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Journal of Applied Mathematics
Volume 2012, Article ID 136254, 22 pages
http://dx.doi.org/10.1155/2012/136254
Research Article

Intuitionistic Fuzzy Normalized Weighted Bonferroni Mean and Its Application in Multicriteria Decision Making

1International Business School, Yunnan University of Finance and Economics, Kunming 650221, China
2School of Economics and Management, Southeast University, Nanjing 211189, China

Received 15 April 2012; Revised 28 June 2012; Accepted 10 July 2012

Academic Editor: Tak-Wah Lam

Copyright © 2012 Wei Zhou and Jian-min He. 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 Bonferroni mean (BM) was introduced by Bonferroni six decades ago but has been a hot research topic recently since its usefulness of the aggregation techniques. The desirable characteristic of the BM is its capability to capture the interrelationship between input arguments. However, the classical BM and GBM ignore the weight vector of aggregated arguments, the general weighted BM (WBM) has not the reducibility, and the revised generalized weighted BM (GWBM) cannot reflect the interrelationship between the individual criterion and other criteria. To deal with these issues, in this paper, we propose the normalized weighted Bonferroni mean (NWBM) and the generalized normalized weighted Bonferroni mean (GNWBM) and study their desirable properties, such as reducibility, idempotency, monotonicity, and boundedness. Furthermore, we investigate the NWBM and GNWBM operators under the intuitionistic fuzzy environment which is more common phenomenon in modern life and develop two new intuitionistic fuzzy aggregation operators based on the NWBM and GNWBM, that is, the intuitionistic fuzzy normalized weighted Bonferroni mean (IFNWBM) and the generalized intuitionistic fuzzy normalized weighted Bonferroni mean (GIFNWBM). Finally, based on the GIFNWBM, we propose an approach to multicriteria decision making under the intuitionistic fuzzy environment, and a practical example is provided to illustrate our results.

1. Introduction

Multicriteria decision making is the pervasive phenomenon in modern life, which is to select the best or optimal alternative from several alternatives or to get their ranking by aggregating the performances of each alternative under some criteria, in which the aggregation operators play an important role. As many different types of criteria relationships exist in the real world there is a need for many types of formal aggregation operations to enable the modeling of these numerous types of relationships. In response to this need a formal mathematical discipline called aggregation theory is emerging [14]. In this paper, we contribute to this theory by looking at the Bonferroni mean (BM), proposing the normalized weighted BM (NWBM) and the generalized normalized weighted BM (GNWBM) and developing the generalized intuitionistic fuzzy normalized weighted BM (GIFNWBM) and its application in multicriteria decision making.

Bonferroni [5] originally introduced a mean-type aggregation operator called the Bonferroni mean, which can provide for the aggregation lying between the max and min operators and logical “oring” and “anding” operators. A prominent characteristic of BM is that it cannot only consider the importance of each criterion but also reflect the interrelationship of the individual criterion. Recently, Yager [6] further studied the BM and provided an interpretation of BM as involving a product of each argument with the average of the other arguments, and where the BM was shown to be suitable for modeling various concepts, such as hard and soft partial conjunction and disjunction [7] and boundedness similar to k-intolerance [8, 9]. Furthermore, Yager [6] extends the BM replacing the simple average by other mean-type operators, such as the Choquet integral [10] and the ordered weighted averaging operator [11], as well as associates differing importance with the arguments. Mordelová and Rückschlossová [12] also investigated the generalizations of BM referred to as ABC-aggregation functions. Beliakov et al. [1] further extended the BM by considering the correlations of any three aggregated arguments instead of any two and proposed the generalized Bonferroni mean (GBM). Nevertheless, the arguments suitable to be aggregated by the BM and GBM can only take the forms of crisp numbers rather than any other types of arguments, which restrict the potential applications of the BM to more extensive areas. In the real world, due to the increasing complexity of the socioeconomic environment and the lack of knowledge and data, crisp data are sometimes unavailable. Thus, the input arguments may be more suitable with representation of fuzzy formats, such as fuzzy number [13], interval-valued fuzzy number [14], intuitionistic fuzzy value [15], interval-valued intuitionistic fuzzy value [16], and hesitant fuzzy element [17, 18]. Therefore, Xu and Yager [19] applied the BM to intuitionistic fuzzy environment and introduced the intuitionistic fuzzy Bonferroni mean (IFBM) and the intuitionistic fuzzy weighted Bonferroni mean (IFWBM), Xu and Chen [20] further applied the BM to interval-valued intuitionistic fuzzy environment and introduced the interval-valued intuitionistic fuzzy Bonferroni mean (IIFBM) and the interval-valued intuitionistic fuzzy weighted Bonferroni mean (IIFWBM).

It is noted that the BM, GBM, IFBM, and IIFBM ignore the weight vector of the aggregated arguments, although the IFWBM and the IIFWBM consider this issue, we cannot, respectively, obtain IFBM and IIFBM when all the weights of the aggregated arguments are the same, that is, these two operators have not reducibility, which seems to be counterintuitive. To deal with this issue, Xia et al. [21] proposed the revised BM and revised generalized weighted Bonferroni mean (GWBM), which take into the weight vector and reducibility and extended them to intuitionistic fuzzy environment. However, a question arises, that is, the revised BM and the GWBM just reflect the correlationship between the individual criterion and all criteria, which is not an interrelationship between the individual criterion and other criteria represented in the BM. Therefore, to further develop BM, we propose the normalized weighted BM (NWBM) and the generalized normalized weighted BM (GNWBM). The main advantage of the NWBM and the GNWBM is that they can not only consider weight vector and interrelationship of the individual criterion which is similar to the IFWBM and the IIFWBM but also have the reducibility like the GWBM. Based on the NWBM and GNWBM operators, we develop the intuitionistic fuzzy normalized weighted Bonferroni mean (IFNWBM) and the generalized intuitionistic fuzzy normalized weighted Bonferroni mean (GIFNWBM), on the basis of which an approach to multicriteria decision making is also proposed.

The remainder of this paper is organized as follows. We briefly review some basic concepts and operations of the IFV and BM, in Section 2. Section 3 proposes the NWBM and GNWBM operators and studies their desirable properties. Section 4, the IFNWBM operator is proposed, and then its corresponding generalized form is also given. A practical example is provided in Section 5 to demonstrate the application of the generalized intuitionistic fuzzy normalized weighted Bonferroni mean. The paper ends in Section 6 with concluding remarks.

2. Basic Concepts and Operations

In this section, we introduce some basic notions and operations related to the intuitionistic fuzzy value and the Bonferroni mean.

2.1. Intuitionistic Fuzzy Values

Definition 2.1 (see [15]). Let 𝑋=(𝑥1,𝑥2,,𝑥𝑛) be fixed. An intuitionistic fuzzy set (IFS) 𝐴 in 𝑋 can be defined as 𝑥𝐴=𝑖𝑥,𝜇𝑖𝑥,𝜈𝑖𝑥𝑖𝑋,(2.1) where 𝜇(𝑥𝑖)[0,1] and 𝜈(𝑥𝑖)[0,1] satisfy 0𝜇(𝑥𝑖)+𝜈(𝑥𝑖)1 for all 𝑥𝑖𝑋 and 𝜇(𝑥𝑖) and 𝜈(𝑥𝑖) are, respectively, called the degree of membership and the degree of nonmembership of the element 𝑥𝑖𝑋 to 𝐴.
Furthermore, 𝜋(𝑥𝑖)=1𝜇(𝑥𝑖)+𝜈(𝑥𝑖) is called the hesitation degree of 𝑥𝑖 to 𝐴, which represents the indeterminacy degree. For computational convenience, Xu [22] named the pair (𝜇𝛼,𝜈𝛼) an intuitionistic fuzzy value (IFV) denoted as 𝛼 with the conditions 0𝜇𝛼1,0𝜈𝛼1 and 0𝜇(𝑥𝑖)+𝜈(𝑥𝑖)1. The set of IFVs is denoted as Ω. To compare and calculate the IFVs, Chen and Tan [23] introduced the score function 𝑠(𝛼)=𝜇𝛼𝜈𝛼 to get the score value of 𝛼, and Hong and Choi [24] defined the accuracy function (𝛼)=𝜇𝛼+𝜈𝛼 to evaluate the accuracy degree of 𝛼. Based on the score function and the accuracy function, Xu and Yager [25] gave a total order relation between two IFVs 𝛼 and 𝛽, as follows:if 𝑠(𝛼)<𝑠(𝛽), then 𝛼<𝛽;if 𝑠(𝛼)=𝑠(𝛽), then
(i) if (𝛼)=(𝛽), then 𝛼=𝛽; (ii) if (𝛼)<(𝛽), then 𝛼<𝛽.

Definition 2.2 (see [22, 25]). Let 𝛼=(𝜇𝛼,𝜈𝛼),𝛼1=(𝜇𝛼1,𝜈𝛼1), and 𝛼2=(𝜇𝛼2,𝜈𝛼2) be three IFVs, then following operational laws are valid:(1)𝛼1𝛼2=(𝜇𝛼1+𝜇𝛼2𝜇𝛼1𝜇𝛼2,𝜈𝛼1𝜈𝛼2);(2)𝛼1𝛼2=(𝜇𝛼1𝜇𝛼2,𝜈𝛼1+𝜈𝛼2𝜈𝛼1𝜈𝛼2);(3)𝜆𝛼=(1(1𝜇𝛼)𝜆,𝜈𝛼𝜆),𝜆>0;(4)𝛼𝜆=(𝜇𝛼𝜆,1(1𝜈𝛼)𝜆),𝜆>0.

2.2. Bonferroni Means

The Bonferroni mean was originally introduced by Bonferroni [5] and intensively investigated by Yager [6], which was defined as follows.

Definition 2.3 (see [5]). Let 𝑝,𝑞0 and 𝑎𝑖(𝑖=1,2,,𝑛) be a collection of nonnegative numbers. If BM𝑝,𝑞𝑎1,𝑎2,,𝑎𝑛=1𝑛(𝑛1)𝑛𝑖,𝑗=1𝑖𝑗𝑎𝑝𝑖𝑎𝑞𝑗1/(𝑝+𝑞)(2.2) then BM𝑝,𝑞 is called the Bonferroni mean (BM).
One interpretation of the Bonferroni Mean is as a kind of combined “anding” and “averaging” operator [6]. Then, here we see that 𝑎𝑝𝑖𝑎𝑞𝑗 indicates the degree to which both criteria 𝐴𝑖 and 𝐴𝑗 are satisfied under the given conditions and the special case when 𝑝=𝑞=1. There exists another interesting way to view this aggregation operator and described as follows. BM𝑝,𝑞𝑎1,𝑎2,,𝑎𝑛=1𝑛𝑛𝑖=1𝑎𝑝𝑖1𝑛1𝑛𝑗=1𝑗𝑖𝑎𝑞𝑗1/(𝑝+𝑞).(2.3)
We see that the term (1/(𝑛1))𝑛𝑗=1,𝑗𝑖𝑎𝑞𝑗 is the power average satisfaction of all criteria except 𝐴𝑖. We will denote this as 𝑢𝑞𝑖. Thus BM𝑝,𝑞𝑎1,𝑎2,,𝑎𝑛=1𝑛𝑛𝑖=1𝑎𝑝𝑖𝑢𝑞𝑖1/(𝑝+𝑞).(2.4)
Here then 𝑢𝑞𝑖 is the power average satisfaction to all criteria except 𝐴𝑖 and 𝑎𝑝𝑖𝑢𝑞𝑖 represent the interrelationship between 𝐴𝑖 and other criteria 𝐴𝑗, which is also the prominent characteristic of the BM. Based on the BM, Beliakov et al. [1] further extended and generalized the BM to the generalized Bonferroni mean (GBM) by considering the correlations of any three aggregated arguments instead of any two.

Definition 2.4 (see [1]). Let 𝑝,𝑞,𝑟0 and 𝑎𝑖(𝑖=1,2,,𝑛) be a collection of nonnegative numbers. If GBM𝑝,𝑞,𝑟𝑎1,𝑎2,,𝑎𝑛=1𝑛(𝑛1)(𝑛2)𝑛𝑖,𝑗,𝑘=1𝑖𝑗𝑘𝑎𝑝𝑖𝑎𝑞𝑗𝑎𝑟𝑘1/(𝑝+𝑞+𝑟)(2.5) then GBM𝑝,𝑞,𝑟 is called the generalized Bonferroni mean (GBM).
It is obvious that the GBM reduces to the BM if 𝑟=0, and the GBM can represent the interrelationship of any three criteria. Here, we see that the term 1/(𝑛1)𝑛𝑗=1,𝑖𝑗(1/(𝑛2)𝑛𝑘=1,𝑘𝑖𝑗𝑎𝑞𝑗𝑎𝑟𝑘) is the power average satisfaction of all criteria correlationship except 𝐴𝑖, denote as 𝑣𝑖𝑞,𝑟. Thus GBM𝑝,𝑞,𝑟𝑎1,𝑎2,,𝑎𝑛=1𝑛𝑛𝑖=1𝑎𝑝𝑖𝑣𝑖𝑞,𝑟1/(𝑝+𝑞+𝑟).(2.6)
The above BM and GBM can only deal with the situation that the arguments are represented by real number but are invalid if the aggregation information is given in other forms, such as the IFV, which is a widely used technique to deal with uncertainty and vagueness. To deal with this issue, Xu and Yager [19] extended the BM to intuitionistic fuzzy environment and gave the following definition.

Definition 2.5 (see [19]). Let 𝑝,𝑞,𝑟0, and 𝛼𝑖(𝑖=1,2,,𝑛) be a collection of intuitionistic fuzzy values. The intuitionistic fuzzy Bonferroni mean (IFBM) and the intuitionistic fuzzy weighted Bonferroni mean (IFWBM) are, respectively, defined as IFBM𝑝,𝑞𝛼1,𝛼2,,𝛼𝑛=1𝑛(𝑛1)𝑛𝑖,𝑗=1,𝑖𝑗𝛼𝑝𝑖𝛼𝑞𝑗1/(𝑝+𝑞),IFWBM𝑝,𝑞𝛼1,𝛼2,,𝛼𝑛=1𝑛(𝑛1)𝑛𝑖,𝑗=1,𝑖𝑗𝑤𝑖𝛼𝑝𝑖𝑤𝑗𝛼𝑞𝑗1/(𝑝+𝑞).(2.7)
However, it is noted that the BM, GBM, and IFBM ignore the weight vector of the aggregated arguments, although the IFWBM considers this issue, we cannot obtain the IFBM when all the weights of the aggregated arguments are the same, that is, these IFWBM operator has not the reducibility, which seems to be counterintuitive. Therefore, to deal with these issues, Xia et al. [21] proposed the generalized weighted Bonferroni mean (GWBM) based on the GBM and described as follows.

Definition 2.6 (see [21]). Let 𝑝,𝑞0, and 𝑎𝑖(𝑖=1,2,,𝑛) be a collection of nonnegative numbers with the weight vector 𝑤=(𝑤1,𝑤2,,𝑤𝑛) such that 𝑤𝑖0 and 𝑛𝑖=1𝑤𝑖=1. If GWBM𝑝,𝑞,𝑟𝑎1,𝑎2,,𝑎𝑛=𝑛𝑖,𝑗,𝑘=1𝑤𝑖𝑤𝑗𝑤𝑘𝑎𝑝𝑖𝑎𝑞𝑗𝑎𝑟𝑘1/(𝑝+𝑞+𝑟)(2.8) then GWBM𝑝,𝑞,𝑟 is called the generalized weighted Bonferroni mean (GWBM).
If 𝑤=(1/𝑛,1/𝑛,,1/𝑛), then the GWBM reduces to the revised GBM, that is, GBM𝑝,𝑞,𝑟𝑎1,𝑎2,,𝑎𝑛=1𝑛3𝑛𝑖,𝑗,𝑘=1𝑎𝑝𝑖𝑎𝑞𝑗𝑎𝑟𝑘1/(𝑝+𝑞+𝑟),(2.9) which reflects the reducibility.
Similarly, we can transform the GWBM into the following form: GBM𝑝,𝑞,𝑟𝑎1,𝑎2,,𝑎𝑛=1𝑛𝑛𝑖=1𝑎𝑝𝑖1𝑛𝑛𝑗=1𝑎𝑞𝑗1𝑛𝑛𝑘=1𝑎𝑟𝑘1/(𝑝+𝑞+𝑟).(2.10)
However, a question arises, that is, the GWBM just considers the whole correlationship between the criterion 𝑎𝑖 and all criteria 𝑛𝑗=1𝑎𝑞𝑗𝑛𝑘=1𝑎𝑟𝑘 and cannot reflect the interrelationship between the individual criterion 𝑎𝑖 and other criteria 𝑣𝑖𝑞,𝑟 which is the main advantage of the BM. To further overcome this drawback, we propose the following NWBM and GNWBM operators.

3. Normalized Weighted BM and Generalized Normalized Weighted BM

The classical BM and GBM ignore the weight vector of aggregated arguments, the general weighted BMs (WBM) have not reducibility, and the revised generalized BM (GWBM) cannot reflect the interrelationship between the individual criterion and other criteria. To deal with these issues, in the following subsections, we propose the normalized weighted versions of BM and GBM, that is, the normalized weighted BM (NWBM) and the generalized normalized weighted BM (GNWBM).

3.1. NWBM

Definition 3.1. Let 𝑝,𝑞0 and 𝑎𝑖(𝑖=1,2,,𝑛) be a collection of nonnegative numbers with the weight vector 𝑤=(𝑤1,𝑤2,,𝑤𝑛) such that 𝑤𝑖0, and 𝑛𝑖=1𝑤𝑖=1. If NWBM𝑝,𝑞𝑎1,𝑎2,,𝑎𝑛=𝑛𝑖,𝑗=1𝑖𝑗𝑤𝑖𝑤𝑗1𝑤𝑖𝑎𝑝𝑖𝑎𝑞𝑗1/(𝑝+𝑞),(3.1) then NWBM𝑝,𝑞 is called the normalized weighted Bonferroni mean (NWBM).
Then, we can transform the NWBM into the interrelationship NWBM form as follows: NWBM𝑝,𝑞𝑎1,𝑎2,,𝑎𝑛=𝑛𝑖=1𝑤𝑖𝑎𝑝𝑖𝑛𝑗=1𝑗𝑖𝑤𝑗1𝑤𝑖𝑎𝑞𝑗1/(𝑝+𝑞).(3.2)
We see that the term 𝑛𝑗=1,𝑗𝑖(𝑤𝑗/(1𝑤𝑖))𝑎𝑞𝑗 is the weighted power average satisfaction of all criteria except 𝐴𝑖 and 𝑛𝑗=1,𝑗𝑖𝑤𝑗/(1𝑤𝑖)=1. We denote the term as 𝑢𝑞𝑖. Thus NWBM𝑝,𝑞𝑎1,𝑎2,,𝑎𝑛=𝑛𝑖=1𝑤𝑖𝑎𝑝𝑖𝑢𝑞𝑖1/(𝑝+𝑞).(3.3)
Here then 𝑢𝑞𝑖 is the weighted power average satisfaction to all criteria except 𝐴𝑖, and NWBM𝑝,𝑞 represents the interrelationship between the individual criterion 𝑎𝑖 and other criteria 𝑎𝑗(𝑗𝑖) which is similar to the BM.

Moreover, the NWBM has the following properties.

Property 1 (Reducibility). Let 𝑝,𝑞0 and 𝑎𝑖(𝑖=1,2,,𝑛) be a collection of nonnegative numbers with the weight vector 𝑤=(1/𝑛,1/𝑛,1/𝑛), then NWBM𝑝,𝑞𝑎1,𝑎2,,𝑎𝑛=BM𝑝,𝑞𝑎1,𝑎2,,𝑎𝑛.(3.4)

Proof. Since 𝑤=(1/𝑛,1/𝑛,1/𝑛), then by Definition 3.1, we have NWBM𝑝,𝑞𝑎1,𝑎2,,𝑎𝑛=𝑛𝑖,𝑗=1𝑖𝑗𝑤𝑖𝑤𝑗1𝑤𝑖𝑎𝑝𝑖𝑎𝑞𝑗1/(𝑝+𝑞)=1𝑛(𝑛1)𝑛𝑖,𝑗=1𝑖𝑗𝑎𝑝𝑖𝑎𝑞𝑗1/(𝑝+𝑞)=BM𝑝,𝑞𝑎1,𝑎2,,𝑎𝑛,(3.5) which complete the proof of the property.

Property 2 (Idempotency). Let 𝑝,𝑞0 and 𝑎𝑖(𝑖=1,2,,𝑛) be a collection of nonnegative numbers with the weight vector 𝑤=(𝑤1,𝑤2,,𝑤𝑛), such that 𝑤𝑖0, and 𝑛𝑖=1𝑤𝑖=1. If all 𝑎𝑖(𝑖=1,2,,𝑛) are equal, that is, 𝑎𝑖=𝑎, for all 𝑖, then NWBM𝑝,𝑞𝑎1,𝑎2,,𝑎𝑛=𝑎.(3.6)

Proof. Since 𝑎𝑖=𝑎(𝑖=1,2,,𝑛), then NWBM𝑝,𝑞𝑎1,𝑎2,,𝑎𝑛=𝑛𝑖=1𝑤𝑖𝑎𝑝𝑛𝑗=1𝑗𝑖𝑤𝑗1𝑤𝑖𝑎𝑞1/(𝑝+𝑞)=𝑎𝑛𝑝+𝑞𝑖=1𝑤𝑖𝑛𝑗=1𝑗𝑖𝑤𝑗1𝑤𝑖1/(𝑝+𝑞)=𝑎,(3.7) which complete the proof of the property.

Property 3 (Monotonicity). Let 𝑝,𝑞0, 𝑎𝑖 and 𝑏𝑖(𝑖=1,2,,𝑛) be two collections of nonnegative numbers with the weight vector 𝑤=(𝑤1,𝑤2,,𝑤𝑛), such that 𝑤𝑖0 and 𝑛𝑖=1𝑤𝑖=1. If 𝑎𝑖𝑏𝑖, for all 𝑖, then NWBM𝑝,𝑞𝑎1,𝑎2,,𝑎𝑛NWBM𝑝,𝑞𝑏1,𝑏2,,𝑏𝑛.(3.8)

Proof. Since 𝑎𝑖𝑏𝑖 for all 𝑖, and 𝑝,𝑞0, then 𝑎𝑝𝑖𝑎𝑞𝑗𝑏𝑝𝑖𝑏𝑞𝑗(𝑖,𝑗=1,2,,𝑛),𝑛𝑖,𝑗=1𝑖𝑗𝑤𝑖𝑤𝑗1𝑤𝑖𝑎𝑝𝑖𝑎𝑞𝑗𝑛𝑖,𝑗=1𝑖𝑗𝑤𝑖𝑤𝑗1𝑤𝑖𝑏𝑝𝑖𝑏𝑞𝑗.(3.9) Therefore, NWBM𝑝,𝑞𝑎1,𝑎2,,𝑎𝑛NWBM𝑝,𝑞𝑏1,𝑏2,,𝑏𝑛,(3.10) which complete the proof.

Property 4 (Boundedness). Let 𝑝,𝑞0 and 𝑎𝑖(𝑖=1,2,,𝑛) be a collection of nonnegative numbers with the weight vector 𝑤=(𝑤1,𝑤2,,𝑤𝑛), such that 𝑤𝑖0, and 𝑛𝑖=1𝑤𝑖=1, then min𝑖𝑎𝑖NWBM𝑝,𝑞𝑎1,𝑎2,,𝑎𝑛max𝑖𝑎𝑖.(3.11)

Proof. By Property 2, we can get NWBM𝑝,𝑞min𝑖𝑎𝑖,min𝑖𝑎𝑖,,min𝑖𝑎𝑖=min𝑖𝑎𝑖,NWBM𝑝,𝑞max𝑖𝑎𝑖,max𝑖𝑎𝑖,,max𝑖𝑎𝑖=max𝑖𝑎𝑖.(3.12) Since min𝑖{𝑎𝑖}𝑎𝑖max𝑖{𝑎𝑖}(𝑖=1,2,,𝑛), then based on Property 3, we have min𝑖𝑎𝑖NWBM𝑝,𝑞𝑎1,𝑎2,,𝑎𝑛max𝑖𝑎𝑖,(3.13) which complete the proof of the theorem.

3.2. GNWMB

In this subsection, we further extend the NWBM to the generalized normalized weighted Bonferroni mean (GNWBM) by considering the correlation of any three aggregated arguments instead of any two based on the GBM

Definition 3.2. Let 𝑝,𝑞,𝑟0, and 𝑎𝑖(𝑖=1,2,,𝑛) be a collection of nonnegative numbers with the weight vector 𝑤=(𝑤1,𝑤2,,𝑤𝑛) such that 𝑤𝑖0, and 𝑛𝑖=1𝑤𝑖=1. If GNWBM𝑝,𝑞,𝑟𝑎1,𝑎2,,𝑎𝑛=𝑛𝑖,𝑗,𝑘=1𝑖𝑗𝑘𝑤𝑖𝑤𝑗𝑤𝑘1𝑤𝑖1𝑤𝑖𝑤𝑗𝑎𝑝𝑖𝑎𝑞𝑗𝑎𝑟𝑘1/(𝑝+𝑞+𝑟),(3.14) then GNWBM𝑝,𝑞,𝑟 is called the generalized normalized weighted Bonferroni mean (GNWBM).
Furthermore, we can transform the GNWBM into the interrelationship GNWBM form as follows: GNWBM𝑝,𝑞,𝑟𝑎1,𝑎2,,𝑎𝑛=𝑛𝑖=1𝑤𝑖𝑎𝑝𝑖𝑛𝑗=1𝑗𝑖𝑤𝑗1𝑤𝑖𝑎𝑞𝑗𝑛𝑘=1𝑘𝑖𝑗𝑤𝑘1𝑤𝑖𝑤𝑗𝑎𝑟𝑘1/(𝑝+𝑞+𝑟).(3.15)
We see that the term 𝑛𝑗=1,𝑗𝑖(𝑤𝑗/(1𝑤𝑖))𝑎𝑞𝑗 is the weighted power average satisfaction of all criteria except 𝐴𝑖, with 𝑛𝑗=1,𝑗𝑖(𝑤𝑗/(1𝑤𝑖))=1. The term 𝑛𝑘=1,𝑘𝑖𝑗(𝑤𝑘/(1𝑤𝑖𝑤𝑗))𝑎𝑟𝑘 is the weighted power average satisfaction of all criteria except 𝐴𝑖 and 𝐴𝑗, with 𝑛𝑘=1,𝑘𝑖𝑗(𝑤𝑘/(1𝑤𝑖𝑤𝑗))=1. Here then NWBM𝑝,𝑞 represents the interrelationship between any three aggregated arguments, which is similar to the GBM. Especially, if 𝑟0, then the GNWBM reduces to the NWBM.

Moreover, the GNWBM has the following properties.

Property 5 (Reducibility). Let 𝑝,𝑞,𝑟0 and 𝑎𝑖(𝑖=1,2,,𝑛) be a collection of nonnegative numbers with the weight vector 𝑤=(1/𝑛,1/𝑛,1/𝑛), then GNWBM𝑝,𝑞,𝑟𝑎1,𝑎2,,𝑎𝑛=GBM𝑝,𝑞,𝑟𝑎1,𝑎2,,𝑎𝑛.(3.16)

Proof. Since 𝑤=(1/𝑛,1/𝑛,1/𝑛), then by Definition 3.2, we can get GNWBM𝑝,𝑞,𝑟𝑎1,𝑎2,,𝑎𝑛=𝑛𝑖,𝑗,𝑘=1𝑖𝑗𝑘1𝑎𝑛(𝑛1)(𝑛2)𝑝𝑖𝑎𝑞𝑗𝑎𝑟𝑘1/(𝑝+𝑞+𝑟)=1𝑛(𝑛1)(𝑛2)𝑛𝑖,𝑗,𝑘=1𝑖𝑗𝑘𝑎𝑝𝑖𝑎𝑞𝑗𝑎𝑟𝑘1/(𝑝+𝑞+𝑟),GNWBM𝑝,𝑞,𝑟𝑎1,𝑎2,,𝑎𝑛=GBM𝑝,𝑞,𝑟𝑎1,𝑎2,,𝑎𝑛,(3.17) which complete the proof of the property.

Property 6 (Idempotency). Let 𝑝,𝑞,𝑟0 and 𝑎𝑖(𝑖=1,2,,𝑛) be a collection of nonnegative numbers with the weight vector 𝑤=(𝑤1,𝑤2,,𝑤𝑛), such that 𝑤𝑖0 and 𝑛𝑖=1𝑤𝑖=1. If all 𝑎𝑖(𝑖=1,2,,𝑛) are equal, that is, 𝑎𝑖=𝑎, for all 𝑖, then GNWBM𝑝,𝑞,𝑟𝑎1,𝑎2,,𝑎𝑛=𝑎.(3.18)

Proof. The proof of Property 6 is similar to Property 2.

Property 7 (Monotonicity). Let 𝑝,𝑞,𝑟0𝑎𝑖 and 𝑏𝑖(𝑖=1,2,,𝑛) be two collections of nonnegative numbers with the weight vector 𝑤=(𝑤1,𝑤2,,𝑤𝑛), such that 𝑤𝑖0 and 𝑛𝑖=1𝑤𝑖=1. If 𝑎𝑖𝑏𝑖, for all 𝑖, then GNWBM𝑝,𝑞,𝑟𝑎1,𝑎2,,𝑎𝑛GNWBM𝑝,𝑞,𝑟𝑏1,𝑏2,,𝑏𝑛.(3.19)

Proof. The proof of Property 7 is similar to Property 3.

Property 8 (Boundedness). Let 𝑝,𝑞,𝑟0 and 𝑎𝑖(𝑖=1,2,,𝑛) be a collection of nonnegative numbers with the weight vector 𝑤=(𝑤1,𝑤2,,𝑤𝑛), such that 𝑤𝑖0 and 𝑛𝑖=1𝑤𝑖=1, then min𝑖𝑎𝑖GNWBM𝑝,𝑞,𝑟𝑎1,𝑎2,,𝑎𝑛max𝑖𝑎𝑖.(3.20)

Proof. The proof of Property 8 is similar to Property 4.

4. Intuitionistic Fuzzy Normalized Weighted BM and Generalized Intuitionistic Fuzzy Normalized Weighted BM

To aggregate the intuition fuzzy correlated information, Xu and Yager [19] proposed the IFBM and IFWBM, and Xia et al. [21] proposed the GIFWBM. However, according to the aforementioned analysis, there are some drawbacks in the IFWBM and the GIFWBM, respectively. To solve these issues, and motivated by the GBM, we propose the intuitionistic fuzzy normalized weighted BM (IFNWBM) and the generalized intuitionistic fuzzy normalized weighted BM (GIFNWBM) based on the NWBM and GNWBM and describe as follows.

4.1. IFNWBM

Definition 4.1. Let 𝑝,𝑞0 and 𝛼𝑖=(𝜇𝑖,𝜈𝑖)(𝑖=1,2,,𝑛) be a collection of IFVs with the weight vector 𝑤=(𝑤1,𝑤2,,𝑤𝑛) such that 𝑤𝑖0 and 𝑛𝑖=1𝑤𝑖=1. If IFNWBM𝑝,𝑞𝛼1,𝛼2,,𝛼𝑛=𝑛𝑖,𝑗=1,𝑖𝑗𝑤𝑖𝑤𝑗1𝑤𝑖𝛼𝑝𝑖𝛼𝑞𝑗1/(𝑝+𝑞)(4.1) thenIFNWBM𝑝,𝑞 is called the intuitionistic fuzzy normalized weighted Bonferroni mean (IFNWBM).

On the basis of the operational laws of IFVs, we have the following theorem.

Theorem 4.2. Let 𝑝,𝑞0 and 𝛼𝑖=(𝜇𝑖,𝜈𝑖)(𝑖=1,2,,𝑛) be a collection of IFVs with the weight vector (𝑤1,𝑤2,,𝑤𝑛), such that 𝑤𝑖0 and 𝑛𝑖=1𝑤𝑖=1, then the aggregated value by using the IFNWBM is also an IFV and IFNWBM𝑝,𝑞𝛼1,𝛼2,,𝛼𝑛=1𝑛𝑖,𝑗=1𝑖𝑗1𝜇𝑝𝑖𝜇𝑞𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞),11𝑛𝑖,𝑗=1𝑖𝑗11𝜈𝑖𝑝1𝜈𝑗𝑞𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞).(4.2)

Proof. By the operational laws for IFVs, we have 𝛼𝑝𝑖=𝜇𝑝𝑖,11𝜈𝑖𝑝,𝛼𝑞𝑗=𝜇𝑞𝑗,11𝜈𝑗𝑞,𝛼𝑝𝑖𝛼𝑞𝑗=𝜇𝑝𝑖𝜇𝑞𝑗,11𝜈𝑖𝑝1𝜈𝑗𝑞,(4.3) then 𝑛𝑖,𝑗=1,𝑖𝑗𝑤𝑖𝑤𝑗1𝑤𝑖𝛼𝑝𝑖𝛼𝑞𝑗=1𝑛𝑖,𝑗=1𝑖𝑗1𝜇𝑝𝑖𝜇𝑞𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖),𝑛𝑖,𝑗=1𝑖𝑗11𝜈𝑖𝑝1𝜈𝑗𝑞𝑤𝑖𝑤𝑗/(1𝑤𝑖)IFNWBM𝑝,𝑞𝛼1,𝛼2,,𝛼𝑛=𝑛𝑖,𝑗=1,𝑖𝑗𝑤𝑖𝑤𝑗1𝑤𝑖𝛼𝑝𝑖𝛼𝑞𝑗1/(𝑝+𝑞),=1𝑛𝑖,𝑗=1𝑖𝑗1𝜇𝑝𝑖𝜇𝑞𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞),11𝑛𝑖,𝑗=1𝑖𝑗11𝜈𝑖𝑝1𝜈𝑗𝑞𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞).(4.4) In addition, since 01𝑛𝑖,𝑗=1𝑖𝑗1𝜇𝑝𝑖𝜇𝑞𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞)1,011𝑛𝑖,𝑗=1𝑖𝑗11𝜈𝑖𝑝1𝜈𝑗𝑞𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞)1,(4.5) then 1𝑛𝑖,𝑗=1𝑖𝑗1𝜇𝑝𝑖𝜇𝑞𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞)+11𝑛𝑖,𝑗=1𝑖𝑗11𝜈𝑖𝑝1𝜈𝑗𝑞𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞)1+1𝑛𝑖,𝑗=1𝑖𝑗11𝜈𝑖𝑝1𝜈𝑗𝑞𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞)1𝑛𝑖,𝑗=1𝑖𝑗11𝜈𝑖𝑝1𝜈𝑗𝑞𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞)=1,(4.6) which completes the proof of the theorem.

Moreover, the IFNWBM also has the following properties.

Property 9. If all 𝛼i(𝑖=1,2,,𝑛) are equal, that is, 𝛼𝑖=𝛼, for all 𝑖, then IFNWBM𝑝,𝑞𝛼1,𝛼2,,𝛼𝑛=𝛼.(4.7)

Proof. Since 𝛼𝑖=(𝜇𝑖,𝜈𝑖)=𝛼, we have IFNWBM𝑝,𝑞𝛼1,𝛼2,,𝛼𝑛=𝑛𝑖,𝑗=1,𝑖𝑗𝑤𝑖𝑤𝑗1𝑤𝑖(𝛼𝑝𝛼𝑞)1/(𝑝+𝑞)=(𝛼𝑝𝛼𝑞)𝑛𝑖,𝑗=1𝑖𝑗𝑤𝑖𝑤𝑗1𝑤𝑖1/(𝑝+𝑞)=𝛼𝑛𝑖𝑤𝑖𝑛𝑗=1𝑗𝑖𝑤𝑗1𝑤𝑖1/(𝑝+𝑞)=𝛼,(4.8) which completes the proof.

Property 10. Let 𝛼𝑖=(𝜇𝛼𝑖,𝜈𝛼𝑖) and 𝛽𝑖=(𝜇𝛽𝑖,𝜈𝛽𝑖)(𝑖=1,2,,𝑛) be two collections of IFVs, if 𝜇𝛼𝑖𝜇𝛽𝑖 and 𝜈𝛼𝑖𝜈𝛽𝑖, for all 𝑖, then IFNWBM𝑝,𝑞𝛼1,𝛼2,,𝛼𝑛IFNWBM𝑝,𝑞𝛽1,𝛽2,,𝛽𝑛.(4.9)

Proof. Since 𝜇𝛼𝑖𝜇𝛽𝑖 and 𝜈𝛼𝑖𝜈𝛽𝑖, for all 𝑖, then 𝜇𝑝𝛼𝑖𝜇𝑞𝛼𝑗𝜇𝑝𝛽𝑖𝜇𝑞𝛽𝑗𝑛𝑖,𝑗=1𝑖𝑗1𝜇𝑝𝛼𝑖𝜇𝑞𝛼𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)𝑛𝑖,𝑗=1𝑖𝑗1𝜇𝑝𝛽𝑖𝜇𝑞𝛽𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1𝑛𝑖,𝑗=1𝑖𝑗1𝜇𝑝𝛼𝑖𝜇𝑞𝛼𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1𝑛𝑖,𝑗=1𝑖𝑗1𝜇𝑝𝛽𝑖𝜇𝑞𝛽𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1𝑛𝑖,𝑗=1𝑖𝑗1𝜇𝑝𝛼𝑖𝜇𝑞𝛼𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞)1𝑛𝑖,𝑗=1𝑖𝑗1𝜇𝑝𝛽𝑖𝜇𝑞𝛽𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞),1𝜈𝑝𝛼𝑖1𝜈𝑞𝛼𝑗1𝜈𝑝𝛽𝑖1𝜈𝑞𝛽𝑗𝑛𝑖,𝑗=1𝑖𝑗11𝜈𝑝𝛼𝑖1𝜈𝑞𝛼𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)𝑛𝑖,𝑗=1𝑖𝑗11𝑣𝑝𝛽𝑖1𝜈𝑞𝛽𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1𝑛𝑖,𝑗=1𝑖𝑗11𝜈𝑝𝛼𝑖1𝜈𝑞𝛼𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞)1𝑛𝑖,𝑗=1𝑖𝑗11𝜈𝑝𝛽𝑖1𝜈𝑞𝛽𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞),(4.10) then 11𝑛𝑖,𝑗=1𝑖𝑗11𝜈𝑝𝛼𝑖1𝜈𝑞𝛼𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞)11𝑛𝑖,𝑗=1𝑖𝑗11𝜈𝑝𝛽𝑖1𝜈𝑞𝛽𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞).(4.11) Therefore, 1𝑛𝑖,𝑗=1𝑖𝑗1𝜇𝑝𝛼𝑖𝜇𝑞𝛼𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞)1+1𝑛𝑖,𝑗=1𝑖𝑗11𝜈𝑝𝛼𝑖1𝜈𝑞𝛼𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞)1𝑛𝑖,𝑗=1𝑖𝑗1𝜇𝑝𝛽𝑖𝜇𝑞𝛽𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞)+11𝑛𝑖,𝑗=1𝑖𝑗11𝜈𝑝𝛽𝑖1𝜈𝑞𝛽𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞).(4.12)
Let 𝛼=IFNWBM𝑝,𝑞(𝛼1,𝛼2,,𝛼𝑛) and 𝛽=IFNWBM𝑝,𝑞(𝛽1,𝛽2,,𝛽𝑛) and set 𝑠(𝛼) and 𝑠(𝛽) be the score values of 𝛼 and 𝛽, then (4.12) is equal to 𝑠(𝛼)𝑠(𝛽). Now we discuss the following two cases.
Case  1. If 𝑠(𝛼)>𝑠(𝛽), then by the total order relation between two IFVs, we have IFNWBM𝑝,𝑞𝛼1,𝛼2,,𝛼𝑛>IFNWBM𝑝,𝑞𝛽1,𝛽2,,𝛽𝑛.(4.13)
Case  2. If 𝑠(𝛼)=𝑠(𝛽), then 1𝑛𝑖,𝑗=1𝑖𝑗1𝜇𝑝𝛼𝑖𝜇𝑞𝛼𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞)1+1𝑛𝑖,𝑗=1𝑖𝑗11𝜈𝑝𝛼𝑖1𝜈𝑞𝛼𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞)=1𝑛𝑖,𝑗=1𝑖𝑗1𝜇𝑝𝛽𝑖𝜇𝑞𝛽𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞)+11𝑛𝑖,𝑗=1𝑖𝑗11𝜈𝑝𝛽𝑖1𝜈𝑞𝛽𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞).(4.14) Since 𝜇𝛼𝑖𝜇𝛽𝑖 and 𝜈𝛼𝑖𝜈𝛽𝑖, for all 𝑖, we can get 1𝑛𝑖,𝑗=1𝑖𝑗1𝜇𝑝𝛼𝑖𝜇𝑞𝛼𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞)=1𝑛𝑖,𝑗=1𝑖𝑗1𝜇𝑝𝛽𝑖𝜇𝑞𝛽𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞),11𝑛𝑖,𝑗=1𝑖𝑗11𝜈𝑝𝛼𝑖1𝜈𝑞𝛼𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞)=11𝑛𝑖,𝑗=1𝑖𝑗11𝜈𝑝𝛽𝑖1𝜈𝑞𝛽𝑗𝑤𝑖𝑤𝑗/(1𝑤𝑖)1/(𝑝+𝑞).(4.15) Therefore, (𝛼)=(𝛽) and IFNWBM𝑝,𝑞𝛼1,𝛼2,,𝛼𝑛=IFNWBM𝑝,𝑞𝛽1,𝛽2,,𝛽𝑛,(4.16) which complete the proof of the property.

Property 11. Let 𝛼𝑖=(𝜇𝑖,𝜈𝑗)(𝑖=1,2,,𝑛) be a collection of IFVs, and (𝛼1,𝛼2,,𝛼𝑛) is any permutation of (𝛼1,𝛼2,,𝛼𝑛), then IFNWBM𝑝,𝑞𝛼1,𝛼2,,𝛼𝑛=IFNWBM𝑝,𝑞𝛼1,𝛼2,,𝛼𝑛.(4.17)

Proof. Since (𝛼1,𝛼2,,𝛼𝑛) is any permutation of (𝛼1,𝛼2,,𝛼𝑛), then 𝑛𝑖,𝑗=1𝑖𝑗𝑤𝑖𝑤𝑗1𝑤𝑖𝛼𝑝𝑖𝛼𝑞𝑗1/(𝑝+𝑞)=𝑛𝑖,𝑗=1𝑖𝑗𝑤𝑖𝑤𝑗1𝑤𝑖𝛼𝑝𝑖𝛼𝑞𝑗1/(𝑝+𝑞).(4.18) Therefore, IFNWBM𝑝,𝑞𝛼1,𝛼2,,𝛼𝑛=IFNWBM𝑝,𝑞𝛼1,𝛼2,,𝛼𝑛,(4.19) which complete the proof.

Property 12. Let 𝛼𝑖=(𝜇𝑖,𝜈𝑗)(𝑖=1,2,,𝑛) be a collection of IFVs, and 𝛼=min𝑖𝜇𝑖,max𝑖𝜈𝑖,𝛼+=max𝑖𝜇𝑖,min𝑖𝜈𝑖,(4.20) then 𝛼IFNWBM𝑝,𝑞𝛼1,𝛼2,,𝛼𝑛𝛼+.(4.21)

Proof. Since 𝜇𝑖min𝑖{𝜇𝑖} and 𝜈𝑖max𝑖={𝜈𝑖}, then based on Properties 9 and 10, we have IFNWBM𝑝,𝑞𝛼1,𝛼2,,𝛼𝑛IFNWBM𝑝,𝑞(𝛼,𝛼,,𝛼)=𝛼.(4.22) Likewise, we can get IFNWBM𝑝,𝑞𝛼1,𝛼2,,𝛼𝑛IFNWBM𝑝,𝑞𝛼+,𝛼+,,𝛼+=𝛼+,(4.23) which complete the proof of the property.

4.2. GIFNWBM

Definition 4.3. Let 𝑝,𝑞,𝑟0 and 𝛼𝑖=(𝜇𝑖,𝜈𝑖)(𝑖=1,2,,𝑛) be a collection of IFVs with the weight vector 𝑤=(𝑤1,𝑤2,,𝑤𝑛) such that 𝑤𝑖0, and 𝑛𝑖=1𝑤𝑖=1. If GIFNWBM𝑝,𝑞,𝑟𝛼1,,𝛼𝑛=𝑛𝑖,𝑗,𝑘=1𝑖𝑗𝑘𝑤𝑖𝑤𝑗𝑤𝑘1𝑤𝑖1𝑤𝑖𝑤𝑗𝛼𝑝𝑖𝛼𝑞𝑗𝛼𝑟𝑘1/(𝑝+𝑞+𝑟),(4.24) then GIFNWBM𝑝,𝑞,𝑟 is called the generalized intuitionistic fuzzy normalized weighted Bonferroni mean (GIFNWBM).

On the basis of the operational laws of IFVs, we can drive the following theorem.

Theorem 4.4. Let 𝑝,𝑞,𝑟0 and 𝛼𝑖=(𝜇𝑖,𝜈𝑖)(𝑖=1,2,,𝑛) be a collection of IFVs with the weight vector 𝑤=(𝑤1,𝑤2,,𝑤𝑛), such that 𝑤𝑖0 and 𝑛𝑖=1𝑤𝑖=1, then the aggregated value by using the GIFNWBM is also an IFV and IFNWBM𝑝,𝑞,𝑟𝛼1,𝛼2,,𝛼𝑛=1𝑛𝑖,𝑗,𝑘=1𝑖𝑗𝑘1𝜇𝑝𝑖𝜇𝑞𝑗𝜇𝑟𝑘𝑤𝑖𝑤𝑗𝑤𝑘/((1𝑤𝑖)(1𝑤𝑖𝑤𝑗))1/(𝑝+𝑞+𝑟),11𝑛𝑖,𝑗,𝑘=1𝑖𝑗𝑘11𝜈𝑖𝑝1𝜈𝑗𝑞1𝜈𝑘𝑟𝑤𝑖𝑤𝑗𝑤𝑘/((1𝑤𝑖)(1𝑤𝑖𝑤𝑗))1/(𝑝+𝑞+𝑟).(4.25)

Proof. The proof of Theorem 4.4 is similar to Theorem 4.2.

Furthermore, the GIFNWBM also has the following properties.

Property 13. If all 𝛼𝑖(𝑖=1,2,,𝑛) are equal, that is, 𝛼𝑖=𝛼, for all 𝑖, then GIFNWBM𝑝,𝑞,𝑟𝛼1,𝛼2,,𝛼𝑛=𝛼.(4.26)

Proof. The proof of Property 13 is similar to Property 9.

Property 14. Let 𝛼𝑖=(𝜇𝛼𝑖,𝜈𝛼𝑖) and 𝛽𝑖=(𝜇𝛽𝑖,𝜈𝛽𝑖)(𝑖=1,2,,𝑛) be two collections of IFVs, if 𝜇𝛼𝑖𝜇𝛽𝑖 and 𝜈𝛼𝑖𝜈𝛽𝑖, for all 𝑖, then GIFNWBM𝑝,𝑞,𝑟𝛼1,𝛼2,,𝛼𝑛GIFNWBM𝑝,𝑞,𝑟𝛽1,𝛽2,,𝛽𝑛.(4.27)

Proof. The proof of Property 14 is similar to Property 10.

Property 15. Let 𝛼𝑖=(𝜇𝑖,𝜈𝑗)(𝑖=1,2,,𝑛) be a collection of IFVs and (𝛼1,𝛼2,,𝛼𝑛) is any permutation of (𝛼1,𝛼2,,𝛼𝑛), then GIFNWBM𝑝,𝑞,𝑟𝛼1,𝛼2,,𝛼𝑛=GIFNWBM𝑝,𝑞,𝑟𝛼1,𝛼2,,𝛼𝑛.(4.28)

Proof. The proof of Property 15 is similar to Property 11.

Property 16. Let 𝛼𝑖=(𝜇𝑖,𝜈𝑗)(𝑖=1,2,,𝑛) be a collection of IFVs, and 𝛼=min𝑖𝜇𝑖,max𝑖𝜈𝑖,𝛼+=max𝑖𝜇𝑖,min𝑖𝜈𝑖,(4.29) then 𝛼GIFNWBM𝑝,𝑞,𝑟𝛼1,𝛼2,,𝛼𝑛𝛼+.(4.30)

Proof. The proof of Property 16 is similar to Property 12.

5. An Approach to Intuitionistic Fuzzy Multicriteria Decision Making

In what follows, we apply the GIFNWBM operator to intuitionistic fuzzy multicriteria decision making, which involves the following steps.

Step  1. For a multicriteria decision making problem, set 𝑌={𝑦1,𝑦2,,𝑦𝑛} be a set of 𝑛 alternatives, 𝐶={𝑐1,𝑐2,,𝑐𝑚} be a set of 𝑚 criteria, whose weight vector is 𝑤=(𝑤1,𝑤2,,𝑤𝑚), satisfying 𝑤𝑗>0(𝑗=1,2,,𝑚), and 𝑚𝑗=1𝑤𝑗=1, where 𝑤𝑗 denotes the important degree of 𝑐𝑗. The performance of 𝑦𝑖 with respect to 𝑐𝑗 is measured by an IFV 𝑏𝑖𝑗=(𝜇𝑖𝑗,𝜈𝑖𝑗), where 𝜇𝑖𝑗 indicates the degree that 𝑦𝑖 satisfies 𝑐𝑗 and 𝜈𝑖𝑗 indicates the degree that 𝑦𝑖 does not satisfy 𝑐𝑗, such that 0𝜇𝑖𝑗1, 0𝜈𝑖𝑗1 and 𝜇𝑖𝑗+𝜈𝑖𝑗1, and the intuitionistic fuzzy decision matrix 𝐵=(𝑏𝑖𝑗)𝑛×𝑚 contains all 𝑏𝑖𝑗=(𝜇𝑖𝑗,𝜈𝑖𝑗)(𝑖=1,2,,𝑛;𝑗=1,2,,𝑚). If all the criteria 𝑐𝑗 are the benefit type, then the performance values do not need normalization. Whereas there are, generally, benefit criteria (the bigger the performance values the better) and cost criteria (the smaller the performance values the better) in multicriteria decision making, in such cases, we may transform the performance values of the cost type into the performance values of the benefit type. by Xu and Hu’s approach [26]. Then, the intuitionistic fuzzy decision matrix 𝐵=(𝑏𝑖𝑗)𝑛×𝑚 can be transformed into the normalization matrix 𝑅=(𝑟𝑖𝑗)𝑛×𝑚 where 𝑟𝑖𝑗𝑛×𝑚=𝜇𝑖𝑗,𝜈𝑖𝑗=𝑏𝑖𝑗,forbenetcriterion𝑐𝑖,𝑏𝑖𝑗,forcostcriterion𝑐𝑖,(5.1) where 𝑖=1,2,,𝑛,𝑗=1,2,,𝑚, and 𝑏𝑖𝑗 is the complement of 𝑏𝑖𝑗 such that 𝑏𝑖𝑗=(𝜈𝑖𝑗,𝜇𝑖𝑗).

Step  2. Utilize the GIFNWBM operator: 𝑟𝑖=𝜇𝑖,𝜈𝑖=GIFNWBM𝑝,𝑞,𝑟𝑟𝑖1,𝑟𝑖2,,𝑟𝑖𝑚.(5.2) to aggregate all the preference values 𝑟𝑖𝑗(𝑗=1,2,,𝑚) of the 𝑖th line and get the overall performance value 𝑟𝑖 corresponding to the alternative 𝑦𝑖.

Step  3. Calculate the score valued and the accuracy degree of the overall performance value 𝑟𝑖 and utilize the total order relation between two IFVs to rank the overall performance value 𝑟𝑖(𝑖=1,2,,𝑛).

Step  4. Rank all the alternatives 𝑦𝑖(𝑖=1,2,,𝑛) in accordance with 𝑟𝑖(𝑖=1,2,,𝑛) in descending order, and then, select the most desirable alternative with the largest score value.

Let us give a practical example to illustrate the proposed approach in the intuitionistic fuzzy multicriteria decision making procedure.

Example 5.1. There is an investment company, which wants to invest a sum of money in the best option (adapted from Herrera and Herrera-Viedma [27]). There is a panel with four possible alternatives {𝑦1,𝑦2,𝑦3,𝑦4} to invest the money, in which 𝑦1 is a car company, 𝑦2 is a fast food chains company, 𝑦3 is an arms company, 𝑦4 is a software company. The investment company must make a decision according to five criteria: 𝑐1 is the growth analysis, 𝑐2 is the environment impact analysis, 𝑐3 is the risk analysis, 𝑐4 is the social impact analysis, 𝑐5 is the profitability analysis. The weight vector of the criteria {𝑐1,𝑐2,𝑐3,𝑐4,𝑐5} is 𝑤=(0.15,0.12,0.24,0.18,0.31). Assume that the characteristics of the alternatives 𝑦𝑖(𝑖=1,2,3,4) with respect to the criteria 𝑐𝑗(𝑗=1,2,3,4,5) are represented by the IFVs 𝑏𝑖𝑗=(𝜇𝑖𝑗,𝜈𝑖𝑗), where 𝜇𝑖𝑗 indicates the degree that the alternative 𝑦𝑖 satisfies the criterion 𝑐𝑗 and 𝜈𝑖𝑗 indicates the degree that the alternative 𝑦𝑖 does not satisfy the criterion 𝑐𝑗.
To get the optimal alternative(s), the following steps are given.

Step 1. Based on (5.1), we normalize 𝑏𝑖𝑗(𝑖=1,2,3,4;𝑗=1,2,3,4,5) to 𝑟𝑖𝑗 and construct the normalization intuitionistic fuzzy decision matrix 𝑅=(𝑟𝑖𝑗)4×5 (see Table 1).

tab1
Table 1: Normalization intuitionistic fuzzy decision matrix 𝑅=(𝑟𝑖𝑗)4×5.

Step 2. Aggregate all the preference values 𝑟𝑖𝑗(𝑗=1,2,3,4,5) of the 𝑖th line, and get the overall performance value 𝑟𝑖 corresponding to the alternative 𝑦𝑖 by the GIFNWBM operator (here we let 𝑝=𝑞=𝑟=1): 𝑟1=(0.6331,0.2910),𝑟2𝑟=(0.5891,0.3692),3=(0.6056,0.3233),𝑟4=(0.6049,0.3222).(5.3)

Step 3. Calculate the score of the overall performance value 𝑟𝑖(𝑖=1,2,3,4): 𝑠𝑟1𝑟=0.3421,𝑠2𝑟=0.2199,𝑠3𝑟=0.2823,𝑠4=0.2827.(5.4)

Step 4. Rank all the alternatives 𝑦𝑖(𝑖=1,2,3,4) in accordance with 𝑟𝑖. Since 𝑠(𝑟1)>𝑠(𝑟4)>𝑠(𝑟3)>𝑠(𝑟2), then by the total order relation between two IFVs, we have the ranking of the IFVs: 𝑦1>𝑦4>𝑦3>𝑦2. Hence, 𝑦1 is the best option.
In Step 2, if we take 𝑝=𝑞=𝑟=2, we can get 𝑟1=(0.6974,0.2335),𝑟2𝑟=(0.6510,0.3079),3=(0.6770,0.2627),𝑟4=(0.6697,0.2604).(5.5)
Then, we calculate the score values of all the alternatives: 𝑠𝑟1𝑟=0.4640,𝑠2𝑟=0.3430,𝑠3𝑟=0.4143,𝑠4=0.4093.(5.6) Therefore, 𝑦1>𝑦3>𝑦4>𝑦2, and 𝑦1 is still the optimal alternative.
By the aforementioned numeral results, the optimal investment decision is the car company 𝑦1. It should be noted out that the whole ranking of the alternatives has changed. The GIFNWBM1,1,1 produces the ranking of all the alternatives as 𝑦1>𝑦4>𝑦3>𝑦2, which is slightly different from the ranking of alternatives 𝑦1>𝑦3>𝑦4>𝑦2, derived by the GIFNWBM2,2,2, that is, the ranking of 𝑦3 and 𝑦4 is reversed while the ranking of the other alternatives is kept unchanged. Therefore, we can see that the value derived by the GIFNWBM operator depends on the choice of the parameters 𝑝,𝑞, and 𝑟, and these parameters are not robust. In general, the bigger parameters 𝑝,𝑞, and 𝑟, the more the calculation effort needed, and in the special case where at least two of these parameters take the value of zero, the GIFNWBM cannot capture the interrelationship of the individual arguments. As a result, in practical applications, we generally take the values of these parameters as 𝑝=𝑞=𝑟=1, which is not only intuitive and simple but also the interrelationship of the individual argument can be fully taken into account [21].

6. Concluding Remarks

To aggregate the intuitionistic fuzzy information, a lot of aggregation operators have been developed and investigated, especially, the ones which reflect the interrelationship of the aggregated arguments are the hot research topics, among which the Bonferroni mean (BM) is an important aggregation technique. The desirable characteristic of the BM is its capability to capture the interrelationship between the input arguments. To further develop the BM, we have proposed the normalized weighted Bonferroni mean (NWBM) and the generalized normalized weighted Bonferroni mean (GNWBM) whose characteristics are to reflect the preference and interrelationship of the aggregated arguments and can satisfy the basic properties of the aggregation techniques simultaneously. To aggregate the IFVs, the intuitionistic fuzzy normalized weighted Bonferroni mean (IFNWBM) operator and the generalized intuitionistic fuzzy normalized weighted Bonferroni mean (GIFNWBM) operator have been developed and discussed. Furthermore, some desirable properties of the IFNWBM operator and the GIFNWBM operator are investigated in detail. To deal with the situation that the criteria have connections in intuitionistic fuzzy multicriteria decision making, an approach has been proposed on the basis of the GIFNWBM operator. It is worth noting that the results of this paper can be extended to the interval-valued intuitionistic fuzzy environment and the hesitant fuzzy environment.

Acknowledgments

This work was supported by National Natural Science Foundation of China (no. 71071034), National Basic Research Program of China (973 Program, no. 2010 CB328104-02), Funding of Jiangsu Innovation Program for Graduate Education (no. CXZZ-0183), and Academic New Artist Ministry of Education Doctoral Post Graduate.

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