Question

How do i generate a gomory cut from this optimal dictionary form, expressing all the cuts in terms of the variables x1 and x2:

max 15 - 4x3 - 2x4

s.t. x1 = 11/4 - (5/2)x3 + (1/4)x4

x2 = 2 + 2x3

x1, x2, x3, x4 >= 0

Answer 1

x1 + (5/2)x3 - (1/4)x4 = 11/4

or, (1+0)x1 + (2+ 1/2)x3 + (-1 + 3/4)x4 = (2 + 3/4)

Now consider the fractional parts only and replace the = sign with >=  

(1/2)x3 + (3/4)x4 >= 3/4 which is the Gomory cut equation

In the tableau, it will become, (1/2)x3 + (3/4)x4 + G = 3/4 i.e. we will write a slack term G to establish equality for the tableau.

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x2 - 2x3 = 2

or, (1+0) + (-2+0)x3 = (2+0)

If we consider the fractional parts here, we will get 0+0=0; so no cut eqaution can be generated here.