Abstract
The purpose of this study is to generalize the concept of -hesitant fuzzy sets and soft set theory to -hesitant fuzzy soft sets. The -hesitant fuzzy set is an admirable hybrid property, specially developed by the new generalized hybrid structure of hesitant fuzzy sets. Our goal is to provide a formal structure for the -polar -hesitant fuzzy soft (MQHFS) set. First, by combining m-pole fuzzy sets, soft set models, and -hesitant fuzzy sets, we introduce the concept of MQHFS and apply it to deal with multiple theories in -algebra. We then develop a framework including MQHFS subalgebras, MQHFS ideals, closed MQHFS ideals, and MQHFS exchange ideals in -algebras. Furthermore, we prove some relevant properties and theorems studied in our work. Finally, the application of MQHFS-based multicriteria decision-making in the Ministry of Health system is illustrated through a recent case study to demonstrate the effectiveness of MQHFS through the use of horizontal soft sets in decision-making.
1. Introduction
Many authors are already interested in modeling ambiguity. The specific reason for this is that the concepts we encounter in our daily life are vague rather than precise. In real-life scenarios, our world is characterized by uncertainty, imprecision, and ambiguity in many fields, including economics, engineering, environmental science, and social science. The uncertainty that arises in these problems cannot be solved successfully by classical methods in mathematics. In addition, probability theory, interval mathematics, and rough set theory can be viewed as mathematical tools that help manage uncertainty. However, all of these theories have their own difficulties. Fuzzy sets, introduced by Zadeh [1], deal with uncertainties related to inaccuracies in states, perceptions, and preferences. After the introduction of fuzzy sets by Zadeh, fuzzy set theory has been actively researched in various fields such as medical and life sciences, engineering, business and social sciences, computer networks, decision making, artificial intelligence, pattern recognition, and robotics. Later, many authors studied some generalizations of the main basic concepts of fuzzy sets in different directions. In many real-world problems, information can come from multiple sources, and there is a lot of multiattribute data that cannot be handled using fuzzy sets. In 2014, Chen et al. [2] introduced the -polar fuzzy set, an extension of the fuzzy set.
Molodtsov [3] first proposed the concept of soft set theory. Soft set theory became a general tool for dealing with uncertainty. The utility of soft set theory is that it does not present difficulties that affect existing methods. Then, Maji et al. combined fuzzy theory with soft sets in [4] and proposed the concept of fuzzy soft sets. After defining soft set theory, Maji et al. [4] also subtly introduced soft sets into decision problems. In 2010, Torra [5] introduced the concept of hesitant fuzzy sets, which help to express people’s hesitation in life. Hesitant fuzzy sets are very useful tools for dealing with uncertainty. Described from the perspective of a decision maker can be nuanced. In 2013, Babitha and John [6] introduced the concept of hesitant fuzzy soft sets.
Algebraic structures have broad and interdisciplinary applications and have a profound impact on mathematics. As a result, algebraic structures provide academics with sufficient motivation to explore a wide range of concepts and come out of the field of abstract algebra in a more inclusive fuzzy context. Imai and Is [7] first introduced the -algebra, an algebraic structure of universal algebra, in 1966. The -algebraic theory was given a soft set theory treatment by Jun [8]. Later, he used subalgebras and the ideals of -algebras to apply the idea of hesitant fuzzy soft sets in [9–11]. Since one-dimensional membership functions cannot be utilized to process data in a two-dimensional ensemble, all the research studies discussed are described as representations of one-dimensional membership functions. At [12], Adem and Hassan introduced the idea of a -fuzzy soft set. A number of authors, including [13–19], discussed various decision-making techniques.
The following are the primary contributions of this study:(1)A generalized concept of -hesitant fuzzy sets and soft set theory to -hesitant fuzzy soft sets is presented(2)A formal structure for the MQHFS set is provided(3)The concept of MQHFS is applied to deal with multiple theories in -algebra(4)A framework including MQHFS subalgebras, MQHFS ideals, closed MQHFS ideals, and MQHFS exchange ideals in -algebras is discussed(5)An application of MQHFS-based multicriteria decision-making in the Ministry of Health system is illustrated
The remainder of this research is structured as follows. We outline the fundamental ideas behind linked literature research in Section 2. The ideas of -pole -hesitant fuzzy sets (MQHFs), subalgebras of MQHFS, and ideals such closed exchange ideals are generalized in Section 3 before some of their features are explored. We provide an example (MQHFS) application to choice issues in Section (4). In Section 5, we discuss some overall conclusions and potential lines of inquiry.
2. Preliminaries
In this section, we recall some basic definitions which will be used in our work.
For other symbols, applications, and concepts, the reader is advised to refer [20–30].
Definition 1 (see [9]). Let be a -algebra. A hesitant fuzzy seton is called a hesitant fuzzy subalgebra of if it satisfies , and
Definition 2 (see [9]). Let be a -algebra. A hesitant fuzzy set,on is called a hesitant fuzzy ideal of if it satisfies , and
Definition 3 (see [10]). Let P be a set of parameters, for a subset of P, A hesitant fuzzy soft set (Z, ) over is called a hesitant fuzzy soft subalgebra based on if the hesitant fuzzy set,is a hesitant fuzzy subalgebra of .
Definition 4 (see [10]). Let P be the set of parameters, for a subset of P, A hesitant fuzzy soft set (Z, ) over is called a hesitant fuzzy soft ideal based on if the hesitant fuzzy set,is a hesitant fuzzy ideal of .
Definition 5 (see [9]). Let be a nonempty finite universe and be a nonempty set. A -hesitant fuzzy set is a set given by the following expression:where .
Definition 6 (see [16]). An MPQHF set on a nonempty set is the mapping . The membership value of every element is denoted by the following expression:where is the i-th projection for all .
3. -Polar -Hesitant Fuzzy Soft Subalgebra and Ideals
Definition 7. Let be a -algebra. The -polar -hesitant fuzzy (MPQHF) set:on is called MPQHF subalgebra of if it satisfies andfor all .
Example 1. Let be a -algebra with Cayley Table 1.
As , we have.
We verify that it is a 2-polar -hesitant fuzzy subalgebra as given in Table 2.
Proposition 8. Every MPQHF subalgebra satisfies the following inequality:(1)It is given as(2)If , then
Definition 9. Let P be a set of parameters. For a subset of P, a MPQHF soft set is called a MPQHF soft subalgebra based on if the MPQHF seton is a -hesitant fuzzy soft subalgebra of , for all .
Example 2. Let be a -algebra set and consider the operation on defined by Cayley Table 3.
Then, is a -algebra. Consider the set and a parameters set , let , which is described in Table 4.
Table 4 shows that it is a 3-polar -hesitant fuzzy soft subalgebra over based on the parameters.
Proposition 10. If is a MPQHF soft subalgebra over , then andwhere e is any parameter in and .
Proof. For any and , we have the following expression:where and . This completes the proof.
Proposition 11. If every MPQHF soft subalgebra of satisfies the following inequality: andfor all , then
Proof. Since . From -algebra definition, it follows that and , we have the following expression:It follows by Proposition 11 thatLet us begin with the definition of the -hesitant fuzzy ideal.
Definition 12. Letbe a MPQHF set in . Then, is called a MPQHF ideal of if it satisfies the following conditions: and .(1)Here, (2)Also,
Example 3. The MPQHF subalgebra that we described in Example 1 is not a MPQHF ideal because of the following reason.
If we take the two MPQHF sets:on , then
Definition 13. Let be a MPQHF soft set over where is a subset of the parameters set P. If , is called a MPQHF soft ideal based on if the following MPQHF setis hesitant fuzzy ideal of for all .
Example 4. Consider a company wants to buy two types of beauty products from two brands, and they are so interested to hear the opinion about these products from two specialists M = 2. Let be the set of products. We consider the operation defined in Cayley Table 5.
Then, is -algebra.
Let be a set of brands, and the parameter set is , and the parameter Z stands for “,” “,” and “,” which is described in Table 6.
Thus, it is a 2-polar -hesitant fuzzy soft ideal.
Theorem 14. For any -algebra , every MPQHF soft ideal is a MPQHF soft subalgebra.
Proof. Let , and is a MPQHF soft ideal over for , , and for all , then is a MPQHF soft subalgebra over . This ends the proof.
Proposition 15. Every MPQHF soft ideal over satisfies the following: and .
Proof. Let , , and such that , then and soand it follows thatThis completes the proof.
Proposition 16. Every MPQHF soft ideal over -algebra satisfies the following: and .
Proof. Let be a MPQHF soft ideal, then for , and for all .
Proposition 17. Every MPQHF soft ideal satisfies the following conditions: and .(1)If , then(2)Also,(3)If , then
Proof. Let , , and .(1)If , then since is a MPQHF soft ideal of .(2)Since from (1), we have . Hence,(3)Ifthen we have
Definition 18. A MPQHF idealof -algebra is said to be closed iffor all and .
Example 5. The MPQHF subalgebra which is described in Example 6 is a closed MPQHF ideal.
Definition 19. A MPQHF soft ideal over a -algebra based on a parameter is said to be closed if the MPQHF soft seton is closed hesitant fuzzy ideal of , for all .
Example 6. A region municipality nominated two types of asphalts (, ) which is described in the set , and then they sent three engineers to the main contractor who is working on the roads to inspect his works on those types of asphalts. His work needs to rely on water level the connectivity between the old and new asphaltThen, the parameter set .
Table 7 shows that is a -algebra.
Then, it is easy to see from Table 8 the closed 3-polar -hesitant fuzzy soft ideal.
Theorem 20. Let be a MPQHF soft set. Then, a closed MPQHF soft ideal of BCI-algebra based on a parameter is a MPQHF soft subalgebra over based on the same parameter.
Proof. Let be a closed MPQHF soft ideal over based on the parameter , thenfor all and it follows thatfor all . Therefore, is a MPQHF soft subalgebra over based on the parameter for all .
Theorem 21. Let be a MPQHF soft ideal of a BCI-algebra based on a parameter , Then, we say it is closed if and only if satisfies and
Proof. Assume that is a closed MPQHF soft ideal over a BCI-algebra based on a parameter , since for all .for all , and . Conversely, let be a MPQHF soft ideal over a BCI-algebra based on a parameter sincefor all , , we have the following expression:for all . Therefore, is a closed MPQHF soft ideal over a -algebra based on a parameter for all .
We introduce the MPQHF soft-commutative ideal and prove some of its propositions and theorems. First, let us define the -polar -hesitant fuzzy exchange ideal.
Definition 22. A MPQHF setin -algebra is called a MPQHF commutative ideal of if it satisfies the following condition:(1)For all ,(2)For all and
Definition 23. Let be a MPQHF soft set over , where is a subset of the parameters set P. If ∈ , is called a MPQHF soft-commutative ideal if the MPQHF setis a hesitant fuzzy commutative ideal of , for all .
Example 7. A study showed that the intensity of light is related to focusing. Therefore, the ministry of education decided to purchase a lighting control device that can also help reduce costs from a specialized company. The Ministry of Health hired two technicians working in the ministry of electricity to ensure the quality of the product .
The product is selected based on the following several factors:(1)Price(2)Production speed(3)Easy maintenanceLet the set be the number of devices and the set be the selected specialized company, we also have the set of parameters , which stands for ().
First, the binary operation on is defined in Cayley Table 9.
It shows that is a -algebra, let.
Table 10 shows that it is a 2-polar -hesitant fuzzy soft-commutative ideal.
Theorem 24. Every MPQHF soft-commutative ideal is a MPQHF soft ideal of .
Proof. Let , , and . Let be a MPQHF soft-commutative ideal of .for all , and . Hence, is a MPQHF soft ideal.
Theorem 25. Let be a MPQHF soft ideal, then is a MPQHF soft-commutative ideal if and only if it satisfies,, and.
Proof. Let , , and .
is a MPQHF soft-commutative ideal of . Taking , then Conversely, assume that satisfiesthenand combining (44) and (45), we obtain the following expression:, , and . Hence, is a MPQHF soft-commutative ideal.
4. Application of MQHFS in Decision-Making Problems
Definition 26. Let and be normalized MQHFS over X. For every , , we can express it in the following matrix form:The relative weight of parameters z is , . Then, we defined the induced -hesitant fuzzy set as follows:Then, we give the algorithm based on MQHFS as follows.
For illustrating the efficiency of the proposed algorithm, we adopt the following example.
Example 8. The COVID-19 pandemic has brought much chaos and grief to human life around the world, and it has presented unprecedented challenges to public health, food systems, and people’s jobs. Millions of people have lost their lives there, and nearly half of the world’s 3.3 billion global workforce is at risk of losing resources for life. Many children, especially those in the poorest parts of the world, are at risk of dying from pneumonia, diarrhoeal disease, malaria, HIV, and other preventable diseases due to the spread of the virus.
To combat the worldwide spread of the COVID-19 virus, the Ministry of Health has decided to offer three different types of COVID-19 vaccines. They went to two labs to compare the vaccines they produced and asked two scientists to help make a decision (M = 2). The factors influencing the decision are as follows: “,” “,” “,” and “.”
represents the type of COVID-19 vaccine. Let be the laboratory set and the parameter set be . Step 1: -polar Q-hesitant fuzzy soft set can describe the characteristics of the candidates types of the COVID-19 vaccines under the -polar Q-hesitant fuzzy information, which is shown in Table 11. Step 2: calculate the score of each -polar Q-hesitant fuzzy element and obtain the induced fuzzy soft set which is shown in Table 12. Step 3: the weight of the parameters in Z: the weight of the parameter is , and the weight of the parameter is , the weight of the parameter is , and the weight of the parameter is . Therefore, the induced fuzzy soft set and its tables are represented in Tables 2 and 3. As an adjustable approach, different rules (or thresholds) can be used in decision problems. For example, by handling this problem using a midlevel decision rule and a midthreshold , we obtain the following fuzzy set (see Algorithm 1): The midlevel soft set L of and the selected values for tabular representation are shown in Table 13. Step 4: from Table 14, it is clear that both scientists believe that the optimal choice is , which they can obtain from the proposed laboratory. Therefore, after assigning weights to different parameters, the Ministry of Health should choose as the best COVID-19 vaccine.
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5. Conclusion
In this study, we neatly achieved our main goal. We established the concept of (MQHS). Furthermore, we explain the concept of (MQHFS). Furthermore, we cleverly point out the concepts of the relationship between MQHFS subalgebras, MQHFS ideals, closed MQHFS ideals, and MQHFS exchange ideals. Therefore, in this study, we provide some illustrative examples and real-world studies of the Ministry of Health system in an effort to defeat the spread of the COVID-19 virus. Furthermore, our work focuses on exploring the application of MQHFS in decision-making. However, MQHFS can be applied to various applications such as forecasting and data analysis. In the future, our work can be further explored from two directions. First, it can be extended to some algebraic structures. BCH-algebras, BCC-algebras, B-algebras, and BRK-algebras can also be extended to several ideals, for example, p-ideal, q-ideal, and a-ideal. The polarities of the ideals we truncated in this master thesis will be an essential step in the future study of other ideals in different algebraic structures. The second direction is to study the “AND,” “OR,” “union,” and “intersection” operations between any two MQHFS sets, which can be considered as another promising research topic.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
The authors would like to thank the Deanship of Scientific Research, Qassim University, for funding publication of this project.