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homework-6-solutions

homework-6-solutions - MthSc810 Mathematical Programming...

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MthSc810 – Mathematical Programming Fall 2011, Solutions to Homework #6 Problem 1. Implement, in AMPL, a solution to the following problem (it might be useful in the next midterm): A classroom can be thought of as a set of n two-dimensional vectors, in- dicating each a seat , i.e., position in the classroom’s x and y coordinates. Let us assume that, as happens in M-104, these n points are fixed (i.e., they are given as an input and are not a variable that we want to find) as the tables can’t be moved. Each seat is then represented by a vector ( a i ,b i ), for i = 1 , 2 ...,n . Given k <n students, where should they sit? We seek a subset of cardi- nality k of the set N of n seats, or a vector of n variables, each variable x i tells us whether seat i has a student or not. In order to avoid cheating, we want to determine the placement that maximizes the minimum (Euclidean) distance between pairs of seats where both seats have a student. In other words, in every assignment S N = { 1 , 2 ...,n } a pair of students will have minimum distance. This minimum distance is a function d ( S ). We want to find the assign- ment S that maximizes d ( S ). Then solve it for M-104, on four instances where k is 10, 15, 20, and 25 re- spectively. Approximate the position of each seat in M-104. Finally, represent graphically the choices for k = 10 and k = 15.
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