#### Abstract

This paper studies the multiattribute decision making (MADM) problems in which the attribute values take the form of dual hesitant fuzzy triangular linguistic elements and the weights of attributes take the form of real numbers. Firstly, to solve the situation where the membership degree and the nonmembership degree of an element to a triangular linguistic variable, the concept, operational laws, score function, and accuracy function of dual hesitant fuzzy triangular linguistic elements (DHFTLEs) are defined. Then, some dual hesitant fuzzy triangular linguistic geometric aggregation operators are developed for aggregating the DHFTLEs, including dual hesitant fuzzy triangular linguistic weighted geometric (DHFTLWG) operator, dual hesitant fuzzy triangular linguistic ordered weighted geometric (DHFTLOWG) operator, dual hesitant fuzzy triangular linguistic hybrid geometric (DHFTLHG) operator, generalized dual hesitant fuzzy triangular linguistic weighted geometric (GDHFTLWG) operator, and generalized dual hesitant fuzzy triangular linguistic ordered weighted geometric (GDHFTLOWG) operator. Furthermore, some desirable properties of these operators are investigated in detail. Based on the proposed operators, an approach to MADM with dual hesitant fuzzy triangular linguistic information is proposed. Finally, a numerical example for investment alternative selection is given to illustrate the application of the proposed method.

#### 1. Introduction

Multiattribute decision making (MADM) has become a hot research topic, which is to select the most desirable solution from a finite set of feasible alternatives with respect to conflicting attributes, both quantitative and qualitative. Due to the increasing complexity of the socioeconomic environment and the lack of knowledge or data about the decision making problems domain, the attributes involved in the decision problems are not always expressed as crisp numbers, and many researchers have utilized the fuzzy theory to deal with investment alternative selection [1–4]. The fuzzy set theory originally proposed by Zadeh [5] is a very useful tool to describe uncertain information. However, in some real decision situations the fuzzy set is imprecise resulting from characterizing the fuzziness just by a membership degree. On the basis of the fuzzy set theory, Atanassov [6, 7] proposed the intuitionistic fuzzy set characterized by a membership function and a nonmembership function. Obviously, the intuitionistic fuzzy set can describe and characterterize the fuzzy essence of the objective world more exquisitely, and it has received more and more attention since its appearance [8–13].

However, in the real world, decision makers usually cannot completely express their opinions by quantitative numbers, and some of them are more appropriately described by qualitative linguistic terms. Since linguistic variables [14] have been proposed, so far, a number of linguistic approaches have been defined such as 2-tuple linguistic [15], interval-valued 2-tuple linguistic [16], uncertain linguistic [17], and trapezoid fuzzy linguistic [18]. In order to express the uncertainty and ambiguity as accurate as possible, Wang and Li [19] proposed the concept of intuitionistic linguistic set based on linguistic variables and intuitionistic fuzzy set, which can overcome the defects for intuitionistic fuzzy set which can only roughly represent criteria’s membership and nonmembership to a particular concept, such as “good” and “bad,” and for linguistic variables which usually implies that membership degree is 1, and the nonmembership degree and hesitation degree of decision makers cannot be expressed. Furthermore, Liu and Jin [20] and Liu [21] proposed the intuitionistic uncertain linguistic variables and the interval-valued intuitionistic uncertain linguistic variables as well as some decision making methods.

In real decision making process, we often encounter such situation that the decision makers are hesitant among a set of possible values which makes the outcome of decision making inconsistent. To solve this problem, the hesitant fuzzy set (HFS), an extension of fuzzy set [5], was proposed by Torra and Narukawa [22] and Torra [23], which permits the membership degree of an element to a given set to be represented by several possible numerical values. To accommodate more complex environment, several extensions of HFS have been presented, such as interval-valued hesitant fuzzy set (IVHFS) [24, 25] and hesitant fuzzy linguistic term set (HFLS) [26]. Especially, considering that the human judgments including preference information may be stated which permits the membership having a set of possible hesitant fuzzy linguistic values or hesitant fuzzy uncertain linguistic values, Lin et al. [27] proposed the concepts of hesitant fuzzy linguistic set (HFLS) and hesitant fuzzy uncertain linguistic set (HFULS). Furthermore, some aggregation operators hesitant fuzzy linguistic weighted average (HFLWA) operator, hesitant fuzzy linguistic ordered weighted average (HFLOWA) operator, hesitant fuzzy uncertain linguistic weighted average (HFULWA) operator, hesitant fuzzy uncertain linguistic ordered weighted average (HFULOWA) operator, and an approach are proposed for MADM problems. However, the above hesitant fuzzy decision making methods just provide the membership degrees and neglect the importance of the nonmembership degrees. In fact, the nonmembership plays the same important role as the membership in describing the vague decision making information, which indicates that the possible degrees of one element do not belong to a fixed set. To assess the attribute values more precisely, Zhu et al. [28] developed the dual hesitant fuzzy set (DHFS), taking much more information into account given by decision makers, in which the membership degree and the nonmembership degree are in the form of sets of values in . Then correlation coefficient [29, 30] and aggregation operators [31] are proposed to deal with MADM problems under dual hesitant fuzzy environment. Furthermore, Ju et al. [32] proposed the definition of the interval-valued dual hesitant fuzzy set (IVDHFS).

To the best of our knowledge, the existing approaches under the hesitant fuzzy environment are not suitable for dealing with MADM problems with dual hesitant fuzzy triangular linguistic information. Therefore, motivated by the idea of HFLS and HFULS, based on the triangular linguistic term set and the DHFS, in this paper, we define a new concept called the dual hesitant fuzzy triangular linguistic set composed of a triangular linguistic term, a set of membership degrees, and a set of nonmembership degrees, which can overcome the shortcomings of the HFLS. For example, for a predefined linguistic set = = extremely low, = very low, = low, = medium, = high, = very high, = extremely high}, we can evaluate the “growth” of a company by a dual hesitant fuzzy triangular linguistic element (DHFTLE) , , ], {0.4, 0.5, 0.6}, {0.2, 0.3, 0.4. This is the motivation of our study. The main advantages of DHFTLE include (1) triangular linguistic term which can describe the uncertainty more precisely and objectively than linguistic term and uncertain linguistic term in qualitative; (2) a set of membership degrees and a set of nonmembership degrees are complements of the triangular linguistic terms, which can explain how much degree that an attribute value belongs to and not belong to a triangular linguistic term in quantitative.

The remainder of this paper is organized as follows. Some basic definitions of triangular linguistic term set, hesitant fuzzy set, and dual hesitant fuzzy set are briefly reviewed in Section 2. In Section 3, the concept, operational laws, score function, and accuracy function of the dual hesitant fuzzy triangular linguistic elements are defined. In Section 4, some dual hesitant fuzzy triangular linguistic geometric aggregation operators are proposed, and then some desirable properties of the proposed operators are investigated. In Section 5, we develop an approach for multiple attribute decision making with dual hesitant fuzzy triangular linguistic information based on the proposed operators. In Section 6, a numerical example is given to illustrate the application of the proposed method. The paper is concluded in Section 7.

#### 2. Preliminaries

To facilitate the following discussion, some basic definitions related to triangular linguistic term set, hesitant fuzzy set, and dual hesitant fuzzy set are briefly reviewed in this section.

##### 2.1. Triangular Linguistic Term Set

Let be a finite linguistic term set with odd cardinality, where represents a possible value for a linguistic term and is the cardinality of . For example, when , a set of seven terms could be given as follows: = = extremely low, = very low, = low, = medium, = high, = very high, = extremely high}.

In general, for any linguistic term set , it is required that and satisfy the following properties [33, 34]:(1)the set is ordered: , if and only if ;(2)there is the negation operator: Neg() = , such that ;(3)maximum operator: max, = , if ;(4)minimum operator: min, = , if .

*Definition 1 (see [35]). *Let , be the continuous form of , , , , and ; then can be called a triangular linguistic variable, and is called a triangular linguistic term set.

Especially, if , then the triangular linguistic variable reduces to a linguistic variable; if or , then the triangular linguistic variable reduces to an uncertain linguistic variable.

Let be a set of triangular linguistic variables; for any two triangular linguistic variables and , , , the operational laws are shown as follows [35]:(1);(2);(3), ;(4), .

##### 2.2. Hesitant Fuzzy Set

*Definition 2 (see [22, 23]). *Let be a fixed set; then a hesitant fuzzy set (HFS) on is in terms of a function that when applied to returns a subset of , which can be represented as the following mathematical symbol:
where is a set of some values in , denoting the possible degrees of the element to the set . For convenience, Xia and Xu [36] called a hesitant fuzzy element (HFE) and the set of all hesitant fuzzy elements (HFEs).

*Definition 3 (see [36]). *Let be a HFE; then the score function of is determined as follows:
where is the number of the elements in . For two HFEs and , if , then ; if , then .

##### 2.3. Dual Hesitant Fuzzy Set

*Definition 4 (see [28]). *Let be a fixed set; then a dual hesitant fuzzy set (DHFS) on is described as
in which and are two sets of some values in , denoting the possible membership degrees and nonmembership degrees of the element to the set , respectively, with the conditions
where , , , and for all . For convenience, the pair is called a dual hesitant fuzzy element (DHFE) denoted by .

Obviously, if there is only one element in both and , the DHFE reduces to an intuitionistic fuzzy number [6].

*Definition 5 (see [28]). *Let () be a collection of DHFEs, then the score function and the accuracy function of () can be defined by (5) and (6), respectively;
where and are the numbers of values in and , respectively.

#### 3. Dual Hesitant Fuzzy Triangular Linguistic Set

Based on the triangular linguistic term set and the dual hesitant fuzzy set, we propose the definition of the dual hesitant fuzzy triangular linguistic set, the operational laws, score function, and accuracy function in what follows.

*Definition 6. *Let be a fixed set; then a dual hesitant fuzzy triangular linguistic set (DHFTLS) on is described as
in which , and are two sets of some values in , denoting the possible membership degrees and nonmembership degrees of the element to the triangular linguistic variable , respectively, with the conditions
where , , , and for all . For convenience, the 3-tuples is called a dual hesitant fuzzy triangular linguistic element (DHFTLE) denoted by .

Especially, if , then the dual hesitant fuzzy triangular linguistic element reduces to a dual hesitant fuzzy linguistic element; if or , then the dual hesitant fuzzy triangular linguistic element reduces to a dual hesitant fuzzy uncertain linguistic element.

*Definition 7. *Let and be two DHFTLEs; then the operational laws are defined as(1), ;(2), ;(3), ;(4).

Theorem 8. *Let and be two DHFTLEs; the operational laws of DHFTLEs are defined as follows:*(1)*;*(2)*;*(3)*, ;*(4)*, .*

*Definition 9. *Let be a DHFTLE; then the score function of is defined as follows:
where and are the numbers of values in and , respectively. () is the cardinality of .

*Definition 10. *Let be a DHFTLE; then the accuracy function of is defined as follows:
where and are the numbers of values in and , respectively. () is the cardinality of .

Theorem 11. *Let and be two DHFTLEs; they can be compared by the following rules:*(1)*if , then ;*(2)*if , then if , then ; if , then .*

#### 4. Some Dual Hesitant Fuzzy Triangular Linguistic Geometric Aggregation Operators

In what follows, based on the operational laws of DHFTLEs, we will develop some geometric aggregation operators for aggregating the dual hesitant fuzzy triangular linguistic information.

*Definition 12. *Let () be a collection of DHFTLEs, and ; then the dual hesitant fuzzy triangular linguistic weighted geometric (DHFTLWG) operator can be defined as
in which is a dual hesitant fuzzy triangular linguistic set and is the weight vector of (), such that and .

Especially, if , then the DHFTLWG operator reduces to the dual hesitant fuzzy triangular linguistic geometric (DHFTLG) operator.

Theorem 13. *Let () be a collection of DHFTLEs and let be the weight vector of (), such that and . Then their aggregated value by the DHFTLWG operator is still a DHFTLE, and
*

Theorem 14 (boundedness). *Let () be a collection of DHFTLEs, for the DHFTLWG operator, if , , , , , , , = , , , for all ; then one can obtain
*

*Proof. *Since , , , , then for any and , , we have
then
That is,
where and are the numbers of values in the membership degrees and nonmembership degrees of , respectively.

Since , we have
Similarly, we have

Therefore, according to Definition 9 and Theorem 11, we obtain
Similarly,
Therefore,

However, the DHFTLWG operator is not idempotent, which can be illustrated by the following example.

*Example 15. *Let and = = be two DHFTLEs; is the weight vector of ();then by the DHFTLWG operator, we have

It is clear that , therefore, the idempotency is not hold.

Lemma 16 (see [37]). *Let , , , and ; then
**
with equality if and only if .*

Theorem 17. *Let () be a collection of DHFTLEs; is the weight vector of (), such that and . One has
**
where denotes the dual hesitant fuzzy triangular linguistic weighted average (DHFTLWA) operator of () proposed by Ju and Yang [38].*

*Proof. *According to Lemma 16, for any (), we have
then, we have
where and are the numbers of values in the membership degrees and nonmembership degrees of , respectively; and are the numbers of values in the membership degrees and nonmembership degrees of , respectively.

Therefore, based on Definition 9 and Theorem 11, we can obtain
which implies that .

*Definition 18. *Let () be a collection of DHFTLEs, and ; then the dual hesitant fuzzy triangular linguistic ordered weighted geometric (DHFTLOWG) operator can be defined as
in which is a dual hesitant fuzzy triangular linguistic set and is the aggregation-associated weight vector, such that and . is the th largest element in ().

Especially, if = , then the DHFTLOWG operator reduces to the dual hesitant fuzzy triangular linguistic geometric (DHFTLG) operator.

Theorem 19. *Let () be a collection of DHFTLEs; is the aggregation-associated weight vector, such that and . Then their aggregated value by the DHFTLOWG operator is still a DHFTLE, and
*

Theorem 20 (boundedness). *Let () be a collection of DHFTLEs, for the DHFTLOWG operator, if , , , , , , , = , , , for all ; then we can obtain
**
which can be proven to be similar to Theorem 14.*

Theorem 21 (commutativity). *Let () be a collection of DHFTLEs, if is any permutation of ; then
*

*Proof. *Since is a permutation of , we have , for all . Then, based on Definition 18, we obtain
which completes the proof of Theorem 21.

Theorem 22. *Let () be a collection of DHFTLEs; is the aggregation-associated weight vector, such that and . One has
**
where is the dual hesitant fuzzy triangular linguistic ordered weighted average (DHFTLOWA) operator of () proposed by Ju and Yang [38]. This theorem can be proven to be similar to Theorem 17.*

*Definition 23. *Let () be a collection of DHFTLEs, and ; then the dual hesitant fuzzy triangular linguistic hybrid geometric (DHFTLHG) operator can be defined as
in which is a dual hesitant fuzzy triangular linguistic set, is the aggregation-associated weight vector, such that and . is the th largest element in (, ), is the weight vector of (), such that and , and is the balancing coefficient.

Especially, if , then the DHFTLHG operator reduces to the DHFTLWG operator in (11). If , then the DHFTLHG operator reduces to the DHFTLOWG operator in (28).

Theorem 24. *Let () be a collection of DHFTLEs and let be the aggregation-associated weight vector, such that and . Then their aggregated value by the DHFTLHG operator is still a DHFTLE, and
*

Theorem 25 (boundedness). *Let () be a collection of DHFTLEs, for the DHFTLHG operator; if , , , , , , , = , , , for all , then one can obtain
**
which can be proven to be similar to Theorem 14.*

Theorem 26. *Let () be a collection of DHFTLEs and let be the aggregation-associated weight vector, such that and . One has
**
where is the dual hesitant fuzzy triangular linguistic hybrid average (DHFTLHA) operator of () proposed by Ju and Yang [38]. This theorem can be proven similar to Theorem 17.*

*Definition 27. *Let () be a collection of DHFTLEs, and ; then the generalized dual hesitant fuzzy triangular linguistic weighted geometric (GDHFTLWG) operator can be defined as
in which is a dual hesitant fuzzy triangular linguistic set and is the weight vector of (), such that and , .

Especially, if , then the GDHFTLWG operator reduces to the DHFTLWG operator in (11).

Theorem 28. *Let () be a collection of DHFTLEs and let be the weight vector of (), such that and , . Then their aggregated value by the GDHFTLWG operator is still a DHFTLE, and
*

Theorem 29 (boundedness). *Let () be a collection of DHFTLEs, for the GDHFTLWG operator; if , , , , , , , = , , , for all ; then one can obtain
**
which can be proven to be similar to Theorem 14.*

Theorem 30. *Let () be a collection of DHFTLEs and let be the weight vector of (), such that and , . One has
**
where is the generalized dual hesitant fuzzy triangular linguistic weighted average (GDHFTLWA) operator of () proposed by Ju and Yang [38]. This theorem can be proven to be similar to Theorem 17.*

*Definition 31. *Let () be a collection of DHFTLEs, and ; then the generalized dual hesitant fuzzy triangular linguistic ordered weighted geometric (GDHFTLOWG) operator can be defined as
in which is a dual hesitant fuzzy triangular linguistic set and is the aggregation-associated weight vector, such that and , . is the th largest element in ().

Especially, if , then the GDHFTLOWG operator reduces to the DHFTLOWG operator in (28).

Theorem 32. *Let () be a collection of DHFTLEs, and let be the aggregation-associated weight vector, such that and , . Then their aggregated value by the GDHFTLOWG operator is still a DHFTLE, and*

Theorem 33 (*boundedness*). *Let () be a collection of DHFTLEs, for the GDHFTLOWG operator; if , , , , , , , = , , , for all , then one can obtain
**
which can be proven to be similar to Theorem 14.*

Theorem 34 (commutativity). *Let () be a collection of DHFTLEs; if is any permutation of , then
**
which can be proven to be similar to Theorem 21.*

Theorem 35. *Let () be a collection of DHFTLEs; is the aggregation-associated weight vector, such that and , . One has
**
where is the generalized dual hesitant fuzzy triangular linguistic ordered weighted average (GDHFTLOWA) operator of () proposed by Ju and Yang [38]. This theorem can be proven to be similar to Theorem 17.*

#### 5. A Method for MADM Problems with Dual Hesitant Fuzzy Triangular Linguistic Information

In this section, we will utilize the dual hesitant fuzzy triangular linguistic geometric aggregation operators to MADM for investment alternatives selection with dual hesitant fuzzy triangular linguistic information.

Let be a finite set of alternatives, the set of attributes, a finite linguistic term set, and the weight vector of attributes (), with and . Suppose that is a dual hesitant fuzzy triangular linguistic decision matrix, where is in the form of DHFTLEs given for the alternative () with respect to the attribute (), with , and . Then, to determine the most desirable alternative(s), the DHFTLWG operator is utilized to develop a MADM method with dual hesitant fuzzy triangular linguistic information, which involves the following steps.

*Step 1. *Transform the dual hesitant fuzzy triangular linguistic decision matrix (, ) into normalized dual hesitant fuzzy triangular linguistic decision matrix . The normalized method is given as follows:where and are the sets of benefit attributes and cost attributes, respectively, and is the cardinality of linguistic term set .

*Step 2. *Aggregate all assessment values of the alternative () on all attributes () into the overall assessment values , based on the DHFTLWG operator in (48):
where and .

*Step 3. *Calculate the score values of the ove