Decision making based on rough approximations generalized by j -fuzzy neighborhoods

Abstract

Rough set and fuzzy set theory are two distinct but related mathematical techniques, and both theories deal with uncertain and incomplete information in data analysis. Approximation operators and accuracy measures are the key points of rough set theory. Various generalizations on rough approximations have been proposed by many authors to reduce the boundary region and increase the accuracy percentage. One of these generalizations is the neighborhood system. The concept of the neighborhood system is a mathematical tool for handling inconsistent and limited knowledge. It can be applied to find the approximations to resolve the vagueness of data collection. This paper aims to examine $j$ -neighborhoods in the context of a binary fuzzy relation and to introduce the concept of $j$ -fuzzy neighborhoods. $j$ -fuzzy approximations are proposed and studied using these neighborhoods. Further, an example related to heart disease is discussed to demonstrate the significance of the proposed work and the result is compared with existing ones. A medical application is described using $j$ -fuzzy neighborhoods to categorise patients based on their COVID-19 infection.

Keywords

Fuzzy rough set neighborhood system rough approximations accuracy measures fuzzy neighborhoods decision making

1 Introduction

Rough set theory was introduced by Pawlak [40] for handling ambiguous data in information systems. A Key fact of this theory is an equivalence relation that restricts the range of applications of rough set theory. To address the problems from equivalence relations, Yao developed the idea of rough approximation using arbitrary binary relations [37-39]. One of the well-known areas of study for generalizing rough set theory is the neighborhood (nbd) system. The primary goal of generalization is to reduce the boundary region and increase the positive region. Nbd systems have been applied to achieve this objective. Researchers have suggested nbd systems such as left and right nbds [37, 39], minimal nbds [1], $j$ -adhesion nbds [14], E-nbds [33], Containment-nbds [32], Subset-nbds [30], Maximal nbds [28], Union nbds [18] can be used to manage incomplete knowledge. Al-Shami [30, 32] provided an approach to prevent exposure to coronavirus in subset and containment nbds.

Approximations via topology are another generalization of rough set theory. Many researchers have investigated the topological characteristics of rough sets and extended the theory of approximations [15 , 29]. For data analysis and knowledge computing, rough topology is a foundation [24]. Several recent studies have explored the use of topological concepts to update the theories of rough sets. For instance, Al-Shami and Alshammari [27] employed supra topology to study generalized rough approximation spaces effectively. Al-Shami and Mhemdi [25] examined rough approximation operators using infra-topology, which strengthens the role of topological generalization in explaining the fundamental idea of a rough set. In [2] , numerous concepts and connections between the rough set model and maximal rough nbd have been presented. Moreover, Mustafa et al. [9] utilized containment nbds and ideals, while Al-Shami and Mhemdi [26] applied the idea of subset nbds and ideals to generate topological structure for generalizing rough approximations.

The focus of study has been to decrease the bound ary region and increase the accuracy measure. To achieve this, Kandil et al. [3] presented a method for obtaining approximations based on the concept of ideals in 2013. In recent advancements, Honsy et al. [16 , 31] proposed different types of generalized rough set models inspired by maximal nbds and ideals and applied them to solve the real-life problems. The constructed nbds and generalized approximations were applied to eliminate the uncertainty in the information system.

The rough approximations are extended using fuzzy set (FS) theory. The main goal of this extension is to expand the object from crisp set to FS. Dubois and Prade [7] approximated the FS using rough set to introduce the fuzzy rough approximations. Wei-Zhi Wu et al. [36] generalized the concept of fuzzy rough set (FRS). Wei-Zhi Wu and Wen-Xiu Zhang [35] further generalized the FRS theory using axiomatic and constructive methods. This method defines a collection of dual generalized fuzzy approximation operators based on any given fuzzy (or crisp) relation. Another interesting field of research in FRS theory is the relationship between FS and fuzzy topology. K. Qin and Z. Pei [11] discussed the topological properties of FRS and also many researchers explored the concept of fuzzy topology induced by fuzzy approximation and binary fuzzy relation (BFR) [5 , 23].

1.1 Research gap and motivation

$j$ -nbds generalize the rough approximations and enhance the accuracy measures; this approach requires crisp set, which has only two outcomes. In-spite of all the advancements made in the rough set theory, the approximation results need further improvement. To address this, membership values are added to create new nbds known as $j$ -fuzzy nbds ( $j$ -FNs), resulting in new fuzzy rough approximations. Abd El-Monsef et al. [15] proposed the concept of $j$ -nbd space. Every element of the universal set in the nbd system is related to its family of non-empty subsets, and each member of this family is referred to as an element’s nbd, with the family itself being called the nbd system. The concept of the nbd system is a mathematical technique for handling inconsistent and limited knowledge. It can be applied to find the approximations to resolve the vagueness of data collection. This approach can lead to better accuracy measures and is helpful in decision-making. Many authors have generalized nbd system to obtain more accurate results [14 , 33]. Motivated by studying nbd systems, $j -$ FNs and $j$ -fuzzy approximations are proposed.

1.2 Contribution

Introduced eight types of fuzzy neighborhoods under binary fuzzy relation.

Introduced fuzzy topologies induced by $j$ -FNs.

Proposed $j$ -fuzzy lower and upper approximations.

Discussed the method of decision-making with the help of medical examples and algorithm.

1.3 Significance

The accuracy measures obtained from previous nbds are compared with our proposed $j$ -FNs. We have obtained a better accuracy percentage.

This strategy can be applied by categorizing people according to the severity of contagious infections.

1.4 Layout

The manuscript is organised as follows: basic definitions of nbds and the characteristics of FRSs are discussed in section 2. In section 3, the $j$ -fuzzy nbd ( $j$ -FN) system is introduced, its properties are investigated, and the fuzzy topology induced by $j$ -FNs is discussed. $j$ -FRS model is defined, and its properties are examined in section 4. In section 5, medical applications and an algorithm for decision- making based on $j$ -FNs, are illustrated with examples and a comparative analysis of our proposed method is given. The conclusion and future work are discussed in the final section.

2 Preliminaries

Some basic definitions of BFR, fuzzy rough approximations, topology generated by BFRs, and other terms are provided in this section.

Definition 1. [13, 34]

Let $P$ and $Q$ be any two FSs of universal set $U$ , then for all $ξ \in U$ ,

$(P \cup Q) (ξ) = P (ξ) \lor Q (ξ)$ and $(P \cap Q) (ξ) = P (ξ) \land Q (ξ)$

Definition 2. [10, 12] A FS $K$ is called BFR if it maps each element from $P \times Q$ to [0, 1],

i.e. $K = {((ξ, ς), μ_{K} (ξ, ς)) | (ξ, ς) \in P \times Q}$ is a BFR in $P \times Q$ .

$K$ is reflexive if $μ_{K} (ξ, ξ) = 1$ and anti-reflexive if $μ_{K} (ξ, ξ) = 0$ ,

$K$ is symmetric if $μ_{K} (ξ, ς) = μ_{K} (ς, ξ),$

$K$ is transitive if $μ_{K} (ξ, ϱ) \geq \max_{ς} \min [μ_{K} (ξ, ς), μ_{K} (ς, ϱ)], \forall ξ, ς, ϱ \in K$ ,

$i . e ., K \circ K \subset K$ . where, $K \circ K = K^{2}$ is called max-min composition.

Definition 3. [6] Let $P$ be any fuzzy subset of universal set $U$ and $(U, τ)$ be any fuzzy topological space, then $P$ is said to be the fuzzy nbd of a FS $A,$ if there exists a fuzzy open set $G$ such that $A \subseteq G \subseteq P$ .

Remark 1. In crisp set, we consider nbd of points where as in fuzzy set, we consider nbd of FSs.

Definition 4. [5, 7] For any BFR $K$ and any fuzzy subset $P$ of $U$ , fuzzy approximation space is represented by $(U, K)$ . Fuzzy lower (FL) and fuzzy upper (FU) approximations of $P$ are defined as:

$\underline{K} (P) (ξ) = \land_{ς \in U} (P (ς) \lor (1 - K (ξ, ς))),$ $\bar{K} (P) (ξ) = \lor_{ς \in U} (P (ς) \land K (ξ, ς)),$ respectively, where $ξ, ς \in U$ and $(\underline{K} (P), \bar{K} (P))$ is the FRS of $P$ .

Definition 5. [1 , 39] Let $K$ be a binary relation on $U$ , then $j = {r, l, i, u, < r >, < l >, , }$

-nbds of $ξ \in U$ are denoted by $N_{j} (ξ)$ and defined as:

$r -$ nbd: $N_{r} (ξ) = {ς \in U, ξ K ς}$ .

$l -$ nbd: $N_{l} (ξ) = {ς \in U, ς K ξ}$ .

$< r > -$ nbd: $N_{< r >} (ξ) = ⋂_{ξ \in N_{r} (ς)} N_{r} (ς)$ .

$< l > -$ nbd: $N_{< l >} (ξ) = ⋂_{ξ \in N_{l} (ς)} N_{l} (ς)$ .

$i -$ nbd: $N_{i} (ξ) = N_{r} (ξ) \cap N_{l} (ξ)$ .

$u -$ nbd: $N_{u} (ξ) = N_{r} (ξ) \cup N_{l} (ξ)$ .

$ -$ nbd: $N_{} (ξ) = N_{< r >} (ξ) \cap N_{< l >} (ξ)$ .

$ -$ nbd: $N_{} (ξ) = N_{< r >} (ξ) \cup N_{< l >} (ξ)$ .

Definition 6. [15, 21] Let $ς_{j} : U \to 2^{U}$ be a mapping defined for each value of $U$ , then $(U, K, ς_{j})$ is known as $j$ -NS and the topology on $U$ is $Z_{j} = {E \subseteq U : \forall ξ \in E, N_{j} (ξ) \subseteq E}$ .

3 $j$ -FN spaces and induced fuzzy topologies

$j$ -FNs, $j$ -FN space, fuzzy topologies induced by these nbds are proposed here, and the relation between fuzzy nbds and their induced topologies are investigated.

Definition 7. Let $K$ be a BFR on universal set $U$ . Then $j$ -FNs of $ξ \in U$ is denoted by $N_{F_{j}} (ξ)$ , where $j = {r, l, i, u, < r >, < l >, , }$ and are defined as:

$r -$ fuzzy nbd: $N_{F_{r}} (ξ) = {(ς, μ_{ς}) : ς \in U, ξ K ς}$ .

$l -$ fuzzy nbd: $N_{F_{l}} (ξ) = {(ς, μ_{ς}) : ς \in U, ς K ξ}$ .

$< r > -$ fuzzy nbd:

$N_{F_{< r >}} (ξ) = ⋂_{(ξ, μ_{ξ}) \in N_{F_{r}} (ς)} N_{F_{r}} (ς) : N_{F_{r}} (ς)$ containing ξ with membership value other than 0.

$< l > -$ fuzzy nbd:

$N_{F_{< l >}} (ξ) = ⋂_{(ξ, μ_{ξ}) \in N_{F_{l}} (ς)} N_{F_{l}} (ς) : N_{F_{l}} (ς)$ containing ξ with membership value other than 0.

$i -$ fuzzy nbd: $N_{F_{i}} (ξ) = N_{F_{r}} (ξ) \cap N_{F_{l}} (ξ)$ .

$u -$ fuzzy nbd: $N_{F_{u}} (ξ) = N_{F_{r}} (ξ) \cup N_{F_{l}} (ξ)$ .

$ -$ fuzzy nbd:

$N_{F_{}} (ξ) = N_{F_{< r >}} (ξ) \cap N_{F_{< l >}} (ξ)$ .

$ -$ fuzzy nbd:

$N_{F_{}} (ξ) = N_{F_{< r >}} (ξ) \cup N_{F_{< l >}} (ξ)$ .

Remark 2. $N_{F_{j}} (ξ)$ are explained in Example 1.

Definition 8. Let $K$ be a BFR on universal set $U, F (U)$ be the family of fuzzy subsets of $U$ and $Λ_{j} : U \to F (U)$ be a mapping that determines the $j$ -FN in $F (U), \forall ξ \in U$ . Then, $(U, K, Λ_{j})$ is said to be $j$ -FNS.

$τ_{N_{j}} = {P \subseteq F (U) : \forall (ξ, μ_{ξ}) \in P, N_{F_{j}} (ξ) \subseteq P}$ is fuzzy topology induced by $j$ -FN, $\forall j$ .

Proposition 1. $(U, K, Λ_{j})$ be a $j$ -FNS. Then,

$N_{F_{i}} (ξ) \subseteq N_{F_{r}} (ξ) \subseteq N_{F_{u}} (ξ)$ ,

$N_{F_{i}} (ξ) \subseteq N_{F_{l}} (ξ) \subseteq N_{F_{u}} (ξ)$ ,

$N_{F_{}} (ξ) \subseteq N_{F_{< r >}} (ξ) \subseteq N_{F_{}} (ξ)$ ,

$N_{F_{}} (ξ) \subseteq N_{F_{< l >}} (ξ) \subseteq N_{F_{}} (ξ)$ , $\forall ξ \in U$ .

Proof. Obvious from Definition-1

Proposition 2. Let $(U, K, Λ_{j})$ be a $j$ -FNS and $K$ be a symmetric BFR, then,

$N_{F_{i}} (ξ) = N_{F_{r}} (ξ) = N_{F_{l}} (ξ) = N_{F_{u}} (ξ)$ and

$N_{F_{}} (ξ) = N_{F_{< r >}} (ξ) = N_{F_{< l >}} (ξ) = N_{F_{}} (ξ)$ .

Proof. Since, $K$ is a symmetric BFR, therefore $N_{F_{r}} (ξ) = N_{F_{l}} (ξ)$ .

Hence, $N_{F_{i}} (ξ) = N_{F_{r}} (ξ) = N_{F_{l}} (ξ) = N_{F_{u}} (ξ)$ and $N_{F_{}} (ξ) = N_{F_{< r >}} (ξ) = N_{F_{< l >}} (ξ) = N_{F_{}} (ξ)$ (using Definition 7). □

Example 1. Let $P = {\frac{. 5}{ξ} + \frac{. 7}{ς} + \frac{. 6}{ϱ}}$ be a fuzzy subset of $U = {ξ, ς, ϱ}$ and $K = [\begin{matrix} . 3 & 0 & 0 \\ . 5 & 0 & . 4 \\ 0 & . 2 & . 1 \end{matrix}]$ be BFR on $U$ . Then $j$ -FNs are calculated in Table 1.

Table 1
$j$ -FNs for all components of $U$

ξ ς ϱ

$N_{F_{r}}$ ${\frac{. 3}{ξ} + \frac{0}{ς} + \frac{0}{ϱ}}$ ${\frac{. 5}{ξ} + \frac{0}{ς} + \frac{. 4}{ϱ}}$ ${\frac{0}{ξ} + \frac{. 2}{ς} + \frac{. 1}{ϱ}}$

$N_{F_{l}}$ ${\frac{. 3}{ξ} + \frac{. 5}{ς} + \frac{0}{ϱ}}$ ${\frac{0}{ξ} + \frac{0}{ς} + \frac{. 2}{ϱ}}$ ${\frac{0}{ξ} + \frac{. 4}{ς} + \frac{. 1}{ϱ}}$

$N_{F_{i}}$ ${\frac{. 3}{ξ} + \frac{0}{ς} + \frac{0}{ϱ}}$ ${\frac{0}{ξ} + \frac{0}{ς} + \frac{. 2}{ϱ}}$ ${\frac{0}{ξ} + \frac{. 2}{ς} + \frac{. 1}{ϱ}}$

$N_{F_{u}}$ ${\frac{. 3}{ξ} + \frac{. 5}{ς} + \frac{0}{ϱ}}$ ${\frac{. 5}{ξ} + \frac{0}{ς} + \frac{. 4}{ϱ}}$ ${\frac{0}{ξ} + \frac{. 4}{ς} + \frac{. 1}{ϱ}}$

$N_{F_{< r >}}$ ${\frac{. 3}{ξ} + \frac{0}{ς} + \frac{0}{ϱ}}$ ${\frac{0}{ξ} + \frac{. 2}{ς} + \frac{. 1}{ϱ}}$ ${\frac{0}{ξ} + \frac{0}{ς} + \frac{. 1}{ϱ}}$

$N_{F_{< l >}}$ ${\frac{. 3}{ξ} + \frac{. 5}{ς} + \frac{0}{ϱ}}$ ${\frac{0}{ξ} + \frac{. 4}{ς} + \frac{0}{ϱ}}$ ${\frac{0}{ξ} + \frac{0}{ς} + \frac{. 1}{ϱ}}$

$N_{F_{}}$ ${\frac{. 3}{ξ} + \frac{0}{ς} + \frac{0}{ϱ}}$ ${\frac{0}{ξ} + \frac{. 2}{ς} + \frac{0}{ϱ}}$ ${\frac{0}{ξ} + \frac{0}{ς} + \frac{. 1}{ϱ}}$

$N_{F_{}}$ ${\frac{. 3}{ξ} + \frac{. 5}{ς} + \frac{0}{ϱ}}$ ${\frac{0}{ξ} + \frac{. 4}{ς} + \frac{. 1}{ϱ}}$ ${\frac{0}{ξ} + \frac{0}{ς} + \frac{. 1}{ϱ}}$

	ξ	ς	ϱ
$N_{F_{r}}$	${\frac{. 3}{ξ} + \frac{0}{ς} + \frac{0}{ϱ}}$	${\frac{. 5}{ξ} + \frac{0}{ς} + \frac{. 4}{ϱ}}$	${\frac{0}{ξ} + \frac{. 2}{ς} + \frac{. 1}{ϱ}}$
$N_{F_{l}}$	${\frac{. 3}{ξ} + \frac{. 5}{ς} + \frac{0}{ϱ}}$	${\frac{0}{ξ} + \frac{0}{ς} + \frac{. 2}{ϱ}}$	${\frac{0}{ξ} + \frac{. 4}{ς} + \frac{. 1}{ϱ}}$
$N_{F_{i}}$	${\frac{. 3}{ξ} + \frac{0}{ς} + \frac{0}{ϱ}}$	${\frac{0}{ξ} + \frac{0}{ς} + \frac{. 2}{ϱ}}$	${\frac{0}{ξ} + \frac{. 2}{ς} + \frac{. 1}{ϱ}}$
$N_{F_{u}}$	${\frac{. 3}{ξ} + \frac{. 5}{ς} + \frac{0}{ϱ}}$	${\frac{. 5}{ξ} + \frac{0}{ς} + \frac{. 4}{ϱ}}$	${\frac{0}{ξ} + \frac{. 4}{ς} + \frac{. 1}{ϱ}}$
$N_{F_{< r >}}$	${\frac{. 3}{ξ} + \frac{0}{ς} + \frac{0}{ϱ}}$	${\frac{0}{ξ} + \frac{. 2}{ς} + \frac{. 1}{ϱ}}$	${\frac{0}{ξ} + \frac{0}{ς} + \frac{. 1}{ϱ}}$
$N_{F_{< l >}}$	${\frac{. 3}{ξ} + \frac{. 5}{ς} + \frac{0}{ϱ}}$	${\frac{0}{ξ} + \frac{. 4}{ς} + \frac{0}{ϱ}}$	${\frac{0}{ξ} + \frac{0}{ς} + \frac{. 1}{ϱ}}$
$N_{F_{< i >}}$	${\frac{. 3}{ξ} + \frac{0}{ς} + \frac{0}{ϱ}}$	${\frac{0}{ξ} + \frac{. 2}{ς} + \frac{0}{ϱ}}$	${\frac{0}{ξ} + \frac{0}{ς} + \frac{. 1}{ϱ}}$
$N_{F_{< u >}}$	${\frac{. 3}{ξ} + \frac{. 5}{ς} + \frac{0}{ϱ}}$	${\frac{0}{ξ} + \frac{. 4}{ς} + \frac{. 1}{ϱ}}$	${\frac{0}{ξ} + \frac{0}{ς} + \frac{. 1}{ϱ}}$

Fuzzy topology induced by $r$ -FNs is calculated using Definition-8 for all the fuzzy subsets of $P$ and is given as:

$τ_{N_{r}} = {{\frac{0}{ξ} + \frac{0}{ς} + \frac{0}{ϱ}}, {\frac{1}{ξ} + \frac{1}{ς} + \frac{1}{ϱ}}, {\frac{. 5}{ξ} + \frac{0}{ς} + \frac{0}{ϱ}}, {\frac{. 5}{ξ} + \frac{. 7}{ς} + \frac{. 6}{ϱ}}}$ .

Similarly, induced fuzzy topologies are calculated for each $j$ .

Definition 9. Let $(U, K, Λ_{j})$ be a $j$ -FNS and the collection $τ_{N_{j}}$ be the fuzzy topology induced by FN for each $j$ . Then, $G \in F (U)$ is said to be $j$ -fuzzy open set if $G \in τ_{N_{j}}$ and $G^{c}$ is known as $j$ -fuzzy closed set.

Proposition 3. Let $(U, K, Λ_{j})$ be a $j$ -FNS, then

$τ_{N_{u}} \subseteq τ_{N_{r}} \subseteq τ_{N_{i}}$ .

$τ_{N_{u}} \subseteq τ_{N_{l}} \subseteq τ_{N_{i}}$ .

$τ_{N_{}} \subseteq τ_{N_{< r >}} \subseteq τ_{N_{}}$ .

$τ_{N_{}} \subseteq τ_{N_{< l >}} \subseteq τ_{N_{}}$ .

Proof.

Let $G \in τ_{N_{u}}$ , then $\forall (ξ, μ_{ξ}) \in G, N_{F_{u}} (ξ) \subseteq G$ $\Rightarrow N_{F_{r}} (ξ) \cup N_{F_{l}} (ξ) \subseteq G$ $\Rightarrow N_{F_{r}} (ξ) \subseteq G$ and $N_{F_{l}} (ξ) \subseteq G$ $\Rightarrow G \in τ_{N_{r}}$ and $G \in τ_{N_{l}}$ . $\Rightarrow τ_{N_{u}} \subseteq τ_{N_{r}}$ and $τ_{N_{u}} \subseteq τ_{N_{l}}$ .

Again, let $G \in τ_{N_{r}}$ , then $\forall (ξ, μ_{ξ}) \in G, N_{F_{r}} (ξ) \subseteq G$ $\Rightarrow N_{F_{r}} (ξ) \cap N_{F_{l}} (ξ) \subseteq G$ .

$\Rightarrow N_{F_{i}} (ξ) \subseteq G$ $\Rightarrow G \in τ_{N_{i}}$ $\Rightarrow τ_{N_{r}} \subseteq τ_{N_{i}}$ . Hence, $τ_{N_{u}} \subseteq τ_{N_{r}} \subseteq τ_{N_{i}}$ and proof of $τ_{N_{u}} \subseteq τ_{N_{l}} \subseteq τ_{N_{i}}$ is similar.

Remaining proof are similar.

□

Remark 3. In $j$ -NS, $τ_{< r >}$ is dual topology of $τ_{< l >}$ . It may not be true in proposed nbds (Example 1).

j

-FN space based fuzzy approximations

Here, we have introduced the new fuzzy rough approximations called $j$ -FL and $j$ -FU approximations and investigated their properties on considering fuzzy subsets $P, Q \in F (U), \forall ξ, ς \in U$ . Example is given for illustration.

Definition 10. Let $(U, K, Λ_{j})$ be a $j$ -FNS, then, $j$ -FL and $j$ -FU approximations of any FS $P \in F (U)$ are defined as:

$\underline{K_{N_{j}}} (P) (ξ) = \land_{ς \in U} (P (ς) \lor (1 - K (N_{F_{j}} (ξ, ς)))),$ $\bar{K_{N_{j}}} (P) (ξ) = \lor_{ς \in U} (P (ς) \land K (N_{F_{j}} (ξ, ς))),$ respectively, where $ξ, ς \in U$ and $(\underline{K_{N_{j}}}, \bar{K_{N_{j}}})$ is called $j$ -FRS of $P$ for each $j$ .

Example 1 (cont.): $r$ -FL and $r$ -FU approximations for the fuzzy subset $P$ are calculated by the formula:

$\underline{K_{N_{r}}} (P) (ξ) = \land_{ς \in U} (P (ς) \lor (1 - K (N_{F_{r}} (ξ, ς)))),$ $\bar{K_{N_{r}}} (P) (ξ) = \lor_{ς \in U} (P (ς) \land K (N_{F_{r}} (ξ, ς))),$

respectively, where $ξ, ς \in U$ and

$K (N_{F_{r}} (ξ, ς)) = [\begin{matrix} . 3 & 0 & 0 \\ . 5 & 0 & . 4 \\ 0 & . 2 & . 1 \end{matrix}]$ .

$\Rightarrow \underline{K_{N_{r}}} (P) = {\frac{. 7}{ξ} + \frac{. 5}{ς} + \frac{. 8}{ϱ}}$ and : $\bar{K_{N_{r}}} (P) = {\frac{. 3}{ξ} + \frac{. 5}{ς} + \frac{. 2}{ϱ}}$ .

Similarly, $j - FL$ and $j$ -FU approximations for $j = {l, i, u, < r >, < l >, , }$ are calculated.

Theorem 4. Let $(U, K, Λ_{j})$ be a $j$ -FNSs, then $\forall j$ :

$\underline{K_{N_{j}}} (U) = U$

If $P \subseteq Q$ , then $\underline{K_{N_{j}}} (P) \subseteq \underline{K_{N_{j}}} (Q)$

$\underline{K_{N_{j}}} (P \cap Q) = \underline{K_{N_{j}}} (P) \cap \underline{K_{N_{j}}} (Q)$

$\underline{K_{N_{j}}} (P^{c}) = (\bar{K_{N_{j}}} (P))^{c}$ .

Proof.

It can be proved by using Definition 10.

Let $P \subseteq Q$ , then,

$\land_{ς \in U} (P (ς) \lor (1 - K (N_{F_{j}} (ξ, ς)))) \leq \land_{ς \in U} (Q (ς) \lor (1 - K (N_{F_{j}} (ξ, ς))))$ .

$\Rightarrow \underline{K_{N_{j}}} (P) (ξ) \leq \underline{K_{N_{j}}} (Q) (ξ)$

$\Rightarrow \underline{K_{N_{j}}} (P) \subseteq \underline{K_{N_{j}}} (Q)$ .

Since, $P \cap Q \subseteq P$ and $P \cap Q \subseteq Q$ .

Therefore, using (2), we get,

$\underline{K_{N_{j}}} (P \cap Q) \subseteq \underline{K_{N_{j}}} (P)$ and

$\underline{K_{N_{j}}} (P \cap Q) \subseteq \underline{K_{N_{j}}} (Q)$ . $\Rightarrow \underline{K_{N_{j}}} (P \cap Q) \subseteq \underline{K_{N_{j}}} (P) \cap \underline{K_{N_{j}}} (Q) .$ (1) Now, let $(ξ, μ_{ξ}) \in \underline{K_{N_{j}}} (P) \cap \underline{K_{N_{j}}} (Q)$

$\Rightarrow μ_{ξ} \leq \underline{K_{N_{j}}} (P) (ξ)$ and $μ_{ξ} \leq \underline{K_{N_{j}}} (Q) (ξ)$ .

$\Rightarrow μ_{ξ} \leq \land_{ς \in U} (P (ς) \lor (1 - K (N_{F_{j}} (ξ, ς))))$ and $μ_{ξ} \leq \land_{ς \in U} (Q (ς) \lor (1 - K (N_{F_{j}} (ξ, ς))))$ .

$\Rightarrow μ_{ξ} \leq (\land_{ς \in U} (P (ς) \lor (1 - K (N_{F_{j}} (ξ, ς))))) \land (\land_{ς \in U} (Q (ς) \lor (1 - K (N_{F_{j}} (ξ, ς)))))$ .

$\Rightarrow μ_{ξ} \leq \land_{ς \in U} ((P (ς) \land Q (ς)) \lor (1 - K (N_{F_{j}} (ξ, ς))))$ .

$\Rightarrow μ_{ξ} \leq \underline{K_{N_{j}}} (P \cap Q) (ξ)$

$\Rightarrow (ξ, μ_{ξ}) \in \underline{K_{N_{j}}} (P \cap Q)$ . $\Rightarrow \underline{K_{N_{j}}} (P) \cap \underline{K_{N_{j}}} (Q) \subseteq \underline{K_{N_{j}}} (P \cap Q) .$ (2) From Equations (1) and (2), we get,

$\underline{K_{N_{j}}} (P \cap Q) = \underline{K_{N_{j}}} (P) \cap \underline{K_{N_{j}}} (Q)$ .

Let $(ξ, μ_{ξ}) \in \underline{K_{N_{j}}} (P^{c}) \Leftrightarrow μ_{ξ} \leq \underline{K_{N_{j}}} (P^{c}) (ξ) \Leftrightarrow μ_{ξ} \leq \land_{ς \in U} ((P (ς))^{c} \lor (1 - K (N_{F_{j}} (ξ, ς)))) \Leftrightarrow μ_{ξ} \leq \land_{ς \in U} ((P (ς))^{c} \lor (K (N_{F_{j}} (ξ, ς)))^{c}) \Leftrightarrow μ_{ξ} \leq \land_{ς \in U} (P (ς) \land K (N_{F_{j}} (ξ, ς)))^{c}$ (using De Morgan’s Law).

$\Leftrightarrow μ_{ξ} \leq (\lor_{ς \in U} (P (ς) \land K (N_{F_{j}} (ξ, ς)))^{c} \Leftrightarrow μ_{ξ} \leq (\bar{K_{N_{j}}} (P))^{c} (ξ) \Leftrightarrow (ξ, μ_{ξ}) \in (\bar{K_{N_{j}}} (P))^{c} . \Leftrightarrow \underline{K_{N_{j}}} (P^{c}) = (\bar{K_{N_{j}}} (P))^{c}$ .

□

Theorem 5. Let $(U, K, Λ_{j})$ be a $j$ -FNSs, then $\forall j$ :

$\bar{K_{N_{j}}} (φ) = φ$

If $P \subseteq Q$ : then $\bar{K_{N_{j}}} (P) \subseteq \bar{K_{N_{j}}} (Q)$ .

$\bar{K_{N_{j}}} (P \cup Q) = \bar{K_{N_{j}}} (P) \cup \bar{K_{N_{j}}} (Q)$ .

$\bar{K_{N_{j}}} (P^{c}) = (\underline{K_{N_{j}}} (P))^{c}$ .

Proof. Proof is identical to that of Theorem 4 □

Proposition 6. Let $(U, K, Λ_{j})$ be a $j$ -FNSs, $\forall j$ :

$\bar{K_{N_{j}}} (P \cap Q) \subseteq \bar{K_{N_{j}}} (P) \cap \bar{K_{N_{j}}} (Q)$

$\underline{K_{N_{j}}} (P) \cup \underline{K_{N_{j}}} (Q) \subseteq \underline{K_{N_{j}}} (P \cup Q)$ .

Proof.

Let $(ξ, μ_{ξ}) \in \bar{K_{N_{j}}} (P \cap Q) . \Rightarrow μ_{ξ} \leq \bar{K_{N_{j}}} (P \cap Q) (ξ) . \Rightarrow μ_{ξ} \leq \lor_{ς \in U} ((P \cap Q) (ς) \land K (N_{F_{j}} (ξ, ς))) . \Rightarrow μ_{ξ} \leq \lor_{ς \in U} ((P (ς) \land Q (ς)) \land K (N_{F_{j}} (ξ, ς))) . \Rightarrow μ_{ξ} \leq \lor_{ς \in U} ((P (ς) \land K (N_{F_{j}} (ξ, ς))) \land (Q (ς) \land K (N_{F_{j}} (ξ, ς)))) . \Rightarrow μ_{ξ} \leq (\lor_{ς \in U} (P (ς) \land K (N_{F_{j}} (ξ, ς)))) \land (\lor_{ς \in U} (Q (ς) \land K (N_{F_{j}} (ξ, ς)))) . \Rightarrow μ_{ξ} \leq (\bar{K_{N_{j}}} (P) (ξ) \land \bar{K_{N_{j}}} (Q) (ξ)) . \Rightarrow (ξ, μ_{ξ}) \in \bar{K_{N_{j}}} (P) \cap \bar{K_{N_{j}}} (Q) . \Rightarrow \bar{K_{N_{j}}} (P \cap Q) \subseteq \bar{K_{N_{j}}} (P) \cap \bar{K_{N_{j}}} (Q)$ .

Similar to Proof of Prop.-6 (1).

□

Theorem 7. Let $(U, K, Λ_{j})$ and $(U, S, Ω_{j})$ be two $j$ -FNSs, then for $j = {r, l, i, u}$ :

If $K \subseteq S$ , then $\underline{S_{M_{j}}} (P) \subseteq \underline{K_{N_{j}}} (P)$ ,

If $K \subseteq S$ , then $\bar{K_{N_{j}}} (P) \subseteq \bar{S_{M_{j}}} (P)$ ,

$\underline{(K_{N_{j}} \cup S_{M_{j}})} (P) = \underline{K_{N_{j}}} (P) \cap \underline{S_{M_{j}}} (P)$ ,

$\bar{(K_{N_{j}} \cup S_{M_{j}})} (P) = \underline{K_{N_{j}}} (P) \cup \underline{S_{M_{j}}} (P)$ ,

where, $M_{F_{j}} (ξ)$ and $N_{F_{j}} (ξ)$ are $j$ -FNs defined by BFRs $S$ and $K$ respectively.

Proof.

Let $(ξ, μ_{ξ}) \in \underline{S_{M_{j}}} (P) \Rightarrow μ_{ξ} \leq \underline{S_{M_{j}}} (P) (ξ) = \land_{ς \in U} (P (ς) \lor (1 - S (M_{F_{j}} (ξ, ς))))$ .

Since, $K \subseteq S \Rightarrow S^{c} \subseteq K^{c} \Rightarrow \land_{ς \in U} (P (ς) \lor (1 - S (M_{F_{j}} (ξ, ς)))) \leq \land_{ς \in U} (P (ς) \lor (1 - K (N_{F_{j}} (ξ, ς)))) . \Rightarrow μ_{ξ} \leq \land_{ς \in U} (P (ς) \lor (1 - K (N_{F_{j}} (ξ, ς)))) . \Rightarrow μ_{ξ} \leq \underline{K_{N_{j}}} (P) (ξ) \Rightarrow (ξ, μ_{ξ}) \in \underline{K_{N_{j}}} (P) \Rightarrow \underline{S_{M_{j}}} (P) \subseteq \underline{K_{N_{j}}} (P) .$

Similar to Proof of Theorem-7 (1).

$\underline{(K_{N_{j}} \cup S_{M_{j}})} (P) (ξ) = \land_{ς \in U} (P (ς) \lor (1 - ((K (N_{F_{j}}) \cup S (M_{F_{j}})) (ξ, ς)))) = \land_{ς \in U} (P (ς) \lor (1 - (K (N_{F_{j}} (ξ, ς)) \lor S (M_{F_{j}} (ξ, ς))))) = \land_{ς \in U} (P (ς) \lor (K (N_{F_{j}} (ξ, ς)) \lor S (M_{F_{j}} (ξ, ς)))^{c}) = \land_{ς \in U} (P (ς) \lor ((K (N_{F_{j}} (ξ, ς)))^{c} \land (S (M_{F_{j}} (ξ, ς)))^{c}))$ (using De Morgan’s Law)

$= \land_{ς \in U} ((P (ς) \lor (K (N_{F_{j}} (ξ, ς)))^{c}) \land (P (ς) \lor (S (M_{F_{j}} (ξ, ς)))^{c}))$ $= (\land_{ς \in U} (P (ς) \lor (K (N_{F_{j}} (ξ, ς)))^{c})) \land (\land_{ς \in U} (P (ς) \lor (S (M_{F_{j}} (ξ, ς)))^{c})) = (\land_{ς \in U} (P (ς) \lor (1 - K (N_{F_{j}} (ξ, ς))))) \land (\land_{ς \in U} (P (ς) \lor (1 - S (M_{F_{j}} (ξ, ς))))) = \underline{K_{N_{j}}} (P) (ξ) \land \underline{S_{M_{j}}} (P) (ξ)$ . Hence, $\underline{(K_{N_{j}} \cup S_{M_{j}})} (P) = \underline{K_{N_{j}}} (P) \cap \underline{S_{M_{j}}} (P)$ .

Similar to Proof of Theorem-7 (3).

□

Theorem 8. Let $(U, K, Λ_{j})$ and $(U, S, Ω_{j})$ be two $j$ -FNSs, then for $j = {r, l, i, u}$ :

$\underline{(K \cup S)_{N_{j}}} (P) = \underline{K_{N_{j}}} (P) \cap \underline{S_{N_{j}}} (P) .$

$\bar{(K \cup S)_{N_{j}}} (P) = \underline{K_{N_{j}}} (P) \cup \underline{S_{N_{j}}} (P)$ .

Proof. Similar to the Proof of (3) and (4) of Theorem 7. □

Proposition 9. Let $(U, K, Λ_{j})$ be $j$ -FNS and $K$ be a reflexive BFR, then, for $j = {r, l, i, u}$ :

$\underline{K_{N_{j}}} (φ) = φ$ .

$\bar{K_{N_{j}}} (U) = U$ .

$\underline{K_{N_{j}}} (P) \subseteq P$ .

$P \subseteq \bar{K_{N_{j}}} (P)$ .

$\underline{K_{N_{j}}} (\underline{K_{N_{j}}} (P)) \subseteq \underline{K_{N_{j}}} (P)$ .

$\bar{K_{N_{j}}} (P) \subseteq \bar{K_{N_{j}}} (\bar{K_{N_{j}}} (P))$ .

Proof. Since, $K$ is reflexive BFR, therefore $N_{F_{j}} (ξ, ξ) = 1$ and $(1 - N_{F_{j}} (ξ, ξ)) = 0$ .

$\underline{K_{N_{j}}} (φ) (ξ) = \land_{ς \in U} (φ (ς) \lor (1 - K (N_{F_{j}} (ξ, ς))))$ $= \land_{ς \in U} (1 - K (N_{F_{j}} (ξ, ς))) = φ (ξ)$ .

$\Rightarrow \underline{K_{N_{j}}} (φ) = φ$ .

$\bar{K_{N_{j}}} (U) (ξ) = \lor_{ς \in U} (U (ς) \land K (N_{F_{j}} (ξ, ς))) = \lor_{ς \in U} (K (N_{F_{j}} (ξ, ς))) = U (ξ)$ .

$\Rightarrow \bar{K_{N_{j}}} (U) = U$ .

$\underline{K_{N_{j}}} (P) (ξ) = \land_{ς \in U} (P (ς) \lor (1 - K (N_{F_{j}} (ξ, ς)))) \leq P (ξ)$ (since $K (N_{F_{j}} (ξ, ς))$ is reflexive and its complement is anti reflexive). Hence, $\underline{K_{N_{j}}} (P) \subseteq P$ .

Proof of Prop.-9(4) is similar to Prop.-9(3). Prop.-9(5) and Prop.-9(6) can be proved by using Prop.-9(3) and Prop.-9(4) respectively.

□

Proposition 10. Let $(U, K, Λ_{j})$ be $j$ -FNS and $K$ be a transitive BFR, then, for $j = {r, l, i, u}$ :

$\underline{K_{N_{j}}} (P) \subseteq \underline{K_{N_{j}}} (\underline{K_{N_{j}}} (P))$ .

$\bar{K_{N_{j}}} (\bar{K_{N_{j}}} (P)) \subseteq \bar{K_{N_{j}}} (P)$ .

Proof.

$K \circ K \subseteq K$ ( $K$ is transitive BFR).

$\Rightarrow \underline{K_{N_{j}}} (P) \subseteq \underline{(K \circ K)_{N_{j}}} (P)$ (using (1) of Theorem 7) $\Rightarrow \underline{K_{N_{j}}} (P) \subseteq \underline{K_{N_{j}}} (\underline{K_{N_{j}}} (P))$

$K \circ K \subseteq K$ ( $K$ is transitive BFR).

$\Rightarrow \bar{(K \circ K)_{N_{j}}} (P) \subseteq \bar{K_{N_{j}}} (P)$ (using (2) of Theorem 7) $\Rightarrow \bar{K_{N_{j}}} (\bar{K_{N_{j}}} (P)) \subseteq \bar{K_{N_{j}}} (P)$ .

□

Proposition 11. Let $(U, K, Λ_{j})$ be a $j$ -FNS, $K$ be a reflexive and transitive BFR, then, for $j = {r, l, i, u}$ :

$\underline{K_{N_{j}}} (P) = \underline{K_{N_{j}}} (\underline{K_{N_{j}}} (P))$ .

$\bar{K_{N_{j}}} (\bar{K_{N_{j}}} (P)) = \bar{K_{N_{j}}} (P)$ .

Proof. It can be proved by using Proposition 10 and Proposition 9. □

Remark 4. All classical properties are satisfied by proposed fuzzy approximations, however,

$\underline{K_{N_{r}}} ((\underline{K_{N_{r}}} (P))^{c}) = (\bar{K_{N_{r}}} (P))^{c}$

and $(\bar{K_{N_{r}}} (\bar{K_{N_{r}}} (P))^{c}) = (\underline{K_{N_{r}}} (P))^{c}$ are not satisfied by $j$ - fuzzy approximations.

5 Medical application in decision-making

A method for decision-making using $j$ -FNs is presented in this section. The decision set of data is split into two parts, one with the presence and the other with the absence of symptoms, using $j$ -FN. The accuracy of the information system is examined in each part. $j$ -FN system classifies patient data based on the severity of the infection. Two different medical examples are explained here.

In 5.1, $j$ -FNs are obtained from the information system of heart disease. The importance of $j$ -FNs is demonstrated with a comparative analysis of existing methods.

5.1 Example on heart disease

Consider twelve patients $U = {ξ_{a}, ξ_{b}, ξ_{c}, ξ_{d}, ξ_{e}, ξ_{f}, ξ_{g}, ξ_{h}, ξ_{k}, ξ_{l}, ξ_{m}, ξ_{n}}$ having five symptoms

( $D_{1}$ =breathlessness, $D_{2}$ =orthopnea, $D_{3}$ =paroxy-

smal nocturnal dyspnea,

D_{4}

=reduced exercise tolerance,

D_{5}

=ankle swelling) of heart disease. The choice

D

regarding heart failure has two possible outcomes: ‘±’ denotes the presence of symptoms, while ‘∓’ denotes the absence of symptoms given in Table 2 [30, 32].

Table 2
Original medical information system

$U$	_ξa	_ξb	_ξc	_ξd	_ξe	_ξf	_ξg	_ξh	_ξk	_ξl	_ξm	_ξn
$D_{1}$	±	∓	±	∓	±	∓	±	±	±	∓	±	±
$D_{2}$	±	∓	±	∓	∓	∓	±	±	∓	∓	∓	∓
$D_{3}$	±	∓	±	∓	∓	∓	±	∓	±	∓	±	∓
$D_{4}$	±	±	±	±	±	±	±	±	±	±	±	±
$D_{5}$	∓	±	±	∓	±	∓	±	±	∓	±	∓	±
$D$	±	∓	±	∓	∓	∓	±	±	±	∓	±	∓

In order to show the similarities between patients’ symptoms, we transform the conditions characteristics (

D_{1}, D_{2}, D_{3}, D_{4}, D_{5}

) into qualitative terms, where the degree of similarity _ξij for patients is determined by:

$ξ_{ij} = \frac{\sum_{k = 1}^{n} (D_{k} (ξ_{i}) = D_{k} (ξ_{j}))}{number of symptoms}$ .

The degree of similarity obtained in Table 3 is considered as membership values and can be represented by BFR $K$ .

Therefore, $j -$ fuzzy rough approximations on the basis of $j -$ fuzzy nbd ( $N_{F_{j}}$ ) are calculated for $j = {r, l, i, u}$ as:

$P = \frac{{\sum_{k = 1}^{5} D_{k} (ξ); D_{k} (ξ) \in \pm}}{number of symptoms} = {\frac{. 8}{ξ_{a}} + \frac{. 4}{ξ_{b}} + \frac{1}{ξ_{c}} + \frac{. 2}{ξ_{d}} + \frac{. 6}{ξ_{e}} + \frac{. 2}{ξ_{f}} + \frac{1}{ξ_{g}} + \frac{. 8}{ξ_{h}} + \frac{. 6}{ξ_{k}} + \frac{. 4}{ξ_{l}} + \frac{. 6}{ξ_{m}} + \frac{. 6}{ξ_{n}}}$ .

$j - FL$ and $j$ -FU approximations for $P$ are:

Table 3

Similarities between symptoms of patients

	_ξa	_ξb	_ξc	_ξd	_ξe	_ξf	_ξg	_ξh	_ξk	_ξl	_ξm	_ξn
_ξa	1	0.2	0.8	0.4	0.4	0.4	0.8	0.6	0.8	0.2	0.8	0.4
_ξb	0.2	1	0.4	0.8	0.8	0.8	0.4	0.6	0.4	1	0.4	0.8
_ξc	0.8	0.4	1	0.2	0.6	0.2	1	0.8	0.6	0.4	0.6	0.6
_ξd	0.4	0.8	0.2	1	0.6	1	0.2	0.4	0.6	0.8	0.6	0.6
_ξe	0.4	0.8	0.6	0.6	1	0.6	0.6	0.8	0.6	0.8	0.6	1
_ξf	0.4	0.8	0.2	1	0.6	1	0.2	0.4	0.6	0.8	0.6	0.6
_ξg	0.8	0.4	1	0.2	0.6	0.2	1	0.8	0.6	0.4	0.6	0.6
_ξh	0.6	0.6	0.8	0.4	0.8	0.4	0.8	1	0.4	0.6	0.4	0.8
_ξk	0.8	0.4	0.6	0.6	0.6	0.6	0.6	0.4	1	0.4	1	0.6
_ξl	0.2	1	0.4	0.8	0.8	0.8	0.4	0.6	0.4	1	0.4	0.8
_ξm	0.8	0.4	0.6	0.6	0.6	0.6	0.6	0.4	1	0.4	1	0.6
_ξn	0.4	0.8	0.6	0.6	1	0.6	0.6	0.8	0.6	0.8	0.6	1

$\underline{K_{N_{j}}} (P) = {\frac{. 6}{ξ_{a}} + \frac{. 2}{ξ_{b}} + \frac{. 6}{ξ_{c}} + \frac{. 2}{ξ_{d}} + \frac{. 4}{ξ_{e}} + \frac{. 2}{ξ_{f}} + \frac{. 6}{ξ_{g}} + \frac{. 4}{ξ_{h}} + \frac{. 4}{ξ_{k}} + \frac{. 2}{ξ_{l}} + \frac{. 4}{ξ_{m}} + \frac{. 4}{ξ_{n}}}$ ,

$\bar{K_{N_{j}}} (P) = {\frac{. 8}{ξ_{a}} + \frac{. 6}{ξ_{b}} + \frac{1}{ξ_{c}} + \frac{. 6}{ξ_{d}} + \frac{. 8}{ξ_{e}} + \frac{. 6}{ξ_{f}} + \frac{1}{ξ_{g}} + \frac{. 8}{ξ_{h}} + \frac{. 8}{ξ_{k}} + \frac{. 6}{ξ_{l}} + \frac{. 8}{ξ_{m}} + \frac{. 8}{ξ_{n}}}$ respectively.

Decision value δ _k on the basis of $j -$ fuzzy approximations is defined as:

$δ_{k} = \underline{K_{N_{j}}} (P) (ξ_{k}) + \bar{K_{N_{j}}} (P) (ξ_{k})$ , where k = {1, 2,..., 12 } [4]. Therefore, decision set is:

$\begin{matrix} K_{N_{j}} (δ) = {\frac{1.4}{ξ_{a}} + \frac{. 8}{ξ_{b}} + \frac{1.6}{ξ_{c}} + \frac{. 8}{ξ_{d}} + \frac{1.2}{ξ_{e}} + \frac{. 8}{ξ_{f}} + \\ \frac{1.6}{ξ_{g}} + \frac{1.2}{ξ_{h}} + \frac{1.2}{ξ_{k}} + \frac{. 8}{ξ_{l}} + \frac{1.2}{ξ_{m}} + \frac{1.2}{ξ_{n}}} . \end{matrix}$ (3)

5.1.1 Results

The decision fuzzy set $K_{N_{j}} (δ)$ is divided into two sets on the basis of decision value δ _k, i.e., if δ _k ≥ 1, then the set $C_{1} = {ξ_{a}, ξ_{c}, ξ_{e}, ξ_{g}, ξ_{h}, ξ_{k}, ξ_{m} ξ_{n}}$ represent the presence of symptoms, and, if δ _k < 1, then the set $C_{2} = {ξ_{b}, ξ_{d}, ξ_{f}, ξ_{h}}$ represent absence of symptoms (using equation 3).

According to decision attribute ( $D$ ) of Table 2, $P_{1} = {ξ_{a}, ξ_{c}, ξ_{g}, ξ_{h}, ξ_{k}, ξ_{m}}$ and $P_{2} = {ξ_{b}, ξ_{d}, ξ_{e}, ξ_{f}, ξ_{l}, ξ_{n}}$ are the sets of presence and absence of symptoms, respectively. Comparing the set $P_{1}$ with $C_{1}$ and $P_{2}$ with $C_{2}$ , it is apparent that both sets $C_{1}$ and $C_{2}$ have an accuracy of 75% .

5.1.2 Comparative analysis of our approach with existing methods

Considering data of Table 2, $N_{j}$ - nbds [15], $U_{j}$ - nbds [18], $M_{j}$ -nbds [28], $C_{j}$ -nbds [32], $E_{j}$ -nbds [33] have been calculated. A binary relation is defined for similarities between patients’ symptoms with a threshold value 0.7 (Table 3). The nbds mentioned in Table 4 are all based on a crisp set where as the proposed nbds are defined by a fuzzy set. The accuracy obtained by the proposed method is better than that of all nbds except $C_{j}$ -nbds. In Table 4, LA and UA stand for lower and upper approximations.

Table 4
Comparative analysis of accuracy of existing nbds

$P_{1}$ $P_{2}$

LA UA accuracy LA UA accuracy

$N_{j}$ - nbds {_ξa, _ξc, _ξg, _ξk, _ξm} {_ξa, _ξc, _ξe, _ξg, _ξh, _ξk, _ξm, _ξn} 62.5% {_ξb, _ξd, _ξf, _ξl} {_ξb, _ξd, _ξe, _ξf, _ξh, _ξl, _ξn} 57.14%

$E_{j}$ -nbds {_ξa, _ξk, _ξm} {_ξa, _ξb, _ξc, _ξe, _ξg, _ξh, _ξk, _ξl, _ξm, _ξn} 30% {_ξd, _ξf} {_ξb, _ξc, _ξd, _ξe, _ξf, _ξg, _ξh, _ξl, _ξn} 22%

$C_{j}$ -nbds {_ξa, _ξc, _ξg, _ξh, _ξk, _ξm} {_ξa, _ξc, _ξg, _ξh, _ξk, _ξm} 100% {_ξb, _ξd, _ξe, _ξf, _ξl, _ξn} {_ξb, _ξd, _ξe, _ξf, _ξl, _ξn} 100%

$M_{j}$ -nbds {_ξa, _ξk, _ξm} {_ξa, _ξb, _ξc, _ξe, _ξg, _ξh, _ξk, _ξl, _ξm, _ξn} 30% {_ξd, _ξf} {_ξb, _ξc, _ξd, _ξe, _ξf, _ξg, _ξh, _ξl, _ξn} 22%

$U_{j}$ - nbds {_ξa, _ξc, _ξg, _ξk, _ξm} {_ξa, _ξc, _ξe, _ξg, _ξh, _ξk, _ξm, _ξn} 62.5% {_ξb, _ξd, _ξf, _ξl} {_ξb, _ξd, _ξe, _ξf, _ξh, _ξl, _ξn} 57.14%

Proposed nbds 75% 75%

( $N_{F_{j}}$ -nbds)

$P_{1}$	$P_{2}$
	LA	UA	accuracy	LA	UA	accuracy
$N_{j}$ - nbds	{_ξa, _ξc, _ξg, _ξk, _ξm}	{_ξa, _ξc, _ξe, _ξg, _ξh, _ξk, _ξm, _ξn}	62.5%	{_ξb, _ξd, _ξf, _ξl}	{_ξb, _ξd, _ξe, _ξf, _ξh, _ξl, _ξn}	57.14%
$E_{j}$ -nbds	{_ξa, _ξk, _ξm}	{_ξa, _ξb, _ξc, _ξe, _ξg, _ξh, _ξk, _ξl, _ξm, _ξn}	30%	{_ξd, _ξf}	{_ξb, _ξc, _ξd, _ξe, _ξf, _ξg, _ξh, _ξl, _ξn}	22%
$C_{j}$ -nbds	{_ξa, _ξc, _ξg, _ξh, _ξk, _ξm}	{_ξa, _ξc, _ξg, _ξh, _ξk, _ξm}	100%	{_ξb, _ξd, _ξe, _ξf, _ξl, _ξn}	{_ξb, _ξd, _ξe, _ξf, _ξl, _ξn}	100%
$M_{j}$ -nbds	{_ξa, _ξk, _ξm}	{_ξa, _ξb, _ξc, _ξe, _ξg, _ξh, _ξk, _ξl, _ξm, _ξn}	30%	{_ξd, _ξf}	{_ξb, _ξc, _ξd, _ξe, _ξf, _ξg, _ξh, _ξl, _ξn}	22%
$U_{j}$ - nbds	{_ξa, _ξc, _ξg, _ξk, _ξm}	{_ξa, _ξc, _ξe, _ξg, _ξh, _ξk, _ξm, _ξn}	62.5%	{_ξb, _ξd, _ξf, _ξl}	{_ξb, _ξd, _ξe, _ξf, _ξh, _ξl, _ξn}	57.14%
Proposed nbds	75%	75%
( $N_{F_{j}}$ -nbds)

Remark 5. $j$ -FNs can be examined by applying the concept of alpha cut to a binary fuzzy relation, making them similar to $j$ -nbds

In 5.2, the method of $j$ -FNs is applied to the data of COVID-19 [8], and the result is described in three categories with high, low and no risk of infection. The proposed method is implemented using Python Code to assess its performance.

5.2 Example on COVID-19

COVID-19 is a contagious viral infection that affects individuals. Some infected individuals experience no or mild symptoms, while others require medical attention due to severe infection. As a result, it is essential to make an appropriate diagnosis to avoid the further transmission of the infection.

In this part, a data set of 5434 persons with 20 attributes (Breathing Problem, Fever, Dry-Cough, Sore throat, Running-Nose, Asthma, Chronic Lung Disease, Headache, Heart Disease, Diabetes, Hyper Tension, Fatigue, Gastrointestinal, Abroad travel, Contact with COVID Patient, Attended Large Gathering, Visited Public Exposed Places, Family working in Public Exposed Places, Wearing Masks, Sensitization from Market) of COVID-19 [8] are examined using $j -$ FNs.

The calculations performed on this data are identical to that provided in 5.1, and the result are analysed based on the two different threshold values α _a and α _b [Table 5].

Table 5
Accuracy based on Threshold values

Individuals Actual α _a = 0.85 Accuracy(α _a) α _b = 0.86 Accuracy(α _b)

COVID-19 Positive 4386 4729 92.17 4268 97.30

COVID-19 Negative 1048 705 67.27 1166 88.74

Individuals	Actual	α _a = 0.85	Accuracy(α _a)	α _b = 0.86	Accuracy(α _b)
COVID-19 Positive	4386	4729	92.17	4268	97.30
COVID-19 Negative	1048	705	67.27	1166	88.74

5.2.1 Results

Following this calculation, the total number of people can be divided into three distinct categories as:

First category: 4268 people belonging to α _b are positive and at high risk of infection.

Second category: The variation between α _b and α _a of COVID-19 Negative, 461 individuals are at low risk.

Third category: 705 individuals in α _a are COVID-19 negative and classified as not infected.

Individuals can be quarantined, and medical care can be given according to three categories.

Algorithm 1 is provided to classify the individuals according to the severity of the infection.

Algorithm 1 The algorithm for calculating the risk of contamination

Require: Data set of medical information of individuals (ξ) with attributes $(D)$ and assign $U$ as the set of individuals.

Ensure: The categorization of $U$ on the basis of threshold value.

1: Classify $U$ of individual into two subsets: $P_{1}$ and $P_{2}$ as infected and not infected individuals respectively.

2: Define the binary relation $R$ , assign yes or ± as 1 and no or ∓ as 0 in data-set.

3: Compute degree of similarity between the patients: $ξ_{ij} = \frac{\sum_{k = 1}^{n} (D_{k} (ξ_{i}) = D_{k} (ξ_{j}))}{number of attributes}$ .

4: Define similarity matrix $K = [ξ_{ij}]$ of order n, where n is the no. of patients in data set.

5: Compute $j -$ fuzzy nbds for $j = {r, l, u, i}$ from BFR $K$ by using Definition 7.

6: Compute fuzzy set $P = \frac{{\sum_{k = 1}^{n} D_{k} (ξ); D_{k} (ξ) \in yes}}{number of attributes}$ .

7: Compute $j -$ fuzzy lower and upper approximations for the fuzzy subset $P$ by using Definition 10.

8: Compute decision value δ _k = sum of $j -$ fuzzy lower and upper approximations of the fuzzy subset $P$ .

9: Decide threshold values δ _k according to data set.

10: Classify the elements of $P$ with respect to δ _k.

11: End

6 Conclusion and future work

In a rough set, minimising the boundary area to raise the decision-making accuracy measure is an important aspect. Nbd system is one method that can be used to achieve the goal. Eight types of $j$ -FNs are proposed in the fuzzy nbd system, and their properties are studied in this paper. $j$ - fuzzy nbd system is used to define the $j$ -FL and $j$ -FU approximations, and their properties are investigated. To demonstrate the importance of our strategy, two medical applications are described by using Python Programming. $j$ -FNs are applied to examine the dengue disease information system and compared it with previous methods. The established $j$ -fuzzy approximation space yields greater accuracy concerning the decision given in Table 2. The proposed method allows us to address practical issues under binary fuzzy relations and works better for handling incomplete data. Based on the results obtained in Section 5.2, our technique classifies group of individuals in three categories and ensures to keep them safe from infection. It minimizes the burden on the healthcare system. It has been concluded that $j$ -FNs provide more accurate results compared to existing nbds, with an accuracy of 75%, making it a promising tool for future research.

Our future work includes applying the j-FNs for data reduction, investigating the topological approach for j-fuzzy approximations, and exploring the proposed work for different types of nbds such as containment nbds, subset nbds, etc.

References

Allam

A.A.

, Bakier

M.Y.

and Abo-Tabl

E.A.

, New approach for closurespaces by relations, Acta Mathematica Academiae PaedagogicaeNyiregyháziensis 22(3) (2006), 285–304.

Azzam

A.A.

and Al-shami

T.M.

, Five generalized rough approximationspaces produced by maximal rough neighborhoods, Symmetry 15(3) (2023), 751.

Kandil

, Yakout

M.M.

and Zakaria

, Generalized rough sets viaideals, Annals of Fuzzy Mathematics and Informatics 5(3) (2013), 525–532.

Sun

and Ma

, Soft fuzzy rough sets and its application indecision making, Artificial Intelligence Review 41(2014), 67–80.

Qin

, Fuzzy approximating spaces, Journal of Appliedmathematics 2014 (2014), 405802.

Chang

C.L.

, Fuzzy topological spaces, Journal of MathematicalAnalysis and Applications 24(1) (1968), 182–190.

Dubois

and Prade

, Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems 17 (1990), 191–209.

Harikrishnan

Symptoms and COVID Presence (May 2020 data), Kaggle. https://www.kaggle.com/datasets/hemanthhari/symptoms-and-covid-presence(2020).

Mustafa

H.I.

, Al-shami

T.M.

and Wassef

, Rough set paradigms viacontainment neighborhoods and ideals, Filomat 37(14) (2023), 4683–4702.

10.

Gupta

K.C.

and Gupta

R.K.

, Fuzzy equivalence relation redefined, Fuzzy Sets and Systems 79(2) (1996), 227–233.

11.

Qin

and Pei

, On the topological properties of fuzzy roughsets, Fuzzy Sets and Systems 151(3) (2005), 601–613.

12.

Zadeh

L.A.

, Similarity relations and fuzzy orderings, Journalof Information Sciences 3(2) (1971), 177–200.

13.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

14.

Atef

, Khalil

A.M.

, Li

S.G.

, Azzam

A.A.

, El

and El Atik

, Comparison of six types of rough approximations based onj-neighborhood space and j-adhesion neighborhood space, Journalof Intelligent and Fuzzy Systems 39(3) (2020), 4515–4531.

15.

Abd El-Monsef

M.E.

, Embaby

and El-Bably

M.K.

, Comparison betweenrough set approximations based on different topologies, International Journal of Granular Computing, Rough Sets andIntelligent Systems 3 (2014), 292–305.

16.

Hosny

, Al-shami

T.M.

and Mhemdi

, Rough approximation spacesvia maximal union neighborhoods and ideals with a medicalapplication, J. Math. 2022 (2022), Art. ID 5459796, 17 pages..

17.

Hosny

, Al-shami

T.M.

and Mhemdi

, Novel approaches ofgeneralized rough approximation spaces inspired by maximalneighbourhoods and ideals, Alexandria Engineering Journal 69 (2023), 497–520.

18.

Madan

and Tripathi

, Rough set model based on unionneighborhood, J. Int. Acad. Phys. Sci.27(3) (2023), 207–223.

19.

Priti and A. Tripathi, Fuzzy Approximation based on

τ - K

Fuzzy Open (Closed) Sets, MathematicalFoundations of Computing 6(3) (2023), 558–572.

20.

Priti and A. Tripathi, Fuzzy topology induced by binary fuzzy relation, In: Emerging Advancements in Mathematical Sciences, Chamola B P, Kumari P and Kaur L., ed., Nova Science Publishers (2022).

21.

Abu-Gdairi

, El-Gayr

M.A.

, Al-shami

T.M.

, El-Bably

M.K.

and Nawar

A.S.

, Some topological approaches for generalized rough sets andtheir decision-making applications, Symmetry. 14(1) (2022), 95.

22.

Mareay

, Generalized rough sets based on neighborhood systems andtopological space, Journal of the Egyptian MathematicalSociety 24(4) (2022), 603–608.

23.

Mishra

and Srivastava

, Fuzzy topologies generated by fuzzyrelations, Soft Computing 22 (2018), 373–385.

24.

Sheeja

T.K.

and Sunny

, Kuriakose, Rough topology on approximation spaces, International Journal of Advanced Research in Computer Science 8(9) (2017), 379–384.

25.

Al-shami

T.M.

and Mhemdi

, Approximation operators and accuracymeasures of rough sets from an infra-topology view, SoftComput. 27 (2023), 1317–1330.

26.

Al-shami

T.M.

and Mhemdi

, Approximation spaces inspired by subsetrough neighborhoods with applications, DemonstratioMathematica 56(1) (2023), 20220223.

27.

Al-shami

T.M.

and Alshammari

, Rough sets models inspired bysupra-topology structures, Artif. Intell. Rev. 56(2023), 6855–6883.

28.

Al-shami

T.M.

Maximal rough neighborhoods with a medical application, Journal of Ambient Intelligence and Humanized Computing (2022).

29.

Al-shami

T.M.

, Topological approach to generate new rough setmodels, Complex and Intelligent System. 8 (2022), 4101–4113.

30.

Al-shami

T.M.

and Ciucci

, Subset neighborhood rough sets, Knowledge Based System 237(2022), 107868.

31.

Al-shami

T.M.

and Hosny

, Improvement of approximation spacesusing maximal left neighborhoods and ideals, IEEE Access 10 (2022), 79379–79393.

32.

Al-shami

T.M.

, New An improvement of rough sets accuracy measureusing containment neighborhoods with medical application, Information Sciences 569 (2021), 110–124.

33.

Al-shami

T.M.

, Fu

W.Q.

and Abo-Tabl

E.A.

, New rough approximationsbased on E-neighborhoods, Complexity 2021 (2021), 6666853.

34.

Tang

, Wu

and Zheng

, On fuzzy rough sets and theirtopological structures, Mathematical Problems in Engineering 2014 (2014), 546372.

35.

W.Z.

and Zhang

W.X.

, Constructive and axiomatic approaches offuzzy approximation operators, Information Sciences 159(3-4) (2004), 233–254.

36.

W.Z.

, Mi

J.S.

and Zhang

W.X.

, Generalized fuzzy rough sets, Information Sciences 151 (2003), 263–28.

37.

Yao

Y.Y.

Generalized rough set models, In: Rough sets in knowledge discovery 1 Eds. Polkowski L and Skowron A. Heidelberg, Physica Verlag, (1998), 286–318.

38.

Yao

Y.Y.

, Relational interpretations of neighborhood operators andrough set approximation operators, Journal of InformationSciences 111(1-4) (1998), 239–259.

39.

Yao

Y.Y.

, Two views of the theory of rough sets in finite universes, International Journal of Approximate Reasoning 15(4) (1996), 291–317.

40.

Pawlak

, Rough set, International Journal of Computer andInformation Sciences 11 (1982), 341–356.

Decision making based on rough approximations generalized by j -fuzzy neighborhoods

Abstract

Keywords

1 Introduction

1.1 Research gap and motivation

1.2 Contribution

1.3 Significance

1.4 Layout

2 Preliminaries

3 j -FN spaces and induced fuzzy topologies

5.1 Example on heart disease

Table 2 Original medical information system

5.1.2 Comparative analysis of our approach with existing methods

Table 5 Accuracy based on Threshold values Individuals Actual α a = 0.85 Accuracy(α a ) α b = 0.86 Accuracy(α b ) COVID-19 Positive 4386 4729 92.17 4268 97.30 COVID-19 Negative 1048 705 67.27 1166 88.74

6 Conclusion and future work

References

3 $j$ -FN spaces and induced fuzzy topologies

Table 2
Original medical information system

Table 5
Accuracy based on Threshold values

Individuals Actual α _a = 0.85 Accuracy(α _a) α _b = 0.86 Accuracy(α _b)

COVID-19 Positive 4386 4729 92.17 4268 97.30

COVID-19 Negative 1048 705 67.27 1166 88.74