Advanced IE521 Optimization Example Set 3
Fall 2011
1. Consider the tableau given below for a maximization problem. Give conditions on the unknowns a1 , a2 , a3 , b, c that make the following statements true. 10 4 1 b -c -1 a2 a3 2 a1 -4 3 0 1 0 0 0 0 1 0 0 0 0 1
(a) Current solution is optimal. (b) The current solution is optimal and there are alternative optimal solutions. (c) The LP is unbounded (assume b 0). 2. Consider the tableau given below for a maximization problem. Give conditions on the unknowns a1 , a2 , a3 , b, c1 , c2 that make the following statements true. 10 b 2 3 c1 4 -1 a3 c2 a1 -5 -3 0 1 0 0 0 0 1 0 0 a2 -1 -4 0 0 0 1
(a) The current solution is optimal and there are alternative optimal solutions. (b) The current solution is not a basic feasible solution. (c) The current solution is a degenerate basic feasible solution. (d) The current solution is feasible, but the LP is unbounded. (e) The current basic solution is feasible, but the objective function value can be improved replacing by x6 as basic variable with x1 . 3. Solve the following LP using 2 phase simplex method. min s.t. 2x1 + 3x2 2x1 + x2 4 x1 - x2 -1 x1 , x2 0
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4. solve the following LP using big M method. max s.t. 3x1 + x2 x1 + x2 3 2x1 + x2 4 x1 + x2 = 3 x1 , x2 0 5. Find the inverse of the following matrix using row operations: B = 0 3 2 1
6. Recall that every LP can have one of the following: Case 1. The LP has a unique optimal solution. Case 2. The LP has alternative or multiple optimal solutions. Two or more extreme points are optimal and the LP will have an infinite number of optimal solutions. Case 3. The LP is infeasible. Case 4. The LP is unbounded (the feasible region is not empty). Identify which of Cases 1-4 apply to each of the following LPs: (a) max s.t. x1 + x2 x1 + x2 4 x1 - x2 5 x1 , x2 0 (c) max s.t. -x1 + 3x2 x1 - x2 4 x1 + 2x2 4 x1 , x2 0 (d) max s.t. (b) max s.t. 4x1 + x2 8x1 + 2x2 16 5x1 + 2x2 12 x1 , x2 0 3x1 + x2 2x1 + x2 6 x1 + 3x2 9 x1 , x2 0
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