HW1-Sol - MS&E111 Spring 2006 Introduction to...

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Unformatted text preview: MS&E111 Spring 2006 Introduction to Optimization May 1, 2006 Prof. Amin Saberi Solutions to Assignment 1 1. a) Let x i be the number of agents working i th time shift, i.e. x 2 means the number of working agents during 8am-4pm and so on. Then we can formulate the situation as follows: min z = 180 x 1 + 170 x 2 + 160 x 3 + 190 x 4 + 200 x 5 + 210 x 6 + 225 x 7 + 210 x 8 subject to x 1 + x 8 68 x 1 + x 2 90 x 1 + x 2 + x 3 56 x 1 + x 2 + x 3 + x 4 107 x 2 + x 3 + x 4 + x 5 80 x 3 + x 4 + x 5 + x 6 93 x 4 + x 5 + x 6 62 x 5 + x 6 56 x 6 + x 7 40 x 7 + x 8 15 x i , i = 1 , . . . , 8 b) Excel Solver gives the following optimal solution: variable optimal solution x 1 53 x 2 37 x 3 31 x 4 6 x 5 16 x 6 40 x 7 x 8 15 z 36680 2. a) z * = 550 at (7 . 5 , 5). See the figure below: Figure 1: Problem 2(a) b) An optimal solution occurs at (7.5,5), which is the intersection of 2 x 1 + 3 x 2 = 30 and 2 x 1 + x 2 = 20. When is small, the optimal solution would occur at the same...
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HW1-Sol - MS&E111 Spring 2006 Introduction to...

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