Multicriteria decision making method based on the higher order hesitant fuzzy soft set.

1. Introduction

Nowadays, many different extensions of FSs are known: L-fuzzy sets (L-FSs) , interval-valued fuzzy sets (IVFSs) , vague sets (VSs) , intuitionistic fuzzy sets (IFSs) , interval-valued intuitionistic fuzzy sets (IVIFSs) , linguistic fuzzy sets (LFSs) , type-2 fuzzy sets (T2FSs) , type-n fuzzy sets (TnFSs) , and fuzzy multisets (FMSs) .

An interesting extension of FSs is the hesitant fuzzy set (HFS)  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) , generalized hesitant fuzzy sets (G-HFSs) , and hesitant fuzzy linguistic term sets (hFlTSs) .

However, HFS  and its recent generalization G-HFS  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  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  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 , data analysis , and forecasting  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 , intuitionistic fuzzy soft set , interval-valued fuzzy soft set , vague fuzzy soft set , and multifuzzy soft set .

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 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 . Assume that X = {[x.sub.1], [x.sub.2], [x.sub.3], [x.sub.4], [x.sub.5]} is the universe set indicating the set of five kinds of colored drawing paper for engineering. Suppose the parameter set A = {[e.sub.1], [e.sub.2], [e.sub.3]} is a set of three criteria to evaluate the performance of these papers, where [e.sub.1] stands for thickness which includes three levels: thick, average, and thin, [e.sub.2] stands for color which consists of three levels: red, green, and blue, and [e.sub.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 , this decision problem can be analyzed using multifuzzy soft set. But if this decision problem is reconsidered slightly different from the original problem as [e.sub.1] stands for thickness which includes three levels:thick, average, and thin, [e.sub.2] stands for color which consists of three levels: red, green, and blue, and [e.sub.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.

2. 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  as the higher order hesitant fuzzy set (HOHFS). Actually, the HOHFS is a generalization of hesitant fuzzy set (HFS) introduced by Torra .

An ordinary fuzzy set (FS) A in X is defined  as A = {<x, A(x)>: x [member of] X}, where A: X [right arrow] [0,1] and the real value A(x) represents the degree of membership of x in A.

Definition 1. Let X be the universe of discourse. A generalized type of fuzzy set (G-Type FS) on X is defined as

[??] = {<x, [??](x)>: x [member of] X}, (1)

where

[??]: X [right arrow] [psi]([0,1]). (2)

Here, [psi]([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, 

(i) if [psi]([0,1]) = [0,1], then the G-Type FS [??] reduces to an ordinary FS;

(ii) if [psi]([0,1]) = [epsilon]([0,1]) denoting the set of all closed intervals, then the G-Type FS [??] reduces to an IVFS;

(iii) if [psi]([0,1]) = F([0,1]) denoting the set of all ordinary FSs, then the G-Type FS [??] reduces to a T2FS;

(iv) if [psi]([0,1]) = L denoting a partially ordered lattice, then the G-Type FS [??] reduces to a L-FS.

Definition 2 (see ). Let X be the universe of discourse. A hesitant fuzzy set (HFS) on X is symbolized by

H = {<x,h(x)>: x [member of] X}, (3)

where h(x), 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 x [member of] X to the set H.

Example 3. If X = {[x.sub.1], [x.sub.2], [x.sub.3]} is the universe of discourse, h([x.sub.1]) = {0.2, 0.4, 0.5}, h([x.sub.2]) = {0.3,0.4}, and h([x.sub.3]) = {0.3, 0.2, 0.5, 0.6} are the HFEs of [x.sub.i] (i = 1,2,3) to a set H, respectively. Then H can be considered as a HFS; that is,

H = {<[x.sub.1], {0.2, 0.4, 0.5}>, <[x.sub.2], {0.3, 0.4}>, <[x.sub.3], {0.3, 0.2, 0.5.0.6}>}. (4)

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 ). Let X be the universe of discourse. A higher order hesitant fuzzy set (HOHFS) on X is defined in terms of a function that when applied to X returns a set of G-Type FSs. A HOHFS is denoted by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (5)

where [??](x), 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 x [member of] X to the set [??]. In this regards, the HOHFS [??] is also represented as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

where all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are G-Type FSs on X.

Example 5. If X = {[x.sub.1], [x.sub.2], [x.sub.3]} is the universe of discourse,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (7)

are the HOHFEs of [x.sub.i] (i = 1,2,3) to a set [??], respectively, where G-Type FSs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are intuitionistic fuzzy set (IFS)  such that 0 [less than or equal to] [[mu].sub.ki], [v.sub.ki] [less than or equal to] 1 and 0 [less than or equal to] [[mu].sub.ki] + [v.sub.ki] [less than or equal to] 1 for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then [??] can be considered as a HOHFS; that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

In view of Definition 4, it is easily deduced that each HOHFS becomes a T2FS if all its G-Type FSs are the same and are in the form ofT2FS. That is, if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any x [member of] X, then the HOHFS [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] reduces to a T2FS.

It is noteworthy that the notions of interval-valued hesitant fuzzy set (IVHFS)  and interval type-2 fuzzy set (IT2FS)  both are special cases of HOHFSs. A HOHFS [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] reduces to an IVHFS, when all G-Type [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any x [member of] X are considered as closed intervals of real numbers in [0,1]. Furthermore, an IVHFS [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] reduces to an IT2FS, when all intervals satisfy [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any x [member of] X.

Recently, HFSs are extended by intuitionistic fuzzy sets (IFSs) and referred to them as generalized hesitant fuzzy sets (G-HFSs) . It is stated in  that FSs, IFSs, and HFSs are special cases of G-HFSs. Obviously, a [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is also an special case of HOHFS where all G-Type FSs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for any x [member of] X are considered as IFSs. This implies that HOHFSs are more useful than G-HFSs to deal with decision making, clustering, pattern recognition, image processing, and so forth, when experts have a hesitation among several possible memberships for an element to a set.

Having introduced HOHFEs of a HOHFS [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we now turn our attention to the interpretation of HOHFS [??] as the union of all HOHFEs; that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

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.

Definition 6. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be two HOHFSs. We give the definition of operation [dot in a circle] on HOHFSs based on its counterpart on HOHFEs as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

where the last operation 0 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.

Notice in the case that G-Type FSs in HOHFSs are reduced to crisp values, the G-Type FS operator in (10) is not the usual sum between real numbers because 0.6 + 0.5 = 1.1 [not member of] [0,1], and indeed the G-Type FS operator is the algebraic sum where 0.6 [direct sum] 0.5 = 0.6 + 0.5 - 0.6 x 0.5 = 0.3 [member of] [0,1]. Thus, the HOHFS operator is performed as the well-known HFS operator [29, 30].

Theorem 7. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be two HOHFEs. If the operation [dot in a circle] on G-Type FSs has a property, denoted by ([??]), then the operation [dot in a circle] on HOHFEs has the counterpart of property ([??]), denoted by ([??]).

Proof. The proof is obvious from Definition 6. Suppose that, for example, the operation [dot in a circle] on G-Type FSs is commutative; that is, for any G-Type FSs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the following equality holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

Then, one can see from Definition 6 that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

This means that the operation [dot in a circle] 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.

As a corollary of Theorem 7, it can be observed that HOHFS operators inherit subsequently all operational properties of G-Type FS operators.

Corollary 8. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be two HOHFSs. If the operation [dot in a circle] on HOHFEs has a property, denoted by ([??]), then the operation [dot in a circle] on HOHFSs has the counterpart of property ([??]), denoted by ([??]).

Example 9. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be two HOHFSs with HOHFEs in the form of IFSs. It can be shown for any IFSs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (13)

where (see e.g., )

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

Accordingly, from Corollary 8, it is deduced that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

3. 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 X, P(X), and E are referred to as the initial universe set, the power set of X, and a set of parameters, respectively.

Definition 10 (see ). A pair (F, A) is called a soft set over X, where F is a mapping given by F: A [subset] E [right arrow] P(X).

Indeed, a soft set over X is not a set but a parameterized family of subsets of X.

Example 11. Suppose that X = {[x.sub.1], [x.sub.2], [x.sub.3], [x.sub.4]} is a set of houses under consideration and A = {[e.sub.1], [e.sub.2], [e.sub.3]} is a set of parameter, where [e.sub.1] = expensive, [e.sub.2] = beautiful, and [e.sub.3] = in the good location. Let

F ([e.sub.1]) = {[x.sub.1]}, F([e.sub.2]) = {[x.sub.1], [x.sub.3]}, F([e.sub.3]) = {[x.sub.2], [x.sub.4]}. (16)

We use here the soft set (F, A) to describe the "attractiveness of the houses." In this regard, F([e.sub.1]) stands for "houses (expensive)" whose function value is the set {[x.sub.1]}, F([e.sub.2]) means "houses (beautiful)" whose function value is the set {[x.sub.1], [x.sub.3]}, and F([e.sub.3]) implies "houses (in the good location)" whose function value is the set {[x.sub.2], [x.sub.4]}.

Definition 12 (see ). Let [??]: A [snubset] E [right arrow] [??](X). A pair ([??], A) is called a fuzzy soft set over X where [??](X) is the set of all fuzzy subsets of X.

Example 13. Reconsider Example 11. We can describe the "attractiveness of the houses" under the fuzzy circumstances using the fuzzy soft set ([??], A), where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (17)

This example shows that if we let the corresponding membership degrees of all elements e [member of] E-A be zero, then fuzzy soft sets ([??], A) and ([??], E) bear the same meaning and have the unified forms. This will bring the next discussions great convenience.

Definition 14. Let [??]: E [right arrow] [??](X). A pair ([??], E) is called a higher order hesitant fuzzy (HOHF) soft set over X where [??](X) is the set of all higher order hesitant fuzzy subsets of X. In this regard, for any e [member of] E, we denote

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (18)

where all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are G-Type FSs on X. That is to say, that the higher order hesitant fuzzy (HOHF) soft set over X can be denoted by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

We hereafter denote all HOHF soft sets over X by HOHFS(X).

Example 15. Suppose that X = {[x.sub.1], [x.sub.2], [x.sub.3], [x.sub.4]} is a set of color cloths under consideration and A = {[e.sub.1], [e.sub.2], [e.sub.3]} is a set of parameter where [e.sub.1] = color which consists of red and blue, [e.sub.2] = ingredient which is made from wool, cotton, and acrylic, and [e.sub.3] = price which can be considered as high and low. In this situation, we should consider a higher order hesitant fuzzy soft set ([??], E) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (20)

where the above G-Type FSs, denoted by ([mu], v), are all in the form of IFS on X.

Definition 16. A higher order hesitant fuzzy soft set ([??], E) over X is called a null higher order hesitant fuzzy soft set over X denoted by [??], if for any e [member of] E

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (21)

where [??] is the zero G-Type FS on X.

Definition 17. A higher order hesitant fuzzy soft set ([??], E) over X is called an absolute higher order hesitant fuzzy soft set over X denoted by [??], if for any e [member of] E

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (22)

where [??] is the one G-Type FS on X.

Definition 18 (see ). Consider two HOHFSs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The definition of distance measure for HOHFSs is given by

d([[??].sub.1], [[??].sub.2]) = [1/n] [n.summation over (i=1)][d.sub.H] ([[??].sub.1]([x.sub.i]), [[??].sub.2]([x.sub.i])), (23)

where for each [x.sub.i] [member of] X

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (24)

Here, D(*, *) is a distance measure defined on G-Type FSs.

Notice that to have a correct comparison between two HOHFSs [[??].sub.1] and [[??].sub.2], 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 [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. ([??], E) is said to be a higher order hesitant fuzzy soft subset of ([??], E) if

(i) (in the ordered case) for all e [member of] E and for all x [member of] X it holds

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (25)

where [??] stands for a partial order defined on G-Type FSs;

(ii) (in the disordered case) for all e [member of] E and for all x [member of] X one has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (26)

where [d.sub.H](*,*) is that given in Definition 18 and [??] is the absolute higher order hesitant fuzzy soft set over X.

The inclusion relation between [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 20. Two higher order hesitant fuzzy soft sets [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are said to be higher order hesitant fuzzy soft equal sets if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 21. Suppose that ([??], E) [member of] HOHFS(X). ([??], E) is said to be a higher order hesitant fuzzy soft complementary set of ([??], E) if for all e [member of] E and for all x [member of] X

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (27)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] means the complement of the corresponding G-Type FSs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Definition 22. Union of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is defined for all e [member of] E as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (28)

Definition 23. Intersection of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], is defined for all e [member of] E as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

Theorem 24. Let ([??], E), ([??], E) [member of] HOHFS(X). Then

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof.

Part (1). Assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In this regard, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. On the other hand, from Definition 22 one gets

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (30)

Part (2). The proof is similar to that of part (1).

Theorem 25. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. The proof is straightforward.

Theorem 26. Let ([??], E) [member of] HOHFS(X), let [??] be the absolute higher order hesitant fuzzy soft set over X, and let [??] be the null higher order hesitant fuzzy soft set over X. Then

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

(6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Proof. The proof is straightforward.

4. 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 [lambda]-level higher order hesitant fuzzy soft set.

Definition 27. Let ([??], E) [member of] HOHFS(X) where [absolute value of X] = n and [absolute value of E] = m. The higher order hesitant fuzzy soft matrix [??] is defined in the concrete form of

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (31)

We denote all higher order hesitant fuzzy soft matrix [??] over X by HOHFSM(X).

As can be seen from Definitions 27 and 14, there exists a bijective mapping from HOHFS(X) to HOHFSM(X), implying that any higher order hesitant fuzzy soft set ([??], E) [member of] HOHFS(X) can be equivalent with its higher order hesitant fuzzy soft matrix [??] [member of] HOHFSM(X). This equivalent relationship allows us to identify any ([??], E) [member of] HOHFS(X) with its [??] [member of] HOHFSM(X), interchangeably.

Definition 28. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Some arithmetic operations on HOHFSM(X) can be defined as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (32)

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 any [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (33)

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(X) given by experts to obtain a collective [??] [member of] HOHFSM(X). 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 [lambda]-level higher order hesitant fuzzy soft set, we define a threshold vector over parameter set E as [lambda](E) = ([lambda]([e.sub.1]), ..., [lambda]([e.sub.m])) where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Obviously, the HOHFEs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are changed dependently on the different parameter set E.

Consider again the higher order hesitant fuzzy soft matrix [??] given in the concrete form of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In order to get reasonably a HOHFE for each parameter [e.sub.j] [member of] E from [??], we can employ some aggregation operators.

We define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (34)

where Agg(*) is an aggregation operator on HOHFEs.

In view of the latter setting, the threshold vector over parameter set E is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (35)

Definition 29. A real function I: HOHFE(X) x HOHFE(X) [right arrow] [0,1] is called the inclusion measure of the HOHFEs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], if I(*,*) satisfies the following properties.

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (36)

where the last inclusion operator [subset or equal to] is that introduced on G-Type FSs.

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let us now define the following two-valued function i(*, *) on G-Type FSs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (37)

and then, the following function is an inclusion measure of the HOHFEs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (38)

Definition 30. We define the [lambda](E)-level higher order hesitant fuzzy soft set of [??] [member of] HOHFSM(X), denoted by L([??], [lambda](E)), as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (39)

where I(*, *) is the inclusion measure introduced by (38).

As a result from Definition 30 and (35), the [[lambda].sub.Agg](E)-level higher order hesitant fuzzy soft set of [??] [member of] HOHFSM(X) is characterized by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (40)

Once the [lambda](E)-level higher order hesitant fuzzy soft set of [??] is achieved, we calculate the inclusion measure values [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] given by (38) to rank all the alternatives. The optimal decision is then selected as [x.sub.k] if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (41)

It is noteworthy to say that if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] reduce to IFSs, then the inclusion measure values [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are coincided to choice values [C.sub.i] of [x.sub.i] (i = 1, ..., n) introduced in .

Now we are in a position to present an algorithm for solving higher order hesitant fuzzy soft set based decision making problem by using [lambda](E)-level higher order hesitant fuzzy soft sets initiated in this study.

Algorithm 31. Consider the following.

Step 1. Input the higher order hesitant fuzzy soft matrix [??] over a finite initial universe X and a finite parameter set E.

Step 2. Compute the threshold vector [lambda](E) = ([lambda]([e.sub.1]), ..., [lambda]([e.sub.m])) corresponding to [??].

Step 3. Calculate inclusion measure value of all alternatives to determine their ranking and then selecting the optimal decision by the use of (41).

5. 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 X = {[x.sub.1], [x.sub.2], [x.sub.3], [x.sub.4]} is a set of color cloths under consideration and A = {[e.sub.1], [e.sub.2], [e.sub.3]} is a set of parameter where [e.sub.1] = color which consists of red and blue, [e.sub.2] = ingredient which is made from wool, cotton, and acrylic, and [e.sub.3] = price which can be considered as high and low. In this situation, we should consider a higher order hesitant fuzzy soft set ([??], E) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (42)

where the above G-Type FSs, denoted by ([mu], v), are all in the form of IFS on X.

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 matrix [??] is given in Table 1.

To perform the second step of Algorithm 31, we need to compute a proper threshold vector over parameter set E. 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) . Based on the aggregation AAO, the threshold vector over parameter set E is defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (43)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (44)

Needless to say that the operator [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is that defined on G-Type FSs.

In this example, operator [direct sum] considered on IFSs is given for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by 

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (45)

Furthermore, based on the aggregation GAO, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (46)

where for IFSs [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (47)

Proposition 33. Assume that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (48)

Then, for any [x.sub.i] [member of] X

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (49)

where I(*, *) is the inclusion measure given by (38).

Proof. The proof is straightforward from the fact that for any indices i, j, k

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (50)

The above proposition demonstrates that the threshold vectors over parameter set E defined in accordance with the aggregations AAO and GAO are not efficient, because indifferent inclusion measure values for the whole alternatives [x.sub.i] [member of] X 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  instead of AAO and GAO. The threshold vector based on median operator can be computed as follows.

Let us denote any IFS [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Suppose that membership degree (resp., nonmembership degree) of all alternatives in X is arranged in increasing order, and then [x.sub.[sigma](i)] is referred to as the alternative with ith largest membership degree (resp., nonmembership degree). In this regards, we define

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (51)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (52)

Now, the threshold vector [[lambda].sub.med](E) 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 [x.sub.2]. The results obtained through Step 3 of Algorithm 31 are summarized in Table 3.

6. 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.

http://dx.doi.org/10.1155/2014/873454

Conflict of Interests

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

Acknowledgments

The author thanks the referees and the editor for their helpful suggestions which improved the presentation of the paper.

References

 J. A. Goguen, "L-fuzzy sets," Journal of Mathematical Analysis and Applications, vol. 18, no. 1, pp. 145-174, 1967.

 I. B. Turksen, "Interval valued fuzzy sets based on normal forms," Fuzzy Sets and Systems, vol. 20, no. 2, pp. 191-210, 1986.

 H. Bustince and P. Burillo, "Vague sets are intuitionistic fuzzy sets," Fuzzy Sets and Systems, vol. 79, no. 3, pp. 403-405, 1996.

 K. T. Atanassov, "Intuitionistic fuzzy sets," Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87-96, 1986.

 K. Atanassov and G. Gargov, "Interval valued intuitionistic fuzzy sets," Fuzzy Sets and Systems, vol. 31, no. 3, pp. 343-349, 1989.

 Z. S. Xu, "Deviation measures of linguistic preference relations in group decision making," Omega, vol. 33, no. 3, pp. 249-254, 2005.

 M. Mizumoto and K. Tanaka, "Some properties of fuzzy sets of type 2," Information and Control, vol. 31, no. 4, pp. 312-340, 1976.

 D. Dubois and H. Prade, Fuzzy Sets and Systems: Theory and Applications, Kluwer Academic, New York, NY, USA, 1980.

 S. Miyamoto, "Fuzzy multisets and their generalizations," in Multiset Processing, vol. 2235 of Lecture Notes in Computer Science, pp. 225-235, Springer, Berlin, Germany, 2001.

 V. Torra, "Hesitant fuzzy sets," International Journal of Intelligent Systems, vol. 25, no. 6, pp. 529-539, 2010.

 B. Zhu, Z. Xu, and M. Xia, "Dual hesitant fuzzy sets," Journal of Applied Mathematics, vol. 2012, Article ID 879629, 13 pages, 2012.

 G. Qian, H. Wang, and X. Feng, "Generalized hesitant fuzzy sets and their application in decision support system," Knowledge-Based Systems, vol. 37, pp. 357-365, 2013.

 R. M. Rodriguez, L. Martinez, and F. Herrera, "Hesitant fuzzy linguistic term sets for decision making," IEEE Transactions on Fuzzy Systems, vol. 20, no. 1, pp. 109-119, 2012.

 B. Farhadinia, "Distance and similarity measures for higher order hesitant fuzzy sets," Knowledge-Based Systems, vol. 55, pp. 43-48, 2014.

 D. A. Molodtsov, "Soft set theory--first results," Computers and Mathematics with Applications, vol. 37, no. 4-5, pp. 19-31, 1999.

 N. Qagman and S. Enginoglu, "Soft matrix theory and its decision making," Computers and Mathematics with Applications, vol. 59, no. 10, pp. 3308-3314, 2010.

 Y. Zou and Z. Xiao, "Data analysis approaches of soft sets under incomplete information," Knowledge-Based Systems, vol. 21, no. 8, pp. 941-945, 2008.

 Z. Xiao, K. Gong, and Y. Zou, "A combined forecasting approach based on fuzzy soft sets," Journal of Computational and Applied Mathematics, vol. 228, no. 1, pp. 326-333, 2009.

 P. K. Maji, R. Biswas, and A. R. Roy, "Fuzzy soft sets," Journal of Fuzzy Mathematics, vol. 9, no. 3, pp. 589-602, 2001.

 J. Mao, D. Yao, and C. Wang, "Group decision making methods based on intuitionistic fuzzy soft matrices," Applied Mathematical Modelling, vol. 37, no. 9, pp. 6425-6436, 2013.

 Y. B. Jun and X. B. Yang, "A note on the paper "combination of interval-valued fuzzy set and soft set" [Comput. Math. Appl. 58 (2009) 521-527]," Computers and Mathematics with Applications, vol. 61, no. 5, pp. 1468-1470, 2011.

 W. Xu, J. Ma, S. Wang, and G. Hao, "Vague soft sets and their properties," Computers and Mathematics with Applications, vol. 59, no. 2, pp. 787-794, 2010.

 Y. Yang, X. Tan, and C. Meng, "The multi-fuzzy soft set and its application in decision making," Applied Mathematical Modelling, vol. 37, no. 7, pp. 4915-4923, 2013.

 L. A. Zadeh, "Fuzzy sets," Information and Control, vol. 8, no. 3, pp. 338-353, 1965.

 G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice Hall, Upper Saddle River, NJ, USA, 1995.

 Z. Xu and M. Xia, "Distance and similarity measures for hesitant fuzzy sets," Information Sciences, vol. 181, no. 11, pp. 2128-2138, 2011.

 N. Chen, Z. S. Xu, and M. M. Xia, "Interval-valued hesitant preference relations and their applications to group decision making," Knowledge-Based Systems, vol. 37, pp. 528-540, 2013.

 J. M. Mendel, Rule-Based Fuzzy Logic Systems: Introduction and New Directions, Prentice Hall, Upper Saddle River, NJ, USA, 2001.

 B. Farhadinia, "A novel method of ranking hesitant fuzzy values for multiple attribute decision-making problems," International Journal of Intelligent Systems, vol. 28, no. 8, pp. 752-767, 2013.

 B. Farhadinia, "Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets," Information Sciences, vol. 240, pp. 129-144, 2013.

 F. Feng, Y. B. Jun, X. Liu, and L. Li, "An adjustable approach to fuzzy soft set based decision making," Journal of Computational and Applied Mathematics, vol. 234, no. 1, pp. 10-20, 2010.

 R. R. Yager, "Generalized OWA aggregation operators," Fuzzy Optimization and Decision Making, vol. 3, no. 1, pp. 93-107, 2004.

 K. Atanassov, Intuitionistic Fuzzy Sets: Theory and Applications, Physica, New York, NY, USA, 1999.

Department of Mathematics, Quchan University of Advanced Technology, Quchan, Iran

Received 7 April 2014; Revised 11 May 2014; Accepted 13 May 2014; Published 21 August 2014

```TABLE 1: Higher order hesitant fuzzy soft matrix [??].

[e.sub.1]      [e.sub.2]      [e.sub.3]

[x.sub.1]   {(0.1,0.2),    {(0.5,0.2),    {(0.5,0,2),
(0.3,0.2)}    (0.1, 0.2),    (0.4,0.1)}
(0.3,0.2)}
[x.sub.2]   {(0.2,0.3),    {(0.2,0.4),    {(0.3,0.7),
(0.3,0.1)}     (0.6,0.1),    (0.6,0.1)}
(0.3,0.1)}
[x.sub.3]   {(0.3,0.3),    {(0.3,0.3),    {(0.4,0.4),
(0.4,0.6)}     (0.4,0.4),    (0.4,0.6)}
(0.4,0.6)}
[x.sub.4]   {(0.1, 0.3),   {(0.1, 0.3),   {(0.8,0.2),
(0.3,0.1)}     (0.3,0.1),    (0.2,0.5)}
(0.2,0.5)}

TABLE 2: Higher order hesitant fuzzy soft matrix [??]
and the threshold vector [[lambda].sub.med](E).

[e.sub.1]      [e.sub.2]      [e.sub.3]

[x.sub.1]       {(0.1,0.2),    {(0.5,0.2),    {(0.5,0,2),
(0.3,0.2)}    (0.1, 0.2),     (0.4,0.1)}
(0.3,0.2)}
[x.sub.2]       {(0.2,0.3),    {(0.2,0.4),    {(0.3,0.7),
(0.3,0.1)}     (0.6,0.1),     (0.6,0.1)}
(0.3,0.1)}
[x.sub.3]       {(0.3,0.3),    {(0.3,0.3),    {(0.4,0.4),
(0.4,0.6)}     (0.4,0.4),     (0.4,0.6)}
(0.4,0.6)}
[x.sub.4]       {(0.1,0.3),    {(0.1, 0.3),   {(0.8,0.2),
(0.3,0.1)}     (0.3,0.1),     (0.2,0.5)}
(0.2,0.5)}
[<[[??].sub.    {(0.15,0.3),   {(0.25,0.3),   {(0.35,0.3),
[lambda](e)>.   (0.3,0.15)}    (0.35,0.15),    (0.4,0.3)}
sub.med]                       {(0.3,0.3)}

TABLE 3: Inclusion measure values corresponding
to [[lambda].sub.med] (E)

[e.sub.l]   [e.sub.2]   [e.sub.3]   Total

[x.sub.1]       0          2/3          1        5/3
[x.sub.2]       1          2/3         1/2      13/6
[x.sub.3]      1/2         1/3          0        5/6
[x.sub.4]      1/2          0          1/2        1
```
Title Annotation: Printer friendly Cite/link Email Feedback Research Article Farhadinia, B. International Scholarly Research Notices Report Jan 1, 2014 6622 Low temperature synthesis of Belite cement based on silica fume and lime. Single dose of levofloxacin versus three dosages for prophylaxis in prostate biopsy. Decision making Decision-making Fuzzy algorithms Fuzzy logic Fuzzy systems Mathematical research