#### Abstract

We investigate the multiple attribute decision-making (MADM) problems with hesitant triangular fuzzy information. Firstly, definition and some operational laws of hesitant triangular fuzzy elements are introduced. Then, we develop some hesitant triangular fuzzy aggregation operators based on Bonferroni means and discuss their basic properties. Some existing operators can be viewed as their special cases. Next, we apply the proposed operators to deal with multiple attribute decision-making problems under hesitant triangular fuzzy environment. Finally, an illustrative example is given to show the developed method and demonstrate its practicality and effectiveness.

#### 1. Introduction

Fuzzy set (FS), proposed by Zadeh in 1965 [1], has achieved a great success in various fields since it appears. As extensions of FS, the intuitionistic fuzzy set and interval-valued intuitionistic fuzzy set have received much attention [2–6]. Furthermore, Torra [7] generalized FS to hesitant fuzzy set (HFS), which allows the membership to a set represented by several possible values. HFS is very useful to express people’s hesitancy in daily life and a series of aggregation operators for hesitant fuzzy information have been developed [8–13]. Although HFS is a powerful tool to deal with uncertainty, it still has inherent drawbacks. HFS only permits the membership having a set of possible exact and crisp values. However, due to the increasing complexity of the socioeconomic environment and the vagueness of inherent subjective nature of human think, the information provided by a decision-maker is often imprecise or uncertain, so exact and crisp values are usually insufficient to model real-life decision. Chen et al. [14] introduced the interval-valued hesitant fuzzy set, based on which Wei et al. [15] proposed the hesitant triangular fuzzy set. As we all know, triangular fuzzy number is a very suitable tool to express uncertainty. Hesitant triangular fuzzy set (HTFS), whose membership degrees are expressed by several possible triangular fuzzy numbers, is more adequate or sufficient to solve real-life decision problem than real numbers. For example [16], suppose three reviewers are to estimate the degrees that a candidate satisfies the criterion of honest. As they have not met each other before, the evaluation is uncertain. The first reviewer thinks the most possible of the candidate satisfying the criterion of honest is 0.8, the minimum possible is 0.7, and the maximum possible is 0.9. Then, he can give the evaluation by a triangular fuzzy number . Similarly, the second reviewer and the third reviewer give their evaluations as and , respectively. As a result, this comprehensive evaluation can be expressed by a hesitant triangular fuzzy element . In this case, HTFS describes the dilemma vividly.

In [15], Wei et al. also developed some hesitant triangular fuzzy aggregation operators such as the hesitant triangular fuzzy weighted averaging (HTFWA) operator, hesitant triangular fuzzy weighted geometric (HTFWG) operator, hesitant triangular fuzzy ordered weighted averaging (HTFOWA) operator, hesitant triangular fuzzy ordered weighted geometric (HTFOWG) operator, hesitant triangular fuzzy hybrid averaging (HTFHA) operator, and hesitant triangular fuzzy hybrid geometric (HTFHG) operator.

However, the above operators, which are the extensions of the average mean (AM) and the geometric mean (GM), only consider the situations where all the elements are independent. Luckily, the Bonferroni mean (BM), which is originally introduced by Bonferroni [17] and generalized by Yager [18], can capture the interrelationships among arguments [17, 19–24]. Moreover, the Choquet integral is also an important tool to consider the correlations among attributes [6, 25–27]. With the analysis above, we attempt to develop new hesitant triangular fuzzy aggregation operators based on the BM and the Choquet integral so as to capture both the interrelationships between input arguments and the correlations among the attributes.

To facilitate our discussion, the remainder of this paper is organized as follows. In the next section, we review some basic concepts. Hesitant triangular fuzzy geometric Bonferroni mean operator and its properties are studied in Section 3. In Section 4, families of hesitant triangular fuzzy aggregation operators based on BM are studied. The relations between these new operators and the existing operators are also investigated. In Section 5, we develop a method for multiple attribute decision-making based on new operators under hesitant triangular fuzzy environment. An illustrative example is also given to show the effectiveness of the developed approach in Section 6. In Section 7, we conclude the paper and give some remarks.

#### 2. Preliminaries

##### 2.1. Triangular Fuzzy Numbers and Hesitant Triangular Fuzzy Set

FS was first proposed by Zadeh [1] in 1965.

*Definition 1 (see [1]). *Let be an universe of discourse; then a fuzzy set is defined as which is characterized by a membership function , where denotes the degree of membership of the element to the set .

Torra [7] generalized FSs to HFSs as follows.

*Definition 2 (see [7]). *Let be a reference set; then one defines hesitant fuzzy set on in terms of a function that when applied to returns a sunset of .

To be easily understood, Xia and Xu [8] express the HFS by a mathematical symbol: , where is a set of some values in , denoting the possible membership degree of the element to the set . For convenience, Xia and Xu [8] call a hesitant fuzzy element (HFE) and the set of all HFEs when there is no confusion.

Triangular fuzzy number, which is proposed by van Laarhoven and Pedrycz [28], is also a useful tool to express uncertainty.

*Definition 3 (see [28]). *A triangular fuzzy number can be defined by a triplet . The membership function is defined as
where , and stand for the lower and upper values of the support of , respectively, and stands for the modal value.

In [28], basic operational laws related to triangular fuzzy numbers were also given as ; ; , .

In order to compare triangular fuzzy numbers, many ranking methods have been proposed and each method has its advantages as well as drawbacks [29]. We adopt one of them as below.

*Definition 4 (see [30]). *Let and be two triangular fuzzy numbers; then the degree of possibility of is defined as
where the value is an index of rating attitude. It reflects the decision-maker’s risk-bearing attitude. If , the decision-maker is risk lover. If , the decision-maker is neutral to risk. If , the decision-maker is risk averter.

From this definition, we can get the following results easily:(1); (2). Especially, .

Wei et al. [15] generalized the HFS to HTFS as follows.

*Definition 5 (see [15]). *Let be a fixed set; a hesitant triangular fuzzy set (HTFS) on is in terms of a function when applied to each in and returns a subset of values in .

To be easily understood, Wei et al. [15] express the HTFS by a mathematical symbol: , where is a set of some possible triangular fuzzy values in , denoting the possible membership degrees of the element to the set . For convenience, they also call a hesitant triangular fuzzy element (HTFE) and the set of all HTFEs.

Given three HTFEs , , and , Wei et al. [15] defined their operations as follows: (1);
(2);
(3);
(4).

In order to compare two HTFEs, the score function was defined as follows.

*Definition 6 (see [15]). *For a HTFE , is called the score function of , where is the number of the triangular fuzzy values in and is a triangular fuzzy value belonging to . For two HTFEs and , if , then .

##### 2.2. Choquet Integral and Bonferroni Mean

In order to weight the elements in , a fuzzy measure was defined as follows.

*Definition 7 (see [31]). *A fuzzy measure on the set is a set function satisfying the following axioms and is the set of all subsets of : (1);
(2) implies , for all ;(3), for all and , where .

Especially, if , then condition (3) reduces to the axiom of additive measure: , for all and . If all the elements in are independent, then we have .

The discrete Choquet integral is a linear expression up to a reordering of the elements.

*Definition 8 (see [32]). *Let be a positive real-valued function on , and let be a fuzzy measure on . The discrete Choquet integral of with respect to is defined by
where is a permutation of , such that for all , , for , and .

As extensions of the arithmetic average and the geometric mean, the Bonferroni mean (BM) and the geometric Bonferroni mean (GBM) are very practical aggregation operators, which consider the interrelationships among arguments. We review the two operators as follows.

*Definition 9 (see [17, 33]). *Let and let be a collection of nonnegative numbers. Then
are called Bonferroni mean (BM) [17] and geometric Bonferroni mean (GBM) [33], respectively.

##### 2.3. The Existing Hesitant Triangular Fuzzy Operators

Here, we will briefly recall the existing hesitant triangular fuzzy operators. To see more details, we can refer to [15, 34].

Let be a collection of HTFEs, let , and let be the weight of , where denotes the importance degree of , satisfying and . Then the hesitant triangular fuzzy weighted averaging (HTFWA) operator was defined as [15] and the hesitant triangular fuzzy weighted geometric (HTFWG) operator was defined as [15] . Furthermore, suppose is a permutation of , such that for all . Then, the hesitant triangular fuzzy ordered weighted averaging (HTFOWA) operator was defined as [15] and the hesitant triangular fuzzy ordered weighted geometric (HTFOWG) operator was defined as [15] .

Suppose is the th largest element of the hesitant triangular fuzzy arguments (); then is called the hesitant triangular fuzzy hybrid average (HTFHA) operator [15].

Suppose is the th largest element of the hesitant triangular fuzzy arguments (); then is called the hesitant triangular fuzzy hybrid geometric (HTFHG) operator [15].

Let be a fuzzy measure on ; is called the hesitant triangular fuzzy Choquet ordered averaging (HTFCOA) operator [34].

#### 3. Hesitant Triangular Fuzzy Geometric Bonferroni Mean

Based on Definition 5, we can easily verify the following distributive properties.

Theorem 10. *Let be two HTFEs; then *(1)*;*(2)*.*

We can extend the GBM operator to hesitant triangular fuzzy set.

*Definition 11. *Let be a collection of HTFEs. For any , if
then HTFGB is called the hesitant triangular fuzzy geometric Bonferroni mean (HTFGBM).

Especially, if hesitant triangular fuzzy set reduces to hesitant fuzzy set, then the HTFGBM reduces to the I-revised geometric Bonferroni mean developed by Sun and Liu [23].

Based on the operational laws of HTFEs, we can derive the following easy-to-prove theorem whose proof is omitted.

Theorem 12. *Let and let be a collection of HTFEs; then the aggregated value by using the HTFGBM is also a HTFE and *

In order to capture the connections between two hesitant fuzzy elements (HFEs), Zhu et al. [22, 35] constructed the hesitant Bonferroni element (HBE) and hesitant fuzzy geometric Bonferroni element (HFGBE), which can be used as calculation units. Inspired by this idea, we can rewrite HTFGBM in another way.

Theorem 13. *Let and let be a collection of HTFEs; then the HTFGBM can be rewritten as
*

From Definition 11, its proof is straightforward.

Based on the operational laws of HTFEs, we can derive the following theorem.

Theorem 14. *Let and let be a collection of HTFEs; then the aggregated value by using the HTFGBM is a HTFE and
**
where .*

*Proof. *By the operational laws of HTFEs and Theorem 10, we get is also a HTFE and
By Theorem 10, we further obtain
Thus
where
Here, we call a hesitant triangular fuzzy geometric Bonferroni element (HTFGBE). The HTFGBE can take much more information into account and can fully represent the connections between two HTFEs. As the basic calculation unit of the HTFGBM, HTFGBE has some desirable properties as follows. As their proofs are straightforward, we omit them here.

Proposition 15. *Let and be two collections of HTFEs, and . If, for any and , we have and , then .*

Proposition 16. *Let be a collection of HTFEs, and , , , ; then
*

Proposition 17. *Exchanging and , we have .**This indicates that the parameters and are symmetric in .*

Proposition 18. *If one takes and , respectively, the corresponding results are or .*

Based on the studies above, we can investigate some basic properties of HTFGBM as below.

Theorem 19 (Monotonicity). *Let and be two collections of HTFEs; if, for any and , one has and , then
*

*Proof. *By Proposition 15, we get , , .

Then
By Definition 6, we acquire

Theorem 20 (boundness). *Let be a collection of HTFEs, , and , ; then
*

*Proof. *By Proposition 16, we have
So
By Definition 6, we complete the proof.

Theorem 21 (commutativity). *Let be a collection of HTFEs and let be any permutation of ; then
**
where and , .*

Theorem 22. *Let be a collection of HTFEs; then
*

Theorem 23. *Let be a collection of HTFEs; if , one has . If , then .*

#### 4. Families of Hesitant Triangular Fuzzy Aggregation Operators Based on Bonferroni Means

In practical society, the decision-makers may have different needs. In order to meet the different needs, we develop various hesitant triangular fuzzy aggregation operators based on Bonferroni means in this section. As their properties are similar to HTFGBM, we omit them for the sake of simplicity.

Based on Definition 9, we can develop hesitant triangular fuzzy Bonferroni mean as below.

*Definition 24. *Let be a collection of HTFEs. For any , one calls
the hesitant triangular fuzzy Bonferroni mean (HTFBM).

*Remark 25. *Especially, if hesitant triangular fuzzy set reduces to triangular fuzzy set, then HTFBM reduces to the triangular fuzzy Bonferroni mean developed by Zhu et al. [24]. Furthermore, let , ; then HTFBM reduces to . Besides, if hesitant triangular fuzzy set reduces to hesitant fuzzy set, then HTFBM reduces to the hesitant fuzzy Bonferroni mean proposed by Zhu and Xu [22].

In some practical applications, we have to weight the hesitant triangular fuzzy arguments. Then, by giving weights to each attribute, we can develop the weighted operators as below.

*Definition 26. *Let be a collection of HTFEs, , and the weight of , where denotes the importance degree of , satisfying and . Then
are called the hesitant triangular fuzzy weighted geometric Bonferroni mean (HTFWGBM) and the hesitant triangular fuzzy weighted Bonferroni mean (HTFWBM), respectively.

*Remark 27. *Suppose there is only one triangular fuzzy value in each and let , ; then and .

Sometimes, we may need to weight the ordered positions of the hesitant triangular fuzzy arguments instead of weighting the arguments themselves. In this case, we can develop the ordered weighted operators as follows.

*Definition 28. *Let be a collection of HTFEs, , and the associated weight vector such that and . is a permutation of , such that for all . Then
are called the hesitant triangular fuzzy ordered weighted geometric Bonferroni mean (HTFOWGBM) and the hesitant triangular fuzzy ordered weighted Bonferroni mean (HTFOWBM), respectively.

*Remark 29. *Suppose there is only one triangular fuzzy value in each and let , ; then and . If , then HTFOWGBM and HTFOWBM reduce to HTFWGBM and HTFWBM, respectively.

Inspired by Xu [36], when we want to not only weight the hesitant triangular fuzzy arguments but also weight the ordered positions of the hesitant triangular fuzzy arguments, we can propose the following hybrid average operators.

*Definition 30. *Let be a collection of HTFEs, , and the associated weight vector such that and . Let be the th largest element of the hesitant triangular fuzzy arguments (). Then, one calls
the hesitant triangular fuzzy hybrid geometric Bonferroni mean (HTFHGBM).

*Definition 31. *Let be a collection of HTFEs, , and the associated weight vector such that and . Let be the th largest element of the hesitant triangular fuzzy arguments (). Then, one calls
the hesitant triangular fuzzy hybrid Bonferroni mean (HTFHBM).

*Remark 32. *If there is only one triangular fuzzy value in each and letting , then , .

However, the above aggregation operators are based on the assumption that the attributes are independent. In real decision-making problems, these is usually interaction among attributes. As we all know, the Choquet integral [37] can depict the correlations of attributes. Combining the BM and the Choquet integral, Zhu et al. [35] developed a hesitant fuzzy Choquet geometric Bonferroni mean. Motivated by their idea, we develop the hesitant triangular fuzzy Choquet ordered Bonferroni mean as follows.

*Definition 33. *Let be a collection of HTFEs on , a fuzzy measure on , and . Then, one calls
the hesitant triangular fuzzy Choquet ordered Bonferroni mean (HTFCOBM), where is a permutation of , such that for all , , for , and .

*Remark 34. *If , , then HTFCOBM reduces to HTFWBM. Let , , then HTFCOBM reduces to HTFOWBM. In addition, suppose there is only one triangular fuzzy value in each and let , , then . This is the so-called hesitant triangular fuzzy Choquet ordered averaging operator proposed by Zhong and Xu [34].

#### 5. An Approach to Multiple Attribute Decision Making with Hesitant Triangular Fuzzy Information

In this section, we shall utilize the proposed operators to multiple attribute decision-makings under hesitant triangular fuzzy environment. As their procedures are similar, we only consider the HTFCOBM operator here.

The following assumptions or notations are used to represent the MADM problems for evaluation of theses with hesitant triangular fuzzy information. Let be a set of alternatives and a set of attributes. If the decision-makers provide values for the alternative under the attribute with anonymity, these values can be considered as a hesitant triangular fuzzy element . In the case where two decision-makers provide the same value, the value emerges only once in . Suppose that the decision matrix is the hesitant triangular fuzzy decision matrix, where are in the form of HTFEs.

In the following, we apply the HTFCOBM operator to the MADM problems for evaluation of theses with hesitant triangular fuzzy information.

*Step **1.* Confirm the fuzzy measures of attributes of and attributes sets of .

*Step **2.* We utilize the decision information given in matrix and the HTFCOBM operator
to derive the overall preference values of the alternative .

*Step **3.* Calculate the scores of the overall hesitant triangular fuzzy values by Definition 6.

*Step **4.* Compare each with all the by Definition 4. For convenience, we let ; then we develop a complementary matrix as , where . Summing all the elements in each line of matrix , we have .

*Step **5.* Rank all the alternatives in accordance with the values of and select the best one(s).

*Step **6.* End.

*Remark 35. *The advantages of our method lie in four aspects.

First, with the aid of fuzzy measure , the HTFCOBM operator can deal with the situation where the attributes are correlative. The weight vectors can be obtained by the source decision information in our method. Traditional additive aggregation operators, such as HTFWA and HTFWG operators, are all based on the assumption that the attributes are independent and each attribute is given a fixed weight representing its importance during the decision process. As a result, they cannot get reasonable results when the attributes are correlative.

Second, as we all know, the desirable characteristic of the BM is its ability to capture the interrelation among the input arguments. As a result, the HTFCOBM operator can deal with the situation where the input arguments are correlative.

Third, the HTFCOBM operator can accommodate situations in which the input arguments are hesitant triangular fuzzy information. As hesitant triangular fuzzy set is a comprehensive set containing FS and HFS as special cases, our method can be widely used.

Fourth, the HTFCOBM operator has additional parameters which control the power. If the parameters take different values, the HTFCOBM operator can be viewed as extensions of some exiting operators under certain conditions. The decision-makers can choose different parameters according to their preferences and interests, which makes decision-making more flexible.

#### 6. Numerical Example

In this section, we will present a numerical example (adapted from [38]) to show evaluation of theses with hesitant triangular fuzzy information in order to illustrate the proposed method.

Suppose there are five theses and we want to select the best one. Four attributes are selected by experts to evaluate the theses: (1) is the language of a thesis; (2) is the innovation; (3) is the rigor; (4) is the structure of the thesis. Perhaps the author who has accurate language also pays great attention to rigorous reasoning. That is to say, there are interactions between these attributes. In order to avoid influencing each other, the experts are required to evaluate the five theses under the above four attributes in anonymity and the decision matrix is presented in Table 1, where are in the form of HTFEs. In the review process, if the thesis has beautiful language, an expert may give better score to the structure of the thesis due to the previous good impression. In other words, there are interrelationships between input arguments. Thus, the HTFCOBM operator is a good choice here. The fuzzy measure of attribute and attribute sets of are as follows: , , , , , , , , , , , , , , and .

##### 6.1. The Decision-Making Steps

Next, we apply the developed approach to evaluate these theses with hesitant triangular fuzzy information.

*Step **1.* We use the decision information given in matrix and the HTFCOBM operator (here, we take ) to obtain the overall preference values of the thesis . Due to the large amount of data, we omitted these results here. When assigning different values to the parameters and , we can obtain different results. Please see Table 2.

*Step **2.* Calculate the scores of the overall hesitant triangular fuzzy preference values by Definition 6: .

*Step **3.* Comparing each with all the by Definition 4, then we develop a complementary matrix as . Summing all the elements in each line of the matrix , we have .

*Step **4.* Rank all the alternatives in accordance with the values of : . Note that means “preferred to.” Thus, the best thesis is .

##### 6.2. Discussion

From Table 2, we find that the values obtained by the HTFCOBM operator change as the parameters vary. Moreover, the rankings are different when we choose different values of . As the two parameters are symmetrical, we can fix one of them and change the other. Here, we set , for example. The trends are shown in Figure 1 as the parameter ranges from 0 to 18. From Figure 1, it can be clearly seen that(1)when , the ranking of the five theses is ;(2)when , the ranking of the five theses is ;(3)when , the ranking of the five theses is ;(4)when , the ranking of the five theses is .

Apparently, different parameters can be chosen according to decision-makers’ interests, which makes the decision more flexible.

##### 6.3. Comparative Analysis

In order to show the merit of the proposed method, we utilized some existing methods proposed by Wei et al. [15] and Zhong and Xu [34] to solve this illustrate example. For simplicity, we omit the calculation process and only list the results in Tables 3 and 4.

From Tables 2, 3, and 4, we can compare these methods as follows.

(1) During the calculation, we can find that the weight vectors can be obtained by the source decision information in our method. As a result, different decision data will acquire different weight vectors automatically. However, for other operators such as HTFWA and HTFWG, the weight vectors must be given by experts in advance. Thus, the proposed method is more reasonable and objective.

(2) Comparing Tables 3 and 4, we can find the relations such as In this case, we can view HTFWG as a special case of HTFWGBM. This has been mentioned in Section 4.

(3) From Table 4, we can find the relations such as Thus, the HTFOWBM can reduce to HTFWBM under certain conditions, which has also been pointed out in Section 4.

(4) We find that the rankings in Table 3 are different from Table 4. The reason may be that there are interdependent phenomena among attributes or input arguments in this numerical example. For example, also tells us that the attributes are correlative. The HTFCOBM operator can perform aggregation of attributes when they are correlative and it allows argument values to support each other in the aggregation process. However, the existing operators, such as HTFWA and HTFWG, always suppose that the attributes are independent and each attribute is given a fixed weight subjectively. So the HTFCOBM operator is a better choice here.

(5) When we change the parameters , we get different rankings in Table 2. This indicates that the HTFCOBM operator can meet the needs of different types of decision-makers.

#### 7. Conclusion

In this paper, we have investigated the multiple attribute decision-making (MADM) problems based on the HTFCOBM operator with hesitant triangular fuzzy information. Firstly, some basic concepts related to hesitant triangular fuzzy set have been reviewed. Then, motivated by the ideal of BM and Choquet integral, some new hesitant triangular fuzzy aggregation operators such as HTFCOBM have been developed. The prominent advantage of HTFCOBM is that it can consider the correlations between the attributes and among the input arguments, which makes it more feasible and practical. At the same time, we have discussed their basic properties. As different parameters can be chosen in these new operators, the decision becomes more flexible. Furthermore, we have discussed the families of new operators. Under certain conditions, they can be seen as extensions of the existing operators. Next, we have applied the HTFCOBM operator to multiple attribute decision-making problems in which attribute values take the form of hesitant triangular fuzzy information. Finally, an illustrative example for evaluation of theses has been given to demonstrate the proposed method. There are some other generalizations of Bonferroni mean such as the generalized hesitant fuzzy Bonferroni mean [21] and normalized geometric Bonferroni operators [23], which can also be used to construct new operators for hesitant triangular fuzzy set. The researches on these new operators may be interesting and meaningful.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

We would like to thank the editor and the anonymous referees for their valuable comments, which have been very helpful in improving the paper. This work was supported by the National Natural Science Foundation of China (no. 11371130, no.11201490) and Research Fund for the Doctoral Program of Higher Education of China (no. 20120161110017).