Numerical solution of a fuzzy linear system by using PSO method

In this paper a numerical algorithm for solving a fuzzy linear system of equations (FLS) is considered. This system would be changed into an optimazition problem which is based on Particle Swarm obtimization (PSO) algorithm. The efficiency of algorithm is illustrated by some numerical examples.


Introduction
Fuzzy systems are employed in various areas such as mathematics, physics, statistics, engineering and etc. since system's parameters and measurements have uncertainity in mathematical models.First, Zadeh niteroduced fuzzy numbers and fuzzy arithmetic [18].Then it is investigated by Dubois and Prade [8].A general model for solving a fuzzy linear system with crisp coefficient matrix and the right-hand side column an arbitrary fuzzy number vector is introduced by Friedman et al. [9].They replaced a fuzzy n × n linear systems AX = B by a crisp 2n × 2n matrix.Different approaches proposed to solve this system.For example, LU decomposition method is applied to general fuzzy linear systems or symmetric fuzzy linear systems [1,2].Muzziloi et al. [14] considered fully fuzzy linear systems of the form A 1 x + b 1 = A 2 x + b 2 with A 1 , A 2 square matrices of fuzzy coefficients and b 1 , b 2 fuzzy number vectors and Dehghan et al. [8] considered fully fuzzy linear systems of the form Ax = b where A and b are a fuzzy matrix and a fuzzy vector, respectively and then discussed the iterative solution of fully fuzzy linear systems.In this study, we focus on solving fuzzy linear systems with fuzzy variations by Particle Swarm optimazation.The mentioned method is based on Particle Swarm Theory.Particle Swarm obtimization , has recently attracted by many researches, since it's applicable in various problems such as classification [3].The rest of this paper is as follows.Some background of fuzzy numbers and fuzzy differential equations which will be applied are brought in the next section.In Section 3. Some numerical examples to illustrate this method is presented in Section 4. Finally, Section 5 presents concluding remarks.

Preliminaries and notations
In this section, some definitions and features of fuzzy numbers and fuzzy differential equations which will be used throughout the paper, will reviewed.Definition 2.1.( [15]).A fuzzy number ũ is completely determined by an ordered pair of functions ũ = [u(r), u(r)], 0 ≤ r ≤ 1, satisfy the following requirements: 1. 1. u(r) is a bounded, monotonic, increasing (non decreasing) left-continuous function for all r ∈ (0, 1] and rightcontinuous for r = 0.
For every ũ = [u(r), u(r)], ṽ = [v(r), v(r)] and k > 0 addition and multiplication have the following properties: The collection of all fuzzy numbers with addition and multiplication as defined by Eqs.(1) − (3) is denoted by E 1 .For 0 < r ≤ 1 we define the r-cuts of fuzzy number ũ with ≥ r} and support of ũ is defined as respectively.ũ is a L-R fuzzy number if its membership function be as the following form: This definition is very general and covers quite different type of information.For example, fuzzy number ũ is trapezoidal fuzzy number when m < n and L( m are linear functions, ũ denotes triangular fuzzy number and we write ũ = (m, α, β ).
between fuzzy numbers is given by: where, for an interval [a, b], the norm is It is easy to see that D is a metric in E 1 and has the following properties ( [4]).
International Scientific Publications and Consulting Services Lemma 2.1.For u, v, w, e ∈ E 1 and k ∈ R, we have the following results where the coefficient matrix A = (a i j ), 1 ≤ i, j ≤ n is a crisp n × n matrix and y i ∈ E 1 is called a FSLEs.
3 Main section

Particle Swarm Optmization
PSO optimizes a problem by having a population of candidate solutions, here dubbed particles, and moving these particles around in the search-space according to simple mathematical formulae.The movements of the particles are guided by the best found positions in the search-space which are updated as better positions are found by the particles.PSO algorithm works by having a population (called a swarm) of candidate solutions (called particles).These particles are moved around in the search-space according to a few simple formulae.The movements of the particles are guided by their own best known position in the search-space as well as the entire swarm's best known position.When improved positions are being discovered these will then it will guide the movements of the swarm.The process is repeated and satisfactory solution will be discovered.PSO Variants: Various variants of a basic PSO algorithm are possible.New and some more sophisticated PSO variants are continually being introduced in an attempt to improve optimization performance.There is a trend in that research; one can make a hybrid optimization method using PSO combined with other optimization techniques [10,16].Applications: The first practical application of PSO was in the field of neural network training and was reported together with the algorithm itself (Kennedy and Eberhart 1995).Many more areas of application have been explored ever since, including telecommunications, control, data mining, design, combinatorial optimization, power systems, signal processing, and many others.PSO algorithms have been developed to solve:

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Implementation Algorithm: The PSO algorithm is simple in concept, easy to implement and computational efficient.Original PSO was implemented in a synchronous manner (Fig 2) but improved convergence rate is achieved by asynchronous PSO algorithm Figure 2 [6,17].In the PSO algorithm each individual is called a "particle", and is subject to a movement in a multidimensional space that represents the belief space.Particles have memory, thus retaining part of their previous state.There is no restriction for particles to share the same point in belief space, but in any case their individuality is preserved.Each particle's movement is the composition of an initial random velocity and two randomly weighted influences: individuality, the tendency to return to the particle's best previous position, and sociality, the tendency to move towards the neighborhood's best previous position.

The Continuous PSO
There are two versions of the basic PSO algorithm.The "continuous" version uses a real-valued multidimensional space as belief space, and evolves the position of each particle in that space using the following equations: • p i : Best position achieved so long by particle .
• p g : Best position found by the neighbors of particle .
The particle used to calculate p g depends on the type of neighborhood selected.In the basic algorithm either a global (gbest) or local (lbest) neighborhood is used.In the global neighborhood, all the particles are considered when calculating p g .In the case of the local neighborhood, neighborhood is only composed by a certain number of particles among the whole population.The local neighborhood of a given particle does not change during the iteration of the algorithm.A constraint (v max ) is imposed on v id 4 to ensure convergence.Its value is usually kept within the interval [−x max id , x max id ], being x max id the maximum value for the particle position [11].A large inertia weight (w ) favors global search, while a small inertia weight favors local search.If inertia is used, it is sometimes decreased linearly during the iteration of the algorithm, starting at an initial value close to [11,17].An alternative formulation of Eq. 1 adds a constriction coefficient that replaces the velocity constraint ( v max ) [10].The PSO algorithm requires tuning of some parameters: the individual and sociality weights (c 1 , c 2 ), and the inertia factor ( w).Both theoretical and empirical studies are available to help in selection of proper values [5,10,6,16,11,17].

PSO algorithm for solving fuzzy linear systems
In this section we used PSO algorithms for solving a fuzzy linear system by crisp coefficients.Consider Equation (2.4)We change this equation to an optimization problem by using Definition (2.3) as follows: min D(A X, Ỹ ) (3.7)

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There A = (a i j ),  3.9), we can solve this fuzzy linear system equation.In the next section we will give a numerical example using this method.

Numerical examples
In this section we present a numerical example.
Example 4.1.Consider the following fuzzy linear system 2 × 2 by crisp coefficients: We used PSO algorithm for (4.11).Numerical solution and exact solution are shown in Figure 1.Also the approximations of ( x1 , x2 ) and their errors, by this method are given in Table

Conclusion
In this paper we used PSO for solving a system of fuzzy linear equations with crisp coefficients.The obtained results shows the small amount of error.One advantage of this algorithm is that we can use it without knowing the exact solution and obtainig an approximation with small error.

Figure 1 :
Figure 1: Exact solution and numerical solution.
a 11 x 1 a 12 x 2 . . .a 1n x n = y 1 a 21 x 1 a 22 x 2 . . .a 2n x n = y 2 [7] Definition 2.4.([7])Let x, y ∈ E 1 .If there exists z ∈ E 1 such that x + y = z, then z is called the H−difference of x, y and it is denoted by x ⊖ y.