Big M method-A variant of simplex method

1. Big M Method
A Variant of Simplex Method
Presented By : Luckshay Batra
luckybatra17@gmail.com

2. Content
• Introduction
• Algorithm
• Points to Remember
• Example
• Analysis of Big M Method
• Drawbacks
• Conclusion
• References

3. Introduction
 A method of solving linear programming problems.
 It is one of the oldest LP techniques.
 Big M refers to a large number associated with the
artificial variables.
 The Big M method introduces surplus and artificial
variables to convert all inequalities into standard
form.

4. Algorithm
 Add artificial variables in the model to obtain a
feasible solution.
 Added only to the ‘>’ type or the ‘=‘
constraints.
 A value M is assigned to each artificial variable.
 The transformed problem is then solved using
simplex eliminating the artificial variables.

5. Points To Remember
Solve the modified LPP by simplex method, until
any one of the three cases may arise:-
 If no artificial variable appears in the basis and the
optimality conditions are satisfied.
 If at least one artificial variable in the basis at zero level
and the optimality condition is satisfied .
 If at least one artificial variable appears in the basis at
positive level and the optimality condition is satisfied, then
the original problem has no feasible solution.

6. Example
 Maximize Z = x1 + 5x2
Subject to 4x1 + 4x2 ≤ 6
x1 + 3x2 ≥ 2
x1 , x2 ≥ 0
Solution : Introducing slack & surplus variables :
4x1 + 4x2 + S1 = 6
x1 + 3x2 - S2 = 2
where
S1 is a slack variable
S2 is a surplus variable
The surplus variable S2 represents the extra units.

7.  Now if we let x1 & x2 equal to zero in the initial solution , we will
have S1=6 , S2=-2 , which is not possible because a surplus variable
cannot be negative . Therefore , we need artificial variables.
Introducing an artificial variable , say A1.
 Standard Form :
Maximize Z = x1 + 5x2+ 0s1 + 0s2 – M(A1)
Subject to 4x1 + 4x2 +S1 = 6
x1 + 3x2 –S2 +A1 = 2
x1 , x2 , S1 , S2 , A1≥0

10. Analysis of Big M Method
 Problem P : Minimize cx
Subject to Ax = b
x≥ 0
 Problem P(M) : Minimize cx + M s
Subject to Ax + s = b
x , s ≥ 0
where,
“s” is an artificial variable

11. Analysis of Big M Method
Solve P(M) for a
very large
positive M
Optimal is finite
s=0. Optimal
solution of P is
found
s≠0. P has no
feasible
solutions
Optimal is
unbounded
s=0. Optimal
solution of P is
unbounded
s≠0. P is
infeasible

12. Drawbacks
 How large should M be?
 If M is too large, serious numerical difficulties in a
computer.
 Big-M method is inferior than 2 phase method.
 Here feasibility is not known until optimality.
 Never used in commercial codes.

13. Conclusion
The application of the M technique requires that M
approaches infinity but to computerize the solution
algorithm , M must be finite while being “sufficiently
large.”
The pitfall in this case is, however, if M is too large it can
lead to substantial round-off error yielding an incorrect
optimal solution . For this reason, most commercial LP
solvers do not apply the M-method but use, rather, an
artificial variable method called the two-phase
method.

14. References
 http://cbom.atozmath.com/CBOM/Simplex.aspx
 http://businessmanagementcourses.org/Lesson09TheBigMMethod.
pdf
 http://www.slideshare.net/NiteshSinghPatel/big-m-32360766
 http://en.wikipedia.org/wiki/Big_M_method
 Linear Programming & Network Flows by Mokhtar S. Bazaraa , Hanif
D. Sherali , John J. Jarvis
 Operation Research An Introduction by H. A. Taha