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11_hw1_sol

# 11_hw1_sol - MS&E 211 Fall 2011 Linear and Nonlinear...

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Fall 2011 Linear and Nonlinear Optimization Oct 11, 2011 Prof. Yinyu Ye Homework Assignment 1: Sample Solution Problem 1 Let x 1 j = tons of waste sent to incinerator j from Palo Alto , x 2 j = tons of waste sent to incinerator j from Stanford, and y ij = tons of debris sent to landﬁll j from incinerator i . The LP model can be written as: min 3(60 x 11 + 9 x 12 + 35 x 21 + 41 x 22 + 5 . 5 y 11 + 7 . 5 y 12 + 9 . 5 y 21 + 6 . 5 y 22 ) +40( x 11 + x 21 ) + 30( x 12 + x 22 ) s.t. x 11 + x 12 = 1000 x 21 + x 22 = 700 y 11 + y 12 = 0 . 2( x 11 + x 21 ) y 21 + y 22 = 0 . 2( x 12 + x 22 ) x 11 + x 21 1000 x 12 + x 22 1000 y 11 + y 21 400 y 12 + y 22 400 x ij , y ij 0 , i = 1 , 2; j = 1 , 2 . Problem 2. Let x ijg be the number of grade g students in neighborhood i assigned to school j . Then we have: min I i =1 J j =1 d ij ( G g =1 x ijg ) s.t. I i =1 x ijg C jg , j, g J j =1 x ijg = S ig , i, g x ijg 0 , i, j, g. Problem 3.

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11_hw1_sol - MS&E 211 Fall 2011 Linear and Nonlinear...

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