Solving of the Quadratic Fractional Programming Problems by a Modified Symmetric Fuzzy Approach

The aims of the paper are to present a modified symmetric fuzzy approach to find the best workable compromise solution for quadratic fractional programming problems (QFPP) with fuzzy crisp in both the objective functions and the constraints. We introduced a modified symmetric fuzzy by proposing a procedure, that starts first by converting the quadratic fractional programming problems that exist in the objective functions to crisp numbers and then converts the linear function that exists in the constraints to crisp numbers. After that, we applied the fuzzy approach to determine the optimal solution for our quadratic fractional programming problem which is supported theoretically and practically. The computer application for the algorithm was tested


Introduction
Mathematical programming finds many applications in the field of management. Optimization of resources in any organization is mostly addressed with the use of mathematical programming. QFPP, which deals with situations where a ratio between two mathematical

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242 functions is either maximized or minimized, is an important class of mathematical programming problems that has attracted a lot of research and interest because it is useful in production planning, economic and financial planning, medical services, and healthcare planning.
There are many managerial decision-making situations where the uncertainties in working situations are best explained by use of fuzzy set theory. The nation of fuzzy set theory is developed by [1], since then a considerable number of scholars have expressed their interest in the application of fuzzy set theory. [2] established the notion of decision making in a fuzzy environment and their concept of fuzzy decision making is used in mathematical programming [3]. Many authors have discussed the use of fuzzy set theory in QFPP e.g., [4, 5 ,6 ,7].
The fuzzy system plays a very significant role in the theory of linear programming (LP). Thus, researchers have shown their interest in the concept of the script for a linear program under a fuzzy environment as well e.g., [8], their works formulates a quadratic fractional bi-level (QFBL) programming problem with probabilistic constraints in both the first (ruler) and second (supporter) levels, with two-parameter exponential random variables with recognized probabilistic, and fuzziness as a triangular and trapezoidal fuzzy number. Using a chanceconstrained programming approach, the issue is first turned into an a like deterministic quadratic fractional fuzzy bi-level programming model in the suggested model. Second, each objective function of the bi-level QFP issue has its own nonlinear membership function in the proposed model. In [9], the fuzzy environment with parabolic concave membership functions is used to investigate a certain form of convex fractional programming issue and its dual. An ambition level technique is used to determine appropriate duality outcomes. It differs from other research in that it uses parabolic concave membership functions to reflect the decision-level maker's of pleasure. The Fully Fuzzy Quadratic Fractional Programming problem (FFQFPP) is solved using a solution process in which all variables and parameters are triangular fuzzy integers. In this case, FFQFPP was converted into a Multi-Objective Quadratic Fractional Programming issue (MOQFPP). Using a mathematical programming method, MOQFPP is then turned into an analogous multi-objective quadratic programming problem. A fuzzy goal programming paradigm for handling bilevel programming issues with decision-makers' objectives in quadratic fractional form. The membership functions for the defined fuzzy goals of the decision-makers' objectives at both levels are established first in the suggested approach. Then, in order to find the most satisfactory answer in the decision-making environment, a fuzzy goal programming model is constructed to minimize the group regret of the degree of satisfaction of both decision-makers. In [10], the linear constraints of quadratic fractional objective programming problem (QFOPP) have been established and developed. The Wolfe's technique and a modified simplex approach were used to solve the specific case for this problem.
To extend this work, we study and introduce a modified symmetric fuzzy approach to solve QFPP. An algorithm and practical technique with theoretical support are suggested, also solving such a problem by a modified simplex approach, and test the validity will be compared against both outcomes.

Fundamentals of Fuzzy set, and Fuzzy Algebra Operation
Some fundamentals and algebra operations of the fuzzy environment that are utilized in this research are listed [11,12]:

Fundamentals of Fuzzy set
Fuzzy set: A fuzzy set Α in Χ is a set of ordered pairs: Α = {( , Α ( ))| ∈ Χ}, Α ( ) is titled by the membership function of in Α. If Χ is a assemble of contraption defined generally by . The support: (Α) is the crisp set of all ∈ Χ if Α ( ) > 0 is called by the support of fuzzy set Α. -level set :The fragile set of elements that pertinence to the fuzzy set Α at the lower to the grade is named by -level set: Α = { ∈ Χ| Α ( ) ≥ }, Α ′ = { ∈ Χ| Α ( ) > } is defined by "stronglevel set" or "strong -cut".

Linear Programming models
A special kind of decision model introduced by [13], is considered as Linear programming models, the restrictions define the decision space; the objective function defines the "target" (utility function), and the kind of choice is decision making under conditions. The standard linear programming model is as follows: With , ∈ , ∈ , ∈ ×

Linear Fractional Programming Formula
The linear fractional programming problem (LFPP) introduced by [14] can be stated in the following manner: (1)

Quadratic Programming Problem (QPP)
Quadratic Programming is a particular kind of mathematical optimization problem It is a problem of reducing or maximizing a quadratic function of numerous variables under linear constraints.
The QPP can be written as follows:: Max. (or Min. z) = + + Subject to: is scalar and = ( ) × is assumed negative definite of the problem in maximization, and positive definite of problem in minimization, The objective function is quadratic, whereas the restrictions are linear. [16].

Quadratic Fractional Programming Problem (QFPP)
This section discusses a specific situation of QFPP's mathematical form, which contains a quadratic function in the numerator and a denominator that can be represented as the product of two linear components, as the following: Subject to: where is × 1 column vector of decision variables, 1 , 2 , are × 1 column vectors of constants, is × 1 column vectors of constants. , , and are scalars, the prime ( ) over the vectors 1 , 2 , 1 and 2 denoted the transpose of the vectors.

Symmetric Fuzzy linear programming
We will suppose that the decision-maker can select an ambition level, z, for the value of the objective function he or she wishes to reach, and that each restriction is modelled as a fuzzy set in model (1). Our fuzzy album then transforms into:

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245 Find x such that ≧ ≦ ≥ 0 Here ≦ signifies the fuzzified form of ≤ and has the linguistic interpretation "essentially smaller than or equal to." ≧ signifies the fuzzified form of ≥ and has the linguistic interpretation "essentially greater than or equal to." The objective function of a model (1) might have to be written as a minimizing goal to consider z as an upper bound founded by [13].

Zimmermann's Approach: A Symmetric Model
Principle for determining the fuzzy choice to solve the problem's fuzzy system of inequalities (FSI) (1). As a result, the following neat linear programming issue arises.
subject to,

Modified Simplex Approach Development
Dantzig invented the simplex approach in 1947. The Simplex technique is a systematic approach that involves moving from one basic variable solution (on the vertex) to another in a predefined order in order to elicit the value of the objective function. This leaping from vertex to vertex operation is repeated. If the goal function improves with each leap, no basis will ever be duplicated, and there will be no need to return to the vertex because it has already been covered. The algorithm must lead to the optimum vertex in a finite number of steps since the number of vertices is finite.

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246 For addressing linear programming problems, the Simplex method is an iterative (step-by-step) approach. It entails the following: Having a fundamental workable solution to constraint equations.
ii) Determine whether it is the best option.
iii) Using a set of criteria to improve the initial trial solution and continuing the procedure until an ideal solution is found. For more details.

Modified Simplex Approach to Solve Quadratic Fractional Programming Problem
This section deals with the solution of the quadratic fractional programming problem developed by [17], which the solution procedure has some similarity. This technology can be used to high-speed computing with success. This strategy can be used if the problem's constraints are linear functions. i.e., the issue is of the following existence: Subject to: To apply simplex process, first ∆ 1 and ∆ 2 need to be found from the coefficients of the first linear vector and second linear vector of objective function respectively, by using the following formula: in this approach the formula is defined to find ∆ from ∆ 1 , ∆ 2 , 1 , 2 following:

Modify Symmetric Fuzzy Approach (MSFA) to Solve Quadratic Fractional Programming Problems
Our modified approach depends on the adopted "fuzzy" version of formula (4) is: find such that ( 3 )( 4 ) ≲ ≧ 0 here 1 , 2 , 1 and 2 are the vector of coefficients of numerator and denominator respectively of the ratio of the goal function, denoted as a vector of constraints, and is the factor of matrix. The sign " ≲" indicate the fuzzified type of " ≦ " and read out "basically smaller than or commensurate to". Note that (7) is wholly symmetric with observance to goal function and constraints, To solve the above problem, First choose an appropriate membership function for each of the fuzzy inequality of (7). In particular, let 0 denote the membership function for objective function and , = 1, … , denote the membership function for constraint, let 0 and , = 1, … , be the permissible tolerances for objective function and constraint and let 0 and , = 1, … , be a continuous and nondecreasing linear membership which is explained below: We have two cases that include only for 0 :  are × 1 column vectors for = 1,2 By using Simplex algorithm, we can solve problem (10 or 13), the optimum result to (10 or 13) is the optimal solution as well to (7) and as well to (4).  (7) is assumed by the parametric solution of the parametric fractional linear programming problem.
. We have to solve the following problem to obtain the solution of (7). ( ) ( , ) subject to Remark 5.1 If the problem (7) has fuzzy as well as crisp constraints, then in the alike (crisp) linear fractional programming problem, the original crisp constraints will not have any modification as for them the tolerances are nil.

Algorithm to Solve QFPP by Using MSFA
To find the result of optimal solution for the QFPP of formulation (4), a procedure is given as below: Step One: Convert the main problem to the fuzzy version as formula 7 Step Two: Find the value of 0 and , = 1, … , which are the permissible tolerances for objective function and constraint by using formula (8,11) and (9,12), respectively.

An Illustrative Numerical Example and Results:
In this section, we solve an example through MSFA and compared with MODIFIED SIMPLEX APPROACH.
In additional, will solve the such numerical example by computational procedure of quadratic fractional algorithm introduced by [ subject to:
The fuzzy and unfuzzy results are shown in the following: We conclude from the solution we have same profit under regard all constraints.

Conclusion
This paper used modify symmetric fuzzy approach. After converting to crisp linear programming, the maximum value of QFPP was discovered and compared to the modified simplex technique. The values of the objective function are used to compare these techniques, the study found that . resulted of those two methods are same but modify symmetric fuzzy solve the problem directly without iteration while modified simplex approach take two iterations. Therefore, we can solve special case of QFPP by these two methods under our approach and algorithm. consequently, reliable results have been found.