Mathematical Problems in Engineering.
Multiple criteria decision making approach with multi-valued neutrosophic linguistic normalized weighted Bonferroni mean Hamacher operator Bao-lin Li1，2, Juan-ru Wang1 , Li-hua Yang2 , Xue-tao Li2 1.
School of Management, Northwestern Polytechnical University, Xi’an 710072, China;
2.
School of Economics and Management, Hubei University of Automotive Technology,
Shiyan 442002, China Corresponding author: Juan-ru Wang E-mail: wjuanru@nwpu.edu.cn Abstract The neutrosophic set and linguistic term set are widely applied in recently years. Motivated by the advantages of them, we co mbine the mult i-valued neutrosophic set and linguistic set and define the concept of the mult i-valued neutrosophic linguistic set (MVNLS). Furthermore, Hamacher operation is an extension of the Algebraic and Einstein operation. Additionally, the normalized weighted Bonferroni mean (NW BM) operator can consider the weight of each argument as well as capture the interrelationship of different arguments . Therefore, the co mbination of NWBM operator and Hamacher operation is more valuable and agile. Firstly, M VNLS and mult i-valued neutrosophic linguistic number (M VNLN) are defined, then some new operational ru les of MVNLNs on account of Hamacher operations are developed, and the comparison functions for MVNLNs are given. Secondly, mu lti-valued neutrosophic linguistic normalized weighted Bonferroni mean Hamacher o perator (M VNLNW BMH) is proposed, and a number of expected characteristics of new operator are investigated. Meanwhile, some special cases of different parameters p ,q and are analy zed. Th ird ly, the approach utilizing the MVNLNW BMH operator is introduced to manage multip le criteria decision making issue (MCDM) in mu lt i-valued neutrosophic linguistic environ ment. Ultimately, a practical example is presented and a comparative analysis is carried out, which validate the effectiveness and generalization of the novel approach. Keywords Multi-valued neutrosophic linguistic, NWBM, Hamacher, Multiple criteria decision making
1 Introduction In real world, due to the complexity of decision informat ion, the fuzzy theory has attracted widespread attentions and has been developed in various fields. Zaheh [1] firstly proposed the notion of fuzzy sets (FSs). Then, Atanassov [2] introduced the intuitionistic fuzzy sets (IFSs), which overco me the weakness of non-membership degrees. Subsequently, in order to address the hesitation degree of decision-makers, Torra [3] defined hesitant fuzzy sets (HFSs). Fuzzy set theory has gained well pro moted, but it still cannot manage the inconsistent and indeterminate information. Under this circu mstance, Smarandache [4] proposed Neutrosophic Sets (NSs), whose indeterminacy degree is independent on both true and false membership. NS is an extension of IFS, and makes decision-makers express their preference more accurately, so some achievements on NSs and its extensions have been undertaken. Some various concepts of different NSs are defined. For examp le, Smarandache [54] and Wang et al. [5] introduced single-valued neutrosophic sets (SVNs) to facilitate its application. Ye [6] pointed out the concept of simp lified neutrosopic sets (SNSs).Wang [7] developed the concept of interval neutrosopic sets (INSs). However, under certain conditions, the decision makers likely give different evaluation numbers for expressing their hesitant. Subsequently, the definition of single-valued neutrosophic hesitant fuzzy sets (SVNHFSs) was firstly proposed by Ye [8] in 2014, then Wang [9] also proposed mult i-valued neutrosophic sets (MVNSs) in 2015. Actually, the notions of SVNHFSs and MVNSs are equal. For simplicity, we adapt the term of MVNSs in this paper.
1
Mathematical Problems in Engineering. On the other hand, the aggregation operators, comparison method for Neutrosophic numbers are also been studied. For SVNSs, Liu [10] employed NWBM operator to solve mult iple criteria problem in single-valued neutrosophic environment. Ye [11] gave the definitions of cross-entropy and correlation coefficient. Fo r INSs, Zhang [12] developed some aggregation operators. Liu [13] not only provided the definition of interval neutrosophic hesitant fuzzy sets (INHFSs), but also discussed the generalized hybrid weighted average operator. Broumi and Smarandache [14-16] studied the correlation coefficients, cosine similarity measure and some new operations. Ye [17] proposed similarity measures between interval neutrosophic sets. For M VNSs, Ye [8] developed SVNHFWA and SVNHFW G operators for MCDM problem. Peng [18, 19] extended power aggregation operators and defined some outranking relat ions under MVNS environ ment. Ji et al. [20] analy zed a novel TODIM method for MVNSs. In real life, o wing to the amb iguity of decision makers’ thinking, people prefer to utilize linguistic variables for describing their assessment value rather than the quantization value. Therefore, linguistic variable has attracted widespread attention in the field of MCDM. The linguistic variab le was firstly proposed by Zadeh [21] and applied fo r the fuzzy reasoning. After that, a series of works on it have been made, Wang [22-24] presented a new approach in view of hesitant fuzzy linguistic informat ion. Meng [25] developed linguistic hesitant fuzzy sets and studied hybrid weighted operator. Tian [26] defined gray linguistic weighted Bonferroni mean operator for MCDM. In order to indicate the true, indeterminate and false extents concerning a linguistic term, the NSs and linguistic set (LS) are co mb ined. Several neutrosophic linguistic sets and its corresponding operator are defined, for examp le single valued o r simp lified neutrosophic linguistic sets and trapezoid linguistic sets [27-30], interval neutrosophic certain or uncertain linguistic sets [31-33].However, due to the hesitancy of people’s thinking, the true of a linguistic term may be given several values, and the case is similar to the false and indeterminate extents. The existing literatures don’t consider this perspective. Therefore, the mult i-valued neutrosophic linguistic set (MVNLS) and mult i-valued neutrosophic linguistic number (M VNLN) in this article are proposed in order to better express the information. Aggregation operator which can fuse multiple arguments into a single comprehensive value is an important tool for MCDM p roblem. Many researchers have developed some efficient operators [34-41], for instance, the weighted geometric average (W GA ) or averag ing (WA) operator, prioritized aggregation (PA) operator, Maclaurin symmetric mean and Bonferroni mean (BM) operator. BM operator was originally defined by Bonferroni [42], which has attracted widespread attentions because of its characteristics of capture interrelat ionship among arguments. Some ach ievements have been made on it [43-49].In order to aggregate neutrosophic linguistic informat ion, some researches on aggregation operators under neutrosophic linguistic and neutrosophic uncertain linguistic environ ments are also been applied [27-33,50].Until to now, BM and NWBM fail to accommodate aggregation informat ion for mult i-valued neutrosophic linguistic environ ment. Motivated by this limitat ion, we will extend the NWBM operator to MVNLS in this article. T-norms and t-conorms are two functions that satisfy certain conditions respectively. The Archimedean t-conorms and t-norms are well-known, which include algebraic, Einstein and Hamacher, Hamacher operation is an extension of algebraic and Einstein. Generally, the algebraic operators are commonly, there are also a few aggregation operations based on Einstein operations. Due to Hamacher operator is more general, Liu [51, 52] discussed the Hamacher operational rules. So far, there is no research for M VNLS based on Hamacher operations. Since it is better for M VNLS to depict the actual situation, NWBM operator can capture the interrelationship among arguments, and Hamacher operations are more general, it is of great meaning to study the NWBM Hamacher operators under multi-valued neutrosophic linguistic environment for MCDM problems. The main purposes of the paper are presented in the following: 1.
To be better exp ress people’s hesitant, comb ining the M VNS and LS, we give the notions of MVNLS and M VNLN, besides, the score, accuracy and certainty functions are also investigated to compare MVNLNs.
2
Mathematical Problems in Engineering. 2.
Due to the generalizat ion of Hamacher operational rules, we define new operations of M VNLNs based on Hamacher operational rules, and discuss their operational relation s.
3.
The NWBM considering the interrelationship of different argu ments has gained widespread concerns, we extend NWBM operator to M VNLN environ ment, the M VNLNW BMH operator is defined, moreover, some desirable characteristics are also studied.
4.
In order to verify the effectiveness, an examp le for M CDM problem utilizing M VNLNWBM H operator is illustrated and conduct a comparative analysis. We also analyze the influence s of different parameter values for the final outcomes, the results demonstrate the operator proposed is more general and flexible. The article is arranged in this way. In Section 2, we review a nu mber o f notions and operations for MVNS, LS, NW BM operator and Hamacher. In Sect ion 3, we propose the definitions of MVNLS and MVNLN, and develop the operations of M VNLNs on the basis of Hamacher t -conorms and t-norms. Meanwhile, the A lgebraic as well as Einstein for M VNLNs are also presented, which are special cases of Hamacher operation. Moreover, the comparison method of M VNLNs is also defined. In Section 4, we propose the M VNLNW BMH operator and investigate its properties. Furthe rmo re, when corresponding parameters are assigned different values, the special examp les are also discussed. In Section 5, we establish the MCDM procedure on account of the proposed aggregation operators with MVNLS in formation. Section 6 presents a concrete example, as well as a comparison analysis is provided to show the practicability utilizing our method. Finally, in Section 7, some results are presented.
2 Preliminaries Some notions and operation are introduced in this section, which will be useful in the latter analysis. 2.1 Linguistic term sets Suppose that S s1 ,s2 ,
,s is an ordered and finite linguistic set, in which s j denotes a
linguistic variable value and is an odd value. When is equal to seven, the corresponding linguistic set are provided in the following: S s1 ,s2 ,s3 ,s 4 ,s5 ,s6 ,s7 extremely poor,very poor,poor,medium,good,very good,extremely good . In order to avoid the linguistic information loss, the set above is expanded, that is a contiguous set,
S s R .
Definiti on 1 [53] Let si and s j be any two linguistic variables, the corresponding operations are presented:
(1) si s i , 0; (2)si s j si j ; (3)si s j si j ; (4) si
si .
2.2 Multi-valued neutrosophic sets Definition 2 [8,9] Suppose that X is a collection of objects, MVNSs A on X is defined by
A
x ,T (x ),I (x ),F (x ) A
A
x X ,
A
Where TA(x ) TA(x ) ,I A(x ) I A(x ) , FA(x ) FA(x ) ,
TA(x ),I A(x ),and FA(x ) are three collections of crisp numbers belonging to 0,1 , representing the probable true-membership degree, indeterminacy-membership degree and falsity- membership degree,
3
Mathematical Problems in Engineering. where x in X belonging to A , respectively, satisfying these conditions 0 , , 1, and 0 sup TA(x ) sup I A(x ) sup FA(x ) 3. If there is only one element in X , A is indicated by the three tuple A TA(x ),I A(x ),FA(x ) , that is known as a mu lti-valued neutrosophic number (M VNN).Generally, M VNSs is considered as the generalizations of the other sets, such as FSs, IFSs, HFSs, DHFs, and SVNSs.
2.3 Normalized weighted Bonferroni mean Definiti on 3 [42] Let
p ,q 0,as well as ai(i 1,2,
n ) be a set of nonnegative
values, then the BM is defined as
n 1 ,an ) aip ajq n(n 1) i ,j 1, i j
BM p ,q(a1 ,a2 ,
Definiti on 4 [45]Let
1
p q
p ,q 0,and ai(i 1,2,
n ) be a set of nonnegative values, and
the corresponding NWBM can be expressed as below: 1
NWBM Where
p ,q
(a1 ,a2 ,
n ww p q p q i j ,an ) 1w (ai aj ) i ,j 1, i i j
w ( w1 , w2 , , wn )
of ai i 1,2,
represents
the
corresponding
weighted
vector
n
, n , satisfying w i 0 and w i 1 . The weight vector can be i 1
given by decision-makers in real problem. Obviously, the NWBM operator possesses a few characteristic s such as commutativity, reducibility, monotonicity, boundedness, and idempotency. 2.4 Hamacher operations
We know aggregation operator is given in accordance with different t-norms and tconorms, there are some exceptional circumstances listed in the following: (1) Algebraic t-norm and t-conorm a b ab,a b a b ab;
(2) Einstein t-norm and t-conorm a b
ab a b ,a b ; 1 ab 1 1 a 1 b
(3) Hamacher t-norm and t-conorm
a b
ab a b ab (1 )ab ,a b , 0 1 (1 )ab 1 a b ab
4
Mathematical Problems in Engineering.
In special, when 1, 2, the Algebraic and Einstein operations are the simplifications of Hamacher t-norm and t-conorm.
3 Multi-valued neutrosophic linguistic set 3.1 MVNLS and its Hamacher operations Definition 5 Let X be a set of points, an MVNLS A in X is defined as follows:
A
x ,s
, TA(x ),I A(x ),FA(x ) x X ,
( x )
Where s(x ) S , TA(x ) TA(x ) ,I A(x ) I A(x ) , FA(x ) FA(x ) ,
TA(x ),I A(x ), and FA(x ) are three sets of crisp values in 0,1 , denoting three degrees of
x in X belonging to s(x ), , that are true, indeterminacy and falsity, satisfying these conditions 0 , , 1,and 0 sup TA(x ) sup I A(x ) sup FA(x ) 1.
Definiti on 6 Let A
x ,s
, TA(x ),I A(x ),FA(x ) x X be an MVNLS, supposing there is
( x )
only one element in X , then tuple
s(x ),TA(x ),I A(x ),FA(x )
is depicted as a mu lt i-valued
neutrosophic linguistic number (MVNLN).For simplicity, the MVNLN can also be represented as
A
s ,T (x ),I (x ),F (x ) ( x )
A
A
A
x X
Definiti on 7 Let a1 s(a1 ), T(a1 ),I(a1 ),F(a1 )
and a2 s(a2 ), T(a2 ),I(a2 ),F(a2 )
be two
MVNLNs, and 0,then the operations of MVNLNs can be defined on the basis of Hamacher operations. (1) a1 a2
s(a ) (a ), 1
2
1 T (a1 ), 2 T (a2 )
1 I (a1 ), 2 I (a2 )
1 2 1 2 (1 ) 1 2 , 1 (1 ) 1 2
1 2 , (1 )(1 2 1 2 )
1 F (a1 ),2 F (a2 )
12 ; (1 )(1 2 12 )
(2) a1 a2 s(a )(a ), 1
2
1 T (a1 ), 2 T (a2 )
1 I (a1 ), 2 I (a2 )
1 F (a1 ),2 F (a2 )
1 2 , (1 )( 1 2 1 2 )
1 2 1 2 (1 )1 2 , 1 (1 )1 2 1 2 12 (1 )12 ; 1 (1 )12
5
Mathematical Problems in Engineering.
(3) a1 s (a ), 1
1 T (a1 )
1 I (a1 )
1 F (a1 )
(1 ( 1) 1 ) (1 1 ) , (1 ( 1) 1 ) ( 1)(1 1 )
1 , (1 ( 1)(1 1 )) ( 1)1 1 ; (1 ( 1)(1 1 )) ( 1)1
(4) a1 s (a ), 1
1 T (a1 )
1 I (a1 )
1 F (a1 )
1 , (1 ( 1)(1 1 )) ( 1) 1
(1 ( 1)1 ) (1 1 ) , (1 ( 1)1 ) ( 1)(1 1 ) (1 ( 1)1 ) (1 1 ) . (1 ( 1)1 ) ( 1)(1 1 )
If 1,then the operations based on Hamacher operational ru les in Definit ion 7 will simp lified to the Algebraic operational rules in the following: (5) a1 a2
s(a ) (a ),
1
2
, , ;
1 T (a1 ), 2 T (a2 ) 1 I (a1 ), 2 I (a2 ) 1 F (a1 ),2 F (a2 )
1
2
1 2
1 2
1 2
(6) a1 a2 s(a )(a ),
1
2
, , ;
1 T (a1 ), 2 T (a2 ) 1 I (a1 ), 2 I (a2 ) 1 F (a1 ),2 F (a2 )
1 2
1
2
1 2
1
2
1 2
(7) a1 s (a ),
1
1 (1 ) , , ;
1 T (a1 )
1
1 I (a1 )
1
1 F (a1 )
1
6
Mathematical Problems in Engineering.
(8) a1 s (a ),
1
, 1 (1 ) , 1 (1 ) .
1 T (a1 )
1
1 I (a1 )
1
1 F (a1 )
1
Supposing T (a1 ),I(a1 ),F (a1 ),T (a2 ),I(a2 ), and F (a 2 ) contain only one value, then the operations defined above can be reduced to the operations of SVNLNs based on Algebraic operations proposed by Ye [27]. If 2,then the operations based on Hamacher operational ru les in Defin ition 7 will simp lified to the Einstein operations of MVNLNs presented below: (9) a1 a2
s(a ) (a ), 1
2
1 T (a1 ), 2 T (a2 )
1 I (a1 ), 2 I (a2 )
1 F (a1 ),2 F (a2 )
1 2 , 1 1 2
1 2 , 2 1 2 1 2 12 ; 2 1 2 12
(10) a1 a2 s(a )(a ), 1
2
1 T (a1 ), 2 T (a2 )
1 I (a1 ), 2 I (a2 )
1 F (a1 ),2 F (a2 )
1 2 , 2 1 2 1 2
1 2 , 1 1 2 1 2 ; 1 12
(11) a1 s (a ), 1
1 T (a1 )
1 I (a1 )
1 F (a1 )
(1 1 ) (1 1 ) , (1 1 ) (1 1 )
21 , (2 1 ) 1 21 ; (2 1 ) 1
7
Mathematical Problems in Engineering.
(12) a1 s (a ), 1
1 T (a1 )
1 I (a1 )
1 F (a1 )
2 1 , (2 1 ) 1
(1 1 ) (1 1 ) , (1 1 ) (1 1 ) (1 1 ) (1 1 ) . (1 1 ) (1 1 )
Supposing T (a1 ),I(a1 ),F (a1 ),T (a2 ),I(a2 ),and F (a 2 ) contain only one value, then the operations defined above can be reduced to the operations of SVNLNs based on Einstein operations.
Theorem 1 Let a1 s(a ),T(a1 ),I(a1 ),F(a1 ) , 1
a2 s(a ),T(a2 ),I(a2 ),F(a2 ) , and 2
a3 s(a ),T(a3 ),I(a3 ),F(a3 ) be any three M VNLNs, and ,1 ,2 0, then the properties 3
below are correct: (1) a1 a2 a2 a1; (2) a1 a2 a2 a1;
(3) a1 a2 a1 a2; (4) 1a1 2a1 (1 2 )a1; (5) a11 a12 a11 2 ;
(6) a1 a2 a1 a2 ;
a a
(7) (8)
1
a2 a3 a1 a2 a3 ;
1
a2 a3 a1 a2 a3 .
Then, equation (4) will be proved as follows: Proof (4) Since 1 ,2 0,
1a1 2a1 s 1(a1 ) 2(a1 ),
1 T (a1 )
1 I (a1 )
1 F (a1 )
(1 ( 1) 1 )1 (1 1 )1 (1 ( 1) 1 )2 (1 1 )2 (1 ( 1) 1 )1 (1 1 )1 (1 ( 1) 1 )2 (1 1 )2 (1 ( 1) 1 )1 (1 1 )1 (1 ( 1) 1 )2 (1 1 )2 (1 ) (1 ( 1) 1 )1 ( 1)(1 1)1 (1 ( 1) 1)2 ( 1)(1 1)2 (1 ( 1) 1 ) 1 ( 1)(1 1 ) 1 (1 ( 1) 1) 2 ( 1)(1 1) 2 (1 ( 1) 1) 1 ( 1)(1 1) 1 (1 ( 1) 1 ) 2 ( 1)(1 1 ) 2 , 1 1 2 2 (1 ( 1) 1 ) (1 1 ) (1 ( 1) 1 ) (1 1 ) 1 (1 ) (1 ( 1) 1 )1 ( 1)(1 1 )1 (1 ( 1) 1 )2 ( 1)(1 1 )2
11 12 (1 ( 1)(1 1 ))1 ( 1)11 (1 ( 1)(1 1))2 ( 1)12 , 1 2 1 2 1 1 1 1 (1 )( ) 1 1 2 2 1 1 2 2 (1 ( 1)(1 1 )) ( 1)1 (1 ( 1)(1 1 )) ( 1)1 (1 ( 1)(1 1 )) ( 1)1 (1 ( 1)(1 1)) ( 1)1 11 12 (1 ( 1)(1 1 ))1 ( 1)11 (1 ( 1)(1 1 ))2 ( 1)12 1 2 1 2 1 1 1 1 (1 )( ) 1 1 2 2 1 1 2 2 (1 ( 1)(1 1 )) ( 1)1 (1 ( 1)(1 1 )) ( 1)1 (1 ( 1)(1 1)) ( 1)1 (1 ( 1)(1 1)) ( 1)1
s(1 2 )(a1 ),
1 T (a1 )
1 I (a1 )
1 F (a1 )
(1 ( 1) 1 ) 1 2 (1 1 ) 1 2 , 1 2 ( 1)(1 1 )1 2 (1 ( 1) 1 )
11 2 , 1 2 ( 1)11 2 (1 ( 1)(1 1 )) 11 2 1 2 ( 1)11 2 (1 ( 1)(1 1 ))
(1 2 )a1
Therefore, equation (4) 1a1 2a1 (1 2 )a1 can be obtained.
8
Mathematical Problems in Engineering.
Similarly, the other equations in Theorem1 are easily certified in the light of Definition 7. 3.2 Comparison method The sco re, accu racy , and certainty fun ct ions are impo rtant inde xes to ran k M VNLNs , an d its corresponding definition is given below: Definiti on 8 Let
a s(a ),T (a ),I(a ),F(a ) be an M VNLN, and the sco re, accuracy , and
certainty functions are achieved as below.
1 1 1 (1) E(a ) ( ) s(a ) T ( a ), I ( a ), F ( a ) 3 T (a ) I (a ) F (a ) s 1 1 1 ) (a ) T (a ), I (a ), F (a )( 3 T (a ) I (a ) F (a )
1 (2) H(a ) ( ) s(a ) T (a ), F (a ) T (a ) F (a ) s 1 T (a ), F (a )( )(a ) T (a ) F (a )
1 (3) C(a ) T (a ) s 1 T (a ) (a ) T (a )
T (a )
s(a )
Where T (a ),I (a ), and F (a ) are the numbers of the values in TA(x ),I A(x ), and FA(x ) respectively. The linguistic variable s(a ) is important for an M VNLN. Therefore, the co mparison functions defined above in Definition 8 are denoted as the linguistic variable. The bigger the truth degree T (a ) concerning the variable s(a ) is, meanwh ile, the smaller the indeterminacy degree I(a ) as well as the false degree F(a ) concerning the linguistic variable s(a ) are, then the higher the MVNLN is. Relating to the function of score, the greater corresponding to s(a ) is, the higher the affirmat ive statement is. Relating to the function of accuracy, the greater minus is, the certain the statement is. Regarding to the function of certainty, the bigger is, the certain the statement is. Based on Definition 8, the comparison method between MVNLNs are obtained. Definiti on 9 Supposing a1 and a2 are two MVNLNs, the compared approach is achieved in the following: (1) Supposing that E(a1 ) E(a2 ) , then a1 is greater than a2 , represented as a1
a2 ;
(2) Supposing that E(a1 ) E(a2 ) , and H (a1 ) H (a2 ) ,then a1 is greater than a2 , represented as a1
a2 ;
(3) Supposing that E(a1 ) E(a2 ) , H (a1 ) H (a2 ) , and C(a1 ) C(a2 ) ,then a1 is greater than a2 , represented as a1
a2 ;
(4) Supposing that E(a1 ) E(a2 ) , H (a1 ) H (a2 ) , and C(a1 ) C(a2 ) ,then a1 equals a2 , represented as a1
a2 ;
9
Mathematical Problems in Engineering. 4 The multi-val ued neutrosophic linguistic normalized weighted Bonferroni mean Hamacher operator The NWBM operator can not only take into account the advantages of BM and WBM, but also has the property of reducibility and idempotency. However, the NW BM operator has not been applied to the cases where the input arguments are MVNLNs. Definiti on
10
Let
ai(i 1,2,
,n )
be
a
ai s(ai ),T(ai ),I(ai ),F(ai ) , p ,q 0, and 1 ,2 , for ai ,i 0,1 and
n
i i
space
of
M VNLNs,
,n be the weighted vector
1.Then the operator of M VNLNWBM H is achieved as below, the
1
aggregation result is still an MVNLN. 1
MVNLNWBMH (a1 ,a2 ,
n p q i j p q an ) (ai aj ) i ,j 1 1 i i j
According to the operational laws in Definition 7, the results are derived below:
10
Mathematical Problems in Engineering.
MVNLNWBMH (a1 ,a2 ,
an )
s n i j p (ai ) q (aj i ,j 1 1 i i j
i T (ai ), j T (aj )
i I (ai ), j I (aj )
i F (ai ),j F (aj )
x y x 1 y 1 p q x y 1 1 1 x 1 y
1 1
1
1
1
1
p q p q u u 1 1 1 v 1 u v 1 u
1 1 g 2 1 h p q g h p q 1 1 g 2 1 h p q 1 g h p q
1 1 v 2 1 u p q v u p q 1 1 v 2 1 u p q 1 v u p q
n
i j , 1
,
,
1
1 x y p q 1 1 x 2 1 y p q 1 x y p q
,
p q )
Where x
1
p q p q u u 1 1 1 v 1 u v 1 u
i F (ai ),j F (aj )
1
1
1
p q
p q p q h h 1 1 1 g 1 h g 1 h
i I (ai ), j I (aj )
1 x x y 1 y
1
s
i T (ai ), j T (aj )
1
p q
p q p q h h 1 1 1 g 1 h g 1 h
n i j p (ai ) q (aj i ,j 1 1 i i j
,
1
p q )
1 i
p
1 j
q
,
,
(1)
2 1 ip jq
i j 1 i
,
i j
11
Mathematical Problems in Engineering. n
i ,j 1 i j n
i ,j 1 i j
n
n
1 i
1
1 j
( 1) 1 i
1 i
1
1 j
1 i
1 i
1
1 j
( 1) 1 i
1 i
1
1 j
1 i
1
i ,j 1 i j
u
1
i ,j 1 i j
v
1
g
h
1 i
n
1
i ,j 1 i j
p
p
p
p
1 j
q
q
q
q
q
i j p
i j
y
p
q 1 i
,
p
i j q 1 i
1
p
2
j
i j q 1 i
1 j
2
p
, i j q 1 i
1
p
j
i j q 1 i
1 j
,
.
,
Proof. According to the operational rules for MVNLNs, the results below can be gained
ai
p
s p (a ), i
i p p i T (ai ) 1 1 1 i 1 i p
,
p p 1 1 i 1 i p i I (ai ) p 1 1 i 1 1 i
p p 1 1 i 1 i p i F (ai ) p 1 1 i 1 1 i
aj q
s q (a ), j
jq q j T (aj ) 1 1 1 j 1 j q
,
q q 1 1 j 1 j q j I (aj ) q 1 1 j 1 1 j
q q 1 1 j 1 j q j F (aj ) q 1 1 j 1 1 j
,
,
12
Mathematical Problems in Engineering. , q i ) (aj )
ai p aj q
i T (ai ), j T (aj ) 1 i
s p (a
i p j q
1 j
p
q
1 i p j q
,
p q q p 1 1 i 1 1 j 1 i 1 j p q i I (ai ),j I (aj ) q p 1 1 i 1 1 j 1 1 i 1 j
p q q p 1 1 i 1 1 j 1 i 1 j p q i F (ai ),j F (aj ) q p 1 1 i 1 1 j 1 1 i 1 j
,
Firstly, we need to testify the mathematical formula below. n
i ,j 1 i j
i j 1 i
(ai p aj q )
s n , i j p q 1 i (ai ) (aj ) i ,j 1 i j
i j i j n n p q p q 1 1 i 1 i 1 j i p j q 1 i 1 j 2 1 i p j q i i i ,j 1 ,j 1 i j i j i j i T (ai ), j T (aj ) n n p q p q 1 i 1 j 2 1 i p j q 1 1 i 1 j i p j q 1 i i ,j 1 ii ,jj1 i j
i I (ai ),j I (aj ) n 1 1 i ii ,jj1
i F (ai ),j F (aj ) n 1 1 i ii ,jj1
n
i ,j 1 i j
1 p
1 j
q
1 1
p
i
( 2 1) 1 i
p
1
1 j
i j q 1 i
1 j
q
1 i
1
n
p
i j 1 i
i j q 1 i
1 j
1 i j , 1
1 i
1 p
1 j
q
1 i
p
i j q 1 i
1 j
i j
n
i ,j 1 i j
1 p
1 j
q
1 1 i
( 2 1) 1 i
p
p
1
1 j
i j q 1 i
1 j
q
1 i
1
n
i j q 1 i
1
p
j
1 i j , 1
1 i
1 p
1 j
q
1 i
p
i j q 1 i
1 j
i j
, ,
(2) The
mathematical
induction
n
on
is
adopt
to
prove
Eq.(2)
(1) Supposing n 2,the equation below is obtained. 2
i j 12 (ai p aj q ) (a1p a2q ) 2 1 (a2 p a1q ) 1 1 1 2 1 1 i
i j ,
i j
s
1 2 (a1 ) (a2 ) 2 1 p (a2 ) q (a1 ) 1 1 1 2 p
q
i I (ai ), j I (aj )
,
i T (ai ), j T (aj )
x 1x 2 y 1y 2 , x 1x 2 1 y 1y 2
m1m 2 , z z 1 m 1m 2 1 2
v 1v 2 i F (ai ),j F (aj ) u u 1 v 1v 2 1 2 Where x 1 1 1 1 1
1p 2q
p
1 2
q
1 1p 2q
12 1 1
,
13
.
Mathematical Problems in Engineering.
x 2 1 1
2 p 1q
1 2
y 1 1
1 1
p
1 2
q
2 p 1q
1 2
p
1 1
q
1 1
q
1 1p 2q
1 1
p
,
p q 1 2 1
q
21 1 2
,
p
q
1
2
q
1
p
2
q
1
2
1 1 1 1 1 1 1 1 1 1 1
m 2
1
p
2
1 2
p
q
p
1
p
q
2
1
1 1
p
1
q
p
q
1
1 1
2
1
p
1
p
p
q
1
2
q
p
2
q
1
p
q
1
p
u1 1 1 1
2
p
q
2
1
q
p
1
1
q
2
q
2
1 2
q
q
2
2
2
p
1 1 1 1 1 1 1 1 1 1 1
q
1
p
p
p
1
q
2
2
1 2
1
,
1
1 1 1 1 1 1 1 1 1 1 1
v 2
21 1 2
1 1 1 1 1 1 1 1 1 1 1
,
p
2
1
v 1
q
p
1
1
12 1 1
1 1 1 1 1 1 1 1 1 1 1
z2 1 1 1
2
1
z1 1 1 1
q
q
2
p
1
,
12 1 1
1 1 1 1 1 1 1 1 1 1 1
m 1
1
p
1p 2q
y 2 1
p q 1 2 1
21 1 2
2
q
1
1
q
2
p
,
p
q
1
2
1
q
2
p q p q 1 1 2 1 1 1 1 2 1 1 u 2 1 1 1 p q p q 1 1 2 1 1 1 1 1 2 1 1
,
21 1 2
p
21 1 2
,
q
2
,
12 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1
p
1
12 1 1
12 1 1
,
21 1 2
,
14
Mathematical Problems in Engineering. Then, i j
2
1 i
i ,j 1 i j
s
2
(ai p aj q )
12 1 1
,
i j p (ai ) q (aj ) 1 i
i ,j 1 i j
21
(a1p a2q )
(a2 p a1q )
1 2
i j i j 2 2 p q p q 1 i 1 i 1 i 1 j 2 1 i p j q 1 i 1 j i p j q i ,j 1 i ,j 1 i j i j i j i T (ai ), j T (aj ) 2 2 p q p q 2 p q 1 i 1 j 1 i j 1 1 i 1 j i p j q 1 i i ,j 1 ii ,jj1 i j
i I (ai ),j I (aj ) 2 1 1 i ii ,jj1
i F (ai ),j F (aj ) 2 1 1 i ii ,jj1
2
i ,j 1 i j
1 p
1 j
2
i ,j 1 i j
1 p
1 j
1 1
p
1
1 j
i j q 1 i
j
p
i
p
1 j
i j q 1 i
1 j
1 i
q
1
1
1
1 1
( 2 1) 1 i
q
p
i
( 2 1) 1 i
q
1 i
q
1
p
1
1 i
1 j
p
i j 1 i
q
1 i
p
i j q 1 i
1 j
1 i
1 p
1 j
q
1 i
p
i j q 1 i
1 j
,
,
i j q 1 i
j
, 1
1
1 i j 2
i j q 1 i
j
2
1
p
1
i ,j 1 i j
i j
.
We can achieve Eq. (2) is right when n k . (2) Supposing n k , the equation below is right, then i j
k
i ,j 1 i j
1 i
(ai p aj q )
s k , i j p q 1 i (ai ) (aj ) i ,j 1 i j
i j i j k k p q p q 1 i 1 i 1 i 1 j 2 1 i p j q 1 i 1 j i p j q i ,j 1 i ,j 1 i j i j i j i T (ai ), j T (aj ) k k p q p q 2 p q 1 i 1 j 1 i j 1 1 i 1 j i p j q 1 i i ,j 1 i ,j 1 i j i j
i I (ai ),j I (aj ) k 1 1 i i ,j 1 i j
i F (ai ),j F (aj ) k 1 1 i ii ,jj1
k
i ,j 1 i j
1 p
1 j
q
k
i ,j 1 i j
1 p
1 j
q
1 1
p
i
1 j
i j q 1 i
j
1 1 i
( 2 1) 1 i
1
p
p
i j q 1 i
1 j
1 j
1 i
q
1
1
1
p
( 2 1) 1 i
k
1
i j 1 i
p
1 i
1
1
1 j
p
1 i
q
i j q 1 i
1
p
j
i j
q 1 i
j
1 k
1 i
i ,j 1 i j
1
1 j
p
If n k 1 , we need to calculate the equation below. k 1 k k i j (ai p aj q ) i j (ai p aj q ) i k 1 (ai p ak 1q ) 1 1 1 i i ,j 1 i ,j 1 i 1 i i i j
i j q 1 i
j
1 i
q
1
p
1
i ,j 1 i j
i j
k
1 i
q
k 1j
1 k j 1
p
i j q 1 i
1 j
(ak 1p aj q )
1
The mathematical induction on k are used to testify the equations below. i k 1
k
(a 1 i i i 1
p
ak 1q )
sk
i k 1
i 1
1 i
p (ai ) q (ak 1 )
,
i k 1 i k 1 k k p q p q 1 i 1 i 1 i 1 k 1 2 1 i p k 1q 1 i 1 k 1 i p k 1q i 1 i 1 i k 1 i k 1 i T (ai ), k 1 T (ak 1 ) k k p q p q 1 i 1 i 1 k 1 2 1 i p k 1q 1 1 i 1 k 1 i p k 1q 1 i i 1 i 1
i I (ai ), k 1 I (ak 1 ) k 1 1 i i 1
i F (ai ),k 1 F (ak 1 ) k 1 1 i i 1
k
1 1 i i 1
1 p
1 k 1
q
( 2 1) 1 i k
p
p
1 p
1 k 1
q
( 2 1) 1 i
1
1 k
q
1
p
1 1 i i 1
p
i k 1 1 i
q
1
q
1 i k
1 k 1
i k 1 1 i
p
q
1
k
p
1 p
1 k
q
1
1 1 1 i i 1
1 k
1 i
,
i k 1 q 1 i
1 1 1 i i 1
1
1 k
1 k 1
q
1 i
p
1 k
q
1
i k 1 1 i
i k 1 1 i
1 p
1 k 1
1 k 1
q
1 i
p
1 k
q
1
i k 1 1 i
,
15
, ,
.
Mathematical Problems in Engineering. k 1j
k
1 k j 1
(ak 1p aj q )
sk
1 k 1k j1 p (ak 1 ) q (aj )
1
j 1
,
k 1j k 1j k k p q p q 1 k 1 1 k 1 1 k 1 1 j 2 1 k 1p j q 1 k 1 1 j k 1p j q j 1 j 1 k 1j k 1j k 1 T (ak 1 ), j T (aj ) k k p q p q 1 k 1 1 k 1 1 j 2 1 k 1p j q 1 1 k 1 1 j k 1p j q 1 k 1 j 1 j 1
k 1 I (ak 1 ),j I (aj ) k 1 1 k 1 j 1
1 1 k 1 j 1
1 p
k 1 F (ak 1 ),j F (aj ) k 1 1 k 1 j 1
1 j
( 2 1) 1 k 1
q
1 p
1 j
q
( 2 1) 1 k 1
1 j
k 1j q 1 k 1
j
1 1 k 1 j 1
1
p
1
p
k
k
p
k
1 j
j
1 k 1
q
1 j
k
k 1j
1
p
p
1
1 j
q
1 k 1
k 1j q 1 k 1
1
p
j
k 1j
q 1 k 1
j
1 1 1 k 1 j 1
,
q 1 k 1
p
1 k 1
1 1 1 k 1
k 1j q 1 k 1
1
q
j 1
1
p
1 p
1 j
q
1 k 1
p
k 1j q 1 k 1
1 j
,
Therefore, k 1
i ,j 1 i j
i j 1 i
(ai p aj q )
k
i ,j 1 i j
s k 1 , 1 i ji p (ai ) q (aj ) i ,j 1 i j
i j 1 i
i k 1
k
(a 1 i i i
ak 1q )
p
1
k 1j
k
1 k j 1
(ak 1p aj q )
1
i j i j k 1 k 1 p q p q 1 1 i 1 i 1 j i p j q 1 i 1 j 2 1 i p j q i i i ,j 1 ,j 1 i j i j i j i T (ai ), j T (aj ) k 1 k 1 p q p q 2 p q 1 i 1 j 1 i j 1 1 i 1 j i p j q 1 i i ,j 1 ii ,jj1 i j
i I (ai ),j I (aj ) k 1 1 1 i ii ,jj1
i F (ai ),j F (aj ) k 1 1 1 i ii ,jj1
(ai p aj q )
k 1
i ,j 1 i j
1 p
1 j
q
1 1
p
i
p
( 2 1) 1 i
1
1 j
i j q 1 i
1 j
q
1 i
1
k 1
p
i j 1 i
i j q 1 i
1 j
1 i j , 1
1 i
1 p
1 j
q
1 i
p
i j q 1 i
1 j
i j
k 1
i ,j 1 i j
1 p
1 j
q
1 1 i
( 2 1) 1 i
p
p
1 i j q 1 i
1 j
1 j
q
1 i
1
k 1
p
1 i j , 1
i j q 1 i
1 j
1 i
1 p
1 j
q
1 i
i j
p
i j q 1 i
1 j
, ,
That is, If n k 1, Eq. (2) is right. Therefore, for all n , Eq. (2) is right. Then, Eq. (1) is right.
In the following, the properties of MVNLNWBMH operator will be proved. 1 1 1 , , , . (1) Reducibility. Let n n n MVNLNWBMH (a1 ,a2 , an ) MVNLBMH (a1 ,a2 , an ).
1 1 Proof. Since , , n n
Then
1 , , then according to the operations in Defin ition 10, the n
result below can be obtained.
16
.
Mathematical Problems in Engineering. 1
n p q i j p q an ) (a aj ) i ,j 1 1 i i i j
MVNLNWBMH (a1 ,a2 ,
1
p q 1 n 2 n (ai p aj q ) i ,j 1 1 1 i j n 1
n p q 1 p q (a aj ) i ,j 1 n(n 1) i i j 1
p q n 1 p q ( a a ) i j n(n 1) i ,j 1 i j MVNLBMH (a1 ,a2 , an ). (2) Idempotency. Let ai a i 1,2,
,n .Then MVNLNWBMH (a1 ,a2 ,
an ) a.
Proof. For each i, owing to ai a , the formu la belo w is obtained on the basis of Eq. (5) in Theorem 1. 1
n p q i j p q an ) (a aj ) i ,j 1 1 i i i j
MVNLNWBMH (a1 ,a2 ,
1
n p q i j (a p a q ) i ,j 1 1 i i j 1
p q n i j p q a i ,j 1 1 i i j
n a i j i ,j 1 1 i i j a.
1
p q
(3) Commutativity. Let
a ,a , 1
MVNLNWBMH a1 ,a2 , Proof.
Owing to
a ,a , 1
2
2
an be any permutation of a1 ,a2 ,
an MVNLNWBMH (a1 ,a2 ,
an is permutation of (a1 ,a2 ,
an .Then
an ). an ) , then the equation below can be
obtained.
17
Mathematical Problems in Engineering. 1
n p q i j p q an ) (a aj ) i ,j 1 1 i i i j
MVNLNWBMH (a1 ,a2 ,
1
n p q i j (ai p aj q ) i ,j 1 1 i i j
(4) Monotonicity.
Suppose
ai s(ai ),T(ai ),I(ai ),F(ai ) (i 1,2,
,n )
bi s(bi ),T(bi ),I(bi ),F(bi ) (i 1,2,
,n ) are
and two
sets
of
MVNLNs,
when s(ai ) s(bi ), T (ai ) T (bi ), I(ai ) I(bi ) and F (ai ) F (bi ) for each i, then
an ) MVNLNWBMH (b1 ,b2 ,
MVNLNWBMH (a1 ,a2 ,
bn ).
Proof. (I) Linguistic term part Owing to p ,q 0, and s(a ) s(b ) for each i, the result below is gained. i
(ai ) (bi ) p
i
(ai ) (bi ) q
p
and
q
p(ai ) q(ai ) p(bi ) q(bi )
i j p (ai ) q(ai ) i j p(bi ) q(bi ) 1 i 1 i
i j
n
,
i j p (ai ) q(ai ) 1 1 i
i j
n
i j p (bi ) q(bi ) 1 1 i
i j ,
i j 1
1
n p q n p q i j i j p q p q (ai ) (ai ) (bi ) (bi ) i ,j 1 1 i ,j 1 1 i i i j i j (II) True, indeterminacy and falsity membership parts Owing to T (ai ) T (bi ), I(ai ) I(bi ) and F (ai ) F (bi ) for each i, p ,q 0, then the following results can be proved easily.
x ai y ai
g g
ai
ai
ai
2 1 ha
ai
2
1 p q
i
2 1 ua
i
1 ua 2
1 p q
i
1 ha
v v
x ai 2 1 y ai
1 p q
i
1 p q
1
1 x a y a
g a ha i
i
i
i
i
i
v a ua i
i
1 p q
g
bi
1
i
1 p q
bi
bi
2 1 hb
bi
2
1 p q
i
2 1 ub
1 p q
i
1 p q
1
p q
1 x b y b
g b hb i
i
i
i
1 p q
i
i
v b ub i
i
1
p q
p q
1
i
1
Where the corresponding x ,y ,g ,h ,u and v are defined in Definition 10. (III) Comparing MVNLNWBMH (a1 ,a2 , an )with MVNLNWBMH (b1 ,b2 , bn ) Suppose
a s(a ),T (a ),I(a ),F(a ) MVNLNWBMH (a1 ,a2 ,
an )
;
p q
1 v b ub i
1
p q
1 p q
1 g b hb
i
1 ub 2
1 p q
i
1 hb
v v
x bi 2 1 y bi
g
p q
1 v a ua i
1 p q
1 p q
1 g a ha 1 p q
x bi y bi
p q
and
18
.
;
Mathematical Problems in Engineering.
b s(b ),T(b ),I(b ),F(b ) MVNLNWBMH (b1 ,b2 , bn ). Because s(a ) s(b ),T (a ) T (b ),I(a ) I(b )and F(a ) F(b ),thus a b .
an ) MVNLNWBMH (b1 ,b2 ,
Then MVNLNWBMH (a1 ,a2 ,
bn ).
In the following, a few special examples for MVNLNWBMH operator regarding different values , p and q will be explored. (1) If q 0, then the MVNLNWBMH operator defined by equation (1) will be reduced to the generalized multi-valued neutrosophic linguistic Hamacher weighted average (GMVNLHWA) operator shown as below. 1
n p q i j p q an ) (a aj ) i ,j 1 1 i i i j
GMVNLHWA(a1 ,a2 ,
1
n p i j p a i ,j 1 1 i i i j
n i 1 i p ai i 1 1 i
1
p
1
n p i ai p i 1
s
,
1
n i p (ai ) i 1
i T (ai ), j T (aj )
p
1 x y p 1 1 2 p 1 x y p x 1 y
1 1 g 2 1 h p g h p 1 1 2 p 1 g h p g 1 h
i I (ai ), j I (aj )
1 1 2 p v 1 u v u p 1 1 2 p 1 v u p v 1 u
i F (ai ),j F (aj )
Where x
n
1 1 i
h 1 1 y
n
i 1
1 i
n
i 1
1 i 1
,
i
p
p
i p
i
p
2 1 i p
,
i
,
,
1 i
p
i
,
19
Mathematical Problems in Engineering.
1 u 1 1 v 1 1 n
1 i
g
1
n
n
p
p
1 i
p
2 1 1 i
i
i 1
p
2 1 1 i
i
i 1
p
i
p
i
i
,
,
i
(2) When p 1,q 0, then the operator of MVNLNWBMH defined by equation (1)
will be reduced to the multi-valued neutrosophic linguistic Hamacher weighted arithmetic average (MVNLHWAA) operator shown as below. n
i ai i
an )
MVNLHWAA(a1 ,a2 ,
1
sn
i(ai )
,
i 1
n n i 1 1 i 1 i i i 1 i 1 n n i 1 1 i 1 1 i i i 1 i 1
i T (ai )
i I (ai )
i F (ai )
n i i i 1 n n 1 i i 1 i i 1 i 1
n i i i 1 n n 1 i i 1 i i 1 i 1
i
i
,
,
If 1, the MVNLHWAA operator will be reduced to MVNLWAA operator shown
as below. an ) s n
MVNLWAA(a1 ,a2 ,
i(ai )
,
i 1
i T (ai )
1
n
1 i i 1
i
,
i i , i 1 n
i I (ai )
i F (ai )
n i i i 1
If 2, the MVNLHWAA operator will be reduced to the multi-valued neutrosophic
linguistic Einstein we ighted arithmetic average (MVNLEWAA) operator shown in the following.
20
Mathematical Problems in Engineering.
an )
MVNLWAA(a1 ,a2 ,
sn
i(ai )
,
i 1
i 1 i i 1 n 1 i i i 1 n
i T (ai )
2 i i i 1 n 2 i i i 1
n
1 i i
i
1
n
1 i i
i
1
n
i I (ai )
i F (ai )
n 2 i i i 1 n 2 i i i 1
n
i i
i
1
n
i i
i
1
,
,
(3) If p 0,q 0, then the MVNLNWBMH operator defined by equation (1) will be
reduced to the multi-valued neutrosophic linguistic Hamacher weighted geometric average (MVNLHWGA) operator shown as below. an )
MVNLHWGA(a1 ,a2 ,
n
ai i
i
1
s
n
i (ai )
,
i 1
i T (ai )
n i i i 1 n n 1 i i 1 i i 1 i 1
,
n n i 1 1 1 i i i i 1 i 1 n n i 1 1 i 1 1 i i i 1 i 1
i F (ai )
i
n n i 1 1 i 1 i i i 1 i 1 n n i 1 1 i 1 1 i i i 1 i 1
i I (ai )
,
If 1, the MVNLHWGA operator will be reduced to MVNLWGA operator shown
as below.
21
Mathematical Problems in Engineering.
MVNLHWGA(a1 ,a2 ,
an )
s
n
i (ai )
,
i 1
i i , i 1 n
i T (ai )
i I (ai )
i F (ai )
1 1
n
,
1 i i
i
1
n
1 i i
i
1
If 2, the MVNLHWGA operator will be reduced to the multi-valued neutrosophic
linguistic Einstein weighted geometr ic average (MVNLEWGA) operator shown as below. MVNLHWGA(a1 ,a2 ,
an ) s
n
i (ai )
,
i 1
2 i i i 1 n 2 i i i 1 n
i T (ai )
i I (ai )
i n 1 i i 1 n 1 i i i 1 i 1 i i 1 n 1 i i i 1 n
i F (ai )
n
i i
i
1
n
,
1 i i
i
1
n
1 i i
i
1
n
1 i i
i
1
n
1 i i 1
i
,
(4) If 1, then the MVNLNWBMH operator defined by equation (1) will be reduced
to the MVNLNWBM operator shown as below.
22
Mathematical Problems in Engineering.
an )
MVNLNWBM (a1 ,a2 ,
s
p q )
i T (ai ), j T (aj )
i I (ai ), j I (aj )
i F (ai ),j F (aj )
1
,
1
n i j p (ai ) q (aj 1 i i ,j 1 i j
1 i p j q n
1 p q ,
i j 1 i
i ,j 1 i j
1 1
1 1 i 1 j i j
1 1
1 1 i 1 j i j
n
i j
q 1 i
p
, 1
i j
n
i j
q 1 i
p
, 1
i j
1 p q , 1 p q
(5) If 2, then the M VNLNW BM H operato r d efined by equat ion (1) will be reduced to th e M VNLNW BM E, that is the s imp lificat io n o f mu lt i-v alued neut rosoph ic lingu ist ic NW BM Einstein.
an )
MVNLNWBMH (a1 ,a2 ,
s
1
,
p q n i j p (ai ) q (aj ) i ,j 1 1 i i j
i T (ai ), j T (aj )
i I (ai ), j I (aj )
i F (ai ),j F (aj )
Where x
y
1 2 x y p q 1 1 x 3y p q x y p q
1 1 g 3h p q g h p q 1 1 g 3h p q g h p q 1 1 v 3u p q v u p q 1 1 v 3u p q v u p q
2 i 2 j n
q
p
i ,j 1 i j
h
1 i
n
i ,j 1 i j
p
q
i j
q 1 i
,
i j
1 i 1 j
i ,j 1 i j
p
3 i j
i j
q
p
,
p
2 i 2 j n
,
q 1 i
p
, i j q 1 i
1 j
,
23
Mathematical Problems in Engineering.
g
1 i 1 j n
3 1 i
q
p
i ,j 1 i j
u
n
i ,j 1 i j
v
1 i
p
1 j 1 i q
1 i 1 j n
p
q
i ,j 1 i j
p
3 1 i
p
i j q 1 i
1 j
i j q 1 i
1 j p
,
i j q 1 i
1 j
,
.
From the above analysis, we can obtain the MVNLNWBMH operator is more generalized. 5 The multiple criteria decision making approach based on the MVNLNWB MH operator The proposed MVNLNW BMH operator is presented to cope with MCDM problem under mu ltivalued neutrosophic linguistic environment in this subsection. Suppose that A A1 ,A2 , ,Am represent m alternatives, and C C 1 ,C 2 , criteria. where
Let
1 ,2 ,
j 0(j 1,2,
,n
,n ), and
be n
j j
the
1
corresponding .The
evaluation
,C n represent n
weights value
of of
the
criteria, criteria
1
C j (j 1,2,
,n ) regarding the alternative Ai (i 1,2,
,m ) is provided by experts. Each value
is represented by MVNLNNs. Suppose that R aij m n is the mu lti-valued neutrosophic linguistic
decision matrix, aij s(aij ), T (aij ),I(aij ),F (aij )
is the evaluation information wh ich
represents the assessment value of alternative Ai(i 1,2,
,m ) on criteria C j (j 1,2,
,n )with
respect to the linguistic value s(aij ),where T (aij ) indicates the satisfaction degree, I(aij ) indicates the indeterminacy degree and F (aij ) indicates the dissatisfaction degree. Then, the main method for ranking and selecting the best alternative is presented in the following. Step1. The decision matrix is normalized. Generally, criteria in MCDM problems consist of two types: maximu m type and minimu m type, the min imu m type should be transformed into the maximu m type for eliminat ing the influence of distinguished types. Suppose that R aij m n is the original decision matrix, which can be normalized
as follows: aij , for maximizing criteria bij s , T (aij ),I(aij ),F(aij ) ,for maximizing criteria (aij ) Thus, the normalized matrix B bij m n is gained.
Step2. The comprehensive value of each alternative is calculated. The comprehensive value represented by ai(i 1,2, ,m )can be obtained by utilizing the MVNLNWBM H operator in Definit ion 10, which can aggregate the overall value fo r each alternative with respect to all criteria. Step3. The compared values of three functions are calculated. According to the equations given in Defin ition 8, the score value denoted by E(ai ), the accuracy value denoted by H(ai ) and the certainty value denoted by C(ai )can be obtained.
24
Mathematical Problems in Engineering. Step4. The alternatives are selected. Based on Defin ition 9, all alternatives Ai(i 1,2,
,m ) can be ranked on the basis of
E(ai ), H(ai ) and C(ai ), and the best alternative(s) can be selected.
6 An numberical example In order to validate the effect iveness and practical of the novel approach, an investment project is adapted from Ye. An investment company wants to expand its business. Four alternatives will be chosen, A1 represents auto corporation, A2 represents food corporation, A3 represents computer company corporation), A4 represents weapon corporation. Each alternative is evaluated under three criteria,
C 1 denotes risk, C 2 denotes growth, C 3 denotes the impact of environ ment, where C 3 is the minimizing criteria. The corresponding weighted vector is 0.35,0.25,0.4 . In real situation, the decision maker may hesitant and give several possible value for the satisfaction, indeterminacy and dissatisfaction regarding the alternative Ai corresponding to the criteria C j under the linguistic term set S . Therefore, the assessment value is given in the form of M VNLNs, and the linguistic term set is emp loyed as S s1 ,s2 ,s3 ,s 4 ,s5 ,s6 ,s7 extremely poor,very poor,poor,medium,good,very good,extremely good . The multi-valued neutrosophic linguistic decision matrix R aij 4 3 s ,0.3,0.4,0.5 ,0.1 ,0.3,0.4 5 s 6 ,0.6,0.7 ,0.1,0.2 ,0.2,0.3 s 6 ,0.5,0.6 ,0.4 ,0.2,0.3 s 4 ,0.7,0.8 ,0.1 ,0.1,0.2
R aij 4 3 is shown as follows.
s 6 ,0.5,0.6 ,0.2,0.3 ,0.3,0.4 s5 ,0.6,0.7 ,0.1 ,0.3 s5 ,0.6 ,0.3 ,0.4 s 4 ,0.6,0.7 ,0.1 ,0.2
s5 ,0.2,0.3 ,0.1,0.2 ,0.5,0.6 s 5 ,0.6,0.7 ,0.1,0.2 ,0.1,0.2 s 4 ,0.5,0.6 ,0.1 ,0.3 s 6 ,0.3,0.5 ,0.2 ,0.1,0.2,0.3
6.1 The procedure using the proposed aggregation operator Step1. The decision matrix is normalized. Because C 3 is the min imizing criteria, wh ich should be converted to the maximizing criteria, then the normalized decision matrix B bij m n .can be obtained as follows: B bij 4 3
s , 0.3,0.4,0.5 ,0.1 ,0.3,0.4 5 s 6 , 0.6,0.7 ,0.1,0.2 ,0.2,0.3 s 6 , 0.5,0.6 ,0.4 ,0.2,0.3 s 4 , 0.7,0.8 ,0.1 ,0.1,0.2
s 6 ,0.5,0.6 ,0.2,0.3 ,0.3,0.4
s 2 ,0.2,0.3 ,0.1,0.2 ,0.5,0.6
s5 ,0.6,0.7 ,0.1 ,0.3
s 2 ,0.6,0.7 ,0.1,0.2 ,0.1,0.2
s5 ,0.6 ,0.3 ,0.4
s 3 ,0.5,0.6 ,0.1 ,0.3
s 4 ,0.6,0.7 ,0.1 ,0.2
s1 ,0.3,0.5 ,0.2 ,0.1,0.2,0.3
Step2. The comprehensive value of each alternative is calculated. Derive the comprehensive value ai(i 1,2, ,m ) of each alternative Ai(i 1,2,
,m )
by using the MVNLNWBM H operator presented in Definition 10. Here let p q 1, 1. The MVNLNWBMH operator is shown as below:
25
Mathematical Problems in Engineering.
an )
MVNLNWBM (a1 ,a2 ,
s
1
n i j (ai )(aj 1 i i ,j 1 i j
i T (ai ), j T (aj )
i I (ai ), j I (aj )
i F (ai ),j F (aj )
1
n
1 i j
i ,j 1 i j
1 1 1 1
i j 1 i
,
2 )
1 2 ,
1 1 i 1 j n
i j 1 i
i ,j 1 i j
1 1 i 1 j i j n
, 1
i j
i j 1 i
1 2 , 1 2
And we have a1 s 3.9529 ,
0.3088,0.3515,0.3309,0.3752,0.3429,0.3869,0.3671,0.4118,0.3746,0.4199,0.4009,0.4463 , 0.1255,0.1645,0.1482,0.19 , 0.3768,0.4171,0.4049,0.446,0.4109,0.4511,0.4382,0.4783
;
a2 s 3.9904 ,
0.6,0.6368,0.6275,0.6648,0.6345,0.6727,0.6622,0.7 , 0.1,0.1363,0.1335,0.1714 , 0.1882,0.2268,0.2261,0.2629
;
a3 s 4.4850 ,
0.5272,0.5644,0.5618,0.6 , 0.263 , 0.2918,0.3273
;
Step3. The compared values of three functions are calculated. By using equations in Defin ition 8, we can obtain E(ai ) , H(ai ) and C(ai ),(i 1,2,3,4) as follows: E(a1 ) s2.3605 ,H (a1 ) s 0.2036 ,C(a1 ) s1.4879;
E(a2 ) s3.0440 ,H (a2 ) s1.6912 ,C(a2 ) s 2.5930; E(a3 ) s2.9763 ,H (a3 ) s1.1382 ,C(a3 ) s 2.5267; E(a4 ) s2.0002 ,H (a4 ) s1.0641 ,C(a4 ) s1.5461; Step4. The alternatives are selected.
By using the compared approach in Definition 9, E(a2 )
E(a3 )
E(a1 )
E(a4 ) can
26
Mathematical Problems in Engineering.
be obtained, so the final raking of alternatives is A2
A3
A1
A4 . Apparently, A2 is the
best one, and A4 is the worst one. We don’t need to co mpare the other functions because the values of score function differ. 6.2 Comparison analysis We t ake d ifferen t value into cons iderat ion in step 2 to select the alternat ive fo r d iscussing th e impact of different values p ,q and . The comparisons are presented in Table 1 and Table 2. Table 1 Ranking of alternatives utilizing different p ,q and 1
1,p ,q
Score function E(ai ),(i 1,2,3,4)
Ranking
p 0,q 0
E(a1 ) s2.1214 ,E(a2 ) s2.8131 ,E(a3 ) s 2.8732 ,E(a4 ) s1.7094
A3
A2
A1
A4
p 0.01,q 0
E(a1 ) s2.1907 ,E(a2 ) s2.8460 ,E(a3 ) s 2.9575 ,E(a4 ) s1.7629
A3
A2
A1
A4
p 0.1,q 0
E(a1 ) s2.2165 ,E(a2 ) s2.8790 ,E(a3 ) s 2.9713 ,E(a4 ) s1.8005
A3
A2
A1
A4
p 1,q 0
E(a1 ) s2.4697 ,E(a2 ) s3.1970 ,E(a3 ) s3.1084 ,E(a4 ) s2.1573
A2
A3
A1
A4
p 2,q 0
E(a1 ) s2.7154 ,E(a2 ) s3.4882 ,E(a3 ) s3.2507 ,E(a4 ) s 2.4519
A2
A3
A1
A4
p 5,q 0
E(a1 ) s3.1768 ,E(a2 ) s3.9655 ,E(a3 ) s3.5749 ,E(a4 ) s 2.8445
A2
A3
A1
A4
p 10,q 0
E(a1 ) s3.5203 ,E(a2 ) s 4.2619 ,E(a3 ) s3.8673 ,E(a4 ) s3.0415
A2
A3
A1
A4
p 0,q 1
E(a1 ) s2.6253 ,E(a2 ) s3.3082 ,E(a3 ) s3.1563 ,E(a4 ) s 2.2904
A3
A2
A1
A4
p 0.01,q 1
E(a1 ) s2.6151 ,E(a2 ) s3.2980 ,E(a3 ) s3.1492 ,E(a4 ) s 2.2778
A2
A3
A1
A4
p 0.1,q 1
E(a1 ) s2.5377 ,E(a2 ) s3.2202 ,E(a3 ) s3.0964 ,E(a4 ) s 2.1839
A2
A3
A1
A4
p 1,q 1
E(a1 ) s2.3605 ,E(a2 ) s3.0440 ,E(a3 ) s 2.9763 ,E(a4 ) s 2.0002
A2
A3
A1
A4
p 2,q 1
E(a1 ) s2.4824 ,E(a2 ) s3.1918 ,E(a3 ) s3.0464 ,E(a4 ) s 2.1664
A2
A3
A1
A4
p 5,q 1
E(a1 ) s2.9208 ,E(a2 ) s3.6644 ,E(a3 ) s3.3532 ,E(a4 ) s 2.5870
A2
A3
A1
A4
p 10,q 1
E(a1 ) s3.3239 ,E(a2 ) s 4.0427 ,E(a3 ) s3.6946 ,E(a4 ) s 2.8680
A2
A3
A1
A4
p 0,q 2
E(a1 ) s2.8597 ,E(a2 ) s3.5712 ,E(a3 ) s3.2871 ,E(a4 ) s 2.5589
A2
A3
A1
A4
p 0.01,q 2
E(a1 ) s2.8519 ,E(a2 ) s3.5630 ,E(a3 ) s3.2820 ,E(a4 ) s2.5494
A2
A3
A1
A4
p 0.1,q 2
E(a1 ) s2.7889 ,E(a2 ) s3.4959 ,E(a3 ) s3.2400 ,E(a4 ) s 2.4731
A2
A3
A1
A4
p 1,q 2
E(a1 ) s2.5228 ,E(a2 ) s3.2087 ,E(a3 ) s3.0542 ,E(a4 ) s 2.1890
A2
A3
A1
A4
p 2,q 2
E(a1 ) s2.5363 ,E(a2 ) s3.2305 ,E(a3 ) s3.0440 ,E(a4 ) s 2.2430
A2
A3
A1
A4
p 5,q 2
E(a1 ) s2.8535 ,E(a2 ) s3.5768 ,E(a3 ) s3.2558 ,E(a4 ) s 2.5581
A2
A3
A1
A4
p 10,q 2
E(a1 ) s3.2306 ,E(a2 ) s3.9375 ,E(a3 ) s3.5836 ,E(a4 ) s 2.8212
A2
A3
A1
A4
Table 2 Ranking of alternatives utilizing different p ,q and 2
2,p ,q
Score function E(ai ),(i 1,2,3,4)
Ranking
p 0,q 0
E(a1 ) s2.1344 ,E(a2 ) s2.8137 ,E(a3 ) s 2.8863 ,E(a4 ) s1.7187
A3
A2
A1
A4
p 0.01,q 0
E(a1 ) s2.1944 ,E(a2 ) s2.8478 ,E(a3 ) s 2.9631 ,E(a4 ) s1.7661
A3
A2
A1
A4
p 0.1,q 0
E(a1 ) s2.2171 ,E(a2 ) s2.8803 ,E(a3 ) s 2.9752 ,E(a4 ) s1.8021
A3
A2
A1
A4
p 1,q 0
E(a1 ) s2.4553 ,E(a2 ) s3.1938 ,E(a3 ) s3.0982 ,E(a4 ) s 2.1483
A2
A3
A1
A4
p 2,q 0
E(a1 ) s2.7142 ,E(a2 ) s3.4825 ,E(a3 ) s3.2361 ,E(a4 ) s 2.4453
A2
A3
A1
A4
27
Mathematical Problems in Engineering.
p 5,q 0
E(a1 ) s3.2212 ,E(a2 ) s3.9697 ,E(a3 ) s3.5908 ,E(a4 ) s 2.8608
A2
A3
A1
A4
p 10,q 0
E(a1 ) s3.5770 ,E(a2 ) s 4.2923 ,E(a3 ) s3.9217 ,E(a4 ) s3.0737
A2
A3
A1
A4
p 0,q 1
E(a1 ) s2.6103 ,E(a2 ) s3.3048 ,E(a3 ) s3.1457 ,E(a4 ) s 2.2818
A2
A3
A1
A4
p 0.01,q 1
E(a1 ) s2.6011 ,E(a2 ) s3.2948 ,E(a3 ) s3.1394 ,E(a4 ) s 2.2700
A2
A3
A1
A4
p 0.1,q 1
E(a1 ) s2.5312 ,E(a2 ) s3.2188 ,E(a3 ) s3.0925 ,E(a4 ) s 2.1809
A2
A3
A1
A4
p 1,q 1
E(a1 ) s2.3702 ,E(a2 ) s3.0464 ,E(a3 ) s2.9849 ,E(a4 ) s 2.0055
A2
A3
A1
A4
p 2,q 1
E(a1 ) s2.4994 ,E(a2 ) s3.1940 ,E(a3 ) s3.0556 ,E(a4 ) s 2.1751
A2
A3
A1
A4
p 5,q 1
E(a1 ) s2.9637 ,E(a2 ) s3.6734 ,E(a3 ) s3.3833 ,E(a4 ) s 2.6091
A2
A3
A1
A4
p 10,q 1
E(a1 ) s3.3787 ,E(a2 ) s 4.0718 ,E(a3 ) s3.7516 ,E(a4 ) s 2.9010
A2
A3
A1
A4
p 0,q 2
E(a1 ) s2.8577 ,E(a2 ) s3.5652 ,E(a3 ) s3.2720 ,E(a4 ) s 2.5525
A2
A3
A1
A4
p 0.01,q 2
E(a1 ) s2.8506 ,E(a2 ) s3.5572 ,E(a3 ) s3.2675 ,E(a4 ) s 2.5435
A2
A3
A1
A4
p 0.1,q 2
E(a1 ) s2.7919 ,E(a2 ) s3.4917 ,E(a3 ) s3.2304 ,E(a4 ) s 2.4707
A2
A3
A1
A4
p 1,q 2
E(a1 ) s2.5398 ,E(a2 ) s3.2108 ,E(a3 ) s3.0635 ,E(a4 ) s2.1976
A2
A3
A1
A4
p 2,q 2
E(a1 ) s2.5586 ,E(a2 ) s3.2343 ,E(a3 ) s3.0581 ,E(a4 ) s 2.2559
A2
A3
A1
A4
p 5,q 2
E(a1 ) s2.8950 ,E(a2 ) s3.5871 ,E(a3 ) s3.2878 ,E(a4 ) s 2.5824
A2
A3
A1
A4
p 10,q 2
E(a1 ) s3.2843 ,E(a2 ) s3.9650 ,E(a3 ) s3.6392 ,E(a4 ) s 2.8549
A2
A3
A1
A4
In tab le 1，we t ake the p arameter value 1,which is based on Algebraic operation, and t he MVNLNWBM operator is applied. In t ab le 2，we take th e p arameter valu e 2,which is based on Einstein operation, and t he MVNLNW BM E operator is applied. As we can see from table 1 and table 2, the ordering of alternatives taking different p arameters p ,q and may
be different, because the different parameters will cause d ifferent sco re fun ct ion valu e. Ho wever, A2 or A3 is always the best select ion , and A4 is always th e wo rst select ion. Wh ether
1 or 2, the same ran king results is obt ained with reg ard to th e same p arameter q except fo r on e s itu at ion value p and the same paramet er value in wh ich p 0,q 1 and 1 .Specially , if p 0,q 0, M VNLNW BM H will reduce to M VNLHW GA ope rato r. If p 0,q 0 and 1 , M VNLHW GA will reduce to M VNLW GA operator. When p 1,q 0 , M VNLNW BMH will reduce to M VNLHWAA operator. If p 1,q 0 and 1 , M VNLHWAA will reduce to MVNLWAA operator. If 1 in Table 1, the ranking results on the basis of the M VNLW GA and M VNLWAA operators differ, wh ich due to the two operators emphasis on different major points, and the same situation happens in Table 2.When the parameters q 0 and p 0,p 0.01,p 0.1,respectively, the rankings are identical in t wo A2 A1 A4 . When p and q are assign the other values in Tables, the ranking order is always A3 two Tables, the ran king order is changed, and the result is A2
A3
A1
A4 .That is, the best
selection is from A3 to A2 except for one situation where p 0,q 1 in Table 1. For illustrating the effective and flexib le of the novel approach, the method in literature is adopted in mu lti-valued neutrosophic linguistic environ ment in this paper, and the same ranking orders are obtained in ref [8] where the SVNHFWA and SVNHFW G operators are adopted to fuse single-valued neutrosophic hesitant fuzzy information. When 1,p 0,q 0 and 1,p 0,q 0, the two operators are special cases of M VNLNWBM H operator .Therefore, novel operator in this paper has better flexib ility and generalizat ion. In actual cases, the decision makers can assign different
28
Mathematical Problems in Engineering. parameter values , p and q .. Generally, for convenience, we can set p 1,which can not only simplify the calculation, but also consider the interrelationship of multiple values. 7 Conclusions In this paper, the MVNLS are proposed by combining the M VNS and LS, wh ich not only describe linguistic terms, but also give the quantitative value of three membership degrees concerning the linguistic variables, which has better flexibility to express the decision informat ion. Moreover, NW BM is a useful operator which has the trait of taking into account the interrelationship of different arguments, and overcome drawbacks of non-reducib ility and non-idempotency. Hamacher operations are the extension of Algebraic and Einstein operations, which is more general. Considering these advantages, we have developed Hamacher operational laws for M VNLNs and extended the NWBM to fuse MVNL in formation. Thus, the M VNLNW BMH operator is proposed, which is appropriate to deal with M VNL info rmation. So me desirable properties of the novel operator are discuss ed in detail, and some special cases are analy zed. Furthermore, the co mparison method for M VNLNs is also studied, and the rankings of alternatives affecting by d ifferent parameters p ,q and are also compared. Fo r verify ing the novel approach, we successfully applied the approach to an example. The results show the novel approach has the follo wing advantages. The M VNLNWBM H operator is more flexib le and more general, and which can capture the interrelationship among argu ments and express decision informat ion more practical, the decision makers can assign appropriate values according to the real situation. In future, we will exp lore to apply the operator to the different do mains, for instance, fault diagnosis, machine learning and medical diagnosis. As a future possible research, we will extend our research by using the refined neutrosophic set [61], i.e.
the
truth
value
T
is
refined
into
types
of sub -truths
such
as
T1,
T2, etc.,
similarly indeterminacy I is refined into types of sub-indeterminacies I1, I2, etc., and the sub-falsehood F is split into F1, F2, etc. Acknowledgement This paper is supported by the National Natural Science Foundation of Ch ina (No. 71371154), the Hu manities and Social Sciences Research Pro ject of Min istry of Education of China (No. 16YJCZH049), Th inking Bank Project of Hubei (No. HBSXK2017055) and the Hu manit ies and social Sciences foundation of Depart ment of Education of Hubei (No. 17Q121).The authors also would like to express appreciation to the anonymous reviewers and Editor for their helpfu l co mments that imp roved the paper. Compliance with ethical standards Conflict of interest ：The authors declare that they have no conflict of interests regarding the publication of this paper.
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