Presentation in EPAM

1. “Soft* IF Rough Approximation Space and its Application on medical field”
Sharmistha Bhattacharya (Halder)
Dept. of Mathematics
Tripura University
And
Piyali Debnath
Dept. of Mathematics
Tripura University

2. Introduction
In 1965 Zadeh [14] introduced the concept of fuzzy set. This set contains only a membership function lying between 0 and 1.
Atanassov in 1986 [2] introduced the concept of IF set. Atanassov named it intuitionistic fuzzy set. But nowadays a problem arose due
to the already introduced concept of intuitionistic logic. Hence, instead of intuitionistic fuzzy set, throughout this paper we are using
the nomenclature IF set. In 1999, Molodtsov [7] introduced the novel concept of soft sets, which is a new mathematical approach to
vagueness. In recent years the researchers have contributed a lot towards fuzzification of soft set theory. In 2011, Neog and Sut [8] put
forward a new notion of complement of a soft set and accordingly some important results have been studied in their work.
Theory of rough sets is presented by Pawlak [9]. In order to handle vagueness and imprecision in the data equivalence relations
play an important role in this theory. It is well known that an equivalence relation on a set partitions the set into disjoint classes and
vice versa. In general a subset of a set is not definable; however it can be approximated by two definable subsets called lower and
upper approximations of the set.
Data mining is one of the areas in which rough set is widely used. Data mining is the process of automatically searching large
volumes of data for patterns using tools such as classifications, association, rule mining, and clustering. The rough set theory is well
understood format framework for building data mining models in the form of logic rules on the bases of which it is possible to issue
predictions that allow classifying new cases.
In general whenever data are collected they are linguistic variables. Not only this, the answers are not always in Yes/No form. So,
in this case to deal with such type of data IF set is a very important tool. Data are in most of the cases a relation between object and
attribute. Soft set is an important tool to deal with such types of data. B. Davvaz and S. Bhattacharya (Halder) [2] in 2011 introduce
the concept of IF soft rough approximation space which is very useful in data mining and decision making processes. In this paper our
aim to introduce the concept of the soft* IF lower and upper rough approximation and some of its properties are studied. Here we also
show an application on soft* IF lower and upper rough approximation using matlab programming.

3. 1. Preliminaries
In the current section we will briefly recall the notions of soft set, rough sets, soft rough approximation spaces etc.
Definition 2.1:-[1] Let X be a nonempty set, and let I be the unit interval [0, 1]. According to [1], an IF set U is an
object having the form
{ , ( ), ( ) : }
U U
U x x x x X
 
    , where the functions U
 : X → [0, 1] and U
 : X → [0, 1] denote,
respectively, the degree of membership and the degree of non- membership of each element x ∈ X to the set U, and
0 ( ) ( ) 1
U U
x x
 
   for each x ∈ X.
Lemma 2.2:- [1] Let X be a nonempty set and let IFS‘s U and V be in the following forms:
{ , ( ), ( ) : }
U U
U x x x x X
 
    ,
{ , ( ), ( ) : }
V V
V x x x x X
 
   
(1) { , ( ), ( ) : }
C
U U
U x x x x X
 
    ,
(2) { , ( ) ( ), ( ) ( ) : }
U V U V
U V x x x x x x X
   
       ,
(3) { , ( ) ( ), ( ) ( ) : }
U V U V
U V x x x x x x X
   
       ,
(4) ~
0 { , 0,1 : }
x x X
    , ~
1 { ,1, 0 : }
x x X
    ,

4. (5) ( )
C C
U U
 , ~ ~
0 1
C
 , ~ ~
1 0
C

Definition 2.3:-[9] Let U be a finite nonempty set, called universe and R an equivalence relation on U, called
indiscernibility relation. The pair (U, R) is called an approximation space. By R(x) we mean that the set of all y
such that xRy, that is, R(x) = [x]R is containing the element x. Let X be a subset of U. We want to characterize
the set X with respect to R. According to Pawlak‘s paper [9], the lower approximation of a set X with respect to
R is the set of all objects, which surely belong to X, that is, R*(X) = {x : R(x) ⊆ X}, and the upper
approximation of X with respect to R is the set of all objects, which are partially belonging to X, that is, R*
(X) =
{x : R(x) ∩ X φ}. For an approximation space (U, θ), by a rough approximation in (U, θ) we mean a mapping
Apr : (U) (U) (U) defined by for every X ∈ (U).
Apr(X) = *
*
( ( ), ( ))
R X R X
Given an approximation space (U, R), a pair (A, B) ∈ (U) (U) is called a rough set in
(U, R) if (A, B) = Apr(X) for some X ∈ (U).
Fuzzy set is defined by employing the fuzzy membership function, whereas rough set is defined by
approximations. The difference of the upper and the lower approximation is a boundary region. Any rough set has
a nonempty boundary region whereas any crisp set has an empty boundary region. The lower approximation is
called interior, and the upper approximation is called closure of the set.

5. Definition 2.4 [5]:- Let U be a universal set and let E be a set of parameters. According to [5], a pair (F, A) is
called a soft set over U, where A ⊆ E and F: A (U) the power set of U, is a set-valued mapping.
Let (U, R) be a Pawlak approximation space. For a fuzzy set μ ∈ F(U), the lower and upper rough
approximations of μ in (U, R) are denoted by ( )
R  and ( )
R  , respectively, which are fuzzy sets defined by
( )( ) { ( ) : [ ] }
R
R x y y x
 
   ,
( )( ) { ( ) : [ ] }
R
R x y y x
 
   ,
for all x ∈ U. The operators R and R are called the lower and upper rough approximation operators on fuzzy sets.
If R = R the fuzzy set μ is said to be definable, otherwise μ is called a rough fuzzy set.
Definition 2.5 [5]:- A soft set (F, A) over U is called a full soft set if ( )
a A
F a U

  . Let (F,A) be a full soft set over
U, and let S = (U, E) be a soft approximation space. For a fuzzy set μ ∈ F(U) the lower and upper soft rough
approximations of μ with respect to S are denoted by ( )
S
sap  and ( )
S
sap  respectively, which are fuzzy sets in U
given by
( )( ) { ( ) : ,{ , } ( )},
S
sap x y a A x y F a
 
    
( )( ) { ( ) : ,{ , } ( )},
S
sap x y a A x y F a
 
    
for all x ∈ U. The operators ( )
S
sap  and ( )
S
sap  are called the lower and upper soft rough approximation operator
on fuzzy sets. If both the operators are the same then μ is said to be soft definable, otherwise μ is said to be soft
rough fuzzy set.

6. Definition.2.6 [7] Let  
,
F A
  be a full soft set [A soft set  
,
F A overU is called a full soft set if  
a A
f a U


 ] over
U and  
,
S U
  be a soft approximation space. For a IF set  
, F U
 
   the lower and upper soft rough
approximations of ,
 
  with respect to S are denoted by ,
S
sap  
  and ,
S
sap  
  respectively which are fuzzy
sets in U given by
       
 
 
, ( , ) : ,
S
sa p x y a A x y F a
   
        
       
 
 
, ( , ) : ,
S
sa p x y a A x y F a
   
        
x U
  . The operators S
sap and S
sap are called the lower and upper soft rough approximation operators on IF sets
respectively. If    
S
S
sap sap
 
 then ,
 
  is called soft definable; otherwise ,
 
  is called a soft rough IF set.

7. Definition 2.7: [4] Let  
,
F A
  be a fuzzy soft set over U . The pair  
,
SF U
  is called a soft fuzzy
approximation space. For a fuzzy set  
F U
  the lower and upper soft fuzzy rough approximations of  with
respect to S F are denoted by  
SF
A pr  and  
SF
A pr  respectively which are fuzzy sets in U given by
       
     
   
 
1 1
S F
a A y U
A p r x F a x F a y y
 
 
 
 
    
 
 
 
 
 
             
 
S F
a A y U
A p r x F a x F a y y
 
 
 
 
  
 
 
 
 
 
x U
  . The operators SF
A pr and SF
A pr are called the lower and upper soft fuzzy rough approximation operators on
fuzzy sets respectively. If    
SF
SF
A pr A pr
 
 then  is called soft fuzzy definable; otherwise  is called a soft
fuzzy rough set.

8. On soft* IF rough approximation space
In this section, we introduce the concept of the lower and upper soft* IF rough approximations. Some of its
properties are studied and examples are presented. The main focus of this paper is to show the scope of this newly
introduced concept in the field of data mining and decision making.
Definition 3.1. Let  
,
F A
  be a IF soft set over U . The pair  
,
SF U
  is called a soft IF approximation space.
For a IF set ,
 
  the lower and upper soft IF rough approximations of ,
 
  with respect to S F are denoted by
 
,
SF
A pr   and  
,
SF
A pr   respectively which are IF sets in U given by
     
 
*
, ( ) ( ) ( ) ( ) ,
C C
S F
a A y U
A p r x F a x F a y y
   
 
 
 
      
 
 
 
 
 
             
 
*
, ,
S F
a A y U
A p r x F a x F a y y
   
 
 
 
      
 
 
 
 
 
x U
  . The operators *
SF
A pr and
*
SF
A pr are called the lower and upper soft IF rough approximation operators on IF
sets respectively. If    
*
*
, ,
S F
S F
A pr A pr
   
     then ,
 
  is called soft IF definable; otherwise ,
 
  is called a
soft fuzzy rough set.

9. Example 3.2. Suppose that U is the set of dresses under consideration; E is the set of parameters where each
parameter is a fuzzy word or a sentence involving fuzzy words, E= {cotton, woolen, synthetic, beautiful, cheap and
expensive}. The IF soft set (F, E) describes ―attractiveness of the dresses‖ which Mr. X(say) is going to buy.
Suppose that there are five dresses in the universe U given that 1 2 3 4 5
{ , , , , }
U d d d d d
 and 1 2 3 4 5
{ , , , , }
E e e e e e

where 1
e stands for the parameter ‗cotton‘, 2
e stands for the parameter ‗woolen‘, 3
e stands for the parameter
‗synthetic‘, 4
e stands for the parameter ‗cheap‘, 5
e stands for the parameter ‗expensive‘.
Suppose that, 1
1 2 3 4 5
0 .5, 0 .3 0 .1, 0 .4 0 .4, 0 .3 0 .2, 0 .6 0 .3, 0 .5
( ) , , , ,
F e
d d d d d
 
         
  
 
,
2
1 2 3 4 5
0.1, 0.2 0.3, 0.5 0.5, 0.2 0.3, 0.4 0.7, 0.2
( ) , , , ,
F e
d d d d d
 
         
  
 
3
1 2 3 4 5
0 .2, 0 .6 0 .4, 0 .3 0 .6, 0 .1 0 .9, 0 0 .5, 0 .3
( ) , , , ,
F e
d d d d d
 
         
  
 
4
1 2 3 4 5
0.3, 0.5 0.4, 0.3 0.2, 0.7 0.8, 0.1 0.6, 0.2
( ) , , , ,
F e
d d d d d
 
         
  
 
5
1 2 3 4 5
0.7, 0.1 0.6, 0.3 0.4, 0.5 0.3, 0.2 0.8, 0.1
( ) , , , ,
F e
d d d d d
 
         
  
 
1 2 3 4 5
0.4, 0.3 0.6, 0.1 0.2, 0.5 0.7, 0.3 0.1, 0.4
, , , , ,
d d d d d
 
 
         
    
 

10. The IF soft set (F, E) is a parameterized family {F (ei), i = 1,2,….8} of all IF sets of the set U and gives us
a collection of approximate description of an object.
Now let us consider
1 2 3 4 5
0 .2, 0 .3 0 .4, 0 .5 0 .1, 0 .4 0 .8, 0 .2 0 .5, 0 .2
, , , , ,
d d d d d
 
 
         
    
 
*
1 2 3 4 5
0 .2, 0 .5 0 .3, 0 .5 0 .1, 0 .4 0, 0 .4 0 .2, 0 .5
, , , , ,
S F
A p r
d d d d d
 
 
         
    
 
*
1 2 3 4 5
0 .3, 0 .3 0 .5, 0 .3 0 .6, 0 .2 0 .8, 0 .2 0 .6, 0 .2
, , , , ,
S F
A p r
d d d d d
 
 
         
    
 
Theorem 3.3. Let  
,
F A
  be a IF soft set overU ,  
,
SF U
  be a soft fuzzy approximation space and ,
 
  ,
,
 
  be two IF sets. Then we have
(i) If * *
, , ( , ) ( , )
SF SF
A pr A pr
       
          
(ii) If
* *
, , ( , ) ( , )
SF SF
A pr A pr
       
          
(iii)
* * *
( , , ) ( , ) ( , )
SF SF SF
A pr A pr A pr
       
          
(iv) * * *
( , , ) ( , ) ( , )
SF SF SF
A pr A pr A pr
       
          
(v)
* * *
( , , ) ( , ) ( , )
SF SF SF
A pr A pr A pr
       
          
(vi) * * *
( , , ) ( , ) ( , )
SF SF SF
A pr A pr A pr
       
          

11. (vii)  
*
*
( , ) ( , )
C
C
S F
S F
A p r A p r
   
     and  
* *
( , ) ( , )
C
C
SF SF
A pr A pr
   
    
Definition 3.4: Let  
,
F A
  be an soft IF set over the universe U. The pair  
,
S U
  is called a soft IF
approximation space. For an IF set ,
 
  the soft* IF lower and upper rough approximations of ,
 
  with respect
to S are denoted by  
 
*
,
S
A S   and  
 
*
,
S
A S   respectively which are IF sets in U and are defined by
     
 
*
, ( )( ) ( ) ( ) ,
C
S
a A y U
A S x F a x F a y y
   
 
 
 
      
 
 
 
 
 
             
 
*
, ,
C
S
a A y U
A S x F a x F a y y
   
 
 
 
      
 
 
 
 
 
x U
  . The operators *
S
A S and
*
S
A S are called the so ft

IF lower and upper rough approximation operators on IF
sets respectively. If  
   
 
*
*
, ,
S
S
A S A S
   
 then ,
 
  is called soft IF definable, otherwise ,
 
  is called a soft
IF rough set.

12. An Application of decision making by soft* IF rough approximation space
In this section, we approach to decision making problem based on the soft* IF rough set model. This
approach will using the data information provided by decision making problem only and does not need any
additional available information provided by decision making makers or other ways. So, it can avoid the effect of
the subjective information for the decision results. Therefore, the results could be more objectively and also could
the paradox results for the same decision problem since the effect of the subjective factors by the different experts.
3.1Steps for decision making based on soft* IF rough sets:
We first construct an normal decision object E on the evaluation universe A.
1
m ax ( )
:
A
i
i
i i
F e
E e A
e

 
 , i.e.  
( ) m ax ( ) ( ) :
i i j j
A e F e u u U
 
We define a new fuzzy subset E of the parameter set by maximum operator for the IF sub set
( )
( ) : ,
i j
i i j
i
F e u
F e e A u U
e
 
  
 
 
, 1, 2,......,
j U
 of the universe U.
We calculate the soft* IF lower rough approximation  
*
,
S
A s  
  and soft* IF upper rough approximation
 
*
,
S
A s  
  of the optimum (ideal) normal decision object A. On the other hand, the lower and upper rough
approximation are two most close to the approximated set of the universe. Therefore, we obtain two most close
values    
*
,
S i
A s x
 
  and    
*
,
S i
A s x
 
  to the decision alternative i
x U
 of the universe U by the soft* IF rough
lower and upper approximations of the IF subset A.

13. So, we redefine the choice value i
 , which is used by the existing decision making based soft IF set, for the
decision alternative i
x U
 as follows:
       
*
*
, , ,
S
S
i i i i
A s x A s x x
    
       U
   
 
*
*
( ), ( ) : , , , ,
S
S
i j i j i i j j
A s and A s
           
          
Finally, taking the object i
x  U in universe U with the maximum membership value and minimum non
membership value of i
 as the optimum decision for the given decision making problem.
B. Sun [11] gave an application of soft fuzzy rough in decision making problems. In this paper, we used so ft

IF rough approximation space and aim to obtain the optimal choice for applying biopsy to the patients with prostate
cancer risk by using the PSA, fPSA, PV and age data of patients. We determine the risk of prostate cancer. Our aim
is to help the doctor to determine whether the patient needs biopsy or not. We collect the data from S. Yuksel‘s
[13] paper. They choose 78 patients from Selcuk University Medicine Faculty with prostate complaint as the data.
Out of 78 patient we choose 10 patient for checking that whether they need to biopsy.
Let us consider the universe 1 5 2 1 3 0 4 6 5 1 5 4 7 1 7 4 7 8
{ , , , , , , , , }
U u u u u u u u u u u
 and 𝐴 = {PSA, fPSA, PV, Age} the
parameter set. Now the parameterized subsets of the universe are given. The patients whose PSA in blood is 50 and
higher than 50, fPSA is 12 and bigger than 12, PV is 20 and bigger than 20, and age is 54 and older than 54 are
chosen with doctor‘s suggestion. We generate the soft IF set G = (𝐹, 𝐴) which is based on PSA, fPSA, PV, and age
values of patients over 𝑈 (Table 1).

14. U P S A fPSA P V age
1
u 76 17 30 65
5
u 39 7 48 64
2 1
u 39 9 52 68
3 0
u 27 7 28 51
4 6
u 88 19 37 77
5 1
u 46 12 62 71
5 4
u 42 10 59 80
7 1
u 52 12 35 65
7 4
u 51 12 78 67
7 8
u 41 13 79 80
Table 1 The input PSA, fPSA, PV and age value of several patient.

15. Let us consider the soft IF set  
,
  which is based on PSA, fPSA, PV and age values of patients over 𝑈:
 
0.8, 0.1 0.6, 0.3 0.7, 0.2 0.5, 0.3
, , , ,
P SA fP SA P V age
 
 
       
  
 
3.2Soft* IF rough approximation space algorithm:
In this subsection, we present the algorithm for the approach to the decision making problem based on
soft* IF rough set model.
Let  
,
F A
  be a IF soft set over the universe of the discourse U , the pair  
,
S U
  is a soft IF approximation
space and ,
 
  be an IF set. Then we present the decision algorithm for the soft* IF rough set as follows:
1. Input the soft IF set  
,
F A .
2. Compute the optimum (or ideal) normal decision object E .
3. Compute the soft* IF rough lower approximation *
( , )
S
A s  
  and the soft IF upper rough approximation
*
( , )
S
A s  
  .
4. Compute the choice value i

5. The decision K
x U
 if m ax , 1, 2, ...,
k i
i
i U
 
  .

16. 6. If K has more than one value, then any of K
x may be choice.
U P S A fPSA P V age
1
u 0 .8 , 0 .1
  0.8, 0.2
  0.5, 0.4
  0.6, 0.2
 
5
u 0.2, 0.7
  0.1, 0.7
  0.7, 0.3
  0.6, 0.3
 
2 1
u 0.2, 0.7
  0.2, 0.7
  0 .7 , 0 .1
  0 .7 , 0 .1
 
3 0
u 0.1, 0.8
  0.1, 0.8
  0.5, 0.4
  0.1, 0.7
 
4 6
u 0 .9 , 0
  0 .9 , 0
  0.6, 0.3
  0 .8 , 0 .1
 
5 1
u 0.5, 0.4
  0.5, 0.3
  0 .8 , 0 .1
  0.8, 0.2
 
5 4
u 0.4, 0.5
  0.5, 0.5
  0.8, 0.2
  0 .9 , 0 .1
 
7 1
u 0 .7 , 0 .1
  0.6, 0.2
  0.6, 0.2
  0.6, 0.2
 
7 4
u 0.7, 0.2
  0.6, 0.3
  0 .9 , 0 .1
  0.7, 0.2
 
7 8
u 0.3, 0.6
  0 .7 , 0 .1
  0 .9 , 0
  0 .9 , 0
 
Table 2: Tabular representation of the soft IF set of ten patients.

17. U P S A fPSA P V age
*
S
A S
*
S
A S Choice
value
i

1
u 0 .8 , 0 .1
  0.8, 0.2
  0.5, 0.4
  0.6, 0.2
  0 .8 , 0 .1
  0.5, 0.3
  1 .3, 0 .2
 
5
u 0.2, 0.7
  0.1, 0.7
  0.7, 0.3
  0.6, 0.3
  0.7, 0.3
  0.5, 0.3
  1 .2, 0
 
2 1
u 0.2, 0.7
  0.2, 0.7
  0 .7 , 0 .1
  0 .7 , 0 .1
  0.7, 0.2
  0.5, 0.3
  1.2, 0.1
 
3 0
u 0.1, 0.8
  0.1, 0.8
  0.5, 0.4
  0.1, 0.7
  0.5, 0.4
  0.7, 0.2
  1 .2, 0 .2
 
4 6
u 0 .9 , 0
  0 .9 , 0
  0.6, 0.3
  0 .8 , 0 .1
  0 .8 , 0 .1
  0.5, 0.3
  1 .3, 0 .2
 
5 1
u 0.5, 0.4
  0.5, 0.3
  0 .8 , 0 .1
  0.8, 0.2
  0.7, 0.2
  0.5, 0.3
  1.2, 0.1
 
5 4
u 0.4, 0.5
  0.5, 0.5
  0.8, 0.2
  0 .9 , 0 .1
  0.7, 0.2
  0.5, 0.3
  1.2, 0.1
 
7 1
u 0 .7 , 0 .1
  0.6, 0.2
  0.6, 0.2
  0.6, 0.2
  0 .7 , 0 .1
  0.5, 0.3
  1 .2, 0 .2
 

18. 7 4
u 0.7, 0.2
  0.6, 0.3
  0 .9 , 0 .1
  0.7, 0.2
  0.7, 0.2
  0.5, 0.3
  1.2, 0.1
 
7 8
u 0.3, 0.6
  0 .7 , 0 .1
  0 .9 , 0
  0 .9 , 0
  0.7, 0.2
  0.5, 0.3
  1.2, 0.1
 
A 0 .9 , 0
  0 .9 , 0
  0 .9 , 0
  0 .9 , 0
 
Table 4: Tabular representation of the result of the decision algorithm
By using the algorithm for soft* IF rough set based on decision making presented in this section, we obtain the
results as the Table 4 From the above results table, it is clear that the maximum membership value and minimum
non membership vale of choice value is 1 .2, 0 .2
i
    , scored by 1
u , 4 6
u He is under the highest risk. By the result
another seven patient 5 2 1 3 0 5 1 5 4 7 4
, , , , ,
u u u u u u and 7 8
u have the score 1.2, 0.1
  are also potential cancer patients and
they need biopsy exactly. The other patients are under low risk and they do not need the biopsy

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21. Thank you