Homework+1+Solutions - Solutions for Homework 1(104 points...

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Solutions for Homework 1 (104 points) 1. (24 points, 4 points each) 1. Feasible region for the problem is depicted below. 2. 3. Using the graph in previous part you can see that problem is unbounded when when we switch to maximization. 4. The nontrivial objective functions for which the problem has multiple optima are 2x + y and y-x. x - y = 1 2x + y = 6 x (1,1) Optimal at point (x,y) = (7/3, 4/3) x - y = 1 2x + y = 6 x 3 1 -1 y
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5. There are infinitely many constraints that make the problem infeasible. The idea is that new constraint should be violated by the points that satisfy the constraints we currently have. An example is x + y ≤ 2. 6. A, B and C are the extreme points of the feasible region. 2. (20 points, 5 points each) 1. True. 2. True. 3. True. 4. False. Suppose that objective function is x + y and we have one constraint which is x - y = 1 2x + y = 6 x (1,1) Optimal at point (x,y) = (5, 4) A B C y = 4 x - y = 1 2x + y = 6 x x + y = 2
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x – y = 1. Then, both minimization and maximization problems are unbounded. 3. (35 points, 5 points each) a) Decision variables are: x1: # of times process 1 is run. x2: # of times process 2 is run. x3: # of times process 3 is run. Using these decision variables, optimization problem can be formulated as the following LP: Max 7000*(20x1 + 20x3) + 10000*(10x1 + 30x2) + 24000*(5x3) – 20000*(2x1 + 3x2)
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