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 theory^{1}, rough set theory^{2} 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 nvalued 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, nvalued interval neutrosophic sets and their applications play a vital role in medical diagnosis.
In 1965, fuzzy set theory was firstly given by Zadeh^{1} 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. Atanassov^{3} introduced the intuitionistic fuzzy sets which consider both truthmembership and falsitymembership. De et al.^{4} presented an application of intuitionistic fuzzy set in medical diagnosis. Ye^{5 }introduced the concept of cosine similarity measures for intuitionistic fuzzy sets. Miaoying^{6} 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 Smarandache^{7 }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 Samanta^{9}. 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 Ye^{11}. Interval valued neutrosophic soft sets were introduced by Deli^{12}.
Sebastian and Ramakrishnan^{13 }studied a new concept called fuzzy multi sets (FMS), which is the generalization of multi sets. Shinoj and Sunil^{14 }extended the concept of fuzzy multi sets by introducing intuitionistic fuzzy multi sets (IFMS). Rajarajeswari and Uma^{15 }proposed the normalized hamming similarity measure between them. However, the concepts of FMS and IFMS are not capable of dealing with indeterminacy. Ye and Ye^{16 }introduced the concept of single valued neutrosophic multi sets. Distance based similarity measures between them were introduced by Ye et al.^{17}. Smarandache^{18} extended the classical neutrosophic logic to nvalued refined neutrosophic logic, by refining each neutrosophic component T, I, F into respectively, T_{1}, T_{2},…, T_{m}, l_{1}, l_{2},…, l_{p} and F_{1}, F_{2},…, F_{T}. Deli et al.^{19} studied a new concept called neutrosophic refined sets. Broumi and Deli^{20} proposed the correlation measure between them. Broumi et al.^{21 } generalize the concept of nvalued neutrosophic sets to the case of nvalued interval neutrosophic sets.
In this study, using the notion of nvalued interval neutrosophic set was provided an exemplary for medical diagnosis. In order to make this, various methods were implemented.
PRELIMINARIES
Interval neutrosophic set^{10}: 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 truthmembership function T_{A}, indeterminacymembership function I_{A} and falsitymembership function F_{A}. For each point x in with the condition that
Interval neutrosophic relation^{10}: Let X and Y be two nonempty 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 T_{R}(x, y), the indeterminacy membership function I_{R}(x, y) and the falsity membership function F_{R}(x, y), where x∈X and y∈Y and T_{R}(x, y), I_{R}(x, y), F_{R}(x, y)⊆[0, 1].
Supstar composition^{10}: Let X and Y be two nonempty 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 supstar composition of R and S:
where, x∈X and y∈Y and T_{R}(x, y), I_{R}(x, y), F_{R}(x, y)⊆[0, 1].
Nvalued interval neutrosophic set^{21}: Let X be a universe, a nvalued interval neutrosophic set on X can be defined as follows:
Where:
Such that:
Inclusion^{21}: A nvalued interval neutrosophic set A is contained in the other nvalued interval neutrosophicset B, denoted by A⊆B if and only if:
PROPOSED DEFINITIONS
The proposed definitions are as follows:
Grade function: Let be an interval neutrosophic number, a grade function E of an interval neutrosophic value, based on the truthmembership degree, indeterminacymembership degree and falsitymembership degree is defined as:
Proposition 1:
E(A) ≥ 0
Proof: The proof is straightforward.
Theorem 1: Let:
and:
be two interval neutrosophic numbers. If A⊆B then E(A)>E(B).
Proof: By‘(3)’:
and:
Since A⊆B, a_{1}<a_{2}, b_{1}<b_{2}, c_{1}>c_{2}, d_{1}>d_{2}, e_{1}>e_{2} and f_{1}>f_{2}:
Hence E(A)E(B)>0.
Similarity grade function: Let be an interval neutrosophic number, a similarity grade function N of an interval neutrosophic value, based on the truthmembership degree, indeterminacymembership degree and falsitymembership degree is defined as:
Proposition 2:
Proof: The proof is straightforward.
Theorem 2: Let:
and:
be two interval neutrosophic numbers. If A⊆B then N(A)<N(B).
Proof: By‘(4)’:
and:
Since A⊆B, a_{1}<a_{2}, b_{1}<b_{2}, c_{1}>c_{2}, d_{1}>d_{2}, e_{1}>e_{2} and f_{1}>f_{2}:
Hence N(A)N(B)<0.
Logarithmic distance: Let:
and:
Be two nvalued interval neutrosophic sets then the logarithmic distance:
Proposition 3:
• 
LD_{NVINS} (A, B)>0 
• 
LD_{NVINS} (A, B) = LD_{NVINS} (B, A) 
• 
If A⊆B⊆C then LD_{NVINS} (A, C)>LD_{NVINS} (A, B) and LD_{NVINS} (A, C) >LD_{NVINS} (B, C) 
Proof:
• 
The proof is straightforward 
• 
It was well known that: 
= LD_{NVINS} (B, A)
Hence:
Here, the logarithmic distance is an increasing function:
∴ LD_{NVINS} (A, C)>LD_{NVINS} (A, B) and LD_{NVINS} (A, C) >LD_{NVINS} (B, C)
Exponential measure: Let:
and:
be two nvalued interval neutrosophic sets then the exponential measure:
Proposition 4:
• 
EM_{NVINS} (A, B)>0 
• 
EM_{NVINS} (A, B) = EM_{NVINS} (B, A) 
• 
If A⊆B⊆C then EM_{NVINS} (A, C)<EM_{NVINS} (A, B) and EM_{NVINS} (A, C)<EM_{NVINS} (B, C) 
Proof:
• 
The proof is straightforward 
• 
It was well known that: 
Hence:
Here, the exponential measure is a decreasing function:
∴ EM_{NVINS} (A, C)<EM_{NVINS} (A, B) and EM_{NVINS} (A, C)<EM_{NVINS} (B, C)
Similarity measure: Let:
and:
be two nvalued interval neutrosophic sets then the similarity measure:
Proposition 5:
• 
SM_{NVINS} (A, B)<1 
• 
SM_{NVINS} (A, B) = SM_{NVINS} (B, A) 
• 
If A⊆B⊆C then SM_{NVINS} (A, C)<SM_{NVINS} (A, B) and SM_{NVINS} (A, C)<SM_{NVINS} (B, C) 
Proof:
• 
The proof is straightforward 
• 
It was well known that: 
Hence:
Here, the similarity measure is a decreasing function:
∴ SM_{NVINS} (A, C)<SM_{NVINS} (A, B) and SM_{NVINS} (A, C)<SM_{NVINS} (B, C)
Logarithmic function: Let:
and:
be two nvalued interval neutrosophic sets. Then, the logarithmic function based on similarity measure formula:
Proposition 6:
• 
l_{NVINS} (A, B)<1 
• 
l_{NVINS} (A, B) = l_{NVINS} (B, A) 
• 
If A⊆B⊆C then l_{NVINS} (A, C)<l_{NVINS} (A, B) and l_{NVINS} (A, C)<l_{NVINS} (B, C) 
Proof:
• 
The proof is straightforward 
• 
Since SM_{NVINS} (A, B) = SM_{NVINS} (B, A): 
Hence:
Here, the logarithmic function is a decreasing function:
∴ l_{NVINS} (A, C)<l_{NVINS} (A, B) and l_{NVINS} (A, C)<l_{NVINS} (B, C)
Definition: Let:
and:
be two nvalued interval neutrosophic sets. Then the exponential function based on similarity measure formula:
Proposition 7:
• 
e_{NVINS} (A, B)>0 
• 
e_{NVINS} (A, B) = e_{NVINS} (B, A) 
• 
If A⊆B⊆C then e_{NVINS} (A, C)<e_{NVINS} (A, B) and e_{NVINS} (A, C)<e_{NVINS} (B, C) 
Proof:
• 
The proof is straightforward 
• 
Since, SM_{NVINS} (A, B) = SM_{NVINS} (B, A): 
Hence:
Here, the exponential function is a decreasing function:
∴ e_{NVINS} (A, C)<e_{NVINS} (A, B) and e_{NVINS} (A, C)<e_{NVINS} (B, C)
METHODOLOGY
In this section, it was presented an application of nvalued 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 anvalued 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 nvalued interval neutrosophic sets and interval neutrosophic sets 
• 
Determination of diagnosis on the basis of various computation techniques of nvalued interval neutrosophic sets 
Algorithm:
Step 1: 
The symptoms of the patients are given to obtain the patientsymptom 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 symptomdisease 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 STUDY^{21}
Let there be three patients P = {P_{1}, P_{2}, P_{3}} and the set of symptoms S = {S_{1} = Temperature, S_{2} = Cough, S_{3} = Throat pain, S_{4} = Headache, S_{5} = Body pain}. The nvalued interval neutrosophic relation Q(P→S) is given as in Table 1. Let the set of diseases D = {D_{1} = Viral fever, D_{2} = Tuberculosis, D_{3} = Typhoid, D_{4} = 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 P_{1} and P_{3} suffers from Viral fever and P_{2} suffers from Throat disease.
Table 1:  Patientsymptom relation (using step1) 

Table 2:  Symptomdisease relation (using step2) 

Table 3:  Average for patientsymptom relation 

Table 4:  Supstar composition between symptomdisease relation and average for patientsymptom relation 

Table 5:  Grade function and similarity grade function (using step 5 and step 7) 

Table 6:  Logarithmic distance, exponential measure, similarity measure, logarithmic function and exponential function(using step 6 and step 7) 

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.