Hesitant Triangular Fuzzy Information Aggregation Operators Based on Bonferroni Means and Their Application to Multiple Attribute Decision Making

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 (0.7,0.8,0.9). Similarly, the second reviewer and the third reviewer give their evaluations as (0.4,0.5,0.6) and (0.6,0.9,1.0), respectively. As a result, this comprehensive evaluation can be expressed by a hesitant triangular fuzzy element {(0.7,0.8,0.9),(0.4,0.5,0.6),(0.6,0.9,1.0)}. 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 𝑋 be an universe of discourse; then a fuzzy set is defined as 𝐴={⟨𝑥,𝜇_𝐴(𝑥)⟩∣𝑥∈𝑋} which is characterized by a membership function 𝜇_𝐴∶𝑋→[0,1], where 𝜇_𝐴 denotes the degree of membership of the element 𝑥 to the set 𝐴.

Torra [7] generalized FSs to HFSs as follows.

Definition 2 (see [7]). Let 𝑋 be a reference set; then one defines hesitant fuzzy set on 𝑋 in terms of a function that when applied to 𝑋 returns a sunset of [0,1].

To be easily understood, Xia and Xu [8] express the HFS by a mathematical symbol: 𝐸=(⟨𝑥,ℎ_𝐸(𝑥)⟩∣𝑥∈𝑋), where ℎ_𝐸(𝑥) is a set of some values in [0,1], denoting the possible membership degree of the element 𝑥∈𝑋 to the set 𝐸. For convenience, Xia and Xu [8] call ℎ=ℎ_𝐸(𝑥) a hesitant fuzzy element (HFE) and 𝐻 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 𝑎 can be defined by a triplet (𝑎^𝐿,𝑎^𝑀,𝑎^𝑈). The membership function 𝜇_𝑎(𝑥) is defined as 𝜇_𝑎(𝑥)=0,𝑥<𝑎^𝐿,𝑥−𝑎^𝐿𝑎^𝑀−𝑎^𝐿,𝑎^𝐿≤𝑥≤𝑎^𝑀,𝑥−𝑎^𝑈𝑎^𝑀−𝑎^𝑈,𝑎^𝑀≤𝑥≤𝑎^𝑈,0,𝑥≥𝑎^𝑈, (1) where 0<𝑎^𝐿≤𝑎^𝑀≤𝑎^𝑈, 𝑎^𝐿 and 𝑎^𝑈 stand for the lower and upper values of the support of 𝑎, respectively, and 𝑎^𝑀 stands for the modal value.

In [28], basic operational laws related to triangular fuzzy numbers were also given as 𝑎⊕𝑏̃=[𝑎^𝐿,𝑎^𝑀,𝑎^𝑈]⊕[𝑏^𝐿,𝑏^𝑀,𝑏^𝑈]=[𝑎^𝐿+𝑏^𝐿,𝑎^𝑀+𝑏^𝑀,𝑎^𝑈+𝑏^𝑈]; 𝑎⊗𝑏̃=[𝑎^𝐿,𝑎^𝑀,𝑎^𝑈]⊗[𝑏^𝐿,𝑏^𝑀,𝑏^𝑈]=[𝑎^𝐿𝑏^𝐿,𝑎^𝑀𝑏^𝑀,𝑎^𝑈𝑏^𝑈]; 𝜆⊗𝑎=𝜆⊗[𝑎^𝐿,𝑎^𝑀,𝑎^𝑈]=[𝜆𝑎^𝐿,𝜆𝑎^𝑀,𝜆𝑎^𝑈],  𝜆>0.

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 𝑏̃=[𝑏^𝐿,𝑏^𝑀,𝑏^𝑈] and 𝑎=[𝑎^𝐿,𝑎^𝑀,𝑎^𝑈] be two triangular fuzzy numbers; then the degree of possibility of 𝑎≥𝑏̃ is defined as 𝑝𝑏̃𝑏𝑎≥=𝜆max1−max^𝑀−𝑎^𝐿𝑎^𝑀−𝑎^𝐿+𝑏^𝑀−𝑏^𝐿+()𝑏,0,01−𝜆×max1−max^𝑈−𝑎^𝑀𝑎^𝑈−𝑎^𝑀+𝑏^𝑈−𝑏^𝑀,0,0, (2) where the value 𝜆 is an index of rating attitude. It reflects the decision-maker’s risk-bearing attitude. If 𝜆>0.5, the decision-maker is risk lover. If 𝜆=0.5, the decision-maker is neutral to risk. If 𝜆<0.5, the decision-maker is risk averter.

From this definition, we can get the following results easily:(1)0≤𝑝(𝑎≥𝑏̃)≤1,0≤𝑝(𝑏̃≥𝑎)≤1; (2)𝑝(𝑎≥𝑏̃)+𝑝(𝑏̃≥𝑎)=1. Especially, 𝑝(𝑎≥𝑎)=𝑝(𝑏̃≥𝑏̃)=0.5.

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

Definition 5 (see [15]). Let 𝑋 be a fixed set; a hesitant triangular fuzzy set (HTFS) on 𝑋 is in terms of a function when applied to each 𝑥 in 𝑋 and returns a subset of values in [0,1].
To be easily understood, Wei et al. [15] express the HTFS by a mathematical symbol: 𝐸={⟨𝑥,ℎ_𝐸(𝑥)⟩∣𝑥∈𝑋}, where ℎ_𝐸(𝑥) is a set of some possible triangular fuzzy values in [0,1], denoting the possible membership degrees of the element 𝑥∈𝑋 to the set 𝐸. For convenience, they also call ℎ_𝐸(𝑥) a hesitant triangular fuzzy element (HTFE) and 𝐻 the set of all HTFEs.
Given three HTFEs ℎ, ℎ₁, ℎ₂ and 𝜆>0, Wei et al. [15] defined their operations as follows: (1)ℎ^𝜆=∪_ℎ𝛾∈{((𝛾^𝐿)^𝜆,(𝛾^𝑀)^𝜆,(𝛾^𝑅)^𝜆)}; (2)𝜆ℎ=∪_ℎ𝛾∈{(1−(1−𝛾^𝐿)^𝜆,1−(1−𝛾^𝑀)^𝜆,1−(1−𝛾^𝑅)^𝜆)}; (3)ℎ₁⊕ℎ₂=∪_{𝛾1∈ℎ1,𝛾2∈ℎ2}{(𝛾^𝐿₁+𝛾^𝐿₂−𝛾^𝐿₁𝛾^𝐿₂,𝛾^𝑀₁+𝛾^𝑀₂−𝛾^𝑀₁𝛾^𝑀₂,𝛾^𝑅₁+𝛾^𝑅₂−𝛾^𝑅₁𝛾^𝑅₂)}; (4)ℎ₁⊗ℎ₂=∪_{𝛾1∈ℎ1,𝛾2∈ℎ2}{(𝛾^𝐿₁𝛾^𝐿₂,𝛾^𝑀₁𝛾^𝑀₂,𝛾^𝑅₁𝛾^𝑅₂)}.

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

Definition 6 (see [15]). For a HTFE ℎ, 𝑠(ℎ)=(1/♯ℎ)∑_ℎ𝛾∈𝛾 is called the score function of ℎ, where ♯ℎ is the number of the triangular fuzzy values in ℎ and 𝑠(ℎ) is a triangular fuzzy value belonging to [0,1]. For two HTFEs ℎ₁ and ℎ₂, if 𝑠(ℎ₁)≥𝑠(ℎ₂), then ℎ₁≥ℎ₂.

2.2. Choquet Integral and Bonferroni Mean

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

Definition 7 (see [31]). A fuzzy measure 𝜇 on the set 𝑋 is a set function 𝜇∶𝜃(𝑋)→[0,1] satisfying the following axioms and 𝜃(𝑋) is the set of all subsets of 𝑋: (1)𝜇(𝜙)=0,𝜇(𝑋)=1; (2)𝐴⊆𝐵 implies 𝜇(𝐴)≤𝜇(𝐵), for all 𝐴,𝐵⊆𝑋;(3)𝜇(𝐴∪𝐵)=𝜇(𝐴)+𝜇(𝐵)+𝜌𝜇(𝐴)𝜇(𝐵), for all 𝐴,𝐵⊆𝑋 and 𝐴∩𝐵=𝜙, where 𝜌∈(−1,∞).

Especially, if 𝜌=0, then condition (3) reduces to the axiom of additive measure: 𝜇(𝐴∪𝐵)=𝜇(𝐴)+𝜇(𝐵), for all 𝐴,𝐵⊆𝑋 and 𝐴∩𝐵=𝜙. If all the elements in 𝑋 are independent, then we have 𝜇(𝐴)=∑_{𝑥𝑖∈𝐴}𝜇({𝑥_𝑖}),∀𝐴⊆𝑋.

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

Definition 8 (see [32]). Let 𝑓 be a positive real-valued function on 𝑋, and let 𝜇 be a fuzzy measure on 𝑋. The discrete Choquet integral of 𝑓 with respect to 𝜇 is defined by 𝐶_𝜇𝑓=^𝑛_𝑖=1𝑓_𝜎(𝑖)𝜇𝐴_𝜎(𝑖)−𝜇𝐴_{𝜎(𝑖−1)}, (3) where (𝜎(1),𝜎(2),…,𝜎(𝑛)) is a permutation of (1,2,…,𝑛), such that 𝑓_{𝜎(𝑖−1)}≥𝑓_𝜎(𝑖) for all 𝑖=2,3,…,𝑛, 𝐴_𝜎(𝑘)={𝑥_𝜎(𝑗)∣𝑗≤𝑘}, for 𝑘≥1, and 𝐴_𝜎(0)=𝜙.

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 𝑝,𝑞≥0 and let 𝑎_𝑖(𝑖=1,2,…,𝑛) be a collection of nonnegative numbers. Then BM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=1𝑛(𝑛−1)^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑎^𝑝_𝑖𝑎^𝑞_𝑗^{1/(𝑝+𝑞)}GBM^𝑝,𝑞𝑎₁,𝑎₂,…,𝑎_𝑛=1𝑝+𝑞^𝑛_{𝑖≠𝑗,𝑖,𝑗=1}𝑝𝑎_𝑖+𝑞𝑎_𝑗^{1/𝑛(𝑛−1)}, (4) 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 ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of HTFEs, let 𝑝,𝑞≥0, and let 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 be the weight of ℎ_𝑖, where 𝑤_𝑖 denotes the importance degree of ℎ_𝑖, satisfying 𝑤_𝑖>0 and ∑^𝑛_𝑖=1𝑤_𝑖=1. Then the hesitant triangular fuzzy weighted averaging (HTFWA) operator was defined as [15] HTFWA(ℎ₁,ℎ₂,…,ℎ_𝑛)=⨁^𝑛_𝑗=1(𝑤_𝑗ℎ_𝑗) and the hesitant triangular fuzzy weighted geometric (HTFWG) operator was defined as [15] HTFWG(ℎ₁,ℎ₂,…,ℎ_𝑛)=⨂^𝑛_𝑗=1(ℎ_𝑗)^𝑤𝑗. Furthermore, suppose (𝜎(1),𝜎(2),…,𝜎(𝑛)) is a permutation of (1,2,…,𝑛), such that ℎ_{𝜎(𝑗−1)}≥ℎ_𝜎(𝑗) for all 𝑗=2,3,…,𝑛. Then, the hesitant triangular fuzzy ordered weighted averaging (HTFOWA) operator was defined as [15] HTFOWA(ℎ₁,ℎ₂,…,ℎ_𝑛)=⨁^𝑛_𝑗=1(𝑤_𝑗ℎ_𝜎(𝑗)) and the hesitant triangular fuzzy ordered weighted geometric (HTFOWG) operator was defined as [15] HTFOWG(ℎ₁,ℎ₂,…,ℎ_𝑛)=⨂^𝑛_𝑗=1(ℎ_𝜎(𝑗))^𝑤𝑗.

Suppose ℎ_𝜎(𝑗) is the 𝑗th largest element of the hesitant triangular fuzzy arguments (ℎ_𝑗=(𝑛𝑤_𝑗)ℎ_𝑗,𝑗=1,2,…,𝑛); then HTFHA(ℎ₁,ℎ₂,…,ℎ_𝑛)=⨁^𝑛_𝑗=1(𝑤_𝑗ℎ_𝜎(𝑗)) is called the hesitant triangular fuzzy hybrid average (HTFHA) operator [15].

Suppose ℎ_𝜎(𝑗) is the 𝑗th largest element of the hesitant triangular fuzzy arguments (ℎ_𝑗=(ℎ_𝑗)^𝑛𝑤𝑗,𝑗=1,2,…,𝑛); then HTFHG(ℎ₁,ℎ₂,…,ℎ_𝑛)=⨂^𝑛_𝑗=1(ℎ_𝜎(𝑗))^𝑤𝑗 is called the hesitant triangular fuzzy hybrid geometric (HTFHG) operator [15].

Let 𝜇 be a fuzzy measure on 𝑋; HTFCOA_𝜇(ℎ₁,ℎ₂,…,ℎ_𝑛)=⨁^𝑛_𝑗=1(𝜇(𝐴_𝜎(𝑗))−𝜇(𝐴_{𝜎(𝑗−1)})ℎ_𝜎(𝑗) 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 ℎ₁,ℎ₂ be two HTFEs; then (1)𝜆(ℎ₁⊕ℎ₂)=𝜆ℎ₁⊕𝜆ℎ₂;(2)(ℎ₁⊗ℎ₂)^𝜆=ℎ^𝜆₁⊗ℎ^𝜆₂.

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

Definition 11. Let ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of HTFEs. For any 𝑝,𝑞≥0, if HTFGBM^𝑝,𝑞ℎ₁,ℎ₂,…,ℎ_𝑛=1𝑝+𝑞^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑝ℎ_𝑖⊕𝑞ℎ_𝑗^{1/𝑛(𝑛−1)}, (5) then HTFGBM^𝑝,𝑞 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 𝑝,𝑞≥0 and let ℎ_𝑗(𝑗=1,2,…,𝑛) be a collection of HTFEs; then the aggregated value by using the HTFGBM is also a HTFE and 𝐻𝑇𝐹𝐺𝐵𝑀^𝑝,𝑞ℎ₁,ℎ₂,…,ℎ_𝑛=1𝑝+𝑞^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑝ℎ_𝑖⊕𝑞ℎ_𝑗^{1/𝑛(𝑛−1)}=_{𝛾𝑖∈ℎ𝑖,𝛾𝑗∈ℎ𝑗}1−1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−1−𝛾^𝐿_𝑖^𝑝1−𝛾^𝐿_𝑗^{𝑞1/𝑛(𝑛−1)1/(𝑝+𝑞)},1−1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−1−𝛾^𝑀_𝑖^𝑝1−𝛾^𝑀_𝑗^{𝑞1/𝑛(𝑛−1)1/(𝑝+𝑞)},1−1−^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}1−1−𝛾^𝑅_𝑖^𝑝1−𝛾^𝑅_𝑗^{𝑞1/𝑛(𝑛−1)1/(𝑝+𝑞)}. (6)

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 𝑝,𝑞≥0 and let ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of HTFEs; then the HTFGBM can be rewritten as 𝐻𝑇𝐹𝐺𝐵𝑀^𝑝,𝑞ℎ₁,ℎ₂,…,ℎ_𝑛=1𝑝+𝑞^𝑛_{𝑖,𝑗=1,𝑖<𝑗}𝑝ℎ_𝑖⊕𝑞ℎ_𝑗⊗𝑝ℎ_𝑗⊕𝑞ℎ_𝑖^{1/𝑛(𝑛−1)}. (7)

From Definition 11, its proof is straightforward.

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

Theorem 14. Let 𝑝,𝑞≥0 and let ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of HTFEs; then the aggregated value by using the HTFGBM is a HTFE and 𝐻𝑇𝐹𝐺𝐵𝑀^𝑝,𝑞ℎ₁,ℎ₂,…,ℎ_𝑛=_{𝜀𝑖𝑗∈𝜏𝑖𝑗,𝑖<𝑗}1−1−^𝑛_{𝑖,𝑗=1;𝑖<𝑗}𝜀^{1/𝑛(𝑛−1)}_𝑖𝑗^{1/(𝑝+𝑞)}, (8) where 𝜏_{𝑖𝑗,𝑖<𝑗}=(𝑝ℎ_𝑖⊕𝑞ℎ_𝑗)⊗(𝑝ℎ_𝑗⊕𝑞ℎ_𝑖),𝑖,𝑗=1,2,…,𝑛.

Proof. By the operational laws of HTFEs and Theorem 10, we get 𝜏_{𝑖𝑗,𝑖<𝑗}=(𝑝ℎ_𝑖⊕𝑞ℎ_𝑗)⊗(𝑝ℎ_𝑗⊕𝑞ℎ_𝑖) is also a HTFE and HTFGBM^𝑝,𝑞ℎ₁,ℎ₂,…,ℎ_𝑛=1𝑝+𝑞^𝑛_{𝑖,𝑗=1,𝑖<𝑗}𝜏^{1/𝑛(𝑛−1)}_𝑖𝑗. (9) By Theorem 10, we further obtain ^𝑛_{𝑖,𝑗=1,𝑖<𝑗}𝜏^{1/𝑛(𝑛−1)}_𝑖𝑗=^𝑛_{𝑖,𝑗=1,𝑖<𝑗}𝜏_𝑖𝑗^{1/𝑛(𝑛−1)}=_{𝜀𝑖𝑗∈𝜏𝑖𝑗,𝑖<𝑗}^𝑛_{𝑖,𝑗=1;𝑖<𝑗}𝜀_𝑖𝑗^{1/𝑛(𝑛−1)}=_{𝜀𝑖𝑗∈𝜏𝑖𝑗,𝑖<𝑗}^𝑛_{𝑖,𝑗=1;𝑖<𝑗}𝜀^{1/𝑛(𝑛−1)}_𝑖𝑗. (10) Thus HTFGBM^𝑝,𝑞ℎ₁,ℎ₂,…,ℎ_𝑛=1𝑝+𝑞^𝑛_{𝑖,𝑗=1,𝑖<𝑗}𝜏^{1/𝑛(𝑛−1)}_𝑖𝑗=_{𝜀𝑖𝑗∈𝜏𝑖𝑗,𝑖<𝑗}1−1−^𝑛_{𝑖,𝑗=1;𝑖<𝑗}𝜀^{1/𝑛(𝑛−1)}_𝑖𝑗^{1/(𝑝+𝑞)}, (11) where 𝜀_𝑖𝑗=𝑝ℎ_𝑖⊕𝑞ℎ_𝑗⊗𝑝ℎ_𝑗⊕𝑞ℎ_𝑖=_{𝛾𝑖∈ℎ𝑖,𝛾𝑗∈ℎ𝑗}1−1−𝛾^𝐿_𝑖^𝑝1−𝛾^𝐿_𝑗^𝑞,1−1−𝛾^𝑀_𝑖^𝑝1−𝛾^𝑀_𝑗^𝑞,1−1−𝛾^𝑅_𝑖^𝑝1−𝛾^𝑅_𝑗^𝑞⊗_{𝛾𝑖∈ℎ𝑖,𝛾𝑗∈ℎ𝑗}1−1−𝛾^𝐿_𝑖^𝑞1−𝛾^𝐿_𝑗^𝑝,1−1−𝛾^𝑀_𝑖^𝑞1−𝛾^𝑀_𝑗^𝑝,1−1−𝛾^𝑅_𝑖^𝑞1−𝛾^𝑅_𝑗^𝑝. (12) Here, we call 𝜏_𝑖𝑗 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 ℎ_𝛼𝑖 and ℎ_𝛽𝑖 be two collections of HTFEs, 𝜏_{𝛼𝑖𝑗,𝑖<𝑗}=(𝑝ℎ_𝛼𝑖⊕𝑞ℎ_𝛼𝑗)⊗(𝑝ℎ_𝛼𝑗⊕𝑞ℎ_𝛼𝑖) and 𝜏_{𝛽𝑖𝑗,𝑖<𝑗}=(𝑝ℎ_𝛽𝑖⊕𝑞ℎ_𝛽𝑗)⊗(𝑝ℎ_𝛽𝑗⊕𝑞ℎ_𝛽𝑖). If, for any 𝛾_𝛼𝑖∈ℎ_𝛼𝑖 and 𝛾_𝛽𝑗∈ℎ_𝛽𝑗(𝑖,𝑗=1,2,…,𝑛;𝑖≠𝑗), we have 𝛾_𝛼𝑖≤𝛾_𝛽𝑖 and 𝛾_𝛼𝑗≤𝛾_𝛽𝑗, then 𝜏_{𝛼𝑖𝑗,𝑖<𝑗}≤𝜏_{𝛽𝑖𝑗,𝑖<𝑗}.

Proposition 16. Let ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of HTFEs, 𝜏_{𝑖𝑗,𝑖<𝑗}=(𝑝ℎ_𝑖⊕𝑞ℎ_𝑗)⊗(𝑝ℎ_𝑗⊕𝑞ℎ_𝑖) and ℎ⁻_𝑖=∪_{𝛾𝑖∈ℎ𝑖}(min{𝛾^𝐿_𝑖},min{𝛾^𝑀_𝑖},min{𝛾^𝑅_𝑖}), ℎ⁺_𝑖=∪_{𝛾𝑖∈ℎ𝑖}(max{𝛾^𝐿_𝑖},max{𝛾^𝑀_𝑖},max{𝛾^𝑅_𝑖}), 𝑖,𝑗∈{1,2,…, 𝑛}; then _{𝛾−∈ℎ−𝑖}1−1−𝛾^{−𝐿𝑝+𝑞2},1−1−𝛾^{−𝑀𝑝+𝑞2},1−1−𝛾^{−𝑅𝑝+𝑞2}≤𝜏_{𝑖𝑗,𝑖<𝑗}≤_{𝛾+∈ℎ+𝑖}1−1−𝛾^{+𝐿𝑝+𝑞2},1−1−𝛾^{+𝑀𝑝+𝑞2},1−1−𝛾^{+𝑅𝑝+𝑞2}. (13)

Proposition 17. Exchanging 𝑝 and 𝑞, we have 𝜏_{𝑖𝑗,𝑖<𝑗}=(𝑝ℎ_𝑖⊕𝑞ℎ_𝑗)⊗(𝑝ℎ_𝑗⊕𝑞ℎ_𝑖)=(𝑞ℎ_𝑖⊕𝑝ℎ_𝑗)⊗(𝑞ℎ_𝑗⊕𝑝ℎ_𝑖).
This indicates that the parameters 𝑝 and 𝑞 are symmetric in 𝐻𝑇𝐹𝐺𝐵𝐸.

Proposition 18. If one takes ℎ=ℎ_𝑖=ℎ_𝑗={(0,0,0)} and ℎ=ℎ_𝑖=ℎ_𝑗={(1,1,1)}, respectively, the corresponding results are 𝜏_{𝑖𝑗,𝑖<𝑗}={(0,0,0)} or 𝜏_{𝑖𝑗,𝑖<𝑗}={(1,1,1)}.

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

Theorem 19 (Monotonicity). Let ℎ_𝛼𝑖 and ℎ_𝛽𝑖(𝑖=1,2,…,𝑛) be two collections of HTFEs; if, for any 𝛾_𝛼𝑖∈ℎ_𝛼𝑖 and 𝛾_𝛽𝑗∈ℎ_𝛽𝑗(𝑖,𝑗=1,2,…,𝑛;𝑖≠𝑗), one has 𝛾_𝛼𝑖≤𝛾_𝛽𝑖 and 𝛾_𝛼𝑗≤𝛾_𝛽𝑗, then 𝐻𝑇𝐹𝐺𝐵𝑀^𝑝,𝑞ℎ_𝛼1,ℎ_𝛼2,…,ℎ_𝛼𝑛≤𝐻𝑇𝐹𝐺𝐵𝑀^𝑝,𝑞ℎ_𝛽1,ℎ_𝛽2,…,ℎ_𝛽𝑛. (14)

Proof. By Proposition 15, we get 𝜀_𝛼𝑖𝑗≤𝜀_𝛽𝑖𝑗, 𝑖,𝑗∈{1,2,…,𝑛}, 𝑖≠𝑗.
Then 1−1−^𝑛_{𝑖,𝑗=1;𝑖≠𝑗}𝜀_𝛼^{1/𝑛(𝑛−1)}_𝑖𝑗^{1/(𝑝+𝑞)}≤1−1−^𝑛_{𝑖,𝑗=1;𝑖≠𝑗}𝜀_𝛽^{1/𝑛(𝑛−1)}_𝑖𝑗^{1/(𝑝+𝑞)}. (15) By Definition 6, we acquire HTFGBM^𝑝,𝑞ℎ_𝛼1,ℎ_𝛼2,…,ℎ_𝛼𝑛=1𝑝+𝑞^𝑛_{𝑖,𝑗=1,𝑖<𝑗}𝜏_𝛼^{1/𝑛(𝑛−1)}_𝑖𝑗=_{𝜀𝛼𝑖𝑗∈𝜏𝑖𝑗,𝑖<𝑗}1−1−^𝑛_{𝑖,𝑗=1;𝑖<𝑗}𝜀_𝛼^{1/𝑛(𝑛−1)}_𝑖𝑗^{1/(𝑝+𝑞)}≤_{𝜀𝛽𝑖𝑗∈𝜏𝑖𝑗,𝑖<𝑗}1−1−^𝑛_{𝑖,𝑗=1;𝑖<𝑗}𝜀_𝛽^{1/𝑛(𝑛−1)}_𝑖𝑗^{1/(𝑝+𝑞)}=1𝑝+𝑞^𝑛_{𝑖,𝑗=1,𝑖<𝑗}𝜏_𝛽^{1/𝑛(𝑛−1)}_𝑖𝑗=HTFGBM^𝑝,𝑞ℎ_𝛽1,ℎ_𝛽2,…,ℎ_𝛽𝑛. (16)

Theorem 20 (boundness). Let ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of HTFEs, ℎ⁻_𝑖=∪_{𝛾𝑖∈ℎ𝑖}(min{𝛾^𝐿_𝑖},min{𝛾^𝑀_𝑖},min{𝛾^𝑅_𝑖}), and ℎ⁺_𝑖=∪_{𝛾𝑖∈ℎ𝑖}(max{𝛾^𝐿_𝑖},max{𝛾^𝑀_𝑖},max{𝛾^𝑅_𝑖}), 𝑖,𝑗∈{1,2,…,𝑛}; then _{𝛾−∈ℎ−𝑖}1−1−1−1−𝛾^{−𝐿𝑝+𝑞21/(𝑝+𝑞)},1−1−1−1−𝛾^{−𝑀𝑝+𝑞21/(𝑝+𝑞)},1−1−1−1−𝛾^{−𝑅𝑝+𝑞21/(𝑝+𝑞)}≤𝐻𝑇𝐹𝐺𝐵𝑀^𝑝,𝑞ℎ₁,ℎ₂,…,ℎ_𝑛≤_{𝛾+∈ℎ+𝑖}1−1−1−1−𝛾^{+𝐿𝑝+𝑞21/(𝑝+𝑞)},1−1−1−1−𝛾^{+𝑀𝑝+𝑞21/(𝑝+𝑞)},1−1−1−1−𝛾^{+𝑅𝑝+𝑞21/𝑝+𝑞}. (17)

Proof. By Proposition 16, we have _{𝛾−∈ℎ−𝑖}1−1−𝛾^{−𝐿𝑝+𝑞2},1−1−𝛾^{−𝑀𝑝+𝑞2},1−1−𝛾^{−𝑅𝑝+𝑞2}≤𝜀_{𝑖𝑗,𝑖<𝑗}≤_{𝛾+∈ℎ+𝑖}1−1−𝛾^{+𝐿𝑝+𝑞2},1−1−𝛾^{+𝑀𝑝+𝑞2},1−1−𝛾^{+𝑅𝑝+𝑞2}. (18) So 1−1−1−1−𝛾^{−𝐿𝑝+𝑞21/(𝑝+𝑞)},1−1−1−1−𝛾^{−𝑀𝑝+𝑞21/(𝑝+𝑞)},1−1−1−1−𝛾^{−𝑅𝑝+𝑞21/(𝑝+𝑞)}≤1−1−_{𝑖,𝑗=1;𝑖≠𝑗}𝜀^{1/𝑛(𝑛−1)}_{𝑖𝑗,𝑖<𝑗}^{1/(𝑝+𝑞)}≤1−1−1−1−𝛾^{+𝐿𝑝+𝑞21/(𝑝+𝑞)},1−1−1−1−𝛾^{+𝑀𝑝+𝑞21/(𝑝+𝑞)},1−1−1−1−𝛾^{+𝑅𝑝+𝑞21/(𝑝+𝑞)}. (19) By Definition 6, we complete the proof.

Theorem 21 (commutativity). Let ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of HTFEs and let (ℎ₁,ℎ₂,…,ℎ_𝑛) be any permutation of (ℎ₁,ℎ₂,…,ℎ_𝑛); then 𝐻𝑇𝐹𝐺𝐵𝑀^𝑝,𝑞ℎ₁,ℎ₂,…,ℎ_𝑛=1𝑝+𝑞^𝑛_{𝑖,𝑗=1,𝑖<𝑗}𝜏^{1/𝑛(𝑛−1)}_𝑖𝑗=1𝑝+𝑞^𝑛_{𝑖,𝑗=1,𝑖<𝑗}𝜏_𝑖𝑗^{1/𝑛(𝑛−1)}=𝐻𝑇𝐹𝐺𝐵𝑀^𝑝,𝑞ℎ₁,ℎ₂,…,ℎ_𝑛, (20) where 𝜏_{𝑖𝑗,𝑖<𝑗}=(𝑝ℎ_𝑖⊕𝑞ℎ_𝑗)⊗(𝑝ℎ_𝑗⊕𝑞ℎ_𝑖) and 𝜏_{𝑖𝑗,𝑖<𝑗}=(𝑝ℎ_𝑖⊕𝑞ℎ_𝑗)⊗(𝑝ℎ_𝑗⊕𝑞ℎ_𝑖), 𝑖,𝑗∈{1,2,…,𝑛}.

Theorem 22. Let ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of HTFEs; then 𝐻𝑇𝐹𝐺𝐵𝑀^𝑝,𝑞ℎ₁,ℎ₂,…,ℎ_𝑛=𝐻𝑇𝐹𝐺𝐵𝑀^𝑞,𝑝ℎ₁,ℎ₂,…,ℎ_𝑛. (21)

Theorem 23. Let ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of HTFEs; if ℎ₁=ℎ₂=⋯=ℎ_𝑛=ℎ={(0,0,0)}, one has 𝐻𝑇𝐹𝐺𝐵𝑀^𝑝,𝑞(ℎ₁,ℎ₂,…,ℎ_𝑛)={(0,0,0)}. If ℎ₁=ℎ₂=⋯=ℎ_𝑛=ℎ={(1,1,1)}, then 𝐻𝑇𝐹𝐺𝐵𝑀^𝑝,𝑞(ℎ₁,ℎ₂,…,ℎ_𝑛)={(1,1,1)}.

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 ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of HTFEs. For any 𝑝,𝑞≥0, one calls HTFBM^𝑝,𝑞ℎ₁,ℎ₂,…,ℎ_𝑛=1𝑛(𝑛−1)^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}(ℎ^𝑝_𝑖⊗ℎ^𝑞_𝑗)^{1/(𝑝+𝑞)} (22) 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 𝑝=1, 𝑞=0; then HTFBM reduces to HTFBM^1,0(ℎ₁,ℎ₂,…,ℎ_𝑛)=(1/𝑛)(⊕^𝑛_𝑖=1ℎ_𝑖). 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 ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of HTFEs, 𝑝,𝑞≥0, and 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 the weight of ℎ_𝑖, where 𝑤_𝑖 denotes the importance degree of ℎ_𝑖, satisfying 𝑤_𝑖>0 and ∑^𝑛_𝑖=1𝑤_𝑖=1. Then HTFWGBM^𝑝,𝑞ℎ₁,ℎ₂,…,ℎ_𝑛=1𝑝+𝑞^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑝ℎ^𝑤𝑖_𝑖⊕𝑞ℎ^𝑤𝑗_𝑗^{1/𝑛(𝑛−1)},HTFWBM^𝑝,𝑞ℎ₁,ℎ₂,…,ℎ_𝑛=1𝑛(𝑛−1)^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}(𝑤_𝑖ℎ^𝑝_𝑖⊗𝑤_𝑗ℎ^𝑞_𝑗)^{1/(𝑝+𝑞)} (23) 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 ℎ_𝑖(𝑖=1,2,…,𝑛) and let 𝑝=1, 𝑞=0; then HTFWGBM^1,0(ℎ₁,ℎ₂,…,ℎ_𝑛)=⨂^𝑛_𝑖=1(ℎ^𝑤𝑖_𝑖)^1/𝑛=(HTFWG(ℎ₁,ℎ₂,…,ℎ_𝑛))^1/𝑛 and HTFWBM^1,0(ℎ₁,ℎ₂,…,ℎ_𝑛)=(1/𝑛)⨁^𝑛_𝑖=1(𝑤_𝑖ℎ_𝑖)=(1/𝑛)HTFWA(ℎ₁,ℎ₂,…,ℎ_𝑛).

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 ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of HTFEs, 𝑝,𝑞≥0, and 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 the associated weight vector such that 𝑤_𝑖>0 and ∑^𝑛_𝑖=1𝑤_𝑖=1. (𝜎(1),𝜎(2),…,𝜎(𝑛)) is a permutation of (1,2,…,𝑛), such that ℎ_{𝜎(𝑗−1)}≥ℎ_𝜎(𝑗) for all 𝑗=2,3,…,𝑛. Then HTFOWGBM^𝑝,𝑞ℎ₁,ℎ₂,…,ℎ_𝑛=1𝑝+𝑞^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑝ℎ^𝑤𝑖_𝜎(𝑖)⊕𝑞ℎ^𝑤𝑗_𝜎𝑗^{1/𝑛(𝑛−1)},HTFOWBM^𝑝,𝑞ℎ₁,ℎ₂,…,ℎ_𝑛=1𝑛(𝑛−1)^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑤_𝑖ℎ^𝑝_𝜎(𝑖)⊗𝑤_𝑗ℎ^𝑞_𝜎𝑗^{1/(𝑝+𝑞)} (24) 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 ℎ_𝑖(𝑖=1,2,…,𝑛) and let 𝑝=1, 𝑞=0; then HTFOWGBM^1,0(ℎ₁,ℎ₂,…,ℎ_𝑛)=⨂^𝑛_𝑖=1(ℎ^𝑤𝑖_𝜎(𝑖))^1/𝑛=(HTFOWG(ℎ₁,ℎ₂,…,ℎ_𝑛))^1/𝑛 and HTFOWBM^1,0(ℎ₁,ℎ₂,…,ℎ_𝑛)=(1/𝑛)⨁^𝑛_𝑖=1(𝑤_𝑖ℎ_𝜎(𝑖))=(1/𝑛)HTFOWA(ℎ₁,ℎ₂,…,ℎ_𝑛). If ℎ₁≥ℎ₂≥⋯≥ℎ_𝑛, 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 ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of HTFEs, 𝑝,𝑞≥0, and 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 the associated weight vector such that 𝑤_𝑖>0 and ∑^𝑛_𝑖=1𝑤_𝑖=1. Let ℎ_𝜎(𝑗) be the 𝑗th largest element of the hesitant triangular fuzzy arguments (ℎ_𝑗=(ℎ_𝑗)^𝑛𝑤𝑗,𝑗=1,2,…,𝑛). Then, one calls HTFHGBM^𝑝,𝑞ℎ₁,ℎ₂,…,ℎ_𝑛=1𝑝+𝑞^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑝ℎ^𝑤𝑖_𝜎(𝑖)⊕𝑞ℎ^𝑤𝑗_𝜎𝑗^{1/𝑛(𝑛−1)} (25) the hesitant triangular fuzzy hybrid geometric Bonferroni mean (HTFHGBM).

Definition 31. Let ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of HTFEs, 𝑝,𝑞≥0, and 𝑤=(𝑤₁,𝑤₂,…,𝑤_𝑛)^𝑇 the associated weight vector such that 𝑤_𝑖>0 and ∑^𝑛_𝑖=1𝑤_𝑖=1. Let ℎ_𝜎(𝑗) be the 𝑗th largest element of the hesitant triangular fuzzy arguments (ℎ_𝑗=(𝑛𝑤_𝑗)ℎ_𝑗,𝑗=1,2,…,𝑛). Then, one calls HTFHBM^𝑝,𝑞ℎ₁,ℎ₂,…,ℎ_𝑛=1𝑛(𝑛−1)^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝑤_𝑖ℎ^𝑝_𝜎(𝑖)⊗𝑤_𝑗ℎ^𝑞_𝜎𝑗^{1/(𝑝+𝑞)} (26) the hesitant triangular fuzzy hybrid Bonferroni mean (HTFHBM).

Remark 32. If there is only one triangular fuzzy value in each ℎ_𝑖(𝑖=1,2,…,𝑛) and letting 𝑝=1,𝑞=0, then HTFHGBM^1,0(ℎ₁,ℎ₂,…,ℎ_𝑛)=⨂^𝑛_𝑖=1(ℎ^𝑤𝑖_𝜎(𝑖))^1/𝑛=(HTFHG(ℎ₁,ℎ₂,…,ℎ_𝑛))^1/𝑛, HTFHBM^1,0(ℎ₁,ℎ₂,…,ℎ_𝑛)=(1/𝑛)(⨁^𝑛_𝑖=1(𝑤_𝑖ℎ_𝜎(𝑖)))=(1/𝑛)HTFHA(ℎ₁,ℎ₂,…,ℎ_𝑛).

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 ℎ_𝑖(𝑖=1,2,…,𝑛) be a collection of HTFEs on 𝑋, 𝜇 a fuzzy measure on 𝑋, and 𝑝,𝑞≥0. Then, one calls HTFCOBM_𝜇^𝑝,𝑞ℎ₁,ℎ₂,…,ℎ_𝑛=1𝑛(𝑛−1)^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝜇𝐴_𝜎(𝑖)−𝜇𝐴_{𝜎(𝑖−1)}ℎ^𝑝_𝜎(𝑖)⊗𝜇𝐴_𝜎(𝑗)−𝜇𝐴_{𝜎(𝑗−1)}ℎ^𝑞_𝜎𝑗^{1/(𝑝+𝑞)} (27) the hesitant triangular fuzzy Choquet ordered Bonferroni mean (HTFCOBM), where (𝜎(1),𝜎(2),…,𝜎(𝑛)) is a permutation of (1,2,…,𝑛), such that ℎ_{𝜎(𝑗−1)}≥ℎ_𝜎(𝑗) for all 𝑗=2,3,…,𝑛, 𝐴_𝜎(𝑘)={𝑥_𝜎(𝑗)∣𝑗≤𝑘}, for 𝑘≥1, and 𝐴_𝜎(0)=𝜙.

Remark 34. If 𝜇({𝑥_𝜎(𝑗)})=𝜇({𝐴_𝜎(𝑗)})−𝜇({𝐴_{𝜎(𝑗−1)}}), 𝑗=1,2,…,𝑛, then HTFCOBM reduces to HTFWBM. Let 𝑤_𝑗=𝜇({𝐴_𝜎(𝑗)})−𝜇({𝐴_{𝜎(𝑗−1)}}), 𝑗=1,2,…,𝑛, then HTFCOBM reduces to HTFOWBM. In addition, suppose there is only one triangular fuzzy value in each ℎ_𝑖(𝑖=1,2,…,𝑛) and let 𝑝=1, 𝑞=0, then HTFCOBM_𝜇^1,0(ℎ₁,ℎ₂,…,ℎ_𝑛)=(1/𝑛)(⨁^𝑛_𝑖=1(𝜇(𝐴_𝜎(𝑖))−𝜇(𝐴_{𝜎(𝑖−1)})ℎ_𝜎(𝑖))=(1/𝑛)HTFCOA_𝜇(ℎ₁,ℎ₂,…,ℎ_𝑛). 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 𝐴={𝐴₁,𝐴₂,…,𝐴_𝑚} be a set of 𝑚 alternatives and 𝐺={𝐺₁,𝐺₂,…,𝐺_𝑛} a set of 𝑛 attributes. If the decision-makers provide values for the alternative 𝐴_𝑖 under the attribute 𝐺_𝑗 with anonymity, these values can be considered as a hesitant triangular fuzzy element ℎ_𝑖𝑗. In the case where two decision-makers provide the same value, the value emerges only once in ℎ_𝑖𝑗. Suppose that the decision matrix 𝐻=(ℎ_𝑖𝑗)_𝑚×𝑛 is the hesitant triangular fuzzy decision matrix, where ℎ_𝑖𝑗(𝑖=1,2,…,𝑚,𝑗=1,2,…,𝑛) 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 𝜇 of attributes of 𝐺 and attributes sets of 𝐺.

Step 2. We utilize the decision information given in matrix 𝐻 and the HTFCOBM operator ℎ_𝑘=HTFCOBM_𝜇^𝑝,𝑞ℎ_𝑘1,ℎ_𝑘2,…,ℎ_𝑘𝑛=1𝑛(𝑛−1)^𝑛_{𝑖,𝑗=1,𝑖≠𝑗}𝜇𝐴_𝜎(𝑖)−𝜇𝐴_{𝜎(𝑖−1)}ℎ^𝑝_{𝜎()𝑘𝑖}⊗𝜇𝐴_𝜎(𝑗)−𝜇𝐴_{𝜎(𝑗−1)}ℎ^𝑞_𝜎𝑘𝑗^{1/(𝑝+𝑞)} (28) to derive the overall preference values ℎ_𝑘(𝑘=1,2,…,𝑚) of the alternative 𝐴_𝑘.

Step 3. Calculate the scores 𝑆(ℎ_𝑘)(𝑘=1,2,…,𝑚) of the overall hesitant triangular fuzzy values ℎ_𝑘 by Definition 6.

Step 4. Compare each 𝑆(ℎ_𝑖) with all the 𝑆(ℎ_𝑗)(𝑖,𝑗=1,2,…,𝑚) by Definition 4. For convenience, we let 𝑝_𝑖𝑗=𝑝(𝑆(ℎ_𝑖)≥𝑆(ℎ_𝑗)); then we develop a complementary matrix as 𝑃=(𝑝_𝑖𝑗)_𝑚×𝑚, where 𝑝_𝑖𝑗≥0,𝑝_𝑖𝑗+𝑝_𝑗𝑖=1,𝑝_𝑖𝑖=0.5,𝑖,𝑗=1,2,…,𝑚. Summing all the elements in each line of matrix 𝑃, we have 𝑝_𝑖=∑^𝑚_𝑗=1𝑝_𝑖𝑗,𝑖=1,2,…,𝑚.

Step 5. Rank all the alternatives 𝐴_𝑖(𝑖=1,2,…,𝑚) in accordance with the values of 𝑝_𝑖 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 𝜇, 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 𝑝,𝑞 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 𝐴_𝑖(𝑖=1,2,3,4,5) and we want to select the best one. Four attributes are selected by experts to evaluate the theses: (1) 𝐺₁ is the language of a thesis; (2) 𝐺₂ is the innovation; (3) 𝐺₃ is the rigor; (4) 𝐺₄ 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 𝐴_𝑖(𝑖=1,2,3,4,5) under the above four attributes in anonymity and the decision matrix 𝐻=(ℎ_𝑖𝑗)_5×4 is presented in Table 1, where ℎ_𝑖𝑗(𝑖=1,2,3,4,5,𝑗=1,2,3,4) 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 𝐺_𝑗(𝑗=1,2,…,4) and attribute sets of 𝐺 are as follows: 𝜇(𝐺₁)=0.30, 𝜇(𝐺₂)=0.35, 𝜇(𝐺₃)=0.30, 𝜇(𝐺₄)=0.22, 𝜇(𝐺₁,𝐺₂)=0.70, 𝜇(𝐺₁,𝐺₃)=0.60, 𝜇(𝐺₁,𝐺₄)=0.55, 𝜇(𝐺₂,𝐺₃)=0.50, 𝜇(𝐺₂,𝐺₄)=0.45, 𝜇(𝐺₃,𝐺₄)=0.40, 𝜇(𝐺₁,𝐺₂,𝐺₃)=0.82, 𝜇(𝐺₁,𝐺₂,𝐺₄)=0.87, 𝜇(𝐺₁,𝐺₃,𝐺₄)=0.75, 𝜇(𝐺₂,𝐺₃,𝐺₄)=0.60, and 𝜇(𝐺₁,𝐺₂,𝐺₃,𝐺₄)=1.00.

Table 1 

Hesitant triangular fuzzy decision matrix 𝐻.

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 𝐻 and the HTFCOBM operator (here, we take 𝑝=𝑞=1) to obtain the overall preference values ℎ_𝑖 of the thesis 𝐴_𝑖(𝑖=1,2,3,4,5). Due to the large amount of data, we omitted these results here. When assigning different values to the parameters 𝑝 and 𝑞, we can obtain different results. Please see Table 2.

Table 2 

Scores for alternatives obtained by HTFCOBM operator.

Step 2. Calculate the scores 𝑆(ℎ_𝑖)(𝑖=1,2,3,4,5) of the overall hesitant triangular fuzzy preference values ℎ_𝑖 by Definition 6: 𝑆(ℎ₁)=(0.096,0.134,0.211),𝑆(ℎ₂)=(0.132,0.180,0.261),𝑆(ℎ₃)= (0.083,0.142,0.209),𝑆(ℎ₄)=(0.106,0.180,0.262),𝑆(ℎ₅)=(0.121,0.181,0.287).

Step 3. Comparing each 𝑆(ℎ_𝑖) with all the 𝑆(ℎ_𝑗)(𝑖,𝑗=1,2,…,5) by Definition 4, then we develop a complementary matrix as 𝑃=(𝑝_𝑖𝑗)_5×5. Summing all the elements in each line of the matrix 𝑃, we have 𝑝₁=1.478,𝑝₂=3.277,𝑝₃=1.558,𝑝₄=2.924,𝑝₅=3.263.

Step 4. Rank all the alternatives 𝐴_𝑖(𝑖=1,2,…,𝑚) in accordance with the values of 𝑝_𝑖: 𝐴₂≻𝐴₅≻𝐴₄≻𝐴₃≻𝐴₁. Note that ≻ means “preferred to.” Thus, the best thesis is 𝐴₂.

6.2. Discussion

From Table 2, we find that the values obtained by the HTFCOBM operator change as the parameters 𝑝,𝑞 vary. Moreover, the rankings are different when we choose different values of 𝑝,𝑞. As the two parameters are symmetrical, we can fix one of them and change the other. Here, we set 𝑞=2, for example. The trends are shown in Figure 1 as the parameter 𝑝 ranges from 0 to 18. From Figure 1, it can be clearly seen that(1)when 𝑝∈(0,2.3898], the ranking of the five theses is 𝐴₂≻𝐴₅≻𝐴₄≻𝐴₃≻𝐴₁;(2)when 𝑝∈(2.3898,4.46315], the ranking of the five theses is 𝐴₂≻𝐴₄≻𝐴₅≻𝐴₃≻𝐴₁;(3)when 𝑝∈(4.46315,5.77833], the ranking of the five theses is 𝐴₂≻𝐴₄≻𝐴₅≻𝐴₁≻𝐴₃;(4)when 𝑝∈(5.77833,18], the ranking of the five theses is 𝐴₄≻𝐴₂≻𝐴₅≻𝐴₁≻𝐴₃.

Figure 1 

Scores for theses obtained by HTFCOBM operator.

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.

Table 3 

Scores for theses obtained by the exsting operators. (Let 𝑤=(0.3,0.4,0.12,0.18)^𝑇).

Table 4 

Scores for theses obtained by new operators based on BM or GBM.

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 HTFWGBM^1,0ℎ₃₁,ℎ₃₂,ℎ₃₃,,ℎ₃₄=HTFWGℎ₃₁,ℎ₃₂,ℎ₃₃,,ℎ₃₄^1/4. (29) 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 HTFOWBM^1,1ℎ₃₁,ℎ₃₂,ℎ₃₃,,ℎ₃₄=HTFWBM^1,1ℎ₃₁,ℎ₃₂,ℎ₃₃,,ℎ₃₄. (30) 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, 𝜇𝐺₁+𝜇𝐺₂+𝜇𝐺₃+𝜇𝐺₄𝐺=0.30+0.35+0.30+0.22>1=𝜇₁,𝐺₂,𝐺₃,𝐺₄ (31) 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 𝑝,𝑞, 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).