1.      Introduction

 The objective of linear fractional programming (LFP) is to find the optimal (maximum or minimum) value of a linear fractional objective function subject to linear constraints on the given variables. The constraints may be either equality or inequality constraints. From the point of view of real world applications, LFP possesses as many nice and extremely useful features, as linear programming (LP). If we have a problem formulated as an LP, we can reformulate this problem as LFP by replacing an original linear objective function with a ratio (fraction) of two linear functions (Bajalinov, 2003).

The fractional programming problems are particularly useful in the solution of economic problems in which various activities use certain resources in various proportions, while the objective is to optimize a certain indicator (Nawkhass and Sulaiman, 2022). Usually the most favorable return on allocation ratio subject to the constraint imposed on the availability of goods. Examples of such situations are financial and corporate planning, production planning (Stancu-Minasian, 1992).  Many proposed methods were presented to get a solution for fuzzy programming FP issue such as: in (Charness and Cooper, 1962) showed that by a simple transformation the original LFP problem can be reduced to an (LP) problem that can therefore be solved using a regular simplex method for a linear programming. In (Sapan and Tarni, 2017) a proposed method with ranking function is presented. In (Malathi and Umadevi, 2018), a new technique for solving special type of fuzzy programming is suggested. (Deb and De 2015), introduced a ranking function for solving fully fuzzy linear fractional programming problem with objective function and constraints are trapezoidal fuzzy numbers. Also, (Rasha, 2021) solved FLFP problem using α-cut method. Furthermore, (Deepak at el. 2021), suggested a new ranking function of trapezoidal fuzzy number, for solving fully fuzzy linear fractional programming problem with the objective function and constraints are trapezoidal fuzzy numbers. A new method to find a fuzzy optimal solution of FLFP problems with inequality constraints (Sapan and Tarni, 2017). The objective of this paper is to propose an algorithm on a new ranking function to solve FLFP problem using triangular and trapezoidal fuzzy numbers. This paper contains five sections: in section two we review some concepts of fuzzy set theory, in section three a suggested ranking function was presented for triangular and trapezoidal fuzzy numbers, and study some properties, in section four a new algorithm for solving this problem was applied, in section five different numerical examples are applied and compared with some ranking functions Material and methods

2.      Preliminaries

 In this section, we will give some basic concepts of fuzzy sets and fuzzy numbers.

Definition (1) (Nalla et al., 2020): Let  be universe of discourse. A fuzzy set  in  can be defined as a set of ordered pairs:

where    and  is called membership function.

Definition (2) (Hari and Jayakumar, 2014): A fuzzy set , which is both convex and normal,  is called fuzzy number.

Definition (3) ( Al Thabhawi 2019): A fuzzy number ,  is called a triangular fuzzy number (TFN) if membership function  is describe as:

Definition (4) (Rasha, 2016): A fuzzy number ,  is called a trapezoidal fuzzy number (TrFN) if membership function  is describe as:

3. Ranking Function of Triangular and Trapezoidal Fuzzy Numbers by Geometric Average

 Several approaches for the ranking of fuzzy numbers have been proposed in the literatures. An efficient approach for comparing the fuzzy numbers is by the use of a ranking function. We defined the geometric average in descriptive statistics for triangular and trapezoidal fuzzy numbers  as following

Let  where  and , be a triangular fuzzy numbers defined  the ranking function  as:

Let  where  and , be a trapezoidal fuzzy numbers defined the ranking function  as:

Let  and  be two arbitrary fuzzy numbers the ranking is:

1. if and only if
2. if and only if
3. if and only if

Remark 1: Our ranking function is able to rank the crisp fuzzy numbers, whereas Cheng’s Distance method (Cheng, 1998), (Wang et al., 2006),  and (Chu and Tsao, 2002) do not.

3.1 Proposition 1

1. If is a ranking function of then  belongs to
2. If then
3. If then .

Proof:  part (1).  Let  be trapezoidal fuzzy number. By definition (4) we have

Hence, we get , ,  and , since  for all i=1,2,…,4,  and by the property of inequality, add 1 to both sides of inequalities , , ,  and   [by ], so,  Take the fourth root (4^th) for both sides of the inequality.

Get , subtract both side by (-1), then   and by equation (2),  . By analogue manner . Therefore, .

Example 1: Consider two triangular fuzzy numbers  and

Since  therefore .

Example 2: Consider two trapezoidal fuzzy numbers  and

Since  therefore .

4.      Algorithm to Solve FLFP Problem using Proposed Ranking Function

 The technique is suggested to solve a problem of fuzzy fractional programming utilizing fuzzy programming technique where the coefficients of the objective function are fuzzy numbers. The ranking approach based on geometric average which is used for fuzzy linear fractional programming problem (FLFPP). The technique converts it to a crisp linear fractional programming (CLFP) problem. The following are summarizes of the algorithm. Consider FLFP problem

Maximize

s.t.

,

where  is an 𝑚 by 𝑛 matrix,  ,  𝑎𝑟𝑒 𝑠𝑐𝑎𝑙𝑎𝑟𝑠.

The ideas can be summarized as follows:

Step 1: Convert the FLFP problem into the following LFP problem by a new ranking function of fuzzy number

Maximize

Subject to,

.

Step 2: Transform the obtained LFP problem into a LP problem by using Charnes-Cooper  transformation method

Maximize

s.t.

.

Step 3: Find the optimal solution  in Step 2.

Step 4: Obtain the optimal solution  using the value  in Step 2.

Step 5: Compare the optimal solution with other exiting ranking functions.

5. Numerical Examples

 In this section, we illustrate two numerical FLFP problems with triangular and trapezoidal fuzzy numbers, with the help of the recommended ranking functions. The FLFP problem is transformed into a crisp programming problem.

Example 3: Consider the fuzzy linear fractional programming problem

s.t.

Apply the proposed algorithm:

Step 1: Convert objective function from fuzzy numbers to crisp value by proposed ranking function 

            as:

The objective function becomes FLP problem (Triangular)

s.t.

Step 2: Transformed this LFP problem into LP problem by using transformation of Charnes Cooper, the model programming problem is:

s.t.

Step 3: The problem is in standard form of programming problem and we can find optimal solution by using simplex method, the optimal solution here is    and t = 0.0361.

Step 4: Find the optimal solution  using the value  as: .  Now, the value of   .

Step 5: Using ranking function (Rasha, 2021), and (Iden and Anfal, 2015)  for comparison with the proposed method, from Table 1

Table 1: Comparison proposed method with existing triangular ranking methods

Example 4: Consider the FLFP problem (Trapezoid)

s.t

Apply the proposed algorithm:

Step 1: Convert objective function from fuzzy numbers to crisp value by proposed ranking function

as:

.

The objective function becomes LFP problem

s.t.

Step 2: Transformed this LFP problem into linear programming problem by using transformation of

Charnes-Cooper, the model programming problem as:

s.t,

Step 3: the problem is standard form of linear programming problem and we can find optimal solution by using simplex  method, optimal solution here is    and  .

Step 4: find the optimal solution  using the value  as:  Now, Z .

Step 5: Using ranking function (Yager, 1981), and (Deepak and Priyank 2021). Compare with proposed method.

Table 2: Comparison proposed method with existing trapezoidal ranking methods