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Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis



A. Edward Samuel and R. Narmadhagnanam
 
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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.

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  How to cite this article:

A. Edward Samuel and R. Narmadhagnanam, 2017. Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis. Journal of Applied Sciences, 17: 429-440.

DOI: 10.3923/jas.2017.429.440

URL: https://scialert.net/abstract/?doi=jas.2017.429.440
 
Received: August 01, 2017; Accepted: November 06, 2017; Published: November 29, 2017


Copyright: © 2017. 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.

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. 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,…, Tm, l1, l2,…, lp and F1, F2,…, FT. Deli et al.19 studied a new concept called neutrosophic refined sets. Broumi and Deli20 proposed the correlation measure between them. Broumi et al.21 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.

PRELIMINARIES

Interval neutrosophic set10: Let X be a space of points (objects), with a generic element in X denoted by x. A`n interval neutrosophic set A in X is characterized by the truth-membership function TA, indeterminacy-membership function IA and falsity-membership function FA. For each point x in Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis with the condition that Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Interval neutrosophic relation10: Let X and Y be two non-empty crisp sets. An interval neutrosophic relation R(X, Y) is a subset of product space X×Y and is characterized by the truth membership function TR(x, y), the indeterminacy membership function IR(x, y) and the falsity membership function FR(x, y), where x∈X and y∈Y and TR(x, y), IR(x, y), FR(x, y)⊆[0, 1].

Sup-star composition10: Let X and Y be two non-empty crisp sets. An interval neutrosophic relation R(X, Y) is a subset of product space X×Y and is characterized by the membership functions for the composition of interval neutrosophic relations R(X, Y) and S(Y, Z) are given by the interval neutrosophic sup-star composition of R and S:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis
(1)

where, x∈X and y∈Y and TR(x, y), IR(x, y), FR(x, y)⊆[0, 1].

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

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Where:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Such that:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Inclusion21: A n-valued interval neutrosophic set A is contained in the other n-valued interval neutrosophicset B, denoted by A⊆B if and only if:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis
(2)

PROPOSED DEFINITIONS

The proposed definitions are as follows:

Grade function: Let Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis be an interval 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:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis
(3)

Proposition 1:

E(A) ≥ 0

Proof: The proof is straightforward.

Theorem 1: Let:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

and:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

be two interval neutrosophic numbers. If A⊆B then E(A)>E(B).

Proof: By‘(3)’:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

and:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

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

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Hence E(A)-E(B)>0.

Similarity grade function: Let Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis 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:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis
(4)

Proposition 2:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Proof: The proof is straightforward.

Theorem 2: Let:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

and:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

be two interval neutrosophic numbers. If A⊆B then N(A)<N(B).

Proof: By‘(4)’:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

and:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

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

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

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

Logarithmic distance: Let:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

and:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Be two n-valued interval neutrosophic sets then the logarithmic distance:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis
(5)

Proposition 3:

LDNVINS (A, B)>0
LDNVINS (A, B) = LDNVINS (B, A)
If A⊆B⊆C then LDNVINS (A, C)>LDNVINS (A, B) and LDNVINS (A, C) >LDNVINS (B, C)

Proof:

The proof is straightforward
It was well known that:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

= LDNVINS (B, A)

By ‘(2)’:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Hence:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Here, the logarithmic distance is an increasing function:

∴ LDNVINS (A, C)>LDNVINS (A, B) and LDNVINS (A, C) >LDNVINS (B, C)

Exponential measure: Let:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

and:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

be two n-valued interval neutrosophic sets then the exponential measure:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis
(6)

Proposition 4:

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

Proof:

The proof is straightforward
It was well known that:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

By ‘(2)’:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Hence:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Here, the exponential measure is a decreasing function:

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

Similarity measure: Let:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

and:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

be two n-valued interval neutrosophic sets then the similarity measure:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis
(7)

Proposition 5:

SMNVINS (A, B)<1
SMNVINS (A, B) = SMNVINS (B, A)
If A⊆B⊆C then SMNVINS (A, C)<SMNVINS (A, B) and SMNVINS (A, C)<SMNVINS (B, C)

Proof:

The proof is straightforward
It was well known that:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

By ‘(2)’:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Hence:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Here, the similarity measure is a decreasing function:

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

Logarithmic function: Let:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

and:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

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

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis
(8)

Proposition 6:

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

Proof:

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

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

By ‘(2)’:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Hence:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Here, the logarithmic function is a decreasing function:

∴ lNVINS (A, C)<lNVINS (A, B) and lNVINS (A, C)<lNVINS (B, C)

Definition: Let:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

and:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

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

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis
(9)

Proposition 7:

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

Proof:

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

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

By ‘(2)’:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Hence:

Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Here, the exponential function is a decreasing function:

∴ eNVINS (A, C)<eNVINS (A, B) and eNVINS (A, C)<eNVINS (B, C)

METHODOLOGY

In this section, it was presented an application of n-valued interval neutrosophic set in medical diagnosis. In a given pathology, suppose S is a set of symptoms, D is a set of diseases and P is a set of patients and let Q be an-valued interval neutrosophic relation from the set of patients to the symptoms. i.e., Q(P→S) and R be an interval neutrosophic relation from the set of symptoms to the diseases i.e., R(S→D) and then the methodology involves three main jobs:

Determination of symptoms
Formulation of medical knowledge based on n-valued interval neutrosophic sets and interval neutrosophic sets
Determination of diagnosis on the basis of various computation techniques of n-valued interval neutrosophic sets

Algorithm:

Step 1: The symptoms of the patients are given to obtain the patient-symptom relation and are noted in Table 1
Step 2: The medical knowledge relating the symptoms with the set of diseases under consideration are given to obtain the symptom-disease relation and are noted in Table 2
Step 3: Table 3 is obtained by calculating average values for Table 1
Step 4: Table 4 is obtained by applying ‘(1)’ between Table 2 and 3
Step 5: The computation T of the relation of patients and diseases is found using ‘(3)’and ‘(4)’in Table 4 and are noted in Table 5
Step 6: The computation T of the relation of patients and diseases is found ‘(5)’, ‘(6)’,‘(7)’, ‘(8)’ and ‘(9)’and are noted in Table 6
Step 7: Finally, the minimum value from Table 5 (grade function) and Table 6 (logarithmic distance) and maximum value from Table 5 (similarity grade function and Table 6 (exponential measure, similarity measure, logarithmic function and exponential function) of each row were selected to find the possibility of the patient affected with the respective disease and then it was concluded that the patient was suffering from the disease

CASE STUDY21

Let there be three patients P = {P1, P2, P3} and the set of symptoms S = {S1 = Temperature, S2 = Cough, S3 = Throat pain, S4 = Headache, S5 = Body pain}. The n-valued interval neutrosophic relation Q(P→S) is given as in Table 1. Let the set of diseases D = {D1 = Viral fever, D2 = Tuberculosis, D3 = Typhoid, D4 = Throat disease}. The interval neutrosophic relation R(S→D) is given as in Table 2.

From Table 5 and 6,it is obvious that, if the doctor agrees, then P1 and P3 suffers from Viral fever and P2 suffers from Throat disease.

Table 1:Patient-symptom relation (using step1)
Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Table 2:Symptom-disease relation (using step2)
Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Table 3:Average for patient-symptom relation
Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Table 4:Sup-star composition between symptom-disease relation and average for patient-symptom relation
Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Table 5:Grade function and similarity grade function (using step 5 and step 7)
Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

Table 6:Logarithmic distance, exponential measure, similarity measure, logarithmic function and exponential function(using step 6 and step 7)
Image for - Innovative Approaches for N-valued Interval Neutrosophic Sets and their Execution in Medical Diagnosis

CONCLUSION

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

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.

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