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

Wang, Jin-Ying; Wang, Yan-Ping; Liu, Lei

doi:https://doi.org/10.1155/2020/6496030

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Volume 2020 | Article ID 6496030 | https://doi.org/10.1155/2020/6496030

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

Jin-Ying Wang,¹Yan-Ping Wang,¹and Lei Liu¹

Guest Editor: Jing Na

Received15 May 2020

Revised21 Jun 2020

Accepted22 Jul 2020

Published17 Aug 2020

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

Theorem 9. Let ; then,(1).(2).

Proof. (1)Let ; then,Again letThus, it getswhere .
Obviously, we can obtainHence, it yields

Theorem 10. Let , , ; then,(1).(2).

Proof. (1)Let ; then, andLet ; then, we getAgain let ; then, it results in andLet ; then, it deducesThus, we can obtainHence, it hasthat is,

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

In order to conveniently describe the content, throughout this section, let , , and .

Definition 17. The fuzzy soft set is called as the score matrix of the hesitant bipolar-valued fuzzy soft set , where the membership degree of each element of the fuzzy set is the score function of each element of the hesitant bipolar-valued fuzzy set .

Definition 18. (see [22]). Let be a fuzzy soft set on and be threshold vector on ; the -level soft set is defined as , where , .
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. 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 (Algorithm 1).
Note: for the calculation method of how to choose the value of in the above algorithm, refer to [22].

Step 1	Input .
Step 2	Calculate the score matrix corresponding to .
Step 3	Calculating the average value assigned to each parameter, the threshold vector can be obtained.
Step 4	According to the threshold vector, calculate the average-level soft set and the choice value of every alternative.
Step 5	Rank the objects in accordance with and then select the optimal decision as if .
Step 6	If the value of is not the one in the previous step, then any one of 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, . The evaluation of the house is represented by five indicators in the hesitant bipolar-valued fuzzy soft set , where = {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 corresponding to , which is shown in Table 2.
Calculate the average value assigned to each parameter, that is,Then, we obtain the threshold vector asCalculate the average-level soft set and the value of , which is shown in Table 3.
Ranking all the houses in accordance with the choice value , we can easily see . Thus, select the optimal decision as .
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.

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.

Data Availability

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

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This study was supported in part by the National Natural Science Foundation of China under grant no. 61803190 and in part by the Liaoning Provincial Education Department Fund (JJL201915403).

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Copyright © 2020 Jin-Ying Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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