On Solving Linear Fractional Programming Problems

A new method namely, denominator objective restriction method based on simplex method is proposed for solving linear fractional programming problems. Further, another method namely, decomposition-restriction method based on decomposition principle and the denominator objective restriction method is proposed for obtaining an optimal fuzzy solution to the fully fuzzy linear fractional programming problem. The procedures for the proposed methods are illustrated with the numerical examples.


Introduction
Linear fractional programming (LFP) problems are a special type of non-linear programming problems in which the objective function is a ratio of linear functions and the constraints are linear functions.In real life situations, linear fractional models arise in decision making such as construction planning, economic and commercial planning, health care and hospital planning.In the literature, several methods (Bajalinov, 2003;Stancu-Minasian, 1997, 2006) have been recommended to solve LFP problems.Isbell and Marlow (1956) first identified an example of LFP problem and solved it by a sequence of linear programming problems.Charnes and Cooper (1962) considered variable transformation method to solve LFP and the updated objective function method were developed for solving the LFP problem by Bitran and Novaes (1973).Gilmore and Gomory (1963), Martos (1964), Swarup (1965), Wagner and Yuan (1968), Pandey and Punnen (2007) and Sharma et al. (1980) solved the LFP problem by various types of solution procedures based on the simplex method developed by Dantzig (1962).Tantawy (2007Tantawy ( , 2008) ) proposed two different approaches namely; a feasible direction approach and a duality approach to solve the LFP problem.Mojtaba Borza et al. (2012) solved the LFP problem with interval coefficients in objective function which is based on Charnes and Cooper technique (1962).Odior (2012) solved the LFP problem by algebraic approach which depends on the duality concept and the partial fractions.
In real life model, the possible values of coefficients of a linear programming problem are obviously unclear and vague.In fuzzy decision making problems, the idea of maximizing decision was anticipated by Bellman and Zadeh (1970).The theory of fuzzy linear programming on general level was initially proposed by Tanaka et al. (1973).Buckley and Feuring (2000) measured the fully fuzzified linear programming problem (FFLP) by transformation to multiple objective deterministic non-linear programming problems.Li and Chen (1996) solved a fuzzy linear fractional programming form by fuzzy coefficients, using the concept as well as mathematical definition of the fuzzy optimal.Jayalakshmi and Pandian (2012) have proposed a method namely, bound and decomposition method to a fully fuzzy linear programming (FFLP) problem to obtain an optimal fuzzy solution.Hashemi et al. (2006) planned a two-phase approach based on the evaluation of mean plus standard deviation of fuzzy numbers to get the optimal solutions of the FFLP problem.Mikaeilvand et al. (2008) projected a method to solve FFLP through defuzzifying with a linear ranking function.Pop andStancu Minasian (2008) andBogdana Stanojevi'ca andStancu-Minasianb (2012) used deterministic multiple objective linear programming problem by quadratic constraints to work out FFLP problems.Nachammai et al. (2012) considered FFLFP problem by using ranking method based on metric distance.
In this paper, we propose a new method namely, denominator objective restriction method for finding an optimal solution to LFP problems.In this proposed method, we construct two linear programming problems from the given LFP problem such that one is of maximization type and the other is of minimization type.Then, we attain an optimal solution to the given LFP problem from the solutions of the two constructed linear programming problems.The proposed method is based only on the simplex method which differs totally from transformation method introduced by Charnes and Copper (1962) and the fractional simplex method introduced by Swarup (1965).Further, based on the decomposition principle and the denominator objective restriction method, we develop a new method namely, decomposition-restriction method to the FFLFP problem.In the decomposition-restriction method, the fuzzy ranking function, the transformation technique and multi-objective non-linear programming technique are not used.Numerical examples are given for better understanding the solution procedures of the proposed methods.

Preliminaries
We require the following definitions of the basic arithmetic operators and partial ordering relations on fuzzy triangular numbers based on the function principle which can be established in Bellman and Zadeh (1970), Jayalakshmi and Pandian (2012) and are used in section 4.

Definition 2.1 A fuzzy number a
 is a triangular fuzzy number denoted by ( , , ) a a a where , a a and a are real numbers and its membership function x a a a for a x a x a x a a for a x a 0 o t h e r w i s e a a a and ( , , ) b b b be two triangular fuzzy numbers.Then, .  , ),  , ,  ( , , ) Let ( ) F R be the set of all real triangular fuzzy numbers.

Linear Fractional Programming Problems
Consider the following LFP problem: where X, C, D are 1 n  vectors, Bis an 1 m  vector and ,   are scalars.
It is assumed that the set of feasible solutions to the problem (P), is non-empty and bounded.Now, we can construct two single objective linear programming problem from the given problem (P) as follows: Now, we prove the following two theorems connecting the solutions of the problem (P), the problem (N) and the problem (D) which are used in the proposed method.
Theorem 3.1 Let  X be an optimal solution to the problem (N).If { } n X is a sequence of basic feasible solutions to the problem (D) such that ( ) ( ) Z X Z X by simplex method considering the solution  X as an initial feasible solution, then n X is an optimal solution to the problem (P).
Proof: It is obvious that n X is a feasible solution to the problem (P) Let U be a feasible solution to the problem (P).
This implies ( ) k 0 1 2 n 1 and ( and the problem (D) is of minimization type, we have ( ) ( ) X is an optimal solution to the problem (P).
Case 2: ( ) and also, the problem (D) is of minimization type, we can conclude that ( ) ( ) X is an optimal solution to the problem (P).

Thus, n
X is an optimal solution to the problem (P).
Thus, the theorem is proved.
Theorem 3.2 Let X  be an optimal solution to the problem ( is an optimal solution to the problem (D) by simplex method considering the solution X  as an initial feasible solution, then is an optimal solution to the problem (P).
is a feasible solution to the problem (P).
Let V be a feasible solution to the problem (P).
X is an optimal solution to the problem (D), we have ( ) ( ) X is an optimal solution to the problem (D), we can conclude that ( ) ( ) is an optimal solution to the problem (P).
Hence, the theorem is proved.Now, we introduce a new method namely, denominator objective restriction method for finding an optimal solution to the LFP problem (P).
The proposed method proceeds as follows: Step 1: Construct two single objective linear programming problems namely, the problem (N) as well as the problem (D) from the given problem (P).
Step 2: Compute the optimal solution to the problem (N) by means of the simplex method.Let the optimal solution to the problem (N) be X  and Max. ( ) Step 3: Using the optimal table of the problem (N) as an initial simplex table to the problem (D), continue to find a sequence of improved basic feasible solutions { } n X to the problem (D) and the value of Z at each of the improved basic feasible solution by the simplex method.
Step 4: for some n, stop the computation process and then, go to Step 5.
Step 4: X is an optimal solution to the problem (D) for some n, stop the computation process and then, go to Step 6.
Step 5: n X is an optimal solution to the problem (P) and Max. ( ) ( )  n Z X Z X by the Theorem 3.1.
Step 6:  n 1 X is an optimal solution to the problem (P) and Max. ) by the Theorem 3.2.
Remark 3.1 The maximum value for (n +1) is the number of the iterations to get an optimal solution to the problem (D) using simplex method.
The proposed method for solving the LFP problem is illustrated through the following examples.
Example 3.1 Consider the following LFP problem: The following two LP problems can be obtained from the given problem: Now, the optimal solution to the problem (N), by the simplex method, is given by the following table: Therefore, the optimal solution to the problem and the value of 51 40  Z .

Now, by
Step 3 of the proposed method, the initial simplex table to the problem (D) is given below: Now, the variable 2 s enters into the basis and the variable 1 x leaves from the basis.

Ist iteration table:
Z and by the Step 4(a) of the proposed method, the optimal solution to the given linear fractional Example 3.2 Consider the following LFP problem: The following two LP problems can be obtained from the given problem: Now, by the simplex method, the optimal solution to the problem (N) is given in the following table: Therefore, the optimal solution to the problem (N) is 1 2 and the value of 3 2  Z .

Now, by
Step 3 of the proposed method, the initial simplex table to the problem (D) is given below: Now, the variable 2 s enters into the basis and the variable 1 x leaves from the basis.

Since the I
st iteration table is optimal and by Step 4(b) of the proposed method, the optimal solution to the given linear fractional programming problem is 1 0

Fully Fuzzy Linear Fractional Programming Problem
Consider the following FFLFP problems having m fuzzy constraints and n fuzzy variables: ( ) , ( ) , for all 1 j n   and 1 i m   .
Let the parameters ,     , , , ,  be the triangular fuzzy numbers 1 2 3 ( , , ) a b c and ( , , ) Then, the problem (FP) can be written as follows: , , ) ( , , ) , , n j j j j j j j n j j j j j j j p q r x y t z z z m n l x y t x y t is a triangular fuzzy number, then j j j x y t   , j =1,2,…,m. (4.1) The relation (4.1) is called bounded constraints.
Now, using the arithmetic operations and partial ordering relations, we decompose the given FFLFP problem as follows: , , ) ( , , ) , , n j j j j j j j n j j j j j j j p q r x y t m n l x y t , , , , , , ) ( , , ) , , n j j j j j j j n j j j j j j j p q r x y t m n l x y t and all decision variables are non-negative.
From the above decomposition problem, we construct the following three crisp LFP problems namely, middle level problem ) (MLP , upper level problem ) (ULP and lower level problem ) (LLP as follows: subject to constraints in the decomposition problem in which at least one decision variable of the ) (MLP occurs and all decision variables are non-negative.
j j j j j n j j j j j j j p q r x y t m n l x y t constraints in the decomposition problem in which at least one decision variable of the ) (ULP occurs and are not used in ) (MLP ; all variables in the constraints and objective function in ) (ULP must satisfy the bounded constraints; replacing all values of the decision variables which are obtained in ) (MLP and all decision variables are non-negative.

And
) constraints in the decomposition constraints in which at least one decision variable of the ) (LLP occurs which are not used in ) (MLP and ) (ULP ; all variables in the constraints and objective function in ) (LLP must satisfy the bounded constraints; replacing all values of the decision variables which are obtained in the ) (MLP and ) (ULP and all decision variables are non-negative, where 2 z  is the optimal objective value of ) (MLP .Now, we prove the following theorem which is used in the proposed method to solve the FFLFP problem.  , for all feasible solution of the problem (P). Therefore,  is an optimal fuzzy solution to the given problem (FP) where each one of 1 2 3 , and , 1, 2,..., Hence the theorem is proved.
Remark 4.1 In the case of LFP problem involving trapezoidal fuzzy numbers and variables, we decompose it into four crisp LFP problems and then, we solve the middle level problems (second and third problems) first.Then, we solve the upper level and lower level problems and then, we obtain an optimal fuzzy solution to the given LFP problem involving trapezoidal fuzzy numbers and variables.
Remark 4.2 In the case of LFP problem involving interval fuzzy numbers and/or interval variables, we decompose it into two crisp LP problems and then, we solve the upper level problem first.Then, we solve the lower level problem and then, we attain the interval optimal solution to the given LFP problem involving interval numbers and / or interval variables.Now, we introduce a new method namely, decomposition-restriction method to the FFLFP problem (FP) for obtaining an optimal fuzzy solution, based on the decomposition principle and the denominator objective restriction method.
The proposed method proceeds as follows: Step 1: Construct three crisp LFP problems namely Middle level problem, Upper level problem and Lower level problem from the given FLFP problem.
Step 2: Solve the Middle level problem by the denominator objective restriction method.
Step 3: Using the results of Step 2 and the denominator objective restriction method, solve the Upper level problem.
Step 4: Using the results of Step 2, Step 3 and the denominator objective restriction method, solve the Lower level problem.
Step 5: Using the results of Step 2, Step 3 and Step 4, obtain an optimal fuzzy solution to the given FFLFP problem by the Theorem 4.1.
The proposed method is illustrated with the following example.
Example 4.1 Consider the following FFLFP problem: x y  and 1 2 , 0 x x  , we can conclude that 2 0 x  and 1 0 x x  where 0 [0,1] x  .

Conclusion
In this paper, the denominator objective restriction method is developed for solving LFP problems based only on simplex method in.It is easy to understand, compute and also, to interpret.In the second part of the proposed method, that is, solving the problem (D), each iteration solution is a next level accepted solution to the LFP problem that may be used by decision makers according to their situations if they need.Further, the decomposition-restriction method is used to solve the FFLFP problem in which the fuzzy ranking function, transformation technique and multi-objective non-linear programming technique are not used, but it is based on the decomposition principle and the denominator objective restriction method.Both the methods can serve decision makers by providing an appropriate best solution to a variety of linear fractional programming models having crisp or fuzzy parameters and variables in a simple and effective manner.
In Bogdana Stanojevi'ca and Stancu-Minasianb (2012), the optimal fuzzy solution to the FFLFP problem (Example 4.1.)is 1 based on Charnes-Copper method and multiobjective nonlinear programming, but by the decomposition-restriction method, the optimal fuzzy solution to the same FFLFP problem is