hw01solutions

hw01solutions - x 2 without bound which means that the...

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IEOR 162 Linear Programming Spring 2007 Homework 1 Solutions 3.1.4. (5 points) Decision variables: x 1 = number of Type 1 trucks to produce each day x 2 = number of Type 2 trucks to produce each day LP Formulation: max 300 x 1 + 500 x 2 s.t. 1 800 x 1 + 1 700 x 2 1 (painting) 1 1500 x 1 + 1 1200 x 2 1 (assembly) x 1 , x 2 0 3.2.6 (5 points) Decision variables: x 1 = number of acres of wheat x 2 = number of acres of corn LP Formulation: max 200 x 1 + 300 x 2 s.t. x 1 + x 2 45 (available land) 3 x 1 + 2 x 2 100 (workers) 2 x 1 + 4 x 2 120 (fertilizer) x 1 , x 2 0 1
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3.2.4 (3 points) This question is actually poorly worded, since the objective function increases along many directions. What they really meant to ask is which direction does the objective increase the most, which is the direction of the gradient. a. Direction of increase: d = ± 4 - 1 ² b. Direction of increase: d = ± - 1 2 ² c. Direction of increase: d = ± - 1 - 3 ² 3.3.3 (3 points) This LP is unbounded. Set x 1 = 0 and note that for any value x 2 2, all constraints are satisfied. Thus we can increase
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Unformatted text preview: x 2 without bound which means that the objective can increase without bound as well. 3.3.5 (2 points) True. Consider the contrapositive of the statement, namely if the LPs feasible region is bounded, then the LPs objective is also bounded. Clearly this statement is true, for if the feasible region is bounded, then every feasible solution (which includes the optimal solution) has nite values for all variables. Any linear function of nite values is also nite, so the LPs objective is bounded. 3.3.6 (2 points) False. A counterexample to this statement is the following very simple LP: min x 1 + x 2 s.t. x 1 , x 2 This LPs feasible region is unbounded since x 1 and x 2 can take any non-negative value. However the optimal solution is clearly x 1 = 0 , x 2 = 0. 2...
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hw01solutions - x 2 without bound which means that the...

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