Hesitant Bipolar-Valued Fuzzy Soft Sets and Their Application in Decision Making

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 bipolarvalued 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. (e effectiveness of the proposed method is illustrated by an example.


Introduction
e 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. e 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][4][5][6][7][8][9][10][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. e corresponding basic procedures and properties are also discussed. Based on the above results, some scholars made a further study about the uncertainty measures [12][13][14][15][16][17][18][19][20][21] and the decision-making methods [22][23][24][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 intervalvalued 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]. e 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 intervalvalued 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][27][28][29][30][31][32][33][34][35][36][37][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 bipolarvalued hesitant fuzzy sets. Both of them were applied to solve decision-making problems [41]. Wei discussed the decisionmaking 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 bipolarvalued neutrosophic sets and introduced an original decision-making method based on the novel model [46]. ose 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. e model is a new mathematical tool for uncertainty problem. It can make up for the inadequacy of the parametric tools. erefore, the model is more suitable to describe the uncertainty problem. e 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.

Preliminaries
For convenience, throughout the paper, let U be an initial limited universe set, E be the set of all possible parameters with respect to U, and D ⊆ E.
Definition 1 (see [1]). Let P(U) denote the power set of U. A pair (F, D) is called a soft set on U, where F is a mapping given by F: D ⟶ P(U).
Definition 2 (see [41]). A hesitant bipolar-valued fuzzy set H on U is defined as Definition 3 (see [41]). Let Z 1 � (Z P 1 , Z N 1 ) and Z 2 � (Z P 2 , Z N 2 ) be two HBVFEs; then, Similarly, we can also obtain □ Definition 4 (see [41]). Let Z � (Z P , Z N ) be a HBVFE and the score function S(Z) is defined as where l Z P and l Z N are the numbers of the elements in Z P and Z N , respectively.
Definition 5. Two special hesitant bipolar-valued fuzzy sets are defined as (1) Empty set: (2) Full set: Definition 6. Let H 1 , H 2 ∈ HBVFS(U) and ∀u ∈ U; then, where Intersection: where Proof. It can be easily obtained from eorem 1 and Definition 7. us, it is omitted here.

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

Hesitant Bipolar-Valued Fuzzy Soft Sets and Operation Properties
In this section, we propose the concept of hesitant bipolarvalued fuzzy soft sets as well as some basic operations.
is called as a hesitant bipolarvalued fuzzy soft set on U, where J is a mapping given by J: D ⟶ HBVFS(U) and We will use the notation HBVFSS(U) to denote all hesitant bipolar-valued fuzzy soft sets on U. Note: , then (J, D) degenerates into a hesitant fuzzy soft set.
In this case, we define (J, Obviously, D 2 ⊆D 1 ; by calculating the score function, we can obtain en, it has us, we can have Hence, we can conclude (K, D 2 ) ⊑(J, D 1 ).  for ∀ε ∈ D.

Further, it yields
(2) It is similar to the proof of conclusion 1.
and ∀ε ∈ D 3 ; one has From eorem 3, we can obtain us, we obtain en, together with Definition 9, we have Similarly, we can also prove (2)-(4).
and L is a mapping given by L: In this case, define (J, D 1 )∧ (K, D 2 ). where and L is a mapping given by L: In this case, define (J, D 1 ) ∨ (K, D 2 ).

Hesitant Bipolar-Valued Fuzzy Soft Sets in Decision-Making Problems
In order to conveniently describe the content, throughout this section, let U � u 1 , u 2 , . . . , u n , D � ε 1 , ε 2 , . . . , ε m , and (J, D) ∈ HBVFSS(U). Definition 18 (see [22]). Let (F, D) be a fuzzy soft set on U and δ(D) � δ(ε 1 ), δ(ε 2 ), . . . , δ(ε m ) be threshold vector on In Definition 18, by using different selection methods of attribute threshold, we may obtain different level soft sets. In [22], we can find some different selection methods of threshold such as maximum, minimum, and average. We determine the threshold with the method of average in this study, that is, Next, a multiattribute decision-making approach for hesitant bipolar-valued fuzzy soft sets is given. is approach demonstrates that hesitant bipolar-valued fuzzy soft sets can be converted into simpler fuzzy soft sets by considering the score matrix. At the same time, the fuzzy soft set is converted into a classic soft set, and each attribute is set to be a threshold according to soft sets; then, the optimal plan can be acquired by calculating the choice value of plan correspondingly (Algorithm 1).
Note: for the calculation method of how to choose the value of C in the above algorithm, refer to [22].

Example 8.
ere are three members in Mr. Zhang's family; they want to buy a house, and they decide to choose one of the five houses after preliminary screening, U � u 1 , u 2 , · · · , u 5 . e evaluation of the house is represented by five indicators in the hesitant bipolar-valued fuzzy soft set (J, D), where D � ε 1 , ε 2 , . . . , ε 5 � {location; price; surrounding environment; building structure; property management}. It is shown in Table 1. Let us help them to make the optimal decision by using the above algorithm.
Calculate the score matrix (J S , D) corresponding to (J, D), which is shown in Table 2.
Calculate the average value assigned to each parameter, that is, en, we obtain the threshold vector as Calculate the average-level soft set (J Sδ , D) and the value of C, which is shown in Table 3.
Ranking all the houses in accordance with the choice value C, we can easily see u 1 > u 5 > u 2 > u 3 � u 4 . us, select the optimal decision as u 1 .
e above examples not only consider the bipolarity and nonuniqueness of element membership but also solve the problem of insufficient parameter chemical tools. e conclusion is in line with objective reality and shows the superiority of the proposed algorithm.
Step 2 Calculate the score matrix (J S , D) corresponding to (J, D).
Step 3 Calculating the average value assigned to each parameter, the threshold vector δ(D) can be obtained.
Step 4 According to the threshold vector, calculate the average-level soft set (J Sδ , D) and the choice value C of every alternative.
Step 5 Rank the objects in accordance with C and then select the optimal decision as u k if C k � max C j .
Step 6 If the value of k is not the one in the previous step, then any one of u k can be chosen for the optimal decision making.

Conclusion
In this paper, the theory of hesitant bipolar-valued fuzzy soft sets is firstly proposed based on soft sets and hesitant bipolar-valued fuzzy sets. en, some basic operations and corresponding properties are defined. A multiattribute decision-making approach for hesitant bipolar-valued fuzzy soft sets is proposed. By using the thought of score function and level soft set, the hesitant bipolar-valued fuzzy soft set is converted into a fuzzy soft set. Since the fuzzy soft set is transformed into a classic soft set, the optimal decision making is obtained. e method not only considers the bipolarity and nonuniqueness of element membership but also solves the insufficient parameterization problem. Finally, a practical problem is solved to demonstrate the capability of the proposed method. Our future work is to study the parameterization reduction or the uncertainty measurement of hesitant bipolar-valued fuzzy soft sets, such as distance, similarity, and entropy, in which a new multiattribute decision-making method will be constructed.

Data Availability
e simulation data used to support the findings of the study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.  Table 3: Average-level soft set and choice value.