Solving Linear Fractional Programming Problem in symmetric trapezoidal fuzzy environment

Fuzzy set theory has been applied on different important fields, such as managements, operations research, and control theory. In this paper, we provided a new algorithm to solving a fuzzy fractional linear programming problem (FFlPP) in full fuzzy environments. All the variables and coefficients of both the constraints and the objective function are symmetric trapezoidal fuzzy numbers. The fuzzy linear fractional programming problem (FLFP) has been switched to fuzzy linear programming (FLP) problems, where both of constraint and objective function involve fuzzy numbers as variables and coefficients. Further, using the fuzzy simplex method algorithm to get an optimal fuzzy solution. Finally, we provide illustrative numerical examples.


Introduction
The significance of fractional linear programming cames from the fact that many real-world problems are based on the ratio on financial, economic or physical values such as (cost/volume, profit /cost or cost/time) in production and financial planning [1].Various application problems, can be derived as mathematical programming problems model, may be formulated with uncertainty.For several cases, the coefficients contained in either the objective or constraint functions are imprecise in nature and have to be considered as fuzzy numbers to reverse the real life situation the emanating mathematical problem is therefore indicated to as a fuzzy mathematical program-mix problem.The concept of using fuzzy numbers in mathematical programming was suggested in general for the first time by Tanaka et al. (1974a) [2] in the framing of the fuzzy decision of Bellman and Zadeh (1970) [3] when they presented building on fuzzy environment.There are several fractional programming model applications in fuzzy numbers as T.Peric, Z. Basic and S.Resic [4] ZIMMERMANN (1978) [5] proposed the first formulation of fuzzy linear programming and constructed a model of the problem also depend on the fuzzy concept of Bellman and Zadeh Fang and Hu [6] contemplated linear programming which its constraint coefficients are fuzzy numbers.Vasant and et al [7] seek linear programming which its variables are fuzzy numbers for decision making in industrial production planning.
Ranking fuzzy number is also play important role in decision to making statistical analysis economic systems and operations research.So A.Nchammai and P.Thangaraj [8] solving intuitionist fuzzy linear programming by using metric distance ranking.Also H.M.Nehi and H.Hajmohamadi [9] provide a ranking function method used for solving fractional linear programming problems with multi -objective function.This paper is displayed as follows; section 1,provied the introduction .insection 2, we provide the definition of both of fuzzy set and trapezoidal fuzzy numbers and some related properties .In section 3, we provide the arithmetic operation on symmetric fuzzy numbers and some other related definitions.In section 4, we give a definition of fuzzy fractional linear programming (FFLP) problem and the duality.In section 5, we explain the dual simplex method algorithm and finally we give some numerical example and our work conclusion.

Preliminaries
In this section, we have introduced some fundamental concepts of fuzzy sets and some kind of fuzzy numbers such as trapezoidal number, which was very useful in this paper.

Fuzzy Sets
Zadeh suggested a chain of membership functions that could be classified into two classes: those made up of straight lines being "linear" ones, and the "curved" represent "nonlinear" ones.In addition, the nonlinear functions rise the time of computation.Therefore, in practice, most applications use linear fit functions.We will now continue to take some types of membership functions construct on Klir, at al [9]

Remarks:
1) A fuzzy set ̃ is called convex set if and only if all of the non-empty -cut are convex.
2) A fuzzy number ̃ is a normal and convex fuzzy set.
3) There are some kinds of fuzzy numbers for instance a trapezoidal fuzzy number which it defined as follow: Definition 2.5: Trapezoidal function defined by its minimum limit and its maximum limit d, and the lower and upper values of its nucleus, b and c respectively.
The fuzzy number with the above membership is called trapezoidal fuzzy number and it denoted by ̃ ] is the support of ̃ and is a set of all symmetric trapezoidal fuzzy numbers, where ̃is a membership of ̃

Arithmetical Operation on Symmetric Fuzzy
Where and

5.
Inverse: ) where Where and And

6.
Division: Where and And:

3.3: Definitions:
We introduce the notation of a fuzzy matrix: Where ̃be the (ij)th minor of the matrix

Definition3.3.6: the ranking function is defined as a function which maps for each fuzzy number in to the real line denoted by By Maleki ranking function suggestion [10]
Let ̃ be a trapezoidal fuzzy number then where ̃ be a symmetric trapezoidal fuzzy number Remark2: let ̃ ̃ be two trapezoidal fuzzy numbers then:

Fully fuzzy fractional linear programming problems:
A fully fuzzy fractional linear programming (FFFLP) problem with symmetric trapezoidal fuzzy numbers can be defined as: Where

Conclusion
The main aim of this paper is ,is to solve (FFFLP) using the complementary development method to convert fuzzy fractional Linear programming problems to fuzzy Linear programming (FLP) problems and the method of dual simplex with the ranking of fuzzy number(FN) which is used partially in solving fuzzy linear programming.

Definition 2.1.1: Support
of a fuzzy set ̃ defind as a set of all ∈ X with ̃  >0.

Definition 2.1.2: Core
of a fuzzy set ̃ is a set of all ∈ with ̃  =1.
Definition 2.1.3:A fuzzy set with non empty core set is called normal fuzzy set Definition 2.2: An -cut of a fuzzy set ̃ is a crisp set that is defined as ̃ = {∈ | ̃  ≥ } and it is also called (level subset -).Definition 2.3: A strong -cut (strong level subset-) is defined as ̃ >= {∈ | ̃  > }.Definition 2.4: A fuzzy set ̃ called convex fuzzy set if for any ∈ and ∈ [0,1] then: