Switching perspectives: Investigating the relationships and applications of q-fractional fuzzy sets with other fuzzy set classes in decision making

Abstract

We explore switching techniques between q-fractional fuzzy sets (qFr sets) and various other classes of fuzzy sets to establish connections and provide a comprehensive framework. In particular, we examine the relationships between qFr sets and interval-valued fuzzy sets (IVFS), type 2 fuzzy sets(T2FS), intuitionistic fuzzy sets(IFS), Pythagorean fuzzy sets(PFS), q-rung orthopair fuzzy sets (q-ROFS), and linear diophantine fuzzy sets(LDFS). By examining these interconnections, we aim to understand better qFr sets and their applications in a wide range of fuzzy systems. It is possible to convert qFr sets into other fuzzy set models using the derived switching techniques, facilitating the utilization of existing methods and algorithms. The versatility of qFr sets, combined with the bridging techniques presented, holds promise for addressing complex problems in decision-making, pattern recognition, and other applications where uncertainty and imprecision play significant roles. Through case studies and practical applications, we illustrate the effectiveness and usefulness of the proposed switching techniques, showcasing their potential impact on real-world scenarios.

Keywords

q-fractional fuzzy sets fuzzy set interval-valued fuzzy sets type 2 fuzzy sets intuitionistic fuzzy sets Pythagorean fuzzy sets q-rung orthopair fuzzy sets linear diophantine fuzzy sets switching techniques uncertainty imprecision

1 Introduction

Zadeh [1], the inventor of fuzzy logic and fuzzy sets (FSs) or Type 1 FSs, initially developed this concept to deal with imprecise or vague information. In FSs, an element can have a degree of membership in a set between 0 and 1, indicating the degree to which it belongs to the set. Type 1, FSs represent uncertain information through a degree of membership between 0 and 1, which extend the concept of boolean membership i.e. either 0 or 1. However, in many real-world applications, the uncertainty is not only due to imprecision in membership but also due to imprecision in the boundaries of the set. Zaheh gave the concept of Type 2 fuzzy sets (T2FSs) [2], which allow for the representation of such imprecision by allowing the membership function to be fuzzy, meaning that the degree of membership is a function of both the input and an additional parameter representing the degree of uncertainty in the membership function. In other words, T2FSs deal with the uncertainty in an element’s membership degree to an FS rather than assuming that the degree of membership is a precise value. This makes T2FSs particularly useful in situations where the boundaries of the FS are not well-defined, or there needs to be more information about the membership function itself. Zadeh further gave the idea of interval valued fuzzy sets (IVFSs) [2] represent imprecision in the membership function by using intervals instead of precise values for the membership grades. In this case, the membership function assigns an interval of possible membership grades to each element rather than a single value. This approach is practical when there is uncertainty in the membership grade. Thus Zadeh initiated the concepts of FSs,T2FSs and IVFSs in a sequential order to capture the uncertainty in a better way.

Zadeh did not consider non-membership grades in FSs because he focused on modeling uncertainty, imprecision, and vagueness rather than explicitly dealing with non-membership. Zadeh’s approach was to capture the degree of membership of an element to a set, which he believed was more important and helpful in modeling uncertainty and vagueness in real-world problems. In addition, he considered that non-membership information could be derived implicitly from the membership grades by subtracting the membership grade from 1. But Atanassov [3], proposed the idea of intuitionistic fuzzy sets (IFSs) in 1986 as an extension of FSs that represent two types of uncertainty: degree of membership and degree of non-membership. IFSs address this limitation by introducing a third parameter, the degree of hesitation, which represents the degree of uncertainty or lack of knowledge about an element’s membership and non-membership grades to a set. Based on the same approach, some generalizations of IFS have been presented, like [4, 5] (Pythagorean Fuzzy Sets) (PFS), [6] q-rung orthopair fuzzy sets (qRFS), and [7] Linear Diophantine Fuzzy Sets (LDFS). But the problem with all these IFSs is that they do not allow membership or non-membership grades to 1. The data of the form $\begin{matrix} X = {(a; 1, 0.2), (b; 1, 1), (c; 0.9, 1)} \end{matrix}$ remains unattended by the use of all versions of IFSs and LDFS.

To handle this type of data we developed the new idea of the q-fractional fuzzy sets (qfrFS) [8], which can not only handle this type of data but also minimized the dependency condition between membership or non-membership grades.

The idea of switching between T2FSs and IFSs with applications have been reported in the literature by Own [9]. Later Kumar and Pandey [10] discuss some improvements in the switching techniques provided by Own [9]. In these switching some similarity measures have been used so for more details about the similarity measures we refer the reader [11 –13].

Motivation and Organization of the Paper: The motivation behind this concept is that sometimes we are not able to capture the uncertainty in one environment so it would be better to handle the same uncertainty in the other environment. After introducing the idea of q-fractional fuzzy sets (qfrFSs) [8], we are curious to develop a linkage between q-fractional fuzzy sets and other classes of Fuzzy sets. We are inspired by the switching technique presented in [9] by Own. The motivation behind exploring switching techniques between qfrFSs and other types of FSs lies in the desire to establish connections and synergies among different FS models. By enabling the conversion between these models, we can leverage their unique characteristics, expand the range of FS representations, and enhance the applicability of FSs in various domains. We aim to provide a comprehensive framework for qfrFSs by establishing the switching techniques between qfrFSs and IFSs, T2FSs, IFSs, PFSs, q-ROFS, and LDFSs. By leveraging these connections, we seek to enhance the understanding, utilization, and practical applications of qfrFSs in decision-making and other fuzzy systems. After the introduction, we reviewed some basic definitions of FSs and different classes of FSs in section 2. We study switching techniques between qfrFSs and other versions of FSs in section 3. A discussion of decision-making problems is presented in section 4. In Section 5, we concludes our study and provide some future research directions.

2 Basic concepts

In this section, we recall some basic definitions for the main section 3.

Definition 1. [1] A FS is the structure of the form Æ= {(x, Æ⁺ (x)) |x ∈ X, Æ⁺ (x) ∈ [0, 1]} . The FS Æ can be written as Æ= {(x, Æ⁺ (x) , 1-Æ⁺ (x)) |x ∈ X, Æ⁺ (x) ∈ [0, 1]} such that H (Æ) =1 - st (Æ) =1 - (Æ⁺ (x) + (1-Æ⁺ (x))) =0 where H (Æ) denoted hesitancy of Æ and st (Æ) = (Æ⁺ (x) + (1-Æ⁺ (x) =1 denotes the strength of the FS Æ. The set Æ is also called T1FS.

Definition 2. [2]Let X be a non-empty set. By a T2FS we mean a structure of the form

$\begin{matrix} £ = \sum_{x \in X} \sum_{μ \in Jx} [1 / (£^{+}) + 0 / (£^{-})] / x . \end{matrix}$ The set of all T2FS of X is denoted by T2F (X).

Definition 3. [11 –13] Let Æ= ∑_x∈X∑_μ∈Jx [1/(Æ⁺) +0/(Æ^-)]/x and £ = ∑_x∈X∑_μ∈Jx [1/(£ ⁺) +0/(£ ^-)]/x be two T2FSs

$\begin{matrix} S (Æ, £) = \frac{1}{n} \sum_{i = 1}^{n} \\ \frac{\sum_{Æ_{i}^{+}, £_{i}^{+}, Æ_{i}^{-}, £_{i}^{-}} (Æ_{i}^{+} £_{i}^{+} + Æ_{i}^{-} £_{i}^{-})}{\sum_{Æ_{i}^{+}, £_{i}^{+}, Æ_{i}^{-}, £_{i}^{-}} {(max {Æ_{i}^{+}, £_{i}^{+}})^{2} + (max {Æ_{i}^{-}, £_{i}^{-}})^{2}}} . \end{matrix}$

Definition 4. [2] An IVFS is the structure of the form Æ $= {(x, \tilde{Æ} (x)) | x \in X, \tilde{Æ} (x) \subset D [0, 1]}$ where D [0, 1] denotes the set of sub-intervals of [0, 1] .

Definition 5. [3] By an IFS we mean a structure of the form Æ= {(x, Æ⁺ (x) , Æ^- (x)) |x ∈ X, Æ⁺ (x) , Æ^- (x) ∈ [0, 1]} with the condition 0≤Æ⁺ (x) +Æ^- (x) ≤1 for x ∈ X . The degree of hesitancy H (x) =1 - (Æ⁺ (x) +Æ^- (x)) .

Definition 6. [4, 5] By a PFS we mean a structure of the form Æ= {(x, Æ⁺ (x) , Æ^- (x)) |x ∈ X, Æ⁺ (x) , Æ^- (x) ∈ [0, 1]} with the condition 0 ≤ (Æ⁺ (x)) ² + (Æ^- (x)) ²) ≤1 for x ∈ X for x ∈ X . The degree of hesitancy is expressed as $H (x) = \sqrt{1 - (Æ^{+} (x))^{2} + (Æ^{-} (x))^{2})} .$

Definition 7. [6] By a q-ROFS we mean a structure of the form Æ= {(x, Æ⁺ (x) , Æ^- (x)) |x ∈ X, Æ⁺ (x) , Æ^- (x) ∈ [0, 1]} with the condition 0 ≤ (Æ⁺ (x)) ^q + (Æ^- (x)) ^q) ≤1 for x ∈ X for x ∈ X, q ≥ 1 . The degree of hesitancy $H (x) = \sqrt[q]{1 - (Æ^{+} (x))^{q} + (Æ^{-} (x))^{q})} .$ The geometrical interpretation of q-ROFS has been shown in Fig. 1.

Fig. 1

q-ROFSs for different values of q.

Definition 8. [7] By a LDFS we mean a structure in the form of Æ= {(x, (Æ⁺ (x) , Æ^- (x)) , (α, β)) |x ∈ X, Æ⁺ (x) , Æ^- (x) , α, β ∈ [0, 1]} with the condition 0 ≤ α (Æ^- (x)) ^q + β (Æ^- (x)) ^q ≤ 1 for x ∈ X, , α, β ∈ [0, 1] such that α + β ≤ 1 .

3 Switching between qfrFSs and other FSs

In this section we discuss different switching techniques between qfrFSs and other FSs.

3.1 Switching from qfrFSs to FS

In the following, we proposed the switch from qfrFS to FS and constructed a qfrFS from FS. As we know, the notion of qfrFS of X is represented by membership and non-membership grades, and both such grading are independent, so we define the following switch for both cases independently using the different types of negations. But we see that we have valid output only in the case of standard negation as c (x) =1 - x; the rest are not valid in this mutual switch.

Definition 9. [8] Let X be a non-empty set. By a qfrFS we mean a structure of the form $\begin{matrix} Æ = {< x; Æ^{+} (x), Æ^{-} (x) >_{q} | x \in X} \end{matrix}$ where Æ⁺ (x) , Æ^- (x) ∈ [0, 1] considered as membership and non-membership (not dependent to each other) values such that $\begin{matrix} 0 \leq \frac{1}{q} (Æ^{+} (x)) + \frac{1}{q} (Æ^{-} (x)) \leq 1 with q \geq 2 . \end{matrix}$ For simplicity we denote a qfrFS as Æ=<Æ⁺, Æ^- > _q . The set of all qfrFS of X is denoted by qFr (X). The geometrical interpretation has been shown in Fig. 2.

Fig. 2

qfrFSs for different values of q.

Definition 10. [8] Let Æ=<Æ⁺, Æ^- > _q be any qfrFS. Then we define;

(i) Strength; St (Æ $) = \frac{1}{q} ($ Æ⁺ +Æ^-) ,

(ii) Hesitancy; H (Æ) =1 - St (Æ) , and St (Æ) + H (Æ) =1,

(iii) Complement; (Æ) ^c = <1-Æ⁺, 1-Æ^- > _q .

Definition 11. Let Æ=<Æ⁺, Æ^- > _q be a qfrFS of X .

(i) (Membership case using the standard negation as c (x) =1 - x) Let define a mapping f_α : qFr (X) → F (X) by $\begin{matrix} f_{α} (Æ) & = < (Æ^{+} + α H (Æ)), c (Æ^{+} + α H (Æ)) > \\ < (Æ^{+} + α H (Æ)), (1 - (Æ^{+} + α H (Æ))) > \\ = < (Æ^{+} + α (1 - \frac{(Æ^{+} + Æ^{-})}{q})), \\ (1 - (Æ^{+} + α (1 - \frac{(Æ^{+} + Æ^{-})}{q}))) > \\ = < (Æ^{+} + \frac{α}{q} (q - (Æ^{+} + Æ^{-}))), \\ (1 - (Æ^{+} + \frac{α}{q} (q - (Æ^{+} + Æ^{-}))) > \end{matrix}$ such that ( $Æ^{+} + \frac{α}{q} (q - ($ Æ⁺ +Æ^-))) + (1 - ( $Æ^{+} + \frac{α}{q} (q - ($ Æ⁺ +Æ^-))) =1 where α ∈ [0, 1] , thus f_α (Æ) ∈ F (X) .

(ii) (Non-membership case standard negation as c (x) =1 - x)) Let Æ=<Æ⁺, Æ^- > _q be a qfrFS of X . Let define a mapping f_α : qFr (X) → F (X) by $\begin{matrix} f_{α} (Æ) = & < (Æ^{-} + α H (Æ)), (1 - (Æ^{-} + α H (Æ))) > \\ = & < (Æ^{-} + α (1 - \frac{(Æ^{+} + Æ^{-})}{q})), \\ (1 - (Æ^{-} + α (1 - \frac{(Æ^{+} + Æ^{-})}{q}))) > \\ = & < (Æ^{-} + \frac{α}{q} (q - (Æ^{+} + Æ^{-}))), \\ (1 - (Æ^{-} + \frac{α}{q} (q - (Æ^{+} + Æ^{-}))) > \end{matrix}$ such that ( $Æ^{-} + \frac{α}{q} (q - ($ Æ⁺ +Æ^-))) + (1 - ( $Æ^{+} \frac{α}{q} (q - ($ Æ⁺ +Æ^-))) =1 where α ∈ [0, 1] , thus f_α (Æ) ∈ F (X) .

(iii) (Membership case using the Yager’s negation as $c (x) = ((1 - x^{w}))^{\frac{1}{w}}$ where w > 1) Let define a mapping f_α : qFr (X) → F (X) by $\begin{matrix} f_{α} (Æ) & = < (Æ^{+} + α H (Æ)), c (Æ^{+} + α H (Æ)) > \\ < (Æ^{+} + α H (Æ)), ((1 - (Æ^{+} + α H (Æ)^{w}))^{\frac{1}{w}}) > \end{matrix}$

such that (Æ⁺ + αH (Æ)) + ((1 - (Æ⁺ + αH (Æ $)^{w}))^{\frac{1}{w}} > 1$ where α ∈ [0, 1] , w ≥ 1 thus f_α (Æ) ∉ F (X) .

(ii) (Non-membership case using the Yager’s negation as $c (x) = ((1 - x^{w}))^{\frac{1}{w}}$ where w > 1) Let Æ=<Æ⁺, Æ^- > _q be a qfrFS of X . Let define a mapping f_α : qFr (X) → F (X) by $\begin{matrix} f_{α} (Æ) & = < (Æ^{-} + α H (Æ)), c (Æ^{-} + α H (Æ)) > \\ < (Æ^{-} + α H (Æ)), (1 - (Æ^{-} + α H (Æ)^{w}))^{\frac{1}{w}} > \end{matrix}$

such that (Æ^- + αH (Æ)) + (1 - (Æ^- + αH (Æ $)^{w}))^{\frac{1}{w}} > 1$ where α ∈ [0, 1] , thus f_α (Æ) ∉ F (X) .

(v) (Membership case using the Sugeno’s negation as $c (x) = \frac{1 - x}{1 - λ x}$ where -1 < λ < 0) Let define a mapping f_α : qFr (X) → F (X) by $\begin{matrix} f_{α} (Æ) & = < (Æ^{+} + α H (Æ)), c (Æ^{+} + α H (Æ)) > \\ < (Æ^{+} + α H (Æ)), (\frac{1 - (Æ^{+} + α H (Æ)}{1 - λ (Æ^{+} + α H (Æ)}) > \end{matrix}$

such that (Æ⁺ + αH (Æ $)) + (\frac{1 - (Æ^{+} + α H (Æ)}{1 - λ (Æ^{+} + α H (Æ)}) > 1$ where α ∈ [0, 1] , thus f_α (Æ) ∉ F (X) .

(vi) (Non-membership case using the Sugeno’s negation as $c (x) = \frac{1 - x}{1 - λ x}$ where -1 < λ < 0) Let Æ=<Æ⁺, Æ^- > _q be a qfrFS of X . Let define a mapping f_α : qFr (X) → F (X) by $\begin{matrix} f_{α} (Æ) & = < (Æ^{-} + α H (Æ)), c (Æ^{-} + α H (Æ)) > \\ < (Æ^{-} + α H (Æ)), (\frac{1 - (Æ^{-} + α H (Æ)}{1 - λ (Æ^{-} + α H (Æ)}) > \end{matrix}$ such that (Æ^- + αH (Æ $)) + (\frac{1 - (Æ^{-} + α H (Æ)}{1 - λ (Æ^{-} + α H (Æ)}) > 1$ where α ∈ [0, 1] , thus f_α (Æ) ∉ F (X) .

Example 1. (i) (Membership case) Let Æ= <1, 1 > _q be a qfrFS and let α = 1, q = 2 . Then f_α (Æ $) = < 1 + \frac{1}{2} (2 - (2)), 1 - (1 + \frac{1}{2} (2 - (2))) > = < 1, 0 >$ . Thus f_α (Æ) ∈ F (X) .

(ii) (Non-membership case) Let Æ= <1, 0.5 > _q be a qfrFS and let α = 1, q = 2 . Then f_α (Æ $) = < 0.5 + \frac{1}{2} (2 - (1.5)), 1 - (1 + \frac{1}{2} (2 - (1.5)) > = < 0.75, 0.25 >$ . Thus f_α (Æ) ∈ F (X) .

(iii) Let Æ= <0.9, 1 > _q be a qfrFS and let α = 1, q = 2 . Then H (Æ) =0.05 and f_α (Æ) =<0.95, 0.31 > which implies that 0.95 + 0.31 = 1.26 > 1. Thus f_α (Æ) ∉ F (X) .

(iv) Let Æ= <1, 0.5 > _q be a qfrFS and let α = 1, q = 2 . Then H (Æ) =0.25 and f_α (Æ) =<0.75, 0.66 > which implies that 0.75 + 0.66 = 1.41 > 1. Thus f_α (Æ) ∉ F (X) .

(v) Let Æ= <0.9, 1 > _q be a qfrFS and let α = 1, q = 2, λ = -0.5 . Then H (Æ) =0.05 and f_α (Æ) =<0.95, 0.033 > which implies that 0.95 + 0.033 = 0.983 ≠ 1. Thus f_α (Æ) ∉ F (X) but f_α (Æ) ∈ IF (X) .

(vi) Let Æ= <1, 0.5 > _q be a qfrFS and let α = 1, q = 2, λ = -0.5 . Then H (Æ) =0.25 and f_α (Æ) =<0.75, 0.018 > which implies that 0.75 + 0.018 = 0.93 ≠ 1. Thus f_α (Æ) ∉ F (X) but f_α (Æ) ∈ IF (X) .

Remark 1. From the above example and switching technique

(i) we observe that both membership and non-membership functions have been used independently which is the main theme used in the construction of qfrFS.

(ii) The mapping defined in (i) and (ii) are valid while others are not valid so we concluded that the switching mapping is not unique.

(iii) From (iii) and (iv) we observed that output is not a FS but it is a q-ROFS.

(iv) From (v) and (vi) we observed that output is not a FS but it is an IFS.

(v) For the reverse relation we proposed the following construction.

3.2 Constructing a qfrFSs from the FS

In qfrFS, we observed that membership and non-membership functions are generated without any mutual relationship. Thus, we are using two FSs: one is for the membership grade and the other for the non-membership grade. Let we have the universe of discourse as "The population of the world". Let us consider a FS Æ₁ = height of the male and a FS Æ₂ =height of the female in the given universe of discourse. Both FSs are not dependent to each other and the combination of Æ₁ and Æ₂ will give us a new qfrFS as Æ=<Æ₁, Æ₂ > . Thus we define the following mapping,

Definition 12. Let Æ₁ and Æ₂ be two FSs of X . Define a mapping f_q : F (X) × F (X) → qFr (X) by $\begin{matrix} Æ = f_{Æ} (Æ_{1}, Æ_{2}) = < Æ_{1}, Æ_{2} >_{q} \end{matrix}$ such that $0 \leq \frac{Æ_{1} + Æ_{2}}{q} \leq 1$ where q ≥ 2 .

Example 2. Let us consider two FSs as $\begin{matrix} Æ_{1} = {(x, Æ^{+} (x) | x \in X, Æ^{+} (x) \in [0, 1]} = {(x, 1)}, \end{matrix}$ $\begin{matrix} Æ_{2} = {(x, Æ^{-} (x) | x \in X, Æ^{-} (x) \in [0, 1]} = {(x, 0.5)} . \end{matrix}$ Then Æ= f (Æ₁, Æ₂) = <1, 0.5 > _q ∈ qFr (X).

3.3 Switching from qfrFSs to IVFS

Definition 13. Let Æ=<Æ⁺, Æ^- > _q be a qfrFS of X .

(i) (Membership case) Let define a mapping f_α : qFr (X) → IVF (X) by $\begin{matrix} f_{α} (Æ) = [\frac{Æ^{+}}{q}, | (Æ^{+} - α H (Æ)) |] \end{matrix}$ such that $\frac{Æ^{+}}{q} \leq | ($ Æ⁺ - αH (Æ)) |≤Æ⁺ where q ≥ 2, α ∈ [0, 1] , thus f_α (Æ) ∈ IVF (X) .

(ii) (Non-membership case) Let define a mapping f_α : qFr (X) → IVF (X) by $\begin{matrix} f_{α} (Æ) = [\frac{Æ^{-}}{q}, | (Æ^{-} - α H (Æ)) |] \end{matrix}$ such that $\frac{Æ^{-}}{q} \leq | ($ Æ^- - αH (Æ)) |≤Æ^- where q ≥ 2, α ∈ [0, 1] , thus f_α (Æ) ∈ IVF (X) .

Thus from both cases we get an IVFS.

Example 3. Let Æ= {< x₁, 0.9, 1 > , < x₂, 1, 0.2 >} be a qfrFS of X . Then from

(i) (Membership case) f_α (Æ) = {< x₁, [0.45, 0.85] , < x₂, [0.5, 0.6] >} ∈ IVF (X) with α = 1 .

(ii) (non-membership case) f_α (Æ) = {< x₁, [0.5, 0.95] , < x₂, [0.1, 0.2] >} ∈ IVF (X) with α = 1 .

3.4 Switching from IVFS to qfrFSs

Definition 14. Let ${\tilde{Æ}}_{1} = [a_{1}^{_}, a_{1}^{+}]$ and ${\tilde{Æ}}_{2} = [a_{2}^{_}, a_{2}^{+}]$ be two IVFSs of X . Define a mapping f_q : IVF (X) × IVF (X) → qFr (X) by $\begin{matrix} Æ = f_{q} ({\tilde{Æ}}_{1}, {\tilde{Æ}}_{2}) = < Æ_{1} = max (a_{1}^{_}, a_{1}^{+}), \\ Æ_{2} = max (a_{2}^{_}, a_{2}^{+}) >_{q} \end{matrix}$ such that $0 \leq \frac{max (a_{1}^{_}, a_{1}^{+}) + max (a_{2}^{_}, a_{2}^{+})}{q} \leq 1$ where q ≥ 2 .

Example 4. Let ${\tilde{Æ}}_{1} = [0.3, 1]$ and ${\tilde{Æ}}_{2} = [0.5, 0.9]$ be two IVFSs of X . Then Æ $= f_{q} ({\tilde{Æ}}_{1}, {\tilde{Æ}}_{2}) = <$ Æ₁ = max(0.3, 1) , Æ₂ = max(0.5, 0.9) > _q = <1, 0.9 > ∈ qFr (X) such that $0 \leq \frac{1 + 0.9}{2} \leq 1 .$

3.5 Switching from qfrFSs and q-ROFS

In this subsection we discuss two types of mutual switching, one from the qfrFS to q-ROFS and other from the q-ROFS to qfrFS.

3.6 Switching from qfrFSs to q-ROFS

Definition 15. Let Æ=<Æ⁺, Æ^- > _q be a qfrFS of X . Define a mapping $\begin{matrix} £ = f_{q} (Æ) = {\begin{matrix} < Æ^{+} - α H (Æ), Æ^{-} - α H (Æ) > {if Æ}^{+}, Æ \neq 1 \\ < \frac{Æ^{+} - α H (Æ)}{q}, Æ^{-} - α H (Æ) > {if Æ}^{+} = 1 and 0 < Æ^{-} < 1 \\ < Æ^{+} - α H (Æ), \frac{Æ^{-} - α H (Æ)}{q} > if 0 < Æ^{+} < 1 {and Æ}^{-} = 1 \\ < \frac{Æ^{+} - α H (Æ)}{q}, \frac{Æ^{-} - α H (Æ)}{q} > {if Æ}^{+}, Æ = 1 \end{matrix} \end{matrix}$ where q ≥ 2, thus f_q (Æ) = £ ∈ qrungF (X) .

Example 5. Let Æ= {< x₁ ; 0.9, 1 > _q, < x₂ ; 1, 0.8 > _q, < x₃ ; 1, 1 > , < x₄ ; 0.8, 0.9} be a qfrFS of X = {x₁, x₂, x₃, x₄} . Then £ = f_q (Æ) = {< x₁ ; 0.85, 0.475 > , < x₂ ; 0.45, 0.7 > , < x₃ ; 0.5, 0.5 > , < x₄ ; 0.65, 0.75 >} ∈ qrungF (X) .

Remark 2. The proposed mapping is not unique.

3.7 Switching from q-ROFS sets to qfrFSs

Definition 16. Let £ =< £ ⁺, £ ^- > be a q-rung orthopair fuzzy (qrungF (X)) of X . Define a mapping $\begin{matrix} Æ = f_{q} (£) = {\begin{matrix} < 1, £^{-} >_{q} if £^{+} > £^{-} \\ < £^{+}, 1^{-} >_{q} if £^{-} > £^{+} \\ < 1, 1 >_{q} if £^{+} = £^{-} \end{matrix} \end{matrix}$ where q ≥ 2, thus f (£) =Æ∈of X .

Example 6. Let £ = {< x₁ ; 0.9, 0.5 > , < x₂ ; 0.5, 0.8 >} ∈ qrungF (X) . Then Æ= {< x₁ ; 1, 0.5 > _q, < x₂ ; 0.5, 1 > _q} be a qfrFS of X .

Remark 3. The proposed mapping is not unique. One can define it in a different way.

3.8 Mutual Switching with qfrFSs and LDFS

In this subsection we discuss two types of mutual switching, one from the qfrFS to LDFS and other from the LDFS to qfrFS.

3.9 Switching from qfrFSs to LDFS

Definition 17. Let Æ=<Æ⁺, Æ^- > _q be a qfrFS of X . Define a mapping $\begin{matrix} £ = f_{α, β, q} (Æ) = (< \frac{Æ^{+}}{q}, \frac{Æ^{-}}{q} >, < α, β >) \end{matrix}$ such that $0 \leq α (\frac{Æ^{+}}{q}) + β ({\frac{Æ}{q}}^{-}) \leq 1,$ and $H (£) = 1 - (α (\frac{Æ^{+}}{q}) + β (\frac{Æ^{-}}{q}))$ where q ≥ 2, α, β ∈ (0, 1) with 0 ≤ α + β ≤ 1. Then f_α,β,q (Æ) = £ ∈ LDFS (X) .

Example 7. Let Æ= {< x₁ ; 0.9, 1 > _q, < x₂ ; 1, 0.8 > _q} be a qfrFS of X = {x₁, x₂} . Consider α = 0.5 and β = 0.2 such that 0 ≤ 0.5 + 0.2 ≤ 1. Then for q = 2, £ = f_α,β,q (Æ) = f_0.5,0.2,2 (Æ) = {(< x₁ ; 0.45, 0.5 > , <0.5, 0.2 >) , (< x₂ ; 0.5, 0.4 > , <0.5, 0.2 >)} ∈ LDFS (X) .

3.10 Switching from LDFS to qfrFSs

Definition 18. Let £ = (< £ ⁺, £ ^- > , < α, β >) be a LDFS of X such that there exist α, β ∈ (0, 1) with 0 ≤ α + β ≤ 1 such that 0 ≤ α £ ⁺ + β £ ^- ≤ 1 . Define a mapping $\begin{matrix} Æ & = & f (£) = {\begin{matrix} < £^{+}, 1 > if α £^{+} - β £^{-} > 0 \\ < 1, £^{-} > if α £^{+} - β £^{-} < 0 \end{matrix} \\ or & < & α £^{+}, 1^{-} >_{q}} \end{matrix}$ where q ≥ 2, thus f (£) =Æ∈of X .

Example 8. Let £ = {(< x₁ ; 0.45, 0.5 > , <0.5, 0.2 >) , (< x₂ ; 0.5, 0.4 > , <0.5, 0.2 >)} ∈ LDFS (X) . Then Æ= {< x₁ ; 0.45, 1 > _q, < x₂ ; 1, 0.4 > _q} be a qfrFS of X .

Remark 4. The proposed mapping is not unique.

3.11 Linkage between qfrFSs and T2FS

The motivation behind this concept is that sometimes we are not able to capture the uncertainty in one environment so it would be better to handle the same uncertainty in the other environment. Thus in this section we presented the idea of mutual shift between qfrFS and T2FS.

3.12 Switching from qfrFSs to T2FS

Definition 19.Let Æ=<Æ⁺, Æ^- > _q be a qfrFS of X . Let define a mapping f_β : qFr (X) → T2F (X) by $\begin{matrix} £ & = & f_{β} (Æ) \\ = & \sum_{x \in X} \sum_{Æ \in Jx} [1 / | (Æ^{+} - β H (Æ)) | \\ + 0 / | (Æ^{-} - β H (Æ)) |] / x, Jx \subseteq [0, 1], \end{matrix}$ where β ∈ [0, 1] , thus f_β (Æ) = £ ∈ T2F (X) .

Example 9. Let Æ= <1, 1 > _q be a qfrFS and let β = 1, q = 2 . Then f₁ (£) =1/1 +0/1 = < x, 1, 1 > . Thus f₁ (£) ∈ T2F (X) .

Example 10. Let Æ= {< x₁ ; 0.5, 1 > _q, < x₂ ; 1, 1 > _q, < x₃ ; 1, 0.7 > _q}be a qfrFS and let β = 1, q = 2 . Then f₁ (Æ $) = \frac{\frac{1}{0.25} + \frac{0}{0.75}}{x_{1}} + \frac{\frac{1}{1} + \frac{0}{1}}{x_{2}} + \frac{\frac{1}{0.85} + \frac{0}{0.55}}{x_{3}} \in T 2 F (X) .$

Example 11. Let Æ= {< x₁ ; 1, 0.1 > _q, < x₂ ; 0.2, 1 > _q, < x₃ ; 1, 0.01 > _q} be a qfrFS and let £=1, q = 2 . Then f₁ (Æ $) = \frac{\frac{1}{0.55} + \frac{0}{0.35}}{x_{1}} + \frac{\frac{1}{0.2} + \frac{0}{0.6}}{x_{2}} + \frac{\frac{1}{0.495} + \frac{0}{0.495}}{x_{3}} \in T 2 F (X) .$

Proposition 1. Let Æ₁ = ${<Æ}_{1}^{+},$ $Æ_{1}^{-} >_{q}$ and Æ₂ = ${<Æ}_{2}^{+},$ $Æ_{2}^{-} >_{q}$ be two qfrFSs of X which are switched to £₁ and £₂ ∈ T2F (X) by using the Definition 1. Then if Æ₁≤Æ₂ if and only if £₁ ≤ £ ₂ .

Proof. As £₁ and £₂ ∈ T2F (X) are switched using the Definition 1 so we have $\begin{matrix} £_{1} & = & f_{β} (Æ_{1}) = < x, | Æ_{1}^{+} - β H (Æ_{1}) |, \\ | Æ_{1}^{-} - β H (Æ_{1}) | > \\ £_{2} & = & f_{β} (Æ) = < x, | Æ_{2}^{+} - β H (Æ_{2}) |, \\ | Æ_{2}^{-} - β H (Æ_{2}) | > \end{matrix}$ Assume that Æ₁≤Æ₂ this implies ; $Æ_{1}^{+} \leq$ $Æ_{2}^{+}$ and $Æ_{1}^{-} \leq$ $Æ_{2}^{-} .$ Which further implies that | $Æ_{1}^{+} - β H ($ Æ₁) | ≤ | $Æ_{2}^{+} - β H ($ Æ₂) | and | $Æ_{1}^{-} - β H ($ Æ₁) | ≤ | $Æ_{2}^{-} - β H ($ Æ₂) | . Thus £₁ ≤ £ ₂ .

3.13 Switching from T2FS to qfrFSs

Definition 20.Let £ be a T2F (X) defined as $\begin{matrix} £ = \sum_{x \in X} \sum_{μ \in Jx} [1 / (Æ^{+}) + 0 / (Æ^{-})] / x, \end{matrix}$ Let define a mapping f_q : T2F (X) → qFr (X) by $\begin{matrix} Æ = f_{q} (£) = < x, Æ^{+} + β H (£), Æ^{-} + β H (£) > \end{matrix}$ where β ∈ [0, 1] , and $H (£) = \frac{| 1 - (Æ^{+} + Æ^{-}) |}{1 + 2 β}$ thus f_β (£) =Æ∈qFr (X) .

Proposition 2. The sum of membership, Æ⁺ + βH (£) , non-membership (Æ^- + βH (£) and hesitancy function $H (£) = \frac{| 1 - (Æ^{+} + Æ^{-}) |}{1 + 2 β}$ is equal to 1.

Proof. Straightforward.

Example 12. Let Æ $= \frac{\frac{1}{0.25} + \frac{0}{0.25}}{x_{1}} + \frac{\frac{1}{1} + \frac{0}{0}}{x_{2}} + \frac{\frac{1}{0.85} + \frac{0}{0.55}}{x_{3}} \in T 2 F (X)$ . Then using the defined switch 3.13 we have f_β (Æ) = {< x₁ ; 0.41, 0.41 > , < x₂ ; 1, 0 > , < x₃ ; 0.98, 0.68 >} ∈ qFr (X) .

4 Applications

The above-developed switching techniques help us in:

(i) Representation of Uncertainty.

(ii) Compatibility with Existing Models and Algorithms.

(iii) Enhanced Expressiveness and Granularity.

(iv) Integration of Diverse Perspectives.

(v) Expanded Applicability.

Overall, the switching techniques between q-fractional and other types of fuzzy sets lie in the desire to leverage the unique characteristics, enhance the representation of uncertainty, and broaden the applicability of FSs in diverse domains. These techniques enable the integration of different perspectives, promote compatibility with existing models and algorithms, and provide a flexible framework for addressing complex problems involving uncertainty. In the following section, we utilize the switching techniques in real-life applications.

Example 13.(i) Let X = {x₁, x₂, x₃} be a finite set and let there are three patterns defined as follows as qfrFSs $\begin{matrix} P_{1} & = {< x_{1}; 1, 1 >_{q}, < x_{2}; 1, 0.5 >_{q}, < x_{3}; 0.8, 1 >_{q}}, \\ P_{2} & = {< x_{1}; 0.8, 1 >_{q}, < x_{2}; 1, 0.7 >_{q}, < x_{3}; 0.8, 1 >_{q}}, \\ P_{3} & = {< x_{1}; 0.9, 1 >_{q}, < x_{2}; 0.7, 1 >_{q}, < x_{3}; 1, 1 >_{q}}, \end{matrix}$

having the classifications as C₁, C₂, C₃ respectively. Let there exist another unknown pattern Q which is also defined in the form of qfrFS as under $\begin{matrix} Q = {< x_{1}; 0.5, 1 >_{q}, < x_{2}; 1, 1 >_{q}, \\ < x_{3}; 1, 0.7 >_{q}} \end{matrix}$ We need to classify Q in the given three classifications C₁, C₂, C₃ using the pattern informations as P₁, P₂, P₃ . For it we use the principle of minimum discrimination information as $\begin{matrix} K = {arg}_{k} min {D (P_{k}, Q)} \end{matrix}$ where $\begin{matrix} D (P_{k}, Q) = \sum_{i = 1}^{n} [P_{i}^{+} ln \frac{P_{i}^{+}}{\frac{1}{2} (P_{i}^{+} + Q_{i}^{+})} \\ + P_{i}^{-} ln \frac{P_{i}^{-}}{\frac{1}{2} (P_{i}^{-} + Q_{i}^{-})}] + \sum_{i = 1}^{n} [Q_{i}^{+} ln \frac{Q_{i}^{+}}{\frac{1}{2} (Q_{i}^{+} + P_{i}^{+})} \\ + Q_{i}^{-} ln \frac{Q_{i}^{-}}{\frac{1}{2} (Q_{i}^{-} + P_{i}^{-})}] \end{matrix}$ Thus D (P₁, Q) = -0.2182, D (P₂, Q) = -0.1121, D (P₃, Q) = -0.1326 . Comparing these values, we classify Q into C₂. The resulting graph will displaying the discrimination values for each classification (C₁, C₂, C₃) as a bar plot in Fig. 3.

Fig. 3

Ranking of patterns.

(ii) Using the switching techniques from qfrFSs to T2FS we have $\begin{matrix} P_{1} & = & \frac{\frac{1}{1} + \frac{0}{1}}{x_{1}} + \frac{\frac{1}{0.75} + \frac{0}{0.25}}{x_{2}} + \frac{\frac{1}{0.7} + \frac{0}{0.9}}{x_{3}}, \\ P_{2} & = & \frac{\frac{1}{0.7} + \frac{0}{0.9}}{x_{1}} + \frac{\frac{1}{0.85} + \frac{0}{0.55}}{x_{2}} + \frac{\frac{1}{0.7} + \frac{0}{0.9}}{x_{3}}, \\ P_{3} & = & \frac{\frac{1}{0.85} + \frac{0}{0.95}}{x_{1}} + \frac{\frac{1}{0.55} + \frac{0}{0.85}}{x_{2}} + \frac{\frac{1}{1} + \frac{0}{1}}{x_{3}}, \\ Q & = & \frac{\frac{1}{0.25} + \frac{0}{0.75}}{x_{1}} + \frac{\frac{1}{1} + \frac{0}{1}}{x_{2}} + \frac{\frac{1}{0.85} + \frac{0}{0.55}}{x_{3}} . \end{matrix}$ After following the same procedure we found that D (P₁, Q) = -3.4138, D (P₂, Q) = -3.4510, D (P₃, Q) =0.5643 . Therefore, based on the principle of minimum discrimination information, Q should be classified in the classification C₂. C₂ shown in Fig. 4. The comparison of both cases has been shown in Fig. 5.

Fig. 4

Ranking of patterns.

Fig. 5

Comparison of patterns ranking.

4.1 Interpretation:

Case (i): Patterns P₂ and P₃ have negative discrimination distances, indicating that they are more similar to the unknown pattern Q. Among these, pattern P₂ has the smallest (in absolute value) negative discrimination distance, making it the most similar to Q. Pattern P₁ also has a negative discrimination distance but is slightly larger (in absolute value) than P₂ and P₃. It’s less similar to Q compared to P₂ and P₃. Overall, in Case (i), the patterns are ranked as P₂ (most similar) >P₃ > P₁ (least similar).

Case (ii): Pattern P₃ has a positive discrimination distance, indicating that it is less similar to the unknown pattern Q than patterns with negative discrimination distances. Patterns P₁ and P₂ both have large negative discrimination distances, making them more similar to Q compared to P₃. Between these two, pattern P₂ has a slightly larger (in absolute value) negative discrimination distance, indicating it’s more similar to Q. In Case (ii), the patterns are ranked as P₂(most similar) >P₁ > P₃ (least similar).

Overall Comparison: Comparing both cases:

Pattern P₂ consistently ranks as the most similar pattern to the unknown pattern Q in both cases. Patterns P₁ and P₃ have varying rankings between the two cases due to the differences in the unknown pattern Q and their respective discrimination distances.

Algorithm 1. Based on the provided example, the algorithm for classifying an unknown pattern Q into one of the given classifications C₁, C₂, C₃ using the principle of minimum discrimination information can be outlined as follows:

(i) Define the given patterns P₁, P₂, P₃ and the unknown pattern Q, along with their q-fractional fuzzy set representations.

(ii) Initialize a variable K to store the classification with minimum discrimination information.

(iii) Calculate the discrimination information values D (P₁, Q) , D (P₂, Q), and D (P₃, Q) using the given formula: $\begin{matrix} D (P_{k}, Q) \\ = \sum_{i = 1}^{n} [P_{i}^{+} ln \frac{P_{i}^{+}}{\frac{1}{2} (P_{i}^{+} + Q_{i}^{+})} + P_{i}^{-} ln \frac{P_{i}^{-}}{\frac{1}{2} (P_{i}^{-} + Q_{i}^{-})}] \\ + \sum_{i = 1}^{n} [Q_{i}^{+} ln \frac{Q_{i}^{+}}{\frac{1}{2} (Q_{i}^{+} + P_{i}^{+})} + Q_{i}^{-} ln \frac{Q_{i}^{-}}{\frac{1}{2} (Q_{i}^{-} + P_{i}^{-})}] \end{matrix}$ (iv) Compare the discrimination information values and identify the classification with the smallest value: K = arg _k min {D (P_k, Q)} ,

(v) Classify pattern Q into the identified classification C_k.

Remark 5. The algorithm described above can be applied to any general case where the patterns and their q-fractional fuzzy set representations are provided. You can plug in the specific values for the patterns and calculate the discrimination information to classify the unknown pattern.

Example 14.Let X = {P₁, P₂, P₃} be the set of patients and assume that they have some common symptoms S = {s₁, s_2,s₃} . Let we have D = {d₁, d₂, d₃} be the set of diseases associated with S . The information received from a given source is provided in the following Table 1, 2,

	s ₁	s ₂	s ₃
p ₁	< 1, 0.1 >_q	< 1, 0.9 >_q	< 1, 1 >_q
p ₂	< 0.7, 0.9 >_q	< 0.9, 0.5 >_q	< 0.9, 1 >_q
p ₃	< 0.8, 1 >_q	< 0.5, 1 >_q	< 1, 0.1 >_q

Now using the switch

\begin{matrix} P & = & f_{β} (Q) \\ = & \sum_{x \in X} \sum_{Q \in Jx} [1 / (Q^{+} - β H (Q)) + 0 / (Q^{-} \\ - β H (Q))] / x, Jx \subseteq [0, 1], \end{matrix}

for the patients X = {P₁, P₂, P₃} , where β = 1,

H (Q) = 1 - (\frac{Q^{+} + Q^{-}}{2})

we have,

\begin{matrix} P_{1} & = & {[1 / (0.55) + 0 / (0.55)] / s_{1}, [1 / (0.95) \\ + 0 / (0.85)] / s_{2}, [1 / (1) + 0 / (1)] / s_{3}}, \\ P_{2} & = & {[1 / (0.5) + 0 / (0.7)] / s_{1}, [1 / (0.6) \\ + 0 / (0.2)] / s_{2}, [1 / (0.85) + 0 / (0.95)] / s_{3}}, \\ P_{3} & = & {[1 / (0.7) + 0 / (0.9)] / s_{1}, [1 / (0.25) \\ + 0 / (0.75)] / s_{2}, [1 / (0.45) + 0 / (0.45)] / s_{3}}, \end{matrix}

and similarly the role of symptom for disease in the form of type-2 fuzzy using the same switch is set

\begin{matrix} d_{1} & = & {[1 / (0.95) + 0 / (0.85)] / s_{1}, [1 / (0.15) \\ + 0 / (0.15)] / s_{2}, [1 / (0.65) + 0 / (0.75)] / s_{3}}, \\ d_{2} & = & {[1 / (1) + 0 / (1)] / s_{1}, [1 / (0.8) + 0 / (0.4)] / \\ s_{2}, [1 / (0.55) + 0 / (0.05)] / s_{3}}, \\ d_{3} & = & {[1 / (0.75) + 0 / (0.25)] / s_{1}, [1 / (0.2) \\ + 0 / (0.6)] / s_{2}, [1 / (1) + 0 / (1)] / s_{3}}, \end{matrix}

Now using the similarity measure between patients and disease as

\begin{matrix} S (P_{i}, d_{i}) = \frac{1}{n} \sum_{i = 1}^{n} \\ \frac{\sum_{p_{i}^{+}, d_{i}^{+}, p_{i}^{-}, d_{i}^{-}} (p_{i}^{+} d_{i}^{+} + p_{i}^{-} d_{i}^{-})}{\sum_{p_{i}^{+}, d_{i}^{+}, p_{i}^{-}, d_{i}^{-}} {(max {p_{i}^{+}, d_{i}^{+}})^{2} + (max {p_{i}^{-}, d_{i}^{-}})^{2}}} \end{matrix}

we have

	d ₁	d ₂	d ₃
p ₁	0.4433	0.4667	0.5417
p ₂	0.4092	0.3752	0.4317
p ₃	0.3689	0.395	0.3583

The resulting graph shown in Fig. 6 are displaying the similarity values between patients (P₁, P₂, P₃) and diseases (d₁, d₂, d₃) as a heat map. The color intensity represents the degree of similarity, with higher values appearing darker in color.

Fig. 6

Patients vs diseases ranking.

4.2 Interpretation:

(i) Similarity Patterns: Among the patients (rows), P₁ shows the highest similarity values for diseases d₁ and d₃. This suggests that P₁ has a higher degree of similarity with these diseases compared to P₂ and P₃. P₂ and P₃ show slightly lower similarity values for diseases d₁ and d₃. This indicates that they are less similar to these diseases compared to P₁.

(ii) Comparing Diseases: For disease d₁, all patients have relatively higher similarity values, indicating that all patients share some level of similarity with this disease. Disease d₂ has lower similarity values for all patients, suggesting that the patients are less similar to this disease. Disease d₃ has varied similarity values among patients. P₁ has the highest similarity value for this disease, followed by P₂, and than P₃.

(iii) Overall Patterns: P₁ generally shows higher similarity values with diseases compared to P₂ and P₃, indicating that it might be more closely associated with the considered diseases in this dataset. P₂ and P₃ show more variable similarity values, suggesting that they might have a mixed level of association with the diseases.

5 Comapirson analysis

Each of these fuzzy set types (T1FSs, T2FSs, IVFSs, IFSs, and qfrFS) addresses aspects of uncertainty and imprecision differently. While T1FSs, T2FSs, and IVFSs capture varying degrees of membership uncertainty, IFSs introduce the concept of non-membership and hesitation. The new qfrFS concept is tailored to handle specific data scenarios that existing fuzzy set types might not cover adequately. The choice of which type to use depends on the nature of the uncertainty and the specific requirements of the problem. In the Examples 13 and 14, the provided data can not be handled using the previous versions of fuzzy sets. So, in this case, the developed switching techniques can be used.

6 Conclusion and future work

This research has explored and investigated the switching techniques and relationships between qfrFSs and IVFS, T2FS, IFS, PFS, q-ROFS, and LDFS. Our analysis demonstrated that qfrFSs offer a versatile framework for representing uncertain information by decoupling membership and non-membership values. The switching techniques developed in this research allow for the conversion between qfrFSs and other FS models, enabling the utilization of existing methods and algorithms in fuzzy systems. The findings of this research contribute to the advancement of FS theory and provide a comprehensive framework for qfrFSs. The developed switching techniques enable the conversion between qfrFSs and other classes of FSs, opening up new possibilities for addressing complex problems in decision-making, pattern recognition, and other applications where uncertainty and imprecision play significant roles. Future research directions include the development of novel algorithms and methodologies based on the switching techniques between qfrFSs and other FS models. Additionally, empirical studies and real-world applications can further validate the effectiveness and practical utility of the proposed switching techniques in diverse domains. In conclusion, exploring switching techniques and relationships between qfrFSs and various other classes of FSs paves the way for enhanced uncertainty modeling and decision support, contributing to the advancement of fuzzy systems and their applications in real-world scenarios.

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