AN ITERATIVE METHOD FOR SOLVING LINEAR FRACTION PROGRAMMING (LFP) PROBLEM WITH SENSITIVITY ANALYSIS

In this paper an iterative method for solving linear fraction programming (LFP) problem is proposed, it will be shown that this method can be used for sensitivity analysis when a scalar parameter is introduced in the objective function coefficients and our task of this sensitivity investigation is to maintain the optimality of the problem under consideration. A simple example is given to illustrate this proposed method.

Linear fraction maximum problems (i.e.ratio objective that have numerator and denominator) have attracted considerable research and interest, since they are useful in production planning, financial and corporate planning, health care and hospital planning.Several methods for solving this problem are proposed in [5], their method depends on transforming this (LFP) to an equivalent linear program.Another method is called updated objective function method derived from [3] 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 .Also some aspects concerning duality and sensitivity analysis in linear fraction program was discussed by [6] and [6] in his paper made a useful study about the optimality condition in fractional programming.More recent works on fractional programming theory and methods can be found in [1,2].The suggested method in this paper depends mainly on the updated method in iterative manner then the optimality condition for a given basic feasible solution of (LFP) is defined.Also we may be interested in the effects of small changes in the objective function coefficient on the optimality condition for a given optimal solution and the task of this sensitivity investigation is to maintain the optimality of the given problem.In section 2 some notations and definitions of the (LFP) problem is given while in section 3 the problem of sensitivity analysis when scalar parameter is introduced into the objective function coefficient is discussed followed by a simple example in section 4 to illustrate this sensitivity study and finally a conclusion about this sensitivity investigation is made in section 5.

2-NOTAIONS AND DEFINITIONS
Linear fraction program (LFP) problem arises when a linear ratio function has to be maximized (minimized) on the set X ={x; Ax = b; x ≥ 0}, hence the (LFP) problem can be written as c T x +α Maximize Z = (1) Here c and d are vectors in R n represent the objective function coefficients, α and β are scalars, A is an m х n matrix and b ∈ R m , also it is assumed that d T x + β > 0 for every x ∈ X.For the (LFP) problem we have (c T -Z d T ) x = β Z -α represents the Z -level curve of the linear fraction objective function and when Z is arbitrary it is seen that each level curve of (LFP) is linear over X, and so one extreme of X has to be the optimal solution for (LFP).Consider the linear programming problem Here Z 0 is a given constant; hence we have the following proposition Proposition 2-1 if x 0 solves the linear fractional programming (LFP) defined by (1) with optimal value Z 0 then x 0 solves the linear programming (2) with optimal value Z* = βZ 0 -α .Proof: Since the level curve of the objective function of the (LFP) defined by ( 2) is in the form (c T -Z d T ) x = β Z -α , the proof is Straight forward.Suppose x is a basic feasible solution of Ax = b, x ≥ 0 with corresponding basic decomposition then Ax = b, x ≥ 0 can be written Bx B + Nx N = b, x B ≥ 0 and x N ≥ 0. If we contract a tableau for x in the form

3-SENSITIVITY ANAYSIS IN LINEAR FRACTION PROGRAMMING
Introducing a scalar parameter in the objective function coefficient of the (LFP) problem defined by (1) may affect the optimality of the given problem due to some changes in the data of the objective function.In this case the (LFP) problem defined by (1) becomes x ∈ X, where c , d are given vectors in R n and µ is scalar parameter introduced to the objective function coefficients, hence to maintain the optimality of a given basic feasible solution determined when µ = 0 the condition ( (7) Must be satisfied, solving (7) for µ will give a range on µ such that the optimal basic feasible solution doesn't change.

An application on production example
Let us consider a company that manufactures two kinds of products A 1, A 2 with a profit of 4, 2 dollar per unit respectively..However the cost for each one unit of the above products is 1, 2 dollar.It is assumed that a fixed cost of 5 dollars is added to the cost function due to expected duration through the process of production and also a fixed amount of 10 dollar is added to the profit function.If the objectives of this company is to maximize the profit in return to the total cost, provided that the company has a raw materials for manufacturing and suppose the material needed per pounds are 1, 3 and the supply for this raw material is restricted to 30 pounds, it is also assumed that twice of production of A 2 is more than the production of A 1 at most by 5 units .In this case if we consider x 1 and x 2 to be the amount of units of A 1 , A 2 to be produced then the above problem can be formulation as For µ = 0 the optimal basic feasible solution is given by

CONCLUDING REMARKS
Sensitivity analysis in linear fraction program due to some changes in the objective function coefficient is investigated to maintain the optimality of the given problem also we hope to extend this method to the case of multiple objective linear fraction programming problems because in many real problems the data of the objective functions are dynamically changes.

5 )Remark 2 - 2 . 1 Bx
Suppose x B is a basic feasible solution for the linear program problem defined by(2) with corresponding tableau T(x), if any r i ∈ r; i ∈ {1,2,…n-m} is negative then pivoting in r i yields a new basic feasible solution and the optimality condition occurs when r ≥ 0 (Remark 2-1.Condition (5) reduces to the well known optimality condition in linear programming problem when d is the zero vectors.In this method pivoting in r i ∈ r; i ∈ {1,2,…n-m} yields a new basic feasible solution with an increment in the objective function value given by the case of the ordinary linear when d = 0 the increment of the objective function value equalsθ r i .

+
scalar parameter is introduced to the objective function coefficient, then the corresponding From the above tableau we have the set of inequalities in the form 8 be the set of all parameters for which the optimal basic feasible solution remains optimal.