xu [8] developed the Choquet integrals of weighted IFNs. Ye [9] gave some Cosine similarity measures for intuitionistic fuzzy sets (IFSs). Li and Ren [10] considered the amount and reliability of IFNs for MADM. Wei [11] proposed some induced geometric aggregation operators with IFNs. Wei and Zhao [12] gave some induced correlated aggregating operators with IFNs. Wei [13,14] developed the gray relational analysis method for MADM with IFNs. Zhao and Wei [15] defined some Einstein hybrid aggregation operators with IFNs. Garg [16] proposed the generalized interactive geometric interaction operators using Einstein T-norm and T-conorm with IFNs. Chu et al. [17] gave a MAGDM model that considered both the additive consistency and group consensus with IFNs. Wan et al. [18] researched a novel risk attitudinal ranking method for MADM with IFNs. Zhao et al. [19] proposed the VIKOR (VIseKriterijumska Optimizacija I KOmpromisno Resenje) method using IFSs. Liu [20] proposed MADM methods with normal intuitionistic fuzzy interaction operators. Shi [21] developed some constructive methods for intuitionistic fuzzy implication operators. Otay et al. [22] studied the multi-expert performance evaluation of healthcare institutions with intuitionistic fuzzy Analytic Hierarchy Process (AHP) and Data Envelopment Analysis (DEA) methodology. Ai and Xu [23] proposed the multiple definite integrals of intuitionistic fuzzy calculus and isomorphic mappings. Montes et al. [24] defined the entropy measures for IFNs based on divergence. Liu et al. [25] evaluated the commercial bank counterparty credit risk management with IFNs. Some similarity measures and information aggregating operators between intuitionistic fuzzy sets [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41] and their extension [42,43,44,45,46,47,48,49,50,51,52,53] have been proposed.
Dombi [54] proposed the operations of the Dombi T-norm and T-conorm. Following this, Liu et al. [55] proposed the Dombi operations with IFNs. Chen and Ye [56] proposed the Dombi weighted arithmetic average and geometric average fuse the single-valued neutrosophic numbers (SVNNs). Wei and Wei [57] gave some dombi prioritized weighting aggregation operators with single-valued neutrosophic numbers.
Through existing studies, we can see that the combination Hamy mean (HM) operator [58,59] and Dombi operations are not extended to IFNs so far. In order to develop Hamy mean operators and Dombi operations for IFNs, the main purposes of this study are (1) to develop some Dombi Hamy mean aggregating operators for IFNs and to investigate their properties, and (2) to propose two models to solve the MADM problems based on these operators with IFNs.
To do so, the rest of this paper is organized as follows. In the next section, we introduce some basic concepts of IFSs, Dombi operations and HM operators. In section three we propose some intuition fuzzy Hamy mean operators based on Dombi T-norm and T-conorm. In section four, we have applied these operators to solve the MAGDM problems with IFNs. In section five, a practical example for the selection of a car supplier is given. In section six, we conclude the paper and give some remarks.

2. Preliminaries

In this section, we introduce the concept of IFS, HM operator, and Dombi T-conorm and T-norm.

2.1. Intuitionistic Fuzzy Sets

Definition 1.
Let X be a fixed set, with a generic in X denoted by x . An intuitionistic fuzzy set (IFS) I in X is following [1,2]:
I = { 〈 x , μ i ( x ) , ν i ( x ) 〉 x ∈ X }  
where μ i ( x ) is the membership function, and ν i ( x ) is the non-membership function. For each point x in X , we have μ i ( x ) , ν i ( x ) ∈ [ 0 , 1 ] and 0 ≤ μ i ( x ) + ν i ( x ) ≤ 1 .
For each IFS I in X , let π i ( x ) = 1 − μ i ( x ) − ν i ( x ) , ∀ x ∈ X , and we call π i ( x ) the indeterminacy degree of the element x to the set I . It can be easily proved that 0 ≤ π i ( x ) ≤ 1 , ∀ x ∈ X . For convenience, we call k = ( μ k , ν k ) an IFN, where μ k ∈ [ 0 , 1 ] , ν k ∈ [ 0 , 1 ] , and 0 ≤ μ k + ν k ≤ 1 .
Definition 2.
Let k 1 = ( μ 1 , ν 1 ) and k 2 = ( μ 2 , ν 2 ) be two IFNs, then operational laws are defined [4,5].
1.
k 1 ⊕ k 2 = ( μ 1 + μ 2 − μ 1 μ 2 , ν 1 ν 2 )
2.
k 1 ⊗ k 2 = ( μ 1 μ 2 , ν 1 + ν 2 − ν 1 ν 2 )
3.
λ k 1 = ( 1 − ( 1 − μ 1 ) λ , ν 1 λ ) ,   λ > 0
4.
k 1 λ = ( μ 1 λ , 1 − ( 1 − ν 1 ) λ ) ,   λ > 0
Example 1.
Suppose that k 1 = ( 0.4   0.5 ) ,   k 2 = ( 0.6   0.4 ) , and λ = 4 , then we have
1.
k 1 ⊕ k 2 = ( 0.4 + 0.6 − 0.4 × 0.6   0.5 × 0.4 ) = ( 0.7600   0.2000 )
2.
k 1 ⊗ k 2 = ( 0.4 × 0.6   0.5 + 0.4 − 0.5 × 0.4 ) = ( 0.2400   0.7000 )
3.
λ k 1 = ( 1 − ( 1 − 0.4 ) 4 , 0.5 4 ) = ( 0.8704 , 0.0625 )
4.
k 1 λ = ( 0.4 4 , 1 − ( 1 − 0.5 ) 4 ) = ( 0.0256 , 0.9375 )
Definition 3.
Let k = ( μ k , ν k ) be an IFN, then a score function is [60]:
S ( k ) = μ k − ν k  
where S ( k ) ∈ [ − 1 , 1 ] , from (2), we can give the comparison method of IFNs on the basis of the above score function. For the difference μ k − ν k , the larger the S ( k ) is, the greater the IFN k is.
Example 2.
Let k 1 = ( 0.5   0.4 ) , k 2 = ( 0.6   0.2 ) be two IFNs, we can get the scores of k 1 and k 2 . S ( k 1 ) = 0.5 − 0.4 = 0.1 , S ( k 2 ) = 0.6 − 0.2 = 0.4 , since S ( k 2 ) > S ( k 1 ) , we get k 2 > k 1 .
Definition 4.
Let k = ( μ k , ν k ) be an IFN, then an accuracy function H of k can be defined as follows [61]:
H ( k ) = μ k + ν k  
where H ( k ) ∈ [ 0 , 1 ] , for the difference μ k + ν k , the larger the H ( k ) is, the greater the IFN k is.
Xu and Yager [5] develop a comparison method of IFNs.
Definition 5.
Let k 1 = ( μ 1 , ν 1 ) and k 2 = ( μ 2 , ν 2 ) be two IFNs, S ( k 1 ) and S ( k 2 ) are the score function of k 1 and k 2 respectively, H ( k 1 ) and H ( k 2 ) are the score function of k 1 and k 2 respectively. Then,
(1) 
If S ( k 1 ) > S ( k 2 ) , then k 1 > k 2 ;
(2) 
If S ( k 1 ) = S ( k 2 ) , then
(3) 
If H ( k 1 ) > H ( k 2 ) , then k 1 > k 2 ;
(4) 
If H ( k 1 ) = H ( k 2 ) , then k 1 = k 2 .
Example 3.
Let k 1 = ( 0.6 , 0.3 ) , k 2 = ( 0.4 , 0.1 ) be two IFNs, we can get the scores and the accuracy of k 1 and k 2 . S ( k 1 ) = 0.6 − 0.3 = 0.3 , S ( k 2 ) = 0.4 − 0.1 = 0.3 . Since S ( k 1 ) = S ( k 2 ) , we can’t get the difference of k 1 and k 2 , then H ( k 1 ) = 0.6 + 0.3 = 0.9 , H ( k 2 ) = 0.4 + 0.1 = 0.5 , since H ( k 1 ) > H ( k 2 ) , we can get k 1 > k 2 .

2.2. HM Operator

Definition 6.
The HM operator is defined as follows [58]:
HM ( x ) ( k 1 , k 2 , ⋯ k n ) = ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( ∏ j = 1 x k i j ) 1 x C n x  
where x is a parameter and x = 1 , 2 , ⋯ , n , i 1 , i 2 , ⋯ i x are x integer values taken from the set { 1 , 2 , ⋯ , n } of k integer values, C n x denotes the binomial coefficient and C n x = n ! x ! ( n − x ) ! . The properties of the operator are shown as follows:
(i) 
When k i = k ( i = 1 , 2 , ⋯ , n ) , HM ( x ) ( k 1 , k 2 , ⋯ k n ) = k ;
(ii) 
When k i ≤ π i ( i = 1 , 2 , ⋯ , n ) , HM ( x ) ( k 1 , k 2 , ⋯ k n ) ≤ HM ( x ) ( π 1 , π 2 , ⋯ π n ) ;
(iii) 
When min { k i } ≤ HM ( x ) ( k 1 , k 2 , ⋯ k n ) ≤ max { k i } .
Two particular cases of the HM operator are given as follows.
(i) 
When x = 1 , HM ( 1 ) ( a 1 , a 2 , ⋯ a n ) = 1 n ∑ i = 1 n k i , it becomes the arithmetic mean operator.
(ii) 
When x = k , HM ( k ) ( k 1 , k 2 , ⋯ k k ) = ( ∏ i = 1 n k i ) 1 n , it becomes the geometric mean operator.

2.3. Dombi T-Conorm and T-Norm

Dombi operations involve the Dombi product and Dombi sum, which are special cases of T-norms and T-conorms, respectively.
Definition 7.
Suppose M = { 〈 x , μ M ( x ) , ν M ( x ) 〉 } and N = { 〈 x , μ N ( x ) , ν N ( x ) 〉 } are any two IFNs, then the generalized intersection and generalized union are proposed as follows [54]:
M ∩ T , T * N = { 〈 x , T ( μ M ( x ) , μ N ( x ) ) , T * ( ν M ( x ) , ν N ( x ) ) 〉 x ∈ X }  
M ∪ T , T * N = { 〈 x , T * ( μ M ( x ) , μ N ( x ) ) , T ( ν M ( x ) , ν N ( x ) ) 〉 x ∈ X }  
where T denotes a T-norm and T * denotes a T-conorm.
Dombi proposed a generator to produce Dombi T-norm and T-conorm which are shown as follows.
T D , λ ( x , y ) = 1 1 + ( ( 1 − x x ) λ + ( 1 − y y ) λ ) 1 λ  
T D , λ * ( x , y ) = 1 − 1 1 + ( ( x 1 − x ) λ + ( y 1 − y ) λ ) 1 λ  
where λ > 0 , x , y ∈ [ 0 , 1 ] .
Based on the Dombi T-norm and T-conorm, we can give the operational rules of IFNs as follows. Suppose k 1 = ( μ 1 , ν 1 ) and k 2 = ( μ 2 , ν 2 ) are any two IFNs, then operational laws of IFNs based on the Dombi T-norm and T-conorm can be defined as follows ( λ > 0 ) :
  • k 1 ⊕ k 2 = ( 1 − 1 1 + ( ( μ 1 1 − μ 1 ) λ + ( μ 2 1 − μ 2 ) λ ) 1 λ , 1 1 + ( ( 1 − ν 1 ν 1 ) λ + ( 1 − ν 2 ν 2 ) λ ) 1 λ )
  • k 1 ⊗ k 2 = ( 1 1 + ( ( 1 − μ 1 μ 1 ) λ + ( 1 − μ 2 μ 2 ) λ ) 1 λ , 1 − 1 1 + ( ( ν 1 1 − ν 1 ) λ + ( ν 2 1 − ν 2 ) λ ) 1 λ )
  • n k 1 = ( 1 − 1 1 + ( n ( μ 1 1 − μ 1 ) λ ) 1 λ , 1 1 + ( n ( 1 − ν 1 ν 1 ) λ ) 1 λ )   ( n > 0 )
  • k 1 n = ( 1 1 + ( n ( 1 − μ 1 μ 1 ) λ ) 1 λ , 1 − 1 1 + ( n ( ν 1 1 − ν 1 ) λ ) 1 λ )   ( n > 0 )
Example 4.
Suppose that k 1 = ( 0.6 , 0.1 ) , k 2 = ( 0.7 , 0.3 ) , and λ = 2 , n = 3 , then we have
(1) 
k 1 ⊕ k 2 = ( 1 − 1 1 + ( ( 0.6 1 − 0.6 ) 2 + ( 0.7 1 − 0.7 ) 2 ) 1 2 , 1 1 + ( ( 1 − 0.1 0.1 ) 2 + ( 1 − 0.3 0.3 ) 2 ) 1 2 ) = ( 0.7350 , 0.0384 )
(2) 
k 1 ⊗ k 2 = ( 1 1 + ( ( 1 − 0.6 0.6 ) 2 + ( 1 − 0.7 0.7 ) 2 ) 1 2 , 1 − 1 1 + ( ( 0.1 1 − 0.1 ) 2 + ( 0.3 1 − 0.3 ) 2 ) 1 2 ) = ( 0.5579 , 0.3069 )
(3) 
n k 1 = ( 1 − 1 1 + ( 3 × ( 0.6 1 − 0.6 ) 2 ) 1 2 , 1 1 + ( 3 × ( 1 − 0.1 0.1 ) 2 ) 1 2 ) = ( 0.7221 , 0.0603 )
(4) 
k 1 n = ( 1 1 + ( 3 × ( 1 − 0.6 0.6 ) 2 ) 1 2 , 1 − 1 1 + ( 3 × ( 0.1 1 − 0.1 ) 2 ) 1 2 ) = ( 0.4641 , 0.1613 )

3. Intuition Fuzzy Hamy Mean Operators Based on Dombi T-Norm and T-Conorm

In this section, we propose the intuitionistic fuzzy Dombi Hamy mean (IFDHM) operator and intuitionistic fuzzy weighted Dombi Hamy mean (IFWDHM) operator.

3.1. The IFDHM Operator

Definition 8.
Let k i = ( μ i j , ν i j ) ( i = 1 , 2 , ⋯ , n ) be a collection of IFNs, then we can define IFDHM operator as follows:
IFDHM ( x ) ( k 1 , k 2 , ⋯ k n ) = 1 C n x ( ⊕ 1 ≤ i 1 < ⋯ < i x ≤ n ( ⊗ j = 1 x k i j ) 1 x )  
where x is a parameter and x = 1 , 2 , ⋯ , n , i 1 , i 2 , ⋯ i x , are x integer values taken from the set { 1 , 2 , ⋯ , n } of n integer values, C n x denotes the binomial coefficient and C n x = n ! x ! ( n − x ) ! .
Theorem 1.
Let k i = ( μ i j , ν i j ) ( i = 1 , 2 , ⋯ , n ) be a collection of IFNs, then the aggregate result of Definition 8 is still an IFN, and have
IFDHM ( x ) ( k 1 , k 2 , ⋯ k n ) = 1 C n x ( ⊕ 1 ≤ i 1 < ⋯ < i x ≤ n ( ⊗ j = 1 x k i j ) 1 x ) = ( 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ ,   1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ )
Proof. 
  • First of all, we prove (10) is kept. According to the operational laws of IFNs, we have
    ⊗ j = 1 x k i j = ( 1 1 + ( ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ , 1 − 1 1 + ( ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ )  
    ( ⊗ j = 1 x k i j ) 1 x = ( 1 1 + ( 1 x ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ , 1 − 1 1 + ( 1 x ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ )  
    Moreover,
    ⊕ 1 ≤ i 1 < ⋯ < i x ≤ n ( ⊗ j = 1 x k i j ) 1 x = ( 1 − 1 1 + ( ∑ 1 ≤ i 1 < ⋯ < i x ≤ n x ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ , 1 1 + ( ∑ 1 ≤ i 1 < ⋯ < i x ≤ n x ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ )
    Furthermore,
    IFDHM ( x ) ( k 1 , k 2 , ⋯ k n ) = 1 C n x ( ⊕ 1 ≤ i 1 < ⋯ < i x ≤ n ( ⊗ j = 1 x k i j ) 1 x ) = ( 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ , 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n x ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ )
  • Next, we prove (10) is an IFN.
    Let
    a = 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ , b = 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n x ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ  
    Then we need to prove that the following two conditions which are satisfied,
    (i) 0 ≤ a ≤ 1 , 0 ≤ b ≤ 1 ;
    (ii) 0 ≤ a + b ≤ 1 .
    (i) Since μ i j ∈ [ 0 , 1 ] , we can get
    1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ ≥ 1 ⇒ 1 / ( 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ ) ∈ [ 0 , 1 ] ⇒ 1 − 1 / ( 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ ) ∈ [ 0 , 1 ]
    Therefore, 0 ≤ a ≤ 1 . Similarly, 0 ≤ b ≤ 1 .
    (ii) Obviously, 0 ≤ a + b ≤ 1 , then
    1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ + 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n x ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ  
    ≤ 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( ν i j ν i j ) λ ) 1 λ + 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n x ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ = 1  
We get 0 ≤ a + b ≤ 1 , so the aggregated result of Definition 8 is still an IFN. Next we will discuss about some of the properties of the IFDHM operator. □
Property 1 (Idempotency).
If k i ( 1 , 2 ⋯ , n ) and k are IFNs, and k i = k = ( μ i , ν i ) for all i = 1 , 2 ⋯ , n , then we get
IFDHM ( x ) ( k 1 , k 2 , ⋯ k n ) = k  
Proof. 
Since k = ( μ , ν ) , based on Theorem 1, we have
IFDHM ( x ) ( k 1 , k 2 , ⋯ k n ) = 1 C n x ( ⊕ 1 ≤ i 1 < ⋯ < i x ≤ n ( ⊗ j = 1 x k i j ) 1 x ) = ( 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ , 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ )
= ( 1 − 1 1 + ( 1 C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ( 1 − μ i μ i ) λ ) 1 λ , 1 1 + ( 1 C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ( ν i 1 − ν i ) λ ) 1 λ ) = ( 1 − 1 1 + ( 1 ( 1 − μ i μ i ) λ ) 1 λ , 1 1 + ( 1 ( ν i 1 − ν i ) λ ) 1 λ ) = ( 1 − 1 1 + 1 1 − μ i μ i , 1 1 + 1 ν i 1 − ν i ) = ( μ i , ν i ) = ( μ , ν ) = k
 □
Property 2 (Monotonicity).
Let k i = ( μ i j , ν i j ) , π i = ( μ θ j , ν θ j ) ( i = 1 , 2 , ⋯ , n ) be two sets of IFNs. If μ i j ≥ μ θ j , ν i j ≤ ν θ j , for all j , then
IFDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) ≥ IFDHM ( x ) ( π 1 , π 2 , ⋯ , π n )  
Proof. 
Since x ≥ 1 , μ i j ≥ μ θ j ≥ 0 , ν θ j ≥ ν i j ≥ 0 , then
∑ j = 1 x ( 1 − μ i j μ i j ) λ ≤ ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ ⇒ 1 / ∑ j = 1 x ( 1 − μ i j μ i j ) λ ≥ 1 / ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ  
⇒ x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 / ∑ j = 1 x ( 1 − μ i j μ i j ) λ ≥ x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 / ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ  
⇒ 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ ≥ 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ ) 1 λ ⇒ 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ ≤ 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ ) 1 λ ⇒ 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ ≥ 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ ) 1 λ
Similarly, we have
∑ j = 1 x ( ν i j 1 − ν i j ) λ ≤ ∑ j = 1 x ( ν θ j 1 − ν θ j ) λ ⇒ 1 / ∑ j = 1 x ( ν i j 1 − ν i j ) λ ≥ 1 / ∑ j = 1 x ( ν θ j 1 − ν θ j ) λ  
⇒ x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 / ∑ j = 1 x ( ν i j 1 − ν i j ) λ ≥ x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 / ∑ j = 1 x ( ν θ j 1 − ν θ j ) λ  
⇒ 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ ≥ 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( ν θ j 1 − ν θ j ) λ ) 1 λ  
⇒ 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ ≤ 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( ν θ j 1 − ν θ j ) λ ) 1 λ  
Let k = IFDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) , π = IFDHM ( x ) ( π 1 , π 2 , ⋯ , π n ) and S ( k ) , S ( π ) be the score values of k and π respectively. Based on the score value of IFN in (2) and the above inequality, we can imply that S ( k ) ≥ S ( π ) , and then we discuss the following cases:
(1)
If S ( k ) > S ( π ) , then we can get IFDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) > IFDHM ( x ) ( π 1 , π 2 , ⋯ , π n ) .
(2)
If S ( k ) = S ( π ) , then
1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ + 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ = 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ ) 1 λ + 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( ν θ j 1 − ν θ j ) λ ) 1 λ
Since μ i j ≥ μ θ j ≥ 0 , ν θ j ≥ ν i j ≥ 0 , we can deduce that
1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ = 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ ) 1 λ  
and
1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ = 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( ν θ j 1 − ν θ j ) λ ) 1 λ  
Therefore, it follows that
H ( k ) = 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ + 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ = 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ ) 1 λ + 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( ν θ j 1 − ν θ j ) λ ) 1 λ = H ( π )
IFDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) = IFDHM ( x ) ( π 1 , π 2 , ⋯ , π n )
 □
Property 3 (Boundedness).
Let k i = ( μ i j , ν i j ) , k + = ( μ max i j , ν max i j ) ( i = 1 , 2 , ⋯ , k ) be a set of IFNs, and k − = ( μ min i j , ν min i j ) then
k − < IFDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) < k +  
Proof. 
Based on Properties 1 and 2, we have
IFDHM ( x ) ( k 1 , k 2 , ⋯ , k k ) ≥ IFDHM ( x ) ( k − , k − , ⋯ , k − ) = k − , IFDHM ( x ) ( k 1 , k 2 , ⋯ , k k ) ≤ IFDHM ( x ) ( k + , k + , ⋯ , k + ) = k + .
Then we have k − < IFDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) < k + . □
Property 4 (Commutativity).
Let k i = ( μ i j , ν i j ) , π i = ( μ θ j , ν θ j ) ( i = 1 , 2 , ⋯ , n ) be two sets of IFNs. Suppose ( π 1 , π 2 , ⋯ , π n ) is any permutation of ( k 1 , k 2 , ⋯ , k n ) , then
IFDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) = IFDHM ( x ) ( π 1 , π 2 , ⋯ , π n )  
Proof. 
Because ( π 1 , π 2 , ⋯ , π n ) is any permutation of ( k 1 , k 2 , ⋯ , k n ) , then 1 C n x ( ⊕ 1 ≤ i 1 < ⋯ < i x ≤ n ( ⊗ j = 1 x k i j ) 1 x ) = 1 C n x ( ⊕ 1 ≤ i 1 < ⋯ < i x ≤ n ( ⊗ j = 1 x π i j ) 1 x ) , thus
IFDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) = IFDHM ( x ) ( π 1 , π 2 , ⋯ , π n ) .
 □
Example 5.
Let k 1 = ( 0.6 , 0.3 ) , k 2 = ( 0.5 , 0.1 ) , k 3 = ( 0.7 , 0.2 ) , k 4 = ( 0.8 , 0.1 ) be four IFNs. Then we use the proposed IFDHM operator to aggregate four IFNs (suppose x = 2 , λ = 2 ).
Let
k = IFDHM ( 2 ) ( k 1 , k 2 , ⋯ , k n ) = ( 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ , 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ ) = ( 1 − 1 1 + ( 1 3 × ( 1 ( 1 − 0.6 0.6 ) 2 + ( 1 − 0.5 0.5 ) 2 + 1 ( 1 − 0.6 0.6 ) 2 + ( 1 − 0.7 0.7 ) 2 + 1 ( 1 − 0.6 0.6 ) 2 + ( 1 − 0.8 0.8 ) 2 + 1 ( 1 − 0.5 0.5 ) 2 + ( 1 − 0.7 0.7 ) 2 + 1 ( 1 − 0.5 0.5 ) 2 + ( 1 − 0.8 0.8 ) 2 + 1 ( 1 − 0.7 0.7 ) 2 + ( 1 − 0.8 0.8 ) 2 ) ) 1 2 , 1 1 + ( 1 3 × ( 1 ( 0.3 1 − 0.3 ) 2 + ( 0.1 1 − 0.1 ) 2 + 1 ( 0.3 1 − 0.3 ) 2 + ( 0.2 1 − 0.2 ) 2 + 1 ( 0.3 1 − 0.3 ) 2 + ( 0.1 1 − 0.1 ) 2 + 1 ( 0.1 1 − 0.1 ) 2 + ( 0.2 1 − 0.2 ) 2 + 1 ( 0.1 1 − 0.1 ) 2 + ( 0.1 1 − 0.1 ) 2 + 1 ( 0.2 1 − 0.2 ) 2 + ( 0.1 1 − 0.1 ) 2 ) ) 1 2 = ( 0.5831 , 0.3355 ) )
At last, we get IFDHM ( 2 ) ( k 1 , k 2 , k 3 , k 4 ) = ( 0.5831 , 0.3355 ) .

3.2. The IFWDHM Operator

The weights of attributes play an important role in practical decision making, and they can influence the decision result. Therefore, it is necessary to consider attribute weights in aggregating information. It is obvious that the IFWDHM operator fails to consider the problem of attribute weights. In order to overcome this defect, we propose the IFWDHM operator.
Definition 9.
Let k i = ( μ i j , ν i j ) ( i = 1 , 2 , ⋯ , n ) be a group of IFNs, ω = ( ω 1 , ω 2 , ⋯ ω n ) T be the weight vector for k i   ( i = 1 , 2 , ⋯ , n ) , which satisfies ω i ∈ [ 0.1 ] and ∑ i = 1 n ω i = 1 , then we can define IFWDHM operator as follows:
IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = { ⊕ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) ( ⊗ j = 1 x k i j ) 1 x C n − 1 x ( 1 ≤ x < n ) ⊗ i = 1 x k i 1 − ω i n − 1 ( x = n )  
Theorem 2.
Let k i = ( μ i j , ν i j ) ( i = 1 , 2 , ⋯ , n ) be a group of IFNs, and their weight vector meet ω i ∈ [ 0.1 ] and ∑ i = 1 n ω i = 1 then the result from Definition 9 is still an IFN, and have
IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = ⊕ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) ( ⊗ j = 1 x k i j ) 1 x C n − 1 x = ( 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ , 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ )   ( 1 ≤ x < n )
or
IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = ⊗ i = 1 x k i 1 − ω i n − 1 = ( 1 1 + ( ∑ i = 1 x ( 1 − ω i n − 1 ) ( 1 − μ i μ i ) λ ) 1 λ , 1 − 1 1 + ( ∑ i = 1 x ( 1 − ω i n − 1 ) ( ν i 1 − ν i ) λ ) 1 λ )   ( x = k )
Proof. 
(1) First of all, we prove that (20) and (21) are kept. For the first case, when ( 1 ≤ x < n ) , according to the operational laws of IFNs, we get
⊗ j = 1 x k i j = ( 1 1 + ( ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ , 1 − 1 1 + ( ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ )  
( ⊗ j = 1 x k i j ) 1 x = ( 1 1 + ( 1 x ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ , 1 − 1 1 + ( 1 x ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ )  
Thereafter,
( 1 − ∑ j = 1 x ω i j ) ( ⊗ j = 1 x k i j ) 1 x = ( 1 − 1 1 + ( ( 1 − ∑ j = 1 x ω i j ) x ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ , 1 1 + ( ( 1 − ∑ j = 1 x ω i j ) x ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ )  
Moreover,
⊕ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) ( ⊗ j = 1 x k i j ) 1 x = ( 1 − 1 1 + ( ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) x ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ , 1 1 + ( ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) x ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ )  
Therefore,
1 C n − 1 x ⊕ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) ( ⊗ j = 1 x k i j ) 1 x = ( 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) x ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ , 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) x ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ )
For the second case, when ( x = n ) , we get
k i 1 − ω i n − 1 = ( 1 1 + ( ( 1 − ω i n − 1 ) ( 1 − μ i μ i ) λ ) 1 λ , 1 − 1 1 + ( ( 1 − ω i n − 1 ) ( ν i 1 − ν i ) λ ) 1 λ )  
Then,
⊗ i = 1 x k i 1 − ω i n − 1 = ( 1 1 + ( ∑ i = 1 x ( 1 − ω i n − 1 ) ( 1 − μ i μ i ) λ ) 1 λ , 1 − 1 1 + ( ∑ i = 1 x ( 1 − ω i n − 1 ) ( ν i 1 − ν i ) λ ) 1 λ )  
(2) Next, we prove the (20) and (21) are IFNs. For the first case, when 1 ≤ x < n ,
Let
a = 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) x ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ  
b = 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) x ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ  
Then we need prove the following two conditions.
(I) 0 ≤ a ≤ 1 , 0 ≤ b ≤ 1 . (II) 0 ≤ a + b ≤ 1 .
(I) Since a ∈ [ 0 , 1 ] , we can get
1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ ≥ 1  
⇒ 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ ∈ [ 0 , 1 ]  
⇒ 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ ∈ [ 0 , 1 ]  
Therefore, 0 ≤ a ≤ 1 . Similarly, we can get
1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ ∈ [ 0 , 1 ]  
Therefore, 0 ≤ b ≤ 1 .
(II) Since 0 ≤ a + b ≤ 1 , we can get the following inequality.
1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ + 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ ≤ 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ + 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ = 1
For the second case, when x = n , we can easily prove that it is kept. So the aggregation result produced by Definition 9 is still an IFN. Next, we shall deduce some desirable properties of IFWDHM operator. □
Property 5 (Idempotency).
If k i ( i = 1 , 2 , ⋯ , n ) are equal, i.e., k i = k = ( μ , ν ) , and weight vector meets ω i ∈ [ 0 , 1 ] and ∑ i = 1 k ω i = 1 then
IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = k  
Proof. 
Since k i = k = ( μ , ν ) , based on Theorem 2, we get
(1) For the first case, when 1 ≤ x < n .
IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = ( 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ , 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ ) = ( 1 − 1 1 + ( 1 C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ( 1 − μ μ ) λ ) 1 λ , 1 1 + ( 1 C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ( ν 1 − ν ) λ ) 1 λ ) = ( 1 − 1 1 + ( 1 C n − 1 x ( C n x − ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( ∑ i = 1 x ω i ) ) 1 ( 1 − μ μ ) λ ) 1 λ , 1 1 + ( 1 C n − 1 x ( C n x − ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( ∑ i = 1 x ω i ) ) 1 ( ν 1 − ν ) λ ) 1 λ ) = ( 1 − 1 1 + ( 1 C n − 1 x ( C n x − C n − 1 x − 1 ∑ i = 1 x ω i ) 1 ( 1 − μ μ ) λ ) 1 λ , 1 1 + ( 1 C n − 1 x ( C n x − C n − 1 x − 1 ∑ i = 1 x ω i ) 1 ( ν 1 − ν ) λ ) 1 λ )
Since ∑ i = 1 k ω i = 1 , we can get
IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = ( 1 − 1 1 + ( 1 C n − 1 x ( C n x − C n − 1 x − 1 ) 1 ( 1 − μ μ ) λ ) 1 λ , 1 1 + ( 1 C n − 1 x ( C n x − C n − 1 x − 1 ) 1 ( ν 1 − ν ) λ ) 1 λ ) = ( 1 − 1 1 + ( 1 ( n − 1 ) ! x ! ( k − 1 − x ) ! ( n − 1 ) ! x ! ( k − 1 − x ) ! 1 ( 1 − μ μ ) λ ) 1 λ , 1 1 + ( 1 ( n − 1 ) ! x ! ( k − 1 − x ) ! ( n − 1 ) ! x ! ( k − 1 − x ) ! 1 ( ν 1 − ν ) λ ) 1 λ ) = ( 1 − 1 1 + ( 1 ( 1 − μ μ ) λ ) 1 λ , 1 1 + ( 1 ( ν 1 − ν ) λ ) 1 λ ) = ( μ , ν ) = k
(2) For the second case, when x = n ,
IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = ( 1 1 + ( ∑ i = 1 x ( 1 − ω i n − 1 ) ( 1 − μ i μ i ) λ ) 1 λ , 1 − 1 1 + ( ∑ i = 1 x ( 1 − ω i n − 1 ) ( ν i 1 − ν i ) λ ) 1 λ ) = ( 1 1 + ( n − 1 n − 1 ( 1 − μ μ ) λ ) 1 λ , 1 − 1 1 + ( n − 1 n − 1 ( ν 1 − ν ) λ ) 1 λ ) = ( μ , ν ) = k
which proves the idempotency property of the IFWDHM operator. □
Property 6 (Monotonicity).
Let k i = ( μ i j , ν i j ) , π i = ( μ θ j , ν θ j ) ( i = 1 , 2 , ⋯ , n ) be two sets of IFNs. If μ i j ≥ μ θ j , ν i j ≤ ν θ j for all j , and weight vector meets ω i ∈ [ 0 , 1 ] and ∑ i = 1 k ω i = 1 , the k and π are equal, then we have
IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) ≥ IFWDHM ω ( x ) ( π 1 , π 2 , ⋯ , π n )  
Proof. 
Since x ≥ 1 , μ i j ≥ μ θ j ≥ 0 , ν θ j ≥ ν i j ≥ 0 , then
∑ j = 1 x ( 1 − μ i j μ i j ) λ ≤ ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ ⇒ 1 / ∑ j = 1 x ( 1 − μ i j μ i j ) λ ≥ 1 / ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ ⇒ ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( 1 − μ i j μ i j ) λ ≥ ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ ⇒ ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( 1 − μ i j μ i j ) λ ≥ ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ ⇒ x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( 1 − μ i j μ i j ) λ ≥ x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ ⇒ 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ ≥ 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ ) 1 λ ⇒ 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ ≤ 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ ) 1 λ ⇒ 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ ≥ 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ ) 1 λ
Similarly, we have
∑ j = 1 x ( ν i j 1 − ν i j ) λ ≤ ∑ j = 1 x ( ν θ j 1 − ν θ j ) λ ⇒ 1 / ∑ j = 1 x ( ν i j 1 − ν i j ) λ ≥ 1 / ∑ j = 1 x ( ν θ j 1 − ν θ j ) λ ⇒ ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( ν i j 1 − ν i j ) λ ≥ ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( ν θ j 1 − ν θ j ) λ . ⇒ x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( ν i j 1 − ν i j ) λ ≥ x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( ν θ j 1 − ν θ j ) λ ⇒ 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ ≥ 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( ν θ j 1 − ν θ j ) λ ) 1 λ 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ ≤ 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( ν θ j 1 − ν θ j ) λ ) 1 λ
Let a = IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) , π = IFWDHM ω ( x ) ( π 1 , π 2 , ⋯ , π n ) and S ( k ) , S ( π ) be the score values of a and π respectively. Based on the score value of IFN in (2) and the above inequality, we can imply that S ( k ) ≥ S ( π ) , and then we discuss the following cases:
(1) If S ( k ) > S ( π ) , then we can get
IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) > IFWDHM ω ( x ) ( π 1 , π 2 , ⋯ , π n )  
(2) If S ( k ) = S ( π ) , then
1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ = 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ ) 1 λ − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( ν θ j 1 − ν θ j ) λ ) 1 λ  
Since μ i j ≥ μ θ j ≥ 0 , ν θ j ≥ ν i j ≥ 0 , and based on the Equations (2) and (3), we can deduce that
  1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ = 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − μ θ j μ θ j ) λ ) 1 λ  
and
1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ = 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( ν θ j 1 − ν θ j ) λ ) 1 λ  
Therefore, it follows that H ( k ) = H ( π ) , the IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = IFWDHM ω ( x ) ( π 1 , π 2 , ⋯ , π n ) , When x = n , we can prove it in a similar way. □
Property 7 (Boundedness).
Let k i = ( μ i j , ν i j ) , k + = ( μ max   i j , ν max   i j ) ( i = 1 , 2 , ⋯ , n ) be a set of IFNs, and k − = ( μ min   i j , ν min   i j ) , and weight vector meets ω i ∈ [ 0 , 1 ] and ∑ i = 1 n ω i = 1 then
k − ≤ IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) ≤ k +  
Proof. 
Based on Properties 5 and 6, we have
IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) ≥ IFWDHM ω ( x ) ( k − , k − , ⋯ , k − ) = k − IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) ≤ IFWDHM ω ( x ) ( k + , k + , ⋯ , k + ) = k +
Then we have k − ≤ IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) ≤ k + . □
Property 8 (Commutativity).
Let k i = ( μ i j , ν i j ) , π i = ( μ θ j , ν θ j ) ( i = 1 , 2 , ⋯ , n ) be two sets of IFNs. Suppose ( π 1 , π 2 , ⋯ , π n ) is any permutation of ( k 1 , k 2 , ⋯ , k n ) , and weight vector meets ω i ∈ [ 0 , 1 ] and ∑ i = 1 k ω i = 1 , the k and π are equal, then we have
IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = IFWDHM ω ( x ) ( π 1 , π 2 , ⋯ , π n )  
Proof. 
Because ( π 1 , π 2 , ⋯ , π n ) is any permutation of ( k 1 , k 2 , ⋯ , k n ) , then
⊕ 1 ≤ i 1 < ⋯ < i x ≤ k ( 1 − ∑ j = 1 x ω i j ) ( ⊗ j = 1 x k i j ) 1 x C k − 1 x = ⊕ 1 ≤ i 1 < ⋯ < i x ≤ k ( 1 − ∑ j = 1 x ω i j ) ( ⊗ j = 1 x π i j ) 1 x C k − 1 x ( 1 ≤ x < k ) ⊗ i = 1 x k i 1 − ω i k − 1 = ⊗ i = 1 x π i 1 − ω i k − 1 ( x = k )
Thus, IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = IFWDHM ω ( x ) ( π 1 , π 2 , ⋯ , π n ) .  □
Example 6.
Let k 1 = ( 0.8 , 0.2 ) , k 2 = ( 0.6 , 0.1 ) , k 3 = ( 0.7 , 0.3 ) , k 4 = ( 0.4 , 0.2 ) be four IFNs, the weighting vector of attributes is ω = { 0.2 , 0.3 , 0.4 , 0.1 } . Then we use the proposed IFWDHM operator to aggregate four IFNs (suppose x = 2 , λ = 2 ).
Let
IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = ⊕ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) ( ⊗ j = 1 x k i j ) 1 x C n − 1 x = ( 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ , 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ ) = ( 1 − 1 1 + ( 2 C 4 − 1 2 ∑ 1 ≤ i 1 < ⋯ < i 2 ≤ 4 ( 1 − ∑ j = 1 2 ω i j ) 1 ∑ j = 1 2 ( 1 − μ i j μ i j ) 2 ) 1 2 , 1 1 + ( 2 C 4 − 1 2 ∑ 1 ≤ i 1 < ⋯ < i 2 ≤ 4 ( 1 − ∑ j = 1 2 ω i j ) 1 ∑ j = 1 2 ( ν i j 1 − ν i j ) 2 ) 1 2 )   = ( 1 − 1 1 + ( 2 3 ( 1 − 0.2 − 0.3 ( 1 − 0.8 0.8 ) 2 + ( 1 − 0.6 0.6 ) 2 + 1 − 0.2 − 0.4 ( 1 − 0.8 0.8 ) 2 + ( 1 − 0.7 0.7 ) 2 + 1 − 0.2 − 0.1 ( 1 − 0.8 0.8 ) 2 + ( 1 − 0.4 0.4 ) 2 + 1 − 0.3 − 0.4 ( 1 − 0.6 0.6 ) 2 + ( 1 − 0.7 0.7 ) 2 + 1 − 0.3 − 0.1 ( 1 − 0.6 0.6 ) 2 + ( 1 − 0.4 0.4 ) 2 + 1 − 0.4 − 0.1 ( 1 − 0.7 0.7 ) 2 + ( 1 − 0.4 0.4 ) 2 ) ) 1 2 , 1 − 1 1 + ( 2 3 ( 1 − 0.2 − 0.3 ( 0.2 1 − 0.2 ) 2 + ( 0.1 1 − 0.1 ) 2 + 1 − 0.2 − 0.4 ( 0.2 1 − 0.2 ) 2 + ( 0.3 1 − 0.3 ) 2 + 1 − 0.2 − 0.1 ( 0.2 1 − 0.2 ) 2 + ( 0.2 1 − 0.2 ) 2 + 1 − 0.3 − 0.4 ( 0.1 1 − 0.1 ) 2 + ( 0.3 1 − 0.3 ) 2 + 1 − 0.3 − 0.1 ( 0.1 1 − 0.1 ) 2 + ( 0.2 1 − 0.2 ) 2 + 1 − 0.4 − 0.1 ( 0.3 1 − 0.3 ) 2 + ( 0.2 1 − 0.2 ) 2 ) ) 1 2 ) = ( 0.5302 , 0.1952 )
At last, we get IFWDHM ω ( 2 ) ( k 1 , k 2 , k 3 , k 4 ) = ( 0.5302 , 0.1952 ) .

3.3. The IFDDHM Operator

Wu et al. [59] proposed the dual Hamy mean (DHM) operator.
Definition 10.
The DHM operator is defined as follows [59]:
DHM ( x ) ( k 1 , k 2 , ⋯ , k n ) = ( ∏ 1 ≤ i 1 < … < i x ≤ n ( ∑ j = 1 x k i j x ) ) 1 C n x  
where x is a parameter and x = 1 , 2 , … , n , i 1 , i 2 , … , i n are x integer values taken from the set { 1 , 2 , … , n } of n integer values, C n x denotes the binomial coefficient and C k x = k ! x ! ( k − x ) ! .
In this section, we will propose the intuitionistic fuzzy Dombi dual Hamy mean DHM (IFDDHM) operator.
Definition 11.
Let k i = ( μ i j , ν i j ) ( i = 1 , 2 , ⋯ , n ) be a collection of IFNs, then we can define IFDDHM operator as follows:
IFDDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) = ( ⊗ 1 ≤ i 1 < ⋯ < i x ≤ n ( ⊕ j = 1 x k i j x ) ) 1 C n x  
where x is a parameter and x = 1 , 2 , … , n , i 1 , i 2 , … , i n are x integer values taken from the set { 1 , 2 , … , n } of n integer values, C n x denotes the binomial coefficient and C k x = k ! x ! ( k − x ) ! .
Theorem 3.
Let k i = ( μ i , ν i ) ( i = 1 , 2 , ⋯ , n ) be a collection of the IFNs, then the aggregate result of Definition 10 is still an IFNs, and have
IFDDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) = ( ⊗ 1 ≤ i 1 < ⋯ < i x ≤ n ( ⊕ j = 1 x k i j x ) ) 1 C n x = ( 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ , 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ )
Proof. 
(1) First of all, we prove (42) is kept. According to the operational laws of IFNs, we get
⊕ j = 1 x k i j = ( 1 − 1 1 + ( ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ , 1 1 + ( ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ )  
⊕ j = 1 x k i j x = ( 1 − 1 1 + ( 1 x ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ , 1 1 + ( 1 x ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ )  
Moreover,
⊗ 1 ≤ i 1 < ⋯ < i x ≤ n ( ⊕ j = 1 x k i j x ) = ( 1 1 + ( ∑ 1 ≤ i 1 < ⋯ < i x ≤ n x ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ , 1 − 1 1 + ( ∑ 1 ≤ i 1 < ⋯ < i x ≤ n x ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ )
Furthermore,
IFDDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) = ( ⊗ 1 ≤ i 1 < ⋯ < i x ≤ n ( ⊕ j = 1 x k i j x ) ) 1 C n x = ( 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ , 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ )
(2) Next, we prove (42) is an IFN.
Let
a = 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n x ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ , b = 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ  
Then we need to prove that the following two conditions which are satisfied.
(i) 0 ≤ a ≤ 1 , 0 ≤ b ≤ 1 ;
(ii) 0 ≤ a + b ≤ 1 .
(i) Since μ i j ∈ [ 0 , 1 ] , we can get
1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n x ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ ≥ 1 ⇒ 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n x ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ ∈ [ 0 , 1 ]  
Therefore, 0 ≤ a ≤ 1 . Similarly, we can get 0 ≤ b ≤ 1 .
(ii) Obviously, 0 ≤ a + b ≤ 1 , then
1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n x ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ + 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ ≤ 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n x ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ + 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ = 1
We get 0 ≤ a + b ≤ 1 . so the aggregated result of Definition 10 is still an IFN. Next we will discuss some properties of IFDDHM operator. □
Property 9 (Idempotency).
If k i ( 1 , 2 ⋯ , n ) and k are IFNs, and k i = k = ( μ i , ν i ) for all i = 1 , 2 ⋯ , n , then we get
IFDDHM ( x ) ( k 1 , k 2 , ⋯ k n ) = k  
Proof. 
Since k = ( μ , ν ) , based on Theorem 3, we have
IFDDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) = ( ⊗ 1 ≤ i 1 < ⋯ < i x ≤ n ( ⊕ j = 1 x k i j x ) ) 1 C n x = ( 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ , 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ ) = ( 1 1 + ( 1 C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ( μ i j 1 − μ i j ) λ ) 1 λ , 1 − 1 1 + ( 1 C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ( 1 − ν i j ν i j ) λ ) 1 λ ) = ( 1 1 + ( 1 ( μ i 1 − μ i ) λ ) 1 λ , 1 − 1 1 + ( 1 ( 1 − ν i ν i ) λ ) 1 λ ) = ( 1 1 + 1 μ i 1 − μ i , 1 − 1 1 + 1 1 − ν i ν i ) = ( μ i , ν i ) = ( μ , ν ) = k
 □
Property 10 (Monotonicity).
Let k i = ( μ i j , ν i j ) , π i = ( μ θ j , ν θ j ) ( i = 1 , 2 , ⋯ , n ) be two sets of IFNs. If μ i j ≥ μ θ j , ν i j ≤ ν θ j , for all j , then
IFDDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) ≥ IFDDHM ( x ) ( π 1 , π 2 , ⋯ , π n )  
Proof. 
Since x ≥ 1 , μ i j ≥ μ θ j ≥ 0 , ν θ j ≥ ν i j ≥ 0 , then
∑ j = 1 x ( μ i j 1 − μ i j ) λ ≤ ∑ j = 1 x ( μ θ j 1 − μ θ j ) λ ⇒ 1 / ∑ j = 1 x ( μ i j 1 − μ i j ) λ ≥ 1 / ∑ j = 1 x ( μ θ j 1 − μ θ j ) λ ⇒ x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 / ∑ j = 1 x ( μ i j 1 − μ i j ) λ ≥ x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 / ∑ j = 1 x ( μ θ j 1 − μ θ j ) λ ⇒ 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 / ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ ≥ 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 / ∑ j = 1 x ( μ θ j 1 − μ θ j ) λ ) 1 λ
Similarly, we have
∑ j = 1 x ( 1 − ν i j ν i j ) λ ≥ ∑ j = 1 x ( 1 − ν θ j ν θ j ) λ ⇒ 1 / ∑ j = 1 x ( 1 − ν i j ν i j ) λ ≤ 1 / ∑ j = 1 x ( 1 − ν θ j ν θ j ) λ ⇒ x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 / ∑ j = 1 x ( 1 − ν i j ν i j ) λ ≤ x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 / ∑ j = 1 x ( 1 − ν θ j ν θ j ) λ ⇒ 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 / ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ ≤ 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 / ∑ j = 1 x ( 1 − ν θ j ν θ j ) λ ) 1 λ ⇒ 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 / ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ ≥ 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 / ∑ j = 1 x ( 1 − ν θ j ν θ j ) λ ) 1 λ ⇒ 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 / ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ ≤ 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 / ∑ j = 1 x ( 1 − ν θ j ν θ j ) λ ) 1 λ
Let k = IFDDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) , π = IFDDHM ( x ) ( π 1 , π 2 , ⋯ , π n ) and S ( k ) , S ( π ) be the score values of k and π respectively. Based on the score value of IFN in (2) and the above inequality, we can imply that S ( k ) ≥ S ( π ) , and then we discuss the following cases:
(1) If S ( k ) > S ( π ) , then we get IFDDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) > IFDDHM ( x ) ( π 1 , π 2 , ⋯ , π n ) .
(2) If S ( k ) = S ( π ) , then
1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ + 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ = 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( μ θ j 1 − μ θ j ) λ ) 1 λ + 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − ν θ j ν θ j ) λ ) 1 λ
Since μ i j ≥ μ θ j ≥ 0 , ν θ j ≥ ν i j ≥ 0 , we can deduce that
1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ = 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( μ θ j 1 − μ θ j ) λ ) 1 λ  
and
1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ = 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − ν θ j ν θ j ) λ ) 1 λ  
Therefore, it follows that
H ( k ) = 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ + 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ = 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( μ θ j 1 − μ θ j ) λ ) 1 λ + 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − ν θ j ν θ j ) λ ) 1 λ = H ( π )
Then IFDDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) = IFDDHM ( x ) ( π 1 , π 2 , ⋯ , π n ) .  □
Property 11 (Boundedness).
Let k i = ( μ i j , ν i j ) , k + = ( μ max   i j , ν max   i j ) ( i = 1 , 2 , ⋯ , k ) be a set of IFNs, and k − = ( μ min   i j , ν min   i j ) then
k − < IFDDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) < k +  
Proof. 
Based on Properties 9 and 10, we have
IFDDHM ( x ) ( k 1 , k 2 , ⋯ , k k ) ≥ IFDDHM ( x ) ( k − , k − , ⋯ , k − ) = k − , IFDDHM ( x ) ( k 1 , k 2 , ⋯ , k k ) ≤ IFDDHM ( x ) ( k + , k + , ⋯ , k + ) = k + .
Then we have k − < IFDDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) < k + . □
Property 12 (Commutativity).
Let k i = ( μ i j , ν i j ) , π i = ( μ θ j , ν θ j ) ( i = 1 , 2 , ⋯ , n ) be two sets of IFNs. Suppose ( π 1 , π 2 , ⋯ , π n ) is any permutation of ( k 1 , k 2 , ⋯ , k n ) , then
IFDDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) = IFDDHM ( x ) ( π 1 , π 2 , ⋯ , π n )  
Proof. 
Because ( π 1 , π 2 , ⋯ , π n ) is any permutation of ( k 1 , k 2 , ⋯ , k n ) , then
( ⊗ 1 ≤ i 1 < ⋯ < i x ≤ n ( ⊕ j = 1 x k i j x ) ) 1 C n x = ( ⊗ 1 ≤ i 1 < ⋯ < i x ≤ n ( ⊕ j = 1 x π i j x ) ) 1 C n x  
thus
IFDDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) = IFDDHM ( x ) ( π 1 , π 2 , ⋯ , π n )
 □
Example 7.
Let k 1 = ( 0.7 , 0.3 ) , k 2 = ( 0.4 , 0.1 ) , k 3 = ( 0.5 , 0.2 ) , k 4 = ( 0.6 , 0.2 ) be four IFNs. Then we use the proposed IFDDHM operator to aggregate four IFNs (suppose x = 2 , λ = 2 ).
Let
IFDDHM ( x ) ( k 1 , k 2 , ⋯ , k n ) = ( ⊗ 1 ≤ i 1 < ⋯ < i x ≤ n ( ⊕ j = 1 x k i j x ) ) 1 C n x = ( 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ , 1 − 1 1 + ( x C n x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n 1 ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ ) = ( 1 1 + ( 2 C 4 2 ∑ 1 ≤ i 1 < ⋯ < i 2 ≤ 4 1 ∑ j = 1 4 ( μ i j 1 − μ i j ) 2 ) 1 2 , 1 − 1 1 + ( 2 C 4 2 ∑ 1 ≤ i 1 < ⋯ < i 4 ≤ 4 1 ∑ j = 1 4 ( 1 − ν i j ν i j ) 4 ) 1 2 ) = ( 1 1 + ( 2 6 × ( 1 ( 0.7 1 − 0.7 ) 2 + ( 0.4 1 − 0.4 ) 2 + 1 ( 0.7 1 − 0.7 ) 2 + ( 0.5 1 − 0.5 ) 2 + 1 ( 0.7 1 − 0.7 ) 2 + ( 0.6 1 − 0.6 ) 2 + 1 ( 0.4 1 − 0.4 ) 2 + ( 0.5 1 − 0.5 ) 2 + 1 ( 0.4 1 − 0.4 ) 2 + ( 0.6 1 − 0.6 ) 2 + 1 ( 0.5 1 − 0.5 ) 2 + ( 0.6 1 − 0.6 ) 2 ) ) 1 2 , 1 − 1 1 + ( 2 6 × ( 1 ( 1 − 0.3 0.3 ) 2 + ( 1 − 0.1 0.1 ) 2 + 1 ( 1 − 0.3 0.3 ) 2 + ( 1 − 0.2 0.2 ) 2 + 1 ( 1 − 0.3 0.3 ) 2 + ( 1 − 0.2 0.2 ) 2 + 1 ( 1 − 0.1 0.1 ) 2 + ( 1 − 0.2 0.2 ) 2 + 1 ( 1 − 0.1 0.1 ) 2 + ( 1 − 0.2 0.2 ) 2 + 1 ( 1 − 0.2 0.2 ) 2 + ( 1 − 0.2 0.2 ) 2 ) ) 1 2 ) = ( 0.5617 , 0.1596 )
At last, we get IFDDHM ( 4 ) ( k 1 , k 2 , k 3 , k 4 ) = ( 0.5617 , 0.1596 ) .

3.4. The IFWDDHM Operator

The weights of attributes play an important role in practical decision making, and they can influence the decision result. Therefore, it is necessary to consider attribute weights in aggregating information. It is obvious that the IFWDDHM operator fails to consider the problem of attribute weights. In order to overcome this defect, we propose the IFWDDHM operator.
Definition 12.
Let k i = ( μ i j , ν i j ) ( i = 1 , 2 , ⋯ , n ) be a group of IFNs, ω = ( ω 1 , ω 2 , ⋯ ω n ) T be the weight vector for ( i = 1 , 2 , ⋯ , n ) , which satisfies ω i ∈ [ 0.1 ] and ∑ i = 1 n ω i = 1 , then we can define IFWDDHM operator as follows:
IFWDDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = { ⊗ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) ( ⊕ j = 1 x k i j ) 1 x C n − 1 x ( 1 ≤ x < n ) ⊕ i = 1 x k i 1 − ω i n − 1 ( x = n )  
Theorem 4.
Let k i = ( μ i j , ν i j ) ( i = 1 , 2 , ⋯ , n ) be a group of IFNs, and their weight vector meet ω i ∈ [ 0 , 1 ] , then the result from Definition 12 is still an IFN, and has
IFWDDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = ⊗ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) ( ⊕ j = 1 x k i j ) 1 x C n − 1 x = ( 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) 1 ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ , 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) 1 ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ )   ( 1 ≤ x < n )
or
IFWDDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = ⊕ i = 1 x k i 1 − ω i n − 1 = ( 1 − 1 1 + ( ∑ i = 1 x ( 1 − ω i n − 1 ) ( μ i 1 − μ i ) λ ) 1 λ , 1 1 + ( ∑ i = 1 x ( 1 − ω i n − 1 ) ( 1 − ν i ν i ) λ ) 1 λ ) ( x = n )
Proof. 
⊕ j = 1 x k i j = ( 1 − 1 1 + ( ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ , 1 1 + ( ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ )  
( ⊕ j = 1 x k i j ) 1 x = ( 1 − 1 1 + ( 1 x ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ , 1 1 + ( 1 x ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ )  
Thereafter,
( 1 − ∑ j = 1 x ω i j ) ( ⊕ j = 1 x k i j ) 1 x = ( 1 1 + ( ( 1 − ∑ i = 1 x ω i ) x ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ , 1 − 1 1 + ( ( 1 − ∑ i = 1 x ω i ) x ∑ j = 1 x ( 1 − μ i j μ i j ) λ ) 1 λ )  
Moreover,
⊗ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) ( ⊕ j = 1 x k i j ) 1 x = ( 1 1 + ( ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) x ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ , 1 − 1 1 + ( ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) x ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ )  
Therefore,
1 C n − 1 x ⊗ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) ( ⊕ j = 1 x k i j ) 1 x = ( 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) x ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ , 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) x ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ )
For the second case, when ( x = n ) , we get
k i 1 − ω i n − 1 = ( 1 1 + ( ( 1 − ω i n − 1 ) ( 1 − μ i μ i ) λ ) 1 λ , 1 − 1 1 + ( ( 1 − ω i n − 1 ) ( ν i 1 − ν i ) λ ) 1 λ )  
Then,
⊕ i = 1 x k i 1 − ω i n − 1 = ( 1 − 1 1 + ( ∑ i = 1 x ( 1 − ω i n − 1 ) ( μ i 1 − μ i ) λ ) 1 λ , 1 1 + ( ∑ i = 1 x ( 1 − ω i n − 1 ) ( 1 − ν i ν i ) λ ) 1 λ )  
Next, we prove the (52) and (53) are IFNs. For the first case, when 1 ≤ x < n ,
Let
a = 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) x ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ  
b = 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) x ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ  
Then we need prove the following two conditions.
(I) 0 ≤ a ≤ 1 , 0 ≤ b ≤ 1 ;
(II) 0 ≤ a + b ≤ 1 .
(I) Since a ∈ [ 0 , 1 ] , we can get
1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) x ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ > 1 ⇒ 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) x ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ ∈ [ 0 , 1 ]
Therefore, 0 ≤ a ≤ 1 . Similarly, we can get
1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) 1 ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ ∈ [ 0 , 1 ]  
Therefore, 0 ≤ b ≤ 1 .
(II) Since 0 ≤ a + b ≤ 1 , we can get the following inequality.
1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) 1 ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ + 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) 1 ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ ≤ + 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) 1 ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ + 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) 1 ∑ j = 1 x ( ν i j 1 − ν i j ) λ ) 1 λ = 1
For the second case, when x = n , we can easily prove that it is kept. So the aggregation result produced by Definition 9 is still an IFN. Next, we shall deduce some desirable properties of IFWDDHM operator. □
Property 13 (Idempotency).
If k i ( i = 1 , 2 , ⋯ , n ) are equal, i.e., k i = k = ( μ , ν ) , and weight vector meets ω i ∈ [ 0 , 1 ] and ∑ i = 1 k ω i = 1 then
IFWDDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = k  
Proof. 
Since k i = k = ( μ , ν ) , based on Theorem 4, we get
(1) For the first case, when 1 ≤ x < n .
IFWDDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = ( 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) 1 ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ , 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) 1 ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ ) = ( 1 1 + ( 1 C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) 1 ( μ 1 − μ ) λ ) 1 λ , 1 − 1 1 + ( 1 C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ i = 1 x ω i ) 1 ( 1 − ν ν ) λ ) 1 λ )   = ( 1 1 + ( 1 C n − 1 x ( C n x − C n − 1 x − 1 ∑ i = 1 x ω i ) 1 ( μ 1 − μ ) λ ) 1 λ , 1 − 1 1 + ( 1 C n − 1 x ( C n x − C n − 1 x − 1 ∑ i = 1 x ω i ) 1 ( 1 − ν ν ) λ ) 1 λ )  
Since ∑ i = 1 k ω i = 1 , we can get
IFWDDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = ( 1 1 + ( 1 C n − 1 x ( C n x − C n − 1 x − 1 ) 1 ( ν 1 − ν ) λ ) 1 λ , 1 − 1 1 + ( 1 C n − 1 x ( C n x − C n − 1 x − 1 ) 1 ( 1 − μ μ ) λ ) 1 λ ) = ( 1 1 + ( 1 ( n − 1 ) ! x ! ( k − 1 − x ) ! ( n − 1 ) ! x ! ( k − 1 − x ) ! 1 ( μ 1 − μ ) λ ) 1 λ , 1 − 1 1 + ( 1 ( n − 1 ) ! x ! ( k − 1 − x ) ! ( n − 1 ) ! x ! ( k − 1 − x ) ! 1 ( 1 − ν ν ) λ ) 1 λ ) = ( 1 1 + ( 1 ( μ 1 − μ ) λ ) 1 λ , 1 − 1 1 + ( 1 ( 1 − ν ν ) λ ) 1 λ ) = ( μ , ν ) = k
(2) For the second case, when x = n ,
IFWDDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = ( 1 − 1 1 + ( ∑ i = 1 x ( 1 − ω i n − 1 ) ( μ i 1 − μ i ) λ ) 1 λ , 1 1 + ( ∑ i = 1 x ( 1 − ω i n − 1 ) ( 1 − ν i ν i ) λ ) 1 λ ) = ( 1 − 1 1 + ( n − 1 n − 1 ( μ 1 − μ ) λ ) 1 λ , 1 1 + ( n − 1 n − 1 ( 1 − ν ν ) λ ) 1 λ ) = ( μ , ν ) = k
which proves the idempotency property of the IFWDDHM operator. □
Property 14 (Monotonicity).
Let k i = ( μ i j , ν i j ) , π i = ( μ θ j , ν θ j ) ( i = 1 , 2 , ⋯ , n ) be two sets of IFNs. If μ i j ≥ μ θ j , ν i j ≤ ν θ j for all j , and weight vector meets ω i ∈ [ 0 , 1 ] and ∑ i = 1 k ω i = 1 , the k and π are equal, then we have
IFWDDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) ≥ IFWDDHM ω ( x ) ( π 1 , π 2 , ⋯ , π n )  
Proof. 
Since x ≥ 1 , μ i j ≥ μ θ j ≥ 0 , ν θ j ≥ ν i j ≥ 0 , then
∑ j = 1 x ( μ i j 1 − μ i j ) λ ≥ ∑ j = 1 x ( μ θ j 1 − μ θ j ) λ ⇒ 1 / ∑ j = 1 x ( μ i j 1 − μ i j ) λ ≤ 1 / ∑ j = 1 x ( μ θ j 1 − μ θ j ) λ ⇒ ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( μ i j 1 − μ i j ) λ ≤ ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( μ θ j 1 − μ θ j ) λ ⇒ x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( μ i j 1 − μ i j ) λ ≤ x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( μ θ j 1 − μ θ j ) λ ⇒ 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ ≤ 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( μ θ j 1 − μ θ j ) λ ) 1 λ 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ ≥ 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( μ θ j 1 − μ θ j ) λ ) 1 λ
Similarly, we have
∑ j = 1 x ( 1 − ν i j ν i j ) λ ≥ ∑ j = 1 x ( 1 − ν θ j ν θ j ) λ ⇒ 1 / ∑ j = 1 x ( 1 − ν i j ν i j ) λ ≤ 1 / ∑ j = 1 x ( 1 − ν θ j ν θ j ) λ ⇒ ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( 1 − ν i j ν i j ) λ ≤ ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( 1 − ν θ j ν θ j ) λ ⇒ ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( 1 − ν i j ν i j ) λ ≤ ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( 1 − ν θ j ν θ j ) λ ⇒ x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( 1 − ν i j ν i j ) λ ≤ x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( 1 − ν θ j ν θ j ) λ ⇒ 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ ≤ 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 / ∑ j = 1 x ( 1 − ν θ j ν θ j ) λ ) 1 λ
Let a = IFWDDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) , π = IFWDDHM ω ( x ) ( π 1 , π 2 , ⋯ , π n ) and S ( k ) , S ( π ) be the score values of a and π respectively. Based on the score value of IFN in (2) and the above inequality, we can imply that S ( k ) ≥ S ( π ) , and then we discuss the following cases:
(1) If S ( k ) > S ( π ) , then we can get
IFWDDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) > IFWDDHM ω ( x ) ( π 1 , π 2 , ⋯ , π n )  
(2) If S ( k ) = S ( π ) , then
  1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ − ( 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ ) = 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( μ θ j 1 − μ θ j ) λ ) 1 λ − ( 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − ν θ j ν θ j ) λ ) 1 λ )  
Since μ i j ≥ μ θ j ≥ 0 , ν θ j ≥ ν i j ≥ 0 , and based on the Equations (2) and (3), we can deduce
1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ = 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( μ θ j 1 − μ θ j ) λ ) 1 λ  
and
  1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ = 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − ν θ j ν θ j ) λ ) 1 λ  
Therefore, it follows that H ( k ) = H ( π ) , IFWDDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = IFWDDHM ω ( x ) ( π 1 , π 2 , ⋯ , π n ) , When x = n , we can prove it in a similar way. □
Property 15 (Boundedness).
Let k i = ( μ i j , ν i j ) , k + = ( μ max i j , ν max i j ) ( i = 1 , 2 , ⋯ , n ) be a set of IFNs, and k − = ( μ min i j , ν min i j ) , and weight vector meets ω i ∈ [ 0 , 1 ] and ∑ i = 1 n ω i = 1
k − ≤ IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) ≤ k +  
Proof. 
Based on Properties 13 and 14, we have
IFWDDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) ≥ IFWDDHM ω ( x ) ( k − , k − , ⋯ , k − ) = k − IFWDDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) ≤ IFWDDHM ω ( x ) ( k + , k + , ⋯ , k + ) = k +
Then we have k − ≤ IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) ≤ k + . □
Property 16 (Commutativity).
Let k i = ( μ i j , ν i j ) , π i = ( μ θ j , ν θ j ) ( i = 1 , 2 , ⋯ , n ) be two sets of IFNs. Suppose ( π 1 , π 2 , ⋯ , π n ) is any permutation of ( k 1 , k 2 , ⋯ , k n ) , and weight vector meets ω i ∈ [ 0 , 1 ] and ∑ i = 1 k ω i = 1 , the k and π are equal, then we have
IFWDDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = IFWDDHM ω ( x ) ( π 1 , π 2 , ⋯ , π n )  
Proof. 
Because ( π 1 , π 2 , ⋯ , π n ) is any permutation of ( k 1 , k 2 , ⋯ , k n ) , then
⊗ 1 ≤ i 1 < ⋯ < i x ≤ k ( 1 − ∑ j = 1 x ω i j ) ( ⊕ j = 1 x k i j ) 1 x C k − 1 x = ⊗ 1 ≤ i 1 < ⋯ < i x ≤ k ( 1 − ∑ j = 1 x ω i j ) ( ⊕ j = 1 x π i j ) 1 x C k − 1 x ( 1 ≤ x < k ) ⊕ i = 1 x k i 1 − ω i k − 1 = ⊕ i = 1 x π i 1 − ω i k − 1 ( x = k )
Thus, IFWDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = IFWDHM ω ( x ) ( π 1 , π 2 , ⋯ , π n ) . □
Example 8.
Let k 1 = ( 0.6 , 0.1 ) , k 2 = ( 0.5 , 0.4 ) , k 3 = ( 0.8 , 0.3 ) , k 4 = ( 0.7 , 0.2 ) be four IFNs. Then we use the IFWDDHM operator to fuse four IFNs, the weighting vector of attributes be ω = { 0.2 , 0.3 , 0.4 , 0.1 } (suppose x = 2 , λ = 2 ).
Let
IFWDDHM ω ( x ) ( k 1 , k 2 , ⋯ , k n ) = ⊗ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) ( ⊕ j = 1 x k i j ) 1 x C n − 1 x = ( 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( μ i j 1 − μ i j ) λ ) 1 λ , 1 − 1 1 + ( x C n − 1 x ∑ 1 ≤ i 1 < ⋯ < i x ≤ n ( 1 − ∑ j = 1 x ω i j ) 1 ∑ j = 1 x ( 1 − ν i j ν i j ) λ ) 1 λ )   = ( 1 1 + ( 2 C 3 2 ∑ 1 ≤ i 1 < ⋯ < i 2 ≤ 4 ( 1 − ∑ j = 1 2 ω i j ) 1 ∑ j = 1 2 ( μ i j 1 − μ i j ) 2 ) 1 2 , 1 − 1 1 + ( 2 C 3 2 ∑ 1 ≤ i 1 < ⋯ < i 2 ≤ 4 ( 1 − ∑ j = 1 2 ω i j ) 1 ∑ j = 1 2 ( 1 − ν i j ν i j ) 2 ) 1 2 )   = ( 1 1 + ( 2 3 × + ( 1 − 0.2 − 0.3 ( 0.6 1 − 0.6 ) 2 + ( 0.5 1 − 0.5 ) 2 + 1 − 0.2 − 0.4 ( 0.6 1 − 0.6 ) 2 + ( 0.8 1 − 0.8 ) 2 + 1 − 0.2 − 0.1 ( 0.6 1 − 0.6 ) 2 + ( 0.7 1 − 0.7 ) 2 + 1 − 0.3 − 0.4 ( 0.5 1 − 0.5 ) 2 + ( 0.8 1 − 0.8 ) 2 + 1 − 0.3 − 0.1 ( 0.5 1 − 0.5 ) 2 + ( 0.7 1 − 0.7 ) 2 + 1 − 0.4 − 0.1 ( 0.8 1 − 0.8 ) 2 + ( 0.7 1 − 0.7 ) 2 ) ) , 1 − 1 1 + ( 2 3 × ( 1 − 0.2 − 0.3 ( 1 − 0.1 0.1 ) 2 + ( 1 − 0.4 0.4 ) 2 + 1 − 0.2 − 0.4 ( 1 − 0.1 0.1 ) 2 + ( 1 − 0.3 0.3 ) 2 + 1 − 0.2 − 0.1 ( 1 − 0.1 0.1 ) 2 + ( 1 − 0.2 0.2 ) 2 + 1 − 0.3 − 0.4 ( 1 − 0.4 0.4 ) 2 + ( 1 − 0.3 0.3 ) 2 + 1 − 0.3 − 0.1 ( 1 − 0.4 0.4 ) 2 + ( 1 − 0.2 0.2 ) 2 + 1 − 0.4 − 0.1 ( 1 − 0.3 0.3 ) 2 + ( 1 − 0.2 0.2 ) 2 ) ) ) = ( 0.6592 , 0.2153 )
At last, we get IFWDDHM ω ( 2 ) ( k 1 , k 2 , k 3 , k 4 ) = ( 0.6592 , 0.2153 ) .

4. A MAGDM Approach Based on the Proposed Operators

In this section, we will apply the proposed IFWDHM (IFWDDHM) operator to cope with the MAGDM problem with IFNs. Let X = { x 1 , x 2 , ⋯ , x m } be a set of alternatives, and C = { c 1 , c 2 , ⋯ , c n } be a set of attributes, the weighting vector of attributes be ω = { ω 1 , ω 2 ⋯ , ω n } , meet ω j ∈ [ 0 , 1 ] , j = 1 , 2 , ⋯ , n , ∑ j = 1 n ω j = 1 . There are experts Y = { y 1 , y 2 , ⋯ y z } who are invited to give the evaluation information, and their weighting vector is w = { w 1 , w 2 , ⋯ w z } T with w t ∈ [ 0 , 1 ] , ( t = 1 , 2 , ⋯ , z ) , ∑ t = 1 z w t = 1 . The expert y t evaluates each attribute c j of each alternative x i by the form of IFN a i j t = ( μ i j t , ν i j t ) ( i = 1 , 2 , ⋯ , m , j = 1 , 2 , ⋯ , n ) , and then the decision matrix A t = ( a ˜ i j t ) m × n = ( ( μ i j t , ν i j t ) ) m × n ( t = 1 , 2 , ⋯ , z ) is constructed. The ultimate goal is to give a ranking of all alternatives.
Then, we will give the steps for solving this problem.
Step1: Calculate the collective evaluation value of each attribute for each alternative by
a ˜ i j t = IFWDHM ω ( x ) ( a ˜ i j 1 , a ˜ i j 2 , ⋯ a ˜ i j z )   and   a ˜ i j t = IFWDDHM ω ( x ) ( a ˜ i j 1 , a ˜ i j 2 , ⋯ a ˜ i j z )  
Step2: Calculate the overall value of each alternative with the IFWDHM (IFWDDHM) operator
a ˜ i = IFWDHM ω ( x ) ( a ˜ i 1 , a ˜ i 2 , ⋯ a ˜ i n ) ,   a ˜ i = IFWDDHM ω ( x ) ( a ˜ i 1 , a ˜ i 2 , ⋯ a ˜ i n )  
Step3: Calculate the S ( a ˜ ) and H ( a ˜ ) .
Step4: Sort all alternatives { x 1 , x 2 , ⋯ , x m } and choose the best one.

5. An Illustrate Example

In this section, we give an example to explain the proposed method. A transportation company wants to pick a car and there are four cars as candidates M i = ( M 1 , M 2 , M 3 , M 4 ) . We evaluate each supplier from four aspects E i = ( E 1 , E 2 , E 3 , E 4 ) , which are “production price”, “production quality”, “production’s service performance”, and “risk factor”. The weight vector of attributes is ω = ( 0.1 , 0.4 , 0.3 , 0.2 ) T . There are four experts, and the weight vector of the experts is ( 0.3 , 0.4 , 0.2 , 0.1 ) T . Then the decision matrix R t = ( a i j t ) 4 × 4 ( t = 1 , 2 , 3 , 4 ) are shown in Table 1, Table 2, Table 3 and Table 4, and our goal is to rank four cars and select the best one.

5.1. Decision-Making Processes

Step 1: Since the four attributes are of the same type, thus, we don’t need to normalize the matrix R 1 ~ R 4 .
Step 2: Use IFWDHM operator to fuse four decision matrix R t = ( a i j t ) m × n into a collective matrix R = ( a i j t ) m × n which is shown in Table 5 (suppose x = 2 , λ = 2 ).
Use IFWDDHM operator to aggregate four decision matrixes R t = ( a i j t ) m × n into a collective matrix R = ( a i j t ) m × n , which is shown in Table 6 (suppose x = 2 , λ = 2 ).
Step 3: Use the IFWDHM (IFWDDHM) operator to aggregate all the attribute values a i j , a ′ i j ( j = 1 , 2 , 3 , 4 ) and get the comprehensive evaluation value (suppose x = 2 , λ = 2 ).
a 1 = ( 0.0694 , 0.4051 ) , a 2 = ( 0.5357 , 0.2264 ) , a 3 = ( 0.1464 , 0.3736 ) , a 4 = ( 0.0330 , 0.6366 ) . a ′ 1 = ( 0.8010 , 0.0103 ) , a ′ 2 = ( 0.9380 , 0.0011 ) , a ′ 3 = ( 0.8584 , 0.0087 ) , a ′ 4 = ( 0.6690 , 0.0290 ) .  
Step 4: Obtain the score values.
S ( a 1 ) = − 0.3357 , S ( a 2 ) = 0.3093 , S ( a 3 ) = − 0.2272 , S ( a 4 ) = − 0.6036. S ( a ′ 1 ) = 0.7907 , S ( a ′ 2 ) = 0.9369 , S ( a ′ 3 ) = 0.8497 , S ( a ′ 4 ) = 0.6399.  
Step 5: Rank all alternatives. a 2 ≻ a 3 ≻ a 1 ≻ a 4 , then the best choice is a 2 .
Considering the different parameter values of an IFWDHM operator that may have an impact on the ranking results, we calculated the scores produced from the different x and the results are listed in Table 7.
Considering the different parameter values of an IFWDDHM operator that may have an impact on the ordering results, we calculated the scores with different x and the results are listed in Table 8.
From Table 7 and Table 8, we get following conclusions.
When x = 1 , the sorting of alternatives is a 2 ≻ a 3 ≻ a 1 ≻ a 4 , and the best choice is a 2 .
When x = 2 , 3 , 4 , the sorting of alternatives is a 2 ≻ a 3 ≻ a 1 ≻ a 4 , and the best choice is a 2 .
Although there is the same best selection, the ranking is different. When x = 1 , the interrelationship between the attributes is not considered, and when x = 2 , 3 , 4 , we can consider the interrelationship for different number of attributes. So these results are reasonable for these two conditions.

5.2. Comparative Analysis

Following this, we compare the proposed method with IFWA operator [4], IFWG operator [5], IFWMM operator [62], and IFDWMM operator [62] and the comparative results are depicted in Table 9.
From above analysis, we arrived at the same results. However, the existing operators, such as IFWA operator and IFWG operator do not consider the relationship between arguments, and thus cannot eliminate the corresponding influence of unfair arguments on decision result. The IFWMM operator, IFDWMM operator, IFWDHM and IFWDDHM operators consider the relationship among the arguments.

6. Conclusions

In this paper, we investigated the MADM problems with IFNs. Following this, we utilized the HM operator, DHM operator, DDHM operator, WDHM operator, and WDDHM operator to develop some novel operators with IFNs: Intuitionistic fuzzy DHM (IFDHM) operator, intuitionistic fuzzy WDHM operator, intuitionistic fuzzy DDHM (IFDDHM) operator, and intuitionistic fuzzy WDDHM (IFWDDHM) operator. The prominent characteristic of these proposed operators were studied. Moreover, we have utilized these operators to develop some models to solve the MAGDM problems with IFNs. Finally, a practical example for the selection of a car car supplier was given. In the future, the application of the IFNs needs to be explored in decision-making processes [63,64,65,66,67,68,69,70,71,72], risk analysis [73,74], and other fuzzy environments [75,76,77,78,79,80].

Author Contributions

Z.L., H.G. and G.W. conceived and worked together to achieve this work, Z.L. compiled the computing program by Matlab and analyzed the data, Z.L. and G.W. wrote the paper. Finally, all the authors have read and approved the final manuscript.

Funding

The work was supported by the National Natural Science Foundation of China under Grant No. 71571128 and the Humanities and Social Sciences Foundation of Ministry of Education of the People’s Republic of China (17XJA630003) and the Construction Plan of Scientific Research Innovation Team for Colleges and Universities in Sichuan Province (15TD0004).

Conflicts of Interest

The authors declare no conflict of interest.

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Table
Table 1. Decision matrix R 1 .
Table 1. Decision matrix R 1 .
E 1   E 2   E 3   E 4  
M 1 (0.5 0.3)(0.6 0.3)(0.5 0.2)(0.6 0.4)
M 2 (0.7 0.3)(0.9 0.1)(0.8 0.1)(0.7 0.2)
M 3 (0.7 0.2)(0.5 0.4)(0.6 0.1)(0.4 0.2)
M 4 (0.5 0.3)(0.3 0.4)(0.5 0.4)(0.4 0.5)
Table
Table 2. Decision matrix R 2 .
Table 2. Decision matrix R 2 .
E 1   E 2   E 3   E 4  
M 1 (0.6 0.2)(0.7 0.1)(0.6 0.2)(0.6 0.3)
M 2 (0.9 0.1)(0.8 0.2)(0.7 0.1)(0.6 0.4)
M 3 (0.5 0.2)(0.6 0.3)(0.7 0.2)(0.8 0.1)
M 4 (0.7 0.2)(0.4 0.3)(0.5 0.5)(0.6 0.3)
Table
Table 3. Decision matrix R 3 .
Table 3. Decision matrix R 3 .
E 1   E 2   E 3   E 4  
M 1 (0.6 0.4)(0.7 0.2)(0.6 0.3)(0.5 0.4)
M 2 (0.8 0.6)(0.7 0.1)(0.6 0.4)(0.9 0.1)
M 3 (0.5 0.2)(0.4 0.5)(0.4 0.3)(0.5 0.4)
M 4 (0.2 0.5)(0.5 0.4)(0.7 0.2)(0.5 0.4)
Table
Table 4. Decision matrix R 4 .
Table 4. Decision matrix R 4 .
E 1   E 2   E 3   E 4  
M 1 (0.6 0.2)(0.5 0.4)(0.6 0.4)(0.4 0.5)
M 2 (0.7 0.3)(0.8 0.1)(0.6 0.2)(0.9 0.1)
M 3 (0.6 0.4)(0.3 0.6)(0.2 0.6)(0.5 0.3)
M 4 (0.3 0.5)(0.2 0.7)(0.5 0.4)(0.3 0.6)
Table
Table 5. The collective decision matrix R .
Table 5. The collective decision matrix R .
G 1   G 2   G 3   G 4  
A 1 (0.2976 0.3875)(0.3504 0.2771)(0.2156 0.2689)(0.3996 0.3304)
A 2 (0.5818 0.2103)(0.5000 0.1638)(0.4172 0.1781)(0.4554 0.2073)
A 3 (0.3282 0.2872)(0.4554 0.3079)(0.3095 0.2316)(0.2857 0.3472)
A 4 (0.2411 0.5299)(0.3671 0.2684)(0.1813 0.2504)(0.0371 0.7363)
Table
Table 6. The collective decision matrix R .
Table 6. The collective decision matrix R .
G 1   G 2   G 3   G 4  
A 1 (0.5372 0.0288)(0.6100 0.0063)(0.5618 0.0043)(0.6535 0.0116)
A 2 (0.7000 0.0021)(0.6667 0.0006)(0.6422 0.0009)(0.6646 0.0012)
A 3 (0.6300 0.0066)(0.6344 0.0105)(0.6066 0.0025)(0.5969 0.0196)
A 4 (0.5827 0.1360)(0.6729 0.0052)(0.5818 0.0035)(0.4172 0.6485)
Table
Table 7. Score and ranking of the alternatives with different parameter values x .
Table 7. Score and ranking of the alternatives with different parameter values x .
x Score   of   S ( a ˜ i )   Ranking
x = 1 S ( a 1 ) = − 0.0073 , S ( a 2 ) = 0.0352 , S ( a 3 ) = − 0.0002 , S ( a 4 ) = − 0.0383. a 2 ≻ a 3 ≻ a 1 ≻ a 4
x = 2 S ( a 1 ) = − 0.3357 , S ( a 2 ) = 0.3093 , S ( a 3 ) = − 0.2272 , S ( a 4 ) = − 0.6036 a 2 ≻ a 3 ≻ a 1 ≻ a 4
x = 3 S ( a 1 ) = − 0.2988 , S ( a 2 ) = − 0.1265 , S ( a 3 ) = − 0.2595 , S ( a 4 ) = − 0.4853. a 2 ≻ a 3 ≻ a 1 ≻ a 4
x = 4 S ( a 1 ) = 0.1391 , S ( a 2 ) = 0.3007 , S ( a 3 ) = 0.1860 , S ( a ˜ 4 ) = − 0.1440. a 2 ≻ a 3 ≻ a 1 ≻ a 4
Table
Table 8. Score and order of the alternatives with different parameter values x .
Table 8. Score and order of the alternatives with different parameter values x .
x Score   of   S ( a i )   Ranking
x = 1 S ( a 1 ) = 0.0437 , S ( a 2 ) = 0.0915 , S ( a 3 ) = 0.0548 , S ( a 4 ) = 0.0110. a 2 ≻ a 3 ≻ a 1 ≻ a 4
x = 2 S ( a 1 ) = 0.7907 , S ( a 2 ) = 0.9369 , S ( a 3 ) = 0.8497 , S ( a 4 ) = 0.6399. a 2 ≻ a 3 ≻ a 1 ≻ a 4
x = 3 S ( a 1 ) = 0.2059 , S ( a 2 ) = 0.3597 , S ( a 3 ) = 0.2397 , S ( a 4 ) = 0.0320. a 2 ≻ a 3 ≻ a 1 ≻ a 4
x = 4 S ( a 1 ) = 0.2398 , S ( a 2 ) = 0.3695 , S ( a 3 ) = 0.2477 , S ( a 4 ) = 0.0424. a 2 ≻ a 3 ≻ a 1 ≻ a 4
Table
Table 9. Ordering of the green suppliers.
Table 9. Ordering of the green suppliers.
Ordering
IFWA a 2 ≻ a 3 ≻ a 1 ≻ a 4
IFWG a 2 ≻ a 3 ≻ a 1 ≻ a 4
IFWMM a 2 ≻ a 3 ≻ a 1 ≻ a 4
IFDWMM a 2 ≻ a 3 ≻ a 1 ≻ a 4

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Li, Z.; Gao, H.; Wei, G. Methods for Multiple Attribute Group Decision Making Based on Intuitionistic Fuzzy Dombi Hamy Mean Operators. Symmetry 2018, 10, 574. https://doi.org/10.3390/sym10110574

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Li Z, Gao H, Wei G. Methods for Multiple Attribute Group Decision Making Based on Intuitionistic Fuzzy Dombi Hamy Mean Operators. Symmetry. 2018; 10(11):574. https://doi.org/10.3390/sym10110574

Chicago/Turabian Style

Li, Zengxian, Hui Gao, and Guiwu Wei. 2018. "Methods for Multiple Attribute Group Decision Making Based on Intuitionistic Fuzzy Dombi Hamy Mean Operators" Symmetry 10, no. 11: 574. https://doi.org/10.3390/sym10110574

APA Style

Li, Z., Gao, H., & Wei, G. (2018). Methods for Multiple Attribute Group Decision Making Based on Intuitionistic Fuzzy Dombi Hamy Mean Operators. Symmetry, 10(11), 574. https://doi.org/10.3390/sym10110574

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