International Journal of Trend in Scientific Research and Development (IJTSRD) Volume 5 Issue 6, September-October 2021 Available Online: www.ijtsrd.com e-ISSN: 2456 – 6470 @ IJTSRD | Unique Paper ID – IJTSRD47531 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 220 Concept on Rough Soft Set and Its Application in Decision Making Manamohan Maharana 1 ; Dr. Debadutta Mohanty 2 1Department of Mathematics, M.P.C. College, Baripada, Odisha, India 2Department of Mathematics, Seemanta Mahavidyalaya, Jharpokharia, Mayurbhanj, Odisha, India ABSTRACT The notion of full rough soft set, upper Soft set and lower Soft set are introduced here in a different manner. Various properties on rough soft set are studied with examples. We define rough soft subset and rough soft equal set with examples. Moreover rough soft equality relation are introduced and properties are presented. Finally measure of roughness of soft set is defined and an algorithm is presented to solve decision making problem by the application of rough soft set. KEYWORDS: rough set, soft set, rough soft set, upper soft set and lower soft set, rough soft subset, rough soft equity relation, measure of roughness How to cite this paper: Manamohan Maharana | Dr. Debadutta Mohanty "Concept on Rough Soft Set and Its Application in Decision Making" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-5 | Issue-6, October 2021, pp.220-226, URL: www.ijtsrd.com/papers/ijtsrd47531.pdf Copyright © 2021 by author (s) and International Journal of Trend in Scientific Research and Development Journal. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (CC BY 4.0) (http://creativecommons.org/licenses/by/4.0) 1. INTRODUCTION Methods in classical mathematics are not successful to solve various types of uncertainties problems in economics, engineering, environmental science and social science. Probability theory, fuzzy set theory (Zedah 1965), rough set the-ory (Pawlak 1982) and other mathematical tools are often useful approaches to solve uncertainties. In 1999 Molodtov introduced the concepts of Soft set based on parameters for dealing with uncertainties. Now work on soft set theory are progressing rapidly. Several operations on soft set and theoretical study on the theory of soft set are introduced (Maji et al. 2003). Ali et al (2009) introduced some new operations on soft set and the notion of complement of soft set. Soft set theory, fuzzy set theory and Rough set theory are all mathe- matical tools to deal with uncertainty or vagueness. Maji (2012) introduced the notion of reduct soft set and described the application of soft set theory to a decision making problem using Rough set. Malodtsov (2004) has shown several applications of soft set in many eld like economics, engineering, medical sciences etc. The soft set theory now become a very good source of research for many scholars of mathematics and computer science due to its wide range of applicability. Several interesting new concepts such as Rough soft set, Soft rough set are emerged by combining Rough set and soft set. Many researchers has shown several application of this theory in many elds like economics, engineering, medical sciences etc. In the present work the lower soft set, upper soft set, lower and upper rough approximation operators on soft set are introduced in a different approach. The notion of Rough soft set is introduced in a different approach and few results are investigated in this context. We de ne rough soft subset and rough soft equal set with examples. Rough soft equality relation are studied and properties are presented. Measure of roughness of soft set is de ned and an algorithm is presented to solve decision making problem by the application of rough soft set. The paper is completed with some concluding remarks. 2. Preliminaries: Difinition 2. 1 (Ali et al 2009):A soft set is a pair (F, A)of non empty nite set U and A where U be an universe of objects, A is a subset of pa-rameter set E IJTSRD47531 International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47531 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 221 ⊆ ⊆ → and F is a function from A to power set of U. For each x 2 A, F (x) is considered as the set of x- approximate elements in the soft set. we denote FA= (F, A) Soft Subset: Let (F,A) and (G,B)be two soft set over U, A,B⊆E;(F,A) is a subset of (G,B)lf (i)A ⊆ B (ii)F(e)⊆ G(A) ∀ e ϵ A Symbolically written as (F,A) ⊆ (G,B) Equal Soft set:(F,A) and (G,B)be two soft set over U, A,B ⊆E;(F,A) is equal to (G,B), symbolically written as (F,A) =(G,B)lf (F,A) ⊆(G,B) and (G,B)⊆ (F,A) Null Soft set: A soft set (F,A) over U is called a null soft set denoted by (φ,A)=φA if for each xϵ A, F(x)=φ Whole Soft set: A soft set (F,A) over U is called a whole soft set denoted by (U,A)=UA if for each x ϵ A, F(x)=U Definition: The restricted Intersection of (F,A) and (G,B)denoted by (F,A)∩R(G,B) =(F,A)∩(G,B) is defined as the soft set (H,C), where C=A∩B and H(x)=F(x)∩G(x) ∀xϵ C Definition: The restricted union of (F,A) and (G,B)denoted by (F,A) ∪ R(G,B) =(F,A) ∪(G,B) is defined as the soft set (H,C), where C=A B and H(x)=F(x)∪G(x) ∀ xϵC Equivalence relation: A relation R on a nonempty set U is said to be an equivalence relation if for x, y.z ϵ U 1. R is reflexive i.e. (x,x)ϵR ∀ xϵ U 2. R is symmetric i.e. (x,y)ϵR ⇒ (y,x)ϵR 3. R is transitive i.e. (x,y)ϵR;(y,z)ϵR ⇒ (x,z)ϵR Equivalence class: The equivalence class of an element x ϵ U with respect to the equivalence relation R, denoted by [x]R= { yϵ U| yRx } Definition: Let R be an equivalence relation on U, U be a non empty finite set (Universe).U/R denote the family of equivalence class of R. For any arbitrary set X⊆U and X is the union of some equivalence classes. We approximate X by a pair of Lower and Upper approximations RX = {x ϵ U | [x]R ⊆ X} RX= { xϵ U|[x]R ∩ X≠Ф } RX and RX are called R-lower and R-upper approximations of X with respect to R. If RX =RX we say X is R-definable We say X is rough with respect to R iff RX ≠ RX The boundary region of X with respect to R denoted by BRX=RX - RX it is clear that RX⊆ X ⊆ RX 3. Rough soft set: Let U be the set of objects called universe and (F,A) be a soft set over U where A is the set of parameters and F: A →P(U) is a mapping. If R be an equivalence relation on U and (U,R)be a pawlak approximation space w.r.to A. Define two soft sets FA and FA , where FA: A→P(U),FA: A→P(U)called lower soft set and upper soft set over U as FA(x) = R(F (x))= {y ϵ U | [y]R ⊆ F (x)} FA(x) = R(F (x))= {y ϵ U | [y]R ∩ F (x) ≠ Φ} ∀ xϵ A Definition: We say (F,A), a rough soft set with respect to parameter x ϵA if FA(x) ≠F¯ A(x) (F, A) is said to be definable or crip soft set with respect to parameter x ϵ A if FA(x) =F¯ A(x). (F, A) is said to be a full rough soft set or rough soft set if F_A(x) ≠F¯ A(x) ∀ xϵA and denote rough soft set w.r.to e as RSFA(e) (F, A) is said to be a full rough soft set or rough Example 3.1 Table 1 contains some observed symptoms of 6 patients suffering from certain diseases. here U = {p1, p2, ....p6} ,condition attributes = temperature, muscular pain, blood from mouth , decision attributes = dengue and R be an equivalence relation ’diagnosis of diseases’. Table 1 patient temp. muscular pain blood from mouth dengue p1 normal no yes yes p2 high yes yes yes p3 high yes yes yes p4 normal no yes no p5 normal yes no yes p6 high no no no IND(R)={{p1, p4}, {p2, p3}, {p5}, {p6}} [p1]R = {p1, p4} = [p4]R,[p2]R = {p2, p3} = [p3]R,[p5]R = {p5},[p6]R = {p6} (F,A)be a soft set representing system of patients given below ,where A={e1, e2, e3} and e1 denotes temp.,e2 denotes muscular pain,e3 denotes blood from mouth, F:A → P(U) such that International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47531 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 222 F(e1) ={p1, p3, p4},F(e2) ={p2, p3, p5}, F(e3) ={p1, p2, p3, p4} FA(e1)={p1, p4} ,FA(e1)={p1, p2, p3, p4} FA(e2)={p2, p3, p5} ,FA(e2)={p2, p3, p5} FA(e3)={p1, p2, p3, p4} ,FA(e3)={p1, p2, p3, p4} (F,A) is a rough soft set with respect to parameter e1 but not e2 and e3 Definition: Let (F,A) and (G,B) be two rough soft sets. (G,B) is said to be rough soft subset of (F,A) written as RSGB ⊆ RSFA if (1)B ⊆ A (2)GB(b)⊆ FA(b) ,GB(b)⊆FA(b) bϵ B Defination: Let (F,A) and (G,B) be two rough soft sets, are said to be equal written as RSGB = RSFA if RSGB ⊆ RSFA and RSFA ⊆ RSGB Example:3.2 In example 3.1 suppose A={e1, e2, e3} , F:A −→ P(U) defined F(e1) ={p1, p3, p4, p5},F(e2) ={p2, p3, p4, p6}, F(e3) ={p1, p3, p4} FA(e1)={p1, p4, p5} ,FA(e1)={p1, p2, p3, p4, p5} FA(e2)={p2, p3, p6} ,FA(e2)={p1p2, p3, p4, , p6} FA(e3)= p1, p4 ,FA(e3)= p1, p2, p3, p4 (F,A) is a rough soft set suppose B={e2, e3} , G:B → P(U)defined by G(e2) ={p1, p2, p3},G(e3) ={p1, p3, p4} GB (e2)={p2, p3} ,GB(e2)={p1, p2, p3, p4} GB (e3)={p1, p4} ,GB(e3)={p1, p2, p3, p4} (G,B) is also a rough soft set and B ⊆ A GB(e2) ⊆ FA(e2) ,GB(e2) ⊆ FA(e2) GB(e3) ⊆ FA(e3) ,GB(e3) ⊆ FA(e3) so RSGB(e) ⊆ RSFA(e) ∀ e⊆ B Proposition 3.1:If (G,B) is soft subset of (F,A)then RSGB ⊆ RSFA Proof: (G,B) is soft subset of (F,A) so (1)B ⊆ A (2)G(b)⊆ F(b)∀b ⊆ B ⇒ GB(b) ⊆ FA(b) ,GB(b) ⊆ FA(b) ∀ b ϵ B Therefore RSGB(e) ⊆ RSFA(e) ∀ e ϵ B Example:3.3 In example 3.1 suppose A={e1, e2, e3} , F: A → P(U) defined F(e1) ={p1, p3, p4, p5},F(e2) ={p2, p3, p4, , p6}, F(e3) ={p1, p3, p4} suppose B={e2, e3} , G:B → P(U)defined by G(e2) ={p2, p3, p4},G(e3) ={p3, p4} here (G,B) ⊆ (F,A) ∀ e ϵ B FA(e2)={p2, p3, p6} ,FA(e2)={p1p2, p3, p4, p6} GB (e2)={p2, p3} ,GB(e2)={p1, p2, p3, p4} FA(e3)={p1, p4} ,FA(e3)={p1, p2, p3, p4} GB (e3)=φ ,GB(e3)={p1, p2, p3, p4} GB (e2) ⊆ FA(e2) ;GB(e2)⊆ FA(e2) GB (e3) ⊆ FA(e3) ;GB(e3)⊆ FA(e3) so RSGB(e)⊆ RSFA(e) eϵ B Definition: (Measure of roughness of soft set) Let (F,A) be a soft set over U and (U,R) be a pawlak approximation space w.r.to A. Measure of roughness of (F,A) with respect to parameter e ϵ A denoted by RFA(e)=| FA(e) | / | FA(e) | where 0 ≤ RFA(e) ≤ 1 Example:3.4 In example 3.1 RFA(e1)=1/2 ;RFA(e2)=1 ;RFA(e3)=1 Proposition 3.2: If FA= (F,A) is a soft set over the universe X and R be an equivalence relation on X then 1. FA⊆ FA⊆ F A 2. φA=φA=φ¯ A 3. XA=X A=XA Proof: i) let yϵ FA(x) =R(F (x)) ⇒ [y]R ⊆ F(x) ∀xϵ A ⇒ [y]R ∩ F (x) = [y]R ≠ φ ⇒ y ϵ R(F (x)) R(F (x)) ⊆ R(F (x)) let yϵ R(F (x)) ⇒ [y]R ⊆ F(x) ⇒ yϵ F(x) R(F (x))⊆ F(x) ∀∈A let yϵ F(x) ⇒ y ϵ [y]R ∩ F (x) ⇒ [y]R ∩ F (x) ≠ φ ⇒ y ϵ R(F (x)) F(x)⊆ R(F (x)) R(F (x))⊆ F(x)⊆ R(F (x)) xϵA R(φ(x)) ⊆ φ(x) = φ by (i) but φ(x)⊆ R(φ(x)) R(φ(x)) = φ(x) φA=φ A International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47531 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 223 ∪ ∩ ƒ ∩ ∪ Assume R¯ (φ(x)) ≠ φ(x) then there exist y such that yϵ R(φ(x)) so [y]R φ(x) ≠ φ but [y]R φ = φ which contradict the assumption therefore R(φ(x)) = φ(x) φA=φ A iii) By (i)R(X(x)) ⊆ X(x) let y⊆ X(x) [y]R⊆ X y⊆ R(X(x)) X(x) ⊆ R(X(x)) so R(X(x))=X(x) By (i) X(x)⊆ R(X(x)) R(X(x))⊆ X(x) obvious because X is the universal set so X(x) =R(X(x)) therefore XA=X A=XA Proposition 3.3 If FA and GA are two soft set, ∪ and ∩ denote restricted union and restricted intersection then 1. FA ∪ GA =FA ∪ GA 2. FA ∩ GA =FA ∩ GA Proof: i) let y⊆ R((F ∪ G)(x)) ⇔ [y]R ∩ (F ∪ G)(x) ≠ φ ⇔ ([y]R ∩ F (x)) ∪ ([y]R ∩ G(x)) ≠ φ ⇔ {[y]R ∩ F (x) ≠ φ}or{[y]R ∩ G(x) ≠ φ} ⇔ {y ϵ R(F (x))}or {y ϵ R(G(x))} ⇔ y ϵ R(F (x)) ∪ R(G(x)) R((F ∪ G)(x))=R(F (x)) ∪ R(G(x)) FA ∪ GA =FA ∪ GA ii) let yϵ R((F ∩ G)(x)) ⇔ [y]R ⊆ (F ∩ G)(x) ⇔ {[y]R ⊆ F (x)}and{([y]R ⊆ G(x))} ⇔ {y ϵ R(F (x))}and{y ϵ R(G(x))} ⇔ y ϵ R(F (x)) ∩ R(G(x)) R((F ∩ G)(x))=R(F (x)) ∩ R(G(x)) FA ∩ GA =FA ∩ GA Proposition 3.4 If FA and GA are two soft set such that FA⊆ GA then 1. FA ⊆ GA 2. F A ⊆ GA Proof:i]FA⊆ GA ⇒ FA ∩ GA = FA ⇒ FA ∩ GA = FA ⇒ FA ∩ GA = FA by Proposition 3.2 (i) FA ⊆ GA ii) FA⊆ GA ⇒ FA ∪ GA = GA ⇒ FA ∪ GA = GA ⇒ FA ∪ GA = GA by Proposition 3.2 (ii) FA ⊆ GA Proposition 3.5 If FA and GA are two soft set, and denote restricted union and restricted intersection then 1. FA ∪ GA ⊇ FA ∪ GA 2. FA ∩ GA ⊆ FA ∩ GA Proof: i] since FA ⊆ FA ∪ GA ⇒ FA ⊆ FA ∪ GA by Proposition 3.3 (i) GA ⊆ FA ∪ GA ⇒ GA ⊆ FA ∪ GA by Proposition 3.3 (i) hence FA ∪ GA ⊆ FA ∪ GA (ii) since FA ∩ GA ⊆ FA ⇒ FA ∩ GA ⊆ FA by Proposition 3.3 (ii) FA ∩ GA ⊆ GA ⇒ FA ∩ GA ⊆ GA by Proposition 3.3 (ii) hence FA ∩ GA ⊆ FA ∩ GA Proposition 3.6 If FA= (F,A) is a soft set over the universe U and R be an equivalence relation on U then 1. FA=FA 2. (ii) F A=FA 3. (iii)F A=FA 4. (iv)FA=FA Proof: i) FA ⊆ FA by Proposition 3.1 let yϵ FA = R(F (x)) ⇒ [y]R ⊆ F (x) ⇒ R[y]R ⊆ R(F (x)) ⇒ [y]R ⊆ R(F (x)) ⇒ y ϵ R(R(F (x))) ⇒ y ϵ FA FA ⊆ F¯ A hence FA=FA (ii)F¯ A ⊆ F¯ A¯ by Proposition 3.1 let y ⊆ F¯ A¯ = R¯ (R¯ (F (x))) ⇒ [y] ∩ R(F (x)) ≠ φ and for y ϵ [y]R and y ϵ R(F (x)) hence [y]R ∩ F (x) ≠ φ ⇒ y ϵ FA FA ⊆ FA (iv)FA ⊆ FA by Proposition 3.1 let y ⊆ FA = R(F (x)) ⇒ [y]R ∩ F (x)≠φ if z ϵ [y]R then [z]R ∩ F (x) = [y]R ∩ F (x) ≠φ ⇒ z ϵ R(F (x)) hence [y]R ⊆ R(F (x)) ⇒ y ϵ R(R(F (x))) ⇒ y ϵ FA International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47531 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 224 ≈ ⇒ FA ⊆ FA therefore FA=FA 4. Definition :( Rough soft equal relation) Let (F, A) be a soft set over U and (U,R) be a pawlak approximation space w.r.to A, we define F (e1) ∼A F (e2) iff FA(e1) =FA(e2) i.e .R(F (e1)) =R(F (e2)) where e1 ,e2 ϵ A F (e1) A F (e2) iff FA(e1) =FA(e2) i.e. R(F (e1)) =R(F (e2)) where e1 ,e2 ϵ A F (e1) A F (e2) iff FA(e1) =FA(e2) and FA(e1) =FA(e2) The above binary relation are called lower rough soft equal relation, upper rough soft equal relation and rough soft equal relation respectively. Proposition 4.1The above binary relation are all equivalence relation over A Exammple: 4.1 In example 3.1 (F, A)be a soft set representing system of patients given below ,where A={e1, e2, e3} and e1 denotes temp.,e2 denotes muscular pain,e3 denotes blood from mouth, F:A → P(U) such that F(e1) ={p1, p3, p4},F(e2) ={p1, p2, p4} FA(e1)={p1, p4} ;FA(e2)={p1, p4} So F(e1) ∼A F(e2) FA(e1)= { p1, p2, p3, p4 ;FA(e2)= {p1, p2, p3, p4 so F(e1) A F(e2) hence F(e1) ≈A F(e2) Proposition 4.2Let FA be a rough soft set with respect to parameter e ϵA then 1. F(e1) A F(e2) iff F(e1) A F(e1)∪ F(e2) A F(e2) 2. F(e1) ∼A F(e2) iff F(e1) ∼A F(e1)∩ F(e2) ∼A F(e2) 3. F(e1) A F(e2) ; F(e3) A F(e4)⇒ F(e1)∪ F(e3) A F(e2)∪ F(e4) 4. (iv)F(e1) ∼A F(e2) ; F(e3) ∼A F(e4)⇒ F(e1)∩ F(e3) ∼A F(e2)∩ F(e4) (v)F(e1) A F(e2) ⇒F(e1) ∪(U- F(e2)) A U 5. F(e1) ∼A F(e2) ⇒F(e1) ∩(U-F(e2)) ∼A φ 6. F(e1) ⊆ F(e2) ; F(e2) A φ ⇒F(e1) A φ 7. F(e1) ⊆ F(e2) ; F(e2) ∼A φ ⇒F(e1) ∼A φ 8. F(e1) ⊆ F(e2) ; F(e1) A U ⇒F(e2) A U 9. F(e1) ⊆ F(e2) ; F(e1) ∼A U ⇒F(e2) ∼A U where e1,e2,e3,e4 A and F(e1),F(e2),F(e3),F(e4) U Proof:(i) F(e1) A F(e2) ⇒FA(e1) =FA(e2) i.e.R(F (e1)) =R(F (e2)) R(F (e1) ∪ F (e2))=R(F (e1)) ∪ R(F (e2))=R(F (e1))=R(F (e2)) F(e1) A F(e1)∪ F(e2) A F(e2) ii) F(e1) ∼A F(e2) ⇒FA(e1) =FA(e2) i.e .R(F (e1)) =R(F (e2)) R(F (e1) ∩ F (e2))=R(F (e1)) ∩ R(F (e2))=R(F (e1))=R(F (e2)) F(e1) ∼A F(e1)∩ F(e2) ∼A F(e2) iii) F(e1) A F(e2) ; F(e3) A F(e4) ⇒R(F (e1)) =R(F (e2)) ;R(F (e3)) =R(F (e4)) R(F (e1) ∪ F (e3))=R(F (e1)) ∪ R(F (e3))=R(F (e2)) ∪ R(F (e4))=R(F (e2) ∪ F(e4)) F(e1)∪ F(e3) A F(e2)∪ F(e4) iv)F(e1) ∼A F(e2) ; F(e3) ∼A F(e4) ⇒R(F (e1)) =R(F (e2)) ;R(F (e3)) =R(F (e4)) R(F (e1) ∩ F (e3))=R(F (e1)) ∩ R(F (e3))=R(F (e2)) ∩ R(F (e4))=R(F (e2) ∩ F (e4)) F(e1) F(e3) A F(e2) F(e4) v)F(e1) A F(e2) ⇒ R(F (e1) =R(F (e2)) R(F (e1) ∪ (U − (F (e2)) =R(F (e1) ∪R(U − (F (e2)) but U=F(e2) ∪( U-F(e2)) R(U )=R(F (e2) ∪ (U − F (e2)))=R(F (e2)) ∪ R(U − F (e2)) so R(F (e1) ∪ (U − F (e2)))=R(F (e1)) ∪ R(U − F (e2))=R(F (e2)) ∪ R(U − F (e2)) =R (U) F(e1) ∪(U-F(e2)) A U vi) F(e1) ∼A F(e2) ⇒ R(F (e1) =R(F (e2)) but φ=F(e2)∩( U-F(e2)) R(φ)=R(F (e2) ∩ (U − F (e2)))=R(F (e2)) ∩ R(U − F (e2)) so R(F (e1) ∩ (U − F (e2)))=R(F (e1)) ∩ R(U − F (e2))=R(F (e2)) ∩ R(U − F (e2))=R(φ) F(e1) ∩(U-F(e2)) ∼A φ vii) F(e1) ⊆ F(e2) ; F(e2) A φ i.e.R(F (e2) =R(φ) ⇒ R(F (e1) ⊆ R(F (e2))=R(φ) but φ ⊆ F (e1) ⇒ R(φ) ⊆ R(F (e1) therefore R(F (e1)=R(φ) F (e1) A φ viii)similarly as per (vii) ix)F(e1) ⊆ F(e2)⇒ R(F (e1) ⊆ R(F (e2)) F(e1) AU ⇒ R(F (e1)=R(U ) ⇒ R(F (e1) ⊆ R(F (e2)) International Journal of Trend in Scientific Research and Development @ www.ijtsrd.com eISSN: 2456-6470 @ IJTSRD | Unique Paper ID – IJTSRD47531 | Volume – 5 | Issue – 6 | Sep-Oct 2021 Page 225 ⇒ ⇒ { } { } ⇒ R(U ) ⊆ R¯ (F (e2)) but F(e2) ⊆U R(F (e2))⊆R(U ) therefore R(F (e2)) = R(U ) F(e2) AU x)similarly as per (ix) 5. Application of rough soft sets in decision making: Let U be the set of objects called universe ,R be an equivalence relation on U and (U,R) be a pawlak approximation space w.r.to A where A is the set of parameters. Suppose (F.A) be a soft set over U and F is a mapping from A to P (U) The decision algorithm for rough soft set as follows 1. Input pawlak approximation space (U, R) 2. Input soft set (F, A) 3. compute FA(ei) and FA(ei) for each ei ϵ A 4. compute measure of roughness of (F,A)w. r. to parameter ei ϵ A where RFA (ei) =| FA(ei) | / | FA(ei) | 5. Find maximum value RFA(ek) of RFA(ei) where RFA(ek)= Max RFA(ei) 6)The decision is F(ek) Example: 5 A scientist detected five materials denoted by e1, e2, e3, e4, e5.After experiment he comes to know all the materials contain one or more molecules out of molecules m1, m2, m3, m4, m5, m6, m7.He want to choice the closest to the molecular structures out of five materials. According to chemical proper- ties he observed, some molecules are equivalent. Suppose A= {e1, e2, e3, e4, e5} and U={ m1, m2, m3, m4, m5, m6, m7} Let R denote the molecules whose chemical properties are equivalent .According to scientist, R={{m1, m3}; {m2, m4, m6}; {m5, m7}} [m1]R={m1,m3}=[m3]R [m2]R={m2,m4,m6}=[m4]R=[m6]R [m5]R= {m5, m7 }=[m7]R Consider (U, R) be the pawlak approximation space where R is an equivalence relation on U .Each kind of materials containing molecules F (e1) = {m1, m3, m4} F (e2) = {m1, m2, m4} F (e3) = {m3, m4, m6} F (e4) = {m2, m5, m7} F (e5) = {m1, m3, m5, m6, m7} respectively Consider (F, A) denote soft set on U defined on above FA(e1)={m1, m3} FA(e2)=φ FA(e3)=φ FA(e4)={m5, m7} FA(e5)={m1, m3, m5, m7} FA(e1)={m1, m2, m3, m4, m6} FA(e2)={m1, m2, m3, m4, m6} FA(e3)={m1, m2, m3, m4, m6} FA(e4)={m2, m4, m5, m6, m7} FA(e5)={m1, m2, m3, m4, m5, m6, m7} RFA(e1)=| FA(e1) | / | F¯ A(e1) | =2/5=.4 RFA(e2)=0/5=0 RFA(e3)=0/5=0 RFA(e4)=2/5=.4 RFA(e5)=4/7=.57 Thus maximum value of RFA(ei) is RFA(e5) that means F(e5)is the anticipated material. i.e. e5 is the closest one in all of the materials. Remarks: 1. It is a new decision making method for rough soft sets. 2. In this method we will get which is the best parameter of given soft set (F,A) 3. We obtain the most expected material on soft set w.r.t. an equivalence relation on the universe U in example 5 In this method, decision makers have adapted various criterions in order to reach the conformation suggesting actual situation. 4. I hope this method may be outfit for our real life decisions making. Remarks: The decision parameter carries distinct importance to various people. It is a difficult task to solve decision making problem. To obtain decision making parameters in a decision making problem, we can use application of rough soft sets. By using above algorithm, we will obtain the key parameter which is necessary for decision making. In this method decision maker will select the goods/candidates/materials by using the key parameters. 6. 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