OPTIMIZATION ALGORITHM BASED ON THE EULER METHOD FOR SOLVING FUZZY NONLINEAR EQUATIONS

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Introduction
The paper [1][2][3] developed the concept of fuzzy numbers and arithmetic operations on them, which was expanded in [4]. Later, the work [5] contributed significantly by developing the main thought of LR fuzzy numbers and then presented a computational formula toward fuzzy number operations. The solution of the mentioned equations, the main parameters of which are fuzzy numbers has emerged as one of the key areas for the application of fuzzy numbers as the theory of fuzzy numbers has progressed. The solution of fuzzy equations is necessary in diverse fields such as chemistry, economics, physics, and others.
Take a look at the set of j nonlinear equations: The general form of the nonlinear equation for j = 1 can be stated simply according to a value for the variable x, which is computed as follows: where G denotes any nonlinear function of χ and g i denotes a mapping from R n to R n . The value of X is then referred to the root or solution of this equation, which could be one of several. The above-mentioned equation appears frequently in engineering, natural and social sciences. Several approaches have been proposed and used to gain its solution. However, fuzzy numbers are used to represent the parameters of these structures of nonlinear equations instead of clear numbers. Thus, the final results are dependent on the roots of the fuzzy equation [6]. Some standard analytical techniques [7][8][9] are not suitable for solving systems of nonlinear equations of the form: where χ, h, k, f, r, u, and w are fuzzy numbers. Therefore, we need to suggest the metaheuristic methods to find the solution to this type of equation. Therefore, studies that are devoted to solving fuzzy nonlinear equations and finding the optimal solution to the equations with the fewest number of iterations and in record time compared to the GWO method are of scientific relevance.

Literature review and problem statement
Actually, the Euler method calculates the penchant and solves ordinary differential equations (ODE) using a special Euler formulation [10]. OBE's fundamental notion is grounded on the Euler method's suggested calculated penchant concept. The OBE employs the calculated penchant as searching reasoning to find the most convenient area in the area of search and construct a group of statutes for the development of a population group based on the logic of the swarm-based optimization technique. The next subsections go over the mathematical formula for OBE. Some researchers have solved fuzzy nonlinear equations by some methods. In 2004, the work [6] proposed Newton's method. The disadvantage of the method is that it requires the calculation of the Hessian matrix. In 2006, the steepest descent method was proposed [11]. One of the disadvantages of the method is that it is slow. In 2008, the Harmonic Newton method was proposed [12], which is an ineffective way to get to the roots. In 2010, they used the Broyden's method [13]. Broyden's method requires calculating the inverse of the Hessian matrix. Also in 2010, the work [14] used the general iterative method for an ineffective method with all kinds of equations. In 2011, the work [15] used the iterative secant method. One of the disadvantages of the method is that it sometimes does not reach the solution. In 2016, some iterative methods were used to solve fuzzy equations such as the Bisection method [16], the False Position method [17], and the secant method modified in [18]. One of the disadvantages of these methods is that they are very slow to reach a solution. In 2018, a new class of paired gradient method [19] and Barzilai-Borwein gradient method were used [20]. These methods need to calculate the gradient vector at each iteration. In the same year, some numerical methods were proposed [21]. One of the disadvantages of these methods is that they are old, traditional and need an initial point. Stirling's-like method [22] does not guarantee reaching a solution. In 2019, the Chord Newton method was used [23], which needs to compute a Hessian matrix for each iteration. In 2020, the accelerated method was proposed [24]. This method does not guarantee reaching a solution. The quasi-Newton method was proposed to solve nonlinear fuzzy equations [25]. One of the disadvantages of the method is that it requires calculating the inverse Hessian matrix for each iteration. In 2021, they proposed four numerical methods for solving fuzzy equations [26]. Traditional numerical methods need an initial point. A Special Iterative Algorithm [27], the Free Levenberg-Marquardt method [28] were proposed. One of the disadvantages of the method is that it requires calculating the inverse Hessian matrix for each iteration. The Shamanskii method was used in [29]. One of the disadvantages of the method is that it does not guarantee reaching a solution. Spectral CG algorithm for solving fuzzy nonlinear equations was developed in [30]. This method does not reach a global solution.
All this allows us to assert that it is expedient to conduct a study on metaheuristic methods for solving fuzzy nonlinear equations, which is an effective optimization algorithm method that depends on the Euler method.

The aim and objectives of the study
The aim of the study is to develop metaheuristic methods for solving nonlinear fuzzy equations by using numerical methods such as Euler iterative method. The OBE technique is easy to use and does not require many complications, as is the case of traditional numerical methods.
To achieve this aim, the following objectives are accomplished: -to find the minimum of fuzzy equations; -to get the best value for variables; -to get the best value for functions.

Materials and methods
In this paper, fuzzy nonlinear equations are solved using a stochastic swarm-based model. The proposed OBE approach is expressed in a non-metaphorical manner, focusing on the mathematical essence as a set of active rules that are applied at the right time. It is not recommended to use metaphors with the population-based models because the only advantage is that it hides the basic structure of the equations used by the optimizers. As a result, OBE accounts for the Euler method's core logic as well as a throng of the agent's population-based evolution.
This stage's logic is to construct an initial swarm that will evolve throughout the specified number of iterations. For a population of size N, N sites are generated at random in OBE. Each individual in the population, X n (n = 1,2,…, N), is a D dimensional optimization problem solution. The initial positions are created at random in general using the following concept: The problem's 1th variable l = 1,2,…, D has lower and upper limits L l and U l , respectively, and is a random number range [0, 1]. Only a small number of solutions are generated by this rule.
The proposed work uses the Euler method to explore the solution space and construct an appropriate global and local search. Furthermore, it is used to determine the proposed OBE search mechanism. The SM formula is defined as follows: where rand 1 and rand 2 are two random values ranging from 0 to 1. Delta X is defined by the following formula: where X w and X b are determined by the following: In the next step, we explain the solution update for the above technique.
The OBE algorithm uses a search mechanism (SM) based on the Euler method to update the current solution location at each iteration, end where r is an integer that can be either 1 or -1, g is a random number between 0 and 2, SF is a coping mechanism, where μ is a number chosen at random. The formula of SF is as follows: where maxi stands for the largest number of iterations. The formulas of X c and X m are as follows: where ϑ is a random number between 0 and 1. X best has shown to be the most effective solution so far. At each iteration, X lbest represents the best position. Enhanced Solution Quality (ESQ) is used by the OBE algorithm to improve the quality of the solutions and then override the local improvement on each iteration. ESQ is used to generate the answer (X new 2 ) using the following technique: if randm<0. 5 if w<1 where k is a random value between 0 and 1, and in this paper, c is a random number that equals 5×rand, x best is the better solution found so far, r is an integer number that can be 1, 0, or -1. The present solution (i.e., f (X new 2 )>f(X n )) may not have the best fitness than the solution determined in this section ( end where ν is a random number with a value of two multiplied by rand [31]. Position χ best should be updated. j = j+1 end Phase Three. Return χ best .
Here we reach the end of a detailed explanation of the method, and in the next part we deal with the numerical results of the OBE technique [31].
All computations are performed in MATLAB 2021b on a Windows 10 HP system with an Intel Core i5 CPU, 4 GB of RAM, and 500 GB of hard disk space, with a maximum iteration count of 500 for solution stopping.
The limits of work in solving fuzzy nonlinear equations have been narrowed, since only nonlinear equations have been solved without solving exponential or trigonometric equations.

Numerical results of the examples
Throughout this section, we will use numerical examples to demonstrate the efficiency and applicability of the OBE technique, which is applied to solve fuzzy non linear equations. The following are some instances from [8]. Table 1 shows some examples of fuzzy nonlinear equations solved by the Optimization Algorithm Based on the Euler Me thod (OBE) technique. Table 1 Examples of fuzzy nonlinear equations solved by the new algorithm  Table 1 gives examples of fuzzy nonlinear equations that have been solved by the Optimization Algorithm Based on the Euler Method (OBE) technique.

1. Efficiency of the Optimization Algorithm Based on the Euler Method (OBE) technique
The minimum fuzzy nonlinear equations were found as shown in the fourth column of Table 2, where the Optimization Algorithm Based on the Euler Method (OBE) technique was very effective in reaching the global solution to the above problems.

2. F-Best
The aim of the paper was to find the best value for the given functions, and by using the Optimization Algorithm Based on the Euler Method (OBE) technique, the best values for the fuzzy nonlinear equations were obtained, as found in the fourth column of Table 2, thus achieving the desired purpose of the Optimization Algorithm Based on the Euler Method (OBE) technique.
At the end of this section, we note that the Optimization Algorithm Based on the Euler Method (OBE) technique is an effective way to find roots for equations, reach the global, and avoid the local solution of the given functions.

3. Best Root Value
We got the roots of the given equations where the best value was found for each variable as shown in the third column of Table 2 and also shown in Fig. 1-3, where these graphics are figures of the values of the variables in the three problems. Where Fig. 1 refers to plotting the solution to the first equation, Fig. 2 refers to plotting the solution to the second equation. The OBE algorithm has been compared with the Grey Wolf Optimizer (GWO) algorithm [32]. There are previous papers that dealt with the topic of solving fuzzy nonlinear equations using swarms, for more see [33,34]. Fig. 3 shows the solution of three equations when using the Optimization Algorithm Based on the Euler Method (OBE) technique. The plot reaches the peak when the value is 0.92 and it starts to decrease reaching zero when the value is 1.05. In this paper, one of the metaheuristic methods for solving fuzzy nonlinear equations is proposed, which is an effective optimization algorithm method that depends on the Euler method. This method was proposed because it is very effective in finding the roots of fuzzy nonlinear equations and has glo bal convergence, unlike some numerical algorithms whose convergence is local and needs an initial point close to the exact solution and may diverge from the approximate solution if the initial point is outside the range of the appro ximate solution.
At the beginning of the paper, it deals with the issue of local and global convergence, and after obtaining the numerical results in Table 2 and Fig. 1-3, we prove that the Optimization Algorithm Based on the Euler Method (OBE) technique has global convergence compared to the GWO algorithm. The researchers did not address the solution of fuzzy nonlinear equations with metaheuristic algorithms before. Table 2 indicates the numerical results of the new technique, where the best value of the variables and the best value of the functions were recorded, as the method converged towards the global in all the given functions compared to the GWO algorithm.
The limits of the study are the minimization of functions to obtain the required roots. One of the difficulties that we encountered in the paper is how to program the OBE algorithm and the problems using Matlab.
One of the disadvantages of the algorithm is that it needs many iterations to reach the solution, and we will overcome this problem in the future.

Conclusions
1. The new method is more efficient than the intelligent swarming methods because the intelligent swarming methods work in a random way.
2. We obtained encouraging results and convergence towards global in solving all the fuzzy nonlinear equations, as the results show that the solution of the equations reached the global minimum of the equations.
3. We can develop a method to use in artificial neural networks for data classification and approximate functions, and we can get encouraging results based on solving nonlinear equations, and they can be applied also in several other areas, including video cameras, smart cars, air conditioners, boilers, ships, and automatic washing machines.