Elsevier

Computers & Industrial Engineering

Volume 128, February 2019, Pages 605-621
Computers & Industrial Engineering

Fuzzy β-covering based (I,T)-fuzzy rough set models and applications to multi-attribute decision-making

https://doi.org/10.1016/j.cie.2019.01.004Get rights and content

Highlights

  • Four types of fuzzy coverings-based (I,T)-FRS models are proposed.

  • The relationships among new proposed models are investigated.

  • Two approaches to MADM problems based on the proposed models are presented.

  • Efficient algorithms to solve MADM problems are developed.

Abstract

Multi-attribute decision-making (MADM) can be regarded as a process of selecting the optimal one from all alternatives. Traditional MADM problems with fuzzy information are mainly focused on a fundamental tool which is a fuzzy binary relation. However some complicated problems cannot be effectively solved by a fuzzy relation. For this reason, in order to solve these issues, we set forth two decision-making methods that are stated in terms of novel and flexible fuzzy rough set models. For the purpose of defining these models we employ a fuzzy implication operator I and a triangular norm T. With these adaptable tools we design four kinds of fuzzy β-coverings based (I,T)-fuzzy rough set models. The elements that make these models different are the combination of fuzzy β-neighborhoods that intervene in the definitions of the lower (upper) approximations. Then, we discuss the relationships among these given four types of models. Finally, we propose two novel methodologies to solve MADM problems with evaluation of fuzzy information, which rely on these models. Through the analysis of the ranking results of these two methods, we observe that the optimal selected alternative is the same, which means that these two decision-making methods are reasonable. In addition, by comparing the ranking results of these two methods and the existing traditional methods (WA operator and TOPSIS), we observe that our proposed methods can solve the ranking problems that traditional methods cannot solve, which means that our proposed methods are superior to traditional methods.

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 β-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 β-coverings instead of fuzzy coverings, where the β parameter lies in the interval (0,1]. The models that arise are more general because when β=1, a fuzzy β-covering reduces to a fuzzy covering. With the help of fuzzy β-coverings, Ma (2016) defined another class of fuzzy β-covering based fuzzy rough sets (FCFRCs) by a fuzzy β-neighborhood. Yang and Hu, 2016, Yang and Hu, 2017 further developed the ideas of a fuzzy β-minimum description and a fuzzy complementary β-neighborhood. Based on the insights of Ma, 2016, Yang and Hu, 2017 also constructed three kinds of fuzzy β-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 β-covering based fuzzy rough set models. Therefore, in order to fill that gap we intend to combine fuzzy β-covering based (I,T)-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 U={u1,u2,u3,u4} represent four engineers and C={C1,C2,C3,C4} represent four attributes, then in the standard notation we describe as follows:C1=0.3u1+0.3u2+0.4u3+0.6u4,C2=0.5u1+0.5u2+0.4u3+0.2u4,C3=0.4u1+0.5u2+0.6u3+0.4u4,C4=0.5u1+0.4u2+0.3u3+0.5u4.

Suppose that β is the threshold value and β=0.5, then, according to (Ma, 2016), (U,C) is a fuzzy 0.5-covering approximation space. We utilize the attributes weight vector W={0.3,0.3,0.2,0.2} that expresses the importance of these four attributes. According to (Yager, 1998), the ranking result when we apply the WA operator method is u1u2u3u4, 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 β-coverings based (I,T)-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 β-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 β-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 β-coverings based (I,T)-fuzzy rough set models, and discuss the basic properties of each model. We mainly explore the mutual relations among four types of fuzzy β-coverings based (I,T)-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

Radzikowska and Kerre, 2002

  • 1.

    For a mapping T:[0,1]×[0,1][0,1], if it verifies the commutative, associative and increasing laws with the boundary condition T(1,u)=u for each u[0,1], then we refer to T as a t-norm of [0,1].

    Well-known continuous t-norms include:

    • (1)

      The min operator TM(u,v)=uv,

    • (2)

      The algebraic product TP(u,v)=uv,

    • (3)

      The Łukasiewicz t-norm TL(u,v)=0(u+v-1).

  • 2.

    For a mapping S:[0,1]×[

Four fuzzy β-coverings based (I,T)-fuzzy rough set models

We now proceed to introduce four different fuzzy β-coverings based (I,T)-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 (U,C) is an FCAS and C={C1,C2,,Cm} is a fuzzy β-covering of U, where β(0,1]. When AF(U), the following are true:

  • (1)

    C4-(A)C1-(A)C3-(A), provided that I satisfies the left monotonicity,

  • (2)

    C4-(A)C2-(A)C3-(A), 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 α. 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 β-coverings based (I,T)-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 β-neighborhoods, fuzzy complement β-neighborhoods and fuzzy logical operators (fuzzy implicators and t-norm operators), we have defined four kinds of fuzzy β-coverings based (I,T)-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).

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