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Most of the current methods for solving linear fractional programming (LFP) problems depend on the simplex type method. In this paper, we present a new approach for solving linear fractional programming problem in which the objective function is a linear fractional function, while constraint functions are in the form of linear inequalities. This approach does not depend on the simplex type method. Here first we transform this LFP problem into linear programming (LP) problem and hence solve this problem algebraically using the concept of duality. Two simple examples to illustrate our algorithm are given. And also we compare this approach with other available methods for solving LFP problems.

The linear fractional programming (LFP) problem has attracted the interest of many researches due to its application in many important fields such as production planning, financial and corporate planning, health care and hospital planning.

Several methods were suggested for solving LFP problem such as the variable transformation method introduced by Charnes and Cooper [

In this paper, our main intent is to develop an approach for solving linear fractional programming problem which does not depend on the simplex type method because method based on vertex information may have difficulties as the problem size increases; this method may prove to be less sensitive to problem size. In this paper, first of all, a linear fractional programming problem is transformed into linear programming problem by choosing an initial feasible point and hence solves this problem algebraically using the concept of duality.

A linear fractional programming problem occurs when a linear fractional function is to be maximized and the problem can be formulated mathematically as follows:

Maximize

Subject to,

where c, d and

We point out that the nonnegative conditions are included in the set of constraints and that

To transform the LFP problem into LP problem, we choose a feasible point

is a given constant vector computed at a given feasible point

Hence the linear programming problem is as follows:

Maximize

Subject to,

If

Now rewrite the LP problem (3) in the form

Maximize

Subject to,

where,

Now consider the dual problem for the linear program (4) in the form

Minimize

Subject to,

Since the set of constraints of this dual problem is written in the matrix form hence we can multiply both side by a matrix

Thus this implies

If we define

where

Minimize

Subject to,

with

Maximize

Subject to,

Note: The set of constraints of the above linear programming problem will give the maximum value

The method for solving LFP problems summarize as follows:

Step 1: Select a feasible point

Step 2: Find the level curve of objective function

Hence find the LP problem (2) which can be rewritten as (3).

Step 3: Compute

Step 4: Find the matrix

Step 5: Find the LP problem (8) and dual of this LP (9). Use the LP (9) to find the optimal value

Step 6: Find the dual variables

Step 7: Solve a

Choose

The level curve is

Then

Find

Compute

Formulate, Maximize

Subject to,

Find

Then

Compute

Here we illustrate two examples to demonstrate our method.

Example 1: Consider the linear fractional programming (LFP) problem

Maximize

Subject to,

Solution:

Step 1: Let

Step 2: Therefore we have the following LP problem

Maximize

Subject to,

Dual problem for this LP problem is

Minimize

Subject to,

Step 3: Compute

And the matrix

Step 4: Compute nonnegative matrix

Also compute

Step 5: We get the LP problem of the form

Maximize

Subject to,

For this LP problem we get that the first constraint is the only active constraint and this active constraint shows that the maximum optimal value is

Step 6: Compute

This indicates that in the original set of constraints the first and the second constraints are the only active constraints.

Step 7: Solve the system of linear equations

We get the optimal solution

Finally we get our desired optimal solution of the given LFP problem is

Example 2: Consider the linear fractional programming (LFP) problem

Maximize

Subject to,

Solution:

Step 1: Let

Step 2: Therefore we have the following LP problem

Maximize

Subject to,

Dual problem for this LP problem is

Minimize

Subject to,

Step 3: Compute

And the matrix

Step 4: Compute nonnegative matrix

Also compute

Step 5: We get the LP problem of the form

Maximize

Subject to,

For this LP problem we get that the first constraint is the only active constraint and this active constraint shows that the maximum optimal value is

Step 6: Compute

This indicates that in the original set of constraints the first and the third constraints are the only active constraints.

Step 7: Solve the system of linear equations

We get the optimal solution

Finally we get our desired optimal solution of the given LFP problem is

Bitran and Novea | Swarup | Tantawy | Our Method | |
---|---|---|---|---|

Example 1 | 3 iterations with lots of calculations | 3 iterations with clumsy calculations | 2 iterations | 1 iterations with simple calculations |

Example 2 | 3 iterations | 3 iterations | 2 iterations | 1 iterations |

Now different methods can be compared with our method and all the methods give the same results for Example 1 and Example 2.

In this Section, we find that our method is better than any other available method. The reason can be given as follows:

§ Any type of LFP problem can be solved by this method.

§ The LFP problem can be transformed into LP problem easily with initial guess.

§ In this method, problems are solved by algebraically with duality concept. So that it’s computational steps are so easy from other methods.

§ The final result converges quickly in this method.

§ In some cases of numerator and denominator, other existing methods are failed but our method is able to solve any kind of problem easily.

In this paper, we give an approach for solving linear fractional programming problems. The proposed method differs from the earlier methods as it is based upon solving the problem algebraically using the concept of duality. This method does not depend on the simplex type method which searches along the boundary from one feasible vertex to an adjacent vertex until the optimal solution is found. In some certain problems, the number of vertices is quite large, hence the simplex method would be prohibitively expensive in computer time if any substantial fraction of the vertices had to be evaluated. But our proposed method appears simple to solve any linear fractional programming problem of any size.

Simi, F.A. and Talukder, Md.S. (2017) A New Approach for Solving Linear Fractional Programming Pro- blems with Duality Concept. Open Journal of Optimization, 6, 1-10. https://doi.org/10.4236/ojop.2017.61001