A topological approach for improving accuracy in decision-making via bi-ideal approximation

Abstract

The present paper proposes a novel version of inducing nano topology by using new kinds of approximation operators via two ideals with respect to a general binary relation. This approach improves the accuracy of the approximation quite significantly. These newly defined approximations constitute the generalized version of rough sets defined by Pawlak in 1982. A comparison is drawn between the suggested technique and the already existing ones to demonstrate the significance of the proposed ideology. In addition, the standard notion of nano topology, based on an equivalence relation is generalized to the binary relation, which can have a broader scope when applied to intelligent systems. Also, the significance of this approach is demonstrated by an example where an algorithm is given to find the key factors responsible for the profit of a company along with the comparison to the previous notions. Likewise, the proposed algorithm can be used in all fields of science to simplify complex information systems in extracting useful data by finding the core.

Keywords

Nano topology rough sets ideals bi-ideal approximation core

1 Introduction

The concepts of rough sets and topology have successfully been applied not only in mathematics but to various fields of science such as physics, chemistry, biochemistry, environmental sciences, etc. Research in these fields reveals that these theories hold a huge potential in simplifying and interpreting practical-life problems and situations which rely on multiple attributes. Intriguingly, the approximation operators, which play the key role in rough set theory, have some inbuilt topological properties like interior and closure, induced by a relation on a finite set known as the universe. So, the conjoint study of rough set theory and topology can be utilized in the mathematical modeling of qualitative and quantitative data. The notion of rough set [38] was first introduced by Pawlak as an extended version of set theory in which upper and lower approximation operators were introduced to define a set, which has been proven to be quite beneficial in investigating intelligent systems. This theory has been applied to a broad range of real-life problems like decision-making, data mining, machine learning, and image processing [1 , 37]. Nano topology is an emerging branch of mathematics, which is motivated by the rough set theory and provides an interdisciplinary forum, emphasizing the advancements in engineering as well as medical sciences. In the past few decades, the generalizations of rough set theory [5 , 30] were given to overcome the drawback of equivalence relation in generating granularity (neighborhood base). Also, numerous topological approaches were given to improve the accuracy of the approximation of rough sets [17 , 32–36]. Similarly, the generalizations of the standard definition of nano topology [4 , 29] were given to investigate novel approaches that can be more appropriate or suitable, when applied to real-life situations.

A new technique of approximation via two ideals was proposed by A. Kandil, which has been proven better than the previous ones as it has served in reducing the boundary region and hence increasing the degree of accuracy in rough set theory [6]. Over the past few years, many theories have been given to induce a nano topology through different mathematical tools like neutrosophic sets [26], fuzzy sets [20], Pythagorean sets [12], ideals [4, 7], graphs [3 , 29], etc.

Although nano topology is a dwarf topology, it has numerous applications. One of these applications is decision-making by finding the core [14, 18]. Nano topology, although small in size, has successfully been applied to medical and other fields. Interestingly, nano topology in association with the graph theory has simplified many biological processes such as blood circulation in the human heart, fetal circulation, urinary system, etc [4, 29]. It has also helped in detection of the diseases like lung disease and covid-19 [22 , 31]. Nano topology has helped in discovering the medical significance of a plant named Couroupita guianensis Abul and also the covid-19 most effective preventive measures by using the concept of a nano topological space [11]. Nano topology has helped in the reduction of electric transmission lines [2].

Many researchers have worked on weak open forms of nano sets [19, 22] using the approach of general topology. In this paper, a new nano topology is generated by using lower and upper approximation operators, defined via two ideals w.r.t a binary relation. Novel terminology is coined in terms of this newly defined space. Some properties and results are investigated. Remarkably, this theory of approximation can further be generalized to ’n’ ideals. At last, a comparison of this approach is drawn with the already existing ones, which implies that the significance of instituting this amalgamation is that the computational techniques, based on the single ideal won’t give the best results but a blend of two or more can often accomplish so. Furthermore, a real-life application of our proposed theory is discussed and the accuracy measure as compared to the previous notion is verified to be better as illustrated by an example of a company’s profit or loss.

2 Preliminaries

This section comprises the already existing definitions and notations that will be used in this paper. It is to be noted that throughout this paper, NT, n . t . s, a . s, n . o, n . c, b . i . n . o, b . i . n . c, BINT, b . i . n . t . s are the abbreviations for the nano topology, nano topological space, approximation space, nano open, nano closed, bi-ideal nano open, bi-ideal nano closed, bi-ideal nano topology and bi-ideal nano topological space respectively.

Definition 2.1: [27] Let U be the universe and $S$ be an equivalence relation which is generally referred as an indiscernibility relation. $S (z)$ is the set of all the elements related to z. The pair $(U, S)$ is known as approximation space (a . s .) and Z ⊆ U, then

$L_{S} (Z) = ⋃_{z \in U} {S (z) : S (z) \subseteq Z}$ .

This is known as a lower approximation of Z w.r.t $S$ .

$U_{S} (Z) = ⋃_{z \in U} {S (z) : S (z) \cap Z \neq \emptyset}$ .

This is known as an upper approximation of Z w.r.t $S$ .

$B_{S} (Z) = U_{S} (Z) - L_{S} (Z)$ .

This is known as the boundary region of Z w.r.t $S$ .

Let $τ_{S} (Z) = {U, \emptyset, L_{S} (Z), U_{S} (Z), B_{S} (Z)} .$

Then,

τ_{S} (Z)

is a topology on U, which is called a nano topology (NT) on U w.r.t Z. The pair

(U, τ_{S} (Z))

is known as a nano topological space (n . t . s). Elements of

(U, τ_{S} (Z))

are called the nano open (n . o .) sets. Complements of n . o . sets are called the nano closed (n . c .) sets.

B = {U, L_{S} (Z), B_{S} (Z)}

is a basis for this topology.

Remark 2.2: [27] If L ⊆ U, then the union of all n . o . subsets of L is called nano interior of L, written as ηint (L) and the intersection of all n . c . sets containing L is called nano closure of L, written as ηcl (L). Nano interior of L is the largest n . o . subset of L and nano closure of L is the smallest n . c . set containing L.

Definition 2.3: [6] An ideal Id on a set U is a non-empty collection of subsets of U which satisfies:

(i) $M \in Id$ and $N \subseteq M$ implies N ∈ Id.

(ii) $M \in Id$ and $N \in Id$ implies $M \cup N \in Id .$

Definition 2.4: [6] Let Id₁, Id₂ be two ideals on U and the set generated by two ideals Id₁, Id₂ be denoted by <Id₁, Id₂>. It is mathematically written as: $< {Id}_{1}, {Id}_{2} > = {M \cup N : M \in {Id}_{1}, N \in {Id}_{2}} .$

Remark 2.5: [6] If Id₁, Id₂ are two ideals on a non-empty set U and $M, N$ are two subsets of U, then the collection <Id₁, Id₂> satisfies the following conditions:

<Id₁, Id₂ > ≠ ∅ .

$M \in < {Id}_{1}, {Id}_{2} >, N \subseteq M$

$\Rightarrow N \in < {Id}_{1}, {Id}_{2} > .$

$M, N \in < {Id}_{1}, {Id}_{2} >$

$\Rightarrow M \cup N \in < {Id}_{1}, {Id}_{2} >$ .

Clearly, the collection <Id₁, Id₂> is an ideal on U.

Definition 2.6: [6] If U is the universe and $S$ is a binary relation on U, a set $< k > S$ is the intersection of all after sets containing ’k’ and $S < k >$ is the intersection of all fore sets containing ’k’.

Expressing mathematically,

$< k > S = {⋂_{k \in z S} (z S)$ , if ( $\exists z : k \in z S$ ) and ∅, otherwise } and $S < k > = {⋂_{k \in S z} (S z)$ , (if $\exists z : k \in S z$ ) and ∅, otherwise } . Also, $S < k > S = (< k > S) \cap (S < k >)$ .

Here, $S z$ is called the fore set of z defined as $S z = {y \in U : y S z}$ and $z S$ is called the after set of z defined by $z S = {y \in U : z S y}$ .

Definition 2.7: [6] Let U be a non-empty finite set of objects known as the universe and Z ⊆ U. Then, $(U, S, Id)$ is an ideal a . s . If $S$ is a binary relation on U and Id is an ideal on U, then the lower and upper approximation operators are defined as: ${\underline{\underline{S}}}_{Id} (Z) = {z \in Z : S < z > S \cap Z^{c} \in Id} .$ ${\bar{\bar{S}}}_{Id} (Z) = Z \cup {z \in U : S < z > S \cap Z \notin Id} .$

Definition 2.8: [6] Let U be a non-empty finite set of objects known as the universe and Z ⊆ U. Then, $(U, S, {Id}_{1}, {Id}_{2})$ is a bi-ideal a . s .. If $S$ is a binary relation on U and Id₁ and Id₂ are two ideals on U, then the lower and upper approximations are defined as: $\begin{matrix} {\underline{\underline{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z) = {z \in Z : & S < z > S \cap Z^{c} \\ \in < {Id}_{1}, {Id}_{2} >} . \end{matrix}$ $\begin{matrix} {\bar{\bar{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z) = Z \cup {z \in U & : S < z > S \cap Z \\ \notin < {Id}_{1}, {Id}_{2} >} . \end{matrix}$

3 Generation of nano topology via bi-ideal approximation space

In this section, we generate the bi-ideal nano topology using the two ideals.

Definition 3.1: Let $(U, S, {Id}_{1}, {Id}_{2})$ be a bi-ideal a . s where U is a non-empty finite set known as the universe and Z is a subset of U. Let $S$ be a binary relation on U. If Id₁ and Id₂ are two ideals on U, the lower and upper approximations are ${\underline{\underline{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z)$ and ${\bar{\bar{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z)$ as defined in Definition 2.8. B_{<Id₁,Id₂>} (Z) is the boundary, given by the difference of upper and lower approximation. $B_{< {Id}_{1}, {Id}_{2} >} (Z) = {\bar{\bar{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z) - {\underline{\underline{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z)$ .

Now, define the collection $τ_{< {Id}_{1}, {Id}_{2} >} (Z) = {\emptyset, U, {\bar{\bar{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z), {\underline{\underline{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z), B_{< {Id}_{1}, {Id}_{2} >} (Z)} .$

This collection forms a topology as it satisfies all three properties of topology. This topology is called a bi-ideal nano topology (BINT). The pair (U, τ_{<Id₁,Id₂>} (Z)) is called a bi-ideal nano topological space (b . i . n . t . s) w.r.t Z. All elements of this space are called bi-ideal nano open (b . i . n . o) sets and complements are called bi-ideal nano closed (b . i . n . c) sets. Complements of b . i . n . o sets, together form the dual bi-ideal NT.

Note: If Id₁ = Id₂ = Id, this definition reduces to a n . t . s induced by a single ideal, that is τ_Id (Z), where approximation operators are as defined in Definition 2.7. In addition, the bi-ideal NT can be extended to any finite number of ideals to form a multi -ideal nano topology.

Example 3.2: Let U be {y, z, x, r} and Z be {r, y}

Also, let Id₁ = {∅ , {r}} and Id₂ = {∅ , {z}}.

So, <Id₁, Id₂ > = {∅ , {r} , {z} , {r, z}} . If $S = {{x, x}, {y, y}, {z, z}, {r, r}, {y, x}, {x, y}}$ . Clearly, here $S$ is an equivalence relation.

Then τ_{<Id₁,Id₂>} (Z) = {∅ , U, {x, y} , {x, r, y} , {r}} and (U, τ_{<Id₁,Id₂>} (Z)) is a b . i . n . t . s.

Example 3.3: Let U = {b, c, a, d} and Z = {d, b}.

Also, let Id₁ = {∅ , {a}} and Id₂ = {∅ , {a} , {d} , {d, a}}.

So, <Id₁, Id₂ > = {∅ , {a} , {d} , {d, a}}. If $S = {{a, a}, {a, d}, {c, a}, {b, c}, {d, b}, {a, c}, {d, d}}$ .

Clearly, here $S$ is not an equivalence relation.

Then τ_{<Id₁,Id₂>} (Z) = {∅ , U, {b, d}} and

(U, τ_{<Id₁,Id₂>} (Z)) is a b . i . n . t . s.

Remark 3.4: Let U be a non-empty finite universe and Z be a subset of U. If Id₁, Id₂ are two ideals on U and $S$ is a binary relation on U, then the following hold:

The collection τ_{<Id₁,Id₂>} (Z) = {∅ , U} is an indiscrete BINT.

If ${\bar{\bar{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z) = {\underline{\underline{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z) = Z$ , then $τ_{< {Id}_{1}, {Id}_{2} >} (Z) = {\emptyset, U, {\bar{\bar{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z)}$ .

If ${\bar{\bar{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z) \neq U, {\underline{\underline{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z) = \emptyset$ , then $τ_{< {Id}_{1}, {Id}_{2} >} (Z) = {\emptyset, U, {\bar{\bar{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z)}$ .

If ${\bar{\bar{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z) = U, {\underline{\underline{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z) \neq \emptyset$ ,

then $τ_{< {Id}_{1}, {Id}_{2} >} (Z) = {\emptyset, U, {\underline{\underline{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z), B_{< {Id}_{1}, {Id}_{2} >} (Z)}$ .

If ${\bar{\bar{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z) \neq {\underline{\underline{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z), {\underline{\underline{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z) \neq \emptyset, {\bar{\bar{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z) \neq U$ ,

then $τ_{< {Id}_{1}, {Id}_{2} >} (Z) = {\emptyset, U, {\bar{\bar{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z), {\underline{\underline{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z), B_{< {Id}_{1}, {Id}_{2} >} (Z)}$ .

(This collection forms a discrete BINT).

Proposition 3.5: Let (U, τ_{<Id₁,Id₂>} (Z)) be a b . i . n . t . s w.r.t. any subset Z of U. If L is any subset of U, then the bi-ideal nano interior and bi-ideal nano closure may be defined as:

Bi-ideal nano-interior: It is union of all the b . i . n . o . subsets of L and it is denoted by

η_{<Id₁,Id₂>}int (L). This is the largest b . i . n . o . subset of L.

Bi-ideal nano closure: It is the intersection of all the b . i . n . c . sets containing L and is denoted by η_{<Id₁,Id₂>}cl (L). It is the smallest b . i . n . c . set containing L.

Example 3.6: Following are the examples of bi-ideal nano interior and bi-ideal nano closure:

Let U be {y, z, x, r} and Z be {r, y}.

If Id₁ = {∅ , {r}} and Id₂ = {∅ , {z}},

then, <Id₁, Id₂ > = {∅ , {r} , {z} , {r, z}} . If $S = {{x, x}, {y, y}, {z, z}, {r, r}, {x, y}, {y, x}}$ ,

then τ_{<Id₁,Id₂>} (Z) = {∅ , U, {y, x} , {y, r, x} , {r}}.

(U, τ_{<Id₁,Id₂>} (Z)) is a b . i . n . t . s w.r.t Z. Also, the collection of b . i . n . o sets is {∅ , U, {x, y} , {r, y, x} , {r}} and the collection of b . i . n . c sets is {∅ , U, {z, r} , {x, y, z} , {z}}.

If L ={x, y} (b . i . n . o), then η_{<Id₁,Id₂>}int (L) = L .

If L = {x, r}, then η_{<Id₁,Id₂>}int (L) = {r} .

If L = {y, x, z} (b . i . n . c set), then η_{<Id₁,Id₂>}cl (L) = L .

If L = {x, z}, then η_{<Id₁,Id₂>}cl (L) = {x, y, z} .

Theorem 3.7: If (U, τ_{<Id₁,Id₂>} (Z)) is a b . i . n . t . s w.r.t Z where Z is a subset of U, then x∈ η_{<Id₁,Id₂>}cl (L) ⇔ P ∩ L ≠ ∅ for every b . i . n . o . set P containing x, where L ⊆ U.

Proof: First consider, z ∈ η_{<Id₁,Id₂>}cl (L) .

Let P be a b . i . n . o . set containing z.

Since P is b . i . n . o, U - P is b . i . n . c set. If L∩ P = ∅, then L ⊆ U - P, U - P is a b . i . n . c set containing L. Thus, η_{<Id₁,Id₂>}cl (L) ⊆ U - P, which is not true due to the fact that z ∈ η_{<Id₁,Id₂>}cl (L) but z ∉ U - P. Therefore, L∩ P ≠ ∅ for every b . i . n . o set G containing z.

Conversely, let L∩ P ≠ ∅ for every b . i . n . o set P having z. If z ∉ η_{<Id₁,Id₂>}cl (L), then z ∈ U - η_{<Id₁,Id₂>}cl (L). As U - η_{<Id₁,Id₂>}cl (L) is a b . i . n . o . set and thus (U - η_{<Id₁,Id₂>}cl (L))∩ L ≠ ∅ (By assumption). As L⊆ η_{<Id₁,Id₂>}cl (L) ⇒U - η_{<Id₁,Id₂>}cl (L) ⊆ U - L . ⇒ (U - η_{<Id₁,Id₂>}cl (L)) ∩ L ⊆ (U - L) ∩ L = ∅, which is a contradiction.

Hence, z ∈ η_{<Id₁,Id₂>}cl (L).

Theorem 3.8: Let (U, τ_{<Id₁,Id₂>} (Z)) be a b . i . n . t . s w.r.t Z where Z is a subset of U. If L is a subset of U, then the following statements hold:

U - η_{<Id₁,Id₂>}int (L)= η_{<Id₁,Id₂>}cl (U - L) .

Proof: Let z ∈ U - η_{<Id₁,Id₂>}int (L) . ⇒z ∉ η_{<Id₁,Id₂>}int (L). Clearly, by the definition of bi-ideal nano interior in Proposition 3.5, PnotsubseteqL for every b . i . n . o . set P containing x. Because PnotsubseteqL⇒P⊆ U - L, P ∩ U - L ≠ ∅ for every b . i . n . o . set P containing z. By Theorem 3.6, P∩ U - L ≠ ∅ for every bi-ideal nano set P having z.

⇔z ∈ η_{<Id₁,Id₂>}cl (U - L) .

Hence, z ∈ U - η_{<Id₁,Id₂>}int (L) ⇒z ∈ η_{<Id₁,Id₂>}cl (U - L) .

So, U - η_{<Id₁,Id₂>}int (L) ⊆ η_{<Id₁,Id₂>}cl (U - L) .

Conversely, let z ∈ η_{<Id₁,Id₂>}cl (U - L) . Then P∩ (U - L) ≠ ∅ for each b . i . n . o . set P containing z, i.e., PnotsubseteqL for every b . i . n . o . set P containing z, i.e., z ∉ η_{<Id₁,Id₂>}int (L) .

Therefore, z ∈ U - η_{<Id₁,Id₂>}int (L) .

⇒ η_{<Id₁,Id₂>}cl (U - L) ⊆ U - η_{<Id₁,Id₂>}int (L)

Hence, U - η_{<Id₁,Id₂>}int (L) = η_{<Id₁,Id₂>}cl (U - L) .

U - η_{<Id₁,Id₂>}cl (L)= η_{<Id₁,Id₂>}int (U - L) .

Proof: Let z ∈ U - η_{<Id₁,Id₂>}cl (L) ⇒z ∈ U and z ∉ η_{<Id₁,Id₂>}cl (L).

Since, z ∉ η_{<Id₁,Id₂>}cl (L), so by Theorem 3.7, for every b . i . n . o . set P containing z, P ∩ L = ∅ , P ⊆ U, L ⊆ U⇒z ∉ L, z ∈ P ⊆ U⇒z ∈ U - L. So, z ∈ η_{<Id₁,Id₂>}int (U - L).

∴ U - η_{<Id₁,Id₂>}cl (L) ⊆ η_{<Id₁,Id₂>}int (U - L).

Conversely, if z ∈ η_{<Id₁,Id₂>}int (U - L) ⇒z ∈ (U - L) ⇒z ∉ L, z ∈ U, then for each b . i . n . o . set P containing z, P∩ L = ∅ for P ⊆ U, L ⊆ U . By Theorem 3.7, z ∉ η_{<Id₁,Id₂>}cl (L) but z ∈ P ⊆ U.

⇒z ∈ U - η_{<Id₁,Id₂>}cl (L).

⇒η_{<Id₁,Id₂>}int (U - L) ⊆ U - η_{<Id₁,Id₂>}cl (L).

Therefore, U - η_{<Id₁,Id₂>}cl (L)= η_{<Id₁,Id₂>}int (U - L) .

Remark 3.9: Taking complements both sides in the Theorem 3.8,

η_{<Id₁,Id₂>}int (L) = U - η_{<Id₁,Id₂>}cl (U - L) .

η_{<Id₁,Id₂>}cl (L) = U - η_{<Id₁,Id₂>}int (U - L) .

Theorem 3.10: Let (U, τ_{<Id₁,Id₂>} (Z)) be a b . i . n . t . s w.r.t Z where Z ⊆ U. If L and M are subsets of U, then the following statements hold:

L ⊆ η_{<Id₁,Id₂>}cl (L).

Proof: (By the definition of bi-ideal nano closure, as defined in Proposition 3.5.)

L is b . i . n . c . iff η_{<Id₁,Id₂>}cl (L) = L.

Proof: If L is b . i . n . c ., then L is the smallest b . i . n . c . set containing itself.

Therefore, η_{<Id₁,Id₂>}cl (L) = L.

Conversely, if η_{<Id₁,Id₂>}cl (A) = L, then L is the smallest b . i . n . c . set containing itself and hence L is b . i . n . c ..

η_{<Id₁,Id₂>}cl (∅) = ∅ and η_{<Id₁,Id₂>}cl (U) = U.

Proof: As U and ∅ are b . i . n . c .. So by (2), we have η_{<Id₁,Id₂>}cl (∅) = ∅.

η_{<Id₁,Id₂>}cl (U) = U.

L ⊆ M⇒η_{<Id₁,Id₂>}cl (L) ⊆ η_{<Id₁,Id₂>}cl (M).

Proof: Now, M ⊆ η_{<Id₁,Id₂>}cl (M). Therefore, η_{<Id₁,Id₂>}cl (M) is a b . i . n . c . set containing L. Hence, η_{<Id₁,Id₂>}cl (L) ⊆ η_{<Id₁,Id₂>}cl (M).

η_{<Id₁,Id₂>}cl (L ∪ M) = η_{<Id₁,Id₂>}cl (L) ∪ η_{<Id₁,Id₂>}cl (M).

Proof: Since L ⊆ L ∪ M and M ⊆ L ∪ M,

so, by (4), η_{<Id₁,Id₂>}cl (L) ⊆ η_{<Id₁,Id₂>}cl (L ∪ M).

Also, η_{<Id₁,Id₂>}cl (M) ⊆ η_{<Id₁,Id₂>}cl (L ∪ M).

This implies, η_{<Id₁,Id₂>}cl (L) ∪ η_{<Id₁,Id₂>}cl (M) ⊆ η_{<Id₁,Id₂>}cl (L ∪ M) .

As L ∪ M ⊆ η_{<Id₁,Id₂>}cl (L) ∪ η_{<Id₁,Id₂>}cl (M) and η_{<Id₁,Id₂>}cl (L ∪ M) is the smallest b . i . n . c . set containing L ∪ M.

So, η_{<Id₁,Id₂>}cl (L ∪ M) ⊆ η_{<Id₁,Id₂>}cl (L) ∪ η_{<Id₁,Id₂>}cl (M).

Thus, η_{<Id₁,Id₂>}cl (L ∪ M) = η_{<Id₁,Id₂>}cl (L) ∪ η_{<Id₁,Id₂>}cl (M).

η_{<Id₁,Id₂>}cl (L ∩ M) ⊆ η_{<Id₁,Id₂>}cl (L) ∩ η_{<Id₁,Id₂>}cl (M)

Proof: Since L ∩ M ⊆ L and L ∩ M ⊆ M,

so, by (4), η_{<Id₁,Id₂>}cl (L ∩ M) ⊆ η_{<Id₁,Id₂>}cl (L).

Also, η_{<Id₁,Id₂>}cl (L ∩ M) ⊆ η_{<Id₁,Id₂>}cl (M).

Therefore, η_{<Id₁,Id₂>}cl (L ∩ M) ⊆ η_{<Id₁,Id₂>}cl (L) ∩ η_{<Id₁,Id₂>}cl (M).

η_{<Id₁,Id₂>}cl (η_{<Id₁,Id₂>}cl (L)) = η_{<Id₁,Id₂>}cl (L).

Proof: As η_{<Id₁,Id₂>}cl (L) is b . i . n . c .,

η_{<Id₁,Id₂>}cl (η_{<Id₁,Id₂>}cl (L)) = η_{<Id₁,Id₂>}cl (L).

(by (2)).

Theorem 3.11: Let (U, τ_{<Id₁,Id₂>} (Z)) be a b . i . n . t . s w.r.t Z where Z ⊆ U. If L and M are subsets of U, then:

L is b . i . n . o . iff η_{<Id₁,Id₂>}int (L) = L.

Proof: If L is b . i . n . o . ⇔U - L is b . i . n . c . ⇔η_{<Id₁,Id₂>}cl (U - L) = U - L .

(by Theorem 3.10 (2))

⇔U - η_{<Id₁,Id₂>}cl (U - L) = L . ⇔ η_{<Id₁,Id₂>}int (L) = L

(by Theorem 3.9).

η_{<Id₁,Id₂>}int (∅) = ∅ and η_{<Id₁,Id₂>}int (U) = U.

Proof: Since U and ∅ are b . i . n . o . sets, so by (1), we have η_{<Id₁,Id₂>}int (∅) = ∅ and η_{<Id₁,Id₂>}int (U) = U.

L ⊆ M⇒η_{<Id₁,Id₂>}int (L) ⊆ η_{<Id₁,Id₂>}int (M).

Proof: L ⊆ M⇒U - M ⊆ U - L.

Hence, by Theorem 3.10 (4),

η_{<Id₁,Id₂>}cl (U - M) ⊆ η_{<Id₁,Id₂>}cl (U - L) . ⇒U - η_{<Id₁,Id₂>}cl (U - L) ⊆ U - η_{<Id₁,Id₂>}cl (U - M).

⇒η_{<Id₁,Id₂>}int (L) ⊆ η_{<Id₁,Id₂>}int (M)

(by Theorem 3.8).

η_{<Id₁,Id₂>}int (L) ∪ η_{<Id₁,Id₂>}int (M) ⊆ η_{<Id₁,Id₂>}int (L ∪ M).

Proof: Since L ⊆ L ∪ M and M ⊆ L ∪ M,

by (3), η_{<Id₁,Id₂>}int (L) ⊆ η_{<Id₁,Id₂>}int (L ∪ M).

Also, η_{<Id₁,Id₂>}int (M) ⊆ η_{<Id₁,Id₂>}int (L ∪ M).

This implies, η_{<Id₁,Id₂>}int (L) ∪ η_{<Id₁,Id₂>}int (M) ⊆ η_{<Id₁,Id₂>}int (L ∪ M).

η_{<Id₁,Id₂>}int (L ∩ M) = η_{<Id₁,Id₂>}int (L) ∩ η_{<Id₁,Id₂>}int (M).

Proof: Since L ∩ M ⊆ L and L ∩ M ⊆ M,

so, by (3), η_{<Id₁,Id₂>}int (L ∩ M) ⊆ η_{<Id₁,Id₂>}int (L) ∩ η_{<Id₁,Id₂>}int (M).

By the fact that

η_{<Id₁,Id₂>}int (L) ∩ η_{<Id₁,Id₂>}int (M) ⊆ L ∩ M

and η_{<Id₁,Id₂>}int (L ∩ M) is the largest b . i . n . o set contained in L ∩ M, then

η_{<Id₁,Id₂>}int (L) ∩ η_{<Id₁,Id₂>}int (M) ⊆ η_{<Id₁,Id₂>}int (L ∩ M).

So, η_{<Id₁,Id₂>}int (L ∩ M) = η_{<Id₁,Id₂>}int (L) ∩ η_{<Id₁,Id₂>}int (M).

η_{<Id₁,Id₂>}int (η_{<Id₁,Id₂>}int (L)) = η_{<Id₁,Id₂>}int (L).

Proof: As η_{<Id₁,Id₂>}int (L) is b . i . n . o .,

η_{<Id₁,Id₂>}int (η_{<Id₁,Id₂>}intL)) = η_{<Id₁,Id₂>}int (L) (by (1)).

4 Comparison of this approach to previous ones

The concept of measurement of accuracy plays an imperative role in decision-making when the rough set theory or nano topology is applied to intelligence systems or the physical world. It is evident that this approach using two ideals to induce topology is a generalization of the previous one because induction of nano topology via a single ideal is the special case of this technique, wherein Id₁ and Id₂, both coincide. As lower approximation increases and the boundary region decreases, thereby resulting in better precision in investigating the interrelation between condition attributes and decision attributes, so it has a huge potential for further research and physical significance. Also, as this theory generalizes the standard definition applicable to equivalence relation to a binary relation which is a broader concept, so it has a wider range and scope in future perspective. The following remarks are illustrated to mark the distinction between current ideology and the already established ones:

${\bar{\bar{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z) \subseteq {\bar{\bar{S}}}_{I_{i}} (X) \subseteq U_{S} (Z)$ , for i=1,2.

$L_{S} (Z) \subseteq {\underline{\underline{S}}}_{{Id}_{i}} (Z) \subseteq {\underline{\underline{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z)$ for all i=1,2.

$B_{< {Id}_{1}, {Id}_{2} >} (Z) \subseteq B_{{Id}_{i}} (Z) \subseteq B_{S} (z)$ for i=1,2.

μ (Z) ≤ μ_{Id
_i} (Z) ≤ μ_{<Id₁,Id₂>} (Z) for i=1,2.

Here, μ stands for accuracy measure, that is, the ratio of lower to upper approximation.

τ_{<Id₁,Id₂>} (Z) ⊈ τ_{Id_i(Z)} for i=1,2 and vice - versa.

In general, bi-ideal NT is not comparable with the topology induced via single ideal.

$τ_{< {Id}_{1}, {Id}_{2} >} (Z) ⊈ τ_{S} (Z))$ and vice - versa.

In general, bi-ideal NT is not comparable with the standard definition of NT.

Bi-ideal approximations coincide with the single ideal approximations if Id₁ = Id₂, so bi-ideal NT is a generalisation of the NT, generated by a single ideal, τ_Id (Z).

If the relation is equivalence and Id₁ = Id₂ =∅, then the bi-ideal NT is same as the standard definition of NT, that is $τ_{S} (Z)$ .

Illustration 4.1: Let U = {n, m, o},

$S = {{m, m}, {n, n}, {m, n}, {n, m}, {o, o}}$ .

Id₁ = {∅ , {m}}, Id₂ = {∅ , {n}}.

So, <Id₁, Id₂ > = {∅ , {m} , {n} , {n, m}}.

The comparison of the proposed theory with the already existing ones is shown in Tables 1 –5:

Table 1
The comparison of the lower approximation operators of a rough set Z w.r.t relation $S$ of the proposed topology with the existing ones

Z $L_{S} (Z)$ ${\underline{\underline{S}}}_{{Id}_{1}} (Z)$ ${\underline{\underline{S}}}_{{Id}_{2}} (Z)$ ${\underline{\underline{S}}}_{< {Id}_{1}, {Id}_{2} >} (z)$

{m} ∅ ∅ {m} {m}

{n} ∅ {n} ∅ {n}

{o} {o} {o} {o} {o}

{n, m} {n, m} {n, m} {n, m} {n, m}

{o, n} {o} {n, o} {o} {o, n}

{m, o} {o} {o} {m, o} {m, o}

{n, m, o} {n, m, o} {n, m, o} {n, m, o} {n, m, o}

Z	$L_{S} (Z)$	${\underline{\underline{S}}}_{{Id}_{1}} (Z)$	${\underline{\underline{S}}}_{{Id}_{2}} (Z)$	${\underline{\underline{S}}}_{< {Id}_{1}, {Id}_{2} >} (z)$
{m}	∅	∅	{m}	{m}
{n}	∅	{n}	∅	{n}
{o}	{o}	{o}	{o}	{o}
{n, m}	{n, m}	{n, m}	{n, m}	{n, m}
{o, n}	{o}	{n, o}	{o}	{o, n}
{m, o}	{o}	{o}	{m, o}	{m, o}
{n, m, o}	{n, m, o}	{n, m, o}	{n, m, o}	{n, m, o}

Table 2

The comparison of the upper approximation operators of a rough set Z w.r.t relation $S$ of the proposed topology with the existing ones

Z	$U_{S} (z)$	${\bar{\bar{S}}}_{{Id}_{1}} (Z)$	${\bar{\bar{S}}}_{{Id}_{2}} (Z)$	${\bar{\bar{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z)$
{m}	{m}	{m}	{m}	{m}
{n}	{n, m}	{n, m}	{n}	{n}
{o}	{o}	{o}	{o}	{o}
{n, m}	{n, m}	{n, m}	{m, n}	{n, m}
{o, n}	{n, m, o}	{n, m, o}	{n, o}	{o, n}
{m, o}	{n, m, o}	{m, o}	{m, o, n}	{m, o}
{n, m, o}	{n, m, o}	{n, m, o}	{n, m, o}	{n, m, o}

Table 3

The comparison of the boundaries of a rough set Z w.r.t relation $S$ of the proposed topology with the existing ones

Z	$B_{S} (Z)$	B_{Id ₁} (Z)	B_{Id ₂} (Z)	B_{<Id₁,Id₂>} (Z)
{m}	{m}	{m}	∅	∅
{n}	{n, m}	{m}	{n}	∅
{o}	∅	∅	∅	∅
{n, m}	∅	∅	∅	∅
{o, n}	{n, m}	{m}	{n}	∅
{m, o}	{n, m}	{m}	{n}	∅
{n, m, o}	∅	∅	∅	∅

Table 4

The comparison of the bi-ideal nano topology with the existing ones

Z	$τ_{S} (Z)$	τ_{Id ₁} (Z)	τ_{<Id₁,Id₂>} (Z)
{m}	{∅ , {m} , {n, m, o}}	{∅ , {m} , {n, m, o}}	{∅ , {m} , {n, m, o}}
{n}	{∅ , {n, m} , {n, m, o}}	{∅ , {m} , {n} , {n, m} , {n, m, o}}	{∅ , {n} , {n, m, o}}
{o}	{∅ , {o} , {n, m, o}}	{∅ , {o} , {n, m, o}}	{∅ , {o} , {n, m, o}}
{n, m}	{∅ , {n, m} , {n, m, o}}	{∅ , {n, m} , {n, m, o}}	{∅ , {n, m} , {n, m, o}}
{o, n}	{∅ , {n, m} , {o} , {n, m, o}}	{∅ , {o, n} , {m} , {n, m, o}}	{∅ , {o, n} , {n, m, o}}
{m, o}	{∅ , {n, m} , {o} , {n, m, o}}	{∅ , {m} , {o} , {m, o} {n, m, o}}	{∅ , {m, o} , {n, m, o}}
{n, m, o}	{∅ , {n, m, o}}	{∅ , {n, m, o}}	{∅ , {n, m, o}}

Table 5

The comparison of the accuracy measures of approximation of the proposed topology with the existing ones

Z	μ (Z)	μ_{Id ₁} (Z)	μ_{Id ₂} (Z)	μ_{<Id₁,Id₂>} (z)
{m}	0	0	1	1
{n}	0	$\frac{1}{2}$	0	1
{o}	1	1	1	1
{n, m}	1	1	1	1
{o, n}	$\frac{1}{3}$	$\frac{2}{3}$	$\frac{1}{2}$	1
{m, o}	$\frac{1}{3}$	$\frac{1}{2}$	$\frac{2}{3}$	1
{n, m, o}	1	1	1	1

5 An application

In this section, we will discuss an algorithm for finding the key factors responsible for any decision using the concept of finding core through a bi-ideal NT. A real-life example has also been discussed, where the concept of bi-ideal NT has been applied to find the key factors responsible for a company’s profit/loss of a company, thereby, illustrating the significance of the following algorithm:

5.1 Algorithm

Step 1: Provided a finite universal set U, a finite set A of attributes which may further be classified as two categories, A₁ of conditional attributes and A₂ of decision attributes, an indiscernibility relation $S$ on U corresponds to A₁ and a subset Z of U, represent the data in a tabular form, columns of which are labeled by the given set of attributes, rows are given by objects and entries of the table are attribute values according to the domain of each. Also, Id₁, Id₂ are the ideals on U, which represent the reports of two external experts.

Step 2: Find the lower approximation, upper approximation, and the boundary region of Z w.r.t $S$ , according to the notion of bi-ideal a . s, as defined in Definition 2.8.

Step 3: Induce the bi-ideal NT τ_{<Id₁,Id₂>} (Z) on U.

Step 4: Eliminate an attribute z from A₁. Then, find the lower, upper approximations and the boundary region of Z again w.r.t the modified equivalence relation on A₁ - {z}.

Step 5: Generate the new bi-ideal NT $τ_{< {Id}_{1}, {Id}_{2} >}^{'} (Z)$ on U w.r.t. the modified relation $S^{'}$ .

Step 6: Repeat steps 4 and 5 for each attribute in A₁.

Step 7: The attributes in A₁ for which $τ_{< {Id}_{1}, {Id}_{2} >} (Z) \neq τ_{< {Id}_{1}, {Id}_{2} >}^{'} (Z)$ forms the Core( $S$ ).

5.2 Example

The performance of a company or an organization is governed by the concept of profit and loss. The profit of a company depends on various factors. In this example, the proposed algorithm based on the bi-ideal nano topological model is used to find the key factors responsible for the success of a firm or business. Let U = {Q₁, Q₂, Q₃, Q₄, Q₅, Q₆, Q₇, Q₈, Q₉, Q₁₀} be the set of 10 companies. This set is the universe. The Table 6 represents the data of the ten companies, with respect to the conditional attributes namely, Brand Image (B.I.), Quality of product (Q.O.P.), marketing/advertisement (M./A.), competition (Comp.), revenue (Rev.), product packaging (P.P.).

Table 6
The data of the ten companies, with respect to various conditional attributes governing profit/loss

Company B.I. Q.O.P M./A. COMP. REV. P.P. Decision

Q ₁ Good Excellent Average High Great Attractive Profit

Q ₂ Bad Superior Poor Low Great Unattractive Loss

Q ₃ Good Inferior Good High Great Attractive Loss

Q ₄ Bad Inferior Average High Moderate Normal Profit

Q ₅ Bad Superior Poor Low Moderate Unattractive Loss

Q ₆ Good Inferior Good High Great Attractive Loss

Q ₇ Good Inferior Average High Great attractive profit

Q ₈ Bad Superior Good Low Moderate Unattractive Loss

Q ₉ Bad Superior Poor Low Moderate Unattractive Profit

Q ₁₀ Good Inferior Good High Great Attractive Profit

Company	B.I.	Q.O.P	M./A.	COMP.	REV.	P.P.	Decision
Q ₁	Good	Excellent	Average	High	Great	Attractive	Profit
Q ₂	Bad	Superior	Poor	Low	Great	Unattractive	Loss
Q ₃	Good	Inferior	Good	High	Great	Attractive	Loss
Q ₄	Bad	Inferior	Average	High	Moderate	Normal	Profit
Q ₅	Bad	Superior	Poor	Low	Moderate	Unattractive	Loss
Q ₆	Good	Inferior	Good	High	Great	Attractive	Loss
Q ₇	Good	Inferior	Average	High	Great	attractive	profit
Q ₈	Bad	Superior	Good	Low	Moderate	Unattractive	Loss
Q ₉	Bad	Superior	Poor	Low	Moderate	Unattractive	Profit
Q ₁₀	Good	Inferior	Good	High	Great	Attractive	Profit

Also, let Z = {Q₁, Q₄, Q₇, Q₉, Q₁₀} be the five companies that have gained profit in a particular time-frame by selling a product and Z^C = {Q₂, Q₃, Q₅, Q₆, Q₈} be the set of the companies which have undergone loss by the sale of a product in that time-frame.

Here, Id₁ = {∅ , {Q₁} , {Q₃} , {Q₁, Q₃}} and Id₂ = {∅ , { Q₅} , { Q₃} , { Q₉} , { Q₃, Q₉} , {Q₅, Q₉} , {Q₃, Q₉} , { Q₃, Q₅, Q₉}} . <Id₁, Id₂ > = {∅ , { Q₁} , { Q₃} , { Q₅} , {Q₉} , {Q₃, Q₅} , {Q₅, Q₉} , {Q₃, Q₉} , { Q₁, Q₃} , { Q₁, Q₉} , {Q₁, Q₅} , {Q₁, Q₃, Q₅} , {Q₁, Q₃, Q₉} , {Q₁, Q₅, Q₉} , {Q₃, Q₅, Q₉} , {Q₁, Q₃, Q₅, Q₉}} .

Here Id₁ and Id₂ represent the reports of companies by external experts 1 and 2. <Id₁, Id₂> is the ideal generated by Id₁ and Id₂ means collective report of both experts.

Here, the decision attributes are profit or loss, i.e., domain of decision attribute = { profit, loss}.

Domain of attribute (B.I.) = {good, bad}.

Domain of attribute (Q.O.P) = {excellent, superior, inferior}.

Domain of marketing/advertisement (M./A.) = {average, good, poor}.

Domain of competition (COMP.) = {high, low}.

Domain of revenue (REV.) = {great, moderate}.

Domain of product packaging (P.P.) = {attractive, normal, unattractive}. Now, if $S$ is the equivalence relation w.r.t. which we say that two companies are related if they are indiscernible w.r.t collection of a particular set of attributes. Also, $S$ represents the notation for equivalence classes w.r.t $S$ .

If Id₁, Id₂ are given and Z ⊆ U, then the bi-ideal NT can be given by $τ_{< {Id}_{1}, {Id}_{2} >} (Z) = {\emptyset, U, {\underline{\underline{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z), {\bar{\bar{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z), B_{< {Id}_{1}, {Id}_{2} >} (Z)} .$

Then, if we remove a certain attribute {X₁}, $S$ gets modified to $S^{'}$ .

New notation or equivalence classes w.r.t $S^{'}$ is $S^{'}$ , i.e., $S^{'} = S - {X_{1}}$ and the new bi-ideal NT induced by this relation $S^{'}$ is $τ_{< {Id}_{1}, {Id}_{2} >}^{'} (Z) = {\emptyset, U, {\underline{\underline{S^{'}}}}_{< {Id}_{1}, {Id}_{2} >} (Z), {\bar{\bar{S^{'}}}}_{< {Id}_{1}, {Id}_{2} >} (Z), B_{< {Id}_{1}, {Id}_{2} >}^{'} (Z)}$ . Now, it is evident that when all the six conditional attributes are taken all together, then we have $S = {{Q_{1}}, {Q_{2}}, {Q_{3}, Q_{6}, Q_{10}}, {Q_{4}}, {Q_{5}, Q_{9}}, {Q_{7}}, {Q_{8}}}$ .

Then, ${\underline{\underline{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z) = {Q_{1}, Q_{4}, Q_{7}, Q_{9}}$ ,

${\bar{\bar{S}}}_{< {Id}_{1}, {Id}_{2} >} (Z) = {Q_{1}, Q_{3}, Q_{4}, Q_{6}, Q_{7}, Q_{9}, Q_{10}}$

and B_{<Id₁,Id₂>} (Z) = {Q₃, Q₆, Q₁₀}.

Hence, bi-ideal NT is τ_{<Id₁,Id₂>} (Z) = {∅ , U, {Q₁, Q₄, Q₇, Q₉} , {Q₁, Q₃, Q₄, Q₆, Q₇, Q₉, Q₁₀} , {Q₃, Q₆, Q₁₀}} .

CASE-I: If we remove the attribute “Brand Image”(B.I.) from the set of attributes, then we have $S^{'} = S -$ {B.I.}. So, $S^{'} = S -$ {B.I.} which implies that $S^{'} = {{Q_{1}}, {Q_{2}}, {Q_{3}, Q_{6}, Q_{10}}, {Q_{4}}, {Q_{5}, Q_{9}}, {Q_{7}}, {Q_{8}}} = S$ . Hence, $τ_{< {Id}_{1}, {Id}_{2} >}^{'} (Z)$ induced by $S^{'} = S -$ {B.I.} is equal to τ_{<Id₁,Id₂>} (Z), i.e., $τ_{< {Id}_{1}, {Id}_{2} >}^{'} (Z) = τ_{< {Id}_{1}, {Id}_{2} >} (X)$ as $S^{'} = S$ .

CASE-II: If we remove the attribute “Quality of Product"(Q.O.P) from the set of attributes, then we have $S^{'} = S -$ {Q.O.P}. So, $S^{'} = S -$ {Q.O.P} which implies that $S^{'} = {{Q_{1}, Q_{7}}, {Q_{2}}, {Q_{3}, Q_{6}, Q_{10}}, {Q_{4}}, {Q_{5}, Q_{9}}, {Q_{8}}} \neq S$ . Hence, $τ_{< {Id}_{1}, {Id}_{2} >}^{'} (Z)$ (w.r.t $S^{'}$ ) is given by $τ_{< {Id}_{1}, {Id}_{2} >}^{'} (Z) = {\emptyset, U, {Q_{1}, Q_{4}, Q_{7}, Q_{10}}, {Q_{1}, Q_{3}, Q_{4}, Q_{6}, Q_{7}, Q_{9}, Q_{10}}, {Q_{3}, Q_{6}, Q_{9}}} \neq τ_{< {Id}_{1}, {Id}_{2} >} (Z)$ .

CASE-III: If we remove the attribute

“Marketing/advertisement"(M./A.) from the set of attributes, then we have $S^{'} = S -$ {M./A.}. So, $S^{'} = S -$ {M./A.} which implies that $S^{'} = {{Q_{1}}, {Q_{2}}, {Q_{3}, Q_{6}, Q_{7}, Q_{10}}, {Q_{4}}, {Q_{5}, Q_{8}, Q_{9}}} \neq S$ . Hence, $τ_{< {Id}_{1}, {Id}_{2} >}^{'} (Z) = {\emptyset, U, {Q_{1}}, {Q_{1}, Q_{3}, Q_{4}, Q_{6}, Q_{7}, Q_{9}, Q_{10}}, {Q_{3}, Q_{4}, Q_{6}, Q_{7}, Q_{9}, Q_{10}}} \neq τ_{< {Id}_{1}, {Id}_{2} >} (Z)$ .

CASE-IV: If we remove the attribute “Competition"(COMP.) from the set of attributes, then we have $S^{'} = S -$ {COMP.}. So, $S^{'} = S -$ {COMP.} which implies that $S^{'} = {{Q_{1}}, {Q_{2}}, {Q_{3}, Q_{6}, Q_{10}}}, {Q_{4}}, {Q_{5}, Q_{9}}, {Q_{7}}, {Q_{8}}} = S$ . Hence, $τ_{< {Id}_{1}, {Id}_{2} >}^{'} (Z)$ is induced by $S^{'}$ = $S$ -{COMP.} is equal to τ_{<Id₁,Id₂>} (Z), i.e., $τ_{< {Id}_{1}, {Id}_{2} >}^{'} (Z) = τ_{< {Id}_{1}, {Id}_{2} >} (Z)$ as $S^{'} = S$ .

CASE-V: If we remove the attribute “Revenue"(REV.) from the set of attributes, then we have $S^{'} = S -$ {REV.}. So, $S^{'} = S -$ {REV.} which implies that $S^{'} = {{Q_{1}}, {Q_{2}, Q_{5}, Q_{9}}, {Q_{3}, Q_{6}, Q_{10}}, {Q_{4}}, {Q_{7}}, {Q_{8}}} \neq S$ . Hence, $τ_{< {Id}_{1}, {Id}_{2} >}^{'} (Z) = {\emptyset, U, {Q_{1}, Q_{4}, Q_{7}}, {Q_{1}, Q_{3}, Q_{4}, Q_{6}, Q_{7}, Q_{9}, Q_{10}}, {Q_{3}, Q_{6}, Q_{9}, Q_{10}}} \neq τ_{< {Id}_{1}, {Id}_{2} >} (Z)$ .

CASE-VI: If we remove the attribute “Product Packaging"(P.P.) from the set of attributes, then we have $S^{'} = S -$ {P.P.}. So, $S^{'} = S -$ {P.P.} which implies that $S^{'} = {{Q_{1}}, {Q_{2}}, {Q_{3}, Q_{6}, Q_{10}}, {Q_{4}}, {Q_{5}, Q_{9}}, {Q_{7}}, {Q_{8}}} = S$ .

Hence, $τ_{< {Id}_{1}, {Id}_{2} >}^{'} (Z)$ is induced by $S^{'} = S -$ {P.P.} is equal to τ_{<Id₁,Id₂>} (Z), i.e., $τ_{< {Id}_{1}, {Id}_{2} >}^{'} (Z) = τ_{< {Id}_{1}, {Id}_{2} >} (Z)$ as $S^{'} = S$ .

OBSERVATION: We can conclude that the set of attributes for which $τ_{< {Id}_{1}, {Id}_{2} >}^{'} (Z) \neq τ_{< {Id}_{1}, {Id}_{2} >} (Z)$ is {Quality of Product, Advertisement, revenue}. So these factors, being the most significant ones constitute the key attributes governing the profit/loss of a company when compared with another set of companies selling some product in a specific time frame. So, companies should emphasize on these three factors for success.

CORE(R)={Quality of Product(Q.O.P.), Marketing/Advertisement(M./A.) and revenue(REV.)}.

Let μ stand for an accuracy measure. A comparison of the suggested approach with existing ones is shown in Table 7:

Table 7

The comparison of the accuracy measures of a rough set Z w.r.t relation $S$ of the proposed topology with the existing ones in the discussed application

Relation $S$	μ (Z)	μ_{Id ₁} (Z)	μ_{<Id₁,Id₂>} (Z)
$S^{'} = S$	$\frac{3}{8}$	$\frac{3}{7}$	$\frac{4}{7}$
$S^{'} = S$ - B.I.	$\frac{3}{8}$	$\frac{3}{7}$	$\frac{4}{7}$
$S^{'} = S$ - Q.O.P.	$\frac{3}{8}$	$\frac{3}{8}$	$\frac{4}{7}$
$S^{'} = S$ - M.A.	$\frac{2}{9}$	$\frac{2}{9}$	$\frac{2}{7}$
$S^{'} = S$ - COMP.	$\frac{3}{8}$	$\frac{3}{7}$	$\frac{4}{7}$
$S^{'} = S$ - REV.	$\frac{1}{3}$	$\frac{1}{3}$	$\frac{3}{7}$
$S^{'} = S$ - P.P.	$\frac{3}{8}$	$\frac{3}{7}$	$\frac{4}{7}$

It is remarkable that $μ (Z) \leq μ_{{Id}_{1}} (Z) \leq μ_{< {Id}_{1}, {Id}_{2} >} (Z) .$ Our purpose was to compare the accuracy with the existing notions. From Table 7, it is clear that the proposed approach is better than the single-ideal approach and the standard approach. It is to be noted that the approximations form the base for the generation of nano topology and the objective of approximating in rough sets is the reduction of boundary and increase in the accuracy measure. So, the value of accuracy measure μ is better using two ideals instead of one, as shown above, so we conclude that this technique is more appropriate in forming an approximation space as it is more precise in analyzing the objective and making the correct judgment. This implies that the decision-making process will give more precise results when bi-ideal nano topology is used.

6 Future scope

The motivation of this study is that this concept can be used to determine a decision based on multiple opinions, rather than one. Also, the main advantage of this ideology is that the accuracy using this approach is significantly greater than the already existing ones. As the bi-ideal approach assimilates together the notion of combined opinions through two ideals, it can further be generalized to ‘n’ ideals (‘n’ opinions) and hence a mathematical model can be designed to study the interdependence of various factors and various situations, which arise in fields of physics, chemistry, and biology. This approach can help in simplification and easy interpretation of any information system by removing unwanted data. Hence, it can contribute a lot to studying the qualitative and quantitative properties of various elements, compounds, and species. This study has a great scope in medicine, biochemistry, and engineering.

7 Conclusion

The term “nano topology” was coined by M.L Thivagar [27], which was a revolutionary idea as it served in the fields of engineering, sciences, and technology. Over the past decade, various theories have been given by many researchers in which nano topology was generated by approximations in rough sets via ideals, graphs, neutrosophic sets, fuzzy sets, Pythagorean sets, etc. This paper extends the idea of approximation via a single ideal to two ideals, which may further be generalized to any finite no. of ideals, which can help in framing an algorithm to deal with condition attributes and decision attributes via a collaborated viewpoint of multiple opinions. A study on its range of applications is currently in progress.

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