Problem 3. Consider the LP max 2x1 + 3x2 s.t -3x1 + x2 51 X1 + 2x2 5 9 X1 + X2 57 X1, X2 20 It is known that the optimal tableau of this LP is 0 0 BVS1z z = 16 1 x2 = 2 0 x = 50 x3 = 140 0 0 1 1 0 0 2 0 0 0 1) Formulate the dual problem of this LP using the direct method". (6 points) 2) What is the optimal solution and optimal value of the dual problem? (6 points) 3) If I want to change the first coefficient in the objective function (now, it is "2"), what is the range of this coefficient that keeps the optimal basis? Use the simplex method to answer this question. Steps are required. (10 points) 4) If I want to change the RHS of the first constraint (now, it is "1"), what is the range of the RHS that keeps the optimal basis? Use the simplex method to answer this question. Steps are required. (10 points) 5) Show the shadow prices of all constraints (if zero, clearly indicate whether it's a "-0" or not). (6 points) 6) Redo parts 3), 4) and 5) by Excel. (10 points) 7) Verify the Strong Duality Theorem based on this LP by Excel (i.e., to solve the dual LP by Excel and verify that its optimal value equals to that of the primal LP). (6 points) 8) Randomly find a non-optimal feasible solution (not necessarily BFS) to the primal and dual LP, respectively. Then verify the Weak Duality Theorem. (8 points)