Homework 8 %26 9 Solutions

# Homework 8 %26 9 Solutions - 1 CE 398 Introduction to Civil...

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1 CE 398 Introduction to Civil Engineering Systems Design, FALL 2011 Homework 8. Due Thursday, October 27, 2011 SOLUTIONS QUESTION 1. Consider the following constraint set: x 1 - 2 x 2 ≤ 2 (1) 2 x 1 + x 2 ≤ 9 (2) -3 x 1 + 2 x 2 ≤ 3 (3) x 1 is unrestricted in sign (4) x 2 ≥ 0 (5) (a) Clearly show the feasible region on a graph for the constraint set. (b) Label all extreme points (i.e., vertices). (c) If a new constraint: x 1 ≥ 0 is imposed, show the new feasible region on a separate graph and label the extreme points. Would you describe the new constraint as a binding constraint or a redundant constraint? Give reasons for your answer. (d) If the objective function is to maximize W = 2 x + 3 y , find the optimal solution subject to the constraints (1) to (5). SOLUTION

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2 Is a binding constraint because its inclusion changes the feasible region. (d) Point x y 2x+3y (0, 1.5) 0 1.5 2(0) + 3(1.5) = (2, 0) 2 0 2(2) + 3(0) = (4, 1) 4 1 2(4) + 3(1) = (15/7, 33/7) 15/7 33/7 2(15/7) + 3(33/7) = The point corresponding top the maximum value of W is … Therefore the optimal solution is: W opt = x opt = y opt =
3 QUESTION 2 (Sample GAMS input files and Excel Solver application, are provided on Blackboard, for your convenience) For construction of a civil engineering facility, a contractor has found natural reserves of sand and gravel at Bloomingdale and Valley Springs where she may purchase such material. The unit cost including delivery from Bloomingdale and Valley Springs is \$5 and \$7, respectively. After the material is brought to the site it is mixed thoroughly and uniformly, and contract specification state that the mix should contain a minimum of 30% sand.

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Homework 8 %26 9 Solutions - 1 CE 398 Introduction to Civil...

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