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INTUITIONISTIC FUZZY EQUATIONS AND ITS

APPLICATION ON RELIABILITY EVALUATION

1A. Dhanalakshmi, 2G. Michael Rosario

1,2 PG Department of Mathematics, Jayaraj Annapackiam College for Women(Autonomous),

Tamilnadu, (India)

ABSTRACT

Reliability analysis using Intuitionistic Fuzzy Numbers (IFNs) is studied by many researchers, because of its

importance in wide range applications in real world. In this paper, due to the lack of a well-established theory of

Intuitionistic Fuzzy Equations, we intend to characterize some properties of Intuitionistic Fuzzy Equations by

discussing equations of two very simple types : where : are

Intuitionistic Fuzzy Numbers and is an unknown Intuitionistic Fuzzy Number for which either of the equations is

to be satisfied. An approach to evaluate the unknown components of system failure using Intuitionistic Fuzzy

Numbers is presented by using Intuitionistic Fuzzy Fault tree analysis.

Keywords : Fuzzy set, Intuitionistic Fuzzy Number, Intuitionistic Fuzzy Equations, System Reliability

I INTRODUCTION

Research on the theory of fuzzy sets has been growing steadily since the inception of the theory by L.A. Zadeh [1]

and has meaningful applications in many fields like engineering, medical science, social science, graph theory etc.

In fuzzy set theory, the membership of an element to a fuzzy set is a single value in [0, 1] and it represents the

degree of belongingness of the element to the fuzzy set. However in reality, it may not always be true that the

degree of non-membership of an element in a fuzzy set is equal to one minus the membership degree because there

may be some hesitation degree. Therefore, a generalization of fuzzy sets was proposed by K. Atanassov [2] as

Intuitionistic fuzzy sets. Intuitionistic fuzzy sets (IFS) are sets whose elements have degrees of membership and

non-membership which is an extension of L. Zadeh’s [1] notion of fuzzy set. Atanassov (1999, 2012) [2-8] carried

out rigorous research based on the theory and applications of Intuitionistic fuzzy sets. Szmidt and Kacrzyk [9],

Cornelis, Deschrijver and Kerre[10], Buhaesku [11], Stoyanova and Atanassov [12], Stoyanova [13], Deschrijver

and Kerre [14] are also contributed much in Intuitionistic Fuzzy Sets. Burillo [15] et al proposed definition of

Intuitionistic Fuzzy Number (IFN) and studied perturbations of IFN and the first properties of the correlation

between these numbers. Mitchell [16] considered the problem of ranking a set of intuitionistic fuzzy numbers to

define a fuzzy rank and a characteristic vagueness factor for each intuitionistic fuzzy number.

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In real world problems, the collected data or system parameters are often imprecise because of incomplete or nonobtainable

information, and the probabilistic approach to the conventional reliability analysis is inadequate to

account for such built-in uncertainties in data. Therefore concept of fuzzy reliability has been introduced and

formulated either in the context of the possibility measures or as a transition from fuzzy success state to fuzzy failure

state [17]. Cheng and Mon [18] considered that components are with different membership functions, then interval

arithmetic and α-cuts were used to evaluate fuzzy system reliability. G. S. Mahapatra and T. K. Roy[19] introduced

intuitionistic fuzzy number and its arithmetic operations based on extension principle of intuitionistic fuzzy sets.

They also presented that the arithmetic operation of two or more IFN is again an IFN.

The Intuitionistic theory is a relatively new branch of the fuzzy set theory and so there are many unsolved or

unformulated problems in it. In the Intuitionistic Fuzzy Theory, there are a bunch of open problems. One area of

such theory in which IFNs and arithmetic operations on IFNs play a fundamental role are Intuitionistic fuzzy

equations. In this paper, Intuitionistic Fuzzy Equations according to the approach of arithmetic operations on IFNs

is presented. Trapezoidal intuitionistic fuzzy number (TrIFN) is defined and simple Intuitionistic Fuzzy Equations

(IFE) are solved. The difficulty in solving such equations arises due to the fact that arithmetic operations on these

equation does not lead to the exact solution. The example presented by G. S. Mahapatra and T. K. Roy [20]

‘Application of system failure using Intuitionistic Fuzzy number’ is used to verify the concept presented in this

paper. Intuitionistic fuzzy equations using TrIFNs are used to evaluate the unknown components of imprecise

system failure by Intuitionistic fuzzy fault tree analysis.

II BASIC CONCEPT OF INTUTIONISTIC FUZZY NUMBERS

Definition : 2.1 An intuitionistic fuzzy set [Attanassov, 1986] on X is given by

with and such that

for all .

The value is a lower bound on the degree of membership of derived from the evidence for and is

a lower bound on the negation of derived from the evidence against . We will call ,

, , the intuitionistic index of . It is the hesitancy of in , and expressed

lack of knowledge of whether or not.

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An IFS in X is characterized by a membership function and non-membership function . Here

and are associated with each point in X, a real number in [0,1] with the value of and at

X representing the grade of membership and non-membership of x in . Thus closure the value of to unity

and the value of to zero; higher the grade of membership and lower the grade of non-membership of x.

When is an ordinary set its membership function or non-membership function can take on only two values 0 and

1. If and the element x does not belong to . An IFS becomes a fuzzy set when

but .

Definition : 2.2 A set of (α, β) – cut, generated by IFS , where α, β are fixed numbers such that α + β

is defined as

(α, β) – level interval or (α, β) – cut, denoted by , is defined as the crisp set of elements x which

belong to atleast to the degree α and which does belong to at most to the degree β.

Definition 2.3 Intuitionistic Fuzzy Number (IFN):

An IFN is defined as follows:

i) An intuitionistic fuzzy subset of the real line.

ii) Normal, i.e., there is any such that (so )

iii) A convex set for the membership function , i.e.,

iv) A concave set for the non – membership function , i.e.,

.

,

Fig. 2 Membership and non membership functions of

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Definition 2.4 Trapezoidal Intuitionistic Fuzzy Number (TrIFN):

A TrIFN (Fig.1) is a subset of IFS in R with membership function and non - membership function as follows

and

Where and TrIFN is denoted by

Note : Here (x) increases with constant rate for and decreases with constant rate for

but decreases with constant rate for and increases with constant rate for .

III ARITHMETIC OPERATIONS AND SOME PROPERTIES ON INTUITIONISTIC FUZZY

NUMBERS

In this section, the arithmetic operations of IFNs based on intuitionistic fuzzy extension principle and

approximation ((α,β)-cuts) method introduced by G.S. Mahapatra, T.K. Roy [15] is presented.

3.1 Arithmetic Operations of Intuitionistic Fuzzy Numbers Based on Extension Principle

The arithmetic operation (*) of two IFNS is a mapping of an input vector X=[ ]T define in the Cartesian product

space RxR onto an output y define in the real space R. If are IFN then their outcome of arithmetic

operation is also a IFN determined with the formula

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.

To calculate the arithmetic operation of IFNs it is sufficient to determine the membership function and nonmembership

function as follows

and .

3.2 Arithmetic Operations of Intuitionistic Fuzzy Numbers Based on (α, β)-cuts Method

If is an IFN, then (α, β) - cut is given by

with α + β 1.

Here (i) > 0, < 0 for all α (0, 1), and

(ii) < 0, > 0 for all (0, 1),

It is expressed as , α + β , α, β [0, 1].

For instance, if is a TrIFN, then (α, β)-level intervals or (α, β)-cuts is

, α + β , α, β [0, 1]

Where ;

G.S. Mahapatra and T.K. Roy [20] presented that the arithmetic operation of two or more intuitionistic fuzzy

number is again an intuitionistic fuzzy number. The properties introduced by them are mentioned below which

forms the basic for this paper.

Property 3.1

(a) If TrIFN and y = ka (with k>0), then is a TrIFN

.

(b) If y = ka (with k<0, i.e., k is negative), then is a TrIFN

.

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Property 3.2

If and are two TrIFNs, then

is also TrIFN = .

Note: If we have the transformation ( are real numbers, not all zero), then the IFS

is the following TrIFN:

(i)

(ii)

(iii)

(iv)

Property 3.3

If and are two TrIFNs, then

is also TrIFN = .

Property 3.4

If and are two TrIFNs, then

is also TrIFN = .

IV INTUITIONISTIC FUZZY EQUATIONS

Intuitionistic fuzzy equations are equations in which coefficients and unknowns are Trapezoidal Intuitionistic fuzzy

number (TrIFN) and formulas are constructed by arithmetic operations on Intuitionistic fuzzy number. Here, we

only intend to characterize some properties of Intuitionistic fuzzy equations by discussing equations of two very

simple types

and

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where are TrIFNs and X is an unknown Trapezoidal Intuitionistic fuzzy number for which

either of the equations is to be satisfied.

Property 4.1 The equation …(1) has a solution if and only if

Where and are two TrIFNs.

Proof. We first prove that is not the solution.

i.e.,

substituting the value for in (1) we get

( ), ( ), ( ), ( );

( ), ( ), ( ), ( ) )

=

i.e., whenever

is not a solution of the equation.

Let Then the Intuitionistic fuzzy equation can be given by

.

From this we get the solution, as

Since we have as a TrIFN, we should have .

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i.e., the equation has a solution if and only if

Property 4.2: The equation …(2) has a solution if and only if

Where and are two TrIFNs.

Proof. We first prove that is not the solution.

i.e.,

substituting the value for in (2) we get,

=

i.e., is not a solution of the equation (2).

Let Then the Intuitionistic fuzzy equation can be given by

.

From this we get the solution, as

Since we have as a TrIFN, we should have .

i.e., the equation has a solution if and only if

In addition, Any Intuitionistic Fuzzy Number can be uniquely represented by its (α, β) cuts. Hence the described

procedure can be applied to (α, β) cuts of arbitrary TrIFNs.

i.e., The (α, β) cut of the Intuitionistic fuzzy equation, has a solution,

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, α + β , α, β

Where , ,

and .

4.1 An Example to Find The Solution of Intuitionistic fuzzy equations:

Let , be two TrIFNs whose membership an non-membership functions are given by

, ,

,

Solve the following Intuitionistic fuzzy equations for and find the membership and non-membership function for

a)

b)

Solution:

From the membership and non-membership function of and . We can formulate the TrIFN as

First, let us consider the Intuitionistic fuzzy equation --------- (1)

Here the solution

is possible iff

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Hence (1) has a solution as

Now, Let us consider the Intuitionistic fuzzy equation ---------- (2)

Substituting the solution in (2) and by means of arithmetic operations on TrIFN we can calculate as

Hence the membership and non-membership function of is given by

,

Note: The cut of the Intuitionistic fuzzy equation is given by

where

Let , then

Hence

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V EVALUATING UNKNOWN COMPONENTS IN SYSTEM FAILURE USING

INTUITIONISTIC FUZZY EQUATIONS & FUZZY FAULT TREE ANALYSIS

Starting failure of an automobile depends on different facts which is briefly explained by G.S. Mahapatra and T.K.

Roy [15]. The facts are battery low charge, ignition failure and fuel supply failure. There are two sub-factors of

each of the facts. The fault tree of failure to start of the automobile is shown below.

Suppose we are given the data such that the fuzzy failure to start an automobile and other facts are presented and we

are intiated to calculate or compute the value of inner components such as the value for ignition failure, the value for

battery internal short, the value for spark plug failure and the value for fuel pump failure. In such case, it is difficult

to follow the arithmetic operations defined on TrIFNs as it deviates from actual value. Hence Intuitionistic Fuzzy

equations play a vital role in such situations to bring out the appropriate value for the unknown TrIFNs.

represents the system failure to start of automobile.

represents the failure to start of automobile due to Battery Low Charge.

represents the failure to start of automobile due to Ignition Failure.

represents the failure to start of automobile due to Fuel Supply Failure.

represents the failure to start of automobile due to Low Battery Fluid.

TrIFN representing the system failure to start an automobile

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represents the failure to start of automobile due to Battery Internal Short.

represents the failure to start of automobile due to Wire Harness Failure.

represents the failure to start of automobile due to Spark Plug Failure.

represents the failure to start of automobile due to Fuel Injector Failure.

represents the failure to start of automobile due to Fuel Pump Failure.

The intuitionistic fuzzy failure to start of an automobile can be calculated when the failures of the occurrence of

basic fault events are known. The numerical explanation for starting failure of the automobile using fault tree

analysis with intuitionistic fuzzy failure rate is presented below. The components failure rates as TrIFN are given by

,

,

,

,

.

Failure to start of an automobile can be evaluated by using the following steps:

Step 1.

……………………………………..(1)

Using equation (1) and the numerical data presented above, we can compute the unknown value of

as follows.

=

Then equation (1) becomes ……………….(1a)

Using the arithmetic operations, we can compute the value of , then we substitute the value

in (1a) and using the concept of Intuitionistic fuzzy equations we can compute the value of as

.

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Step 2.

By similar procedure explained above, we can able to compute the unknown values of using the

concept of intuitionistic fuzzy equations as

Hence, we can able to compute the unknown values of a inner component of fuzzy system failure to start a

system by means of intuitionistic fuzzy equations to get the appropriate value.

VI CONCLUSION

In this paper, Intuitionistic fuzzy equations are defined using the properties of IFNs. The difficulty in solving IFE

arise due the fact that normal arithmetic operations defined on IFN does not lead us to the exact or appropriate

solution. IFSs separate the positive and negative evidence for the membership of an element in a set. Finally, the

result is verified by using the example given by G.s. Mahapatra, T.K. Roy. They computed the Intuitionistic fuzzy

failure to start of an automobile when the failures of the occurrence of basic fault events are known. In this paper,

we have taken the intuitionistic fuzzy failure to start of an automobile as known and evaluated the unknown basic

fault events such as Ignition failure, Battery internal shortage, Spark plug failure and fuel pump failure using

Intuitionistic Fuzzy equations. Our computational procedure is very simple to implement for calculations in

intuitionistic fuzzy environment for all field of engineering and sciences where vagueness occur.

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