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Unformatted text preview: IOE512 Midterm Solution 1. (50 points) Your state legislature has a total of R representatives, and the state population is divided into N districts (e.g., towns, cities, or counties), N < R . Let P i = population in district i , i = 1 ,...,N ; P = total population in the state, i.e., P = ∑ N i =1 P i . Clearly, if fractional solution is allowed then district i would get RP i /P representatives. But, we want the number of representatives in each district to be an integer. Let x i be the number of representatives in district i . A statewide debate agreed that the best way is to allocate the number of representatives is to find x i such that the following is minimized: min N X i =1 x i R P i P , where, of course, ∑ N i =1 x i = R . (a) (25 points) Formulate a dynamic program for finding the number of rep resentatives for each district. (b) (25 points) Solve the problem numerically with data R = 4, N = 3, P 1 = 1 . 2 ,P 2 = 6 and P 3 = 2 . 8. 1 Solution: (a) Let f n ( y ) denote the optimal value of the following objective function: min n X i =1  x i R P i P  s.t....
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 Spring '12
 Richard
 Optimization, optimization problem, optimal value, max Li, dynamic program

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