A New Approach of Solving Linear Fractional Programming Problem ( LFP ) by Using Computer Algorithm

In this paper, we study a new approach for solving linear fractional programming problem (LFP) by converting it into a single Linear Programming (LP) Problem, which can be solved by using any type of linear fractional programming technique. In the objective function of an LFP, if β is negative, the available methods are failed to solve, while our proposed method is capable of solving such problems. In the present paper, we propose a new method and develop FORTRAN programs to solve the problem. The optimal LFP solution procedure is illustrated with numerical examples and also by a computer program. We also compare our method with other available methods for solving LFP problems. Our proposed method of linear fractional programming (LFP) problem is very simple and easy to understand and apply.


Introduction
Various optimization problems in engineering and economics involve maximization (or minimization) of the ratio of physical or economic function, for instances cost/time, cost/volume, cost/benefit, profit/cost or other quantities measuring the efficiency of the system.Naturally, there is a need for generalizing the simplex technique for linear programming to the ratio of linear functions or to the case of the ratio of quadratic functions in such a situation.All these problems are fragments of a general class of optimization problems, termed in the literature as fractional programming problems.This field of LFP was developed by Hungarian mathematician Matros [1] [2] in 1960.Several methods are proposed to solve this problem.Charnes and Kooper [3] have proposed their method depended on transforming this (LFP) to an equivalent linear program, they say the feasible region X is nonempty and bounded, cx α + and ax β + do not vanish simultaneously in S then they used the variable transformation , 0 y tx t = ≥ in such a way that dt β γ + = where 0 γ ≠ is a specified number and transform LFP to an LP problem.Multiplying the numerator and denominator and the system of inequalities by t and , 0 y tx t = ≥ , they obtain two equivalent LP problems and name them as EP and EN.If EP or EN has an optimal solution and other is inconsistence, then LFP also has an optimal solution.If any of the two problems EP or EN is unbound, then LFP is also unbound.So if the first problem is not unbound, one needs to solve the other.That's why one needs to solve two LPs by Big-M or two-phase simplex method, which is a very lengthy process.On the other hand, the simplex type algorithm is introduced by Swarup [4] and Swarup, Gupta and Mohan [5].
In that method one needs to compute ( ) ( ) in each step and continues this process until the value of j ∆ satisfying the required condition.We see that it has to deal with the ratio of two linear functions, that's why its computational process is complicated and also when the constraints are not in canonical form then it becomes lengthy.Another method is called updated objective function method derived from Bitran and Novaes [6] is used to solve this linear fractional program by solving a sequence of linear programs only re-computing the local gradient of the objective function.But to solve a sequence of problems sometimes may need many iterations and at some cases say, 0 x S ∀ ∈ , Bitran-Novaes method is failed.Singh [7] in his paper makes a useful study about the optimality condition in fractional programming.
Tantawy [8] develops a technique with the dual solution.Hasan and Acharjee [9] also develop a method for solving LFP by converting it into a single LP, but for the negative value of β , their method fails.Tantawy [10] develops another technique for solving LFP which can be used for sensitivity analysis.Effati and Pakdaman [11] propose a method for solving the interval-valued linear fractional programming problem.Pramanik et al. [12] develops a method for solving multi-objective linear plus linear fractional programming problem based on Taylor Series approximation.In this paper, our intent is to develop a new technique for solving any type of LFP problem by converting it into a single linear programming (LP) problem because at some cases in the denominator and numerator when β is negative, available methods are failed to solve the linear fractional problem.We also develop a FOR- TRAN computer program for solving it and analyze the solution by numerical examples.

Mathematical Formulation of LP and LFP
The mathematical expression of a general linear programming problem is as follows: where one and only one of the signs (≤, =, ≥) holds for each constraint and the sign may vary from one constraint to another.Here ( ) , , , : , , , and ( ) . The set S is called the constraints set, feasible set or feasible regionof (LP).
In matrix vector notation the above problem can be expressed as: where A is an m n × matrix, x is an The mathematical formulation of an LFP (in matrix vector notation) is as follows: where A is an m n × matrix, x is an  ( ) where,

Calculation for the Unknown Variable of the LFP
From the above LP, we get This is our required optimal solution.Putting the value of x in the original objective function, we can get the optimal value.
Case II: ) where, , and , Same as above procedure, we have ( ) where, For constraints, following the same procedure as above, we get Same as above procedure, we have ( ) For constraints, following the same procedure as above, we get

Algorithm
If

Numerical Examples
Here we illustrate some numerical examples to demonstrate our method.Example 1: , A b is related to the second constraint and 3 3 , A b is related to the third constraint.So, we have the new objective function.

Minimize
Now for the first constraint, For the third constraint, ( ) ( ) Converting the LP in standard form we have Maximize ( ) ( )

Output:
Minimum value of the Objective Function = −1.090909.X 1 = 7.000000; X 2 = 0.000000.We see that our hand calculation result and computer oriented solution is the same.This shows that our computer program is correct.
Solution: Here we have, ( ) ( ) So, we have the new objective function Maximize ( ) ( ) ( )( ) ( ) Now for the first constraint, For the first constraint, , , , 0 y y s s ≥ Now we get the following tables (Table 3 and Table 4):

Output:
Maximum value of the Objective Function = 3.000000.X 1 = 3.000000; X 2 = 0.000000.Note: This problem cannot be solved by any available method because the of β is negative.Example 3: For the second constraint, ( ) ( ) , , , 0 y y s s ≥ Now we get the following tables (Table 5 and Table 6):

Output:
Maximum value of the Objective Function = 1.800000.X 1 = 0.000000; X 2 = 1.000000.Example 4: Production Problem of a Certain Industry.Suppose an industry has Tk.3,00,00,000/= by which it can produce six different products Palm oil, Coconut oil, Mustard oil, Soyabean oil, Sunflower oil and Dalda.The net refined oil from per metric ton cobra, master seeds, sunflower seeds, palm crude oil, soyabean crude oil are respectively 300 kg, 400 kg, 400 kg, 980 kg, 970 kg.The industry has some production loss for palm oil and soyabean oil, which are respectively 2% and 3%.The industry has a fixed establishment cost is Tk.5,00,000.The management of industry wishes to produce maximum 600 metric tons different types of oil.The cost for different raw materials to produce per metric ton crude oil/ seed/cobra in taka as follows (Table 7).
In addition the industry has the following limitations on expenditures: The objective is to maximize the ratio of return to investment.This leads to a linear fractional program as shown below.
Formulation of Example 4. The three basic steps in constructing an LFP model are as follows: Step 1. Identify the unknown variables to be determined (decision variables) and represent them in terms of algebraic symbols.
Step 2. Identify all the restrictions or constraints in the problem and express them as linear equations or inequalities, which are linear functions of the unknown variables.
Step 3. Identify the objective or criterion and represent it as a ratio of two linear functions of the decision vawhich is to be maximized (or minimized).Now we shall formulate the above problem as follows: Step 1. Identify the decision variables.For this problem the unknown variables are the metric tons of refined oil produced for different product.So, let 1 x = The metric tons of dalda has to be refined; Step 2. Identify the constraints.In this problem constraints are the limited availability of found for different purposes as follows: 1) Since the management of industry wishes to produce maximum 600 metric tons different types of oil, so we have x i =  are not allowed to be negative.That is, we do not make negative quantities of any product.
Step 3. Identify the objective function.
In this case, the objective is to maximize the ratio of total return and investment by different crops.That is ( ) , , , , , 0 x x x x x x ≥ The problem consists of 6 decision variables and 11 constraints.To solve it by hand calculations it involves variables and 11 constraints, which cannot be accommodated in available size of papers.Moreover, in real life, there may be some problems which may be involved with hundreds of constraints and variables and hence these cannot be solved by hand calculations.To overcome difficulties one has to require computer oriented solutions.Now, applying the computer program, we have obtained the following solutions.
We convert the LFP into an LP in the following way assuming that 0 Here we have, Now, we solve the above problem by computer program.
related to the first constraint and 2 2 , A b is related to the second constraint.
Putting this value in the original objective function, By using computer technique we get the following result.
related to the first constraint and 2 2 , A b is related to the second constraint.So we have the new objective function Maximize Converting the LP in standard form, we have Maximize Putting this value in the original objective function, we have Maximum 6 Using computer program, we get the following result.

2 x= 3 x= 4 x= 5 x= 6 x=
The metric tons of coconut oil has to be refined; The metric tons of mustard oil has to be refined; The metric tons of sunflower oil has to be refined; The metric tons of soyabean oil has to be refined; The metric tons of palm oil has to be refined.

Table 1 andTable 2 ) : Table 1 .
Now we get the following table (Initial table for Example 1.

Table 2 .
Final table for Example 1.

Table 3 .
Initial table for Example 2.

Table 4 .
Final table for Example 2.

Table 5 .
Initial table for Example 3.

Table 6 .
Final table for Example 3.

Table 7 .
Cost for different raw materials.
Now we have expressed our problem as a mathematical model.Since the objective function is the ratio of return to investment and all of the constraints functions are linear, the problem can be modeled as the following LFP model: