Artificial Variable Technique (The Big-M Method) ATISH KHADSE

1
Artificial Variable Technique (The Big-M
Method)ATISH KHADSE
2
Big-M Method of solving LPP

• The Big-M method of handling instances with
  artificial
• variables is the commonsense approach.
  Essentially, the
• notion is to make the artificial variables,
  through their
• coefficients in the objective function, so costly
  or unprofitable
• that any feasible solution to the real problem
  would be
• preferred....unless the original instance
  possessed no feasible
• solutions at all. But this means that we need to
  assign, in the
• objective function, coefficients to the
  artificial variables that are
• either very small (maximization problem) or very
  large
• (minimization problem) whatever this value,let
  us call it Big M.
• In fact, this notion is an old trick in
  optimization in general we
• simply associate a penalty value with variables
  that we do not
• want to be part of an ultimate solution(unless
  such an outcome
• Is unavoidable).

3
Indeed, the penalty is so costly that unless any
of the respective variables' inclusion is
warranted algorithmically,such variables will
never be part of any feasible solution.This
method removes artificial variables from the
basis. Here, we assign a large undesirable
(unacceptable penalty) coefficients to artificial
variables from the objective function point of
view. If the objective function (Z) is to be
minimized, then a very large positive price
(penalty, M) is assigned to each artificial
variable and if Z is to be minimized, then a very
large negative price is to be assigned. The
penalty will be designated by M for minimization
problem and by M for a maximization problem and
also Mgt0.
4
Example Minimize Z 600X1500X2subject to
constraints,2X1 X2 gtor 80 X12X2 gtor 60
and X1,X2 gtor 0Step1 Convert the LP problem
into a system of linear equations.We do this by
rewriting the constraint inequalities as
equations by subtracting new surplus
artificial variables" and assigning them zero
M coefficientsrespectively in the objective
function as shown below.So the Objective
Function would be Z600X1500X20.S10.S2MA1MA
2subject to constraints, 2X1 X2-S1A1 80
X12X2-S2A2 60 X1,X2,S1,S2,A1,A2 gtor 0
5
Step 2 Obtain a Basic Solution to the
problem.We do this by putting the decision
variables X1X2S1S20,so that A1 80 and
A260. These are the initial values of
artificial variables.Step 3 Form the Initial
Tableau as shown.
6
It is clear from the tableau that X2 will enter
and A2 will leave the basis. Hence 2 is the key
element in pivotal column. Now,the new row
operations are as followsR2(New)
R2(Old)/2R1(New) R1(Old) - 1R2(New)
7
It is clear from the tableau that X1 will enter
and A1 will leave the basis. Hence 2 is the key
element in pivotal column. Now,the new row
operations are as followsR1(New)
R1(Old)2/3R2(New) R2(Old) (1/2)R1(New)
8
Since all the values of (Cj-Zj) are either zero
or positive and also both the artificial
variables have been removed, an optimum solution
has been arrived at with X1100/3 , X240/3 and
Z80,000/3.
9
(No Transcript)