Volume-3, Issue-6, December-2013, ISSN No.: 2250-0758
International Journal of Engineering and Management Research
Available at: www.ijemr.net
Page Number: 73-75
A New Variant of Simplex Method
Dr. R.G. Kedia
Assistant Professor & Head, Department of Statistics, Govt. Vidarbha Institute of Science & Humanities, Amravati, INDIA
ABSTRACT
In this paper a new variant of simplex method is
proposed to solve a class of linear programming problem
having all constraints≥ type. In the proposed method,
instead of using artificial basis techniques we start with
infeasible solution obtained by converting all the
constraints into≤ type. Then by judicious selection of
entering and leaving vector in and from the initial simplex
table, feasible solution is obtained in the first iteration only
and optimal solution then can be obtained using regular
simplex method.
Keywords: Linear programming problem, Simplex
method, artificial variable, dual simplex method,
I.
INTRODUCTION
Linear programming model is extensively used to
solve the variety of problems in Engineering and
Management field particularly for production planning
and optimum resources allocation. To solve linear
programming problem, Dantzig devised simplex method
in 1947. ASimplex algorithm of Dantzig is one of the top
10 algorithms of 20th century [Barry Cipra (3)]. The
computational complexity in simplex method depends on
the number of variables and the number of constraints
and is directly proportional to both. To reduce the
complexity, several variants of simplex method are
developed by various researchers. For details Gass (2)
can be referred. If in the given linear programming
problem, one or more constraints are ≥ type or equality,
simplex method with artificial basis techniques is used.
Big-M method and two phase simplex method are quite
commonly used. In these methods artificial variables are
used to get the standard basis artificially. Artificial
variables are then forced to leave the basis step by step.
Once all the artificial variables are removed from the
basis, optimal solution is then obtained by using regular
simplex method. However, use of artificial variables to
get the standard basis increases the computational
complexity due to the reason mentioned above. To avoid
the use of artificial variables, the≥ type constraints are
converted into ≤ type constraints by multiplying by -1.
Then using slack variables only, initial basic solution is
obtained that is not feasible. In other words, one starts
with infeasible solution. Methods are developed to
solve linear programming problem by starting with
infeasible solution and force the solution to be feasible
as well as optimal at some iteration. One such method
namely dual simplex method devised by Lemke (5) is
most popular. In dual simplex method, starting from
infeasible solution feasible solution is obtained step by
step. Another basis-exchange pivoting algorithm is
the criss-cross algorithm by Terlaky, Tamás (4) .There
are polynomial-time algorithms for linear programming
that
use
interior
point
methods:
These
include Khachiyan's
ellipsoidal
algorithm, Karmarkar's projective algorithm, and pathfollowing algorithms. Interested readers can refer Robert
J. Vanderbeib (1).
In the proposed method in this article, we start
with initial basic solution that is infeasible. Then by
judicious selection of entering and leaving vector in and
from the initial simplex table, feasible solution is
obtained in the first iteration only and optimal solution
then can be obtained using regular simplex method.
Notations: The proposed method can be used to solve
the linear programming problem of the type
For further discussion, following notations will also be
used:
a j will denote the jth column in A matrix. Bi will denote
ith basis vector (row). x Bi will denote ith solution term. s i
will denote the slack variable in ith constraint. Other
notations will have their usual meaning.
II.
PROPOSED METHOD
Let the given linear programming problem be
We assume that there is no equality constraint
Rewrite the given problem as
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This completes the proof.
III.
PROPOSED METHOD
The working of the method is as follows.
The corresponding
problem is
standard
linear
programming
Let the given linear programming problem be
The given problem can be converted into maximization
type as follows:
First we prove the following theorem.
Theorem:
Let x B = -b be the initial basic
infeasible solution to the problem.
Now, select the non-basic vector all the elements
of which are negative to enter the basis all the elements
of which are negative. If there are more than such
vectors then among these, the vector for which z j –c j (= c j , as z j = 0 for all j) is least positive is selected to enter
the next basis. This is because the objective function is
to be minimized and cj with minimum value will
contribute least. Let a s be entering vector. Next, we
compute the ratioӨ i = x Bi /-a is = -b i /-a is = b i /a is . The
basis vector for whichӨ i is most positive is selected to
leave the current basis. ӨLet
r is most positive.
Therefore, r th basis vector i.e. Br leave the current basis
and -a rs is the pivotal element. The elements in the next
iteration are computed as follows:
If there exist at least one non basis vector a j such that all
a ij < 0 then basic feasible solution can be obtained in the
first iteration only.
Proof: Let a s be non basis vector such that all a is < 0.
Select a s to enter in the basis of first iteration. Compute
the ration Ө i = x Bi /-a is = -b i /-a is = bi /a is for basis vector
and select the basis vector to leave the current basis for
which Ө i is maximum positive. Let Br be the leaving
vector. The new solution will be
The solution obtained in first iteration using (3.1) and
(3.2) will be feasible and now regular simplex method
can be used.
The steps in the proposed method can be stated as
follows:
Since we want ⏞
𝑥𝑥 Bi ≥ 0, we must have for all i
R
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Step3. Write down initial simplex table.
Step4. Select non-basic vector a s to enter in the basis of
first iteration such that all the elements of it are negative.
If there exists more than one such non-basic vectors,
then the vector for which z j –c j (= c j , as z j = 0 for all j)
is least positive. Let it be a r .
Step5. For the basis vectors (rows) compute the ratio
Step5. Select the outgoing vector from the current basis
for which Ө i is most positive. Let it be a r . Therefore, a rs is the pivotal element.
Step6. Write down the simplex table for first iteration as
in regular simplex method. The elements in the simplex
table for first iteration will be computed by equations
(3.1), (3.2), (3.3) and (3.4). The new solution is basic
feasible solution.
Step7. Proceed with regular simplex method till optimal
solution is obtained or there is indication that solution
does not exists.
Example: Consider the problem,
Iteration 1:
(Note that the solution in first iteration is feasible. So we
can proceed with regular simplex method. Since z j –c j (7) is most negative for a 1 , it is entering and s 2 is leaving
vector as shown in above table.)
Iteration 2:
Corresponding standard linear programming problem is
REFERENCES
[1] Robert J. Vanderbeib: Linear Programming:
Foundations and Extensions, 3rd ed., International Series
in Operations Research & Management Science, Vol.
114, Springer Verlag, 20, 2008
[2] Gass, S.: Linear programming, methods and
Applications, 5th ed. McGraw-Hill, New York, 2003
[3] Barry Cipra: The Best of the 20th Century: Editors
Name
Top
10
Algorithms
SIAM News, Volume 33, Number 4, page 1, 2000.
[4] Terlaky, Tamás : "A finite crisscross method for
oriented matroids". Journal of Combinatorial Theory.
Series B 42 (3): 319–327, 1987
[5] Lemke, C. E. (1954): The dual simplex method of
solving the linear programming problem, Naval Res.
Logist. Quart., Vol. 1, 36-47.
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