#### Abstract

As an extension of fuzzy sets, hesitant bipolar-valued fuzzy set is a new mathematical tool for dealing with fuzzy problems, but it still has the problem with the inadequacy of the parametric tools. In order to further improve the accuracy of decision making, a new mixed mathematical model, named hesitant bipolar-valued fuzzy soft set, is constructed by combining hesitant bipolar-valued fuzzy sets with soft sets. Firstly, some related theories of hesitant bipolar-valued fuzzy sets are discussed. Secondly, the concept of hesitant bipolar-valued fuzzy soft set is given, and the algorithms of complement, union, intersection, “AND,” and “OR” are defined. Based on the above algorithms, the corresponding results of operation are analyzed and the relevant properties are discussed. Finally, a multiattribute decision-making method of hesitant bipolar-valued fuzzy soft sets is proposed by using the idea of score function and level soft sets. The effectiveness of the proposed method is illustrated by an example.

#### 1. Introduction

The theory of fuzzy sets was put forward by Zadeh in 1965 [1], and the method of soft sets was presented even later by Molodtsov in 1999 [2]. Fuzzy sets as well as soft sets are complementary and important to deal with vague, imprecise, and uncertain knowledge. The integration of the two theories has become one of the popular research directions. In recent years, some scholars presented fuzzy soft sets and various kinds of extended models [3–11], such as fuzzy soft sets, intuitionistic fuzzy soft sets, interval-valued fuzzy soft sets, interval-valued intuitionistic fuzzy soft sets, hesitant fuzzy soft sets, interval-valued hesitant fuzzy soft sets, and generalized hesitant fuzzy soft sets. The corresponding basic procedures and properties are also discussed. Based on the above results, some scholars made a further study about the uncertainty measures [12–21] and the decision-making methods [22–25] of the extended model. For example, Majumdar and Samanta analyzed and compared the similarities and difference of different measures of the fuzzy soft sets [12]. Based on [12], Liu et al. introduced a novel approach to define the similarity measures and the distance measures between two fuzzy soft sets with different parameter sets [13]. Jiang et al. generalized the uncertainty measures of fuzzy soft set, defined the distance measures between intuitionistic fuzzy soft sets, and provided the entropy between intuitionistic fuzzy soft sets and interval-valued fuzzy soft sets [14]. Based on [14], Muthukumar and Sai Sundara Krishnan proposed a novel similarity measure of intuitionistic fuzzy soft sets [15]. In recent years, some scholars have extended the uncertainty measures of fuzzy soft set to another direction. Beg and Rashid defined the distance measures between any two elements in hesitant fuzzy soft sets [16]. A novel distance measure and similarity measure of hesitant fuzzy soft sets were presented by K. Rezaei and H. Rezaei in [17]. Garg and Arora raised novel distance measures and similarity measures of dual hesitant fuzzy soft sets [18]. The correlation measures of hesitant fuzzy soft sets and interval-valued hesitant fuzzy soft sets were discussed by Das et al. in [19]. Garg and Arora made a further study about the weighted average and geometric operators of dual hesitant fuzzy soft sets [20]. Arora and Garg introduced a novel correlation coefficient measure of dual hesitant fuzzy soft sets [21]. Feng et al. introduced a novel way to fuzzy soft set by the idea of level soft sets [22]. Jiang et al. presented a novel adjustable way for decision-making problems of intuitionistic fuzzy soft sets [23]. Xiao et al. studied the decision-making problems of interval-valued fuzzy soft sets in uncertain environment [24]. Zhao et al. introduced an innovative multicriteria ranking approach for decision-making problems of intuitionistic fuzzy soft sets [25]. With the continuous development of fuzzy set, soft set, and uncertainty theory, artificial intelligence has received extensive attention around the world. Related control schemes have also achieved fruitful results [26–38]. More detailed research on the uncertainty measures of fuzzy soft sets and other models can provide a reliable mathematical foundation for the development of cybernetics.

Although the research results of the soft set extension model have been very fruitful, the incompatible bipolarity is inevitable in the objective world. Just as an example of a psychology disease-bipolar disorder patient, who has symptoms of mania and depression, the sum of positive and negative symptoms value is more than one, or even both the two symptoms may reach extreme cases simultaneously. In recent years, the integration of bipolar-valued and hesitant fuzzy sets had drawn the attention of many scholars, and good results were obtained. Han put forward the perception of hesitant bipolar-valued fuzzy sets as well as introduced a hesitant bipolar fuzzy decision-making problem method based on TOPSIS [39]. Ullah et al. presented the concept of bipolar-valued hesitant fuzzy sets, and the aggregation operators were applied to deal with the decision-making problems [40]. Mandal and Ranadive proposed the perception of hesitant bipolar-valued fuzzy sets and bipolar-valued hesitant fuzzy sets. Both of them were applied to solve decision-making problems [41]. Wei discussed the decision-making problems of aggregation operators founded on hesitant bipolar-valued fuzzy information [42]. According to the Einstein operational laws, Mahmood et al. made a further study about the aggregation operators and the decision-making problems for bipolar-valued hesitant fuzzy information [43]. Xu and Wei considered the dual hesitant bipolar fuzzy sets and the decision-making problems [44]. A further study about a few aggregation operators for dual hesitant bipolar fuzzy information was discussed in [45]. Awang et al. presented the perception of hesitant bipolar-valued neutrosophic sets and introduced an original decision-making method based on the novel model [46]. Those models were superior to other fuzzy mathematical tools in describing abilities because of considering the bipolarity and nonuniqueness of the membership degree of an element simultaneously. But like other mathematical tools, it still has the inadequacy problem of the parametric tools. On this basis, the theory of parameters is introduced into the hesitant bipolar-valued fuzzy sets, and the concept of hesitant bipolar-valued fuzzy soft sets is defined. The model is a new mathematical tool for uncertainty problem. It can make up for the inadequacy of the parametric tools. Therefore, the model is more suitable to describe the uncertainty problem.

The rest of this paper is arranged as follows. In Section 2, we retrospect the basic concepts of soft sets and hesitant bipolar-valued fuzzy sets and supplement some correlated knowledge. In Section 3, we put forward the perception of hesitant bipolar-valued fuzzy soft sets and define 2a number of basic procedures. A few corresponding properties are also considered. In Section 4, we propose a multiattribute decision-making approach based on the novel model, and the effectiveness of the proposed methodology is given by practical problems. Finally, the concluding is given.

#### 2. Preliminaries

For convenience, throughout the paper, let be an initial limited universe set, be the set of all possible parameters with respect to , and .

*Definition 1. *(see [1]). Let denote the power set of . A pair is called a soft set on , where is a mapping given by .

*Definition 2. *(see [41]). A hesitant bipolar-valued fuzzy set on is defined aswhere , known as the hesitant fuzzy positive element, is a set of some values in denoting the possible satisfaction degree of the to the corresponding property to the ; , known as the hesitant fuzzy positive element, is a set of some values in denoting the possible satisfaction degree of the to the opposite property to the ; and , known as the hesitant bipolar-valued fuzzy element, is a set of some values in to the , abbreviated as HBVFE.

As a matter of convenience, the HBVFE is abbreviated as and all hesitant bipolar-valued fuzzy sets on are abbreviated as in this study.

*Definition 3. *(see [41]). Let and be two HBVFEs; then,(1)Complement:(2)Union:(3)Intersection:

Theorem 1. *Let , and be three HBVFEs; then, we have some operational laws, such as*(1)*Involutive:*(2)*Commutativity:*(3)*Associative:*(4)*Distributive:*(5)*De Morgan [41]:*

*Proof. * (1)For a HBVFE , we have from Definition 3; then,(2)It can be easily obtained from Definition 3.(3)For three HBVFEs , , and , we have and from Definition 3, we have Similarly, we can also obtain(4)For three HBVFEs , and , we have Similarly, we can also obtain

*Definition 4. *(see [41]). Let be a HBVFE and the score function is defined aswhere and are the numbers of the elements in and , respectively.

*Definition 5. *Two special hesitant bipolar-valued fuzzy sets are defined as(1)Empty set:(2)Full set:

*Definition 6. *Let and ; then,(1)If , then is called as a hesitant bipolar-valued fuzzy quasisubset of , denoted by or .(2)If , then and are called hesitant bipolar-valued fuzzy quasiequal, denoted by .(3)If and have the same elements, then and are equal, denoted by .

*Definition 7. *Let ; then,(1)Complement: where .(2)Union: where .(3)Intersection: where .

Theorem 2. *Let ; then, we have some operational laws, such as*(1)*.*(2)*, .*(3)*, *(4)*, *(5)*, *

*Proof. *It can be easily obtained from Theorem 1 and Definition 7. Thus, it is omitted here.

Theorem 3. *Let ; then, we have*(1)*.*(2)*.*

*Proof. * (1)From Definitions 3 and 7, we have Then, together with Definition 6, one has(2)It is similar to the proof of conclusion 1.

#### 3. Hesitant Bipolar-Valued Fuzzy Soft Sets and Operation Properties

In this section, we propose the concept of hesitant bipolar-valued fuzzy soft sets as well as some basic operations.

*Definition 8. *A pair is called as a hesitant bipolar-valued fuzzy soft set on , where is a mapping given by andWe will use the notation to denote all hesitant bipolar-valued fuzzy soft sets on .

Note:(1)If has only one value, that is , then degenerates into a hesitant bipolar-valued fuzzy set.(2)If and have only one value, that is and , then degenerates into a bipolar-valued fuzzy soft set.(3)If the bipolarity is not considered, that is , then degenerates into a hesitant fuzzy soft set.

*Example 1. *Let be the set of houses, and , where = {location; price; surrounding environment}. is a mapping given by , whereThen, .

*Definition 9. *Let and , is called as a hesitant bipolar-valued fuzzy soft subset of , if and for .

In this case, we define or .

*Example 2. *Let ; it is defined as that in Example 1. Let = {location; price} and ; it is defined asObviously, ; by calculating the score function, we can obtainThen, it hasThus, we can haveHence, we can conclude .

*Definition 10. *Let ; then,(1)If and , then and are called hesitant bipolar-valued fuzzy soft equal, denoted by .(2)If and for , then and are equal, denoted by .Note: is different from . Obviously, can be regarded as a special case of . If holds, then holds; conversely, if holds, then may not hold. For example, let and , , . We can obtain , which implies , but .

*Definition 11. *Let and be two special hesitant bipolar-valued fuzzy soft sets, which are defined as(1) is called as an empty set, if for , denoted by .(2) is called as a full set, if for , denoted by .

*Definition 12. *Let , and is called as the complementary set of the , denoted by , where and for .

*Example 3. *Let ; it is defined as that in Example 2; we obtain as follows:

Theorem 4. *Let ; then,*(1)*.*(2)*.*

*Proof. *It can be easily obtained from Definition 12.

*Definition 13. *Let ; if and ,Then, is called as the union of and , denoted by

*Example 4. *Let be the set of houses, and , where = {location, surrounding environment} and = {location, price}. Let ; it is defined asLet ; it is defined as that in Example 2; then, , where and

*Definition 14. *Let ; if and ,Then, is called as the intersection of and , denoted by

*Example 5. *Let ; it is defined as that in Example 4; then,where and

Theorem 5. *Let ; then,*(1)*.*(2)*.*(3)*.*(4)*.*(5)*.*(6)*.*(7)*.*

*Proof. *It can be obtained from Definitions 12–14.

Theorem 6. *Let ; then,*(1)* iff .*(2)* iff .*(3)*.*(4)*.*

*Proof. *It can be easily proved from Definitions 12–14.

Theorem 7. *Let ; then,*(1)*.*(2)*.*

*Proof. * (1)Let , where and where Let , where Thus, one has and . Further, it yields(2)It is similar to the proof of conclusion 1.

Theorem 8. *Let ; then,*(1)*.*(2)*.*(3)*.*(4)*.*

*Proof. * (1)Let , where , , andThen, it obtainsand ; one hasLet , where , , andFrom Theorem 3, we can obtainThus, we obtainThen, together with Definition 9, we haveSimilarly, we can also prove (2)–(4).

*Definition 15. *Let ; are defined as , whereand is a mapping given by .

In this case, define .

*Example 6. *Let . They are defined in Example 4; then, one getswhere

*Definition 16. *Let . is defined as , whereand is a mapping given by .

In this case, define .

*Example 7. *Let . They are defined as those in Example 4; then, it leads towhere