A Study of Algorithmic Approaches for Fuzzy Pattern Quadratic Fraction

Fuzzy goal programming was used by Pramanik et al. (2011) to solve the quadratic optimization problem. Gani and Kumar (2013) came up with a new way to solve fuzzy quadratic programming problems. They called it the principal pivoting method. Ghadle and Pawar (2016) came up with a new way to solve a Quadratic fractional programming problem in a special case. Kheirfam (2011) came up with a way to solve fully fuzzy Quadratic programming problems. Linear fractional programming problems were solved by Pandian and Jayalakshmi in 2013. Sharma and Singh (2014) came up with a way to solve Quadratic fractional optimization through the FGP method. Jain et al. (2011) came up with a way to solve the fuzzy linear fractional programming problem.A lot of application problems can be thought of as a mathematical problem that can be written with uncertainty. Bellman and Zadeh (1970) came up with the idea of maximising a decision in a fuzzy situation. Tanaka and others came up with a way to solve fuzzy mathematical programming in 1973. If you want to solve an optimization problem, you can use linear programming. This branch of math is used to solve problems where all of the constraints and the goal functions are linear functions. How do you figure out the best answer from a wide range? Linear programming is a method for doing this. Classical linear programming and fuzzy linear programming are two types of linear programming problems. Classical linear programming is when you solve a simple linear programming problem. Fuzzy linear programming is when you solve a fuzzy linear programming problem. Zimmermann came up with the idea of fuzzy linear programming problems (1978). Vansant et al. (2004) used linear programming with fuzzy parameters to help them make decisions about industrial production. It was in 1989 that Campos and Verdegay came up with a way to solve linear programming with fuzzy coefficients in both the matrix and on the right hand side of the constraint, which is why they are so important. Gani et al. (2009) came up with a fuzzy linear programming problem based on the L-R fuzzy number. Jimenez et al. (2007) came up with a way to solve linear programming problems where all the coefficients are fuzzy numbers and the linear ranking method is used. Decision-makers are made up of non-symmetrical trapezoidal fuzzy numbers, and they can be solved by using a ranking function. Hashem (2013) came up with this idea.

membership function of a trapezoidal fuzzy number is given by In particular case if the trapezoidal fuzzy number reduces to a triangular fuzzy number given by The value of 2 a corresponds with the mode or core and [ 1 4 aa, ] with the support. If w=1, then is a normalized triangular fuzzy number a  is a generalized or non-normal triangular fuzzy number if

Let be two trapezoidal fuzzy numbers, then the fuzzy numbers addition and fuzzy numbers subtraction are defined as follows: i. Fuzzy numbers addition of is denoted by and is given by ii. Fuzzy numbers subtraction of is denoted by and is given by

If is a trapezoidal fuzzy number, then the defuzzified value or the crisp number of is given by . We need the following definitions of ordering on the set of the fuzzy numbers based on the magnitude of a fuzzy number. The magnitude of the trapezoidal fuzzy number, Where the function f(r) is a non-negative and increasing function Clearly, the function f(r) can be thought of as a weighting function. In real-life applications, the function f(r) can be chosen based on the situation. The size of a trapezoidal fuzzy number u, which is defined by (1), shows how much information there is about each membership level, and the meaning of this size is visual and natural. The resulting scalar value is used to rank the fuzzy numbers. In the other words Mag is used to rank fuzzy numbers. The larger Mag means the larger fuzzy number. The magnitude of the trapezoidal fuzzy number Let be two trapezoidal fuzzy numbers. The ranking of the set of trapezoidal fuzzy numbers is defined as follows: The ordering ≥ and ≤ between any two trapezoidal fuzzy numbers are defined as follows:

The mathematical properties of the "magnitude" method for ranking fuzzy numbers are shown here. It doesn't require a lot of computer work, and it doesn't need to know what all the other options are before start.

The quadratic programming method is one of the most important ways to improve operations research. Quadratic programming models are usually used in real-world situations to figure out what to do next. In real-life situations, there may be a lot of uncertainty about the parameters. In this case, fuzzy numbers can be used to show the parameters of quadratic programming problems. There is a special quadratic programming problem that we look at. In this case, the quadratic fractional objective function can be changed into a linear fractional objective function with fuzzy variables. Mathematically, consider the special case of quadratic fuzzy fractional programming problem (QFFPP):

Find optimal solution of special case of QFFPP by an alternative method, algorithm is given as follows: Step 1 Convert all fuzzy numbers into crisp value by using proposed ranking method. Step 2 Check objective function of QFFPP is of maximization. If it is to be minimization type than convert it into maximization. Step 3 Convert quadratic fractional objective function to the product of linear fractional objective functions. Step 4 Check whether all bi (RHS) are non- negative. If any bi is negative then convert it to positive. Step 5 Express the given QFFPP in standard form then obtain an initial basic feasible solution. Step 6 Find net evaluation  j for each variable j x by the formula: Step 7 Use simplex method for this table and go to next step. Step 8 Check solution for optimality if all  0 ij , then current solution is an optimal solution, otherwise go to step 6 and repeat the same procedure. Thus optimum solution of special type of QFFPP is obtained.

By using the method for defuzzyfing the trapezoidal fuzzy numbers Now solving the above problem by an alternative method the detailed process of the solution is as follows. QFFPP is in standard form By solving above problem, we obtain the following final simplex table.

interval arithmetic. This way, you can come up with numerical methods that work. In this way, each value is shown as a range of possible values. The following are the basic operation of interval arithmetic for two variable [a,b] and [c,d] that are subsets of the real line (-∞,∞) .

In this section we consider addition, subtraction, multiplication and division of fuzzy numbers using α-cut method. Now for calculate addition, subtraction, multiplication and division of fuzzy numbers using interval arithmetic.

The arithmetic operations of fuzzy numbers using -cut operations discussed in the earlier sections are used below to solve the fuzzy linear programming problem. The steps for the computation of an optimum solution are as follows: Step 1 Find the value of  Step 2 Add the -cuts of using interval arithmetic. Step 3 The values obtained in step 1 and 2 is converted into a crisp linear programming problem and formulated as, The function to be maximized is called an objective function. This is denoted by is called a cost vector. The matrix is called an activity matrix and the vector is called right hand side vector. Step 4 Now solve reduced linear programming problem by two phase simplex method.

obtained in two phases as follows:

In this phase, the simplex method is applied to a specially constructed auxiliary linear programming problem leading to a final simplex table containing a basic feasible solution to the original problem.

• Assign a cost -1 to each artificial variable and a cost 0 to all other variables (in place of their original cost) in the objective function.

• Construct the auxiliary linear programming problem in which the new objective function

is to be maximized subject to the given set of constraints.

• Solve the auxiliary problem by simplex method until either of the following three possibilities do arise.

• Max and at least one artificial vector appear in the optimum basis at a positive level. In this case given problem does not possess any feasible solution.

• Max and at least one artificial vector appears in the optimum basis at zero level or no artificial vector appears in the optimum basis. In this case proceed to phase II.

Now, put the real costs into the variable in the goal function, and put a zero cost into every artificial variable that is in the basis at the zero level. Now, the simplex method is used to try to maximise this new goal function, but only if the given constraints are met. That is, the simplex method is used on the modified simplex table that was found at the end of phase-I until an optimum basic feasible solution (if there is one) has been found. All of the artificial variables that aren't basic at the end of Phase I are taken away.

Fuzzy linear programming problem using interval arithmetic when compared to the earlier approaches. This also can be applied in triangular, hexagonal, octagonal fuzzy numbers and also can be extended to multi objective linear programming and fully fuzzy linear programming problems. Interval arithmetic linear programming problem with trapezoidal fuzzy number is solved by using α-cut operation without converting them into a classical linear programming problem.Fuzzy logic is a computational worldview that gives a numerical apparatus to managing the vulnerability and the imprecision which is ordinary of human thinking. The great trait of fuzzy logic is its principle. The fuzzy set system has been used in a few unique approaches to demonstrate the symptomatic cycle. This part manages the extension of Sanchez's methodology for medical diagnosis utilizing the portrayal of a stretch esteemed fuzzy lattice as a time period fuzzy grids. The vast majority of our genuine issues in medical sciences, designing, the executives, climate and sociologies regularly include data which are not really fresh, exact and deterministic in character because of different vulnerabilities related with these issues. Such vulnerabilities are typically being taken care of with the assistance of the themes like likelihood, fuzzy sets, intuitionistic fuzzy sets, span arithmetic and harsh sets and so on Span esteemed intuitionistic fuzzy sets acquired consideration from specialists for their applications in different fields. The fuzzy set structure has been used in a few distinct approaches to demonstrate the symptomatic interaction.

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Research Scholar, Sardar Patel University