Multicriteria Decision Making Method Based on the Higher Order Hesitant Fuzzy Soft Set

The main goal of this contribution is to introduce the concept of higher order hesitant fuzzy soft set as an extension of fuzzy soft set that encompasses most of the existing extensions of fuzzy soft set as special cases. Furthermore, this new concept provides us with a method for dealing with multicriteria fuzzy decision making problems which are difficult to explain in other existing extensions of fuzzy soft set theory, especially when problems involve parameters with different-dimensional levels.

An interesting extension of FSs is the hesitant fuzzy set (HFS) [10] in which the membership degree of an element is defined as a set of possible values. HFSs are quite suitable to deal with the situation where we have a set of possible values, rather than a margin of error (as in IFSs) or some possibility distribution on the possible values (as in T2FSs). Later, a number of other extensions of HFSs have been developed such as dual hesitant fuzzy sets (DHFSs) [11], generalized hesitant fuzzy sets (G-HFSs) [12], and hesitant fuzzy linguistic term sets (HFLTSs) [13].
However, HFS [10] and its recent generalization G-HFS [12] have their inherent drawbacks, because they express the membership degrees of an element to a given set only by crisp numbers or IFSs. In many practical decision making problems, the information provided by decision makers might often be described by FSs instead of crisp numbers or by other FS extensions instead of IFSs. This makes decision makers uncomfortable to provide exact crisp values or just IFSs for the membership degrees. Such a problem can be avoided by using the higher order HFS (HOHFS) introduced originally by the author in [14] for describing membership degrees. The HOHFS is fit for the situation where the decision makers have a hesitation among several possible memberships for an element. The HOHFS is the actual extension of HFS encompassing not only FSs, IFSs, VSs, T2FSs, FMSs, and HFSs, but also the recent extension of HFSs, called G-HFSs.
Since soft set theory was first initiated by Molodtsov in [15] as a new mathematical tool to deal efficiently with uncertainty and imprecision, many authors have become interested in applying this theory in various fields such as decision making [16], data analysis [17], and forecasting [18] recently. Moreover, a growing number of studies focus on the combination of FS and some extensions of FSs with soft set. Among them are, for instance, fuzzy soft set [19], intuitionistic fuzzy soft set [20], interval-valued fuzzy soft set [21], vague fuzzy soft set [22], and multifuzzy soft set [23].
The rationale behind this study is that on the one hand we tend to propose a more general extension of fuzzy soft set encompassing the existing ones, and on the other hand we try to model those situations where the existing kinds of fuzzy soft set are not applicable. We explain the rationale as follows.
As mentioned above and it will be discussed later, the HOHFS encompasses most of the fuzzy extensions as special cases, and therefore the combination of HOHFS with soft set 2 International Scholarly Research Notices contains most of the existing kinds of fuzzy soft set as special cases, too.
One of more interesting feature of the higher order hesitant fuzzy (HOHF) soft set which will be proposed in this contribution is that we can correspond each parameter to levels with different dimensions, the situation that cannot be modeled by the other kinds of fuzzy soft sets, specially by multifuzzy soft set. To make this clear, let us consider the situation discussed in [23]. Assume that = { 1 , 2 , 3 , 4 , 5 } is the universe set indicating the set of five kinds of colored drawing paper for engineering. Suppose the parameter set = { 1 , 2 , 3 } is a set of three criteria to evaluate the performance of these papers, where 1 stands for thickness which includes three levels: thick, average, and thin, 2 stands for color which consists of three levels: red, green, and blue, and 3 stands for ingredient which is made from three levels: cellulose, hemicellulose, and lignin. A company would like to select one of these papers depending on its performance. As discussed in [23], this decision problem can be analyzed using multifuzzy soft set. But if this decision problem is reconsidered slightly different from the original problem as 1 stands for thickness which includes three levels: thick, average, and thin, 2 stands for color which consists of three levels: red, green, and blue, and 3 stands for ingredient which is made from two levels: cellulose and hemicellulose where the level lignin is not considered, then, obviously, this situation cannot be handled not only by the existing kinds of fuzzy soft set theory, but also by multifuzzy soft set one in which all the parameters have to possess levels with the same dimension.
To deal with such cases, we should employ the concept of the HOHF soft set.
The present paper is organized as follows. An extension of HFSs, which is referred to as the higher order HFS (HOHFS), is introduced in Section 2. Section 3 presents HOHF soft sets and discusses the properties of HOHF soft sets. In Section 4, we apply the proposed HOHF soft sets to multicriteria decision making problems involving parameters whose levels have different dimensions. Section 5 presents an illustrative example. Finally, conclusion is drown in Section 6.

Preliminaries
This section is devoted to review the basic definitions and notions of fuzzy set (FS) and its new generalization which are originally referred to by Farhadinia [14] as the higher order hesitant fuzzy set (HOHFS). Actually, the HOHFS is a generalization of hesitant fuzzy set (HFS) introduced by Torra [10].
An ordinary fuzzy set (FS) in is defined [24] wherẽ: Here, ([0, 1]) denotes a family of crisp or fuzzy sets that can be defined within the universal set [0, 1].
It is noteworthy that most of the existing extensions of ordinary FS are special cases of G-Type FS, for instance, [25]  Definition 2 (see [26]). Let be the universe of discourse. A hesitant fuzzy set (HFS) on is symbolized by where ℎ( ), referred to as the hesitant fuzzy element (HFE), is a set of some values in [0, 1] denoting the possible membership degree of the element ∈ to the set .
As can be seen from Definition 2, HFS expresses the membership degrees of an element to a given set only by several real numbers between 0 and 1, while in many realworld situations assigning exact values to the membership degrees does not describe properly the imprecise or uncertain decision information. Thus, it seems to be difficult for the decision makers to rely on HFSs for expressing uncertainty of an element.
To overcome the difficulty associated with expressing uncertainty of an element to a given set, the concept of higher order hesitant fuzzy set (HOHFS) is introduced here to let the membership degrees of an element to a given set be expressed by several possible G-Type FSs.
Definition 4 (see [14]). Let be the universe of discourse. A higher order hesitant fuzzy set (HOHFS) on is defined in terms of a function that when applied to returns a set of G-Type FSs. A HOHFS is denoted bỹ whereh( ), referred to as the higher order hesitant fuzzy element (HOHFE), is a set of some G-Type FSs denoting the possible membership degree of the element ∈ to the set . In this regards, the HOHFS̃is also represented as where allh (1) ( ), . . . ,h ( ) ( ) are G-Type FSs on .
Having introduced HOHFEs of a HOHFS̃= {⟨ ,h( )⟩: ∈ }, we now turn our attention to the interpretation of HOHFS̃as the union of all HOHFEs; that is, which is of fundamental importance in the study of HOHFS aggregation and the higher order hesitant fuzzy soft matrix that will be introduced within the next parts of the paper.
We give the definition of operation ⊙ on HOHFSs based on its counterpart on HOHFEs as follows: where the last operation ⊙ stands for the operation on G-Type FSs.
Note that if there is no confusion, HOHFS operator, HOHFE operator, and G-Type FS operator are denoted by the same symbol.
If the operation ⊙ on G-Type FSs has a property, denoted by( ), then the operation ⊙ on HOHFEs has the counterpart of property( ), denoted by( ).
Proof. The proof is obvious from Definition 6. Suppose that, for example, the operation ⊙ on G-Type FSs is commutative; that is, for any G-Type FSsh ℎ1 ∈h 1 ,h ℎ2 ∈h 2 , the following equality holds: Then, one can see from Definition 6 that This means that the operation ⊙ on HOHFEs is accordingly commutative.
By similar reasoning, one can easily prove that the HOHFE operator inherits all the properties of the G-Type FS operator.

HOHF Soft Sets and Their Properties
In the following, we first review briefly the concepts of soft set and fuzzy soft set. To do this, it may be helpful to recall that the sets , P( ), and are referred to as the initial universe set, the power set of , and a set of parameters, respectively.
Indeed, a soft set over is not a set but a parameterized family of subsets of .
Example 13. Reconsider Example 11. We can describe the "attractiveness of the houses" under the fuzzy circumstances using the fuzzy soft set (, ), wherȇ This example shows that if we let the corresponding membership degrees of all elements ∈ − be zero, then fuzzy soft sets (, ) and (, ) bear the same meaning and have the unified forms. This will bring the next discussions great convenience.
where the above G-Type FSs, denoted by ( , ]), are all in the form of IFS on .
where0 is the zero G-Type FS on .
where1 is the one G-Type FS on .
Notice that to have a correct comparison between two HOHFSs̃1 and̃2, the two following cases should be taken into account. The first case occurs if all G-Type FSs in each HOHFEs of HOHFS can be arranged increasingly (or decreasingly) by the use of a partial order. We refer to this case as the ordered case. Another case occurs when all G-Type FSs in each HOHFEs of HOHFS cannot be arranged increasingly (or decreasingly) using a partial order. This case is referred to as the disordered case.
In view of the above arguments, we shall represent the definition of a higher order hesitant fuzzy soft subset as follows.
Definition 19. Suppose that (̃, ), (̃, ) ∈ HOHFS( ). (̃, ) is said to be a higher order hesitant fuzzy soft subset of (̃, ) if (i) (in the ordered case) for all ∈ and for all ∈ it holdsh International Scholarly Research Notices where ⪯ stands for a partial order defined on G-Type FSs; (ii) (in the disordered case) for all ∈ and for all ∈ one has where (⋅, ⋅) is that given in Definition 18 and1 is the absolute higher order hesitant fuzzy soft set over .
Part (2). The proof is similar to that of part (1).  Proof. The proof is straightforward.

HOHF Soft Set Based Decision Making
In what follows, we are interested to present an application of HOHF soft set theory in a multiple criterion decision making problem. In order to solve decision making problems involving HOHF soft sets, an approach is proposed based on the concept of -level higher order hesitant fuzzy soft set.
We denote all higher order hesitant fuzzy soft matrixF over by HOHFSM( ).
Definition 28. LetF,G ∈ HOHFSM( ). Some arithmetic operations on HOHFSM( ) can be defined as follows: where the right-hand-side operations are that considered on G-Type FSs in Definition 6.
Accordingly, from Definition 28 and Corollary 8, it can be deduced for anyF,G ∈ HOHFSM( ) that The latter results can be employed whenever some experts are invited to conduct evaluation for one decision making problem. In such a situation, we are required to use aggregation operators on HOHFSM( ) given by experts to obtain a collectiveF * ∈ HOHFSM( ). However, dealing with the aggregation methods based on multiexperts' higher order hesitant fuzzy soft matrix is beyond the scope of this paper. Before introducing the concept of the -level higher order hesitant fuzzy soft set, we define a threshold vector over parameter set as ( ) = ( ( 1 ), . . . , ( )) where ( ) = ℎ ( ) ( = 1, . . . , ). Obviously, the HOHFEsh ( ) ( = 1, . . . , ) are changed dependently on the different parameter set .
Consider again the higher order hesitant fuzzy soft matrix F given in the concrete form ofF = [h̃( ) ( )] × . In order to get reasonably a HOHFE for each parameter ∈ from F, we can employ some aggregation operators.
We define where Agg(⋅) is an aggregation operator on HOHFEs.
In view of the latter setting, the threshold vector over parameter set is defined as (1) (1,0) = 0.
Let us now define the following two-valued function (⋅, ⋅) on G-Type FSsh where (⋅, ⋅) is the inclusion measure introduced by (38).
As a result from Definition 30 and (35), the Agg ( )-level higher order hesitant fuzzy soft set ofF ∈ HOHFSM( ) is characterized by  Step 1. Input the higher order hesitant fuzzy soft matrixF over a finite initial universe and a finite parameter set .
Step 3. Calculate inclusion measure value of all alternatives to determine their ranking and then selecting the optimal decision by the use of (41).

Illustrative Example
To illustrate the idea of Algorithm 31, let us take into consideration the following example which addresses Example 15 with more details.
Example 32. Suppose that = { 1 , 2 , 3 , 4 } is a set of color cloths under consideration and = { 1 , 2 , 3 } is a set of parameter where 1 = color which consists of red and blue, where the above G-Type FSs, denoted by ( , ]), are all in the form of IFS on .
The aim here is to select one of color cloths depending on its performance evaluated with respect to all parameters.
The tabular representation of the above higher order hesitant fuzzy soft matrixF is given in Table 1.
To perform the second step of Algorithm 31, we need to compute a proper threshold vector over parameter set . First of all we evaluate the efficiency of two well-known aggregation operators which are called the arithmetic average operator (AAO) and the geometric average operator (GAO) [32]. Based on the aggregation AAO, the threshold vector over parameter set is defined as International Scholarly Research Notices 9    where Needless to say that the operator ⊕ onh ( ) ( ) ( ) ( = 1, . . . , ) is that defined on G-Type FSs.
Proof. The proof is straightforward from the fact that for any indices , , The above proposition demonstrates that the threshold vectors over parameter set defined in accordance with the aggregations AAO and GAO are not efficient, because indifferent inclusion measure values for the whole alternatives ∈ do not allow us to have a promising judgment about optimal decision. In order to overcome adverse effect of indifferent inclusion measure values, we use, but we are not limited to, median operator [20] instead of AAO and GAO. The threshold vector based on median operator can be computed as follows.
Let us denote any IFSh ( ) ( ) ( ) byh ( ) ( ) ( ) = ( , , ] , ) for 1 ≤ ≤ ,1 ≤ ≤ ,1 ≤ ≤ . Suppose that membership degree (resp., nonmembership degree) of all alternatives in is arranged in increasing order, and then ( ) is referred to as the alternative with th largest membership degree (resp., nonmembership degree). In this regards, we define Now, the threshold vector med ( ) can be obtained as represented in tabular form in Table 2. By applying the third step of Algorithm 31, we find that the optimal decision is to select 2 . The results obtained through Step 3 of Algorithm 31 are summarized in Table 3.

Conclusion
This paper proposed a more general extension of fuzzy soft set that not only encompasses the existing ones as special cases, but also can be applied to model those situations where the existing kinds of fuzzy soft set are not applicable. The proposed higher order hesitant fuzzy soft set provides us with a multicriteria decision making method to deal with problems involving parameters with different-dimensional levels. As future work, we consider the study of aggregation methods based on multiexperts' higher order hesitant fuzzy soft matrix where some experts are invited to conduct evaluation for one decision making problem.

Conflict of Interests
The author declares that there is no conflict of interests regarding the publishing of this paper.