Generalized Ideals of BCK/BCI -Algebras Based on MQHF Soft Set with Application in Decision Making

. Te purpose of this study is to generalize the concept of Q -hesitant fuzzy sets and soft set theory to Q -hesitant fuzzy soft sets. Te Q -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 m -polar Q -hesitant fuzzy soft (MQHFS) set. First, by combining m-pole fuzzy sets, soft set models, and Q -hesitant fuzzy sets, we introduce the concept of MQHFS and apply it to deal with multiple theories in BCK / BCI -algebra. We then develop a framework including MQHFS subalgebras, MQHFS ideals, closed MQHFS ideals, and MQHFS exchange ideals in BCK / BCI -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 efectiveness of MQHFS through the use of horizontal soft sets in decision-making.


Introduction
Many authors are already interested in modeling ambiguity. Te specifc 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 felds, including economics, engineering, environmental science, and social science. Te 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 difculties. 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 felds such as medical and life sciences, engineering, business and social sciences, computer networks, decision making, artifcial intelligence, pattern recognition, and robotics. Later, many authors studied some generalizations of the main basic concepts of fuzzy sets in diferent 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 m-polar fuzzy set, an extension of the fuzzy set.
Molodtsov [3] frst proposed the concept of soft set theory. Soft set theory became a general tool for dealing with uncertainty. Te utility of soft set theory is that it does not present difculties that afect existing methods. Ten, Maji et al. combined fuzzy theory with soft sets in [4] and proposed the concept of fuzzy soft sets. After defning 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 sufcient motivation to explore a wide range of concepts and come out of the feld of abstract algebra in a more inclusive fuzzy context. Imai and Is [7] frst introduced the BCK/BCI-algebra, an algebraic structure of universal algebra, in 1966. Te BCK/BCI-algebraic theory was given a soft set theory treatment by Jun [8]. Later, he used subalgebras and the ideals of BCK/BCI-algebras to apply the idea of hesitant fuzzy soft sets in [9][10][11]. Since onedimensional membership functions cannot be utilized to process data in a two-dimensional ensemble, all the research studies discussed are described as representations of onedimensional membership functions. At [12], Adem and Hassan introduced the idea of a Q-fuzzy soft set. A number of authors, including [13][14][15][16][17][18][19], discussed various decisionmaking techniques.
Te following are the primary contributions of this study: (1) A generalized concept of Q-hesitant fuzzy sets and soft set theory to Q-hesitant fuzzy soft sets is presented (2) A formal structure for the MQHFS set is provided (3) Te concept of MQHFS is applied to deal with multiple theories in BCK/BCI-algebra (4) A framework including MQHFS subalgebras, MQHFS ideals, closed MQHFS ideals, and MQHFS exchange ideals in BCK/BCI-algebras is discussed (5) An application of MQHFS-based multicriteria decision-making in the Ministry of Health system is illustrated Te remainder of this research is structured as follows. We outline the fundamental ideas behind linked literature research in Section 2. Te ideas of m-pole Q-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.

Preliminaries
In this section, we recall some basic defnitions which will be used in our work.
Defnition 1 (see [9]). Let B be a BCK/BCI-algebra. A hesitant fuzzy set on B is called a hesitant fuzzy subalgebra of B if it satisfes ∀α, β ∈ B, and Defnition 2 (see [9]). Let B be a BCK/BCI-algebra. A hesitant fuzzy set, on B is called a hesitant fuzzy ideal of B if it satisfes ∀α, β ∈ B, and Defnition 3 (see [10]). Let P be a set of parameters, for a subset λ of P, A hesitant fuzzy soft set (Z, λ) over B is called a hesitant fuzzy soft subalgebra based on e ∈ λ if the hesitant fuzzy set, is a hesitant fuzzy subalgebra of B.
Defnition 4 (see [10]). Let P be the set of parameters, for a subset λ of P, A hesitant fuzzy soft set (Z, λ) over B is called a hesitant fuzzy soft ideal based on e ∈ λ if the hesitant fuzzy set, is a hesitant fuzzy ideal of B.
Defnition 5 (see [9]). Let B be a nonempty fnite universe and Q be a nonempty set. A Q-hesitant fuzzy set Z Q is a set given by the following expression: where ζ Z Q : B × Q ⟶ [0, 1].
Defnition 6 (see [16]). An MPQHF set on a nonempty set B is the mapping Z Q : B × Q ⟶ [0, 1] m . Te membership value of every element α ∈ B is denoted by the following expression: where

m-Polar Q-Hesitant Fuzzy Soft Subalgebra and Ideals
Defnition 7. Let B be a BCK/BCI-algebra. Te m-polar Q-hesitant fuzzy (MPQHF) set: on B is called MPQHF subalgebra of B if it satisfes ∀α, β ∈ B, q ∈ Q and for all i � 1, 2, . . . , m. As ρ � 2, we have. We verify that it is a 2-polar Q-hesitant fuzzy subalgebra as given in Table 2.
Proposition 8. Every MPQHF subalgebra satisfes the following inequality: (1) It is given as Defnition 9. Let P be a set of parameters. For a subset λ of P, a MPQHF soft set (ρ i ∘ Z, λ) is called a MPQHF soft subalgebra based on e ∈ λ if the MPQHF set on B is a Q-hesitant fuzzy soft subalgebra of B, for all i � 1, 2 . . . , m. 3 be a BCK-algebra set and consider the operation * on B defned by Cayley Table 3. Ten, (B, * , x 1 ) is a BCK-algebra. Consider the set Q � ϕ and a parameters set Z � p 1 , p 2 , let m � 3, which is described in Table 4. Table 4 shows that it is a 3-polar Q-hesitant fuzzy soft subalgebra over B based on the parameters.
be a MPQHF set in B. Ten, ρ i ∘ Z is called a MPQHF ideal of B if it satisfes the following conditions: ∀α, β ∈ B, q ∈ Q and i � 1, 2 . . . , m.
(1) Here, (2) Also, Example 3. Te MPQHF subalgebra that we described in Example 1 is not a MPQHF ideal because of the following reason.
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 B � a, b { } be the set of products. We consider the operation ⊗ defned in Cayley Table 5.
Let Q � α, β be a set of brands, and the parameter set is Z � e 1 , e 2 , e 3 , and the parameter Z stands for "e 1 � Natural," "e 2 � price," and "e 3 � Attractive packaging," which is described in Table 6.
Tus, it is a 2-polar Q-hesitant fuzzy soft ideal.
Example 5. Te MPQHF subalgebra which is described in Example 6 is a closed MPQHF ideal.

Defnition 19.
A MPQHF soft ideal (ρ i ∘ Z, λ) over a BCIalgebra B based on a parameter e ∈ λ is said to be closed if the MPQHF soft set on B is closed hesitant fuzzy ideal of B, for all i � 1, 2, . . . , m.
Example 6. A region municipality nominated two types of asphalts (x 1 � base course, x 2 � wearing course) which is described in the set B � x 1 , x 2 , and then they sent three engineers M � 3 to the main contractor Q � A { } who is working on the roads to inspect his works on those types of asphalts. His work needs to rely on e 1 � water level e 2 � the connectivity between the old and new asphalt Ten, the parameter set Z � e 1 , e 2 . Table 7 shows that B, * , x 1 is a BCK-algebra. Ten, it is easy to see from Table 8   Proof. Let (ρ i ∘ Z, λ) be a closed MPQHF soft ideal over B based on the parameter e ∈ λ, then for all α, β ∈ B, q ∈ Q and i � 1, 2, . . . m, it follows that Proof. (⇒) Assume that (ρ i ∘ Z, λ) is a closed MPQHF soft ideal over a BCI-algebra B based on a parameter e ∈ λ, since (α * β) * α ≤ 0 * β for all α, β ∈ B.
We introduce the MPQHF soft-commutative ideal and prove some of its propositions and theorems. First, let us defne the m-polar Q-hesitant fuzzy exchange ideal.

Example 7.
A study showed that the intensity of light is related to focusing. Terefore, the ministry of education decided to purchase a lighting control device that can also help reduce costs from a specialized company. Te Ministry of Health hired two technicians working in the ministry of electricity to ensure the quality of the product M � 2.
Te product is selected based on the following several factors: (1) Price (2) Production speed (3) Easy maintenance Let the set B � n 1 , n 2 , n 3 be the number of devices and the set Q � ξ { } be the selected specialized company, we also have the set of parameters Z � e 1 , e 2 , e 3 , which stands for (e 1 � price, e 2 � produ ction speed, ande 3 � easy maintenance).
First, the binary operation * on B is defned in Cayley Table 9.
It shows that (B, * , n 1 ) is a BCK-algebra, let. Table 10 shows that it is a 2-polar Q-hesitant fuzzy softcommutative ideal.
for all α, c ∈ B, q ∈ Q and e ∈ λ. Hence,

) is a MPQHF soft-commutative ideal if and only if it
α, β ∈ B, q ∈ Q, and e ∈ λ.

Application of MQHFS in Decision-Making Problems
Te relative weight of parameters z is Z(z) � (Z 1 , Z 2 , . . . , Z m ) T , ( n i�1 Z i � 1). Ten, we defned the induced Q-hesitant fuzzy set ζ Z [z] as follows: Ten, we give the algorithm based on MQHFS as follows.
For illustrating the efciency of the proposed algorithm, we adopt the following example. To combat the worldwide spread of the COVID-19 virus, the Ministry of Health has decided to ofer three diferent types of COVID-19 vaccines. Tey went to two labs to compare the vaccines they produced and asked two scientists to help make a decision (M � 2). Te factors infuencing the decision are as follows: "z 1 � side effects," "z 2 � Price," "z 3 � efficiency," and "z 4 � outputspeed." B � x 1 , x 2 , x 3 represents the type of COVID-19 vaccine. Let Q � A, B { } be the laboratory set and the parameter set be Z � z 1 , z 2 , z 3 , z 4 .
Step 1: m-polar Q-hesitant fuzzy soft set can describe the characteristics of the candidates types of the COVID-19 vaccines under the m-polar Q-hesitant fuzzy information, which is shown in Table 11.
Step 2: calculate the score of each m-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 side effects is Z 1 � (0.5, 0.5), and the weight of the parameter Price is Z 2 � (0.4, 0.6), the weight of the parameter efficiency is Z 3 � (0.6, 0.4), and the weight of the parameter Outputspeed is Z 4 � (0.4, 0.6). Terefore, the induced fuzzy soft set ∆ ρ i ∘ Z and its tables are represented in Tables 2 and 3. As an adjustable approach, diferent rules (or thresholds) can be used in decision problems. For example, by handling this problem using a midlevel decision rule and a midthreshold ∆ ρ i ∘ Z , we obtain the following fuzzy set (see Algorithm 1): Table 9: (B, * , n 1 ) is a BCK-algebra. * n 1 n 2 n 3 n 1 n 1 n 1 n 1 n 2 n 2 n 1 n 2 n 3 n 3 n 3 n 1 Te midlevel soft set L(∆ Z ; mid) of ∆ Z 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 x 1 , which they can obtain from the proposed laboratory. Terefore, after assigning weights to diferent parameters, the Ministry of Health should choose x 1 as the best COVID-19 vaccine.

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. Terefore, in this study, we provide some illustrative examples and real-world studies of the Ministry of Health system in an efort 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, BCCalgebras, B-algebras, and BRK-algebras can also be extended to several ideals, for example, p-ideal, q-ideal, and a-ideal. Te polarities of the ideals we truncated in this master thesis (1) Input the MQHFS (ρ i ∘ Z, λ).
(2) Input the relative weight Z i of parameters.
(4) Compute the induced fuzzy soft set ∆ ρ i ∘ Z � (ρ i ∘ Z, λ) (5) Input the midlevel decision rule (or the threshold fuzzy set φ: λ ⟶ [1, 0]; or give a threshold value u ∈ [0, 1]; or choose the toplevel decision rule) for decision making. (6) Compute the midlevel soft set L(∆ ρ i ∘ Z ; mid) (or the level soft set L(∆ ρ i ∘ Z ; φ) of ∆ ρ i ∘ Z with the respect of the threshold fuzzy set; or the t-level soft set L (∆ ρ i ∘ Z ; u); or the top-level soft set L (∆ ρ i ∘ Z ; max)) (7) Put the midlevel soft set L(∆ ρ i ∘ Z ; mid) (or the level soft set L(∆ ρ i ∘ Z ; φ); or L(∆ ρ i ∘ Z ; t); or L(∆ ρ i ∘ Z ; max)) in tabular form and compute the choice value a i of Z i . (8) Select the optimal alternative Z i if a i � max(a k ) (9) If there are more than one Z i 's then any one of Z i may be chosen.       Journal of Mathematics will be an essential step in the future study of other ideals in diferent algebraic structures. Te 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
Te authors declare that there are no conficts of interest.