midterm_2006_sol_2

# midterm_2006_sol_2 - MS&E 211 Midterm Examination...

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Unformatted text preview: MS&E 211 Midterm Examination - Autumn 2006 Page 1 of 11 M IDTERM E XAMINATION S OLUTIONS MS&E 211 November 2, 2006 INSTRUCTIONS a) Take alternate seating if possible. b) This is an open book and open notes exam. You are not permitted to use computers or any calculator programming functions. c) The examination ends in 75 minutes. d) There are 4 questions plus a bonus problem on this exam. The exam is worth 100 points plus 10 points for a bonus question. HONOR CODE In taking this examination, I acknowledge and accept the Stanford University Honor Code. NAME (signed) NAME (printed) Pr. 1 Pr. 2 Pr. 3 Pr. 4 Bonus Total MS&E 211 Midterm Examination - Autumn 2006 Page 2 of 11 Problem 1 - True or False (20 points) State whether each of the following italicized statements is True or False. Assume all other in- formation given is correct. For each part, only your answer, which should be one of “True” or “False”, will be graded. Explanations will not be read. a) Consider a production problem where a company must decide how many of each of its n products to produce. Production is constrained by the availability of m different types of resources. Assume that we can derive an optimization problem of the following form for this problem where c i is revenue per unit of product i and b j is the amount of resource j available. Maximize c T x subject to Ax ≤ b x ≥ where x ∈ < n , A ∈ < mxn , b ∈ < m , and A is full rank. Assume that m < n and that there is an optimal solution to this problem. True or False: There will be an optimal solution where the company produces at most m products. True - If a linear program has an optimal solution, then it has an optimal basic solution. In this case, a basic solution will have at most m variables which are greater than zero. Thus, there is an optimal solution where the company produces at most m products. b) In the problem above, assume that we find two optimal solutions, x 1 and x 2 (where x 1 6 = x 2 ). Note: x 1 and x 2 are vectors. True or False: There must be an infinite number of optimal solutions. True - Since x 1 and x 2 are optimal, then any convex combination of these two solutions will also be optimal. There are an infinite number of convex combinations. c) Consider the following formulation: Maximize x 1 + x 2 subject to x 1 + x 2 ≥ 3 x 1 ≤ x 2 x 2 ≤ α x 1 ,x 2 ≥ MS&E 211 Midterm Examination - Autumn 2006 Page 3 of 11 True or False: The dual of the problem is infeasible if and only if α < 3 2 False: If α < 3 2 , the primal problem becomes infeasible. Thus, the dual can be either unbounded or infeasible, and in this particular case, it is unbounded....
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midterm_2006_sol_2 - MS&E 211 Midterm Examination...

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