Novel concepts of soft rough set topology with applications

1 Introduction

The intricacy of modeling real world problems is enhancing nowadays. Mostly we are incapable to determine the accurate solutions in some arduous mathematical models. We often encounter with various types of uncertainties, while considering the problems in social science, engineering, economics, environmental science, artificial intelligence and medical science. The information about a problem is usually imperfect and imprecise. This imprecision may be caused by the granularity in the representation of given data. In the field of computer science and mathematics, experts have proposed many beneficial theories including theory of probability, theory of fuzzy sets, theory of rough sets and interval mathematics. These approaches are useful to extract the beneficial information hidden in the voluminous data. Sometimes these theories are incompetent to handle the data involving parameterizations which was pointed out by Molodstov [15]. Fuzzy set proposed by Zadeh in 1965 [31] was the first tentative approach to deal with vagueness or uncertainty. Rough set introduced by Pawlak [16, 17] extract vagueness by means of boundary region instead of membership values. If boundary region is empty then the set under consideration is crisp otherwise it is rough or inexact. In 1999 Molodstov [15] initiated the concept of soft set as a powerful mathematical technique towards uncertainties. He efficiently used this technique in different areas like smoothness of functions, Riemann integration, game theory, theory of measurements and operation research. Ali et al. [1, 2] briefly discussed the connection between soft sets, rough sets and fuzzy soft sets. According to the study of Aktas and Cagman in 2007. [3] soft set, fuzzy set and rough set are closely related approaches. In 2002 Maji et al. [13, 14] presented some basic operations of soft set and effectively used this concept to decision making problem. In 2011 Shabir and Naz [29] and Cagman et al. [5] independently introduced soft topology on soft set. Mathew et al. [12] used multi-modal approach on soft set for medical diagnostic prediction system. Riaz et al. [19 –21] established some results of soft algebra, soft metric spaces and measurable soft mappings. Riaz and Masooma [22 –26] presented fpfs -topology and fpfs -compact spaces with some important applications of fpfs -set to the decision-making problems. They also introduced fns -mappings and fixed points of fns -mapping.

In contrast to other methodologies, the rough set approach depends only on the information presented by the data itself and does not need any outside parameters or assumptions. Rough set used to characterize any subset of the universe by defining two subsets namely lower approximation R ★ and upper approximation R ★ [16, 17]. In rough set theory the equivalence classes of an indiscernibility relation R are considered to generate rough topology. Using equivalence relations, Thivagar et al. [30] established the topology on rough set which includes approximations and boundary region. The SR -approximations and SR -set which based on soft approximation space are introduced by Feng et al. [7, 8]. Zhan et al. [10 , 32–34] introduced several ideas including soft rough covering, soft rough hemirings, z-soft fuzzy rough set and certain hybrid soft set models and efficiently employ these techniques in multi-criteria decision making problems. Zhang et al. [35 –37] introduced covering based fuzzy rough sets and Covering-based generalized IF rough sets with applications to MCDM. Bakier et al. 2016 [4] introduced SR -topology. Applications of soft set theory and rough set theory have been studied by many mathematicians. [2 , 30]. The remainder of the paper is composed as follows: In section 2, we have briefly define notions of rough set R -set and soft rough set SR -set. In Section 3, we explain topological structure of SR -set as defined by Bakier et al. [4]. In section 4 we present some new results of SR -topology related to SR -closure, SR -interior, SR -frontier, SR -neighbourhood and SR -limit point. In section 5 we present use of SR -sets in multicriteria group decision making (MCGDM). In section 6 we modify an algorithm and use SR -topology to find out the deciding factors of the chronic disease Tuberculosis.

2 Preliminaries

This section presents some basic definitions including soft set, rough set and SR -set.

Definition 2.1. Let V be the universal set. Let P ( V ) is collection of subsets of V . A pair ( T , A ) is said to be a soft set over the universe V , where A ⊆ E and T : A → P ( V ) is a set-valued function. We denote soft set as ( T , A ) or T A and mathematically write it as T A = { ( ζ , T ( ζ ) ) : ζ ∈ A , T ( ζ ) ∈ P ( V ) } . For any ζ ∈ A , T ( ζ ) is ζ-approximate elements of soft set T A . The set containing all soft sets over V is denoted by S ( V ) " (See [15]).

Example 2.2. Suppose that V = { s 1 , s 2 , s 3 , s 4 , s 5 } be the set consisting of five pair of shoes and set criterions is given by A = { ζ 1 , ζ 2 , ζ 3 , ζ 4 } = E , where ζ ₁ stands for “modern style”, ζ ₂ stands for “durable”, ζ ₃ stands for “reasonable price”, ζ ₄ stands for “comfortable”.

The soft set T A expresses the “quality of shoes” that Mr. Zeeshan required to purchase. Consider a mapping T : A → P ( V ) such that T ( ζ 1 ) = { s 3 , s 5 } , T ( ζ 2 ) = { s 2 , s 4 , s 5 } , T ( ζ 3 ) = { s 1 , s 2 , s 5 } , T ( ζ 4 ) = { s 2 , s 3 } .

Then the soft set T A is the set of approximations, each approximation is a pair of predicate and an approximate value set.

Definition 2.3. Suppose we have a set of objects under consideration V and an indiscernibility relation R ⊆ V × V which indicates our information about elements of V . For sake of our convenience, we take R as an equivalence relation and denote it as R ( y ) . The pair ( V , R ) is referred as approximation space. A subset Y of V is taken to characterize it w.r.t R .

The union of the granules entirely included in the set Y forms lower approximation of the set Y w.r.t R . mathematically defined as; R ★ ( Y ) = ⋃ y ∈ V { R ( y ) : R ( y ) ⊆ Y } .

The set Y is said to be defined if R ★ ( Y ) = R ★ ( Y ) . The set Y is (imprecise) rough set w.r.t R , if R ★ ( Y ) ≠ R ★ ( Y ) i.e B R ( Y ) ≠ ∅ (See [16]).

Example 2.4. Suppose we have set of patients V = { ϱ 1 , ϱ 2 , ϱ 3 , ϱ 4 , ϱ 5 , ϱ 6 , ϱ 7 , ϱ 8 , ϱ 9 } suffering from certain disease. Consider set of attributes as set of symptoms. Consider Y = { ϱ 1 , ϱ 2 , ϱ 3 , ϱ 4 , ϱ 5 , ϱ 6 , ϱ 7 } ⊆ V and Diabetes as indiscernibility relation. We present the information in tabular form, rows indicate (objects) patients, columns shows attributes and entries of table give attribute values. Such tables are known as information systems. We can see that {ϱ ₁, ϱ ₃} are the patients having diabetes. The table below shows that patient ϱ ₂ is having diabetes while ϱ ₇ is not having diabetes, and they have same symptoms, hence ϱ ₂ and ϱ ₇ lies in boundary region. Hence lower approximation of the set w.r.t relation “diabetes” is R ★ ( Y ) = { ϱ 1 , ϱ 3 } and the upper approximation of this set is the set R ★ ( Y ) = { ϱ 1 , ϱ 2 , ϱ 3 , ϱ 7 } , while boundary region is B R ( Y ) = R ★ ( Y ) \ R ★ ( Y ) ={ϱ ₂, ϱ ₇}.

Definition 2.5. Consider a soft set S = ( T , A ) over the universe V , where A ⊆ E and T is a function given as T : A → P ( V ) . Then the pair P = ( V , S ) is called a soft approximation space. Following the soft approximation space P , we get two approximations to every subset Y ⊆ V given by R P ★ ( Y ) = { v ∈ V : ∃ a ∈ A , v ∈ T ( a ) ⊆ Y } , R P ★ ( Y ) = { v ∈ V : ∃ a ∈ A , v ∈ T ( a ) ∩ Y ≠ ∅ } , which we call soft P-lower approximation and soft P-upper approximationof Y . Generally, R P ★ ( Y ) and R P ★ ( Y ) are called SR -approximations of Y w.r.t P. If R P ★ ( Y ) ≠ R P ★ ( Y ) then Y is said to be soft P-rough set otherwise soft P-definable(See [7]).

Example 2.6. Let V = { s 1 , s 2 , s 3 , s 4 , s 5 } , E = { ζ 1 , ζ 2 , ζ 3 , ζ 4 , ζ 5 , ζ 6 } and A = { ζ 1 , ζ 2 , ζ 3 , ζ 4 } ⊆ E . Let S = ( T , A ) is soft set over V and T ( ζ 1 ) = { s 2 , s 3 } , T ( ζ 2 ) = { s 2 , s 4 , s 5 } , T ( ζ 3 ) = { s 1 , s 2 , s 5 } , T ( ζ 4 ) = { s 3 , s 5 } . The tabular form of soft set ( T , A ) is given in Table 2. Then we obtain soft approximation space P = ( V , S ) . For Y = { s 3 , s 4 , s 5 } ⊆ V , we obtain R P ★ ( Y ) = { s 3 , s 5 } and R P ★ ( Y ) = { s 1 , s 2 , s 3 , s 4 , s 5 } .

Since R P ★ ( Y ) ≠ R P ★ ( Y ) and hence Y is said to be a soft P-rough set.

Definition 2.7. If ⋃ a ∈ A T ( a ) = V then the soft set ( T , A ) over V is said to be full soft set,

Definition 2.8. [7] If { T ( a ) : a ∈ A } forms a partition of V then the soft set ( T , A ) over V is called partition soft set.

Definition 3.1. Let V as universe of objects, Y ⊆ V and P = ( V , S ) be a soft approximation space. Then the collection τ SR ( Y ) = { V , ∅ , R P ★ ( Y ) , R P ★ ( Y ) , Bd P ( Y ) } is called soft rough topology which satisfies the following axioms:

V and ∅ belongs to τ SR ( Y ) .

If τ SR ( Y ) is Soft rough topology then ( V , τ SR ( Y ) , E ) is said to be SR -topological space (See 4).

Definition 3.2. Let ( V , τ SR ( Y ) , E ) be a SR -topological space, then the members of τ SR ( Y ) are known as SR -open sets. A SR -set is known as SR -closed set if its complement belongs to τ SR ( Y ) .

Proposition 3.3 Consider ( V , τ SR ( Y ) , E ) as SR -space over V . Then

V and ∅ are SR -closed sets.

Proof. The proof is straightforward.□

Definition 3.4. Let ( V , τ SR ( Y ) , E ) be a SR -space over V and τ SR ( Y ) = { V , ∅ } then τ SR is called SR -indiscrete topology on V w.r.t Y and corresponding space is said to be a SR -indiscrete space over V .

Example 3.5. 1 Let V = { ϑ 1 , ϑ 2 , ϑ 3 , ϑ 4 , ϑ 5 , ϑ 6 } be the set of cars under consideration and let E = { ζ 1 , ζ 2 , ζ 3 , ζ 4 , ζ 5 } the set of all parameters and A = { ζ 1 , ζ 2 , ζ 3 } ⊆ E . Consider the soft approximation P = ( V , S ) , where S = ( T , A ) is a soft set over U given by: T ( ζ 1 ) = { ϑ 1 , ϑ 3 } , T ( ζ 2 ) = { ϑ 1 , ϑ 3 , ϑ 6 } , T ( ζ 3 ) = { ϑ 2 , ϑ 4 } . For Y = { ϑ 2 , ϑ 3 , ϑ 4 , ϑ 6 } , we obtain R P ★ ( Y ) = { ϑ 2 , ϑ 4 } , R P ★ ( Y ) = { ϑ 1 , ϑ 2 , ϑ 3 , ϑ 4 , ϑ 6 } and Bd P ( Y ) = { ϑ 1 , ϑ 3 , ϑ 6 } . Then τ SR ( Y ) = { V , ∅ , { ϑ 2 , ϑ 4 } , { ϑ 1 , ϑ 2 , ϑ 3 , ϑ 4 , ϑ 6 } , { ϑ 1 , ϑ 3 , ϑ 6 } }

is a SR -topology.

Definition 3.6. Let ( V , τ SR ( Y ) , E ) be a SR -topological space and β SR ⊆ τ SR . If we can write element of τ SR as the union of elements of β SR , then β SR is called SR -basis for the SR -topology τ SR .

Proposition 3.7. [4] If τ SR ( Y ) is a SR -topology on V w.r.t Y the the collection β SR = { V , R P ★ ( Y ) , Bd P ( Y ) } is a base for τ SR ( Y )

Definition 3.8. [4] Let ( U , τ SR ( Y ) , E ) and be two SR -topological spaces. is finer than τ SR ( Y ) , if .

Theorem 3.9. [4] Let ( V , τ SR ( Y ) , E ) and ( V , τ SR ′ ( Y ) , E ) be two SR -topological spaces w.r.t Y and Y respectively. Let β SR and β SR ′ be SR -bases for τ SR and τ SR ′ respectively. If β SR ′ ⊆ β SR , then τ SR is finer than τ SR ′ and τ SR ′ is weaker than τ SR .

Definition 3.10. [4] Let ( V , τ SR ( Y ) , E ) is a SR -topological space and A ⊆ B ⊆ V . Then the collection τ SR A = { B i ∩ A : B i ∈ τ SR , k ∈ K ⊆ N } is called SR -subspace topology on A. Then, ( A , τ SR A ) is called a SR -topological subspace of ( B , τ SR ) .

Theorem 3.11. Let ( V , τ SR ( Y ) , E ) be a SR -topological space. If β SR is a SR -basis for τ SR , then the collection β SR B = { A i ∩ B : A i ∈ β SR , i ∈ I ⊆ ℕ } is a SR -basis for the SR -subspace topology on B.

Proof. Consider A i ∈ τ SR B . By definition of SR -subspace topology, C = D ∩ B, where D ∈ τ SR . Since D ∈ τ SR , it follows that D = ⋃ A i ∈ β SR A i . Therefore, C = ( ⋃ A i ∈ β SR A i ) ∩ B = ⋃ A i ∈ β SR ( A i ∩ B ) . □

Definition 3.12. [4] Let ( V , τ SR ( Y ) , E ) be a SR -topological space w.r.t Y , where Y ⊆ V and B V. The SR -interior of B is union of all SR -open subsets of B and we denote it as Int SR ( B ) .

We can see that Int SR ( B ) is the largest SR -open set contained by B.

4 Main Results

Bakier et al. defined Soft rough topology and presented some properties of SR -approximation. We present some new results of Soft rough topology including SR -interior, SR -closure, SR -frontier, SR -neighbourhood and SR -limit point.

Theorem 4.1. Let ( V , τ SR ( Y ) , E ) be a SR -topological space over V w.r.t Y , A and B are SR -sets over V . Then

Int SR ( ∅ ) = ∅ and Int SR ( V ) = V ;

Proof. (i) and (ii) are obvious. (iii) First, suppose that Int SR ( A ) = A . Since Int SR ( A ) is a SR -open set, it follows that A is SR -open set. For the converse, if A is a SR -open set, then the largest SR -open set that is contained in A is A itself. Thus, Int SR ( A ) = A . (iv) Since Int SR ( A ) is a SR -open set, by part (iii) we get Int SR ( Int SR ( A ) ) = Int SR ( A ) . (v) Suppose that A ⊆ B . By (ii) Int SR ( A ) ⊆ A . Then Int SR ( A ) ⊆ B . Since Int SR ( A ) is SR -open set contained by B . So by definition of SR -interior Int SR ( A ) ⊆ Int SR ( B ) . (vi) By using (ii) Int SR ( A ) ⊆ A and Int SR ( B ) ⊆ B . Then Int SR ( A ) ∪ Int SR ( B ) ⊆ A ∪ B . Since Int SR ( A ) ∪ Int SR ( B ) is a SR -open, it follows that Int SR ( A ) ∪ Int SR ( B ) ⊆ Int SR ( A ∪ B ) . (vii) By using (ii) Int SR ( A ) ⊆ A and Int SR ( B ) ⊆ B . Then Int SR ( A ) ∩ Int SR ( B ) ⊆ A ∩ B . Since Int SR ( A ) ∩ Int SR ( B ) is SR -open, it follows that Int SR ( A ) ∩ Int SR ( B ) ⊆ Int SR ( A ∩ B ) . For the converse, A ∩ B ⊆ A also A ∩ B ⊆ B . Then, Int SR ( A ∩ B ) ⊆ Int SR ( A ) and Int SR ( A ∩ B ) ⊆ Int SR ( B ) . Hence Int SR ( A ∩ B ) ⊆ Int SR ( A ) ∩ Int SR ( B ) .□

Definition 4.2. [4] Let ( V , τ SR ( Y ) , E ) be a SR -topological space w.r.t Y , where Y ⊆ V . Let B V then SR -closure of B is defined to be intersection of all SR -closed supersets of B and is denoted by Cl SR ( B ) .

Example 4.3. Consider the SR -topology given in Example 1, taking B={}. Then Int SR ( B ) = ∅ and Cl SR ( B ) = { ϑ 1 , ϑ 3 , ϑ 5 } .

Theorem 4.4. Let ( V , τ SR ( Y ) , E ) be a SR -topological space over V w.r.t Y , A and B are SR -sets over V . Then

Cl SR ( ∅ ) = ∅ and Cl SR ( V ) = V ;

Proof. (i) and (ii) are straightforward. (iii) First consider A = Cl SR ( A ) . Since Cl SR ( A ) is a SR -closed set, so A is a SR -closed set over V . For the converse, suppose that A be a SR -closed set over V . Then A is SR -closed superset of A . So that A = Cl SR ( A ) . (iv) By definition Cl SR ( A ) is always SR -closed set. Therefore by part (iii) we have Cl SR ( Cl SR ( A ) ) = Cl SR ( A ) . (v) Let A ⊆ B . By (ii) B ⊆ Cl SR ( B ) . Then A ⊆ Cl SR ( B ) . Since Cl SR ( B ) is a SR -closed superset of A , it follows that Cl SR ( A ) ⊆ Cl SR ( B ) . (vi) Since A ⊆ A ∪ B and B ⊆ A ∪ B , by part (v), Cl SR ( A ) ⊆ Cl SR ( A ∪ B ) and Cl SR ( B ) ⊆ Cl SR ( A ∪ B ) . Hence Cl SR ( A ) ∪ Cl SR ( B ) ⊆ Cl SR ( A ∪ B ) . For the converse, suppose that A ⊆ Cl SR ( A ) and B ⊆ Cl SR ( B ) . Then A ∪ B ⊆ Cl SR ( A ) ∪ CL SR ( B ) . Since Cl SR ( A ) ∪ Cl SR ( B ) is a SR -closed superset of A ∪ B . Thus, Cl SR ( A ∪ B ) = Cl SR ( A ) ∪ Cl SR ( B ) . (vii) Since A ∩ B ⊆ A and A ∩ B ⊆ B , by part(5) Cl SR ( A ∩ B ) ⊆ Cl SR ( A ) and Cl SR ( A ∩ B ) ⊆ Cl SR ( B ) . Thus, we obtain Cl SR ( A ∩ B ) ⊆ Cl SR ( A ) ∩ Cl SR ( B ) .□

Definition 4.5. Let ( V , τ SR ( Y ) , E ) be a SR -topological space w.r.t Y , where Y ⊆ V . Let B ⊆ V then SR -frontier or SR -boundary of B is denoted by F r SR ( B ) or b SR ( B ) and mathematically defined as F r SR ( B ) = Cl SR ( B ) ∩ Cl SR ( B c ) . Clearly SR -frontier F r SR ( B ) is a SR -closed set. In simple words, we can say that a point x is in SR -frontier of B if and only if every SR -open set V containing x also contains a point of B and a point of B c .

Example 4.6. Consider the SR -topology given in Example 1, taking B={} then B={}. Following the definition SR -closure, we have Cl SR ( B ) = { ϑ 1 , ϑ 3 , ϑ 5 } and Cl SR ( B c ) = V ,then F r SR ( B ) = Cl SR ( B ) ∩ Cl SR ( B c ) = { ϑ 1 , ϑ 3 , ϑ 5 } ∩ V = { ϑ 1 , ϑ 3 , ϑ 5 } .

Theorem 4.7. Let A be a subset of SR -topological space ( V , τ SR ( Y ) , E ) . Then

Cl SR ( A ) = A ∪ F r SR ( A ) ;

Proof. (i) For any subset A of V , V = A ∪ A c ⊆ A ∪ Cl SR ( A c ) ⊆ V . Then, we have V = A ∪ Cl SR ( A c ) . Hence, we get A ∪ F r SR ( A ) = A ∪ ( Cl SR ( A ) ∩ Cl SR ( A c ) ) = ( A ∪ Cl SR ( A ) ) ∩ ( A ∪ Cl SR ( A c ) ) = Cl SR ( A ) ∩ V = Cl SR ( A ) as required. (ii) We have A \ F r SR ( A ) = A \ ( Cl SR ( A ) ∩ Cl SR ( A c ) ) = ( A \ Cl SR ( A ) ) ∪ ( A \ Cl SR ( A c ) ) = ∅ ∪ ( A \ Cl SR ( A c ) ) = A \ Cl SR ( A c ) = Int SR ( A ) as required.

Corollary 4.8.

For any subset A of ( V , τ SR ( Y ) , E ) , A is SR -open if and only if A ∩ F r SR ( A ) = ∅ .

Proof. (i) and (ii) are obvious. (iii) Suppose that F r SR ( A ) = ∅ . We have Cl SR ( A ) = A ∪ F r SR ( A ) and Int SR ( A ) = A \ F r SR ( A ) . This implies that Int SR ( A ) = A and Cl SR ( A ) = A . Hence A is both SR -open and SR -closed. For the converse, suppose A is both SR -open and SR -closed. Then Int SR ( A ) = A and Cl SR ( A ) = A . Thus, we get A = A \ F r SR ( A ) and A = A ∪ F r SR ( A ) . It is only possible if F r SR ( A ) = ∅ .

Definition 4.9. Let ( V , τ SR ( Y ) , E ) be a SR -topological space. A subset N of V is said to be SR -neighborhood of x ∈ V if there exist a SR -open set W x containing x so that x ∈ W x ⊆ N .

Example 4.10. Consider the SR -topology given in Example 1. For ϑ 6 ∈ V , the SR -open sets containing ϑ ₆ are {ϑ ₁, ϑ ₂, ϑ ₃, ϑ ₄, ϑ ₆}, {ϑ ₁, ϑ ₃, ϑ ₆} and V . The supersets of {ϑ ₁, ϑ ₂, ϑ ₃, ϑ ₄, ϑ ₆} are {ϑ ₁, ϑ ₂, ϑ ₃, ϑ ₄, ϑ ₆} and V . The supersets of {ϑ ₁, ϑ ₃, ϑ ₆} are {ϑ ₁, ϑ ₂, ϑ ₃, ϑ ₆}, {ϑ ₁, ϑ ₃, ϑ ₄, ϑ ₆}, {ϑ ₁, ϑ ₃, ϑ ₅, ϑ ₆}, {ϑ ₁, ϑ ₂, ϑ ₃, ϑ ₄, ϑ ₆}, {ϑ ₁, ϑ ₂, ϑ ₃, ϑ ₅, ϑ ₆}, {ϑ ₁, ϑ ₃, ϑ ₄, ϑ ₅, ϑ ₆} and V . Hence the class of SR -neighborhoods of ϑ ₆ is N _SR (ϑ ₆) = {{ϑ ₁, ϑ ₃, ϑ ₆}, {ϑ ₁, ϑ ₂, ϑ ₃, ϑ ₆}, {ϑ ₁, ϑ ₃, ϑ 5 , ϑ 6 } , { ϑ 1 , ϑ 2 , ϑ 3 , ϑ 4 , ϑ 6 } , { ϑ 1 , ϑ 2 , ϑ 3 , ϑ 5 , ϑ 6 , } , { ϑ 1 , ϑ 3 , ϑ 4 , ϑ 5 , ϑ 6 } , V } .

Definition 4.11. Let ( V , τ SR ( Y ) , E ) be a SR -topological space. A point x of V is said to be a SR -limit point of A if every SR -open set W containing x contains a point of A different from x, i.e., W ∩ ( A \ { x } ) ≠ ∅ .

Definition 4.12. The set of all the SR -limit points of A is known as SR -derived set of A and is denoted by A SR d .

Theorem 4.13. Let A be a subset of ( V , τ SR ( Y ) , E ) . Then

Cl SR ( A ) = A ∪ A SR d : Thus, the Cl _SR (A) of A consists of points of A and its SR -limit points.

Proof. The proof is straightforward.□

5 SR -sets in multi-criteria group decision-making

This section presents the use of SR -sets in object assessment and multi-criteria group decision making. Let V = { α 1 , α 2 , α 3 , . . . , α m } is the set of objects under observation, E is the set of criterions to evaluate the objects in V . Let A = { ξ 1 , ξ 2 , ξ 3 , . . . , ξ n } ⊆ E . We consider a soft set G = ( T , A ) for real world problems, it is convenient to to consider a full soft set over V . Let S = { D 1 , D 2 , D 3 , . . . , D n } be the set consisting of decision makers who evaluate the objects to identify the optimal solution and X _i be the initial assessment derived by experts D _i which is represented by the soft set Ω = ( ω , S ) . To extract more accurate results we find SR -approximation of initial assessment results X _i according to soft approximation space P = ( V , G ) , consequently we get two soft sets Ω ★ = ( ω ★ , S ) and Ω ★ = ( ω ★ , S ) . Following these soft sets define fuzzy sets ν _{Ω ★} (α _k ), ν _Ω ( α _k ) and ν _Ω ^★ (α _k ) defined as: ν Ω ★ ( α k ) = 1 m ∑ ι = 1 m C ω ★ D ι ( α k ) , ν Ω ( α k ) = 1 m ∑ ι = 1 m C ω D ι ( α k ) , ν Ω ★ ( α k ) = 1 m ∑ ι = 1 m C ω ★ D ι ( α k ) . Soft set and fuzzy results are then combined. Suppose that C = { L , M , H } is the set of parameters which represents ‘low satisfaction’, ‘medium satisfaction’ and ‘high satisfaction’. Define fuzzy soft set R ( β , C ) over V as β ( L ) = ν Ω ★ , β ( M ) = ν Ω , and β ( H ) = ν Ω ★ . Calculate the choice value c i corresponding to each object α _i as: c i = ∑ j α ij . At the end we are able to select the optimal alternative having maximum choice value c i . Algorithm 1 The scheme of the algorithm is given as: Input Step-1: Write the soft set G = ( T , A ) which describes the given data. Step-2: Based on initial assessment results of the group of analyst S , define a soft set. Step-3: Obtain SR -approximations in the form of soft sets Ω ★ = ( ω ★ , S ) and Ω ★ = ( ω ★ , S ) . Step-4: Define fuzzy sets ν _{Ω ★}, ν _Ω and ν _Ω ^★ corresponding to the soft sets Ω ★ = ( ω ★ , S ) , Ω = ( ω , S ) and Ω ★ = ( ω ★ , S ) . Step-5: Characterize the satisfaction level of experts in the form of parameter set C = { L , M , H } . Step-6: Define fuzzy soft set R ( β , C ) over V using fuzzy sets ν _{Ω ★}, ν _Ω and ν _Ω ^★. Output Step-7: Calculate choice value c i for each object. Choose the object having maximum choice value.

OPEN IN VIEWER

Fig. 1.

Graphical representation of Algorithm 1.

Example 5.1. Consider a set of Football players V = { ϱ 1 , ϱ 2 , ϱ 3 , ϱ 4 , ϱ 5 , ϱ 6 } for the selection in the national football team. Let A = { ξ 1 , ξ 2 , ξ 3 , ξ 4 } ⊆ E be the set of qualities considered by the selection team where ξ ₁ represents “physical fitness”, ξ ₂ represents “confidence”, ξ ₃ represents “skill application under pressure” and ξ ₄ represents “sportsman spirit”. Construct a soft set G = ( T , A ) which specify the abilities of players given in Table 3.

Following these SR -approximations, we get two soft sets Ω ★ = ( ω ★ , S ) and Ω ★ = ( ω ★ , S ) where, ω ★ ( D i ) = R P ★ ( Xi ) and ω ★ ( D i ) = R P ★ ( Xi ) . Tabular representation of these soft sets are given in Tables 5 and 6.

Now, we define fuzzy soft set ν _{Ω ★} (ϱ _k ), ν _Ω (ϱ _k ) and ν _Ω ^★ (ϱ _k ) as follows: ν Ω ★ ( ϱ k ) = 1 3 ∑ i = 1 3 C ω ★ D i ( P k ) , ν Ω ( ϱ k ) = 1 3 ∑ i = 1 3 C ω D i ( P k ) , ν Ω ★ ( ϱ k ) = 1 3 ∑ i = 1 3 C ω ★ D i ( P k ) .

Thus, we have ν Ω ★ ( ϱ k ) = { ( ϱ 1 , 0 ) , ( ϱ 2 , 1 / 3 ) , ( ϱ 3 , 2 / 3 ) , ( ϱ 4 , 1 / 3 ) , ( ϱ 5 , 1 / 3 ) , ( ϱ 6 , 1 / 3 ) } , ν Ω ( ϱ k ) = { ( ϱ 1 , 1 / 3 ) , ( ϱ 2 , 1 / 3 ) , ( ϱ 3 , 2 / 3 ) , ( ϱ 4 , 1 / 3 ) , ( ϱ 5 , 1 / 3 ) , ( ϱ 6 , 1 / 3 ) } , ν Ω ★ ( ϱ k ) = { ( ϱ 1 , 2 / 3 ) , ( ϱ 2 , 1 / 3 ) , ( ϱ 3 , 2 / 3 ) , ( ϱ 4 , 1 / 3 ) , ( ϱ 5 , 2 / 3 ) , ( ϱ 6 , 2 / 3 ) } .

Let C = { L , M , H } be the set of parameters by experts which represents ‘low satisfaction’, ‘medium satisfaction’ and ‘high satisfaction’. Then we get the fuzzy soft set R ( β , C ) over V by setting β ( L ) = ν Ω ★ , β ( M ) = ν Ω and β ( H ) = ν Ω ★ . Calculating choice value corresponding to each player. Fuzzy soft set R ( β , C ) with assessment values is given in Table 7.

We can arrange all the alternatives according to their choice evaluation values: ϱ 3 ≻ ϱ 5 ≈ ϱ 6 ≻ ϱ 4 ≻ ϱ 1 ≻ ϱ 2 . Thus, ϱ ₃ is the player to be selected for the national football team.

Algorithm 2 The scheme of the algorithm is given as: Step-1: Write the soft set G = ( T , A ) which describes the given data.Step-2: Based on initial assessment results of the group of analyst S , define a soft set. Step-3: Obtain SR -approximations in the form of soft sets Ω ★ = ( ω ★ , S ) and Ω ★ = ( ω ★ , S ) . Step-4: Find choice value for all selected soft sets Ω ★ = ( ω ★ , S ) , Ω = ( ω , S ) and Ω ★ = ( ω ★ , S ) . Step-5: Find the decision set by adding all the choice values of obtained soft sets. Step-6: Characterize the satisfaction level of experts in the form of parameter set C = { L , M , H } .In put the weighting vector W = (w _L , w _M , w _H ) and compute the weighted evaluation value for each object. Step-7: Find the decision set by adding all the weighted values ∑ _i w _i .Choose the object having maximum value.

OPEN IN VIEWER

Fig. 2.

Graphical representation of Algorithm 2.

Example 5.2. Consider Example 1. First three steps same as done by algorithm 1. Find choice value for all selected soft sets Ω ★ = ( ω ★ , S ) , Ω = ( ω , S ) and Ω ★ = ( ω ★ , S ) . Now, we find the decision table by adding choice values for each player.

Let C = { L , M , H } be the set of parameters by experts which represents ‘low satisfaction’, ‘medium satisfaction’ and ‘high satisfaction’. Assume that the weighting vector W = (w _L , w _M , w _H ) = (.2, . 3, . 5). Calculate weighted choice value for each player. Find the decision set by adding all the weighted values ∑ _i w _i .

Choose the player having maximum final weighted choice value.

we can arrange all the players according to their choice evaluation values: ϱ 3 ≻ ϱ 5 ≈ ϱ 6 ≻ ϱ 4 ≻ ϱ 1 ≻ ϱ 2 . Thus, ϱ ₃ is the player to be selected for the national football team.

Remark

It is important to mention that both algorithms as given above give the same optimal result.

The idea of core in the selection of attributes to the rough set was given by Thivagar in [30]. In the next definition we extend this idea of core to the SR -set.

Definition 6.1. Suppose V as set of objects, S = ( T , A ) is the soft set and P = ( V , S ) is the the corresponding soft approximation space. Let R be an indiscernibility relation. Let τ SR be a SR -topology on V and β _SR be the basis defined for τ SR . Let N be the subset of A , is said to be core of R if β N ≠ β SR - ( s ) for each ’s’ in N . i.e. a core of R is the subset of attributes with the condition that if we remove any element from N it will affect the classification power of the attributes. Algorithm: Input Step-1: Consider a universe of discourse (set of objects) V , a set of attributes E which can be classified into two categories D of decision attributes and C of condition attributes. Consider an indiscernibility relation R on V . Construct the soft set in tabular form corresponding to C condition attributes and a subset Y of V . The columns indicate the objects, rows indicate the attributes and entries of table give attribute values. Output Step-2: Classify set Y and find the SR - approximation subsets ( R P ★ ( Y ) , R P ★ ( Y ) and Bd P ( Y ) ) w.r.t R . Step-3: Define Soft Rough Topology τ R on V and find basis β _SR . Step-4: By removing an attribute x from C , find again the SR -approximation subsets ( R P ★ ( Y ) , R P ★ ( Y ) , Bd P ( Y ) ) w.r.t R on C - ( x ) . Step-5: Generate SR -topology τ SR - ( x ) on V ,define its basis β SR - ( x ) . Step-6: Repeat step 4 and step 5 for each attribute in C . Step-7: The attributes for which β SR - ( x ) ≠ β SR gives the core ( R ) .