Solving Intuitionistic Fuzzy Nonlinear Equations

Available online at www.ispacs.com/jfsva Volume 2014, Year 2014 Article ID jfsva-00142, 6 Pages

doi:10.5899/2014/jfsva-00142 Research Article

Solving Intuitionistic Fuzzy Nonlinear Equations

(1) Department of Applied Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran.

Copyright 2014 c⃝ Mohammad Keyanpour and Tahereh Akbarian. This is an open access article distributed under the Creative Commons At-tribution License, which permits unrestricted use, disAt-tribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper a numerical method is proposed to solve intuitionistic fuzzy nonlinear equations. The method is based on the Midpoint Newton’s method in which the intuitionistic fuzzy quantities are presented in parametric form. The efficiency of the proposed method is tested with an illustrative example.

Keywords: Midpoint Newton’s method, Fuzzy set, Intuitionistic fuzzy set, Fuzzy parametric form, Intuitionistic fuzzy nonlinear equations, Fixed point iteration.

1 Introduction

The concept of intuitionistic fuzzy sets is the generalization of the concept of fuzzy sets. The theory of intuitionis-tic fuzzy sets is well suited to dealing with vagueness[11]. Systems of intuitionisintuitionis-tic fuzzy nonlinear equations (IFNE) appeared in a wide variety of research topics in engineering and science. As analytical solutions are seldom available, numerical methods gathered significant attentions among scientists to solve (IFNE). Though solving fuzzy equations and fuzzy nonlinear equations has long been an attractive area of research in fuzzy set theory and many works have been done on these topics, the interested reader is referred to [7, 8, 10, 13, 1], no method is proposed for solving IFNE. The goal of the present paper is to propose a numerical method to solve IFNE. The intuitionistic fuzzy set properties is introduce first, then the Midpoint Newton’s method for solving intuitionistic fuzzy nonlinear equations is presented.

2 Concepts of intuitionistic fuzzy sets

In this section we will review some basic concepts related to intuitionistic fuzzy sets Definition 2.1. [14]. Let X be a universe of discourse, then a fuzzy set is defined as

A ={< x,µA(x) > | x ∈ X}

whereµA(x) : X→ [0,1] is the membership function of the fuzzy set A.

An intuitionistic fuzzy set(IFS) ˜A is defined as follows.

Definition 2.2. [3]. Let X be a universal of discourse. An intuitionistic fuzzy set eA in X is an object having the following form

e

A ={< x,µ_Ae(x),ν_Ae(x) > | x ∈ X}

where the functionsµ_Ae(x) : X→ [0,1] andν_Ae(x) : X→ [0,1] define the degree of membership and degree of

non-membership of the element x∈ X to the set eA⊆ X , respectively, and for every x ∈ X , 0 ≤µ_Ae(x) +ν_e

A(x)≤ 1. Definition 2.3. [4, 5, 6]. For each IFS eA in X , and x∈ X,

πAe(x) = 1−µAe(x)−νAe(x),

is called the degree of indeterminacy of x to eA.

Atanassov [4, 5] extended the r-cut for intuitionistic fuzzy sets, as follows: Definition 2.4. For r∈ (0,1]), r-cut of an IFS ˜A is defined as:

e

Ar={x ∈ R|µAe(x)≥ r, νAe(x)≤ 1 − r}.

We separately define the 0-cut as

e

A0= cl({x ∈ R|µAe(x)≥ 0, νAe(x)≤ 1}).

Definition 2.5. A fuzzy number u in parametric form is a pair u = ([u, u], [u, u]) of function u(r), u(r), u(r), u(r), which

satisfies the following requirements:

1.u(r) is a bounded monotonic increasing left continuous function, 2.u(r) is a bounded monotonic decreasing left continuous function, 3.u(r) is a bounded monotonic increasing left continuous function, 4.u(r) is a bounded monotonic decreasing left continuous function,

5.u(r)≤ u(r),u(r) ≤ u(r),0 ≤ r ≤ 1.

Definition 2.6. Let the numbers b1≤ a1≤ b2≤ a2≤ a3≤ b3≤ a4≤ b4be given. A trapezoidal intuitionistic fuzzy

number is denoted byeu = (b1, a1, b2, a2, a3, b3, a4, b4), and its membership and non-membership function is defined

as follows: µeu(x) =            0 x < a1 x−a1 a2−a1 a1≤ x ≤ a2, 1 a2≤ x ≤ a3 x−a4 a3−a4 a3≤ x ≤ a4, 0 a4≤ x. νeu(x) =              1 x < b1 b2−x b2−b1 b1≤ x ≤ b2, 0 b2≤ x ≤ b3 x−b3 b4−b3 b3≤ x ≤ b4, 1 b4≤ x. (2.1)

Its parametric form is

u(r) = (a2− a1)r + a1, u(r) = (a3− a4)r + a4,

u(r) = (b2− b1)r + b1, u(r) = (b3− b4)r + b4.

Let T F(R) be the set of all trapezoidal fuzzy numbers. The addition and scalar multiplication of fuzzy numbers are de-fined by the extension principle and can be equivalently represented as follows. For arbitrary m = ([m, m], [m, m]), n = ([n, n], [n, n]) and k > 0 we define addition (n + m) and multiplication by scaler k as

n + m = ([n + m, n + m], [n + m, n + m]), kn = ([kn, kn], [kn, kn]),

3 The Midpoint Newton’s method

In this section we obtain a solution for fuzzy nonlinear equation F(x) = c. The parametric form is as follows:        F(x, x, x, x, r) = c(r), F(x, x, x, x, r) = c(r), F(x, x, x, x, r) = c(r), F(x, x, x, x, r) = c(r). (3.2) for all r∈ [0,1].

Suppose that x = (α,α,α,α) is the solution of system (3.2), and x0= (x0, x0, x₀, x0) is an approximate solution of the system. Then there are h(r), k(r), ¯h(r), and ¯k(r) such that,

       α(r) = x₀(r) + h(r), α(r) = x0(r) + k(r), α(r) = x 0(r) + ¯h(r), α(r) = x0(r) + ¯k(r). (3.3)

If we use the Taylor series of F, F, F, F about (x₀, x0, x₀, x0) then we can obtain the approximate solution of Eq.(3.2) by solving the following system:

                                             F(α,α,α,α, r) = F(x₀, x0, x₀, x0, r) + hFx(x0, x0, x₀, x0, r) + kFx(x0, x0, x₀, x0, r) +¯hF_x(x₀, x0, x₀, ¯kx0, r) + ¯kFx(x0, x0, x₀, x0, r) + O((h + k + ¯h + ¯k)2) = c(r), F(α,α,α,α, r) = F(x₀, x0, x₀, x0, r) + hFx(x0, x0, x₀, x0, r) + kFx(x0, x0, x₀, x0, r) +¯hFx(x0, x0, x₀, ¯kx0, r) + ¯kFx(x0, x0, x₀, x0, r) + O((h + k + ¯h + ¯k)2) = c(r), F(α,α,α,α, r) = F(x₀, x0, x₀, x0, r) + hF_x(x0, x0, x₀, x0, r) + kF_x(x0, x0, x₀, x0, r) +¯hF x(x0, x0, x0, ¯kx0, r) + ¯kFx(x0, x0, x0, x0, r) + O((h + k + ¯h + ¯k) 2₎ = c(r), F(α,α,α,α, r) = F(x₀, x0, x₀, x0, r) + hFx(x0, x0, x₀, x0, r) + kFx(x0, x0, x₀, x0, r) +¯hFx(x0, x0, x₀, ¯kx0, r) + ¯kFx(x0, x0, x₀, x0, r) + O((h + k + ¯h + ¯k)2) = c(r). (3.4)

and if x₀, x0, x₀and x0are near toα,α,αandα respectively, then h(r), k(r), ¯h(r) and ¯k(r) are small enough.

Suppose that all partial derivatives exist and are bounded. Therefore for enough small h(r), k(r), ¯h(r) and ¯k(r), we

have                              F(x₀, x0, x₀, x0, r) + hFx(x0, x0, x₀, x0, r) + kFx(x0, x0, x₀, x0, r) +¯hF_x(x₀, x0, x₀, ¯kx0, r) + ¯kFx(x0, x0, x₀, x0, r) = c(r), F(x₀, x0, x₀, x0, r) + hFx(x0, x0, x₀, x0, r) + kFx(x0, x0, x₀, x0, r) +¯hFx(x0, x0, x₀, ¯kx0, r) + ¯kFx(x0, x0, x₀, x0, r) = c(r), F(x₀, x0, x₀, x0, r) + hF_x(x0, x0, x₀, x0, r) + kF_x(x0, x0, x₀, x0, r) +¯hF x(x0, x0, x0, ¯kx0, r) + ¯kFx(x0, x0, x0, x0, r) = c(r), F(x₀, x₀, x 0, x0, r) + hFx(x0, x0, x0, x0, r) + kFx(x0, x0, x0, x0, r) +¯hFx(x0, x0, x₀, ¯kx0, r) + ¯kFx(x0, x0, x₀, x0, r) = c(r). (3.5)

and since h(r), k(r), ¯h(r) and ¯k(r), are unknown quantities, they can be obtained by solving the following equations,

J(x₀, x0, x₀, x0, r)     h(r) k(r) ¯h(r) ¯k(r)     =      c(r)− F(x₀, x0, x₀, x0, r) c(r)− F(x₀, x₀, x 0, x0, r) c(r)− F(x₀, x0, x₀, x0, r) c(r)− F(x₀, x0, x₀, x0, r)     , (3.6)

where J(x₀, x0, x₀, x0, r) =      F_x(x₀, x0, x₀, x0, r) Fx(x0, x0, x₀, x0, r) Fx(x0, x0, x₀, x0, r) Fx(x0, x0, x₀, x0, r) Fx(x0, x0, x₀, x0, r) Fx(x0, x0, x₀, x0, r) Fx(x0, x0, x₀, x0, r) Fx(x0, x0, x₀, x0, r) F x(x0, x0, x0, x0, r) Fx(x0, x0, x0, x0, r) Fx(x0, x0, x0, x0, r) Fx(x0, x0, x0, x0, r) Fx(x0, x0, x₀, x0, r) Fx(x0, x0, x₀, x0, r) Fx(x0, x0, x₀, x0, r) Fx(x0, x0, x₀, x0, r)      (3.7) The next approximations for x(r), x(r), x(r), and x(r) are found by using the recursive scheme as follows:

    x_n+1(r) xn+1(r) x n+1(r) xn+1(r)     =     x_n(r) xn(r) x n(r) xn(r)     + J−1(xn+ zn 2 , xn+ zn 2 , x n+ zn 2 , xn+ zn 2 , r)      c(r)− F(x_n, xn, x_n, xn, r) c(r)− F(x_n, xn, x_n, xn, r) c(r)− F(x_n, xn, x_n, xn, r) c(r)− F(x_n, xn, x_n, xn, r)     , where _    z_n(r) zn(r) z n(r) zn(r)     =     x_n(r) xn(r) x n(r) xn(r)     +     hn(r) kn(r) ¯hn(r) ¯kn(r)    , where n = 1, 2,··· . For initial guess, one can use the intuitionistic fuzzy number

x0= (x(0), x(0), x(1), x(1), x(1), x(1), x(0), x(0)).

Remark 3.1. Sequence{(x_n, xn, x_n, xn)}n=0∞ convergent to{(α,α,α,α)} iff for all r ∈ [0,1], lim_n→∞x_n(r) =α(r), limn→∞xn(r) =α(r), limn→∞x_n(r) =α(r) and limn→∞xn(r) =α(r) . Lemma 3.1. Let

F(α,α,α,α) = (F(α,α,α,α), F(α,α,α,α), F(α,α,α,α), F(α,α,α,α)) = (0, 0, 0, 0)

and if the sequence{(x_n, xn, x_n, xn)}n=0∞ converges to{(α,α,α,α)} according to Midpoint Newton’s method, then

lim n→∞Pn= 0 where Pn= sup 0≤r≤1 max{hn(r), kn(r), hn(r), kn(r)}.

Proof. In the convergent case, we have

lim

n→∞hn(r) = limn→∞kn(r) = limn→∞hn(r) = limn→∞kn(r) = 0, which completes the proof.

Finally, in the following it is shown that, under certain conditions, Midpoint Newton’s method for fuzzy equation

F(x) = 0 is convergent and thr rate of convergence is quadratic.

Theorem 3.1. Let for all r∈ [0,1], the functions F,F,F and F are continuously differentiable with respect to x,x,x

and x at each point of an open neighborhood D⊂ R4_{of x(r) = (α(r),α(r),α(r),α(r)), that is F(x) = F(x) = F(x) =}

F = 0. Let us suppose that J will be Lipschitz continuous on D and nonsingular at x. Then the Midpoint Newton’s

method converges to x = (α,α,α,α), and there exists a constant M > 0 such that,

∥(xn+1, xn+1, x_n+1, xn+1)− (α,α,α,α)∥ ≤ M∥(xn, xn, x_n, xn)− (α,α,α,α)∥2

4 Numerical examples

Here we present an example to illustrate the Midpoint Newton’s method for an ituitionistic fuzzy nonlinear equa-tion.

Example 4.1. Consider an ituitionistic fuzzy nonlinear equation.

(3, 3, 4, 4, 4, 4, 5, 5)x2+ (1, 1, 2, 2, 2, 2, 3, 3)x = (1, 1, 2, 2, 2, 2, 3, 3)

First assume that x is positive, then the parametric form of this equation is as follows:

       (3 + r)x2(r) + (1 + r)x = 1 + r, (5− r)x2_{(r) + (3− r)x = 3 − r,} (3 + r)x2_{(r) + (1 + r)x = 1 + r,} (5− r)x2(r) + (3− r)x = 3 − r. (4.8)

To obtain initial guess we use above system for r = 0 and r = 1, let

x0= (0.4343, 0.4343, 0.5, 0.5, 0.5, 0.5, 0.53066, 0.53066)

be initial guess. After 2 iterations, we obtain the solution by Midpoint Newton’s method with the maximum error less than 10−9. For more details see Figures 1.

Figure 1:Midpoint Newton’s Method for x .

Now suppose x is negative, we have        (3 + r)x2(r) + (1 + r)x = 1 + r, (5− r)x2_{(r) + (3− r)x = 3 − r,} (3 + r)x2_{(r) + (1 + r)x = 1 + r,} (5− r)x2(r) + (3− r)x = 3 − r. (4.9)

For r = 0, we have,x(1) =−.8259 and x(1) = −1.0467, hence x(1) > x(1), therefore negative root does not exist.

5 Conclusion

In this article, the Midpoint Newton’s method used to solve ituitionistic fuzzy nonlinear equations. Initially we wrote ituitionistic fuzzy nonlinear equations in parametric form and then solve them by Midpoint Newton’s method. Finally, an example presented show efficiency and applicability of the proposed method.

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