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hmk1_sol_07 - MS&E 211 Linear and Nonlinear Optimization...

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MS&E 211 Fall 2007 Linear and Nonlinear Optimization Oct 9, 2007 Prof. Yinyu Ye Homework Assignment 1: Sample Solution Problem 1. Assume the sales quotas for special risk insurance and mortgages are x units and y units respectively. a) We can formulate the problem as follows: Maximize 5 x + 2 y subject to 3 x + 2 y 2400 y 800 2 x 1200 x, y 0 . b) see graph 1. c) The optimal solution: x = 600, y = 300. The optimal value is 3600. Problem 2. Let x ijg be the number of grade g students in neighborhood i assigned to school j . Then we have Minimize I X i =1 J X j =1 d ij ( G X g =1 x ijg ) subject to I X 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. 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:
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5x+2y Figure 1: The graph for problem 1. Minimize 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 ) subject to x 11
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