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# Linear Programming : Duality and the Simplex Method

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Consider the following linear programming problem:

Maximize 2x1 + 3x2 + 5x3

Subject to
x1 + 2x2 + 3x3 &#8804; 8
x1 - 2x2 + 2x3 &#8804; 6

x1, x2, x3 &#8805; 0

a. Write the dual problem
b. Solve the foregoing problem by the simplex method (not the dual -simplex). At each iteration, identify the dual variable values and show which dual constraints are violated.
c. At each iteration, identify the dual basic and nonbasic variables, along with the corresponding 3 x 3 dual basis.
d. Show that at each iteration of the simplex method, the dual objective is worsened.
e. Verify that at termination, feasible solutions of both problems are at hand, having equal objective values and with complementary slackness holding.

The solution to the above maximization problem is (Objective value = 15.5, x1 = 7,
x2 = 0.5, x3 = 0)
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