ABSTRACT Anew method which does not depend on the simplex method for solving the linear programming (LP) problems is given. The proposed method is based mainly upon solving this problem algebraically using the concept of duality .The important of this method is that we are not based our work on vertex information which may have difficulties as the problem size increases. A simple example is given to clarify the developed theory of this proposed method

R H Sallam

S F Tantawy

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* Corresponding author:  Tel: + 
E-mail: saidtantawy@hotmail.com 
 
 
 
 
 
 
 
ISSN: 0976-3031 
RESEARCH ARTICLE 
 
A NEW METHOD FOR SOLVING LINEAR PROGRAMMING PROBLEMS 
 
 
Tantawy, S.F and Sallam, R.H 
 
Mathematics Department Faculty of Science,Helwan University (11795) Cairo, Egypt 
 
 
ARTICLE INFO                                                ABSTRACT 
 
 
Anew method which does not depend on the simplex method for solving the linear 
programming (LP) problems is given. The proposed method is based mainly upon solving 
this problem algebraically using the concept of duality .The important of this method is 
that we are not based our work on vertex information which may have difficulties as the 
problem size increases.   A simple example is given to clarify the developed theory of this 
proposed method. 
 
 
 
 
INTRODUCTION  
 
 
The problem of linear programming (LP) is one of the earliest 
formulated problems in mathematical programming where a 
linear function has to be maximized (minimized) over a 
convex constraint polyhedron X. The simplex algorithm was 
early suggested for solving this problem by moving toward a 
solution on the exterior of the constraint polyhedron X. In 
1984, the area of linear programming underwent a 
considerable change of orientation when Karmarker [1984] 
introduced an algorithm for solving (LP) problems which 
moves through the interior of the polyhedron. This algorithm 
of Karmarker's and subsequent additional variants [6,8,9,16   ] 
established a new class of algorithms for solving linear 
programming problems known as the interior point methods 
for more details about interior point methods we refer the 
reader to  [17]  .Also more  works has been made to solve this 
problem [1-3,10-15]. In this paper a new method which does 
not depend on the simplex method for solving linear 
programming problems is given .This method solves this 
problem algebraically using the concept of duality. The 
important of this study is that in some certain problems on 
using the simplex algorithm we have to visit a large number of 
extreme points since we are moving toward a solution on the 
exterior of the constraint polyhedron. In Section 2 we give a 
full description of the problem together with our main results 
while Section 3 contains the steps of our new algorithm with a 
simple example to illustrate the given method followed by our 
conclusion in Section 4. 
 
 
Definitions and theory 
 
 
The linear programming problem (LP) arises when a linear 
function has to be maximized on a convex constraint 
polyhedron X. this problem can be formulated as follows: 
                        Maximize F(x) =    cTx  
                            Subject to  
        x   X = {x, Ax ≤ b}                                                      (1) 
 
where c, x   Rn, A is an (m + n) × n matrix, b   Rm+n, we 
point out that the nonnegative conditions are included in the 
set of constraints. This problem can also be written in the 
form: 
                       Maximize   F(x) = cTx  
                               Subject to  
     a Ti x  bi        i = 1, 2... , m + n.                                      (2)  
 
here a Ti  represents the  i th row of the given matrix A, then 
we have to note that   in the non degenerate case an extreme 
point (vertex) of X lies on some n linearly independent subset 
of  X.  Now consider the dual problem for the linear program 
(1) in the form  
 
                        Minimize    w= uTb   
                             Subject to     uTA   = cT           (3) 
                                  u ≥ 0  
since the set of constraints of this dual problem  is written in 
matrix form hence we can multiply both side by a matrix T = 
(T1 | T2),  where T1 = c (  cT  c )-1  , and the columns of the 
matrix T2 constitute the bases of 
 
N(cT ) = {x; cT  x =0 }                                                             (4) 
 
Then we have        uTAT1 = 1,     uTAT2 = 0    and u ≥ 0.        (5) 
 
If we define   q х (m+n) matrix P of non-negative entries such 
that   PAT2 = 0,  then (5) can be rewritten in the form  
 
vT G= 1,     and v ≥ 0.                                                              (6) 
 
where    G=P AT1 and  vT P= uT  ,   we have to note that 
equation (6)  will play an important  rule for specifying the 
dual values needed for solving the linear programming (1) .  
This dual values satisfied the well known Kuhn-Tucker 
condition [4,5]such that  for a given  point x0 to be an optimal 
solution of the  linear program we must find  
 u ≥ 0 satisfy  A Tr u = c    ,   this equation can be written as 

Available Online at http://www.recentscientific.com 
 International Journal 
of Recent Scientific 
Research 
 
International Journal of Recent Scientific Research 
Vol. 4, Issue, 5, pp. 623 - 625, May, 2013 
 
Article History: 
 
 
Received 15th, March, 2013 
Received in revised form 17th, April, 2013 
Accepted15th, May, 2013 
Published online 28th May, 2013 
 
Key words: 
 
linear program-duality                © Copy Right, IJRSR, 2013, Academic Journals. All rights reserved.                  
 
International Journal of Recent Scientific Research, Vol. 4, Issue, 5, pp. 623 - 625, May, 2013 
624 
 
 u = (ArA
T
r )
-1Arc                                                                       (7)                       
here Ar is a submatrix of the given matrix A containing only 
the coefficients of the set of active constraints at the current 
point x0 . Also from the complementary slackness theorem for 
the above set of active constraints the corresponding dual 
variables must be positive. Using equation (6) then the 
equivalent linear programming of the linear programming   (3) 
can be written in the form   
                                       Minimize    w=vT g   
                                                 Subject to    
                                                     vT G = 1,      
                                                       v ≥   0                                (8                                              
with  G=P AT1,  g=Pb , vT P= uT ,the linear programming (8) 
has the dual linear programming problem in just one unknown 
Z in the form 
Maximize Z 
Subject to 
G Z≤   g                                                                                   (9) 
 
Remark  
 
The set of constraint of the above linear programming problem 
will give the maximum value Z0 and also will define only one 
active constraint for this optimal value. 
 
We  have to note that  from the complementary slackness 
theorem the corresponding dual value will be positive and the 
remaining dual variables will be zeros for the corresponding 
non active constraints 
                    
3- New method for solving linear programming (LP) 
problems 
 
Our method for solving (LP) problems summarize as follows  
 
Step1-Compute T1 = ( cT  c )-1 c , and the matrix T2  as in (4). 
Step 2-Find the matrix P of non-negative entries such that   
PAT2 = 0, 
Step 3-Use the linear programming (9)  to  find the optimal 
value Z0 and also determine the corresponding active 
constraint and use the constraint of (8) to compute  vT   
Step 4- Find the dual variables uT =vT P  , for each positive 
variable  ui   ,i=1,2,3,…,n  
find the corresponding active set of constraint of the matrix A  
Step 5- Solve an nxn system of linear equations for these set of 
active constraints 
(a subset  from an m+n  constraints )  to get the optimal 
solution. 
 
Example: Consider the linear programming problem 
  
                                     Maximize F(x) =    x1 + x2 
                                          Subject to:    
                                          2 x1 + 3x2  6  
                                           x1                 2  
                                                     x2 1                                       
                                    - x1  0, -  x2  0 
 
                                      1/2 -1 
Step1- compute   T1=    1/2       , and the matrix T2 is given by    
 
 
T2=       1       , 
             1                        
 
      we go to step2 to compute the nonnegative matrix P 
Step2 -                                               
1    1   0   0    0 
1    0   0   0    1 
0    1   1   0    0 
          P =                       0    1   0   1    0 
0    0   1   0    1 
0    0   0   1    1 
 
 
Step3-  For the linear programming problem 
 
Max Z 
Subject to 
3 Z≤   8 
2 Z≤   6 
Z  ≤  3 
0 Z≤   2 
0 Z≤   1 
0 Z≤ 0 
                     
The first constraint is the only  active constraint for the above 
linear programming , this active constraint shows that the  
maximum optimal value Z0=8/3 . Use the constraint of (8) to 
get the dual variables vT    =(1/3  0 0 0  0 0 )  and go to step 4 
Step 4- Compute  uT =vT P =   (1/3 1/3  0  0  0). And this   
Indicates that in the original set of constraints the first and the 
second constraint are the only active constraints   
Step5-Solve the system of linear equations in the form 
                                               
2 x1 + 3x2 = 6 
x1            =2   , 
   
 to get  the optimal solution  xT*= (2   2/3),    of the above linear 
programming problem with optimal value F* =8/3  
 
 
Conclusion  
 
In this paper we gave a new method to find the optimal 
extreme point for  the  linear programming problem .Our 
method is based on solving this problem algebraically and 
doesn't depend on the simplex version of linear program .The 
important of this study is that in using the simplex algorithm 
we are moving toward a solution on the exterior of the 
constraint polyhedron and in some certain problems we have to 
visit a large number of extreme points of the constraint 
polyhedron.  
 
References   
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A NEW METHOD FOR SOLVING LINEAR PROGRAMMING PROBLEMS

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A NEW METHOD FOR SOLVING LINEAR PROGRAMMING PROBLEMS

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