Firstly in this paper we introduce a new concept of the 2nd power of a fuzzy number. It is exponent to production (EP) method that provides an analytical and approximate solution for fully fuzzy quadratic equation (FFQE) :
The problem of finding the roots of equations like the quadratic equation has many applications in applied sciences like finance [
The rest of the paper is set out as follows. In the second section some related basic definitions of fuzzy mathematics for the analysis are recalled. In Section
The basic definitions are given as follows.
A fuzzy number is a function
Note: in this paper we consider fuzzy numbers which have a unique
The set of all these fuzzy numbers is denoted by
A fuzzy number
A fuzzy number
For arbitrary Addition: Subtraction: Scalar product:
Multiplication:
If If If If
For two important cases multiplication of two fuzzy numbers is defined by the following terms.
Arithmetics of
Two fuzzy numbers
A crisp number
Let
Therefore
Let FFQE has analytical solution; FFQE does not have analytical solution.
In this case we have
To find
Using the proposed method we obtain the following set of expressions for
if if if if
if if if if
where
In this case we do not have analytical solution, and we do not have
If
Without loss of generality suppose that
To find the optimum parameters
To delimitate maximum error for any
Necessary condition for existence of EP solution with
Since
The EP method with
This lemma is conclusion of Lemma
Suppose
We know
Because of construction of EP method for
Suppose if if if if if if if if
with
We consider only one case to discuss. We consider
In the next examples we use round numbers with approximation less than
Let
We will look for a solution where
Equation
Therefore we have analytical solution and EP solution by (
By (
Letting
Therefore this example does not have analytical solution. We look for EP solution. By (
Now we consider an example with
Letting
Therefore, this example does not have analytical solution. We look for EP solution. By (
Notice that the numerical methods needed
In this paper we introduced a new method to solve a fully fuzzy quadratic equation. To this purpose we found the optimum spreads to decrease maximum error. One of the advantages of this method is that complications do not depend on the sign of the coefficients and variable. It is possible that these equations do not have any analytical solution, but the proposed method gives us an approximate analytical solution.
The authors declare that there is no conflict of interests regarding the publication of this paper.