Complexity

Volume 2017, Article ID 5937376, 16 pages

https://doi.org/10.1155/2017/5937376

## Some Generalized Pythagorean Fuzzy Bonferroni Mean Aggregation Operators with Their Application to Multiattribute Group Decision-Making

^{1}School of Economics and Management, Beijing Jiaotong University, Beijing 100044, China^{2}School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing 100044, China^{3}Department of Industrial Engineering, Tsinghua University, Beijing 100084, China

Correspondence should be addressed to Xiaomin Zhu; nc.ude.utjb@uhzmx

Received 25 April 2017; Accepted 1 June 2017; Published 1 August 2017

Academic Editor: Jurgita Antucheviciene

Copyright © 2017 Runtong Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The Pythagorean fuzzy set as an extension of the intuitionistic fuzzy set characterized by membership and nonmembership degrees has been introduced recently. Accordingly, the square sum of the membership and nonmembership degrees is a maximum of one. The Pythagorean fuzzy set has been previously applied to multiattribute group decision-making. This study develops a few aggregation operators for fusing the Pythagorean fuzzy information, and a novel approach to decision-making is introduced based on the proposed operators. First, we extend the generalized Bonferroni mean to the Pythagorean fuzzy environment and introduce the generalized Pythagorean fuzzy Bonferroni mean and the generalized Pythagorean fuzzy Bonferroni geometric mean. Second, a new generalization of the Bonferroni mean, namely, the dual generalized Bonferroni mean, is proposed by considering the shortcomings of the generalized Bonferroni mean. Furthermore, we investigate the dual generalized Bonferroni mean in the Pythagorean fuzzy sets and introduce the dual generalized Pythagorean fuzzy Bonferroni mean and dual generalized Pythagorean fuzzy Bonferroni geometric mean. Third, a novel approach to multiattribute group decision-making based on proposed operators is proposed. Lastly, a numerical instance is provided to illustrate the validity of the new approach.

#### 1. Introduction

Decision-making is a common and significant activity in daily life. In the past decades, decision-making problems in real life have become increasingly complicated because of the increasing complexity in economic and social management. Given the fuzziness and vagueness in decision-making, crisp numbers are inadequate and insufficient for managing real decision-making problems. In 1965, Zadeh introduced the concept of fuzzy set (FS) [1], which is an effective tool in handling fuzziness and uncertainty. However, FS only has a membership degree, which is unsuitable in managing several real decision-making problems. Atanassov [2] introduced the intuitionistic fuzzy set (IFS), which simultaneously has membership and nonmembership degrees, due to the shortcomings of FS. A few achievements on IFS have been reported [3, 4]. Mao et al. [5] introduced a few new cross-entropy and entropy measures for IFSs and applied them to decision-making. Liu and Teng [6] introduced the normal intuitionistic fuzzy numbers and several new normal intuitionistic fuzzy aggregation operators and applied them to multiattribute group decision-making (MAGDM). Lakshmana et al. [7] proposed a total order on the entire class of intuitionistic fuzzy numbers using upper, lower dense sequence in the interval . Lakshmana et al. [8] introduced a new principle for ordering trapezoidal intuitionistic fuzzy numbers. P. Liu and X. Liu [9] introduced the linguistic intuitionistic fuzzy set and a few linguistic intuitionistic fuzzy power Bonferroni mean (BM) aggregation operators by combining IFS and the linguistic terms set and applied them to MAGDM. Liu et al. [10] introduced the interval-valued intuitionistic fuzzy ordered weighted cosine similarity measure by combining the interval-valued intuitionistic fuzzy cosine similarity measure with the generalized ordered weighted averaging operator. Liu [11] used the Hamacher operations as basis to develop several new aggregation operators to fuse the interval-valued intuitionistic fuzzy information.

IFS is a powerful tool in decision-making. An extension of IFS, which is called the neutrosophic set [12], was introduced in 1999 to effectively address several real decision-making problems. In recent years, a few neutrosophic aggregation operators have been introduced [13–18]. A new extension of IFS, namely, the Pythagorean fuzzy set (PFS) [19], has been developed. The difference between PFS and IFS is that the square sum of the membership and nonmembership degrees is a maximum of one in PFS, whereas the sum of the membership and nonmembership degrees is a maximum of one in IFS. Several studies have been conducted on PFSs. Gou et al. [20] developed a few Pythagorean fuzzy functions and studied their fundamental properties. Zhang and Xu [21] introduced several operations for the Pythagorean fuzzy numbers (PFNs) and extended the technique for order preference by similarity to ideal solution (TOPSIS) method to solve MAGDM problems with Pythagorean fuzzy information. Several Pythagorean fuzzy aggregation operators have been introduced because aggregation operators are vital in decision-making [22]. Yager and Abbasov [23] introduced a few Pythagorean fuzzy aggregation operators, such as the Pythagorean fuzzy weighted averaging (PFWA) operator and Pythagorean fuzzy weighted geometric (PFWG) operator. Ma and Xu [24] introduced new score and accuracy functions of PFNs and developed the symmetric Pythagorean fuzzy weighted averaging (SPFWA) operator and the symmetric Pythagorean fuzzy weighted geometric (SPFWG) operator. Zeng et al. [25] introduced the Pythagorean fuzzy ordered weighted averaging weighted average distance (PFOWAWAD) operator, from which a hybrid TOPSIS method was proposed for the Pythagorean fuzzy MAGDM problems. Garg [26] introduced the Pythagorean fuzzy Einstein operations and developed a few new Pythagorean fuzzy aggregation operators. Peng and Yuan [27] developed a series of Pythagorean fuzzy point operators. However, these aggregation operators cannot consider the correlations among PFNs. Therefore, Peng and Yang [28] developed several Choquet integral-based operators for the Pythagorean fuzzy information.

Several aggregation operators, such as the BM [29] and the Heronian mean (HM) [30], can capture the interrelationship between arguments. These operators have been successfully extended to IFSs [31–33] and hesitant FSs [34–36]. However, BM and HM can only consider the interrelationship between any two arguments. Beliakov et al. [37] introduced the generalized Bonferroni mean (GBM) to overcome the drawback of BM; GBM has also been extended to IFSs [38]. However, to the best of our knowledge, no research has been conducted on GBM in the Pythagorean fuzzy environment. Therefore, it is necessary to extend the GBM to the Pythagorean fuzzy environment. The shortcoming of GBM is that it can only consider the interrelationship among any three arguments. However, the correlations are ubiquitous among all arguments. To overcome the shortcoming of GBM, we introduce a few new extensions of BM, which can consider the interrelationship among all arguments. Therefore, the main objective of this study is to investigate GBM in PFSs. This research aims to develop several new GBM aggregation operators for PFNs and a new approach to MAGDM with Pythagorean fuzzy information.

The rest of this paper is organized as follows. Section 2 briefly reviews a few basic concepts. Section 3 extends GBM to PFSs and develops a few generalized Pythagorean fuzzy BM operators. Section 4 proposes and utilizes several new GBM operators to aggregate PFNs. Section 5 introduces a novel approach to MAGDM. Section 6 provides a numerical example to illustrate the approach. The final section summarizes this study.

#### 2. Basic Concepts

This section reviews a few notions, such as IFS, PFS, and GBM.

##### 2.1. IFS and PFS

In 1986, Atanassov [2] introduced IFS, which simultaneously has membership and nonmembership degrees.

*Definition 1 (see [2]). *Let be an ordinary fixed set. An IFS defined on is expressed as follows: where and represent the membership and nonmembership degrees, respectively, thereby satisfying , , and . For convenience, the pair is called an intuitionistic fuzzy number (IFN) [39], in which , , and . The hesitancy degree is denoted by .

In 2014, Yager [19] introduced PFS, which is a generalization of IFS.

*Definition 2 (see [19]). *Let be an ordinary fixed set; a PFS defined on is expressed as follows:where and are the membership and nonmembership degrees, respectively, thereby satisfying and . Thereafter, the indeterminacy degree is expressed by . Zhang and Xu [21] called the pair a PFN, which can be denoted by .

Peng and Yuan [27] introduced the comparison law for PFNs to compare two PFNs.

*Definition 3 (see [27]). *For any PFN , the score function of* p* is defined as . For any two PFNs, such as and , if , then ; if , then .

Zhang and Xu [21] introduced a few operations for PFNs.

*Definition 4 (see [21]). *Let , , and be any three PFNs and be a positive real number. Thereafter,(1),(2),(3), and(4).

##### 2.2. GBM

Beliakov et al. [37] introduced GBM, which can consider the correlations of any three aggregated arguments because the traditional BM can only determine the interrelationship between any two arguments. Nevertheless, Xia et al. [38] highlighted that the GBM introduced by Beliakov et al. [37] has a drawback. Therefore, Xia et al. [38] introduced a new form of GBM. In the new GBM, the weights of the arguments are also considered.

*Definition 5 (see [38]). *Let and be a collection of nonnegative crisp numbers with the weight vector being , thereby satisfying and . The generalized weighted BM (GWBM) is defined as follows:Xia et al. [38] also introduced the generalized weighted Bonferroni geometric mean (GWBGM).

*Definition 6 (see [38]). *Let and be a collection of nonnegative crisp numbers with the weight vector being , thereby satisfying and . Ifthen is called GWBGM.

#### 3. The Generalized Pythagorean Fuzzy Weighted Bonferroni Mean

This section extends GWBM and GWBGM to fuse the Pythagorean fuzzy information and proposes several new Pythagorean fuzzy aggregation operators.

*Definition 7. *Let and be a collection of PFNs with their weight vector being , thereby satisfying and . Ifthen is called the generalized Pythagorean fuzzy weighted Bonferroni mean (GPFWBM).

We can obtain the following theorem according to Definition 4.

Theorem 8. *Let and be a collection of PFNs. The aggregated value by GPFWBM is also a PFN and*

*Proof. *According to Definition 4, we can obtainThus,Thereafter,Furthermore,Therefore,Hence, (6) is maintained.

Thereafter,Thereafter,thereby completing the proof.

Moreover, GPFWBM has the following properties.

Theorem 9 (idempotency). *If are equal, that is, , then*

*Proof. *

Theorem 10 (monotonicity). *Let and be two collections of PFNs. If and holds for all i, then*

*Proof. *Let and . Given that , we can obtainTherefore,Thus, which means . Similarly, we can obtain . If because , then ; If and , then ; If and , then .Therefore, the proof of Theorem 10 is completed.

Theorem 11 (boundedness). *Let be a collection of PFNs. If and , then*

*Proof. *From Theorem 9, we can obtainFrom Theorem 10, we can obtainTherefore, .

Thereafter, we extend GWBGM to PFSs and introduce the generalized Pythagorean fuzzy weighted Bonferroni geometric mean (GPFWBGM).

*Definition 12. *Let and be a collection of PFNs with their weight vector being , thereby satisfying and . Ifthen is called GPFWBGM.

We can obtain the following theorem based on Definition 4.

Theorem 13. *Let and be a collection of PFNs. The aggregated value by GPFWBGM is also a PFN and *

*Proof. *Through Definition 4, we can obtainThereafter,Therefore,thus, Hence, (24) is maintained.

Thereafter, Therefore,thereby completing the proof.

Similar to GPFWBM, the GPFWBGM has the same properties. The proofs of these properties are similar to that of the properties of GPFWBM. Accordingly, the proofs are omitted to save space.

Theorem 14. *Let and be a collection of PFNs.*

*(1) Idempotency*. If are equal, that is, , then

*(2) Monotonicity*. Let be two collections of PFNs. If and holds for all* i*, then

*(3) Boundedness*. If and , then

#### 4. Dual Generalized Pythagorean Fuzzy Weighted BM

The primary advantage of BM is that it can determine the interrelationship between arguments. However, the traditional BM can only consider the correlations of any two aggregated arguments. Thereafter, Beliakov et al. [37] extended the traditional BM and introduced GBM, which can determine the correlations between any three aggregated arguments. Xia et al. [38] introduced GBWM and GBWGM given that the GBM introduced by Beliakov et al. [37] still has a few drawbacks. However, GBWM and GBWGM can only consider the interrelationship between any three aggregated arguments. We introduce a new generalization of the traditional BM because the correlations are ubiquitous among all arguments. The new generalization of the traditional BM is called the dual GBM (DGBM) to distinguish the new aggregation operator from the GBM introduced by Beliakov et al. [37] and Xia et al. [38]. Furthermore, we develop the dual generalized weighted BM (DGWBM) and dual generalized weighted Bonferroni geometric mean (DGWBGM) to consider the weights of the arguments.

*Definition 15. *Let be a collection of nonnegative crisp numbers with the weight vector being , thereby satisfying and . Ifwhere is the parameter vector with , then is called DGWBM.

Several special cases can be obtained given the change of the parameter vector.

() If , then we obtainwhich is the generalized weighted averaging operator.

() If , then we obtainwhich is the weighted BM.

() If , then we obtainwhich is the GWBM.

*Definition 16. *Let be a collection of nonnegative crisp numbers with the weight vector being , thereby satisfying and . Ifwhere is the parameter vector with , then is called DGWBGM.

Similar to the DGWBM, we can consider some special cases given the change of the parameter vector.

() If , then we obtainwhich is the generalized weighted geometric averaging operator.

() If , then we obtainwhich is the weighted Bonferroni geometric mean.

() If , thenwhich is the GWBGM.

We extend DGWBM and DGWBGM to PFSs, as well as introduce several new aggregation operators for fusing the Pythagorean fuzzy information.

*Definition 17. *Let be a collection of PFNs with their weight vector being , thereby satisfying and . Thereafter, the dual generalized Pythagorean fuzzy weighted Bonferroni mean (DGPFWBM) is defined aswhere is the parameter vector with .

We can derive the following theorem based on Definition 4.

Theorem 18. *Let be a collection of PFNs. Hence, the aggregated value by DGPFWBM is also PFN and*

*Proof. *Through Definition 4, we obtainTherefore,Thus,Therefore,Thus, (45) is maintained.

Thereafter,In addition,thereby completing the proof.

Moreover, DGPFWBM has the following properties.

Theorem 19 (monotonicity). *Let and be two collections of PFNs. If and holds for all , then*

*Proof. *Let and .

Given that , we obtainTherefore,thus, . Similarly, we can obtain . If because , then If and , then ; If and , then .Therefore, and the proof of Theorem 19 is completed.

Theorem 20 (boundedness). *Let be a collection of PFNs. If and , then*

*Proof. *According to Theorem 18, we can obtainAccording to Theorem 19, we can obtainEvidently, the DGPFWBM operator lacks the property of idempotency.

We extend DGBWGM to PFSs and introduce the dual generalized Pythagorean fuzzy weighted Bonferroni geometric mean (DGPFWBGM) operator.

*Definition 21. *Let be a collection of PFNs with their weight vector being , thereby satisfying and . Ifwhere is the parameter vector with ; then is called the DGPFWBGM

We obtain the following theorem based on Definition 4.

Theorem 22. *Let be a collection of PFNs. The aggregated value by the DGPFWBGM operator is also PFN and*

The proof of Theorem 22 is similar to that of Theorem 18; thus, such proof is omitted to save space.

Similar to DGPFWBM, we can obtain the following properties of DGPFWBGM. The proofs of these properties are likewise omitted to save space.

Theorem 23. *Let *