Multiple attribute group decision making based on IVHFPBMs and a new ranking method for interval-valued hesitant fuzzy information

Highlights

• A new ranking method for interval-valued hesitant fuzzy sets is presented.

• We develop the interval-valued hesitant fuzzy power Bonferroni means.

• The properties of the new proposed aggregation operators are investigated.

• New approaches to multiple attributes decision making are investigated.

Abstract

The interval-valued hesitant fuzzy set is a significant tool to express the uncertain information. In this paper, we define the interval-valued hesitant fuzzy 2nd-order central polymerization degree (IVHFCP₂) function and the interval-valued hesitant fuzzy 2nd-order dispersive central polymerization degree (IVHFDCP₂) function to further compare different interval-valued hesitant fuzzy sets. To capture much more information for the multiple attribute group decision making, we combine the Bonferroni mean with the power average operator to accommodate to interval-valued hesitant fuzzy environments and develop the interval-valued hesitant fuzzy power Bonferroni mean (IVHFPBM) and the interval-valued hesitant fuzzy weighted power Bonferroni mean (IVHFWPBM). We investigate the desirable properties of the new interval-valued hesitant fuzzy aggregation operators and discuss their special cases in detail. Finally, the new aggregation operators are applied to interval-valued hesitant fuzzy multiple attribute group decision making and a numerical example is given to illustrate the effectiveness of the presented approaches.

Introduction

Atanassov (1986) introduced the concept of intuitionistic fuzzy set (IFS), which is a generalization of the concept of a fuzzy set (Zadeh, 1965). Each element in the IFS is expressed by an ordered pair, and each ordered pair is characterized by a membership degree and a non-membership degree. When discussing the membership degree of x in A, different decision makers may assign different values, for example, one wants to assign 0.3 while another wants to assign 0.5. But they are not willing to compromise with each other, for which, Torra (2010) and Torra and Narukawa (2009) extended fuzzy sets (Zadeh, 1965) to hesitant fuzzy sets (HFSs), and the above membership of x in A can be presented as {0.3, 0.5}. Torra and Narukawa (2009) further discussed the similarities between HFSs and intuitionistic fuzzy sets (IFSs) (Atanassov, 1986), and showed that the envelope of a hesitant fuzzy set is an intuitionistic fuzzy set. Rodriguez, Martinez, and Hsrrera (2012) combined linguistic sets with hesitant fuzzy sets and proposed hesitant fuzzy linguistic term sets. Wei (2015) combined uncertain linguistic sets with interval-valued hesitant fuzzy sets and proposed interval-valued hesitant fuzzy uncertain linguistic sets.

Other important generalizations of fuzzy sets and their applications can refer to the research on fuzzy graphs (Akram, 2011), linguistic fuzzy sets (Merigó and Gil-Lafuente, 2009, Merigó and Gil-Lafuente, 2013, Merigó et al., 2012, Zadeh, 1975), type-2 fuzzy sets (Castillo and Melin, 2012, Fazel Zarandi et al., 2012, Galluzzo and Cosenza, 2011, Greenfield et al., 2012, Zhai and Mendel, 2011), intuitionistic fuzzy sets (Atanassov, 1994, Akram and Dudek, 2013, Beliakov et al., 2011, Xia et al., 2013, Xu, 2013, Xu and Yager, 2011, Zhao et al., 2010), vague sets (Gau & Buehrer, 1993), hesitant fuzzy sets (Chen et al., 2013, Chen et al., 2015, Wu et al., 2014, Xu and Xia, 2011, Zhu and Xu, 2014), interval-valued hesitant fuzzy sets (Liu et al., 2014, Wang et al., 2014;) and fuzzy multisets (Miyamoto, 2005).

As a significant human activity, multiple attribute decision making (MADM) problems (He et al., 2015, He et al., 2015, Xia et al., 2013, Yager, 1988) are the process of finding the best alternative(s) from all of the feasible alternatives where all the alternatives can be evaluated according to a number of attributes. Information aggregation is one of the core techniques—many papers have investigated this issue (Dubois & Prade, 1980; Gau et al., 1993; He et al., 2014, He and He, 2016, He et al., 2012, Narukawa, 2007, Torra, 2010, Wang and Dong, 2009, Wei, 2012, Xia and Xu, 2011, Xu and Xia, 2011, Yager, 2008, Zhou and Chen, 2011, Zhu and Xu, 2014, Zhu et al., 2012. Yager (2001) originally introduced the power average (PA) operator. Bonferroni (1950) considered the interrelationship of the individual arguments and introduced a mean-type aggregation operator called the Bonferroni mean (BM). Zhu et al. (2012) presented the hesitant fuzzy geometric Bonferroni means. Xia and Xu (2011) proposed some hesitant fuzzy aggregation operators. Wei (2012) developed some prioritized aggregation operators for aggregating hesitant fuzzy information. Zhang (2013) developed a series of hesitant fuzzy power aggregation operators. Liao, Xu, and Xia (2014) investigated the multiplicative consistency of a hesitant fuzzy preference relation. Rodriguez et al. (2012) presented the hesitant fuzzy linguistic term sets. Chen et al. (2013) introduced the concept of interval-valued hesitant fuzzy sets (IVHFSs), permitting the membership degrees of an element to a given set to have a few different interval values.

However, the existing ranking method (Zhang, Wang, Tian, & Li, 2014) for interval-valued hesitant fuzzy sets can’t rank all IVHFSs. For example, assume ${\tilde{h}}_1$ and ${\tilde{h}}_2$ are two interval-valued hesitant fuzzy sets, ${\tilde{h}}_1=\left\{{\tilde{h}}_1^1\text{,}{\tilde{h}}_1^2\right\}$, ${\tilde{h}}_2=\left\{{\tilde{h}}_2^1\text{,}{\tilde{h}}_2^2\right\}$, ${\tilde{h}}_1^1+{\tilde{h}}_1^2={\tilde{h}}_2^1+{\tilde{h}}_2^2$, ${\tilde{h}}_1^1\ne {\tilde{h}}_2^1$, ${\tilde{h}}_1^1\ne {\tilde{h}}_2^2$, and ${\tilde{h}}_1^2\ne {\tilde{h}}_2^1$, ${\tilde{h}}_1^2\ne {\tilde{h}}_2^2$, if we use the comparison method by Zhang et al. (2014), i.e., just considering the score of different IVHFSs, then $h_1=h_2$ for $s\left(h_1\right)=s\left(h_2\right)$. However, $h_1^1\ne h_2^1\text{,}{h}_1^1\ne h_2^2\text{,}{h}_1^2\ne h_2^1\text{,}{h}_1^2\ne h_2^2$, which means it is actually not reasonable to have the result that $h_1=h_2$. If we take account the dispersive central polymerization degree of all values in the interval-valued hesitant fuzzy set, the above weakness can be solved. Therefore, we define the interval-valued hesitant fuzzy 2nd-order dispersive central polymerization degree function, which can be explained as the variance in statistics. Based on this, a new ranking method is presented. Moreover, in many real decision making problems, it may be difficult for decision makers to exactly quantify their opinions with a single crisp number due to the insufficiency in available information, but instead define an interval number in [0, 1]. Motivated by Chen et al., 2013, He et al., 2015, we present the IVHFPBM and the IVHFWPBM, capturing not only the interrelationship between input arguments, but also the relationships between the fusedvalues, providing a new train of thought for multiple attribute group decision making underinterval-valued hesitant fuzzy environments.

The rest of paper is organized as follows. Section 2 reviews some basic concepts and proposes the new ranking methods for different interval-valued hesitant fuzzy sets. Section 3 develops the IVHFPBM and the IVHFWPBM, investigates their desirable properties, and evaluates some special cases. Section 4 applies the new aggregation operators to interval-valued hesitant fuzzy multi-criteria group decision making. Section 5 investigates a numerical example to illustrate the feasibility and effectiveness of the new approaches. Section 6 ends the paper.