Uncertainty affects all the decision of experts and appears in different forms. Uncertainty is an objective fact or just a subjective impression which is closely related to individual person. The choice of an appropriate uncertainty calculus may depend on the cause of uncertainty, quantity and quality of information available, type of information processing required by the respective uncertainty calculus and the language required by the final observer. The concept of information is fully connected with concept of uncertainty. The most fundamental aspect of this connection is that uncertainty involved in any problem-solving situation is a result of some information deficiency, which may be incomplete, imprecise, fragmentary, vague, contradictory and not reliable information. Fuzzy approximate reasoning [1] [2] allows handling such type of uncertainty. Fuzzy logic proposed by L. A. Zadeh in 1965 [3] [4] covers the uncertainty with the help of membership function only. Fuzzy logic provides a mathematical theory to handle the uncertainty associated with the human decision with the help of its membership function. In 1986 the concept of intuitionistic fuzzy sets was proposed by K. Atanassov [5] [6] , which deals with membership function, non-membership function and hesitation part which has the property to incorporate the uncertainty of the information. Intuitionistic fuzzy sets are the generalization of fuzzy sets. IFSs proffer a new criterion to represent impartial knowledge and therefore, to present in a more adequate manner for may real world problems. In 2014 Hajek [7] gave the methods for the defuzzification in the inference system for the Takagi-Sugeno type so that we can observe the crisp values for the outputs.

Many real life applications have given the evidence that intuitionistic fuzzy sets are better than the traditional fuzzy logic. In this list we may also mention some more aspects [8] [9] that may give better results than traditional fuzzy logic i.e. type-I, type-II, interval valued fuzzy sets and vague sets, interval valued vague sets etc. but the intuitionistic fuzzy sets cover the uncertainty caused by membership, non-membership and the hesitation part. Castillo etc. [10] in 2003 gave a new method for the inference for the fuzzy inference based on intuitionistic fuzzy logic that is in this work he explained the importance of favorable and unfavorable cases. Again in 2007 [11] used this concept for plant monitoring system and gave a scheme for the diagnosis of the defects. In 2007 Melin [12] used the mediative fuzzy logic for the contradictory knowledge management and explained how the logic is better than previous ones. Montiel etc. [13] in 2008 gave the concept of mediative fuzzy logic that is a new approach for handling the contradiction in the decision. So we can construct an intuitionistic fuzzy logic controller similar to the fuzzy logic controller given by Jang etc. [14] in neuro-fuzzy and soft computing. 1979 Sanchez [15] used the fuzzy logic and fuzzy relation for medical diagnosis in 2001 Supriya [16] used the extension of fuzzy sets in the form of intuitionistic fuzzy sets in medical diagnosis but What happens if the knowledge base rule changes with the perception of experts give a contradictory, non-contradictory and the incomplete information or the combination of these situations? Intuitionistic fuzzy sets are inadequate to explain these situations. To deal such situations, which inference system be used. Montiel [17] etc. in 2009 gave an algorithm which is able to deal with kind of information for controlling population size, which may be inconsistent, incomplete and contradiction exists. This is a mediate solution. Mediative fuzzy logic can bring down to the intuitionistic and fuzzy logic based on the affirmations and denials are established. In 2018 Iancu [18] used the mediative fuzzy logic in the heart disease using the Mamdani fuzzy inference rule for single input and single output. In our work we have extended this by using Sugeno’s-TSK fuzzy inference system with two inputs and one output. By using this inference system we have developed an algorithm and on the basis of this algorithm we have constructed eight hundred and eighty five rules, from these rules here we have given the fifty rules and their crisp outputs and the firing level.

In this present research paper we have designed a meditative fuzzy inference system in Figure 1 for Sugeno’s-TSK fuzzy controller for the diagnosis of heart disease. The present research paper is divided into six sections. In the second section we have taken some basic definitions on mediative fuzzy logic, contradiction fuzzy sets and intuitionstic fuzzy numbers. In section third of the research paper we have developed an algorithm which is based on Sugeno’s-TSK controller using mediative fuzzy logic. The algorithm contains all the steps which we have proposed for our method including the block diagram. In section four of the research paper we have categorized the outputs and make the membership and non-membership for the stages of the risk about the sickness. In section five we have computed the values of the outputs using the defuzzification methods and the firing level of the observed during that output. In the last section conclusion of the research paper is given.

In disease diagnosis, we often find illogical information that comes from different inference system that does not concede. In case when the classical logic, fuzzy logic or intuitionistic fuzzy logic do not work, then we need to apply meditative fuzzy logic. That will give the better results considering the favorable, unfavorable and the situation where neither the membership nor non-membership help for getting the better results but also the agreement and non-agreement membership function. In this present research paper we will develop a methodology using Sugeno’s fuzzy inference system based on meditative fuzzy logic. We will develop Sugeno’s-TSK meditative fuzzy logic controller and make fuzzy rule base with using two parameters as input variables which variations may cause heart disease in the form of one output value for the diagnosis of heart disease. We have proposed the algorithm as follows:

Step 1: Suppose we have a fuzzy inference rule for conditional and unqualified proposition as hypothetical syllogism which is the generalization of inference for hypothetical syllogism for classical logic. In the present research paper we will use the inference rule for fuzzy logic for conditional fuzzy proposition in the extended form of intuitionistic fuzzy logic. That is for conditional and unqualified proposition the inference rule R is: if X 1 is A 1 and X 2 is A 2 then Y is B. Because we are working with a rule with two inputs, so for the intuitionistic fuzzy set, in case of A 1 using μ —firing level I μ 1 for the membership will be denoted by α 1 and ν —firing level l υ 1 for the non-membership for A 2 will be denoted by β 1 and μ —firing level for I μ 2 and ν —firing level I ν 2 corresponds to rule R will be denoted by α 2 and β 2 respectively. Firing level for membership and non-membership will be given by setting Here we have two input values, we give some input to both the input variable, and we get the value of firing level corresponds to membership and non membership function. Namely α₁ and β₁ for first input and α₂ and β₂ for second input respectively. This rule is represented by lukasiewicz implication and conclusion is inferred using Sugeno’s TSK Fuzzy Model.

Step 2: Using Sugeno’s fuzzy inference model with two input variables and one output variable i.e. an inference for conditional and qualified proposition we will get the firing level for the two antecedents that will result in one consequence for the membership and non-membership. Find I μ = min ( I μ 1 , I μ 2 ) for membership for the functions and the non-membership l υ = min ( l υ 1 , l υ 2 ) for the factors causing the stages of the disease, by setting α = min ( α 1 , α 2 ) and β = min ( β 1 , β 2 ) .

Step 3: After the step second we need to calculate the values for the intuitionistic fuzzy index in the form of hesitation part and contradiction that is in the form of agreement and disagreement by taking ζ = minimum of α and β, and π = 1 − ( α + β ) . The values of α and β will be taken from step 2.

Step 4: For the final Membership Fuzzy System output, the values obtained in step 3 in the form of the conclusions B μ and B ν as follows:

μ B μ ( y ) = I L ( α , μ B ( y ) ) ,         ∀ y ∈ Y = min ( 1 − α + μ B ( y ) , 1 ) (11)

y μ = d e f u z z (Bμ)

μ B ν ( y ) = I L ( β , μ B ( y ) ) ,       ∀ y ∈ Y = min ( 1 − β + μ B ( y ) , 1 ) (12)

y μ = d e f u z z (Bν)

Will be defuzzified into the values for membership fuzzy output and non-membership fuzzy output as y μ and y υ , respectively by using the defuzzification method of Middle of Maxima and Middle of Minima techniques.

Step 5: Finally for getting the output after the final step from Sugeno’s-TSK fuzzy inference system, we will get membership fuzzy system output by the formula

Z = [ 1 − ( 1 − ( α + β ) ) + 1 2 min ( α , β ) ] y μ + [ ( 1 − ( α + β ) ) + 1 2 min ( α , β ) ] y ν (13)

The last step of our proposed algorithm is to get the output in the fuzzy form. First we will obtain the aggregation of the factors and after getting aggregation of we will obtain the fuzzy form of our output. Then the output values obtained from the inference of the input in the form of fuzzy propositions can be classified into four categories. The four categories may be in the combination of qualified, unqualified, conditional and unconditional fuzzy propositions. In the four categories inference from input variables will be divided into four stages. The classification of the stages will be based on the stages to obtain the stages of risk to the patient which are namely classified as stage 1, stage 2, stage 3 and stage 4 and these will take values on the scaling from 1 to 5. Stage 1 patient considered as low risk for heart disease, Stage 2 denotes medium risk, stage 3 denote high risk and stage 4 denotes very high risk for heart disease to the patient. The functions for the output values are shown by using intuitionistic fuzzy numbers.

For membership values

μ stage1 ( x ) = { 1                           if   x < 1.35 2 − x 0.65         if   1.35 ≤ x ≤ 2 , μ stage2 ( x ) = { 1.8 − x 0.8         if   1.8 ≤ x ≤ 2.6 1                                   if   x = 2.6 3.4 − x 0.8         if   2.6 ≤ x ≤ 3.4

μ stage3 ( x ) = { 3.2 − x 0.7         if   3.2 ≤ x ≤ 3.9 1                                     if   x = 3.9 4.6 − x 0.7         if   3.9 ≤ x ≤ 4.6 , μ stage4 ( x ) = { x − 4.3 0.4         if   4.3 ≤ x ≤ 4.7 1                                   if   x > 4.7

Non membership functions

υ stage1 ( x ) = { x − 1.6 0.8         if   1.6 ≤ x ≤ 2.8 1                                   if   x > 2.8 , υ stage2 ( x ) = { 1                                   if   x < 1.6 x − 1.6 1         if   1.6 ≤ x ≤ 2.6 0                                 if   x = 2.6 x − 2.6 0.8         if   2.6 ≤ x ≤ 3.4 1                                   if   x > 3.4

υ stage3 ( x ) = { 1                                     if   x < 2.9 3.9 − x 1.5           if   2.9 ≤ x ≤ 3.9 0                                   if   x = 3.9 x − 3.9 1           if   3.9 ≤ x ≤ 4.9 1                                     if   x > 4.9 , υ stage4 ( x ) = { 1                                   if   x < 4.1 4.8 − x 0.7         if   4.1 ≤ x ≤ 4.8

Fuzzy logic provides a platform to handle the uncertainty associated with human cognizance. The cognizance may be due to reasoning or thinking of the human being. But when the information may be incomplete, vague, fragmentarily reliable that is not fully reliable, there exists contradictory remark about the information then we are not in the position to deal it with fuzzy logic. In the present research paper we will use the inference rule for fuzzy logic for conditional fuzzy proposition in the extended form of intuitionistic fuzzy logic for the input factors which have been shown in Table 1 (effects the heart diseases). That is for conditional and unqualified proposition the inference rule R is: if X 1 is A 1 and X 2 is A 2 then Y is B. In this present paper we have evaluated the firing level and output which have been shown in Table 2 with a rule which included two inputs, so for the meditative fuzzy logic. In the present research paper we have also shown the superiority of meditative fuzzy logic on the previous traditional and intuitional logics. In the present paper we have extended and improved the system by using Sugeno’s-TSK model with the help of meditative fuzzy logic. On the basis of output we can categorize the risk stages. The output of the reasoning system corresponds to the category of sickness.

The first author is grateful to University Grant Commission for the financial assistance.

The authors declare no conflicts of interest regarding the publication of this paper.

Dhiman, N. and Sharma, M.K. (2019) Mediative Sugeno’s-TSK Fuzzy Logic Based Screening Analysis to Diagnosis of Heart Disease. Applied Mathematics, 10, 448-467. https://doi.org/10.4236/am.2019.106032