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.
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 is the universe set indicating the set of five kinds of colored drawing paper for engineering. Suppose the parameter set is a set of three criteria to evaluate the performance of these papers, where stands for thickness which includes three levels: thick, average, and thin, stands for color which consists of three levels: red, green, and blue, and 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 stands for thickness which includes three levels: thick, average, and thin, stands for color which consists of three levels: red, green, and blue, and 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.
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) in is defined  as , where and the real value represents the degree of membership of in .
Definition 1. Let be the universe of discourse. A generalized type of fuzzy set (G-Type FS) on is defined as where Here, denotes a family of crisp or fuzzy sets that can be defined within the universal set .
It is noteworthy that most of the existing extensions of ordinary FS are special cases of G-Type FS, for instance, (i)if , then the G-Type FS reduces to an ordinary FS;(ii)if denoting the set of all closed intervals, then the G-Type FS reduces to an IVFS;(iii)if denoting the set of all ordinary FSs, then the G-Type FS reduces to a T2FS;(iv)if denoting a partially ordered lattice, then the G-Type FS reduces to a L-FS.
Definition 2 (see ). Let be the universe of discourse. A hesitant fuzzy set (HFS) on is symbolized by where , referred to as the hesitant fuzzy element (HFE), is a set of some values in denoting the possible membership degree of the element to the set .
Example 3. If is the universe of discourse, , , and are the HFEs of to a set , respectively. Then can be considered as a HFS; that is,
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 real-world 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 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 by where , 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 HOHFS is also represented as where all are G-Type FSs on .
Example 5. If is the universe of discourse, are the HOHFEs of to a set , respectively, where G-Type FSs are intuitionistic fuzzy set (IFS)  such that and for and . Then can be considered as a HOHFS; that is,
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 of T2FS. That is, if for any , then the HOHFS 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 reduces to an IVHFS, when all G-Type FSs for any are considered as closed intervals of real numbers in . Furthermore, an IVHFS reduces to an IT2FS, when all intervals satisfy for any .
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 G-HFS is also an special case of HOHFS where all G-Type FSs for any 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 , we now turn our attention to the interpretation of HOHFS as 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.
Definition 6. Let and be two HOHFSs. 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.
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 , and indeed the G-Type FS operator is the algebraic sum where . Thus, the HOHFS operator is performed as the well-known HFS operator [29, 30].
Theorem 7. Let and be two HOHFEs. 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 FSs , 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.
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 and be two HOHFSs. If the operation on HOHFEs has a property, denoted by , then the operation on HOHFSs has the counterpart of property , denoted by .
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 , , and are referred to as the initial universe set, the power set of , and a set of parameters, respectively.
Definition 10 (see ). A pair is called a soft set over , where is a mapping given by .
Indeed, a soft set over is not a set but a parameterized family of subsets of .
Example 11. Suppose that is a set of houses under consideration and is a set of parameter, where = expensive, = beautiful, and = in the good location. Let We use here the soft set to describe the “attractiveness of the houses.” In this regard, stands for “houses (expensive)” whose function value is the set , means “houses (beautiful)” whose function value is the set , and implies “houses (in the good location)” whose function value is the set .
Definition 12 (see ). Let . A pair is called a fuzzy soft set over where is the set of all fuzzy subsets of .
Example 13. Reconsider Example 11. We can describe the “attractiveness of the houses” under the fuzzy circumstances using the fuzzy soft set , where 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.
Definition 14. Let . A pair is called a higher order hesitant fuzzy (HOHF) soft set over where is the set of all higher order hesitant fuzzy subsets of . In this regard, for any , we denote where all are G-Type FSs on . That is to say, that the higher order hesitant fuzzy (HOHF) soft set over can be denoted by We hereafter denote all HOHF soft sets over by .
Example 15. Suppose that is a set of color cloths under consideration and is a set of parameter where = color which consists of red and blue, = ingredient which is made from wool, cotton, and acrylic, and = price which can be considered as high and low. In this situation, we should consider a higher order hesitant fuzzy soft set as follows: where the above G-Type FSs, denoted by , are all in the form of IFS on .
Definition 16. A higher order hesitant fuzzy soft set over is called a null higher order hesitant fuzzy soft set over denoted by , if for any where is the zero G-Type FS on .
Definition 17. A higher order hesitant fuzzy soft set over is called an absolute higher order hesitant fuzzy soft set over denoted by , if for any where is the one G-Type FS on .
Definition 18 (see ). Consider two HOHFSs and on . The definition of distance measure for HOHFSs is given by where for each Here, is a distance measure defined on G-Type FSs.
Notice that to have a correct comparison between two HOHFSs and , 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 . is said to be a higher order hesitant fuzzy soft subset of if (i)(in the ordered case) for all and for all it holds 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 and is the absolute higher order hesitant fuzzy soft set over .
The inclusion relation between and is denoted by .
Definition 20. Two higher order hesitant fuzzy soft sets are said to be higher order hesitant fuzzy soft equal sets if and only if and .
Definition 21. Suppose that . is said to be a higher order hesitant fuzzy soft complementary set of if for all and for all where means the complement of the corresponding G-Type FSs .
Definition 22. Union of , denoted by , is defined for all as
Definition 23. Intersection of , denoted by , is defined for all as
Theorem 24. Let . Then (1), (2).
Proof. Part (1). Assume that . In this regard, . On the other hand, from Definition 22 one gets Part (2). The proof is similar to that of part (1).
Theorem 25. Let . Then (1), (2), (3), (4).
Proof. The proof is straightforward.
Theorem 26. Let , let be the absolute higher order hesitant fuzzy soft set over , and let be the null higher order hesitant fuzzy soft set over . Then (1), (2), (3), (4), (5), (6).
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 -level higher order hesitant fuzzy soft set.
Definition 27. Let where and . The higher order hesitant fuzzy soft matrix is defined in the concrete form of We denote all higher order hesitant fuzzy soft matrix over by .
As can be seen from Definitions 27 and 14, there exists a bijective mapping from to , implying that any higher order hesitant fuzzy soft set can be equivalent with its higher order hesitant fuzzy soft matrix . This equivalent relationship allows us to identify any with its , interchangeably.
Definition 28. Let . Some arithmetic operations on 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 any 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 given by experts to obtain a collective . 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 where . Obviously, the HOHFEs are changed dependently on the different parameter set .
Consider again the higher order hesitant fuzzy soft matrix given in the concrete form of . In order to get reasonably a HOHFE for each parameter from , we can employ some aggregation operators.
We define where is an aggregation operator on HOHFEs.
In view of the latter setting, the threshold vector over parameter set is defined as
Definition 29. A real function is called the inclusion measure of the HOHFEs , , and , if satisfies the following properties. (1). (2) iff where the last inclusion operator is that introduced on G-Type FSs.(3)If , then and . Let us now define the following two-valued function on G-Type FSs and as and then, the following function is an inclusion measure of the HOHFEs and given by
Definition 30. We define the -level higher order hesitant fuzzy soft set of , denoted by , as follows: where is the inclusion measure introduced by (38).
Once the -level higher order hesitant fuzzy soft set of is achieved, we calculate the inclusion measure values for and given by (38) to rank all the alternatives. The optimal decision is then selected as if It is noteworthy to say that if HOHFEs reduce to IFSs, then the inclusion measure values are coincided to choice values of 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 -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 and a finite parameter set .
Step 2. Compute the threshold vector 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
Example 32. Suppose that is a set of color cloths under consideration and is a set of parameter where = color which consists of red and blue, = ingredient which is made from wool, cotton, and acrylic, and = price which can be considered as high and low. In this situation, we should consider a higher order hesitant fuzzy soft set as follows: 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 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 . 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 is defined as where Needless to say that the operator on is that defined on G-Type FSs.
In this example, operator considered on IFSs is given for IFSs by  Furthermore, based on the aggregation GAO, we have where for IFSs , we define
Proposition 33. Assume that Then, for any where is the inclusion measure given by (38).
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  instead of AAO and GAO. The threshold vector based on median operator can be computed as follows.
Let us denote any IFS by for . 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 where Now, the threshold vector can be obtained as represented in tabular form in Table 2.
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.
The author thanks the referees and the editor for their helpful suggestions which improved the presentation of the paper.
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