Multiple attribute decision-making approach based on dual hesitant fuzzy Frank aggregation operators

Accepted Manuscript Title: Multiple attribute decision-making approach based on dual hesitant fuzzy Frank aggregation operators Authors: Xiaoan Tang, Shanlin Yang, Witold Pedrycz PII: DOI: Reference:

S1568-4946(18)30191-1 https://doi.org/10.1016/j.asoc.2018.03.055 ASOC 4809

To appear in:

Applied Soft Computing

Received date: Revised date: Accepted date:

16-10-2017 16-2-2018 28-3-2018

Please cite this article as: Xiaoan Tang, Shanlin Yang, Witold Pedrycz, Multiple attribute decision-making approach based on dual hesitant fuzzy Frank aggregation operators, Applied Soft Computing Journal https://doi.org/10.1016/j.asoc.2018.03.055 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Multiple attribute decision-making approach based on dual hesitant fuzzy Frank aggregation operators

Xiaoan Tanga,b,c*, Shanlin Yanga,b, Witold Pedryczc,d,e aSchool bKey

of Management, Hefei University of Technology, Hefei, Box 270, Hefei 230009, Anhui, P.R. China;

Laboratory of Process Optimization and Intelligent Decision-making, Ministry of Education, Hefei, Box 270, Hefei 230009, Anhui,

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P.R. China; cDepartment

of Electrical & Computer Engineering, University of Alberta, Edmonton T6R 2V4 AB Canada;

dDepartment

of Electrical and Computer Engineering, Faculty of Engineering, King Abdulaziz University, Jeddah, 21589, Saudi Arabia;

*

Research Institute, Polish Academy of Sciences, Warsaw, Poland

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eSystems

Corresponding author. Tel: +86 0551 62904930; fax: +86 0551 62905263. E-mail address: sichuanshengxiaoan@163 (Xiaoan Tang).

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Graphical abstract

Highlights  New score and accuracy functions of dual hesitant fuzzy elements are designed.  Some Frank operational rules of dual hesitant fuzzy sets are developed.  Frank hybrid weighted arithmetic and geometric aggregation operators are proposed.  Monotonicity of the two types of aggregation operators regarding the parameter in Frank operations and 1

their relationships are investigated and used to determine the parameter.  A flexible MADM approach with dual hesitant fuzzy information is proposed and applied to an investment evaluation problem.

Abstract

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Real-world decision-making problems are often complex and indeterminate. Thus, uncertainty and hesitancy are usually unavoidable issues being experienced by decision makers. Dual hesitant fuzzy sets (DHFSs) which are described in terms of the two functions, namely the membership hesitancy function and the non-membership hesitancy function, have been developed. In light of their properties, they are considered as a powerful vehicle to express uncertain information in the process of multi-attribute decision-making (MADM). In accordance with the practical demand, this study proposes a new MADM approach with dual hesitant fuzzy (DHF) assessments based on Frank aggregation operators. First, original score and accuracy functions of DHFS are developed to construct a new comparison method of DHFSs. The properties of the developed score and accuracy functions are analyzed. Second, we investigate the generalized operations of DHFS based on Frank t-norm and t-conorm. The generalized operations are then used to build the generalized arithmetic and geometric aggregation operators of DHF assessments in the context of fuzzy MADM. The monotonicity of arithmetic and geometric aggregated assessments with respect to a parameter in Frank t-norm and t-conorm and their relationship are also demonstrated. In particular, the monotonicity is employed to associate the parameter with the risk attitude of a decision maker, by which a method is designed to determine the parameter. A procedure of the proposed MADM method is presented. Finally, an investment evaluation problem is discussed by the proposed approach to demonstrate its applicability and validity. A detailed sensitivity analysis and a comparative study are also conducted to highlight the validity and advantages of the approach proposed in this paper. More importantly, we discuss the situations where Frank aggregation operators are replaced by Hamacher aggregation operators at the second step of the proposed approach, through re-considering the investment evaluation problem.

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Keywords: Multiple attribute decision-making (MADM); Dual hesitant fuzzy sets; Frank aggregation operators; Dual hesitant fuzzy aggregation operators; Investment evaluation

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1. Introduction

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Multiple attribute decision-making (MADM) embraces designing computational and mathematical tools to support the subjective evaluation of performance criteria by decision makers. More specifically, a decision maker usually uses multiple attributes to analyze and evaluate a set of alternatives. However, in some situations, a decision maker may find difficulties in providing precise attribute values or utilities due to the increasing complexity of socio-economic environment and the vagueness of inherent subjective nature of human thinking. Fuzzy sets and various extensions have been proposed [1-5] to deal with these situations. In the last few decades, many attempts have been reported in the application field of fuzzy sets such as fuzzy decision making [6-9] and fuzzy systems for control system and energy prediction [10-15]. However, fuzzy sets and their extensions as outlined above still cannot deal with the cases where there are some difficulties in determining the membership of an element to a set due to doubts between a few different values. For example, when a decision maker has sufficient data, knowledge, and experience to produce possible membership degrees with values in [0,1], the decision maker will not employ the above mentioned extensions to express his preferences. The problem is not solved even if the decision maker takes opinions of multiple experts into account to provide his preferences. To address this problem, Torra [16] introduced a concept of hesitant fuzzy set, which is another generalized form of fuzzy set and permits several possible membership degrees of an element to a set. For instance, let us consider a MADM problem in which some decision makers consider as possible membership degrees of x into the set A the values 0.2, 0.3, and 0.4 replacing just one number or a tuple. Hereinto, the certainty and uncertainty on the possible values are somehow limited, which can reflect the original information 2

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given by the decision makers as much as possible. It is easy to represent the original information by using a hesitant fuzzy assessment, i.e.,({0.2,0.3,0.4}). Subsequently, Zhu et al. [17] proposed the concept of dual hesitant fuzzy set (DHFS), in which membership and non-membership degrees are characterized by two sets of possible values, respectively. The advantage of DHFS is that it can integrate hesitant fuzzy set and intuitionistic fuzzy set in handling ambiguity and imprecision in real-world situations. A significant number of studies have been completed to enrich the field of decision making with DHFS [18-30]. However, there exist two main problems to be resolved in DHFS-based decision making problems. The first issue needs to be addressed in this type of decision-making environment is the ranking of DHFSs. The existing comparison techniques of DHFSs are generally constructed based on score and accuracy functions of DHFS [17,22,25-29]. However, the score and accuracy functions developed in Zhu et al. [17] only take into consideration the mean of possible membership and non-membership degrees, but fail to consider the divergence of the degrees, which cannot fully characterize dual hesitant fuzzy (DHF) information and inevitably may cause information loss. In other words, such functions may be arguable in some situations where there are DHFSs, which exhibit the same means of possible membership and non-membership degrees. Therefore, the first aim of this study is to develop a new ranking method of DHFSs. To do that, the mean and variance of possible membership and non-membership degrees are considered simultaneously, and then applied to construct new score and accuracy functions of DHFS. Compared with existing score and accuracy functions of DHFS, the constructed score and accuracy functions are more reliable and rational because they can more completely characterize the information carried by DHFS. The properties of the new score and accuracy functions are also analyzed and proven. Following the two functions, new comparison mechanisms of DHFSs are developed. The second issue to be addressed in this type of decision-making environment is how to aggregate DHF information. This topic has drawn a great deal of interest [17,21,22,24-32]. For instance, Zhu et al. [17] constructed some basic operations of DHFSs based on the algebraic t-norm and t-conorm, and discussed their properties in detail. Motivated by the idea of arithmetic and geometric operations in [17], Wang et al. [21] proposed a few DHF aggregation operators, such as the DHF hybrid arithmetic aggregation operators and the DHF hybrid geometric aggregation operators, to deal with MADM problems. Yang and Ju [24] proposed several DHF linguistic prioritized aggregation operators based on the prioritized average operator, and investigated some desirable properties and special cases of these operators in detail. Yu et al. [27] extended Heronian mean to DHFSs, and proposed some DHF Heronian mean operators. According to Einstein t-norm and t-conorm, Zhao et al. [22] presented the DHF Einstein weighted arithmetic operators and the DHF Einstein weighted geometric operators to aggregate DHF information. Wang et al. [26] and Yu et al. [28] proposed a wide range of DHF power aggregation operators based on Archimedean t-norm and t-conorm for DHF information, and applied them to solve multiple attribute group decision-making problems. Qu er al. [30] presented a series of induced generalized DHF Shapley operators, such as the induced generalized DHF Shapley hybrid weighted averaging operator and the induced generalized DHF Shapley hybrid geometric mean operator. From the above analysis, we can conclude that most of the aforementioned DHF aggregation operators are based on the algebraic and Einstein operational laws of DHFS to carry the aggregation process. To generate the generalized operations of DHFS, Ju et al. [25] extended Hamacher t-norm and t-conorm to DHF context, and explored the DHF Hamacher weighted arithmetic operator, the DHF Hamacher weighted geometric operator, the DHF Hamacher hybrid arithmetic operator and the DHF Hamacher hybrid geometric operator. These fuzzy Hamacher aggregation operators can reduce to fuzzy algebraic operators and fuzzy Einstein operators respectively when the parameter r in Hamacher t-conorm and t-norm is set as some specific values. However, to the best of our knowledge, the two essential points are omitted in the Hamacher aggregation operators: (1) what is the physical meaning of the parameter r; and (2) how this parameter is determined in the context of fuzzy MADM. The physical meaning of the parameter r is usually not clear and the determination of the parameter is generally done in an arbitrary and subjective manner, which may negatively influence the rationality of decision results. Frank t-norm and t-conorm[33], which have the characteristics of general t-norm and t-conorm such as algebraic, Einstein, and Hamacher t-norm and t-conorm, have recently been extended to aggregate the fuzzy information within the context of fuzzy MADM [34-37]. Sarkoci [38] proved that the Frank t-norms and the Hamacher t-norms come 3

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from the same family after a comparison between the two different t-norms. Yager [39] built some implication operators based on Frank t-norms and utilized them to approximate reasoning. With the use of Frank t-norm and t-conorm, Wang and He [40] proposed a few probability logic operators, which can satisfy both basic axiom of probability measure and the continuous monotone variability. As the only one type of t-norm and t-conorm that satisfies the compatibility, Frank t-norm and t-conorm involve an additional parameter r to determine the attitude of decision makers, which makes the information fusion process more flexible and robust than other t-norms and t-conorms. When the parameter r is set as some specific values, some special operators can be obtained. So far, we have not seen any related studies on aggregation techniques using the Frank operations on DHFSs for aggregating a collection of DHFSs. More importantly, there is a lack of a generalized operation framework of DHFSs to build DHF aggregation operators for attribute combination so as to flexibly handle various real applications. In view of the fact that DHFSs can express the fuzzy decision information more objectively and precisely; and meanwhile Frank t-norm and t-conorm can provide more flexible than other t-norms and t-conorms. Thus, the second aim of this study is to investigate generalized DHF aggregation operators based on the Frank t-norm and t-conorm and their application to MADM with DHF information. Furthermore, similar to the situation where Hamacher aggregation operators are applied to solve MADM problems, the above-mentioned two essential points concerning the parameter r in the Frank aggregation operators are still not investigated in the studies [34,35,37]. Therefore, to address the above two points concerning the parameter r in the Frank aggregation operators when they are applied to solve MADM problems strengthens the motivations of this study. From the analysis made above, the primary motivations behind this study are summarized as follows: 1. Extant comparison techniques of DHFSs are generally constructed based on score and accuracy functions that only take into consideration the mean of possible membership and non-membership degrees. Such functions cannot effectively and totally portray the difference among the possible degrees, which inevitably may cause information loss. Therefore, this study constructs new score and accuracy functions of DHFS that consider the mean and variance of possible membership and non-membership degrees simultaneously, and develops a new ranking method of DHFSs. 2. DHFSs integrate the advantages of hesitant fuzzy sets and intuitionistic fuzzy sets in handling ambiguity and imprecision in real-world situations. Furthermore, Frank t-norm and t-conorm are more flexible and robust than other t-norms and t-conorms, but have not been studied in the context of MADM with DHF information. Thus, this study investigates generalized DHF aggregation operators based on the Frank t-norm and t-conorm and their application to MADM with DHF information. 3. Similar to the situation where Hamacher aggregation operators are applied to solve MADM problems, the two essential points are still not investigated in the existing Frank aggregation operators: (1) what is the physical meaning of the parameter r; and (2) how this parameter is determined in the context of fuzzy MADM. To solve these issues, this study discusses the physical meaning and determination of the parameter in MADM. To achieve the above goals, this study proposes a new fuzzy MADM approach with DHF assessments based on Frank t-norm and t-conorm. First, the generalized operations of DHFSs based on Frank t-norm and t-conorm are defined and their properties are analyzed and theoretically proved. Second, the developed operations are then used to build the generalized arithmetic and geometric aggregation operators of DHF assessments in the context of fuzzy MADM, in which a combination of attribute weights and the ordered weights of DHF assessments on each attribute is applied. Third, the monotonicity and relationship of the score functions of the aggregated assessments, with respect to the parameter r in the proposed Frank arithmetic and geometric aggregation operators are discussed and proven, and are further used to discuss the physical meaning and determination of the parameter in MADM. In particular, the monotonicity is employed to associate the parameter with the risk attitude of a decision maker, by which a method is designed to determine the parameter. Fourth, an approach to MADM based on new ranking method and DHF Frank aggregation operators is developed. Finally, the proposed approach is applied to solve an investment evaluation problem faced by a large Chinese company of iron and steel. A detailed sensitivity analysis and a comparative study are conducted to highlight the validity and advantages of the approach proposed in this paper. In order to address the two essential points concerning the parameter r in Hamacher aggregation operators, we also re-consider the 4

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investment evaluation problem by replacing Frank aggregation operators with Hamacher aggregation operators at the second step of the proposed approach. The paper is organized as follows. In Section 2, we review some basic concepts of DHFS, Frank t-norm and t-conorm, and ordered weighted averaging operator. Section 3 introduces the new ranking method of DHFSs. In Section 4, we study the generalized operations of DHFS based on Frank t-norm and t-conorm. In Section 5, we propose two new DHF Frank aggregation operators for aggregating DHF assessments within the context of fuzzy MADM; furthermore, we also discuss the monotonicity and relationship of arithmetic and geometric aggregated assessments, with respect to the parameter r in the proposed aggregation operators, and the meaning and determination of the parameter. In Section 6, we develop an approach to MADM based on the ranking method introduced in Section 3 and the DHF Frank aggregation operators proposed in Section 5. In Section 7, an investment evaluation problem is investigated to demonstrate the validity and applicability of the proposed approach. Finally, Section 8 concludes this paper. 2. Preliminaries

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In this section, we briefly review the basic concepts of DHFS, Frank t-norm and t-conorm, and ordered weighted averaging operator. 2.1 Dual hesitant fuzzy set

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Zhu et al. [17] introduced the concept of DHFS defined as follows. Definition 1. [17] Given a universe of discourse X, a DHFS on X is defined as

(1)

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M=  x, hM ( x), gM ( x) x  X  ,

g} = {

{ },

 hM ( x )

{} }.

 g M ( x )

 M ( x )hM ( x ),M ( x )gM ( x )

x, hM ( x), g M ( x)

is called a dual hesitant fuzzy element (DHFE) denoted by α={h,

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degrees of x to M. For a specific x,

1   M ( x) M ( x) expresses possible indeterminacy (uncertain)

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Given hM(x) and gM(x), fM(x)=

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where hM(x) and gM(x) denote the set of possible membership degrees and the set of non-membership degrees of x to M, respectively, such that γM(x)  hM(x), ηM(x)  gM(x), γM(x)  [0, 1], ηM(x)  [0, 1], and 0  max{ M ( x)}  max{M ( x)}  1 for all x  X .

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To compare the DHFEs, Zhu et al. [17] presented the following comparison mechanism. Definition 2. [17] Let α = {h, g} = {{  1 , …,   ( h ) }, { 1 , …,  ( g ) }} be a DHFE, where  (h) and  ( g ) are the

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numbers of the elements in h and g, respectively. Then, the score function of α is defined as S(α) = h - g , and the

accuracy of α is defined as H(α) = h + g , where

 h

 ( h) i 1

i

 ( h)

,

 g

 (g) j 1

j

 (g)

.

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For the two DHFEs α1=(h1, g1) and α2=(h2, g2), if S(α1) < S(α2), then α1 is smaller than α2 denoted by α1 < α2, and vice versa; if S(α1) = S(α2), then (1) H(α1)=H(α2) indicates that α1 and α2 represent the same information and are denoted by α1=α2; and (2) H(α1)
1 h1 , 2 h2

{ 1   2   1 2} ,

1 g1 ,2 g2

(2) α1  α2={

1 h1 , 2 h2

{ 1 2} ,

{1  2   1 2 } },

1 g1 ,2 g2

{ 1 2 } },

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(3) λα1={

1 h1

{1  (1   1 ) } ,

(4) 1 ={

 1 h1

{ 1 } ,

1 g1

(5) 1c ={

1 g1

{1} ,

 1 h1

1 g1

{1 } },   0 ,

{1  (1  1 ) } },   0 , and

{ 1} }.

Here, 1c represents the complement of the DHFE α1. 2.2 Frank t-norm and t-conorm

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T-norm and t-conorm are widely applied in fuzzy sets to define the generalized intersection and union operations of fuzzy sets. In Ref. [41], Deschrijver and Kerre gave the definition and condition of t-norm and t-conorm. A continuous t-norm function T(x,y) such that T(x,x)x for all x  (0,1). Strict Archimedean t-norm and t-conorm are strictly increasing for all x,y  (0,1) [41]. As stated by Xia et al. [42], a strictly decreasing function p :[0,1]  [0, ]

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such that p(1)=0 can be used to create a strict Archimedean t-norm T(x,y)=p-1(p(x)+p(y)) and its dual t-conorm S(x,y)=q-1(q(x)+q(y)) such that q(x)=p(1-x). Different forms of p are introduced by Xia et al. [42] to generate different Archimedean t-norms and t-conorms. To generalize operations of DHFS using t-norm and t-conorm, we choose a specific p(x), i.e.,  r 1   , r > 1. x  r 1 

(2)

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p(x)= log 

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(3)

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Frank t-norm and t-conorm [33] are then constructed by the functions p(x) and q(x), which are Sr ( x, y) = (r1 x  1)(r1 y  1) (r x  1)(r y  1) ) and Tr ( x, y) = log r (1  ) , r>1. The chosen p(x) is flexible and easy to r 1 r 1

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calculate, which can cover different t-norms and t-conorms generated by other forms of p(x). Specifically, if r→1, Sr(x,y) and Tr(x,y) reduce to the algebraic t-norm and t-conorm. 2.3 Ordered weighted averaging operator

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Yager [43] defined an alternative combination function to synthesize a set of ordered values—the ordered weighted averaging (OWA) operator, which is simply introduced as follows. Definition 4. [43] An OWA operator with s variables is defined as a mapping F: s   . This mapping is associated

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with a weight vector w = (w1, …, ws) such that 0 ≤ wj ≤ 1 (j = 1, …, s),



s j 1

w j  1 , and F(x1, …, xs) =



s j 1

wj y j ,

where yj is the jth largest element of x1, …, xs. The orness degree also called the attitudinal character of the operator is represented in the form

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1 s (s  j )w j .  s  1 j 1

(4)

It is limited to [0, 1] and used to measure the optimism level of a decision maker. When orness(w)>0.5, orness(w)<0.5, and orness(w)=0.5, the decision maker is considered optimistic, pessimistic, and neutral, respectively [44]. The weights of the OWA operator satisfy the relationships (1) w1 ≥ w2 ≥ ... ≥ ws ≥ 0 if orness(w) > 0.5, (2) 0 ≤ w1 ≤ w2 ≤ ... ≤ ws if orness(w) < 0.5, and (3) w1 = w2 = ... = ws = 1 / s if orness(w) = 0.5. To determine OWA operator weights, Yager [45] designed a function Q: [0, 1] → [0, 1] such that Q(0) = 0, Q(1) = 1, and Q(x) ≥ Q(y) if x > y, which was called regular increasing monotone quantifier. By using this function, w is 6

calculated by j j 1 wj = Q   - Q   , j = 1, …, s. s

(5)

 s 

The function can also determine orness(w), i.e., 1

 Q( x)dx .

orness(w) =

(6)

0

In Section 5.3, Q(x) = xa (a>0) will be employed to generate OWA operator weights. 3. New ranking method of DHFSs

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The score and accuracy functions defined in Definition 2 have been successfully applied to compare alternative assessments denoted by DHFS in the literature [17,22,25-29]. However, such functions may not be suitable in some situations. For example, suppose that there are three DHFEs α1 = {{0.35}, {0.4, 0.6}}, α2 = {{0.3, 0.4}, {0.5}}, α3 = {{0.2, 0.5}, {0.5}}, which have the same score -0.15 and the same accuracy 0.85. In other words, their score and accuracy cannot be used to judge which one is the best. From a point of view of statistics, this issue may be due to lack of considering the variance of possible membership degrees (or possible non-membership degrees), which measures the spread or dispersion of the membership degrees (or the non-membership degrees). 3.1 New score function of DHFE

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Definition 5. Let α = {h, g} = {{  1 , …,   ( h ) }, { 1 , …,  ( g ) }} be a DHFE, where  (h) and  ( g ) symbolize the



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numbers of the elements in h and g, respectively. Then, its score function is defined as



 h

where

 ( h) i 1

i

,

 (g) j 1

j

 (g)

,  ( h) 



 (h) i 1

( i  h )2

, and  ( g ) 

 ( h)

(7)



 (g) j 1

( j  g )2

 (g)

.

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 ( h)

 g

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S ( )  h  (1   (h)2 )  g  (1   ( g )2 )

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If (7) is considered the score function of a DHFE, then the scores of the above three DHFEs can be obtained as -0.145, -0.151, and -0.158, from which it can be reasoned that α1 α2 α3, where the notation ‘ ’ denotes ‘prior to’. The property of the score function in Definition 5 is presented in the following. Property 1. Given a DHFE α = {h, g} = {{  1 , …,   ( h ) }, { 1 , …,  ( g ) }}, its score function S(α) satisfies the

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following relationships

(1) S ( )  min{ i }  max{ j } , and  i h

 j g

(2) S ( )  max{ i }  min{ j } ,  j g

 i h

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where the equality holds if and only if (iff) the DHFE α is reduced to an intuitionistic fuzzy number [46]. Proof.

From

(h  min{ i }) - h 

(7)



 i h

we

 (h) i 1

( i  h )2

 ( h)

can

infer

+ (max{ j }  g ) + g   j g

S ( )  min{ i }  max{ j }

that

 i h



 (g) j 1

( j  g ) 2

 (g)

It can be derived that (h  min{ i }) - h   i h



 (h) i 1

( i  h )2

 ( h) 7

can

 j g

be

transformed

 ≥0, i.e., (h  min{ i }) - h  i 1

 (h)

 i h

( i  h ) ≥0.  ( h)

into

1

 (i 1 ( i  min{ i })  h  i 1 ( i  h )2 )  ( h)

 ( h)

=

 (h)

≥

1  ( h)  ( h)  (i 1 ( i  min{ i })  i 1 ( i  h )2 )   h i  ( h)

= = =

1

 (i 1 ( i  min{ i })2  i 1 ( i  h )2 )  ( h)

 (h)

 ( h)

 i h

2 1  ( h)  i 1 ((min{ i })  (h )2  2 i (min{ i }  h ))   h  i h i  (h)

1

 ( h)



 ( h) i 1

((min{ i }  h )(min{ i }  h  2 i ))  i h

 i h

1 (min{ i }  h )  ( (h)  min{ i }   (h)  h  2 (h)  h )  i h  (h)  i h

IP T

≥

 i h

= (min{ i }  h )2 ≥0.  i h

Thus, S ( )  min{ i }  max{ j } holds. In a similar way we show  i h

SC R

 j g

S ( )  max{ i }  min{ j } can be verified.

□

 j g

 i h

U

3.2 New accuracy function of DHFE

N

For the two DHFEs with the same scores, their accuracy functions are constructed to further compare them, as shown below.

A

Definition 6. Let α = {h, g} = {{  1 , …,   ( h ) }, { 1 , …,  ( g ) }} be a DHFE, where  (h) and  ( g ) symbolize the





H ( )  h  (1   (h)2 )  g  (1   ( g )2 ) ,  ( h) i 1

i

 ( h)

,

 g

 (g) j 1

j

 (g)

,  ( h) 



 (h) i 1

( i  h )2

ED

where

 h

M

numbers of the elements in h and g, respectively. Then, its accuracy function is defined as

 ( h)

, and  ( g ) 

(8)



 (g) j 1

( j  g )2

 (g)

.

The accuracy function in Definition 6 exhibits the following property.

PT

Property 2. Given a DHFEα = {h, g} = {{  1 , …,   ( h ) }, { 1 , …,  ( g ) }}, its accuracy function H(α) satisfies (1) H ( )  min{ i }  min{ j } , and  j g

CC E

 i h

(2) H ( )  max{ i }  max{ j } ,  i h

 j g

A

where the equality holds iff the DHFE α is reduced to an intuitionistic fuzzy number [40]. Property 2 can be directly inferred from Property 1. 3.3 New comparison laws of DHFEs Based on the score function in Definition 5 and the accuracy function in Definition 6, the new comparison laws of two DHFEs are formally defined as follows. Definition 7. Let α1={h1, g1} and α2={h2, g2} be two DHFEs, S(α1) and S(α2) be their score functions, and H(α1) and H(α2) be their accuracy functions. Then, if S(α1) < S(α2), α1 is smaller than α2 denoted by α1 < α2, and vice versa. If S(α1) = S(α2), then (1) H(α1)=H(α2) indicates that α1 and α2 represent the same information and are denoted by α1=α2; and (2) H(α1) < H(α2) reveals that α1 is smaller than α2 represented by α1<α2. 8

4. Generalized operations of DHFS based on Frank t-norm and t-conorm In this section, Frank t-norm and t-conorm given in Section 2.2 are used to generalize operations of DHFS. The properties of the generalized operations are also analyzed. The addition and multiplication operations in Definition 3 are the algebraic sum and algebraic product operational rules of DHFS, respectively, a special pair of dual t-norm and t-conorm. We can extend these operations to obtain more general operations of DHFS in terms of the Frank t-norm and t-conorm, which are defined as follows. Definition 8. Let α1={h1, g1} and α2={h2, g2} be two DHFEs, then we have

={ 1   2

(2)

{Sr ( 1,  2 )},

1 g1 ,2 g2

{Tr (1,2 )} }={

1 h1 , 2 h2

 (r11  1)(r1 2  1)  ) , 1  log r (1   1 h1 , 2 h2 r 1  

=

1 g1 ,2 g2

{

 1 h1 , 2 h2

{Tr ( 1,  2 )},

{q 1 (q(1 )  q(2 ))}

1 g1 ,2 g2

{q 1 (q( 1 )  q( 2 ))} ,

{Sr (1,2 )}

 (r1  1)  )  }, λ>0, log r (1  1 g1 (r  1) 1  

{ p 1 ( p( 1 ))},

 1 h1

U

,

 (r11  1)  ) , 1  log r (1   1 h1 (r  1) 1  

{q 1 (q(1 ))} }={

1 g1

 (r 1  1)  ) , log r (1   1 h1 (r  1) 1  

ED

(4) 1 = {

,

N

1 g1

{ p 1 ( p( 1 )  p( 2 ))}

A

 1 h1

1 h1 , 2 h2

 (r 1  1)(r  2  1)  ) log r (1   1 h1 , 2 h2 r 1  

}={

{ p 1 ( p(1 ))} }={

}=(

M

{q 1 (q( 1 ))},

{ p 1 ( p(1 )  p(2 ))} }

 (r1  1)(r2  1)  )  }, log r (1  1 g1 ,2 g2 r 1  

 (r11  1)(r12  1)  )  }, 1  log r (1   1 g1 , 2 g2 r 1  

(3) λα1 = {

1 g1 ,2 g2

IP T

 1 h1 , 2 h2

SC R

(1) 1   2 = {

(5) 1c = {

{1} ,

 1 h1

{ 1} }.

CC E

1 g1

PT

 (r11  1)  )  }, λ>0, and 1  log r (1  1 g1 (r  1) 1  

Here, 1c represents the complement of the DHFE α1. The generalized operations presented in Definition 8 exhibit

A

the following properties. Theorem 1. Let α1={h1, g1} and α2={h2, g2} be two DHFEs, then they satisfy: (1) α1  α2=α2  α1, (2) α1  α2=α2  α1, (3) λ(α1  α2)=λα1  λα2, λ>0, (4) (α1  α2)λ= 1   2 , λ>0, (5) λ1α1  λ2α1=(λ1+λ2)α1, λ1>0, λ2>0, (6) 1  1  1   , λ1>0, λ2>0, 1

2

1

2

(7) (1c )  (1 )c , λ>0, 9

(8)  (1c )  (1 )c , λ>0, (9) 1c   2c  (1   2 )c , and (10) 1c   2c  (1   2 )c . Proof. We can directly infer from Definition 8 that the properties (1) and (2) hold. For the property (3), it can be inferred from Definition 8 that

 1 h1 , 2 h2

{q 1 (q(q 1 (q( 1 )  q( 2 ))))} , {q 1 ( (q( 1 )  q( 2 )))} ,

λα1  λα2={

{q 1 ( q( 1 ))} ,

 1 h1

2 g2

1 g1 ,2 g2

{ p 1 ( p(1 )  p(2 ))} }

{ p 1 ( p( p 1 ( p(1 )  p(2 ))))} }

{ p 1 ( ( p(1 )  p(2 )))} } and

{ p 1 ( p(1 ))} }  {

1 g1

 2 h2

{q 1 ( q( 2 ))} ,

}={

1 h1 , 2 h2

{q 1 (q(q 1 (q( 1 )))  q(q 1 (q( 2 ))))}

{ p 1 ( p( p 1 ( p(1 )))  p( p 1 ( p(2 ))))} }=

 1 h1 , 2 h2

{q 1 ( (q( 1 )  q( 2 )))} ,

1 g1 ,2 g2

,

{ p 1 ( ( p(1 )  p(2 )))} }.

N

{

1 g1 ,2 g2

{ p 1 ( p(2 ))}

1 g1 ,2 g2

1 g1 ,2 g2

IP T

={

 1 h1 , 2 h2

{ q 1 (q( 1 )  q( 2 ))} ,

SC R

={

 1 h1 , 2 h2

U

λ(α1  α2)={

{q 1 (1q( 1 ))} ,

 1 h1

{ p 1 (1 p(1 ))} }  {

1 g1

{ p 1 (2 p(1 ))} }={

1 g1

{q 1 (2q( 1 ))} ,

 1 h1

M

λ1α1  λ2α1={

A

Therefore, property (3) holds. The satisfaction of property (4) can be demonstrated in the same manner. From Definition 8 we have

{q 1 (q(q 1 (1q( 1 )))  q(q 1 (2q( 1 ))))} ,

1 h1

ED

{ p 1 ( p( p 1 (1 p(1 )))  p( p 1 (2 p(1 ))))} }=

1 g1

{q 1 (1q( 1 )  2q( 1 ))} ,

1 h1

{ p 1 (1 p(1 )  2 p(1 ))} }=(λ1+λ2)α1.

1 g1

PT

{

CC E

It validates property (5) and can be similarly applied to validate the property (6). The properties (7)-(10) can be directly deduced from Definition 8. □ When DHFEs reduce to intuitionistic fuzzy numbers, the properties in Theorem 1 are also proven by Xia et al. [42]. With the above background, concepts and definitions, in the next section, we propose two new DHF Frank aggregation operators.

A

5. Proposed DHF Frank aggregation operators Aggregation functions are used to combine a set of input values into a single representative output. In decision science, various approaches to MADM call for suitable aggregation functions. In this section, we first define two new DHF Frank aggregation operators with the use of Frank operations of DHFS in the context of MADM. The monotonicity and relationship of the two aggregation operators, and the meaning and determination of the parameter r in Frank t-norm and t-conorm are then discussed. To facilitate and motivate this introduction, MADM problems with DHF assessments are introduced. Suppose that a MADM problem includes m alternatives Ai (i = 1, ..., m) and n attributes Cj (j = 1, ..., n). The relative weights of the n attributes are represented by ω = (ω1, ω2, ..., ωn)T such that 0 ≤ ωj ≤ 1 and 10



n i 1

 j  1 where

the notation ‘T’ denotes ‘transpose’. Let Aij = {hij, gij} signify the DHF assessment of an alternative Ai on an attribute Cj. It is clear that the DHF assessment of any alternative Ai on any attribute Cj is represented by a DHFE. 5.1 DHF Frank hybrid arithmetic and geometric aggregation operators To solve a MADM problem, the assessments of alternatives for each attribute should be combined using aggregation functions to generate the aggregated assessments of alternatives. In most combination functions, the ordered positions of assessments Aij (j = 1, ..., n) are generally not taken into account. Differently, by weighing both

IP T

the given DHF assessment and its ordered position, in the following, we design hybrid weights to create DHF aggregation operators. The hybrid weights are designed as follows  j =  ( j )  (1   )w j , j = 1, ..., n.

(9)

Here, the relaxation coefficient θ such that 0 ≤ θ ≤ 1 combines attribute weights and ordered weights to generate

SC R

hybrid weights, and σ(j) (j = 1, …, n) represents a permutation of (1, 2, …, n) such that S ( Ai ( j ) ) ≥ S ( Ai ( j 1) ) (j = 1, …, n-1). The hybrid weights are then used to develop a Frank hybrid weighted arithmetic averaging (FHWAA) operator of assessments signified by DHFEs.  ij hij

{ ij },

ij gij

{ij } } (i = 1, 2, …, m, j = 1, …, n) be assessments of alternatives on

U

Definition 9. Let Aij = {hij, gij} = {

each attribute for a MADM problem denoted by DHFEs, ω = (ω1, ω2, ..., ωn)T be the relative weights of the n

N

attributes, wi = (wi1, …, win)T be the OWA operator weights with respect to Aij , and ij =  ( j )  (1   )wij (j = 1, …, n)

A

be the hybrid weight of Ai ( j ) . Then, a Frank hybrid weighted arithmetic averaging operator is a function n  

M

such that

FHWAA( Ai1 , …, Ain ) = i1 Ai (1)  …  in Ai ( n )

ED

(10)

where  is the set of all Aij denoted by DHFEs.

Based on the Frank operations of DHFS, we derive the following result. { ij },

PT

Theorem 2. Let Aij = {hij, gij} = {

 ij hij

ij gij

{ij } } (i = 1, 2, …, m, j = 1, …, n) be assessments of alternatives on

CC E

each attribute for a MADM problem denoted by DHFEs. Then, the arithmetic aggregated assessment Aia = {hi, gi} (i = 1, 2, …, m) obtained by the FHWAA operator is still a DHFE, and Aia = FHWAA( Ai1 , …, Ain ) =  i ( j ) hi ( j )

q

1



( j 1ij q( i ( j ) )) , n

 i ( j ) gi ( j )

p

1



( j 1ij p(i ( j ) )) } = n

A

{

    (r  1)   ) , {  i ( j ) hi ( j ) 1  log r (1  n r  1 ij   j 1 ( r1 i ( j )  1)   

    r 1   )  }.  i ( j ) gi ( j ) log r (1  n r  1 ij   j 1 ( ri ( j )  1)   

(11)

11

Proof. Based on Definition 8, we have ij Ai ( j ) ={

={

 i ( j ) hi ( j )

{q 1 (ij q( i ( j ) ))} ,

i ( j ) gi ( j )

{ p 1 (ij p(i ( j ) ))} }

1    (r i ( j )  1) ij   ) , 1  log r (1  ij 1  i ( j ) hi ( j )  ( r  1)   

ij   (r i ( j )  1)  ij 1  i ( j ) gi ( j ) log r (1  (r  1)  



  )  }.  

 i (1) hi (1)  i ( 2) hi ( 2)

{

{ p 1 ( p( p 1(i1 p(i (1) )))  p( p 1( i2 p(i (2) ))))} }=

 i (1) hi (1)  i ( 2) hi ( 2)

{q 1 (i1q( i (1) )  i 2q( i (2) ))} ,

i (1) gi (1) i ( 2) gi ( 2)

{ p 1 (i1 p(i (1) )  i 2 p(i (2) ))} }

SC R

i (1) gi (1) i (2) gi (2)

A

N

U

    (r  1)   ={ ) , 1  log r (1  r  1 r  1  i (1)hi (1)  ( 1 i (1) )i1  ( 1 i ( 2 ) )i 2   i ( 2 )hi ( 2 )   r 1 r 1       r 1   )  }. log r (1  r  1 r  1  i (1) gi (1)  ( i (1) )1  ( i ( 2 ) )i 2   i ( 2 ) gi ( 2 )   r 1 r 1  

IP T

In what follows, mathematical induction is applied to prove the conclusion (11). When n = 2, we have FHWAA( Ai1 , Ai 2 ) = i1 Ai (1)  i 2 Ai (2) = { {q 1 (q(q 1 (i1q( i (1) )))  q(q 1 (i 2q( i (2) ))))} ,

M

The conclusion (11) holds. Suppose that the conclusion (11) still holds when n = k, i.e.,

{

{q 1  j 1ij q( i ( j ) )} , k

 i ( j ) hi ( j )

ED

FHWAA( Ai1 , …, Aik ) = i1 Ai (1)  …  ik Ai ( k ) =

{ p 1  j 1ij p(i ( j ) )} } k

i ( j ) gi ( j )

CC E

PT

    (r  1)   ={  i ( j ) hi ( j ) 1  log r (1  ) , k r  1 ij   j 1 ( r1 i ( j )  1)        r 1   )  }.  i ( j ) gi ( j ) log r (1  k r  1 ij   j 1 ( ri ( j )  1)   

A

When n = k + 1, we have

FHWAA( Ai1 , …, Ai ( k 1) ) = i1 Ai (1)  …  ik Ai ( k )  i ( k 1) Ai ( k 1) = q 1 ( j 1ij q( i ( j ) )) , k

{

 i ( j ) hi ( j )

{

 i ( k 1) hi ( k 1)

{

q 1 (i ( k 1) q( i ( k 1) )) ,

p 1 ( j 1ij p(i ( j ) )) }  k

i ( j ) gi ( j )

i ( k 1) gi ( k 1)

p 1 (i ( k 1) p(i ( k 1) )) } =

{q 1 (q(q 1 ( j 1ij q( i ( j ) )))  q(q 1 (i ( k 1) q( i ( k 1) ))))} k

 i ( j ) hi ( j )  i ( k 1) hi ( k 1)

12

,

{ p 1 ( p( p 1 ( j 1ij p(i ( j ) )))  p( p 1 (i ( k 1) p(i ( k 1) ))))} }= n

i ( j ) gi ( j ) i ( k 1) gi ( k 1)

    (r  1)   {  i ( j ) hi ( j ) 1  log r (1  ) , k r  1 r  1 ij ik 1  ( 1 i (k 1) )   j 1 ( 1 i ( j ) )    r 1 r 1  

    (r  1)   = {  i ( j ) hi ( j ) 1  log r (1  ) , k 1 r  1 ij   j 1 ( r1 i ( j )  1)   

    r 1   )  },  i ( j ) gi ( j ) log r (1  k 1 r  1 ij   j 1 ( ri ( j )  1)   

IP T

  k 1   r 1 k 1   { p 1 (ij p(i ( j ) ))} } ) } = { {q 1 ( j 1ij q( i ( j ) ))} ,  i ( j ) gi ( j ) log r (1  k r 1 r 1  i ( j ) gi ( j )  i ( j ) hi ( j ) j 1  ( i (k 1) )ik 1   j 1 ( i ( j ) ) ij    r 1 r 1  

 ij hij

{ ij },

{ij } } (i = 1, 2, …, m, j = 1, …, n) be assessments of alternatives on

M

Definition 10. Let Aij = {hij, gij} = {

A

N

U

SC R

which means that the conclusion (11) holds. This theorem has been proved. □ The FHWAA operator in Theorem 2 involves several special cases: (1) When θ = 1, the operator only takes account of attribute weights; (2) When θ = 0, the operator becomes an OWA operator based on Frank t-norm and t-conorm; (3) When θ = 1 and r→1, the operator is transformed into the weighted arithmetic averaging operator of assessments profiled by DHFEs [21];and (4) When θ = 0 and r→1, the operator reduces to the ordered weighted arithmetic averaging operator of assessments denoted by DHFEs. Similar to the FHWAA operator, a Frank hybrid weighted geometric averaging (FHWGA) operator of assessments symbolized by DHFEs can also be developed using the hybrid weights. ij gij

ED

each attribute for a MADM problem denoted by DHFEs, ω = (ω1, ω2, ..., ωn)T be the relative weights of the n attributes, wi = (wi1, …, win)T be the OWA operator weights with respect to Aij , and ij =  ( j )  (1   )wij (j = 1, …, n)

PT

be the hybrid weight of Ai ( j ) . Then, a Frank hybrid weighted geometric averaging operator is a function n   such that

FHWGA( Ai1 , …, Ain ) = Ai (1)  …  Ai ( n )

CC E

i1

(12)

in

where  is the set of all Aij denoted by DHFEs. Based on the Frank operations of DHFS, we can derive the following result.  ij hij

{ ij },

ij gij

{ij } } (i = 1, 2, …, m, j = 1, …, n) be assessments of alternatives on

A

Theorem 3. Let Aij = {hij, gij} = {

each attribute for a MADM problem denoted by DHFEs. Then, the geometric aggregated assessment Aig = {hi, gi} (i = 1, 2, …, m) obtained by the FHWGA operator is still a DHFE, and Aig = FHWGA( Ai1 , …, Ain ) =

(

 i ( j ) hi ( j )

p

1



( j 1ij p( i ( j ) )) , n

i ( j ) gi ( j )

q

1



( j 1ij q(i ( j ) )) ) = n

13

    r 1   (  i ( j ) hi ( j ) log r (1  ) , n r  1 ij   j 1 ( r  i ( j )  1)        (r  1)   )  ). i ( j ) gi ( j ) 1  log r (1  n r  1 ij   j 1 ( r1i ( j )  1)   

(13)

The proof of Theorem 3 is similar to Theorem 2 and thus we omit it here.

IP T

5.2 Monotonicity and relationship of scores of the aggregated assessments with respect to r in Frank arithmetic and geometric aggregation operators

SC R

In this section, we first show the monotonicity of the score functions of the aggregated assessments, with respect to the parameter r in Frank arithmetic and geometric aggregation operators. Based on this monotonicity, the relationship between the score functions of the aggregated assessments will be discussed and proven. Once the arithmetic and geometric aggregated assessments have been obtained, their scores can be calculated using Definition 5. (a) Given the aggregated assessment Aia = {hi, gi} =  i ( j ) hi ( j )

q

1



( j 1ij q( i ( j ) )) , n

 i ( j ) gi ( j )

p

1



( j 1ij p(i ( j ) )) } generated by the FHWAA operator, where  (hi ) n

U

{

N

and  ( gi ) denote the numbers of the elements in hi and gi, respectively. Then, the score of Aia denoted by S( Aia ) is

h  (1   (h ) )  g  (1   ( g ) ) , 2

1

n j 1

ij q( i ( j ) ))]

 (hi )  ( gi ) j 1

[ p (  1

, g=

[ p 1 ( j 1ij p(i ( j ) ))  g ]2 n

 ( gi )

n

j 1

ij p(i ( j ) ))]

 ( gi )

(14)

,  (hi ) 



 ( hi ) i 1

[q 1 ( j 1ij q( i ( j ) ))  h ]2 n

 (hi )

, and

.

PT

 ( gi ) 

i

M

[ q ( 

where h =



2

i

ED

S( Aia ) =

A

calculated as

(b) Given the aggregated assessment Aig = {hi, gi} =

p

1

 i ( j ) hi ( j )



( j 1ij p( i ( j ) )) , n

CC E

{

i ( j ) gi ( j )

q

1



( j 1ij q(i ( j ) )) } generated by the FHWGA operator, where  (hi ) n

and  ( gi ) stand for the numbers of the elements in hi and gi, respectively. Then, the score of Aig denoted by S( Aig ) is calculated in the following way

A

S( Aig ) =

h  (1   (h ) )  g  (1   ( g ) ) , 2

i

[ p (  1

where h =

 ( gi ) 



n j 1

ij p( i ( j ) ))]

 (hi )  ( gi ) j 1

2

(15)

i

[ q (  1

, g=

[q 1 ( j 1ij q(i ( j ) ))  g ]2

n j 1

ij q(i ( j ) ))]

 ( gi )

,  (hi ) 



 ( hi ) i 1

[ p 1 ( j 1ij p( i ( j ) ))  h ]2 n

 (hi )

, and

n

 ( gi )

.

From (14) and (15) we find that S( Aia ) (or S( Aig ) comprises two parts, which are 14

h  (1   (h ) ) 2

i

and

g  (1   ( g ) ) . 2

i

The definitions of the functions p and q in (2) and (3) indicate that S( Aia ) and S( Aig ) are the

functions with respect to the parameter r. In this context the analysis of the monotonicity of S( Aia ) and S( Aig ) with respect to the parameter r is equivalently transformed into the discussion of the monotonicity of the functions p and q with respect to r. To facilitate this analysis, we first present the following relevant lemma and propositions. Lemma 1. Given a set H = {hi} (i = 1, ..., n) such that 0 ≤ hi ≤ 1, then the function F(h1, ..., hn) = h  (1   (h)2 ) 1 n 1 n  hi and  (h)2  n i 1(hi  h )2 is monotonically increasing with respect to hi ( hi  H ). n i 1

where h 

1 1 n 1 (1   j 1 (h j  h )2 ) + h  (2h  2hi ) n n n

=

1 1 n 1 (1  h 2   j 1 h2j ) + h  (2h  2hi ) n n n

=

1 1 n (1  h 2   j 1 h2j  2h 2  2hhi ) n n

≥

1 (1  3h 2  h  2hhi ) n

≥

1 (1  3h 2  3h ) . n

SC R

=

U

F hi

IP T

Proof. It is clear that F(h1, ..., hn) is a multivariate function. For any variable hi such that hi > h , we have

N

For the function G(x) = 3x2 - 3x + 1 such that 0 ≤ x ≤ 1, it can be known that G x < 0 if 0 ≤ x < 0.5, G x = 0 if x = 0.5, and G x > 0 if 0.5 < x ≤ 1. Thus, we can infer that G(x) ≥ G(0.5) = 0.25, which deduces that F hi ≥

A

0.25/n > 0. On the other hand, for any variable hi such that hi < h , F hi > 0 can be directly deduced from 1 1 n 1 (1   j 1 (h j  h )2 ) + h  (2h  2hi ) . As a whole, F(h1, ..., hn) is a monotonically increasing function n n n

with respect to hi ( hi  H ).

□

M

F hi =

ED

Proposition 1. Given a function M(r) = q 1 ( j 1 j q( j )) with a parameter r such that 0 ≤  j ≤ 1, 0 ≤  j ≤ 1, n

and r  (1, ) . Then, the function is monotonically decreasing with respect to r.

PT

Proof. From (3) we infer that

CC E

 (r1 x  1)  (1  x)r  x (r  1)   r1 x  1   (1  x)r  x  xr1 x  1  q =    =  . 1 x 2 1 x (r  1) r    r 1   (r  1)(r  1) 

Let W1(r) = (1  x)r  x  xr1 x  1 , where r > 1 and 0 < x ≤ 1, W1 = x(1  x)(r  1)r  x 1 > 0, r

A

so we can conclude that W1(r) is monotonically increasing with respect to r, then W1(r) > lim W1 (r ) = 0. r 1

And r-1 > 0, r1 x  1  0 , so

 (1  x)r  x  xr1 x  1  q =   > 0. 1 x r  (r  1)(r  1) 

Similarly, it can be shown on a basis of (3) that

15

 log r log(r  1  e x )  x     x q 1 r =  r  1  e  2 r (log r )        r log r  (r  1  e x )log(r  1  e x )  x (r  1 ex )  . r (r  1  e x )(log r )2  

= 

Let W2(r) = r log r  (r  1  e x )log(r  1  e x ) + x(r  1  e x ) , where r > 1 and 0 < x ≤ 1,

IP T

W2 re x (r  1)e x  e x ) log( ) > 0, = log r  log(r  1  ex )  x = log( = r  1  ex r  1  ex r

so we conclude that W2(r) is monotonically increasing with respect to r, then W2(r) > lim W2 (r ) = 0. r 1

q 1 < 0. r

SC R

And r (r  1  ex )(log r )2  0 , so

It is clear that q(x) and q-1(x) are monotonically increasing and decreasing regarding r, respectively. As such, under

U

the conditions of 0 ≤  j ≤ 1 and 0 ≤  j ≤ 1, the increase in r will increase



about the decrease in q 1 ( j 1 j q( j )) . The conclusion in this proposition is valid.

N

n

n

j 1

 j q( j ) , which further brings

□

Proposition 2. Given a function N(r) = p 1 ( j 1 j p( j )) with a parameter r such that 0 ≤  j ≤ 1, 0 ≤  j ≤ 1,

A

n

Proof. It can be deduced from (2) that

M

and r  (1, ) . Then, the function is monotonically increasing with respect to r.

ED

 (r x  1)  xr x 1 (r  1)   r x  1   (1  x)r x  xr x 1  1  p =    =  . x 2 x (r  1) r    r 1   (r  1)(r  1) 

Let Z1(r) = (1  x)r x  xr x 1  1 , where r > 1 and 0 < x ≤ 1,

PT

Z1 = x(r  1)(1  x)r x  2 > 0, r

so we conclude that Z1(r) is monotonically increasing with respect to r, then

CC E

Z1(r) > lim Z1 (r ) = 0. r 1

And (r x  1)(r  1) > 0, so

p > 0. r

A

We can similarly derive from (2) that p 1 = r

r 1 ex  r  1 ) r log r  (e x  r  1)log( ) x e ex = . r r (e x  r  1)(log r )2

 log r (1 

Let Z2(r) = r log r  (r  1  e x )log(

r  1  ex ) , where r > 1 and 0 < x ≤ 1, ex

Z 2 ex  r  1 e x (r  1)  e x ) = log( ) > 0, = log r  log( x e r  1  ex r

so we conclude that Z2(r) is monotonically increasing with respect to r, then Z2(r) > lim Z 2 (r ) = 0. r 1

16

And r (r  1  ex )(log r )2  0 , so

p 1 > 0. r

These observations indicate that both p(x) and p-1(x) are monotonically increasing functions of r. In the situation of 0 ≤  j ≤ 1 and 0 ≤  j ≤ 1, if r increases,



n j 1

 j p(  j ) will grow, which results in the rise in p 1 ( j 1 j p( j )) . n

□

As a result, it is clear that N(r) is a monotonically increasing function of r.

From (a) and (b), Lemma 1, and Propositions 1 and 2, the monotonicity of S( Aia ) and S( Aig ) can be directly inferred, which is presented in the following proposition.

IP T

Proposition 3. Let S( Aia ) and S( Aig ) be the score functions of the aggregated assessments Aia and Aig , as presented in (14) and (15), respectively. Then, S( Aia ) and S( Aig ) are monotonically decreasing and increasing with respect to r, respectively.

SC R

Based on the monotonicity of S( Aia ) and S( Aig ) with respect to r in Frank arithmetic and geometric aggregation operators, the relationship between S( Aia ) and S( Aig ) is discussed and proven in the following. To facilitate the

n j 1

x j j   j 1  j x j 

n j 1

 j =1, then we have

n

with equality if and only if x1= x2= … = xn.

xb where x > 0 and b ≥ 0, it is monotonically decreasing with respect to x. x

M

Lemma 3. Given a function f(x)= Proof. For this function, we have

ED

f b = 2 ≤ 0. x x

(16)

A





N

Lemma 2. [47] Assume that xj > 0, λj > 0 (j = 1, …, n), and

U

analysis of the relationship between S( Aia ) and S( Aig ), we first present two relevant lemmas.

□

As a result, f(x) is a monotonically decreasing function of x.

PT

Based on these two lemmas, the relationship between S( Aia ) and S( Aig ) is presented in the following theorem.

CC E

Theorem 4. Let S( Aia ) and S( Aig ) be the score functions of the aggregated assessments Aia and Aig , as presented in (14) and (15), respectively. Then, we have S( Aia ) > S( Aig ),

(17)

which is independent of r such that r  (1, ) .

A

Proof. Proposition 3 states that S( Aia ) and S( Aig ) are monotonically decreasing and increasing with respect to r, respectively. Then, the conclusion in (17) can be transformed into lim S ( Aia ) ≥ lim S ( Aig ) . By using Lemma 2, it is r 

further converted into the following two inequalities lim (1  log r (1 

r 

r 1 r 1 )) ≥ lim (log(1  )) and r  n r  1 ij r 1   j 1 ( r1 i ( j )  1)  j 1 ( r  i ( j )  1) ij n

17

r 

r 1 r 1 )) ≥ lim (log(1  )) . r  n r  1 ij r 1   j 1 ( r1i ( j )  1)  j 1 ( ri ( j )  1) ij

lim (1  log r (1 

r 

n

The two inequalities are the same and thus only the first one is proven. We can reason from Lemma 3 that the left-hand side of the first inequality r 1 r )) ≥ lim (1  log r (1  )) = lim r  r  n r  1 ij r   j 1 ( r1 i ( j )  1)  j 1 ( r1 i ( j ) ) ij

lim (1  log r (1 

r 

log

1

r n  j 1ij  i ( j )

(1  r log r

n

)

and the right-hand side of the first inequality  ij  i ( j ) r 1 r log(1  r j 1 ) lim (log r (1  )) ≤ lim (log r (1  )) = lim . r  r  r  n n r  1 ij r ij log r ( ) ( )  j 1 r  i ( j )  1 i 1 r  i ( j )

Based on the L'Hôpital's rule, we have

1

 j 1ij  i ( j )

r 



n j 1

r

1 r )(

1

 j 1ij  i ( j ) n

1  ij  i ( j )  ij  i ( j ) j 1  r (1  r )r j 1 ) n 1  ij  i ( j ) 2 j 1 (1  r ) n

n

ij i ( j )

N

=

n

U

1 r = lim (

 ij  i ( j )  ij  i ( j ) (log(1  r j 1 )) r log(1  r j 1 ) and lim = lim r  r  (log r ) r log r n

n j 1

ij i ( j )  r

A

n

 j 1ij  i ( j ) 1 n

M



 j 1ij  i ( j ) n

1

r 



n j 1

ij i ( j ) .

Thus we have lim (1  log r (1 



r 1 )) ≥ r 1   j 1 ( r1 i ( j )  1) ij n

CC E

r 

PT

=

r

ED

1 r

= lim

lim (1  log r (1 

r 

SC R

r r log( )  (log( )) r n n 1  ij  i ( j ) 1  ij  i ( j ) j 1 j 1 1 r 1 r = lim lim r  r  log r  (log r ) r

IP T

n



n j 1

r 1 )) ≥ r  1 ij ( )  j 1 r1i ( j )  1 n

ij i ( j ) ≥ lim (log(1 



r 

n j 1

r 1 )) r 1   j 1 ( r  i ( j )  1) ij

iji ( j ) ≥ lim (log(1  r 

n

r 1 )) . r  1 ij ( )  j 1 ri ( j )  1 n

As a result, lim S ( Aia ) ≥ lim S ( Aig ) holds, which completes the proof of Theorem 4. r 

□

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r 

It is noted that Proposition 3 and Theorem 4 reveal universal conclusions. We can reason from the proof of Propositions 1 and 2 that Proposition 3 and Theorem 4 always hold when q-1(x) and q(x) are monotonically increasing and decreasing functions, or the converse, and both p-1(x) and p(x) are monotonically increasing or decreasing functions with respect to r. More importantly, the conclusions are still efficacious when the DHF assessments in Proposition 3 and Theorem 4 reduce to intuitionistic fuzzy assessments. 5.3 Meaning and determination of the parameter r in Frank arithmetic and geometric aggregation operators The above analysis indicates that the parameter r in the Frank aggregation operators exhibits a significant effect on 18

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the arithmetic and geometric aggregated assessments and further on the solution to a MADM problem. As such, the meaning of r and how to determine r in the context of MADM need to be elaborated on. Proposition 3 points out that the arithmetic and geometric aggregated assessments are monotonically decreasing and increasing, respectively, with the increase of r. In other words, if a decision maker chooses a large r, then he or she is pessimistic to prefer small arithmetic aggregated assessment and optimistic to favor large geometric aggregated assessment. Therefore, we can associate r with the risk attitude of the decision maker. When the FHWAA operator is applied to combine attributes, small r means that the decision maker is risk seeking. The converse situation will occur when the FHWGA operator is employed. In practice, a decision maker can determine r by his or her preference, like the decision on orness(w). However, the decision maker may make inconsistent decisions on r and orness(w). To avoid this possible inconsistency, we attempt to construct an equivalent relationship between r and orness(w). First of all, the decision maker determines the possible minimal and maximal r denoted by ra and ra for the FHWAA operator and rg and rg for the FHWGA

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operator. For example, it can be set that ra = rg = 1.05 which is close to 1. Differently, ra and rg may be judged by the preference of the decision maker or the score curves of alternatives with a variation in ra and rg, which will be illustrated in Section 7. Secondly, a function Ga(ra) such that Ga( ra ) = 1, Ga( ra ) = 0, and Ga(ra1) ≤ Ga(ra2)

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 ra ≥ ra1 ≥ ra2 ≥ ra is designed to depict the risk attitude of the decision maker when the FHWAA operator is

used. For instance, we can develop a linear function ra  ra , ra ≤ ra ≤ ra ... ra  ra

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Ga(ra) =

(18)

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In the situation of the FHWGA operator, we construct a function Gg(rg) such that Gg( rg ) = 0, Gg( rg ) = 1, and Gg(rg1)

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≥ Gg(rg2)  rg ≥ rg1 ≥ rg2 ≥ rg to portray the risk attitude of the decision maker. Similarly, a linear Gg(rg) is

Gg(rg) =

rg  rg rg  rg

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expressed as , rg ≤ rg ≤ rg .

(19)

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Thirdly, we create an equivalent relationship between ra and orness(w) and that between rg and orness(w), which is profiled by Ga(ra) = orness(w) and Gg(rg) = orness(w). When Ga(ra) and Gg(rg) respectively in (18) and (19) are employed, it can be inferred from Ga(ra) = orness(w) and Gg(rg) = orness(w) that (20)

rg = rg  orness(w)  (rg  rg ) .

(21)

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ra = ra  orness(w)  (ra  ra ) and

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We can further infer from (20) - (21) that under the conditions of ra = rg and ra = rg if orness(w) < 0.5, ra > rg; otherwise, ra ≤ rg. In other words, when orness(w) ≠ 0.5, ra ≠ rg is required to satisfy ra = rg. This section gave an intensive study on DHF information aggregation techniques in the context of MADM. Frank t-norm and t-conorm are used to model the intersection and union of DHFEs. By combining these operational rules of DHFEs with OWA operator, we put forward the DHF Frank aggregation operators. One can find that the proposed operators consider the importance degrees of both the given DHF assessments and their ordered positions. More importantly, the risk attitudes of decision makers can be reflected by the determination of the parameter r. When different choices of the parameters r and orness(w) for the FHWAA (or FHWGA) operator are used, some special 19

cases can be obtained, which allows decision-makers have more choices in MADM problems. During the aggregation process of DHF assessments, one can aggregate all the DHF assessments Aij (j = 1, …, n) into a global assessment Ai by means of the proposed DHF Frank aggregation operators. These operators also provide us with some useful

ways for data or information fusion in data mining, pattern recognition, and DHF cluster analysis. Besides, one can combine individual DHF assessments with their associated weights to generate a collected DHF assessment in group decision making with DHF information. In the following section, we will focus on the development of an MADM approach with the aid of the proposed DHF Frank aggregation operators. 6. An approach to MADM based on new ranking method and DHF Frank aggregation operators

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In this section, the new ranking method of DHFSs and the proposed DHF Frank aggregation operators are used to develop an approach to solve MADM with DHF assessments. For a MADM problem modeled in Section 5, the approach based on FHWAA (or FHWGA) operator and the new ranking method to resolve the MADM problems with DHF assessments mainly involves the following steps: Step 1. The decision maker gives assessments of alternatives on each attribute to generate a decision matrix Amn = ( Aij )m×n, in which all assessments on cost attributes are transformed into benefit assessments using the

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complementary operation of assessments in Definition 8. Step 2. The decision maker specifies orness(w) or the parameter a of the function Q(x) to determine OWA operator weights, as mentioned in Section 2.3, the parameter ra or rg in Frank t-norm and t-conorm by his or her preference or the way presented in Section 5.3, and the relaxation coefficient θ to balance attribute weights and ordered weights (i.e., OWA operator weights), according to data relevant to the decision problem under consideration and his or her knowledge and experience.

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Step 3. The assessments of alternatives Aij (i = 1, …, m, j = 1, …, n) are aggregated using FHWAA (or FHWGA)

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operator presented in Theorem 2 (or Theorem 3) to generate the aggregated assessment Ai . Step 4. The scores of the aggregated assessments Ai (i = 1, …, m), i.e., S( Ai ), are calculated using Definition 5. For

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7. Case study

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the alternatives with the same score, their accuracy functions are calculated using Definition 6. Step 5. A rank-order of the m alternatives can be obtained by using the comparison laws in Definition 7, which is a solution to the MADM problem.

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In this section, an investment evaluation problem is analyzed by the proposed approach to demonstrate its applicability and validity. A detailed sensitivity analysis and a comparative study are also conducted to highlight the validity and advantages of the proposed approach. We also discuss the situations where Hamacher aggregation operators are employed under the framework of the proposed approach, by re-investigating this investment evaluation problem. The schematic diagram of the case study is shown in Figure 1.

7.1 Description of the investment evaluation problem The iron and steel industry is one of the fundamental industries contributing to the economy of China. The industry is not only closely related to upstream and downstream industries and driven largely by the requirements of consumption, but also significantly influences the development of economy and society in China. As an important and necessary source for producing steel, iron directly restricts the development of the iron and steel industry. Unfortunately, iron resources in China are relatively limited and their quality is below international average level. As 20

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such, large Chinese companies in the iron and steel industry have to adopt an international development strategy to seek the optimal iron resource allocation at a global level, which might be the only way for the companies to compete internationally in the near future. However, most domestic companies of iron and steel lack experience of overseas investment which might result in significant economic losses. To overcome this problem, our research institute takes on a project named “research on the key issues about international development strategy of mineral-resource-based enterprises”, which is approved by the Chinese Academy of Engineering (CAE). As one of main tasks in the project, taking a large domestic company in iron and steel as an example, we investigate how to assist the company to invest abroad effectively and reasonably. The general manager of the company acts as the decision maker, and he must decide which of the following five nations to invest in and source iron ores from: Australia (A1), India (A2), Brazil (A3), Canada (A4), and Russia (A5) from Chinese main importing countries of iron by integrating our opinions and the opinions of four experts from CAE and Ministry of Land and Resources (MLR). In the same manner, seven attributes are identified, as depicted in Table 1. The decision maker applies the method discussed in [48] to determine the relative weights of the seven attributes. In detail, he first identifies the most important attribute, i.e., the first attribute, and then compares other attributes with the first one to determine the relative importance of these attributes. Finally, by normalizing the relative weights, we can get ω = (0.35, 0.10, 0.05, 0.10, 0.05, 0.15, 0.20).

7.2 Solution to the investment evaluation problem

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As mentioned above, overseas investment might be the only way for the long-term development of the company. For this reason, decisions on the investment evaluation problem are very important for the company. To make decisions as rational as possible, the decision maker invites us and four experts from CAE and MLR to independently and anonymously evaluate the five nations on each attribute. To provide the estimation of the degrees to which whether alternatives satisfy certain attributes or not, different experts may provide different values; additionally, these experts cannot persuade one another to change their opinions. Then, the decision maker combines the opinions and

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his preference to construct a DHF decision matrix A57 which is shown in Table 2. For example, India (A2) is

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assessed on attribute C2 as {{0.3,0.4},{0.5,0.6}}, which indicates that the degree to which alternative A2 satisfies attribute C2 may be 0.3 or 0.4, and the degree to which alternative A2 does not satisfy attribute C2 may be 0.5 or 0.6. Step 1 has been completed.

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The decision maker uses the method discussed in [45] to determine the OWA weighting vectors. More specifically, in the process of quantifier guided aggregation [45] the decision maker should first provide a linguistic quantifier Q indicating the proportion of attributes he feels is necessary for a good solution. The quantifier is then used to generate an OWA weighting vector w of dimension n. To avoid economic and developmental losses caused by irrational decisions, which are due to less consideration of inferior assessments, the decision maker assumes the quantifier guiding this aggregation to be “more than half” which is defined by Q(x) = x1.5. This means that “more than half” indicating 4/7 attributes are satisfied by a good solution. Then, it can be derived from (5) and (6) that w = (0.05, 0.10, 0.13, 0.15, 0.17, 0.19, 0.21)T. The decision maker also specifies the parameter ra = 2 and the relaxation coefficient θ = 0.5 according to his preference. Step 2 has been completed. From Theorem 2 we obtain 1 = (0.20, 0.15, 0.12, 0.10, 0.16, 0.12, 0.15)T, 2 = (0.20, 0.10, 0.09, 0.15, 0.11, 0.20, 0.15)T, 3 = (0.20, 0.10, 0.09, 0.18, 0.16, 0.14, 0.13)T, 4 = (0.08, 0.22, 0.14, 0.18, 0.11, 0.14, 0.13)T, and 5 = (0.20, 0.15, 0.11, 0.13, 0.16, 0.12, 0.13)T. By using the hybrid weights and the FHWAA operator, the aggregated assessments of the five nations Ai (i = 1, …, 5) are produced, which are not presented to save space. Step 3 has been completed. 21

It can be derived from the aggregated assessments of the five nations and Definition 5 that S( Ai ) (i = 1, …, 5) = (0.3758, 0.1280, 0.1983, 0.2610, 0.1088). Because there are no any two alternatives with the same score, step 4 has been completed. Then, a rank-order of the five nations can be inferred from S( Ai ) (i = 1, …, 5) and Definition 7, which is A1

A4

A3 A2 A5 where the notation ‘ ’ means ‘prior to’. The decision maker obtains the solution to the investment evaluation problem and the optimal choice, i.e., Australia. Step 5 has been completed. Under the conditions of orness(w) = 0.4, rg = 2, and θ = 0.5, the application of the FHWGA operator leads to S( Ai )

7.3 Sensitivity analysis for parameters in Frank aggregation operators

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(i = 1, …, 5) = (0.1569, 0.1255, 0.1532, 0.1795, 0.0616). It is clear that the scores of the five nations become smaller. The solution and the optimal choice are converted into A4 A1 A3 A2 A5 and Canada, respectively, which are considerably different from the results generated by the FHWAA operator. From the above analysis, we can find that the rank orders generated by arithmetic aggregation operators differ from those generated by geometric aggregation operators, which indicates that different types of dual hesitant fuzzy operators can produce different final results. The causes of the difference will be analyzed in the next section.

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Through re-analyzing the investment evaluation problem, we discuss the sole influence of attribute weights and ordered weights on the solution to the problem, the effect of the parameter r on the solution, the impact of orness(w) on the solution, and the affection of different attribute weights on the solution.

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When orness(w) = 0.4, ra = 2, and θ = 1, only attribute weights contribute to the aggregation of Aij (i = 1, …, 5, j

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= 1, …, 7) and further to the solution to the investment evaluation problem. In this situation, we recalculate by using

rank-order of the five nations A1

A4

A3

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the FHWAA operator that S( Ai ) (i = 1, …, 5) = (0.4686, 0.1515, 0.2630, 0.3308, 0.1965), which results in a different A5

A2. In another extreme situation where orness(w) = 0.4, ra

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= 2, and θ = 0, S( Ai ) (i = 1, …, 5) is changed to (0.2535, 0.0938, 0.1073, 0.1821, -0.0048), which indicates that A1

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A4 A3 A2 A5. We can infer from these two extreme situations that except A1, A3 and A4, the other two nations vary in their ranks along with the variation in θ. Under the conditions of orness(w) = 0.4 and rg = 2, the scores of the five nations in the two extreme situations are respectively equal to (0.3629, 0.2035, 0.2931, 0.2847, 0.2283) and (-0.0209, 0.0519, 0.0262, 0.0821, -0.0829), which make two different solutions, i.e., A1 A3 A4 A5 A2 and A4 A2 A3 A1 A5. So, different θ may bring about various ranks of the five nations.

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As presented in Section 5.3, ra and rg can be derived from orness(w) when Ga(ra) and Gg(rg), ra , ra , rg and rg are confirmed. Because ra and rg are close to 1, the difficulty lies in the decision on ra and rg . To make such

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decision, we need to examine the arithmetic score curves of the five nations with a variation in ra and the geometric score curves of the five nations with a variation in rg. In the situation of orness(w) = 0.4 and θ = 0.5, it is assumed that ra and rg move from 1.05 to 10 with a step 0.5. Then, the corresponding scores and rankings of the nations are listed in Tables 3 and 4, respectively. Further, the arithmetic and geometric score curves of the five nations are plotted in Figures 2 and 3, respectively.

It is shown from Figure 2 that the rank-order of the five nations keeps fixed with a variation in ra. As such, the choice of ra has no influence on the rank-order of the five nations but may only contribute to a further analysis of the problem. As for Figure 3, suppose rg = 1.05 and Gg(rg) in (19) are selected. By observing the five curves in Figure 22

3, the decision maker chooses the maximum inflection point in the figure to quantify rg , i.e., rg = 7, beyond which the five curves in Figure 3 have no intersection and become flat. That is, when rg > rg , the solution becomes fixed

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except that the scores of the five nations have slightly changed. Under this assumption, we can infer from (21) that rg = 3.43. Furthermore, Figures 2 and 3 verify Proposition 3 and Theorem 4. Because there are less intersections among the score curves of the five nations in Figure 2 compared with Figure 3, the solutions by the FHWAA operator with a variation in ra can be considered more steady than those by the FHWGA operator with the increase of rg. In the setting of r = 2 and θ = 0.5, suppose that orness(w) is changed from 0.05 to 0.5 with a step 0.05, which is due to the great importance of the problem for the company. The corresponding scores and rankings of the nations with variation in orness(w) are listed in Tables 5 and 6, respectively. The change of arithmetic and geometric scores of the five nations with a variation in orness(w) is plotted in Figures 4 and 5, respectively. Along with the increase of orness(w), scores of the five nations continuously increase. This demonstrates that the decision maker has gradually preferred large scores when the pessimistic degree represented by orness(w) has steadily decreased. Particularly, there are seven intersections when orness(w) is about 0.16, 0.2, 0.22, 0.27, 0.36, 0.41, and 0.47 in Figure 4. The seven intersections reveal that orness(w) has an important effect on both the optimal choice and the rank-order of the five nations. It is noted that Figures 4 and 4 also validate Theorem 4.

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Given orness(w) = 0.4 and ra = 2, two extreme cases indicate that A1 is always the optimal choice no matter whatever θ is specified. Meanwhile, it is verified that different orness(w) can lead to different optimal choices. Also, different attribute weights may cause different optimal choices. Assume that θ = 0.5 and ω = (0.05, 0.05, 0.35, 0.25,

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0.05, 0.15, 0.10)T, then it is recalculated by Theorem 2 that S( Ai ) (i = 1, …, 5) = (0.1532, 0.1400, 0.0506, 0.1240,

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-0.0262). Different from the solution made in Section 7.2, the rank-order of the five nations is changed to A1 A2 A4 A3 A5 and it becomes that India is superior to Canada. When the FHWGA operator is applied under the

to the solution, i.e., A2

A4

A3

A5

A1 and the optimal choice, i.e., India.

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7.4 Comparative analysis

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same assumption, we obtain that S( Ai ) (i = 1, …, 5) = (-0.1281, 0.1010, -0.0280, 0.0167, -0.1068), which gives rise

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In the following, the proposed method is compared with other previous methods including dual hesitant fuzzy hybrid average (DHFHA) operator (or DHFHG operator) [21], Einstein dual hesitant fuzzy weighted averaging (EDHFWA) operator (or EDHFWG operator) [22], dual hesitant fuzzy Hamacher hybrid averaging (DHFHHA) operator (or DHFHHG operator) [25], and the correlation coefficient based-dual hesitant fuzzy MADM method [19] to highlight its validity and advantages. The results of applying all these methods to the practical example aforementioned are presented in Table 7. From the results shown in Table 7, we can see that the method proposed in this paper and the methods developed in [21,22,25] generate the same rank order of the five alternatives: A1 A4 A3 A2 A5 when arithmetic aggregation operators such as FHWAA, DHFHA [11], EDHFWA [22], and DHFHHA [25] are used to combine attribute values. This reflects the validity of the proposed method in this paper. The rank order generated by Ye’s method is A1 A4 A2 A3 A5, where the rankings of A2 and A3 differ from those generated by the proposed method, but the best choice is still A1. When geometric aggregation operators such as FHWGA, DHFHG [21], EDHFWG [22], and DHFHHG [25] are used to combine attribute values, the rank orders generated by these aggregation operator based-methods may be different, but the results obtained by the methods in [21] and [22] can be derived from the proposed method given certain special values of the parameter r in the DHFHHG operator and the FHWGA operator. In other words, the Frank aggregation operators can contain almost of the ones in [21] and [22]. 23

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The above comparative analysis and the sensitive analysis completed in Section 7.3 reveal the following advantages of the proposed method: (1) The aggregation operators developed in Wang et al.[21] and Zhao et al. [22], and many other aggregation operators of intuitionistic fuzzy assessments, which can be extended for DHF assessments, can be regarded as special cases of the dual hesitant Frank aggregation operators in (10) and (12) with different θ and ra (rg). Therefore, the proposed aggregation operators with parameters can provide the decision makers more choices and thus the proposed method is more flexible and general than the existing ones [21,22]. This is also demonstrated in Section 5.1. (2) With different θ and ra (rg), the dynamic variation trend of rank orders of alternatives can be portrayed clearly. So, relative to a static fixed decision result obtained by the methods of Ye [19], Wang et al. [11], and Zhao et al. [22], the dynamic decision result generated by the proposed method can better reflect the inherent variety rule. (3) More importantly, compared with the method in [25], the meaning of ra (rg) in the context of MADM is illustrated and the equivalent relationship between ra (rg) and orness(w) is constructed to determine ra (rg) in the proposed method, which makes the determination of ra (rg) more consistent with the risk attitudes of the decision makers. Furthermore, the computational complexity of the developed method is simpler than DHFHHA (or DHFHHG) operator in [25]. (4) The arithmetic and geometric aggregated assessments are monotonically decreasing and increasing respectively concerning ra and rg, which provide the decision makers to choose the appropriate value based on their risk attitudes. Moreover, the arithmetic aggregated assessment is larger than the geometric aggregated assessment for the same ra and rg, and the relationship between the assessments can be captured by using the proposed aggregation operators, which make the choice of aggregation operator more flexible in accordance with the real decision needs. (5) Both mean and variance of possible membership and non-membership degrees are involved in creating score and accuracy functions of DHF assessments, which provide good differentiation between the assessments. In summary, the sensitive analysis made in Section 7.3 and the comparisons completed above highlight the validity and advantages of the method proposed in this paper. 7.5 Sensitive analysis for parameters in Hamacher aggregation operators

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Section 5.3 discussed the meaning of r and how to determine r in the context of MADM where dual hesitant fuzzy Frank aggregation operators are employed to combine attribute values. These issues can also be discussed in the same manner as they are discussed in MADM where dual hesitant fuzzy Hamacher aggregation operators are employed. In this section, we further discuss the effects of parameters r and orness(w) on the solution through re-considering the investment evaluation problem, under the assumption that the Frank aggregation operators are replaced by Hamacher aggregation operators at the second step of the proposed method. Motivated by the idea of Section 5, and based on the Hamacher operations of DHFEs developed in Ju et al. [25], we first develop a Hamacher hybrid weighted arithmetic averaging (HHWAA) operator of DHF assessments in the following. Proposition 4. Let Aij = {hij, gij} = {

 ij hij

{ ij },

ij gij

{ij } } (i = 1, 2, …, m, j = 1, …, n) be assessments of alternatives

on each attribute for a MADM problem denoted by DHFEs, ω = (ω1, ω2, ..., ωn)T be the relative weights of the n

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attributes, wi = (wi1, …, win)T be the OWA operator weights with respect to Aij , and ij =  ( j )  (1   )wij (j = 1, …, n) be the hybrid weight of Ai ( j ) . Then, the arithmetic aggregated assessment Aia = {hi, gi} (i = 1, 2, …, m) obtained by the HHWAA operator is calculated as Aia = HHWAA( Ai1 , …, Ain ) = i1 Ai (1)  …  in Ai ( n ) =

  n (1  (r  1) i ( j ) )ij   n (1   i ( j ) )ij  j 1 {  i ( j ) hi ( j )  n j 1 n ij  (1  (r  1) i ( j ) )  (r  1) j 1 (1   i ( j ) ) ij    j 1

  ,  

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n    r  j 1iij( j )    }.  i ( j ) gi ( j )  n n ij ij (1  ( r  1)(1   ))  ( r  1)     i ( j ) i ( j )  j  1 j  1  

(22)

Using mathematical induction, the proof of Proposition 5 is similar to Theorem 2 and thus omitted. Similar to the HHWAA operator, a Hamacher hybrid weighted geometric averaging (HHWGA) operator of DHF assessments can also be developed. Proposition 5. Let Aij = {hij, gij} = {

 ij hij

{ ij },

ij gij

{ij } } (i = 1, 2, …, m, j = 1, …, n) be assessments of alternatives

on each attribute for a MADM problem denoted by DHFEs, ω = (ω1, ω2, ..., ωn)T be the relative weights of the n

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attributes, wi = (wi1, …, win)T be the OWA operator weights with respect to Aij , and ij =  ( j )  (1   )wij (j = 1, …, n) be the hybrid weight of Ai ( j ) . Then, the geometric aggregated assessment Aig = {hi, gi} (i = 1, 2, …, m) obtained by the HHWGA operator is calculated in the following way

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Aig = HHWGA( Ai1 , …, Ain ) = Aii1(1)  …  Aiin( n ) = n    r  j 1 iij( j )   {  i ( j ) hi ( j )  n , n ij ij (1  ( r  1)(1   ))  ( r  1)     i ( j ) i ( j )  j  1 j  1  

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(23)

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  n (1  (r  1)i ( j ) )ij   n (1  i ( j ) )ij  j 1 j 1 i ( j ) gi ( j )  n n ij  (1  (r  1)i ( j ) )  (r  1) j 1 (1  i ( j ) ) ij    j 1

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Using mathematical induction, Proposition 5 can be proved and thus omitted. Tang et al. [29] recently carried out a comprehensive analysis of fuzzy Hamacher aggregation operators in the context of uncertain MADM, and discussed the monotonicity of the aggregated assessments, with respect to the parameter r in Hamacher arithmetic and geometric aggregation operators from a theoretical point of view. From the work of Tang et al., we can find that the arithmetic and geometric aggregated assessments are monotonically decreasing and increasing with the increase of the parameter r in Hamacher arithmetic and geometric aggregation operators respectively. As such, the meaning and determination of the parameter r in the HHWAA operator (or HHWGA operator) can be discussed in the same manner as they are analyzed in the FHWAA operator (or FHWGA operator) and thus omitted. Then, we discuss the effects of parameters r and orness(w) on the solution through re-considering the investment evaluation problem, under the assumption that the Frank aggregation operators are replaced by Hamacher aggregation operators at the second step of the proposed method. As presented in Section 5.3, ra and rg can be derived from

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orness(w) when Ga(ra) and Gg(rg), ra , ra , rg and rg are confirmed. Because ra and rg are close to 0 in the HHWAA operator (or HHWGA operator), the difficulty lies in the decision on ra and rg . To make such decision,

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similar to Section 7.3 we need to examine the arithmetic score curves of the five nations with a variation in ra and the geometric score curves of the five nations with a variation in rg. In the situation of orness(w) = 0.4 and θ = 0.5, it is assumed that ra and rg move from 0.05 to 6 with a step 0.25. Then, the arithmetic and geometric score curves of the five nations are plotted in Figures 6 and 7, respectively. Suppose that ra = rg = 0.05, and Ga(ra) and Gg(rg) respectively in (18) and (19) are selected. By observing the five curves in Figures 6 and 7, the decision maker chooses the maximum inflection points in the two figures to quantify ra and rg , i.e., ra = 3.5 and rg = 4, beyond which the five curves in Figure 6 and those in Figure 7 have no intersection and become flat. That is, when ra > ra and rg > rg , the solution becomes fixed except that the 25

scores of the five nations have slightly changed. Under this assumption, we can infer from (20) - (21) that ra = 2.12 and rg = 1.63. To make ra = rg, we have to increase rg . Because there are less intersections among the score curves of the five nations in Figure 6 compared with Figure 7, the solutions by the HHWAA operator with a variation in ra can be considered more steady than those by the HHWGA operator with the increase of rg. More specifically, if there is no intersection in the five curves of Figure 6 and those of Figure 7, the choice of ra and rg has no influence on the rank-order of the five nations but may only contribute to a further analysis of the problem.

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In the setting of r = 2 and θ = 0.5, suppose that orness(w) is changed from 0.05 to 0.5 with a step 0.05, which is due to the great importance of the problem for the company. The change of arithmetic and geometric scores of the five nations with a variation in orness(w) is plotted in Figures 8 and 9, respectively. With the increase of orness(w), scores of the five nations continuously increase. This demonstrates that the decision maker has gradually preferred large scores when the pessimistic degree represented by orness(w) has steadily decreased. Particularly, there are two intersections when orness(w) is about 0.1 and 0.45, respectively. The two intersections reveal that orness(w) has an important effect on both the optimal choice and the rank-order of the five nations.

8. Conclusions and further study

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In this paper, we propose a new MADM method based on Frank t-norm and t-conorm, in which assessments are profiled by DHFEs. Firstly, the score and accuracy functions of DHFS are designed using a combination of mean and variance of membership and non-membership degrees in DHFS. With the use of the new score and accuracy functions, the comparison mechanism of DHFSs is constructed to compare the DHFSs. By using Frank t-norm and t-conorm, we generalize operations of DHFS to cover the operations in [17,21] as special cases. On the basis of the generalized operations of DHFS, arithmetic and geometric attribute aggregation operators are developed and proven. In particular, both attribute weights and ordered weights (i.e., OWA operator weights) play a role in the combination of assessments of alternatives on each attribute, which is more adaptable than the only application of attribute weights and ordered weights in the combination. The monotonicity of the scores of arithmetic and geometric aggregated assessments regarding the parameter in Frank aggregation operators, and their relationship are demonstrated and proven theoretically. Through being associated with the risk attitude of a decision maker, the parameter is interpreted and determined. Finally, an investment evaluation problem is analyzed by the proposed method to demonstrate its applicability and validity. Focusing on θ, ra (rg), orness(w), and ω, we gives a sensitivity analysis of the solution to the problem, followed by a detailed comparative study to highlight the validity and advantages of the method proposed in this paper. More importantly, we discuss the situations where Frank aggregation operators are replaced by Hamacher aggregation operators at the second step of the proposed method, through re-considering the investment evaluation problem. Although the proposed method underlies the generation of a solution to a MADM problem with DHF assessments, it cannot analyze more complex problems where a decision maker gives interval-valued DHF assessments. In the next step, we intend to extend the method to analyze the problems with interval-valued DHF assessments. We also consider ways of combining the proposed method with group decision-making methods to solve multiple attribute group decision-making problems [49-54]. In addition, it is worth future studies to discuss the practical applications of the proposed theoretical approach to other domains such as the evaluation of logistics and supply chain green supplier [27,55,56], cluster analysis [57-59] and data mining and data fusion[60-62]. Acknowledgements This research was supported by the Foundation for Innovative Research Groups of the Natural Science Foundation of China (No. 71521001), the National Key Basic Research Program of China (No. 2013CB329603), the Fundamental Research Funds for the Central Universities (No. JZ2016HGTB0728), and the National Natural Science Foundation of China (Nos. 71303073, 71501056, 71501054 and 71571166). 26

References [1]

G. Deschrijver, E.E. Kerre, On the relationship between some extensions of fuzzy set theory, Fuzzy Sets and Systems 133 (2) (2003) 227–235.

[2]

D. Dubois, H. Prade, Fuzzy Sets and Systems: Theory and Applications, Academic Press, New York, 1980.

[3]

S. Miyamoto, Remarks on basics of fuzzy sets and fuzzy multisets, Fuzzy Sets and Systems 156 (3) (2005) 427–431.

[4]

K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1) (1986) 87–96.

[5]

K. Atanassov, G. Gargov, Interval valued intuitionistic fuzzy sets, Fuzzy Sets and Systems 31 (3) (1989) 343–349.

[6]

X.A. Tang, N.P. Feng, M. Xue, S.L. Yang and J. Wu, The expert reliability and evidential reasoning rule based intuitionistic fuzzy multiple attribute group decision making, Journal of Intelligent & Fuzzy Systems 33 (2017) 1067–1082.

[7]

P. Liu, S.M. Chen, J. Liu, Some intuitionistic fuzzy interaction partitioned Bonferroni mean operators and their application to

[8]

IP T

multi-attribute group decision making, Information Sciences 411 (2017) 98-121.

P. Liu, J. Liu, J.M. Merigó, Partitioned Heronian means based on linguistic intuitionistic fuzzy numbers for dealing with multi-attribute group decision making Applied Soft Computing 62 (2018) 395-422.

[9]

P. Liu, S.M. Chen, Multiattribute group decision making based on intuitionistic 2-tuple linguistic information, Information Sciences

SC R

430 (2018) 599-619.

[10] J. Yoneyama, Stability Analysis of Nonlinear Networked Control Systems via Takagi-Sugeno Fuzzy Model, IFAC-PapersOnLine 50(1) (2017) 2989-2994.

[11] Y. Pan, G.H. Yang, Event-triggered fuzzy control for nonlinear networked control systems, Fuzzy Sets and Systems 329 (2017)

U

91-107.

[12] A.E. Saleh, M.S. Moustafa, K.M. Abo-Al-Ez, A.A. Abdullah, A hybrid neuro-fuzzy power prediction system for wind energy

N

generation, International Journal of Electrical Power & Energy Systems 74 (2016) 384-395. [13] W. Yaïci, E. Entchev, Adaptive Neuro-Fuzzy Inference System modelling for performance prediction of solar thermal energy

A

system, Renewable energy 86 (2016) 302-315.

[14] A. Khosravi, R.N.N. Koury, L. Machado, J.J.G. Pabon, Prediction of wind speed and wind direction using artificial neural network,

M

support vector regression and adaptive neuro-fuzzy inference system, Sustainable Energy Technologies and Assessments 25 (2018) 146-160. Energy 186 (2017) 68-81.

ED

[15] A. Keshtkar, S. Arzanpour, An adaptive fuzzy logic system for residential energy management in smart grid environments, Applied [16] V. Torra, Hesitant fuzzy sets, International Journal of Intelligent Systems 25 (6) (2010) 529–539. [17] B.

Zhu,

Z.S.

Xu,

&

M.M.

Xia,

Dual

hesitant

fuzzy

sets,

Journal

of

Applied

Mathematics,

in

press,

PT

http://dx.doi.org/10.1155/2012/879629.

[18] B. Farhadinia, Correlation for dual hesitant fuzzy sets and dual interval-valued hesitant fuzzy sets, International Journal of Intelligent Systems 29 (2014), 184–205.

CC E

[19] J. Ye, Correlation coefficient of dual hesitant fuzzy sets and its application to multiple attribute decision making, Applied Mathematical Modelling 38 (2) (2014) 659–666. [20] P. Singh, A new method for solving dual hesitant fuzzy assignment problems with restrictions based on similarity measure, Applied Soft Computing 24 (2014) 559-571. [21] H.J. Wang, X.F. Zhao, G.W. Wei, Dual hesitant fuzzy aggregation operators in multiple attribute decision making, Journal of

A

Intelligent & Fuzzy Systems 26 (5) (2014) 2281–2290. [22] H. Zhao, Z.S. Xu, S.S. Liu, Dual hesitant fuzzy information aggregation with Einstein t-conorm and t-norm, Journal of Systems Science and Systems Engineering 26 (2) (2017) 240–264.

[23] Y.F. Chen, X.D. Peng, G.H. Guan, H.D. Jiang, Approaches to multiple attribute decision making based on the correlation coefficient with dual hesitant fuzzy information, Journal of Intelligent & Fuzzy Systems 26 (5) (2014), 2547–2556. [24] S.H. Yang, Y.B. Ju, Dual hesitant fuzzy linguistic aggregation operators and their applications to multi-attribute decision making, Journal of Intelligent & Fuzzy Systems 27 (4) (2014) 1935–1947. [25] Y.B. Ju, W.K. Zhang and S.H Yang, Some dual hesitant fuzzy Hamacher aggregation operators and their applications to multiple attribute decision making, Journal of Intelligent & Fuzzy Systems 27 (2014) 2481–2495. [26] L. Wang, Q.G. Shen, L. Zhu, Dual hesitant fuzzy power aggregation operators based on Archimedean t-norm and t-conorm and 27

their application to multiple attribute group decision making, Applied Soft Computing 38 (2016) 23–50. [27] D.J. Yu, D.F. Li, J.M. Merigo, Dual hesitant fuzzy group decision making method and its application to supplier selection, International Journal of Machine Learning and Cybernetics 7 (5) (2016) 819–831. [28] D.J. Yu, J.M. Merigo, Y.J. Xu, Group Decision Making in Information Systems Security Assessment Using Dual Hesitant Fuzzy Set, International Journal of Intelligent Systems 31 (8) (2016) 786–812. [29] X.A. Tang, C. Fu, D.-L. Xu, S.L. Yang, Analysis of fuzzy Hamacher aggregation functions for uncertain multiple attribute decision making, Information Sciences 387 (2017) 19–33. [30] G.H. Qu, H.P. Zhang, W.H. Qu, Induced generalized dual hesitant fuzzy Shapley hybrid operators and their application in multi-attributes decision making, Journal of Intelligent & Fuzzy Systems 31 (1) (2016) 633–650. [31] D.C. Liang, Z.S. Xu, D. Liu, Three-way decisions based on decision-theoretic rough sets with dual hesitant fuzzy information, Information Sciences 396 (2017) 127–143.

IP T

[32] Z.L. Ren, Z.S. Xu, H.Wang, Dual hesitant fuzzy VIKOR method for multi-criteria group decision making based on fuzzy measure and new comparison method, Information Sciences 388–389 (2017) 1–16.

[33] M.J. Frank, On the simultaneous associativity of F (x y) and x + y − F (x, y), Aequationes mathematicae 19 (1) (1979) 194–226.

SC R

[34] J.D. Qin, X.W. Liu, W. Pedrycz, Frank aggregation operators and their application to hesitant fuzzy multiple attribute decision making, Applied Soft Computing 41 (2016) 428–452.

[35] X. Zhang, P.D. Liu, Y.M. Wang, Multiple attribute group decision making methods based on intuitionistic fuzzy frank power aggregation operators, Journal of Intelligent & Fuzzy Systems 29 (2015) 2235–2246.

[36] P. Ji, J.Q. Wang, H.Y. Zhang, Frank prioritized Bonferroni mean operator with single-valued neutrosophic sets and its application in

U

selecting third-party logistics providers, Neural Computing and Applications (2016), 1–25, DOI 10.1007/s00521-016-2660-6. [37] Z.M. Zhang, Interval-valued intuitionistic fuzzy Frank aggregation operators and their applications to multiple attribute group

N

decision making, Neural Computing and Applications 28(6) (2017) 1471–1501.

[38] P. Sarkoci, Domination in the families of Frank and Hamacher t-norms, Kybernetika 41(3) (2005) 349–360.

A

[39] R.R. Yager, On some new classes of implication operators and their role in approximate reasoning, Information Sciences 167(1) (2004) 193–216.

M

[40] W.S. Wang, H.C. He, Research on flexible probability logic operator based on Frank T/S norms, Acta Electronica Sinica 37(5) (2009) 1141–1145.

[41] G. Deschrijver & E.E. Kerre, A generalization of operators on intuitionistic fuzzy sets using triangular norms and conforms, Fuzzy

ED

Sets and Systems 8(1) (2002) 19–27.

[42] M.M. Xia, Z.S. Xu, B. Zhu, Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm, Knowledge-Based Systems 31 (2012) 78–88.

PT

[43] R.R. Yager, On ordered weighted averaging aggregation operators in multi-criteria decision making, IEEE Transactions on Systems Man and Cybernetics 18 (1) (1988) 183–190. [44] B.S. Ahn, S.H. Choi, Aggregation of ordinal data using ordered weighted averaging operator weights, Annals of Operations

CC E

Research 201 (1) (2012) 1–16.

[45] R.R. Yager, Quantifier guided aggregation using OWA operators, International Journal of Intelligent Systems 11 (1) (1996) 49–73. [46] Z.S. Xu, Intuitionistic fuzzy aggregation operators, IEEE Transactions on Fuzzy Systems 15 (2007) 1179-1187. [47] Z.S. Xu, On consistency of the weighted geometric mean complex judgement matrix in AHP, European Journal of Operational Research 126 (3) (2000) 683–687.

A

[48] A.İ. Ölçer, A.Y. Odabaşi, A new fuzzy multiple attributive group decision making methodology and its application to propulsion/manoeuvring system selection problem, European Journal of Operational Research 166(1) (2005) 93–114.

[49] V. Mohagheghi, S.M. Mousavi, B. Vahdani, Enhancing decision-making flexibility by introducing a new last aggregation evaluating approach based on multi-criteria group decision making and Pythagorean fuzzy sets, Applied Soft Computing 61 (2017) 527–535. [50] J. Wu, F. Chiclana, H. Fujita, E. Herrera-Viedma, A visual interaction consensus model for social network group decision making with trust propagation, Knowledge-Based Systems 122 (2017) 39–50. [51] P.D. Liu, Multiple attribute group decision making method based on interval-valued intuitionistic fuzzy power Heronian aggregation operators, Computers & Industrial Engineering 108 (2017) 199–212. 28

[52] P. Liu, H. Li, Interval-valued intuitionistic fuzzy power Bonferroni aggregation operators and their application to group decision making, Cognitive Computation (2017) 1-19. [53] P. Liu, L. Zhang, X. Liu, P. Wang, Multi-valued neutrosophic number Bonferroni mean operators with their applications in multiple attribute group decision making, International Journal of Information Technology & Decision Making 15(5) (2016) 1181-1210. [54] P. Liu, J. Liu, S. M. Chen, Some intuitionistic fuzzy Dombi Bonferroni mean operators and their application to multi-attribute group decision making, Journal of the Operational Research Society 69(1) (2018) 1-24. [55] G. Büyüközkan, F. Göc¸er, Application of a new combined intuitionistic fuzzy MCDM approach based on axiomatic design methodology for the supplier selection problem, Applied Soft Computing 52 (2017) 1222–1238. [56] R. Krishankumar, K.S. Ravichandran, A.B. Saeid, A new extension to PROMETHEE under intuitionistic fuzzy environment for solving supplier selection problem with linguistic preferences, Applied Soft Computing 60 (2017) 564–576. [57] X.B. Zhu, W. Pedrycz, Z.W. Li, Fuzzy clustering with nonlinearly transformed data, Applied Soft Computing 61 (2017) 364–376.

IP T

[58] C. Li, L. Ledo, M. Delgado, M. Cerrada, F. Pacheco, D. Cabrera, R.-V. Sánchez, J. V. de Oliveira, A Bayesian approach to consequent parameter estimation in probabilistic fuzzy systems and its application to bearing fault classification, Knowledge-Based Systems 129 (2017) 39–60.

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[59] F. Pacheco, M. Cerrada, R.-V. Sánchez, D. Cabrera, C. Li, J.V. de Oliveira, Attribute clustering using rough set theory for feature selection in fault severity classification of rotating machinery, Expert Systems With Applications 71 (2017) 69–86. [60] P. Della Rocca, S. Senatore, V. Loia, A semantic-grained perspective of latent knowledge modeling, Information Fusion 36 (2017) 52-67.

[61] V. Loia, S. Senatore, M.I. Sessa, Similarity-based SLD resolution and its role for web knowledge discovery, Fuzzy Sets and

U

Systems 144(1) (2004) 151-171.

[62] V. Loia, S. Tomasiello, A, Vaccaro, Using fuzzy transform in multi-agent based monitoring of smart grids, Information Sciences

A

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388 (2017) 209-224.

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Figure 1. The schematic diagram of the case study.

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Figure 2. Frank arithmetic score curves of the five nations with a variation in ra.

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Figure 3. Frank geometric score curves of the five nations with a variation in rg.

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Figure 4. Frank arithmetic score curves of the five nations with a variation in orness(w).

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Figure 5. Frank geometric score curves of the five nations with a variation in orness(w).

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Figure 6. Hamacher arithmetic score curves of the five nations with a variation in ra.

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Figure 7. Hamacher geometric score curves of the five nations with a variation in rg.

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Figure 8. Hamacher arithmetic score curves of the five nations with a variation in orness(w).

33

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Figure 9. Hamacher geometric score curves of the five nations with a variation in orness(w).

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Table 1. Description of the seven attributes. Attribute Description C1 Quality and amount of Iron ore resources C2 Situations of legal system, taxation regime, and trade barriers C3 Competition from other overseas investment in iron resources C4 Uncertainty about environmental regulations and availability of skilled labor C5 Infrastructure concerning overseas investments C6 Conditions of socioeconomic agreements/community development C7 Political stability and security level

35

N A M ED PT CC E A

36

A5 {{0.7,0.8}, {0.1,0.2}} {{0.3,0.6}, {0.4,0.3}} {{0.1,0.3}, {0.5,0.6}} {{0.4}, {0.4,0.5}} {{0.1,0.4,0.5} ,{0.4,0.5}} {{0.3,0.4}, {0.4,0.5}} {{0.4,0.6,0.7} ,{0.1,0.2,}}

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A4 {{0.5,0.6,0.7}, {0.1,0.2,0.3}} {{0.8,0.5}, {0.2,0.1}} {{0.3,0.2}, {0.5,0.6}} {{0.2,0.3,0.4}, {0.4,0.5,0.6}} {{0.6,0.5}, {0.2,0.3}} {{0.5,0.6,0.7}, {0.1,0.2,0.3}} {{0.5,0.7}, {0.1,0.3}}

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A3 {{0.7,0.9}, {0.1}} {{0.6,0.5,0.8} ,{0.1,0.2}} {{0.2,0.3,0.4} ,{0.5,0.6}} {{0.3,0.4}, {0.5,0.6}} {{0.5,0.6}, {0.2,0.3}} {{0.4,0.5}, {0.5}} {{0.6,0.5,0.4} ,{0.2,0.3}}

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Table 2. Transpose of the DHF decision matrix. Attribute A1 A2 C1 {{0.9,0.8,0. {{0.6,0.7}, 7},{0.1}} {0.1,0.2}} C2 {{0.7,0.8}, {{0.3,0.4}, {0.1,0.2}} {0.5,0.6}} C3 {{0.1,0.2}, {{0.4,0.6}, {0.6,0.7}} {0.3,0.4}} C4 {{0.1,0.2}, {{0.6,0.5,0.4}, {0.6,0.7}} {0.1,0.2,0.3}} C5 {{0.6,0.8}, {{0.3,0.5}, {0.1,0.2}} {0.2,0.3}} C6 {{0.5,0.6} {{0.3,0.4,0.5,0.6 {0.3,0.2}} },{0.2,0.3,0.4}} C7 {{0.7,0.8,0. {{0.3,0.5}, 9},{0.1}} {0.4,0.5}}

0.1196 0.1135 0.1088 0.1053 0.1026 0.1003 0.0984 0.0968 0.0953 0.0941 0.0929 0.0919 0.0909 0.0901 0.0893 0.0885 0.0878 0.0872 0.0866

ED PT CC E A

37

A1

A4 A2

A3

A5

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0.2689 0.2645 0.2610 0.2584 0.2563 0.2545 0.2530 0.2518 0.2506 0.2496 0.2487 0.2479 0.2471 0.2464 0.2458 0.2452 0.2446 0.2441 0.2436

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0.2098 0.2033 0.1983 0.1945 0.1914 0.1889 0.1868 0.1850 0.1834 0.1819 0.1806 0.1795 0.1784 0.1774 0.1765 0.1757 0.1749 0.1741 0.1734

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0.1353 0.1311 0.1280 0.1256 0.1237 0.1222 0.1209 0.1197 0.1187 0.1179 0.1171 0.1164 0.1157 0.1151 0.1146 0.1140 0.1136 0.1131 0.1127

A

0.3916 0.3828 0.3758 0.3705 0.3662 0.3626 0.3595 0.3569 0.3545 0.3524 0.3505 0.3488 0.3472 0.3457 0.3443 0.3431 0.3419 0.3407 0.3397

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1.05 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Ranking

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Table 3. Scores obtained by the FHWAA operator with variation in r. FHWAA operator r S( A1 ) S( A2 ) S( A3 ) S( A4 ) S( A5 )

0.0504 0.0567 0.0616 0.0652 0.0681 0.0704 0.0724 0.0741 0.0756 0.0769 0.0781 0.0791 0.0801 0.0810 0.0818 0.0826 0.0833 0.0839 0.0845

ED PT CC E A

38

A4

A3 A2

A4

A1 A5

A1 A2

A3

A5

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0.1701 0.1755 0.1795 0.1825 0.1848 0.1867 0.1882 0.1896 0.1907 0.1918 0.1927 0.1935 0.1942 0.1949 0.1955 0.1961 0.1966 0.1971 0.1976

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0.1421 0.1484 0.1532 0.1567 0.1595 0.1617 0.1636 0.1652 0.1666 0.1678 0.1689 0.1699 0.1708 0.1716 0.1724 0.1730 0.1737 0.1743 0.1748

N

0.1187 0.1226 0.1255 0.1276 0.1293 0.1306 0.1318 0.1327 0.1336 0.1343 0.1350 0.1356 0.1361 0.1366 0.1371 0.1375 0.1379 0.1382 0.1386

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0.1317 0.1460 0.1569 0.1650 0.1713 0.1764 0.1807 0.1843 0.1875 0.1904 0.1929 0.1952 0.1972 0.1991 0.2008 0.2024 0.2039 0.2053 0.2066

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1.05 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

Ranking

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Table 4. Scores obtained by the FHWGA operator with variation in r. FHWGA operator r S( A1 ) S( A2 ) S( A3 ) S( A4 ) S( A5 )

A1

A4

A2

A3

A5

0.0095 0.0273 0.0485 0.0681 0.0856 0.1012 0.1153 0.1280 0.1391 0.1488

0.0574 0.0662 0.0823 0.1034 0.1271 0.1515 0.1756 0.1983 0.2190 0.2373

0.0708 0.0909 0.1246 0.1601 0.1924 0.2199 0.2426 0.2610 0.2755 0.2865

-0.0262 -0.0050 0.0143 0.0318 0.0499 0.0691 0.0890 0.1088 0.1278 0.1453

N A M ED PT CC E A

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A1

A4 A2

A3 A5

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0.1079 0.1255 0.1664 0.2153 0.2631 0.3063 0.3438 0.3758 0.4027 0.4249

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0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5

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Table 5. Scores obtained by the FHWAA operator with variation in orness(w). FHWAA operator orness(w) S( A1 ) S( A2 ) S( A3 ) S( A4 ) S( A5 )

Table 6. Scores obtained by the FHWGA operator with variation in orness(w). FHWGA operator orness(w) S( A1 ) S( A2 ) S( A3 ) S( A4 ) S( A5 ) 0.05 0.1 0.15 0.2

-0.1717 -0.1536 -0.1093 -0.0544

-0.0130 0.0071 0.0305 0.0522

-0.0132 -0.0023 0.0169 0.0410

-0.0458 -0.0233 0.0128 0.0515

-0.1109 -0.0864 -0.0608 -0.0371

Ranking A2

A3 A5

A2

A1 A4

A5 0.0022

0.0716

0.0676

0.0881

-0.0139

A4

0.0570

0.0897

0.0954

0.1215

0.0099

A4

0.1075

0.1239

0.1518

0.0350

A5

A4

A3

A2

0.1569 0.2021 0.2444

0.1255 0.1441 0.1633

0.1532 0.1830 0.2132

0.1795 0.2051 0.2291

0.0616 0.0896 0.1188

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A

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0.4 0.45 0.5

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A2

A5

A1

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0.1086

A3

A3

A1 0.35

A1 A2

A1 0.3

A3

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0.25

A4

A4

A5

A1

A2

A1

A5

A4

A2

A3

A5

A3

Table 7. A comparison of the ranking orders of the five nations with the use of different methods. Method Parameter Ranking order A1 A4 A2 A3 A5 Ye [19] None DHFHA operator DHFHG operator Wang et al. [21] None A1 A4 A3 A2 A5 A4 A3 A2 A1 EDHFWA operator EDHFWG operator Zhao et al. [22] None A1 A4 A3 A2 A5 A4 A1 A3 A2

A1

A4

A3

A2

A5

FHWAA operator r= r=2 r=6

A1

A4

A3

A2

A5

A

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N

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The proposed method

1+

41

A5

A1 A2 A2

A5 A5 A5

A2 A2 A2

A5 A5 A5

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Ju et al. [25]

DHFHHG operator A4 A3 A2 A4 A1 A3 A1 A4 A3 FHWGA operator A4 A3 A1 A4 A1 A3 A1 A4 A3

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DHFHHA operator r=1 r=2 r=6

A5