Solving a Fully Rough Integer Linear Fractional Programming Problem

Rough


Introduction
The main interest in fractional programming was generated by the fact that a lot of Borza et al. [3]proposed the method to solve linear fractional programming problem with interval coefficients in objective function.
Jayalakshmi and Pandian, Proposed a new method namely, denominator objective restriction method for finding an optimal solution to linear fractional programming problems [4]. Linear fractional programming problem with interval coefficients in the objective function is introduced [5], It is proved that we can convert an IVLFP to an optimization problem with interval valued objective function which its bounds are linear fractional functions. Rough Set Theory (RST) was initiated by Pawlak [6] in 1982 as a method for ambiguity management. Pandian et al. [7] considered that the transportation problem has all or some parameters as rough integer intervals. Also, proposed a new method named, a slice-sum method to solve Rough constraints are fuzzy rough number [11]. A Large-Scale three level fractional problem is introduced with random rough coefficient in the objective function in [12].

BASIC PRELIMINARIES
In this section a basic notions of interval analysis are given [5]: i. ii.

Basic operations of intervals [5]
Let , be two closed intervals in . When and we have: Definition 2.2. Let , be two closed intervals in . We write . Also it mean that is inferior to or is superior to . . ii. .
Where is the set of all rough intervals in , .

Basic Operations of Rough Intervals
For any two rough intervals we can define the operations on rough intervals [7,9,10] where, and , The integer linear fractional programming problem (1) is transformed into the following problem [2,11]:

Variable Transformation Method
Subject to: , and integer , Firstly we solve the problem (2) Now we know that: , , The problems (6) and (7) can be written as follows:

LILFP(2):
Subject to: and integer interval (9) Now using the arithmetic operations, we decompose the above two problems (8) and (9) as the follows model:

LILFP(3):
Subject to: and integer interval (11) We can write (10) and (11)  Therefore, the set of rough integer intervals is an optimal solution for the ( FRILFP) problem. Hence, the theorem is proved.

Algorithm: Solution for FRILFP problem
The propose algorithm to solve (FRILFP) problem can be summarized in the following steps: Step Step 2. Find the integer optimal solution for (UUILFP) problem with the objective value ( , by the variable transformation method.
Step 3. Solve the (ULILFP) problem by the variable transformation method to obtain the integer optimal solution with the objective value ( .
Step 4. Solve the (LLILFP) problem by the variable transformation method to obtain the integer optimal solution with the objective value ( .
Step 5. Solve the (LUILFP) problem by the variable transformation method to obtain the integer optimal solution with the objective value ( . Step 6. The set of rough integer interval is an integer optimal solution to the given problem (FRILFP) with the objective value by the theorem (3.1).

Numerical example
Consider the following (FRILFP) problem: , and rough integer intervals and integer Using the arithmetic operation of interval we get four crisp integer linear fractional programming problem as the following: Model(1): UUILFP with the maximum objective value .

Conclusion
In this paper, we focused on the solution of the