Hesitant Fuzzy Soft Set and Its Applications in Multicriteria Decision Making

. Molodtsov’s soft set theory is a newly emerging mathematical tool to handle uncertainty. However, the classical soft sets are not appropriate to deal with imprecise and fuzzy parameters. This paper aims to extend the classical soft sets to hesitant fuzzy soft sets which are combined by the soft sets and hesitant fuzzy sets. Then, the complement, “AND”, “OR”, union and intersection operations are defined on hesitant fuzzy soft sets. The basic properties such as DeMorgan’s laws and the relevant laws of hesitant fuzzy soft sets are proved. Finally, with the help of level soft set, the hesitant fuzzy soft sets are applied to a decision making problem and the effectiveness is proved by a numerical example.


Introduction
In the real world, there are many complicated problems in economics, engineering, environment, social science, and management science.They are characterized with uncertainty, imprecision, and vagueness.We cannot successfully utilize the classical methods to deal with these problems because there are various types of uncertainties involved in these problems.Moreover, though there are many theories, such as theory of probability, theory of fuzzy sets, theory of interval mathematics, and theory of rough sets to be considered as mathematical tools to deal with uncertainties, Molodtsov [1] pointed out all these theories had their own limitations.Moreover, in order to overcome these difficulties, Molodtsov [1] firstly proposed a new mathematical tool named soft set theory to deal with uncertainty and imprecision.This theory has been demonstrated to be a useful tool in many applications such as decision making, measurement theory, and game theory.
The soft set model can be combined with other mathematical models.Maji et al. [2] firstly presented the concept of fuzzy soft set by combining the theories of fuzzy set and soft set together.Furthermore, Maji et al. [3][4][5] established the notion of intuitionistic fuzzy soft set which was based on a combination of the intuitionistic fuzzy set [6,7] and soft set models.By combining the interval-valued fuzzy set [8,9] and soft set, Yang et al. [10] presented the concept of the intervalvalued fuzzy soft set.Jiang et al. [11] initiated the concept of interval-valued intuitionistic fuzzy soft sets as an intervalvalued fuzzy extension of the intuitionistic fuzzy soft set theory or an intuitionistic fuzzy extension of the interval-valued fuzzy soft set theory.Recently, Xiao et al. [12] presented the trapezoidal fuzzy soft set and Yang et al. [13] introduced the multifuzzy soft set, respectively, and applied them in decision making problems.Roy and Maji [14], Kong et al. [15], Feng et al. [16], and Jiang et al. [17] also applied the fuzzy soft set in the decision making problems.Jun [18] initiated the application of soft sets in BCK/BCI-algebras and introduced the concept of soft BCK/BCI-algebras.Furthermore, Jun and Park [19] and Jun et al. [20,21] applied the soft sets in ideal theory of BCK/BCI-algebras and d-algebras.Feng et al. [22,23] initiated the concept of rough soft sets, soft rough sets, and soft rough fuzzy sets.
Recently, in order to tackle the difficulty in establishing the degree of membership of an element in a set, Torra and Narukawa [24] and Torra [25] proposed the concept of a hesitant fuzzy set.This new extension of fuzzy set can handle the cases that the difficulty in establishing the membership degree does not arise from a margin of error (as in intuitionistic or interval-valued fuzzy sets) or a specified possibility distribution of the possible values (as in type-2 fuzzy sets), but arises from our hesitation among a few different values [26].Thus the hesitant fuzzy set can more accurately reflect the people's hesitancy in stating their preferences over objects, compared to the fuzzy set and its many classical extensions.The purpose of this paper is to extend the soft set model to the hesitant fuzzy set, and, thus, we establish a new soft set model named hesitant fuzzy soft set.
The rest of this paper is organized as follows.We first review some background on soft set, fuzzy soft set, and hesitant fuzzy set in Section 2. In Section 3, the concepts and operations of hesitant fuzzy soft set are proposed and their properties are discussed in detail.In Section 4, we apply the proposed hesitant fuzzy soft set to a decision making problem and give an explicit algorithm.Finally, we conclude in Section 5.

Preliminaries
2.1.Soft Sets.Suppose that  is an initial universe set,  is a set of parameters, () is the power set of  and  ⊂ .
In other words, a soft set over  is a parameterized family of subsets of the universe .For  ∈ , () may be considered as the set of -approximate elements of the soft set (, ).
Example 2. Suppose that  = {ℎ 1 , ℎ 2 , ℎ 3 , ℎ 4 , ℎ 5 , ℎ 6 } is a set of houses and  = { 1 ,  2 ,  3 ,  4 ,  5 } is a set of parameters, which stands for the parameters "cheap, " "beautiful, " "size, " "location, " and "surrounding environment, " respectively.In this case, a soft set (, ) can be defined as a mapping from parameter set  to the set of all subsets of .In this way, the set of the houses with specific characteristics can be described.

Fuzzy Soft Sets
Definition 3 (see [2]).Let P() be the set of all fuzzy subsets of , a pair ( F, ) is called a fuzzy soft set over , where F is a mapping given by F :  → P().
Example 4. Reconsider Example 2. In real life, the perception of the people is characterized by a certain degree of vagueness and imprecision; thus, when people consider if a house ℎ 1 is cheap the information cannot be expressed with only two crisp numbers 0 and 1.Instead it should be characterized by a membership function  Ã() which associates with each element a real number in the interval [0, 1].Then, fuzzy soft set ( F, ) can describe the characteristics of the house under the fuzzy information.( Similarly, for the purpose of storing a fuzzy soft set in a computer, we could also represent the fuzzy soft set defined in Example 4 in Table 2.

Hesitant Fuzzy Sets
Definition 5 (see [25]).A hesitant fuzzy set (HFS) on  is in terms of a function that when applied to  returns a subset of [0, 1], which can be represented as the following mathematical symbol: where ℎ Ã() is a set of values in [0, 1], denoting the possible membership degrees of the element  ∈  to the set Ã.For convenience, we call ℎ Ã() a hesitant fuzzy element (HFE) and  the set of all HFEs.
Furthermore, Torra [25] defined the empty hesitant set and the full hesitant set.
To extend the shorter one, the best way is to add the same value several times in it [28].In fact, we can extend the shorter one by adding any value in it.The selection of this value mainly depends on the decision makers' risk preferences.Optimists anticipate desirable outcomes and may add the maximum value, while pessimists expect unfavorable outcomes and may add the minimum value.For example, let ℎ 1 = {0.1,0.2, 0.3}, let ℎ 2 = {0.4,0.5}, and let (ℎ 1 ) > (ℎ 2 ).To operate correctly, we should extend ℎ 2 to ℎ  2 = {0.4,0.4, 0.5} until it has the same length of ℎ 1 , the optimist may extend ℎ 2 as ℎ  2 = {0.4,0.5, 0.5} and the pessimist may extend it as ℎ  2 = {0.4,0.4, 0.5}.Although the results may be different if we extend the shorter one by adding different values, this is reasonable because the decision makers' risk preferences can directly influence the final decision.The same situation can also be found in many existing references [29][30][31].In this paper, we assume that the decision makers are all pessimistic (other situations can be studied similarly).We arrange the elements in ℎ Ã() in decreasing order, and let ℎ () Ã () be the th largest value in ℎ Ã().(1) M ⊆ Ñ (2) M ⊇ Ñ which can be denoted by M = Ñ.Given three HFEs, ℎ, ℎ 1 , and ℎ 2 , Torra [25] and Torra and Narukawa [24] defined the following HFE operations: (1) where ℎ − () = min ℎ() and ℎ + () = max ℎ() are the lower bound and upper bound of the given hesitant fuzzy elements, respectively.

The Concept of Hesitant Fuzzy Soft Sets
Definition 10.Let H() be the set of all hesitant fuzzy sets in ; a pair ( F, ) is called a hesitant fuzzy soft set over , where F is a mapping given by A hesitant fuzzy soft set is a mapping from parameters to H().It is a parameterized family of hesitant fuzzy subsets of .For  ∈ , F() may be considered as the set of approximate elements of the hesitant fuzzy soft set ( F, ).
Example 11.Continue to consider Example 2. Mr. X evaluates the optional six houses under various attributes with hesitant fuzzy element; then, hesitant fuzzy soft set ( F, ) can describe the characteristics of the house under the hesitant fuzzy information.
Similarly, we can also represent the hesitant fuzzy soft set in the form of Table 3 for the purpose of storing the hesitant fuzzy soft set in a computer.Definition 12. Let ,  ∈ . ( F, ) and ( G, ) are two hesitant fuzzy soft sets over . ( F, ) is said to be a hesitant fuzzy soft subset of ( G, ) if For all  ∈ , F() ⊆ G().
Definition 14.Two hesitant fuzzy soft sets ( F, ) and ( G, ) are said to be hesitant fuzzy soft equal if ( F, ) is a hesitant fuzzy soft subset of ( G, ) and ( G, ) is a hesitant fuzzy soft subset of ( F, ).
It is worth noting that in the above definition of complement, the parameter set of the complement ( F, ) is still the original parameter set , instead of ¬.  4 and 5, respectively.
Theorem 22 (De Morgan's laws of hesitant fuzzy soft sets).Let ( F, ) and ( G, ) be two hesitant fuzzy soft sets over U; we have

Conclusion
In this paper we consider the notion of hesitant fuzzy soft sets which combine the hesitant fuzzy sets and soft sets.Then we define the complement, "AND", "OR", union and intersection operations on hesitant fuzzy soft sets.The basic properties such as De Morgan's laws and the relevant laws of hesitant fuzzy soft sets are proved.Finally, we apply it to a decision making problem with the help of level soft set.Our research can be explored deeply in two directions in the future.Firstly, we can combine other membership functions and soft sets to make novel soft sets which have different forms; Secondly, we can not only explore the application of hesitant fuzzy soft set in decision making more deeply, but also apply the hesitant fuzzy soft set in many other areas such as forecasting and data analysis.

Table 1 :
The tabular representation of the soft set (, ).

Table 2 :
The tabular representation of the fuzzy soft set ( F, ).

Table 3 :
The tabular representation of the hesitant fuzzy soft set ( F, ).
Definition 15.A hesitant fuzzy soft set ( F, ) is said to be empty hesitant fuzzy soft set, denoted by Φ , if F() = φ for all  ∈ .
Definition 16.A hesitant fuzzy soft set ( F, ) is said to be full hesitant fuzzy soft set, denoted by Ũ , if F() = 1 for all  ∈ .

Table 4 :
The result of "AND" operation on ( F, ) and ( G, ).

Table 10 :
The tabular representation of the midlevel soft set (Δ F; mid) with choice values in Example 35.