A framework for medical diagnosis via fuzzy soft matrices

In this paper we introduce matrice represantation of fuzzy s oft sets. By using the notion of fuzzy soft matrices, we apply fuzzy soft set technology through the well known Sanchez’s [ 8] approach for medical diagnosis. Also we exhibit the techni que with a hypothetical case study.


Introduction
Many complicated problems in economics, engineering, social science, medical science and many other fields involve uncertain data. These problems which one come 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 [1], rough set theory [2] and other mathematical tools. But all of these theories have their inherit difficulties as pointed out by Molodtsov [3]. To overcome these difficulties, Molodtsov introduced the concept of soft set as a new mathematical tool for dealing with uncertainties that is free from the difficulties affecting existing methods. The theory of soft sets has rich potential for applications in several directions, few of which had been demonstrated by Molodtsov in his pioneer work [3]. At present, works on soft set theory are making progress rapidly. Maji et al. [4] 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 [5] 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 complement of a fuzzy soft set meets all the requirements that complement of a set in 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. [6] have studied Sanchez's [7,8] method of medical diagnosis using intuitionistic fuzzy set. Saikia et.al. [9] have extended the method in [6] using intuitionistic fuzzy soft set theory. In [10],Chetia and Das have studied Sanchez's approach of medical diagnosis through IVFSS obtaining an improvement of the same presented in De et.al. [6]. Using the representation of interval valued fuzzy matrix, Meenakshi et.al. [11] 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 fuzzy soft matrices, we apply fuzzy soft set technology through the well known Sanchez's [8] approach for medical diagnosis and we exhibit the technique with a hypothetical case study.

Preliminaries
In this section, we will give some known and useful definitions and notations regarding soft set, fuzzy soft set.The definitions and notions, in this part, may be found in references [1,3,4,12]. Definition 1. [3] Let U be an inital universe set and E be a set of parameters. The power set of U is denoted by P(U) and A is a subset of E. A pair (F, A) is called a soft set over U, where F is a mapping given by F : A → P(U).

Definition 2. [1]
A fuzzy subset µ of U is defined as a map from U to [0,1]. The family of all fuzzy subsets of U is denoted by F (U). Let µ, ν ∈ F (U) and x ∈ U. Then the union and intersection of µ and ν are defined following way:

Definition 4. [4] For two fuzzy soft sets ( F, A) and ( G, B) over a common universe U , we say that ( F, A) is a fuzzy soft subset of ( G, B) if
In this case, we write ( F, A) ⊆( G, B).

Definition 6. [4] (i) A fuzzy soft set ( F, A) is said to be absolute fuzzy soft set over U if F(a) = U for all a ∈ A. (ii) A fuzzy soft set ( F, A) is said to be null fuzzy soft set over U if F(a) = 0 U for all a ∈ A.
Definition 7. [12] Let ( F, A) and ( G, B) be two fuzzy soft sets over a common universe U. Then,

Fuzzy soft matrices
Then using Definition (11), we obtain two new relation matrices T 1 = B · A and T 2 = B · ∼ A c called patient symptom disease matrix and patient symptom non disease matrix respectively. In similarly, we obtain the relation matrices called the patient non symptom disease matrix and patient non symptom non disease matrix respectively.
Using Definition (8), we may obtain the corresponding membership value matrices MV ( T 1 ), MV ( T 2 ), MV ( T 3 ) and MV ( T 4 ). We calculate the diagnosis score S T 1 and S T 2 for and against the disease respectively as ] occurs for exactly (p i , d k ) only, then we would be in a position to accept that diagnostic hypothesis for patient p i is the disease d k . In case there is a tie, the process is repeated for patient p i by reassessing the symptoms.

Algorithm
Step I: Input the fuzzy soft set ( F, D) and compute ( F, D) c . Compute the corresponding matrices A and ∼ A c .
Step II: Input the fuzzy soft set ( G, S) and compute ( G, S) c . Compute the corresponding matrices B and ∼ B c .
Step V: Compute the diagnosis score S T 1 and S T 2 .
Step VI: Then we conclude that the patient p i is suffering from the disease d k . If S k has more than one value, then go to step I and repeat the process by reassessing the symptoms for the patient.

Case study
Suppose there are three patients John, George and Albert in a hospital those who intake over dosage for sensual pleasure which will affect the brain cells lead to the symptoms of hysteria, then the patient who used sleeping pills will have the side affect of headache and stomach pain, then the patient who take birth control pills will have side effect of depression and stroke. We consider the set S = {s 1 , s 2 , s 3 } as universal set where s 1 , s 2 , s 3 represent symptoms of hysteria, headache and stomach pain, depression and stroke problems respectively and the set D = {d 1 , d 2 } where d 1 and d 2 represent the parameters of side effect in the human body, particularly brain and heart problem disease respectively.  Next suppose,