(a) Find the dual problem of the following linear program:

maximize z = 6x1 3x2 2x3 +5x4 (2a)

4x1 +3x2 8x3 +7x4 =11 (2b)

3x1 +2x2 +7x3 +6x4 23 (2c)

7x1 +4x2 +3x3 +2x4 12 (2d)

(x1,x2)0,x3 0 (2e)

(b) Verify that the dual of the dual is the primal.

(c) Solve the dual problem found in Part (a). What does this tell you about the primal problem?

(d) Modify the right-hand side of Eq. (2c) to be equal to 8. Solve both the primal and dual problems and show the property of complementary slackness is true in this case.