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L05_Artificial Variable techniques - Big M method

L05_Artificial Variable techniques - Big M method -...

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Artificial Variable Techniques Big M-method
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Lecture 6 Abstract If in a starting simplex tableau, we don’t have an identity submatrix (i.e. an obvious starting BFS), then we introduce artificial variables to have a starting BFS. This is known as artificial variable technique. There are two methods to find the starting BFS and solve the problem – the Big M method and two-phase method. In this lecture, we discuss the Big M method.
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Suppose a constraint equation i does not have a slack variable. i.e. there is no ith unit vector column in the LHS of the constraint equations. (This happens for example when the ith constraint in the original LPP is either ≥ or = .) Then we augment the equation with an artificial variable R i to form the ith unit vector column. However as the artificial variable is extraneous to the given LPP,
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we use a feedback mechanism in which the optimization process automatically attempts to force these variables to zero level. This is achieved by giving a large penalty to the coefficient of the artificial variable in the objective function as follows: Artificial variable objective coefficient = - M in a maximization problem, = M in a minimization problem where M is a very large positive number.
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Consider the LPP: Minimize 2 1 2 x x z + = Subject to the constraints 0 , 6 9 3 2 1 2 1 2 1 + + x x x x x x
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Putting this in the standard form, the LPP is: 2 1 2 x x z + = Subject to the constraints 1 2 1 1 2 2 1
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