Fuzzy soft set theory applied to medical diagnosis using fuzzy arithmetic operations
© Çelik and Yamak; licensee Springer 2013
Received: 26 November 2012
Accepted: 14 February 2013
Published: 1 March 2013
In this paper, we apply fuzzy soft set theory through well-known Sanchez’s approach for medical diagnosis using fuzzy arithmetic operations and exhibit the technique with a hypothetical case study.
AMS Subject Classification:03E72, 92C50.
1 Introduction and preliminaries
Many complicated problems in economics, engineering, social sciences, medical sciences and many other fields involve uncertain data. These problems, which one comes face to face with in life, cannot be solved using classical mathematic methods. In classical mathematics, a mathematical model of an object is devised and the notion of the exact solution of this model is determined. Because of that, the mathematical model is too complex, the exact solution cannot be found. There are several well-known theories to describe uncertainty. For instance, fuzzy set theory , rough set theory  and other mathematical tools. But all of these theories have their inherit difficulties as pointed out by Molodtsov . To overcome these difficulties, Molodtsov introduced the concept of a soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties affecting the existing methods. The theory of soft sets has rich potential for applications in several directions, a few of which were demonstrated by Molodtsov in his pioneer work . At present, works on soft set theory are making progress rapidly. Maji et al.  initiated the concept of fuzzy soft sets with some properties regarding fuzzy soft union, intersection, complement of a fuzzy soft set, De Morgan’s law etc. Neog and Sut  have reintroduced the notion of fuzzy soft sets and redefined the complement of a fuzzy soft set accordingly. They have shown that the modified definition of the complement of a fuzzy soft set meets all the requirements that the complement of a set in the classical sense really does. Applications of fuzzy soft set theory in many disciplines and real life situations have been studied by many researchers. De et al.  have studied Sanchez’s [7, 8] method of medical diagnosis using an intuitionistic fuzzy set. Saikia et al.  have extended the method in  using intuitionistic fuzzy soft set theory. In , Chetia and Das have studied Sanchez’s approach of medical diagnosis through IVFSS (interval-valued fuzzy soft set) obtaining an improvement of the same set presented in De et al. . Using the representation of interval-valued fuzzy matrix, Meenakshi and Kaliraja  have provided the techniques to study Sanchez’s approach of medical diagnosis of interval-valued fuzzy matrices. In this paper, by using the notion of a fuzzy soft set together with arithmetic operations on fuzzy number, we apply the fuzzy soft set technology through well-known Sanchez’s  approach for medical diagnosis, and we exhibit the technique with a hypothetical case study.
In this section, we will give some known and useful definitions and notations regarding a soft set and a fuzzy soft set. The definitions and notions in this part may be found in references [1, 3, 4, 12–14].
Let U be an initial universal set and E be a set of parameters. The power set of U is denoted by and A is a subset of E. A pair is called a soft set over U, where F is a mapping given by .
Definition 2 
if and only if for all .
Definition 3 
Let U be a common universe, E be a set of parameters and . Then a pair is called a fuzzy soft set over U, where F is a mapping given by .
Definition 4 
for all .
In this case, we write .
Definition 5 
The relative complement of a fuzzy soft set is denoted by and is defined by , where is a mapping given by for all .
It should be noted that denotes the fuzzy complement of .
Definition 6 
A fuzzy soft set is said to be the absolute fuzzy soft set over U, denoted by Ω, if for all .
A fuzzy soft set is said to be the null fuzzy soft set over U, denoted by Φ, if for all .
- (i)The union of fuzzy soft sets and is defined as the fuzzy soft set over U, where and
The restricted intersection of fuzzy soft sets and is defined as the fuzzy soft set over U, where and for all .
The restricted union of fuzzy soft sets and is defined as the fuzzy soft set over U, where and for all .
- (iv)The extended intersection of fuzzy soft sets and is defined as the fuzzy soft set over U, where and
The ∧-intersection of fuzzy soft sets and is defined as the fuzzy soft set over U, where and for all .
The ∨-union of fuzzy soft sets and is defined as the fuzzy soft set over U, where and for all .
Example Let be the set of three houses under consideration and be the set of parameters.
Definition 8 
A fuzzy subset μ on the universe of discourse ℝ (the set of all real numbers) is convex if and only if for , where .
A fuzzy subset μ on the universe of discourse U is called a normal fuzzy subset if there exist such that .
A fuzzy number is a fuzzy subset defined on the universe of discourse ℝ which is both convex and normal.
If the membership function is piecewise linear, then μ is said to be a trapezoidal fuzzy number.
2 Methodology and algorithm
In this section we present an algorithm for medical diagnosis using fuzzy arithmetic operations. Assume that there is a set of m patients, with a set of n symptoms related to a set of k diseases .
We apply fuzzy soft set theory to develop a technique through Sanchez’s method to diagnose which patient is suffering from what disease. For this, construct a fuzzy soft set over S where F is a mapping . This fuzzy soft set gives a relation matrix Q, called patient-symptom matrix, where the entries are fuzzy numbers parameterized by a triplet .
Then construct another fuzzy soft set over D, where G is a mapping . This fuzzy soft set gives a relation matrix (weighted matrix) R, called symptom-disease matrix, where each element denotes the weight of the symptoms for a certain disease. These elements are also taken as triangular fuzzy numbers.
Now if for , then we conclude that the patient is suffering from disease . In case occurs for more than one value of l, , then we can reassess the symptoms to break the tie.
Step I: Input the soft set to obtain the patient-symptom matrix Q.
Step II: Input the soft set to obtain the symptom-disease matrix R.
Step III: Perform the transformation operation to get the patient diagnosis matrix .
Step IV: Defuzzify all the elements of the matrix by (1) to obtain the matrix .
Step V: Find s for which .
Then we conclude that the patient is suffering from disease .
3 Case study
Suppose there are three patients John, George and Albert in a hospital with symptoms temperature, headache, cough and stomach problem. Let the possible diseases relating to the above symptoms be viral fever, typhoid and malaria. Now take as the universal set where , and represent patients John, George and Albert, respectively. Next consider the set as a universal set where , , , represent symptoms temperature, headache, cough and stomach problem, respectively and the set , where , and represent the diseases viral fever, typhoid and malaria, respectively.
The above values are obtained from (∗).
It is clear from the above matrix that patient is suffering from disease and patients and both are suffering from disease .
We have applied the notion of fuzzy soft sets in Sanchez’s method of medical diagnosis. A case study has been taken to exhibit the simplicity of the technique. Future work in this regard would be required to study whether the notions put forward in this paper yield a fruitful result.
Dedicated to Professor Hari M Srivastava.
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