Cmjv01i01p0087

J. Comp. & Math. Sci. Vol. 1(1), 87-91 (2009).

MODIFIED PARAMETRIC APPROACH FOR LINEAR PLUS LINEAR FRACTIONAL PROGRAMMING PROBLEM Sanjay Jain and Adarsh Mangal* Department of Mathematical Sciences, Government College, Ajmer- 305 001 (India) *Department of Mathematics, Government Engineering College, Ajmer-305001 (India) E-mail: drjainsanjay@gmail.com, dradarshmangal1@gmail.com ABSTRACT A new parametric approach has been proposed for solving Linear plus Linear Fractional Programming Problem (LPLFPP). The approach is quite simple because initially the original linear plus linear Fractional Programming Problem is reduced into Linear Fractional Programming Problem (LFPP) by using parametric substitution. Solution of original LPLFPP can be recovered from the solution of LFPP by using programming theorems.

Key words: Linear plus linear fractional programming problem, Optimal solution, Objective function, Constraints, Feasible solution.

1. INTRODUCTION

problem by treating the value of the linear form in the objective function as a parameter.

Earlier Joksch2, Hirche4 and Schaible6 considered the linear plus linear fractional programming problem with objective function as

Max s. t . and

f ( x)  L ( x)  Ax  b x0

1 M ( x) (1.1)

and reduced the problem to a parametric linear

Further Tuteja 1, Sinha and Tuteja 5 considered the linear plus linear fractional programming problem with objective function as the sum of linear function of the form

Max

f ( x )  f1 ( x ) 

s. t.

Ax  b

and

x0

1 g1 ( x) (1.2)

[ 88 ] and obtained its solution by converting it into linear fractional programming problem.

Recently Jain et al. 7 , Jain and Mangal 8 considered the linear plus linear fractional programming problem with objective function as the sum of linear and linear fractional function of the form

Max s. t. and

f ( x)  L ( x)  Ax  b x0

M ( x) N ( x) (1.3)

and obtained its solution by reducing it to a linear fractional programming problem and further reducing to a linear programming problem. Jain et al.9,10 also solve such problem by elimination technique and by using fuzzy approach.

Max s. t .

a iT x  b

and

x0

Max s. t.

(c T x   ) (d T x   ) (2.1) x  X  x, A x  b

f ( x )  (e T x   ) 

where, x  Rn , c, d and e are n vectors, b  Rm+n and ,  and  are scalars, A is an (m+n) x n matrix. Here nonnegative constraints are included in set of constraints. It is assumed that the set of feasible solutions X is bounded and closed and (dTx + ) > 0.

Above problem (2.1) can be put in the standard form as

(c T x   ) (d T x   )

; i  1,2, ..., m  n (2.2)

Here aiT represent the ith row of the given matrix A. In the non-degenerate case, an extreme point of the feasible region X lies on the some n-linearly independent subset of X. If (dTx + )  0, then problem (2.2) can be reformulated as

Max s. t.

2. The Problem Formulation for Parametric Approach : Here we consider the linear plus linear fractional programming problem as:

f ( x )  (e T x   ) 

  dT  x   T f ( x)  (e T x   )   c T    ( d x   )    T  bd  x b  A    (2.3)   (d T x   )  

If

x  y  0 , then problem (d x   ) T

(2.3) can be put in the form

Max

f ( y )  (c T y   ) 

  (1  d T y )

2 d T y  2 eT     (1  d T y ) (d T x   ) s. t.

 bdT  A   

(2.4)

 b  y   

Using the substitutions t =  (1  d T y ) and z = t y in problem (2.4), it takes the following form of fractional programming problem:

[ 89 ] c T z t   t   t 2  2  d T z   2 eT  t Max. F ( z, t )  t2 T s. t.

A b d z bt 0





 t  d z  t z, t  0 T

and

We have assumed that  (1  d T y ) >0 and z = t y, where z1 is an optimal solution of FPP (2.5). Let us put t1 =  (1  d T y ) > 0

2

then (t1z1, t1) is feasible solution for FPP (2.5) and (2.5)





The set S = {  z , t : A   b d T z  b t  0,

 t  d T z  t 2 ; z , t  0 } is clearly a convex

Max. F ( z1t1 , t1 )



c T z1t1   t1   t12  2  d T z1   2 e T  t1 t12

set.

 cT 3. Solution : The following programming theorems in generalized form are useful for the solution of LPLFPP.



Theorem-1: If z s , t s

 is an optimal

solution of reduced fractional programming problem (2.5), then ts>0.

Proof : Suppose zs  0 and ts = 0 be an optimal solution of reduced FPP (2.5) and take an element z from a set S.

z   z   z s is in S of all   0 and

z1  z 1 eT      2 d T 1   2 t1 t1 t1 t1 t1

 cT

z1     T  1 d y  1 d T y











2  d T z1 eT  2    1 dT y  1 dT y









> 0 and finite.

Which contradicts that zs = 0 and ts = 0 is an optimal solution of reduced FPP (2.5). Hence ts > 0.

Theorem-2 : If

z

s



, t s is an optimal

solution of reduced fractional programming

 zs  ts

problem (2.5), then 

  is an optimal solution 

in that case S is unbounded, which contradicts our assumption that S is bounded.

of original fractional programming problem.

Hence zs  0 and ts = 0 cannot be an optimal solution of reduced FPP (2.5).

Proof: Let y1 be an optimal solution of reduced FPP (2.5). Then there exists a t1 namely t1 =  (1  d T y ) > 0, such that (t1z1, t1)

Now we assume that both variables zs = 0 and ts = 0 is an optimal solution of reduced FPP (2.5), then {Max F(z, t) : (z, t)  S} = .



is feasible solution for FPP (2.5). Since z s , t s



is an optimal solution of reduced fractional programming problem (2.5), so we have

[ 90 ] Max. F (z s , t s )

Hence the required optimal solution of original

c T z s t s   t s   t s2  2  d T z s   2 e T  t s  t s2

problem is 

 c T y1   

 zs  ts

 2  d T y1 e T  (3.1)  2 t1 t1 t1

 zs Now by theorem (1), ts > 0; so   ts

  . 

4. CONCLUSION

z 

  is feasible 

As already proved that  s  is a  ts 

solution for LPLFPP and we have

z f  s  ts

Max

z 

 z    (c T s   )  z ts   (1  d T s ) ts

feasible solution of LPLFPP and  s  also  ts   satisfies the objective function of LPLFPP,

zs ts  2 eT    z z  (1  d T s )  (1  d T s ) ts ts 2 d T

z f  s  ts

.

 z  2 d T z s  2 eT    (c T s   )    ts ts ts t s2 

 zs 

which clearly prove by theorem 2 that    ts  is an optimal solution of LPLFPP. Thus for the solution of proposed original LPLFPP, we first solve the reduced FPP (2.5) by any existing method and then using programming theorems.

Thus any local minimum of the FPP (2.5) will be a global minimum of original LPLFPP. 5. Particular Case :

z f  s  ts

.



T

  2 d y1   (c T y1   )   t1 t1 

 2eT  t1

[u sin g (3.1)]

2 d T y1   (c y1   )    (1  d T y1 )  (1  d T y1 ) T

2 T

 e   (1  d T y1 )  F ( y1 ) 

If we substitute cT = 0 and  = 1 in proposed problem (2.2), then it reduces to

Max

f ( x )  (e T x   ) 

s. t.

Ax  b

and

x0

1 (d x   ) T

which is considered by Sinha and Tuteja5.

[ 91 ] REFERENCES 1. G. C. Tuteja, "Programming with sum of a linear and quotient objective function" Opsearch, 37, 177-180 (2000).

2. H.C. Joksch, "Programming with fractional linear objective functions" Naval Research Log. Quart., 65, 197-204 (1964).

3. I.M. Stancu-Minasian, "Fractional Programming: Theory, Methods and Applications" Kluwer Academic Publishers, Dordrecht (1997).

4. J. Hirche, "A note on programming problems with linear - plus linear fractional objective functions" European Journal of Operational Research, 89, 212-214 (1996).

5. S.M. Sinha and G.C. Tuteja, "On fractional programming" Opsearch, 36, 418 - 424 (1999).

6. S. Schaible, "A note on the sum of a linear and linear fractional functions" Naval

Research Log. Quart., 24, 691-693 (1977).

7. S. Jain, A. Mangal and P. R. Parihar, "Linear plus linear fractional programming with non-differentiable function" Journal of Combinatorics, Information and System Sciences, 30, 139-149 (2005).

8. S. Jain and A. Mangal, "Solution of a generalized fractional programming problem" Journal of Indian Academy of Mathematics, 26, 15-21 (2004).

9. S. Jain and K. Lachhwani, "Linear plus fractional multiobjective programming with homogenous constraints using fuzzy approach" Accepted for publication in Iranian Journal of Operations Research (2009).

10. S. Jain, "Modeling of Fourier elimination technique for multiobjective linear programming problem" Accepted for publication in Journal of A. M. S. E. (France) SeriesA (2009).