#### Abstract

Dual hesitant fuzzy geometric Bonferroni mean is defined for dual hesitant fuzzy sets. Different properties of dual hesitant fuzzy geometric Bonferroni mean are discussed. Some special cases are studied in detail for dual hesitant fuzzy geometric Bonferroni mean. In addition, dual hesitant fuzzy weighted geometric Bonferroni mean and dual hesitant fuzzy Choquet geometric Bonferroni mean are proposed. A multicriteria decision-making method is discussed to find the best alternative among different alternatives by using proposed aggregated operators and an illustrated example is also given to understand our proposal.

#### 1. Introduction

Fuzzy sets () initiated by Zadeh [1] are great invention in the field of sciences. The idea of fuzzy sets received great intension for handling uncertainty and vagueness situations. Often the information taken from human preferences has vagueness. Bellman and Zadeh succeeded to develop a useful technique in decision making for these situations by the help of fuzzy sets theory [2]. Due to use of fuzzy sets as the effective tools to collect and represent arguments in different fields like [3–11], several well-known extensions of fuzzy sets have been developed including interval-valued fuzzy sets [12], intuitionistic fuzzy sets [13], interval-valued intuitionistic fuzzy sets [4], hesitant fuzzy set [14], and dual hesitant fuzzy set [15].

In , membership degree of an element is taken as , a single value, but in real life, the degree of nonmembership of is not equal to say because there may exist some hesitation degree. Due to that, intuitionistic fuzzy set which is a generalization of fuzzy sets theory was suggested by Atanassov [13, 16]. integrates the membership degree of hesitation defined as where . as the generalization of motivates the researchers to apply it in different real life applications. Atanassov [16, 17] showed that the information and semantic illustration of intuitionistic fuzzy set become more significant, practical, and applicable due to its degree of belongingness, degree of nonbelongingness, and the hesitation boundary. is a useful representation of uncertainty and ambiguity of an entity and hence is able to be used as a great instrument to express data information under a range of different fuzzy situations. Torra [14] extended the concept of fuzzy sets to hesitant fuzzy sets . In , a decision maker tries to manage those situations where hesitation involves in decision to provide membership value for particular element [18–23]. Zhu et al. initiated to merge both and and introduced the idea of dual hesitant fuzzy sets [15].

Multicriteria decision making is the general observable routine in daily life, which is to select the most suitable selection that possibly takes place from several likely options or to obtain their position by aggregating the performances of each choice under several criteria. Aggregation is a procedure in which different available options are aggregated with different method and return a single value [24, 25]. Dual hesitant fuzzy sets have ability to deal with the circumstances when the assessment of an alternative under each condition is corresponded to several possible values for membership and nonmembership for the same element [15]. In existing literature there is very fewer work existing on aggregation operator for ; particularly no work exists when arguments interrelated with each other. This inadequacy motivated us to develop some aggregative operators for based on geometric Bonferroni mean and Choquet integral. Geometric Bonferroni mean has the properties to capture the interrelationships among arguments [26]. The Choquet integral also is an important tool to consider the correlations among arguments [27]. Both of these operators have ability to aggregate all arguments when they interrelated with each other.

In this research article, we purposed some aggregation operators for as dual hesitant fuzzy geometric Bonferroni mean and dual hesitant fuzzy weighted geometric Bonferroni mean . Further in this research, Choquet integral with geometric Bonferroni mean is used to introduce aggregation operators for called dual hesitant fuzzy Choquet geometric Bonferroni mean

The rest of the paper is organized as follows: Section 2 briefly reviewed some basic concepts including definition of fuzzy set, its extension, and aggregated operators defined on these generalized fuzzy sets. Section 3 defined a ranking method for dual hesitant fuzzy sets and also defined some operational laws for manipulation of . Section 4 defined and discussed different properties of . Section 5 is devoted to discussion on Choquet integral (CI) based geometric Bonferroni mean in dual hesitant fuzzy environment. In Section 6, a multicriteria decision-making approach is discussed based on and an illustrated example is given to understand our method. The research article finished in Section 7 with some concluding remarks.

#### 2. Preliminaries

This section is devoted to some significant basic theories which are important to understand the article.

*Definition 1 (see [28]). *Let and for all Then Bonferroni geometric mean (BGM) is defined as

*Definition 2 (see [1]). *Let be a set of crisp elements, a* fuzzy set * on is a function for any , and the value is said to be the degree of membership of in .

*Definition 3 (see [13]). *Let be a nonempty set; an intuitionistic fuzzy set on is combination of functions and , where and represent the degrees of membership and nonmembership for every element and must satisfy the condition

Let , , and be elements of intuitionistic fuzzy set , then following basic operations introduced by Xu and Yager [29] hold:(1), ;(2), ;(3);(4).

Based on these operation laws intuitionistic fuzzy Bonferroni mean and intuitionistic fuzzy geometric Bonferroni mean are defined as follows.

*Definition 4 (see [11]). *Let for all be the elements of intuitionistic fuzzy set for any , ,

then is said to be intuitionistic fuzzy Bonferroni mean .

*Definition 5 (see [30]). *Let for all be the elements of intuitionistic fuzzy set for any , ,

then is said to be intuitionistic fuzzy Bonferroni Geometric mean .

Torra [14] proposed , which is a more general fuzzy set and permits the membership to include a set of possible values.

*Definition 6 (see [14]). *Let be a fixed set; a hesitant fuzzy set on is defined in terms of a function that when applied to it returns to a finite subset of

To be easily understood, Xia and Xu [31] express the by a mathematical symbol: where is a set of some values in , denoting the possible membership degrees of the element to the set , and called a hesitant fuzzy element .

Let , , and be elements of hesitant fuzzy set , then following basic operations introduced by Xia et al. [28] hold:(1), ;(2);(3);(4).

Based on these operational laws hesitant fuzzy geometric Bonferroni mean is defined as follows.

*Definition 7 (see [32]). *Let for all be a collection of . For any , then hesitant fuzzy geometric Bonferroni mean is defined as

In 2012, Zhu, et al. introduced dual hesitant fuzzy set which has both membership and nonmembership degree of an element in hesitant environment [15]. In 2014, Beg and Rashid [5] further discussed dual hesitant fuzzy set with new name intuitionistic fuzzy hesitant set and defined TOPSIS technique.

*Definition 8 (see [15]). *Let be a nonempty set. A dual hesitant fuzzy set on is defined as pair of functions and , which can return to the subsets of and is represented as For every element , the condition and must be satisfied. is called an dual hesitant fuzzy element , where is the possible membership values while is the possible nonmembership values of

#### 3. Manipulation with Dual Hesitant Fuzzy Elements

Ranking is a basic necessity of decision making in any fuzzy environment; therefore, dual hesitant fuzzy numbers must be ranked before any action is taken by a decision maker. To find order between two , we defined accuracy and similarity function as follows.

*Definition 9. *Let be an where and , then an accuracy function for is defined as

*Definition 10. *If is where and , then a similarity function for is defined as

Order between two and based on (7) and (8) is defined as follows.

*Definition 11. *Let and be two where and , then order between and is as follows:(1)if then ;(2)if and then ;(3)if and then .

*Definition 12. *Let and be collection of , where , and for and with condition andthen the following operations are hold

(1) (2) (3) (4)

Theorem 13. *Let and be and , then*(i)*;*(ii)*.*

Theorem 14. *Let and be two collections of , if for any , one has andand if for all and for all then one has *

*Proof. *As and , we havetake similarly,Let and then by Definition 12 and (18)−(23), we have with and henceis the required result.

Theorem 15. *Let be a collection of and for any , one has **andif and for with and then*

*Proof. *Since and thenandasaccording to Definition 12 and (28)−(32) we have which is required.

*Remark 16. *Let be a collection of then for any we have

#### 4. Dual Hesitant Fuzzy Geometric Bonferroni Operators

In multicriteria decision making, the qualifications of an option under a certain condition may be characterized by many potential options. To aggregate all the potential options of an alternative under some criteria, we presented a generalization of the in dual hesitant fuzzy environments, which is defined as follows.

*Definition 17. *Let , and be the collection of , then dual hesitant fuzzy geometric Bonferroni mean is defined aswith the help of operational laws (11)-(14), defined on , we further derive the following results.

Theorem 18. *Let and be the collection of . Then is an and *

where

*Proof. *From (11)−(14) and Theorem 13 we see that is also an . By (35) and (37) we haveTheorem 13 applies on (38); we obtain by (14) from (12) using (11), we havethe proof is completed.

*Example 19. *Let us consider three , , , and If , then

Theorem 20. *Let be the collection of , then by theorem one has the following results:*(1)*If then *(2)*If then *(3)*If then *

Next we discussed some special cases of by replacing the parameters and as follows.

*Case 1. *If then by (35) we have which is called generalized dual hesitant fuzzy geometric Bonferroni mean (GDHFGBM).

*Case 2. *If and then (35) becomes which is called dual hesitant fuzzy unit geometric Bonferroni mean.

*Case 3. *If and , is called dual hesitant fuzzy square geometric Bonferroni mean.

*Case 4. *If and , let then (35) reduces to the following: which is called an dual hesitant fuzzy interrelated square geometric Bonferroni mean.

Corollary 21. *If , such that for all , then for any is *

where and

*Proof. *From (11)-(14) and Theorem 13 we see that is also an Let be total number of and be the collection of all , then the total number of elements in istakethen

Corollary 22 (monotonicity). *Let and be two collections of ; also , , and with the conditions and , then .*

*Proof. *By (24) and (42) we haveby Definition 12 and (56) and (57) we have hence