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• Introduction• Algorithm• Points to Remember• Example• Analysis of Big M Method• Drawbacks• Conclusion• References
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.
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.
Points To Remember
Solve the modified LPP by simplex method, untilany 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.
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
whereS1 is a slack variable S2 is a surplus variable
The surplus variable S2 represents the extra units.
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
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
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
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.
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.
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
