Fuzzy -covering based -fuzzy rough set models and applications to multi-attribute decision-making
Introduction
As is well known that multi-attribute decision-making (MADM) is a major part of decision-making, which in turn plays an important role in management science, operations research, industrial engineering, and so on. Experts have proposed many direct application methods (He and Xu, 2018, Liang et al., 2018, Sun et al., 2018, Wu et al., 2018, Xu et al., 2015, Xu et al., 2018) for various practical problems. However, in the process of decision-making, we often encounter practical problems with complex information (like fuzzy information and intuitional fuzzy information). In recent years, methods based on aggregation operators (Chaji et al., 2018, Xu, 2008, Xu and Yager, 2006, Yager, 1998, Yager, 1993), on TOPSIS (Arslan and Cunkas, 2012, Boran et al., 2009, Chen, 2000, Vencheh and Mirjaberi, 2014, Wang and Duan, 2018, Yue, 2014) and on TODIM (Fan et al., 2013, Gomes and Rangel, 2009, Krohling et al., 2013, Lourenzutti and Krohling, 2013) in different environments have been proposed and investigated. In order to expand the scope of fuzzy rough set theory in MADM, in this paper we aim at building two new decision-making methods for MADM problems by means of four kinds of novel fuzzy rough set models based on fuzzy -coverings. Let us review the development of fuzzy rough set theory to better situate our research in that context.
In 1982, Pawlak (1982) initiated rough set theory (RST) which is a tool that deals with inaccurate, incomplete information systems. Till now, this uncertain theory has been widely used in knowledge discovery, conflict analysis, physics, particle computing, and so on (see, e.g. (Angiulli and Pizzuti, 2005, Dick et al., 2007, Zadeh, 2005, Zhong et al., 2003)). Originally, this uncertain theory is based on an equivalence relation. But the recourse to an equivalence relation imposes very stringent conditions that may cause to limit the application domain of the Pawlak’s RST. The reason is that agents have a limited ability to deal with that notion (Alcantud, 2002). Therefore, it comes as no surprise that many authors generalized equivalence relations in Pawlak’s design of rough sets to non-equivalence relations (see e.g. (Bonikowski et al., 1998, Dai et al., 2017, D’eer et al., 2016, Sun et al., 2017, Sun et al., 2013, Yao, 1998, Zhan et al., 2018, Zhan et al., 2017, Zhang et al., 2018, Zhu, 2007)).
On the other hand, classical RST has encountered a bottleneck restriction when dealing with quantitative data. In order to overcome this drawback, Zadeh (1965) began to combine fuzzy set theory with RST due to its potential to cope with graded indiscernibility together with vagueness. In their founding contribution in this strand of the literature, and with reference to the Min t-norm and its dual conorm Max, Dubois and Prade (1990) presented the extended notions of fuzzy rough sets (FRSs) and rough fuzzy sets (RFSs) by defining the lower and upper approximations of fuzzy sets w.r.t. a fuzzy similarity relation and a Pawlak approximation space, respectively. Radzikowska and Kerre (2002) developed FRSs based on a fuzzy logical operator and a fuzzy similarity relation, and in particular they constructed three kinds of lower and upper approximation operators w.r.t. three types of implicators (S-implicators, R-implicators and QL-implicators). Shortly thereafter, Wu et al., 2005, Wu et al., 2013, Wu et al., 2016 studied these approximation operators both from a constructive and an axiomatic point of perspective.
Another direction for the generalization of Pawlak’s rough sets consists of replacing the partition induced from the equivalence binary relation by a more general concept named a covering. Zakowski (1983) was the first to propose covering based rough set theory (CRS), which soon attracted wide attention (D’eer et al., 2016, Deng et al., 2007, Ma, 2012, Yao and Yao, 2012, Zhan et al., 2018, Zhu, 2007). With a more general perspective, some authors extended classical CRSs to fuzzy covering based rough sets (FCRSs). For example, by using fuzzy logic operators, Li, Leung, and Zhang (2008) produced two pairs of generalized lower (upper) approximation operators based on fuzzy coverings. In addition, D’eer and Cornelis, 2018, D’eer et al., 2017 explored a variety of fuzzy neighborhood operators based on a fuzzy neighborhood system, a fuzzy minimum description and a fuzzy maximum description. However, the definition of fuzzy coverings is still limited for practical research. Due to this observation, Ma (2016) suggested to utilize fuzzy -coverings instead of fuzzy coverings, where the parameter lies in the interval . The models that arise are more general because when , a fuzzy -covering reduces to a fuzzy covering. With the help of fuzzy -coverings, Ma (2016) defined another class of fuzzy -covering based fuzzy rough sets (FCFRCs) by a fuzzy -neighborhood. Yang and Hu, 2016, Yang and Hu, 2017 further developed the ideas of a fuzzy -minimum description and a fuzzy complementary -neighborhood. Based on the insights of Ma, 2016, Yang and Hu, 2017 also constructed three kinds of fuzzy -coverings based fuzzy rough set models.
To the best of our knowledge, the decision-making application about fuzzy rough sets is rarely (see e.g. (Sun et al., 2018, Zhan et al., 2019)) and the literature provides few decision-making applications for fuzzy -covering based fuzzy rough set models. Therefore, in order to fill that gap we intend to combine fuzzy -covering based -fuzzy rough set models with decision-making applications in the present paper. A possibility that soon comes to mind is the recourse to weighted aggregation (WA) operators in (Yager, 1998) and elsewhere. Indeed we are aware that this notion is of paramount importance in MADM with the fuzzy environment and granular information (Huang, Tseng, & Tang, 2016), as well as other branches (Human Development Index,1 intergenerational equity (Alcantud and Garcí a-Sanz, 2010, Scarborough, 2011), polarization indices (Chakravarty and Maharaj, 2015, Esteban and Ray, 2004), et cetera). Nevertheless, it should not be the unique tool to approach the covering based fuzzy problems. Let us argue by example. To this purpose, assume that a software company wishes to hire a system analysis engineer. Let represent four engineers and represent four attributes, then in the standard notation we describe as follows:
Suppose that is the threshold value and , then, according to (Ma, 2016), is a fuzzy 0.5-covering approximation space. We utilize the attributes weight vector that expresses the importance of these four attributes. According to (Yager, 1998), the ranking result when we apply the WA operator method is , which results into failure to choose a best candidate since they are all indistinguishable.
Motivated by this problem, and based on fuzzy rough set models in Dubois and Prade, 1990, Ma, 2016, Yang and Hu, 2017, fuzzy logical operators (triangular norm operators and fuzzy implication operators) and ideas of decision-making methods (fuzzy TOPSIS methods and OWA operator methods), we are enabled to construct four kinds of fuzzy -coverings based -fuzzy rough set models and to propose suitable decision-making methods that take advantage of their more interesting characteristics. We also explain how these models solve the above issues.
Our research is in continuation of the valuable line of research. Drawing from the ideas above, four models of fuzzy -coverings based fuzzy rough sets will be established in this work by means of fuzzy logical operators. In order to explicitly show the relationship between the fuzzy covering based rough sets investigated in this paper and Pawlaks rough sets and their extensions, we give Fig. 1 below. This figure captures the fact that fuzzy -coverings based fuzzy rough set models are both a combination and an extension of FRS theory and CRS theory.
The outline of the present paper is listed as follows. We review some basic terminologies about fuzzy logical operators and fuzzy coverings in Section 2. In Section 3, we set forth four kinds of fuzzy -coverings based -fuzzy rough set models, and discuss the basic properties of each model. We mainly explore the mutual relations among four types of fuzzy -coverings based -fuzzy rough set models in Section 4. In Section 5, we propose two types of different decision-making methods. Then, numerical analysis, sensitivity analysis and comparison analysis among our proposed methods and other methods are investigated in Section 6. Finally, we conclude our work with a summary of the paper in Section 7. Here we also outline some possibilities for further research.
Section snippets
Preliminaries
Let us introduce some basic concepts. We begin with the definitions of fuzzy logical operators, a powerful tool for the study of fuzzy logic: Definition 2.1 For a mapping , if it verifies the commutative, associative and increasing laws with the boundary condition for each , then we refer to as a t-norm of . Well-known continuous t-norms include: The min operator , The algebraic product , The ukasiewicz t-norm .Radzikowska and Kerre, 2002
For a mapping
Four fuzzy -coverings based -fuzzy rough set models
We now proceed to introduce four different fuzzy -coverings based -fuzzy rough set models. Afterwards we examine their basic properties.
The relationships among FCITFRS models of four types
In Section 3, we have introduced four types of FCITFRS models. We now proceed to establish the relationships among the FCITFRSs that we have constructed, which clearly share some structural similarities.
Our arguments rely on the following list of inclusions: Proposition 4.1 Assume that is an FCAS and is a fuzzy -covering of U, where . When , the following are true: , provided that satisfies the left monotonicity, , provided
Two approaches to MADM with evaluation of fuzzy information based on FCITFRS models
In general, the numerical information of the MADM problem is vague or inaccurate. Moreover, we know that fuzzy rough set theory, as a mathematical theory dealing with inconsistencies in numerical data, has a unique role in dealing with vague or inaccurate data. Therefore, the combination of fuzzy rough sets and MADM methods will have unexpected advantages.
Furthermore, as argued above, the applications of covering based fuzzy rough sets in decision-making are rather limited. In this section, we
Numerical analysis, sensitivity analysis and comparison analysis
In this section, we utilize our proposed methods I and II to solve an MADM problem with fuzzy information. In Section 6.1, we present a related example for these two methods. In Section 6.2, we perform a sensitivity analysis in relation with the values of the parameter . Finally, in Section 6.3, we use an example in Section 6.1 to compare the differences between our proposed methods and the existing traditional decision-making methods. Besides, we also give another practical example to
Conclusion
In this paper we have established four kinds of fuzzy -coverings based -fuzzy rough set models. Besides, we have proposed two decision-making methods based on FCITFRS models. The core content of this paper is summarized as follows:
- (1)
Through a suitable blend of fuzzy -neighborhoods, fuzzy complement -neighborhoods and fuzzy logical operators (fuzzy implicators and t-norm operators), we have defined four kinds of fuzzy -coverings based -fuzzy rough set models.
- (2)
By analysis, the lower
Acknowledgments
The authors are extremely grateful to the editor and three anonymous referees for their valuable comments and helpful suggestions which helped to improve the presentation of this paper.
The research was partially supported by the NNSFC (Grant Nos. 11461025, 11561023, 61573321, 41631179) and the Zhejiang Provincial NSFC (Grant No. LY18F030017).
References (67)
Revealed indifference and models of choice behavior
Journal of Mathematical Psychology
(2002)- et al.
Paretian evaluation of infinite utility streams: an egalitarian criterion
Economic Letters
(2010) - et al.
Extension and intentions in the rough set theory
Information Sciences
(1998) - et al.
A multi-criteria intuitionistic fuzzy group decision making for supplier selection with TOPSIS method
Expert Systems with Applications
(2009) - et al.
Selecting a model for generating OWA operator weights in MAGDM problems by maximum entropy membership function
Computers & Industrial Engineering
(2018) Extensions of the TOPSIS for group decision-making under fuzzy environment
Fuzzy Sets and Systems
(2000)- et al.
Uncertainty measurement for incomplete interval-valued information system based on -weak similarity
Knowledge-Based Systems
(2017) - et al.
A comprehensive study of fuzzy covering-based rough set models: definitions, properties and interrelationships
Fuzzy Sets and Systems
(2018) - et al.
Fuzzy neighborhood operators based on fuzzy coverings
Fuzzy Sets and Systems
(2017) - et al.
A semantically sound approach to Pawlak rough sets and covering based rough sets
International Journal of Approximate Reasoning
(2016)
Neighborhood operators for covering based rough sets
Information Sciences
A novel approach to fuzzy rough sets based on a fuzzy covering
Information Sciences
Regranulation: a granular algorithm enabling communication between granular worlds
Information Sciences
Strategic weight manipulation in multiple attribute decision making
Omega
Extended TODIM method for hybrid multiple attribute decision making problems
Knowledge-Based Systems
A consensus framework with different preference ordering structures and its applications in human resource selection
Computers & Industrial Engineering
Feature extraction using rough set theory in sector application from incremental perspective
Computers & Industrial Engineering
IF-TODIM: An intuitionistic fuzzy TODIM to multi-criteria
Knowledge-Based Systems
A consistency-based approach to multiple attribute decision making with preference information on alternatives
Computers & Industrial Engineering
Generalized fuzzy rough approximation operators based on fuzzy coverings
International Journal of Approximate Reasoning
A study of TODIM in a intuitionistic fuzzy and random environment
Expert Systems with Applications
On some types of neighborhood-related covering rough sets
International Journal of Approximate Reasoning
Two fuzzy covering rough set models and their generalizations over fuzzy lattices
Fuzzy Sets and Systems
A comparative study of fuzzy rough sets
Fuzzy Sets and Systems
Heterogeneous multigranulation fuzzy rough set-based multiple attribute group decision making with heterogeneous preference information
Computers & Industrial Engineering
Three-way decisions approach to multiple attribute group decision making with linguistic information-based decision-theoretic rough fuzzy set
International Journal of Approximate Reasoning
Multigranulation fuzzy rough set over two universes and its application to decision making
Knowledge-Based Systems
A fuzzy rough set approach to emergency material demand prediction over two universes
Applied Mathematical Modelling
Fuzzy rough sets, fuzzy preorders and fuzzy topologies
Fuzzy Sets and Systems
TOPSIS approach for multi-attribute decision making problems based on n-intuitionistic polygonal fuzzy sets description
Computers & Industrial Engineering
On characterizations of -fuzzy rough approximation operators
Fuzzy Sets and Systems
Generalized fuzzy rough approximation operators determined by fuzzy implicators
International Journal of Approximate Reasoning
Hesitant fuzzy linguistic projection model to multi-criteria decision making for hospital decision support systems
Computers & Industrial Engineering
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