Innovative Approaches for N-valued Interval Neutrosophic Sets by Ioan Degău

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Journal of Applied Sciences ISSN 1812-5654 DOI: 10.3923/jas.2017.429.440

Review Article Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis A. Edward Samuel and R. Narmadhagnanam Ramanujan Research Centre, P.G. and Research Department of Mathematics, GAC (A), Kumbakonam, TN, India

Abstract The objective of the study was to find out the relationship between the disease and the symptoms seen within patients and diagnose the disease that impacted the patient using n-valued interval neutrosophic sets. Neoteric methods were devised in n-valued interval neutrosophic sets. Utilization of medical diagnosis was commenced with using prescribed procedures to identify a person suffering from the disease for a considerable period. The result showed that the proposed methods were free from shortcomings that affect the existing methods and found to be more accurate in diagnosing the diseases. It was concluded that the techniques adopted in this study were more reliable and easier to handle medical diagnosis problems. Key words: N-valued interval neutrosophic set, grade function, logarithmic distance, exponential measure, medical diagnosis Citation: A. Edward Samuel and R. Narmadhagnanam, 2017. Innovative approaches for n-valued interval neutrosophic sets and their execution in medical diagnosis. J. Applied Sci., 17: 429-440. Corresponding Author: A. Edward Samuel, Ramanujan Research Centre, P.G. and Research Department of Mathematics, GAC(A), Kumbakonam, TN, India Tel: +918903163860 Copyright: ÂŠ 2017 A. Edward Samuel and R. Narmadhagnanam. This is an open access article distributed under the terms of the creative commons attribution License, which permits unrestricted use, distribution and reproduction in any medium, provided the original author and source are credited. Competing Interest: The authors have declared that no competing interest exists. Data Availability: All relevant data are within the paper and its supporting information files.

J. Applied Sci., 17 (9): 429-440, 2017 By taking one time inspection, there may be error in diagnosis. Hence, multi time inspection, by taking the samples of the same patient at different times gives the best diagnosis. So, n-valued interval neutrosophic sets and their applications play a vital role in medical diagnosis. In 1965, fuzzy set theory was firstly given by Zadeh1 which is applied in many real applications to handle uncertainty. Sometimes membership function itself is uncertain and hard to be defined by a crisp value. So the concept of interval valued fuzzy sets was proposed to capture the uncertainty of grade of membership. Atanassov3 introduced the intuitionistic fuzzy sets which consider both truth-membership and falsity-membership. De et al.4 presented an application of intuitionistic fuzzy set in medical diagnosis. Ye5 introduced the concept of cosine similarity measures for intuitionistic fuzzy sets. Miaoying6 presented the cotangent similarity function for intuitionistic fuzzy sets. Later on, intuitionistic fuzzy sets were extended to the interval valued intuitionistic fuzzy sets. Intuitionistic fuzzy sets and interval valued intuitionistic fuzzy sets can only handle incomplete information not the indeterminate information and inconsistent information which exists commonly in belief systems. So, neutrosophic set (generalization of fuzzy sets, intuitionistic fuzzy sets and so on) defined by Smarandache7 has capability to deal with uncertainty, imprecise, incomplete and inconsistent information which exists in real world from philosophical point of view. Wang et al.8 proposed the single valued neutrosophic set. Similarity and entropy between neutrosophic sets were proposed by Mamjumdar and Samanta9. Wang et al.10 proposed the set theoretic operations on an instance of neutrosophic set is called interval valued neutrosophic set which is more flexible and practical than neutrosophic set. Similarity measures between interval valued neutrosophic sets were proposed by Ye11. Interval valued neutrosophic soft sets were introduced by Deli12. Sebastian and Ramakrishnan13 studied a new concept called fuzzy multi sets (FMS), which is the generalization of multi sets. Shinoj and Sunil14 extended the concept of fuzzy multi sets by introducing intuitionistic fuzzy multi sets (IFMS). Rajarajeswari and Uma15 proposed the normalized hamming similarity measure between them. However, the concepts of FMS and IFMS are not capable of dealing with indeterminacy. Ye and Ye16 introduced the concept of single valued neutrosophic multi sets. Distance based similarity measures between them were introduced by Ye et al.17. Smarandache18 extended the classical neutrosophic logic to n-valued refined neutrosophic logic, by refining each neutrosophic component T, I, F into respectively, T1, T2,â€Ś, Tm, l1, l2,â€Ś, lp and F1, F2,â€Ś, FT. Deli et al.19 studied a new concept called neutrosophic refined

INTRODUCTION Kumbakonam is a thickly populated town. Although underground drainage system is available here, it is yet to cover all the houses in the town. So, open drainage system continues to be in practice in different places of the town. Further this town is racing fast towards total sanitation in all spheres. As a result, Kumbakonam continues to be a repository of all new kinds of diseases. This created an urge to carry out research in the medical field. By introducing innovative methods in the research, the diseases can be diagnosed instantly and infallibly. A number of real life problems in engineering, medical sciences, social sciences, economics etc., involve imprecise data and their solution involves the use of mathematical principles based on uncertainty and imprecision. Such uncertainties are being dealt with the help of topics like probability theory, fuzzy set theory1, rough set theory2 etc., Healthcare industry has been trying to complement the services offered by conventional clinical decision making systems with the integration of fuzzy logic techniques in them. As it is not an easy task for a clinician to derive a fool proof diagnosis, it is advantageous to automate few initial steps of diagnosis which would not require intervention from an expert doctor. Neutrosophic set which is a generalized set possesses all attributes necessary to encode medical knowledge base and capture medical inputs. As medical diagnosis demands large amount of information processing, large portion of which is quantifiable, also intuitive thought process involve rapid unconscious data processing and combines available information by law of average, the whole process offers low intra and inter personal consistency. So contradictions, inconsistency, indeterminacy and fuzziness should be accepted as unavoidable as they are integrated in the behavior of biological systems as well as in their characterization. To model an expert doctor it is imperative that it should not disallow uncertainty as it would be then inapt to capture fuzzy or incomplete knowledge that might lead to the danger of fallacies due to misplaced precision. As medical diagnosis contains lots of uncertainties and increased volume of information available to physicians from new medical technologies, the process of classifying different sets of symptoms under a single name of disease becomes difficult. In some practical situations, there is the possibility of each element having different truth membership, indeterminate and false membership functions. The unique feature of n-valued interval neutrosophic set is that it contains multi truth membership, indeterminate and false membership. 430

J. Applied Sci., 17 (9): 429-440, 2017 sets. Broumi and Deli20 proposed the correlation measure between them. Broumi et al.

  inf T1A  x  ,sup T1A  x   , inf T A2  x  ,sup T A2  x   ,,   ,   x,     inf T pA  x  ,sup T Ap  x         inf I1A  x  ,sup I1A  x   , inf I A2  x  ,sup I A2  x   ,,         , A    p p   inf I A  x  ,sup I A  x       1 1 2 2   inf FA  x  ,sup FA  x   , inf FA  x  ,sup FA  x   ,,    : x  X  p p   inf x ,sup FA  x       FA   

generalize the concept of

n-valued neutrosophic sets to the case of n-valued interval neutrosophic sets. In this study, using the notion of n-valued interval neutrosophic set was provided an exemplary for medical diagnosis. In order to make this, various methods were implemented.

Where: PRELIMINARIES inf T1A  x  ,inf T A2  x  ,,inf T pA  x  , inf I1A  x  ,inf I A2  x  ,,inf I Ap  x 

Interval neutrosophic set10: Let X be a space of points inf F1A  x  ,inf FA2  x  ,,inf FpA  x  , sup T1A  x  ,sup T 2A  x  ,,sup T Ap  x 

(objects), with a generic element in X denoted by x. An interval neutrosophic set A in X is characterized by the

sup I1A  x  ,sup I A2  x  , ,sup I pA  x  , sup F1A  x  ,sup FA2  x  ,,sup FAp  x    0,1

truth-membership function TA, indeterminacy-membership function IA and falsity-membership function FA. For each point x in X, T A  x  , IA  x  , FA  x   0,1 with the condition that 0  T A  x   I A  x   FA  x   3.

Such that: 0  sup T Aj  x   sup I Aj  x   sup FAj  x   3  j  1, 2,3,,p

Interval neutrosophic relation : Let X and Y be two 10

non-empty crisp sets. An interval neutrosophic relation R(X, Y)

Inclusion21: A n-valued interval neutrosophic set A is

is a subset of product space X×Y and is characterized by the

contained in the other n-valued interval neutrosophicset B, denoted by AfB if and only if:

truth membership function TR(x, y), the indeterminacy membership function IR(x, y) and the falsity membership function FR(x, y), where x0X and y0Y and TR(x, y), IR(x, y),

inf T1A  x   inf T1B  x  , inf T A2  x   inf T B2  x  , ,inf T pA  x   inf T Bp  x 

FR(x, y)f[0, 1].

sup T1A  x   sup T1B  x  , sup T A2  x   sup T B2  x  ,,sup T pA  x   sup T Bp  x 

Sup-star composition10: Let X and Y be two non-empty crisp sets. An interval neutrosophic relation R(X, Y) is a subset of

inf I1A  x   inf I1B  x  ,inf I A2  x   inf I B2  x  ,,inf I pA  x   inf I Bp  x 

product space X×Y and is characterized by the membership functions for the composition of interval neutrosophic

sup I1A  x   sup I1B  x  , sup I A2  x   sup I B2  x  ,, sup I pA  x   sup I Bp  x 

relations R(X, Y) and S(Y, Z) are given by the interval neutrosophic sup-star composition of R and S:

inf F1A  x   inf F1B  x  , inf F2A  x   inf F2B  x  ,, inf FpA  x   inf FpB  x  T R S  x,z   sup min  T R  x, y  , TS  y,z   yY

sup F1A  x   sup F1B  x  , sup FA2  x   sup FB2  x  ,, sup FpA  x   sup FpB  x   x  X

I R S  x,z   sup min  I R  x, y  , Is  y,z  

(2)

yY

PROPOSED DEFINITIONS FR S  x,z   inf max  FR  x, y  , FS  y,z   yY

(1)

The proposed definitions are as follows: Grade function: Let A    a,b , c,d  , e,f   be an interval

where, x0X and y0Y and TR(x, y), IR(x, y), FR(x, y)f[0, 1].

neutrosophic number, a grade function E of an interval neutrosophic value, based on the truth-membership degree, indeterminacy-membership degree and falsity-membership degree is defined as:

N-valued interval neutrosophic set21: Let X be a universe, a n-valued interval neutrosophic set on X can be defined as follows: 431

J. Applied Sci., 17 (9): 429-440, 2017 EA 

 1  a   b  c  d  e  f 

Proof: The proof is straightforward.

(3)

Theorem 2: Let: Proposition 1:

A    a1 , b1  ,  c1 ,d1  ,  e1 ,f1   E A  0

and:

Proof: The proof is straightforward.

B    a 2 , b 2  ,  c2 ,d 2  ,  e2 ,f 2  

Theorem 1: Let:

be two interval neutrosophic numbers. If AfB then N(A)<N(B). A    a1 , b1  ,  c1 ,d1  ,  e1 ,f1  

Proof: By (4) :

and:

NA 1 B    a 2 , b 2  ,  c 2 ,d 2  ,  e2 ,f 2  

1  a1   b1  c1  d1  e1  f1

and:

be two interval neutrosophic numbers. If AfB then E(A)>E(B). N  B  1 

1  a 2   b2  c2  d 2  e2  f 2

Proof: By (3) :

EA 

 1  a   b 1

 c1  d1  e1  f1 

Since AfB, a1<a2, b1<b2, c1>c2, d1>d2, e1>e2 and f1>f2:

 (a 2  a1 )  0,(b 2  b1 )  0,(c1  c2 )  0,(d1  d 2 )  0,(e1  e 2 )  0 and (f1  f 2 )  0

and: E  B 

 1  a   b 2

 c2  d 2  e2  f 2 

Hence N(A)-N(B)<0.

Logarithmic distance: Let:

Since AfB, a1<a2, b1<b2, c1>c2, d1>d2, e1>e2 and f1>f2:

  inf T1A  x  ,sup T1A  x   , inf T A2  x  ,sup T A2  x   ,,   ,   x,  p p    inf T A  x  ,sup T A  x         inf I1A  x  ,sup I1A  x   , inf I A2  x  ,sup I A2  x   ,,         , A  p p   inf I A  x  ,sup I A  x       1 1 2 2   inf FA  x  ,sup FA  x   , inf FA  x  ,sup FA  x   ,,    : x  X  p p    inf FA  x  ,sup FA  x    

 (a 2  a1 )  0, (b 2  b1 )  0, (c1  c2 )  0,(d1  d 2 )  0,(e1  e2 )  0 and (f1  f 2 )  0

Hence E(A)-E(B)>0. Similarity grade function: Let A    a,b , c,d  , e,f   be an interval neutrosophic number, a similarity grade function N of an interval neutrosophic value, based on the truth-membership degree, indeterminacy-membership degree and falsity-membership degree is defined as: NA 1

1  a   b  c  d  e  f 6

and:   inf T1B  x  ,sup T1B  x   , inf T 2B  x  ,sup T 2B  x   ,,   ,  x,   p p    inf T B  x  ,sup T B  x         inf I1B  x  ,sup I1B  x   , inf I 2B  x  ,sup I B2  x   ,,         , B    p p   inf I B  x  ,sup I B  x       1 1 2 2   inf FB  x  ,sup FB  x   , inf FB  x  ,sup FB  x   ,,    : x  X  p p    inf FB  x  ,sup FB  x    

(4)

Proposition 2: N  A    0,1

432

J. Applied Sci., 17 (9): 429-440, 2017 Be two n-valued interval neutrosophic sets then the

                     1 p  1 n       log  2 p j1  2n i 1                     

logarithmic distance:                      p n 1 1   LD NVINS  A, B      log   p j1  2n i 1  2                     

inf T Aj  x i   inf T Bj  x i   sup T Aj  x i   sup T Bj  x i        inf IAj  x i   inf IBj  x i     j j j j sup IA  x i   sup I B  x i   inf FA  x i   inf FB  x i       sup FAj  x i   sup FBj  x i    6          

= LDNVINS (B, A)

(5) C

Proposition 3: C

inf T Bj  x i   inf T Aj  x i   sup T Bj  x i   sup T Aj  x i        inf I Bj  x i   inf I Aj  x i     j j j j sup I B  x i   sup I A  x i   inf FB  x i   inf FA  x i          sup F Bj  x i   sup F Aj  x i    6         

By (2) : inf T Aj  x i   inf T Bj  x i   inf T Cj  x i 

LDNVINS (A, B)>0

LDNVINS (A, B) = LDNVINS (B, A)

If AfBfC then LDNVINS (A, C)>LDNVINS (A, B) and LDNVINS (A, C)

sup T Aj  x i   sup T Bj  x i   sup T Cj  x i 

>LDNVINS (B, C)

inf I Aj  x i   inf I Bj  x i   inf ICj  x i 

Proof: sup I Aj  x i   sup I Bj  x i   sup ICj  x i 

The proof is straightforward

It was well known that:

inf FAj  x i   inf FBj  x i   inf FCj  x i 

inf T Aj  x i   inf T Bj  x i   inf T Bj  x i   inf T Aj  x i 

sup FAj  x i   sup FBj  x i   sup FCj  x i 

sup T Aj  x i   sup T Bj  x i   sup T Bj  x i   sup T Aj  x i 

Hence: inf T Aj  x i   inf T Bj  x i   inf T Aj  x i   inf T Cj  x i 

inf I Aj  x i   inf I Bj  x i   inf I Bj  x i   inf I Aj  x i 

sup T Aj  x i   sup T Bj  x i   sup T Aj  x i   sup T Cj  x i 

sup I Aj  x i   sup I Bj  x i   sup I Bj  x i   sup I Aj  x i 

inf I Aj  x i   inf I Bj  x i   inf I Aj  x i   inf ICj  x i 

inf FAj  x i   inf FBj  x i   inf FBj  x i   inf FAj  x i 

sup I Aj  x i   sup I Bj  x i   sup I Aj  x i   sup ICj  x i 

sup FAj  x i   sup FBj  x i   sup FBj  x i   sup FAj  x i 

                      1 p 1 n   LD NVINS  A, B      log   p j1  2n i 1  2                     

inf FAj  x i   inf FBj  x i   inf FAj  x i   inf FCj  x i 

inf T Aj  x i   inf T Bj  x i   sup T Aj  x i   sup T Bj  x i        inf IAj  x i   inf IBj  x i      j j j j sup IA  x i   sup IB  x i   inf FA  x i   inf FB  x i       j j sup FA  x i   sup FB  x i    6          

sup FAj  x i   sup FBj  x i   sup FAj  x i   sup FCj  x i 

Here, the logarithmic distance is an increasing function: ˆ LDNVINS (A, C)>LDNVINS (A, B) and LDNVINS (A, C) >LDNVINS (B, C)

433

J. Applied Sci., 17 (9): 429-440, 2017 inf I Aj  x i   inf I Bj  x i   inf I Bj  x i   inf I Aj  x i 

Exponential measure: Let:   inf T1A  x  ,sup T1A  x   , inf T A2  x  ,sup T A2  x   ,,   ,   x,     inf T pA  x  ,sup T Ap  x         inf I1A  x  ,sup I1A  x   , inf I A2  x  ,sup I A2  x   ,,         , A  p p    inf I A  x  ,sup I A  x       1 1 2 2   inf FA  x  ,sup FA  x   , inf FA  x  ,sup FA  x   ,,     : x  X  p p     inf FA  x  ,sup FA  x    

sup I Aj  x i   sup I Bj  x i   sup I Bj  x i   sup I Aj  x i  inf FAj  x i   inf FBj  x i   inf FBj  x i   inf FAj  x i  sup FAj  x i   sup FBj  x i   sup FBj  x i   sup FAj  x i 

 inf j  x  inf j  x   sup j  x  sup j  x   inf j  x  inf j  x       TB i TA i TB i TA i IB i IA i       j j j j j j    sup I B xi   sup I A  x i   inf F B x i  inf F A  x i   sup F B x i  sup F A  x i           n           1 p  n    e   EM NVINS  A,B       2np j  1  i  1  4              

and:   inf T1B  x  ,sup T1B  x   , inf T 2B  x  ,sup T B2  x   ,,    x,  ,  p p    inf T B  x  ,sup T B  x         inf I1B  x  ,sup I1B  x   , inf I 2B  x  ,sup I B2  x   ,,         , B  p p     inf x ,sup x     I I B B          inf F1B  x  ,sup F1B  x   , inf F2B  x  ,sup F2B  x   ,,    : x  X  p p    inf FB  x  ,sup FB  x    

 inf j  x inf j  x   sup j  x  sup j  x   inf j  x  inf j  x       TA i TB i TA i TB i IA i IB i       j j j j j j    sup I A  x i sup I B x i   inf FA  x i  inf FB x i   sup F A  x i sup F B x i          n             1 p  n      e      2np j  1  i  1  4             

be two n-valued interval neutrosophic sets then the exponential measure:  inf j  x  inf j  x   sup j  x  sup j  x   inf j  x  inf j  x       TA i TB i TA i TB i IA i IB i       j j j j j j    sup I A  x i   sup I B x i   inf F A  x i  inf F B x i   sup F A  x i  sup F B x i           n        p n      1   EM NVINS  A,B      e   2np j  1  i  1  4              

= EMNVINS (B, A) C

By (2) : inf T Aj  x i   inf T Bj  x i   inf T Cj  x i 

(6)

sup T Aj  x i   sup T Bj  x i   sup T Cj  x i 

Proposition 4: C C C

inf I Aj  x i   inf I Bj  x i   inf ICj  x i 

EMNVINS (A, B)>0 EMNVINS (A, B) = EMNVINS (B, A) If AfBfC then EMNVINS (A, C)<EMNVINS (A, B) and EMNVINS (A, C)<EMNVINS (B, C)

sup I Aj  x i   sup I Bj  x i   sup ICj  x i  inf FAj  x i   inf FBj  x i   inf FCj  x i 

Proof: C C

sup FAj  x i   sup FBj  x i   sup FCj  x i 

The proof is straightforward It was well known that:

Hence:

inf T Aj  x i   inf T Bj  x i   inf T Bj  x i   inf T Aj  x i 

inf T Aj  x i   inf T Bj  x i   inf T Aj  x i   inf T Cj  x i 

sup T Aj  x i   sup T Bj  x i   sup T Bj  x i   sup T Aj  x i 

sup T Aj  x i   sup T Bj  x i   sup T Aj  x i   sup T Cj  x i 

434

J. Applied Sci., 17 (9): 429-440, 2017 inf I Aj  x i   inf I Bj  x i   inf I Aj  x i   inf ICj  x i 

sup I Aj  x i   sup I Bj  x i   sup I Aj  x i   sup ICj  x i 

If AfBfC then SMNVINS (A, C)<SMNVINS (A, B) and SMNVINS (A, C)<SMNVINS (B, C)

Proof:

inf FAj  x i   inf FBj  x i   inf FAj  x i   inf FCj  x i 

C C

sup FAj  x i   sup FBj  x i   sup FAj  x i   sup FCj  x i 

The proof is straightforward It was well known that: inf T Aj  x i   inf T Bj  x i   inf T Bj  x i   inf T Aj  x i 

Here, the exponential measure is a decreasing function: sup T Aj  x i   sup T Bj  x i   sup T Bj  x i   sup T Aj  x i 

ˆ EMNVINS (A, C)<EMNVINS (A, B) and EMNVINS (A, C)<EMNVINS (B, C)

inf I Aj  x i   inf I Bj  x i   inf I Bj  x i   inf I Aj  x i 

Similarity measure: Let:

sup I Aj  x i   sup I Bj  x i   sup I Bj  x i   sup I Aj  x i 

  inf T1A  x  ,sup T1A  x   , inf T A2  x  ,sup T A2  x   ,,    x,  ,     inf T pA  x  ,sup T Ap  x         inf I1A  x  ,sup I1A  x   , inf I A2  x  ,sup I A2  x   ,,         , A    p p   inf I A  x  ,sup I A  x       1 1 2 2   inf FA  x  ,sup FA  x   , inf FA  x  ,sup FA  x   ,,    : x  X  p p     inf FA  x  ,sup FA  x    

inf FAj  x i   inf FBj  x i   inf FBj  x i   inf FAj  x i  sup FAj  x i   sup FBj  x i   sup FBj  x i   sup FAj  x i           n  p 1  SM NVINS  A,B     1  2np j  1  i  1           

and:   inf T1B  x  ,sup T1B  x   , inf T 2B  x  ,sup T B2  x   ,,    x,  ,  p p    inf T B  x  ,sup T B  x         inf I1B  x  ,sup I1B  x   , inf I 2B  x  ,sup I B2  x   ,,         , B  p p   inf I B  x  ,sup I B  x       1 1 2 2   inf FB  x  ,sup FB  x   , inf FB  x  ,sup FB  x   ,,    : x  X  p p    inf FB  x  ,sup FB  x    

          1 p  n     1 2np j  1  i 1           

be two n-valued interval neutrosophic sets then the similarity measure:           1 p  n  SM NVINS  A, B     1 2np j  1  i  1            

inf T Aj  x i   inf T Bj  x i   sup T Aj  x i   sup T Bj  x i      inf I Aj  x i   inf I Bj  x i   sup I Aj  x i   sup I Bj  x i       inf F Aj  x i   inf F Bj  x i   sup F Aj  x i   sup F Bj  x i      6       

inf T Bj  x i   inf T Aj  x i   sup T Bj  x i   sup T Aj  x i      inf I Bj  x i   inf I Aj  x i   sup I Bj  x i   sup I Aj  x i      inf FBj  x i   inf FAj  x i   sup F Bj  x i   sup F Aj  x i      6      

= SMNVINS (B, A) C

By (2) : inf T Aj  x i   inf T Bj  x i   inf T Cj  x i 

sup T Aj  x i   sup T Bj  x i   sup T Cj  x i 

(7) Proposition 5: C C

inf T Aj  x i   inf T Bj  x i   sup T Aj  x i   sup T Bj  x i      inf I Aj  x i   inf I Bj  x i   sup I Aj  x i   sup I Bj  x i      inf F Aj  x i   inf F Bj  x i   sup F Aj  x i   sup F Bj  x i      6      

inf I Aj  x i   inf I Bj  x i   inf ICj  x i 

SMNVINS (A, B)<1 SMNVINS (A, B) = SMNVINS (B, A)

sup I Aj  x i   sup I Bj  x i   sup ICj  x i 

435

J. Applied Sci., 17 (9): 429-440, 2017 inf FAj  x i   inf FBj  x i   inf FCj  x i 

l NVINS  A, B  

sup FAj  x i   sup FBj  x i   sup FCj  x i 

 2  SM NVINS  A, B   1 log   2p  2  SM NVINS  A, B  

(8)

Proposition 6: Hence: C C C

inf T Aj  x i   inf T Bj  x i   inf T Aj  x i   inf T Cj  x i  sup T Aj  x i   sup T Bj  x i   sup T Aj  x i   sup T Cj  x i 

lNVINS (A, B)<1 lNVINS (A, B) = lNVINS (B, A) If AfBfC then lNVINS (A, C)<lNVINS (A, B) and lNVINS (A, C)<lNVINS (B, C)

Proof: inf I Aj  x i   inf I Bj  x i   inf I Aj  x i   inf ICj  x i 

C C

The proof is straightforward Since SMNVINS (A, B) = SMNVINS (B, A):

sup I Aj  x i   sup I Bj  x i   sup I Aj  x i   sup ICj  x i  l NVINS  A, B   inf FAj  x i   inf FBj  x i   inf FAj  x i   inf FCj  x i  sup FAj  x i   sup FBj  x i   sup FAj  x i   sup FCj  x i 



Here, the similarity measure is a decreasing function:

1 log 2p

 2  SM NVINS  A, B    2  SM NVINS  A, B 

  

 2  SM NVINS  B, A   1 log   2p  2  SM NVINS  B, A  

= lNVINS (B, A)

ˆ SMNVINS (A, C)<SMNVINS (A, B) and SMNVINS (A, C)<SMNVINS (B, C)

By (2) :

Logarithmic function: Let:

inf T Aj  x i   inf T Bj  x i   inf T Cj  x i 

  inf T1A  x  ,sup T1A  x   , inf T A2  x  ,sup T A2  x      x,  ,     ,, inf T pA  x  ,sup T Ap  x         inf I1A  x  ,sup I1A  x   , inf I A2  x  ,sup I A2  x           A ,  p p    ,, inf I A  x  ,sup I A  x       1 1 2 2   inf FA  x  ,sup FA  x   , inf FA  x  ,sup FA  x       : x  X  p p     ,, inf FA  x  ,sup FA  x    

sup T Aj  x i   sup T Bj  x i   sup T Cj  x i 

inf I Aj  x i   inf I Bj  x i   inf ICj  x i  sup I Aj  x i   sup I Bj  x i   sup ICj  x i  inf FAj  x i   inf FBj  x i   inf FCj  x i 

and:

sup FAj  x i   sup FBj  x i   sup FCj  x i 

  inf T1B  x  ,sup T1B  x   , inf T 2B  x  ,sup T B2  x   ,,    x,  ,     inf T pB  x  ,sup T pB  x         inf I1B  x  ,sup I1B  x   , inf I 2B  x  ,sup I B2  x   ,,         , B  p p     inf x ,sup x IB      I B       1 1 2 2   inf FB  x  ,sup FB  x   , inf FB  x  ,sup FB  x   ,,     : x  X  p p     inf FB  x  ,sup FB  x    

Hence: inf T Aj  x i   inf T Bj  x i   inf T Aj  x i   inf T Cj  x i  sup T Aj  x i   sup T Bj  x i   sup T Aj  x i   sup T Cj  x i  inf I Aj  x i   inf I Bj  x i   inf I Aj  x i   inf ICj  x i 

be two n-valued interval neutrosophic sets. Then, the logarithmic function based on similarity measure formula:

sup I Aj  x i   sup I Bj  x i   sup I Aj  x i   sup ICj  x i 

436

J. Applied Sci., 17 (9): 429-440, 2017 inf FAj  x i   inf FBj  x i   inf FAj  x i   inf FCj  x i 

  2SM NVINS A,B   1   p 2SM NVINS A,B   e NVINS  A, B   e  2n    

sup FAj  x i   sup FBj  x i   sup FAj  x i   sup FCj  x i 

Here, the logarithmic function is a decreasing function:

  2SM NVINS B,A    1   p 2SM NVINS B,A     e  2n    

ˆ lNVINS (A, C)<lNVINS (A, B) and lNVINS (A, C)<lNVINS (B, C) Definition: Let:

= eNVINS (B, A)   inf T1A  x  ,sup T1A  x   , inf T A2  x  ,sup T A2  x   ,,    x,  ,  p p    inf T A  x  ,sup T A  x         inf I1A  x  ,sup I1A  x   , inf I A2  x  ,sup I A2  x   ,,         , A  p p     inf x ,sup x     I I    A A       inf F1A  x  ,sup F1A  x   , inf FA2  x  ,sup FA2  x   ,,    : x  X  p p    inf FA  x  ,sup FA  x    

By (2) : inf T Aj  x i   inf T Bj  x i   inf T Cj  x i 

sup T Aj  x i   sup T Bj  x i   sup T Cj  x i 

inf I Aj  x i   inf I Bj  x i   inf ICj  x i 

and: sup I Aj  x i   sup I Bj  x i   sup ICj  x i 

  inf T1B  x  ,sup T1B  x   , inf T 2B  x  ,sup T B2  x   ,,    x,  ,     inf T pB  x  ,sup T pB  x         inf I1B  x  ,sup I1B  x   , inf I 2B  x  ,sup I B2  x   ,,         , B    p p   inf I B  x  ,sup I B  x       1 1 2 2   inf FB  x  ,sup FB  x   , inf FB  x  ,sup FB  x   ,,    : x  X  p p     inf FB  x  ,sup FB  x    

inf FAj  x i   inf FBj  x i   inf FCj  x i  sup FAj  x i   sup FBj  x i   sup FCj  x i 

Hence: inf T Aj  x i   inf T Bj  x i   inf T Aj  x i   inf T Cj  x i 

be two n-valued interval neutrosophic sets. Then the exponential function based on similarity measure formula:   2SM NVINS(A,B)   1   p 2SM NVINS A,B   e NVINS  A,B   e  2n    

sup T Aj  x i   sup T Bj  x i   sup T Aj  x i   sup T Cj  x i 

(9)

inf I Aj  x i   inf I Bj  x i   inf I Aj  x i   inf ICj  x i  sup I Aj  x i   sup I Bj  x i   sup I Aj  x i   sup ICj  x i 

Proposition 7: C C C

inf FAj  x i   inf FBj  x i   inf FAj  x i   inf FCj  x i 

eNVINS (A, B)>0 eNVINS (A, B) = eNVINS (B, A) If AfBfC then eNVINS (A, C)<eNVINS (A, B) and eNVINS (A, C)<eNVINS (B, C)

sup FAj  x i   sup FBj  x i   sup FAj  x i   sup FCj  x i 

Here, the exponential function is a decreasing

Proof:

function: C C

The proof is straightforward Since, SMNVINS (A, B) = SMNVINS (B, A):

ˆ eNVINS (A, C)<eNVINS (A, B) and eNVINS (A, C)<eNVINS (B, C) 437

J. Applied Sci., 17 (9): 429-440, 2017 METHODOLOGY

Step 4: Table 4 is obtained by applying (1) between Table 2 and 3

In this section, it was presented an application of

Step 5: The computation T of the relation of patients and

n-valued interval neutrosophic set in medical diagnosis. In

diseases is found using (3) and (4) in Table 4 and are

a given pathology, suppose S is a set of symptoms, D is a

noted in Table 5

set of diseases and P is a set of patients and let Q be

Step 6: The computation T of the relation of patients and

an-valued interval neutrosophic relation from the set of

diseases is found (5) , (6) , (7) , (8) and (9) and are

patients to the symptoms. i.e., Q(P6S) and R be an interval

noted in Table 6

neutrosophic relation from the set of symptoms to the

Step 7: Finally, the minimum value from Table 5 (grade

diseases i.e., R(S6D) and then the methodology involves three

function) and Table 6 (logarithmic distance) and

main jobs:

maximum value from Table 5 (similarity grade function and Table 6 (exponential

measure,

Determination of symptoms

similarity measure, logarithmic function and

Formulation of medical knowledge based on n-valued

exponential function) of each row were selected to

interval neutrosophic sets and interval neutrosophic sets

find the possibility of the patient affected with the

Determination of diagnosis on the basis of various

respective disease and then it was concluded that

computation

the patient was suffering from the disease

techniques

n-valued

interval

neutrosophic sets CASE STUDY21 Algorithm: Let there be three patients P = {P1, P2, P3} and the set of Step 1: The symptoms of the patients are given to obtain

symptoms S = {S1 = Temperature, S2 = Cough, S3 = Throat

the patient-symptom relation and are noted in

pain, S4 = Headache, S5 = Body pain}. The n-valued interval

Table 1

neutrosophic relation Q(P6S) is given as in Table 1. Let the

Step 2: The medical knowledge relating the symptoms with

set of diseases D = {D1 = Viral fever, D2 = Tuberculosis,

the set of diseases under consideration are given to

D3 = Typhoid, D4 = Throat disease}. The interval neutrosophic

obtain the symptom-disease relation and are noted

relation R(S6D) is given as in Table 2.

in Table 2

From Table 5 and 6,it is obvious that, if the doctor agrees,

Step 3: Table 3 is obtained by calculating average values for

then P1 and P3 suffers from Viral fever and P2 suffers from Throat disease.

Table 1 Table 1: Patient-symptom relation (using step1) Q

Temperature

Cough

Throat pain

Headache

Body pain

[0.3,0.4], [0.4,0.5],

[0.1,0.2], [0.3,0.6],

[0.0,0.5],

[0.2,0.3],

[0.0,0.4],

[0.3,0.7]

[0.6,0.8]

[0.2,0.6], [0.0,0.4]

[0.3,0.5], [0.0,0.7]

[0.6,0.7], [0.2,0.5]

[0.0,0.3], [0.1,0.3],

[0.0,0.5], [0.4,0.7],

[0.3,0.4],

[0.4,0.5],

[0.2,0.4],

[0.0,0.5]

[0.4,0.5]

[0.2,0.3], [0.3,0.4]

[0.4,0.7], [0.3,0.6]

[0.4,0.5], [0.1,0.2]

[0.0,0.6], [0.4,0.5],

[0.2,0.3], [0.0,0.5],

[0.0,0.7],

[0.2,0.6],

[0.1,0.3],

[0.3,0.4]

[0.4,0.6]

[0.3,0.7], [0.3,0.5]

[0.0,0.6], [0.3,0.4]

[0.1,0.3], [0.2,0.3]

[0.2,0.03],

[0.5,0.7], [0.0,0.4],

[0.5,0.6],

[0.2,0.5],

[0.2,0.4],

[0.4,0.5], [0.1,0.2]

[0.7,0.8]

[0.0,0.6], [0.2,0.3]

[0.5,0.6], [0.1,0.5]

[0.4,0.6], [0.1,0.4]

[0.4,0.5], [0.2,0.5],

[0.6,0.7], [0.0,0.5],

[0.4,0.7],

[0.2,0.3],

[0.0,0.5],

[0.0,0.3]

[0.4,0.5]

[0.4,0.6], [0.3,0.4]

[0.2,0.5], [0.5,0.6]

[0.2,0.4], [0.5,0.6]

[0.6,0.7], [0.4,0.5],

[0.4,0.6], [0.2,0.7],

[0.1,0.3],

[0.5,0.7],

[0.4,0.5]

[0.0,0.3]

[0.2,0.3], [0.5,0.7]

[0.3,0.4], [0.4,0.5]

[0.0,0.7], [0.2,0.4]

[0.1,0.3], [0.0,0.5],

[0.2,0.3], [0.0,0.7],

[0.2,0.4],

[0.2,0.3],

[0.0,0.6],

[0.4,0.6]

[0.1,0.4]

[0.3,0.6], [0.0,0.6]

[0.5,0.6], [0.4,0.5]

[0.4,0.7], [0.2,0.3]

[0.1,0.2], [0.3,0.4],

[0.5,0.6], [0.0,0.3],

[0.4,0.5],

[0.2,0.4],

[0.2,0.3],

[0.2,0.5]

[0.3,0.5]

[0.0,0.3], [0.3,0.4]

[0.0,0.4], [0.2,0.7]

[0.2,0.3], [0.1,0.2]

[0.2,0.4], [0.4,0.5],

[0.3,0.5], [0.2,0.5],

[0.5,0.7],

[0.4,0.5],

[0.0,0.6],

[0.3,0.7]

[0.4,0.6]

[0.4,0.6], [0.3,0.7]

[0.2,0.3], [0.3,0.5]

[0.2,0.4], [0.4,0.6]

438

J. Applied Sci., 17 (9): 429-440, 2017 Table 2: Symptom-disease relation (using step2) R Temperature Cough Throat pain Headache Body pain

Viral fever [0.2,0.4], [0.3,0.5], [0.3,0.7] [0.2,0.4], [0.2,0.3], [0.0,0.5] [0.0,0.4], [0.2,0.4], [0.2,0.4] [0.4,0.7], [0.0,0.3], [0.3,0.5] [0.1,0.4], [0.2,0.5], [0.3,0.4]

Tuberculosis [0.1,0.4], [0.2,0.6], [0.6,0.7] [0.3,0.4], [0.2,0.5], [0.7,0.8] [0.0,0.2], [0.3,0.6], [0.6,0.7] [0.1,0.2], [0.0,0.5], [0.0,0.6] [0.5,0.7], [0.4,0.5], [0.2,0.5]

Table 3: Average for patient-symptom relation Q Temperature Cough [0.1,0.43], [0.3,0.43], [0.1,0.33], [0.23,0.6], P1 [0.2,0.53] [0.46,0.63] [0.4,0.5], [0.33,0.5], [0.5,0.66], [0.06,0.53], P2 [0.16,0.33] [0.36,0.53] [0.13,0.3], [0.23,0.46], [0.33,0.46], [0.06,0.5], P3 [0.3,0.6] [0.26,0.5]

Typhoid [0.0,0.3], [0.4,0.6], [0.0,0.2] [0.3,0.4], [0.2,0.3], [0.1,0.2] [0.1,0.2], [0.4,0.5], [0.3,0.4] [0.3,0.4], [0.2,0.3], [0.2,0.5] [0.2,0.3], [0.2,0.4], [0.2,0.3]

Throat pain [0.1,0.53], [0.23,0.53], [0.2,0.43] [0.33,0.53], [0.2,0.5], [0.33,0.46] [0.36,0.53], [0.23,0.5], [0.2,0.56]

Throat disease [0.3,0.4], [0.2,0.5], [0.0,0.6] [0.4,0.5], [0.1,0.3], [0.0,0.5] [0.2,0.4], [0.2,0.5], [0.3,0.7] [0.0,0.3], [0.3,0.6], [0.2,0.5] [0.0,0.4], [0.1,0.2], [0.1,0.3]

Headache [0.26,0.46], [0.23,0.6], [0.2,0.56] [0.16,0.36], [0.33,0.5], [0.33,0.53] [0.26,0.4], [0.23,0.43], [0.3,0.56]

Body pain [0.1,0.36], [0.36,0.5], [0.16,0.33] [0.23,0.53], [0.2,0.56], [0.26,0.46] [0.06,0.5], [0.26,0.46], [0.23,0.36]

Table 4: Sup-star composition between symptom-disease relation and average for patient-symptom relation T P1 P2 P3

Viral fever [0.26,0.46], [0.3,0.5], [0.2,0.4] [0.2,0.4], [0.3,0.5], [0.3,0.46] [0.26,0.4], [0.23,0.46], [0.2,0.4]

Tuberculosis [0.1,0.4], [0.36,0.53], [0.2,0.5] [0.3,0.53], [0.2,0.5], [0.26,0.5] [0.3,0.5], [0.26,0.5], [0.23,0.5]

Typhoid [0.26,0.4], [0.3,0.5], [0.2,0.33] [0.3,0.4], [0.33,0.5], [0.16,0.33] [0.3,0.4], [0.23,0.5], [0.23,0.36]

Table 5: Grade function and similarity grade function (using step 5 and step 7) T Viral fever Tuberculosis Typhoid Throat disease Grade function 0.8100 0.8464 P3 0.6642 Similarity grade function P1 P2

P1 P2 P3

0.7000 0.6933 0.7283

0.8930 0.7140 0.7140

0.8190 0.7744 0.7225

0.9604 0.6400 0.7482

0.6850 0.7183 0.7183

0.6983 0.7066 0.7166

0.6733 0.7333 0.7116

diseases and employed seven methods (grade function, similarity grade function, logarithmic distance, exponential measure, similarity measure, logarithmic function, exponential function) to find out the disease possibly affected the patient. The techniques considered in this study were more reliable to handle medical diagnosis problems quiet comfortably. The proposed methods had more accuracy than the others and they could handle the limitations and drawbacks of the previous works well.

Table 6: Logarithmic distance, exponential measure, similarity measure, logarithmic function and exponential function(using step 6 and step 7) T Viral fever Tuberculosis Typhoid Throat disease Logarithmic distance 0.1180 0.1208 P3 0.1174 Exponential measure

0.1208 0.1217 0.1210

0.1196 0.1201 0.1187

0.1193 0.1191 0.1180

0.8108 0.7907 P3 0.8145 Similarity measure

0.7906 0.7827 0.7884

0.7991 0.7952 0.8059

0.8019 0.8038 0.8126

0.4226 0. 4115 P3 0.4247 Logarithmic function

0.4115 0.4078 0.4106

0.4161 0.4142 0.4197

0.4175 0.4184 0.4226

0.0310 P2 0.0302 P3 0.0312 Exponential function

0.0302 0.0299 0.0301

0.0305 0.0304 0.0308

0.0306 0.0307 0.0310

0.1658 0.1655 0.1657

0.1662 0.1661 0.1665

0.1663 0.1664 0.1668

P1 P2

P1 P2 P3

0.1668 0.1658 0.1670

Throat disease [0.1,0.4], [0.23,0.6], [0.16,0.33] [0.4,0.5], [0.3,0.5], [0.16,0.46] [0.33,0.46], [0.23,0.5], [0.23,0.36]

SIGNIFICANCE STATEMENTS This study discovers the relationship between the symptoms found within patients and set of diseases. This study will help the researcher to find out the diseases accurately that impacted the patients. The methods employed are free from the limitations that are commonly found in other studies. Without such limitations, in this study a new theory on image processing, cluster analysis etc., has been developed. REFERENCES 1. 2. 3. 4.

CONCLUSION In this study, it was analyzed that the relationship between the set of symptoms found within patients and set of

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