AN APPLICATION OF INTERVAL-

VALUED FUZZY SOFT SETS IN

MEDICAL DIAGNOSIS

Guide:Dr. Sunil Jacob John Jobish VD

M090054MA

Contents.

1. Preliminaries.

2. Application of interval valued fuzzy

soft set in medical diagnosis.

3. Algorithm.

4. Case Study.

1. Preliminaries.

Definition 1.1[3]:

Let U - initial universe set

E - set of parameters.

P (U) - power set of U. and,

A - non-empty subset of E.

A pair (F, A) is called a soft set over U,

where F is a mapping given by F: A P (U).

Example 1.1;

Let U={c1,c2,c3} - set of three cars.

E ={costly(e1), metallic color (e2), cheap (e3)}

- set of parameters. A={e1,e2} ⊂ E. Then;

(F,A)={F(e1)={c1,c2,c3},F(e2)={c1,c3}}

“ attractiveness of the cars” which Mr. X is going to buy .

Definition 1.2[3]:

Let U - universal set,

E - set of parameters and A ⊂ E.

Let F (U) - set of all fuzzy subsets of U.

Then a pair (F,A) is called fuzzy soft set over

U, where F :A F (U).

Example 1.2;

Let U = {c1,c2,c3} - set of three cars.

E ={costly(e1),metallic color(e2) , getup (e3)}

- set of parameters.

A={e1,e2 } ⊂ E.

Then;

(G,A) = { G(e1)={c1/.6, c2/.4, c3/.3},

G(e2)={c1/.5, c2/.7, c3/.8} }.

- fuzzy soft set over U.

Describes the “ attractiveness of the cars” which

Mr. S want.

.

Definition 1.3[3]: An interval-valued fuzzy

sets X on the universe U is a mapping

such that;

X : U → Int ([0,1]).

where; Int ([0,1]) - all closed sub-intervals

of [0,1].

The set of all interval-valued fuzzy sets on U is

denoted by F (U).

.1)()(0

X x tomembership of degreeupper )(

X x tomembership of degreelower )(

X toelement x an of

-membership of degree T he)](),([)(

),(~ˆ

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Definition 1.7 [4]:

Let U universal set.

E set of parameters.

and A ⊂E.

set of all interval-valued fuzzy sets on

U.

Then a pair (F, A) is called interval-valued fuzzy

soft set over U.

where F : A

)(~

UF

).(~

UF

Definition 1.8[5]: The complement of a

interval valued fuzzy soft set (F,A) is,

(F,A)C = (FC,¬A),

where ∀α ∈ A ,¬α = not α .

FC: ¬A F ( U ).

FC(β ) = (F ( ¬β ))C , ∀β ∈ ¬A

Example2.3:

Let U={c1,c2,c3} set of three cars.

E ={costly(e1), grey color(e2),mileage (e3)},

set of parameters.

A={e1,e2} ⊂ E. Then,

(G,A) = {

G(e1)=⟨c1,[.6,.9]⟩,⟨c2,[.4,.6]⟩,⟨c3,[.3,.5]⟩,

G(e2)= ⟨c1,[.5,.7]⟩, ⟨c2,[.7,.9]⟩ ⟨c3,[.6,.9]⟩

}

“ attractiveness of the cars” which Mr. X want.

Example 2.4:

In example 2.3,

(G,A)C = {

G(¬e1)=⟨c1,[0.1,0.4]⟩, ⟨c2,[0.4,0.6]⟩,⟨c3,[0.5,0.7]⟩,

G(¬e2)=⟨c1,[0.3,0.5]⟩, ⟨c2,[0.1,0.3]⟩⟨c3,[0.1,0.4]⟩

}.

2. Application –

in

medical diagnosis.

S - Symptoms, D – Diseases, and P - Patients.

Construct an I-V fuzzy soft set (F,D) over S

F:D→

A relation matrix say, R1 - symptom-disease

matrix- constructed from (F,D).

Its complement (F,D)c gives R2 - non

symptom-disease matrix.

We construct another I-V fuzzy soft set (F1,S)

over P, F1:S→

).(~

SF

).(~

PF

We construct another I-V fuzzy soft set (F1,S)

over P, F1:S→

Relation matrix Q - patient-symptom matrix-

from (F1,S).

Then matrices,

T1=Q R1 - symptom-patient matrix, and

T2= Q R2 - non symptom-patient matrix.

).(~

PF

The membership values are calculated by,

)},,(),({ b

)},,,(),({

],[),(

1

1

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inf

kjR

U

jiQ

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)},,(),({ y

)},,,(),({

],[),(

2

2

2

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inf

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The membership values are calculated by,

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)},(),({

),(

11

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jiTU

ijTL

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)},(),({

),(

22

22

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dpdps

tsdpS

3. Algorithm.

1. Input the interval valued fuzzy soft sets (F,D)

and (F,D)c over the sets S of symptoms, where

D -set of diseases.

2. Write the soft medical knowledge R1 and R2

representing the relation matrices of the

IVFSS (F,D) and (F,D)c respectively.

3. Input the IVFSS (F1,S) over the set P of

patients and write its relation matrix Q.

4. Compute the relation matrices T1=Q R1 and

T2=Q R2.

5. Compute the diagnosis scores ST1 and ST2

6. Find Sk= maxj { ST1 (pi , dj) ─ ST2 (pi,┐dj)}.

Then we conclude that the patient pi is

suffering from the disease dk.

4. Case Study.

Patients - p1, p2 and p3.

Symptoms (S) - Temperature, Headache, Cough

and Stomach problem

S={ e1,e2,e3,e4} as universal set.

D ={d1,d2}.

d1 - viral fever, and

d2 - malaria.

Suppose that,

F(d1) ={ ⟨e1, [0.7,1]⟩, ⟨e2, [0.1,0.4]⟩,

⟨e3, [0.5,0.6]⟩, ⟨e4,[0.2,0.4]⟩) }.

F(d2) ={ ⟨e1,[0.6,0.9] ⟩, ⟨e2,[0.4,0.6] ⟩,

⟨e3,[0.3,0.6] ⟩, ⟨e4,[0.8, 1] ⟩ }.

IVFSS - (F,D) is a parameterized family

={ F(d1), F(d2) }.

IVFSS - (F,D) can be represented by a relation

matrix R1 - symptom-disease matrix- given by,

R1 d1 d2

e1 [0.7, 1.0 ] [ 0.6, 0.9 ]

e2 [0.1, 0.4 ] [0.4, 0.6 ]

e3 [0.5, 0.6 ] [0.3, 0.6 ]

e4 [0.2, 0.4 ] [0.8, 1.0 ]

The IVFSS - (F, D)c also can be represented by

a relation matrix R2, - non symptom-disease

matrix, given by-

R2 d1 d2

e1 [0 , 0.3 ] [ 0.1, 0.4 ]

e2 [0.6, 0.9 ] [0.4, 0.6 ]

e3 [0.4, 0.5 ] [0.4, 0.7 ]

e4 [0.6, 0.8 ] [0 , 0.2 ]

We take P = { p1, p2, p3} - universal set .

S = { e1, e2, e3, e4} - parameters.

Suppose that,

F1(e1)={⟨p1, [.6, .9]⟩, ⟨p2, [.3,.5]⟩,⟨p3, [.6,.8]⟩},

F1(e2)={ ⟨p1, [.3,.5] ⟩, ⟨p2, [.3,.7] ⟩, ⟨p3, [.2,.6] ⟩},

F1(e3)={⟨p1, [.8, 1]⟩, ⟨p2, [.2,.4]⟩,⟨p3, [.5,.7]⟩} and

F1(e4)={⟨p1, [.6,.9] ⟩,⟨p2, [.3,.5] ⟩, ⟨p3, [.2,.5] ⟩},

IVFSS - (F1,S) is a parameterized family

={ F1(e1), F1(e2), F1(e3), F1(e4) }.

gives a collection of approximate

description of the patient-symptoms in the

hospital.

Q e1 e2 e3 e4

p1 [0.6, 0.9] [0.3, 0.5] [0.8, 1] [0.6, 0.9]

p2 [0.3, 0.5] [0.3, 0.7] [0.2, 0.4] [0.3, 0.5]

p3 [0.6, 0.8] [0.2, 0.6] [0.5, 0.7] [0.2, 0.5]

(F1,S) - represents a relation a relation matrix

Q - patient-symptom matrix - given by;

Combining the relation matrices R1 and R2

separately with Q. we get,

T1=Q o R1 - patient-disease matrix.

T2=Q o R2 - patient-non disease -

matrix.

T1 d1 d2

p1 [0.1 ,0.9] [0.3 ,0.9]

p2 [0.1 ,0.5] [0.2 ,0.6]

p3 [0.1 ,0.8] [0.2 ,0.8]

T2 d1 d2

p1 [0 , 0.8] [0 , 0.7]

p2 [0 , 0.7] [0 , 0.6]

p3 [0 , 0.6] [0 , 0.7]

ST1-ST2 d1 d2

p1 0.2 0.6

p2 -0.7 -0.4

p3 0.5 -0.1

Now we calculate,

The patient p3 is suffering from the disease d1.

Patients p1 and p2 are both suffering from

disease d2.

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