# HW3 - MS&amp;E 211 Linear &amp; Nonlinear Optimization Fall...

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MS&E 211 Linear & Nonlinear Optimization Fall 2011 Prof Yinyu Ye Homework Assignment 3: Due Tuesday November 1 st at 6:00pm Problem 1. Sensitivity Analysis: (22 points) You have rented a metal detector for two and a half hours. You can spend your time with it searching for valuables in one of three locations: the sand, the grass, or the hills. The metal detector has three settings—one for each location. The “sand” setting uses the battery power at only one-third the normal rate. The “grass” setting uses the battery power at two-thirds the normal rate. And the “hills” setting uses the batter power at full blast. You have only enough battery life to operate the machine at full blast for one hour and twenty minutes. If you estimate that people find about five dollars per hour on the beach, nine dollars per hour on the grass, and ten dollars per hour on the hills, how should you allocate your time to maximize expected earnings? Hint: read all the questions before starting. a) What is the optimal allocation? Hint: Show that the optimal basis is “grass and sand”. b) Clearly formulate the primal of the linear program of this problem. c) Write out the final simplex tableau. d) How can you use the hint in the first question (a) to speed up the previous question (c)? Explain why your answer would still be trustable. e) Clearly formulate the dual linear program. f) If the battery had been fully charged at two-hours life, would the optimal basis still be “grass and sand”? g) Explain how you could have answered the previous question (f) by only looking at the final simplex tableau and the primal & dual formulations. h)

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## HW3 - MS&amp;E 211 Linear &amp; Nonlinear Optimization Fall...

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