Multiple attribute group decision making based on IVHFPBMs and a new ranking method for interval-valued hesitant fuzzy information
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 and are two interval-valued hesitant fuzzy sets, , , , , , and , , if we use the comparison method by Zhang et al. (2014), i.e., just considering the score of different IVHFSs, then for . However, , which means it is actually not reasonable to have the result that . 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.
Section snippets
Preliminaries
In this section, we first define some important notations. Then we review the basic concepts, including the power average (PA) operator (Yager, 2001), interval-valued hesitant fuzzy sets (IVHFSs) (Chen et al., 2013) and some basic operational laws on interval-valued hesitant fuzzy elements (IVHFEs). Then we propose an improved comparison law for IVHFEs.
Interval-valued hesitant fuzzy power Bonferroni means
In this Section, we shall investigate the Bonferroni mean (BM) under interval- valued hesitant fuzzy environments considering the interactions of different IVHFEs.
Methods for multiple attribute group decision making under interval-valued hesitant fuzzy environments
For a multiple attribute decision making problem, assume are a set of alternatives, are a set of attributes with the associated weighting vector , and are a group of experts with the associated weighting vector . Suppose that the characteristics of the alternative under attribute by are expressed by interval-valued hesitant fuzzy information . If the attributes are benefit attributes, the bigger
Illustrative example
Example 2 Revised from Zhou and Chen (2011) let us consider a bid inviting process through which the employer or investor is trying to find out the optimal bidding scheme. In order to keep pace with the development of modern iron and steel industry as well as to improve the environmental equality of the city, Steel and Iron Works wants to construct a pelletizing plant in the primary iron ore production area where the production capacity reaches 1.20 million tons per year. According to the characteristics
Conclusions
In this paper, we define interval-valued hesitant fuzzy 2nd-order central polymerization degree function and interval-valued hesitant fuzzy 2nd-order dispersive central polymerization degree function to compare different interval-valued hesitant fuzzy sets further, which is similar to the variance in statistics. We develop the interval-valued hesitant fuzzy power Bonferroni mean (IVHFPBM) and the interval-valued hesitant fuzzy weighted power Bonferroni mean (IVHFWPBM), and investigate their
Acknowledgment
The work was supported by National Natural Science Foundation of China (Nos. 71225006, 71532008).
Yingdong He received the M.S. degree from School of Mathematical Sciences, Anhui University, Hefei, China, in 2014. He is currently working towards the Ph.D. degree at the College of Management & Economics, Tianjin University, Tianjin, China.
He has contributed about twenty articles to professional journals as the first author, such as IEEE Transactions on Fuzzy Systems, IEEE Transactions on Cybernetics, Information Sciences, Applied Soft Computing, Journal of the Operational Research Society,
References (56)
Bipolar fuzzy graphs
Information Sciences
(2011)- et al.
Intuitionistic fuzzy hypergraphs with applications
Information Sciences
(2013) Intuitionistic fuzzy sets
Fuzzy Sets and Systems
(1986)New operations defined over the intuitionistic fuzzy sets
Fuzzy sets and Systems
(1994)- et al.
On averaging operators for Atanassov’s intuitionistic fuzzy sets
Information Sciences
(2011) - et al.
Optimization of type-2 fuzzy systems based on bio-inspired methods: A concise review
Information Sciences
(2012) - et al.
Interval-valued hesitant preference relations and their applications to group decision making
Knowledge-Based Systems
(2013) - et al.
A type-2 fuzzy c-regression clustering algorithm for Takagi–Sugeno system identification and its application in the steel industry
Information Sciences
(2012) - et al.
Control of a non-isothermal continuous stirred tank reactor by a feedback-feed forward structure using type-2 fuzzy logic controllers
Information Sciences
(2011) - et al.
The sampling method of defuzzification for type-2 fuzzy sets: Experimental evaluation
Information Sciences
(2012)
Multi-attribute decision making based on neutral averaging operators for intuitionistic fuzzy information
Applied Soft Computing
Intuitionistic fuzzy geometric interaction averaging operators and their application to multi-criteria decision making
Information Sciences
A robust desirability function method for multi-response surface optimization considering model uncertainty
European Journal of Operational Research
The induced generalized OWA operator
Information Sciences
Induced 2-tuple linguistic generalized aggregation operators and their application in decision-making
Information Sciences
Remarks on basics of fuzzy sets and fuzzy multisets
Fuzzy Sets and Systems
Interval-valued hesitant fuzzy linguistic sets and their applications in multi-criteria decision-making problems
Information Sciences
Hesitant fuzzy prioritized operators and their application to multiple attribute decision making
Knowledge-Based Systems
Hesitant fuzzy information aggregation in decision making
International Journal of Approximate Reasoning
Geometric Bonferroni means with their application in multi-criteria decision making
Knowledge-Based Systems
Distance and similarity measures for hesitant fuzzy sets
Information Sciences
Prioritized aggregation operators
International Journal of Approximate Reasoning
Fuzzy set
Information Control
The concept of a linguistic variable and its application to approximate reasoning. Parts 1
Information Sciences
Uncertainty measures for general Type-2 fuzzy sets
Information Sciences
Hesitant fuzzy power aggregation operators and their application to multiple attribute group decision making
Information Sciences
Induced generalized hesitant fuzzy operators and their application to multiple attribute group decision making
Computers & Industrial Engineering
Continuous generalized OWA operator and its application to decision making
Fuzzy Sets and Systems
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2019, Engineering Applications of Artificial IntelligenceCitation Excerpt :Hesitant fuzzy sets (HFS): Hesitant fuzzy sets can be used as a functional tool allowing many potential membership degrees of an element to a set. These fuzzy sets let several membership degrees of an element to be possible between zero and one (Kutlu Gündoğdu et al., 2018; Qin et al., 2016; He et al., 2016; Farhadinia, 2016; Zhu et al., 2016; Wang and Xu, 2016; Liao et al., 2015; Peng et al., 2015; Zhou and Xu, 2015; Meng et al., 2015; Liao et al., 2014; Zhang and Wu, 2014; RodríGuez et al., 2013; Xu and Zhang, 2013; Qian et al., 2013; Zhang and Wei, 2013; Xia and Xu, 2011). Pythagorean fuzzy sets: Atanassov’s intuitionistic fuzzy sets of second type (IFS2) or Yager’s Pythagorean fuzzy sets are characterized by a membership degree and a nonmembership degree satisfying the condition that the square sum of membership and non-membership degrees is at most equal to one, which is a generalization of Intuitionistic Fuzzy Sets (IFS).
Group decision making with multiplicative interval linguistic hesitant fuzzy preference relations
2019, Information SciencesCitation Excerpt :Although HFVs are convenient to express the hesitancy judgments, Chen et al. [4] noted that HFVs cannot show the DMs’ uncertain hesitancy and proposed interval hesitant fuzzy variables (IHFVs) to address this problem. With respect to IHFVs, the aggregation operators are studied in [4,6,20], the cross-entropy and similarity measures of IHFVs are studied in [10], and the correlation coefficients of IHFVs are discussed in [21]. The above mentioned fuzzy sets need the DMs to provide quantitative information.
Yingdong He received the M.S. degree from School of Mathematical Sciences, Anhui University, Hefei, China, in 2014. He is currently working towards the Ph.D. degree at the College of Management & Economics, Tianjin University, Tianjin, China.
He has contributed about twenty articles to professional journals as the first author, such as IEEE Transactions on Fuzzy Systems, IEEE Transactions on Cybernetics, Information Sciences, Applied Soft Computing, Journal of the Operational Research Society, Expert Systems with Applications, Soft Computing, International Journal of Fuzzy Systems, etc. His current research interests include fuzzy mathematics, information fusion, group decision making, quality management and control, Six Sigma and the multiple response analysis.
He received the first prize in the tenth National Postgraduate Mathematical Contest in Modeling in 2013, the National Scholarship for Postgraduates of China Awards in 2013 and the National Scholarship for Ph.D. Students of China Awards in 2014 and 2015. Email: [email protected] ([email protected]); Tel.: +8615620057691.
Zhen He received the Ph.D. degree in management sciences and engineering from Tianjin University, Tianjin, China, in 2001.
He is a professor with the College of Management & Economics, Tianjin University, Tianjin, China. His main research interests include quality management and control, Six Sigma and Lean Production. He has published over 150 journal papers in IEEE Transactions on Fuzzy Systems, IEEE Transactions on Cybernetics, European Journal of Operational Research, Quality Progress, Total Quality Management & Business Excellence, International Journal of Production Research, International Journal of Production Economics, Quality and Reliability Engineering International, Applied Soft Computing, Journal of the Operational Research and other international and domestic academic journals.
He is an Academician of the International Academy for Quality, the recipient of NSFC (National Natural Science Foundation of China) Outstanding Young Scholars and the recipient of the New Century Excellent Talents program, Ministry of Education of China. Email: [email protected]
Liangxing Shi received the Ph.D. degree in management sciences and engineering from Tianjin University, Tianjin, China, in 2008.
He is an associate professor with the College of Management & Economics, Tianjin University, Tianjin, China. In 2012, he was a visiting scholar with the Department of Statistics at the Pennsylvania State University, US. His main research interests include quality management and control, Six Sigma and decision making. He has published about thirty articles in European Journal of Operational Research, IEEE Transactions on Knowledge and Data Engineering, Quality and Reliability Engineering International, Asia Journal of Quality, International Journal of Fuzzy Systems, etc. He is also a paper reviewer of many professional journals. He has directed and completed over ten research projects. For further information, see his webpage: http://come.tju.edu.cn/jxsz/xysz/S/201301/t20130112_169531.htm
Shanshan Meng is currently working towards the Ph.D. degree at the College of Management & Economics, Tianjin University, Tianjin, China.