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Hesitant Fuzzy Soft Sets with Application in Multicriteria Group Decision Making Problems
Soft sets have been regarded as a useful mathematical tool to deal with uncertainty. In recent years, many scholars have shown an intense interest in soft sets and extended standard soft sets to intuitionistic fuzzy soft sets, interval-valued fuzzy soft sets, and generalized fuzzy soft sets. In this paper, hesitant fuzzy soft sets are defined by combining fuzzy soft sets with hesitant fuzzy sets. And some operations on hesitant fuzzy soft sets based on Archimedean t-norm and Archimedean t-conorm are defined. Besides, four aggregation operations, such as the HFSWA, HFSWG, GHFSWA, and GHFSWG operators, are given. Based on these operators, a multicriteria group decision making approach with hesitant fuzzy soft sets is also proposed. To demonstrate its accuracy and applicability, this approach is finally employed to calculate a numerical example.
Since the fuzzy set (FS) was proposed by Zadeh in 1965 , it has been widely studied, developed, and successfully applied in various fields, such as multicriteria decision making (MCDM) [2, 3], fuzzy logic and approximate reasoning , and pattern recognition . In real MCDM cases, due to the fuzziness and uncertainty of decision making problems, the criteria’s weights and evaluation values of alternatives can be inaccurate, uncertain, or incomplete. For the problems like those, FSs, especially fuzzy numbers, can provide good solutions. However, in FSs the membership degree of the element is represented by a single value between zero and one, and a major drawback of FSs is that single values cannot convey information precisely.
In practice, the information regarding alternatives, when referring to a fuzzy concept, may be incomplete; that is, the sum of the membership and nonmembership degree of an element in the universe can be less than one. The FS fails when it comes to managing the insufficient understanding of membership degrees. Thus, Atanassov’s intuitionistic fuzzy sets (IFSs), interval-valued intuitionistic fuzzy sets (IVIFSs), and trapezoidal or triangular intuitionistic fuzzy sets, as the extensions of Zadeh’s FSs, were introduced [6–11]. IFSs and IVIFSs have been widely applied in solving MCDM problems [10, 12–14].
However, in some cases, the membership degree of an element is neither a single value nor an interval, but a set of possible values. To manage such situations where decision-makers are hesitant in expressing their preferences over alternatives, hesitant fuzzy sets (HFSs), another extension of traditional FSs, provide a useful reference. HFSs are first introduced by Torra [15, 16] and permit the membership degree of an element to be a set of several possible values between 0 and 1. HFSs are tremendously useful in handling the situations where people have hesitancy in providing their preferences over objects in a decision making process. The aggregation operators of HFSs were studied and applied to MCDM problems in [17–20]. Besides, Wang et al.  provided an outranking approach with HFSs to solve MCDM problems. Yu et al.  and Chen et al.  discussed the correlation coefficients of HFSs and their applications to clustering analysis. Xu and Xia [24, 25] discussed the distance and correlation measures for HFSs.
However, in some practical cases, FSs and their extensions failed to model uncertain data being of various types because of their inadequacy as a parameterization tool. To overcome this difficulty, Molodtsov  proposed soft sets (SSs), which were considered as a useful mathematical tool for dealing with uncertainties which is free from the difficulties affecting the exiting methods. After that, many scholars have shown an intense interest in this. Maji et al.  gave a theoretical study of SSs. They  also described the application of SSs in a decision making problem. In addition, the operations of SSs were extended in [29–31]. Min  proposed the similarity measures between SSs. Aktaş and Çağman  gave a definition of soft groups and discussed the basic properties. Acar et al.  introduced soft rings. Gong et al.  proposed bijective SSs and the corresponding operations. Furthermore, the relations and functions of SSs were studied by Babitha and Sunil . Ali et al. studied the algebraic structures of SSs  and also discussed the idea of reduction of parameters in case of SSs . Çağman and Enginoğlu  defined the products of SSs and uni-int decision function and then constructed a uni-int decision making method. Feng et al.  improved Çağman and Enginoğlu’s uni-int decision making method based on choice value being in the form of SSs.
But situations in real world may be complex because of the fuzzy nature of parameters. To deal with such situation, Maji et al.  extended classical SSs to fuzzy soft sets (FSSs). Borah et al.  discussed some operations of FSSs. Guan et al.  gave a new order relation of FSSs and Feng et al.  studied the decomposition of FSSs with finite value spaces. Similar to FS theory, several new extensional concepts based on FSSs are given. For example, Maji et al.  introduced intuitionistic fuzzy soft sets (IFSSs) by integrating SSs with IFSs. To overcome the difficulties of representation of parameter’s vagueness, Xu et al.  introduced vague soft set (VSSs) and Zhou and Li  defined generalized vague soft sets (GVSSs). By combining IVFSs and SSs, Yang et al.  proposed interval-valued fuzzy soft sets (IVFSSs). Ma et al.  analyzed the parameter reduction of IVFSSs and Jiang et al.  studied the entropy of IFSSs and IVFSSs. In addition, Majumdar and Samanta  defined generalized fuzzy soft sets (GFSSs). Moreover, FSSs have been also successfully applied in MCDM problems in recent years. Roy and Maji  gave an approach to decision making problems. By means of level soft sets, Feng et al.  presented an adjustable approach to FSSs based decision making. Kong et al.  presented a decision making algorithm of FSSs based on grey theory. Mitra Basu et al.  proposed a balanced solution of FSSs in medical science. Xiao et al.  integrated the fuzzy cognitive map and FSSs for solving the supplier selection problem. In [57, 58], the approaches to MCDM based on IFSSs are given. Jiang et al.  presented an adjustable approach to IFSSs-based decision making by using level soft sets of IFSSs. To deal with the problems of subjective evaluation and uncertain knowledge, Xiao et al.  proposed an evaluation method based on GFSSs and its application in medical diagnosis problem. They also  extended classical SSs to trapezoidal fuzzy soft sets (TFSSs) and applied them to MCDM problems. Zhang et al.  applied generalized TFSSs to medical diagnosis. Zhang  presented a rough set approach to IFSSs-based decision making.
IFSSs , VSSs , IVFSSs , and GFSSs  are all proposed to deal with uncertainties by taking advantages of SSs. In those extensions of SSs, the value of membership is either a single value or an interval. But in fact, the membership degree may be a set of possible values in a SS, so the purpose of this paper is to deal with this situation by combining HFSs with FSSs. To do this, a new kind of SSs, hesitant fuzzy soft sets (HFSSs), can be defined. HFSSs can represent various different preferences from different decision-makers and avoid overlooking any subjective intentions of decision-makers. Babitha and John  introduced HFSSs and analyzed some basis operations. However, the MCDM method proposed by Babitha and John  was not persuadable. In fact, HFSSs are more suitable for multiple criteria group decision making (MCGDM) problems. In this paper, a further study of HFSSs and their application in MCGDM problems are given.
The rest of this paper is organized as follows. In Section 2, some basic concepts of t-norm, t-conorm, SSs, FSSs, and HFSs are briefly reviewed. In Section 3, the concept of HFSSs and the corresponding operations are introduced. In Section 4, some aggregation operators of HFSSs are given. Based on hesitant fuzzy soft numbers (HFSNs), a MCGDM approach is proposed in Section 5. An illustrative example is given in Section 6 and the conclusions are provided in Section 7.
In this section, some basic concepts of t-norm, t-conorm, SSs, FSSs, and HFSs are reviewed.
2.1. T-Norm and T-Conorm
Definition 3 (see [65, 66]). A t-norm function is called Archimedean t-norm if it is continuous and, , . If , is strictly increasing, can be called strictly Archimedean t-norm. A t-conorm function is called Archimedean t-conorm if it is continuous and, , . If , is strictly increasing, can be called strictly Archimedean t-conorm.
It is well known  that a strictly Archimedean t-norm is generated by its additive generator as and is a strictly decreasing function: such that . Let , and then Archimedean t-conorm can be expressed as .
If we use specific forms to represent , then some t-norms and t-conorms can be obtained.(1)Assuming , then , , and . Algebraic t-conorm and t-norm  can be obtained: , and .(2)Assuming , then , , and . Einstein t-conorm and t-norm  can be obtained: , and .
2.2. Soft Sets and Fuzzy Soft Sets
In this subsection, the definitions of SSs and FSSs are introduced.
Definition 4 (see ). Let be an initial universe and let be a set of parameters. A pair is called soft set (SS) over , where is a mapping of into the set of all subsets of .
In other words, any SS is a parameterized family of subsets of the set . , may be considered the set of elements of the sets , or the set of -approximate elements of the SS. To illustrate this idea, let us consider the following example.
Example 5. Suppose that is a set of houses and is a set of parameters, which stand for being beautiful, being cheap and being in the green surroundings, respectively. Consider the mapping from parameter set to the set of all subsets of . Then SS can describe an “attractive” house that Mr. X is going to buy:
Thus, we can view the SS as a collection of approximations as follows: beautiful houses = , cheap houses = , in the green surroundings houses = .
Definition 6 (see ). Let be the set of all fuzzy subsets of , and then a pair is called a fuzzy soft set (FSS) over , where is a mapping denoted by
Example 7. Consider Example 5. If Mr. X thinks is a little expensive and this fuzzy information cannot be expressed only by two crisp numbers, that is, 0 and 1, a membership degree can be used instead, which is associated with each element and represented by a real number in the interval . Then FSS can describe the “attractive” house that Mr. X is going to buy under the fuzzy information:
2.3. Hesitant Fuzzy Sets
Definition 8 (see ). Let be a universal set, and then a hesitant fuzzy set (HFS) on is defined in terms of a function that when applied to returns a finite subset of . A HFS can be represented by where is a set of values in , denoting the possible membership degrees of the element to the set E. is called a hesitant fuzzy element (HFE) , and is the set of all HFEs. In particular, if has only one element, we call a hesitant fuzzy number (HFN), briefly denoted by . The set of all hesitate fuzzy numbers is represented as HFNs.
Definition 9 (see ). Let , and three operations are defined as follows:(1);(2);(3).
The arithmetical operations of HFNs based on Archimedean t-norm and Archimedean t-conorm are defined as follows.
Definition 10 (see ). Let , , and be three HFNs, and . Four operations are defined as follows:(1);(2);(3);(4). In particular, if , then(5);(6);(7);(8). If , then(9);(10);(11);(12).
Definition 11 (see ). For , is called the score function of , where is the number of elements in . For two HFNs and , if , then ; if , then .
It is clear that Definition 11 does not consider the situation where two HFNs and have the same score, but their deviation degrees may be different. The deviation degree of all elements with respect to the average value in a HFN reflects how elements are consistent with each other, that is, whether they have a higher consistency or not. To better represent this issue, Chen et al.  defined the deviation degree as follows.
Definition 12 (see ). For , is defined as the variance of , where is the score function of , and denotes the deviation degree of .
Definition 13 (see ). Let and be two HFNs; if , then .
If , then(1)if , then ;(2)if , then ;(3)if , then .
3. Hesitant Fuzzy Soft Sets and Their Operations
In this section, a HFSS is defined by combining HFSs and SSs; some operations on HFSSs based on Archimedean t-norm and Archimedean t-conorm are also given.
3.1. Definition of Hesitant Fuzzy Soft Sets
HFSs and SSs are integrated as mentioned in Section 2. The concept of HFSSs is defined as follows.
Definition 14. Let be an universe, let be a set of parameters, and let be the set of all hesitant fuzzy subsets of . A pair is called a HFSS over , where is a mapping denoted by
A HFSS is a parameterized family of hesitant fuzzy subsets of , that is, . For all is referred to as the set of -approximate elements of the HFSS . It can be written as
Since HFE can represent the situation, in which different membership functions are considered possible , is a set of several possible values, which is the hesitant fuzzy membership degree. In particular, if has only one element, can be called a hesitant fuzzy soft number (HSSN). For convenience, a HSSN is denoted by .
Example 15. Consider Example 7. Mr. X found it was hard to give a single value to express his opinion about the houses with respect to different criteria. For example, Mr. X thinks that the degree of house satisfies that criterion “beautiful” is 0.3 or 0.2. Then a HFSS can be used to describe the “attractive” house that Mr. X is going to buy:
3.2. Operations on Hesitant Fuzzy Soft Sets
In this subsection, some operations on HFSNs are defined.
Definition 16. Let and be two HFSNs and . Then the following can be defined:(1);(2);(3);(4);(5).
Theorem 17. Let and be two HFSNs and . Then the following are true:(1);(2);(3);(4);(5);(6).
4. Aggregation Operators of Hesitant Fuzzy Soft Sets
In this section, some aggregation operators are defined.
Definition 18. Let be the elements of the HFSS , and let be the weight vector of , where indicates the importance degree of , satisfying and . Then the HFSWA operator can be called hesitant fuzzy soft weighted averaging operator and defined as
Theorem 19. Let be the elements of the HFSS , and let be the weight vector of , where indicates the importance degree of , satisfying and . Then the aggregated value by using the HFSWA operator is still a HFSN, and
Proof. Use the mathematical introduction of to complete this proof.
For , we have
Suppose (10) holds for ; that is,
Now, (10) holds for , and thus (10) holds for all .
Subsequently, some desirable properties of the operator are investigated.
Property 1. If all are equal, that is, for all , then
Proof. Let , and then
Property 2. Let be the elements of the HFSS , and let be the weight vector of such that and . If is a hesitant fuzzy soft element (HFSE), then
Proof. Since , then Therefore, HFSWA HFSWA .
Property 3. Let be the elements of the HFSS , and let be the weight vector of such that and . If , then
Property 4. Let be all the elements of the HFSS , and let be the weight vector of , such that and . If and is a HFSE, then
Property 5. Let be the elements of the HFSS , let be the elements of HFSS , and let be the weight vector of them, satisfying and . Then
Definition 20. Let be the elements of the HFSS , and let be the weight vector of , where indicates the importance degree of , satisfying and . Then the operator can be called hesitant fuzzy soft weighted geometric operator and defined as
Theorem 21. Let be the elements of the HFSS , and let be the weight vector of , where indicates the importance degree of , satisfying and . Then the aggregated value by using HFSWG operator is a HFSN, and
Similarly, it is clear that the operator also satisfies the properties that operator has, and the relative details are omitted here.
Definition 22. Let be the elements of the HFSS , and let be the weight vector of , where indicates the importance degree of , satisfying and . Then the operator can be called the generalized hesitant fuzzy soft weighted averaging operator and defined as
In particular, if , then the operator is reduced to the operator.
Theorem 23. Let be the elements of the HFSS , and let be the weight vector of , where indicates the importance degree of , satisfying and . Then the aggregated value by using GHFSWA operator is a HFSN, and
Definition 24. Let be the elements of the HFSS , and let be the weight vector of , where indicates the importance degree of , satisfying and . Then the GHFSWG operator can be called the generalized hesitant fuzzy soft weighted geometric operator and defined as
In particular, if , then the operator is reduced to the operator.
Theorem 25. Let be the elements of the HFSS , and let be the weight vector of , where indicates the importance degree of , satisfying and . Then the aggregated value by using GHFSWG operator is a HFSN, and
Similarly, it is clear that the GHFSWA and GHFSWG operators possess the same properties that the operator has.
Example 26. Suppose and are the elements of the HFSS , and is the weight vector of , which indicate the corresponding importance degrees.