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

Authors: Jin-Ying Wang; Yan-Ping Wang; Lei Liu
Journal: Complexity
 (2020)

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## 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.


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## Body

## 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 Section2, 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, letU 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 PU 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⟶PU.Definition 2.
(see [41]). A hesitant bipolar-valued fuzzy set ℍ on U is defined as(1)ℍ=u,ℏℍu=ℏℍPu,ℏℍNuu∈U,where ℏℍPu, known as the hesitant fuzzy positive element, is a set of some values in 0,1 denoting the possible satisfaction degree of the u∈U to the corresponding property to the ℍ; ℏℍNu, known as the hesitant fuzzy positive element, is a set of some values in −1,0 denoting the possible satisfaction degree of the u∈U to the opposite property to the ℍ; and ℏℍu=ℏℍPu,ℏℍNu, known as the hesitant bipolar-valued fuzzy element, is a set of some values in 0,1×−1,0 to the ℍ, abbreviated as HBVFE.
As a matter of convenience, the HBVFEℏℍu=ℏℍPu,ℏℍNu is abbreviated as ℏℍ=ℏℍP,ℏℍN and all hesitant bipolar-valued fuzzy sets on U are abbreviated as HBVFSU in this study.Definition 3.
(see [41]). Let ℏ1=ℏ1P,ℏ1N and ℏ2=ℏ2P,ℏ2N be two HBVFEs; then,(1)
Complement:(2)ℏ1c=∪γ1P∈ℏ1P1−γ1P,∪γ1N∈ℏ1N−1−γ1N.(2)
Union:(3)ℏ1∪ℏ2=∪γ1P∈ℏ1P,γ2P∈ℏ2Pmaxγ1P,γ2P,∪γ1N∈ℏ1N,γ2N∈ℏ2Nminγ1N,γ2N.(3)
Intersection:(4)ℏ1∩ℏ2=∪γ1P∈ℏ1P,γ2P∈ℏ2Pminγ1P,γ2P,∪γ1N∈ℏ1N,γ2N∈ℏ2Nmaxγ1N,γ2N.Theorem 1.
Letℏ1=ℏ1P,ℏ1N, ℏ2=ℏ2P,ℏ2N and ℏ3=ℏ3P,ℏ3N be three HBVFEs; then, we have some operational laws, such as(1)
Involutive:(5)ℏ1cc=ℏ1.(2)
Commutativity:(6)ℏ1∪ℏ2=ℏ2∪ℏ1,ℏ1∩ℏ2=ℏ2∩ℏ1.(3)
Associative:(7)ℏ1∪ℏ2∪ℏ3=ℏ1∪ℏ2∪ℏ3,ℏ1∩ℏ2∩ℏ3=ℏ1∩ℏ2∩ℏ3.(4)
Distributive:(8)ℏ1∪ℏ2∩ℏ3=ℏ1∪ℏ2∩ℏ1∪ℏ3,ℏ1∩ℏ2∪ℏ3=ℏ1∩ℏ2∪ℏ1∩ℏ3.(5)
De Morgan [41]:(9)ℏ1∪ℏ2c=ℏ1c∩ℏ2c,ℏ1∩ℏ2c=ℏ1c∪ℏ2c.Proof.
(1)
For a HBVFEh˜1=h˜1P,h˜1N, we have(10)ℏ1c=∪γ1P∈ℏ1P1−γ1P,∪γ1N∈ℏ1N−1−γ1N,from Definition3; then,(11)ℏ1cc=∪γ1P∈ℏ1P1−γ1P,∪γ1N∈ℏ1N−1−γ1Nc=∪γ1P∈ℏ1P1−1−γ1P,∪γ1N∈ℏ1N−1−−1−γ1N=∪γ1P∈ℏ1Pγ1P,∪γ1N∈ℏ1Nγ1N=ℏ1.(2)
It can be easily obtained from Definition3.(3)
For three HBVFEsℏ1=ℏ1P,ℏ1N, ℏ2=ℏ2P,ℏ2N, and ℏ3=ℏ3P,ℏ3N, we have(12)ℏ1∪ℏ2=∪γ1P∈ℏ1P,γ2P∈ℏ2Pmaxγ1P,γ2P,∪γ1N∈ℏ1N,γ2N∈ℏ2Nminγ1N,γ2N,and from Definition3, we have(13)ℏ1∪ℏ2∪ℏ3=∪γ1P∈ℏ1P,γ2P∈ℏ2P,γ3P∈ℏ3Pmaxmaxγ1P,γ2P,γ3P,∪γ1N∈ℏ1N,γ2N∈ℏ2N,γ3N∈ℏ3Nminminγ1N,γ2N,γ3N=∪γ1P∈ℏ1P,γ2P∈ℏ2P,γ3P∈ℏ3Pmaxγ1P,γ2P,γ3P,∪γ1N∈ℏ1N,γ2N∈ℏ2N,γ3N∈ℏ3Nminγ1N,γ2N,γ3N=∪γ1P∈ℏ1P,γ2P∈ℏ2P,γ3P∈ℏ3Pmaxγ1P,maxγ2P,γ3P,∪γ1N∈ℏ1N,γ2N∈ℏ2N,γ3N∈ℏ3Nminγ1N,minγ2N,γ3N=ℏ1∪ℏ2∪ℏ3.Similarly, we can also obtain(14)ℏ1∩ℏ2∩ℏ3=ℏ1∩ℏ2∩ℏ3.(4)
For three HBVFEsℏ1, ℏ2 and ℏ3, we have(15)ℏ1∪ℏ2∩ℏ3=∪γ1P∈ℏ1P,γ2P∈ℏ2P,γ3P∈ℏ3Pmaxγ1P,minγ2P,γ3P,∪γ1N∈ℏ1N,γ2N∈ℏ2N,γ3N∈ℏ3Nminγ1N,maxγ2N,γ3N=∪γ1P∈ℏ1P,γ2P∈ℏ2P,γ3P∈ℏ3Pminmaxγ1P,γ2P,maxγ1P,γ3P,∪γ1N∈ℏ1N,γ2N∈ℏ2N,γ3N∈ℏ3Nmaxminγ1N,γ2N,minγ1N,γ3N=∪γ1P∈ℏ1P,γ2P∈ℏ2Pmaxγ1P,γ2P,∪γ1N∈ℏ1N,γ2N∈ℏ2Nminγ1N,γ2N∩∪γ1P∈ℏ1P,γ3P∈ℏ3Pmaxγ1P,γ3P,∪γ1N∈ℏ1N,γ3N∈ℏ3Nminγ1N,γ3N=ℏ1∪ℏ2∩ℏ1∪ℏ3.Similarly, we can also obtain(16)ℏ1∩ℏ2∪ℏ3=ℏ1∩ℏ2∪ℏ1∩ℏ3.Definition 4.
(see [41]). Let ℏ=ℏP,ℏN be a HBVFE and the score function Sℏ is defined as(17)Sℏ=121lℏP∑γP∈ℏPγP−1lℏN∑γN∈ℏNγN,where lℏP and lℏN are the numbers of the elements in ℏP and ℏN, respectively.Definition 5.
Two special hesitant bipolar-valued fuzzy sets are defined as(1)
Empty set:(18)Ø=u,ℏØuu∈U,whereℏØu=0,0.(2)
Full set:(19)U=u,ℏUuu∈U,whereℏUu=1,−1.Definition 6.
Letℍ1,ℍ2∈HBVFSU and ∀u∈U; then,(1)
IfSℏℍ1u≤Sℏℍ2u, then ℍ1 is called as a hesitant bipolar-valued fuzzy quasisubset of ℍ2, denoted by ℍ1⋐ℍ2 or ℍ2⋑ℍ1.(2)
IfSℏℍ1u=Sℏℍ2u, then ℍ1 and ℍ2 are called hesitant bipolar-valued fuzzy quasiequal, denoted by ℍ1≈ℍ2.(3)
Ifℏℍ1u and ℏℍ2u have the same elements, then ℍ1 and ℍ2 are equal, denoted by ℍ1=ℍ2.Definition 7.
Letℍ1,ℍ2∈HBVFSU; then,(1)
Complement:(20)ℍ1c=u,ℏℍ1cuu∈Uwhereℏℍ1c=ℏℍ1c.(2)
Union:(21)ℍ1⋓ℍ2=u,ℏℍ1⋓ℍ2uu∈U,whereℏℍ1⋓ℍ2=ℏℍ1∪ℏℍ2.(3)
Intersection:(22)ℍ1⋒ℍ2=u,ℏℍ1⋒ℍ2uu∈U,whereℏℍ1⋒ℍ2=ℏℍ1∩ℏℍ2.Theorem 2.
Letℍ1,ℍ2,ℍ3∈HBVFSU; then, we have some operational laws, such as(1)
ℍ1cc=ℍ1.(2)
ℍ1⋓ℍ2=ℍ2⋓ℍ1, ℍ1⋒ℍ2=ℍ2⋒ℍ1.(3)
ℍ1⋓ℍ2⋓ℍ3=ℍ1⋓ℍ2⋓ℍ3, ℍ1⋒ℍ2⋒ℍ3=ℍ1⋒ℍ2⋒ℍ3.(4)
ℍ1⋓ℍ2⋒ℍ3=ℍ1⋓ℍ2⋒ℍ1⋓ℍ3, ℍ1⋒ℍ2⋓ℍ3=ℍ1⋒ℍ2⋓ℍ1⋒ℍ3.(5)
ℍ1⋓ℍ2c=ℍ1c⋒ℍ2c, ℍ1⋒ℍ2c=ℍ1c⋓ℍ2c.Proof.
It can be easily obtained from Theorem1 and Definition 7. Thus, it is omitted here.Theorem 3.
Letℍ1,ℍ2∈HBVFSU; then, we have(1)
ℍ1⋒ℍ2⋐ℍ1⋐ℍ1⋓ℍ2.(2)
ℍ1⋒ℍ2⋐ℍ2⋐ℍ1⋓ℍ2.Proof.
(1)
From Definitions3 and 7, we have(23)Sℏℍ1⋒ℍ2≤Sℏℍ1≤Sℏℍ1⋓ℍ2.Then, together with Definition6, one has(24)ℍ1⋒ℍ2⋐ℍ1⋐ℍ1⋓ℍ2.(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 pairJ˜,D is called as a hesitant bipolar-valued fuzzy soft set on U, where J˜ is a mapping given by J˜:D⟶HBVFSU and(25)J˜ε=u,ℏJ˜εuu∈U,for∀ε∈D.
We will use the notationHBVFSSU to denote all hesitant bipolar-valued fuzzy soft sets on U.
Note:(1)
IfD has only one value, that is D=ε, then J˜,D degenerates into a hesitant bipolar-valued fuzzy set.(2)
IfℏJ˜εP and ℏJ˜εN have only one value, that is ℏJ˜εP=μP and ℏJ˜εN=μN, then J˜,D degenerates into a bipolar-valued fuzzy soft set.(3)
If the bipolarity is not considered, that isℏJ˜ε=ℏJ˜εP, then J˜,D degenerates into a hesitant fuzzy soft set.Example 1.
LetU=u1,u2,u3 be the set of houses, and D1⊆E, where D1=ε1,ε2,ε3 = {location; price; surrounding environment}. J˜ is a mapping given by J˜:D1⟶HBVFSU, where(26)J˜ε1=u1,0.7,0.8,−0.6,−0.8,u2,0.3,0.4,0.6,−0.7,u3,0.6,0.7,−0.5,−0.6,−0.8,J˜ε2=u1,0.4,0.7,−0.2,−0.4,u2,0.5,0.6,−0.3,−0.5,u3,0.6,0.7,0.8,−0.3,−0.4,J˜ε3=u1,0.3,0.4,−0.6,−0.7,u2,0.6,0.7,0.9,−0.4,−0.5,u3,0.8,−0.4,−0.6,−0.7.
Then,J˜,D1∈HBVFSSU.Definition 9.
LetJ˜,D1,K˜,D2∈HBVFSSU and D1,D2⊆E, J˜,D1 is called as a hesitant bipolar-valued fuzzy soft subset of K˜,D2, if D1⊆D2 and J˜ε⋐K˜ε for ∀ε∈D1.
In this case, we defineJ˜,D1⊑K˜,D2 or K˜,D2⊒J˜,D1.Example 2.
LetJ˜,D1∈HBVFSSU; it is defined as that in Example 1. Let D2=ε1,ε2 = {location; price} and K˜,D2∈HBVFSSU; it is defined as(27)K˜ε1=u1,0.3,0.4,0.6,−0.5,−0.7,u2,0.1,0.3,−0.2,−0.4,u3,0.6,0.8,−0.3,−0.7,K˜ε2=u1,0.4,0.6,−0.2,−0.3,u2,0.3,−0.2,−0.5,−0.6,u3,0.4,0.5,−0.3,−0.4.
Obviously,D2⊆D1; by calculating the score function, we can obtain(28)SℏJ˜ε1u1=0.725,SℏK˜ε1u1=0.5167;SℏJ˜ε1u2=0.5667,SℏK˜ε1u2=0.25;SℏJ˜ε1u3=0.6417,SℏK˜ε1u3=0.6;SℏJ˜ε2u1=0.425,SℏK˜ε2u1=0.375;SℏJ˜ε2u2=0.475,SℏK˜ε2u2=0.3667;SℏJ˜ε2u3=0.525,SℏK˜ε2u3=0.4.
Then, it has(29)sℏJ˜εiuj≥sℏK˜εiuj.
Thus, we can have(30)K˜ε⋐J˜ε,for∀ε∈D2.
Hence, we can concludeK˜,D2⊑J˜,D1.Definition 10.
LetJ˜,D1,K˜,D2∈HBVFSSU; then,(1)
IfJ˜,D1⊑K˜,D2 and K˜,D2⊑J˜,D1, then J˜,D1 and K˜,D2 are called hesitant bipolar-valued fuzzy soft equal, denoted by J˜,D1≙K˜,D2.(2)
IfD1=D2 and J˜ε=K˜ε for ∀ε∈D1, then J˜,D1 and K˜,D2 are equal, denoted by J˜,D1=K˜,D2.
Note:J˜,D1=K˜,D2 is different from J˜,D1≙K˜,D2. Obviously, J˜,D1=K˜,D2 can be regarded as a special case of J˜,D1≙K˜,D2. If J˜,D1=K˜,D2 holds, then J˜,D1≙K˜,D2 holds; conversely, if J˜,D1≙K˜,D2 holds, then J˜,D1=K˜,D2 may not hold. For example, let U=u and D1=D2=ε, J˜ε=u,0.5,0.6,−0.3,−0.4, K˜ε=u,0.6,0.7,−0.2,−0.3. We can obtain SℏJ˜ε=0.45=SℏK˜ε, which implies J˜,D1≙K˜,D2, but J˜,D1≠K˜,D2.Definition 11.
LetJ˜,D∈HBVFSSU and u∈U be two special hesitant bipolar-valued fuzzy soft sets, which are defined as(1)
J˜,D is called as an empty set, if J˜ε=Ø for ∀ε∈D, denoted by Ø˜,D.(2)
J˜,D is called as a full set, if J˜ε=U for ∀ε∈D, denoted by U˜,D.Definition 12.
LetJ˜,D∈HBVFSSU, and J˜c,D is called as the complementary set of the J˜,D, denoted by J˜,Dc, where J˜c:D⟶HBVFSU and J˜cε=J˜εc for ∀ε∈D.Example 3.
LetK˜,D2∈HBVFSSU; it is defined as that in Example 2; we obtain K˜,D2c as follows:(31)K˜cε1=u1,0.4,0.6,0.7,−0.3,−0.5,u2,0.7,0.9,−0.6,−0.8,u3,0.2,0.4,−0.3,−0.7,K˜cε2=u1,0.4,0.6,−0.7,−0.8,u2,0.7,−0.4,−0.5,−0.8,u3,0.5,0.6,−0.6,−0.7.Theorem 4.
LetØ˜,D,U˜,D∈HBVFSSU; then,(1)
Ø˜,Dc=U˜,D.(2)
U˜,Dc=Ø˜,D.Proof.
It can be easily obtained from Definition12.Definition 13.
LetJ˜,D1,K˜,D2∈HBVFSSU; if D=D1∪D2 and ∀ε∈D,(32)L˜ε=J˜ε,ifε∈D1−D2,K˜ε,ifε∈D2−D1,J˜ε⋓K˜ε,ifε∈D1∩D2.
Then,L˜,D is called as the union of J˜,D1 and K˜,D2, denoted by(33)J˜,D1⊔K˜,D2=L˜,DExample 4.
LetU=u1,u2,u3 be the set of houses, and D1,D2⊆E, where D1=ε1,ε3 = {location, surrounding environment} and D2=ε1,ε2 = {location, price}. Let J˜,D1∈HBVFSSU; it is defined as(34)J˜ε1=u1,0.7,0.8,−0.6,−0.8,u2,0.3,0.4,0.6,−0.7,u3,0.6,0.7,−0.5,−0.6,−0.8,J˜ε3=u1,0.3,0.4,−0.6,−0.7,u2,0.6,0.7,0.9,−0.4,−0.5,u3,0.8,−0.4,−0.6,−0.7.
LetK˜,D2∈HBVFSSU; it is defined as that in Example 2; then, J˜,D1⊔K˜,D2=L˜,D, where D=ε1,ε2,ε3 and(35)L˜ε1=u1,0.7,0.8,−0.6,−0.7,−0.8,u2,0.3,0.4,0.6,−0.7,u3,0.6,0.7,0.8,−0.5,−0.6,−0.7,−0.8,L˜ε2=u1,0.4,0.6,−0.2,−0.3,u2,0.3,−0.2,−0.5,−0.6,u3,0.4,0.5,−0.3,−0.4,L˜ε3=u1,0.3,0.4,−0.6,−0.7,u2,0.6,0.7,0.9,−0.4,−0.5,u3,0.8,−0.4,−0.6,−0.7.Definition 14.
LetJ˜,D1,K˜,D2∈HBVFSSU; if D=D1∩D2 and ∀ε∈D,(36)L˜ε=J˜ε⋒K˜ε.
Then,L˜,D is called as the intersection of J˜,D1 and K˜,D2, denoted by(37)J˜,D1⊓K˜,D2=L˜,D.Example 5.
LetJ˜,D1,K˜,D2∈HBVFSSU; it is defined as that in Example 4; then,(38)J˜,D1⊓K˜,D2=L˜,D,where D=ε1 and(39)L˜ε1=u1,0.3,0.4,0.6,−0.5,−0.6,−0.7,u2,0.1,0.3,−0.2,−0.4,u3,0.6,0.7,−0.3,−0.5,−0.6,−0.7.Theorem 5.
LetJ˜,D∈HBVFSSU; then,(1)
J˜,Dcc=J˜,D.(2)
J˜,D⊔J˜,D=J˜,D.(3)
J˜,D⊓J˜,D=J˜,D.(4)
J˜,D⊔Ø˜,D=J˜,D.(5)
J˜,D⊓Ø˜,D=Ø˜,D.(6)
J˜,D⊔U˜,D=U˜,D.(7)
J˜,D⊓U˜,D=J˜,D.Proof.
It can be obtained from Definitions12–14.Theorem 6.
LetJ˜,D1∈HBVFSSU; then,(1)
Ø˜,D⊔J˜,D1=J˜,D1 iff D⊆D1.(2)
U˜,D⊔J˜,D1=U˜,D iff D1⊆D.(3)
Ø˜,D⊓J˜,D1=Ø˜,D∩D1.(4)
U˜,D⊓J˜,D1=J˜,D∩D1.Proof.
It can be easily proved from Definitions12–14.Theorem 7.
LetJ˜,D,K˜,D∈HBVFSSU; then,(1)
J˜,D⊔K˜,Dc=J˜,Dc⊓K˜,Dc.(2)
J˜,D⊓K˜,Dc=J˜,Dc⊔K˜,Dc.Proof.
(1)
LetJ˜,D⊔K˜,D=ℙ˜,D, where(40)ℙ˜ε=J˜ε⋓K˜ε,for∀ε∈D,and(41)J˜,D⊔K˜,Dc=ℙ˜,Dc=ℙ˜c,D,where(42)ℙ˜cε=J˜ε⋓K˜εc=J˜cε⋒K˜cε,for∀ε∈D.LetJ˜,Dc⊓K˜,Dc=J˜c,D⊓K˜c,D=ℚ˜,D, where(43)ℚ˜ε=J˜cε⋒K˜cε,for∀ε∈D.Thus, one hasℙ˜cε=ℚ˜ε and ℙ˜c,D=ℚ˜,D.Further, it yields(44)J˜,D⊔K˜,Dc=J˜,Dc⊓K˜,Dc.(2)
It is similar to the proof of conclusion 1.Theorem 8.
LetJ˜,D1,K˜,D2∈HBVFSSU; then,(1)
J˜,D1⊔K˜,D2c⊑J˜,D1c⊔K˜,D2c.(2)
J˜,D1⊓K˜,D2c⊒J˜,D1c⊓K˜,D2c.(3)
J˜,D1⊔K˜,D2c⊒J˜,D1c⊓K˜,D2c.(4)
J˜,D1⊓K˜,D2c⊑J˜,D1c⊔K˜,D2c.Proof.
(1)
LetJ˜,D1⊔K˜,D2=ℙ˜,D3, where D3=D1∪D2, ∀ε∈D3, and(45)ℙ˜ε=J˜ε,ifε∈D1−D2,K˜ε,ifε∈D2−D1,J˜ε⋓K˜ε,ifε∈D1∩D2.
Then, it obtains(46)J˜,D1⊔K˜,D2c=ℙ˜,D3c=ℙ˜c,D3,and ∀ε∈D3; one has(47)ℙ˜cε=J˜cε,ifε∈D1−D2,K˜cε,ifε∈D2−D1,J˜cε⋒K˜cε,ifε∈D1∩D2.
LetJ˜,D1c⊔K˜,D2c=J˜c,D1⊔K˜c,D2=ℚ˜,D4, where D4=D1∪D2, ∀ε∈D4, and(48)ℚ˜ε=J˜cε,ifε∈D1−D2,K˜cε,ifε∈D2−D1,J˜cε⋓K˜cε,ifε∈D1∩D2.
From Theorem3, we can obtain(49)J˜cε⋒K˜cε⋐J˜cε⋓K˜cε,for∀ε∈D1∩D2.
Thus, we obtain(50)D3=D4=D1∪D2,ℙ˜cε⋐ℚ˜ε,for∀ε∈D1∪D2.
Then, together with Definition9, we have(51)J˜,D1⊔K˜,D2c⊑J˜,D1c⊔K˜,D2c.
Similarly, we can also prove (2)–(4).Definition 15.
LetJ˜,D1,K˜,D2∈HBVFSSU; J˜,D1andK˜,D2 are defined as L˜,D1×D2, where(52)L˜ξ,η=J˜ξ⋒K˜η,for∀ξ,η∈D1×D2,and L˜ is a mapping given by L˜:D1×D2⟶HBVFSU.
In this case, defineJ˜,D1∧¯K˜,D2.Example 6.
LetJ˜,D1,K˜,D2∈HBVFSSU. They are defined in Example 4; then, one gets(53)J˜,D1∧¯K˜,D2=L˜,D1×D2,where(54)L˜ε1,ε1=u1,0.3,0.4,0.6,−0.5,−0.6,−0.7,u2,0.1,0.3,−0.2,−0.4,u3,0.6,0.7,−0.3,−0.5,−0.6,−0.7,L˜ε1,ε2=u1,0.4,0.6,−0.2,−0.3,u2,0.3,−0.2,−0.5,−0.6,u3,0.4,0.5,−0.3,−0.4,L˜ε3,ε1=u1,0.3,0.4,−0.5,−0.6,−0.7,u2,0.1,0.3,−0.2,−0.4,u3,0.6,0.8,0.3,−0.4,−0.6,−0.7,L˜ε3,ε2=u1,0.3,0.4,−0.2,−0.3,u2,0.3,−0.2,−0.4,−0.5,u3,0.4,0.5,−0.3,−0.4.Definition 16.
LetJ˜,D1,K˜,D2∈HBVFSSU. J˜,D1orK˜,D2 is defined as L˜,D1×D2, where(55)L˜ξ,η=J˜ξ⋓K˜η,for∀ξ,η∈D1×D2,and L˜ is a mapping given by L˜:D1×D2⟶HBVFSU.
In this case, defineJ˜,D1∨¯K˜,D2.Example 7.
LetJ˜,D1,K˜,D2∈HBVFSSU. They are defined as those in Example 4; then, it leads to(56)J˜,D1∨¯K˜,D2=L˜,D1×D2,where(57)L˜ε1,ε1=u1,0.7,0.8,−0.6,−0.7,−0.8,u2,0.3,0.4,0.6,−0.7,u3,0.6,0.7,0.8,−0.5,−0.6,−0.7,−0.8,L˜ε1,ε2=u1,0.7,0.8,−0.6,−0.8,u2,0.3,0.4,0.6,−0.7,u3,0.6,0.7,−0.5,−0.6,−0.8,L˜ε3,ε1=u1,0.3,0.4,0.6,−0.6,−0.7,u2,0.6,0.7,0.9,−0.4,−0.5,u3,0.8,−0.4,−0.6,−0.7,L˜ε3,ε2=u1,0.4,0.6,−0.6,−0.7,u2,0.6,0.7,0.9,−0.4,−0.5,−0.6,u3,0.8,−0.4,−0.6,−0.7.Theorem 9.
LetJ˜,D1,K˜,D2∈HBVFSSU; then,(1)
J˜,D1∧¯K˜,D2c=J˜,D1c∨¯K˜,D2c.(2)
J˜,D1∨¯K˜,D2c=J˜,D1c∧¯K˜,D2c.Proof.
(1)
LetJ˜,D1∧¯K˜,D2=ℙ˜,D1×D2; then,(58)J˜,D1∧¯K˜,D2c=ℙ˜c,D1×D2.
Again let(59)J˜,D1c∨¯K˜,D2c=J˜c,D1∨¯K˜c,D2=ℚ˜,D1×D2.
Thus, it gets(60)ℙ˜cξ,η=J˜ξ⋒K˜ηc=J˜cξ⋓K˜cη,ℚ˜ξ,η=J˜cξ⋓K˜cη,where ∀ξ,η∈D1×D2.
Obviously, we can obtain(61)ℙ˜cξ,η=ℚ˜ξ,η.
Hence, it yields(62)J˜,D1∧¯K˜,D2c=J˜,D1c∨¯K˜,D2c.Theorem 10.
LetJ˜,D1, K˜,D2, L˜,D3∈HBVFSSU; then,(1)
J˜,D1∧¯K˜,D2∧¯L˜,D3=J˜,D1∧¯K˜,D2∧¯L˜,D3.(2)
J˜,D1⊻¯K˜,D2⊻¯L˜,D3=J˜,D1⊻¯K˜,D2⊻¯L˜,D3.Proof.
(1)
LetK˜,D2∧¯L˜,D3=ℙ˜,D4; then, D4=D2×D3 and(63)ℙ˜η,ζ=K˜η⋒L˜ζ,for∀η,ζ∈D4.
LetJ˜,D1∧¯ℙ˜,D4=ℚ˜,D5; then, we get(64)D5=D1×D4=D1×D2×D3,ℚ˜ξ,η,ζ=J˜ξ⋒ℙ˜η,ζ=J˜ξ⋒K˜η⋒L˜ζ,for∀ξ,η,ζ∈D5.
Again letJ˜,D1∧¯K˜,D2=M˜,D6; then, it results in D6=D1×D2 and(65)M˜ξ,η=J˜ξ⋒K˜η,for∀ξ,η∈D6.
LetM˜,D6∧¯L˜,D3=ℕ˜,D7; then, it deduces(66)D7=D6×D3=D1×D2×D3,ℕ˜ξ,η,ζ=M˜ξ,η⋒L˜ζ=J˜ξ⋒K˜η⋒L˜ζ,for∀ξ,η,ζ∈D7.
Thus, we can obtain(67)D5=D7=D1×D2×D3,ℚ˜ξ,η,ζ=ℕ˜ξ,η,ζ,for∀ξ,η,ζ∈D5=D7.
Hence, it has(68)ℚ˜,D5=ℕ˜,D7,that is,(69)J˜,D1∧¯K˜,D2∧¯L˜,D3=J˜,D1∧¯K˜,D2∧¯L˜,D3.


## 4. Hesitant Bipolar-Valued Fuzzy Soft Sets in Decision-Making Problems

In order to conveniently describe the content, throughout this section, letU=u1,u2,…,un, D=ε1,ε2,…,εm, and J˜,D∈HBVFSSU.Definition 17.
The fuzzy soft setJ˜S,D is called as the score matrix of the hesitant bipolar-valued fuzzy soft set J˜,D, where the membership degree of each element of the fuzzy set J˜Sεi is the score function of each element of the hesitant bipolar-valued fuzzy set J˜εii=1,2,…,m.Definition 18.
(see [22]). Let F,D be a fuzzy soft set on U and δD=δε1,δε2,…,δεm be threshold vector on D; the δD-level soft set is defined as Fδ,D, where Fδε=uj∈U:μFεuj≥δε, ∀ε∈D.
In Definition18, 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,(70)δε=1n∑j=1nμFεujfor∀ε∈D.
Next, a multiattribute decision-making approach for hesitant bipolar-valued fuzzy soft sets is given. This 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 (Algorithm1).
Note: for the calculation method of how to choose the value ofC in the above algorithm, refer to [22].Algorithm 1:Multiattribute decision making of HBVFSS.
Step 1
InputJ˜,D.Step 2
Calculate the score matrixJ˜S,D corresponding to J˜,D.Step 3
Calculating the average value assigned to each parameter, the threshold vectorδD can be obtained.Step 4
According to the threshold vector, calculate the average-level soft setJ˜Sδ,D and the choice value C of every alternative.Step 5
Rank the objects in accordance withC and then select the optimal decision as uk if Ck=maxCj.Step 6
If the value ofk is not the one in the previous step, then any one of uk can be chosen for the optimal decision making.Example 8.
There 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=u1,u2,⋯,u5. The evaluation of the house is represented by five indicators in the hesitant bipolar-valued fuzzy soft set J˜,D, where D=ε1,ε2,…,ε5 = {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 matrixJ˜S,D corresponding to J˜,D, which is shown in Table 2.
Calculate the average value assigned to each parameter, that is,(71)δε1=15∑j=15μJ˜Sε1uj=0.4533,δε2=15∑j=15μJ˜Sε2uj=0.4650,δε3=15∑j=15μJ˜Sε3uj=0.4200,δε4=15∑j=15μJ˜Sε4uj=0.4217,δε5=15∑j=15μJ˜Sε5uj=0.4267.
Then, we obtain the threshold vector as(72)δD=δε1,δε2,δε3,δε4,δε5=0.4533,0.4650,0.4200,0.4217,0.4267.
Calculate the average-level soft setJ˜Sδ,D and the value of C, which is shown in Table 3.
Ranking all the houses in accordance with the choice valueC, we can easily see u1>u5>u2>u3=u4. Thus, select the optimal decision as u1.
The above examples not only consider the bipolarity and nonuniqueness of element membership but also solve the problem of insufficient parameter chemical tools. The conclusion is in line with objective reality and shows the superiority of the proposed algorithm.Table 1
Hesitant bipolar-valued fuzzy soft set.
ε1ε2ε3ε4ε5u10.6,0.5,0.4,−0.3,−0.4,−0.50.9,0.8,0.6,−0.3,−0.40.8,0.7,0.4,−0.3,−0.40.7,0.6,−0.2,−0.4,−0.60.6,0.5,0.4,−0.2,−0.3,−0.6u20.8,0.6,−0.2,−0.3,−0.60.7,0.6,0.5,−0.1,−0.30.4,0.3,−0.3,−0.4,−0.60.6,0.4,0.3,−0.1,−0.20.9,0.7,−0.1,−0.3,−0.4u30.6,0.4,0.3,−0.1,−0.3,−0.60.8,0.7,0.6,−0.1,−0.30.6,0.4,0.3,−0.2,−0.40.6,0.2,−0.2,−0.4,−0.70.8,0.6,0.5,−0.1,−0.4u40.9,0.7,0.6,−0.1,−0.20.6,0.3,−0.1,−0.2,−0.30.7,0.6,0.3,−0.1,−0.20.8,0.6,0.5,−0.3,−0.40.6,0.4,0.3,−0.1,−0.2,−0.4u50.7,0.6,0.4,−0.3,−0.40.9,0.8,−0.1,−0.3,−0.6,0.8,0.7,0.5,−0.3,−0.40.7,0.4,0.3,−0.2,−0.3,−0.40.3,0.2,−0.3,−0.6,−0.7Table 2
Score matrix.
ε1ε2ε3ε4ε5u10.45000.55830.49170.52500.4333u20.53330.40000.39170.29170.5333u30.38330.45000.36670.41670.4417u40.44170.32500.34170.49170.3333u50.45830.59170.50830.38330.3917Table 3
Average-level soft set and choice value.
ε1ε2ε3ε4ε5Cu1011114u2100012u3000011u4000101u5111003


## 5. 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. Then, 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. The 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.



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