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# homework-2 - maximum number of employees to be allocated on...

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MthSc810 – Mathematical Programming Fall 2011, Homework #2. Due Tuesday, September 6, 2011. Exercise 1 Solve the following problem using the graphical method: min x 1 + 4 x 2 s.t. x 1 + 2 x 2 10 x 1 + x 2 6 x 1 x 2 2 x 2 0 . Then write the problem in standard form. Exercise 2 Given two convex sets S 1 and S 2 in R n prove the following: S 1 S 2 is convex; S 1 S 2 = { x 1 + x 2 : x 1 S 1 , x 2 S 2 } is convex; S 1 S 2 = { x 1 x 2 : x 1 S 1 , x 2 S 2 } is convex. Exercise 3 Your company has the problem of deciding what projects, among a set of n = 5 projects, to pursue for the next few years. For all of them, we are given an expected return r i (in M\$, millions of dollars), an initial investment s i (in M\$), and the number of employees to be allocated q i , as in the table below: Project A B C D E r i 19 41 15 22 30 s i 12 29 9 14 18 q i 25 22 31 25 30 The total initial investment cannot be larger than a budget B = 28, and the
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Unformatted text preview: maximum number of employees to be allocated on all projects is W = 50. 1. Formulate an Integer Linear Programming model for the problem of maxi-mizing the total expected return. Reminder: clearly state the variables and their meaning, the constraints, and the objective function. There can be in-teger variables, but all constraints, apart from the integrality ones, must be linear. 2. What can be said about project B, even if we don’t know how to solve the problem? Explain. Exercise 4 Prove that following functions are convex: – f ( x ) = max { f 1 ( x ) , f 2 ( x ) . . . , f k ( x ) } , with f i ( x ) convex for i = 1 , 2 . . . , k . – f ( x ) = | x | . Exercise 5 Solve problems 1.5 and 2.2 of the textbook....
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