An Efficient Algorithm for Fuzzy Linear Fractional Programming Problems via Ranking Function

In many applications such as production, planning, the decision maker is important in optimizing an objective function that has fuzzy ratio two functions which can be handed using fuzzy fractional programming problem technique. A special class of optimization technique named fuzzy fractional programming problem is considered in this work when the coefficients of objective function are fuzzy. New ranking function is proposed and used to convert the data of the fuzzy fractional programming problem from fuzzy number to crisp number so that the shortcoming when treating the original fuzzy problem can be avoided. Here a novel ranking function approach of ordinary fuzzy numbers is adopted for ranking of triangular fuzzy numbers with simpler and easier calculations as well as shortening in the procedures. The fuzzy fractional programming problem is the first reduced to a fractional programming problem and then solved with the technique to obtain the optimal solution. It has a power to give a best solution for supporting the solution theory proposed in this work, some numerical fuzzy fractional programming problem are included to ensure the advantage, efficiency and accuracy of the suggested algorithm. In addition, this research paper describes a comparison between our optimal solutions with other existing solutions for inequalities constrains fuzzy fractional program.


Introduction:
Fractional programming problem (FPP) is a special kind of non-linear programming problems in which the objective function is a ratio two functions with the constraints. Fractional model arises in decision making such as construction planning, economic, and commercial planning, health care and hospital planning 1 .
Recently, Charnes and Kooper 2 established a method for transforming the fractional programming (FP) to an equivalent model program. Effati and Pakdaman 3 introduced a technique obtain the solution of the interval valued fractional programming (FP). Tantawy 4 proposed an iterative method for treating fraction programming problem with sensitivity analysis. The fractional programming (FP) problem with interval coefficients in the objective function was invented by Borza1 et al. 5 . Many researches proposed some techniques depending on integer solution for FP Problem 6-8 while multi-objective integer FP problems are treated with the help of an improved method by Mehdi et al. 9 . The fuzzy fractional programming problem is interested and has applications in many important fileds 10,11 . Some researchers suggested some algorithms to obtain the approximate solution for fuzzy programming (FP) problem such as proposed technique introduced by Kabiraj et al. 12 . A new method presented by Malathi et al. 13 to solve special fuzzy programming (FP) problem.
Another way to solve the fuzzy programming (FP) problem is established by using ranking function to convert the fuzzy programming (FP) problem into an equivalent crisp programming problem 14,15 . Furthermore, the ranking function is utilized to solve fuzzy fractional programming (FFP) problem by 16 , fully fuzzy multi-objective programming problem (FFMOPP), fuzzy rough fractional programming (FP) problem, fuzzy fractional transportation problem and multi-objective fractional programming (MOFP) are also solved using different method [17][18][19][20][21][22][23] .
Here, an efficient ranking method is suggested to obtain an optimal solution for FFP problem by reducing it into a crisp programming. The presented technique is explained through illustrations. This article is organized as follows: some important preliminaries concerning triangular fuzzy number and ranking function are listed is Section 2. In Section 3, a ranking function is adopted while its application to solve FFP problem is derived in Section 4. Two examples of FFP problem are included in Section 5. In order to demonstrate the efficiency of the proposed approach, a comparison with other obtains result is also listed in this section. Section 6 describes the conclusion of our obtain results.

Preliminaries
Some necessary background and definitions are recalled throughout this section. Definition 1 24 Let be a collection of objects denoted by . A fuzzy set ̃ in can be defined as a set of ordered pairs: ̃ = { ( ,̃ ( ) ) ∶ ∈ } where ̃ ( ) is named the membership function of x in a set ̃. The function ̃ ( ) maps each element of a set X to a membership grade (or membership value) between 0 and 1.

Definition 2 24
A fuzzy number ̃ is a convex normalized fuzzy set on R such that: is called a triangular fuzzy number if ̃( X) is defined by In general, the fuzzy number ̃ = (a, b, c; w) can be defined to be a generalized triangular fuzzy number if ̃( X) is defined by

The Suggested New Ranking Function
In fact, an efficient scheme for ordering the elements is to give a ranking function : ( ) → which maps for each fuzzy number into , and exists a natural order. The orders on is defined by:

The proposed ranking of fuzzy numbers can be defined as
Consider the generalized triangular membership function 3. Find the optimal solution y in step 2. 4. Obtain the optimal solution x using the value y in step 2. 5. The above algorithm is illustrated by numerical examples given in the next section a mathematical programming will utilize to get the optimal solution. 6. Compare a new ranking function with other ranking function type triangular fuzzy numbers to obtain the optimal solution of problems.
Afterwards, comparing optimized solution in Table  2. Note that, the new ranking function is more optimized than the suggested ranking function in 26 .

Conclusion
An efficient algorithm is proposed to solve the fuzzy fractional programming problem together with triangular fuzzy numbers. In this paper a novel ranking function has been used to convert fuzzy fractional programming problems to crisp fractional programming problems with simpler and easier calculations.