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The Scientific World Journal
Volume 2014 (2014), Article ID 156203, 6 pages
The Solution of Fully Fuzzy Quadratic Equation Based on Optimization Theory
Department of Mathematics, Islamic Azad University, Science and Research Branch, Tehran, Iran
Received 16 April 2014; Accepted 17 May 2014; Published 9 June 2014
Academic Editor: Mohsen Vaez-ghasemi
Copyright © 2014 T. Allahviranloo and L. Gerami Moazam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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) : , where . To use the mentioned EP method, at first the 1-cut solution of FFQE as a real root is obtained and then unknown manipulated unsymmetrical spreads are allocated to the core point. To this purpose we find and as optimum values which construct the best spreads. Finally to illustrate easy application and rich behavior of EP method, several examples are given.
The problem of finding the roots of equations like the quadratic equation has many applications in applied sciences like finance [1, 2], economy [3–6], and mechanics . Sevastjanov and Dymova  proposed a new method for solving interval and fuzzy linear equations. In [9–11] Buckly discussed solving fuzzy equations. Abbasbandy and Otadi in  obtained the real valued roots of fuzzy polynomials using fuzzy neural networks. In [13–16] the authors have introduced numerical and neural net solutions to solve fuzzy equations. In the current paper we propose a new method to solve that the complication of arithmetics does not depend on , , and being positive or negative. This method solves some problems that have no analytical solution and this is an advantage of EP method because, as we know, numerical methods need initial guess to continue and the new method provides this neediness.
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 3, a new method, called EP, for solving fully fuzzy quadratic equation is presented. In Section 4, this method is used for an analytical approximate solution of FFQE. In Section 5, the conclusions are drawn.
2. Basic Concepts
The basic definitions are given as follows.
Definition 1 (see [17–20]). A fuzzy number is a function which satisfies the following:(1) is upper semicontinuous on ;(2) is normal; that is, with ;(3) is convex fuzzy set;(4) is compact, where denotes the closure of .
Note: in this paper we consider fuzzy numbers which have a unique with .
The set of all these fuzzy numbers is denoted by . Obviously, . For , we define -cut of fuzzy number as and . In  from (4)–(10) it follows that is a bounded closed interval for each . We denote the -cut of fuzzy number as .
Definition 2 (see ). A fuzzy number is positive (negative) if for all .
Definition 3 (see [21, 22]). A fuzzy number in parametric form is a pair of functions and , , which satisfy the following requirements:(1) is a bounded nondecreasing left continuous function in ;(2) is a bounded nonincreasing left continuous function in ;(3), .
Definition 4 (see ). For arbitrary and , , and scalar , we define addition, subtraction, and scalar product by and multiplication is, respectively, as follows. Addition: ; . Subtraction: ; . Scalar product: Multiplication: For two important cases multiplication of two fuzzy numbers is defined by the following terms. If and , then and . If and , then and . If and , then and . If and , then and .
Definition 5 (see ). Two fuzzy numbers and are said to be equal, if and only if and , for each .
A crisp number in parametric form is , . A triangular fuzzy number is popular and represented by , where and , which has the parametric form as follows:
Definition 6 (see ). Let and let = be the Hausdorff distance between fuzzy numbers, where and . The following properties are well known:(1), for all , , ;(2), for all and , ;(3), for all , , , .Therefore is a complete metric space.
3. Exponent to Production Method
Let , , , , and where . In this method, we convert the 2nd exponent of a fuzzy number to a product of two fuzzy numbers in parametric form. By this conversion we obtain an analytical and approximate solution for a FFQE. In EP method, at first, we find , as a real root of crisp 1-cut equation and then we get as solution of (4) and apply the approximation , in which , , , and . To find , for , we consider two cases:(1)FFQE has analytical solution;(2)FFQE does not have analytical solution.
In this case we have . Therefore we construct conditions that provide a good approximation. These conditions are as follows:(1), for ;(2) and .To find , , at first we substitute = and ≅, in parametric form of (4), and then we obtain Set . Using condition and , , , and , we obtain for each ; that means By this conclusion we decide to get Using condition we have Up to now we have two equations and three unknown , , and . The third equation comes from the equality below, for , Now we can find , , and . To construct solution of the new method, in the above parametric form, let and ; then, by solving a system via and , we obtain and we set as a solution of (4). Notice that always we have real spreads, which means , because and are the roots of .
Using the proposed method we obtain the following set of expressions for .
Case (1). (1)if and , then (2)if and , then (3)if and , then (4)if and , then
Case (2). (1)if and , then (2)if and , then (3)if and , then (4)if and , then where in which and .
In this case we do not have analytical solution, and we do not have and in which ; therefore we propose that because of Lemma 7:
Lemma 7. If is real core point of and we do not have analytical solution, then is the best choice in EP method.
Proof. Without loss of generality suppose that . Let
In EP method we use the approximation .
To find the optimum parameters and , we must solve the minimization problem as follows: where .
To delimitate maximum error for any , we choose . This completes the proof.
Lemma 8. Necessary condition for existence of EP solution with is
Proof. Since is sum of and and we want and to be nonnegative, then necessary condition for existence of EP solution with is and since , for all , it is sufficient to have . Considering denominators of (13)–(20) we find that (24) and (25) hold if , and this completes the proof.
Lemma 9. The EP method with does not have solution,
Proof. This lemma is conclusion of Lemma 8.
Lemma 10. Suppose in EP method and , then sufficient condition for existence of solution is .
Proof. We know and are nonnegative if and .
Because of construction of EP method for and Lemma 8, it is obvious that we must have .
Lemma 11. Suppose and in EP method, then we have solution with as follows: (1)if and , then (2)if and , then (3)if and , then (4)if and , then with ;(1)if and , then (2)if and , then (3)if and , then (4)if and , then
Proof. We consider only one case to discuss. We consider , , and and solve (11) via and in ; we obtain This is obvious that , if the numerator is nonnegative and this completes the proof.
4. Numerical Examples
In the next examples we use round numbers with approximation less than .
Example 1. Let , , , and .
We will look for a solution where .
Equation becomes in parametric form as follows: The real roots of 1-cut and 0-cut equations are , , and .
Therefore we have analytical solution and EP solution by (24) and (29).
By (10) and (14), we find , , and In this example, with , Hausdorff metric is .
Example 2. Letting , , and , we have
The real roots of 1-cut and 0-cut equations are , , and .
Therefore this example does not have analytical solution. We look for EP solution. By (26) we find that this example does not have EP solution with too.
Now we consider an example with , , and .
Example 3. Letting , , and , we have
The real roots of 1-cut and 0-cut equations are , , , and , .
Therefore, this example does not have analytical solution. We look for EP solution. By (25), (30), and (15), we find that this example has EP solution with as follows: that and 0-cut of EP solution is and . In this example by using Hausdorff metric we have .
Notice that the numerical methods needed and to obtain initial guess and often these values achieve analytical solution, but in Example 3 these values achieve EP method because in this example we do not have analytical solution.
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.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
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