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Article

Research on Construction Engineering Project Risk Assessment with Some 2-Tuple Linguistic Neutrosophic Hamy Mean Operators

1
School of Business, Sichuan Normal University, Chengdu 610101, China
2
School of Finance, Yunnan University of Finance and Economics, Kunming 650221, China
*
Authors to whom correspondence should be addressed.
Sustainability 2018, 10(5), 1536; https://doi.org/10.3390/su10051536
Submission received: 14 April 2018 / Revised: 24 April 2018 / Accepted: 4 May 2018 / Published: 11 May 2018
(This article belongs to the Special Issue Sustainability in Construction Engineering)

Abstract

:
In this paper, we expand the Hamy mean (HM) operator, weighted Hamy mean (WHM), dual Hamy mean (DHM) operator, and weighted dual Hamy mean (WDHM) operator with 2-tuple linguistic neutrosophic numbers (2TLNNs) to propose a 2-tuple linguistic neutrosophic Hamy mean (2TLNHM) operator, 2-tuple linguistic neutrosophic weighted Hamy mean (2TLNWHM) operator, 2-tuple linguistic neutrosophic dual Hamy mean (2TLNDHM) operator, and 2-tuple linguistic neutrosophic weighted dual Hamy mean (2TLNWDHM) operator. Then, the multiple attribute decision-making (MADM) methods are proposed with these operators. Finally, we utilize an applicable example in risk assessment for construction engineering projects to prove the proposed methods.

1. Introduction

Neutrosophic sets (NSs), which were proposed originally by Smarandache [1,2], have attracted the attention of many scholars, and NSs have acted as a workspace in depicting indeterminate and inconsistent information. A NS has more potential power than other modeling mathematical tools, such as fuzzy set [3], intuitionistic fuzzy set (IFS) [4] and interval-valued intuitionistic fuzzy set (IVIFS) [5]. But, it is difficult to apply NSs to solve real life problems. Therefore, Wang et al. [6,7] defined single valued neutrosophic sets (SVNSs) and interval neutrosophic sets (INS), which are characterized by a truth membership, an indeterminacy membership and a falsity membership. Hence, SVNSs and INSs can express much more information than fuzzy sets, IFSs and IVIFSs. Ye [8] proposed a multiple attribute decision-making (MADM) method with correlation coefficients of SVNSs. Broumi and Smarandache [9] defined the correlation coefficients of INSs. Biswas et al. [10] proposed the Technique for Order Preference by Similarity to an Ideal Solution(TOPSIS) method with SVNNs. Liu et al. [11] defined the generalized neutrosophic number Hamacher aggregation for SVNSs. Sahin and Liu [12] defined the maximizing deviation model under a neutrosophic environment. Ye [13] developed some similarity measures of INS. Zhang et al. [14] defined some aggregating operators with INNs. Ye [15] defined a simplified neutrosophic set (SNS). Peng et al. [16] developed aggregation operators under SNS. Peng et al. [17] investigated the outranking approach with SNS, and then Zhang et al. [18] extended Peng’s approach. Liu and Liu [19] proposed a power averaging operator with SVNNs. Deli and Subas [20] discussed a novel method to rank SVNNs. Peng et al. [21] proposed multi-valued neutrosophic sets. Zhang et al. [22] gave the improved weighted correlation coefficient for interval neutrosophic sets. Chen and Ye [23] proposed Dombi operations for SVNSs. Liu and Wang [24] proposed the MADM method based on a SVN-normalized weighted Bonferroni mean. Wu et al. [25] proposed a cross-entropy and prioritized an aggregation operator with SNSs in MADM problems. Li et al. [26] developed SVNN Heronian mean operators in MADM problems. Zavadskas et al. [27] proposed a model for residential house elements and material selection using the neutrosophic MULTIMOORA method. Zavadskas et al. [28] studied the sustainable market valuation of buildings using the SVN MAMVA method. Bausys and Juodagalviene [29] investigated the garage location selection for residential houses using the WASPAS-SVNS method. Wu et al. [30] proposed some Hamacher aggregation operators under an SVN 2-tuple linguistic environment for MAGDM.
Although SVNS theory has been successfully applied in some areas, the SVNS is also characterized by truth membership degree, indeterminacy membership degree, and falsity membership degree information. However, all the above approaches are unsuitable for describing the truth membership degree, indeterminacy membership degree, and falsity membership degree information of an element of a set by linguistic variables on the basis of the given linguistic term sets, which can reflect a decision maker’s confidence level when they are making an evaluation. In order to overcome this limit, we propose the concept of a 2-tuple linguistic neutrosophic numbers set (2TLNNSs) to solve this problem based on SVNS [6,7] and a 2-tuple linguistic information processing model [31]. Thus, how to aggregate these 2-tuple linguistic neutrosophic numbers is an interesting topic. To solve this issue, in this paper, we develop aggregation operators with 2TLNNs based on the traditional operator [32]. In order to do so, the remainder of this paper is set out as follows. In the next section, we propose the concept of 2TLNNSs. In Section 3, we propose Hamy mean (HM) operators with 2TLNNs. In Section 4, we give a numerical example for risk assessment of a construction engineering projects. Section 5 concludes the paper with some remarks.

2. Preliminaries

In this section, we propose the concept of using 2-tuple linguistic neutrosophic sets (2TLNSs) based on SVNSs [6,7] and 2-tuple linguistic sets (2TLSs) [31].

2.1. 2TLSs

Definition 1.
Let S = { s i | i = 0 , 1 , , t } be a linguistic term set with an odd cardinality. Any label, s i , represents a possible value for a linguistic variable, and S can be defined as:
S = { s 0 = e x t r e m e l y   p o o r ,   s 1 = v e r y   p o o r ,   s 2 = p o o r ,   s 3 = m e d i u m , s 4 = g o o d ,   s 5 = v e r y   g o o d ,   s 6 = e x t r e m e l y   g o o d . }
Herrera and Martinez [27,28] developed the 2-tuple fuzzy linguistic representation model based on the concept of symbolic translation. It is used for representing the linguistic assessment information by means of a 2-tuple ( s i , ρ i ) , where s i is a linguistic label for predefined linguistic term set S and ρ i is the value of symbolic translation, and ρ i [ 0.5 , 0.5 ) .

2.2. SVNSs

Let X be a space of points (objects) with a generic element in a fixed set, X , denoted by x . An SVNS, A , in X is characterized as the following [6,7]:
A = { ( x , T A ( x ) , I A ( x ) , F A ( x ) ) | x X }
where the truth membership function, T A ( x ) , indeterminacy-membership, I A ( x ) , and falsity membership function, F A ( x ) , are single subintervals/subsets in the real standard [ 0 , 1 ] , that is, T A ( x ) : X [ 0 , 1 ] , I A ( x ) : X [ 0 , 1 ] and F A ( x ) : X [ 0 , 1 ] . In addition, the sum of T A ( x ) , I A ( x ) and F A ( x ) satisfies the condition 0 T A ( x ) + I A ( x ) + F A ( x ) 3 . Then, a simplification of A is denoted by A = { ( x , T A ( x ) , I A ( x ) , F A ( x ) ) | x X } , which is a SVNS.
For a SVNS { ( x , T A ( x ) , I A ( x ) , F A ( x ) ) | x X } , the ordered triple components, ( T A ( x ) , I A ( x ) , F A ( x ) ) , are described as a single-valued neutrosophic number (SVNN), and each SVNN can be expressed as A = ( T A , I A , F A ) , where T A [ 0 , 1 ] , I A [ 0 , 1 ] , F A [ 0 , 1 ] and 0 T A + I A + F A 3 .

2.3. 2TLNSs

Definition 2.
Assume that φ = { φ 0 , φ 1 , , φ t } is a 2TLSs with an odd cardinality, t + 1 . If φ = ( s T , α ) , ( s I , β ) , ( s F , γ ) is defined for ( s T , α ) , ( s I , β ) , ( s F , γ ) φ and α , β , γ [ 0 , t ] , where ( s T , α ) , ( s I , β ) and ( s F , γ ) express independently the truth degree, indeterminacy degree, and falsity degree by 2TLSs, then 2TLNSs is defined as follows:
φ j = ( s T j , α j ) , ( s I j , β j ) , ( s F j , γ j )
where 0 Δ 1 ( s T j , α j ) t , 0 Δ 1 ( s I j , β j ) t , 0 Δ 1 ( s F j , γ j ) t , and 0 Δ 1 ( s T j , α j ) + Δ 1 ( s I j , β j ) + Δ 1 ( s F j , γ j ) 3 t .
Definition 3.
Let φ 1 = ( s T 1 , α 1 ) , ( s I 1 , β 1 ) , ( s F 1 , γ 1 ) be a 2TLNN in φ . Then, the score and accuracy functions of φ 1 are defined as follows:
S ( φ 1 ) = Δ { ( 2 t + Δ 1 ( s T 1 , α 1 ) Δ 1 ( s I 1 , β 1 ) Δ 1 ( s F 1 , γ 1 ) ) 3 } , S ( φ 1 ) [ 0 , t ]
H ( φ 1 ) = Δ { Δ 1 ( s T 1 , α 1 ) Δ 1 ( s F 1 , γ 1 ) } , H ( φ 1 ) [ t , t ] .
Definition 4.
Let φ 1 = ( s T 1 , α 1 ) , ( s I 1 , β 1 ) , ( s F 1 , γ 1 ) and φ 2 = ( s T 2 , α 2 ) , ( s I 2 , β 2 ) , ( s F 2 , γ 2 ) be two 2TLNNs, then
(1)
if S ( φ 1 ) < S ( φ 2 ) , then φ 1 < φ 2 ;
(2)
if S ( φ 1 ) > S ( φ 2 ) , then φ 1 > φ 2 ;
(3)
if S ( φ 1 ) = S ( φ 2 ) , H ( φ 1 ) < H ( φ 2 ) , then φ 1 < φ 2 ;
(4)
if S ( φ 1 ) = S ( φ 2 ) , H ( φ 1 ) > H ( φ 2 ) , then φ 1 > φ 2 ;
(5)
if S ( φ 1 ) = S ( φ 2 ) , H ( φ 1 ) = H ( φ 2 ) , then φ 1 = φ 2 .
Definition 5.
Let φ 1 = ( s T 1 , α 1 ) , ( s I 1 , β 1 ) , ( s F 1 , γ 1 ) and φ 2 = ( s T 2 , α 2 ) , ( s I 2 , β 2 ) , ( s F 2 , γ 2 ) be two 2TLNNs, ζ > 0 , then
(1)
φ 1 φ 2 = { Δ ( t ( Δ 1 ( s T 1 , α 1 ) t + Δ 1 ( s T 2 , α 2 ) t Δ 1 ( s T 1 , α 1 ) t · Δ 1 ( s T 2 , α 2 ) t ) ) , Δ ( t ( Δ 1 ( s I 1 , β 1 ) t · Δ 1 ( s I 2 , β 2 ) t ) ) , Δ ( t ( Δ 1 ( s F 1 , γ 1 ) t · Δ 1 ( s F 2 , γ 2 ) t ) ) } ;
(2)
φ 1 φ 2 = { Δ ( t ( Δ 1 ( s T 1 , α 1 ) t · Δ 1 ( s T 2 , α 2 ) t ) ) , Δ ( t ( Δ 1 ( s T 1 , β 1 ) t + Δ 1 ( s T 2 , β 2 ) t Δ 1 ( s T 1 , β 1 ) t · Δ 1 ( s T 2 , β 2 ) t ) ) , Δ ( t ( Δ 1 ( s F 1 , γ 1 ) t + Δ 1 ( s F 2 , γ 2 ) t Δ 1 ( s F 1 , γ 1 ) t · Δ 1 ( s F 2 , γ 2 ) t ) ) } ;
(3)
ζ φ 1 = { Δ ( t ( 1 ( 1 Δ 1 ( s T 1 , α 1 ) t ) ζ ) ) , Δ ( t ( Δ 1 ( s I 1 , β 1 ) t ) ζ ) , Δ ( t ( Δ 1 ( s F 1 , γ 1 ) t ) ζ ) } , ζ > 0 ;
(4)
( φ 1 ) ζ = { Δ ( t ( Δ 1 ( s T 1 , α 1 ) t ) ζ ) , Δ ( t ( 1 ( 1 Δ 1 ( s I 1 , β 1 ) t ) ζ ) ) , Δ ( t ( 1 ( 1 Δ 1 ( s F 1 , γ 1 ) t ) ζ ) ) } , ζ > 0 .

2.4. HM Operator

Definition 6 [32].
The Hamy mean (HM) operator is defined as follows:
HM ( x ) ( φ 1 , φ 2 , , φ n ) = 1 i 1 < < i x n ( j = 1 x φ 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 ) ! .

3. Some 2TLNHM Operators

3.1. 2TLNHM Operator

In this section, we will combine HM and 2TLNNs and propose the 2-tuple linguistic neutrosophic Hamy mean (2TLNHM) operator.
Definition 7.
Let φ j = ( s T j , α j ) , ( s I j , β j ) , ( s F j , γ j ) ( j = 1 , 2 , , n ) be a set of 2TLNNs. The 2TLNHM operator is
2 TLNHM ( x ) ( φ 1 , φ 2 , , φ n ) = 1 i 1 < < i x n ( j = 1 x φ i j ) 1 x C n x .
Theorem 1.
Let φ j = ( s T j , α j ) , ( s I j , β j ) , ( s F j , γ j ) ( j = 1 , 2 , , n ) be a set of 2TLNNs. The aggregated value from the 2TLNHM operators is also a 2TLNN where
2 TLNHM ( x ) ( φ 1 , φ 2 , , φ n ) = 1 i 1 < < i x n ( j = 1 x φ i j ) 1 x C n x = { Δ ( t ( 1 ( 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s T j , α j ) t ) ) 1 x ) ) 1 C n x ) ) , Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s I j , β j ) t ) ) 1 x ) ) 1 C n x ) , Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s F j , γ j ) t ) ) 1 x ) ) 1 C n x ) } .
Proof: 
j = 1 x φ i j = { Δ ( t j = 1 x ( Δ 1 ( s T j , α j ) t ) ) , Δ ( t ( 1 j = 1 x ( 1 Δ 1 ( s I j , β j ) t ) ) ) , Δ ( t ( 1 j = 1 x ( 1 Δ 1 ( s F j , γ j ) t ) ) ) } .
Thus,
( j = 1 x φ i j ) 1 x = { Δ ( t ( j = 1 x ( Δ 1 ( s T j , α j ) t ) ) 1 x ) , Δ ( t ( 1 ( j = 1 x ( 1 Δ 1 ( s I j , β j ) t ) ) 1 x ) ) , Δ ( t ( 1 ( j = 1 x ( 1 Δ 1 ( s F j , γ j ) t ) ) 1 x ) ) }
Thereafter,
1 i 1 < < i x n ( j = 1 x φ i j ) 1 x = { Δ ( t ( 1 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s T j , α j ) t ) ) 1 x ) ) ) , Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s I j , β j ) t ) ) 1 x ) ) ) , Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s F j , γ j ) t ) ) 1 x ) ) ) } .
Therefore,
2 TLNHM ( x ) ( φ 1 , φ 2 , , φ n ) = 1 i 1 < < i x n ( j = 1 x φ i j ) 1 x C n x = { Δ ( t ( 1 ( 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s T j , α j ) t ) ) 1 x ) ) 1 C n x ) ) , Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s I j , β j ) t ) ) 1 x ) ) 1 C n x ) , Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s F j , γ j ) t ) ) 1 x ) ) 1 C n x ) }
Hence, (7) is kept.
Then we need to prove that (7) is a 2TLNN. We need to prove two conditions, as follows:
0 Δ 1 ( s T , α ) t , 0 Δ 1 ( s I , β ) t , 0 Δ 1 ( s F , γ ) t .
0 Δ 1 ( s T , α ) + Δ 1 ( s I , β ) + Δ 1 ( s F , γ ) 3 t .
Let
Δ 1 ( s T , α ) t = 1 ( 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s T j , α j ) t ) ) 1 x ) ) 1 C n x Δ 1 ( s I , β ) t = ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s I j , β j ) t ) ) 1 x ) ) 1 C n x . Δ 1 ( s F , γ ) t = ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s F j , γ j ) t ) ) 1 x ) ) 1 C n x
Proof: 
① Since 0 Δ 1 ( s T j , α j ) t 1 , we get
0 j = 1 x ( Δ 1 ( s T j , α j ) t ) 1   and   0 1 ( j = 1 x ( Δ 1 ( s T j , α j ) t ) ) 1 x 1
Then,
0 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s T j , α j ) t ) ) 1 x ) 1 ,
0 1 ( 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s T j , α j ) t ) ) 1 x ) ) 1 C n x 1 .
That means 0 Δ 1 ( s T , α ) t . Similarly, we can get 0 Δ 1 ( s I , β ) t , 0 Δ 1 ( s F , γ ) t so ① is maintained. ② Since 0 Δ 1 ( s T , α ) t , 0 Δ 1 ( s I , β ) t , 0 Δ 1 ( s F , γ ) t . , 0 Δ 1 ( s T , α ) + Δ 1 ( s I , β ) + Δ 1 ( s F , γ ) 3 t .
Example 1.
Let ( s 5 , 0 ) , ( s 2 , 0 ) , ( s 1 , 0 ) , ( s 4 , 0 ) , ( s 3 , 0 ) , ( s 4 , 0 ) , ( s 2 , 0 ) , ( s 5 , 0 ) , ( s 1 , 0 ) and ( s 5 , 0 ) , ( s 1 , 0 ) , ( s 3 , 0 ) be four 2TLNNs, and suppose x = 2 , then according to (4), we have
2 TLNHM ( 2 ) ( φ 1 , φ 2 , , φ n ) = 1 i 1 < < i x n ( j = 1 x φ i j ) 1 x C n x = { Δ ( 6 × ( 1 ( ( 1 ( 5 6 × 4 6 ) 1 2 ) × ( 1 ( 5 6 × 2 6 ) 1 2 ) × ( 1 ( 5 6 × 5 6 ) 1 2 ) × ( 1 ( 4 6 × 2 6 ) 1 2 ) × ( 1 ( 4 6 × 5 6 ) 1 2 ) × ( 1 ( 2 6 × 5 6 ) 1 2 ) ) 1 C 4 2 ) ) , Δ ( 6 × ( ( 1 ( ( 1 2 6 ) × ( 1 3 6 ) ) 1 2 ) × ( 1 ( ( 1 2 6 ) × ( 1 5 6 ) ) 1 2 ) × ( 1 ( ( 1 2 6 ) × ( 1 1 6 ) ) 1 2 ) × ( 1 ( ( 1 3 6 ) × ( 1 5 6 ) ) 1 2 ) × ( 1 ( ( 1 3 6 ) × ( 1 1 6 ) ) 1 2 ) × ( 1 ( ( 1 5 6 ) × ( 1 1 6 ) ) 1 2 ) ) 1 C 4 2 ) , Δ ( 6 × ( ( 1 ( ( 1 1 6 ) × ( 1 4 6 ) ) 1 2 ) × ( 1 ( ( 1 1 6 ) × ( 1 1 6 ) ) 1 2 ) × ( 1 ( ( 1 1 6 ) × ( 1 3 6 ) ) 1 2 ) × ( 1 ( ( 1 4 6 ) × ( 1 1 6 ) ) 1 2 ) × ( 1 ( ( 1 4 6 ) × ( 1 3 6 ) ) 1 2 ) × ( 1 ( ( 1 1 6 ) × ( 1 3 6 ) ) 1 2 ) ) 1 C 4 2 ) } = ( s 4 , 0.0235 ) , ( s 3 , 0.1556 ) , ( s 2 , 0.2489 )
Now, we will give some properties of a 2TLNHM operator.
Property 1.
(Idempotency) If φ j = ( s T j , α j ) , ( s I j , β j ) , ( s F j , γ j ) ( j = 1 , 2 , , n ) are equal, then
2 TLNHM ( x ) ( φ 1 , φ 2 , , φ n ) = φ .
Proof: 
Since φ j = φ = ( s T , α ) , ( s I , β ) , ( s F , γ ) , then
2 TLNHM ( x ) ( φ 1 , φ 2 , , φ n ) = 1 i 1 < < i x n ( j = 1 x φ i j ) 1 x C n x = { Δ ( t ( 1 ( 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s T , α ) t ) ) 1 x ) ) 1 C n x ) ) , Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s I , β ) t ) ) 1 x ) ) 1 C n x ) , Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s F , γ ) t ) ) 1 x ) ) 1 C n x ) }
= { Δ ( t ( 1 ( ( 1 ( ( Δ 1 ( s T , α ) t ) x ) 1 x ) 1 C n x ) 1 C n x ) ) , Δ ( t ( ( 1 ( ( 1 Δ 1 ( s I , β ) t ) x ) 1 x ) 1 C n x ) 1 C n x ) , Δ ( t ( ( 1 ( ( 1 Δ 1 ( s F , γ ) t ) x ) 1 x ) 1 C n x ) 1 C n x ) } = ( s T , α ) , ( s I , β ) , ( s F , γ ) = φ
Property 2.
(Monotonicity) Let φ a j = ( s T a j , α a j ) , ( s I a j , β a j ) , ( s F a j , γ a j ) ( j = 1 , 2 , , n ) and φ b j = ( s T b j , α b j ) , ( s I b j , β b j ) , ( s F b j , γ b j ) ( j = 1 , 2 , , n ) be two sets of 2TLNNs. If Δ 1 ( s T a j , α a j ) Δ 1 ( s T b j , α b j ) , Δ 1 ( s I a j , β a j ) Δ 1 ( s I b j , β b j )   and   Δ 1 ( s F a j , γ a j ) Δ 1 ( s F b j , γ b j ) hold for all j , then
2 TLNHM ( x ) ( φ a 1 , φ a 2 , , φ a n ) 2 TLNHM ( x ) ( φ b 1 , φ b 2 , , φ b n )
Proof: 
Let φ a j = ( s T a j , α a j ) , ( s I a j , β a j ) , ( s F a j , γ a j ) and φ b j = ( s T b j , α b j ) , ( s I b j , β b j ) , ( s F b j , γ b j ) , given that Δ 1 ( s T a j , α a j ) Δ 1 ( s T b j , α b j ) , we can obtain
j = 1 x ( Δ 1 ( s T a j , α a j ) t ) j = 1 x ( Δ 1 ( s T b j , α b j ) t ) ,
1 ( j = 1 x ( Δ 1 ( s T a j , α a j ) t ) ) 1 x 1 ( j = 1 x ( Δ 1 ( s T b j , α b j ) t ) ) 1 x .
Thereafter,
( 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s T a j , α a j ) t ) ) 1 x ) ) 1 C n x ( 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s T b j , α b j ) t ) ) 1 x ) ) 1 C n x
Furthermore,
1 ( 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s T a j , α a j ) t ) ) 1 x ) ) 1 C n x 1 ( 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s T b j , α b j ) t ) ) 1 x ) ) 1 C n x .
That means Δ 1 ( s T a , α a ) Δ 1 ( s T b , α b ) . Similarly, we can obtain Δ 1 ( s I a , β a ) Δ 1 ( s I b , β b ) and Δ 1 ( s F a , γ a ) Δ 1 ( s F b , γ b ) .
If Δ 1 ( s T a , α a ) < Δ 1 ( s T b , α b ) , Δ 1 ( s I a , β a ) Δ 1 ( s I b , β b ) and Δ 1 ( s F a , γ a ) Δ 1 ( s F b , γ b ) ,
2 TLNHM ( x ) ( φ a , φ a , , φ a ) < 2 TLNHM ( x ) ( φ b , φ b , , φ b )
If Δ 1 ( s T a , α a ) = Δ 1 ( s T b , α b ) , Δ 1 ( s I a , β a ) > Δ 1 ( s I b , β b )   and   Δ 1 ( s F a , γ a ) > Δ 1 ( s F b , γ b ) ,
2 TLNHM ( x ) ( φ a , φ a , , φ a ) < 2 TLNHM ( x ) ( φ b , φ b , , φ b )
If Δ 1 ( s T a , α a ) = Δ 1 ( s T b , α b ) , Δ 1 ( s I a , β a ) = Δ 1 ( s I b , β b )   and   Δ 1 ( s F a , γ a ) = Δ 1 ( s F b , γ b ) ,
2 TLNHM ( x ) ( φ a , φ a , , φ a ) = 2 TLNHM ( x ) ( φ b , φ b , , φ b )
So, Property 2 is right. □
Property 3.
(Boundedness) Let φ j = ( s T j , α j ) , ( s I j , β j ) , ( s F j , γ j ) ( j = 1 , 2 , , n ) be a set of 2TLNNs. If φ i + = ( max i ( s T j , α j ) , min i ( s I j , β j ) , min i ( s F j , γ j ) ) and φ i + = ( max i ( s T j , α j ) , min i ( s I j , β j ) , min i ( s F j , γ j ) ) , then
φ 2 TLNHM ( x ) ( φ 1 , φ 2 , , φ n ) φ + .
From Property 1,
2 TLNHM ( x ) ( φ 1 , φ 2 , , φ n ) = φ 2 TLNHM ( x ) ( φ 1 + , φ 2 + , , φ n + ) = φ + .
From Property 2,
φ 2 TLNHM ( x ) ( φ 1 , φ 2 , , φ n ) φ + .

3.2. The 2TLNWHM Operator

In an actual MADM, it is important to consider attribute weights. This section proposes a 2-tuple linguistic neutrosophic weighted Hamy mean (2TLNWHM) operator as follows.
Definition 8.
Let φ j = ( s T j , α j ) , ( s I j , β j ) , ( s F j , γ j ) ( j = 1 , 2 , , n ) be a set of 2TLNNs with a weight vector, w i = ( w 1 , w 2 , , w n ) T , thereby satisfying w i [ 0 , 1 ] and i = 1 n w i = 1 . Then, we can define the 2TLNWHM operator as follows:
2 TLNWHM w ( x ) ( φ 1 , φ 2 , , φ n ) = 1 i 1 < < i x n ( j = 1 x ( φ i j ) w i j ) 1 x C n x .
Theorem 2.
Let φ j = ( s T j , α j ) , ( s I j , β j ) , ( s F j , γ j ) ( j = 1 , 2 , , n ) be a set of 2TLNNs. The aggregated value determined using a 2TLNWHM operator is also a 2TLNN, where
2 TLNWHM w ( x ) ( φ 1 , φ 2 , , φ n ) = 1 i 1 < < i x n ( j = 1 x ( φ i j ) w i j ) 1 x C n x = { Δ ( t ( 1 ( 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s T j , α j ) t ) w i j ) 1 x ) ) 1 C n x ) ) , Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s I j , β j ) t ) w i j ) 1 x ) ) 1 C n x ) , Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s F j , γ j ) t ) w i j ) 1 x ) ) 1 C n x ) } .
Proof: 
From Definition 5, we can obtain,
( φ i j ) w i j = { Δ ( t ( Δ 1 ( s T j , α j ) t ) w i j ) , Δ ( t ( 1 ( 1 Δ 1 ( s I j , β j ) t ) w i j ) ) , Δ ( t ( 1 ( 1 Δ 1 ( s F j , γ j ) t ) w i j ) ) } .
Thus,
j = 1 x ( φ i j ) w i j = { Δ ( t ( j = 1 x ( Δ 1 ( s T j , α j ) t ) w i j ) ) , Δ ( t ( 1 j = 1 x ( 1 Δ 1 ( s I j , β j ) t ) w i j ) ) , Δ ( t ( 1 j = 1 x ( 1 Δ 1 ( s F j , γ j ) t ) w i j ) ) }
Therefore,
( j = 1 x ( φ i j ) w i j ) 1 x = { Δ ( t ( j = 1 x ( Δ 1 ( s T j , α j ) t ) w i j ) 1 x ) , Δ ( t ( 1 ( j = 1 x ( 1 Δ 1 ( s I j , β j ) t ) w i j ) 1 x ) ) , Δ ( t ( 1 ( j = 1 x ( 1 Δ 1 ( s F j , γ j ) t ) w i j ) 1 x ) ) } .
Thereafter,
1 i 1 < < i x n ( j = 1 x ( φ i j ) w i j ) 1 x = { Δ ( t ( 1 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s T j , α j ) t ) w i j ) 1 x ) ) ) , Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s I j , β j ) t ) w i j ) 1 x ) ) ) , Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s F j , γ j ) t ) w i j ) 1 x ) ) ) }
Furthermore,
2 TLNWHM w ( x ) ( φ 1 , φ 2 , , φ n ) = 1 i 1 < < i x n ( j = 1 x ( φ i j ) w i j ) 1 x C n x . = { Δ ( t ( 1 ( 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s T j , α j ) t ) w i j ) 1 x ) ) 1 C n x ) ) , Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s I j , β j ) t ) w i j ) 1 x ) ) 1 C n x ) , Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s F j , γ j ) t ) w i j ) 1 x ) ) 1 C n x ) } .
Hence, (23) is kept.
Then we need to prove that (23) is a 2TLNN. We need to prove two conditions as follows:
0 Δ 1 ( s T , α ) t , 0 Δ 1 ( s I , β ) t , 0 Δ 1 ( s F , γ ) t .
0 Δ 1 ( s T , α ) + Δ 1 ( s I , β ) + Δ 1 ( s F , γ ) 3 t .
Let
Δ 1 ( s T , α ) t = 1 ( 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s T j , α j ) t ) w i j ) 1 x ) ) 1 C n x Δ 1 ( s I , β ) t = ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s I j , β j ) t ) w i j ) 1 x ) ) 1 C n x
Δ 1 ( s F , γ ) t = ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s F j , γ j ) t ) w i j ) 1 x ) ) 1 C n x .
Proof. 
① Since 0 Δ 1 ( s T j , α j ) t 1 , we get
0 j = 1 x ( Δ 1 ( s T j , α j ) t ) w i j 1   and   0 1 ( j = 1 x ( Δ 1 ( s T j , α j ) t ) w i j ) 1 x 1
Then,
0 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s T j , α j ) t ) w i j ) 1 x ) 1
0 1 ( 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s T j , α j ) t ) w i j ) 1 x ) ) 1 C n x 1
That means 0 Δ 1 ( s T , α ) t . Similarly, we can get 0 Δ 1 ( s I , β ) t , 0 Δ 1 ( s F , γ ) t . So, ① is maintained; ② Since 0 Δ 1 ( s T , α ) t , 0 Δ 1 ( s I , β ) t , 0 Δ 1 ( s F , γ ) t . 0 Δ 1 ( s T , α ) + Δ 1 ( s I , β ) + Δ 1 ( s F , γ ) 3 t . □
Example 2.
Let ( s 5 , 0 ) , ( s 2 , 0 ) , ( s 1 , 0 ) , ( s 4 , 0 ) , ( s 3 , 0 ) , ( s 4 , 0 ) , ( s 2 , 0 ) , ( s 5 , 0 ) , ( s 1 , 0 ) and ( s 5 , 0 ) , ( s 1 , 0 ) , ( s 3 , 0 ) be four 2TLNNs, w = ( 0.2 , 0.3 , 0.4 , 0.1 ) and suppose x = 2 , then according to (23), we have
2 TLNHM ( 2 ) ( φ 1 , φ 2 , , φ n ) = 1 i 1 < < i x n ( j = 1 x ( φ i j ) w i j ) 1 x C n x = { Δ { 6 × { 1 { ( 1 ( ( 5 6 ) 0.2 × ( 4 6 ) 0.3 ) 1 2 ) × ( 1 ( ( 5 6 ) 0.2 × ( 2 6 ) 0.4 ) 1 2 ) × ( 1 ( ( 5 6 ) 0.2 × ( 5 6 ) 0.1 ) 1 2 ) × ( 1 ( ( 4 6 ) 0.3 × ( 2 6 ) 0.4 ) 1 2 ) × ( 1 ( ( 4 6 ) 0.3 × ( 5 6 ) 0.1 ) 1 2 ) × ( 1 ( ( 2 6 ) 0.4 × ( 5 6 ) 0.1 ) 1 2 ) } 1 C 4 2 } } , Δ { 6 × { ( 1 ( ( 1 ( 2 6 ) 0.2 ) × ( 1 ( 3 6 ) 0.3 ) ) 1 2 ) × ( 1 ( ( 1 ( 2 6 ) 0.2 ) × ( 1 ( 5 6 ) 0.4 ) ) 1 2 ) × ( 1 ( ( 1 ( 2 6 ) 0.2 ) × ( 1 ( 1 6 ) 0.1 ) ) 1 2 ) × ( 1 ( ( 1 ( 3 6 ) 0.3 ) × ( 1 ( 5 6 ) 0.4 ) ) 1 2 ) × ( 1 ( ( 1 ( 3 6 ) 0.3 ) × ( 1 ( 1 6 ) 0.1 ) ) 1 2 ) × ( 1 ( ( 1 ( 5 6 ) 0.4 ) × ( 1 ( 1 6 ) 0.1 ) ) 1 2 ) } 1 C 4 2 } , Δ { 6 × { ( 1 ( ( 1 ( 1 6 ) 0.2 ) × ( 1 ( 4 6 ) 0.3 ) ) 1 2 ) × ( 1 ( ( 1 ( 1 6 ) 0.2 ) × ( 1 ( 1 6 ) 0.4 ) ) 1 2 ) × ( 1 ( ( 1 ( 1 6 ) 0.2 ) × ( 1 ( 3 6 ) 0.1 ) ) 1 2 ) × ( 1 ( ( 1 ( 4 6 ) 0.3 ) × ( 1 ( 1 6 ) 0.4 ) ) 1 2 ) × ( 1 ( ( 1 ( 4 6 ) 0.3 ) × ( 1 ( 3 6 ) 0.1 ) ) 1 2 ) × ( 1 ( ( 1 ( 1 6 ) 0.4 ) × ( 1 ( 3 6 ) 0.1 ) ) 1 2 ) } 1 C 4 2 } } . = ( s 5 , 0.3604 ) , ( s 1 , 0.0344 ) , ( s 1 , 0.3963 )
Now, we will discuss some properties of the 2TLNWHM operator.
Property 4.
(Monotonicity) Let φ a j = ( s T a j , α a j ) , ( s I a j , β a j ) , ( s F a j , γ a j ) ( j = 1 , 2 , , n ) and φ b j = ( s T b j , α b j ) , ( s I b j , β b j ) , ( s F b j , γ b j ) ( j = 1 , 2 , , n ) be two sets of 2TLNNs. If Δ 1 ( s T a j , α a j ) Δ 1 ( s T b j , α b j ) , Δ 1 ( s I a j , β a j ) Δ 1 ( s I b j , β b j )   and   Δ 1 ( s F a j , γ a j ) Δ 1 ( s F b j , γ b j ) hold for all j , then
2 TLNWHM ( x ) ( φ a 1 , φ a 2 , , φ a n ) 2 TLNWHM ( x ) ( φ b 1 , φ b 2 , , φ b n )
The proof is similar to 2TLNWHM; it is omitted here.
Property 5.
(Boundedness) Let φ j = ( s T j , α j ) , ( s I j , β j ) , ( s F j , γ j ) ( j = 1 , 2 , , n ) be a set of 2TLNNs. If φ i + = ( max i ( s T j , α j ) , min i ( s I j , β j ) , min i ( s F j , γ j ) ) and φ i + = ( max i ( s T j , α j ) , min i ( s I j , β j ) , min i ( s F j , γ j ) ) , then
φ 2 TLNWHM ( x ) ( φ 1 , φ 2 , , φ n ) φ + .
From theorem 2, we get
2 TLNWHM w ( x ) ( ( φ 1 , φ 2 , , φ n ) ) = 1 i 1 < < i x n ( j = 1 x ( min φ i j ) w i j ) 1 x C n x = { Δ ( t ( 1 ( 1 i 1 < < i x n ( 1 ( j = 1 x ( min Δ 1 ( s T j , α j ) t ) w i j ) 1 x ) ) 1 C n x ) ) , Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 max Δ 1 ( s I j , β j ) t ) w i j ) 1 x ) ) 1 C n x ) , Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 max Δ 1 ( s F j , γ j ) t ) w i j ) 1 x ) ) 1 C n x ) } ,
2 TLNWHM w ( x ) ( ( φ 1 + , φ 2 + , , φ n + ) ) = 1 i 1 < < i x n ( j = 1 x ( max φ i j ) w i j ) 1 x C n x = { Δ ( t ( 1 ( 1 i 1 < < i x n ( 1 ( j = 1 x ( max Δ 1 ( s T j , α j ) t ) w i j ) 1 x ) ) 1 C n x ) ) , Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 min Δ 1 ( s I j , β j ) t ) w i j ) 1 x ) ) 1 C n x ) , Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 min Δ 1 ( s F j , γ j ) t ) w i j ) 1 x ) ) 1 C n x ) } .
From property 4, we get
φ 2 TLNWHM ( x ) ( φ 1 , φ 2 , , φ n ) φ +
It is obvious that the 2TLNWHM operator lacks the property of idempotency.

3.3. The 2TLNDHM Operator

Based on the Hamy mean (HM) operator [32], we propose the dual Hamy mean (DHM) operator.
Definition 9.
The DHM operator is defined as follows:
DHM ( x ) ( φ 1 , φ 2 , , φ n ) = ( 1 i 1 < < i x n ( j = 1 x φ i j x ) ) 1 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 ) ! .
In this section, we propose the 2-tuple linguistic neutrosophic DHM (2TLNDHM) operator.
Definition 10.
Let φ j = ( s T j , α j ) , ( s I j , β j ) , ( s F j , γ j ) ( j = 1 , 2 , , n ) be a set of 2TLNNs. The 2TLNDHM operator is:
2 TLNDHM ( x ) ( φ 1 , φ 2 , , φ n ) = ( 1 i 1 < < i x n ( j = 1 x φ i j x ) ) 1 C n x .
Theorem 3.
Let φ j = ( s T j , α j ) , ( s I j , β j ) , ( s F j , γ j ) ( j = 1 , 2 , , n ) be a set of 2TLNNs. The aggregated value determined using 2TLNDHM operators is also a 2TLNN where
2 TLNDHM ( x ) ( φ 1 , φ 2 , , φ n ) = ( 1 i 1 < < i x n ( j = 1 x φ i j x ) ) 1 C n x = { Δ ( t ( 1 i 1 < < i x n ( 1 ( j = 1 x ( 1 Δ 1 ( s T j , α j ) t ) ) 1 x ) ) 1 C n x ) , Δ ( t ( 1 ( 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s I j , β j ) t ) ) 1 x ) ) 1 C n x ) ) , Δ ( t ( 1 ( 1 i 1 < < i x n ( 1 ( j = 1 x ( Δ 1 ( s F j , γ j ) t ) ) 1 x ) ) 1 C n x ) ) }
Proof: 
j = 1 x φ i j = { Δ ( t ( 1 j = 1 x ( 1 Δ 1 ( s T j , α j ) t ) ) ) , Δ ( t ( j = 1 x ( Δ 1 ( s I j , β j ) t ) ) )